Simulation of Optimal Movements Using the 
Minimum-Muscle-Tension-Change Model. 
Menashe Dornay* 
Yoji Uno" 
Mitsuo Kawato' 
Ryoji Suzuki" 
*Cognitive Processes Department, ATR Auditory and Visual Perception Research 
Laboratories, Sanpeidani, Inuidani, Seika-Cho, Soraku-Gun, Kyoto 619-02 Japan. 
**Department of Mathematical Engineering and Information Physics, Faculty of 
Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113 Japan. 
Abstract 
This work discusses various optimization techniques which were 
proposed in models for controlling arm movements. In particular, the 
minimum-muscle-tension-change model is investigated. A dynamic 
simulator of the monkey's arm, including seventeen single and double 
joint muscles, is utilized to generate horizontal hand movements. The 
hand trajectories produced by this algorithm are discussed. 
1 INTRODUCTION 
To perform a voluntary hand movement, the primate nervous system must solve the 
following problems: (A) Which trajectory (hand path and velocity) should be used while 
moving the hand from the initial to the desired position. () What muscle forces should 
be generated. Those two problems are termed "ill-posed" because they can be solved in 
an infinite number of ways. The interesting question to us is: what strategy does the 
nervous system use while choosing a specific solution for these problems ? The chosen 
solutions must comply with the known experimental data: Human and monkey's free 
horizontal multi-joint hand movements have straight or gently curved paths. The hand 
velocity profiles are always roughly bell shaped (Bizzi & Abend 1986). 
627 
628 Dornay, Uno, Kawato, and Suzuki 
1.1 THE MINIMUM-JERK MODEL 
Flash and Hogan (1985) proposed that a global kinematic optimization approach, the 
minimum-jerk model, defines a solution for the trajectory determination problem (problem 
A). Using this strategy, the nervous system is choosing the (unique) smoothest trajectory 
of the hand for any horizontal movement, without having to deal with the structure or 
dynamics of the arm. The minimum-jerk model produces reasonable approximations for 
hand trajectories in unconstrained point to point movements in the horizontal plane in 
front of the body (Flash & Hogan 1985; Morasso 1981; Uno et al. 1989a). It fails to 
describe, however, some important experimental findings for human arm movements (Uno 
et al. 1989a). 
1.2 THE EQUILIBRIUM-TRAJECTORY HYPOTHESIS 
According to the equilibrium-trajectory hypothesis (Feldman 1966), the nervous system 
generates movements by a gradual change in the equilibrium posture of the hand: at all 
times during the execution of a movement the muscle forces defines a stable posture 
which acts as a point of attraction in the configurational space of the limb. The actual 
hand movement is the realized trajectory. The realized hand trajectory is usually different 
from the attracting pre-planned virtual trajectory (Hogan 1984). Simulations by Flash 
(1987), have suggested that realistic multi-joint arm movements at moderate speed can be 
generated by moving the hand equilibrium position along a pre-planned minimum-jerk 
virtual trajectory. The interactions of the dynamic properties of the arm and the attracting 
virtual trajectory create together the actual realized trajectory. Flash did not suggest a 
solution to problem I. 
A static local optimization algorithm related to the equilibrium-trajectory hypothesis and 
called backdriving was proposed by Mussa-Ivaldi et al. (1991). This algorithm can be used 
to solve problem I only after the virtual trajectory is known. The virtual trajectory is not 
necessarily a minimum-jerk trajectory. Driving the arm from a current equilibrium position 
to the next one on the virtual trajectory is performed by two steps: 1) simulate a passive 
displacement of the arm to the new position and 2) update the muscle forces so as to 
eliminate the induced hand force. A unique active change (step 2) is chosen by finding 
these muscle forces which minimize the change in the potential energy stored in the 
muscles. Using a static model of the monkey's arm, the first author has analyzed this 
sequential computational approach, including a solution for both the trajectory 
determination (A) and the muscle forces (I) problems (Dornay 1990, 1991a, 1991b). 
The equilibrium-trajectory hypothesis which is using the minimum-jerk model was 
criticized by Katayama and Kawato (in preparation). According to their recent findings, 
the values of the dynamic stiffness used by Flash (1987) are too high to be realistic. They 
have found that a very complex virtual trajectory, completely different from the one 
predicted by the minimum-jerk model, is needed for coding realistic hand movements. 
Simulation of Optimal Movements Using the Minimum-Muscle-Tension-Change Model 629 
2 GLOBAL DYNAMIC OPTIMIZATIONS 
A set of global dynamic optimizations have been proposed by Uno et al. (1989a, 1989b). 
Uno et al. suggested that the dynamic properties of the arm must be considered by any 
algorithm for controlling hand movements. They also proposed that the hand trajectory 
and the motor commands (joint torques, muscle tensions, etc.,) are computed in parallel. 
2.1 THE MINIMUM-TORQUE-CHANGE MODEL 
Uno et al. (1989a) have proposed the minimum-torque-change model. The model proposes 
that the hand trajectory and the joint torques are determined simultaneously, while the 
algorithm minimizes globally the rate of change of the joint torques. The minimum-torque- 
change model was criticized by Flash (1990), saying that the rotary inertia used was not 
realistic. If Flash's inertia values are used then the hand path predicted by the minimum- 
torque-change model is curved (Flash 1990). 
2.2 THE MINIMUM-MUSCLE-TENSION-CHANGE MODEL 
The minimum-muscle-tension-change model (Uno et al. 1989b, Dornay et al. 1991) is a 
parallel dynamic optimization approach in which the trajectory determination problem (A) 
and the muscle force generation problem (II) are solved simultaneously. No explicit 
trajectory is imposed on the hand, but that it must reach the final desired state (position, 
velocity, etc.) in a pre-specified time. The numerical solution used is a "penalty" method, 
in which the controller minimizes globally by iterations an energy function E: 
e = o + (1) 
E is the energy that must be minimized in iterations. E o is a collection of hard 
constraints, like, for example that the hand must reach the desired position at the specified 
time. E s is a smoothness constraint, like the minimum-muscle-tension-change model.  
is a regularization function, that needs to become smaller and smaller as the number of 
iterations increases. This is a key point because the hard constraints must be strictly 
satisfied at the end of the iterative process.  is a small rate term. The smoothness 
constraint E s , is the minimum-muscle-tension-change model, defined as: 
Es = O.S f E ( df i / dt) 2 clt 
i--1 
t o 
(2) 
f. is the tension of muscle i, n is the total number of muscles, t o is the initial time and 
t is the final time of the movement. 
Preliminary studies have shown (Uno et al. 1989b) that the minimum-muscle-tension- 
change model can simulate reasonable hand movements. 
630 Dornay, Uno, Kawato, and Suzuki 
3 THE MONKEY'S ARM MODEL 
The model used was recently described (Domay 1991a; Domay et al. 1991). It is based 
on anatomical study using the Rhesus monkey. Attachments of 17 shoulder, elbow and 
double joint muscles were marked on the skeleton. The skeleton was cleaned and 
reassembled to a natural configuration of a monkey during horizontal arm movements 
(Fig. 1). X-ray analysis was used to create a simplified horizontal model of the arm (Fig. 
1). Effective origins and insertions of the muscles were estimated by computer 
simulations to ensure the postural stability of the hand at equilibrium (Dornay 199 la). The 
simplified dynamic model used in this study is described in Dornay et al. (1991). 
I-BRA 
o 
E :*W 
o 
o 
- ! 
\ o SHOULDER 
O-BRA O I-PMA-CAPS O-PM -CAPS 
  CLAVICLE 
.' \\ 
'\ 
o \ , 
STERNUM   
 ;, 
Figure 1: The Monkey's Arm Model. Top left is a ventral view of the skeleton. Middle 
right is a dorsal view. The bottom shows a top-down X-ray projection of the skeleton, 
with the axes marked on it. The photos were taken by Mr. H.S. Hall, MIT. 
Simulation of Optimal Movements Using the Minimum-Muscle-Tension-Change Model 631 
4 THE BEHAVIORAL TASK 
We tried to simulate the horizontal arm movements reported by Uno et al. (1989a) for 
human subjects, using the monkey's model. Fig. 2 (left) shows a top view of the hand 
workspace of the monkey (light small dots). We used 7 hand positions defined by the 
following shoulder and elbow relative angles (in degrees): T 1 {14,122}; T 2 {67,100}; T 3 
{75,64}; T 4 {63,45}; T s {35,54}; T 6 {-5,101} and T 7 {-25,45}. The joint angles used by 
Uno et al. (1989a) for T 4 and T 7, {77,22} and {0,0}, are out of the workspace of the 
monkey's hand (open circles in Fig 2, left). We approximated them by our T 4 and T 7 
(filled circles). The behavioral task that we simulated using the minimum-muscle-tension- 
change model consisted of the 4 trajectories shown in Fig. 2 (right). 
5 SIMULATION RESULTS 
Figure 2 (right) shows the paths (T2->T6), (T3->T), (Tn->T1), and (T7->Ts). The paths T2- 
>T6, T3->T6 and T7->T 5 are slightly convex. Slightly convex paths for T=->T6 were 
reported in human movements by Flash (1987), Uno et al. (1989a) and Morasso (1981). 
Human T3->T6 paths have a small tendency to be slightly convex (Uno et al. 1989a; Flash 
(1987). In our simulations, T=->T6 and T3->T6 have slightly larger curvatures than those 
reported in humans. Human large movements from the side of the body to the front of the 
body similar to our T7->T 5 were reported by Uno et al. (1989a). The path of these 
movements is convex and similar to our simulation results. The simulated path of Tn->T 1 
is slightly curved to the left and then to the right, but roughly straight. The human's T n- 
>T paths look slightly straighter than in our simulations (Uno et al. 1989a; Flash 1987). 
Y 
L 03 
 0.2 
E 
O.' 
0.0 
+ % 
-C'.2 -0 1 0.0 0.1 0.2 0 3 0 
X (meters) 
Figure 2. The Behavioral Task. The left side shows the hand workspace (small dots). The 
shoulder position and origin of coordinates (0,0) is marked by +. The elbow location 
when the hand is on position T 1 is marked by E. The right side shows 4 hand paths 
simulated by the minimum-muscle-tension-change model. Arrows indicate the directions 
of the movements. 
632 Dornay, Uno, Kawato, and Suzuki 
Fig. 3 shows the corresponding simulated hand velocities. The velocity profiles have a 
single peak and are roughly bell shaped, like those reported for human subjects. The left 
side of the velocity profile of Ta->T x looks slightly irregular. 
The hand trajectories simulated here are in general closer to human data than those 
reported by us in the past (Dornay et al. 1991). In the current study we used a much 
slower protocol for reducing  than in the previous study, and we think that we are closer 
now to the optimal solution of the numerical calculation than in the previous study. 
Indeed, the hand velocity profiles and muscle tension profiles look smoother here than in 
the previous study. It is in general very difficult to guarantee that the optimal solution is 
achieved, unless an unpractical large number of iterations is used. Fig. 4 (top,left) shows 
the way E o and E s of equation 1 are changing as a function 3 for the trajectory T7->T 5. 
Ideally, both should reach a plato when the optimal solution is reached. The muscle 
tensions simulated for T7->T 5 are shown in Fig. 4. They look quite smooth. 
T2 .-T 6 
Figure 3. The Hand Tangential Velocity. 
o 
Tinn(;} 
6 DISCUSSION 
Various control strategies have been proposed to explain the roughly straight hand 
trajectory shown by primates in planar reaching movements. The minimum-jerk model 
(Flash & Itogan 1985) takes into account only the desired hand movement, and 
completely ignores the dynamic properties of the ann. This simplified approach is a good 
approximation for many movements, but cannot explain some experimental evidence (Uno 
et al. 1989a). A more demanding approach, the minimum-torque-change model (Uno et 
al. 1989a), takes into account the dynamics of the arm, but emphasizes only the torques 
at the joints, and completely ignores the properties of the muscles. This model was 
criticized to produce unrealistic hand trajectories when proper inertia values are used 
(Flash 1990). A third and more complicated model is the minimum-muscle-tension-change 
model (Uno et al. 1989b, Dornay et al. 1991). The minimum-muscle-tension-change model 
was shown here to produce gently curved hand movements, which although not identical, 
are quite close to the primate behavior. In the current study the initial and final tensions 
of the muscles were assumed to be zero. This is not a realistic assumption since even a 
static hand at an equilibrium is expected to have some stiffness. Using the minimum- 
muscle-tension-change model with non-zero initial and final muscle tensions is a logical 
Simulation of Optimal Movements Using the Minimum-Muscle-Tension-Change Model 633 
1.2 
0 
0 ' -[og(.)  o 
Muscle 
Sel 
Forces 
Seg 
Se3 
Se4 
Se5 
Ef12 
Tmme(s) 
S f9 
Ell3 
EelO 
Ef14 
Eell 
, Del5 
....' 
Figure 4. Numerical Analysis and Muscle Tensions For T7->T s. S=shoulder, E=elbow, 
D=double-joint muscle, e=extensor, f=flexor. 
study which we intend to test in the near future. Still, the minimum-muscle-tension-change 
model considers only the muscle moment-arms (p.) and momvels (0p./00) and ignores 
the muscle length-tension curves. A more complicated model which we are studying now 
is the minimum-motor-command-change model, which includes the length-tension curves. 
634 Dornay, Uno, Kawato, and Suzuki 
Acknowledgements 
M. Domay and M. Kawato would like to thank Drs. K. Nakane and E. Yodogawa, ATR, 
for their valuable help and support. Preparation of the paper was supported by Human 
Frontier Science Program grant to M. Kawato. 
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PART X 
APPLICATIONS 
