2.1. Problem formulation

For cable-driven robotic arms, the pull of the cable provides the joint wrench.

(1) \begin{equation} dq_{MS}\cdot \tau _{MS}=dl_{M}\cdot f_{M} \end{equation}

(2) \begin{equation} \tau _{\mathrm{MS}}=dl_{M}/dq_{MS}\cdot f_{M}=J_{c}\!\left(q_{MS}\right)f_{M} \end{equation}

(3) \begin{equation} \dot{l}_{M}=J_{c}\!\left(q_{\mathrm{MS}}\right)\dot{q}_{\mathrm{MS}} \end{equation}

(4) \begin{equation} \ddot{l}_{M}=\dot{J}_{c}\!\left(q_{\mathrm{MS}}\right)\dot{q}_{\mathrm{MS}}+J_{c}\!\left(q_{\mathrm{MS}}\right)\ddot{q}_{\mathrm{MS}} \end{equation}

where $l_{\mathbf{M}},\dot{l}_{M},\ddot{l}_{M}\in \mathrm{\mathbb{R}}^{m}$ are the position, velocity, and acceleration vectors of the muscle, respectively, $dq_{MS}$ is the differential of the joint state, $dl_{M}$ is the differential of the cable displacement, and the Jacobi matrix $J_{c}(q_{\mathrm{MS}})\in \mathrm{\mathbb{R}}^{m\times n}$ relates to the muscle force vector $f_{M}\in \mathrm{\mathbb{R}}^{m}$ and the system torque $\tau _{MS}$ . The muscle force is limited by a positive feasible muscle force set $0\leq f_{\min }\leq\, f_{{M_{i}}}\leq\, f_{\max }, \forall i\colon f_{{M_{i}}}\in\, f_{M}$ , where $f_{\min }$ is the minimum force and $f_{\max }$ is the maximum force.

The robotic arm can be modeled as a multi-linked cable-driven robot. The following equations can describe the system dynamics of an n-DOF musculoskeletal robot with m muscles:

(5) \begin{equation} D_{M}\!\left(q_{MS}\right)\ddot{q}_{MS}+C\!\left(q_{MS},\dot{q}_{MS}\right)\dot{q}_{MS}+G\!\left(q_{MS}\right)=\tau _{MS}-T_{f} \end{equation}

where $q_{MS}\in \mathrm{\mathbb{R}}^{n}$ is the generalized coordinate, $D_{M}(q_{MS})\in \mathrm{\mathbb{R}}^{n\times n}$ is the inertia matrix, $C(q_{\mathrm{MS}},\dot{q}_{MS})\in \mathrm{\mathbb{R}}^{n}$ is the centrifugal Coriolis vector, $G(q_{\mathrm{MS}})\in \mathrm{\mathbb{R}}^{n}$ is the gravitational vector, $\tau _{MS}\in \mathrm{\mathbb{R}}^{n}$ is the system wrench, and $T_{f}\in \mathrm{\mathbb{R}}^{n}$ is the error of the wrench acting on the system. The variable $T_{f}$ contains the effects of cable friction $T_{p}$ , rigid friction $T_{\textit{friction}}$ , and muscle antagonistic relaxation $\tau _{r}$ on the system.

The number of muscles is nine, and the number of joints is four. It indicates the degree of muscle contraction when the joint moves in a particular direction when the joint angle is $q_{MS}$ . The muscle Jacobi matrix $J_{c}(q_{MS})$ is denoted as $J_{c}(q_{MS})=dl_{M}/dq_{MS}$ . When the target joint angle is $\hat{q}_{MS}$ , and the current joint angle is $q_{MS}, J_{c}(q_{MS})\cdot (\hat{q}_{MS}-q_{MS})$ indicates whether each muscle is an agonist or an antagonist’s muscle when moving to that position. In the literature [Reference Kawaharazuka, Kawamura, Makino, Asano, Okada and Inaba16], antagonistic muscle relaxation $\tau _{r}$ was expressed as:

(6) \begin{align} \begin{cases} \tau _{r}=K_{s}\cdot \left(l_{M}-\hat{l}_{M}\right) & J_{c}\!\left(q_{\mathrm{MS}}\right)\cdot \frac{\hat{q}_{\mathrm{MS}}-q_{\mathrm{MS}}}{\left| \hat{q}_{\mathrm{MS}}-q_{\mathrm{MS}}\right| }\lt C_{\mathrm{MS}}\\[5pt] \tau _{r}=0 & \text{otherwise} \end{cases} \end{align}

where $hatl_{M}$ is the desired cable length, $l_{M}$ is the actual cable length, $hatq_{M}$ is the angle of the desired joint, and $q_{MS}$ is the angle of the actual joint. $C_{MS}$ is a constant.

The connection between the output $(T_{\text{output}})$ and the input $(T_{in})$ is $T_{\mathrm{in}}=T_{\text{output}}/r_{\text{winch}}$ . The output of the sensor is $\mathrm{T}_{\mathrm{in}}=\mathrm{F}_{\text{sensor}}$ . Theoretically, the relationship between the length of the tensioned cable and the angle of the joint is $\hat{l}_{M}={\unicode{x03D5}} (\hat{q}_{MS})$ , and its differentiation is $\hat{i}_{M}=d{\unicode{x03D5}} (\hat{q}_{MS})/dt\cdot \hat{q}_{MS}=J_{c}(\hat{q}_{MS})\cdot \hat{q}_{MS}$ . Assuming that the radius of the winch on the motor is constant, the relationship between the displacement of the cable and the speed of the motor is $\hat{l}_{M}=\int \omega _{m}\cdot r_{\text{winch}}\ dt$ , and the relationship between cable and sensor is $v=\dot{l}_{M}=\omega _{m}\cdot r_{\text{winch}}$ . The relationship between the output torque of the winch and the tension obtained by the sensor is $T_{\text{output}}=F_{\text{sensor}}\cdot r_{\text{winch}}$ , where $r_{\text{winch}}$ is the radius of the winch and $F_{\text{sensor}}$ is the cable’s tension obtained from the sensor. According to the above experience, the friction can be expressed as:

(7) \begin{equation} T_{p}+T_{\text{friction}}+\tau _{r}=T_{f} \end{equation}

Referring to the effects of cable friction $T_{p}$ [Reference Wei, Qiu and Sheng9–Reference Kim, Jung, Jeong, Park, Jung, Cheong, Choi, Do and Park11], rigid friction $T_{\text{friction}}$ [Reference Lischinsky, Canudas-de-Wit and Morel12–Reference Katz, Carlo and Kim14], and muscle antagonistic relaxation $\tau _{r}$ on the system [Reference Kawaharazuka, Kawamura, Makino, Asano, Okada and Inaba16], a concise expression that can satisfy multiple action error wrenches is

(8) \begin{equation} T_{f}=\tau _{f}\!\left(\left[U_{A},U_{B},U_{C},U_{D},U_{E},U_{F}\right]\right) \end{equation}

where $${{\rm{U}}_A} = [{q_{MS}},{{\dot q}_{MS}},{{\ddot q}_{MS}}],\cdot{{\rm{U}}_B} = {{\hat l}_{\rm{M}}} - {l_{\rm{M}}},\cdot{{\rm{U}}_C} = sign({{\dot q}_{MS}}),\cdot{{\rm{U}}_D} = \exp ( - sign({{\dot q}_{MS}}))\cdot sign({{\dot q}_{MS}}),{{\rm{U}}_E} =$$ $$\exp ( - sign({{\dot l}_{MS}})),\,\, {\textrm{and}} \;{{\rm{U}}_F} = {F_{{\rm{sensor}}}}$$

2.2. Neural network fitting

Artificial neural networks are widely used in solving many engineering problems. The main applications are in common problems such as pattern classification, clustering, function approximation, prediction, and control. Artificial neural networks are often applied in situations where analytical relations for nonlinear functions are difficult to derive and their solutions are not easy to compute [Reference Runge, Wiese and Raatz19].

A three-layer neural network is used to fit the unknown state. The network consists of a hidden layer, an input layer, and an output layer:

(9) \begin{equation} y={W}_{MS}^{T}{\unicode{x03D5}} \!\left({S}_{MS}^{T}x+S_{MS0}\right)+W_{MS} \end{equation}

where $x=[\mathrm{U}_{A},\mathrm{U}_{B},\mathrm{U}_{C},\mathrm{U}_{D},\mathrm{U}_{B},\mathrm{U}_{F}]$ is the input and $y=T_{f}$ is the output. The inputs are associated with weighting factors $S_{MS}$ and $\cdot W_{MS}\cdot$ Add $\cdot$ offsets $\cdot W_{MS0}$ and $W_{MS0}\cdot$ to the components of the resulting vector to generate the network output.

In terms of network parameter selection, referring to the contributions by Su et al. [Reference Su, Qi, Hu, Sandoval, Zhang, Schmirander, Chen, Aliverti, Knoll, Ferrigno and De Momi18], similarly, the artificial neural network consists of 42 input nodes in the first layer, 150 hidden nodes in the second layer, and four output nodes. The output quantity is the compensation error torque value $\Delta \tau$ . This error-fitting model is used for feedforward control compensation. The four joints of the robot arm are Joint1, Joint2, and Joint3 of the shoulder joint and Joint4 of the elbow joint.

Define the error z and its percentage r of the trajectory as:

(10) \begin{equation} z=p-p_{d} \end{equation}

(11) \begin{equation} r_{p}=z/{p}_{0}^{\mathrm{end}}\times 100\% \end{equation}

where $p$ is the of the final end-effector position, $p_{d}$ is the desired position of $p$ , and ${p}_{0}^{end}$ is the distance of the desired position from the initial point.

There are two common evaluation indices to measure the performance of the built feedforward compensation models, namely root mean square error (RMSE) and its percentage r of the trajectory as:

(12) \begin{equation} RMSE=\sqrt{\left({\sum }_{i=1}^{t}\!\left(l_{Mi}-\hat{l}_{Mi}\right)^{2}/t\right)} \end{equation}

(13) \begin{equation} r_{Mi}=RMSE_{i}/l_{Mdi}\times 100\% \end{equation}

where t can be regarded as the number of observations, $l_{Mi}$ is the length of the muscle at the i-th observations, $hatl_{Mi}$ is the expected length of the muscle at the i-th observations, and $r_{Mi}$ is the percentage of the RMSE to the maximum muscle travel.

2.3. Muscle model

The Hill-type muscle model is utilized in this experiment, which is excellent in expressing biological accuracy, calculation speed, and muscle control [Reference Yuan, Wu, Wang and Qiao17, Reference Mizuuchi, Yoshikai, Nakanishi, Sodeyama, Yamamoto, Miyadera, Niemela, Hayashi, Urata and Inaba20]. Mizuuchi and Wu used the Hill-type muscle model [Reference Mizuuchi, Yoshikai, Nakanishi, Sodeyama, Yamamoto, Miyadera, Niemela, Hayashi, Urata and Inaba20, Reference Wu, Chen and Qiao21]. The muscle model was used for the control of the MS. In the literature [Reference Wu, Chen and Qiao21, Reference Hill22], Hill-type muscle model was expressed as:

(14) \begin{equation} F_{TD}=F_{m}\cos \alpha =\left(af_{l}\!\left(l_{M}\right)f_{v}\!\left(\dot{l}_{M}\right)+f_{PE}\!\left(l_{M}\right)\right)\cos \alpha \end{equation}

(15) \begin{equation} \dot{l}_{M}={f}_{v}^{-1}\!\left(\left(f_{TD}\!\left(l_{TD}\right)/\cos \alpha -f_{PE}\!\left(l_{M}\right)\right)/af_{l}\!\left(l_{M}\right)\right) \end{equation}

where $F_{TD}$ is the normalized tendon force. $f_{PE}$ is the normalized passive force of the muscle fiber, whereas $f_{l}$ and $f_{v}$ are the force–length and force–velocity functions, respectively. $l_{M}$ is the normalized fiber length by $l_{M0}$ and $dotl_{M}$ is the normalized velocity of the muscle fiber.

Table I. Target expectation of joint angle for the second experiment.