Biomimetic Control of Myoelectric Prosthetic Hand Based on a Lambda-type Muscle Model

First, based on impedance control [11], the characteristics of joint motion is represented by the tension balance between flexors and extensors (Fig. 1). The equation of motion around each joint $j~{}(j=1,~{}2,~{}\dots,~{}J)$ of the prosthetic hand is defined as

$\displaystyle I_{j}\ddot{\theta_{j}}+B_{j}(\alpha_{j})\dot{\theta_{j}}+K_{j}(\alpha_{j})(\theta_{j}-\theta^{0}_{j})=\tau_{j\mathrm{f}}-\tau_{j\mathrm{e}},$

(1)

where $I_{j}$, $B_{j}(\alpha_{j})$, and $K_{j}(\alpha_{j})$ are the inertia, viscosity, and stiffness, respectively. $\theta_{j}$ and $\theta^{0}_{j}$ are the joint angle and its equilibrium angle for stiffness element only. Furthermore, $\tau_{j\mathrm{f}}$ and $\tau_{j\mathrm{e}}$ are the joint torque generated by the flexor and extensor, respectively. The change in the characteristics associated with muscle activity is expressed by defining the stiffness and viscosity as functions of the muscle contraction level $\alpha_{j}$ $(0\leq\alpha_{j}\leq 1)$.

	$\displaystyle K_{j}(\alpha_{j})$	$\displaystyle=k_{j,1}\alpha_{j}^{k_{j,2}}+k_{j,3},$		(2)
	$\displaystyle B_{j}(\alpha_{j})$	$\displaystyle=b_{j,1}\alpha_{j}^{b_{j,2}}+b_{j,3},$		(3)

The equilibrium angle $\theta_{j}^{\mathrm{eq}}$ of the complete system can be calculated as the joint angle $\theta_{j}$ when $\ddot{\theta}$ and $\dot{\theta}=0$.

$$\theta_{j}^{\mathrm{eq}}=\frac{1}{K_{j}(\alpha_{j})}(\tau_{j\mathrm{f}}-\tau_{j\mathrm{e}})+\theta_{j}^{0}$$

(4)

From (4), the torque $\tau_{j}(t)$ and the equilibrium angle $\theta_{0}$ of the joint angle can be varied adaptability to control the equilibrium angle of the model dynamically, according to the muscle contraction state. This study designs these variables to build a control model satisfying the following specifications.

• •

  Specification 1: When the muscle relaxes, the equilibrium position is maintained at the value just before relaxation.

• •

  Specification 2: Continuity of the equilibrium position is maintained even when the direction of motion changes from flexion to extension or extension to flexion.

When designing the model, it is desirable to consider all the muscle information for the joint movement. However, various muscles act when a person performs a complicated operation, such as flexing their fingers, and it is thus difficult to estimate all the muscle information on the joint motion from the EMG signals. Therefore, we switch the muscle contraction level depending on the classified motion as follows:

$$\alpha_{j}(t)=\begin{cases}\alpha_{j\mathrm{f}}(t)&(\mathrm{Flexion\ motion})\\ \alpha_{j\mathrm{e}}(t)&(\mathrm{Extension\ motion})\end{cases}.$$

(5)

Furthermore, the torque $\tau_{j\rm{f}},~{}\tau_{j\rm{e}}$ can be expressed as a function of the muscle contraction level $\alpha(t)$:

$\displaystyle\tau_{ji}(t)=\alpha_{ji}(t)\tau_{ji}^{\rm{max}},$

(6)

where $\tau_{ji}^{\rm{max}}$ is the maximum value of each predetermined torque, and the subscript $i\in\{\mathrm{f},\mathrm{e}\}$ indicates flexor and extensor. Henceforth, we omit $(t)$ to indicate a function of time, except when highlighting the temporal context.

We first design a model satisfying the specification 1. In the model design, we adopt the concept of the $\lambda$-model [14], which is a model of human muscle movements. The $\lambda$-model assumes that the brain achieves motion by controlling the muscle length $x$, while the intensity of the motor commands determines the threshold $\lambda$ of the muscle length. Motion conditions of the muscle are formulated as follows:

$\displaystyle x-\lambda>0.$

(7)

The muscle is active if the condition of (7) expression is satisfied; otherwise, the muscle activity is stopped, and the current state is maintained. Here, the muscle activity level $s$ in the $\lambda$-model is defined as a function of the muscle length $x$ and threshold $\lambda$. Moreover, the muscle activity level is assumed to be a simple linear function that can be written as follows:

$\displaystyle s=s(x,~{}\lambda)=G(x-\lambda)+h,$

(8)

where $G$ and $h$ are constants. Equation (8) can be rewritten as

$\displaystyle x-\lambda=\frac{1}{G}(s-h).$

(9)

Here, we assume that the constant $G$ is the maximum value $s^{\rm{max}}$ of $s$, and $s/G$ in (9) can be regarded as the muscle contraction level $\alpha_{ji}$. Furthermore, if the constant $h/G$ is regarded as the threshold value $\widetilde{\lambda}_{ji}$ of the muscle contraction level, from the conditional expression (7), the muscle length $x$ can be replaced by the muscle contraction level $\alpha_{ji}$, and the muscle length threshold $\lambda$ can be replaced by the muscle contraction level threshold $\widetilde{\lambda}_{ji}$. Subsequently, the muscle activity condition in the control model is defined as follows:

$\displaystyle\alpha_{ji}-\widetilde{\lambda}_{ji}>0.$

(10)

Here, we introduce a function expressing the active state of the muscle using the activity condition (10) as follows:

$\displaystyle C^{+}=[\alpha_{ji}-\widetilde{\lambda}_{ji}]^{+}=\begin{cases}1&(\alpha_{ji}-\widetilde{\lambda}_{ji}>0)\\ 0&(\alpha_{ji}-\widetilde{\lambda}_{ji}\leq 0)\end{cases},$

(11)

$$C^{-}=1-C^{+},$$

(12)

where $C^{+}$ is a function expressing the active state of the muscle and becomes $C_{i}^{+}=1$ when $\alpha_{ji}$ is larger than the threshold $\widetilde{\lambda}_{ji}$. Meanwhile, $C_{i}^{-}$ is a function expressing the inactive state of the muscle and becomes $C_{i}^{-}=1$ when $\alpha_{ji}$ is equal to or less than the threshold $\widetilde{\lambda}_{ji}$. We consider that the joint angle varies when the muscular activity condition is satisfied and is maintained when the muscle is in an inactive state, according to the concept of the $\lambda$-model (Fig. 2).

Then, using (11) and (12), we define $\tau_{j\rm{f}},~{}\tau_{j\rm{e}}$, and $\theta^{0}_{ji}$ as follows:

	$\displaystyle\tau_{ji}$	$\displaystyle=$	$\displaystyle C^{+}_{i}\tau_{ji}^{\rm{max}}\alpha_{ji},$		(13)
	$\displaystyle\theta^{0}_{ji}(t)$	$\displaystyle=$	$\displaystyle C_{i}^{-}\theta^{\mathrm{eq}}_{ji}(t-\Delta t),$		(14)

where $\theta^{\mathrm{eq}}_{ji}(t-\Delta t)$ is the equilibrium angle before the sampling time $\Delta t$.

Subsequently, we preform the control using muscle information only in the flexion direction or extension direction based on the constraint condition mentioned above. A separate model is defined for each direction, and control is achieved by switching between models. Under this condition, when a model is applied in a specific direction, the muscle contraction in the other direction is zero. Therefore, when the control is performed based on (13) and (14), the torque and the equilibrium position are discontinuous before and after switching the models. The continuity of the equilibrium position can be maintained by complementing the muscle contraction level in the other direction with the muscle contraction level in the classified direction. In cases other than co-contraction, it is assumed that one muscle contraction level gradually decreases as the other muscle contraction level increases. Based on this assumption, when the flexion direction is classified, the muscle contraction level $\alpha_{j\rm{e}}$ in the extension direction is expressed using the muscle contraction level $\alpha_{j\rm{f}}$ in the flexion direction:

$\displaystyle\alpha_{j\rm{e}}=A_{\rm{e}}(1-\alpha_{j\rm{f}}),$

(15)

where $A_{\rm{e}}$ is a constant. Here, substituting (15) into (13) and expressing torque in the flexion direction, the control model is

$\displaystyle\begin{cases}\tau_{j\rm{f}}=C^{+}_{\rm{f}}\tau_{j\rm{f}}^{\rm{max}}\alpha_{j\rm{f}}\\ \tau_{j\rm{e}}=C^{+}_{\rm{f}}\tau_{j\rm{e}}^{\rm{max}}A_{\rm{e}}(1-\alpha_{j\rm{f}})\end{cases}\mbox{(Flexion direction)}.$

(16)

Similarly, when the extension direction is classified, the muscle contraction level $\alpha_{j\rm{f}}$ in the flexion direction is expressed using the muscle contraction level $\alpha_{j\rm{e}}$ in the extension direction:

$\displaystyle\alpha_{j\rm{f}}=A_{\rm{f}}(1-\alpha_{j\rm{e}}),$

(17)

where $A_{\rm{f}}$ is a constant. Thus, the control model in the extension direction is

$\displaystyle\begin{cases}\tau_{j\rm{f}}=C^{+}_{\rm{e}}\tau_{j\rm{f}}^{\rm{max}}A_{\rm{f}}(1-\alpha_{j\rm{e}})\\ \tau_{j\rm{e}}=C^{+}_{\rm{e}}\tau_{j\rm{e}}^{\rm{max}}\alpha_{j\rm{e}}\end{cases}\mbox{(Extension direction)}.$

(18)

From the above equations, considering the muscle activity condition based on the $\lambda$-model for muscle motion, we can construct a model that maintains the equilibrium position of the joint angle even when the muscle relaxes.

We next consider the continuity of the equilibrium position at the time of changing the model to satisfy the specification 2. We consider that a model is applied in the extension direction at time $t_{\rm e}$ and then in the flexion direction at time $t_{\rm e}+\Delta t$. At this time, the equation of motion of the model in the extension direction applied at $t_{\rm e}$ becomes

	$\displaystyle I_{j}\ddot{\theta_{j}}$	$\displaystyle+B_{j}(\alpha_{j\rm{e}}(t_{\rm{e}}))\dot{\theta_{j}}+K_{j}(\alpha_{j\rm{e}}(t_{\rm{e}}))\theta_{j}$
		$\displaystyle=\tau_{j\rm{f}}^{\rm{max}}A_{\rm{f}}(t_{\rm{e}})(1-\alpha_{j\rm{e}}(t_{\rm{e}}))-\tau_{j\rm{e}}^{\rm{max}}\alpha_{j\rm{e}}(t_{\rm{e}}).$		(19)

The equation of motion of the model in the flexion direction applied at $t_{\rm{e}}+\Delta t$ immediately thereafter is given as

	$\displaystyle I_{j}\ddot{\theta_{j}}$	$\displaystyle+B_{j}(\alpha_{j\rm{f}}(t_{\rm{e}}+\Delta t))\dot{\theta_{j}}+K_{j}(\alpha_{j\rm{f}}(t_{\rm{e}}+\Delta t))\theta_{j}$
		$\displaystyle=\tau_{j\rm{f}}^{\rm{max}}\alpha_{j\rm{f}}(t_{\rm{e}}+\Delta t)$
		$\displaystyle\quad-\tau_{j\rm{e}}^{\rm{max}}A_{\rm{e}}(t_{\rm{e}}+\Delta t)(1-\alpha_{j\rm{f}}(t_{\rm{e}}+\Delta t)).$		(20)

The equilibrium positions of (19) and (20) must be equal to maintain continuity at the time the model switches. Here, we assume that the state of motion is the steady state $(\ddot{\theta_{j}},~{}\dot{\theta}_{j}=0)$ just before switching the direction of motion. The equilibrium positions in (19) and (20) are

	$\displaystyle\theta_{j}(t_{\rm e})=\frac{1}{K_{j}(\alpha_{j\rm{e}}(t_{\rm{e}}))}\{$	$\displaystyle\tau_{j{\rm f}}^{\rm max}A_{\rm f}(t_{\rm e})(1-\alpha_{j{\rm e}}(t_{\rm e}))$
		$\displaystyle-\tau_{j{\rm e}}^{\rm max}\alpha_{j{\rm e}}(t_{\rm e})\}$		(21)

	$\displaystyle\theta_{j}(t_{\rm e}+\Delta t)$	$\displaystyle=\frac{1}{K_{j}(\alpha_{j\rm{f}}(t_{\rm{e}}+\Delta t))}\{\tau_{j{\rm f}}^{\rm max}\alpha_{j{\rm f}}(t_{\rm e}+\Delta t)$
		$\displaystyle\quad-\tau_{j{\rm e}}^{\rm max}A_{\rm e}(t_{\rm e}+\Delta t)(1-\alpha_{j{\rm f}}(t_{\rm e})+\Delta t)\}.$		(22)

Solving the above equations for $A_{\rm{e}}$ and organizing it by substituting in equation (16), we obtain the following torque $\tau_{j\rm{f}}$ expression, when the flexion direction is determined.

$\displaystyle\begin{cases}\tau_{j\rm{f}}=C^{+}_{\rm{f}}\tau_{j\rm{f}}^{\rm{max}}\alpha_{j\rm{f}}\\ \tau_{j\rm{e}}=C^{+}_{\rm{f}}\frac{1-\alpha_{j\rm{f}}}{1-\alpha_{j\rm{f}}^{\rm{post}}}(\tau_{j\rm{f}}^{\rm{max}}\alpha_{j\rm{f}}^{\rm{post}}-K_{j}(\alpha_{j{\rm f}}^{\rm post})\theta_{j\rm{e}}^{\rm{pre}})\end{cases},$

(23)

where $\theta_{j\rm{e}}^{\rm{pre}}=\theta_{j\rm{e}}(t_{\rm{e}})$ is the equilibrium angle just before switching, and $\alpha_{j\rm{f}}^{\rm{post}}=\alpha_{j\rm{f}}(t_{\rm{e}}+\Delta t)$ is the muscle contraction level in the flexion direction immediately after switching. Similarly, when the model is applied in the flexion direction at time $t_{\rm{f}}$ and there is a switch to the model in the extension direction at time $t_{\rm{f}}+\Delta t$, the torques $\tau_{j\rm{f}}$ and $\tau_{j\rm{e}}$ in the extension direction can be written as follows:

$\displaystyle\begin{cases}\tau_{j\rm{f}}=C^{+}_{\rm{e}}\frac{1-\alpha_{j\rm{e}}}{1-\alpha_{j\rm{e}}^{\rm{post}}}(\tau_{j\rm{e}}^{\rm{max}}\alpha_{j\rm{e}}^{\rm{post}}+K_{j}(\alpha_{j\rm e}^{\rm post})\theta_{j\rm{f}}^{\rm{pre}})\\ \tau_{j\rm{e}}=C^{+}_{\rm{e}}\tau_{j\rm{e}}^{\rm{max}}\alpha_{j\rm{e}}\end{cases},$

(24)

where $\theta_{j\rm{f}}^{\rm{pre}}=\theta_{j\rm{f}}(t_{\rm{f}})$ is the equilibrium angle just before switching, and $\alpha_{j\rm{e}}^{\rm{post}}=\alpha_{j\rm{e}}(t_{\rm{f}}+\Delta t)$ is the muscle contraction level in the extension direction immediately after switching. Finally, using (1), (23), and (24), the control model for the direction $i\in\{\mathrm{f},\mathrm{e}\}$ is obtained as follows:

	$\displaystyle I_{j}\ddot{\theta_{j}}(t)+B_{j}(\alpha_{ji})\dot{\theta_{j}}(t)+K_{j}(\alpha_{ji})(\theta_{j}(t)-\theta_{ji}^{0}(t))$
	$\displaystyle\quad\qquad\qquad\qquad=C^{+}_{i}\delta_{i}\tau_{ji}^{\rm{max}}(\alpha_{ji}(t)-V_{i}(t)\alpha_{ji}^{\rm{post}}),$		(25)
	$\displaystyle\theta_{ji}^{0}(t)=C_{i}^{+}\frac{K_{j}(\alpha_{ji}^{\rm post})}{K_{j}(\alpha_{ji})}V_{i}(t)\theta_{ji}^{\rm{pre}}+C_{i}^{-}\theta^{\mathrm{eq}}_{ji}(t-\Delta t),$		(26)

where $\delta_{\mathrm{f}}=1$, $\delta_{\mathrm{e}}=-1$, and

$\displaystyle V_{i}(t)=\frac{1-\alpha_{ji}(t)}{1-\alpha_{ji}^{\rm{post}}}.$

(27)

Numerical simulation using artificial data was conducted to study whether the control model satisfies the design specifications. For simplicity, the number of joints was $j=1$, and the change in the equilibrium angle $\theta^{\mathrm{eq}}$ was confirmed for the two types of input signals.
Input 1

	$\displaystyle\alpha_{\rm{f}}$	$\displaystyle=\begin{cases}t/10&\mathrm{if}\ 0\leq t<5~{}\rm{s}\\ (t-20)/10&\mathrm{if}\ 20\leq t\leq 30~{}\rm{s}\\ 0&\mathrm{otherwise}\end{cases},$		(28)
	$\displaystyle\alpha_{\rm{e}}$	$\displaystyle=\begin{cases}(t-10)/10&\mathrm{if}\ 10\leq t<15~{}\rm{s}\\ 0&\mathrm{otherwise}\end{cases}.$		(29)

Input 2

	$\displaystyle\alpha_{\rm{f}}$	$\displaystyle=\frac{1}{2}\sin(0.2\pi t-\frac{\pi}{2})+\frac{1}{2},$		(30)
	$\displaystyle\alpha_{\rm{e}}$	$\displaystyle=\begin{cases}0&\mathrm{if}\ 0\leq t<5~{}\rm{s}\\ \frac{1}{2}\sin(0.2\pi t-\frac{3}{2}\pi)+\frac{1}{2}&\mathrm{if}\ 5\leq t\leq 30~{}\rm{s}\end{cases}.$		(31)

Here, $\alpha_{\rm f}$ and $\alpha_{\rm e}$ were assumed to be the muscle contraction level obtained from the flexor and extensor muscles, respectively. The threshold $\widetilde{\lambda}_{i}$ was defined as the maximum value of the muscle contraction level in the classification section. In addition, the section where $\alpha_{\rm f}>\alpha_{\rm e}$ represents the classification section of the flexor muscle while the section where $\alpha_{\rm f}<\alpha_{\rm e}$ represents the classification section of the extensor muscle. In this experiment, only the muscle contraction level of the classified muscle was used as the input of the control model. The impedance parameters were set to the values of the wrist joints (Table I) measured in a previous study [11].

Fig. 3 shows the artificially generated muscle contraction level $\alpha_{i}$ and the equilibrium angle $\theta_{0}$ of the joint obtained from the numerical simulation for (a) Input 1 and (b) Input 2. The direction of joint extension is defined as positive, and the shaded area in the figure represents the section of muscle relaxation.

In Fig. 3, when flexors are classified, the equilibrium angle transitions into a negative direction. In contrast, when extensors are classified, the equilibrium angle transitions into a positive direction. When considering the muscle relaxation section, the equilibrium angle was maintained at the same level as before muscle relaxation. Furthermore, the equilibrium angle changes continuously and smoothly before and after the switching of the motion. The equilibrium angle presents a periodic waveform following the muscle contraction level, even for inputs where the flexor muscle and extensor muscle are intermittently switched as shown in Fig. 3(b). Hence, the continuity of the equilibrium position is maintained successfully. These results suggest that the control model satisfies the design specifications and maintains the equilibrium position even when the muscle relaxes.

Furthermore, the equilibrium angle shifts in a curved manner with respect to a linear input (Fig. 3(a)). The reason is that the proposed method was constructed based on the impedance model, so that the impedance characteristics of a human are reflected in the transition of the angle. The simulation results revealed that the proposed control method can replicate the motion characteristics of the muscle based on the muscle model and the motion characteristics of the joint by the impedance control, realizing a behavior approaching that of humans.

To evaluate the applicability of the proposed method to the prosthetic hand control, we constructed a myoelectric prosthetic hand system and performed control experiments using EMG signals. All measurement experiments were approved by the Hiroshima University Ethics Committee (registration number E-840).

Fig. 5 shows examples of joint angle calculation for each task. The upper, middle, and lower panels represent the classified motions, muscle contraction levels, and joint angles, respectively. The shaded area in the figure corresponds to the section where the participants were instructed to relax.

Fig. 6 shows the average RMSE over participants for each task. The statistical test results of a paired $t$-test (significance level of 5%) with Holm adjustment are also shown.

Fig. 7 shows the results of the control experiments. The upper, middle, and lower panels represent the classified motions, muscle contraction levels, and motor output angles, respectively. The opening motion of the prosthetic hand was set as the initial state, implying that the initial angle was zero. Shaded areas represent sections of muscle relaxation.

In the joint angle calculation experiment, the RMSE was used to evaluate the error between the actual human joint angle and the calculated joint angle based on the control models, using muscle contraction information. The RMSE of the proposed method was smaller than that of the comparison methods in both tasks, suggesting that the proposed method can realize more human-like wrist motions with relatively high accuracy. Although this tendency was similar for all sections, it was more remarkable in the relaxation section. Moreover, the proposed method had significantly smaller RMSEs.

In the conventional impedance control method, as the origin of the joint angle is always fixed to the initial position, the equilibrium and joint angles converge to the initial position when the muscle is relaxed. Therefore, the error of the impedance control method is higher in the section where the wrist joint angle is maintained in the muscle relaxation. The proportional control method showed lower RMSEs than the impedance control method in the muscle relaxation section. The reason is that the angle saturation in the proportional control method, and the RMSE becomes zero when the target angle coincides with the saturation angle, as shown in Fig. 5(b). In the active sections, the RMSEs of the proportional control method were higher than those of the proposed model and the impedance control method. The proportional control method cannot calculate the joint angle by considering the movement characteristics of a human, resulting in errors in the process of transition of the joint angle. In contrast, the proposed method controls the equilibrium angle based on the muscle model dynamically. The error concerning the measured joint angle was small because it was maintained, even when the muscle was relaxed. Moreover, as the control model involves the impedance characteristics of the human, the human-like smoothness was replicated in the change of the joint angle; hence, the error during the transition was small. These results suggest the effectiveness of the proposed method to dynamically control the equilibrium position, even during muscle relaxation, in reproducing human joint movements.

Furthermore, we verified the applicability of the proposed control method through an experiment considering the control of the prosthetic hand. In Fig. 7, the joint angle of the prosthetic hand is smoothly controlled according to the operator’s movements. We can also confirm that the motor angle is maintained during muscle relaxation by dynamically changing the equilibrium position. Therefore, by using the proposed control method, the state of the muscle, such as contraction or relaxation, was revealed to be reflected in the motion of the prosthetic hand, and that human motion characteristics can be reproduced.

In the conventional impedance control method, maintaining the muscles contracted through invariance of the equilibrium angle is necessary to maintain the joint angle, which may increase the burden on the user. The EMG signal pattern is known to change due to muscle fatigue when the muscles are contracted for a long time, affecting the operability of myoelectric prosthetic hands [19, 20]. In contrast, in the proposed method, the equilibrium angles are dynamically controlled, so that the motion can be maintained even when the muscle is relaxed. Therefore, the proposed method has the potential to reduce muscle fatigue and improve the operability of myoelectric prosthetic hands during long-term use.