Hydraulic machines have been widely used in a variety of high power
applications. Typically these machines are operated by an operator. In past
research the ...
UNIVERSITY OF MINNESOTA
This is to certify that I have examined this bound copy of a doctoral thesis by
Kailash Krishnaswamy
and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made.
Perry Y. Li Name of Faculty Adviser
Signature of Faculty Adviser
Date
GRADUATE SCHOOL
Passive teleoperation of hydraulic systems
A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY
Kailash Krishnaswamy
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Perry Y. Li, Adviser
May 2004
c Kailash Krishnaswamy May 2004
Dedication
To ma, dad and K
i
Abstract Hydraulic machines have been widely used in a variety of high power applications. Typically these machines are operated by an operator. In past research the interaction safety between the operator and the machine was seldom investigated. This dissertation addressess the interaction stability issue by exploiting the safety guarantees offered by passive systems. The hydraulic machine considered in this dissertation is a 2-DOF hydraulic backhoe teleoperated by a 2-DOF, electro-mechanical, force-feedback capable joystick (master). The impediment to applying the concept of passivity to hydraulics is that hydraulic systems, in general, do not behave like passive systems. An algorithm was presented which ensured that single-stage proportional valves, a key element in any hydraulic system, behave like passive devices. Using such a passive valve, a passive 2-DOF hydraulic backhoe (slave) is constructed by cascade interconnection of passive subsystems. Given a passive master and a passive slave, this dissertation proposes a systematic approach to design teleoperation control systems (based on bondgraphs) which ensure that the master and slave satisfy certain coordination and passivity requirements. Bondgraphs are used as they offer benefit of representing power flow in a dynamic system. Bondgraph based control is designed in three stages. In the first stage, a coordination control law is determined which ensures that the master and slave systems remain coordinated when teleoperated. The control law is determined based on system inversion ideas. By representing the coordination control law using bondgraphs it is possible to determine the power necessary to maintain coordination of the master and slave systems. In the second stage, desired dynamics of the coordinated teleoperated system are chosen based on the application. The desired dynamics (desired locked system) should be passive with respect to a power supply rate. In the third stage, a passive control law is determined so as to ensure that the coordinated system behaves dynamically like the desired locked system hence in turn ensuring that the teleoperated system is passive with respect to a power supply rate. The proposed passive control ensures that the joystick teleoperated backhoe 1) is coordinated thus guaranteeing performance, 2) is passive with respect to an appropriate power input thus guaranteeing interaction safety and 3) can behave like 4th order system or a 2nd order system based on the choice of desired dynamics. The proposed passive control algorithms is experimentally verified and the benefits achieved by dynamically tuning the passive teleoperated system are documented. It is also shown that the proposed control design method can be extended to ensure passive teleoperation of arbitrary order passive master and passive slave systems. ii
Acknowledgments Foremost, I would like to thank my advisor Prof. Perry Li for being patient, motivating and providing guidance when I was in need. I am also grateful to the huge team of engineers and scientists at Eaton Inc. without whose involvement, experimental verification of the controllers would not have been possible. Specifically, I would like to thank Linda Wolhowe and Steve Zumbusch for driving and providing for the development of the Hydraulic Backhoe at the Fluid Power Control Laboratory. I thank NSF (CMS-0088964) and the Graduate School Doctoral Dissertation Fellowship (DDF) for supporting me through the years and providing a conducive research atmosphere. Gratitude are also in order to Peter Zimmerman and Bob Nelson with the ME Machine Shop who helped with bringing together the mechanical structure of the Backhoe well in time. Finally, I am very grateful to my wife, Pavithra for understanding my academic committments and constantly urging / reminding me to finish this Dissertation.
iii
Contents
Chapter 1
Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Dissertation Contributions . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Dissertation Structure . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Chapter 2 2.1
Literature review and problem formulation
6
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.1
Teleoperation and operation of hydraulic machines
. . . . . .
6
2.1.2
Passive systems and passive teleoperation
. . . . . . . . . . .
8
2.1.3
Passivity of hydraulic valves . . . . . . . . . . . . . . . . . . .
11
2.1.4
Control of hydraulic actuators . . . . . . . . . . . . . . . . . .
13
2.1.5
Bondgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.1.5.1
Model-based inversion for control design: An example, (Ngwompo et al., 2001a) . . . . . . . . . . . . .
17
2.2
Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Chapter 3 3.1
System models and identification
Subsystem models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
23 23
3.2
3.1.1
2-DOF Master motorized joystick . . . . . . . . . . . . . . . .
24
3.1.2
Single-stage passive valve . . . . . . . . . . . . . . . . . . . . .
25
3.1.3
Passification of the Vickers Proportional valve . . . . . . . . .
28
3.1.4
Dynamic model of the hydraulic actuator . . . . . . . . . . . .
30
3.1.5
Kinematic model of the hydraulic actuator . . . . . . . . . . .
31
3.1.6
Backhoe dynamics . . . . . . . . . . . . . . . . . . . . . . . .
31
Control Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Chapter 4
Passive teleoperation of the kinematic modeled hydraulic backhoe
4.1
4.2
38
Passive teleoperation controller design using the Interconnection Lemma 39 4.1.1
Passive controller structure . . . . . . . . . . . . . . . . . . . .
39
4.1.2
Coordination controller . . . . . . . . . . . . . . . . . . . . . .
40
4.1.3
Haptic behavior of the teleoperated backhoe . . . . . . . . . .
44
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Chapter 5
Passive teleoperation of the hydraulic backhoe using bondgraphs 47
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
5.2
Bondgraph, Model and Control objective . . . . . . . . . . . . . . . .
48
5.3
Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.3.1
Coordination control design . . . . . . . . . . . . . . . . . . .
54
5.3.2
Closed loop passivity property: perfect model . . . . . . . . .
55
5.3.3
Locked System Design : 4th Order, 2-DOF Haptic Behavior .
57
5.3.4
Locked System Design : 2nd Order, 2-DOF Haptic Behavior .
64
5.3.5
Closed loop passivity property: imperfect system model . . . .
69
v
5.3.5.1
Coordination analysis . . . . . . . . . . . . . . . . .
69
5.3.5.2
Passive Locked System Design : 4th Order, 2-DOF Haptic Behavior . . . . . . . . . . . . . . . . . . . .
71
Passive Locked System Design : 2nd Order, 2-DOF Haptic Behavior . . . . . . . . . . . . . . . . . . . .
78
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.3.5.3
5.4
Chapter 6
Passive teleoperation of cascade passive systems using bondgraphs 86
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
6.2
Bondgraph, Model and Control objective . . . . . . . . . . . . . . . .
87
6.3
Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
6.3.1
Coordination control design . . . . . . . . . . . . . . . . . . .
90
6.3.2
Closed loop passivity property . . . . . . . . . . . . . . . . . .
93
6.3.3
Locked System Design : m-Order, p-DOF Haptic Behavior . .
94
6.3.4
Locked System Design : 2-Order, p-DOF Haptic Behavior . . 100
6.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Chapter 7
Experimental results
107
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.2
Kinematic Teleoperation . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.3
Dynamic Teleoperation . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.4
7.3.1
4th order locked system . . . . . . . . . . . . . . . . . . . . . 112
7.3.2
2nd order locked system . . . . . . . . . . . . . . . . . . . . . 114
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
vi
Chapter 8
Conclusions and future work
118
8.1
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
vii
List of Figures 1.1
The concept of passive hydraulic teleoperation . . . . . . . . . . . . .
4
2.1
Interconnection of two passive two-port systems . . . . . . . . . . . .
9
2.2
A fluid power circuit of directional control valve controlling a hydraulic actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.3
Schematic of the Valve - Actuator setup . . . . . . . . . . . . . . . .
12
2.4
The directional control valve represented as an active electrical circuit
13
2.5
Bondgraph subsystems . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.6
Bicausal bondgraph subsystems . . . . . . . . . . . . . . . . . . . . .
16
2.7
Spring-mass-damper system schematic . . . . . . . . . . . . . . . . .
17
2.8
Bondgraph of the spring-mass-damper system with the shortest causal paths (dotted lines) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Inverse bondgraph of the spring mass damper system . . . . . . . . .
19
2.9
2.10 The teleoperation setup consists of a 1) master (motorized joystick), 2) electrohydraulic passified valves, 3) single-ended actuators, 4) slave (backhoe) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3.1
The teleoperation setup considered in this dissertation
. . . . . . . .
23
3.2
Electro-mechanical joystick. Each link is about 10 inches long . . . .
24
3.3
Single stage valve connected to a single-rod actuator
. . . . . . . . .
26
3.4
The Vickers proportional valve mounted on a manifold. . . . . . . . .
29
viii
3.5
3.6
Experimental (solid) and modeled fit (dashed) for the Vickers proportional valve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
Hydraulic backhoe in the Fluid Power Control Laboratory. Link 1 is 44 inches long. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
3.7
Modelled (dashed) and measured forces (solid) of the hydraulic backhoe 36
5.1
Bondgraphs of Master and Slave systems. . . . . . . . . . . . . . . . .
49
5.2
Bondgraph of coordinate transformed teleoperator . . . . . . . . . . .
52
5.3
˙ 54 Inverse dynamics bondgraph of the teleoperator, input Fq and output E
5.4
Bondgraph of the master and slave systems with coordination control law involving accurate estimates . . . . . . . . . . . . . . . . . . . . .
56
5.5
Fourth order desired locked system . . . . . . . . . . . . . . . . . . .
58
5.6
Second order desired locked system . . . . . . . . . . . . . . . . . . .
64
5.7
Bondgraph of the master and slave systems with coordination control law involving inaccurate estimates . . . . . . . . . . . . . . . . . . . .
70
5.8
Choice of BE and KE . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
6.1
Bondgraphs of the Master and Slave systems . . . . . . . . . . . . . .
87
6.2
Bondgraph of teleoperator after coordinate transformation . . . . . .
89
6.3
Inverse dynamics bondgraph of the teleoperator, input u2 and output e 91
6.4
Bondgraph of the master and slave systems with coordination control law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
6.5
m-order desired locked system . . . . . . . . . . . . . . . . . . . . . .
95
6.6
Second order desired locked system . . . . . . . . . . . . . . . . . . . 101
7.1
Schematic of the experimental setup
7.2
Picture of the backhoe and joystick in the Fluid Power Control Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 ix
. . . . . . . . . . . . . . . . . . 108
7.3
Displacement trajectories (Scaled joystick - solid, Backhoe - dashed) during a digging task. . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.4
Haptic torque (Fq ) trajectories (Stick - solid, Bucket - dashed) during a digging task. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.5
Displacement trajectories for 4th Order locked system (Joystick - Solid, Backhoe - dashed) during a digging task. Sequentially from top : 1)-2) ρ = 12, 3)-4) ρ = 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.6
Locked system Force (ρTq − Fe ) trajectories during a digging task for 4th Order locked system (Stick - solid, Backhoe - dashed). . . . . . . 114
7.7
Displacement trajectories for 2nd Order locked system (Joystick - Solid, Backhoe - dashed) during a digging task. Sequentially from top : 1)-2) ρ = 12, 3)-4) ρ = 18. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.8
Locked system force (ρTq − Fe ) trajectories during a digging task for 2nd Order locked system (Stick - solid, Backhoe - dashed). . . . . . . 116
x
Chapter 1 Introduction 1.1
Motivation
Hydraulics are used in a variety of applications as power source and transmission, owing to their high power density, flexibility and portability. Many of these applications involve direct human operation. Examples include earth moving equipment or construction machinery. The human (operator) operates levers in his cab to control the hydraulic machine. Typically the operator uses only visual information to perform the task. Providing the operator with feedback can greatly enhance operation productivity (Sheridan, 1987). To cater to this need, human-in-the-loop, teleoperation control has been developed for earth moving machines where in a controller ensured that human motion captured by a master manipulator is mimicked by a hydraulic slave manipulator, (Parker et al., 1993), (Lawrence et al., 1995). In contrast, there have also been research efforts directed towards development of fully autonomous machines by removing the operator, (Stentz et al., 1999), (Bradley and Seward, 1998). Teleoperated machines which are typically commanded by a human operator are favorable as they are 1) more responsive to the environment 2) cheaper to build 3) utilize human decision making abilities in unforeseen circumstances when compared to the autonomous machines. The broad class of teleoperated hydraulic machines interact physically with the environment. For safety reasons it is necessary that the interaction be stable. In previous research, this interaction stability was not always guaranteed, especially when the environment was uncertain or when the environment model was unknown. In the case of human operated machines, it will be useful to provide the operator with a haptic feel of the environment which is currently nonex1
istent. The safety and human friendliness aspects of human-operated machines can be greatly enhanced if the machine can be shown to be passive. Roughly speaking, a passive system does not generate any energy of its own but only transmits, stores or releases it. The key passivity theorem (Vidyasagar, 1993) ensures the interaction stability of a passive system and a strictly passive system. Since the environment with which a machine interacts is often strictly passive, a passive machine can operate without going unstable in a broad class of environments. In addition to providing interaction stability, passive machines also permit information to be transfered naturally between the human and work environment via physical power. For this reason, the haptics / teleoperation community recognize passivity as a desirable closed loop property to have. The inherent safety of passive machines has been exploited in many systems that require human-machine interaction such as, smart exercise machines (Li and Horowitz, 1997), bilateral teleoperation manipulators (Li, 1998), (Li and Lee, 2000), (Lee and Li, 2001), (Itoh et al., 2000), Cobots (Colgate et al., 1996) and passive trajectory enhancing robot (Gomes and Book, 1997). The concept of passivity in the context of hydraulic control systems had not been investigated until recently (Li, 2000) when it was shown that a hydraulic control valve, which is a key element in any hydraulic system, is not inherently passive but can be passified (made passive) by modifying its design. This dissertation focuses on the development of control methodologies for human teleoperated hydraulic machines so that they behave like passive systems and are natural for the human to operate. Control development is accomplished in two stages. In the first stage individual passive subsystems are designed and interconnected in cascade thus rendering the overall system passive. In the second stage, a teleoperation control algorithm is determined which ensures that a) the master and slave systems of the teleoperated hydraulic machine are coordinated at all times and b) the controlled hydraulic machine is passive with respect to an appropriate power input. The intuition for control design is derived by investigating the power flow in the coordinated hydraulic machine using bondgraphs. By ensuring closed loop passivity of the teleoperated passive hydraulic machine, it is expected that the operator can perform any task in an efficient manner and safety to environment and operator will be guaranteed.
2
1.2
Dissertation Contributions
This dissertation proposes to develop safer and more human friendly hydraulic machines by designing control laws that render the machines passive. A general idea of the benefit of such controllers is shown in Fig. 1.1. The specific contributions of this dissertation are as follows: 1. Hydraulic subsystems were investigated for their passivity properties. Components that are not inherently passive, particularly control valves, were rendered passive using feedback control. Two hydraulic machines differing by their complexity are built using the passified valve; a) A simpler, linear, kinematic modeled hydraulic backhoe and b) A complex, nonlinear, hydraulic backhoe. Both hydraulic machines are 2-DOF and are meant to be teleoperated by a 2-DOF force feedback capable joystick. 2. A passive teleoperation controller is designed for the simpler kinematic modeled hydraulic backhoe. The controller ensures a) coordination of the master (joystick) and slave (kinematic modeled backhoe), b) passivity of the teleoperated machine with respect to a desired power supply rate. Moreover, the controller is itself passive with respect to the control power input to the master and slave thus ensuring desirable structural robustness to parametric uncertainty. The dynamics of the coordinated teleoperated backhoe (locked system dynamics) have desirable haptic behavior. 3. The key contribution of this dissertation is the development of a general systematic teleoperation control algorithm based on Bondgraphs. The control algorithm ensures a) coordinated behavior of passive master and slave systems, b) passivity property of the coordinated teleoperator with respect to a particular supply rate and c) the locked system dynamics can be appropriately designed based on the task at hand. 4. The passive teleoperation controller which was designed for the kinematic modeled hydraulic backhoe was experimentally tested and the results are documented. 5. The bondgraph based teleoperation control algorithm is applied to the complex, nonlinear hydraulic backhoe and is experimentally verified to provide the teleoperator behavior (locked system dynamics) it was designed for. 3
Operator 2 DOF Master Arm
2 DOF Slave Arm
Environment
Controller Hydraulic Actuators Motors
Common Rigid Passive Mechanical Tool Operator Environment
Figure 1.1: The concept of passive hydraulic teleoperation The developed control algorithms address the ‘safety-of-interaction’ and the ‘human friendliness’ issues in teleoperation of hydraulic machines. Unlike previous hydraulic teleoperation architectures (Salcudean et al., 1999), the proposed passive control methodology guarantees interaction stability without having to model the environment. This methodology also provides the operator with a haptic feedback of the environment thus improving productivity. Moreover, using the proposed methodology it is possible to also design the dynamics of the passive coordinated (closed-loop) teleoperator thus ensuring task specific dynamic abilities.
1.3
Dissertation Structure
Chapter 2 surveys previous work in hydraulic teleoperation, passive teleoperation, hydraulic control and system inversion based control design using bondgraphs. The passive teleoperation problem being considered in this dissertation is defined and passive teleoperator goals are specified. Chapter 3 describes the various subsystems of the hydraulic teleoperation setup. These include the motorized joystick (master), the hydraulic actuated backhoe (slave), the actuators and the proportional electrohydraulic valves. The system models are validated experimentally. The teleoperation control problem is formulated mathe4
matically. Chapter 4 presents a control algorithm for passive teleoperation based on a simplified kinematic model of the backhoe. The developed control is intrinsically passive and guarantees passivity of the teleoperator in the presence of parametric uncertainties of the teleoperated system. However, the teleoperator performance deteriorates in operating conditions which cannot be described by a kinematic model of the backhoe, namely fluid compressibility and inertial effects. Since, this occurs quite frequently during backhoe operation, it is necessary to develop a better controller which considers the nonlinear dynamic behavior of the backhoe. Chapter 5 presents a control algorithm for passive teleoperation of the nonlinear dynamic modeled backhoe that include fluid compressibility and inertial effects which are neglected in the kinematic model developed in chapter 4. The proposed control technique utilizes the physical intuition offered by Bondgraphs to suggest a control architecture. In addition to defining coordination and passivity requirements, the control engineer can also specify desired dynamics of the teleoperated backhoe. Chapter 6 generalizes the bondgraph based control design applied to the hydraulic backhoe to teleoperation of arbitrary relative degree passive systems. Chapter 7 presents the experimental results of implementation of the various control schemes. The performance of the control techniques are compared. Chapter 8 contains concluding remarks and suggestions for future work.
5
Chapter 2 Literature review and problem formulation 2.1 2.1.1
Background Teleoperation and operation of hydraulic machines
Researchers have adopted at least two distinct architectures in the operation of hydraulic machines, in order to improve safety and productivity. At one end of the spectrum researchers aim to develop fully autonomous machines. In these efforts, the humans are removed from direct operation of the machine and play only the role of remote supervisors. At the other end, humans remain in direct control of the machine. With enhanced feedback, teleoperation architectures were developed so as to improve performance and stability of the machine. In this dissertation, we choose a teleoperation architecture as it includes the operator’s decision making ability which we hope will be beneficial in coimplex and uncertain environments. The research group at Carnegie-Mellon University, PA directed their effort towards the development of autonomous earth moving machines. They proposed integrating the excavator with intelligence in the form of visual sensing, image processing, soil interaction dynamic modeling and hence proceed towards autonomous digging and loading (Stentz et al., 1999), (Singh, 1997). In such an approach it is important a) to work in a moderately clear work environment to facilitate accurate image processing b) to ensure the properties of soil are well-characterized by the model, failing which, stability of the excavation process may be under jeopardy. Notice that in the case of 6
autonomous excavation, there is no guarantee of the terrain in which the excavation is to be performed. A supervisor overviewing the excavation task is only provided with 2-D images of the process and hence lacks awareness of the excavation environment. To partially remedy these issues, the authors in (Lipsett et al., 1998) developed a virtual environment using integration of hi-tech electronics and an excavator so as to provide a sense of ‘realism’ for a human supervisor. In another effort towards autonomous excavation the authors in (Bradley and Seward, 1998) develop an artificial intelligence based controller for excavation of a rectangular trench. In all the above approaches, the controllers can perform only a pre-programmed task as coded in them and are not able to cope with unforeseen situations. There is an inherent difficulty in developing autonomous machines that can operate in highly uncertain, unstructured environment. Another approach to improve productivity and safety of earth moving equipment is to enhance the human ability to control the machine via a teleoperated manipulator. This approach is superior to autonomous operation as it takes advantage of the human’s superior decision making capabilities in an unstructured environment. Significant research in teleoperation control of excavators has been undertaken by researchers at University of British Columbia, Canada. Their goal is to install a transparent mapping between the human’s interaction with the machine and the machines interaction with the work environment so that the teleoperation system compensates for the lack of physical ‘feel’ usually associated with teleoperation. Their approach has been primarily based on impedance control. Parker et al. (1993) and Lawrence et al. (1995) develop a force feedback teleoperation system for a dynamic model of an excavator using a 6-DOF magnetically levitated hand controller. Later Salcudean et al. (1999), Salcudean et al. (1998), Salcudean et al. (1997) and Salcudean et al. (2000) used a four channel bilateral teleoperation architecture, (which uses feedback of the force and velocity of both master and slave manipulators to guarantee teleoperation stability) and demonstrated transparency under certain conditions. Differential PWM valves were used by Tafazoli et al. (1996) to drive the excavator. In most of the above cases, it was necessary to dynamically estimate the inertial and friction parameters of the master and the slave. To provide these, Tafazoli et al. (1999) proposed estimation algorithms and much recently, Zhu and Salcudean (2000) developed an adaptation scheme for the models of the environment and the human to guarantee stability of the teleoperation. In most of the above research, there was not much focus on the control of hydraulics itself. Simple inner loop control techniques were 7
used to ensure that the hydraulic actuators emulated velocity sources. Also, passivity of the teleoperation system was not guaranteed. Hence in the presence of an poorly characterized environment, the stability of machine-environment interaction cannot be ensured. 2.1.2
Passive systems and passive teleoperation
A teleoperation system consists of a master manipulator, a controller and a slave manipulator. The controller ensures that the motion of the master manipulator operated by the human is mimicked by the slave manipulator. Some teleoperation controllers also aim to reflect the environment force experienced by the slave onto the human operator. Such teleoperation systems form closed loop feedback connections with the human and the work environment. To guarantee the stability of this closed loop system, researchers have utilized the models of the environment and the human (Kazerooni and Moore, 1997), (Hannaford, 1989) in the control design. Since the environments are subject to change, (e.g. the dynamics of a huge football player and a child are significantly different) the designed teleoperation controllers would also need to address the issue of robustness. However, the environments are often strictly passive systems. This fact can be exploited by control designers to guarantee interaction stability with a broad class of environment if the machine can be shown to be passive itself. This is because the coupling of a passive and a strictly passive system is necessarily stable. The concept of Passivity Consider the following dynamic system: x˙ = f (x, u) y = g(x, u)
(2.1)
where x, u, y are the state vector, input and output of the system, respectively. The system (2.1) is said to be passive with respect to the supply rate s(u, y) for some scalar function s : (u, y) → s(u, y) ∈ R if for all initial state, there exists an initial energy, c2 so that, for all admissible inputs, u(·) and for all t, −
t
s(u(τ ), y(τ ))dτ ≤ c2 .
t0
8
(2.2)
u1Π1 yΠ1 1
Π1
yΠ2 1 =: u1Π2 u2Π1 := yΠ1 2
Π2
u2Π2 yΠ2 2
Figure 2.1: Interconnection of two passive two-port systems Here, the supply rate s(u, y) is a scalar function of the input and output that generalizes the concept of physical power into a system. If u denotes the effort variable into the system and y denotes the flow variable out of the system then s = uT y is exactly the physical power input into the system. If the supply rate is the physical power into a sytem then (2.2) means that for a passive system, the total energy that can be extracted from it is at most the initial energy, c2 . Although supply rates can be arbitrarily defined (subject to a technical assumption that it must be a L1e function), we are mostly interested in supply rate’s that are related to physical power in this research. The concept of passivity is applicable to single port systems (with one pair of input and output) or to multiple port systems (with multiple pairs of inputs and outputs). A useful property of passive systems is that the cascade interconnection of passive systems is passive. This is formally stated in the following lemma. Lemma 1. The interconnection lemma, (Li and Krishnaswamy, 2001) Consider two two-port systems Π1 and Π2 with respective port variables, {(u1Π1 , yΠ1 1 ), (u2Π1 , yΠ2 1 )}, {(u1Π2 , yΠ1 2 ), (u2Π2 , yΠ2 2 )} as illustrated in Fig. 2.1. Suppose that system Π1 and Π2 are passive with supply rates: sΠ1 ((u1Π1 , yΠ1 1 ), (u2Π1 , yΠ2 1 )) = u1Π1 · yΠ1 1 + γ1 u2Π1 · yΠ2 1 sΠ2 ((u1Π2 , yΠ1 2 ), (u2Π2 , yΠ2 2 )) = −u1Π2 · yΠ1 2 + γ2 u2Π2 · yΠ2 2 respectively. The first terms in the two supply rates is the power input to the left ports of the two systems Π1 and Π2 respectively. Similarly the second terms in the two equations are the power input from the environment into the right ports of the two systems. The scalars γ1 > 0 and γ2 > 0 represent the power scalings of the two individual systems. The interconnection given by u2Π1 := yΠ1 2 and u1Π2 := yΠ2 1 is passive with respect to the following supply rate: sΠ1 Π2 ((u1Π1 , yΠ1 1 ), (u2Π2 , yΠ2 2 )) = u1Π1 · yΠ1 1 + γ1 γ2 u2Π2 · yΠ2 2
9
(2.3)
This lemma states that a passive system can be obtained by interconnecting a number of passive subsystems in cascade. It is necessary that these passive subsystems be compatible in supply rates hence allowing the cascade interconnection. The overall interconnected system will now be passive with respect to the power input (2.3) into the overall interconnected system. The power scaling of this overall system is a product of the individual power scalings, γ1 γ2 as can be seen in (2.3). The inherent safety of passive machines has been exploited in many applications that involve human-machine interaction e.g., smart exercise machines (Li and Horowitz, 1997), bilateral electro-mechanical teleoperated manipulators (Li, 1998), (Li and Krishnaswamy, 2001), (Li and Lee, 2000), (Lee and Li, 2001), (Itoh et al., 2000), Cobots (Colgate et al., 1996) and Passive Trajectory Enhancing Robots (PTER) (Gomes and Book, 1997). These systems are designed to be closed loop passive with respect to a supply rate defined by the physical power input to the system from the environment. The safety of passive systems is also exploited in the Passive Velocity Field Control (PVFC) methodology (Li and Li, 2000), (Li and Horowitz, 1996), (Li and Horowitz, 1998a), (Li and Horowitz, 1998b), (Li and Horowitz, 1999) which ensures that the closed loop system is passive and enables mechanical systems to interact safely with the often ill-characterized environment while achieving certain coordination goals. All the above mentioned passive machines involved inherently passive components. For example, the electro-mechanical linked manipulators used in bilateral teleoperated manipulators or mechanical brakes and clutches used in PTER are all inherently passive components. In other uses of passivity theory researchers have developed a passivity based analysis / design tool which guarantees global asymptotic stability of a class of minimum phase nonlinear systems, (Byrnes et al., 1991). Jankovic et al. (1999) developed another passivity based approach to stabilize a class of nonlinear systems. Their approach involves 1) decomposing a single system to a virtual master and a virtual slave, 2) passifying the master and slave systems individually and 3) interconnecting the passified master and slave systems. The passive interconnection theorem (Willems, 1973), guarantees the global stability of such interconnections. The authors in (Adams and Hannaford, 1999) use passivity theory to guarantee stability of a haptic interface between a human operator and a virtual wall. Kim et al. (1992) used passivity theory to analyze the stability of teleoperation systems with time delay.
10
QL QL
Pressure Gages DIRECTIONAL CONTROL VALVE
U Solenoid
PL Motor
Pump
M
HYDRAULIC ACTUATOR
Hoses
Ps Reservoir
Figure 2.2: A fluid power circuit of directional control valve controlling a hydraulic actuator 2.1.3
Passivity of hydraulic valves
An impediment to ensuring passivity of machines that use hydraulics is that hydraulic control valves are inherently non-passive. In order to use the interconnection lemma to construct passive hydraulic machines, it is necessary to ensure passivity of the hydraulic control valve driven hydraulic machine. The concept of passivity applied to the hydraulic control valve was only recently investigated, (Li, 2000). Consider the fluid power circuit of a typical hydraulic system shown in Fig. 2.2. Hydraulic fluid is pumped by the electric motor driven variable displacement pump. A solenoid stroked directional control valve regulates the flow rate and the direction of flow into the hydraulic actuator. We now review the dynamics of a solenoid stroked directional control valve for a single degree of freedom configuration. Consider a 4-way, critically centered, directional control valve shown in Fig. 2.3. As a solenoid strokes the spool downward, flow is metered into the actuator through port 1 of the valve. Return flow from the actuator is metered into the tank through port 3. Let xv be the spool displacement. Ignoring flow saturation, the output flow, QL of the valve is given by, (Merritt, 1967): Cd w QL (xv , PL) = √ xv Ps − sgn(xv )PL ρh
(2.4)
where Cd > 0 is the orifice coefficient, ρh > 0 is the fluid density, w > 0 is the gradient 11
u Actuator
P1
4
P1 Supply
Ap
QL
1
xp
PL = P1 − P2
Ps 2
P2 QL Return
P2
3
0 xv
Spool
Fenv Figure 2.3: Schematic of the Valve - Actuator setup of the orifice area with respect to the spool position, Ps is a constant supply pressure. Following (Li, 2000) and (Krishnaswamy, 2000), (2.4) can be written as an ideal flow source (Kq xv ) combined with leakage flow across an impedance (1/Kt (xv , PL)) as follows: QL (xv , PL ) = Kq xv − Kt (xv , PL )PL
(2.5) (2.6)
where Kt (xv , PL ) = √
Ps
√
Kq |xv | Ps + Ps − sgn(xv )PL
(2.7)
It is significant to note that Kt (xv , PL ) > 0 whenever xv = 0 and |PL| < Ps which is often the case. The equation (2.5) which characterizes the valve flow can be represented in an analogical electrical circuit as shown in Fig. 2.4. The passivity properties of such a valve was analyzed by Li (2000). It is clear that such a valve is not passive as it contains an ideal current source which continuously supplies energy to the output port. Hence, as long as the valve spool is open (xv = 0), the load can draw any 12
QL
Kq xv 1/Kt PL
(Output Port to Load)
Figure 2.4: The directional control valve represented as an active electrical circuit amount of energy through the valve. Two modification techniques were suggested to ensure that this valve behaves like a passive two-port device, (Li, 2000). In the first technique, structural modification involving additional leakage, spring stroked spool actuation and differential pressure feedback were suggested so that the valve was passive with respect to a physical power input from the environment. In the second technique, a feedback control method was suggested. The second technique, which will be explained in detail later, is used for the development of passive hydraulic machines in this research. 2.1.4
Control of hydraulic actuators
Literature on control of electrohydraulic systems generally focus on the problems of position or force trajectory tracking. The tracking problem of interest in this dissertation is that of coordinating the positions of the master (joystick) and slave (backhoe) systems. The control of hydraulic actuators is riddled with difficulties of fluid compressibility, nonlinear dynamics, valve dead zone and significant friction. Upto a few years ago, most researchers used a linearized model of the nonlinear dynamics of a hydraulic system and developed control. Such control laws perform well and guarantee only local stability. The first nonlinear approach to control an electrohydraulic servo system was the standard dynamic feedback linearization technique, (Vossoughi and Donath, 1995). Details of such a technique can be found in Khalil (1995) and Slotine and Li (1991). A similar technique was also developed by Re and Isidori (1995). This technique ensures that the nonlinear system emulates a linear closed loop system by canceling the nonlinearities of the system in feedback. In the event of an inaccurate dynamic model, the technique alone could fail to guarantee the stability of the system. To guarantee stability of the hydraulic system in the presence of inaccurately modeled 13
dynamics, Alleyne and Hedrick (1995) used sliding mode control, (Utkin, 1978). Their technique guaranteed convergence of the hydraulic actuator position to a desired trajectory. Optimal and robust control approaches which involved formulating a nonlinear H∞ problem and subsequent solution using the Hamilton-Jacobi equation (van der Schaft, 1992) was developed by McLain and Beard (1998) and McLain and Beard (1997) respectively. Another often used technique for the control of nonlinear systems is the back-stepping approach, (Krstic et al., 1995). This Lyapunov based approach was adopted for control design of hydraulic servo-systems by Sirouspour and Salcudean (2000). Later, Sirouspour and Salcudean (2001) extended the design to control of a hydraulically actuated stewart platform robot while integrating adaptive robust methodologies to handle parametric uncertainties. A Lyapunov based back-stepping design was developed to achieve position tracking of a loaded actuator, (Sohl and Bobrow, 1999). Another version of the back-stepping control technique was used to develop an Adaptive Robust Control (ARC) scheme (Yao, 1997) for a class of nonlinear systems. Later, a discontinuous version of the approach was applied to single-rod hydraulic actuators, (Yao et al., 2000). The control method guaranteed asymptotic tracking of an inertial load in the presence of parametric uncertainties, uncompensated friction forces and external disturbances. The above mentioned research approaches specifically target the tracking problem of a hydraulic actuator. While this is not exactly the problem that arises in the proposed research the ideas associated with control of mismatched nonlinear systems (systems with relative degree greater than 0) are used in the design of coordination and passive control laws for the hydraulic backhoe. 2.1.5
Bondgraphs
Bondgraphs are a means to represent the dynamic behavior of physical systems using graphic methods, (Karnopp and Rosenberg, 1975). They capture, graphically, the power transfer associated within the various subsystems in a dynamical system. It is for this reason that bondgraphs are used in this dissertation. Using bondgraphs we will be able to identify certain passive and non-passive properties of the dynamical system. Bondgraph representation of physical systems is composed of a few basic subsystems interconnected in a systematic manner. The subsystems are shown in Fig. 2.5. Every 14
Se
Sf I
e1 f1 e2 f2 e3 f3
C
e4 f4
R
e5 f5
m TF
e6 f6
f15
e15
e7 f7
e14 f14
r e10 e11 GY f10 f11
0 e16 f16 e18 f18
e17 f17
1
e19 f19
Figure 2.5: Bondgraph subsystems half arrowed bond indicates flow of effort and flow in opposite directions. The half arrow indicates the direction of power. The stroke at one end of a half arrow is called the causal stroke. It determines the causality behavior of the bond. The variable on the side of the causal stroke has an effort imposed on it by the system and returns a flow to the system. In the figure, only the effort variable associated with each bond is written ei . There are implicit flow’s associated with each effort. Let the flow associated with effort ei be defined as fi . Then, the dynamics of the various subsystems shown in the figure are given as follows. Effort Source: Se e1 := Se Flow Source: Sf f2 := Sf Inductor: I f3 := I
−1
Capacitor: C e4 := C
−1
e3 (τ )dτ
f4 (τ )dτ
Resistor: R f5 := R(e5 ) Transformer: T F e7 := me6 ; f6 := 15
1 f7 m
m SS1
SS2
e21
e21
e24
f21
f21
f24
e29
e25
TF
f25
e28 f28
f29 0 e30 f30
e22 e22 f22
f22
e26
r GY
f26
e32
e27 f27
e23 I1
e31 f31
f32 1 e33 f33
f23
Figure 2.6: Bicausal bondgraph subsystems Gyrator: GY 1 1 f11 := e10 ; f10 := e11 r r Flow sum: 0 e15 = e16 = e14 ; f16 := f14 − f15 Effort sum: 1 f19 = f17 = f18 ; e18 := e17 + e19 Researchers, (Gawthrop, 1995) have also proposed bicausal bonds. While a power bond in bondgraphs always has one causal stroke, bicausal bonds have two causal strokes. The use of bicausal bonds is explained a little later. Unlike regular bonds which are unicausal, bicausal bonds characterize systems which can exert both effort and flow on other systems. Some examples of bicausal bonds are presented by Gawthrop (1995) and are shown in Fig. 2.6. Effort and Flow Sensor: SS1 e21 := e21 ; f21 := f21 Effort and Flow Source: SS2 e22 := e22 ; f22 := f22 Linear Inductance: I
1 I := f23
e23 (τ )dτ
16
y1
y2 k2
k1
u1 m1
m2
m3
u2
b
Figure 2.7: Spring-mass-damper system schematic Transformer: T F e25 := me24 ; f25 :=
1 f24 m
Gyrator: GY 1 e26 := re27 ; f26 := f27 r Flow sum: 0 e30 = e28 = e29 ; f30 := f29 + f28 Effort sum: 1 f33 = f32 = f31 ; e33 := e32 + e31 Bicausal bonds are typically used to represent inverse systems. Inversion of system dynamics (system inversion) as a technique for control design using bondgraphs has been proposed and implemented by many researchers, (Gawthrop, 1995), (Ngwompo and Gawthrop, 1999), (Ngwompo et al., 2001a), (Ngwompo et al., 2001b), (Ngwompo et al., 1996). Given a dynamic system, system inversion based on bondgraphs is used to determine an appropriate control input to ensure the regulation of a particular state (effort or flow). This technique is summarized in the following subsection using an example presented by Ngwompo et al. (2001a). 2.1.5.1 Model-based inversion for control design: An example, (Ngwompo et al., 2001a) Consider the bondgraph of spring-mass-damper system whose schematic is shown in Fig. 2.7 and its bondgraph is shown in Fig. 2.8. The dynamics of the spring mass
17
C:1/k2 SS:y1 u1
e=0 1 I:m1
C:1/k2 SS:y2 0
e=0 1 I:m2
R:b 1 0
1
u2
I:m3
Figure 2.8: Bondgraph of the spring-mass-damper system with the shortest causal paths (dotted lines) damper system as derived from the bondgraph can be written as follows: ⎞⎡ ⎤ ⎛ ⎡ ⎤ ⎛ 0 −k1 p1 1 p˙1 0 0 0 ⎢ ⎜ ⎟ ⎥ ⎢ ⎥ ⎜ 1 0 − m12 0 0 ⎟ ⎢ q1 ⎥ ⎜0 ⎢q˙1 ⎥ ⎜ m1 ⎟⎢ ⎥ ⎜ ⎢ ⎥ ⎜ b ⎢ ⎥ ⎜ ⎢p˙2 ⎥ = ⎜ 0 k1 − m2 −k2 mb3 ⎟ ⎟ ⎢p2 ⎥ + ⎜0 ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎜ ⎢ ⎥ ⎜ 1 0 0 − m13 ⎠ ⎣ q2 ⎦ ⎝0 ⎣q˙2 ⎦ ⎝ 0 m2 b 0 p˙3 0 0 k1 − mb3 p3 m2 ⎡ ⎤ p1 ⎥ ⎢ ⎢q1 ⎥ 1 ⎥ y1 0 0 0 0 ⎢ ⎢p2 ⎥ = m2 ⎥ y2 0 0 m12 0 0 ⎢ ⎢ ⎥ ⎣q2 ⎦ p3
⎞ 0 ⎟ 0⎟ ⎟ u1 0⎟ ⎟ u ⎟ 2 0⎠ 1
(2.8)
The control objective is to determine the control forces, u1 and u2 so that y1 and y2 can be regulated to 0. The Sequential Causality Assignment Procedure for Inversion (SCAPI) as outlined by Ngwompo et al. (2001a), is written below. 1. Determine the shortest causal path between the output and the input. The shortest causal path by definition is the one which minimizes the difference between the number of integral and differential elements along the path. In the present case, the shortest causal paths are shown in the Fig. 2.8 as a dotted line. 2. Represent, using bicausal bonds, the inverse bondgraph of the original bondgraph. By starting at the output, say y2 , we need the bondgraph to indicate that y2 imposes a flow on the 1-junction, thus making it an input to the inverse system. This is represented using an Effort and Flow source (SS1 in Fig 2.6). In order to ensure the correct causal representation of the 1-junction, 1) the I bond switches causality 18
C:1/k2 SS:y1 u1
e=0 1
C:1/k2 SS:y2 0
I:m1
e=0 1 I:m2
R:b 1 0
1
u2
I:m3
Figure 2.9: Inverse bondgraph of the spring mass damper system (represented by switching sides of the causal stroke) and 2) effort is summed and imposed on the adjacent 0-junction along the causal path. Proceeding in this manner, the inverse bondgraph is constructed and is shown in Fig. 2.9. The dynamics of the inverse bondgraph system are now given by: r˙1 0 0 r1 1 −1 y1 = k1 + r˙2 − kb2 r2 y2 0 − m2 d b dt b k1 0 r1 u1 = m3 k1 k2 m3 k22 u2 r2 − k1 − b2 2 b 0 m1 dtd d + 3 k2 − m3bk1 m3bk1 + m2 + m3 − m2 m + 2 b dt
y1 m2 m3 d2 b dt2
y2
(2.9)
Notice that by virtue of system inversion techniques, it is possible to design control laws which control variables (y1 and y2 in this particular case). Using the above outlined procedure, a coordination control algorithm will be designed in chapter 5 for teleoperation of the hydraulic backhoe. The method is powerful as it suggests the control law in a natural and systematic way. As a result it is possible to determine, in general, a control law which ensures the coordination behavior of arbitrary order master and slave passive systems that can be represented as bondgraphs. The general procedure is presented in chapter 6.
2.2
Problem Formulation
From the previous section it is apparent that none of the current work in teleoperation of hydraulic machines guarantee its passivity property. On the other hand, researchers have exploited the passivity property of mechanical systems and developed appropriate coordination and passivity control laws. Keeping the benefits of passive machines 19
q˙
Fx1 Passifying control
x1v
Tq
Teleoperation CONTROL 1
Fx2
q˙
Passifying control
Fq Master Force Feedback Joystick
Passified Electrohydraulic Valve 1
x2v Teleoperation CONTROL 2
Passified Electrohydraulic Valve 2
Q1c Pc1 Q1r Pr1 Q2c Pc2 Q2r Pr2
FL1
x˙
x˙ 1 Hydraulic Actuator 1
FL2
−Fe
x˙ 2 Hydraulic Actuator 2
Environment Slave Backhoe
Figure 2.10: The teleoperation setup consists of a 1) master (motorized joystick), 2) electrohydraulic passified valves, 3) single-ended actuators, 4) slave (backhoe) in mind, this research is directed at developing a passive teleoperation framework for multiple degree of freedom hydraulic actuated systems. The schematic of a 2-DOF teleoperation setup which is considered in this dissertation is shown in Fig. 2.10. The teleoperation setup consists of a master manipulator which is a 2-Degree of Freedom motorized joystick, two electrohydraulic valves which are four-way single-stage directional control valves, two hydraulic single-ended actuators and a 2-Degree of Freedom inertial slave arm (the backhoe). The primary research objective is towards seeking a controller that acts on both the master joystick (through the motor) and the slave manipulator (through the hydraulic actuators) to achieve coordinated motion of the two systems. The control inputs to the joystick are the force feedback motor torque, Fq and the control inputs to the valves are the exogenous (will be explained later) spool actuation forces Fx1 , Fx2 . Often the range of master manipulator motions are smaller compared to the range of motions expected of the slave manipulator. Hence, the controller needs to be able to scale the motion of the operator while achieving coordination. This is acheived by scaling the joystick position and then defining a coordination error. Another objective of the proposed research is to ensure that the closed loop system ˙ (Human input torque, joystick velocinteracting with the human via the pair (Tq , q) ity) and the environment via the pair (−Fe , x) ˙ (Environment input torque, actuator velocity) in Fig. 2.10 is passive with respect to a supply rate which is the sum of the scaled human power input and the environment power input. A power scaling ρ(> 1) 20
effectively amplifies the physical power that the human exerts on the virtual rigid, passive mechanical system that the teleoperation system will mimic. In view of the coordinating and passivity requirements of the controller, the problem is formulated as follows. To find a controller which guarantees Coordination
Passivity
αq(t) → x(t) as t → ∞ 0
∞
q(τ ˙ )T ρTq (τ ) − x(τ ˙ )T Fe (τ ) dτ ≥ −c2 ,
(2.10)
(2.11)
s((Tq ,Fe ),(q, ˙ x)) ˙
where q(t) and q(t) ˙ are the angular position and velocity of the master manipulator, x(t) and x(t) ˙ are the position and velocity of the slave manipulator, Tq (t) is the Torque input by the human operator on the joystick, Fe (t) is the environment force experienced by the slave manipulator, ρ ∈ R++ 1 and α ∈ R++ are power and kinematic scalings. If ρ > 1 the human will have to input less power than the environment to balance the power supply rate and vice versa if ρ < 1.
2.3
Summary
In this chapter, past research on the control of hydraulic machines was presented. It was shown that researchers aimed at improving interaction ability between the operator and hydraulic machine by proposing autonomous machines with higher level, remote human operation on one hand and teleoperated machines on the other. The deficiencies with the former architecture is that autonomous control of the hydraulic machine might break down in an unstructured, uncertain environment. The deficiencies with the latter teleoperation architectures is that poorly modeled operator / work environment may potentially destabilize the machine. The research approach in this dissertation is to combine the high performance behavior of hydraulic systems with the stability benefits offered by passive systems to ensure stable and safe human operated hydraulic machines. In order to achieve this, it is necessary to not only consider the mechanical dynamics of the machines but also consider the underlying hydraulic dynamics of the machines. Past research in a) control of hydraulic systems and b) passive control of systems will be used to ensure that the proposed passive hydraulic teleoperated machine will have desirable performance properties. Both the coordination and passive control methodologies will be developed by exploiting the 1
R++ = (0, ∞)
21
benefits offered by bondgraph modeling of dynamic systems.
22
Chapter 3 System models and identification In this chapter, the models of the various subsystems of the hydraulic teleoperation setup are presented. A schematic of the Two Degree of Freedom teleoperation setup considered in this dissertation is shown in Fig. 3.1. ti represents the variable associated with link i. Bold face T represent vectors / matrices with the variable as components i.e., T = Diag[t1 , t2 , . . . , tn ] or T = [t1 , t2 , . . . , tn ] as the case may be.
3.1
Subsystem models
The teleoperation setup consists of: 1. A two-port 2-DOF motorized joystick which acts as the master during teleoperation. q˙
Fx1 Passifying control
x1v
Tq
Teleoperation CONTROL 1
Passified Electrohydraulic Valve 1
Fx2
q˙
Passifying control
Fq Master Force Feedback Joystick
x2v Teleoperation CONTROL 2
Passified Electrohydraulic Valve 2
Q11 P11 Q12 P21 Q21 P12 Q22 P22
FL1
x˙
x˙ 1 Hydraulic Actuator 1
FL2
−Fe
x˙ 2 Hydraulic Actuator 2
Environment Slave Backhoe
Figure 3.1: The teleoperation setup considered in this dissertation 23
Figure 3.2: Electro-mechanical joystick. Each link is about 10 inches long 2. Two single-stage proportional valves which are feedback controlled to behave like passive two-port devices. 3. Two passive two-port hydraulic actuators. 4. A two-port 2-DOF mechanical backhoe which acts as the slave during teleoperation. 3.1.1
2-DOF Master motorized joystick
The joystick is modeled as a planar, horizontal, rigid 2-link robotic system. It is shown in Fig. 3.2. Its dynamics are given by: q + Cq (q, q) ˙ q˙ = Fq + Tq , Mq (q)¨ where Mq (q) = Mq (q)T > 0 and
d Mq (q)−2Cq (q, q) ˙ dt
(3.1)
is skew-symmetric, Fq and Tq
are the motor actuated control torque and the human input torque on the joystick. q = [q1 , q2 ] is a vector of the link angular positions. Notice that gravity does not affect the dynamics as the joystick motion is constrained to the horizontal plane. The joystick is a passive 2-port subsystem with human operator interacting at one-port and the motor providing feedback torque at the other port. The joystick behaves like
24
a passive 2-port system with respect to the supply rate, sj ((Fq , q), ˙ (Tq , q)) ˙ = q˙ T (Fq + Tq ). which can be verified by differentiating the kinetic energy storage function, (Spong and Vidyasagar, 1989) 1 Wjoystick = q˙ T Mq (q)q˙ 2
(3.2)
and substituting the dynamics (3.1). 3.1.2
Single-stage passive valve
The schematic of a conventional single-stage valve similar to the one we use is shown in Fig. 3.3. The spool is stroked with a force U by a solenoid, resulting in valve displacement xv . As a result, high pressure hydraulic fluid flow Q1 is metered through orifice 1 into the capside chamber of the hydraulic actuator. Return flow Q2 is metered through orifice 3 from the rodside chamber. This combination of flow results in downward extension of the actuator piston, x. Assuming symmetric, matched spool and incompressible flow in the valve, the flow rates, Q1 and Q2 are given by Qi1 = Ai1 (KQi xiv − KTi (|xiv |, FLi ) · FLi )
(3.3)
Qi2 = −Ai2 (KQi xiv − KTi (|xiv |, FLi ) · FLi ).
(3.4)
where FLi = Ai1 P1i − Ai2 P2i is the differential hydraulic load force with subscript ‘1’ refering to the capside parameter and subscript ‘2’ refering to the rodside parameter, Ai1 and Ai2 are the capside and rodside piston area, KQi is the no load flow gain of the valve, KTi (|xiv |, FLi ) is the shunt conductance that represents the loss of the valve in the presence of a load FLi . The noload flow gain and the shunt conductance are given as follows: KQi =
Kqi
As Ps i3 Ai3 1 + A2 K i |xi | Q v KTi (|xiv |, FLi ) = √ A P As Ps − sign(xiv )FLi ⎧ s s ⎨Ai , xi ≥ 0 1 v As = ⎩Ai , xi < 0 2
v
25
(3.5) (3.6)
(3.7)
Ui Actuator
P1i
4
Supply Psi
1
Ai1
Qi1
P1i
xi
FLi = Ai1 P1i − Ai2 P2i
2
i P2i −Q2
Return 0
Ai2
P2i
3
xiv
Spool
Fei Figure 3.3: Single stage valve connected to a single-rod actuator Notice that KTi (|xiv |, FLi ) · FLi2 ≥ 0 for all xiv , FLi . Throughout this paper, the superscript i of the symbols is used to represent the variable corresponding to link i of the backhoe or joystick (as applicable). Eqs. (3.3)-(3.7) are merely rearrangements of the 4 way valve equations taking into account the single rod actuator, and separating the no-load behavior and the loaded behavior. Dynamics of the valve spool are given by i x¨iv = U i
(3.8)
where i is the spool inertia and U i is the solenoid stroking force. It was shown (Li, 2000) that, in general, single stage valves (3.3)-(3.8) behave like non passive devices with respect to a supply rate that includes the hydraulic power withdrawn from the hydraulic ports of the valve. This means that if the valve is stroked to a non-zero position, (xv = 0), it is possible to extract infinite energy from the valve resulting in its non-passive behavior. An active passification control technique was proposed, (Li, 2000) which ensured that single-stage proportional valves behaved like passive two-port hydraulic elements with respect to the supply rate sv
Kq xv ), (PL, QL ) (Fx , A
26
= Fx ·
Kq xv − PL · QL , A
where Fx is an exogenous control input to the passified valve and KAq xv is the corresponding flow variable. The first term in the supply rate is the fictitious power input to the passive valve by the exogenous control input (implemented using the stroking solenoid) (port 1) and the second term represents the actual hydraulic power extracted by an attached load (like an actuator) (port 2). However, passification of this valve by Li (2000) assumed that flow metered out of the valve is equal to the flow rate metered into the reservoir which is true only when the valve is driving a symmetric double-rod actuator. The algorithm presented in Li (2000) is modified to include situations where the ratio of flow rates coming in and out of the valve are related by the ratio of the cross-sectional areas of the control volumes within the hydraulic load (single-rod actuator in this case). The basic idea of passification control remains the same, viz. to incorporate load pressure feedback onto the valve spool. The control law which renders the valve as a passive three port device is stated in the following lemma. Lemma 2. The choice of valve spool stroking force control input in (3.8) i U i = −B i x˙ iv − γ i FLi + Fxi + Fact ,
(3.9)
where γ i > 0 is the load force feedback gain and $ i " # i# i i i i ˙ ˙ = − µ + i Fx − γ FL − g2 (t)sgn(z ) , B 1 i i i z := x˙ v − i Fx − γ i FLi , B " $ # # i i ˙ i − γ iF ˙ i }sgn(z i ), g2 (t) > {[F˙x − γ i F˙Li ] − F x L i Fact
i i
B i xiv
(3.10) (3.11) (3.12)
# is the estimate of the argument (·), ensures that the single-stage valve dewhere (·) scribed by Eqs. (3.3), (3.4), (3.8) behaves like a passive 3-port device with respect to the supply rate: sv
(Fxi , xiv ), (P1i , Qi1 ), (P2i, Q2 )
KQi i i = i Fx xv − P1i Qi1 − P2i Qi2 . γ
(3.13)
Proof: Consider the storage function Wvalve
& % Kqi i 2 1 Kqi B i i 2 = (xv ) + i i (z ) . 2 γi γµ
27
(3.14)
Upon differentiating the storage function w.r.t. time and substituting the dynamics (3.11) for x˙ iv and (3.9) in (3.8) for z˙ i , we get: ˙ valve W
i Kqi B i i i i Kqi i 1 B i i Fact i i i i i = x (B z + Fx − γ FL ) + i i z − i z + − i (Fx − γ FL ) . γi v γµ i B
i Upon substituting for Fact and g2i using (3.10) and (3.12), respectively, and for FLi using (3.3) and (3.4), we get:
Ki ˙ valve ≤ sv (F i , xi ), (P i, Qi ), (P i , Qi ) = Q F i xi − P i Qi − P iQi . W x v 1 1 2 2 1 1 2 2 γi x v Integrating this equation proves that the valve behaves like a passive 3-port device with respect to the supply rate (3.13). The first term in the supply rate is the power input by the exogenous control Fxi , the sum of the second and the third terms is the net hydraulic by $the " power withdrawn $ " # # ˙ ˙ ˙ ˙L is hydraulic load (single-rod actuator). If the estimate of Fx − ΓFL , Fx − ΓF accurate, then the passive valve dynamics are given by:
0 B Fx − ΓFL xv Bx˙ v = + z˙ −Bµ −B z 0
(3.15)
where xv (z) = [x1v (z 1 ), . . . , xnv (z n )]T , Fx = [Fx1 , . . . , Fxn ]T , FL = [FL1 , . . . , FLn ]T , Ω1 = Diagi [ω1i ], Γ = Diagi [γ i ], Bµ = Diagi [µi B i ], B = Diagi [B i ] and B = Diagi [B i /i ]. Notice that the passification algorithm decreases the relative degree between the new control input Fx to the output xv from 2 in (3.8) to 1 in (3.15). In practice, it is natural for proportional valves to behave like first order systems (between stroking force and spool displacement) due to integrated spool position control electronics. In such an event, the passification algorithm can be modified to include only the load force feedback on the spool. 3.1.3
Passification of the Vickers Proportional valve
Vickers manufactured KBFDG4V-5 series proportional valves used for experimentation of the algorithms presented in this dissertation behave like first order systems upto 50 Hz due to the integrated valve electronics. A picture of the valve is shown in Fig. 3.4. In order to determine the dynamic behavior of the valve, they were tested to a pseudo random binary sequence input. A frequency response plot of the experimentally observed data with the modeled fit (least square estimation of data collected 28
Figure 3.4: The Vickers proportional valve mounted on a manifold. over multiple experiments) is shown in Fig. 3.5. It was observed that all the three valves which are used in the experiments in this dissertation behave similarly and the plot shown is for one run of one of the valves. The dynamics of the proportional valve are modeled as: x˙ v = −Ω1 xv + Ω12 U. In order to ensure that such a valve behaves like a passive valve w.r.t. the supply rate (3.13) it is enough to choose the proportional valve control to implement pressure feedback on the spool as follows: Ω12 U = Fx − ΓFL .
(3.16)
The fundamental difference between the proposed valve passifying control (3.9) and the actual valve passifying control (3.16) is due to relative degree of the spool dynamics. In the former case, the valve spool dynamics were modeled as a second order system (3.8). However, in reality, the dynamics were observed to be of first order and so it is not necessary to implement any feedforward control as does valve control (3.9). Moreover, first order stable dynamics are naturally passive w.r.t. a supply rate which is the product of input (U) and output (xv ). Hence, all that is necessary to ensure passivity of the first order Vickers valve is to ensure passivity w.r.t. the appropriate supply rate given in (3.13). This is accomplished by choosing the valve passifying control as given in (3.16). The passive valve dynamics are then given by: x˙ v = −Ω1 xv + Fx − ΓFL
29
(3.17)
10
dB
5
0
−5
−10
1
2
10
10
50
degrees
0
−50
−100
1
2
10
10 rad/sec
Figure 3.5: Experimental (solid) and modeled fit (dashed) for the Vickers proportional valve 3.1.4
Dynamic model of the hydraulic actuator
The pressure dynamics of a hydraulic actuator associated with link i are given by: V1i ˙ i P = Qi1 − Ai1 x˙ i , β 1
V2i ˙ i P = Qi2 + Ai2 x˙ i β 2
(3.18)
where V1i , V2i are the total volumes of the capside and the rodside chambers including the hose volume, respectively, β is the fluid compressibility, P1i, Qi1 , P2i , Qi2 are the pressures and flows in the actuator chambers, Ai1 , Ai2 are the cap and rod side piston cross-section areas and xi is the piston position. Although the actuator volumes V1i , V2i are dependant on the piston position xi , it is assumed that β β β ≈ ≈ i i Vj min(Vj ) max(Vji ) This is because the actuator chamber volumes are predominantly comprised of a constant hose volume. The hydraulic actuator is a passive 3-port subsystem with
30
respect to the supply rate: sa (P1i , Qi1 ), (P2i , Q2 ), (FLi , x˙ i ) = P1i Qi1 + P2i Qi2 − FLi x˙ i .
(3.19)
This supply rate is the difference between the cumulative input hydraulic power at the cap, (P1iQi1 ) and rod, (P2i Qi2 ) side ports and the output mechanical power, (FLi x˙ i ). The passivity property can be verified by differentiating the energy storage function: Wactuators =
' %V i 1
β
i
(P1i)2
Vi + 2 (P2i)2 β
& (3.20)
and substituting the dynamics (3.18). 3.1.5
Kinematic model of the hydraulic actuator
The hydraulic actuator can also be modeled as a velocity source in the absence of a driving load. In the absence of any leakages, neglecting the inertia of the actuator itself and assuming incompressible flow, the flow-velocity and pressure-force relationships for a single-rod actuator are given by: Ai2 x˙ i = −Qi2
(3.21)
Fe = Diagi [FLi ].
(3.22)
Ai1 x˙ i = Qi1 ,
where Fe is the environment force acting on the actuators through the backhoe. By multiplying the LHS and RHS of the above two equations, it can be seen that the kinematic modeled actuator is also passive w.r.t. (3.19). Henceforth, the hydraulic backhoe model with kinematically modeled actuators is refered to as the kinematic modeled backhoe. 3.1.6
Backhoe dynamics
A figure of the backhoe is shown in Fig. 3.6. The backhoe is modeled as a planar (in the vertical plane), rigid 2-link robotic system. Its dynamics are given by, x + Cx (x, x) ˙ x˙ = FL − (Gr (x) + Fr (x)) ˙ , Mx (x)¨
(3.23)
Fe
˙ is the inertial matrix where Mx (x) = Mx (x)T > 0 is the inertial matrix and Cx (x, x) due to coriolis forces, FL = A1 P1 − A2 P2 is the differential hydraulic force acting 31
Figure 3.6: Hydraulic backhoe in the Fluid Power Control Laboratory. Link 1 is 44 inches long. on each backhoe link, x˙ is a vector of the actuator piston velocities and Fe is the net environment force acting on the backhoe which includes the gravitational torque Gr and the frictional torque Fr . It is well known, (Spong and Vidyasagar, 1989) that d Mx (x) − 2Cx (x, x) ˙ is skew-symmetric. The environment force includes effects due dt to friction (links and actuators) and gravity. The mechanical backhoe is a passive 2-port subsystem with respect to the supply rate, sb ((FL , x), ˙ (−Fe , x)) ˙ = x˙ T (FL − Fe ), which can be verified by differentiating its kinetic energy storage function 1 Wbackhoe = x˙ T Mx (x)x˙ 2
(3.24)
and substituting the dynamics (3.23). System identification tests performed on the various subsystems of the hydraulic backhoe led to the following choice of values for the various hydraulic parameters. The joystick is modeled as a 2-DOF robotic system. Its inertial parameters were
32
Ps (psi.) Ac (sq in) Ar (sq in) KQc (1/sec) KQr (1/sec) β (psi) Γ (N/volt) Ω1 (rad/sec) Ω12 (rad/sec)
Bucket link 1000 3.14 1.67 1.85 1.79 165000 3 282 280
Stick link Boom link 1000 1000 3.14 4.35 1.67 2.17 1.85 1.85 1.79 1.79 165000 165000 3 282 282 280 280
Table 3.1: Parameters necessary for implementation of control laws modeled by Lee and Li (2002) as: Mq (q) = Cq (q, q) ˙ =
0.08894 + 0.0142 cos(q2 ) 0.01125 + 0.0071 cos(q2 ) 0.01125 + 0.0071 cos(q2 ) 0.01125 −0.0071 sin(q˙2 ) −0.0071 sin(q˙2 + q˙1 ) 0.0071 sin(q˙1 )
0
The Backhoe is modeled as a 3-DOF robotic system. Its inertial, gravitational and frictional parameters are modeled as: ⎛ ∗ ψ1 + ψ2 + ψ3 + 2l1 c2 φ1 + 2(l2 c3 + l1 c23 )φ2 ⎜ Mx (x) = ⎝ ψ1 + ψ2 + l1 c2 φ1 + (2l2 c3 + l1 c23 )φ2 ψ1 + ψ2 + 2l2 c3 φ2 ψ1 + (l2 c3 + l1 c23 )φ2 Mx (x) = Mx (x)T ⎛ ⎞ C11 C12 C13 ⎜ ⎟ Cx (x, x) ˙ = ⎝C21 C22 C23 ⎠ C31 C32 0 ⎞ ⎛ c1 φ3 + c12 φ1 + c123 φ2 ⎟ ⎜ Gr (x) = ⎝ c12 φ1 + c123 φ2 ⎠g c123 φ2 ⎞ ⎛ δ1 sgn(x˙ 1 ) − δ2 x˙ 1 ⎟ ⎜ ˙ = ⎝δ3 sgn(x˙ 2 ) − δ4 x˙ 2 ⎠ , Fr (x) δ5 sgn(x˙ 3 ) − δ6 x˙ 3
33
ψ1 + l2 c3 φ2
⎞ ∗ ⎟ ∗⎠ ψ1
φ1 φ2 φ3 ψ1 ψ2 ψ3 δ1 δ2 δ3 δ4 δ5 δ6
0.014 0.0015 0.128 0.063 0.5759 4.3355 -4.56 3.814 -1.1272 4.99 -2.33 0.93
Table 3.2: Inertial, gravitational and frictional parameters of the hydraulic backhoe where C11 = −l1 s2 x˙ 2 φ1 − (l1 s23 x˙ 23 + l2 s3 x˙ 3 )φ2 C12 = −l1 s2 x˙ 12 φ1 − (l1 s23 x˙ 123 + l2 s3 x˙ 3 )φ2 C13 = −(l1 s23 + l2 s3 )x˙ 123 φ2 C21 = l1 s2 x˙ 1 φ1 + (l1 s23 x˙ 1 − l2 s3 x˙ 3 )φ2 C22 = −l2 s3 x˙ 3 φ2 C23 = −l2 s3 x˙ 123 φ2 C31 = (l1 s23 x˙ 1 + l2 s23 x˙ 12 )φ2 C32 = l2 s3 x˙ 12 φ2 l1 = 46in; l2 = 28in; l3 = 17in cijk = cos(xijk ); sijk = sin(xijk ) xijk = xi + xj + xk , and the various inertial parameters are given in Tab. 3.2. In order to determine the inertial properties of the backhoe, unconstrained sinosoidal trajectory tracking experiments were conducted about various geometric link configurations (‘pose’). The comparison of estimated load force (computation of Mx (x)¨ x+ ˙ x˙ + Fe in (3.23)) and measured load forces (measured FL in (3.23) using Cx (x, x) pressure transducers) of the backhoe is shown in Fig. 3.7. Notice that the boom estimated load force (solid line) has a characteristic high frequency oscillatory signal. 34
This is due to the structural vibrations induced by insufficient stiffness of the backhoe arm (boom + stick + bucket).
3.2
Control Objective
The central objective of the controllers designed in this dissertation is to ensure that the teleoperation system shown in Fig. 3.1 has the following properties. • It is passive with respect to the supply rate: stele ((ρTq , q), ˙ (Fe , x)) ˙ = ρq˙ T Tq − x˙ T Fe .
(3.25)
where ρ is the desired power scaling factor. In Eq. (3.25), the first term is the scaled human power and the second term is the environment power input to the teleoperated backhoe. Human power input to the teleoperator is scaled using ρ. Satisfying the above passivity condition will ensure that the joystick teleoperated backhoe will behave like a passive 2-port device and thus guarantee stable interaction with the passive power scaled human and work environment. • In addition, the teleoperation control should also ensure that the scaled coordination error E := αq − x → 0,
(3.26)
asymptotically, where α is a kinematic scaling. This ensures that the backhoe motion mimics the scaled joystick motion. The above objectives can be achieved using at least two different approaches. The first approach is with the aid of the interconnection lemma, (Li and Krishnaswamy, 2001) (also explained in chapter 2) which states that the cascade interconnection of two passive systems results in a passive system with a modified supply rate. By virtue of this property, notice from Fig. 3.1 that the passive valve, hydraulic actuators and the backhoe form a cascade interconnected passive system. In order to ensure the passivity property of the teleoperated machine, we need to determine a teleoperation controller which is intrinsically passive with respect to the supply rate, sc ((Fq , q), ˙ (Fx , xv )) = −ρq˙ T Fq − xv T KQ Γ−1 Fx . 35
(3.27)
7
6
Modeled and Actual Torques
5
Boom
4
3
2 Stick 1
0 Bucket −1
0
100
200
300
400
500
600
400
500
600
pose 6
5 Boom
Modeled and Actual Torques
4
3
2
1 Stick
0 Bucket −1
0
100
200
300 pose
Figure 3.7: Modelled (dashed) and measured forces (solid) of the hydraulic backhoe
36
This will ensure that the cascade interconnection of the joystick, controller, passive valve, hydraulic actuators and backhoe (as shown in Fig. 3.1) is also passive w.r.t. the overall system supply rate (3.25). The controller will also need to ensure that the master and slave systems are coordinated. Another approach to designing a passive teleoperation controller is by using bondgraphs to analyze the passivity property of the coordinated teleoperation system. In this approach, the controller is first designed to ensure coordination of the teleoperated system. Bond graphs are then used to analyze the energy flow in the coordinated system and the controller is modified to ensure passivity of the teleoperated system w.r.t. the supply rate (3.25) while maintaining coordination betweem the master and the slave systems. Although this approach ensures passivity of the teleoperated system, the controller itself is not necessarily passive w.r.t. its supply rate sc as given in (3.27). As a result, it will not be possible to ensure that the cascade interconnection as shown in Fig. 3.1 is passive. However, the overall interconnected system will still be passive with respect to the same supply rate, (3.25). In chapter 4, an Intrinsically Passive Controller is designed for the teleoperated backhoe with kinematic modeled hydraulic actuators described by Eq. (3.21). This controller is designed using the first approach described above. In chapter 5 a controller is designed for the nonlinear modeled (nonlinearity due to modeling fluid compressibility and inertial effects) hydraulic backhoe (described by Eqs. (3.18), (3.23)) using the second approach (bondgraph based) detailed above. The bond graph based approach is used to design a passive teleoperation controller for a generic cascade passive master and slave systems in chapter 6.
37
Chapter 4 Passive teleoperation of the kinematic modeled hydraulic backhoe In this chapter a controller for passive teleoperation of a multi degree of freedom simplified kinematic modeled teleoperator is proposed in this chapter. The teleoperated system which is considered in this chapter consists of: • A master force feedback joystick whose dynamics are given by (3.1). • A single stage passified valve whose dynamics are given by (3.17). • A kinematic modeled hydraulic backhoe which consists of a) kinematic modeled hydraulic actuators and b) Backhoe with neglected inertia. The dynamics of the kinematic modeled backhoe are given by(3.21). The teleoperation controller ensures passive two-port behavior of the teleoperated backhoe w.r.t. a scaled power input as its supply rate. The passivity property of the hydraulic teleoperator ensures interaction stability with any human operator and the work environment that could be modelled passive. The intrinsically passive control ensures two-port rigid coordinated behavior of the passive valves driven teleoperated backhoe.
38
4.1
Passive teleoperation controller design using the Interconnection Lemma
In previous research a teleoperation controller was designed for a similar kinematic modeled slave hydraulic system, (Li and Krishnaswamy, 2001), (Krishnaswamy, 2000). The differences being that a) the passive valve was approximated by its low-frequency behavior, b) the passive teleoperated system was of a single DOF. The design procedure involved a) choosing a control architecture which ensured that the teleoperated system was passive with respect to a particular supply rate and b) choosing the design variables in the passive control architecture which ensured coordination of the master and slave systems. The teleoperation control algorithm presented in this chapter is similar to the previously developed algorithm, (Li and Krishnaswamy, 2001), (Krishnaswamy, 2000). The first stage involves choosing the control structure which ensures that the teleoperated system is passive with respect to the supply rate (3.25). In the second stage, the control ‘design variables’ are determined so as to ensure the coordination of the joystick and the backhoe configurations, (3.26). 4.1.1
Passive controller structure
Consider the storage function of the teleoperated backhoe, Wtele =
1 ρ · q˙ T Mq (q)q˙ + xv T KQ Γ−1 Bxv + zT KQ Γ−1 µ−1 z+ 2 & i ' %V i V 1 (P1i )2 + 2 (P2i)2 + x˙ T Mx (x)x˙ + f˙ T Mf f˙ β β i
(4.1)
Notice that the first term is the kinetic energy of the joystick system scaled by ρ, the second and third terms are the storage function of the passive valve, the fourth term is the energy of the compressed fluid in the hydraulic actuators, the fifth term is the kinetic energy of the backhoe and the last term in (4.1) is the kinetic energy of the fictitious flywheel. In order to ensure that the controller ensures passivity of the teleoperator, robustly, a fictitious energy storage element that mimics a flywheel is used. Its dynamics are given by: Mfi f¨i = Ffi ,
(4.2)
where Mfi is the flywheel inertia, f i is the flywheel angular velocity and Ffi is the flywheel control force. 39
In order to determine the passivity property of the teleoperated system, the power flow through the system is analyzed. Hence, taking the time derivative of the energy stored, (4.1), and substituting the dynamics of the joystick (3.1), passive valve (3.17), hydraulic actuator chambers (3.18), backhoe (3.23) and flywheel (4.2), and simplifying: ˙ tele ≤ ρq˙ T Tq − x˙ T Fe + ρq˙ T Fq + xv T KQ Γ−1 Fx + f˙ T Ff . W The following choice of a skew-symmetric control structure: ⎛ ⎡ ⎤ ⎞⎡ ⎤ ρI 0 0 q˙ Fq ⎥ ⎜ ⎥ ⎢ ⎟ ⎢ ⎝ 0 KQ Γ−1 0⎠ ⎣Fx ⎦ = Ψ(t) ⎣xv ⎦ f˙ Ff 0 0 I
Ψ(t) + Ψ(t)T = 0.
(4.3)
where Ψ(t) is to be designed, ensures that ˙ tele ≤ stele ((ρTq , q), W ˙ (Fe , x)) ˙ ,
(4.4)
so that on integration the desired passivity property given by Eq. (3.25) is achieved. Remark 1. Notice that the controller structure (4.3) guarantees the passivity property of the teleoperated backhoe regardless of the choice of the design parameters within Ψ(t), as long as it is bounded and satisfies the skew-symmetric property. Such a control implementation will ensure passivity property in the presence of uncertain system parameters. Remark 2. The controller is Intrinsically Passive with respect to the supply rate sc ((Fq , q), ˙ (Fx , xv )) = −ρq˙ T Fq − xv T KQ Γ−1 Fx . This property can be verified by differentiating the controller storage energy, 0.5 · f˙ T Mf f˙ (= flywheel storage energy) and substituting Eq. (4.3). The intrinsically passive nature of the controller ensures structural robustness to parameteric uncertainties of the teleoperation control implementation, (4.3). The skew-symmetric matrix Ψ(t) is determined to ensure that the joystick and the backhoe are coordinated. 4.1.2
Coordination controller
To determine the coordination control law, three assumptions are made. 40
1. Fluid is incompressible. This assumption leads to the following relation between spool position xv and piston extension x. x˙ = Q1 A1 −1 = Q2 A2 −1 .
(4.5)
By substituting for the Q1 and Q2 as given by (3.3) and (3.4) and simplifying, the above equation can also be written as ¯ Q (sign(xv ), FL ) · xv , x˙ = K ¯ Qi (sign(xiv ), FLi )] ¯ Q (sign(xv ), FL ) = Diag[K K ¯ Qi (sign(xiv ), FLi ) = √ K
K i sign(xiv ) Q · FLi . i i As Ps As Ps − sign(xv )FL
(4.6) (4.7)
The above relation can be verified using (3.18), (3.3) and (3.4). It is important ¯ Q ≥ 0 if F i ≤ As Ps which is the usual mode to note that the load flow gain K L of hydraulic operation. 2. Backhoe inertia is negligible. This assumption leads to the following relation between hydraulic differential force, FL and the environment force, Fe . FL = Fe .
(4.8)
The above relation can be verified using (3.23). 3. Negligible coriolis forces leading to decoupled joystick interia. This means that the joystick dynamics are approximated to be: Mq q ¨ = Fq + Tq .
(4.9)
It is important to note that all three assumptions are made only to determine the coordination control law. Passivity property of the teleoperated backhoe is guaranteed without the aforementioned assumptions. If the valve and joystick control inputs, Fx and Fq respectively, are designed to satisfy ¯ Q B−1 Fx = −D(t) αMq −1 Fq − K " $ −1 −1 ˙ ¯ ¯ ˙ D(t) = αMq Tq − KQ xv − KQ (z − ΓB FL ) + Λ1 E + Λ0 E ,
41
(4.10)
¨ + Λ1 E ˙ + Λ0 E = 0, hence guaranteeing where Λ0 = ΛT0 > 0, Λ1 = ΛT1 > 0 then E asymptotic coordination of the kinematically scaled (α) joystick and the backhoe. This can be verified by differentiating the coordination error twice and substituting (4.6)-(4.9) and (3.17). In order to implement (4.10), we choose Ψ(t) in (4.3) as follows: (the argument (t) has been dropped to avoid clutter): ⎛
0 0
⎜ Ψ(t) = ⎝
0 0
⎞ ¯ Q Ξ)−1 D −αφ(K ⎟ Ξ−1 φD ⎠,
¯ Q Ξ)−1 D]T −[Ξ−1 φD]T α[φ(K
(4.11)
0
¯ Q B−1 −1 and ¯ −1 + ρK where Ξ(t) = Diag [ξ i (t)], φ(t) = ρKQ Γ−1 α2 Mq −1 KQ Γ−1 K Q i
ξ (t) =
⎧ ⎨f˙i (t),
f˙i (t) ≥ fthreshold ∈ +
⎩f
f˙i (t) < fthreshold
threshold ,
.
(4.12)
The above choice of Ψ ensures that the coordination control law (4.10) is implemented when the flywheel velocity f˙i > fthreshold. In order to preserve passivity, if the flywheel velocity falls below the threshold, the coordination control law (4.10) is modified by virtue of the definition of ξ i(t) in (4.12). This property of the Ψ maintains the passivity property of the teleoperation system at the possible expense of coordination performance deterioration. We will now show that it is possible to ensure that the flywheel velocity will be always above the threshold. Notice that D(t) in (4.10), (4.11) consists of exogenous inputs Tq , Fe and states xv , z. Assuming bounded inputs and bounded states, D(t) is bounded. If the flywheel is initialized such that Wf lywheel (0) > δ +
∞
0
¯ −1D(τ )E(τ ˙ )|dτ |φK Q
˙ → 0 exponenand since the coordination control law, (4.3), (4.11) will ensure that E tially, Wf lywheel (t) > δ > 0, ∀t. ˙ also ensures that the initial flywheel energy Wf lywheel (0) Asymptotic convergence of E is lower bounded. As a result of initialization of the flywheel, the teleoperation control (4.3), (4.11) only uses bounded initial energy to ensure coordination. In the event that 42
˙ → 0 then ξ i(t) in the implementation f˙ falls below the threshold fthreshold before E structure (4.11)-(4.12) is used to limit the energy extracted from the flywheel for coordination purposes. This will lead to some loss in performance but passivity of the teleoperator will be preserved. The above analysis is summarized in the following theorem: Theorem 1. The teleoperation control algorithm given by (4.3), (4.11) ensures that the cascade interconnection of the force-feedback joystick (3.1) teleoperated passive backhoe (3.3), (3.4), (3.17)-(3.23) behaves like a passive two-port device w.r.t. the supply rate: ˙ (Fe , x)) ˙ = ρq˙ T Tq − x˙ T Fe . stele ((ρTq , q), Moreover, if the flywheel initial energy Wf lywheel (0) is appropriately initialized such that ∞ ¯ −1D(τ )E(τ ˙ )|dτ |φK Wf lywheel (0) > δ + Q 0
then the coordination error E := αq − x tends to 0, exponentially. Proof: Differentiate the following teleoperator storage function Wtele =
1 ρ · q˙ T Mq (q)q˙ + xv T KQ Γ−1 Bxv + zT KQ Γ−1 µ−1 z 2 & i ' %V i V 1 (P1i )2 + 2 (P2i)2 + x˙ T Mx (x)x˙ + f˙ T Mf f˙ + β β i
w.r.t. time and substitute the dynamics of the motorised joystick, (3.1), the passified valve, (3.17), the dynamic modeled hydraulic actuators (3.18) and the Backhoe, (3.23), to get: ˙ tele ≤ ρq˙ T Tq − x˙ T Fe + ρq˙ T Fq + xv T KQ Γ−1 Fx + f˙ T Ff . W Substitution of the passive control law (4.11) in the above equation leads to ˙ tele ≤ stele ((ρTq , q), ˙ (Fe , x)) ˙ = ρq˙ T Tq − x˙ T Fe W integrating which provides the desired passivity result. 43
To prove the asymptotic behavior of the coordination error, consider its second derivate; ¨ = α¨ ¨. E q−x Substitute the assumed joystick dynamics and the passive valve dynamics along with the kinematic actuator, (4.6)-(4.9) and (3.17) to get: ¯˙ Q xv − K ¨ = αMq −1 Fq − K ¯ Qz − K ¯ Q B−1 Fx + αMq −1 Tq + K ¯ Q B−1 ΓFe E Substitute the joystick and valve control input Fq and Fx in the above equation as given in (4.11) and assuming that the flywheel is appropriately initialized so that ξ i (t) = f˙i (t) to get: ¨ = −Λ1 E ˙ − Λ0 E, E ˙ → (0, 0), exponentially. hence proving that (E, E) To prove that the flywheel velocity f˙i (t) > 0 for all t, consider the flywheel storage energy 1 Wf lywheel = Σni=1 Mfi [f˙i (t)]2 . 2 Differentiating and substituting the flywheel control as given in (4.11) leads to: ˙ f lywheel = f˙ T φ(K ¯ QΞ)−1 D(t)E ˙ ≤ |φK ¯ −1 D(t)E|. ˙ W Q This is because, by definition, Ξ ≤ I. If the flywheel initial energy Wf lywheel is initialized such that Wf lywheel (0) > δ +
∞
0
¯ −1D(τ )E(τ ˙ )|dτ, |φK Q
¯ Q (t) are bounded and E ˙ → 0 exponentially then, since D(t), φ(t), K Wf lywheel (t) > δ, completing the proof.
4.1.3
Haptic behavior of the teleoperated backhoe
In the above theorem a passive teleoperation controller for a simplified kinematic modeled backhoe was designed. The controller uses a fictitious storage element to 44
ensure robustness of the passive teleoperator. The controller ensures that the teleoperator behaves like a tool operated by the human operator. The dynamics of the closed loop passive teleoperator (haptic behavior) are presented. Upon substituting the control (4.3) and (4.11) in the joystick dynamics, (3.1) and assuming that f˙i (t) ≥ fthreshold and E → 0, we get: % 2 & & α α2 −1 −2 −1 −3 ¯ ˙ ¯ ¯ KQ Γ BKQ KQ q˙ ¨− Mq + KQ Γ BKQ q ρ ρ
%
Mh (t)
−Ch (t)
α ¯ −1z − α KQ K ¯ −1 Fe + Tq , = KQ Γ−1 BK Q Q ρ ρ ¯ −1q˙ − B z. z˙ = −αµBK Q
(4.13)
˙ h (t) − 2Ch (t) is skew-symmetric (K ¯ Q is diagonal) thus indicating that the where M haptic behavior of the passive teleoperator mimic that of a coupled mechanical multiDOF system. These dynamics explain the behavior of the coordinated passive teleoperated backhoe. Notice that the operator feels a different inertia, Mh (t), some dissipation in the valve due to the term related to z and a scaled environment force, Fe . Notice that z manifests as a damping because at low frequency oper¯ −1 ˙ The dynamics ensure rigid mechanical damped tool ation, z ≈ −B−1 αµBKQ q. behavior of the teleoperated backhoe.
4.2
Summary
In this chapter, a control algorithm for passive teleoperation of a kinematic modeled backhoe was designed. The passive teleoperator ensured two-port rigid, damped interaction between an operator and the environment. However, the control designed included certain simplifications of the backhoe. It assumed 1) that the hydraulic flow was incompressible, 2) that the backhoe inertia was negligible and 3) that the joystick dynamics are decoupled. These assumptions led to teleoperator performance deterioration under some conditions (for example: large amplitude simultaneous motion of the two DOF). In the next chapter (chapter 5) the above assumptions will be removed and a teleoperation control algorithm will be designed for the full order nonlinear dynamic modeled backhoe operated by the nonlinear modeled inertial motorized joystick. The algorithm that will be presented is based on bond graphs. Bond graphs are used as 45
they provide physical insight about the flow of power in the coordinated teleoperator. The bond graph technique can be generalized to teleoperation of ararbitrary order master / slave cascade passive systems. The generalized result is presented in chapter 6.
46
Chapter 5 Passive teleoperation of the hydraulic backhoe using bondgraphs 5.1
Introduction
In this chapter, a systematic control design approach for passive teleoperation of a dynamic modeled hydraulic backhoe is proposed. The control procedure will be generalized for passive teleoperation teleoperation of passive systems in chapter 6. The passive systems under consideration consist of cascade interconnection of a master passive system, the teleoperation controller and a slave passive system. The master and slave systems are themselves permitted to be a cascade interconnection of passive systems (hence are passive w.r.t. a modified supply rate (Vidyasagar, 1993)). The concept is explained using a motorized joystick (2-DOF, 2nd order master system) teleoperating a hydraulic backhoe(2-DOF, 4th order slave system). The main differences between the teleoperated system considered in this chapter and chapter 4 are: • A nonlinear mechanical 2-DOF model of the force feedback joystick is considered. (Chap. 4 assumed decoupled linear dynamics) • A nonlinear mechanical 2-DOF model of the hydraulic backhoe is considered. (Chap. 4 neglected the backhoe inertial dynamics)
47
• Hydraulic actuator dynamics due to fluid compressibility are considered. (Chap. 4 neglected effects due to fluid compressibility and assumed a kinematic modeled actuator) Under the assumptions made in chapter 4 the master and slave systems had the same relative degree (= 2) between the joystick and valve control inputs Fq , Fx and the master and slave positions (outputs of interest), q, x, respectively. However, by removing the assumptions as stated above, the master system has a relative degree of 2 from joystick torque input Fq to joystick position q and the slave has a relative degree of 4 from passive valve control input Fx to backhoe position x. This mismatch in relative degree poses a difficulty in balancing the control power input to the master (ρq˙ T Fq ) and the slave (xv T KQ Γ−1 Fx ) which is required to ensure Intrinsic Passivity of the controller (described in Eqs. (4.3), (4.11)) proposed in chapter 4. To resolve the problem, a bondgraph based approach is proposed in the chapter. While the controller is not Intrinsically Passive, it succeeds in ensuring the passivity of the teleoperated backhoe. The control design procedure consists of the following steps in sequence. • The dynamics of the teleoperated hydraulic backhoe will be represented using bondgraphs. • Bondgraph based system inversion ideas will be used to determine a coordination control law which will ensure that the master and slave systems of the backhoe remain coordianted • The power necessary to ensure coordination will be represented on the bondgraph of the hydraulic teleoperated backhoe and a suitable coordinate transformation and a passive control law will be determined so that the coordinated teleoperator (locked system) will be passive w.r.t. a desired sypply rate. • Coordination and passivity control laws will also be determined assuming imperfect model of the teleoperated backhoe.
5.2
Bondgraph, Model and Control objective
The bondgraph representation of the passive motorized joystick with power scaling (3.1) and the passive hydraulic backhoe (3.17), (3.18), (3.3), (3.4) and (3.23) is shown 48
SLAVE (BACKHOE) DYNAMICS
C : V 2β −1 −1 Q A2 ) I : KQ Γ−1 (K P1 0 TF
KQ Γ−1 Fx : Se
xv
¯1 R:Ω
R : KT A22 R : KT A21
GY
1
TF
(KQ A1
)−1
A−1
2 TF
P2
I : Mx (x) 1 x˙
Se : −Fe
TF
0
A−1 1
C : V 1β −1 I : ρMq (q) ρFq : Se
1 q˙
Se : ρTq MASTER (JOYSTICK) DYNAMICS
Figure 5.1: Bondgraphs of Master and Slave systems. in Fig. 5.1. Notice that the bondgraph representation involves two disjoint graphs which are individually passive w.r.t. their respective supply rates. For the top graph (hydraulic backhoe), the supply rate is sb = xv T KQ Γ−1 Fx − x˙ T Fe . For the bottom graph (motorized joystick) the supply rate is sj = ρq˙ T Fq + ρq˙ T Tq .
49
The dynamics of the master and slave systems represented in the above bond graph, Fig. 5.1 are given by (5.1). ⎞ ⎡ ⎤ ⎛ 0 0 0 0 q˙ ρMq (q) ⎟ ⎢ ⎥ ⎜ Mx (x) 0 0 0 ⎟ ⎢ x˙ ⎥ ⎜ 0 ⎟ d ⎢ ⎥ ⎜ −1 ⎟ ⎢P1 ⎥ ⎜ 0 β 0 0 0 V 1 ⎟ dt ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎜ 0 0 V2 β −1 0 ⎠ ⎣P2 ⎦ ⎝ 0 xv 0 0 0 0 KQ Γ−1 ⎞⎡ ⎤ ⎛ ˙ 0 0 0 0 −ρCq (q, q) q˙ ⎟⎢ ⎥ ⎜ ˙ A1 −A2 0 ⎟ ⎢ x˙ ⎥ 0 −Cx (x, x) ⎜ ⎟⎢ ⎥ ⎜ 2 ⎜ ⎢ ⎥ =⎜ −A1 KT A1 A2 KT A1 KQ ⎟ 0 −A1 ⎟ ⎢P1 ⎥ ⎟⎢ ⎥ ⎜ 0 A2 A1 A2 KT −A22 KT −A2 KQ ⎠ ⎣P2 ⎦ ⎝ ¯1 0 0 −A1 KQ A2 KQ −Ω xv ⎤ ⎡ ⎤ ⎡ ρTq ρFq ⎥ ⎢ ⎥ ⎢ ⎥ ⎢−Fe ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + ⎢ 0 ⎥, +⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ⎦ ⎣ 0 ⎦ ⎣ KQ Γ−1 Fx 0
(5.1)
¯ 1 = KQ Γ−1 Ω1 . where Ω In order to simplify the passive control synthesis and analysis, the dynamics of the master and slave systems (5.1) are transformed using an orthogonal decomposition (w.r.t. the metric M) similar to the one proposed by Lee and Li (2002), conceptually. The difference is that the following decomposition is proposed for N-DOF fourth order hydraulic system whereas the decomposition proposed by Lee and Li (2002) is applicable to N-DOF second order systems. The metric, M of the orthogonal decomposition is given by ⎛ 0 0 0 ρMq (q) ⎜ Mx (x) 0 0 ⎜ 0 ⎜ M := ⎜ 0 0 V1 β −1 ⎜ 0 ⎜ 0 0 V2 β −1 ⎝ 0 0
0
0
50
0
0 0 0 0 KQ Γ−1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and the decomposition is given in Eq. (5.2). ⎛ ⎤ ˙ αI −I 0 0 E ⎜ ⎢ ⎥ 0 0 ⎜I − αΨ Ψ ⎢ q˙ L ⎥ ⎜ ⎢ ⎥ ⎢ FL ⎥ := ⎜ 0 0 A1 −A2 ⎜ ⎢ ⎥ ⎜ ⎢ ⊥⎥ Φ 0 A−1 ⎝ 0 ⎣FL ⎦ 2 (I − A1 Φ) ⎡
0
xv
0
0
0
⎞⎡ ⎤ q˙ 0 ⎟⎢ ⎥ 0⎟ ⎢ x˙ ⎥ ⎟⎢ ⎥ ⎢ ⎥ 0⎟ ⎟ ⎢P1 ⎥ ⎟⎢ ⎥ 0⎠ ⎣P2 ⎦ I xv
(5.2)
where Ψ = α(ρMq (q)Mx (x)−1 + α2 I)−1
(5.3)
−1 Φ = A1 + A2 V1 β −1 A2 A−1 . 1 V2 β
(5.4)
The above decomposition transforms the master-slave dynamics into dynamics in˙ 2) average motion of the master and volving the states; 1) coordination error, E, ˙ = 0, 3) load force acting on the backhoe, FL and slave systems, q˙ L = x˙ = αq, ˙ if E 4) its perpendicular component, F⊥ L . Using the above decomposition, the dynamics of the master and slave are transformed into ⎞ ⎡ ⎤ ⎛ ˙ E ME 0 0 0 ⎟ ⎢ ⎥ ⎜ ⎢ ⎥ ⎜ 0 ML 0 0 ⎟ ⎟ d ⎢ q˙ L ⎥ ⎜ = ⎢ ⎥ ⎜ 0 0 ∆1 0 ⎟ ⎠ dt ⎣FL ⎦ ⎝ 0
0
0
KQ Γ−1
xv
⎤ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ˙ E ρΨT Fq ρΨT TE −CE −CEL (αΨT − I) 0 ⎜ ⎥ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎜ −CLE ⎟ ⎢ q˙ L ⎥ ⎢ ρFq ⎥ ⎢ρTq − Fe ⎥ C αI 0 L ⎜ ⎥ ⎟⎢ ⎥+ ⎢ ⎥+⎢ ⎜(I − αΨ) −αI ⎥ ⎟ ⎢F ⎥ ⎢ ⎥ ⎢ −K K 0 0 T Q ⎠ ⎣ L⎦ ⎣ ⎝ ⎦ ⎦ ⎣ −1 ¯ 0 0 −KQ −Ω1 xv KQ Γ Fx 0 d ∆2 F⊥ = 0, dt L (5.5) ⎛
51
I : ML q˙ L I : KQ Γ−1
C : ∆1 KQ −1
KQ Γ−1 Fx : Se
xv
1
TF
0
FL R : KT
¯1 R:Ω
Se : ρTq − Fe
1
I/α
TF GY : CEL −T Ψ TF TF
Se : ρFq
(αΨT −I)−1
˙ E
1
SS I : ME
Se : ρΨT TE
Figure 5.2: Bondgraph of coordinate transformed teleoperator where ME (x, q) = ρΨT Mq (q)Ψ + (αΨT − I)Mx (x)(αΨ − I) ˙ x, x) ˙ = ρΨT Cq (q, q)Ψ ˙ + (αΨT − I)Cx (x, x)(αΨ ˙ − I) CE (q, q, CEL (q, q, ˙ x, x) ˙ = ρΨT Cq (q, q) ˙ + α(αΨT − I)Cx (x, x) ˙ ML (q, x) = ρMq (q) + α2 Mx (x) ˙ x, x) ˙ = ρCq (q, q) ˙ + α2 Cx (x, x) ˙ CL (q, q, ˙ CLE (q, q, ˙ x, x) ˙ = ρCq (q, q)Ψ ˙ + αCx (x, x)(αΨ ˙ − I) + ML Ψ 1 TE = Tq − (αI − ΨT )Fe ρ M˙ E − 2CE = −[M˙ E − 2CE ]T M˙ L − 2CL = −[M˙ L − 2CL ]T CEL = −CLE T −1 −1 ∆1 = ΦT V1 β −1 Φ + (A1 ΦT − I)A−T A2 (A1 Φ − I) 2 V2 β
∆2 = A2 V1 β −1 A2 + A1 V2 β −1 A1 and the bond graph is given in Fig. 5.2. Notice that the decomposition (5.2) preserves the supply rate, (5.6) and energy (5.7) of the teleoperator. That is, the untransformed
52
teleoperator shown in Fig. 5.1 is passive w.r.t. the supply rate stele1 ((·), (·)) = ρq˙ T Tq + ρq˙ T Fq − x˙ T Fe + xv T KQ Γ−1 Fx
(5.6)
which can be verified by differentiating the storage function W1 =
1 T ρq˙ Mq (q)q˙ + x˙ T Mx x˙ + β −1 (V1 P1 + V2 P2 ) + xv T KQ Γ−1 xv , 2
(5.7)
and substituting the dynamics (5.1). In the new coordinates as given by (5.2), the transformed teleoperator shown in Fig. 5.2 is passive w.r.t. the supply rate ˙ T ΨT TE + xv T KQ Γ−1 Fx , stele2 ((·), (·)) = q˙ TL (ρTq − Fe ) + ρq˙ TL Fq + ρE
(5.8)
which is exactly the same as (5.6) and can be verified by differentiating the storage function W2 =
1 T T ⊥ −1 T ˙ + FL T ∆1 FL + F⊥ ˙ T ME E ∆ F + x K Γ x q˙ L ML q˙ L + E 2 L v Q v , L 2
(5.9)
and substituting the dynamics (5.5). Notice that W2 in (5.9) is exactly the same as W1 in (5.7). By transforming the dynamics, notice that a) the internal state F⊥ L is eliminated, b) coordination error is a state thus simplifying design and analysis and c) bondgraph based system inversion techniques can be easily applied. Our control objective is to synthesize passive valve control inputs Fx and the joystick control input Fq so that: • E → 0, asymptotically. This requirement ensures that the master and the slave systems remain coordinated at all time. • The teleoperator in Fig. 5.2 is passive with respect to the two-port supply rate: stele ((Tq , −Fe ), (q˙ L , q˙ L )) = q˙ TL (ρTq − Fe ).
(5.10)
This requirement ensures that the closed loop teleoperator behaves as it it does not generate any energy of its own but only acts as a ‘transformer’ (with some dissipation) between the human interaction port, (Tq , q˙ L ) and the environment interaction port, (Fe , q˙ L ). Such passive systems are guaranteed to provide stable feedback interconnections with strictly passive systems, (Vidyasagar, 1993). 53
I : ML q˙ L I : KQ Γ−1
C : ∆1 KQ −1
KQ Γ−1 Fx : Se
xv
1
TF
0
FL R : KT
¯1 R:Ω
Se : ρTq − Fe
1
I/α
TF GY : CEL −T Ψ (αΨT −I)−1 TF TF ˙ E
Se : ρFq
1
SS I : ME
Se : ρΨT TE
˙ Figure 5.3: Inverse dynamics bondgraph of the teleoperator, input Fq and output E
5.3
Control Design
Passive control is designed in two steps. In the first step, a bondgraph inversion method (Ngwompo et al., 2001a), (Ngwompo et al., 2001b) is used to determine the coordination control action which will ensure asymptotic convergence of E to 0. In the second step, the coordination control action is represented as an extension to the master-slave bondgraph (Fig. 5.2) and a suitable coordinate transformation (Energy) and passive control is determined which will ensure that the coordinated bondgraph (locked system) behaves like a passive system w.r.t. the supply rate (5.10). 5.3.1
Coordination control design
˙ and a control input is of length 1 to The shortest causal path between the output E the joystick control input Fq . Using bicausal bonds (Gawthrop, 1995), the inverse ˙ and the output Fq is represented as shown dynamics bondgraph between the input E in Fig. 5.3. The inverse dynamics from the bondgraph can be written as follows:
54
⎤ ⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎡ ⎤ ⎡ 0 αI 0 q˙ L ML 0 q˙ L −CL ρTq − Fe ⎥ ⎜ ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎢ 0 ⎠ ⎣FL ⎦ = ⎝ −αI −KT KQ ⎠ ⎣FL ⎦ + ⎣ 0 ⎦ ⎝ 0 ∆1 dt ¯1 xv 0 −KQ −Ω xv 0 0 KQ Γ−1 0 ⎤ ⎡ ˙ ρFq − CLE E ⎢ ˙ ⎥ + ⎣ (I − αΨ)E ⎦ KQ Γ−1 Fx $ −T d " ˙ − TE + Ψ CEL q˙ L − (αΨT − I)FL . ME E Fq = dt ρ
(5.11)
The following choice of joystick control Fq Ψ−T T ˙ CEL q˙ L − (αΨ − I)FL − KE E − BE E , Fq = −TE + ρ
(5.12)
where BE = BE T > 0 and KE = KE T > 0 ensures asymptotic coordination error dynamics, ¨ + CE ((x, q), (x, ˙ = −BE E ˙ − KE E. ME (x, q)E ˙ q)) ˙ E
(5.13)
In the above particular case, the shortest causal path led to the joystick control input, Fq . In a general case the shortest causal path would be associated with the system (master or slave) with lower relative degree. In the next section we determine the energy necessary to implement the coordination control law (5.12) and a passive control so that the coordinated teleoperator is passive w.r.t. the supply rate (5.10). This is achieved by first graphically representing the coordination control in the bondgraph Fig. 5.2 and analyzing the power flow. 5.3.2
Closed loop passivity property: perfect model
The coordination control (5.12) is represented in the bondgraph shown in Fig. 5.4. Notice the use of a signal bond to indicate the flow of effort from FL to q˙ L . Other researchers have used regular power bonds with one of the port variables set to 0 (Effort (AE) or Flow Amplifiers (AF) in Ngwompo and Gawthrop (1999)). The idea behind using either AE or AF or a signal bond is to be able to indicate unidirectional information exchange which cannot be done using regular power bonds. Power bonds represent the transfer of both effort in one direction and flow in the other. Signal (active) bonds are used here as they visually certifying the non-passive property of 55
I : ML
Se : ρTq − Fe
q˙ L
1
I : KQ Γ−1
I/α
TF
C:I KQ −1
KQ Γ−1 Fx : Se
xv
1
TF
Se : EF1 (Ψ−T − I/α)
0
FL ˙ Se : (αΨT − I)−1 E
¯1 R:Ω
R : KT
LOCKED SYSTEM DYNAMICS
R : KE R : BE C:I
˙ E
E 0
1
I : ME COORDINATION ERROR DYNAMICS
Figure 5.4: Bondgraph of the master and slave systems with coordination control law involving accurate estimates the bond graph. In Fig. 5.4, EF1 is given by:
EF1
˙ ˙ − Ψ−T CEL q˙ L = −ρTE − ρ KE E + BE E − CLE E
The bondgraph shown in Fig. 5.4 represents the ‘Locked System’ behavior; i.e., the behavior of the coordinated teleoperator and ‘Shape System’ behavior; i.e., the behavior of the coordination dynamics of the teleoperator. The bottom disconnected bond graph in the figure represents the dynamics of the coordination error. The top disconnected bond graph in the figure represents the dynamics of the locked system and determines the haptics experienced by the human operator. Unlike the locked system in Lee and Li (2002) which is the result of an isometric energetic decomposition, the desired locked systems (introduced in the following section) proposed in this dissertation is a design variable. This approach has the advantage of letting the control designer choose the haptic behavior of the teleoperator while ensuring the control objectives listed above. Notice that the locked system shown in Fig. 5.4 is not passive w.r.t. the desired 56
supply rate (5.10) due to the presence of 1) signal bonds (uni-directional arrows) and 2) additional effort sources Se in the bond graph Fig. 5.4. The task now is to determine the remaining control input Fx so that the bondgraph shown in Fig. 5.4 is passive w.r.t. the supply rate (5.10). As stated above, the desired locked system structure is up to the control system engineer. Towards that end, two locked system designs are presented in this dissertation. The first design shown in Fig. 5.5 proposes a fourth order locked system. The second design shown in Fig. 5.6 proposes a second order locked system. 5.3.3
Locked System Design : 4th Order, 2-DOF Haptic Behavior
We need to determine the passive valve exogenous control input Fx in Fig. 5.4 so that the closed loop dynamics of the actual locked system shown in the figure behaves like the desired locked system dynamics shown in Fig. 5.5. The design process involves comparing the dynamics of the actual and desired locked systems and choosing the appropriate control input Fx so that the actual dynamics mimic the desired dynamics. The dynamics of the actual locked system shown in Fig. 5.4 are: ⎛
ML 0 ⎜ ⎝ 0 ∆1 0
0
⎤ ⎛ ⎞⎡ ⎤ ⎡ ⎤ 0 q˙ L q˙ L −CL Ψ−T ρTq − Fe ⎟d ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎢ ⎥ 0 ⎠ ⎣FL ⎦ = ⎝ −αI −KT KQ ⎠ ⎣FL ⎦ + ⎣ ⎦ dt −1 ¯1 KQ Γ 0 xv 0 −KQ −Ω xv ⎤ ⎡ ˙ − CLE E ˙ + Ψ−T CEL q˙ L −ρTE − ρ KE E + BE E ⎥ ⎢ ⎥ ˙ +⎢ (I − αΨ) E ⎦ ⎣ −1 KQ Γ Fx 0 0
⎞
⎡
(5.14) The dynamics of the desired locked system shown in Fig. 5.5 are: ⎞ ⎡ ⎤ ⎛ ⎞⎡ ⎤ ⎡ ⎤ 0 0 q˙ L q˙ L ML 0 −CL Ψ−T ρTq − Fe ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎜ 0 ⎠ ⎣ z1 ⎦ = ⎝ −Ψ −KT KQ ⎠ ⎣ z1 ⎦ + ⎣ 0 ⎦ ⎝ 0 ∆1 dt −1 ¯1 z2 0 −KQ −Ω z2 0 0 KQ Γ 0 ⎛
(5.15) Note that z1 , z2 are different from FL , xv even though they have the same dynamics. We will determine the relationship between them later. In general, the desired locked dynamics (5.15) need not have the same parameters as the actual locked system (5.14). However, the proposed locked system minimizes additional control effort Fx 57
I : ML
4th ORDER LOCKED SYSTEM
q˙ L I : KQ Γ−1
Ψ−T
TF
C:I KQ −1
z2
1
TF
Se : ρTq − Fe
1
0
z1
¯1 R:Ω
R : KT R : KE R : BE ˙ E
E
C:I
0
1
I : ME DISSIPATIVE
Figure 5.5: Fourth order desired locked system which would be necessary to modify the natural behavior of the coordinated hydraulic teleoperator and hence the actual locked system. The passive control objective is to design the passive valve control Fx in (5.14) (also in bondgraph in Fig. 5.4) so that the actual locked system behaves like the desired locked system in Eq. (5.15) (also in bondgraph in Fig. 5.5). By doing so, we can ensure that the locked system is passive with respect to the supply rate (5.10). In order to achieve this, compare the actual dynamics of q˙ L in (5.14) and the desired dynamics as given by (5.15), using a procedure similar to the procedure proposed by Li and Ngwompo (2002) where a bondgraph approach to passification of a single-stage valve is presented. Notice that the actual locked system dynamics (5.14) and desired dynamics (5.15) will be identical if and only if z1 := FL + ΨT D(t). ˙ − CLE E ˙ + Ψ−T CEL q˙ L . By substituting this where D(t) = −ρTE − ρ KE E + BE E transformation for z1 into the desired dynamics (5.15) and comparing the actual and desired dynamics of FL it can be noted that the dynamics will be identical if and 58
only if % −1
z2 := KQ (Ψ − αI)q˙ L + xv + KQ T
−1
& d T KT Ψ D(t) + [∆1 Ψ D(t)] . dt T
The above analysis leads to the following coordinate transformation [q˙ L FL xv ]T → [q˙ L z1 z2 ]T : ⎞⎡ ⎤ ⎡ ⎤ ⎛ q˙ L q˙ L I 0 0 ⎟⎢ ⎥ ⎢ ⎥ ⎜ 0 I 0⎠ ⎣FL ⎦ ⎣ z1 ⎦ = ⎝ z2 xv KQ −1 (Ψ−1 − αI) 0 I ⎤ ⎡ 0 ⎥ ⎢ +⎣ ΨT D(t) ⎦, −1 d T T KT Ψ D(t) + dt [∆1 Ψ D(t)] KQ
(5.16)
˙ − CLE E ˙ + Ψ−T CEL q˙ L . This transformation where D(t) = −ρTE − ρ KE E + BE E (Eq. (5.16)) when applied to the actual locked system dynamics (5.14) results in the following transformed dynamics: ⎞ ⎡ ⎤ ⎛ ⎞⎡ ⎤ 0 0 q˙ L q˙ L ML 0 −CL Ψ−T ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎜ T 0 ⎠ ⎣ z1 ⎦ = ⎝−Ψ −KT KQ ⎠ ⎣ z1 ⎦ ⎝ 0 ∆1 dt ¯1 z2 0 −KQ −Ω z2 0 0 KQ Γ−1 ⎤ ⎡ ρTq − Fe ⎥ ⎢ +⎣ 0 ⎦, KQ Γ−1 Fx + D2 (t) ⎛
(5.17)
where D2 (t) = KQ [z1 FL ] + KQ Γ
−1
d Ω[z2 xv ] + [z2 xv ] , dt
z1 FL := ΨT D(t), −1
z2 xv := KQ (Ψ
−1
− αI)q˙ L + KQ
−1
(5.18)
d T T KT Ψ D(t) + [∆1 Ψ D(t)] . dt
Notice that the signals z1 FL and z1 xv are dependant on D(t) and its successive ˙ If the derivatives which are functions of external variables TE , q˙ L , FL , E and E.
59
locked system control Fx in (5.17) is chosen as follows: KQ Γ−1 Fx = −D2 (t),
(5.19)
then the locked system is passive w.r.t. the supply rate (5.10) and its dynamics are given by: ⎛
ML 0 ⎜ ⎝ 0 ∆1 0
0
⎤ ⎛ ⎞⎡ ⎤ 0 q˙ L q˙ L −CL Ψ−T ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ T ⎠ ⎣ z1 ⎦ = ⎝−Ψ −KT KQ ⎠ ⎣ z1 ⎦ dt ¯1 KQ Γ−1 z2 0 −KQ −Ω z2 ⎤ ⎡ ρTq − Fe ⎥ ⎢ +⎣ 0 ⎦. 0 0
⎞
⎡
(5.20)
0 The above analysis is summarized in the following theorem. Theorem 2. 4th Order, 2-DOF Locked System, perfect model: The teleoperated hydraulic backhoe composed of a) the passive valve (3.17), b) the dynamically modeled hydraulic actuators (3.18), c) the mechanical backhoe (3.23) and d) the motorized joystick (3.1) which is concisely represented in Eq. (5.5) transforms to an interconnected locked system (5.17) under the choice of joystick control input Fq as given in (5.51) and the coordinate transformation (5.16). The interconnected locked system (5.17) is passive with respect to the supply rate: stele ((Tq , −Fe ), (q˙ L , q˙ L )) = q˙ TL (ρTq − Fe ),
(5.21)
if the passive valve control input, Fx is chosen as follows: Fq = −TE +
$ Ψ−T " ˙ , CEL q˙ L − (αΨT − I)FL − KE E − BE E ρ
Fx = −ΓKQ −1 D2 (t),
(5.22)
where D2 (t) is given by (5.18). The choice of joystick control input Fq in (5.22) ensures that the coordination error E → 0 asymptotically. Proof: Consider the dynamics of the coordinate transformed teleoperated hydraulic backhoe given by Eq. (5.5). The choice of coordination control Fq as given by Eq. (5.22) transforms the teleoperator dynamics as given by Eq. (5.14). The bondgraph based coordinate transformation derived in (5.16) transforms the actual locked system 60
(5.14) into (5.17). The choice of valve control Fx as given in (5.22) ensures that the locked system behaves like the desired 4-Order passive locked system given by (5.20). In order to prove that the locked system is passive w.r.t. the supply rate (5.21), consider the storage function of the locked system 1 Wlocked4 = xL T ML4 xL , 2
(5.23)
where xL = [q˙ L z1 z2 ]T and ⎛
ML4
ML 0 ⎜ = ⎝ 0 ∆1
0 0
0
KQ Γ−1
0
⎞ ⎟ ⎠.
Differentiating the storage function w.r.t. time and substituting the controlled locked system (after implementing Fq , Fx ) dynamics (5.14) leads to the following: xL T ML4
d xL = xL T Ω4 xL + q˙ TL (ρTq − Fe ), dt
(5.24)
where ⎛ −CL Ψ−T ⎜ Ω4 = ⎝−ΨT −KT 0
−KQ
⎞ 0 ⎟ KQ ⎠ . ¯1 −Ω
Since Ω4 is negative semi-definite, the above equation simplifies to d Wlocked4 ≤ q˙ TL (ρTq − Fe ). dt Integrating the above equation provides the desired passivity result. ˙ T → 0 consider the lyapunov function: To prove that [E E] V =
1˙T ˙ + ET (KE + 1 BE )E + 1 E ˙ T ME E , E ME E 2
where 1 ∈ R++ , KE = KE T > 0, BE = BE T > 0 can be appropriately chosen to ensure that V is positive definite. Differentiating V w.r.t. time and substituting for the dynamics of q ¨ and x ¨ as given by (3.1) and (3.18) respectively followed by
61
substitution of the joystick control Fq as given in (5.22), leads to ˙ − 1 ET KE E, ˙ T (BE − 1 ME )E V˙ = −E ˙ is globally exponentially stable. The negative definiteness of V˙ ensures that (E, E) Consider the storage energy associated with the error dynamics: V =
1˙T ˙ + ET (KE + 1 BE )E + 1 E ˙ T ME E . E ME E 2
(5.25)
˙ → (0, 0) (as proved in theorem 2), the Since the coordination error states (E, E) energy associated with the error dynamics V → 0. Owing to this property, ensuring that the locked system given by Eq. (5.20) is passive w.r.t. the supply rate (5.21) implies that the teleoperated backhoe is passive w.r.t. the same supply rate, (5.21). Moreover, ensuring that the teleoperated backhoe is passive w.r.t. the 1-port supply rate stele1 ((Tq , −Fe ), (q˙ L , q˙ L )) = q˙ TL (ρTq − Fe ), also ensures passivity of the teleoperated backhoe w.r.t. the 2-port supply rate W orkenv.inputpower
stele2 ((Tq , −Fe ), (q, ˙ x)) ˙ =
ρq˙ T Tq
−
x˙ T Fe
.
(5.26)
Operatorinputpower
This is significant as, the environment consisting of the human and the work environment interact with the teleoperated backoe via a supply rate given by stele2 and not stele1 . Hence, ensuring passivity of the teleoperated backhoe (5.5) w.r.t. the supply rate (5.26) not only has better physical relevance but is also necessary to ensure 2port passivity property of the teleoperated backhoe. Ensuring that the teleoperated backhoe behaves like a 2-port passive device is in turn beneficial it is then possible to analyze the stability of interconnection of the passive backhoe with other passive systems by invoking properties of interconnection of passive systems stated in chapter 2. The above analysis is proved in the following theorem. Theorem 3. The teleoperated hydraulic backhoe given by Eq. (5.5) whose locked system dynamics are represented by (5.20) is passive w.r.t. the 2-port supply rate given in Eq. (5.26). Proof: Consider the storage energy of the teleoperated system given by the sum of 62
the locked system energy (5.23) and the energy due to coordination error (5.25). Wlocked4 + V =
1 T ˙ T ME E ˙ + ET (KE + 1 BE )E + 1 E ˙ T ME E . xL ML4 xL + E 2
Differentiating the above equation and noting the following results from theorem 2: ˙ locked4 ≤ stele ((Tq , −Fe ), (q˙ L , q˙ L )), W ˙ T (BE − 1 ME )E ˙ − 1 ET KE E < 0 V˙ = −E ensures that ˙ locked4 + V˙ < stele ((Tq , −Fe ), (q˙ L , q˙ L ) W ˙ locked4 + V˙ < q˙ TL (ρTq − Fe ). =⇒ W Substituting for q˙ L = (I − αΨ)q˙ + Ψx˙ as given in Eq. (5.2) and further using the ˙ = αq˙ − x, definition of E ˙ ensures that ˙ locked4 + V˙ < ρq˙ T Tq − x˙ T Fe − E ˙ T ΨT (ρTq − Fe ) W ∞ stele2 ((Tq , −Fe ), (q, ˙ x))dτ ˙ =⇒ −Wlocked4 (0) − V (0) < 0∞ ˙ T ΨT (ρTq − Fe )dτ E + 0 ∞ stele2 ((Tq , −Fe ), (q, ˙ x))dτ ˙ =⇒ −Wlocked4 (0) − V (0) − c2 < 0
˙ → 0, exponentially as shown in theorem 2, hence completing the proof of as E passivity of the teleoperated backhoe. Theorems 2-3 presented a passive teleoperation control design which ensures that the master-slave teleoperator mimics a passive 4th order 2-DOF system operated upon by the human operator and the environment. Since the locked system dynamics are of fourth order, the teleoperator behaves like a fourth system when operated upon by the operator, interacting with the environment. While the system order has certain advantages in terms of filtering high frequency content of environment forces, it may be disadvantageous in other circumstances when the operator requires an agile teleoperator. Hence, it would be benefecial to control (using passive valve control Fx ) the locked system to behave like a lower order dynamic system. Such a method is proposed in the following section where, the teleoperator behaves like a second order locked system. 63
I : ML q˙ L
I : KQ Γ−1
C:I KQ
Se : Ψ−T z1
−1
TF
z3 1
Se : ρTq − Fe
1
DISSIPATIVE
2nd ORDER LOCKED SYSTEM
0
z1
¯1 R:Ω
R : KT R : KE R : B E ˙ E
E
C:I
0
1
I : ME DISSIPATIVE
Figure 5.6: Second order desired locked system 5.3.4
Locked System Design : 2nd Order, 2-DOF Haptic Behavior
The locked system control objective remains the same which is to determine Fx in Fig. 5.4 so that the closed loop dynamics of the actual locked system shown in the figure behaves like the desired 2nd order locked system dynamics as shown in Fig. 5.6. The design process is similar to that outlined in the previous section and involves comparing the dynamics of the actual and desired locked systems and choosing the appropriate control input Fx so that the actual dynamics mimic the desired dynamics. The dynamics of the actual locked system shown in Fig. 5.4 were given by Eq. (5.14). The dynamics of the desired locked system shown in Fig. 5.6 are: ⎛
ML 0 ⎜ ⎝ 0 ∆1 0
0
⎤ ⎛ ⎞⎡ ⎤ ⎡ ⎤ 0 q˙ L q˙ L −CL Ψ−T ρTq − Fe ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎢ ⎥ 0 −KT KQ ⎠ ⎣ z1 ⎦ + ⎣ ⎠ ⎣ z1 ⎦ = ⎝ 0 ⎦ dt −1 ¯1 KQ Γ 0 z3 0 −KQ −Ω z3 0 0
⎞
⎡
(5.27) where ML is the locked system inertia, q˙ L is the locked system velocity, ρTq − Fe is the external torque operating on the locked system, z1 and z3 are the internal states of the locked system. The benefit of this locked system (Fig. 5.6) is that the 64
operator and environment interact with a second order mechanical system instead of the fourth order hydraulic-mechanical system as in the fourth order locked system (Fig. 5.5). Since the locked system dynamics is essentially the haptics property of the teleoperated system it is desirable to have minimum order in such a system. The passive control objective is to design the passive valve control Fx in (5.14) (also in bondgraph in Fig. 5.4) so that the actual locked system behaves like the desired locked system in Eq. (5.27) (also in bondgraph in Fig. 5.6). Comparison of the bondgraph representations shown in Fig. 5.6 and Fig. 5.4 as explained in detail in the previous subsection, suggests the following coordinate transformation [q˙ L FL xv ]T → [q˙ L z1 z3 ]T : ⎞⎡ ⎤ ⎡ ⎤ ⎛ q˙ L q˙ L I 0 0 ⎟⎢ ⎥ ⎢ ⎥ ⎜ 0 I 0⎠ ⎣FL ⎦ ⎣ z1 ⎦ = ⎝ −1 z3 0 I xv −αKQ
⎤ 0 ⎥ ⎢ +⎣ ΨT D(t) ⎦, −1 d KT ΨT D(t) + dt [∆1 ΨT D(t)] KQ ⎡
(5.28) ˙ − CLE E ˙ + Ψ−T CEL q˙ L . This transformation where D(t) = −ρTE − ρ(KE E + BE E) (Eq. (5.28)) when applied to the actual locked system dynamics (5.14) results in the following transformed dynamics: ⎛
ML 0 ⎜ ⎝ 0 ∆1 0
0
⎞
⎤ ⎛ ⎞⎡ ⎤ 0 q˙ L q˙ L −CL Ψ−T ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ −KT KQ ⎠ ⎣ z1 ⎦ ⎠ ⎣ z1 ⎦ = ⎝ 0 dt −1 ¯1 KQ Γ z3 0 −KQ −Ω z3 ⎡ ⎤ ρTq − Fe ⎢ ⎥ +⎣ 0 ⎦, −1 KQ Γ Fx + D3 (t) 0 0
⎡
(5.29)
where D3 (t) = KQ [z1 FL ] + KQ Γ
−1
z1 FL := ΨT D(t), −1
z3 xv := −αKQ q˙ + KQ
d Ω[z3 xv ] + [z3 xv ] , dt
−1
(5.30)
d T KT Ψ D(t) + [∆1 Ψ D(t)] . dt T
The signals z1 FL and z1 xv are dependant on D(t) and its successive derivatives ˙ If the locked system which are functions of external variables TE , q˙ L , FL , E and E.
65
control Fx in (5.29) is chosen to satisfy: KQ Γ−1 Fx = −D3 (t),
(5.31)
then the locked system is passive w.r.t. the supply rate (5.10) and its dynamics are given by: ⎛
ML 0 ⎜ ⎝ 0 ∆1 0
0
⎤ ⎛ ⎞⎡ ⎤ 0 q˙ L q˙ L −CL Ψ−T ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ −KT KQ ⎠ ⎣ z1 ⎦ ⎠ ⎣ z1 ⎦ = ⎝ 0 dt −1 ¯1 KQ Γ z3 0 −KQ −Ω z3 ⎤ ⎡ ρTq − Fe ⎥ ⎢ +⎣ 0 ⎦. 0 0
⎞
⎡
(5.32)
0 Since the shunt conductance KT (|xv |, FL ) > 0 (given in (3.5)) for all xv and P1 , P2 < Ps I (Ps is the hydraulic system pressure defined in (3.7)), the states [z1 z3 ]T → 0, asymptotically. Hence, the locked system (5.20) asymptotically approaches the second order locked system given by: ML (x, q)¨ qL + CL ((x, q), (x, ˙ q)) ˙ q˙ L = ρTq − Fe .
(5.33)
The passive coordination analysis presented above is now summarized in the following theorem. Theorem 4. 2nd Order, 2-DOF Locked System, perfect model: The teleoperated hydraulic backhoe composed of a) the passive valve (3.17), b) the dynamically modeled hydraulic actuators (3.18), c) the mechanical backhoe (3.23) and d) the motorized joystick (3.1) which is concisely represented in Eq. (5.5) transforms to an interconnected locked system (5.29) under the choice of joystick control input Fq as given in (5.35) and the coordinate transformation (5.28). The interconnected system (5.29) is passive with respect to the supply rate: stele ((Tq , −Fe ), (q˙ L , q˙ L )) = q˙ TL (ρTq − Fe ),
66
(5.34)
if the passive valve control input, Fx is chosen as follows: Ψ−T T ˙ CEL q˙ L − (αΨ − I)FL − KE E − BE E , Fq = −TE + ρ Fx = −ΓKQ −1 D3 (t),
(5.35)
where D3 (t) is given by (5.30). The choice of joystick control input Fq in (5.35) ensures that the coordination error E → 0 asymptotically. Proof: Consider the dynamics of the coordinate transformed teleoperated hydraulic backhoe given by Eq. (5.5). The choice of coordination control Fq as given by Eq. (5.35) transforms the teleoperator dynamics as given by Eq. (5.14). The bondgraph based coordinate transformation derived in (5.28) transforms the actual locked system (5.14) into (5.29). The choice of valve control Fx as given in (5.35) ensures that the locked system behaves like the desired 2-Order passive locked system given by (5.32). In order to prove that the locked system is passive w.r.t. the supply rate (5.34), consider the storage function of the locked system Wlocked2 = xL T2 ML2 xL2 ,
(5.36)
where xL2 = [q˙ L z1 z3 ]T and ⎛
ML
⎜ ML2 = ⎝ 0 0
0
0
⎞
⎟ ∆1 0 ⎠. 0 KQ Γ−1
Differentiating the storage function w.r.t. time and substituting the transformed locked system dynamics (5.29) and further substituting the locked system control Fx as given in (5.35) leads to the following: xL T2 ML2
d xL 2 = [z1 T z3 T ]T Ω2 [z1 z3 ] + Ψ−T z1 + q˙ TL (ρTq − Fe ), dt
(5.37)
where Ω2 =
KQ −KT (|xv |, FL ) , −KQ −KQ Γ−1 Ω
is negative semidefinite (Note that KT (|xv |, FL ) > 0 as given by (3.5)). Hence, the 67
above equation simplifies to d Wlocked2 ≤ q˙ TL Ψ−T z1 + q˙ TL (ρTq − Fe ). dt Under the assumption that q˙ L (t) is bounded by q˙ L and since the states [z1 z3 ]T → 0 exponentially 1 , integrating the above equation leads to: Wlocked2(T ) − Wlocked2(0) T T −T q˙ L (τ )Ψ(τ ) z1 (τ )dτ + stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ ≤ 0 0 T T −ω2 τ ≤ q˙ L Ψz1 (0) e dτ + stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ 0 0 T ≤c+ stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ 0
where ω2 = σ(Ω2 ), c ∈ R+ . Hence
T 0
stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ ≥ −Wlocked2 (0) − c,
completing the proof for passivity of the locked system. The proof for convergence of ˙ T is exactly the same as in theorem 2. [E E] Using an argument similar to the one made in theorem 3, we can show that the hydraulic backhoe with 2nd order locked system is also passive w.r.t. the 1-port supply rate (5.34) thus ensuring that the teleoperated backhoe is passive w.r.t. the 2-port supply rate stele2 ((Tq , −Fe ), (q, ˙ x)) ˙ = ρq˙ T Tq − x˙ T Fe .
(5.38)
Theorem 5. The teleoperated hydraulic backhoe given by Eq. (5.5) whose locked system dynamics are represented by (5.32) is passive w.r.t. the 2-port supply rate given in Eq. (5.38). Proof: Consider the storage energy of the teleoperated system given by the sum of the locked system energy (5.36) and the energy due to coordination error (5.25). Wlocked2 + V =
1 T ˙ T ME E ˙ + ET (KE + 1 BE )E + 1 E ˙ T ME E . xL 2 ML2 xL2 + E 2
1
Can be proved by differentiating the lyapunov function V = 12 [z1 T z3 T ]T Diag [∆1 , KQ Γ−1 ][z1 z3 ] and substituting the dynamics (5.32).
68
Differentiating the above equation and noting the following results from theorem 4: −Wlocked (0) − c1 ≤
0
T
stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ
˙ T (BE − 1 ME )E ˙ − 1 ET KE E < 0 V˙ = −E ensures that −Wlocked2 (0) − V (0) − c1 ≤
∞
0
stele ((Tq , −Fe ), (q˙ L , q˙ L )dτ.
hence ensuring passivity of the teleoperated backhoe w.r.t. the 1-port supply rate (5.34). Invoking theorem 3 completes the proof ensuring passivity w.r.t. the 2-port supply rate (5.38). Notice in both theorems 2 and 4, the valve and joystick control forces Fx and Fq in (5.22) and (5.35) assume accurate knowledge of Ψ, TE which depend on the backhoe / joystick inertia and external forces respectively. In reality, it may not be possible to know these parameters exactly. As a result it is necessary to analyze the passivity and coordination property of the proposed algorithm in the presence of such uncertainties, which is presented in the following section. 5.3.5
Closed loop passivity property: imperfect system model
5.3.5.1 Coordination analysis The coordination control law derived in Eq. (5.12) requires accurate knowledge of the backhoe and joystick inertia Mx (x), Mq (q) and the disturbance force TE (given in Eq. (5.5)) which may not be exactly known. One possible modification to the coordination control law is given by (5.39). ( −T # ) #E + Ψ ( −T − I)FL − KE E − BE E ˙ , (5.39) ˙ − ge (t)s(E) C Fq = −T ˙ L − (αΨ EL q ρ # is an estimate of the argument (·), 2) s(E) ˙ is a smooth approximation to where 1) (·) ˙ and one choice of the function ge (t) is as follows: the signum function sgn(E) #E || ˙ ≥ α1 (q, x)||[q˙ x]|||| qe (t)sgn(E) ˙ q˙ L || + α2 (q, x)||FL || + ρα3 (q, x)||TE − T
69
I : ML
Se : ρTq − Fe
q˙ L
1
I : KQ Γ−1
I/α
KQ −1
KQ Γ−1 Fx : Se
xv
1
TF
Se : EF2
TF
C:I
( −T − I/α) (Ψ
0
FL
¯1 R:Ω
˙ Se : (αΨT − I)−1 E
R : KT
LOCKED SYSTEM DYNAMICS
R : KE R : BE C:I
˙ E
E 0
1
Se : EF3
I : ME COORDINATION ERROR DYNAMICS
Figure 5.7: Bondgraph of the master and slave systems with coordination control law involving inaccurate estimates and αi (·) are such that: ( x)−T CEL α1 (·) > ||Ψ(q, x)T Ψ(q, (q, x) − CEL (q, x)|| ( x)−T || α2 (·) > ||I − Ψ(q, x)T Ψ(q, α3 (·) > ||Ψ(q, x)T ||.
(5.40)
The first three terms in (5.39) are estimates of the terms in (5.12). The last three ˙ The bondgraph representation terms are to ensure exponential convergence of E, E. of the modified coordination control law (5.39) applied to the teleoperator is shown in Fig. 5.7. In Fig. 5.7, EF2 , EF3 are given by: # #E − ρe(E, E) ( −T C ) ˙ − CLE E ˙ +Ψ EF2 = −ρT ˙L EL q # ( −T C ) ( −T )FL − ge (t)s(E) #E ) + ΨT Ψ ˙ . EF3 = ρΨT (TE − T ˙ L − CEL q˙ L + (I − ΨT Ψ EL q (5.41)
70
where ˙ := KE E + BE E ˙ ˙ + ge (t)s(E). e(E, E) The above coordination control is represented and the resulting locked system is shown in Fig. 5.7. The error dynamics are given by: ¨ + CE ((x, q), (x, ˙ = −BE E ˙ − KE E + EF3 . ME (x, q)E ˙ q)) ˙ E
(5.42)
The locked system bondgraph (Fig. 5.7) is not passive w.r.t. the desired supply rate (5.10) due to the presence of 1) signal bonds (uni-directional arrows) and 2) additional effort sources Se in the bond graph Fig 5.7. As in passive control design with perfect model case, the task now is to determine the remaining control input Fx so that the bondgraph shown in Fig. 5.7 is passive w.r.t. the supply rate (5.10). Two locked system designs are proposed. The first design similar to the one shown in Fig. 5.5 proposes a fourth order locked system. The second design similar to the one shown in Fig. 5.6 proposes a second order locked system. 5.3.5.2 Passive Locked System Design : 4th Order, 2-DOF Haptic Behavior We need to determine the passive valve exogenous control input Fx in Fig. 5.7 so that the closed loop dynamics of the actual locked system shown in the figure behaves like the desired locked system dynamics shown in Fig. 5.5. However, notice that the error dynamics (lower bond graph in Fig. 5.7) whose dynamics are given by Eq. (5.42) has an additional effort source EF3 . This manifests due to the inaccurate model estimates employed by the coordination control law given in Eq. (5.39). As a result it is necessary to consider the energy of the coordination dynamics when analyzing the passivity of the entire teleoperated system. This analysis will be presented in theorems 6 and 8. The design of a passive control law remains the same as outlined in section 5.3.3 which is to compare the dynamics of the actual and desired locked systems and choose the appropriate control input Fx so that the actual dynamics mimic the desired dynamics.
71
The dynamics of the actual locked system shown in Fig. 5.7 are: ⎞ ⎡ ⎤ ⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎛ ( −T 0 0 q˙ L q˙ L ML 0 −CL Ψ ρTq − Fe ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎜ 0 ⎠ ⎣FL ⎦ = ⎝ −αI −KT KQ ⎠ ⎣FL ⎦ + ⎣ 0 ⎦ ⎝ 0 ∆1 dt −1 ¯ xv 0 −KQ −Ω1 xv 0 0 KQ Γ 0 ⎤ ⎡ ) # #E − ρe(E, E) ( −T C ˙ − CLE E ˙ +Ψ ˙L −ρT EL q ⎥ ⎢ ˙ +⎣ (I − αΨ)E ⎦ KQ Γ−1 Fx (5.43) The dynamics of the desired locked system similar to the one shown in Fig. 5.5 are: ⎛
ML
⎜ ⎝ 0 0
0 ∆1 0
⎤ ⎛ ⎞⎡ ⎤ ⎡ ⎤ ( −T −CL Ψ q˙ L ρTq − Fe q˙ L 0 ⎟ d ⎢ ⎥ ⎜ (T ⎟⎢ ⎥ ⎢ ⎥ 0 ⎠ ⎣ z(1 ⎦ = ⎝−Ψ −KT KQ ⎠ ⎣ z(1 ⎦ + ⎣ 0 ⎦ dt −1 ¯1 z(2 z(2 0 −KQ −Ω KQ Γ 0 0
⎞
⎡
(5.44) The passive control objective is to design the passive valve control Fx in (5.43) (also in bondgraph in Fig. 5.7) so that the actual locked system behaves like the desired locked system in Eq. (5.44). By doing so, we can ensure that the locked system is passive with respect to the supply rate (5.10). As seen in section 5.3.3, the first step is to determine a coordinate transformation from [q˙ L FL xv ]T → [q˙ L z(1 z(2 ]T . Following the steps outlined in section 5.3.3, we can determine the coordinate transformation which is given as: ⎞⎡ ⎤ ⎤ ⎛ q˙ L q˙ L I 0 0 ⎟⎢ ⎥ ⎢ ⎥ ⎜ 0 I 0⎠ ⎣FL ⎦ ⎣ z(1 ⎦ = ⎝ −1 ( −1 z(2 xv KQ (Ψ − αI) 0 I ⎤ ⎡ 0 ⎥ ⎢ ( T D(t) ⎥ Ψ +⎢ ⎦ , ⎣ ( T D(t) + d [∆1 Ψ ( T D(t)] KQ −1 KT Ψ ⎡
(5.45)
dt
# ) #E − ρe(E, E) ( −T C ˙ − CLE E ˙ +Ψ where D(t) = −ρT ˙ L . This transformation when apEL q plied to the actual locked system dynamics (5.43) results in the following transformed
72
dynamics: ⎞ ⎡ ⎤ ⎛ ⎞⎡ ⎤ ( −T 0 0 q˙ L q˙ L ML 0 −CL Ψ ⎟ d ⎢ ⎥ ⎜ (T ⎟⎢ ⎥ ⎜ 0 ⎠ ⎣ z(1 ⎦ = ⎝−Ψ −KT KQ ⎠ ⎣ z(1 ⎦ ⎝ 0 ∆1 dt ¯1 z(2 z(2 0 −KQ −Ω 0 0 KQ Γ−1 ⎤ ⎡ ρTq − Fe ⎥ ⎢ +⎣ 0 ⎦, KQ Γ−1 Fx + D4 (t) ⎛
(5.46)
where D4 (t) = KQ [z(1 FL ] + KQ Γ
−1
d Ω[z(2 xv ] + [z(2 xv ] , dt
(5.47)
( T D(t), z(1 FL := Ψ z(2 xv := KQ
−1
d −1 T T −1 ( ( ( (Ψ − αI)q˙ L + KQ KT Ψ D(t) + [∆1 Ψ D(t)] . dt
Notice that the signals z(1 FL and z(1 xv are available for differentiation. If the locked system control Fx in (5.46) is chosen as follows: KQ Γ−1 Fx = −D4 (t),
(5.48)
then the locked system is passive w.r.t. the supply rate (5.10) and its dynamics are given by: ⎛
ML
⎜ ⎝ 0 0
0 ∆1 0
⎞
⎡
⎤ ⎛ ⎞⎡ ⎤ ( −T q˙ L 0 q˙ L −CL Ψ ⎟ d ⎢ ⎥ ⎜ (T ⎟⎢ ⎥ 0 ⎠ ⎣ z(1 ⎦ = ⎝−Ψ −KT KQ ⎠ ⎣ z(1 ⎦ dt ¯1 z(2 z(2 0 −KQ −Ω KQ Γ−1 ⎤ ⎡ ρTq − Fe ⎥ ⎢ +⎣ 0 ⎦. 0 0
(5.49)
The above analysis is summarized in the following theorem. Theorem 6. 4th Order, 2-DOF Locked System, imperfect system model: The teleoperated hydraulic backhoe composed of a) the passive valve (3.17), b) the dynamically modeled hydraulic actuators (3.18), c) the mechanical backhoe (3.23) and d) the motorized joystick (3.1) which is concisely represented in Eq. (5.5) transforms to an interconnected system (5.46) under the choice of joystick control input Fq as 73
given in (5.51) and a coordinate transformation (5.45). The interconnected system (5.46) is passive with respect to the supply rate: stele ((Tq , −Fe ), (q˙ L , q˙ L )) = q˙ TL (ρTq − Fe ),
(5.50)
if the passive valve control input, Fx is chosen as follows: ( −T #E + Ψ # ) ( −T − I)FL − KE E − BE E ˙ , ˙ − ge (t)s(E) Fq = −T ˙ L − (αΨ C EL q ρ ( 4 (t) − gx (t)sgn(z(2 ) , Fx = −ΓKQ −1 D (5.51) # is an estimate of the argument (·), 2) s(E) ˙ is a smooth approximation where 1) (·) ˙ 3) D4 (t) is given by (5.47) and 4) one choice of the to the signum function sgn(E), functions ge (t) and gx (t) are such that: #E ||, ˙ ≥ α1 (q, x)||[q˙ x]|||| qe (t)sgn(E) ˙ q˙ L || + α2 (q, x)||FL || + ρα3 (q, x)||TE − T ( 4 ||, (5.52) gx (t)sgn(z2 ) ≥ ||D4 − D and αi (·) are such that ( x)−T CEL α1 (·) > ||Ψ(q, x)T Ψ(q, (q, x) − CEL (q, x)|| ( x)−T || α2 (·) > ||I − Ψ(q, x)T Ψ(q, α3 (·) > ||Ψ(q, x)T ||.
(5.53)
The choice of joystick control input Fq in (5.51) ensures that ∃, ∃t1 > 0 such that ˙ < where V (E, E) ˙ is positive definite. ∀t > t1 ; 0 ≤ V (E, E) Proof: Consider the dynamics of the coordinate transformed teleoperated hydraulic backhoe given by Eq. (5.5). The choice of coordination control Fq as given by Eq. (5.51) transforms the teleoperator dynamics as given by Eq. (5.43). The bondgraph based coordinate transformation derived in (5.45) transforms the actual locked system (5.43) into (5.46). The choice of valve control Fx as given in (5.51) ensures that the locked system behaves like the desired 4-Order passive locked system given by (5.49). In order to prove the passivity of the locked system, consider its storage function Wlocked4 = xL T ML4 xL , 74
(5.54)
where xL = [q˙ L z(1 z(2 ]T and ⎛
ML4
ML 0 ⎜ = ⎝ 0 ∆1
0 0
0
KQ Γ−1
0
⎞ ⎟ ⎠.
Differentiating the storage function w.r.t. time and substituting the transformed locked system dynamics (5.46) and further substituting the locked system control Fx as given in (5.51)-(5.52) leads to the following: xL T ML4
$ " d ( 4 (t) − gx (t)sgn(z(2 ) + q˙ T (ρTq − Fe ), xL = xL T Ω4 xL + z(2 D4 (t) − D L dt (5.55)
where ⎞ ⎛ ( −T 0 −CL Ψ ⎟ ⎜ (T Ω4 = ⎝−Ψ −KT KQ ⎠ . ¯1 0 −KQ −Ω Consider the second term on the $ R.H.S. of the equality in (5.55). When " ( 4 (t) − gx (t)sgn(z(2 ) < 0 by definition of gx (t) in (5.52) and 0, D4 (t) − D " $ ( 4 (t) − gx (t)sgn(z(2 ) ≥ 0 hence ensuring that the second z(2 ≤ 0, D4 (t) − D $ " ( z(2 D4 (t) − D4 (t) − gx (t)sgn(z(2 ) ≤ 0, ∀z(2 . Moreover, since Ω4 is negative definite, the above equation simplifies to
z(2 > when term, semi-
d Wlocked4 ≤ q˙ TL (ρTq − Fe ). dt Integrating the above equation provides the desired passivity result. ˙ remain bounded, consider the lyapunov function: To prove that [E E] 1˙T T T ˙ ˙ V = E ME E + E (KE + 1 BE )E + 1 E ME E , 2 where, given KE = KE T > 0 and BE = BE T > 0, 1 ∈ R++ can be appropriately chosen to ensure that V is positive definite. Differentiating V w.r.t. time and substituting for the dynamics of q ¨ and x ¨ as given by (3.1) and (3.18) respectively followed
75
˙ Ψ(q)E Θ1 (t)
Ψ(q)E ( −T (BE − 1 ME (q))E(t) ˙ Ψ(q)
˙ E(t)
Θ2 (t)
( −T KE E(t) Ψ(q)
E(t) Figure 5.8: Choice of BE and KE
by substitution of the joystick control Fq as given in (5.51), leads to ( ( ˙ T ΨT Ψ−T ˙ − 1 ET ΨT Ψ−T (BE − 1 ME )E KE E+ V˙ = −E ( −T C ) ( −T )FL + ρΨT (TE − T #E ) − ˙ T (ΨT Ψ E ˙ L + (I − ΨT Ψ EL − CEL )q ( −T ge (t)sgn(E) ( −T ge (t)(sgn(E) ˙ T ΨT Ψ ˙ +E ˙ T ΨT Ψ ˙ − s(E)). ˙ E Due to the definition of ge (t) as given in (5.52) and following the explanation as detailed in the passivity proof and under the assumption stated in Eq. 5.53, the sum of third and fourth terms in the above equation is negative. Hence the above equation simplifies to ( −T (BE − 1 ME )E ( −T KE E ˙ T ΨT Ψ ˙ − 1 ET ΨT Ψ V˙ ≤ −E ( −T ge (t)(sgn(E) ˙ T ΨT Ψ ˙ − s(E)). ˙ +E
(5.56)
( are both positive definite it is possible to choose BE = BE T > 0 and Since Ψ, Ψ KE = KE T > 0 as shown in Fig. 5.8 so that the first two terms of the above ˙ are vectors in Euclidean space). equation are negative definite (assuming that E, E Geometrically, the appropriate BE , KE will be the ones which ensure the following inequality: ( −T (BE − 1 ME )E ˙ T ΨT E ˙ TΨ ˙ ˙ E E , ≤ ˙ ˙ ( T (BE − 1 ME )E| ˙ Ψ ˙ |E||Ψ E| |E|| ( −T KE E ET Ψ E T ΨT E 0≤ ≤ . ( T KE E| |E||ΨE| |E||Ψ 0≤
Since Ψ is positive definite it is possible to choose BE = BE T > 0 and KE = KE T > 0 as shown in Fig. 5.8 so that the first two terms of the above equation are negative definite. The remainder of the proof follows Lee and Li (2002).
76
Equation (5.56) can be written as √ ¯ V˙ ≤ −δ1 V + δ2 V (t)D, ( −T ge (t)(sgn(E) ¯ ≥ ||ΨT Ψ ˙ − s(E))||, ˙ ˙ for some δi > 0 and D ∀t > 0. Hence if E(0), E(0) √ √ ¯ then ∃t1 > 0, such that V (t) < δ2 D ¯ =⇒ 0 < are such that V (0) > δδ21 D δ1 2 ˙ ≤ δ22 D¯2 , ∀t > t1 . V (E, E) δ1
˙ 0, necessarily (as proved in theorem 6), Since the coordination error velocity E the energy associated with the error dynamics V given in (5.25) is such that V 0, necessarily. Owing to this property, ensuring that the locked system given by Eq. (5.48) is passive w.r.t. the supply rate (5.50) does not imply that the teleoperated backhoe is passive w.r.t. the same supply rate, (5.50). It is however possible to show that the teleoperated backhoe is nearly passive w.r.t. the 1-port supply rate (5.50) and hence is nearly passive w.r.t. the 2-port supply rate (5.57): stele3 ((Tq , −Fe ), (q, ˙ x)) ˙ = ρq˙ T Tq − x˙ T Fe .
(5.57)
A dynamic system x˙ = f (x, u) y = g(x, u) where x, u, y are the state vector, input and output of the system, respectively is said to be nearly-passive with respect to the supply rate s(u, y) ∈ L1e for some scalar function s : (u, y) → s(u, y) ∈ R if for all initial states, there exists an initial energy, c2 so that, for all admissible inputs, u(·) and for all t < Tf , −
0
t
s(u(τ ), y(τ ))dτ ≤ c2 + δ()t,
(5.58)
where δ() can be made arbitrarily small. Theorem 7. The teleoperated hydraulic backhoe given by Eq. (5.5) whose locked system dynamics are represented by (5.49) is nearly-passive w.r.t. the supply rate given in Eq. (5.57). Proof: Consider the storage energy of the teleoperated system given by the sum of
77
the locked system energy (5.54) and the energy due to coordination error (5.25). Wlocked4 + V =
1 T ˙ T ME E ˙ + ET (KE + 1 BE )E + 1 E ˙ T ME E . xL ML4 xL + E 2 (5.59)
˙ in Eq. (5.42) can be chosen such that given > 0 arbitrarily small, Notice that s(E) ˙ = sgnE, ˙ ∀E ˙ ≥ . As a result, the storage function V (E, E) ˙ in theorem 6 can be s(E) 2 ¯ 2 , ∀t > t1 . ˙ ≤ δ22 D() ensured to satisfy the property that ∃t1 > 0, such that V (E, E) δ1
Differentiating Eq. (5.59) and noting from theorem 6 that: ˙ locked4 ≤ stele ((Tq , −Fe ), (q˙ L , q˙ L )), W √ ¯ V˙ < δ2 V (t)D ensures that √ ˙ locked4 + V˙ < stele ((Tq , −Fe ), (q˙ L , q˙ L )) + δ2 V (t)D(). ¯ W However, from theorem 6 it is shown that V is ultimately bounded by ∃δ3 > 0 such that the above equation can be written as:
δ22 ¯ D()2 . δ12
Hence
¯ 2 (), ˙ locked4 + V˙ < stele ((Tq , −Fe ), (q˙ L , q˙ L )) + δ3 D W t ¯ stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ + δ3 D()t =⇒ −Wlocked4 (0) − V (0) < 0
thus ensuring that the teleoperated backhoe is nearly passive w.r.t. the 1-port supply rate (5.50). The remainder of the proof of passivity w.r.t. the 2-port supply rate (5.57) follows the proof of theorem 3. Theorem 6 summarized the passive and coordination control designs if the master and slave systems are not exactly known. The theorem ensures 4-Order, 2-DOF locked system behavior of the hydraulic teleoperator. We now present the passive coordination control design in order to ensure 2-Order, 2-DOF haptic behavior of the hydraulic teleoperator in the presence of imperfect models. 5.3.5.3 Passive Locked System Design : 2nd Order, 2-DOF Haptic Behavior The locked system control objective remains the same which is to determine Fx in Fig. 5.4 so that the closed loop dynamics of the actual locked system shown in the figure behaves like the desired 2nd order locked system dynamics similar to the one shown 78
in Fig. 5.6. The design process is similar to that outlined in the previous section and involves comparing the dynamics of the actual and desired locked systems and choosing the appropriate control input Fx so that the actual dynamics mimic the desired dynamics. The dynamics of the actual locked system shown in Fig. 5.7 are given by Eq. (5.43). The dynamics of the desired locked system similar to the one shown in Fig. 5.6 are: ⎛
ML
⎜ ⎝ 0 0
0 ∆1 0
⎞
⎡
⎤ ⎛ ⎞⎡ ⎤ ⎡ ⎤ ( −T q˙ L 0 q˙ L −CL Ψ ρTq − Fe ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎢ ⎥ −KT KQ ⎠ ⎣ z(1 ⎦ + ⎣ 0 ⎠ ⎣ z(1 ⎦ = ⎝ 0 0 ⎦ dt −1 ¯1 z(3 z(3 0 −KQ −Ω KQ Γ 0 0
(5.60) where ML is the locked system inertia, q˙ L is the locked system velocity, ρTq − Fe is the external torque operating on the locked system, z(1 and z(3 are the internal states of the locked system. The passive control objective is to design the passive valve control Fx in (5.43) so that the actual locked system behaves like the desired locked system in Eq. (5.60). Comparison of the bondgraph representations shown in Fig. 5.6 and Fig. 5.4 as explained in detail in the previous subsection, suggests the following coordinate transformation [q˙ L FL xv ]T → [q˙ L z(1 z(3 ]T : ⎡
⎤
⎛
⎞⎡
⎤
⎡
0 ( T D(t) Ψ
q˙ L q˙ L I 0 0 ⎢ ⎟⎢ ⎥ ⎢ ⎥ ⎜ 0 I 0⎠ ⎣FL ⎦ + ⎢ ⎣ z(1 ⎦ = ⎝ ⎣ −1 −1 ( T D(t) + KQ KT Ψ z(3 0 I xv −αKQ
⎤
⎥ ⎥ ⎦ , d ( T D(t)] [∆1 Ψ dt (5.61)
# ) ( −T C #E − ρe(E, E) ˙ − CLE E ˙ +Ψ ˙ L . This transformation (Eq. where D(t) = −ρT EL q (5.61)) when applied to the actual locked system dynamics (5.14) results in the following transformed dynamics: ⎞ ⎡ ⎤ ⎛ ⎞⎡ ⎤ ( −T 0 0 q˙ L q˙ L ML 0 −CL Ψ ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎜ 0 ⎠ ⎣ z(1 ⎦ = ⎝ 0 −KT KQ ⎠ ⎣ z(1 ⎦ ⎝ 0 ∆1 dt ¯1 z(3 z(3 0 −KQ −Ω 0 0 KQ Γ−1 ⎤ ⎡ ρTq − Fe ⎥ ⎢ +⎣ 0 ⎦, −1 KQ Γ Fx + D5 (t) ⎛
79
(5.62)
where D5 (t) = KQ [z(1 FL ] + KQ Γ
−1
( T D(t), z(1 FL := Ψ −1
z(3 xv := −αKQ q˙ L + KQ
−1
d Ω[z(3 xv ] + [z(3 xv ] , dt
(5.63)
d T T ( D(t) + [∆1 Ψ ( D(t)] . KT Ψ dt
The signals z(1 FL and z(3 xv are dependant on D(t) and its successive derivatives ˙ If the locked system which are functions of external variables TE , q˙ L , FL , E and E. control Fx in (5.62) is chosen to satisfy: KQ Γ−1 Fx = −D5 (t),
(5.64)
then the locked system is passive w.r.t. the supply rate (5.10) and its dynamics are given by: ⎛
ML 0 ⎜ ⎝ 0 ∆1 0
0
⎤ ⎛ ⎞⎡ ⎤ ( −T 0 q˙ L q˙ L −CL Ψ ⎟ d ⎢ ⎥ ⎜ ⎟⎢ ⎥ −KT KQ ⎠ ⎣ z(1 ⎦ ⎠ ⎣ z(1 ⎦ = ⎝ 0 dt ¯1 z(3 z(3 KQ Γ−1 0 −KQ −Ω ⎤ ⎡ ρTq − Fe ⎥ ⎢ +⎣ 0 ⎦. 0 0
⎞
⎡
(5.65)
0 Since the shunt conductance KT (|xv |, FL ) > 0 (given in (3.5)) for all xv and P1 , P2 < Ps I (Ps is the hydraulic system pressure defined in (3.7)), the states [z(1 z(3 ]T → 0, asymptotically. Hence, the locked system (5.20) asymptotically approaches the second order locked system given by: ML (x, q)¨ qL + CL ((x, q), (x, ˙ q)) ˙ q˙ L = ρTq − Fe .
(5.66)
The passive coordination analysis presented above is now summarized in the following theorem. Theorem 8. 2nd Order, 2-DOF Locked System, imperfect system model: The teleoperated hydraulic backhoe composed of a) the passive valve (3.17), b) the dynamically modeled hydraulic actuators (3.18), c) the mechanical backhoe (3.23) and d) the motorized joystick (3.1) which is concisely represented in Eq. (5.5) transforms to an interconnected system (5.62) under the choice of joystick control input Fq as 80
given in (5.68) and a coordinate transformation (5.61). The interconnected system (5.62) is passive with respect to the supply rate: stele ((Tq , −Fe ), (q˙ L , q˙ L )) = q˙ TL (ρTq − Fe ),
(5.67)
if the passive valve control input, Fx is chosen as follows: ( −T #E + Ψ # ) ( −T − I)FL − KE E − BE E ˙ , ˙ − ge (t)s(E) Fq = −T ˙ L − (αΨ C EL q ρ ( 5 (t) − gx (t)sgn(z(3 ) , Fx = −ΓKQ −1 −D (5.68) # is an estimate of the argument (·), 2) s(E) ˙ is a smooth approximation where 1) (·) ˙ 3) D5 (t) is given by (5.63) and 4) one choice of the to the signum function sgn(E), functions ge (t) and gx (t) are such that: #E ||, ˙ ≥ α1 (q, x)||[q˙ x]|||| qe (t)sgn(E) ˙ q˙ L || + α2 (q, x)||FL || + ρα3 (q, x)||TE − T ( 5 (t||, (5.69) gx (t)sgn(z2 ) ≥ ||D5 (t) − D and αi (·) are such that ( x)−T CEL α1 (·) > ||Ψ(q, x)T Ψ(q, (q, x) − CEL (q, x)|| ( x)−T || α2 (·) > ||I − Ψ(q, x)T Ψ(q, α3 (·) > ||Ψ(q, x)T ||.
(5.70)
The choice of joystick control input Fq in (5.68) ensures that ∃, ∃t1 > 0 such that ˙ < where V (E, E) ˙ is positive definite. ∀t > t1 ; 0 ≤ V (E, E) Proof: Consider the dynamics of the coordinate transformed teleoperated hydraulic backhoe given by Eq. (5.5). The choice of coordination control Fq as given by Eq. (5.68) transforms the teleoperator dynamics into Eq. (5.43). The bondgraph based coordinate transformation derived in (5.45) transforms the actual locked system (5.43) into (5.62). The choice of valve control Fx as given in (5.68) ensures that the locked system behaves like the desired 2-Order passive locked system given by (5.65). In order to prove the passivity of the locked system, consider its storage function: Wlocked2 = xL T2 ML2 xL2 , 81
(5.71)
where xL2 = [q˙ L z(1 z(3 ]T and ⎛
ML2
ML 0 ⎜ = ⎝ 0 ∆1
0 0
0
KQ Γ−1
0
⎞ ⎟ ⎠.
Differentiating the storage function w.r.t. time and substituting the transformed locked system dynamics (5.62) and further substituting the locked system control Fx as given in (5.68)-(5.69) leads to the following: xL T2 ML2
$ " d ( 5 (t) − gx (t)sgn(z(3 ) xL2 = [z(1 T z(3 T ]T Ω2 [z(1 z(3 ] + z(3 D5 (t) − D dt ( −T FL + q˙ T (ρTq − Fe ), +Ψ (5.72) L
where Ω2 =
−KT (|xv |, FL ) KQ , −KQ −KQ Γ−1 Ω
is negative semidefinite. By definition of gx (t) in (5.69) and following the explanation as given in theorem 6, the above equation simplifies to d ( −T FL + q˙ T (ρTq − Fe ). Wlocked2 ≤ q˙ TL Ψ L dt Under the assumption that (q˙ L , FL ) are bounded by (q˙ L , FL ) and since the states [z(1 z(3 ]T → 0 asymptotically 2 , integrating the above equation leads to: Wlocked2 (T ) − Wlocked2 (0) T T −T ( ≤ q˙ L (τ )Ψ(τ ) FL (τ )dτ + stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ 0 0 T T −ω τ 2 ( L (0) ≤ q˙ L ΨF e dτ + stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ 0 0 T ≤c+ stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ 0
2
Can be proved by differentiating the lyapunov function T T T −1 V = 12 [# z1 z# ][# z1 z# 3 ] Diag [∆1 , KQ Γ 3 ] and substituting the dynamics (5.65).
82
where ω2 = σ(Ω2 ), c ∈ R+ . Hence
T 0
stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ ≥ −Wlocked2 (0) − c,
completing the proof for passivity of the locked system. The proof for boundedness ˙ is exactly the same as in theorem 6. of [E E] Since the locked system behavior directly influences the haptics of the teleoperator, the proposed algorithm is benefecial as it permits the designer to choose the appropriate haptics, given the application of the teleoperator. As in the case with theorems 6 and theorem 7, it is possible to show that the teleoperated hydraulic backhoe is nearly passive w.r.t. the supply rate stele4 ((Tq , −Fe ), (q, ˙ x)) ˙ = ρq˙ T Tq − x˙ T Fe .
(5.73)
This is presented in the following theorem. Theorem 9. The teleoperated hydraulic backhoe given by Eq. (5.5) whose locked system dynamics are represented by (5.65) is nearly-passive w.r.t. the supply rate given in Eq. (5.73). Proof: Consider the storage energy of the teleoperated system given by the sum of the locked system energy (5.71) and the energy due to coordination error (5.25). Wlocked4 + V =
1 T ˙ T ME E ˙ + ET (KE + 1 BE )E + 1 E ˙ T ME E . xL ML4 xL + E 2 (5.74)
˙ in Eq. (5.42) can be chosen such that given > 0 arbitrarily small, Notice that s(E) ˙ = sgnE, ˙ ∀E ˙ ≥ . As a result, the storage function V (E, E) ˙ in theorem 8 can be s(E) 2 ¯ 2 , ∀t > t1 . ˙ ≤ δ22 D() ensured to satisfy the property that ∃t1 > 0, such that V (E, E) δ1
Differentiating Eq. (5.74) and noting from theorem 8 and 7 that: ˙ locked2(0) − c1 ≤ W
0
T
stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ,
√ ¯ V˙ < δ2 V (t)D, ¯ ∃ δ3 > 0 such that V˙ < δ3 D()
83
ensures that ˙ locked2(0) − c1 − V (0) < W
t 0
¯ stele ((Tq , −Fe ), (q˙ L , q˙ L ))dτ + δ3 D()t
thus ensuring that the teleoperated backhoe is nearly passive w.r.t. the 1-port supply rate (5.67). The remainder of the proof of passivity w.r.t. the 2-port supply rate (5.73) follows the proof of theorem 3. Theorems 6 and 8 present the passivity and coordination results of the proposed passive teleoperation algorithms. It is shown that the passivity property of the teleoperation system does not depend on the coordination performance of the teleoperation control. This is because in order to preserve the passivity property of the teleoperator it is necessary for the valve control Fx to know exactly the joystick control Fq and its successive derivatives. However, the joystick control which is designed to ensure coordination of the teleoperator need not exactly achieve asymptotic convergence of the coordination error in order to guarantee the passivity property of the teleoperator. The theorems also demonstrate the ability of the control design method in achieving different order locked systems and hence designing the haptic behavior of the passive teleoperator. This is beneficial as the locked system behavior is exactly the haptic dynamics of the teleoperator. Being able to flexibly design the order of the teleoperator will improve the efficiency of the teleoperator depending on its purpose. For example, if the teleoperator is used in tasks like micro-surgery it may be beneficial to design a low order teleoperator in order to avoid sluggishness associated with higher order dynamics. On the other hand, if the teleoperator is used in a harsh environment like hammering, it might be beneficial to design a higher order teleoperator so that the external forces get sufficiently filtered by the natural dynamics of the locked system before contacting with the operator.
5.4
Summary
In this chapter a bond graph based approach for passive teleoperation of a hydraulic backhoe was presented. The teleoperation control algorithm proposed in this chapter is superior to the method proposed in chapter 4 as it was possible to systematically account for the nonlinear dynamics of the slave. It will be shown that the proposed algorithm can be scaled to control arbitrary order master and slave systems in the next chapter with mismatched relative degree between the master system and the slave system. Using the intrinsically passive control algorithm proposed in the pre84
vious chapter, it is not known if the chosen skew-symmetric control architecture can be scaled to ensure passivity and coordination of arbitrary order master and slave systems. The proposed bondgraph based passive coordination control algorithm consists of the following key steps. • Transforming the master and slave systems to include the coordination error, E as state. • Identification of the shorted causal path between a control input Fq and the ˙ coordination variable E. • Determination, using inverse dynamics, the coordination control law which en˙ → 0. sures that E • Representation of the coordination control law Fq in the original bond graph of the master and slave systems using Effort / Flow sources and any state feedback as necessary. • Determination of 1) a coordinate transformation and 2) a control input, Fx which will ensure the actual active bond graph behaves like a desired passive bond graph hence achieving a desired locked system behavior. In the next chapter, the above developed passive teleoperation design procedure will be extended to passive teleoperation of arbitrary degree cascade passive systems.
85
Chapter 6 Passive teleoperation of cascade passive systems using bondgraphs 6.1
Introduction
In this chapter, a systematic control design approach for passive teleoperation of cascade passive master and slave systems is proposed. The control algorithm presented in chapter 5 is generalized in this chapter. The controller operates bilaterally between a master system and a slave system (both can be a cascade interconnection of passive systems). Without Loss Of Generality (WLOG), it is assumed that the master system is a cascade interconnection of n(odd) (< m(even)), p-Degrees Of Freedom systems and the slave system is a cascade interconnection of m p-DOF systems. Although the master and slave sytems are assumed to be a cascade interconnection of first order passive systems, the algorithm can also scale to arbitrary order passive systems.
86
N1m u1
1 xm
N1m_1 TFm_1
Lm
N13
...
0 xm_1
TF2
x3
Lm_1
...
0
1
On
0
TF4
O3
T1
x1 L1
N22
y3
1
TF1
L2
N23
yn
0 x2
L3
N2n u2
1
N11
N12
N21 TF3
y2 O2
1
T2
y1 O1
Figure 6.1: Bondgraphs of the Master and Slave systems
6.2
Bondgraph, Model and Control objective
The dynamics of the master and slave systems are given in (6.1). The bond graph representation is shown in Fig. 6.1 ⎛ N11 ⎜ ⎜ 0 ⎜ ⎜ : ⎝ 0 ⎛ N21 ⎜ ⎜ 0 ⎜ ⎜ : ⎝ 0
0 ... N12 . . . : 0
... ...
0 ... N22 . . . : 0
... ...
⎞⎡ ⎤ ⎛ ⎞⎡ ⎤ ⎡ ⎤ x˙ 1 −L1 M1 x1 T1 0 ... 0 ⎢ ⎢ ⎟⎢ ⎥ ⎜ ⎟ ⎥ ⎥ ⎢ x˙ 2 ⎥ ⎜−M1T −L2 ⎟ ⎢ x2 ⎥ ⎢ 0 ⎥ 0 ⎟ . . . 0 ⎟⎢ ⎥ = ⎜ ⎟⎢ ⎥ + ⎢ ⎥ ⎢ : ⎥ ⎜ : ⎟⎢ : ⎥ ⎢ : ⎥ ... ⎟ : . . . . . . ⎠⎣ ⎦ ⎝ ⎠⎣ ⎦ ⎣ ⎦ T N1m 0 0 −Mm−1 −Lm u1 x˙ m xm ⎞⎡ ⎤ ⎛ ⎞⎡ ⎤ ⎡ ⎤ 0 ... 0 y˙ 1 −O1 K1 y1 T2 ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎢ ⎥ T ⎢ ⎥ ⎜ ⎢ ⎥ ⎢ ⎥ 0 ⎟ ... 0 ⎟ ⎟ ⎢ y˙ 2 ⎥ = ⎜−K1 −O2 ⎟ ⎢ y2 ⎥ + ⎢ 0 ⎥ ⎢ ⎥ ⎜ ⎢ ⎥ ⎢ ⎥ ... ⎟ : ... ... ⎟ ⎠⎣ : ⎦ ⎝ : ⎠⎣ : ⎦ ⎣ : ⎦ T N2n 0 0 −Kn−1 u2 −On y˙ n yn (6.1)
where xi , yi , Tj , uk ∈ Rp , p is the number of degrees of freedom, Nij , Li , Oj are positive definite. Notice that the bondgraph representation involves two disjoint bondgraphs which are individually passive w.r.t. their respective supply rates. For the top graph, the slave system, the supply rate is ss = xTm u1 + xT1 T1 . For the bottom graph the supply rate is sm = ynT u2 + y1T T2 .
87
In order to simplify the passive control synthesis and analysis, the states x1 and y1 of the slave and master systems (6.1) are transformed using a decomposition similar to the one proposed in chapter 5. It is given in Eq. (6.2).
e I −I x1 := y1 e⊥ I −Ψ Ψ
(6.2)
−1 −1 +I is invertible. The above decomposition transforms the where Ψ = N11 N21 master-slave dynamics into dynamics involving the states; 1) coordination error, e and 2) average motion of the master and slave systems, e⊥ . Using the above decomposition, the dynamics of the master and slave are transformed into ⎡
⎤ ⎡ ⎤ ⎡ ⎤ e˙ e ΨT T1 + (ΨT − I)T2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ e˙⊥ ⎥ ⎢ e⊥ ⎥ ⎢ ⎥ T + T 1 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙ ⎥ ⎢x ⎥ ⎢ ⎥ 0 ⎢ 2⎥ ⎢ 2⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ : ⎥ ⎢ : ⎥ ⎢ ⎥ : ⎥ = Ω1 ⎢ ⎥ + ⎢ ⎥ N1 ⎢ ⎢x˙ ⎥ ⎢x ⎥ ⎢ ⎥ u 1 ⎢ m⎥ ⎢ m⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ y˙ 2 ⎥ ⎢ y2 ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ : ⎥ ⎢ : ⎥ ⎢ ⎥ : ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ u2 y˙n yn
(6.3)
where −1 −1 +I Ψ = N11 N21
(6.4)
N1 = Diag[Ne , Ne⊥ , N12 , . . . , N1m , N22 , . . . , N2n ]
(6.5)
Ne = ΨT N11 Ψ + (ΨT − I)N21 (Ψ − I)
(6.6)
Ne⊥ = N11 + N21
(6.7)
and A B , Ω1 = −B T C
88
(6.8)
N1m_1
N1m u1
1 xm
TFm_1
0 xm_1
N13
u2
0
1
TF2
x3
L2 N22
N23
...
yn
1
TF1
1
TF5
TF6
0
TF4
y3
On
0
T1+T2
x2
L3
N2n
Nep ep
...
Lm_1
Lm
N12
Ne
TF3
y2
O3
L1
1
SS TE
e
O2
O1
Figure 6.2: Bondgraph of teleoperator after coordinate transformation ⎡
−Le 0 ΨT M1 ... ⎢ ⎢ 0 M1 ... −Le⊥ ⎢ T T A=⎢ −L2 ... ⎢−M1 Ψ −M1 ⎢ : : : ⎣ : 0 ... 0 −Mm−1 ⎡ ⎤ (ΨT − I)K1 0 . . . 0 ⎢ ⎥ ⎢ ⎥ K 0 . . . 0 1 ⎥ B=⎢ ⎢ ⎥ 0 0 . . . 0 ⎣ ⎦ : : : ... ⎡ K2 0 ... 0 −O2 ⎢ T ⎢−K2 −O3 K3 ... 0 ⎢ C=⎢ −K3T −O4 ... 0 ⎢ 0 ⎢ : : : ... ⎣ : 0
...
0
T −Kn−1
⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ ... ⎦ −Lm
(6.9)
(6.10) ⎤
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ −On .
(6.11)
The bond graph of the coordinate transformed master and slave systems is given in Fig. 6.2. The passive teleoperation control objective is to synthesize control inputs u1 and u2 so that : • the coordination error e → 0,
(6.12)
asymptotically. Satisfying this requirement ensures that the master and the 89
slave systems are coordinated at all time. • The teleoperator in Fig. 6.2 is passive with respect to the two-port supply rate: stele ((T1 , T2 ), (x1 , y1)) = xT1 T1 + y1T T2 .
(6.13)
This requirement ensures that the closed loop teleoperator does not generate any energy of its own but only acts as a ‘transformer’ (possibly with some dissipation) between one interaction port, (T1 , x1 ) and the other, (T2 , y1 ). Such passive systems are guaranteed to provide stable feedback interconnections with strictly passive systems, (Vidyasagar, 1993).
6.3
Control Design
Passive control is designed in two stages. In the first stage, a bondgraph inversion method (Ngwompo et al., 2001a), (Ngwompo et al., 2001b), an extension to that shown in chapter 5, is used to determine the coordination control action which will ensure asymptotic convergence of e to 0, (6.12). In the second stage, the coordination control action is represented as an extension to the master-slave bondgraph (Fig. 6.2) and a suitable pair (Energy, Passive control) is determined which will ensure that the coordinated bondgraph behaves like a passive system w.r.t. the supply rate (6.13). 6.3.1
Coordination control design
The shortest causal path between the output e and a control input is of length n to the slave control input u2 . Using bicausal bonds (Gawthrop, 1995), the inverse dynamics bondgraph is represented as shown in Fig. 6.3. The inverse dynamics from
90
N1m
N1m_1
N13
N12
Nep ep
u1
1 xm
TFm_1
0 xm_1
Lm
...
1
Lm_1
0
L2 N22
N23
...
pn
1
0
TF4
p3
On
0
TF1
1
T1+T2
x2
L3
N2n u2
TF2
x3
TF5
TF6
Ne
TF3
p2
O3
L1
O2
1
SS TE
e O1
Figure 6.3: Inverse dynamics bondgraph of the teleoperator, input u2 and output e the above bondgraph can be written as follows: ⎡ ⎤ ΨT T1 + (ΨT − I)T2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ p˙2 ⎥ ⎢ p2 ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ p˙ ⎥ ⎢ p ⎥ ⎢ ⎥ 0 ⎢ 3 ⎥ ⎢ 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ : ⎥ ⎢ : ⎥ ⎢ ⎥ : ⎢ ⎢ ⎥ ⎢ ⎥ ⎥ = Ω2 ⎢ +⎢ N2 ⎢ ⎥ ⎥ ⎥ 0 ⎢p˙n−1 ⎥ ⎢pn−1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ e˙⊥ ⎥ ⎢ e⊥ ⎥ ⎢ ⎥ T1 + T2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˙ ⎥ ⎢ x ⎥ ⎢ ⎥ 0 ⎣ 2 ⎦ ⎣ 2 ⎦ ⎣ ⎦ ⎡
e˙
x˙ m
⎤
⎡
e
⎤
u1 xm d T u2 = N2n pn + Kn−1 pn−1 + On pn . dt
(6.14)
(6.15)
where pi are states as defined in the bondgraph shown in Fig. 6.3, N2 = Diag[Ne , N22 , N23 , . . . , Nn−1 , Ne⊥ , N12 , . . . , N1m−1 ] ⎡ ⎤ −Oe (ΨT − I)K1 0 ... 0 0 ΨT M1 0 ⎢ ⎥ ⎢−K1T (Ψ − I) −O2 K2 ... 0 K1 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ T −O . . . 0 0 0 0 0 −K ⎢ ⎥ 3 2 ⎢ ⎥ ⎢ : : : : ... ... ... ... ⎥ ⎢ ⎥. Ω2 = ⎢ ⎥ T 0 0 . . . −K −O K 0 0 ⎢ ⎥ n−1 n−1 n−2 ⎢ ⎥ T ⎢ ⎥ 0 . . . 0 −L M 0 0 −K e 1 1 ⊥ ⎢ ⎥ ⎢ −M T Ψ ⎥ T 0 0 . . . 0 M −L 0 2 ⎣ ⎦ 1 1 0 0 0 ... 0 0... 0 −Lm
91
The following choice of slave control u2 T u2 = −Kn−1 Dn−1 (t) − On Dn (t) − N2n D˙ n (t)
(6.16)
where Dn (t) =
−1 Kn−1
$ " T ˙ N2n−1 Dn−1 (t) − Kn−2 Dn−2 (t) − On−1Dn−1 (t)
(6.17)
:= :
$ " D3 (t) = K2−1 O2 (D2 (t)) − (K1 e⊥ − N22 D˙ 2 (t)) D2 (t) = (ΨT − I)K1 (ΨT T1 + (ΨT − I)T2 + ΨT M1 x2 ).
(6.18)
ensures the following coordination dynamics: ⎡ ⎤ ⎛ ⎞⎡ ⎤ (ΨT − I)K1 0 ... 0 e˙ e −Oe ⎢ ⎥ ⎜ ⎟ ⎥ ⎢ ⎢ e˙ 2 ⎥ ⎜−K1T (Ψ − I) −O2 K2 ... 0 ⎟ ⎢ e2 ⎥ ⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎥ ⎜ ⎢ ⎥ Ne ⎢ −O3 ... 0 ⎟ 0 −K2T ⎢ e˙ 3 ⎥ = ⎜ ⎟ ⎢ e3 ⎥ . ⎢ ⎥ ⎜ ⎟⎢ ⎥ : : ... ... ... ⎠⎣ : ⎦ ⎣:⎦ ⎝ T e˙n en 0 ... 0 −Kn−1 −On
(6.19)
where ⎤ ⎡ ⎤ ⎡ p2 + D2 (t) e2 ⎥ ⎢ ⎥ ⎢ ⎢ e3 ⎥ ⎢ p3 − D3 (t) ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ e4 ⎥ = ⎢ p4 − D4 (t) ⎥ , ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ : ⎦ ⎣:⎦ ⎣ en pn − Dn (t) where Dn (t), D2 (t) are given in Eqs. (6.17)-(6.18) and Ne = Diag[Ne , N22 , . . . , N2n ]. In the particular case u2 was used to control the coordination behavior of the teleoperator. This is because the shortest causal path was determined to be between the coordination variable (e) and the control input (u2 ). In the next section we determine the energy necessary to implement the coordination control law (6.16) so that the coordinated teleoperator is passive w.r.t. the supply rate (6.13). This is achieved by first graphically representing the coordination control in the bondgraph Fig. 6.2 and analyzing the power flow.
92
LOCKED SYSTEM DYNAMICS N1m Seu1
1 xm Lm
N1m_1 TFm_1
0 xm_1
N13
...
On
L2
N23
...
0
TF1
x2
L3
N2n 0
TF2
x3
Lm_1
en
1
Nep
N12
1
EF2
N22 TF4
e3
0
1
L1
T1+T2
EF1
Ne TF3
e2
O3
ep
O2
1 e O1
COORDINATION ERROR DYNAMICS
Figure 6.4: Bondgraph of the master and slave systems with coordination control law 6.3.2
Closed loop passivity property
In the bondgraph, Fig. 6.4, notice that the coordination dynamics are decoupled from the locked system. This decoupling imposes additional power inputs to the locked system. In this previous chapter, additional power to the locked system was represented using effort sources and signal bonds. In the current development, the additional power input is represented using the more general effort sources alone. In Fig. 6.4, EF1 , EF2 are given by: EF1 = K1T y2 EF2 = M1T Ψe. The bondgraph shown in Fig. 6.4 represents the ‘Locked System’ behavior; i.e., the behavior of the coordinated teleoperator. Notice that the bottom disconnected bond graph in the figure represents the dynamics of the coordination error. The top disconnected bond graph in the figure represents the dynamics of the locked system and determines the haptics experienced by the human operator. Unlike the locked system in Lee and Li (2002) which is the result of an isometric energetic decomposition, the locked system proposed here is a design variable. This approach has the advantage of letting the control designer choose the haptic behavior of the teleoperator while ensuring the control obectives listed above. Notice that the locked system is not passive w.r.t. the desired supply rate (6.13) due to the presence of additional effort sources Se in the bond graph Fig 6.4. The task now is to determine the remaining control input u1 so that the bondgraph shown in 93
Fig. 6.4 is passive w.r.t. the supply rate (6.13). As stated above, the desired locked system structure is upto the control system engineer. Towards that end, two locked system designs are presented in this dissertation. The first design shown in Fig. 6.5 proposes a ‘mth’ order, p-DOF locked system. The second design shown in Fig. 6.6 proposes a ‘2nd’ order, p-DOF locked system. 6.3.3
Locked System Design : m-Order, p-DOF Haptic Behavior
We need to determine the control u1 in Fig. 6.4 so that the closed loop dynamics of the actual locked system shown in the figure behaves like the desired locked system dynamics shown in Fig. 6.5. The design process involves comparing the dynamics of the actual and desired locked systems and choosing the appropriate control input u1 so that the actual dynamics mimic the desired dynamics. The dynamics of the actual locked system shown in Fig. 6.4 are: ⎛ 0 Ne⊥ 0 ⎜ ⎜ 0 N12 0 ⎜ ⎜ 0 0 N13 ⎜ ⎜ : ... ⎝ : 0 ... 0
⎞⎡ ⎤ ⎛ ⎞⎡ ⎤ 0 0 ... 0 e˙ ⊥ −Le⊥ M1 e⊥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ T 0 ⎟ ⎢ x˙ 2 ⎥ ⎜−M1 −L2 M2 ... 0 ⎟ ⎢ x2 ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ T ⎟ ⎢ x3 ⎥ ⎢ x˙ 3 ⎥ = ⎜ 0 −M ... 0 ⎟ −L . . . 0 3 2 ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ... ... ⎠⎣ : ⎦ ⎝ : : ... ... ... ⎠⎣ : ⎦ T −Lm x˙ m xm 0 N1m 0 ... 0 −Km−1 ⎤ ⎡ T1 + T2 + K1 y2 ⎥ ⎢ ΨT M1 e ⎥ ⎢ ⎥ ⎢ ⎥. ⎢ +⎢ (6.20) 0 ⎥ ⎥ ⎢ : ⎦ ⎣ u1
... ...
94
m−ORDER DESIRED LOCKED SYSTEM DYNAMICS N1m 1
N1m_1 TFm_1
0 zm_1
zm Lm
N13
...
1
...
ep
1
TF4
0
T1+T2
Ne TF3
e2
O3
1
L1
N22
e3
On
TF1
L2
N23
en
0 z2
L3
N2n 0
TF2
z3
Lm_1
Nep
N12
1 e
O2
O1
DISSIPATIVE DYNAMICS
Figure 6.5: m-order desired locked system The dynamics of the desired locked system shown in Fig. 6.5 are: ⎛ 0 Ne⊥ 0 ⎜ ⎜ 0 N12 0 ⎜ ⎜ 0 0 N13 ⎜ ⎜ : ... ⎝ : 0 ... 0
...
0
... ...
0 0
⎞⎡
e˙ ⊥
⎤
⎛
−Le⊥
M1
0
...
0
⎞⎡
e⊥
⎤
⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ... 0 ⎟ ⎢ z2 ⎥ ⎟ ⎢ z˙2 ⎥ ⎜−M1T −L2 M2 ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎟ ⎢ z˙3 ⎥ = ⎜ 0 ⎢ ⎥ ... 0 ⎟ −M2T −L3 ⎟⎢ ⎥ ⎜ ⎟ ⎢ z3 ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ... ... ⎠⎣ : ⎦ ⎝ : : ... ... ... ⎠⎣ : ⎦ T 0 N1m 0 ... 0 −Km−1 −Lm z˙m zm ⎤ ⎡ T1 + T2 ⎥ ⎢ ⎢ 0 ⎥ ⎥ ⎢ ⎥. (6.21) +⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎣ : ⎦ 0
The passive control objective is to design the control u1 in (6.20) (also in bondgraph in Fig. 6.4) so that the actual locked system behaves like the desired locked system in Eq. (6.21) (also in bondgraph in Fig. 6.5). In order to achieve this, compare the actual dynamics of e⊥ in (6.20) and the desired dynamics as given by (6.21). Notice that the dynamics are identical if and only if z2 := x2 + M1−1 K1 y2 . By substituting this transformation for z2 into the desired dynamics (6.21) and comparing the actual and desired dynamics of x2 it can be noted that the dynamics will
95
be identical if and only if z3 := x3 + M2−1 N12 M1−1 K1 y˙2 + ΨT M1 e + L2 M1−1 K1 y2 . Proceeding in this manner, we can determine the coordinate transformation [e⊥ x2 x3 . . . xm ]T → [e⊥ z2 z3 . . . zm ]T : ⎡
⎤ ⎛ e⊥ I ⎢ ⎥ ⎜ ⎢ z2 ⎥ ⎜0 ⎢ ⎥ ⎜ ⎢ z ⎥ ⎜0 ⎢ 3⎥ ⎜ ⎢ ⎥=⎜ ⎢ z4 ⎥ ⎜0 ⎢ ⎥ ⎜ ⎢ : ⎥ ⎜0 ⎣ ⎦ ⎝ zm 0 ⎡
⎞⎡ ⎤ 0 0 0 ... 0 e⊥ ⎟⎢ ⎥ ⎢ ⎥ I 0 0 . . . 0⎟ ⎟ ⎢ x2 ⎥ ⎢ ⎥ 0 I 0 . . . 0⎟ ⎟ ⎢ x3 ⎥ ⎟⎢ ⎥ ⎢ ⎥ 0 0 I . . . 0⎟ ⎟ ⎢ x4 ⎥ ⎢ ⎥ 0 0 0 . . . 0⎟ ⎠⎣ : ⎦ xm 0 0 0 ... I 0
⎤
⎢ ⎥ M1−1 K1 y2 ⎢ ⎥ ⎢ −1 ⎥ −1 −1 T ⎢M M1 K1 y˙ 2 + Ψ M1 e + L2 M1 K$1 y2 ⎥ ⎢ 2 N12 ⎥ " +⎢ ⎥, −1 T ˙ ⎢ M3 N13 D3 (t) + L3 D3 (t) + M1 D2 (t) ⎥ ⎢ ⎥ ⎢ ⎥ : ⎣ ⎦ Dm (t)
(6.22)
where D2 (t) = M1−1 K1 y2 D3 (t) = M2−1 N12 M1−1 K1 y˙2 + ΨT M1 e + L2 M1−1 K1 y2 : =:
" $ −1 T N1m−1 D˙ m−1 (t) + Lm−1 Dm−1 + Mm−2 Dm−2 . Dm (t) = Mm−1 This transformation (Eq. (6.22)) when applied to the actual locked system dynamics
96
(6.20) results in the following transformed dynamics: ⎛ Ne⊥ 0 0 ⎜ ⎜ 0 N12 0 ⎜ ⎜ 0 0 N13 ⎜ ⎜ : ... ⎝ : 0
...
0
... ... ... ... 0
⎞⎡ ⎤ ⎛ e˙ ⊥ −Le⊥ M1 0 0 ⎟⎢ ⎥ ⎜ 0 ⎟ ⎢ z˙2 ⎥ ⎜−M1T −L2 M2 ⎟⎢ ⎥ ⎜ ⎢ ⎥ ⎜ −M2T −L3 0 ⎟ ⎟ ⎢ z˙3 ⎥ = ⎜ 0 ⎟⎢ ⎥ ⎜ ... ⎠⎣ : ⎦ ⎝ : : ... z˙m
N1m
0
⎡
...
T1 + T2
⎢ ⎢ ⎢ +⎢ ⎢ ⎢ ⎣
0 0 :
⎤
0
... ... ... ... T −Km−1
⎞⎡ ⎤ e⊥ 0 ⎟⎢ ⎥ 0 ⎟ ⎢ z2 ⎥ ⎟⎢ ⎥ ⎢ ⎥ 0 ⎟ ⎟ ⎢ z3 ⎥ ⎟⎢ ⎥ ... ⎠⎣ : ⎦ zm −Lm
⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦
(6.23)
u1 + Dm+1 (t) where " $ T T Dm+1 (t) = Mm−1 Dm−1 . N1m D˙ m (t) + Lm Dm + Mm−1
(6.24)
If the locked system control u2 in (6.23) is chosen as follows: u1 = −Dm+1 (t)
(6.25)
then the locked system is passive w.r.t. the supply rate (6.13) and its dynamics are given by: ⎛ 0 Ne⊥ 0 ⎜ ⎜ 0 N12 0 ⎜ ⎜ 0 0 N13 ⎜ ⎜ : ... ⎝ : 0 ... 0
... ... ... ... 0
0
⎞⎡
e˙ ⊥
⎤
⎛
−Le⊥
M1
0
...
0
⎞⎡
e⊥
⎤
⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎟ ⎢ z˙2 ⎥ ⎜−M1T −L2 M2 ... 0 ⎟ ⎢ z2 ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ T ⎟ ⎢ z˙3 ⎥ = ⎜ 0 ⎟ ⎢ z3 ⎥ −L . . . 0 −M 3 2 ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ : : : . . . . . . . . . ⎠⎣ ⎦ ⎝ ⎠⎣ : ⎦ T z˙m zm N1m 0 ... 0 −Km−1 −Lm ⎤ ⎡ T1 + T2 ⎥ ⎢ ⎢ 0 ⎥ ⎥ ⎢ ⎥ (6.26) +⎢ ⎢ 0 ⎥, ⎥ ⎢ ⎣ : ⎦ 0 0 0 ...
which is exactly the same as the desired locked system (6.21) The above analysis is summarized in the following theorem. 97
Theorem 10. m-Order, p-DOF Locked System: Under the coordination control u1 as defined in Eq. (6.28) and the coordinate transformation (6.22), the teleoperated master and slave systems given by the dynamics (6.3) behave like a mth order passive coordinated teleoperator as shown in Fig. 6.5 with respect to the supply rate: stele ((T1 , T2 ), (x1 , y1)) = xT1 T1 + y1T T2 ,
(6.27)
if the control inputs, u1 is chosen as follows: T u2 = −Kn−1 Dn−1 (t) − On Dn (t) − N2n D˙ n (t)
u1 = −Dm+1 (t),
(6.28)
where Dn (t) = :=
−1 Kn−1
$ " T ˙ N2n−1 Dn−1 (t) − Kn−2 Dn−2 (t) − On−1 Dn−1 (t)
:
" $ D3 (t) = K2−1 O2 (D2 (t)) − (K1 e⊥ − N22 D˙ 2 (t)) D2 (t) = (ΨT − I)K1 (ΨT T1 + (ΨT − I)T2 + ΨT M1 x2 ) " $ N1m D˙ m (t) + Lm Dm + M T Dm−1 . Dm+1 (t) = M T m−1
m−1
(6.29)
(6.30)
The choice of control input u2 in (6.28) ensures that the coordination error e → 0 asymptotically. Proof:
Consider the teleoperator dynamics given in Eq. (6.3). The coordina-
tion control u2 ensures locked system dynamics as given by (6.20). The coordinate transformation given by Eq. (6.22) ensures that the locked system transforms to the dynamics as given by Eq. (6.23). In order to prove that the locked system (6.23) is passive consider the storage function of the locked system 1 Wlocked−m = zlT NL zl , 2
(6.31)
where zl = [e⊥ z1 . . . zm ]T and NL = Diag[Ne⊥ , N12 , . . . , N1m ]. Differentiating the storage function w.r.t. time and substituting the transformed 98
locked system dynamics (6.23) and further substituting the locked system control u1 as given in (6.28) leads to the following: zlT NL
d zl = zlT ΩLm zl + stele , dt
(6.32)
where ⎛
ΩLm
0 −Le⊥ M1 ⎜ T ⎜−M1 −L2 M2 ⎜ =⎜ −M2T −L3 ⎜ 0 ⎜ : ... ⎝ : 0
...
0
... ... ... ... T −Km−1
⎞ 0 ⎟ 0 ⎟ ⎟ 0 ⎟ ⎟. ⎟ ... ⎠ −Lm
Since ΩL4 is negative semi-definite the above equation simplies to d Wlocked−m ≤ stele . dt Integrating the above equation provides the desired passivity result. To prove that e → 0 consider the lyapunov function: 1 V = [e e2 . . . en ]Ne [e e2 . . . en ]T , 2
(6.33)
where Ne = Diag[Ne , N22 , . . . , N2n ]. Differentiating V w.r.t. time and substituting for the dynamics of [e e2 . . . en ] as given by (6.19) leads to V˙ = [e e2 . . . en ]Ωe [e e2 . . . en ]T , where Ωe is negative definite and is given by: ⎛
−Oe
⎜ ⎜−K1T (Ψ − I) ⎜ Ωe = ⎜ 0 ⎜ ⎜ : ⎝ 0
(ΨT − I)K1
0
...
0
−O2 −K2T :
K2 −O3 ...
... ... ...
0 0 ...
...
0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
T −Kn−1 −On
thus ensuring that (e, e1 , . . . , en ) is globally exponentially stable.
Remark 3. The teleoperator given by Eq. (6.3) with locked system dynamics as 99
given by Eq. (6.26) is passive w.r.t. to the 2-port supply rate (6.13). This fact can be proven by considering the sum of the locked system storage energy (Eq. (6.31)) and the error dynamics storage energy (Eq. (6.33)) and following the procedure as outlined in theorem 3. The above theorem presented a passive teleoperation control design which ensures that the master-slave teleoperator mimics a passive mth order system operated upon by the human operator and the environment. Since the locked system dynamics are of ‘m’ order, the teleoperator behaves like a ‘m’ order, p-DOF system when operated upon by the operator, while interacting with the environment. While the system order has certain advantages in terms of filtering high frequency content of environment forces, it may be disadvantageous in other circumstances when the operator requires an agile teleoperator. Hence, it would be benefecial to control (using u1 ) the locked system to behave like a lower order dynamic system. Such a method is proposed in the following section where, the teleoperator behaves like a 2nd order, p-DOF locked system. In general, it is possible to ensure that the locked system behaves like system of order 2 to m using the proposed methodology. 6.3.4
Locked System Design : 2-Order, p-DOF Haptic Behavior
The locked system control objective remains the same which is to determine u1 in Fig. 6.4 so that the closed loop dynamics of the actual locked system shown in the figure behaves like the desired 2nd order locked system dynamics as shown in Fig. 6.6. The design process involves comparing the dynamics of the actual and desired locked systems (as shown above for the mth order locked system) and choosing the appropriate control input u1 so that the actual dynamics mimic the desired dynamics. The dynamics of the actual locked system shown in Fig. 6.4 were given by Eq. (6.20). The dynamics of the desired locked system shown in Fig. 6.6 are:
100
2−ORDER DESIRED LOCKED SYSTEM DYNAMICS
DISSIPATIVE DYNAMICS N1m 1 zbm
N1m_1
...
TFm_1 0 zbm_1
Lm
N13
Lm_1
1 zb3
N12 TF2
0 en On
M1 z¯2
0 zb2
1
N23
...
1 e3
L1 N22
TF4
0 e2
O3
T1+T2
ep
L2
L3 N2n
Nep
Ne TF3
1 e
O2
O1
DISSIPATIVE DYNAMICS
Figure 6.6: Second order desired locked system ⎛ 0 Ne⊥ 0 ⎜ ⎜ 0 N12 0 ⎜ ⎜ 0 0 N13 ⎜ ⎜ : ... ⎝ : 0
...
0
⎤ ⎛ ⎞⎡ ⎤ 0 ... 0 e˙ ⊥ −Le⊥ M1 e⊥ ⎟⎢ ˙ ⎥ ⎜ ⎟⎢ ⎥ −L2 M2 . . . 0 ⎟ ⎢ z¯2 ⎥ ⎜ 0 ... 0 ⎟ ⎢ z¯2 ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎜ ⎢ ⎥ ... 0 ⎟ ... 0 ⎟ −M2T −L3 ⎟ ⎢ z¯˙3 ⎥ = ⎜ 0 ⎟ ⎢ z¯3 ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ... ... ⎠⎣ : ⎦ ⎝ : : ... ... ... ⎠⎣ : ⎦ T z¯˙m z¯m 0 N1m 0 ... 0 −Km−1 −Lm ⎤ ⎡ T1 + T2 ⎥ ⎢ ⎢ 0 ⎥ ⎥ ⎢ ⎥. +⎢ (6.34) 0 ⎥ ⎢ ⎥ ⎢ ⎣ : ⎦ 0
...
0
⎞⎡
The passive control objective is to design the slave control u1 in (6.20) (also in bondgraph in Fig. 6.4) so that the actual locked system behaves like the desired locked system in Eq. (6.34) (also in bondgraph in Fig. 6.6). The comparison of the bondgraph representations shown in Fig. 6.6 and Fig. 6.4 and following the coordination transformation development for the mth order locked system, we get the following
101
coordinate transformation [e⊥ x2 x3 . . . xm ]T → [e⊥ z¯2 z¯3 . . . z¯m ]T : ⎞⎡ ⎤ e⊥ 0 0 ... 0 ⎟⎢ ⎥ I 0 . . . 0⎟ ⎢ x2 ⎥ ⎟⎢ ⎥ ⎢ ⎥ 0 I . . . 0⎟ ⎟ ⎢ x3 ⎥ ⎟⎢ ⎥ 0 0 . . . 0⎠ ⎣ : ⎦ 0 0 0 ... I xm
⎤ ⎛ I e⊥ ⎢ ⎥ ⎜ ⎢ z¯1 ⎥ ⎜0 ⎢ ⎥ ⎜ ⎢ z¯2 ⎥ = ⎜0 ⎢ ⎥ ⎜ ⎢ ⎥ ⎜ ⎣ : ⎦ ⎝0 ⎡
z¯m
⎡
0
⎤
⎢ ⎥ M1−1 K1 y2 ⎢ ⎥ ⎢ −1 ⎥ T T ⎥. +⎢ N M K y ˙ + Ψ M e − M e + L K y 12 1 2 1 ⊥ 2 1 2 2 1 ⎢ ⎥ ⎢ ⎥ : ⎣ ⎦ Dm (t)
(6.35)
" $ −1 T N1m−1 D˙ m−1 (t) + Lm−1 Dm−1 + Mm−2 where Dm (t) = Mm−1 Dm−2 for m = 4, . . . m. This transformation (Eq. (6.35)) when applied to the actual locked system dynamics (6.20) results in the following transformed dynamics: ⎛ 0 Ne⊥ 0 ⎜ ⎜ 0 N12 0 ⎜ ⎜ 0 0 N13 ⎜ ⎜ : ... ⎝ : 0 ... 0
⎤ ⎛ ⎞⎡ ⎤ 0 ... 0 e˙ ⊥ −Le⊥ M1 e⊥ ⎟⎢ ˙ ⎥ ⎜ ⎟⎢ ⎥ −L2 M2 . . . 0 ⎟ ⎢ z¯2 ⎥ ⎜ 0 ... 0 ⎟ ⎢ z¯2 ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ T ⎟ ⎥ ⎢ ⎜ ⎢ ⎥ . . . 0 ⎟ ⎢ z¯˙3 ⎥ = ⎜ 0 ... 0 ⎟ −M2 −L3 ⎟ ⎢ z¯3 ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ... ... ⎠⎣ : ⎦ ⎝ : : ... ... ... ⎠⎣ : ⎦ T 0 N1m 0 ... 0 −Km−1 −Lm z¯˙m z¯m ⎤ ⎡ T1 + T2 ⎥ ⎢ ⎥ 0 ⎢ ⎥ ⎢ ⎥. (6.36) +⎢ 0 ⎥ ⎢ ⎥ ⎢ : ⎦ ⎣ ¯ m+1 u1 + D
...
0
⎞⎡
where " $ T T ¯ m+1 (t) = Mm−1 ¯ m + Mm−1 ¯ m−1 . ¯˙ m (t) + Lm D D D N1m D
(6.37)
If the locked system control u1 in (6.36) is chosen as follows: ¯ m+1 (t) u1 = −D
102
(6.38)
then the locked system is passive w.r.t. the supply rate (6.13) and its dynamics are given by: ⎛ Ne⊥ 0 0 ⎜ ⎜ 0 N12 0 ⎜ ⎜ 0 0 N13 ⎜ ⎜ : ... ⎝ : 0 ... 0
⎞⎡ ⎤ ⎛ ⎞⎡ ⎤ e˙ ⊥ −Le⊥ M1 e⊥ 0 0 ... 0 ⎟⎢ ˙ ⎥ ⎜ ⎟⎢ ⎥ 0 ⎟ ⎢ z¯2 ⎥ ⎜ 0 ... 0 ⎟ ⎢ z¯2 ⎥ −L2 M2 ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ T ⎢ z¯˙3 ⎥ = ⎜ 0 ⎟ ⎢ z¯3 ⎥ −M ... 0 ⎟ −L . . . 0 3 2 ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ⎟⎢ ⎥ ⎜ ⎟⎢ ⎥ ... ... ⎠⎣ : ⎦ ⎝ : : ... ... ... ⎠⎣ : ⎦ T −Lm z¯˙m z¯m 0 N1m 0 ... 0 −Km−1 ⎡ ⎤ T1 + T2 ⎢ ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎥. (6.39) +⎢ 0 ⎢ ⎥ ⎢ ⎥ ⎣ : ⎦ 0
... ...
The states [¯ z2 z¯3 . . . z¯m ]T → 0, asymptotically. Hence, the locked system (6.26) asymptotically approaches the second order locked system given by: Ne⊥ e˙ ⊥ = −Le⊥ e⊥ + T1 + T2 .
(6.40)
The above analysis is now summarized in the following theorem. Theorem 11. 2-Order, p-DOF Locked System: The teleoperated master and slave systems given by the dynamics (6.1) behave like a 2nd order passive coordinated teleoperator as shown in Fig. 6.6 with respect to the supply rate: stele ((T1 , T2 ), (x1 , y1)) = xT1 T1 + y1T T2 ,
(6.41)
if the control inputs, u1 and u2 are chosen as follows: T u2 = −Kn−1 Dn−1 (t) − On Dn (t) − N2n D˙ n (t)
¯ m+1 (t), u1 = −D
(6.42)
103
where $ " −1 T Dn−2 (t) − On−1Dn−1 (t) Dn (t) = Kn−1 N2n−1 D˙ n−1 (t) − Kn−2 := :
(6.43)
$ " D3 (t) = K2−1 O2 (D2 (t)) − (K1 e⊥ − N22 D˙ 2 (t)) D2 (t) = (ΨT − I)K1 (ΨT T1 + (ΨT − I)T2 + ΨT M1 x2 ) " $ T T ¯ m+1 (t) = Mm−1 ¯˙ m (t) + Lm D ¯ m + Mm−1 ¯ m−1 . D D N1m D
(6.44)
The choice of control input u2 in (6.42) ensures that the coordination error e → 0 asymptotically. Proof: Consider the storage function of the locked system 1 Wlocked−2 = z¯lT NL z¯l , 2
(6.45)
where z¯l = [e⊥ z¯1 . . . z¯m ]T and NL = Diag[Ne⊥ , N12 , . . . , N1m ]. Differentiating the storage function w.r.t. time and substituting the transformed locked system dynamics (6.36) and further substituting the locked system control u1 as given in (6.42) leads to the following: z¯lT NL
d z¯l = z¯lT ΩL2 z¯l + eT⊥ M1 z¯2 + stele , dt
(6.46)
where
ΩL2
⎛ 0 0 −Le⊥ ⎜ ⎜ 0 −L2 M2 ⎜ =⎜ −M2T −L3 ⎜ 0 ⎜ : ... ⎝ : 0
...
0
104
...
0
... ... ...
0 0 ...
T −Km−1 −Lm
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
Under the assumption that e⊥ , z¯2 ∈ L∞ and are bounded by (e⊥ , ¯ z2 ) and since T 1 the states [¯ z2 . . . z¯m ] → 0 exponentially , integrating the above equation leads to: Wlocked−2 (T ) − Wlocked−2(0) T T ≤ e⊥ (τ )M1 z¯2 (τ )dτ + stele ((T1 , T2 ), (x1 , y1 ))dτ 0 0 T T −ω2 τ ≤ e⊥ M1 z¯1 (0) e dτ + stele ((T1 , T2 ), (x1 , y1 ))dτ 0 0 T stele ((T1 , T2 ), (x1 , y1 ))dτ ≤c+ 0
where ω2 = σ(ΩL2 ), c ∈ R+ . Hence 0
T
stele ((T1 , T2 ), (x1 , y1 ))dτ ≥ −Wlocked−2 (0) − c,
completing the proof for passivity of the locked system. The proof for convergence of e is exactly the same as in theorem 2.
Remark 4. The teleoperator given by Eq. (6.3) with locked system dynamics as given by Eq. (6.39) is passive w.r.t. to the 2-port supply rate (6.13). This fact can be proven by considering the sum of the locked system storage energy (Eq. (6.45)) and the error dynamics storage energy (Eq. (6.33)) and following the procedure as outlined in theorem 5. Since the locked system behavior directly influences the haptics of the teleoperator, the proposed algorithm is benefecial as it permits the designer to choose the appropriate haptics, given the application of the teleoperator. The two theorems 10 and 11 present the passivity and coordination results of the proposed passive teleoperation algorithms. It is shown that the passivity property of the teleoperation system does not depend on the coordination performance of the teleoperation control. This is because in order to preserve the passivity property of the teleoperator it is necessary for the control u1 to know exactly the other control u2 and its successive derivatives. The theorems also demonstrate the ability of the control design method in achieving 1
Can be proved by differentiating the lyapunov function V = 12 [¯ z1 . . . z¯m ]T Diag [N12 , . . . , N1m ][¯ z1 . . . z¯m ]T and substituting the dynamics (6.39).
105
different order locked systems. This is beneficial as the locked system behavior is exactly the haptic dynamics of the teleoperator. Being able to flexibly design the order of the teleoperator will improve the efficiency of the teleoperator depending on its purpose.
6.4
Summary
In this chapter a bond graph based approach for passive teleoperation of a hydraulic backhoe was presented. The proposed algorithm consists of the following key steps. • Transforming the master and slave systems to include the coordination error, e as state. • Identification of the shorted causal path between a control input u2 and the coordination variable e. • Determination, using inverse dynamics, the coordination control law which ensures that e → 0. • Representation of the coordination control law u2 in the original bond graph of the master and slave systems using Effort / Flow sources and any state feedback as necessary. • Determination of 1)a coordinate transformation and 2) a control input, u1 which will ensure the actual active bond graph behaves like a desired passive bond graph hence achieving a desired locked system behavior.
106
Chapter 7 Experimental results 7.1
Introduction
In this section simulation and experimental results of teleoperation of the hydraulic backhoe using the algorithms presented in the preceding chapters are presented. The teleoperator consists of a master 2-DOF motorized joystick and a slave 3-DOF hydraulic backhoe interconnected by the computer. A schematic of the experimental setup is shown in Fig. 7.1. A picture of the backhoe with the joystick is shown in Fig. 7.2. The motors in the joystick are of the low speed hi torque type. These provide the necessary haptic force to the operator (Fq in chapters 4, 5). The motors are also instrumented with co-axial encoders which provides angular position (q in chapters 4, 5) reading of the joystick links and a JR3 force sensor which measures the operator input force (Tq in chapters 4, 5) on the teleoperator. Since the master has only 2-DOF, the slave hydraulic backhoe is also restricted to operate as a 2-DOF robot. Of the three links of the backhoe, the boom, the stick and the bucket, the boom is constrained and the bucket and stick are teleoperated using the master joystick. The hydraulic actuation system of the backhoe involves 1) A pressure compensated constant flow pump regulated at a fixed 1000 psi of pressure (Ps ), 2) Vickers KBFDG4V-5 series proportional valve with a valve command (U i in chapter 3) to spool position (xv in chapters 4, 5) bandwidth of about 50 Hz, 3) Vickers customized hydraulic actuators instrumented with Hydroline sensors to measure the actuator position x and 4) Pressure sensors to measure the individual actuator chamber pressures (P1i , P2i in chapter 3). The RealTime Operating System (RTOS) used for the experiments is MATLAB xPC Target. The software operates 107
Tq
− Fe
Fq
q
FL
x
MOTORS
ACTUATORS
ENCODERS
LVDT’s
QL Fq
q
FL
VALVE LVDTs U xv AD + DA
xPC TARGET COMPUTER
HOST COMPUTER CONTROL C CODE
Figure 7.1: Schematic of the experimental setup
108
USER
JOYSTICK
HUMAN OPERATOR
BAC KHOE
PC
Figure 7.2: Picture of the backhoe and joystick in the Fluid Power Control Laboratory on both the master and the slave machine. While the necessary control algorithm is designed and built on the master machine using MATLAB and its toolboxes, xPC Target converts the code to RT executable files and downloads it to the slave machine via RS-232, in the present case. The joystick and backhoe are electrically connected only to the slave as shown in Fig. 7.1. The interface used are 1) AD card with 16 channels, 2) DA card with 8 channels, 3) decoder card with 4 channels and 4) Force sensor signal conditioning card. A sample rate of 1 KHz was used for all the command and sensor signals. A list of the various parameters and their values was given in chapter 3 in tables 3.1 and 3.2. Three different teleoperator control algorithms are experimentally verified in this chapter. The conducted experiments are typical of a digging task. The backhoe is teleoperated to dig into a sand box. A wooden box buried in the sandbox is supposed to mimic underground obstacles. The first control algorithm presented in chapter 4 109
presents a teleoperation algorithm assuming that the backhoe dynamics can be approximated by an appropriate kinematic behavior. In this setup compressible fluid dynamics of the hydraulics and the backhoe inertia are neglected. The corresponding experimental results are presented in section 7.2. The second algorithm presented in chapter 5 is developed for the dynamic backhoe which considers the effects due to compressible fluid dynamics and backhoe inertia. The algorithm ensures that the teleoperator behaves like a 4th order locked system when coordinated. The nonlinear dynamics due to a) hydraulic fluid compressibility and b) inertial coupling of the backhoe are considered while designing the algorithm. The third control algorithm is also designed for the nonlinear backhoe except that the teleoperated backhoe behaves like a 2nd order locked system when coordinated. The experimental results are presented in section 7.3.
7.2
Kinematic Teleoperation
In this section, we test the intrinsically passive control law developed in chapter 4 described by Eqs. (4.3) and (4.11). The controller was designed assuming a kinematic model of the the hydraulic actuators on the backhoe. Backhoe inertia was also neglected. The teleoperated trajectories of the bucket and the stick (and their corresponding joystick links) are shown in Fig. 7.3. The corresponding haptic force (Fq ) experienced by the operator while performing the task is shown in Fig. 7.4. A kinematic scaling, α = 5in./rad and a power scaling ρ = 12W att/W att. Notice from Fig. 7.3 that the joystick and backhoe operate in a coordinated manner. The maximum observed coordination error is within 0.5in. The dynamic valve teleoperation algorithm performs better than the previously developed (Li and Krishnaswamy, 2001) algorithm which relies on low frequency operation of the teleoperator. The previously developed algorithm could achieve coordination error of 0.7in for a similar experiment using similar bandwidth closed loop behavior. The performance improvement would be more significant for higher frequency operation. Notice also that when the backhoe hits the woodden box (t = 218 repetitively and 228 seconds repeatedly), the operator experiences the backhoe contact force (Fe ) embedded in the bucket and stick haptic forces (Fq ). Notice that there are severe oscillations in the haptic force experienced by the operator. This is primarily because compressible dynamics of the hydraulic backhoe have been neglected in the control design. Also notice that while the bucket coordination is good, the stick coordination is rather poor. This is again because the coordination control law was designed assuming negligible inertia of the 110
Bucket Displacement (inches)
5
4
3
2
1 215
220
225
220
225
230
235
240
230
235
240
Stick Displacement (inches)
7
6
5
4
3 215
Time (seconds)
Figure 7.3: Displacement trajectories (Scaled joystick - solid, Backhoe - dashed) during a digging task.
1.5
Feedback Torque (N−m)
1
0.5
0
−0.5
215
220
225
230
235
240
time (sec)
Figure 7.4: Haptic torque (Fq ) trajectories (Stick - solid, Bucket - dashed) during a digging task. 111
backhoe. While this is probably true for the bucket, it is not for the stick. The operator can observe chatter during moderately slow teleoperation of the machine. The experiments indicate that the passive teleoperation controller is able to ensure bilateral transfer of power between the human and the work environments. However, there seem to be much scope for improvement in improving the haptic behavior as well as the coordination performance.
7.3
Dynamic Teleoperation
In this section we test the bondgraph based dynamic teleoperation control algorithms designed in chapter 5. The control algorithms were designed for a 2-DOF teleoperated system with included nonlinear models of the joystick and backhoe and compressible effects of the hydraulic actuator. 7.3.1
4th order locked system
In this subsection, we test the control law which ensured that the nonlinear teleoperated system (5.5) behaves like a 4th order locked system. The control law is described in Eqs. (5.51) - (5.52). The teleoperated trajectories of the bucket and the stick (and their corresponding joystick links) are shown in Fig. 7.5. The corresponding locked system force (ρTq − Fe ) experienced by the 4th order locked system while performing the task is shown in Fig. 7.6. A kinematic scaling, α = 5in./rad and two different power scaling ρ = 12W att/W att and ρ = 18W att/W att were used for the experiments. Notice from Fig. 7.5 that the joystick and backhoe operate in a coordinated manner. The maximum observed coordination error is within 0.1in for the backhoe stick. Notice that when the locked system hits the underground woodden box (about t = 18 seconds), the locked system force (Fig. 7.6) does not experience any oscillation as does the Haptic torque in Fig. 7.4. This is because the compressible dynamics of hydraulic fluid are modelled in the dynamic teleoperation scenario and not in the kinematic teleoperation. Also, notice that when the backhoe hits the box, the net force on the locked system shown in Fig. 7.6 is about zero even though the operator is pushing on the joystick (ρTq > 0). This is because, the environement force (Fe ) provides an equal and opposite reaction force on the joystick thus indicating a bilateral transfer of power. The high order (4th) of the teleoperator is felt by the operator during hi frequency motion (oscillatory for example) of the teleoperator. The teleoperator (via the master joystick) feels resistant to instantaneous change in direction of 112
Bucket Displacement (inches)
5 4 3 2 10
15
20
10
15
20
10
15
20
10
15
20
25
30
35
30
35
30
35
30
35
Bucket Displacement (inches)
Stick Displacement (inches)
5 4 3 2 1 25 Time (seconds)
5 4 3 2 1 25
Stick Displacement (inches)
5 4
3 2
1 25 Time (seconds)
Figure 7.5: Displacement trajectories for 4th Order locked system (Joystick - Solid, Backhoe - dashed) during a digging task. Sequentially from top : 1)-2) ρ = 12, 3)-4) ρ = 18.
113
Haptic Force, ρ = 12 (N)
15 10 5 0 −5 −10 −15 10
15
10
15
20
25
30
25
30
Haptic Force, ρ = 18 (N)
15 10 5 0 −5 −10 −15 −20 20 time (sec)
Figure 7.6: Locked system Force (ρTq − Fe ) trajectories during a digging task for 4th Order locked system (Stick - solid, Backhoe - dashed). motion. 7.3.2
2nd order locked system
In this subsection, we test the control law which ensures that the nonlinear teleoperated system (5.5) behaves like a 2nd order locked system. The control law is described in Eqs. (5.68) - (5.69). The teleoperated trajectories of the bucket and the stick (and their corresponding joystick links) are shown in Fig. 7.7. The corresponding locked system force (ρTq −Fe ) experienced by the 2nd order locked system while performing the task is shown in Fig. 7.8. A kinematic scaling, α = 5in./rad and two different power scaling ρ = 12W att/W att and ρ = 18W att/W att were used for the experiments. Notice from Fig. 7.7 that the joystick and backhoe operate in a coordinated manner. The maximum observed coordination error is within 0.01in and significantly better than the 4th order locked system. Notice again, that when the locked system hits the underground woodden box (about t = 16, 19 seconds), the locked system force (Fig. 7.8) does not experience any oscillation as does the Haptic torque in Fig. 7.4. This is because the compressible dynamics of the hydraulic fluid are modelled in the dynamic teleoperation scenario and not in the kinematic teleoperation. Also, notice 114
Bucket Displacement (inches)
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Figure 7.7: Displacement trajectories for 2nd Order locked system (Joystick - Solid, Backhoe - dashed) during a digging task. Sequentially from top : 1)-2) ρ = 12, 3)-4) ρ = 18.
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Haptic Force, ρ = 12 (N)
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Figure 7.8: Locked system force (ρTq − Fe ) trajectories during a digging task for 2nd Order locked system (Stick - solid, Backhoe - dashed). that when the backhoe hits the box, the net force on the locked system shown in Fig. 7.8 is about zero even though the operator is pushing on the joystick (ρTq > 0). This is because, the environement force (Fe ) provides an equal and opposite reaction force on the joystick thus indicating a bilateral transfer of power. Another key factor to notice is that the range of locked system force acting on the 2nd order locked system (Fig. 7.8) is only about 20 % of that acting on the 4th order locked system (Fig. 7.6). This is an indication that the 2nd order locked system is easier (in terms of necessary power) to teleoperate than the 4th order locked system. The teleoperator is easier to manipulate than in the case with 4th order haptics. This 2nd order locked system has the haptic benefits similar to the kinematic teleoperation but with improved performance owing to the richer model (compressible effects + inertial effects) used to design the passive coordination control. Hence, it is very beneficial to be able to not only design passive teleoperators but also design their dynamic behavior as was achieved in this dissertation and proven using experiments.
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7.4
Summary
In this chapter, three different teleoperation control algorithms were experimentally verified and presented. It was shown that the simplest teleoperation control algorithm developed for a kinematic modeled backhoe performed well in operating conditions that did not violate the assumptions, for example, during unconstrained teleoperation. However, the coordination and haptic performance deteriorated when the teleoperator interacted with a hard constraint. To remedy the problem, two different control algorithms which included the nonlinear compressible dynamics of the hydraulic backhoe were experimentally tested. These algorithms’ performance was demonstrated to be superior to the kinematic teleoperation. The two algorithms were themselves different in the haptic performance of the teleoperator. While one ensured that the teleoperator behaves like a 4th order system, the other ensures that the teleoperator behaves like a second order system. The haptic and coordination performance of the 2nd order locked system are superior to the 4th order system.
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Chapter 8 Conclusions and future work 8.1
Conclusions
Hydraulics are used in a variety of applications as power source and transmission, owing to their high power density, flexibility and portability. Many of these applications involve direct human operation. Researchers have devised methods to ensure that such machines are safe to interact with while retaining their power benefits in at least two distinct ways. One method was to provide only supervisory remote control to the operator thus rendering the machines autonomous. Such machines do not perform well in unstructured, uncertain environments. The other method was to provide teleoperation control to the operator. Such machines are beneficial as the operator retains operation decision making abilities while interacting with hostile, uncertain environments. However, environment-machine interaction stability guarantees were seldom addressed. This dissertation addressed the environment-machine interaction stability aspect of human teleoperated hydraulic machines by utilizing the benefits of passive machines. Researchers had demonstrated the benefit of passive electromechanical machines but none investigated the benefits provided by passive hydraulic machines. It was shown that hydraulic machines are inherently non-passive due to the electrohydraulic valve. A recently proposed method to ensure passive behavior of electrohydraulic valves was presented. By invoking the result that the cascade interconnection of passive systems is passive, it was possible to construct passive hydraulic machines by interconnecting passive hydraulic subsystems. In the teleoperation scenario (considered in this dissertation) the passive hydraulic machine (backhoe in the present case) is termed 118
the slave and a force-feedback joystick is termed the master. Given the master and slave systems, control objectives to ensure passive teleoperation were 1) to ensure coordination of the kinematically master and slave systems at all time, 2) to ensure passivity of the coordinated teleoperator w.r.t. an appropriate power supply to the system. Given these objectives, controllers were synthesized in stages. In the first stage, a passive coordinating control was synthesized for a simplified model of the master-slave system. The simplifications were a) approximating the hydraulic actuators on the slave (backhoe) to behave like velocity sources instead of pressure sources by neglecting the compressible fluid dynamics, b) approximating the slave (backhoe) to have insignificant inertia and c) approximating the 2-DOF nonlinear coupled inertial dynamics of the joystick to be decoupled. While the synthesized controller did ensure passivity of the passive teleoperated hydraulic machine, experimentally observed performance was poor. The control performance was remedied by redesigning the controller without making the aforesaid assumptions about the backhoe. In the second stage, a passive coordinating control was synthesized for the nonlinear dynamic model of the master-slave system. Coordination aspect of the control was designed using system inversion ideas based on bondgraphs. Passivity aspect of the control was designed by choosing appropriate energy storage of the coordinated master-slave system (locked system). This was also achieved by drawing physical intuition from bondgraphs. Moreover, it was also possible to design the haptic behavior of the coordinated passive teleoperated backhoe by suitably modifying the storage energy of the locked system. Passive teleoperated hydraulic backhoe with two different haptic properties were designed in this dissertation. Both the systems were experimentally tested and their benefits were demonstrated. It was also shown that the passive coordinating control design methodology based on bondgraphs is scalable. The control design method was applied to arbitrary relative degree, arbitrary order, master and slave systems demonstrating its systematic and generalizable abilities.
8.2
Future Work
While the proposed teleoperation control algorithm ensures that the teleoperated machine is passive with respect to a particular supply rate, it does not guarantee that passivity property of the controller itself. That is the proposed controller is not 119
intrinsically passive with respect to the physical control power input to the joystick (ρq˙ T Fq ) and the passive valve (xv T KQ Γ−1 Fx ). Development of such an intrinsically passive control law has been accomplised for systems electro-mechanical second order systems by (Lee and Li, 2002) but not for passive systems of higher order. Such a controller if developed would ensure passivity property of the teleoperated system in a natural way. The proposed teleoperation control algorithm has been shown to be applicable to teleoperation of arbitrary order passive master and slave systems. However, using the proposed algorithms it may also be possible to ensure that certain high order passive systems be controlled to evolve on lower dimensional spaces thus ensuring superior performance (for example, constraining a high order robotic needle in surgery to a line). This is possible because the control designer can choose the dynamics and order of a desired locked system as outlined in this dissertation. Construction of a 3-DOF master force feedback joystick would permit experimentation of 3-DOF passive teleoperation of the hydraulic backhoe and therefore enhance the use of developing such a passive teleoperated machine.
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Bibliography Adams, R. J. and Hannaford, B., 1999. “Stable haptic interaction with virtual environments.” IEEE Transactions on Robotics and Automation, vol. 15, no. 3, pp. 465–474. Alleyne, A. and Hedrick, J. K., 1995. “Nonlinear adaptive control of active suspensions.” IEEE Transactions on Control Systems Technology, vol. 3, no. 1, pp. 94–101. Bradley, D. and Seward, D. W., 1998. “The development, control and operation of an autonomous robotic excavator.” Journal of Intelligent and Robotic Systems, vol. 21, pp. 73–97. Byrnes, C. I., Isidori, A., and Willems, J. C., 1991. “Passivity, feedback equivalence and the global stabilization of minimum phase nonlinear systems.” IEEE Transactions on Automatic Control, vol. 36(11), pp. 1228–1240. Colgate, J. E., Wannasuphoprasit, W., and Peshkin, M. A., 1996. “Cobots: Robots for collaboration with human operators.” Proceedings of the ASME Dynamic Systems and Control Division, vol. 58, pp. 433–439. Gawthrop, P. J., 1995. “Bicausal bond graphs.” Proceedings of the International Conference on Bond Graph Modeling and simulation (ICBGM), pp. 83–88. Gomes, M. W. and Book, W., 1997. “Control approaches for a dissipative passive trajectory enhancing robot.” IEEE/ASME Conference on Advanced Intelligent Mechatronics. Hannaford, B., 1989. “A design framework for teleoperators with kinesthetic feedback.” IEEE Transactions on Robotics and Automation, vol. 5, no. 4, pp. 426–434. Itoh, T., Kosuge, K., and Fukuda, T., 2000. “Human-machine cooperative telemanipulation with motion and force scaling using task-oriented virtual tool dynamics.” IEEE Transactions on Robotics and Automation, vol. 16, no. 5, pp. 505–516. 121
Jankovic, M., Larsen, M., and Kokotovic, P. V., 1999. “Master-Slave passivity design for stabilization of nonlinear systems.” Proceedings of the American Control Conference, San Diego, CA, pp. 769–773. Karnopp, D. and Rosenberg, R., 1975. “System dynamics: A unified approach.” John Wiley and Sons, NY. Kazerooni, H. and Moore, C. L., 1997. “An approach to telerobotic manipulations.” ASME Journal of Dynamic Systems, Measurement and Control, vol. 119, pp. 431– 438. Khalil, H. K., 1995. “Nonlinear Systems.” Prentice-Hall. Kim, W. S., Hannaford, B., and Bejcky, A. K., 1992. “Force-reflection and shared compliant control in operating teleomanipulators with time delay.” IEEE Transaction on Robotics and Automation, vol. 8, no. 2, pp. 176–185. Krishnaswamy, K., 2000. “Passive teleoperation of a single degree of freedom electrohydraulic actuator.” MS Thesis, University of Minnesota. Krstic, M., Kanellakopoulos, I., and Kokotovic, P. V., 1995. “Nonlinear and adaptive control design.” Wiley, New York, NY. Lawrence, P. D., Salcudean, S. E., Sepehri, N., Chan, D., Bachmann, S., Parker, N., Zhu, M., and Frenette, R., 1995. “Coordinated and force feedback control of hydraulic excavators.” Proceedings, 4th Internationl Symposium on Experimental Robotics, pp. 114–121. Lee, D. and Li, P., 2002. “Passive Coordination Control of Nonlinear Mechanical Teleoperator.” Submitted to IEEE Transactions on Robotics and Automation. Lee, D. J. and Li, P. Y., 2001. “Passive control of bilateral teleoperated manipulators: Theory.” Proceedings of the American Control Conference, vol. 6, pp. 4612–4618, Arlington, VA. Li, J. and Li, P., 2000. “Passive Velocity Field control approach to robot force control and contour following.” 2000 Japan-USA Symposium on Flexible Automation. Li, P. and Lee, D. J., 2000. “Passive feedforward approach to the control of bilateral teleoperated manipulators.” Proceedings of the Symposium on Haptic Interfaces and Teleoperators, at the ASME IMECE. 122
Li, P. Y., 1998. “Passive control of bilateral teleoperated manipulators.” Proceedings of the 1998 American Control Conference, pp. 3838–3842. Li, P. Y., 2000. “Towards safe and human friendly hydraulics: The Passive Valve.” ASME Journal of Dynamic Systems, Measurement and Control, vol. 122, pp. 402– 409. Li, P. Y. and Horowitz, R., 1996. “Application of passive velocity field control to contour following problems.” Proceedings of 1996 IEEE Conference on Decision and Control, vol. 1, pp. 378–385. Li, P. Y. and Horowitz, R., 1997. “Control of smart exercise machines, Part 1: problem formulation and non-adaptive control.” IEEE/ASME Transactions on Mechatronics, vol. 2, pp. 237–247. Li, P. Y. and Horowitz, R., 1998a. “Passive velocity field control, part 1: Geometry and Robustness.” IEEE Transactions on Automatic Control, vol. 1, pp. 1–2. Li, P. Y. and Horowitz, R., 1998b. “Passive velocity field control, part 2: Application to contour following problems.” IEEE Transactions on Automatic Control, vol. 1, pp. 1–2. Li, P. Y. and Horowitz, R., 1999. “Passive velocity field control of mechanical manipulators.” IEEE Transactions on Robotics and Automation, vol. 15, pp. 751–763. Li, P. Y. and Krishnaswamy, K., 2001. “Passive bilateral teleoperation of an electrohydraulic actuator using an electrohydraulic passive valve.” Proceedings of the American Control Conference, vol. 5, pp. 3932–3937. Li, P. Y. and Ngwompo, R. F., 2002. “Passification of electrohydraulic valves using bond graphs.” Proceedings of 15th IFAC World Congress. Lipsett, M. G., Ballantyne, W. J., and Greenspan, M., 1998. “Virtual environments for surface mining operations.” CIM Bulletin, vol. 91, no. 1016, pp. 80–85. McLain, T. W. and Beard, R. W., 1997. “Nonlinear optimal control of a hydraulically actuated positioning system.” FPST, Fluid Power Systems and Technology, ASME, vol. 5, pp. 119–125. McLain, T. W. and Beard, R. W., 1998. “Nonlinear robust control of an electrohydraulic positioning system.” FPST, Fluid Power Systems and Technology, ASME, vol. 5, pp. 119–125. 123
Merritt, H. E., 1967. “Hydraulic Control Systems.” John Wiley & Sons. Ngwompo, R. F. and Gawthrop, P. J., 1999. “Bond graph based simulation of nonlinear inverse systems using physical performance specifications.” Journal of the Franklin Institute, vol. 336, pp. 1225–1247. Ngwompo, R. F., Scavarda, S., and Thomasset, D., 1996. “Inversion of linear time invariant SISO systems modelled by bond graph.” Journal of the Franklin Institute, vol. 333(B), pp. 157–174. Ngwompo, R. F., Scavarda, S., and Thomasset, D., 2001a. “Physical model-based inversion in control systems design using bond graph representation Part 1: theory.” Proceedings of the Institute of Mechanical Engineers, vol. 215, pp. 95–103. Ngwompo, R. F., Scavarda, S., and Thomasset, D., 2001b. “Physical model-based inversion in control systems design using bond graph representation Part 2: applications.” Proceedings of the Institute of Mechanical Engineers, vol. 215, pp. 104–112. Parker, N. R., Salcudean, S. E., and Lawrence, P. D., 1993. “Application of force feedback to heavy duty hydraulic machines.” International Conference on Robotics and Automation, vol. 1, pp. 375–381. Re, L. D. and Isidori, A., 1995. “Performance enhancement of nonlinear drives by feedback linearization of linear-bilinear cascade models.” IEEE Transactions on Control Systems Technology, vol. 3, no. 3, pp. 299–308. Salcudean, S. E., H.-Zaad, K., Tafazoli, S., DiMaio, S. P., and Reboulet, C., 1999. “Bilateral matched-impedance teleoperation with application to excavator control.” IEEE Control Systems Magazine, vol. 19, no. 6, pp. 29–37. Salcudean, S. E., Tafazoli, S., H.-Zaad, K., and Lawrence, P. D., 1998. “Evaluation of impedance and teleoperation control of a hydraulic mini-excavator.” The Fifth International Symposium on Experimental Robotics, pp. 229–240. Salcudean, S. E., Tafazoli, S., Lawrence, P. D., and Chau, I., 1997. “Impedance control of a teleoperated mini excavator.” International Conference on Advanced Robotics, pp. 19–25. Salcudean, S. E., Zhu, M., Zhu, W.-H., and H.-Zaad, K., 2000. “Transparent bilateral teleoperation under position and rate control.” International Journal of Robotics Research, vol. 19, no. 12, pp. 1185–1202. 124
Sheridan, T., 1987. “Telerobotics, automation and human supervisory control.” MIT Press, Cambridge, MA. Singh, S., 1997. “State of the art in the automation of earthmoving.” Journal of Aerospace Engineering, vol. 10, no. 4, pp. 179–188. Sirouspour, M. R. and Salcudean, S. E., 2000. “On the nonlinear control of hydraulic servosystems.” Proceedings of the 2000 IEEE International Conference on Robotics and Automation, San Francisco, CA, pp. 1276–1282. Sirouspour, M. R. and Salcudean, S. E., 2001. “Nonlinear control of hydraulic robots.” IEEE Transactions on Robotics and Automation, vol. 17, no. 2, pp. 173–182. Slotine, J. J. and Li, W., 1991. “Applied nonlinear control.” Prentice Hall, Englewood Cliffs, NJ. Sohl, G. A. and Bobrow, J. E., 1999. “Experiments and simulations on the nonlinear control of a hydraulic servosystem.” IEEE Transactions on Control Systems Technology, vol. 7, no. 2, pp. 238–247. Spong, M. W. and Vidyasagar, M., 1989. “Robot dynamics and control.” Wiley, New York, NY. Stentz, A., Bares, J., Singh, S., and Rowe, P., 1999. “A robotic excavator for autonomous truck loading.” Autonomous Robots, vol. 7, pp. 175–186. Tafazoli, S., Lawrence, P. D., and Salcudean, S. E., 1999. “Identification of inertial and friction parameters for excavator arms.” IEEE Transactions on Robotics and Automation, vol. 15, pp. 966–971. Tafazoli, S., Peussa, P., Lawrence, P. D., Salcudean, S. E., and de Silva, C. W., 1996. “Differential PWM operated solenoid valves in the pilot stage of mini excavators: modeling and identification.” Fluid Power Systems and Technology, Collected pepers ASME, vol. 3, pp. 93–99. Utkin, V. I., 1978. “Sliding modes and their application in variable structure systems.” MIR publishers, Moscow. van der Schaft, A. J., 1992. “L2 -gain analysis of nonlinear systems and nonlinear state feedback H∞ control.” IEEE Transactions on Automatic Control, vol. 37, no. 6, pp. 770–784. 125
Vidyasagar, M., 1993. “Nonlinear Systems Analysis.” Prentice Hall. Vossoughi, G. and Donath, M., 1995. “Dynamic feedback linearization for electrohydraulically actuated control systems.” ASME Journal of Dynamic Systems, Measurement and Control, vol. 117, pp. 468–477. Willems, J. C., 1973. “Qualitative behaviour of interconnected systems.” Ann. Systems Res., vol. 3, pp. 61–80. Yao, B., 1997. “High performance adaptive robust control of nonlinear systems: A general framework and new schemes.” Proceedings of the 36th Conference on Decision and Control, San Diego, CA, pp. 2489–2494. Yao, B., Bu, F., Reedy, J., and Chiu, G. T.-C., 2000. “Adaptive robust motion control of single-rod hydraulic actuators: Theory and experiments.” IEEE/ASME Transaction on Mechatronics, vol. 5, no. 1, pp. 79–91. Zhu, W.-H. and Salcudean, S. E., 2000. “Stability guaranteed teleoperation: An adaptive motion/force control approach.” IEEE Transactions on Automatic Control, vol. 45, no. 11, pp. 1951–1969.
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