2. The Planar Diver Model. ael = p, a constant. We find that p is of the form:

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1 Proceedings of the 34th Conference on Decision & Control New Orleans, LA - December 1995 FP BIOLOGICAL MOTOR CONTROL APPROACHES FOR A PLANAR DIVER Lara S. Crawford and S. Shankar Sastry2 lgraduate Group in Biophysics 2Department of Electrical Engineering and Computer Sciences University of California at Berkeley, Berkeley, CA lara@eecs.berkeley.edu, sastry@eecs.berkeley.edu ABSTRACT A human diver in the air is an example of a nonholonomic system with drift. In this paper, we present a dynamic model of the diver performing planar dives and examine a simple learning approach to the control problem. Finally, we present a proposal for a general, hierarchical, learning controller for systems with many degrees of freedom. 1. Introduction A great deal of work has been done on the control and steering of nonholonomic systems in the past few years. For the most part, the systems that have been discussed have generally been drift-free; when the controls are set to zero, the system stays at rest. Very little work exists concerning general systems with drift, although some specific cases have been addressed (for example, steering left-invariant systems on SO(3) and GL(n) in [2] and controllability of a class of third-order nonlinear systems in [lo]). In the real world, however, systems with drift are common. For example, bodies in free-fall with some initial angular momentum have drift. One instance of this kind of system is a human diver. After the diver has left the board, his angular momentum is conserved, but is generally not zero. While in the air, the diver can change the drift velocity by changing his moment of inertia. He can also convert some of his somersaulting motion into twisting about the body s long axis by performing a throwing maneuver with his arms which shifts the body s angular velocity vector so that it is no longer aligned with the angular momentum. Frohlich [5] has done an extensive analysis of the physics of various diving and trampoline maneuvers. A similar problem, though one without drift, is that of the falling cat; the cat must use internal motions to reorient itself before it reaches the ground (see [ll]). The control goal for our diver system is to execute a certain dive maneuver, which we have chosen to be a forward If somersault pike, and then enter the water in a fully extended, vertical position. Since the diver is falling while executing the maneuver, there is a pre-determined length of time in which the controls can act. Hodgins has developed finite-state-machine controllers for various systems with constraints of this kind, including acrobatic robots [8] and simulations of human diving [18]. This approach combines a detailed planning strategy with rudimentary PD controllers. In this paper, we discuss a simplified, two-dimensional diver model and a learning approach to the associated control problem. The outline of the paper is as follows: in Section 2, we present the model of the diver and derive the nonholonomic constraint for its motion. In Section 3, we perform a preliminary kinematic analysis of the model. In Section 4, we present a simple learning approach to the control problem, and in Section 5, we discuss our proposal for a general learning controller. 2. The Planar Diver Model The two-dimensional diver model is shown in Figure 1. The two-dimensional dynamic model used in simulation was developed from a three-dimensional model with 18 degrees of freedom in the joints, generously shared with us by Jessica Hodgins (see [HI). We model the diver as three linked rigid bodies with five degrees of freedom. The x and z directions do not affect the control except to determine when the diver hits the water, so we ignore them. 82 and 83 are the shape space variables of the diver system; they describe the internal structure of the rotating body. 81 is a position variable of the system describing the overall orientation of the diver. The system has symmetry group S1; that is, the Lagrangian is invariant under changes in 81. By Noether s theorem, then, we have a conserved quantity, namely the angular momentum of the diver: ael = p, a constant. We find that p is of the form: p= [ai + 2P cos 0, + 27 sin 6% + 26 COS e3 + 2~ sin e3 +X c0s(e3 - e,) sin(& - e,)]& +[a2 +pcose2 +ysin& +C c0s(e3 - e,) + 71 sin(03 - e,)]& +[a3 + scoses + sine3 +C COS(B~ - e,) + 7sin(e3 - e,)]e /95 $ IEEE 388 1

2 Figure 1: 2-D diver model, showing the shape space variables and overall orientation variables. The central mass models the body, the upper one the arms and the lower one the legs of the diver. where a1, cq, a3, p, 7, 6, E, C, and q are constants. Conservation of p, the angular momentum, is the single constraint for this system. 3. 1; Somersault Pike - Kinematic Analysis Our goal is to control the diver in a forward 1; somersault pike. In three dimensions, combinations of somersaulting and twisting may be performed, but somersaults are the only dives that are possible for the two-dimensional model. Since there are three degrees of freedom and one constraint, there are two controls available. If we choose two linearly independent vectors 91 and 92 which annihilate the constraint (2) and which allow us to control 62 and 63 directly, we [;;I=[ can write -!Iu1+[ -~Iu2+[ ti] =: 91(62,e3)U1 +92(02,e3)u2 + f(82,63) (3) Note that the drift is a nonlinear function of 62 and 63. Since (b3% - bl% - b2% + bl%)], (4) = {gl, g2, [gl,g2]} spans the space (except at those isolated values of (62,63) where the first entry in [gl, 921 is zero). The system is locally controllable even without making use of terms of the form [f,gl], [f,92], etc., since the Lie bracket (4) is the same as the drift direction. Our initial attempts to control the diver proved unsatisfactory. These included using the dual, Pfaffian exterior differential system approach to generate sinusoidal controls. The diver model is an asymmetric version of the planar skater discussed in [17], which (when drift-free) can reorient itself arbitrarily by moving its two arms sinusoidally and out of phase with each other. This method of steering with sinusoids has also been used for other nonholonomic systems such as cars with trailers and firetrucks [4]. These systems were all drift-free, however; the drift in the diver system complicated this approach considerably. Methods that have been developed for steering left-invariant control systems cannot be applied, since the diver system is not left-invariant. We attempted to make use of the optimal control techniques developed in [12], which minimize i s Iu(t)12dt, but the resulting equations were extreme! y complex and would require numerical solution. Both the sinusoidal steering and the optimal control approach have the drawback that, because of their complexity, the?ontrols generated do not provide any insight into the structure of the system. A further drawback, in our view, is that these methods require a full model of the controlled system to be known. 4. A Simple Learning Algorithm Our current approach to the diving problem is to design an adaptive controller which searches a restricted control space to find a good control. Even a very simple learning algorithm can, if given a suitable space to search, find a control law that will drive the diver through a If somersault pike. The simulation shown in Figures 2 and 3 uses controls of the form = -- dzs- [e A -.&, -e (t;2). This family of controls was chosen as a biologically plausible velocity profile: in single-joint movements, the limb involved typically has a single-peak velocity curve; the same is true for the point of greatest attention in a multi-joint movement (see, for example, [3], Chapter 7; [9], Chapter 1). The controls used in the simulation, shown in Figure 2B, were chosen by an algorithm that searched among the control family (5) by varying the parameter vector around the best value it had found so far. For these steps, each entry in p was restricted to a certain

3 range; Pmax and pmin are vectors of the maximum and minimum allowed values, respectively. The algorithm for updating the parameter vector was gradient descent with the error measurement defined at the end of the dive as E(p) = (total rotation - 540")' +final leg angle2 +final arm angle' + + a)' (7) The last term, in which IC is a constant, provides a penalty for pulling out of the pike too late, since finishing the piked rotations early is considered good diving style. There are thus four constraints to minimize, and four parameters. The gradient descent was preceded by N iterations with random parameters. Each of the four parameters was uniformly distributed within its allowed range. The algorithm began with: Then, for each i < N: p1 = random (8) E1 = WPl) (9) &+l = random (10) Pi+l = best(pi, Fi+l) (11) Ei+l = E(pi+l) (12) where best(pi,@i+l) is determined by which vector has the lower error measurement. For i 2 N, the algorithm performed gradient descent with an esti- 0 (13) EiSZIW Pi - +noise (16) I ISi+l II S is a weighted average of past gradient measurements D, with more recent measurements weighted more heavily so the algorithm can adapt to different regions of the control space. m is thus a parameter describing how long the memory of the estimate is. Each entry in S was restricted to [-3000,3000]. W is a vector scaling factors; W = (pmax - p,i,)/ Some uniformly distributed noise was added to the next parameter choice to avoid getting stuck at local minima or at a parameter maximum or minimum. The noise was proportional to Ei and W. If Ei and pi stayed the same for too long, the constant of proportionality was increased for one step, and in (17) pi+l was always set equal to pi+l in order to move to a different region of the control space. When the error measurement dropped below a cutoff value, the algorithm halted. In the simulations shown in Figures 2 and 3, m = 5, IC = 25, r o i r and the error cutoff was Since the gradient descent starts from the best of several random iterations, the number of steps the algorithm takes to terminate varies widely. This learning algorithm generates movements that are qualitatively similar to those of human divers performing piked somersaults. The control was kinematic, for better comparison to the above, more traditional control methods, but future simulations will feature dynamic control. The torques required to produce the controls (see Figure 2C) are the same order of magnitude as torques humans can produce, but are probably somewhat too large. Since our simulation began at the moment the diver left the board, we have ignored the take-off from the board, which is actually one of the most important things real divers have to learn. The simulation is thus unrealistic in that the initial conditions cannot be changed by the learning algorithm. 5. A General Learning Controller Our goal is to develop a more general learning controller based on principles of biological motor control. In order to organize the complexity of coordinating the motion of several degrees of freedom, such a controller should be hierarchical. It should be able to generalize from known tasks to new tasks (see, for example, [15]). It should also have a compact representation of the learned controls, for efficient storage as well as communication between hierarchical levels. 5.1 Single Joint Controller Our proposed controller, shown schematically in Figure 4, is based on the idea of biological pattern generators. It has been known for some time that rhythmic movements like walking, swimming, breathing, and chewing are controlled in many animals by relatively simple neural networks called central pattern generators. These networks produce periodic bursting patterns which can be modulated by signals descending from higher levels in the control hierarchy. (For a review, see [7].) Recently, several investigators have found evidence for low-level controllers in fast, goal-directed movements (see, for example, [6]). Such a controller would produce a stereotypical fast movement, identifiable by its torque, velocity, and doubleor triple-burst EMG profile. Again, the movement could be tuned by descending commands, but the system would only use control signals from a restricted class. In our scenario (which is based on that proposed in [SI), the low-level controller for each joint is a pulse generator which produces square activation pulses. The pulses are then filtered, in analogy with the action of the motoneuron pool, to produce 3883

4 ,, / Figure 2: Simulation with control parameters chosen by the learning algorithm. A = , u2 = , n = , and = A. 02 (solid) and 83 (dashed). B. 82 (solid) and 83 (dashed). C. Leg (solid) and arm (dashed) torques required to produce the movement. D. 81 with these controls (solid) and with the controls set to zero (dashed) B w, time (seconds) leg and arm torques 4 the control torque. The resulting torque is reminiscent both of the double- or triple-burst EMG pattern common in fast movements and of bang-bang control. In the model in [SI, the width and height of the pulses are changed in response to changing task requirements. Here, we suggest that the pulse generator is a motor schema in the sense of Schmidt [13]. In this sense, a schema is a learned relationship between the input and required output vectors of the controller. Thus, the pulse generator has as inputs the desired end position of the joint, the desired movement duration, and the current state of the joint, and as outputs pulse height and width for each control pulse (two pulses are shown in Figure 4A). Each time a movement is performed, a new data point is added to the schema; the system learns by repetition. The controller can generalize by interpolating and extrapolating or by fitting curves to the available data. Our basic controller can thus learn low-level joint control through parametrization of a class of control functions. No continuous trajectories need to be stored. In order to adapt to external forces, though, the system needs another schema in parallel with the pulse generator, which would learn the relationship between the state of the system and the external joint torques (see [15]; [l]). For the diving problem, this will be essential for dealing with the centripetal forces on the legs and arms while somersaulting. To stabilize the system with respect to disturbances, some feedback is also required. As the controller is learning, then, the system can progress from feedback to feedforward control, much as humans do when learning complex new tasks w rotation angle I I lime (seconds) * Coordinating Controller The next higher level of our proposed learning controller is the coordinating controller, shown in Figure 4B. This controller is motivated by work showing that the most important piece of information when learning new, complex tasks is the relative timing between the different movement segments or the phasing between continuous movements (see, for example, [ls]; [14]; [9], Chapter 1). It is also the impression of divers that once they learn the basic skills, such as how to start the rotation and how to pull out, they can learn new dives just by learning how to put the pieces together. The coordinating controller in our model is a schema which learns the required relationship between the movement command (for example, li somersault pike ) and the desired final positions, movement durations, and relative timing of the joints. The controller sends the desired final position and movement duration to each of the single-joint controllers involved, and then sends start signals at 3884

5 -LL!mm"J- Figure 3: Frames from the simulation shown in Figure 2. The graphical human model is from Viewpoint DataLabs. 3885

6 A, desired generator filter,, current state, end position, desired duration torque state B leg joint Tt command generator end pos, duration controller Figure 4: Schematic of a biologically-motivated learning controller. A. Single-joint controller. B. Coordinating controller (shown for a two-joint system). the appropriate timing intervals. 6. Conclusion Controls for systems like the diver have potential applications in dynamic animation and robotics and may also possibly lead to insights into biological control systems themselves. We believe that a hierarchical learning system like the one presented here will be able to generate controls for many-degree-of-freedom systems like humanoids. We also expect that it will be able to generalize from one task to another, for example, from a 1; somersault dive to a 2; somersault dive, without extensive relearning. The system presented here has the additional advantage that it does not need to store a continuous control signal for each task. Since the controls are chosen from a class of very simple functions, the solutions to the control problem may provide insight into the structure of the system as well. We are currently in the process of implementing our proposed design to control threedimensional, 18-degree-of-freedom diver. 7. Acknowledgments We would like to thank Jessica Hodgins for generously providing us with the physical human model used in this work. This research was supported in part by ARO under grants DAAL03-91-GO171 and DAAHOP 94-GO References C. G. Atkeson and J. M Hollerbach. Kinematic features of unrestrained vertical arm movements. Journal of Neuroscience, 5(9): , R. W. Brockett. Systems theory on group manifolds and coset spaces. SIAM Journal of Control, 10( 2) ~ , V. Brooks. The Neural Basis of Motor Control. Oxford University Press, New York, [4] L. Bushnell, D. Tilbury, and S. Sastry. Extended goursat normal forms with applications to nonholonomic motion planning. In Proceedings of the 32nd IEEE Conference on Deciszon and Control, volume 4, pages , [5] C. Frohlich. Do springboard divers violate angular momentum conservation? American Journal of Physics, 47(7): , July [SI G. L. Gottlieb, D. M. Corcos, and G. C. Agarwal. Strategies for the control of voluntary movements with one mechanical degree of freedom. Behavioral and Brazn Sciences, 12: , [7] S. Grillner. Locomotion in vertebrates: central mechanisms and reflex interaction. Physiological Reviews, 55(2): , April [8] J. K. Hodgins and M. H. Raibert. Biped gymnastics. International Journal of Robotics Research, 9(2): , April [9] M. Jeannerod. The Neural and Behavioral Organizataon of Goal-Directed Movements. Clarendon Press, Oxford, [lo] I. V. Kolmanovsky, N. H. McClamroch, and V. T. Coppola. Controllability of a class of nonlinear systems with drift. In Proceedings of the 33rd Conference on Decision and Control, pages , [ll] R. Montgomery. Isoholonomic problems and some applications. Communzcations in Mathematical Physics, 128(3): , [12] S. S. Sastry and R. Montgomery. The structure of optimal controls for a steering problem. In M. Fliess, editor, Nonlinear Control Systems Design 1992, pages Pergamon Press, Oxford, [131 R. A. Schmidt. A schema theory of discrete motor skill learning. Psychological Review, 82(4): , [14] G. Schoner and J. A. S. Kelso. A synergetic theory of environmentally-specified and learned patterns of movement coordination: I. relative phase dynamics. Biological Cybernetics, 58:71-80, R. Shadmehr and F. A. Mussa-Ivaldi. Adaptive representation of dynamics during learning of a motor task. Journal of Neuroscience, 14(5): , May [16] B. Vereijken, H. T. A. Whiting, and W. J. Beek. A dynamical systems approach to skill acquisition. Quarterly Journal of Experimental Psychology, Section A - Human Experimental Psychology, 45(2): , August [17] G. C. Walsh and S. S. Sastry. On reorienting linked rigid bodies using internal motions. IEEE Tkansactions on Robotics and Automation, 11(1): , February [18] W. L. Wooten and J. K. Hodgins. Simulation of human diving. In Proceedings of Graphics Interface '95, pages 1-9, The complete references for this work are given in a full-length paper available from the authors. 3886

Motion Control of a Bipedal Walking Robot

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