1992, Thring 1983) including stair climbers (Voves 1990) and customized outdoor buggies have been developed to solve specific problems. However, they tend ...
Modeling and Control of a Hybrid Locomotion System Venkat Krovi *
Vijay Kumar
Research Assistant
Associate Professor
Department of Mechanical Engineering University of Pennsylvania Philadelphia, PA 19104
Abstract This paper describes a hybrid mobility system that combines the advantages of both legged and wheeled locomotion. The legs of the hybrid mobility system permit it to surmount obstacles and navigate difficult terrain, while the wheels allow efficient locomotion on prepared surfaces and a reliable passive mechanism for supporting the weight of the vehicle. The paper addresses the modeling, analysis and control of such hybrid mobility systems using the specific example of a wheelchair with two powered rear wheels, two passive front casters, and two powered articulated 2 degree-of-freedom legs. Such systems, typically, rely on multiple frictional contacts with the ground for support and for locomotion. They also exhibit redundancy in actuation, which is used here to actively control and optimize the contact forces. The scheme for active traction optimization developed here minimizes the tendency to slip at the frictional contacts by redistributing the contact forces so as to minimize the largest ratio of tangential to normal forces among all the contacts. Simulation and experimental results obtained for our prototype wheelchair are presented to demonstrate and evaluate the approach.
1. Introduction 1.1 Motivation Most land based mechanized locomotion systems are based on the principle of the wheel because of the simplicity, reliability and efficiency of this technology. In contrast, most land based biological systems rely on legged locomotion for the agility offered in highly unstructured environments. Wheeled systems are attractive for two principal reasons. First, they support the load passively which improves the reliability. Passively supported systems require actuation for
* Contact Author.
1
propulsion alone. Second, rolling contacts between the wheel and ground allow for efficient locomotion on flat, prepared surfaces. However, the locomotion potential of a wheeled vehicle (with fixed wheel radii) degrades rapidly with increasing terrain roughness. In contrast, a legged locomotion system has the potential for navigating environments cluttered with obstacles. The ability to pick footholds and to actively control the distribution of forces between the legs provides superior mobility in a variety of terrain conditions (Kumar and Waldron 1989b, Song and Waldron 1989). Another attractive feature is the inherent active suspension in legged locomotion systems. This enhanced versatility, however, comes at the price of increased cost and complexity and poor efficiency and reliability. The actuators have to support the weight of the vehicle, in addition to providing the tractive force, which translates to high payload/weight ratios for each leg1. The actuators perform isometric work just to keep the vehicle stable which leads to poor energy efficiencies. In addition, actuator sizes scale poorly with increasing payloads causing increased power consumption. As the number of legs increase, the reliability and stability improve at the expense of enhanced complexity of design and control. These factors combine to make it difficult to design a moderately-expensive, compact, legged locomotion system.
Figure 1 An artist’s rendition of a hybrid vehicle. The vehicle has two articulated legs, two powered rear wheels and two passive front wheels (casters).
Motivated by the above observations, we consider a hybrid vehicle with both legs and wheels. An artist’s rendition (Figure 1) shows an example of such a hybrid vehicle, a wheelchair
1Non-backdriveable transmissions can be used to reduce the forces that must be resisted by the actuator. However, this makes it
difficult to control and optimize the distribution of the forces, which is essential for superior mobility.
2
with legs. This vehicle can use its powered wheels for efficient locomotion on prepared surfaces (without deploying the legs). Both the wheels and legs can be used to propel the vehicle thereby providing superior traction. However, the legs are not required to support the entire weight of the vehicle, as in legged locomotion systems. The static stability (without active control) ensures that catastrophic failure of the actuators does not compromise the safety. The legs can also be used as manipulators to interact with the environment, when not deployed for locomotion. Thus, the hybrid system (Kumar et al. 1996) can perform many tasks that can be accomplished by the traditional legged vehicle, and yet be simpler, safer and less expensive. There are many applications for such a locomotion system. The design of a wheelchair with legs for people with disabilities is proposed in Wellman et al. (1995) and further described in this paper. A similar vehicle, but with wheels attached to the articulated legs, was proposed for forestry work by Hiller and Kecskemethy (1987). The ability to surmount obstacles, navigate different terrain at low cost and increased reliability makes such a system attractive for operation in hazardous remote environments and for such applications as unmanned planetary exploration (Kumar and Waldron 1989b). A hybrid locomotion system, such as the one shown in Figure 1, shares two important characteristic features with legged locomotion systems. First, the system relies on frictional contacts for stable support and successful locomotion. Thus, any control scheme must ensure that the frictional contacts are maintained. Second, the concurrent use of the actively controlled wheels and legs makes the system overconstrained. The number of actuators exceeds the number of degrees of freedom of the system. This situation, referred to as actuator redundancy, is typical of most closed chain systems including multifingered grippers and walking machines. It is meaningful to exploit the redundancy in actuation to maintain the frictional contacts. This is achieved by optimizing the force distribution at the frictional point contacts between the locomotion elements (wheels or legs) and the ground. In contrast to multifingered grippers and multilegged walkers, it is not possible to completely control all components of contact forces in a hybrid system. While, it is possible to control all components of the contact force at a foot, hybrid systems (such as the one under study) also possess wheels, both powered and unpowered. It is only possible to control the tangential component of the contact forces at the powered wheels
3
while none of the contact forces at passive, unpowered supports (for example, casters) can be controlled directly. The subject of this paper is the modeling and control of constrained mechanical systems typified by hybrid vehicles with both legs and wheels. In order to optimize the locomotion of such systems, an active traction optimization scheme is developed in this paper. The scheme, minimizes the tendency to slip by minimizing the largest ratio of tangential to normal force. In essence, the actuator forces are redistributed so that the available traction is maximized.
1.2 Previous Work In the past two decades, considerable research interest has been focused on legged locomotion systems. Much of the reported work has focused on statically stable locomotion (Hirose 1984, Mosher 1969, Waldron et al. 1984), characteristic of insects (Wilson 1966), in which the legs maintain the vehicle in static equilibrium. However, dynamically stable locomotion (Raibert 1985) exhibited by galloping four-legged animals and walking (or running) humans has also been studied. A number of proof-of-concept, statically stable, articulated legged vehicles have been designed, built and extensively tested in research laboratories. The design considerations for such vehicles are well known (Okhotsimski et al. 1977, Waldron et al. 1984, Song and Waldron 1989). Configurations of such vehicles included four-legged (Hirose 1984) and six-legged (Okhotsimski et al. 1977, Russell 1983, Waldron et al. 1984) as well as snake like crawling machines (Hirose and Morishima 1990) (which may be thought of as multilegged robots). Dynamically-stable legged systems including bipeds and quadrupeds have also been built and studied (Miura and Shimoyama 1984, Raibert 1985, Wong and Orin 1993). However, in most of these studies, the attempted holistic solution resulted in a complex control task and poor reliability. Actively controlled articulated wheeled vehicles possess many of the advantages of legged vehicles. A three-module vehicle, with six wheels for planetary exploration is described in (Kumar and Waldron 1989b, Yu and Waldron 1991). A similar system has been proposed for inspection and maintenance operations in nuclear plants by Hirose and Morishima (1990). Sreenivasan and Waldron (1996), discuss an n-module version as well as several other kinds of “actively coordinated wheeled rovers”. The characteristic features of the above systems are their closed chains and the multiple
4
frictional contacts between actively coordinated, articulated locomotion elements and the passive terrain. Multifingered grippers, also belong to the same class of systems, i.e. they involve contacts between fingers and a grasped object. For a given load, there is no unique way of distributing it between the different actuators. However, some method of distribution of force inputs is imperative for control. The kinematics and so-called force distribution problem engendered by such systems has been extensively addressed in the literature with specific references to both multifingered grippers (Demmel and Lafferriere 1989, Kerr and Roth 1986, Holzmann and McCarthy 1985, Hsu et al. 1988, Ji and Roth 1988, Nguyen 1987, Salisbury and Craig 1982, Salisbury and Roth 1983, Yoshikawa and Nagai 1987), as well as walking robots (Hirose 1984, Klein and Chung 1987, Kumar and Waldron 1988, Orin and Oh 1981, Orin and Cheng 1989, Waldron et al. 1984). Optimization is a logical choice for analysis of underdetermined systems and several mathematical techniques can be employed for this purpose. Linear programming has been used in (Kerr and Roth 1986, Kumar and Waldron 1989b), and methods based on the pseudo-inverse have been described by (Kumar and Waldron 1988, Kumar and Waldron 1989b). While the former is unsuitable for real time computation, the latter fails to yield solutions that satisfy the frictional constraints. More sophisticated schemes derived from mathematical programming are described in (Hsu et al. 1988, Cheng and Orin 1989). The advantage of this approach is the ease of incorporation of inequality constraints, such as friction and actuator limits, to the general formulation. The minimization of the largest force ratio (tangent of the friction angle) in the presence of constraints is discussed in Johnson (1991) and Johnson et al. (1991). In the special cases of two or three-fingered grasps, this problem can be solved in closed form (Demmel and Lafferriere 1989, Ji and Roth 1988, Mukherjee 1992). More details are given in Section 4.3 during the presentation of the traction ratio. However, we note that for more complicated problems, the principal limitation of this approach is the computational complexity (Nahon and Angeles 1992). The present application as a hybrid wheelchair serves as a good motivation for system for enhancing the mobility of conventional locomotion systems, without sacrificing compactness, reliability or safety. Since this work was motivated by the needs of people with disabilities it is worth mentioning other relevant work in rehabilitation engineering. While motorized
5
wheelchairs with sophisticated controls are available for people with disabilities, such vehicles require prepared surfaces to locomote. Most are unable to surmount common obstacles like steps and curbs (ANSI 1991). Wheelchair users can neither enjoy strolling on beaches and parks nor tackle muddy patches and potholes easily. Many special purpose aids (Houston and Metzger 1992, Thring 1983) including stair climbers (Voves 1990) and customized outdoor buggies have been developed to solve specific problems. However, they tend to be customized to a particular environment. Despite advances in legged locomotion, legs have not been considered a suitable option (with the exception of Song and Waldron (1989) and Zhang and Song (1989)) due to concerns of reliability and safety.
1.3 Scope and Organization of Paper The main contribution of the paper is the modeling, simulation and control of hybrid mobility systems. A proof-of-concept prototype hybrid wheelchair with two powered wheels and two articulated 2 degree-of-freedom legs2 to demonstrate the concept. Another contribution of this paper is the development and evaluation of a new method, called active traction optimization, for resolving the redundancy in actuation for such hybrid systems. This method mitigates the tendency to slip by force redistribution to minimize the largest traction ratio at the contact points. Experimental and simulation results obtained with our prototype wheelchair are used to demonstrate and evaluate the approach. In Section 2, we describe the experimental prototype of a wheelchair with legs and present the main features of the design. The kinematics and dynamics of the system are discussed in Sections 3 and 4. The traction optimization is also developed in Section 4. In Section 5, simulation studies are used to demonstrate the potential benefits of actuator redundancy, and study the effects of actuator limits. Section 6 contains the experimental results from the prototype. The performance of the traction optimization scheme is analyzed and compared with the ideal performance. Section 7 concludes the paper.
2 In this paper we use the term ‘leg’ to refer to the articulated manipulator system attached to the wheeled vehicle. Alternatively
the term ‘arm’ may be substituted for the term ‘leg’ with the corresponding analogies between the first joint/hip/shoulder, the first link/thigh/upperarm, the second joint/knee/elbow, the second link/shank/forearm and the tip/foot/hand.
6
2. Experimental Prototype of a Wheelchair with Legs 2.1 Configuration In this section we discuss the specific example of a wheelchair to highlight some of the distinct advantages offered by a hybrid vehicle over legged or wheeled vehicles. The prototype described here will be used as an experimental testbed in sections 4 and 5. A brief overview of the system design is presented here. Details of the mechanical design, the control system and the user interface can be found in Wellman (1994), Krovi (1995) and Krovi and Kumar (1995).
Figure 2 Forward locomotion maneuver.
Our prototype chair has 2 two degree-of-freedom planar legs with the provision of adding a third degree of freedom for out of plane motions − see Figure 1. When in contact with the ground, the two degree-of-freedom legs constrain the chair to operate in its sagittal plane. Since the vehicle is free to move in the plane of the ground when the legs are not in contact with the ground, this does not pose a serious limitation. All four wheels are independent. The two rear wheels are powered while the two front wheels are casters. The forward (backward) motion sequence shown in Figure 2 may be accomplished by using the powered rear wheels alone or by using the legs to drag the vehicle forward (backward) or both. Since wheeled locomotion is energy efficient, only the powered rear wheels are used for travel on prepared surfaces. However, under conditions of poor traction, the legs can be utilized to improve traction as well as stability and safety.
Figure 3 Complete curb climbing maneuver.
The legs of this hybrid chair are used primarily on uneven terrain. Figure 3 shows the
7
hybrid chair climbing a curb using a five-step procedure3. Note that the wheels must either be powered or be equipped with brakes to accomplish the transition from stage 2 to stage 3 safely. If the wheels are powered (as are the legs), the system is redundantly actuated and the legs and wheels must “cooperate” to climb curbs. Both maneuvers, forward locomotion and curb climbing, can be performed by using the arms/legs alone (without the aid of powered wheels). However, as discussed later, powered wheels enable us to take advantage of the redundant actuation to optimize the contact forces. The legs can be used to step over small obstacles (rocks, toys, potholes) or avoid areas with poor support or traction as seen in Figure 4(A). The legs can also be used to manipulate objects in the environment either individually, as shown in Figure 4(B), or cooperatively as shown in Figure 4(C).
Figure 4 (A) Crossing a ditch (B) Opening a door and (C) Manipulating objects.
2.2 Mechanical Design The chair was designed to be compact in order to operate indoors. It uses electric motors that require a 24V d.c. supply − two standard 12 volt batteries. The width is 0.775 meters (30 inches) and the weight without the amplifiers, batteries or the payload is 28.2 kg (62 lb). The design optimization of the configuration is discussed in Wellman (1994) and Wellman et al. (1995). Simulation and preliminary experimental studies were performed for the combined use of the legs and wheels to climb a 0.3048 meter (12 inch) curb with a 75 kg dummy (ANSI 1991), riding in our chair and the results are reported in Wellman et al. (1995).
3MPEG video footage can be found at our WWW site : http://www.cis.upenn.edu/~venkat/wheel.html.
8
Design Parameters
Values
Weight (without batteries or payload)
62.0 lb. (28.2 kg.)
Overall Width
30.5 in. (0.775m.)
Wheelbase
21.0 in. (0.530 m.)
Radius of Wheel Height of CG
4.5 in. (0.114 m.) 7.5 in. (0.191 m.)
Distance of C. G. from Rear Wheel
13.2 in. (0.335 m.)
Height of Shoulder Joint
12.0 in. (0.305 m.)
Distance of Shoulder Joint from Rear Wheel
10.5 in. (0.267 m.)
Length of Thigh/Upperarm
15.9 in. (0.404 m.)
Weight of Thigh/Upperarm
2.25 lb. (1.02 kg.)
Length of Shank/Lowerarm
18.9 in. (0.480 m.)
Weight of Shank/Lowerarm
1.85 lb. (0.841 kg.)
Maximum Torque (Thigh/Upperarm)
600 in-lb. (67.80 Nm.)
Maximum Torque (Shank/Lowerarm)
600 in-lb. (67.80 Nm.)
Maximum Wheel Torque
35 in-lb. (3.95 Nm.)
Table I Optimal design parameters.
Figure 5 The experimental prototype.
The chassis, the two legs and the motors are shown in Figure 6. Some of the key dimensions are shown in Table I. We note that the legs in the actual prototype do not possess the third degree of freedom required for three-dimensional maneuvers (shown in Figure 4). In addition, the two 12 volt batteries, the amplifiers and the controller are also not shown in any of the figures. Each leg has a reach of approximately 0.81 meters (32 inches). The proximal link is made from four thin walled titanium tubes. This arrangement allows for high stiffness in torsion and in bending with a very small penalty in weight. Power is transmitted to the distal joint (elbow/knee) through a chain and sprocket transmission. The parallel drive configuration allows the drive motors to be placed on the chassis and it requires less power and lower actuator torques than conventional serial chain transmissions (Ulrich and Kumar, 1991). The chain is preloaded to remove backlash. The proximal link is designed so that the chain passes through the neutral axis of the link to minimize bending moments. The distal link is fabricated from an aluminum tube.
Figure 6 AutoCAD® drawings of the wheelchair chassis.
9
At the tip, it is equipped with a compliant ankle which is a linear spring loaded joint on linear bearings. The ankle is instrumented with a linear potentiometer that enables the estimation of the axial foot force. The tension in the chain is sensed by strain gages on the proximal link and this gives us the second component of the foot force. Each joint in the leg is driven with a single DC gear motor (Kollmorgen/PMI, 12FG) capable of exerting 22.6 N-m (200 in-lb) of torque at 26 rpm. An external gear reduction of 3:1 is also accomplished between the gear motor and the joint. Each rear wheel is also driven by a similar gear motor (Kollmorgen/PMI, 9FG) with a rating of 3.95 N-m (35 in-lb) of peak torque. The maximum foot speed is approximately 2.2 meters/second. At the rated wheel speed (78 rpm), the chair moves at 0.93 meters/second (3.4 km/hour). All motors (legs and wheels) are mounted on the chassis (base of the chair) so that their weight is not borne by the moving links.
2.3 Control System The motors are driven by 20 kHz PWM amplifiers (Kollmorgen/PMI, VXA-48-8-8) that operate off a 24 Volt DC source. Although in the laboratory we use transformers and AC line voltage, the vehicle can be operated on two standard 12 volt lead-acid batteries. The amplifiers are configured to clamp to the motor current determined by the control signals received from the control computer. System feedback is accomplished through 500 line resolution incremental optical encoders. Due to the high gearing ratios the encoders provide a resolution of 2500 counts per degree of joint shaft rotation for each leg. Position is measured directly from the encoders mounted on the input side of each motor and the velocity is computed digitally by taking successive derivatives of the position signal. A digital I/O card (Keithley-Metrabyte, DDA06) is used for data acquisition. In addition, as mentioned earlier, both components of foot forces are available for feedback. The circuitry for conditioning and amplifying strain gage signals (Analog Devices, 3B18) is mounted on the distal link close to the strain gages to minimize noise. A data acquisition card (Real Time Devices, ADA2000) is used to convert the analog strain measurements and the ankle displacements. The control computer is an IBM compatible 486 machine with an i860 coprocessor running concurrently. The i860 is used to perform the control computations that are necessary to process the sensory data and coordinate the multiple actuators. The 486 processor performs all system input and output tasks. This includes non-synchronized tasks such as reading and writing
10
to files, processing input from the user and controlling the video display, as well as synchronized tasks such as reading the encoders, providing input data to the i860 and sending control signals to the motors. A shared memory block is used to facilitate exchange of information between the two processors.
3. Kinematics 3.1 Modeling In this section, we analyze the configuration and model the kinematics of the hybrid mobility system typified by the modified wheelchair described in Section 2. The actively controlled legs are capable of exerting forces in the sagittal plane (perpendicular to the wheel axles) of the vehicle. Since, our chief interest is in the concurrent operation of legs and wheels, we consider a planar model of the system. We will consider maneuvers such as the ones shown in Figures 2 and 3 and will ignore out of plane motions and forces for the analysis. Threedimensional maneuvers that require the three degree-of-freedom legs (as shown in Figure 4) are not addressed in this paper. A schematic of the wheelchair sectioned through its sagittal plane is shown in Figure 7. For planar maneuvers, both legs act in concert. (And similarly, both rear wheels.) When the legs are not in contact with the ground the system can be modeled as a 4 degree-of-freedom serial kinematic chain. The tip of the Link 4 (foot) is the end effector while the rear-wheel/ground contact forms the base joint. This joint may be modeled as a rolling/gear pair while the rest of the joints are modeled as revolute joints. When the legs come in contact with the ground, a simple Variables
Description
Variables
Description
OXY
Global inertial frame.
L3c
distance of center of mass of proximal
L3
link from the joint (joint 3) xc
x coordinate of the center of mass
L4
yc
y coordinate of the center of mass
L4c distance of center of mass of distal link from the elbow joint (joint 4)
xw
x coordinate of the wheel axle
a
angle between x-axis and the tangent to
yw
y coordinate of the wheel axle
θ1
wheel rotation angle measured CW
xf
x coordinate of the foot contact point
θ2
the chair lift angle
L4 CG
the wheel at the point of contact
yf
y coordinate of the foot contact point
α
terrain slope
Mc
signed link length from the wheel axle
η
angular location of the CG in the chair
Nc
signed link length from the center of
W to the center of mass C
θ1 Y
θ4
elbow joint angle.
θ2
ξ
xw
shoulder joint angle.
mass C to the shoulder joint of the arm L3 length of the proximal link (upper arm)
η
w.r.t the wheel axle. θ3
θ4
θ3
length of the distal link (forearm).
O
Table II Table of symbols.
α yw
X
xf
Figure 7 The variables assigned to the chair.
11
yf
closed kinematic chain is formed. This kinematic chain is now equivalent to a five bar linkage (with the replacement of the base revolute joint by a rolling/gear pair). Regardless of the nature of the kinematic chain (open/closed) the configuration of the system can be described (redundantly) by the following vector of Lagrangian coordinates: q = [ x w , y w ,θ 1 ,θ 2 ,θ 3 ,θ 4 ]
T
(3.1)
where x w , yw are the Cartesian position of the wheel axle, θ 1 , θ 2 , θ 3 , θ 4 are the angles as defined in Figure 7 and Table II. This choice of a redundant set of coordinates is deliberate. Constraints are introduced later to reduce the Lagrangian coordinates to the suitable number of generalized coordinates (corresponding to the degrees-of-freedom). When used in the framework of the Lagrangian dynamic analysis (with appropriate constraints), it enables us to obtain expressions for the contact forces directly from the corresponding multipliers. Also, as shown in Sarkar (1993), it also facilitates the treatment of holonomic and nonholonomic constraints in a unified framework.
3.2 Trajectory Generation The desired trajectory for the system is composed of trajectory segments generated in the task space. Each trajectory segment is specified by the time required for the motion, the initial and final Cartesian positions for (a) the center of mass of the chair and (b) the tip of the foot. The initial and final velocities and accelerations for each trajectory segment are zero. Each trajectory curve segment is a solution curve to the minimization of the integral-norm of the Cartesian jerk between the initial and final positions. This optimal solution can be expressed as the following non-dimensionalized Cartesian trajectory for the point of interest (which is used to generate the trajectory). − winit w t ,τ = ( 3.2 ) w final − winit T w is the non-dimensionalized Cartesian coordinate, winit , w final are the initial and final w = 6τ 5 − 15τ 4 + 10τ 3
where
w=
positions for the motion and τ is the non-dimensionalized time. The above equation may be differentiated with respect to time to obtain expressions (non-dimensionalized) for the Cartesian velocity and acceleration. This procedure is performed for each coordinate at each point of interest and the resulting trajectories are used by the inverse kinematics routine (which will be described in the next
12
subsection) to obtain the joint trajectory profiles.
3.3 Inverse Kinematics A global inertial frame (OXY) is defined so that the origin coincides with contact point at the beginning of the motion (as shown in Figure 7). The Cartesian position of the center of mass of the chair in this frame is given by: xcg = R θ 1 Cα − R Sα + Mc Cαη 2
(3.3)
ycg = R θ 1 Sα + R Cα + Mc Sαη 2
(3.4)
Similarly, the Cartesian position of the foot contact in this inertial frame is given by: x f = R θ 1 Cα − R Sα + N c Cαξ 2 + L3 Cαξ 23 + L4 Cαξ 234
(3.5)
y f = R θ 1 Sα + R Cα + Nc Sαξ 2 + L3 Sαξ 23 + L4 Sαξ 234
(3.6)
where Cαβ L234 = cos( α
L
+ β+ + θ 2 + θ 3 + θ 4 )
and Sαβ L234 = sin( α + β +L+ θ 2
+θ3 +θ4)
We obtain the Cartesian motion profiles of two points of interest (the center of mass of the chair, (xcg, ycg), and the tip of the foot, (xf, yf)) from the trajectory planning stage. The inverse kinematics computation involves solving the nonlinear system of Equations 3.3-3.6. The computation in the inverse kinematics proceeds in two stages. First, the desired position for the center of mass of the chair (xcg and ycg) and geometric parameters of the chair (R, α, η and Mc) are used in Equations 3.3 and 3.4 to compute the unknown joint angles (θ1 and θ2). Differentiating Equations 3.3 and 3.4 yields Equations 3.7 and 3.8, x& cg = R θ& 1 Cα − Mc Sαη 2θ& 2
(3.7)
y&cg = R θ& 1 Sα + Mc Cαη 2θ& 2
(3.8)
These may be solved simultaneously to obtain values for θ& 1 and θ& 2 for known values of x& cg and y&cg and the geometric parameters of the vehicle. Similarly, further differentiation and simultaneous solution of Equations 3.9-3.10 below, gives the values of θ&& 1 and θ&& 2 for known xcg , && ycg , x& cg and y&cg , the geometric parameters of the vehicle and (previously solved) values of && values of joint angles and their velocities. && xcg
= R θ&& 1 Cα − Mc Sαη 2θ&& 2 − Mc Cαη 2θ& 22
(3.9)
13
&& ycg
= R θ&& 1 Sα + Mc Cαη 2θ&& 2 − Mc Sαη 2θ& 22
(3.10)
Thus, this first stage procedure yields the joint position, velocity and acceleration values for the first two unknowns (θ1 and θ2). In the second stage, the results of the first stage computation, other geometric parameters (Nc, ξ, L3, L4) and the desired motion profile of the foot ( xf, yf and higher derivatives), are used to compute the other two unknown joint angles (θ3 and θ4) from the Equations 3.5-3.6. The joint velocities and accelerations are then computed by differentiating the Equations 3.5-3.6 and solving for the unknowns simultaneously in a fashion similar to that described for the derivatives of θ1 and θ2. We note that the above inverse kinematics procedure is valid only under the assumption that the wheel and the foot do not slip. Violation of this assumption invalidates all the preceding inverse kinematics solution procedure and necessitates a reinitialization of all parameters before continuation.
14
4. Dynamics 4.1 Modeling The dynamic model of the system is developed using the framework of Sarkar (1993), in the state space. The general formalism developed for a mechanical system with “n” Lagrangian coordinates (denoted by q) and a set of “m” constraints of which “m1” are holonomic and “m-m1” are non holonomic is used here. Both types of constraints are functions of the Lagrangian coordinates (and their derivatives). Hence, it is theoretically possible to eliminate “m1” coordinates from the set of “n” coordinates in q using the holonomic constraints (Rosenberg 1977). However, practically it is, often, very difficult to do because of the nonlinear nature of the constraint equations. Hence we choose to differentiate the holonomic constraints and collectively express all the constraints at the velocity (differential) level. x& w + Tα y& w − ( R Cα )θ& 1 − Tα x& w + xw
=0
( 4.1 )
=0
( 4.2 )
+ N c Cαξ 2 + L3 Cαξ 23 + L4 Cαξ 234 = x f = k 1
( 4.3 )
y w + N c Sαξ 2 + L 3 Sαξ 23 + L 4 Sαξ 234 = y f = k 2
( 4.4 )
y& w
L
L
L
where Sα Lδ = sin( α + + δ ), Cα Lδ = cos( α + + δ ), Tα Lδ = tan( α + + δ ), R is the radius of the driving wheel, x& w , y& w , are the Cartesian velocities of the axle and θ& 1 the angular velocity of the wheel respectively. This system has both holonomic and nonholonomic constraints. The non holonomic constraints arise out of the rolling contact condition between the wheels and the ground during any motion. (Note, these non-holonomic equations are nominally integrable in the planar case. We, however, will choose to retain them in the differential form.) Explicitly, Equation 4.1 constrains the velocities of the axle parallel to the ground to satisfy the rolling equation while Equation 4.2 restrains all velocities of the axle perpendicular to the ground (so that contact is not broken). The two holonomic constraints arise out of the kinematic loop closure equations (within the mechanism) and impenetrability of rigid bodies (foot and ground). Explicitly, Equations 4.34.4 constrain the Cartesian position of the foot generated using the forward kinematics to equal k1
15
and k2 (which can in general be time varying functions). If k1 and k2 are required to be constant it constrains the position of the tip of the leg to remain constant. All the Lagrangian coordinates are retained (instead of eliminating a subset using the holonomic constraints). The constraints are incorporated into the dynamics and enforced using Lagrange multipliers. The four constraints (two integrable rolling constraints and two loop closure constraints) are shown in the differential matrix form below. A( q )q& = 0
1 − Tα 1 0
Tα
− R Cα
0
0
1
0
0
0
0
0
a34
a35
1
0
a44
a45
x& w a 36 = − L4 Sαξ 234 a 35 = − L3 Sαξ 23 + a 36 y& w 0 a 34 = − N c Sαξ 2 + a 35 & 0 θ 1 where a 46 = L4 C αξ 234 & = [ 0]6 ×1 a36 θ 2 a 45 = L3 Cαξ 23 + a 46 a46 θ& 3 a 44 = N c Cαξ 2 + a 45 θ& 4
(4.5)
where the A is full rank size m × n matrix and q&& is a n × 1 vector of velocities of the Lagrangian coordinates. The system has four infinitesimal degrees of freedom when the arms are not in contact with the ground and two infinitesimal degrees of freedom when the arms are in contact (by the Grübler Criterion for closed chains). This may also be verified by the following constructive technique for obtaining the feasible motion directions. Let S(q) be a n × (n − m ) full rank matrix defined as the null space (kernel) of A(q) and use the definition of Equation 4.6 below to compute the elements of the matrix S(q). A(q)S(q) = 0
(4.6)
By construction, the columns of S(q) are linearly independent and any linear combination of the columns is a vector that satisfies the constraint equations. Thus, the columns of S(q) span the available motion space and constitute the “feasible” motion directions of the constrained system. We can now define the (n − m ) multipliers, ν = [ν 1 ,⋅ ⋅ ⋅, ν n-m ] , and the general solution to Τ
Equation 4.5 can now be written as,
16
q& = S(q )ν (t )
( 4.7 )
The ν(t) can now be identified as the vector of independent generalized velocities and Equation 4.7 as the specification of the velocities of the Lagrangian coordinates in terms of the independent variables (which describe the system uniquely upto the velocity level). The Cartesian velocities of the center of mass of the chair are chosen to be the two independent generalized velocities. Equation 4.7 can now be differentiated to yield the accelerations of the Lagrangian coordinates (in terms of the independent generalized velocities and accelerations), q&& = S( q ) ν& (t ) + S& (q )ν (t )
( 4.8 )
The dynamic equations of the constrained mechanical system can then be written as, H (q ) q&& + h( q , q& ) = E( q ) τ − AT σ where,
( 4.9 )
H is the 6 × 6 generalized inertia matrix, q&& is the 6 × 1 vector of accelerations of the generalized coordinates, h( q , q& ) is the 6 × 1 vector of Coriolis terms and gravity terms, 0 E = 0 0
0
0 0 0 1 0 0 0 1
T
is a 6 × 3 matrix mapping actuators to the generalized forces, 0 1
1 1 0 0
τ = [τ 1 τ 2 τ 3 ] is the 3 × 1 vector of actuator forces applied to the system, Τ
σ = [σ 1 , σ 2 , σ 3 , σ 4 ] is the 4 × 1 vector of Lagrange multipliers. Τ
Substituting Equation 4.8 into Equation 4.9 and premultiplying the entire equation by ST we obtain, S T HS ν& + S T HS& ν + S T h = S T E( q ) τ − S T AT σ
( 4.10 )
We note that the last term vanishes (By definition S is the null space of A, i.ε. A S = 0 = ST AT). Solving for ν& , the rate of change of independent velocities we obtain,
[(
)
ν& = −(S T HS) −1 S T HS& ν + S T h − S T E τ
]
( 4.11 )
Equation 4.11 gives the reduced order dynamics of the system in terms of the independent generalized velocities and accelerations and can be rewritten as,
ν& 2×1 = K 2 ×1 + L2×3τ 3×1
( 4.12 )
17
[(
where, K 2×1 = − ( S T HS ) −1 S T HS& ν + S T h
)] and
L2×3 = ( S T HS ) −1 S T E are known and
τ 3×1 is the vector of unknown torques. Additionally since H is a 6 × 6 positive definite symmetric matrix it is invertible and Equation 4.9 now can be rewritten as,
[
q&& = H −1 E τ − A T σ − h
]
( 4.13 )
Differentiating Equation 4.5 we get, & & = 0 Aq&& + Aq
( 4.14 )
and substituting for the value of q&& from Equation 4.13 yields,
σ = ( AH −1 AT )
−1
[ ASν − AH &
−1
h + AH −1 E τ
]
( 4.15 )
Equation 4.15 gives the explicit analytical equations for the multipliers and can be separated in terms of known ( P4x1 and Q4x3 ) and unknown ( σ 4× 1 and τ 3×1 ) variables.
σ 4×1 = P4x1 + Q4× 3τ 3× 1 where P4x1 = ( AH −1 AT )
( 4.16 ) −1
[ ASν − AH h] and &
−1
Q4x3 = ( AH −1 AT ) AH −1 E ) are known. −1
This sets the stage for the traction optimization scheme which will be detailed in Section 4.3.
4.2 Actuator Indeterminacy Actuator indeterminacy has been studied extensively by the robotics community especially in the context of multifingered hands (Kerr and Roth 1986, Ji and Roth 1988, Salisbury 1982) and walking vehicles (Orin and Oh 1981, Orin and Cheng 1989, Kumar and Waldron 1988). There are “n-m” equations in “p” unknowns for a given desired trajectory. In general, this is a non square system with “p > n-m”. Specifically, in our case, we have to solve for τ 3× 1 , the three unknown actuator forces using two equations. The resolution of this actuator indeterminacy is imperative for analysis and control. A variety of criteria have been proposed in the past literature which are briefly reviewed here.
18
The Moore Penrose generalized inverse (pseudo-inverse) solution of the wrench matrix to solve for the magnitudes of the wrenches to satisfy the force equilibrium equations was used by Klein et al. (1983), Kumar and Waldron (1988) and Salisbury and Roth (1983). The solution so obtained is the minimum norm solution of the contact forces. However, this solution in general does not satisfy contact constraints. An alternative approach (Kumar and Waldron 1990, Park and Starr 1990, Yoshikawa and Nagai 1987) is to decompose the problem into two subproblems. They find the particular solution (the equilibrating force field) and then performing linear programming on the homogenous solution (the interaction force field) to satisfy frictional and actuator limit constraints. However, the resulting solutions for forces may lie on or very near the edge of the frictional constraints and exhibit a chattering phenomenon that is undesirable for real-time control. Cheng and Orin (1989) avoid the decomposition, instead solving the original problem directly along with the constraints using the Compact Dual LP method to obtain computationally efficient but sub-optimal solutions. A third approach has been to treat this as a calculus of variations problem to minimize an integral cost function that depends on the actuator torques. For example, Zefran and Kumar (1995), obtain optimal force distributions for the grasping and locomotion tasks by minimizing the integral norm of the rate of change of actuator forces. This approach optimizes additional objectives along with providing the solution to the given problem. However, it is computationally intensive, involving the solution of a two point boundary value problem. In addition, it often is simpler to solve the initial problem by assuming a trajectory. Subsequent literature in multifingered grasping has considered the minimization of the contact forces (or at least some component of it) as the primary objective function. Ji and Roth (1988), used geometric reasoning to minimize the interaction forces between the contacts which resulted in equal friction ratios at each contact. Demmel and Lafferriere (1989), simplified and extended that approach to three dimensions. Johnson et al. (1991), and Mukherjee (1992), address the problem of minimization of the maximum of the friction ratios and arrive at the result of equal friction ratios at each contact. Our approach is motivated by the need to optimize traction. In the spirit of the papers mentioned above, we seek to minimize the traction (friction) ratios for a specified trajectory. This
19
is implemented in the form of an additional (optimality) constraint to the undetermined system of force equilibrium equations. However, a solution to this may not always be feasible due to the imposition of actuator limits. The determinacy of the actuator forces (when at their limit) reduces the degree of actuator redundancy and can be considered as equivalent to an apriori specification of a force distribution scheme. These aspects will be discussed in great detail in the next few subsections.
4.3 Traction Optimization In this paper, we develop a method called active traction optimization which exploits the redundancy to minimize the largest traction ratio. The traction ratio (ri) at each contact i = 1,2 is the ratio of the tangential force to the normal force (similar to the criterion proposed by Johnson et al. (1991), Mukherjee (1992)), normalized by the friction coefficient µ.
[
ri = Ft , i ( µ i Fn , i )
]
( 4.17 )
This traction ratio provides a normalized measure of the contact condition as well as an ideal parametrization of the tendency to slip. Slip occurs when ri = 1 . The closer the value of ri is to zero, lesser is the tendency to slip. Using Equations 4.5, 4.9 and the principle of virtual work, we identify the individual components of the σ 4× 1 vector of Lagrange multipliers with the constraint forces associated with each equation of constraint. Specifically, σ1 and σ2 are the tangential and normal wheel forces while σ3 and σ4 correspond to the tangential and normal foot forces. Such an identification enables us to define the traction ratios at each contact. The requisite conditions to maintain point contact without slip are, 1) The normal force must be non-negative at each contact which implies, Fn , i ≥ 0
i = 1,2
( 4.18 )
2) Based on the Coulomb friction model for slip not to occur we require that, Ft , i ≤ µ i Fn , i
i = 1,2
( 4.19 )
To minimize the possibility of slip and to minimize losses due to frictional forces we minimize the larger of the two traction ratios:
20
2 Minimize [ Max i = 1,2 {ri }]
( 4.20 )
subject to H ( q ) q&& + h( q , q& ) = E(q ) τ − AT σ A(q )q& = 0
σ = (Ft, wheel Fn, wheel Ft , foot Fn, foot ) T
[
]
ri = Ft , i ( µ i Fn , i ) Fn , i ≥ 0 Ft , i
≤
i = 1,2
µ i Fn , i
i = 1,2 i = 1,2
In the absence of actuator constraints, this kind of optimization problem has solution (Johnson et. al. 1991, Mukherjee 1992), r1 2 = r2 2
( 4.21 )
We use this result to provide us with an additional equation at the actuator level. We append Equation 4.21 to the underdetermined system in Equation 4.12 and solve for the unknown actuator forces. The resulting solution is a quartic in the torques which has four solutions (of which at most two are real). Amongst these two we choose the solution which can accomplish the motion with the smallest traction ratio. This solution is analytically obtained in contrast with the explicit minimization using mathematical programming.
4.4 Actuator constraints
r2
Q
ri
P
r1
a
c
b
Free Variable (τ ) Figure 8 Variation of ri versus the free variable in the optimization.
21
d
Let τ be the free variable in the optimization problem. The actuator torque limits can be mapped on to limits on τ for a given state q , q& . The figure above illustrates the variation of the traction ratios with τ. Two scenarios are possible. In the first case, the limits on τ are defined by the interval [a, b] and the intersection of the two curves (P) lies within this interval. Thus, in the first case the minimax problem of Equation 4.20 has the solution shown by Equation 4.21 (which occurs at the point P in Figure 8). In the second case the interval typified by [c, d] does not include P, the point of intersection. Thus, the solution to the minimax problem always occurs at one of the limits of τ in the above case at the point Q. In any redundant system, the effect of the actuator reaching its limit is to reduce the degree of actuator redundancy. This is because the actuator under consideration will now be producing a determinate actuation force (equal to its limit). Hence, we can eliminate this variable from the list of unknowns by setting it to its limit . We then solve for the other actuator forces. Thus, the solution to the system of equations (from Equations 4.12 and 4.21) is obtained in two stages. The first stage involves solution of the problem without imposing any actuator limits using the optimizing criterion developed earlier. The actuator forces so obtained are checked for compatibility with the actuator limits (the physical limitations of the actuators). Upon violation of an actuation limit, we set the value of that particular actuation force to its limiting value and recompute the other actuator forces for the reduced order redundant system. As an example, we present the solution to the problem where actuator limits are imposed on the wheel motor. Since the degree of redundancy in our system is 1, the reduced order redundant system is now determinate and we solve for the other 2 actuator forces using a suitably modified version of Equation 4.12 written as, τ 2 τ = 3
L12 L 22
−1
L13 ν& 1 L11 K 1 − τ 1,lim it − L23 ν& 2 L21 K 2
( 4.22)
These aspects will be demonstrated and further explored in the succeeding sections where we compare and contrast the results of the analysis in the absence and presence of actuator limits.
22
4.5 Friction Typically, friction may be ignored by assuming that it is negligible compared to the actuation forces and does not affect the system dynamics significantly. However, in several situations this assumption does not hold and friction has to be modeled. In the absence of good analytic models for complex systems, we use empirical techniques to determine the friction present in our system. We assume a Coulomb like frictional law between the overall system and use experimental techniques to determine the coefficients (presented in the section on the experimental prototype).
5. Simulation 200
18
Wheel Rotation
17
− − − X Position of CG
150
_____Y Position of CG
16
100
14
Angle (deg)
Position (inch)
15
13
50
Chair Angle
0
12
Shoulder Angle
11
−50
10 Elbow Angle
−100 9
8
0
5
10
15
20 Time (sec)
25
30
35
40
−150
0
5
10
15
20 Time (sec)
25
30
35
40
(A) Cartesian trajectory of c.g. (B) Joint trajectories. Figure 9 Desired motion of the center of gravity.
The model described in the previous section was implemented in the form of a Matlab® simulation. The simulation studies were carried out for the motion described in Figure 9. This step climbing maneuver will be the desired motion trajectory for both simulation studies and experimental studies to follow. In Figure 9(A) we show the task space motion profile for the c.g. of the vehicle during a motion which lifts the front casters of the wheelchair onto a 0.3556 meter step in two phases. The first phase consists of lifting the front wheels of the wheelchair 0.3556 meters (14 inches) off the ground and moving it forward by 0.0254 meters (1 inch) in 20 seconds. The second phase then moves the wheelchair a further 0.2286 meters (9 inches) forward while maintaining the same height in the next 20 seconds. The Cartesian trajectories are mapped back into the joint space using the inverse kinematics map and the resulting joint trajectories are shown in Figure 9(B).
23
System without Actuator Redundancy We first analyze the system in the absence of actuator redundancy. As mentioned earlier, we do not require actuated wheel motors to perform a curb climb – the arms alone are sufficient to accomplish the maneuver. The results presented in this subsection are generated without powering the wheels. These results will enable us to analyze the performance of the system without traction optimization. 30
0.9 Foot Ratio
25
0.8
Wheel Normal Force
0.7
20
Ratio
Force ( lbf )
0.6 15
0.5 0.4
10
Foot Normal Force
0.3 5
Foot Tang. Force
0.2 0.1
Wheel Tang. Force
0
Wheel Ratio
0 −5 0
5
10
15
20 Time (sec)
25
30
35
40
−0.1 0
5
10
15
20
25
30
35
40
Time (sec)
(A) Force profile. (B) Traction ratios. Figure 10 Simulation results without actuator redundancy.
As in all the subsequent figures in Section 5.1, the subplot (A) of Figure 10 is the plot of the forces at each contact point (solid lines are the normal forces, dashed lines are the tangential forces). We observe that very large tractive forces are required of the foot to accomplish the maneuver, while the tractive forces at the wheel are zero (as expected at the unactuated wheel). The second subplot (Figure 10(B)) is a plot of the traction ratio r1
=
(Ft1/(µ1Fn1) at the
wheel (dashed line) and the traction ration r2 = (Ft2/(µ2Fn2) at the foot (dash-dot line) vs time. We notice that the lack of wheel actuation results in high traction ratio at the foot which tends towards 1, the point at which slip would occur. System with Actuator Redundancy When the wheels are powered the traction optimization algorithm yields the results shown in Figure 11. When contrasted with the results in Figure 10, we observe that redundant actuation in the system enables a redistribution of the tractive forces between the wheels and legs. However, since the wheel normal force component is significantly higher than the foot normal force, we notice that the bulk of the tractive force is now provided by the wheel. Thus,
24
this lowers the traction ratio at the foot considerably (from about 0.9 in Figure 10(B) to about 0.3 in Figure 11(B)) while raising the traction ratio at the wheel marginally (from 0 in Figure 10(B) to about 0.3 in Figure 11(B)). The low (equal) traction ratios at the wheel and the foot now imply a lowered tendency to slip at both contacts. 1
30
0.9
Wheel Normal Force
25
0.8 0.7
20
Ratio
Force (lbf)
0.6
15
0.5 0.4
10
0.3 Wheel Tang. Force
Foot Force Ratio
0.2
5
Wheel Force Ratio
Foot Normal Force
0.1 Foot Tang. Force
0 0
5
10
15
20
25
30
35
40
0 0
5
10
15
20
Time (sec)
25
30
35
40
Time (sec)
(A) Force profile. (B) Traction ratios. Figure 11 Simulation results with actuator redundancy.
System with Actuator Redundancy and Actuator Limits Due to packaging and weight constraints, during the process of design of the chair, the wheel motors were chosen to be small. The actuator size limits the maximum torque and therefore the maximum tangential force that can be generated by the wheel. While the wheel motors at powered rear wheels are adequate for locomotion on even terrain, they prove inadequate for traction optimization. Thus, it is necessary to study the effects of actuator limits (a possibility in any real-life system) on our traction optimization scheme. 1
30 0.9
25
0.8
Wheel Normal Force
Foot Force Ratio
0.7
20 Ratio
Force (lbf)
0.6
15
0.5 0.4
10
0.3 Foot Normal Force
0.2
5
Foot Tang. Force Wheel Tang. Force
0 0
5
Wheel Force Ratio
0.1
10
15
20
25
30
35
0 0
40
Time (sec)
5
10
15
20
25
Time (sec)
(A) Force profile. (B) Traction ratios. Figure 12 Simulation results with redundant actuation and actuator (wheel torque) limits.
25
30
35
40
In Figure 12, we present the solution to the problem where actuator limits are imposed on the wheel motor. This solution to the system of equations is obtained in two stages. The first stage involves solution of the problem without imposing any actuator limits using the optimizing criterion developed in the earlier (shown in Figure 11). In a second stage, the torques so obtained are checked for compatibility with the actuator limits (the physical limitations of the actuators). Because the wheel torque required exceeds the maximum, we set the actuator torque to its limiting value and recompute all the other actuator torques for the reduced order redundant system. τ 2 τ = 3
L12 L 22
−1
L13 ν& 1 L11 K 1 − τ − 1 it ,lim K L23 ν& 2 L21 2
( 5. 1)
While the wheels still do provide some tractive force, this force is inadequate to achieve complete traction optimization. This scenario forms the medium between the two limiting cases discussed in the prior two sections. We will further explore this effect in the subsequent section.
5.2 Parametric Analysis of Effects of Actuator Limits In this subsection, we vary the limits of one of the actuators (wheel motor) and study its effects on the force distribution. We verify the observation from the results of earlier sections that the traction ratios deviates from the ideal optimal ratio. We will consider the effect of imposition of different torque limits on the wheel motor in the plots below. In Figure 13, the 3D plots are generated by solving Equation 5.1 while the wheel torque τ1 parametrically over a range. Since the degree of redundancy is 1, the reduced order system is determinate and the remaining torques can be computed over all time for the specific value of τ1. The resulting traction ratios are plotted parametrized by time on one axis and by wheel torque on an orthogonal axis. We note that wheel traction ratio increases while the foot traction ratio decreases monotonically with increasing wheel motor torque. The surfaces intersect with each other to give a unique real solution at each time instant. In Figure 13(A), the wheel torque limit of 4.52 N-m (40 in-lb) is adequate to permit the two surfaces to intersect making the ideal solution of Figure 11(B) possible. In Figures 13(B) and 13(C) we reduce the limit to 3.39 N-m (30 in-lb) and 2.26 N-m (20 in-lb) respectively to study the effect on the ratios. Thus, in Figure 13(B) we see that the ideal condition of r1 = r2 is
26
realized only in the latter half of the motion. In Figure 13(C) the two surfaces do not intersect and the ideal conditions are never achieved during the motion. Thus, the problem of computing the optimal traction forces can be reduced to the problem of finding the intersection (or nearest neighbor curves) of algebraic surfaces (Gilbert et al. 1988) both in the presence and absence of actuator limits.
Figure 13 Traction ratio plots with variation of actuator (wheel) torque limits - 4.52, 3.39, 2.26 N-m (40, 30 and 20 in-lb).
27
6. Experiments 6.1 Data Collection And Analysis We measure the force exerted normal to the plane of the ground (Fy in Figure 6.1) using a force plate. A strain-gage sensor is mounted on the chain driving the distal link. The chain is pretensioned and the strain-gage is then calibrated. The strain-gage sensor now measures changes in tension in the chain which is used to infer the torque (τ4 in Figure 6.1) exerted on the link. We also record the joint angles for each sample along with the force and strain-gage data. L3 τ3
τ4
θ4
θ3
L4
Fx
Fy
Figure 14 The arm subsystem.
The data from both sources is collected on a 80286 IBM compatible computer running AtlantisTM data collection software using an A/D conversion board (Real Time Devices, ADA2000). However, since we measure only the torque exerted on the final link and the normal force exerted by the tip of the final link on the ground, post–processing is necessary to resolve the forces into the Cartesian components. Using the variables as shown in Figure 14 we can generate the following relationships between the various bodies and infer the other forces present in the system. τ 3 τ = 4
− L3 S3 − L S 4 4
L3 C3 Fx , foot − g ( M 3 L3c + M 4 L3 ) C3 + L4 C4 Fy , foot − g M 4 L4c C4
( 6.1 )
where Mi is the mass of ith link and Lic is the position of the center of mass of each link from the link origin. Hence in the above set of equations we know Fy, foot and τ4. Thus, we can infer the other two variables Fx, foot and τ3 from Equation 6.1. From these known values for the forces at the foot ( Fx , foot and Fy , foot ) we can find the relationship between the values of the forces at the wheel given the masses of the various
28
components (Mi) as defined in Table II, the Cartesian accelerations ( && y ) and Cartesian x and && components of friction ( λ x and λ y ) in the system as shown below, Fx ,wheel + Fx , foot = λ x − ( M1 + M2 + M3 + M4 ) && x Fy ,wheel + Fy , foot = λ y + ( M1 + M2 + M3 + M4 )( g − && y)
( 6.2 )
6. 2 Estimation of Friction 15 14.5
12 Foot Normal Force
− − − X Position of CG _____Y Position of CG
10 14 Foot Tang. Force
13.5 13 Force(lbf)
Position (inch)
8
12.5
6
12 4 11.5 11 2 10.5 10 0
2
4
6
8
10
12
14
16
18
20
Time (sec)
0 0
2
4
6
8
10
12
14
16
18
20
Time (sec)
(A) Trajectory of the cg. (B) Foot forces. Figure 15 Estimation of the friction in the system.
A controlled experiment is set up for acquiring this data. A forward motion of 0.0889 meters (3.5 inches) in 20 seconds (while maintaining a constant height of the CG) is commanded as shown in Figure 15(A). The wheel motors are turned off and the system uses only the powered legs for this maneuver. The foot forces for the motion are recorded using the force plate and are plotted in Figure 15(B). Since the inertial forces for this motion can be computed we use the obtained data for tangential force at the foot to characterize and estimate the net friction in the system. The tangential force profile during the motion shows the characteristic peak in breakaway static friction which is overcome at about 6 seconds into the maneuver (as can be verified by the motion of xcg in Figure 15(A)). After the motion commences, we note that the tangential force value stabilizes to a roughly constant value which gives us an estimate of the kinetic friction in the system. While our experiment yields only the Cartesian frictional force (net sum of all friction of the system resolved in the Cartesian directions) it is adequate for our modeling and simulation purposes. We include the friction in the form of an external drag force on the chassis along with other external forces like gravity.
29
6.3 Experimental Results A number of experiments were performed with varying step sizes and stride lengths and a sample described here is for the same 0.3556 meters (14 inch) lift discussed in the simulation section. The motion commanded during the trajectory generation phase is shown in Figure 16(A). The center of gravity is controlled in this two phase maneuver to move using along a minimum jerk trajectory as detailed earlier. The first phase consists of lifting the front casters of the wheelchair 0.3556 meters (14 inches) off the ground and moving it forward by 0.0254 meters (1 inch) in 20 seconds. The second phase then moves the wheelchair a further 0.2286 meters (9 inches) forward while maintaining the same height in the next 20 seconds. 18
200 Wheel Rotation
17
− − − X Position of CG _____Y Position of CG
150
16
100
14
50
Angle (deg)
Position (inch)
15
13
0
12
Shoulder Angle
11
−50
10 Elbow Angle
−100 9
8
0
5
10
15
20 Time (sec)
25
30
35
40
−150 0
5
10
15
20
25
30
35
40
Time (sec)
(A) Motion of the center of gravity in task space (B) The joint trajectories − actual vs. desired. Figure 16 The Cartesian motion trajectories.
Figure 16(B) plots both the desired trajectories as well as the actual trajectories realized by the experimental setup in joint space. We note that the wheel rotation lags slightly behind the desired trajectory because the wheel motors have saturated. Figure 17 shows the experimental forces as well as the traction ratios realized during the motion. We remind the reader that only the foot normal force shown here is the true measured value. All the other forces are inferred from the measurements as detailed in the earlier section. The force data plotted here are not used for any control computations. In Figure 17(B) we note that forces that are observed saturate about 5 seconds into the maneuver. This causes the lag in the rotation of the wheel mentioned earlier. We also note that because of the non–collocation and varying system dynamics the inferred wheel torques fluctuate
30
a little. Finally because of varying system dynamic responses of the different motors (not modeled) the responses of the system to changes in trajectory is poor as can be seen at the beginning and end of the motion.
1
30
0.9
Wheel Normal Force
25
0.8 0.7
Foot Force Ratio
20
Ratio
Force(lbf)
0.6
15
0.5 0.4
10
0.3 Foot Normal Force
0.2
Wheel Force Ratio
5 Foot Tang. Force
Wheel Tang. Force
0 0
5
0.1
10
15
20
25
30
35
0 0
40
5
10
15
Time (sec)
20
25
30
35
40
Time (sec)
(A) Experimental forces. (B) Experimental traction ratios. Figure 17 The forces and traction ratios for the motion.
6.4 Experimental and Simulation Results A better contrast of the performance of the simulation and experimental results is obtained when we overlay the two results on top of each other (as can be seen in Figure 18). The force histories obtained from the simulation (dotted) and actual experimental (solid) are shown in Figure 18(A). 1 30
0.9 Wheel Normal Force
0.8
25
Foot Force Ratio
0.7 20 Ratio
Force(lbf)
0.6
15
0.5 0.4
10
0.3 Foot Normal Force
0.2
Wheel Force Ratio
5 Foot Tang. Force
Wheel Tang. Force
0 0
5
0.1
10
15
20
25
30
35
40
0 0
5
10
15
20
25
30
35
40
Time (sec)
Time (sec)
(A) Overlaid forces. (B)Overlaid traction ratios. Figure 18 Experimental results overlaid on simulation results.
Figure 18(B) depicts the comparison between the predicted traction ratio at the foot and the actual traction ratio. We note that the wheel motion lags considerably behind the desired
31
value because of the lower bandwidth for control and the effects of actuator limiting. We also note that initially and when terminating the motion the system is slow to respond to the changes because of the low bandwidth of the wheel motor. We observe that the experimental data agrees with the simulation. Since the experiments were performed with a rubber foot on a carpeted floor, we observe very large traction ratios without any noticeable slip. The wheel normal force can only be inferred by subtracting the foot force from the weight of the vehicle and is therefore not shown.
7. Conclusions In this paper, we propose a completely new approach termed hybrid locomotion for combining the advantages of wheeled and legged locomotion. Our hybrid locomotion system is capable of tackling a variety of terrain conditions with a combination of wheels and legs. Our prototype vehicle has demonstrated the ability to climb 0.3556 meter (14 inch) high obstacles (like curbs), ascend (or descend) 30 degree inclines and locomote omnidirectionally on planar surfaces using wheels and on “difficult” terrain with both wheels and legs. The second main contribution of the paper is an optimal traction scheme for mechanical systems with actuator redundancy and actuator limits. The methodology for controlling and optimizing the performance of such combined legged/wheeled systems is being presented here for the first time. The control scheme for optimal traction control was implemented on our proof of concept experimental prototype. This control scheme was experimentally tested and the results compared with the results of the simulation. The extension of our model to encompass three dimensional motion in the areas of formulation, simulation and subsequent experimental implementation is the logical direction for future research. We note that our system lends itself readily to such an implementation. The addition of a third degree of freedom swivel axis in the vertical direction so as to make each manipulator’s workspace a section of a sphere would achieve the desired purpose.
32
8. References •
ANSI/RESNA WC-01 to WC-11, 1991, “RESNA Standards for Wheelchairs,” American National Standards Institute.
•
Cheng, F.-T, and Orin, D.E., 1990, “Efficient Algorithm for Optimal Force Distribution – The Compact–Dual LP Method, ” IEEE Transactions on Robotics and Automation, 6(2), pp. 178-187.
•
Demmel, J., and Lafferriere, 1989, "Optimal Three Fingered Grasps," Proc. 1989 IEEE International Conference on Robotics and Automation, Scottsdale, Arizona, pp. 936-942.
•
Gilbert, E. G., Johnson, D. W., and Sathiyakeerthi, S., 1988, “A Fast Procedure for Computing the Distance Between Complex Objects in 3–D Space,” IEEE Journal of Robotic Systems, 4(2), pp. 193-203.
•
Hiller, M., and Kecskemethy, A., 1987, “A Computer-oriented Approach for the Automatic Generation and Solution of the Equations of Motion for Complex Mechanisms,” Proc. 7th World Congress on the Theory of Machines and Mechanisms, Sevilla.
•
Hirose, S., 1984, “A Study of Design and Control of a Quadruped Walking,” International Journal of Robotics Research, 3(2), pp. 113-133.
•
Hirose, S., and Morishima, A, 1990, “Design and Control of a Mobile Robot with an Articulated Body,” International Journal of Robotics Research, 9(2), pp. 99-114.
•
Holzmann, W., and McCarthy, J.M., 1985, "Computing the Friction Forces Associated with a Three Fingered Grasp," IEEE Journal of Robotics and Automation, Vol. RA-1, No. 4, pp. 206-210.
•
Houston, T., and Metzger, R., 1992, “Combination Wheelchair and Walker Apparatus”, U.S. Patent 5,137,102.
•
Hsu, P., Li, Z., and Sastry, S. , 1988, “On Grasping and Coordinated Manipulation by a Multi-fingered Robot Hand,” Proc. 1988 IEEE International Conference on Robotics Automation, pp. 384-389.
•
Ji, Z., and Roth, B., 1988,"Direct Computation of Grasping Force for Three-Finger Tip-Prehension Grasps," ASME Journal of Mechansims, Transmissions, and Automation in Design, Vol. 110, No. 4, pp. 405-413.
•
Johnson, L., Kumar, V., and Gardener, J. F., 1991, “Optimization of Contact Forces in Multi-fingered and Multi-legged Robots,” Proc. 2nd Ann. Applied Mechanisms and Robotics Conference, V2.
•
Johnson, L. , 1992, “Optimization of Contact Forces in Multi-fingered and Multi-legged Robots,” Masters Thesis. Department of Mechanical Engineering, University of Pennsylvania.
•
Kerr, J. and Roth, B., 1986, "Analysis of Multifingered Hands," International Journal of Robotics Research, 4(4), pp. 3-17.
•
Klein, C.A., Olson, K.W., and Pugh,D., 1983, “Use of Force and Attitude sensors for Locomotion of a .Legged Vehicle over Irregular Terrain,” International Journal of Robotics Research, 2(2), pp. 3-17
•
Klein, C.A., and Chung, T.S. , 1987, "Force Interaction and Allocation for the Legs of a Walking Vehicle," IEEE Journal of Robotics and Automation, RA-3, No. 6.
•
Krovi, V., Wellman, P., and Kumar, V. , 1994, “Design of a Walking Wheelchair for the Motor Disabled,” Proc. 4th International Conference on Rehabilitation Robotics, pp. 125-131, Wilmington, Delaware.
33
•
Krovi, V., and Kumar, V. , 1995, “Optimal Traction Control in a Wheelchair with Legs and Wheels,” Proc. 4th Ann. Applied Mechanisms and Robotics Conference, AMR- 030, Cincinnati, Ohio.
•
Krovi, V., 1995, “Modeling and Control of a Hybrid Locomotion System,” Masters Thesis. Department of Mechanical Engineering, University of Pennsylvania.
•
Kumar, V. and Waldron, K.J., 1988, "Force Distribution in Closed Kinematic Chains," IEEE Transactions on Robotics and Automation, 4(6), pp. 657-664.
•
Kumar, V., and Waldron, K. J. , 1989a, “Adaptive Gait Control for a Walking Robot,” IEEE Journal of Robotic Systems, 6(1), pp. 49-75.
•
Kumar, V. and Waldron, K. J., 1989b, "Actively Coordinated Vehicle Systems,"
ASME Journal of
Mechansims, Transmissions, and Automation in Design, 111(2), pp. 223-231. •
Kumar, V. and Waldron, K. J., 1990, "Force Distribution in Walking Vehicles," ASME Journal of Mechanical Design, 112(1), pp. 90-99.
•
Kumar, V., Wellman, P., and Krovi, V., 1996, “Adaptive mobility system,” U. S. Patent 5,513,716.
•
Lin, M. C., and Canny, J. F., 1991, “A Fast Algorithm for Incremental Distance Calculation,” Proc. 1991 IEEE International Conference on Robotics and Automation, Sacremento, California, pp. 1008-1014.
•
Miura, H., and Shimoyama, H., 1984, “Dynamic Walk of a Biped”, International Journal of Robotics Research, 3(3), pp. 60-74.
•
Mosher, R. S., 1969, “Exploring the Potential of a Quadruped,”
International Automotive Engineering
Congress, SAE Paper 690191, Detroit, Michigan. •
Mukherjee, S., 1992, “Dextrous Grasp and Manipulation”,
Ph.D. Thesis, Department of Mechanical
Engineering, The Ohio State University. •
Nahon, M., and Angeles, J., 1992, "Real-Time Force Optimization in Parallel Kinematic Chains Under Inequality Constraints", IEEE Transactions on Robotics and Automation, 8(4), pp. 439-450.
•
Nguyen, V.D., 1987, "Constructing Force-Closure Grasps," Proc. 1987 IEEE Int. Conf. on Robotics and Automation, Raleigh, CA, pp. 240-245.
•
Okhotsimski, D. E., Gurfinkel, V. S., Devyanin, E. A., and Platonov, A. K., 1977, “Integrated Walking Robot Development”, Machine Intelligence, V9,. Eds. J.E. Hayes, D.Michie and L.J. Mikulich.
•
Orin, D.E., and Oh, S.Y., 1981, "Control of Force Distribution in Robotic Mechanisms Containing Closed Kinematic Chains," Journal of Dynamic Systems, Measurements, and Control, Vol. 102, No.2, pp. 134-141.
•
Orin, D.E., and Cheng, F-T., 1989, “General Dynamic Formulation of the Force Distribution Equations,” Fourth International Conference on Advanced Robotics (ICAR), May 1989, Columbus, Ohio, pp.525-546,1990.
•
Park, Y., and Starr, G., 1989, “Finger Force Computation for Manipulation of an Object by a Multifingered Hand,” IEEE Journal of Robotics and Automation, August 1989, pp. 930 - 935.
•
Raibert, M. H., 1985, “Legged Robots that Balance,” MIT Press., Cambridge, Massachusetts.
•
Rosenberg, R. M., 1977, “Analytical Dynamics of Discrete Systems,” Plenum Press, New York, London.
•
Russell, M., 1983, "Odex 1: The First Functionoid,” Robotics Age. 5(5), pp. 14-18.
34
•
Salisbury, J. K., 1982, “Kinematic and Force Analysis of Articulated Hands,” Ph.D Thesis, Department of Mechanical Engineering, Stanford University.
•
Salisbury, J. K. and Craig, J. J., 1982, “Analysis of Multifingered Hands : Force Control and Kinematic Issues”, International Journal of Robotics Research, 1(1), pp. 4-17.
•
Salisbury, J. K. and Roth, B. , 1983, “Kinematic and Force Analysis of Articulated Mechanical Hands,” ASME J. Mechanisms, Transmissions and Automation in Design, 105(1), pp. 35-41.
•
Sarkar, N., 1993, “Control of Mechanical Systems with Rolling Contacts: Applications to Robotics,” Ph.D Thesis, Department of Mechanical Engineering, University of Pennsylvania.
•
Song, S. M., and Waldron, K. J., 1989, “Machines that Walk”, MIT Press, Cambridge MA.
•
Sreenivasan, S. V., and Waldron, K. J., 1996, “Displacement Analysis of an Actively Articulated Wheeled Vehicle Configuration With Extensions to Motion Planning on Uneven Terrain,” ASME Journal of Mechanical Design, 118(2), pp.312-320.
•
Thring, M. W., 1983, “Robots and Telechairs: Manipulators with Memory, Remote Manipulators, Machine Limbs for the Handicapped”, Ellis Horwood; New York: Halsted Press.
•
Ulrich, N., and Kumar, V. , 1991,
“Design Methods of Improving Robot Manipulator Performance,”
Advances in Design Automation ASME, DE 32(2). •
Voves, D.R., Prendergast, J.F., and Green, T.J., 1990,
“Stairway Chairlift Mechanism,” U.S. Patent
4,913,264. •
Waldron, K. J., Vohnout, V. J., Pery, A., and McGhee, R. B., 1984, “Configuration Design of the Adaptive Suspension Vehicle,” International Journal of Robotics Research,. 3(2), pp. 37-48.
•
Wellman, P., 1994, “A Hybrid Mobility System,” Masters Thesis Department of Mechanical Engineering, University of Pennsylvania.
•
Wellman, P., Krovi, V., Kumar, V., and Harwin, W., 1995, “Design of a Wheelchair with Legs for People with Motor Disabilities,” IEEE Transactions on Rehabilitation Engineering, 3(4), pp. 343 - 353.
•
Wilson, D. M., 1966, “Insect Walking”, Annual Review Entomology, Vol. 11, pp. 103-122.
•
Wong, H. C., and Orin, D. E.,
“Design and Control of a Mobile Robot with and Articulated Body”,
International Journal of Robotics Research, 12(2), 1993. •
Yoshikawa, T., and Nagai, K., 1987, “Manipulating and Grasping Forces in Manipulation by Multifingered Grippers,” 1987 IEEE International Conference on Robotics and Automation, Raleigh, NC, pp. 1998-2007.
•
Yu, J., and Waldron, K.J., 1991, “Design of Wheeled Actively Articulated Vehicle,” Proc. 2nd Ann. Applied Mechanisms and Robotics Conf., Cincinnati, Ohio, 3-6 November 1991.
•
Zefran, M., Kumar, V., and Yun, X., 1994, “Optimal Trajectories and Force Distribution for Cooperating Arms”, 1994 IEEE International Conference on Robotics and Automation, San Diego, CA, May 1994.
•
Zhang, C-D., and Song, S. M. , 1989, “Gaits and Geometry of a Walking Chair for the Disabled,” Journal of Terramechanics, 26(314), pp. 211-233.
35
