IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 5, SEPTEMBER 2007
891
H
Loop Shaping Bilateral Controller for a Two-Fingered Tele-Micromanipulation System Moussa Boukhnifer and Antoine Ferreira, Member, IEEE
Abstract—Recent developments on micromanipulation and internet technologies allow potential impacts on networked microautomation of small industrial products. However, reliable and efficient microteleoperation systems with haptic feedback over the internet face strong problems due to microenvironment variations, the scaling effect problem between worlds with different physical characteristics, and time delays in communication lines. Taking into account these constraints, a force-reflecting macro-microteleoperator controller is designed based on the framework loop-shaping procedure approach. This approach allows a convenient way to tradeoff the robustness for a prespecified time delay margin, variation of force scaling factors, and uncertainties in environment models. The experimental results clearly show that the proposed approach ensures the stability robustness against time delays and scaling factors. loop-shaping design, microteleoperation, Index Terms— motion and force scaling, robust bilateral controller, time delay.
I. INTRODUCTION
R
ECENT developments in the internet media have significantly increased the human capability to obtain information from remote locations. In parallel, reliable and safe manipulation systems of microsized objects are currently investigated by taking into account the force scaling effects. When combined, the internet and micromanipulation will produce a new technology for humans to sense and act in a remote microenvironment. This new technology has a potential impact on several fields concerning the biological remote control handling or the assembly of microsystems [1], [2]. However, to make these systems efficient and safe, multimedia information should be provided to the operator, which transfers human feeling to a remote microenvironment through macro-microbilateral teleoperation. By definition, teleoperation systems frequently experience significant time delays in the communication channels between local and remote sites. If they are untreated, even small delays (in the order of several hundred milliseconds) can lead to instabilities of current microteleoperation systems due to unwanted power generation in the communications. In addition, the design of a bilateral controller for efficient and robust micromanipulation requires the careful analysis of scaling between micro and macroenvironment forces [3]–[5]. Manuscript received August 30, 2006; revised November 17, 2006. Manuscript received in final form February 1, 2007. Recommended by Associate Editor R. Moheimani. The authors are with the Laboratoire Vision et Robotique, ENSI Bourges-University of Orléans, 18000 Bourges, France (e-mail: moussa.
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2007.902951
Actually, the main key to the success of internet-based remote sensing and manipulation in microenvironments concerns mainly the optimum design of robust bilateral controllers. The commonly proposed approaches to deal with bilateral teleoperation with time varying delays are mainly based on the scattering theory formalism [6], the wave variable concept [7], or the sliding mode control [8]. These previously mentioned frameworks have proven their robustness in presence of time delays of less than 1 s. However, they have difficulty in treating the uncertainty of the plant, disturbances, and measurement noise applied to the system. As previously stated, the stability performance tradeoff is the main determiner of the control design for microteleoperation systems. Recent controller designs using control theory are very effective for macroscale the robust teleoperation. theory in bilateral controller design is proThe use of the posed by Hu et al. in [9] as a four-channel formulation. The suggested controller design is described as a multiple objective optimization problem with no time delay and fixed scaling factors. A parametrization of all the coordinating force transfer functions stabilizing the controller is shown via the Youla parametrization. optimization is used to find the parameters which best shape the closed-loop response but the achievable performance limit with the designed controller has not been proven in practice. optimization in order In [10], Kazerooni et al. applied to shape the relationships between the forces and the positions at the end of the teleoperator. In this design, the approach is based on the minimization of the error between the actual and the desired transfer function when considering an ideal communication channel (i.e., delay-free analysis and constant amplification of position and force). Model reduction techniques were employed (the order of the resulting controller was 9). The experiments were conducted only in the case of constrained movements in continuous contact with the master robot. Yan et al. [11] have proposed a general design strategy based theory with application to a motion scaling teleoperaon tion system. However, the practical implementation of the design presents limitations at microscale. First, the force scaling is not included (no force sensors), only small model uncertainties of the environment impedances are allowed and the time delays are assumed to be negligible. Second, stability in both free and constrained motions is not efficiently guaranteed leading inevitably to discontinuous forces, notably during hard contact tasks. These aspects are reflected in the limited performance of the teleoperator. The implementation of the controller also presents some difficulties. The high order of the closed-loop controller (in the order of 40) is computationally intensive to synthesize and impractical to implement without a model reduction of the overall plant. The proposed designs result in an over conservative controller, limiting the nominal performance.
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 5, SEPTEMBER 2007
A less conservative approach is to use the combined optimal control and the -synthesis framework. Leung et al. [12] proposed to design a teleoperator which would be stable for a prespecified time delay while optimizing performance characteristics for both free motion and constrained motion. No considerations about the force scaling factors and environment uncertainty are reported. In [13] and [14], the authors propose to extend the -synthesis framework for a microteleoperation application. The framework allows a convenient means to tradeoff the optimization of various performance criteria and robustness for prespecified time delay margin while taking into account stability under prespecified ranges of scaling parameters (force, position). In addition, Lee et al. [15] introduced different modeling uncertainties of robots (master and slave) and environment as extra objectives in the -synthesis framework to achieve the robust stability and performance of the teleoperator. In practice, the -synthesis framework requires an important number of weighting functions (varying from 5 to 8). The selection of weighting functions turns out to be a difficult task leading to resultant high order closed-loop controllers. None of the previously mentioned papers solves the problem of robust stability and of the performance of the teleoperator with multiple sources of uncertainties and perturbations at microscale. In this paper, we extend the loop-shaping approach, originally proposed in [16] and further developed in [17] and [18], which incorporates the characteristics of both loop shaping design. Specifically, we make use of the so-called norand malized coprime factor robust stabilization problem which has been solved in [19] and [20] and is equivalent to the gap metric robustness optimization as in [21]. The design technique has two main stages: 1) loop shaping is used to shape the nominal plant singular values to give desired open-loop properties at frequencies of high and low loop gain and 2) the problem mentioned before is normalized coprime factor used to robustly stabilize this shaped plant. Theoretically, the loop-shaping technique is optimal when dealing with unstructured uncertainty described by the gap metric or the gap metric. Practically, it is well-adapted to microscale constraints since it is very effective in cases when the uncertainty has unknown sources (i.e., adhesive microforces, modeling errors of microenvironment impedance, unmodeled dynamics of mi) croactuators varying with time and operating condition, and measurement limitations (signal noises, limited resolution controller design methods of sensors). Compared to other presented in [9]–[12], the loop-shaping design turns the difficult task of external weighting function selection into the relatively easy choice of loop-shaping functions. Furthermore, it eliminates the time consuming iteration, which is required in usual optimization, in the computation of the optimal controller. This paper is organized as follows. Section II begins by giving an overview of standard loop-shaping design procedure. In loop-shaping controller with Section III, the design of the compensation of time delay is discussed. Section IV makes a stability and robustness analysis of the scaled bilateral controller. Then, Section V illustrates our result via a microteleoperation system to prove the effectiveness of the proposed approach. In order to make the presentation clear, we deal with a single degree-of-freedom linear time-invariant teleoperator model.
Fig. 1.
H
standard problem.
Fig. 2. Coprime factor robust stabilization problem.
II. ROBUST CONTROL METHODOLOGY A.
Standard Problem Given and , the standard problem is to find which: • stabilizes the internal loop system in Fig. 1; , where • maintains the norm of is defined as the transfer function of exits entries. according to the
B. Robust Controller Design Using Normalized Coprime Factor An approach was developed by [19] starting from the concept of the coprime factorization of transfer matrix. This approach presents interesting properties and its implementation calls upon traditional notions of automatic control. We define the nominal model of the system to be controlled starting from the coprime . Then, a perturbed factors on the left: model is written (see Fig. 2) as (1) where is a left coprime factorization (LCF) of , and are unknown and stable transfer functions representing the uncertainty. We can then define a family of models with the following expression: (2) represents the margin of maximum stability. The where robust problem of stability is thus to find the greatest value of , such that all the models belonging to can be stabilized by the same corrector . The problem of robust stability amounts finding and stabilizing such as
(3) However, [18] showed that the minimal value
is given by (4)
BOUKHNIFER AND FERREIRA:
LOOP SHAPING BILATERAL CONTROLLER FOR A TWO-FINGERED TELE-MICROMANIPULATION SYSTEM
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Fig. 3. Loop-shaping design procedure.
where indicates the greatest eigenvalue of . Moreover, a corrector stabilizing all the models for any value belonging to is given by
Fig. 4. Controller with communication channel and scaling factors.
III. DESIGN OF THE
(5) where , , and are state matrices of the system defined by function and and are the positive definite matrices and solutions to the Ricatti equation (6) (7)
C. Loop Shaping Design Procedure Contrary to the approach proposed by Glover and Doyle [20], no weight function can be introduced into the problem. The adjustment of performances is obtained by affecting an open modeling (loop shaping) process before calculating the compensator. The design procedure is as follows. • We add a precompensator and/or a post-compensator to the matrix of the system to be controlled. Then, the singular values of the nominal plant are shaped to give a desired open-loop shape. The nominal plant and shaping functions and are combined in order to improve the performances of the system such [see Fig. 3(a)]. In the that monovariable case, this step is carried out by controlling in the Bode plan. the gain and the phase of • From coprime factorizations of , we apply the pre, and then synthesize a stavious results to calculate ensuring a value of slightly lower bilizing controller than
(8) • The final feedback controller is obtained by combining the controller with the shaping functions and such that [see Fig. 3(b)].
LOOP SHAPING CONTROLLER
The bilateral controller to be used for the teleoperation system needs to satisfy optimal performance and stability robustness against multiple disturbances (communication time delays, measurement noise) and uncertainties (microenvironment impedance, online tuning of force scaling parameters). loop-shaping design framework for This section analyzes optimum bilateral controller design. A. Three-Stage Design Methodology optimal controller To design in one step a standard filling out all the specifications appeared to be too conservative in practice leading to small margins of stability and placing restrictions in the choice of high order weighting functions [9]–[11]. A suitable solution is to divide the standard optimization framework into three-stage loop-shaping controllers constituting the standard teleoperator architecture, i.e., the macromaster controller (haptic interface), the microslave controller (micromanipulator), and the bilateral controller (communication channel). The main advantage is that the performance is treated both for free motion and for constrained motion. In a first step, it leads to design controllers separately for the macromaster and the microslave system in free motion. The second step consists in designing an “outer-loop” bilateral controller for the time delayed scaled communication channel in a constrained motion where the slave is in contact with its environment. Furthermore, separate designs of macromaster and microslave provide a convenient framework for specifying individually high precision tracking (force and position) with high bandwidth, compensating the adverse effects such as hysteresis, creep, and friction and achieving robustness of the closed-loop map towards model uncertainties. The loop-shaping time delayed scaled teleoperator is illustrated in Fig. 4. The controller design procedure is described in the following. 1) Design for Free Motion: The first stage only considers free motion when the master and the slave move freely in the ) so micromanipulation workspace (i.e., the case when
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 5, SEPTEMBER 2007
Fig. 5. Bode diagram of
W P W
and
W P W K
.
TABLE I EXPERIMENTAL PARAMETERS
Fig. 6. Implementation scheme for the slave controller for free motion.
an operator force results in motion of the master and the corresponding motion of the slave. The local hand controller is represented by the haptic interface (master) and its poand a human operator which applies a force sition controller represented by . The master device consists of a single degree-of-freedom paddle force-feedback driven by a dc motor. represents the micromanipulator (slave) and The interacting with the microenvironment its position controller via a force . Consider loop-shaping design of controllers for the master and slave manipulators for free motion. and denote free motion controllers for the master Let and slave, respectively. The force/position transfer functions of and slave have been identified at their nommaster inal operating point as
(9) and denote the mass of master and slave, where and the compliance coefficients, and and denote the viscosity coefficients. Table I gives the experimentally identified parameters. The time delayed scaled communication channel is only from master to slave. 2) Design for Constrained Motion: The second stage considers the design of an “outer-loop” bilateral controller in a constrained motion where the slave is in contact with the mi). In this case, the croenvironment (i.e., the case where forces are reflected back to the haptic interface in order to be loop-shaping controller repsensed by the operator. The resents the bilateral controller to be designed. A generalized scaled bilateral manipulator is characterized by a fixed motion scaling factor, , and a fixed-force scaling factor, . Using the
scaling factors, the relationships between the master and slave are (10) (11) and are the position command from the master and where and are the operator the slave position, respectively, and force command and the external force from slave to master. The is characterized by impedance . The slave environment device is a two-fingered piezoelectric microgripper. The intermeasured by the microaction force with the environment gripper is sent back to the operator as a reference signal for the . It should be noticed that haptic interface such that the design of the bilateral controller should not affect the design of the master and slave controllers of the first step. This is due to the fact that for operation in free space (no contact with external environment), there is no force feedback signal from slave system to master system. The time delay from the master to the slave, and vice versa, and . are represented by B.
Loop-Shaping Controller of the Master
The synthesis of the master controller is obtained according to the implementation shown in Fig. 3 using the comof MATLAB -analysis and synthesis toolbox mand is obtained by combining the prefilter [21]. Controller and the post-filter. The prefilter and post-filter are used to shape the open-loop plant to achieve a desired frequency response according to some well-defined design specifications such as bandwidth and steady-state error [22]. In order to ensure a high gain at low frequencies and a low gain at
BOUKHNIFER AND FERREIRA:
Fig. 7. Bode diagram of
W PW
LOOP SHAPING BILATERAL CONTROLLER FOR A TWO-FINGERED TELE-MICROMANIPULATION SYSTEM
and
W PW K.
high frequencies, we add the following weighting functions: . In order to obtain a good compromise between performance and robustness, Fig. 5 shows the frequency responses of the , and master system, the shaped master system . The results show the open-loop system that the open-loop remains close to the step response obtained ensures the after the choice of the shaping functions and correct margins of stability. C.
895
Loop-Shaping Controller of the Microgripper
–stage micromaThe slave system is composed of an nipulator and a two-fingered piezoelectric microgripper. The open-loop operation of the microgripper is not satisfactory for different reasons. The model of the device is uncertain with respect to its operating parameters (voltage, frequency, time). Furthermore, the hysteresis and creep effects are dominant and actually limit the use of the device in high-precision micromanipulation tasks. The design goals of control design are to achieve high precision tracking with high bandwidths, to compensate for the adverse effects of hysteresis and creep. Robustness of the closed-loop system against model uncertainties should be is obtained achieved. The synthesis of the slave controller according to the implementation in Fig. 6 for free motion (i.e., ). The slave position model is adjusted by the case where , . Taking into account the low the shaping functions frequency behavior of the microgripper, we chose the shaping and as functions
and Controller is synthesized by using the command of MATLAB -analysis and synthesis toolbox. Fig. 7 shows the frequency responses of the shaped slave system and the open-loop system ensuring the correct margins of stability. The shaped slave is small at high frequencies where unmodeled dynamics are the primary cause of actuator uncertainty and large at low frequencies where parametric
Fig. 8.
H
loop-shaping bilateral controller
K.
uncertainty due to the nonlinear hysteresis effects is more dominant. Note that the design and implementation of does not and ). affect the performance of the master system (i.e., In the case of operation in a free space (no contact with the environment), there is no feedback signal from the slave to the master system. D.
Loop-Shaping Bilateral Controller
In order to synthesize the robust bilateral controller, we have used a Padé approximation for the fixed communication time delay . Time delay variations are infinite-dimensional in polynomial space and cannot be represented exactly in the model. A Padé all-pass approximation has been used as (12) Approximations using a truncation of this series are most valid at low frequencies and if other frequencies are of interest than a more general Taylor series expansion about a different value might be used. In our study, the microenvironment is modeled as a spring and a damper (the inertial effects are negligible): (13) loop-shaping bilateral controller design, given in The Fig. 8, is defined by the closed-loop transfer function (14)
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 5, SEPTEMBER 2007
Fig. 9. Bode diagram of
where
W GW
and
W GW K .
is the loop transfer function around master (15)
is the loop transfer function around the slave in the presence of the microenvironment (16) is the loop-shaping of master (17) and
and effective method. It turns the difficult task of external weighting function selection into the relatively easy choice of the loop-shaping functions. Furthermore, the weighting functions are chosen according to some well-defined system specifications such as bandwidth and steady-state error requirement. The synthesis of the closed-loop bilateral controller loop-shaping procedure requires only two weighting using functions instead of five and greatly reduces the high-order controllers (around 44) to the 13th order without model reduction techniques. Furthermore, it eliminates the time consuming iteration, which is required in usual optimization, in the computation of the optimal controller. IV. STABILITY AND ROBUSTNESS ANALYSIS OF THE BILATERAL CONTROLLER
is the loop-shaping of slave (18)
Controller is obtained by combining prefilter and post. The prefilter and post-filter are used to shape the openfilter loop plant to achieve a desired frequency response according to some well-defined design specifications such as bandwidth and steady-state error. In order to ensure a high gain at low frequencies and a small gain at high frequencies, we added the following . weighting functions: Fig. 9 shows the frequency responses of the shaped system and the open-loop system with . The appropriate choice of the shaping functions , , and the bilateral controller ensures the correct margins of stability of the bilateral teleoperation system. The introduction of the bilateral controller must not interfere with the operation of the original system when there is no sensed force ). In this case, the output at the slave microgripper (when of controller is also zero and does not affect the operation of the system designed for free motion. robust bilateral controller deIn comparison to other loop-shaping procedure has signs presented in [9]–[12], several advantages. The loop-shaping design presents a simple
The two main objectives of this design are robust stabilization and disturbance rejection. The global stability and robustness analysis is discussed in the following. A. Feasible Scale Region The region of stable operation in the space formed by the position scaling factor and force scaling factor is analyzed through the pole location criteria. In order to study the stability, the loop-transfer functions of master and slave can be expressed as the following symbolic form: (19) where
and (20)
BOUKHNIFER AND FERREIRA:
LOOP SHAPING BILATERAL CONTROLLER FOR A TWO-FINGERED TELE-MICROMANIPULATION SYSTEM
Fig. 10. Root locus for a time delay of
Fig. 11. Stable position and force scaling zone with respect to a communication time delay 2 s.
T
