Computational model of motor learning and perceptual change

Abstract

Motor learning in the context of arm reaching movements has been frequently investigated using the paradigm of force-field learning. It has been recently shown that changes to somatosensory perception are likewise associated with motor learning. Changes in perceptual function may be the reason that when the perturbation is removed following motor learning, the hand trajectory does not return to a straight line path even after several dozen trials. To explain the computational mechanisms that produce these characteristics, we propose a motor control and learning scheme using a simplified two-link system in the horizontal plane: We represent learning as the adjustment of desired joint-angular trajectories so as to achieve the reference trajectory of the hand. The convergence of the actual hand movement to the reference trajectory is proved by using a Lyapunov-like lemma, and the result is confirmed using computer simulations. The model assumes that changes in the desired hand trajectory influence the perception of hand position and this in turn affects movement control. Our computer simulations support the idea that perceptual change may come as a result of adjustments to movement planning with motor learning.

Appendices

Appendix 1: 2-link system dynamics and kinematics

The 2-link dynamics of the arm in the horizontal plane are described as follows:

$$\begin{aligned} {M}(\varvec{\theta }) {\varvec{\ddot{\theta }}} + {C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) {\varvec{\dot{\theta }}} = \varvec{\tau } \end{aligned}$$

(39)

Here, ${M}(\varvec{\theta })$ and $C(\varvec{\theta }, {\varvec{\dot{\theta }}})$ are given as follows:

$$\begin{aligned} {M}(\varvec{\theta })&= \left[ \begin{array}{cc} M_{11} &{} M_{12} \\ M_{21} &{} M_{22} \end{array} \right] \end{aligned}$$

(40)

$$\begin{aligned} {C}(\varvec{\theta }, {\varvec{\dot{\theta }}})&= \left[ \begin{array}{cc} C_{11} &{} C_{12} \\ C_{21} &{} C_{22} \end{array} \right] \end{aligned}$$

(41)

$$\begin{aligned} M_{11}&= I_0 + I_1 + m_0 \ell _0^2 + m_1 L_0^2 +m_1 \ell _1^2\nonumber \\&\quad + 2 m_1 L_0 \ell _1 \cos \theta _1 \end{aligned}$$

(42)

$$\begin{aligned} M_{12} = M_{21}&= I_1 + m_1 \ell _1^2 + m_1 L_0 \ell _1 \cos \theta _1 \end{aligned}$$

(43)

$$\begin{aligned} M_{22}&= I_1 + m_1 \ell _1^2 \end{aligned}$$

(44)

$$\begin{aligned} C_{11}&= - m_1 L_0 \ell _1 \dot{\theta }_1\sin \theta _1 \end{aligned}$$

(45)

$$\begin{aligned} C_{12}&= - m_1 L_0 \ell _1 (\dot{\theta }_0 + \dot{\theta }_1) \sin \theta _1 \end{aligned}$$

(46)

$$\begin{aligned} C_{21}&= m_1 L_0 \ell _1 \dot{\theta }_0\sin \theta _1 \end{aligned}$$

(47)

$$\begin{aligned} C_{22}&= 0 \end{aligned}$$

(48)

As shown in Fig. 2, $\theta _i$ denotes joint angle, $L_i$ is link length, $\ell _i$ is the distance to center of mass of the link, $m_{i}$ is mass of the link, $I_i$ is the moment of inertia about the center of mass, $\tau _i$ denotes joint torque and $i$ distinguishes the shoulder $(i=0)$ and the elbow $(i=1)$. The operation $\dot{}$ denotes time derivative.

Note here that the left side of Eq. (39) can be described as the product of ${Y}_L$ and $\varvec{\sigma }_L$ as shown in (7). In the case of 2-link arm dynamics, they are given as

$$\begin{aligned}&{{Y}_{L}(\varvec{\theta }, {\varvec{\dot{\theta }}}, {\varvec{\dot{\theta }}}_{ r}, {\varvec{\ddot{\theta }}}_{ r})} =\nonumber \\&\left[ \begin{array}{cccc} \ddot{\theta }_{r0} &{} (\ddot{\theta }_{r0} + 2 \ddot{\theta }_{r1}) \cos \theta _1 + \dot{\theta }_{r1} (2 \dot{\theta }_0 +\dot{\theta }_1) \sin \theta _1 &{} \ddot{\theta }_{r1} \\ 0 &{}2 \ddot{\theta }_{r0} \cos \theta _1 - \dot{\theta }_{r0}^2 \sin \theta _1 &{} \ddot{\theta }_{r0} \!+\! \ddot{\theta }_{r1} \end{array}\right] \nonumber \\ \end{aligned}$$

(49)

$$\begin{aligned}&\varvec{\sigma }_L = \left[ \begin{array}{cccc} \sigma _{L1}&\sigma _{L2}&\sigma _{L3} \end{array}\right] ^T \end{aligned}$$

(50)

$$\begin{aligned}&\sigma _{L1} = I_0 + I_1 + m_0 \ell _0^2 + m_1 L_0^2 +m_1 \ell _1^2 \end{aligned}$$

(51)

$$\begin{aligned}&\sigma _{L2} = m_1 L_0 \ell _1 \end{aligned}$$

(52)

$$\begin{aligned}&\sigma _{L3} = I_1 \!+\! m_1 \ell _1^2 \end{aligned}$$

(53)

Then, the following equation holds using ${\varvec{\hat{\sigma }}}_L$:

$$\begin{aligned} {Y}_L (\varvec{\theta }, {\varvec{\dot{\theta }}}, {\varvec{\dot{\theta }}}_r, {\varvec{\ddot{\theta }}}_r) {\varvec{\hat{\sigma }}}_L = {\hat{ M}}(\varvec{\theta }) {\varvec{\ddot{\theta }}}_{r} + {\hat{ C}}(\varvec{\theta }, {\varvec{\dot{\theta }}}) {\varvec{\dot{\theta }}}_{r} \end{aligned}$$

(54)

${\hat{M}}(\varvec{\theta })$ and ${\hat{C}}(\varvec{\theta }, {\varvec{\dot{\theta }}})$, the estimates of ${M}(\varvec{\theta })$ and ${C}(\varvec{\theta },{\varvec{\dot{\theta }}})$, respectively.

On the other hand, the relationship between hand position and joint angles is given as follows:

$$\begin{aligned} x_e&= L_0 \cos \theta _0 + L_1 \cos (\theta _0 + \theta _1) \end{aligned}$$

(55)

$$\begin{aligned} y_e&= L_0 \sin \theta _0 + L_1 \sin (\theta _0 + \theta _1) \end{aligned}$$

(56)

Hand velocity are related to joint velocities by the Jacobian matrix $J(\varvec{\theta })$ as follows:

$$\begin{aligned} \dot{{\varvec{p}}} = {J} (\varvec{\theta }) {\varvec{\dot{\theta }}} \end{aligned}$$

(57)

where

$$\begin{aligned} \varvec{p} = \left[ \begin{array}{cc} x_e&y_e \end{array}\right] ^T \end{aligned}$$

(58)

and

$$\begin{aligned} {J}(\varvec{\theta }) = \left[ \begin{array}{rr} -L_0 \sin \theta _0 - L_1 \sin (\theta _0 + \theta _1) &{} - L_1 \sin (\theta _0 + \theta _1) \\ L_0 \cos \theta _0 + L_1 \cos (\theta _0 + \theta _1) &{} L_1 \cos (\theta _0 + \theta _1) \end{array}\right] \nonumber \\ \end{aligned}$$

(59)

Appendix 2: Calculation of control law

Using (9), ${\varvec{\dot{\theta }}}_r$ becomes

$$\begin{aligned} {\varvec{\dot{\theta }}}_r&= {\varvec{\dot{\theta }}}_d + {K_a} (\varvec{\theta }_d - \varvec{\theta }) \nonumber \\&= {\varvec{\dot{\theta }}}_d^*+ \varDelta {\varvec{\dot{\theta }}}_d^*+ {K_a} (\varvec{\theta }_d^*+ \varDelta \varvec{\theta }_d^*- \varvec{\theta }) \nonumber \\&= {\varvec{\dot{\theta }}}_r^*+ \varDelta {\varvec{\dot{\theta }}}_r^*\end{aligned}$$

(60)

where

$$\begin{aligned} \varDelta {\varvec{\dot{\theta }}}_r^*= \varDelta {\varvec{\dot{\theta }}}_d^*+ {K_a} \varDelta \varvec{\theta }_d^*\end{aligned}$$

(61)

Assuming that all the dynamical parameters are known, the control law (3) can be rewritten as follows:

$$\begin{aligned} \varvec{\tau }&= {M}(\varvec{\theta }) {\varvec{\ddot{\theta }}}_{r} + {C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) {\varvec{\dot{\theta }}}_{ r} - {K}_{ d} ({\varvec{\dot{\theta }}} - {\varvec{\dot{\theta }}}_{ r}) \nonumber \\&= { M}(\varvec{\theta }) {\varvec{\ddot{\theta }}}_{ r}^*+ { C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) {\varvec{\dot{\theta }}}_{r}^*\nonumber \\&+ {M}(\varvec{\theta }) \varDelta {\varvec{\ddot{\theta }}}_{r}^*+ ({C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) + { K}_{ d} )\varDelta {\varvec{\dot{\theta }}}_{ r}^*\nonumber \\&- { K}_{ d} ({\varvec{\dot{\theta }}} - {\varvec{\dot{\theta }}}_{ r}^*) \nonumber \\&= { Y}_{ L}(\varvec{\theta }, {\varvec{\dot{\theta }}}, {\varvec{\dot{\theta }}}_{ r}^*, {\varvec{\ddot{\theta }}}_{ r}^*) \varvec{\sigma }_{ L} \nonumber \\&+ [{ M}(\varvec{\theta }) \varDelta {\varvec{\ddot{\theta }}}_{ r}^*+ ({ C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) + { K}_{ d} )\varDelta {\varvec{\dot{\theta }}}_{ r}^*] \nonumber \\&- { K}_{ d} \varvec{s} \end{aligned}$$

(62)

It can be seen from Eq. (61) that the second term in Eq. (62) is a function of $\varDelta \varvec{\theta }_d^*,\,\varDelta {\varvec{\dot{\theta }}}_d^*$, and $\varDelta {\varvec{\ddot{\theta }}}_d^*$. If we compare Eq. (10) with Eq. (62), the second term of (10) will be written as follows using (13)

$$\begin{aligned}&\varvec{\tau }_{\varDelta }(\varDelta \varvec{\theta }_{ d}^*, \varDelta {\varvec{\dot{\theta }}}_{ d}^*, \varDelta {\varvec{\ddot{\theta }}}_{ d}^*) \nonumber \\&= [{ M}(\varvec{\theta }) \varDelta {\varvec{\ddot{\theta }}}_{ r}^*+ ({ C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) + { K}_{ d} )\varDelta {\varvec{\dot{\theta }}}_{ r}^*] \nonumber \\&= { M}(\varvec{\theta }) (\varDelta {\varvec{\ddot{\theta }}}_{ d}^*+ { K_a} \varDelta {\varvec{\dot{\theta }}}_{ d}^*) \nonumber \\&\quad + ({ C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) + { K}_{ d} ) (\varDelta {\varvec{\dot{\theta }}}_{ d}^*+ { K_a} \varDelta \varvec{\theta }_{ d}^*) \nonumber \\&= { M}(\varvec{\theta }) \varDelta {\varvec{\ddot{\theta }}}_{ d}^*+ ({ M}(\varvec{\theta }) { K_a} + { C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) + { K}_{ d}) \varDelta {\varvec{\dot{\theta }}}_{ d}^*\nonumber \\&+ ({ C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) + { K}_{ d} ) { K_a} \varDelta \varvec{\theta }_{ d}^*\nonumber \\&= { M}({\varvec{\theta }}) {\ddot{{Y}}}_\phi (t) {\varvec{\sigma }}_{ c} + ({ M}(\varvec{\theta }) { K}_{ a} + { C}({\varvec{\theta }}, {\varvec{\dot{\theta }}}) + { K}_{ d}){\dot{{Y}}}_\phi (t) {\varvec{\sigma }}_{ c}\nonumber \\&+ ({ C}(\varvec{\theta }, {\varvec{\dot{\theta }}}) + { K}_{ d} ) { K_a} { Y}_\phi (t) \varvec{\sigma }_{ c} \nonumber \\&= { Y}_\psi (t) {\varvec{\sigma }}_{ c} \end{aligned}$$

(63)

where

$$\begin{aligned} { Y}_\psi (t)&= { M}(\varvec{\theta }) {\ddot{{Y}}}_\phi (t) + ({ M}({\varvec{\theta }}) { K}_{ a} + { C}({\varvec{\theta }}, {\varvec{\dot{\theta }}}) + { K}_{ d}) {\dot{{Y}}}_\phi (t) \nonumber \\&+ ({ C}({\varvec{\theta }}, {\varvec{\dot{\theta }}}) + { K}_{ d}){ K}_{ a} { Y}_\phi (t) \end{aligned}$$

(64)

This gives us Eq. (16).

Appendix 3: Boundedness

Based on assumption A7, $\varvec{\theta }_d^*,\,{\varvec{\dot{\theta }}}_d^*,\,{\varvec{\ddot{\theta }}}_d^*$ are bounded. ${ Y}_\psi (t)$ is also bounded as assumed in Theorem 1. Furthermore, ${ M}(\varvec{\theta })$ is bounded since each element contains only constants or cosine functions.

Because $\dot{V} \le 0,\,V(0) \ge V > 0,\,V$ is bounded, i.e., $\varvec{s}$ and $\bar{\varvec{\sigma }}_c$ are bounded. The boundedness $\varvec{s}$ ensures the boundedness of ${\varvec{\dot{\theta }}}$ and ${\varvec{\dot{\theta }}}_r^*$. The boundedness of ${\varvec{\dot{\theta }}}_r^*$ means that $\varvec{\theta }$ is bounded (see Eq. (11)). Thus, ${ C}(\varvec{\theta }, {\varvec{\dot{\theta }}})$ is also bounded.

We can ensure the boundedness of $\dot{\varvec{s}}$ because of Eq. (23) and the nonsingularity of ${M}(\varvec{\theta })$.

Because we can obtain the boundedness of $\varvec{s}$ and $\dot{\varvec{s}},\,\ddot{V}$ given by (29) becomes bounded.