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The central nervous system stabilizes unstable dynamics by learning optimal impedance

Abstract

To manipulate objects or to use tools we must compensate for any forces arising from interaction with the physical environment. Recent studies indicate that this compensation is achieved by learning an internal model of the dynamics1,2,3,4,5,6, that is, a neural representation of the relation between motor command and movement5,7. In these studies interaction with the physical environment was stable, but many common tasks are intrinsically unstable8,9. For example, keeping a screwdriver in the slot of a screw is unstable because excessive force parallel to the slot can cause the screwdriver to slip and because misdirected force can cause loss of contact between the screwdriver and the screw. Stability may be dependent on the control of mechanical impedance in the human arm because mechanical impedance can generate forces which resist destabilizing motion. Here we examined arm movements in an unstable dynamic environment created by a robotic interface. Our results show that humans learn to stabilize unstable dynamics using the skilful and energy-efficient strategy of selective control of impedance geometry.

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Figure 1: Experimental set-up and learning in a divergent force field (DF).
Figure 2: Error evolution during learning the DF, endpoint force–time profiles after learning for the same subject as in Fig. 1, and forces at the movement midpoint for all subjects.
Figure 3: Adaptation of stiffness geometry to the DF.
Figure 4: Stiffness and stability.

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Acknowledgements

The experiments were performed at ATR. We thank A. Illiesch and T. Flash for discussions. We thank A. Smith, S. Scott, D. Wolpert, Z. Ghahramani, and S. Schaal for giving valuable comments on an earlier version of the manuscript. This research was supported by the Special Coordination Fund for promoting Science and Technology of the Science and Technology agency of the Japanese Government; the Swiss National Science Foundation; the Natural Sciences and Engineering Research Council of Canada; and the Human Frontier Science Program.

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Correspondence to Mitsuo Kawato.

Supplementary information

This note describes the details of how the correlation between stiffness and force was evaluated in the paper "The CNS Skillfully Stabilizes Unstable Dynamics by Learning Optimal Impedance" by E. Burdet et al.

When the arm is constrained in a horizontal plane at shoulder height, it can be modelled as a two-link mechanism. The associated 2x2 joint stiffness matrix K is defined by

(A1)

, i.e. dt=(dts, dte)T is the elastic resistance to an infinitesimal displacement dq=(dqs, dqe)T. K can be estimated in real movements from the restoring force in response to small perturbations relative to a prediction of the undisturbed trajectory (see Burdet et al., J Biomech 33, 1705-1709, 2000).

The difference of stiffness between the DF and NF conditions can be estimated by subtracting from , resulting in

(A2)

where and

Gomi and Osu (1998) have shown that in static posture joint stiffness is well correlated with joint torque. i) The shoulder stiffness Kss is well correlated with the shoulder torque |ts| and weakly correlated with the elbow torque |te|; ii) the bi-articular stiffness elements Kse and Kes are almost equal, well correlated with |te| and weakly correlated with |ts|; iii) the elbow stiffness Kee is well correlated with |te| and weakly with |ts|. Similarly we assume that

(A3)
, 1, 2, 3 constants and

(A4)
, 1, 2, 3 constants where K0 is the passive stiffness. Eqs A4 – A3 yields

(A5)
, 1, 2, 3 constants

To move the arm, the muscles have to produce a force corresponding to the arm dynamics and the force measured at the handle, i.e. t = td + th. As the movement kinematics are not different in the two force fields after learning, we can assume that tDF,d d tNF,d ? td. Using a two link model of the arm dynamics, we have confirmed by simulation that for all subjects |tDF| = |td + tDF,h| d td + tDF,h and |tNF| = |td + tNF,h| d td + tNF,h at the midpoint of the movement. Therefore Eq A5 can be simplified to

(A6)
, 1, 2, 3 constants

tDF,h and tNF,h were measured during movement using the force sensor at the handle of the PFM.

Because endpoint stiffness in the DF was modified principally in the x-direction, this relation was tested for endpoint force in the x-direction Fx. The relation was transformed from joint coordinates to endpoint coordinates using the relationship t = JT F with the Jacobian J:

(A7)

To evaluate the correlation between the change of stiffness in the DF relative to the NF and the corresponding difference of torque, we have used Eq A6 substituted into Eq A7. As the trajectories of movements in the VF and DF after learning were similar to movements in the NF and J(qs, qe) changed little in the region of stiffness measurement, we can assume that J is constant. Therefore, Eq A7 can be simplified to the linear equation:

(A8)

in which g 1 = g11 a1, g2 = g11 a2 + g12 a3, g3 = g12 a2 with .

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Burdet, E., Osu, R., Franklin, D. et al. The central nervous system stabilizes unstable dynamics by learning optimal impedance. Nature 414, 446–449 (2001). https://doi.org/10.1038/35106566

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