- Email: [email protected]
On control design in simulation of human motion
ON CONTROL DESIGN IN SIMULATION OF HUMAN MOTION ...
14th World Congress ofIFAC
C-2a-15-3
Copyright © 1999 IF AC 14th Triennial World Congress. Beiiing. P.R. China
ON CONTROL DESIGN IN SIMULA TION OF HUMAN MOTION
Petko Kiriazov, Hyeongseok Ko
Human Animation Center, School 0/Electrical Engineering. Seoul National University Seoul 151-742, Korea, email: kiriazov(ko)@graphics.snu.ac.kr
Abstract: Efficient simulation of articulated structures is needed in biomechanical engineering, human animation, virtual reality, and robotics. A conceptual framework for control design in constrained and unconstrained motion tasks is developed. In the fIrst case, decentralized sliding-mode controllers with maximum degree of robustness can be designed. In the other case, simple but consistent with Pontryagin' s Maximum principle test functions are employed whose parameters are determined through a fast converging shooting procedure. Besides the analytical guarantees, we verify the effIciency of our approach in dynamic simulation of standing-up motion where both open-loop and cIosedloop controllers are implemented. Copy ri,:{ht © 19991FAC Keywords: dynamics, open/closed loops, controllability, robustness, optimalleaming;
few of them are appropriate for dealing with the complex human dynamics models.
1. INTRODUCTION
Motion simulation of articulated structures such as hwnans is challenging due to the great number of joints and muscles and the complex multibody and actuator dynamics. A very important question is how that large-scale musculo-skeletal systems are kept controllable in dynamic motion tasks. Numerous models have been developed to give insight to the performance criteria and organization of movement control in humans, (Happee, 1992; Alexander, 1997; Dariush et aI, 1998). An optimal control theory is potentially a powerful methodology to evaluate driving forces during movement. Tt can deliver purely predictive results independent of experiments. The predicted time histories of the optimal control forces can be compared with the corresponding measurements and analysed in detail to understand various aspects of multi-joint co-ordination and muscle function. The problem in the aspect of feedforward control is to determine the unique trajectory which yields the best performance based on the selected dynamic model and performance criterion. To solve that problem, there are various control optimization techniques but
Classical optimal control theory, based on Pontryagin's Maximum principle (PMP), gives the necessary conditions for optimality and, if dynamic models are complex, the related two-point boundary value-problems (TPBVP) are very difficult or impossible to solve. As the techniques based on this theory involve gradients of functions, the optimal solutions are very sensitive to modelling errors and may not be global ones. Applying PMP to properly simplified models however, can give us useful data regarding the structure and shape of optimal control laws. For global optimization, dynamic programming is sometimes preferred but it is impractical for dynamic systems with more than two degrees of freedom, mainly due to the computation burden. Control parameter optimization is a unified approach for converting optimal control problems into problems of non linear programming where welldeveloped optimization algorithms exist. It can be applied 10 inverse dynamics models employing test state trajectories (Dariush et ai, 1998), or using test control functions applied to direct dynamics models
1525
Copyright 1999 IF AC
ISBN: 0 08 043248 4
ON CONTROL DESIGN IN SIMULATION OF HUMAN MOTION ...
14th World Congress oflFAC
complex dynamic movement. Finally, some concluding remarks are given and directions for ongoing research are outlined.
(Goh and Teo, 1988), (Pandy et ai, 1992). With inverse dynamics methods, a specific problem is how to defme the test state trajectories because the optimal control laws are, in principle, bang-bang (Kirk, 1970). Having experimental data about the kinematics however, the inverse-dynamics method can be efficient, especially, when complex force constraints (e.g. strength, static and dynamic balancing conditions), are to be satisfied (Ko, 1994).
2. CONCEPTS IN CONTROL DESIGN
A) Simulating human motion means that we have to deal with large-scale, optimal control problems that are difficult or impossible to efficiently solve when full dynamic models are used. • Trade-off is always needed between full dynamic modelling, precise parameter ickntification and optimal system performance
A common problem with most dynamics-based as well as neural nets techniques, is the great number of decision parameters. When applying gradient-based algorithms to large-scale optimization problems, it is very likely to get trapped in local suboptima and we may not be able to find even feasible solutions of the given TPBVP.
B) Finding a proper control structure, Le., input-
output pairs, is a problem of principal meaning for a controlled system (Yeung, 1991; Skogestad and Postlethwaithe, 1996). For the controllability, couplings between single input-output pairs have to be reduced as much as possible. Optimal input-output assignments can be done using appropriate dynamic models and design criteria (Kiriazov, 1994; Kiriazov and Schiehlen, 1997). • Control structure design with most appropriate input-output pairing;
The direct dynamics, control synthesis methods have the capability to cope with systems characterised by control as well as dynamics discontinuities. The latter discontinuities are inherent in human locomotion where, in different motion phases, one has to deal with dynamics models having different structures. And the problem here, is how to parameterise the test control functions and what is the minimum number of control parameters. This problem is solved in the cases ofa manipulator (Marinov and Kiriazov, 1991), and a five-link biped mechanism (Kiriazov and Schiehlen, 1997). Feedback controllers are needed for posture or trajectory tracking stabilization. A feedback control design technique based on necessary and sufficient conditions for robust decentralized controllability is developed in (Kiriazov, 1994; Kiriazov et ai, 1997) . For robotic systems satisfying a very weak condition on the transfer matrix, sliding-mode controllers with maximum degree of robustness can be designed. In simulating, e.g., standing-up, walking, running, constant orientation of the trunk. is to be maintained, along with perfonning optimal point-to-point movements with the legs. It means that, for most locomotion tasks, both open-loop and closed-loop controls have to be applied. Although there are guarantees for the feasibility of our methods for open- and closed-loop control design, it is interesting to see the dynamic perfonnance when both type of controllers are applied at the same time.
C) In voluntary movements, where no space-time or force-time constraints are imposed, it is natural to assume that such movements are controlled in an open-loop manner. Control learning methods, that optimize the performance through repeated forward dynamics simulation and motion evaluation, are to be applied. They should have the following features: a) guarantees for existence of feasible solutions; b); motion synthesis done with as small as possible number of decision parameters; c) the optimization procedure converges with minimum number of trials. • Open-loop control for unconstrained (free) pOint-to-point movements; control synthesis by learning; D) Feedback controllers are to be applied in case of
constrained motion, e.g., for posture or trajectory tracking stabilization; They have to be robust and optimal as the feedback controllers of humans are: • Robust closed-loop control for movements with space-time or force-time constraints;
3. MATHEMATICAL BACKGROUND
The paper is organised as follows. In the next section, we present our concepts in the control design and the necessary mathematical background is briefly described in Section 3. Next, we refer to some physiological results which are very useful for the control structure design and when defining the test control functions. In Section 5, standing-up motion is considered as an illustrative example of combined open-loop and closed-loop control design for a
3.1. Dynamic Modelling
Applying the multi-body system approach (Schiehlen, 1990), and the Lagrange formalism, the dynamic perfonnance of skeletal systems can be, in general, described, by the following system of differential equations
1526
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ON CONTROL DESIGN IN SIMULATION OF HUMAN MOTION...
q=
M(q)-l(BF - C(q,iJ.} + g(q»,
14th World Congress ofTFAC
and bounds of the test control functions. Simple linear-spline control functions of ''bang-pause-bang'' or "bang-slope-bang" types can be used for IlTStorder optimization. Such control laws are consistent with Pontryagin's Maximum principle for linear-incontrol dynamic systems and·· time/energy performance criteria. Parameters Pi' i = 1, ... ,n, n > 1 , describing the test control functions, like switch times, slopes, or pause lengths, mostly influence the reached position and the performance criteria.
(1)
where q is the vector of I generalised co-ordinates (e.g., links' rotation angles or joint angles), M(q) is the inertia matrix, C(q,q) is the vector of velocity forces, g(q) stands for friction and gravitation forces, F is the vector of control forces; matrix B represents the control force distribution, and A
= M(q) -1 B
is the control transfer matrix (TM);
Define the most relevant input-output pairs: For simplicity, we assume that the end-point velocities
As the objective in this section is to present the main features of our open- and closed-loop control design methods, the actuator (muscle/tendon) dynamics will not be addressed here. Such dynamic models can be found in (Hill, 1938; Winter, 1990; Dariush et aI, 1998). That how detailed actuator dynamics models are to be employed should depend on the purpose of human motion simulation: human animation, virtual reality, biomechanical engineering, or robotics. In our numerical considerations, we use the well-known Hill model.
vO , vi are zero and the generalised co-ordinates q are monotonous functions of time during the trial movements. We deftne I controlled outputs to be
Yi = Yi(P) = qj(t(), where if,(tf) = O. Then for each controlled output we assign a control input which mostly influences it. That can be done by a sensitivity analysis using a dynamic model (1) or by engineering intuition. The input-output pairs (Pi'Yi) will be most relevant regarding the controllability if the couplings between these single-input singleoutput subsystems are minimal.
A point-to-point motion task means that the dynamic system (l) has to be transferred from a given initial state {qO, v O} to a required Imal state {qf, v r }. The
Solve shooting equations and perform control parameter optimization. With the above input-output pairing, the given TPBVP is transformed into a
following TPBVP is to be solved
system of shooting equations p
=> qC with decision
variables Pj' i = 1, ... ,1 which is the first level of our control synthesis procedure. At the next level, the other parameters Ph i = 1+ l, ... ,n are varied to optimize J. In principle, one should Imd the minimum time movement fIrst, and then, with given movement execution time, minimise the energy cost.
such that a performance criterion J(q,4,F) (time/energy) is optimised and a set of control and state constraints are satisfied. The control constraints are due to the limited power resources, strength bounds and comfort demands, or due to a task for force interaction with the environment. If we have to pay attention to space or force constraints during motion, we design and apply closed-loop, trajectory tracking controllers. Otherwise, we use open-loop control, and its design is next described.
Following this approach, we have the chance to fmd a satisfactory suboptimal solution with minimum number of control parameters. Existence of feasible solutions and convergence of bisection algorithms , Imding them can be guaranteed, provided that a margin of independent parameter controllability (IPC) can be found. The margin (parallelepiped) of IPC is defmed by:
3.2. Open-loop (feedforwardj control design Unconstrained point-to-point motion tasks are considered as free Imal-time optimal control problems. At the foundation of our and other studies on human motion control lies the notion that such locomotion patterns are, in general, optimized in time, control effort, or energy expenditure (Happee, 1992; Alexander, 1997). The control functions in simulation of such tasks can be efficiently optimized applying the direct-dynamics control parameter approach (Marinov and Kiriazov, 1991; Kiriazov and Schiehlen, 1997), having the following main steps:
De! 1: P
= {p:
Pi e[pi;pt], i
= 1, ... ,/}
is said to
be a margin of IPC iff, for any pair of antipodal points p and P on the boundary of P, there exists an integer i such that (y, (e) - q() (Yj (p) - q() <0 . The existence of feasible solutions is guaranteed by a Fixed-Point theorem (Kiriazov, 1995), that is an extension of the well-known Borsuk-Ulam theorem for antipodal maps (Todd and Wright, 1980). The
Choose a set of appropriate test control fUnctions: The term "appropriate" concerns the structure, shape,
1527
Copyright 1999 IF AC
ISBN: 008 0432484
ON CONTROL DESIGN IN SIMULATION OF HUMAN MOTION...
14th World Congress ofTFAC
segments as well as the muscles spanning more than one joint. We have to take this fact into account whether the movement is voluntary or intended.
dynamic system (1) has an IPC-margin if a set of reasonable control force dominance conditions are satisfied (Kiriazov and Schiehlen, 1997; Marinov and Kiriazov, 1991).
4.1. About Control Structure Design in Voluntary, Unconstrained Movements
3.3. Closed-loop (feedbackj control design
Bemstein (1967) has suggested that synergies among the muscles are established by the nervous system to reduce the number of degrees of freedom that it must independently control. This fact is in accordance with our understanding of decentralization at the lowest level of the control subsystem.
The following reduced model of error dynamics can be used for the purpose of closed-loop control design
e= A(q)F+d
(3)
where e = q - q rer, and the vector d stands here for the model uncertainty and external disturbances. As a measure of tracking precision, we take the absolute value of s = e +;. e, A > 0 .
Further, the patterns of electromyographic activity for a point-to-point movement of a limb are, in general, triphasic (acceleration-pause-deceleration): initial burst of activity in agonist muscles, followed by a pause of agonist activity, and then, activity in antagonist muscles (Gottlieb et al., 1996). The optimal in time/energy, "bang-pause-bang" control laws, mentioned in the previous section, can be considered as a fIrst approximation to such muscle force functions.
We consider decentralized controllers which means that each stabilizing control force F-! depends solely . on the corresponding controlled output si. We use sliding-mode controllers of saturation type (Kiriazov et al, 1997), thus avoiding chattering effects. De! 2: A decentralized controller is robust against random disturbances d with known upper bounds d+ if it gets the local subsystem state (4, q) at each joint to track the desired state (qref ,qref) with maximum allowable absolute values s + of errors s. Theorem (Kiriazov, 1994): Necessary and sufficient condition for a dynamic system (1) to be robustly controlled by a decentralized controller is that matrix A be generalized diagonally dominant (GOD). When matrix A is GOD, there always exists, (Lunze, 1992), a positive vector F+ of control function magnitudes solving the following system Ai/Ft -
.I J"*
.IAijIFj+ = dt, i = 1, ... ,n
Furthermore, there is a linear relationship between the two main joint torques found in ann motion experiments with varied load, direction, speed, angular excursion. Both torques go through extrema and zero crossings almost simultaneously (Gottlieb et al., 1996). Excitation signals are parametric ally modified to fit the requirements of different tasks (Bock et ai, 1996). The control signals for human movements are not calculated anew each time; rather, movements are controlled by stored, pertinent forcetime functions, whose parameter values are generated according to the required motion task. The parameters learned by the human are the control magnitudes and switch times (KarnieJ, and Inbar, 1997), and they are the most important parameters in our open loop control synthesis procedure. Thus in our direct-search control learning approach, test control functions with shapes/structures similar to that of the human control functions are used.
(4)
I
Eqs. (4) present optimal trade-off relations between the bounds of model uncertainties (and external disturbances) and the control force limits. The robustness of so-designed controllers is proven using Lyapunov theory (Kiriazov, 1994).
4. PHYSIOLOGICAL BACKGROUND Movement of any segment in the human body requires activation of muscles across several joints, even if the intent is to move a single joint only. This is due to the mechanical interactions among the body
4.2. Feedback Controllersfor Intended (Space or Force Constrained) Movements The feedback control strategy in this case characterizes the spring-like behaviour of the neuromuscular system in such a way that whenever the limb is disturbed, it will return to its equilibrium posture (Rogan, 1990). The control system stiffness is an important quantity in studies of movement control (Latash, 1993); Dariush et ai, 1998), because
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Copyright 1999 IF AC
ISBN: 008 0432484
ON CONTROL DESIGN IN SIMULATION OF HUMAN MOTION ...
14th World Congress ofTFAC
standing-up motion and the animated figure are shown in Fig. 2. The animation produced from the computed result visualises the standing-up motion in actual speed. The motion picture is available at the web site (ftp:llgraphics.snu.ac.kr/pub/data/animation/ standing-up.mov).
its magnitude determines the resistance to external perturbations.
5. CASE STUDY: CONTROL DESIGN FOR STANDING-UP MOTION Besides the analytical guarantees given in Section 3 and the successful applications to robotic systems, we now verify the practicability of our approach, considering a specific daily-life motion task of a human: dynamic standing-up motion. In such a task, we have to apply both open-loop and closed-loop controls at the same time. Along with checking the convergence of the open-loop control synthesis algorithm and the robustness· of the closed-loop controller, it is interesting to see also if a dynamicsbased motion animation looks natural. Such a performance objective is reasonable because the human eye is very sensitive to "errors" and gets disturbed by irregularities in movements that it perceives every day.
shank
[rad/sl trunk
/ ~
thigh
[5]
Fig. I: Velocity time histories In our experiment, human body is modeled as a three-segment (trunk, thigh, and shank), planar linkage, and the ankle is assumed to be hinged to the ground. The dynamics of such an articulated structure is well described in (Winter, 1990). When the human rises from a chair or a static squatting position, the trunk and shank rotations can be considered as space-time constrained motions and, consecutively we have to apply to them closed-loop control. The rotation of the thigh, with the massive trunk hinged to it, is an unconstrained, relatively long-range motion with open-loop controL The structure of the latter control fimction is triphasic and the control force magnitude in such a voluntary movement is defined according to the Hill model: F = -G + 11 (bl vl+c) , where v in our case is taken to be the speed of the thigh rotation (which is proportional to the velocity of shortening in muscle contraction). Varying the control parameters G, b, and c, we can efficiently optimise the movement execution time as well as control energy criteria. In a neighbourhood of the final upward configuration, we apply closed-loop controllers to all the three links, stabilising, in this way, the body at this vertical posture.
------+-
.~.'::-,
Fig. 2: The stick-diagram and the animated figure
6. CONCLUSION AND ONGOING RESEARCH A unified approach for dynamics-based simulation of complex articulated structures, like humans or legged robots, has been proposed. Constrained and unconstrained motion tasks have been considered, and accordingly, two types of control strategies are to be applied: closed-loop and open-loop. The first one assigns robust sliding-mode controllers, the other involves simple, but consistent with Pontryagin's Maximum principle, test control fimctions. Their parameters can be determined through a shooting procedure which converges within minimum number of trial movements.
The standing-up motion simulation was performed with the numerical data given in (Winter, 1990). With specified values for a, b, and c, the open-loop control synthesis, when using the bisection algorithm, converges within 10 trial movements. For the convergence of this control learning procedure, the robustness of the feedback control of the trunk and shank rotations is of great importance. The time histories of the angular velocities of the three links are depicted in Fig. 1. The stick-diagram of the
Despite the analytical guarantees, the feasibility of our approach has been verified on a full dynamic model of the standing-up task in which both, openloop and closed-loop, controls are applied. The approach can also be efficiently applied in simulating
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Copyright 1999 IF AC
ISBN: 008 0432484
14th World Congress oflFAC
ON CONTROL DESIGN IN SIMULATION OF HUMAN MOTION ...
other tasks like walking, running, and grasping which is a subject of ongoing research. Following the proposed approach, actual problems in biorhechanical engineering, human animation, virtual reality, and robotics can be solved. For example, by introducing realistic dynamics models of the muscle activation, new control techniques in functional neuromuscular stimulation of neuroprosthetic systems can be developed. Knowing how to control certain classes of dynamic motions, we can provide animators/robot operators with a minimum set of movement commands and parameters which completely control the animated figures/robots. The proposed techniques can learn hwnan models or humanoid robots how to locomote automatically, in a way that is inspired by the remarkable ability of real humans to acquire locomotion skills through action and perception.
Acknowledgement: The flllancial support from the Creative Research Initiatives of the Korean Ministry of Science and Technology is gratefully acknowledged.
REFERENCES Alexander, R. McN, (1997). A minimum energy cost hypothesis for human arm trajectories. BiolOgical Cybernetics, 76(2), 97-lO5 . Bernstein, N. (1967). Coordination and Regulation of Movemeni. Oxford: Pergamon Press. Bock 0., D'Eleuterio, G., Lipitkas, J. and Grodski, J. (1996). Parametric motor control: a new approach to the control of point-to-point manipulator movements. J. Robotic Systems, 13(1),35-40. Dariush B., Pamianpour, M, and Hemami, H. (1998) . Stability and a control strategy of a multilink musculoskeletal model with applications in FES. IEEE Transactions on Biomedical Engineering, Vol. 45, No.l, 3-14. Goh, C . and Teo, K (1988) Control parameterization: a unified approach to optimal control problems with general constraints. Automatica, 24: 3 -18. Gottlieb, G., Song, Q., Hong, D., Almeida, G., and Corcos, D. (1996). Coordinating movement at two joints: a principle of linear covariance. J Neurophysiology, 75(5), 1760-1764. Happee, R. (1992). Time-optimality in the control of human movements. BioI. Cybernetics, 66, 357366. Hill, A. (1938). The heat of shortening and dynamic constants of muscle. Proc. Royal Soc. B , 126, 136-195. Kamiel, A. and G. Inbar. (1997). A model for learning human reaching movements . Biological Cybernetics, Val. 77(3),173-183.
Hogan, N . (1990). Mechanical impedance of single and multi-articular systems. In Biomechanics and Movement Organization, I. Winters and S. Woo Eds., Springer-Verlag, ch. 9,149-164. Kiriazov, P. (1995). Controllability of a class of dynamic systems. ZAMM, Vol. 75 SI, 85-86. Kiriazov, P. (1994). A necessary and sufficient condition for robust decentralized controllability of robot manipulators. In Proceedings American Control Conference, Baltimore, MD, 2285-2287. Kiriazov, P., E. Kreuzer, and F. Pinto (1997). Robust feedback stabilization of underwater robotic vehicles . Journal of Robotics and Autonomous Systems. Vot. 21, 415-423. Kiriazov P. and W. Schieh1en . (1997). On directsearch optimization of biped walking. In CISM Courses, Vol. 381, Eds. A. Morecki et al., Springer-Wien-NewYork, 134-140. Kirk, D.E. (1970). Optimal Control Theory: an introduction, Prentice Hall. Ko, H. (1994). Kinematic and Dynamic Techniques for Analyzing, Predicting, and Animating Human Locomotion. Ph.D. Thesis, University of Pennsylvania, USA. Latash, M.L. (1993). Control of Human Movement: Human Kinetics . Chanpaign, IL. Lunze, J. (1992). Feedback Control of Large-Scale Systems. Prentice Hall, UK. Marinov P. and Kiriazov, P . (1992). Point-to-point motion of robotic manipulators: dynamics, control synthesis and optimisation. In: IFAC Symposia Series, Robot Control 1991, Eds. l.Troch & K.Desoyer, 149-152. Pandy, M.G ., Anderson, F.e., and D.G. Hull. (1992). A parameter optimization approach for the optimal control of large-scale musculoskeletal systems. Transactions of ASME, Journal of Biomechanical Engineering., 114. Schiehlen, W. (ed.) (1990) Multibody Systems Handbook. Berlin: Springer. Skogestad, S. and Postlethwaithe, I. (1996). Multivariable Feedback Control: Analysis and Design, Wiley. Todd, M.I. and Wright, A.H. (1980) A variabledimension symplicial algorithm for fixed-point theorems, Numerical Functional Analysis and Optimization, 155-186. Winter, D. (1990). Biomechanics and Motor Control ofHuman Movement, lohnWiley&Sons, Inc. Yeung, L.F. (199\). Optimal input-output variable assignments for multivariable systems, Automatica, Vol. 27, No.4, 733-738.
1530
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
C-2a-15-3
Copyright © 1999 IF AC 14th Triennial World Congress. Beiiing. P.R. China
ON CONTROL DESIGN IN SIMULA TION OF HUMAN MOTION
Petko Kiriazov, Hyeongseok Ko
Human Animation Center, School 0/Electrical Engineering. Seoul National University Seoul 151-742, Korea, email: kiriazov(ko)@graphics.snu.ac.kr
Abstract: Efficient simulation of articulated structures is needed in biomechanical engineering, human animation, virtual reality, and robotics. A conceptual framework for control design in constrained and unconstrained motion tasks is developed. In the fIrst case, decentralized sliding-mode controllers with maximum degree of robustness can be designed. In the other case, simple but consistent with Pontryagin' s Maximum principle test functions are employed whose parameters are determined through a fast converging shooting procedure. Besides the analytical guarantees, we verify the effIciency of our approach in dynamic simulation of standing-up motion where both open-loop and cIosedloop controllers are implemented. Copy ri,:{ht © 19991FAC Keywords: dynamics, open/closed loops, controllability, robustness, optimalleaming;
few of them are appropriate for dealing with the complex human dynamics models.
1. INTRODUCTION
Motion simulation of articulated structures such as hwnans is challenging due to the great number of joints and muscles and the complex multibody and actuator dynamics. A very important question is how that large-scale musculo-skeletal systems are kept controllable in dynamic motion tasks. Numerous models have been developed to give insight to the performance criteria and organization of movement control in humans, (Happee, 1992; Alexander, 1997; Dariush et aI, 1998). An optimal control theory is potentially a powerful methodology to evaluate driving forces during movement. Tt can deliver purely predictive results independent of experiments. The predicted time histories of the optimal control forces can be compared with the corresponding measurements and analysed in detail to understand various aspects of multi-joint co-ordination and muscle function. The problem in the aspect of feedforward control is to determine the unique trajectory which yields the best performance based on the selected dynamic model and performance criterion. To solve that problem, there are various control optimization techniques but
Classical optimal control theory, based on Pontryagin's Maximum principle (PMP), gives the necessary conditions for optimality and, if dynamic models are complex, the related two-point boundary value-problems (TPBVP) are very difficult or impossible to solve. As the techniques based on this theory involve gradients of functions, the optimal solutions are very sensitive to modelling errors and may not be global ones. Applying PMP to properly simplified models however, can give us useful data regarding the structure and shape of optimal control laws. For global optimization, dynamic programming is sometimes preferred but it is impractical for dynamic systems with more than two degrees of freedom, mainly due to the computation burden. Control parameter optimization is a unified approach for converting optimal control problems into problems of non linear programming where welldeveloped optimization algorithms exist. It can be applied 10 inverse dynamics models employing test state trajectories (Dariush et ai, 1998), or using test control functions applied to direct dynamics models
1525
Copyright 1999 IF AC
ISBN: 0 08 043248 4
ON CONTROL DESIGN IN SIMULATION OF HUMAN MOTION ...
14th World Congress oflFAC
complex dynamic movement. Finally, some concluding remarks are given and directions for ongoing research are outlined.
(Goh and Teo, 1988), (Pandy et ai, 1992). With inverse dynamics methods, a specific problem is how to defme the test state trajectories because the optimal control laws are, in principle, bang-bang (Kirk, 1970). Having experimental data about the kinematics however, the inverse-dynamics method can be efficient, especially, when complex force constraints (e.g. strength, static and dynamic balancing conditions), are to be satisfied (Ko, 1994).
2. CONCEPTS IN CONTROL DESIGN
A) Simulating human motion means that we have to deal with large-scale, optimal control problems that are difficult or impossible to efficiently solve when full dynamic models are used. • Trade-off is always needed between full dynamic modelling, precise parameter ickntification and optimal system performance
A common problem with most dynamics-based as well as neural nets techniques, is the great number of decision parameters. When applying gradient-based algorithms to large-scale optimization problems, it is very likely to get trapped in local suboptima and we may not be able to find even feasible solutions of the given TPBVP.
B) Finding a proper control structure, Le., input-
output pairs, is a problem of principal meaning for a controlled system (Yeung, 1991; Skogestad and Postlethwaithe, 1996). For the controllability, couplings between single input-output pairs have to be reduced as much as possible. Optimal input-output assignments can be done using appropriate dynamic models and design criteria (Kiriazov, 1994; Kiriazov and Schiehlen, 1997). • Control structure design with most appropriate input-output pairing;
The direct dynamics, control synthesis methods have the capability to cope with systems characterised by control as well as dynamics discontinuities. The latter discontinuities are inherent in human locomotion where, in different motion phases, one has to deal with dynamics models having different structures. And the problem here, is how to parameterise the test control functions and what is the minimum number of control parameters. This problem is solved in the cases ofa manipulator (Marinov and Kiriazov, 1991), and a five-link biped mechanism (Kiriazov and Schiehlen, 1997). Feedback controllers are needed for posture or trajectory tracking stabilization. A feedback control design technique based on necessary and sufficient conditions for robust decentralized controllability is developed in (Kiriazov, 1994; Kiriazov et ai, 1997) . For robotic systems satisfying a very weak condition on the transfer matrix, sliding-mode controllers with maximum degree of robustness can be designed. In simulating, e.g., standing-up, walking, running, constant orientation of the trunk. is to be maintained, along with perfonning optimal point-to-point movements with the legs. It means that, for most locomotion tasks, both open-loop and closed-loop controls have to be applied. Although there are guarantees for the feasibility of our methods for open- and closed-loop control design, it is interesting to see the dynamic perfonnance when both type of controllers are applied at the same time.
C) In voluntary movements, where no space-time or force-time constraints are imposed, it is natural to assume that such movements are controlled in an open-loop manner. Control learning methods, that optimize the performance through repeated forward dynamics simulation and motion evaluation, are to be applied. They should have the following features: a) guarantees for existence of feasible solutions; b); motion synthesis done with as small as possible number of decision parameters; c) the optimization procedure converges with minimum number of trials. • Open-loop control for unconstrained (free) pOint-to-point movements; control synthesis by learning; D) Feedback controllers are to be applied in case of
constrained motion, e.g., for posture or trajectory tracking stabilization; They have to be robust and optimal as the feedback controllers of humans are: • Robust closed-loop control for movements with space-time or force-time constraints;
3. MATHEMATICAL BACKGROUND
The paper is organised as follows. In the next section, we present our concepts in the control design and the necessary mathematical background is briefly described in Section 3. Next, we refer to some physiological results which are very useful for the control structure design and when defining the test control functions. In Section 5, standing-up motion is considered as an illustrative example of combined open-loop and closed-loop control design for a
3.1. Dynamic Modelling
Applying the multi-body system approach (Schiehlen, 1990), and the Lagrange formalism, the dynamic perfonnance of skeletal systems can be, in general, described, by the following system of differential equations
1526
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ON CONTROL DESIGN IN SIMULATION OF HUMAN MOTION...
q=
M(q)-l(BF - C(q,iJ.} + g(q»,
14th World Congress ofTFAC
and bounds of the test control functions. Simple linear-spline control functions of ''bang-pause-bang'' or "bang-slope-bang" types can be used for IlTStorder optimization. Such control laws are consistent with Pontryagin's Maximum principle for linear-incontrol dynamic systems and·· time/energy performance criteria. Parameters Pi' i = 1, ... ,n, n > 1 , describing the test control functions, like switch times, slopes, or pause lengths, mostly influence the reached position and the performance criteria.
(1)
where q is the vector of I generalised co-ordinates (e.g., links' rotation angles or joint angles), M(q) is the inertia matrix, C(q,q) is the vector of velocity forces, g(q) stands for friction and gravitation forces, F is the vector of control forces; matrix B represents the control force distribution, and A
= M(q) -1 B
is the control transfer matrix (TM);
Define the most relevant input-output pairs: For simplicity, we assume that the end-point velocities
As the objective in this section is to present the main features of our open- and closed-loop control design methods, the actuator (muscle/tendon) dynamics will not be addressed here. Such dynamic models can be found in (Hill, 1938; Winter, 1990; Dariush et aI, 1998). That how detailed actuator dynamics models are to be employed should depend on the purpose of human motion simulation: human animation, virtual reality, biomechanical engineering, or robotics. In our numerical considerations, we use the well-known Hill model.
vO , vi are zero and the generalised co-ordinates q are monotonous functions of time during the trial movements. We deftne I controlled outputs to be
Yi = Yi(P) = qj(t(), where if,(tf) = O. Then for each controlled output we assign a control input which mostly influences it. That can be done by a sensitivity analysis using a dynamic model (1) or by engineering intuition. The input-output pairs (Pi'Yi) will be most relevant regarding the controllability if the couplings between these single-input singleoutput subsystems are minimal.
A point-to-point motion task means that the dynamic system (l) has to be transferred from a given initial state {qO, v O} to a required Imal state {qf, v r }. The
Solve shooting equations and perform control parameter optimization. With the above input-output pairing, the given TPBVP is transformed into a
following TPBVP is to be solved
system of shooting equations p
=> qC with decision
variables Pj' i = 1, ... ,1 which is the first level of our control synthesis procedure. At the next level, the other parameters Ph i = 1+ l, ... ,n are varied to optimize J. In principle, one should Imd the minimum time movement fIrst, and then, with given movement execution time, minimise the energy cost.
such that a performance criterion J(q,4,F) (time/energy) is optimised and a set of control and state constraints are satisfied. The control constraints are due to the limited power resources, strength bounds and comfort demands, or due to a task for force interaction with the environment. If we have to pay attention to space or force constraints during motion, we design and apply closed-loop, trajectory tracking controllers. Otherwise, we use open-loop control, and its design is next described.
Following this approach, we have the chance to fmd a satisfactory suboptimal solution with minimum number of control parameters. Existence of feasible solutions and convergence of bisection algorithms , Imding them can be guaranteed, provided that a margin of independent parameter controllability (IPC) can be found. The margin (parallelepiped) of IPC is defmed by:
3.2. Open-loop (feedforwardj control design Unconstrained point-to-point motion tasks are considered as free Imal-time optimal control problems. At the foundation of our and other studies on human motion control lies the notion that such locomotion patterns are, in general, optimized in time, control effort, or energy expenditure (Happee, 1992; Alexander, 1997). The control functions in simulation of such tasks can be efficiently optimized applying the direct-dynamics control parameter approach (Marinov and Kiriazov, 1991; Kiriazov and Schiehlen, 1997), having the following main steps:
De! 1: P
= {p:
Pi e[pi;pt], i
= 1, ... ,/}
is said to
be a margin of IPC iff, for any pair of antipodal points p and P on the boundary of P, there exists an integer i such that (y, (e) - q() (Yj (p) - q() <0 . The existence of feasible solutions is guaranteed by a Fixed-Point theorem (Kiriazov, 1995), that is an extension of the well-known Borsuk-Ulam theorem for antipodal maps (Todd and Wright, 1980). The
Choose a set of appropriate test control fUnctions: The term "appropriate" concerns the structure, shape,
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segments as well as the muscles spanning more than one joint. We have to take this fact into account whether the movement is voluntary or intended.
dynamic system (1) has an IPC-margin if a set of reasonable control force dominance conditions are satisfied (Kiriazov and Schiehlen, 1997; Marinov and Kiriazov, 1991).
4.1. About Control Structure Design in Voluntary, Unconstrained Movements
3.3. Closed-loop (feedbackj control design
Bemstein (1967) has suggested that synergies among the muscles are established by the nervous system to reduce the number of degrees of freedom that it must independently control. This fact is in accordance with our understanding of decentralization at the lowest level of the control subsystem.
The following reduced model of error dynamics can be used for the purpose of closed-loop control design
e= A(q)F+d
(3)
where e = q - q rer, and the vector d stands here for the model uncertainty and external disturbances. As a measure of tracking precision, we take the absolute value of s = e +;. e, A > 0 .
Further, the patterns of electromyographic activity for a point-to-point movement of a limb are, in general, triphasic (acceleration-pause-deceleration): initial burst of activity in agonist muscles, followed by a pause of agonist activity, and then, activity in antagonist muscles (Gottlieb et al., 1996). The optimal in time/energy, "bang-pause-bang" control laws, mentioned in the previous section, can be considered as a fIrst approximation to such muscle force functions.
We consider decentralized controllers which means that each stabilizing control force F-! depends solely . on the corresponding controlled output si. We use sliding-mode controllers of saturation type (Kiriazov et al, 1997), thus avoiding chattering effects. De! 2: A decentralized controller is robust against random disturbances d with known upper bounds d+ if it gets the local subsystem state (4, q) at each joint to track the desired state (qref ,qref) with maximum allowable absolute values s + of errors s. Theorem (Kiriazov, 1994): Necessary and sufficient condition for a dynamic system (1) to be robustly controlled by a decentralized controller is that matrix A be generalized diagonally dominant (GOD). When matrix A is GOD, there always exists, (Lunze, 1992), a positive vector F+ of control function magnitudes solving the following system Ai/Ft -
.I J"*
.IAijIFj+ = dt, i = 1, ... ,n
Furthermore, there is a linear relationship between the two main joint torques found in ann motion experiments with varied load, direction, speed, angular excursion. Both torques go through extrema and zero crossings almost simultaneously (Gottlieb et al., 1996). Excitation signals are parametric ally modified to fit the requirements of different tasks (Bock et ai, 1996). The control signals for human movements are not calculated anew each time; rather, movements are controlled by stored, pertinent forcetime functions, whose parameter values are generated according to the required motion task. The parameters learned by the human are the control magnitudes and switch times (KarnieJ, and Inbar, 1997), and they are the most important parameters in our open loop control synthesis procedure. Thus in our direct-search control learning approach, test control functions with shapes/structures similar to that of the human control functions are used.
(4)
I
Eqs. (4) present optimal trade-off relations between the bounds of model uncertainties (and external disturbances) and the control force limits. The robustness of so-designed controllers is proven using Lyapunov theory (Kiriazov, 1994).
4. PHYSIOLOGICAL BACKGROUND Movement of any segment in the human body requires activation of muscles across several joints, even if the intent is to move a single joint only. This is due to the mechanical interactions among the body
4.2. Feedback Controllersfor Intended (Space or Force Constrained) Movements The feedback control strategy in this case characterizes the spring-like behaviour of the neuromuscular system in such a way that whenever the limb is disturbed, it will return to its equilibrium posture (Rogan, 1990). The control system stiffness is an important quantity in studies of movement control (Latash, 1993); Dariush et ai, 1998), because
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standing-up motion and the animated figure are shown in Fig. 2. The animation produced from the computed result visualises the standing-up motion in actual speed. The motion picture is available at the web site (ftp:llgraphics.snu.ac.kr/pub/data/animation/ standing-up.mov).
its magnitude determines the resistance to external perturbations.
5. CASE STUDY: CONTROL DESIGN FOR STANDING-UP MOTION Besides the analytical guarantees given in Section 3 and the successful applications to robotic systems, we now verify the practicability of our approach, considering a specific daily-life motion task of a human: dynamic standing-up motion. In such a task, we have to apply both open-loop and closed-loop controls at the same time. Along with checking the convergence of the open-loop control synthesis algorithm and the robustness· of the closed-loop controller, it is interesting to see also if a dynamicsbased motion animation looks natural. Such a performance objective is reasonable because the human eye is very sensitive to "errors" and gets disturbed by irregularities in movements that it perceives every day.
shank
[rad/sl trunk
/ ~
thigh
[5]
Fig. I: Velocity time histories In our experiment, human body is modeled as a three-segment (trunk, thigh, and shank), planar linkage, and the ankle is assumed to be hinged to the ground. The dynamics of such an articulated structure is well described in (Winter, 1990). When the human rises from a chair or a static squatting position, the trunk and shank rotations can be considered as space-time constrained motions and, consecutively we have to apply to them closed-loop control. The rotation of the thigh, with the massive trunk hinged to it, is an unconstrained, relatively long-range motion with open-loop controL The structure of the latter control fimction is triphasic and the control force magnitude in such a voluntary movement is defined according to the Hill model: F = -G + 11 (bl vl+c) , where v in our case is taken to be the speed of the thigh rotation (which is proportional to the velocity of shortening in muscle contraction). Varying the control parameters G, b, and c, we can efficiently optimise the movement execution time as well as control energy criteria. In a neighbourhood of the final upward configuration, we apply closed-loop controllers to all the three links, stabilising, in this way, the body at this vertical posture.
------+-
.~.'::-,
Fig. 2: The stick-diagram and the animated figure
6. CONCLUSION AND ONGOING RESEARCH A unified approach for dynamics-based simulation of complex articulated structures, like humans or legged robots, has been proposed. Constrained and unconstrained motion tasks have been considered, and accordingly, two types of control strategies are to be applied: closed-loop and open-loop. The first one assigns robust sliding-mode controllers, the other involves simple, but consistent with Pontryagin's Maximum principle, test control fimctions. Their parameters can be determined through a shooting procedure which converges within minimum number of trial movements.
The standing-up motion simulation was performed with the numerical data given in (Winter, 1990). With specified values for a, b, and c, the open-loop control synthesis, when using the bisection algorithm, converges within 10 trial movements. For the convergence of this control learning procedure, the robustness of the feedback control of the trunk and shank rotations is of great importance. The time histories of the angular velocities of the three links are depicted in Fig. 1. The stick-diagram of the
Despite the analytical guarantees, the feasibility of our approach has been verified on a full dynamic model of the standing-up task in which both, openloop and closed-loop, controls are applied. The approach can also be efficiently applied in simulating
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other tasks like walking, running, and grasping which is a subject of ongoing research. Following the proposed approach, actual problems in biorhechanical engineering, human animation, virtual reality, and robotics can be solved. For example, by introducing realistic dynamics models of the muscle activation, new control techniques in functional neuromuscular stimulation of neuroprosthetic systems can be developed. Knowing how to control certain classes of dynamic motions, we can provide animators/robot operators with a minimum set of movement commands and parameters which completely control the animated figures/robots. The proposed techniques can learn hwnan models or humanoid robots how to locomote automatically, in a way that is inspired by the remarkable ability of real humans to acquire locomotion skills through action and perception.
Acknowledgement: The flllancial support from the Creative Research Initiatives of the Korean Ministry of Science and Technology is gratefully acknowledged.
REFERENCES Alexander, R. McN, (1997). A minimum energy cost hypothesis for human arm trajectories. BiolOgical Cybernetics, 76(2), 97-lO5 . Bernstein, N. (1967). Coordination and Regulation of Movemeni. Oxford: Pergamon Press. Bock 0., D'Eleuterio, G., Lipitkas, J. and Grodski, J. (1996). Parametric motor control: a new approach to the control of point-to-point manipulator movements. J. Robotic Systems, 13(1),35-40. Dariush B., Pamianpour, M, and Hemami, H. (1998) . Stability and a control strategy of a multilink musculoskeletal model with applications in FES. IEEE Transactions on Biomedical Engineering, Vol. 45, No.l, 3-14. Goh, C . and Teo, K (1988) Control parameterization: a unified approach to optimal control problems with general constraints. Automatica, 24: 3 -18. Gottlieb, G., Song, Q., Hong, D., Almeida, G., and Corcos, D. (1996). Coordinating movement at two joints: a principle of linear covariance. J Neurophysiology, 75(5), 1760-1764. Happee, R. (1992). Time-optimality in the control of human movements. BioI. Cybernetics, 66, 357366. Hill, A. (1938). The heat of shortening and dynamic constants of muscle. Proc. Royal Soc. B , 126, 136-195. Kamiel, A. and G. Inbar. (1997). A model for learning human reaching movements . Biological Cybernetics, Val. 77(3),173-183.
Hogan, N . (1990). Mechanical impedance of single and multi-articular systems. In Biomechanics and Movement Organization, I. Winters and S. Woo Eds., Springer-Verlag, ch. 9,149-164. Kiriazov, P. (1995). Controllability of a class of dynamic systems. ZAMM, Vol. 75 SI, 85-86. Kiriazov, P. (1994). A necessary and sufficient condition for robust decentralized controllability of robot manipulators. In Proceedings American Control Conference, Baltimore, MD, 2285-2287. Kiriazov, P., E. Kreuzer, and F. Pinto (1997). Robust feedback stabilization of underwater robotic vehicles . Journal of Robotics and Autonomous Systems. Vot. 21, 415-423. Kiriazov P. and W. Schieh1en . (1997). On directsearch optimization of biped walking. In CISM Courses, Vol. 381, Eds. A. Morecki et al., Springer-Wien-NewYork, 134-140. Kirk, D.E. (1970). Optimal Control Theory: an introduction, Prentice Hall. Ko, H. (1994). Kinematic and Dynamic Techniques for Analyzing, Predicting, and Animating Human Locomotion. Ph.D. Thesis, University of Pennsylvania, USA. Latash, M.L. (1993). Control of Human Movement: Human Kinetics . Chanpaign, IL. Lunze, J. (1992). Feedback Control of Large-Scale Systems. Prentice Hall, UK. Marinov P. and Kiriazov, P . (1992). Point-to-point motion of robotic manipulators: dynamics, control synthesis and optimisation. In: IFAC Symposia Series, Robot Control 1991, Eds. l.Troch & K.Desoyer, 149-152. Pandy, M.G ., Anderson, F.e., and D.G. Hull. (1992). A parameter optimization approach for the optimal control of large-scale musculoskeletal systems. Transactions of ASME, Journal of Biomechanical Engineering., 114. Schiehlen, W. (ed.) (1990) Multibody Systems Handbook. Berlin: Springer. Skogestad, S. and Postlethwaithe, I. (1996). Multivariable Feedback Control: Analysis and Design, Wiley. Todd, M.I. and Wright, A.H. (1980) A variabledimension symplicial algorithm for fixed-point theorems, Numerical Functional Analysis and Optimization, 155-186. Winter, D. (1990). Biomechanics and Motor Control ofHuman Movement, lohnWiley&Sons, Inc. Yeung, L.F. (199\). Optimal input-output variable assignments for multivariable systems, Automatica, Vol. 27, No.4, 733-738.
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