2.1. Design of MSMS

The skeleton is driven by a combination of active and passive muscles. The DC motor was selected as the power of MSMS in order to meet the requirements of high energy density, mature control, and remote installation. In order to tighten or loosen the cable, the winch is powered by the motor and reducer. Figure 1 shows the structural design of the MC-AM. The length, width, and height are 142 mm, 60 mm, and 40 mm, respectively. DC motors and reducers may be swapped out depending on the demands of various loads. The bracket is made from Nylon NO.7500 (HP3DHR-PA12) which is manufactured by high-precision 3D printing technology. The steel cable will be loose and broken due to wear under the condition of large curvature transmission, so we selected high-strength polyester fiber cable as the transmission medium [Reference Cui, Geng and Ding35]. The sheath is made up of a lubricated polytetrafluoroethylene (PTFE), a metal tube, and a rubber tube. The inner tube and cable are separated by a 0.2-mm gap, allowing the cable to slip freely within. To minimize friction between the cable and the supporting surface, the grease has been applied to the cable’s surface.

Figure 1. Design schematic of MC-AM.

Figure 2(a) shows an application case of the MSMS. The TSPS can be used to transport that tension from MC-AM’s output-end to the distal anchor. The benefit of this design is that the MC-AM is mounted at the remote of the load, reducing the weight of the equipment and increasing the number of configurable muscles on the skeleton. As shown in Fig. 2(b), the motor has four working modes. The active and passive working requirements of the muscles are represented by the first and fourth quadrants, respectively. The MSMS’s circuit control scheme is depicted in Fig. 2(c). The industrial Ether CAT protocol is selected to connect the physical level (motors and various sensors) with the lower level (control system in Matlab/Simulink software) with a communication frequency of 1 kHz. The UDP protocol transmits data between lower level and higher level (development computer) with a communication frequency of 500 Hz. Such a control scheme can meet the information transmission in the case of multiple MSMS and multiple sensors, including muscle tension sensors, angle sensors, and vision sensors.

Figure 2. Artificial muscle system case schematic.

2.2. Friction compensation mathematical model of TSPS

The movement of the skeleton requires the antagonism of multiple MAMSs. TSPS can not only realize the transmission of muscle tension from the distal MC-AM to the skeleton anchor, but also brings negative problems such as nonlinear friction and elastic deformation of the cable. The tension equilibrium may be used to calculate the tension distribution along the cable at a quasi-static condition, producing the well-known Capstan equation [Reference Kaneko, Yamashita and Tanie36].

(1) \begin{equation}dT=F=\mu N\cdot \delta ;\,N=T(p)d\phi\end{equation}

(2) \begin{equation}T_{\mathrm{out}}=T_{\mathrm{in}}e^{-\mu \phi \cdot \delta }\end{equation}

where F is friction tension, T(p) is the cable tension at position p along the cable, N is normal force exists between the cable and the inner wall of the sheath, and $\mu$ is a friction coefficient. $\delta$ is the sign of velocity of the cable relative to the sheath.

Due to factors such as the transmission efficiency of the cable itself, the lubricant viscosity, elastic cable deformation, the practical application of Eq. (2) to the cable drive is inapplicable. To accommodate the real operating circumstances, the friction model needs to be modified. A rudimentary musculoskeletal application scenario for one joint and two MAMSs is shown in Fig. 3. A tension sensor is installed at the output end of the MC-AM, which is a closed-loop control between current and tension. The accuracy tension from MC-AM is assumed in this work. The combined transmission mode of TSPS is used from the MC-AM output end to the distal anchor of the skeleton. The joint angle affects the cable-pulley envelope angle. Considering the energy transfer efficiency of the cable itself, viscous friction, and reference [Reference Wang, Kinugawa and Kosuge25, Reference Jeong and Cho26], we get

(3) \begin{equation}\phi =\sum _{i=1}^{n}\int _{0}^{\theta _{\mathrm{pi}}}d\phi _{p}+\int _{0}^{l_{s}}d\phi _{s}\end{equation}

(4) \begin{equation}v=\frac{dT_{\mathrm{in}}}{dt}\end{equation}

(5) \begin{equation}\mu =\delta (v)\end{equation}

(6) \begin{equation}\theta _{\mathrm{pi}}=H(q_{i},r,n)\end{equation}

(7) \begin{equation}T_{\mathrm{out}}=\eta ^{-\text{sign}(v)}T_{\mathrm{in}}e^{-\mu \phi \cdot \text{sign}(v)}+T_{0}\end{equation}

where $\eta$ is energy efficiency of the cable, $\phi$ is sum of cable bending angles; q is joint angle, $\phi _{s}$ and $\phi _{p}$ is elements of cable bending angle in cable sheath and cable pulley subsystem. Correspondingly, $l_{s}$ is length of sheath, $r$ is radius of pulley, $n$ is number of pulley, $\theta _{pi}$ is pulley wrap angle relative to joint angle, $H(q_{i},r,n)$ is the function of pulley wrap angle related to joint angle and number of pulley, $T_{\mathrm{out}}$ is tension of skeleton remote anchors, $T_{\mathrm{in}}$ is the tension at the output of the MC-AM, $T_{0}$ is pretension of the cable, and $\mu$ is viscosity coefficient function of relative velocity of cable relative to the sheath.

Figure 3. Diagram of friction in the musculoskeletal system.

The feed-forward compensation rule follows the inverse of the Eq. (7):

(8) \begin{equation}T_{r}=\eta ^{\text{sign}\left(v_{s}\right)}T_{s}\left(1-e^{\mu \phi \cdot \text{sign}\left(v\right)}\right)\end{equation}

(9) \begin{equation}v_{s}=\frac{dT_{j}}{dt}\end{equation}

where, $T_{r}$ is compensated friction. $T_{j}$ is tension after filtering at the output of MA-AM (in order to prevent control jitter).

2.3. Tendon elongation mathematical model of TSPS

The elastic deformation of cable would make the inaccurate position control of the system using TSPS. There is a serious problem for applications that require time-varying configurations of the TDPS but without equivalent sensory feedback at the distal end due to location limitations, such as surgical robots and musculoskeletal robot. As the main medium for transmitting tension, the cable will produce elastic deformation under the action of external tension. Incorporate references [Reference Wang, Sun and Phee21, Reference Kaneko, Yamashita and Tanie36], the elongation $\delta _{l}$ of the cable under input tension $T_{s}$ as follows:

(10) \begin{equation}\delta _{l}=\begin{cases} \dfrac{\rho R}{\mu }\eta ^{\text{sign}\left(v\right)}T_{j}\left(1-e^{-\dfrac{\mu l}{R}\cdot \text{sign}\left(v\right)}\right)-\rho T_{0}l_{c} &\mathrm{if} \,l_{c}\lt L_{1}\\[14pt] \dfrac{\rho R}{\mu }\eta ^{\text{sign}\left(v\right)}T_{j}\left(1-e^{-\dfrac{\mu L_{1}}{R}\cdot \text{sign}\left(v\right)}\right)-\rho T_{0}L_{1} &\mathrm{if} \,l_{c}\geq L_{1} \end{cases}\end{equation}

(11) \begin{equation}L_{1}=\min \left\{l_{c}\in T_{j}\left(l_{c}\right)=T_{0}\right\}\end{equation}

where, $l_{c}$ is length of cable. $\rho =\frac{1}{EA}$ , $E$ is the Youngs’ modulus of cable, A is the cross-sectional area of the cable; R is the bending radius of the sheath; $T_{0}$ is pretension of the cable, $L_{1}$ is the maximum length along the tendon until where the input tension can be transmitted.

Figure 4. Multilayer neural network structure diagram.

2.4. FF-MMN parameter identification method

The TSPS do not have standard consistency in design, manufacture, and installation, making friction compensation impossible through conventional model-based algorithms. As a flexible object, the cable will deform when it transmits the tension and generate nonlinear friction with the surface of the contacting object under the tension. It is difficult to complete the nonlinear regression prediction of friction by means of polynomial fitting. This section examines the relationship between the TSPS’s measurable input tension and unmeasurable output tension as well as cable elongation using the FF-MMN, according to the theory depicted in Fig. 4. Although FF-MMN can establish complex functions and nonlinear relationships between a set of input and output data with multiple dimensions, there are still some limitations that need to be addressed, such as overfitting, underfitting, and serious time-consuming issues. In order to improve the prediction performance of the constructed model, such as high precision, stability and high velocity, this work adopts a FF-MNN method to establish the regression model of the TSPS. The principle formula is as follows:

(12) \begin{equation}y=W_{0}\sum _{k=1}^{K}{\Gamma} \left(\sum _{i=1}^{N}\sum _{j=1}^{M}\left(W_{i,jk}\cdot X_{jk}+b_{jk}\right)\right)+b_{0}\end{equation}

(13) \begin{equation}{\Theta} =\underset{{\Theta} }{\text{argmin}} \sum _{t=1}^{N}\left(\hat{y}_{t}-y_{t}\right)^{2}\end{equation}

where, $X_{jk}$ is input of the $ j{\mathrm{th}}$ neuron in the $ k{\mathrm{th}}$ layer, $W_{i,jk}$ and $b_{jk}$ are the weighting factors and bias, respectively. $W_{0}$ and $b_{0}$ are initial value of the weighting factors and bias, respectively. ${\Theta}$ is the optimal parameter, which can be calculated by the minimum least squares between the predicted result $\hat{y}$ and the real value y.

There are three common evaluation indices to measure the performance of the built FF-MNN models, namely mean square error (MSE), root mean square error (RMSE), and Pearson correlation coefficient $\rho$ :

(14) \begin{equation}\mathrm{MSE}=\frac{\sum _{t=1}^{N}\left(\hat{y}_{t}-y_{t}\right)^{2}}{N}\end{equation}

(15) \begin{equation}\text{RMSE}=\sqrt{\mathrm{MSE}}\end{equation}

(16) \begin{equation}\rho =\frac{1}{N-1}\sum _{t=1}^{N}\left(\frac{\hat{y}_{t}-\mu _{\hat{y}}}{\sigma _{\hat{y}}}\right)\left(\frac{y_{t}-\mu _{y}}{\sigma _{y}}\right)\end{equation}

where, the time t can be regarded as the number of observations. $\mu _{\hat{y}}$ and $\sigma _{\hat{y}}$ are the average and standard deviation of $\hat{y}$ , while $\mu _{y}$ and $\sigma _{y}$ are the same values of y. The best score for the Pearson correlation coefficient $\rho$ is 1 while for the other errors, it is 0. The FF-MNN model aims to predict the tension close to the measured value.

Although there are alternative ways to evaluate sheath bending accumulation angle [Reference Jeong and Cho37], they are not available in the musculoskeletal robot. In the musculoskeletal robot, the posture of the sheath and the number of pulleys are fixed, and the wrapping angle of the cable on the pulley is related to the joint angle $q$ . According to Eqs. (8) and (10) we get

(17) \begin{equation}T_{r}=\tau ({\Theta}, T_{s}, l_{s}, v_{c}, q,n)\end{equation}

(18) \begin{equation}l_{r}=l({\Theta}, T_{s}, v_{c}, q, l_{c}, L_{1})\end{equation}

Where, $q$ is the joint angle of the driven equipment, in order to calculate the wrap angle of the cable pulley.

The number of training samples and nodes is critical to the accuracy of the training results and the computation time. According to the experimental experience of TSPS, 500,000 (total time 500 s, sampling time 0.001 s) sample data and 30 nodes were selected for friction compensation and cable elongation rate prediction.

3.1. Experiment design of friction compensation

Figure 5 depicts the minimal MSMS friction experiment bench using two MA-AMs and a TSPS. One of MA-AM acts as the distal load and is coupled to the other end through TSPS. A transmission distance and the length of the sheath are around 1667 mm and 806 mm, correspondingly, which can accommodate most of the musculoskeletal system utilization requirements. Each MA-AM is equipped with a MAXON DC35 motor (rated power 120 W), an RV-type reducer (53:1), and a high-precision tension sensor. The 16 braided Nylon cable with a diameter of 3 mm is selected, and the Young’s modulus of cable is about 1.4 GPa (the maximum allowable tension is about 800 N). The surface of the cable is coated with an appropriate amount of grease (MOLYKOTE G-0052FG, the viscosity is 115 cst at 40°C).

Figure 5. Minimal MSMS friction experiment bench. (a) Designed scheme. (b) Real test bench.

The tendon sheath friction related to the length and bending angle of the sheath have been studied by many scholars [Reference Jeong and Cho26, Reference Jeong and Cho37, Reference Liang, Du, Wang and Sun38]. These principles are applicable to the time-varying sheath bending angle and feedback. The MSMS designed in this study is used under the condition that the sheath’s length and posture remain unchanged. Under larger load and higher contraction velocity conditions, it is not possible to install additional sensors in the sheath. So that according to Eqs. (17) and (18), the experimental variables are designed as muscle tension ( $T_{s}$ and $T_{\mathrm{out}}$ ) and muscle contraction velocity (v). The specific implementation method is as follows:

1. 1. Experiments of active muscle: the motor working in the first quadrant supply a tension. The range of expected muscle tension ( $T_{\mathrm{out}}$ ) and contraction velocity (v) is 40– $\sim$ 200 N and 0.02– $\sim$ 0.12 m/s, respectively.

2. 2. Experiments of passive muscle: the motor working in the fourth quadrant supply a tension. The range of expected muscle tension ( $T_{\mathrm{out}}$ ) and contraction velocity (v) is 40– $\sim$ 200 N and −0.12– $\sim$ −0.02 m/s, respectively.

3.2. Design of LM-Arm

Based on the designed MSMS, we built the LM-Arm platform, which is strikingly comparable to the anatomy of the human arm [Reference Zhong, Chen, Niu, Fu and Qiao39]. As illustrated in Fig. 6, it consists of four skeleton (shoulder blade, humerus, ulna, and radius), seven DOF (shoulder three DOF, elbow one DOF, forearm one DOF, and wrist two DOF), and 15 MSMSs (the shoulder, elbow, forearm, and wrist joint have seven MSMSs, two MSMSs, two MSMSs, and four MSMSs, respectively). The lengths of upper arm, forearm, and hand are, respectively, 0.33 m, 0.39 m, and 0.26 m. It is essential to have a high-precision sensing system, which includes individual muscle tension and joint angle sensors. At least two MSMSs control each DOF, and each MSMSs controls only one joint. This has the benefit of decreasing the coupling between the muscle and the joint space, provided that the antagonistic connection between the multiple muscles is satisfied.

Figure 6. LM-Arm design diagram.

Although friction compensation is carried out for MSMS, it cannot compensate for the manufacturing and installation errors and dynamic model errors. We therefore introduce a straightforward computational muscle force feedback control strategy to improve the effect of controlling LM-Arm. The control amount is accurate to the muscle tension. The kinematic of the skeletons refers to the method of Denavit–Hartenberg Matrix. We can calculate muscle length $l$ according to the following formula.

(19) \begin{equation}l=g(q)\end{equation}

The Jacobian matrix $L(q)$ maps the relationship between muscle and joint space. Combined with the principle of virtual work [Reference De Sapio, Warren, Khatib and Delp40], we get the following relation:

(20) \begin{equation}dl=-L(q)dq\end{equation}

(21) \begin{equation}{\tau} =-L(q)\cdot F\end{equation}

where, q is a vector of joint variables, muscle length matrix $dl=[dl_{1},dl_{2}\ldots dl_{15}]$ ; joint angle matrix $dq=[dq_{1},dq_{2}\ldots dq_{7}];$ $F=[f_{1},f_{2},f_{3}\ldots f_{15}]$ is the muscle tension matrix; ${\tau} =[\tau _{1},\tau _{2},\tau _{3}\ldots \tau _{7}]$ is joints torque matrix. The direction of muscular contraction is indicated by the negative sign. Optimize level of muscular tension under the following conditions:

(22) \begin{equation}\min _{F_{d}} \| F_{d}\| ^{2}\, \text{Subject to}\, \begin{cases} {\tau} =-L(q)\cdot F_{d}\\[5pt] F_{\min }\leq F_{d}\leq F_{\max } \end{cases} \end{equation}

where, $F_{d}$ is optimal muscle tension; $F_{\min }$ and $F_{\max }$ is upper and lower limits of muscle tension.

The dynamic of LM-Arm is calculated with Lagrangian dynamic:

(23) \begin{equation}H [-\dot{L}^{+}(q)\dot{l}-L(q)\ddot{l}]-{\Theta} \mathrm{L}^{+}(q)\dot{l}+G=-L(q)F-d\end{equation}

where $H$ represents a symmetric positive defined inertia matrix, ${\Theta}$ denotes the Centripetal and Coriolis force, G is the gravitational force, F is muscle tension, and $d$ represents the uncertain interference of system friction and model error.

We estimated the compensated muscle acceleration through the error of muscle length and muscle velocity. The formula is as follows:

(24) \begin{equation}{\Delta} l=l_{e}-l\end{equation}

(25) \begin{equation}{\Delta} \dot{l}=\dot{l_{e}}-\dot{l}\end{equation}

(26) \begin{equation}{\Delta} \ddot{l}=A\cdot {\Delta} l-B\cdot {\Delta} \dot{l}\end{equation}

where $l_{e}$ is the desired muscle length; $l$ is the measured muscle length; $\dot{l_{e}}$ is the desired muscle velocity; $\dot{l}$ is the measured muscle velocity; ${\Delta} l$ , ${\Delta} \dot{l}$ are error of muscle length, muscle velocity, respectively. ${\Delta} \ddot{l}$ is the compensation of muscle acceleration; A and B are proportional coefficient and differential coefficient, correspondingly.

To test the flexibility of LM-Arm and the performance of MSMS, we build a sin-function end-point 3D trajectory tracking with two period and an amplitude of 0.15 m. We adjusted the running time to 115 s and repeat the experiment six times.

Figure 7. Experimental results no friction compensation .