SICE Annual Conference in Fukui, August 4-6, 2003 Fukui University, Japan Human impedance characteristics during reaching movements Jun Izawa, Toshiyuki Kondo, Koji Ito Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, 4259 Nagatsuta Midori Yokohama 226-8502, Japan izawa@ito.dis.titech.ac.jp Abstract: This paper addresses the characteristics of human impedance learning during reaching movement. The motor learning model during a reaching movement proposed in our previous work predicts that the adjustment of hand’s impedance must be calculated based on state value of the biological controlled systems in human brain. Several results in psychophysical experiments is shown in this paper to reveal the human impedance characteristics. The experiment was designed by using high-performance manipulandom. Keywords: human movement, impedance learning, reaching movement 1. Introduction that the stiffness control IM is stored in same portion of the brain as torque control IM. The aim of our research presented in this paper is to confirm whether the impedance control is stored in a brain as a IM and what is a coordinate system of impedance control IM. With practice, humans adapt to externally applied force. It is believed that the internal representation is stored in central nerves system for compensating the applied force. This internal representation is called as Internal Model(IM) 1) . The coordinate system that is used to store IM is crucial to understanding the nature of the IM. The motor learning model during a reaching movement proposed in our previous work predicts that the adjustment of hand’s impedance is calculated based on state value of the biological controlled systems 3) . This system could get the robust motion against disturbance force through reaching motion by using reinforcement learning. Our question, here, is whether humans can get the impedance control as a function of states related to reaching motion and which the coordination system of the representation in such a state is. 2. Transfer of Impedance Control across Arm Configurations Here, our question is, if there is a kind of force field at a work space that causes a high impedance for stable reaching motion, whether the force field transfer across workspace locations. To confirm this we propose a new environment named random viscosity field(RVF) shown in following equation. f w = n · B ẋ, 1)
According to Shadmerh and Mussa-Ivaldi(1994) , IM of the force field designed in joint space practiced at left workspace can transfer to the right workspace, although that designed on hand coordinate system cannot transfer across workspace. This study showed that the IM, mapping from states value to motor command, is expressed in join space. B= b11 b21 b12 b22  , where n is a filtered Gaussian noise which variance is 1. Thus, f w is a white noise which variance equals B ẋ. The hand trajectory may be drawn as in Fig.1. The expectation of the hand position in RVF is equal to the hand position in NF. The variance of the hand position in RVF is caused only by f w . Intuitively, if the subjects are asked to reach a hand to target position within a constant duration time decreasing the variance at goal position, they will increase their hand stiffness and hand viscosity to avoid a deviation from mean value. If we consider metabolic cost such as muscle fatigue, we won’t adopt the strategy co-contracting all the muscles. Then, the major axis of ellipse of the stiffness may be parallel to f w . Burdet confirmed that humans could learn to stabilize unstable dynamics through optimizing the stiffness of the hand2) . They concluded that the impedance control is a kind of IM. However, nobody has shown that the impedance control is represented in joint coordinate. Results showing that the impedance control is stored in the memory based on joint space could support a idea of the hypotheses that the impedance control is a kind of IM and this memory has a same characteristic as the torque control IM. Furthermore, it might be said -2106- PR0001/03/0000-2106 ¥400 © 2003 SICE Y Left Workspace E[Xt] Right Workspace Var[Xt] 10cm goal goal F Elbow Displacement R Stiffness ellipse 90o start Shoulder start Force exerted at hand X Displacement Figure 3: Configurations of a two joint arm at two work space Figure 1: The predictive hand trajectory and disturbed force in random viscosity field 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.1 -0.05 0 0.05 0.1 -0.05 -0.1 (a) Period B -0.05 0 0.05 0.1 (b) Period C 0.15 0.1 Figure 2: manipulandom 0.05 0 Suppose the noise is exactly perpendicular to hand velocity, the variance at end point is  V  = 0 -0.05 0 0.05 0.1 (c) Period D T = 0 -0.05 -0.1 σ 2 dt, (1) bvdt, (2) T Figure 4: Hand trajectories in each period = b · L, 3. Methods (3) , where T is duration time of reaching motion, v is a hand velocity b is a coefficient of viscosity field , and L is a distance from starting point to goal point. If the subjects is asked to minimize the variances at end point, the optimal solution is to minimize L. Thus, a path that the subjects will acquire is shortest path. This solution dose not owe to duration time T . The solution is same as that in reaching motion in normal situation. To confirm a generalization of RVF, subjects learn the field f = nB ẋ at left workspace. This field is translation invariant in hand coordinate. Then, they are asked to do reaching motion at right work space. If the reaching motion succeed at right workspace without practice, the internal representation of RVF is based on hand coordination. If not, the representation is based on the other coordination. We summarize a relationship between the results and the hypotheses in Tab.1 A healthy subject (right handed; 23 yr old) is participated in the study. The seated subject grasped a handle of a impedance controlled manipulandom. The manipulandom was controlled so that the subject could feel that the handle was light wight(1.5 kg). The noise exerted at hand is band-pass filtered(5-20Hz). 3.1 Experiment 1 The hand position was recorded during movement at each trial. the subject learned a reaching movement during 5 periods( A, B, C, and D). 150 trials composes each period. In the period B, C and D, the random noise field was introduced at the subject’s hand defined in the following equation.
f w = n · B ẋ, B = -2107- 0 b1 2 b2 1 0  , Table 1: Results and Hypotheses coordinate !-success X-failure results hand joint ! ! ! X X ! X X hypotheses(function of what) context of environment hand coordinate joint coordinate other coordinate(e.g.muscle coordinate) 0.15 0.15 0.1 0.1 0.05 0.05 0 0 -0.05 -0.1 -0.05 0 0.05 0.1 (a) Period A -0.05 -0.1 -0.05 0 0.05 (b) Period D (noise field) 0.15 0.1 0.05 Figure 5: Variance at 400ms 0 -0.05 -0.1 Note that, the null field is exerted in the period A and catch trial is introduced in the period D. -0.05 0 0.05 0.1 (c) Period D (null field) Figure 6: Catch trials and after effects 3.2 Experiment 2 4. Results At first, the subject asked practice at both workspace (shown in Fig.3) in forward directionunder null field. Next, reaching motion under RVF in the forward direction at left workspace(Left-F) practiced. Then, the subject was asked to reach at right workspace in the forward direction(Right-F) and right direction(RightR) under RVF. The practice at left workspace is composed of 400 trials. The test trial at right workspace in each direction is composed of 50 trials. The trajectory error is calculated with the following equation. Error = T 
x∗ (t) − x(t)
, 4.1 Experiment 1 Fig. 4 shows hand paths in each period. The deviation from a normal reaching path decreased as learning. Showing it clearly, the variance of the hand position at 400ms of the reaching movement was calculated. Fig.5 shows the variances at each period. Clearly, the variances decreased as learning and that on period D is same level as that on period A whose environment is a null field. Period D was consist of random force field trials and null field trials. The null field was introduced randomly. Thus the trials of null field is called as catch trial. Fig.6(a) shows hand paths of the null field trial on period A. Fig.6-(b) shows hand paths of the random force field trial on period D. Fig.6-(c) shows hand paths of the (4) 0 where x∗ is optimum trajectory. -2108- 0.1 catch trials. The after effect dose not appearer in the catch trials. Thus, the causes that the variance decreases is deffer from what human could learn in viscosity force field addressed by Shadmher 1) . Namely, what the subject learned is not a mapping from states value to a torque. 0.45 0.45 F F 0.4 L R Y[m] Y[m] 0.4 0.35 0.3 Error 50 B B Right-F Right-R Left-F 60 R 0.3 80 70 L 0.35 0.25 -0.45 -0.4 -0.35 -0.3 0.25 0.25 -0.25 0.3 X[m] 0.35 0.4 0.45 X[m] 40 30 (a) Left Workspace, Cartesian Space 20 (b) Right Workspace , Cartesian Space 10 0 15 20 25 30 35 40 45 50 110 Trial B elbow(degree) 110 Figure 7: Error at each trial R elbow(degree) 10 B 90 L L 90 F R 70 70 70 90 F 110 -20 shoulder(degree) (c) Left Workspace Joint Space 0 20 Shoulder(degree) , (d) Right Joint Space Workspace, Figure 9: RVF at each workspace the controller can transfer in the other direction(Right -R). Note that ,mean error in Right-F and Right-R is different(P < 0.001). Why dose the difference in error correction occur? Fig.9 shows the RVF at each workspace in each direction. Arrow shows force or torque excerted at hand at the point in reaching path. Indeed, motor control strategy in Left-F dose not generalize in Right-F in spite of the similarity in the shape of the RVF(Fig9-(a),(b)). However, the shape of the RVF in joint space is not similar in forward direction(Fig9-(c),(d)). On the other hand, the shape of the RVF in Left-F and Right-R is similar in joint space. This is why the error in Righ-R is smaller that that in Righ-F, which shows that the motor memory in Right-R is affected by practice in LeftF. This suggests that the impedance control could be stored in the memory based on joint coordinate. Figure 8: the mean value of the error at each period 4.2 Experiment 2 Reaching motion under RVF in the forward direction at left workspace(Left-F) practiced. Then, the subject is asked to reach at right workspace in the forward direction(Right-F) and right direction(Right-R) under RVF. Thick dashed line in Fig 7 shows trajectory error at each trials after practice in Left-F. Solid line shows the error in Right-F. The error in Right-F seems to be greater than that in Left-F especially in early trials. Dashed line shows the error in Right-R. The error in Right-R is smaller than that in Right-F. Fig. 8 shows the mean value of the error at each period. The controller stored in memory could not correct the error in Right-F, although Left-F and Right-F is same direction in Cartesian space. On the other hand, 5. Conclusion These results imply that the subject can adjust the control strategy to avoid a deviation from a natural -2109- trajectory of reaching movements against dynamically changed random environment. The IM for preventing from deviation in the trajectory is not a mapping to torque but to a stiffness, because the after effect is not shown in catch trials. Thus, we can conclude that the subjects could study IM of Stiffness control against dynamically changed environment. Moreover, the IM of stiffness control is stored in the memory based on joint coordinate. It is likely that the IM of impedance control has a same characteristic and that is a part of IM which is a mapping from state space not only to torque but also to more general motor commands including impedance controll. Acknowledgment This research was supported in part by Grants from the Japanese Ministry of Education, Culture, Sports, Science and Technology (No.14350227, 14750362) and Mitsutoyo Association for Science and Technology. References [1] Shadmehr R, Mussa-Ivaldi FA(1994) Adaptive representation of dynamics during learning of a motor task. Journal of Neuroscience 14 (5): 32008-2334 [2] Burdet E, Osu R et al.(2001) The central nervous system stabilizes unstable dynamics by learning optimal impedance. Nature 401(22): 446-449 [3] Izawa J, Kondo T, Ito K, (2002) Biological Robot Arm Motion through Reinforcement Learning, Proceedings of International Conferencee on Robotics and Automation:WAII-11-3 -2110- View publication stats