### Experimental setup and protocol

Eight right-handed male participants (23 ± 2 years old, all right-handed), who gave written informed consent prior to the experiment, participated in the study, which was approved by the Ethical Review Board for Epidemiological Studies of the Tokyo Institute of Technology. All research was carried out in accordance with relevant guidelines and regulations.

Participants were seated in front of the KINARM planar robotic manipulator from BKIN Technologies (Fig.1The). Participants held the handle of the robotic interface, which was fixed in position by the robot's motors, so that participants completed the task isometrically with the shoulder abducted to 45° and the elbow flexed so that the hand was 10 cm in front of the hand. midline of the thorax. An Edero Armon was used to support the weight of the arm. Visual feedback was provided on a monitor placed upside down, so that participants saw a reflection of the monitor in a mirror placed above the hand that obscured his and the robot's vision.

Two wireless EMG sensors (picoEMG, Cometa) were placed on the shoulder flexor (pectoralis major) and shoulder extensor (posterior deltoid). The raw signals from the electromyographic sensors were high-pass filtered at 10 Hz, rectified and low-pass filtered at 1 Hz. A second-order Butterworth filter was used for the high-pass and low-pass filters.

In the point-to-point reaching task, participants had to reach and stop at one of five targets {(i), (ii), (iii), (iv), (v)} (Fig.1B). One block contained one instance of each point-to-point target. Participants were presented with randomly drawn targets in each block, experiencing twenty blocks in total. After the point-to-point reaching task, participants were then presented with ten repetitions of the waypoint targets in the order of {(II), (III), (I), (IV)} (Fig.1b), respectively. In total, this totaled 140 range tests.

Participants were instructed to reach and stop within the target circles. In the waypoint task, they were instructed to pass the waypoint before reaching and stopping at the second final target position. The final target radius was 3 cm on the screen, while the waypoint radius was 0.5 cm. Cursor velocity needed to be below 10N/s and within the target circle for the test to be successful, which prevented participants from completing the movement past the target with a high rate of muscle activation.

### EMG normalization

Prior to the main experiment, the filtered EMG was normalized by asking participants to exert a 10N tangential force against the robot using only the flexor or extensor muscle 10 times per muscle. Visual feedback of the tangential force was continuously provided to the participant so that he exerted the force using the correct muscle. The data when the measured tangential force\(\left\| F \right\|\)exceeded 1N was collected and regressed as a function of the filtered EMG activity of each muscle. For example, the strength\(\left\| F \right\|\)was regressed as a function of the filtered flexor muscle activity\(m_{f}\)

$$\esquerda\| F \certo\| = \alpha_{f}m_{f} + \beta_{f}$$

(1)

to identify the tuning parameters\(\left\{{\alpha_{f} ,\beta_{f} } \right\}\)e\(\left\{\alpha_{e} ,\beta_{e} } \right\}\)for the flexor and extensor muscles, respectively. The mean values of the group of parameters were\(\alpha_{f} = 998 \pm 12\),\(\beta_{f} = 6,0 \pm 0,6\),\(\alpha_{e} = 963 \pm 9\),\(\beta_{e} = 5,0 \pm 0,8\). The variance explained by the adjusted parameters was*R*^{2}= 0,81 ± 0,02.

In the main experiment, the strength of each muscle was estimated using

$$\begin{aligned} \hat{F}_{f} \left( t \right) & = \alpha_{f} m_{f} \left( t \right) + \beta_{f} \\ \ chapéu{F}_{e} \left( t \right) & = \alpha_{e} m_{e} \left( t \right) + \beta_{e} \\ \end{aligned}$$

(2)

where*t*It's the time,\(\hat{F}_{f} \left( t \right)\)is the estimated shoulder flexor strength and\(\hat{F}_{e} \left( t \right)\)is the estimated shoulder extensor strength.

### Interface muscular

Participants controlled a cursor (\(X\left( t \right)\)*,*\(S\left( t \right)\)) position on the monitor using reciprocal shoulder activity, defined as the difference between flexor and extensor strength\(\hat{F}_{f} \left( t \right) - \hat{F}_{e} \left( t \right)\)where*t*is the time and co-contraction of the shoulder, defined as the minimum force of the flexor and extensor muscles\({\text{min}}\left( {\hat{F}_{f} \left( t \right),\hat{F}_{e} \left( t \right)} \right) \). The cursor position is given by

$$\left[ {\begin{array}{*{20}c} {X\left( t \right)} \\ {Y\left( t \right)} \\ \end{array} } \right ] = 0,015\left[ {\begin{array}{*{20}c} {\hat{F}_{f} \left( t \right) - \hat{F}_{e} \left( t \right)} \\ {\min \left( {\hat{F}_{f} \left( t \right),\hat{F}_{e} \left( t \right)} \right) } \\ \end{array} } \right]$$

(3)

such that 1N of reciprocal activity or co-contraction would shift the cursor by 1.5 cm on the monitor.

Prior to the main experiment, we confirmed that participants could control the muscle interface by instructing them to push and pull the shoulder joint and then asking them to contract the shoulder. This confirmed whether participants could move the cursor in two-dimensional muscle space.

### Optimal control modeling of the co-contraction reaching task

The task of reaching a force target in a two-dimensional muscle space is simulated using a cushioned mechanical system in which simulated reciprocal activity\(u_{{{\text{RA}}}}\)and the simulated co-contraction\(u_{{{\text{CC}}}}\)are under independent control with second-order muscle dynamics^{15}. The muscle activation command passes through a spring damper to produce muscle force, mimicking the viscoelastic behavior of muscles.^{16}. Flexor and extensor muscles were not simulated.\(u_{{{\text{RA}}}}\)e\(u_{{{\text{CC}}}}\)were scaled by 0.015 to produce the discretized position time series\(\left( {x_{i} ,y_{i} } \right)\)to compare it with the discretized data\(\left( {X_{i} ,Y_{i} } \right)\). The control policy for determining optimal simulated reciprocal activity and simulated co-contraction is derived using a finite horizon control framework^{15}.

The state\({\mathbf{z}} = \left[ {u_{{{\text{RA}}}} ,\dot{u}_{{{\text{RA}}}} ,u_{{{\ texto{CC}}}} ,\ponto{u}_{{{\text{CC}}}} } \direita]^{T}\)evolves discreetly with step size\(dt = 0,01\,{\texto{s}}\)and time index*k*according to damped mechanical dynamics

$${\mathbf{z}}_{k + 1} = {\mathbf{Az}}_{k} + {\mathbf{Bu}}_{k}$$

(4)

where\({\mathbf{u}}_{k} \equiv \left[ {\begin{array}{*{20}c} {\ddot{u}_{{{\text{RA}}}} } \\{\ddot{u}_{{{\text{CC}}}}\\\end{array}}\right]\)e

$${\mathbf{A}} = \left[{\begin{array}{*{20}c}{{\mathbf{A}}_{1}} & {0_{2}} \\{0_ {2}} & {{\mathbf{A}}_{1}}\end{array}}\right],\;{\mathbf{A}}_{1} = \left[{\begin { array}{*{20}c}{1-dt/\tau}&{dt/\tau}\\0&{1-dt/\tau}\\\end{array}}\right],\ ;{ \mathbf{B}} = \left[{\begin{array}{*{20}c}{{\mathbf{B}}_{1}} & {0_{2,1}} \\{ 0_{ 2.1} } & {{\mathbf{B}}_{1}}\end{array}}\right],\;{\mathbf{B}}_{1} = \left[{\begin {array}{*{20}c}0\\{dt/\tau}\\\end{array}}\right]$$

(5)

where\(\to take\)is the time constant of damped muscle dynamics, a free parameter whose influence is examined in more detail in the next subsection. The controller regulates the rate of change of reciprocal activity and the rate of change of co-contraction.

the control command\({\mathbf{u}}_{k}\)minimizes functional cost

$$J = \left( {{\mathbf{z}}_{{\varvec{N}}} - {\mathbf{Z}}_{N}}\right)^{T} {\mathbf{Q }}\left( {{\mathbf{z}}_{{\varvec{N}}} - {\mathbf{Z}}_{N}} \right) + \left( {{\mathbf{z} }_{{\varvec{v}}} - {\mathbf{Z}}_{{\varvec{v}}} } \right)^{T} {\mathbf{Q}}_{{\varvec{ v}}} \left( {{\mathbf{z}}_{{\varvec{v}}} - {\mathbf{Z}}_{{\varvec{v}}} } \right) + \mathop \sum \limits_{i}^{N}{\mathbf{u}}_{i}^{T}{\mathbf{Ru}}_{i}$$

(6)

where*N*= 200 is the total movement duration, chosen to be 2s, equivalent to the average group movement duration across all movements across all participants (2.0 ± 0.4s), and where\(v\)is the index of time at which the waypoint is reached in the simulation, which was identified from the data by choosing the time when the trajectory was closest to the waypoint. The target\({\mathbf{Z}}\)for the point-to-point movement was defined as the mean muscle strength of the population at the end of the movement for each target,\({\mathbf{Z}}_{N} = \frac{1}{0.015}\left[ {X_{N} ,0,Y_{N} ,0} \right]^{T}\), where\(\left( {X_{N} ,Y_{N} } \right)\)is the final position of the cursor at the end of the move. In the waypoint simulation, the target was first set as the waypoint target\({\mathbf{Z}}_{v} = \frac{1}{0.015}\left[ {X_{v} ,0,Y_{v} ,0} \right]^{T}\), where\(\left( {X_{v} ,Y_{v} } \right)\)was the closest position the cursor position passed to the waypoint defined in the experiment. The values of\(\left( {X_{N} ,Y_{N} } \right)\)e\(\left( {X_{v} ,Y_{v} } \right)\)were taken from the data to simulate the behavior of each participant in separate simulations. The cost functional is composed of the distance to the target at the end of the movement in muscle space, the distance to the passing point in time\(v\)and the operating cost of the control inputs.

The control cost matrix is\({\mathbf{R}} = \left[{\begin{array}{*{20}c}{10^{-4}}&0\\0&{10^{-4}}\\\end {array}}\direct]\)and the state cost matrix is\({\mathbf{Q}} = {\mathbf{Q}}_{v} = \left[{\begin{array}{*{20}c} {{\mathbf{Q}}_{e} } &{0_{2}}\\{0_{2}}&{{\mathbf{Q}}_{e}}\\\end{array}}\right]\)where\({\mathbf{Q}}_{e} = \left[{\begin{array}{*{20}c}q&0\\0&0\\\end{array}}\right]\ ). The weight of the cost of the state*q*is a free parameter whose influence on simulated RA and co-contraction is examined in more detail in the next subsection. To simulate point-to-point movements, the via-point cost matrix\({\mathbf{Q}}_{v} = 0\).

The control command that minimizes functional cost*J.*It is

$${\mathbf{u}}_{k} = - {\mathbf{L}}_{k} \left( {{\mathbf{z}}_{k} - {\mathbf{Z}}_ {k}} \sure)$$

(7)

where the target state\({\mathbf{Z}}_{k} = {\mathbf{Z}}_{N}\)is used to simulate point-to-point movement. In simulating the movements of the waypoints,\({\mathbf{Z}}_{k} = {\mathbf{Z}}_{v}\)when\(k \le v\)e\({\mathbf{Z}}_{k} = {\mathbf{Z}}_{N}\)for\(k>v\). the control command\({\mathbf{u}}_{k}\)is calculated backwards in time using

$${\mathbf{L}}_{k} = \left( {{\mathbf{R}} + {\mathbf{B}}^{T} {\mathbf{S}}_{k + 1} {\mathbf{B}}} \right)^{ - 1} {\mathbf{B}}^{T} {\mathbf{S}}_{k + 1} {\mathbf{A}}$$

$${\mathbf{S}}_{k} = {\mathbf{Q}}_{k} + {\mathbf{A}}^{T} {\mathbf{S}}_{k + 1} \left( {{\mathbf{A}} - {\mathbf{BL}}_{k} } \said)$$

(8)

which starts at the beginning with\({\mathbf{S}}_{N} = {\mathbf{Q}}_{N}\), e

$${\mathbf{Q}}_{k} = \left\{{\begin{array}{*{20}c}{{\mathbf{Q}}_{N},\quad k = N} \\{{\mathbf{Q}}_{v},\quad k = v}\\{0,\quad \mathrm{the opposite case}.} \\end{array}} \right.$$

(9)

Once the control policy is calculated, the system is simulated forward in time by iteratively calculating Eq.(4).

### Effect of free parameters τ and q

The simulation has two free parameters*t*e*q*that influence the simulated RA and simulated co-contraction time series in different ways. We systematically observed how the time series of simulated muscle activity changed with each parameter, modifying each parameter while keeping the other constant.

In the simulations of Fig.3, the time constant was selected from the range\(\tau = \left[ {0,16,10} \right]\)s in 0.5s increments while the state cost weight was kept constant at\(q = 1000\). The state cost was selected from the set\(q = \left\{ {10^{ - 6} ,10^{ - 5} ,10^{ - 4} ,10^{ - 3} ,10^{ - 2} ,0,1,1, 10,100, 10^{3} ,10^{4} ,10^{5} ,10^{6} ,10^{7} } \right\}\)while\(\tau = 5,5\)S.

To identify the parameters that best explained the behavioral data, we simulated AR and co-contraction in the two-dimensional parameter space using the intervals of\(\to take\)e*q*described above, and calculated the coefficient of determination\(R^{2} = 1 - \frac{{SS_{{{\text{res}}}} }}{{SS_{{{\text{tot}}}} }}\)where\(SS_{{{\text{res}}}} = \mathop \sum \limits_{i} \left( {X_{i} - x_{i} } \right)^{2} + \left( { Y_{i} - y_{i} } \direita)^{2}\)e\(SS_{{{\text{tot}}}} = \mathop \sum \limits_{i} \left( {X_{i} - \mathop X\limits } \right)^{2} + \left( {Y_{i} - \mathop Y\limites } \direita)^{2}\), which quantified the variability in the data explained by the simulation. The values of\(\to take\)e*q*that minimized\(R^{2}\)were considered the best set of parameters.

## FAQs

### What is the equilibrium point theory of motor control? ›

One of the most compelling features of the equilibrium point hypothesis is the **integration of posture and movement control into a single mechanism**. Thus, at the end of motion and in the absence of external forces, a new posture is achieved without persistent muscle activity.

**Why is the control of movement so complex? ›**

Controlling movement is one of the brain's core functions — and also one of its most complex. **To accomplish even the simplest of tasks, like picking up a glass of water, your neural circuits must precisely coordinate dozens of muscles in perfect synchrony**.

**How is movement controlled? ›**

Efferent connections. Movement is organized in increasingly complex and hierarchical levels. **Reflexes are controlled at the spinal or higher levels**. Stereotypic repetitious movements, such as walking or swimming, are governed by neural networks that include the spinal cord, brain stem, and cerebellum.

**What is the equilibrium point hypothesis? ›**

According to this hypothesis, the neural controller sets a referent configuration of the body (or of salient points on the body), a configuration at which all the muscles are at the threshold of activation via the stretch reflex.