Optimal Coordination ROLF
and Control of Posture and Locomotion
JOHANSSON
Department
of Automatic
Control, Lund Institute of Technology, S-221 00 Lund, Sweden
AND M&S MAGNUSSON Department of Otorhinolaryngology, Received 2 November
Lund University Hospital, S-221 85 Lund, Sweden
1989; revised 16 March 1990
ABSTRACT This paper presents a theoretical model of stability and coordination of posture and locomotion, together with algorithms for continuous-time quadratic optimization of motion control. Explicit solutions to the Hamilton-Jacobi equation for optimal control of rigid-body motion are obtained by solving an algebraic matrix equation. The stability is investigated with Lyapunov function theory, and it is shown that global asymptotic stability holds. It is also shown how optimal control and adaptive control may act in concert in the case of unknown or uncertain system parameters. The solution describes motion strategies of minimum effort and variance. The proposed optimal control is formulated to be suitable as a posture and stance model for experimental validation and verification. The combination of adaptive and optimal control makes this algorithm a candidate for coordination and control of functional neuromuscular stimulation as well as of prostheses.
INTRODUCTION The quantitative knowledge of biped gait and stance is important both for performance evaluation in basic physiology and for improvement of functional neuromuscular stimulation and such human limb substitutes as prostheses [9]. Experimental work has been conducted with several different foci such as purely physical properties (mass, center of gravity, ground reaction forces) and myophysiology. Measurement of mechanical work during walking as a function of speed, step length, and frequency is one such approach [9]. The elementary reflexes of a muscle to control its force, velocity, and length according to sensory feedback derived from various muscle and tendon receptors have been widely studied [43]. MATHEMATICAL
BIOSCIENCES
OElsevier Science Publishing 655 Avenue of the Americas,
103:203-244
203
(1991)
Co., Inc., 1991 New York, NY 10010
0025-5564/91/$03.50
204
ROLF JOHANSSON
AND MiiNS
MAGNUSSON
Control and coordination of locomotion are poorly understood, as compared to the elementary motion reflexes [l, 16, 43, 481. Nashner and colleagues [49] made an important contribution with their formulation of “ankle and hip strategies.” In early literature on postural control (see Chow and Jacobsen [12] and Behtskii and Chudinov [7]), the presence of biological optimization criteria was postulated. The linear optimal control solutions thus derived relied on linearized (approximate) equations with regard to a given operating point. Optimality of energy expenditure is an attractive hypothetical principle of motion coordination investigated by Levine and colleagues [25, 421. A reason to presume that biological organisms might adapt to minimization of mechanical work is that such operation would be closely related to the ability cf maximum effort and performance and to thermodynamic equilibrium. However, it has not yet been experimentally established whether human stance and locomotion do indeed obey an optimality principle. One reason this issue has not been settled would appear to be the few available mathematical results concerning analysis of the optimization problems involved. Apart from approximate solutions, the only approaches that have been explored are suboptimal solutions based on nonlinear equations. Previous attempts to find general optimal control solutions with application to motion equations have failed owing to the position-dependent, nonlinear behavior of mechanical motion. Also, other areas of applied motion such as flight control and robotics have failed to derive explicit control solutions to this problem [40, 51, 54, 61, 621. Experimental investigation of the integrative action in the mechanisms of motor control must be quantitative and must include both static and dynamic components of the motor response [24, 30-331. A prerequisite for quantitative understanding of integrative aspects is obviously a meticulous mathematical investigation on a form suitable for experimental verification. The need has been stressed of suitable identification models as a necessary basis for progress in the understanding of locomotion control, coordination, and adaptation [27-27, 341. As yet, however, mathematical modeling has failed to produce any model that satisfactorily explains the complexity of coordination, stability, control effort, and equilibrium. The absence of results in this respect is due both to experimental conditions and to the difficulties inherent in control systems modeling [34]. The subsystems requiring treatment in this context are - The mechanical motion of multilinked body segments * Control systems modeling of coordination and reflex action - The active adaptation of neural control A methodological
aspect also requires
serious attention,
* The model should allow for system identification with experimental data.
namely,
and model validation
OPTIMAL POSTURE COORDINATION
The rigid-body mechanics with the general equations arguments omitted).
205
of musculoskeletal motion is often formulated obtained from Lagrangian mechanics (time
M(q)~+C(q,4)4+G(s)=r;
M(q)
= W(q)
> 0.
(1)
The position coordinates q E R” with associated velocities 4 and accelerations 4 are controlled with the driving torques T E R”. The (generalized) moment of inertia M(q), the Coriolis, centripetal, and frictional forces C(q,k)cj, and the gravitational forces G(q) all vary along the trajectories. Several models of type (1) varying in biomechanical complexity have been formulated hitherto: for example, a four-segment model of Vukobratovic and Juricic [62], a five-segment model of Hemami and Farnsworth [29], and a 17-segment model of Hatze [23, 241. The coordination of muscular forces may be considered either at the level of muscular activation or at the level of joint torque. The control problem formulated in terms of joint torques is as follows: Find the torques (forces) T such that the linked body segments assume a prescribed final position (or follow a prescribed trajectory), provided that the body mechanics are described by Equations (1). Optimal control solutions always rely on the accuracy of the underlying model in order to remain optimal. Contexts of model uncertainty or model changes pose a need of active adaptation to new conditions to maintain optimality [3]. Consider the problem of multilink coordination of torques and kinematics. The aim is to minimize velocity and position errors (state errors) with a minimum of both the applied torques and the energy consumption. We provide an analytic solution to the optimal motion control problem and formulate the solutions suitable for extensions to adaptive control. The problem of how to identify a mathematical model of this type from experimental data is considered in a special section.
PROBLEM
STATEMENT
The following aspects in the modeling of postural control need to be covered in any attempt to describe the integrative coordination of motor control: * * *
Variance of position and velocity errors Muscular control effort magnitude Mechanical energy consumed by muscular Stability
control
206
ROLF JOHANSSON
Other desirable * * -
The The The The The
model model control model model
modeling
should should effort should should
features
AND MANS MAGNUSSON
include the following:
explain feedback notions. explain quantitative motion coordination. should not tend to violate muscle stiffness. allow for adaptation. be experimentally verifiable.
The desired reference trajectory for the control object to follow, as generated by some motor program (volition or motion pattern generators), is here assumed to be available as a final desired position or as bounded functions of time in terms of generalized positions qr E 8’ in R" and, if specified, its corresponding accelerations 4, and velocities 4,. Define the errors of accelerations, velocities, and position as
01 1 G d 4
= ii4-4, 4 -- 4, 4r
;
~=x_-x,=
( 1 i-t.
r-
(2)
The control objective is to follow the given, bounded reference trajectory c&q,. without position errors < or velocity errors 4’. Consider an optimization criterion J(U) where the matrix S is used for weighting of the cross term between Z and U.
(3a)
The positive definite matrices Q, R and the matrix S = (S,, S,) define the weighting compromises of the optimization. The first term of (3b) penalizes the variances of position and velocities. The second term of (3b) represents the control effort magnitude. A weighted sum of the energy consumed at each joint can be expressed as the integral of urS,G (the instantaneous power), whereas urS,@ penalizes control actions that tend to increase local errors and results in enhancement of local reflex actions. It is the purpose of this paper to present stable, analytic solutions to the problem of quadratic optimal control of motion control with minimization of the applied torques (forces) when velocity and position feedback are available. The optimal control problem is solved with the Hamilton-Jacobi equation, and feedback solutions to the stated optimal motion control problem are presented. The second-stage problem of adaptive control is then solved.
OPTIMAL POSTURE COORDINATION
DYNAMICS
OF SEGMENTED,
ARTICULATED
207 BODIES
We model the motion dynamics as a set of n rigid bodies connected and described by a set of generalized position coordinates q E R”. The derivation of the motion equations (1) in accordance with Lagrange theory [4, 1.51 involves explicit expression of both kinetic energy y and potential energy Q. The Lagrangian _&’ of motion in a space with a velocity-independent gravitation potential is defined by
-e4,4)
= n9,4)
- Q(4) = WM(9)4 - Q’(4).
The Lagrangian is the basis for formulation equations of motion
(4)
of the Euler-Lagrange
(5) where T E R” are the externally applied torques general equations (1) are obtainded from (5) as
and forces. The standard
M(q)@+ C(q,4)4 + G(q) = 7,
(1)
where M(q) is the inertia matrix and C(q,cj)cj represents Coriolis and centripetal forces. It is assumed that the positions q and velocities 4 but not the accelerations Q are available as neural feedback. It is further assumed that the torque vector T is available as the control input. Equation (1) already covers a large model set including the equations of Hatze [23]. Hemami and Farnsworth [29] and Vukobratovic and Juricic [62]. Further, model extensions with various forms of friction and contact forces as well as holonomic constraints may be expressed in equations of the type cl).’ WHAT
CONTROL
EFFORT
SHOULD
BE MINIMIZED?
A natural aim is to minimize velocity and position errors (state errors) with a minimum of applied torque and energy consumption. For a velocityindependent potential energy 9, the Euler-Lagrange equations give
(6) Changes in potential energy due to gravitation are inevitable and can be determined from the start and end points only. Thus, the gravitationdependent energy expenditure cannot be altered, and there is little point in trying to optimize the corresponding torques or forces. Consider, therefore,
208
ROLF JOHANSSON
the applied torques
TV that selectively
AND MANS MAGNUSSON
affect kinetic energy.
(7) To investigate the properties of TV, we introduce the skew-symmetric matrix N(q,G) with elements nEj defined from the components mZj of M(q) as
amik(d ___ %j
1
amjk(q) . aqi
qk3
which verifies
Nq,4)4 = $&,4)4- ;&p14(q)q),
(9)
so that
which contains the force terms associated with inertia (acceleration) and centripetal and Coriolis forces. It is a standard result from Lagrangian mechanics that the third term N(q,cj)g of (10) represents the workless forces of the system. It is straightforward to verify that the work done on the system by the applied forces TV determines the kinetic energy
A modification of (10) is appropriate in cases where the motion is prescribed not only with respect to the final position but also with respect to its intermediary values. During voluntary motion along a prescribed trajectory [q,(t)f 01,it is important that the optimization does not compromise the desired trajectory qr. Instead, we model the optimization so that the motion of body segments is stabilized to follow the desired trajectory with minimal effort. To minimize the necessary forces (torques), we include the control variable TV in the more general definition
(12)
OPTIMAL
POSTURE
COORDINATION
with i and T, introduced
209
via the following state-space
transformation
of 2:
(13) This definition includes forces (torques) affecting kinetic energy (lo), reference trajectories (2), and a state-space transformation (13). The control variable u of (13) can be reduced to rK of (10) for qr = 0, T,, = Inxn, and T,, = 0 so that u=M(q)ii+[filj(q,q)+N(s,4)]4 =7-G(4)+[~k(q,q)+N(q,4)-C(4,4)]4,
(14)
where the last term is zero when no friction forces are present. QUADRATIC
OPTIMIZATION
We therefore embed the motion control problem into the following somewhat more general optimization problem. The assumptions made are a summarized as follows. BASIC ASSUMPTIONS
A.?.
motion equations are M(q)4 + C(k, q)4 + G(q) = 7 with coordinates q and external torques (forces) T. The reference trajectory is given as qr, 4,, @, E L”, and qr E 8’ with the error-state i = (4’ Q7). A state-space transformation is given as
A4.
The control action to minimize
AI.
A2.
The
is
u=T-G(q) or, for nonzero
B=
reference
trajectories
qr,
(16)
210
A5. A6. A7. A8.
ROLF JOHANSSON AND M.&S
MAGNUSSON
Positions and velocities of all segments of rigid link motion available for measurement. The structure of M,C,G is known. The parameters of M,C,G are known. The performance index of optimal control is
are
J(~)=j~;~~(t)Qi(t)+;~~(t)R~(t)+u~(r)S~(t)dt; f0
Q-STR-‘00,
the condition Q - STR-‘S > 0 being imposed to render mization of J(u) a well-posed optimization problem [41].
the mini-
Given the performance index J(U), we find an optimal control u = u* that will transfer from an initial state to a desired state. The control u = U* moves the system from an arbitrary initial state _i!(t,) to the origin of the error space while minimizing J(u). The control variable u is weighted with the matrix R = R T > 0, and the vector of velocity and position errors Z is weighted with the matrix Q = QT > 0. The rate of compensation can be adjusted by choosing proper weights Q. The term uTRu guarantees smoothness of operation. THE HAMILTON-
JACOBI EQUATION
Solutions for optimization problems of type (3) under assumptions Al-A8 are obtained by solving partial differential equations obtained from Hamilton-Jacobi theory, dynamic programming, or the Pontryagin maximum principle [14, 411. As the Hamiltonian of optimization is defined as
a necessary and sufficient condition for optimality value function I/ that satisfies the Hamilton-Jacobi
[14, 411 is to choose a equation.
(18)
OPTIMAL
POSTURE
This minimum nian
COORDINATION
is attained
for the optimal
211 control
u = U* and the Hamilto-
1
H* = min H = min
u
u
(19)
The optimal value function principal function LEMMA
V that solves (18) for u = U* is called Hamilton’s
[14, 151 of the system.
1
The following function V composed of i, q,(t), T,, M, and a symmetric matrix K E Rnxn solves the Hamilton- Jacobi equation of the optimization problem under assumptions Al-
A8,
=7xl-T T, ~(M(0(1)
V(J(t),t)
;)Toi
(20)
for K, T, solving the algebraic matrix equation
(21) The optimal feedback
control law u = u* that minimizes J(u) u*(t)
Proof
STABILITY
See Appendix
= - R-‘(S+BTT&(t).
is
(22)
1.
AND CONTROL
All optimal control generated by the solutions (20)~(22) to the Hamilton-Jacobi equation does not necessarily guarantee stable closed-loop behavior. Only solutions that also guarantee a stable closed-loop behavior are interesting for stance and locomotion. Such a stability condition provides some constraints as to the choice of the weighting matrices Q, R, and S. A sufficient condition for stable, optimal control is that K = K T> 0 in (20) as formulated in the following theorem.
212
ROLF JOHANSSON
AND M&S
MAGNUSSON
THEOREM 1 Let the weighting matrices chosen such that
Q, R with Cholesky factors
QTQ, + Q,‘e, - (QT, +
Q,,Q,,
R,
be
Qu> ’ 0;
R= RT= RTR,>O.
(23)
Let T,,, K be chosen as the matrices
K =
icT= ;(Q;Q,+ Qz’el) - ;(Q?i + Qn) ’
O.
(24)
The optimal control solulion subject to assumptions Al- A8 then prouides an asymptotically stable and an Lz-stable closed-loop system with the optimal feedback
control law u = u*,
= - R-‘(S+
u*(t)
BTT&t).
(25)
The minimal optimization criterion is then obtained as
J( u*) = min/mL(Z,u)d,=(mL(Z,u*)dt=V(P(tO),fO), U to fo where V solves the Hamilton-Jacobi
V(i(t),t)=?x
(26)
equation (18),
’ -T T,~(“‘d”)
$2.
(27)
Moreover, the value function V(x’, t) is a Lyapunov function for the asymptotically stable system. The Lyapunov function derivative f = dV/ dt < 0 for 11.?]1#0, and global asymptotic stability holds for Q - STR-‘S > 0, R > 0. Proof.
See Appendix
2.
The function WE, t) of (27) can be viewed as an aggregate of kinetic energy and the ‘potential energy’ from a set of springs with a stiffness matrix K. The controlled motion keeps stable with an equilibrium on the prescribed reference trajectory as long as V does not grow. This physical
OPTIMAL
POSTURE
213
COORDINATION l
position
q, velocity
dq/dt
position
q, velocity
dqldt
* Motion planning
Reference trajectory
Optimal control
*
U
A
control
--)
object
q r FIG. 1. Algorithmic
organization
of the quadratic
analogy can be mathematically formalized Lyapunov function as stated in Theorem 1. THE CONTROL
optimal
feedback
in a stability
control.
proof
with
a
LAW
The optimal control was given as the feedback
control
U*(t)=-R-l(S+BTT&(t). The appropriate external torques to apply are then calculated and (22). This gives the applied torque r*, which is calculated with assumptions Al-A7 and is optimal in the sense of A8. 7* = M(q)(&
-
T;,‘T,,&-
(22) from (16) in accord
T1;‘M(q)-’
x{[r~(r1,4)+N(q,ci)]BTTo~+u*}) + C(q>d)d+ G(q).
(28)
The torque obtained contains compensation of gravity and Coriolis torques as well as an anticipatory action with respect to d’,. The special choices q, = 0, To = Zznx2,, and the absence of frictional forces provide a simplified interpretation as T* =
SELF-OPTIMIZING
u* + G(q).
(2%
ADAPTATION
As model-based optimal control laws are contingent upon the model accuracy, performance is sensitive to changes of physical parameters or
214
ROLF JOHANSSON
AND M.iNS
MAGNUSSON
other model changes. In cases with uncertain or time-varying parameters of M,C,G, there is a need to adapt the optimal control to the operating conditions. The optimal control algorithm presented here, Equations (28) and (29), is readily modified for self-optimizing adaptive control. Assume that the matrices M,C,G have a known structure (A61, and consider a case of uncertain parameters (cf. A7). Let the optimal control law be expressed in terms of the unknown parameters 0 E RP of M,C,G and the data vector I,!IE Rnxp, I+!I~ E R”. The matrix $ and the vector I/J” contain the terms of T* that can be computed without reference to unknown or uncertain parameters.
The adaptive control law is a modification replaced by an estimate e*
of (241, see Figure
1, with B
=A(q)(ii,-T,,4)-~~(q,g)B=T,f+e(q,g)q+i;(q)+u*. (31) A prerequisite of su:cessful motor learning and adaptation environment is that 0 may be purposely modified. THEOREM
to a changing
2
Assume that the optimal control u* is determined according to Theorem 1. Let the optimal control law be expressed in terms of uncertain parameters 0 E RP of M,C,G and the data matrices $ E Rnxp, t,!rO E R”. The matrices $, & contain terms of r* that may be computed without reference to unknown or uncertain parameters.
Mb&i, - T,,q’)-t~(wi)B=T,,f + C(q,4)4 + G(q) = $0
+ $0.
(32)
The adaptive control law with f3 replaced by an estimate 6~ RP is
7=l$+l),+u*
(33)
with the adaptation law
e^= - K~lJITBTTO~.
(34)
OPTIMAL
POSTURE
215
COORDINATION
The Lyapunov function
V,( d, t) = ;Z7T;
Vx,
(“‘R)
:)
T,$ + ;fjTK,8;
K, = K;>
0,
(35)
with the negative semidefinite deriuative
=-3~T(e-sT~-1s+~,TBR-1~TT,)~:, 0.
which results in the control law r=G(q)-(35.OIs.s Three examples are given to demonstrate
12O.O&s).Z.
(39)
this methodology.
Example 1. Consider the anthropomorphic five-segment model depicted in Figure 2. Assume that the support leg is in contact with the ground at the origin of the coordinate system OX,,. Let all initial angular positions be zero except q&O) f 0. This case simulates the five-link model starting to climb a step. The result is shown in Figure 3. Example 2. This example shows how the five-link model rises from an initial bending position at rest with the initial position coordinates qt(O) = q2(0) = q&O)= q4(0) = - q&O). The result is shown in Figure 4. There is little displacement of the common center of mass initially, compared to Example 1. The ground reaction shear force is therefore smaller, than in Example 1. Example 3. Assume that an additional load of 20 kg is attached at the center of mass of segment 5. This situation simulates the presence of a back load on a human climbing a step. The optimal control is now modified by the adaptation, which estimates the new segment weight and corrects the feedback and antigravity actions. The result is shown in Figure 5. Examples 1 and 2 show that the anthropomorphic model is capable of stable posture and movement in the presence of antigravity action and local error feedback only. Example 3 shows that motor learning and adaptation
OPTIMAL
POSTURE
if,
217
COORDINATION
knee torques [Nm]
I
hip, knee angles [deg]
_:c’l;
ankle torque [Nm]
0
,
10
20
GRF normal Fy [N] 1
0
10
20
0
10
FIG. 3. An anthropomorphic five-link model climbing a step. forces are a shear force F, and a normal force F,. (Left) Torques ankle; (right) angular positions. All graphs versus time in seconds.
20
The ground reaction at the hip, knee, and
are feasible in this context. The mass of the torso and load (m, = 30+ 20 [kg]) is also well estimated, although a small bias may persist in cases where the control performance is good. This is in agreement with Equation (30), where the parameter errors e’ do not appear on the right-hand side. IDENTIFICATION
MODELS
It is sometimes overlooked that quantitative modeling must be experimentally verified not only qualitatively but also quantitatively. The explicit solution to (18) and the associated control law supports the formulation of an identification model similar to (30). Let the torque equation be formulated in terms of the uncertain parameters 0 as the linear estimation model
where
4, ,&. contain
functions
of data (G,, 4,, qr, 4, q) computable
without
218
ROLF
_Fkneetyques[Nm;
0 _f*’
10 torque [Nm]
0
d
10
knee angles
,
20
AND MANS MAGNUSSON
;,
20
,
_rshea;Fx[N]
JOHANSSON
ankle
lirr
[deg]
10 angle
;0 [deg]
normal FYWI
;I
io
i0
FIG. 4. An anthropomorphic five-link model rising from an initial bending position [q&O) = q3(0) = - q&O)]. The ground reaction forces are a shear force F, and a normal force F,. (Left) Torques at the hip, knee, and ankle; (right) angular positions. All graphs versus time in seconds.
reference to the uncertain parameters. It is an easy identification problem to find the unknown @ provided that observations ?-k,f$k,+ko of (40) at times k = 1,2,. . . , N are available. The least squares criterion based on N observations is
where
(42)
OPTIMAL
POSTURE
219
s
COORDINATION
20
I1r
tyques [Nm;
, hip, knee angles [deg]
FIG. 5. An anthropomorphic five-link model climbing a step while the control law adapts to an extra load (20 kg) on the torso. The ground reaction forces are a shear force F, and a normal force Fy. (Left) Torques at the hip, knee, and ankle; (right) angular positions. All graphs versus time in seconds.
The least squares solution
based on N observations
is (43)
which can be obtained by completing the squares of (41) (cf. [58]). This is immediately recognized as a linear regression problem with a least squares solution subject to statistical hypothesis testing by standard methods of variance analysis (x2 tests). This shows that experimental verification of the proposed mathematical
modeling is feasible by standard methods.
Hence, the optimization weighting matrices as well as unknown physical or physiological parameters can be estimated from measurements of . . .. T>4,4,9rT $9 4r. DISCUSSION
AND CONCLUSIONS
We have solved an optimal control problem of posture dynamics with explicit solutions to the Hamilton-Jacobi
and locomotion equation. The
220
ROLF JOHANSSON AND MriNS MAGNUSSON
optimal solution explains asymptotically stable optimal control. Self-optimization providing globally stable adaptive control has been designed to solve the case of uncertain parameters. The optimal control is globally asymptotically stable, whereas the selfoptimizing adaptive control is globally stable in the sense of Lyapunov. Uniform stability in the sense of Lyapunov follows from the existence of a negative semidefinite Lyapunov function derivative as shown in Theorem 2. Finite initial conditions and 4,, 4, E L" mean that the initial value of the Lyapunov function V(Z(t,), to) is bounded. A finite value of the Lyapunov function V implies a finite magnitude of the tracking errors @,i. The L" stability follows from the fact that the Lyapunov function is finite and always decreases with time. The asymptotic stability also implies that the optimal closed-loop system is stable under persistent disturbances [22]. The optimal control algorithm presented here exhibits a certain similarity to the linear quadratic control problem. The closed-loop properties may be effectively determined from the weighting matrices Q, R, and S of A8. Equation (21) and the algebraic Riccati equation are similar, but the solutions are very different. The Riccati equation solution is positive definite, but the present algorithm does not in general provide a symmetric weighting matrix TO. From a mechanical point of view there are several interesting aspects. The proposed solutions contribute to the understanding of the close connections between classical mechanics and optimization theory for motion control. The matrix K of (241, (27) represents a virtual stiffness around the desired position qr, whereas terms containing the inerita matrix M(q) represent kinetic energy. The Hamiltonian GV = Y+ L% of analytical mechanics may be compared with the Hamilton-Jacobi solution V(_?,t) that represents an aggregate of kinetic energy and the ‘potential energy’ of a spring action described by a stiffness matrix K. The virtual spring action established by feedback control thus formally replaces gravitation as the source of potential energy. The method offers a description in any set of relevant coordinates, Cartesian space, or configuration space. Associated optimization criteria in terms of kinetic energy are invariant to coordinate transformations. Furthermore, a matrix T,, of [13] with T,,# 0 renders the controlled system dissipative; that is, as the motion is not energy conservative, the system is able to ‘absorb’ energy of initial conditions and disturbances. There are several advantages to the analytic solution proposed in this paper, as compared to earlier work [7, 12, 25, 621. We avoid approximate solutions as well as exact solutions based on approximate models that may exhibit severe degradation of closed-loop control performance as compared to optimal control. The reason is that all model-based optimal control is contingent upon the accuracy of the underlying model, that is, the bodysegment parameters. A small change of system parameters typically results
OPTIMAL
POSTURE
COORDINATION
221
in a severe degradation of performance as compared to optimal conditions. From the point of view of parameter uncertainty, approximations involved in the linearized or approximated models [lo, 12, 281 as well as approximate solutions [54] to the exact problem make such optimal control error-prone in nonideal situations. The analytic solution of this paper avoids such problems and is valid also for transient motion with velocities and positions away from particular equilibrium points. As many tasks of controlled motion must be solved in finite time, it could be argued that the infinite-time problem is less relevant for physiological motion control. However, the problem with an infinite-time optimization criterion may be viewed as a finite-time problem with a performance optimization together with an end-point condition at t = tf on the closed-loop accuracy and stability.
v(f(t,),r,,)=~:L(i,u*)~t=j;:‘L(i,u’)dr+V(i(tf),tf).
(44)
A similar relation holds for the Lyapunov function VX of (35). Notice that the Lagrangian L is positive so that V(i(t,>, tf) d V(i(t,), to). This makes the possible adaptation and learning action applicable also as a model of finite-time operation with periodic or iterative motion. The self-optimizing adaptation of an optimal trajectory intended for periodic motion may thus be made in a few repetitive trials. Only body motion in the form of segmented and articulated rigid links has been explicitly treated here. For example, considerable neural transmission time delay may place a limit upon the applicability of the feedback model. Several models of the type (1) of various degrees of biomechanical complexity have been formulated [23, 30, 611. Elastic deformation and other structural flexibilities that can be modeled by methods of analytical mechanics may be included in Equations (1) and thus in the optimal control solution. The mathematical description above involves reference trajectories q, corresponding to intended or instructed motion, motor programs, preprogrammed motion, or reference trajectories that are planned by the systems as movement evolves. There is a considerable body of literature on the existence of such reference trajectories, and we restrict ourselves to recent contributions in this field. The evidence of motor programs was reviewed by Grillner [181, and supraspinal and spinal mechanisms were considered by Grillner and Dubuc [19]. Zattara and Bouisset 1651 support the idea of preprogrammed motion in the context of postural adjustment where anticipatory postural movements appear to counteract the disturbing effects of the forthcoming voluntary motion. Anticipatory postural responses in human subjects were demonstrated by Marsden 146-471 and Haas and Diener
222
ROLF JOHANSSON
AND Mb.5
MAGNUSSON
1211. Because of reproducibility and specificity, the anticipatory postural movements can be considered to be preprogrammed. Grillner and Wallen [20] and Sanes and Jennings [53] treated the control programs underlying the motor behavior. Thorstensson et al. [59] demonstrated that trunk movements are generated and controlled by specific patterns of muscle coordination. Anticipatory EMG responses comprising early and late responses, timing, and amplitude modulation were analyzed by Lacquaniti and Maioli [391. Shapovalova et al. [57] experimentally investigated the importance of the caudate nucleus in the neural processes preceding motion. Marsden [46] provided evidence for the statement that patients with Parkinson’s disease are unable to execute learned motor strategies involving the selection, sequencing, and initiation of motor programs. The functional role of the substantia nigra in the initiation and particularly the execution of movements was demonstrated by Viallet et al. [60]. Hence, thorough biological experimental evidence exists for motor programs and preprogramming of motor behavior in such contexts as those assumed in the present paper. The complexity of the reference trajectory is an open question, and our model allows the specification and inclusion of 4, or q,.,Qr or even q,, c&,4,. At least five cases are included in the problem formulation presented: (1) Motor programs with preformulated trajectories (2) Reference trajectories generated and planned as movement evolves (3) Goal determination as an end-point condition only or as a sequence of points in position and/or velocity space (4) Goal determination as an end-point condition only, to be accomplished at a final time ff with an additional end-point condition of stability (5) Motor planning (trajectory planning) to reach a given end point The algorithm provided in this paper may thus solve both the motor planning and feedback control problems of motion control. Execution of the motor programs involved requires a sensory feedback of position and velocity obtainable from the visual, vestibular, and somatosensory subsystems [52]. Moreover, it has been suggested that a “corollary discharge” exists in motor programs [44, 451. This is the postulated internal feedback from motor to sensory structures that indicates to the sensory systems that a limb is about to be moved, in order to permit correct interpretation of the sensory consequences of the movement. It is often claimed that such a system must exist, as otherwise we could never be sure whether we had moved or whether the environment had moved us. The interaction between the motor program and sensory feedback is modeled in the present paper as a comparison between the intended motion qr and the feedback information q that results in an error e = q - q,.
OPTIMAL
POSTURE
223
COORDINATION
The present ‘error feedback’ is thus in harmony with ideas of “corollary organization principles are not discharge,” although other topographical precluded. The coordination of muscular forces may be considered either at the level of muscular activations or at the level of joint torques. The relationship between joint torques and muscular action is a research topic in its own right [6, 91. It should be stressed that there are many degrees of freedom to schedule muscles, a fact that is often described as redundancy or static indeterminacy. The total muscular torque acting on the knee with a certain knee angle is produced by not less than 12 muscles, some of which are diarthric muscles. The force distribution of the II stabilizing forces r over the m individual muscles (m > n) can (formally) be uniquely solved as a quadratic optimization problem as follows. Assume that the optimal control T E R” and the muscular forces of m muscles are described by the vector F E Rm, where m > n. Let p E RnXm denote the matrix of anatomical-biometrical data that describes the torque-force relationship T = pF, that is, the individual muscle action and the load sharing between different muscles as determined by the insertion of muscles and tendons at bone segments (m > n). A quadratic optimization of the muscular forces F-the minimization of FTF subject to the constraint 7 = pF-is easily obtained by completing the squares so that min FTF
=min(FTIZ-p’(ppT)-l~]F+rT(PPT)-lr)
subject to r
= pF = T’( ppT) - 5
with the minimum
obtained
for the unique
(45) solution
F = F* = pT( ppT) - ‘7, which is valid also for position-dependent torque-force relations p = p(q). (Note that the matrix p is of full row rank n for all well-posed problems so that ppT is invertible.) In particular, quadratic optimization to obtain the minimum muscle forces that balance the gravitation forces G(q) can be obtained as F = pT(ppT)-‘G(q). A similar static optimization problem has been treated with linear programming for ad hoc determination of minimum forces [S-56]. The problem is statically indeterminate (m > n), a circumstance that may cause some frustration in engineering approaches to the evaluation of forces [55-561. It is, however, very difficult to motivate why and how any such restrictions should be imposed. From the point of view of optimization of mechanical energy expenditure, stability, and coordination, there is no
224
ROLF JOHANSSON
AND MzkNS MAGNUSSON
reason to prefer a particular set of forces F to another set of forces F that also satisfies r = pF. The scheduling of individual muscles may also presumably depend on many nonmechanical factors such as the metabolic state. We therefore avoid suggesting any unnecessary restrictions of the solution space that would only limit the explanatory power of the model presented. Neuromuscular transmission, length and force relationships, and the correspondence between EMG data and muscular forces [6, 8, 111 constitutes a controversial subject, a debate that we hesitate to enter here. By solving the coordination and control problem we support such research by reducing neuromuscular transmission to “local research topics.” Several problems of muscular physiology- for example, inverse myodynamics, the indeterminacy problem, local feedback, metabolic energy consumption, and heat production-may thus be considered at the local body segment level 161. The proposed solution is sufficient to explain many interesting features of posture control and coordination. The closed-loop properties can be effectively determined over a large range of behaviors from the weighting matrices Q, R, and S of (16). Example 1 shows the coordination for climbing a step based on the information of velocity and position measurements. The control law consists of gravity and Coriolis torque compensations and local error feedback only [corresponding to a diagonal feedback matrix R-‘(S + BTT,,)]. Several hypotheses on the importance and sufficiency of local feedback for postural control are therefore supported with respect to modeling complexity (cf. Houk 132, 3311. In our examples we reproduce “ankle and hip strategies” of Nashner and colleagues [49]. Example 3 demonstrates the adaptation of step climbing when an additional back load of 20 kg is present. The model presented is therefore demonstrated to be of relevance also in the study of motor learning and adaptation. All model-based optimal controls rely on the accuracy of the underlying model, that is, the body-segment parameters. A small change of system parameters typically results in a severe degradation of performance, as compared to optimal conditions. It is therefore necessary to consider the problem of adaptation in the context of optimal control, although adaptation is a difficult topic of research [2, 5, 17, 26, 34-36, 61-641. The adaptation included in this paper is of a type similar to the “MIT rule” [4; p. 81 with a modification to Lyapunov theory similar to that reported in [13l, [38], and [50]. The gradient method (28) is also similar to earlier attempts to describe neural learning mechanisms; compare the “Hebb rule” [26]. The correspondence between the Hebb rule and gradient methods of adaptation has been demonstrated (e.g., [4; p. 492; 641). Adaptivity in neural mechanisms is usually attributed to higher neural centra such as the
OPTIMAL
POSTURE
COORDINATION
225
cerebellum, with a possible involvement of gating mechanisms at a lower, spinal level. The computational topology (Figure 1) of the optimal control adaptation does not disagree with known neural tract topography (cf. [49]). Experimental verification involves the formulation of adequate postural tests with measurements of joint positions and joint velocities of the articulated and segmental model as well as of ground reaction forces. Johansson et al. [37] reported on identification methodologies, though only ankle strategy dynamics was considered. Statistical hypothesis testing based on the proposed linear regression models (37)-(40) is here straightforward. The analysis presented thus supports experimental verification as put forward as a prequisite by Ito [34] and others. The analytic methods presented are applicable to control systems analysis as well as for purposes of control system design and may therefore prove useful for the control of prostheses of functional neuromuscular stimulation. APPENDIX
1.
THE HAMILTON-
Proof of Lemma
1
JACOBI EQUATION
The Lagrangian is given by (3), and the lemma claims that the Hamilton-Jacobi equation
-
av(.c,t)
at
av(i,t)
= min u
[i
a2
T.
1
i+L(.?-,u) I
(18)
is satisfied for a function
(20) The proof consists of five steps:
(1) A state-space description (2) Verification that V = V(Z, t) (3) (4) (5)
Evaluation of partial derivatives of V Derivation of the u that minimizes H of (18) Verification that V solves (18)
A STATE-SPACE
DESCRIPTION
The full error state-space
representation
Z(t) = ($‘(t)g(t)y;
is found as 2
E
R2".
(All)
226
ROLF JOHANSSON AND M&S
MAGNUSSON
The error dynamics of the linked body segments can be obtained and (2) as a state-space description where the derivative of 2 is
L(t) =
i(t) i s”(t)i
l
M-1(4)c(44)
=
from (1)
+
-
Znxn
0,x, 0 nxn
4, - M-‘W(G(d
I
i(t)
+ C(q,(ibi,)
M_,(q)
7
0flX?l (Al .2)
or, with shorter notation,
I(t) = A(q,cj)l(t)+ where r is available for assignment
-
+T,-’
.
i
BM-‘(qh
(A1.3)
of the control law.
M(q)-‘(3wLd+ NqA)) Tl?
xT,Z+T,-’
VERIFICATION
B,(4,,Q,,Q,q)+
u.
0,x, -
Tl
,,, = 0.
13 (’q1
-n32=
=my5=
%Y
tixx = 0,
II II --0,
n2,=-ns2=0 -t(m213f4+m413r4)~34(93+Q4) +&I
n55=0
nl, = - nxl = +mlrI.s,4L - +( m2 + m3 t m4 f m,)l,s,cj, =
+m2r2s2G2
-nx3
=
-
-nx4
=
fm214s4ij4 + im,r,s,g,
nh = -
n 4x
=
93)
iiz,, = 0
m&s343
The matrix N(q, 4) of (9) is obtained
n 3x=
- 4)
=m2r2c2q2
my4
n 33
= m2k2b(Li4
+ Cm2 + m3 + m4 + mJl,s,4,
m,l,s,&
.
- &),
m,r,s,42
-
=mx4=-
MlY
ci,) %2=0
riz33=0,
I
PiI. 35=m53=
=
- &I
m311rss15(ris-cjl),
rnz3 = TiE32= - m213r2.A& ril z5=til
with the elements
m31,r3S~3(43-9,)+(1712+m4+m5)1,/3S13(~3-(il)
?klq
m15 = m51= .
m2x
237
nx2
+m3r3s3&
nsx = - nx5 = - im5r5sSg5
- $(m,
+ m4 + m,)l,s+j,
+&I
238
ROLF JOHANSSON n1y
=
-
ny1
=
-
+mlrlcl(il
n2y
=
-
ny2
=
-
+m2r2c2(i2
n3y
=
-
ny3
=
+m3r3c3d3
n4y
=
-
ny4
=
-
n5y
=
-
ny5
=
+
i(m2
t(
m2
+
+
+
m3
m4
+
+
m4
AND MANS MAGNUSSON +
m5)l,c,cj,
m5)13c3cj3
+m214c4q4 - fm4r4c4G4
+m5r5c5G5,
nxy = - nyx = nxx = nyy = 0.
The Coriolis and centripetal forces can now be calculated (l), (10) and from (A4.12) and (A4.13):
c. Yq
The Euler-Lagrange
1M
M,
CYY
equations
I
according
(1) are thus ‘G\
C,,
0
c,,
G,
(
\
= \
\CY,
mYY
( In cases with no motion equations of motion
to
(A4.14)
‘
My
M,’ mxx mxy i +
\My’ mYX
0
(A4.13)
I
4.15)
in x or y (i.e., i = y = 0, I = j; = 0) this gives
r = M(q)ii
+ C(q>4)4
and the corresponding ground reaction force Fy as follows:
+ G(q)
(~4.16)
forces with a horizontal
= (1’)
shear force
F, and vertical
M$j
+ C,,cj (A4.17)
M;$+Cyq4+Gy Elimination of 4 from (A4.16) provides torques and the ground reaction forces:
a relation
between
the joint
(~4.18)
OPTIMAL
POSTURE
with the matrices
M,, My obtained I mix
1 mzx
IV,=
239
COORDINATION
1713,
/mlr,cl
from (A4.7), (A4.8), - (m2 + m3 + m4 + m5)11c1 m2r2c2
=
-
-(m2
m3r3c3
m4r4c4
m4x
1
,m5x
m,r,sl
-
(m2
+
=
m3Y
_
m3r3s3
> (A4.19)
m5)13c3
m214c4
m3
-
(m2
+
+
m4r4s4
m4Y
I
msy
_
mlrlslQl
+
m4
+
\
m,)l,s,
m4
+
,
m5)13s3
(A4.20)
m214s4
I
m5r5s5
from (A4.12)-(A4.141,
+++ (m2 -
I
I
+
-
CXq,Cy, obtained
cxq=
+
m2r2s2
m2Y
and the matrices
+
m4
- m5rscs
\
mlY
My =
+
+
m3
+
m4
+
m5)llslcil
m2r2S242
m3r3s343 Cm2 _ m214s4G4
m4
-
,
mdl3s343
(A4.21)
m4r4s4G4
m5rswk
mlrlc14, -Cm2+ m3
+ m4 +
mJl,c,4, \
m2r2c242 CY,
=
I
- m3r3c3G3 - (m2 + m4 + m5V3c3ci3 . m314c44, + m4r4c4G4 - m5r5c5&
\
CONDITIONS
I
OF SIMULATION
Initial conditions
of Examples
1, 2, and 3
s,(O)
5.73
right shank
42(O)
5.00
left thigh
43(O)
’ left shank
right thigh torso
(A4.22)
=
5.73
5.73 \ 5 .oo/ , and
[deal.
5.73
q4(0)
5 .oo
5.00
95(O)/
5.73
5.73
I
I (A.23)
240
ROLF JOHANSSON
Weights and moments \
I+
=
0.2
J,
[kg];
=
0.2
J4
10
m4
‘0.2\
J2
10
\%/
[kg.m2].
(A4.24)
[ml.
(A4.25)
0.2
\3
\Js
\30
data 11,’
/
10.40 \ 0.40
12
I,
OPTIMAL
‘J,\
10
m2
Geometrical
of inertia lO\
m3
AND Mz&NS MAGNUSSON
=
\
r1
0.20 0.20
r2
0.40
[ml;
14
0.40
\15,
\ 1.oo I
zz
r3
0.20
r4
0.20
, r5
0.50 /
/
CONTROL
J(u)
The performance index that Q - S%-‘S > 0.
12Z,,,)f
+ u’( 315x5
for the optimal
control
was chosen
+ $U7I5,,Udl,
such
(38)
which results in the control law T = G(q)
120.01,&,
-( 35.OZ5X5
(39’)
or, with more detail [cf. (A4.1111,
-(mlgrl+(m2 + m3+
71 72 73
=
m5)dl)sl\
-(m3gr3+(m2+m4+m5)g13)s3 (m2d4
T4 751
m4 +
m2gr2s2
+ -
\
41’
&
,
‘a’
q2
-35.0
m4gr4b4
m5gr5s5
q2
- 120.0
q3 .
q4
q4
\ q5
,451
(~4.26)
OPTIMAL
POSTURE
241
COORDINATION
As compared to the complexity of the equations of motion, the control law is very simple, with only antigravitation forces and feedback of position and velocity. ADAPTIVE
CONTROL
An unknown value of m5 due to the back load suggests that m5 should be replaced by an estimate e^ and that (A4.26) should be replaced by the adaptive control law -(m,gr,+(m2+m3+m4)gl,)s, m2gr2s2 T’=
-(m,gr3+(m2+m4)gl,)s,)
I- j 0
(m2d4
gr5s5
m4gr4b4
0
\
‘91\
/41’ q2
42
-35.0
+
(is
-
120.0
93
94
q4
\ 95
\ 45
I
where e^(i.e., the estimated
(A4.27)
,
1
m,) adapts
according
to (34), which gives
41’
-g13s3
-K;‘(-gl,s,
-gr5s5)
91
I. 1 9.0
4”
9.0
94
8=
0
0
-35.0
45
4’
q2 q3 94
/
+ -120.0
9s
(A4.
with values in the simulated initial value f?(O)= 10.
examples
chosen
9
as K, = 0.006 and with an
We wish to thank Dr. John Allum for drawing our attention to this area of research. REFERENCES 1 2 3
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JOHANSSON
AND MANS MAGNUSSON
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