Optimal trajectory of human arm reaching movements in dynamical environments Ken Ohta1 Rafael Laboissi`ere1 Mikhail M. Svinin2 ZhiWei Luo2 Shigeyuki Hosoe2 1 Sensorimotor Coordination, Max Planck Institute for Psychological Research, Munich, Germany 2 Bio-Mimetic Control Research Center, RIKEN, Nagoya, Japan

1

Introduction

3

In this study, we examine the optimal trajectory of human arm reaching movement in dynamical environment such as kinematically constrained movement and movement in velocity-dependent field. In these movements there is force interaction with the external environment via hand. One approach to accounting for trajectory formation is via optimization theory. Several optimal models have been proposed for explaining the optimal trajectory of unconstrained reaching movement. However, the trajectory formation of the force-interactive tasks is not clear. To explain the formation of the trajectory in dynamical environment, we also exploit the optimization approach, selecting the conventional criteria which have been successfully used in predicting the unconstrained reaching movements. The analysis shows that the conventional criteria, which are the minimum jerk criterion (Flash & Hogan, 1985) and the minimum motor command change criterion (Uno et al., 1989), do not match the experimental data. Consequently, we came up with a novel form of the optimality criterion, Z J=

T

T T F˙ F˙ + wf˙ f˙ dt

Fig.2 (solid line) shows experimental results after the training. Angular velocity curves are roughly bell shape that consistence with the results of the unconstrained reaching movement. However, the angular velocity shapes have local minimums. If the optimal trajectory of the crank task are defined purely kinematic criterion, as it is in the minimum jerk criteria, the angular velocity profiles should be completely bell shaped and is not able to predict the force trajectory. Fig.2 shows the comparison of minimum muscle force change criterion and experimental results. As can be seen, the minimum muscle force change criterion does not match well the experimental data. A large discrepancy between the theoretical and experimental curves can be seen in observing the predicted hand force. The comparisons between the simulation results of combined criterion (1) and the experimental data are shown in Fig.3. As can be seen, not only the kinematic trajectories are now predicted accurately enough but also the force profiles capture the tendency of human motions. Especially the predicted crank angular velocity captures the features of local minimum of the experimental data. Thus, we conclude that the combined (minimum hand force plus muscle force change) criterion match the experimental data much better than the minimum muscle force change criterion. This indicate that in constrained movements the central nervous system naturally select a moving contact frame as the criterion for the trajectory formation. This combined criterion (1) predicts also well the trajectories of reaching movement in velocity-dependent force filed (see Fig.4). When subjects are exposed to this environment their movements are initially perturbed, but return to approximately their normal pattern after several hundred movements (Shadmehr et al., 1994, 1997). However, after training in the viscous force environment, human subjects move with slightly curved trajectories. Minimum jerk criterion is not able to capture this feature of kinematic trajectory.

(1)

0

where F is the hand contact force vector and f is muscle force vector. This combined criterion has two components. One component is the hand force change and the other is the muscle force. The analysis shows that in constrained point-to-point motions this combined criterion match experimental data much better than the conventional criteria. This criterion predicts even the muscle activities. The criterion is good enough in capturing the basic tendency of the muscle activities.

2

Comparative analysis

Method

To investigate these problems, we examine the trajectories of the human arm in a crank rotation task (Fig.1). Subjects were trained to make movements in horizontal plane while holding the crank handle. The subject’s wrist joint was fixed by a cast in order to exclude the kinematic redundancy of the hand. Subjects sat in a chair with a harness in order to constrain the shoulder joint and grasped the handle of the crank lever with their right hand. AC servo motor was connected to the crank’s shaft for producing viscous frictions. Subjects were asked to make circular movement by their own paces, as comfortable as possible. Each subjects performed 200–300 reaching movements. To examine the muscle force pattern during the movement, surface electromyogram (EMG) signals are recorded from the muscles .

4

Discussion

The results obtained in our research indicate that both the smoothness of the hand force and that of the actuating force are of primary importance in the force interactive tasks where the force perception plays a fundamental role in the trajectory planning. Purely kinematic-based criterion is not appropriate for the criterion as the trajectory formation of the movement in dynamical environment. We suggest that dynamic-based criterion mainly plays a role in the trajectory formation. 1

handle

F θ

q2

a

elbow

Cast

f4

Harness

f6 AC servo motor

f2 τ1

τ2 f3 f5

b

f1 q1 shoulder

.

Figure 1: Experimental setup c Experiment

a Crank angle θ (rad) Joint anglar velocity Crank anglar velocity q (rad/sec) θ (rad/sec)

b

3 2 1

.

-1 -2 -3

0.5

-1 -2 -3

1

Experiment

1.5

f

0.5

1

1.5

0.5

1

1.5

Combined criterion

3 2 . 1 q1 1

0.5

-1 . q2 -2 -3

1.5

Time (s)

g Muscle force (N)

IEMG

3 2 1 -1 -2 -3

1.5 EMG4

e

Experiment Combined

EMG5

1

10 (N)

e

EMG6

0.5

Experiment

-2 -4 -6 -8 -10

.

-2 -4 -6 . -8 -10

c

d

Muscle force change

d

3 2 1

Crank angle θ (rad)

Balancer

Joint anglar velocity Crank anglar velocity q (rad/sec) θ (rad/sec)

r

10 (N)

f6 0 400

f5 0 400

f4 0 400

f3

EMG3 0 400

EMG2 EMG1 0.2

Muscle force change

1

0.6

1

0.8

1.2

1.4

f1 0

0.25

0.5

0.75

1

1.25

1.5

Time (s)

. q1 0.5

0.4

f2 0 400

Figure 3: Comparison of experimental data and prediction by the combined criterion. The last five trajectories of experiment are shown in graph a–c. Each time interval of predicted data is 0.05 s. Movement time T = 1.50 s. A typical force vectors are shown in graph d. a Crank angle. b Crank angular velocity. c Joint angular velocities. d Typical hand contact force vectors of experiment. e Predicted hand contact force vectors. f iEMG data. Each signal is plotted in steps of 2.0 %. g Predicted muscle force trajectories. Each force curve is plotted in steps of 500 N

1.5

. q2 Time (s)

Figure 2: Comparison of experimental data and prediction by the minimum muscle force change criterion. The last five trajectories of experiment are shown in graph a– c. Each time interval of predicted data is 0.05 s. Movement time T = 1.50 s. A typical force vectors are shown in graph d. a Crank angle. b Crank angular velocity. c Joint angular velocities. d Typical hand contact force vectors of experiment. e Predicted hand contact force vectors

References

0.1

[1] T. Flash and N. Hogan, ”The coordination of arm movements”, J. Neuroscience, vol.5, pp.1688-1703, 1985.

y position (m)

0.05

[2] Y. Uno, R. Suzuki and M. Kawato, ”Minimum muscle tension change model which produces human arm movement.”, Proceedings of the 4th Symposium on Biological and Physiological Engineering (in Japanese)”, pp.299-302, 1989.

0

-0.05

[3] R. Shadmehr and F. A. Mussa-Ivaldi, ”Adaptive representation of dynamics during learning of a motor task”, J. of Neuroscience, vol.14, no.5, pp.3208-3224, 1994.

-0.1 -0.1

-0.05

0

0.05

0.1

x position (m)

Figure 4: Trajectories in viscosity field {{0,13},{-13,0}} predicted by combined criterion

[4] R. Shadmehr and T. Brashers-Krug, ”Functional stages in the formation of human long-term motor memory” J. of Neuroscience, vol.17, no.1, pp.409-419, 1997. 2