A new control for the pneumatic muscle bionic legged robot based on neural network

Abstract

The bionic joints composed of pneumatic muscles (PMs) can simulate the motion of biological joints. However, the PMs themselves have non-linear characteristics such as hysteresis and creep, which make it difficult to achieve high-precision trajectory tracking control of PM-driven robots. In order to effectively suppress the adverse effects of non-linearity on control performance and improve the dynamic performance of PM-driven legged robot, this study designs a double closed-loop control structure based on neural network. First, according to the motion model of the bionic joint, a mapping model between PM contraction force and joint torque is proposed. Second, a control strategy is designed for the inner loop of PM contraction force and the outer loop of bionic joint angle. In the inner control loop, a feedforward neuron Proportional-Integral-Derivative controller is designed based on the PM three-element model. In the control outer loop, a sliding mode robust controller with local model approximation is designed by using the radial basis function neural network approximation capability. Finally, it is verified by simulation and physical experiments that the designed control strategy is suitable for humanoid motion control of antagonistic PM joints, and it can satisfy the requirements of reliability, flexibility, and bionics during human–robot collaboration.

1 INTRODUCTION

Legged robots have flexible mobility similar to human bipedal [1]. In terms of adapting to complex terrain, because the legged robot has discrete landing points in time and space, it can overcome obstacles with a size similar to the robot's leg length in a non-contact way. Therefore, the environmental adaptability of legged robots is stronger than that of wheeled robots and crawler robots [2-4].

With the gradual development of hydraulic driven and motor-driven legged robots, the defects of heavy weight, complex structure and high cost have gradually become prominent, while bionic soft robots have strong adaptability to unstructured environments and higher human–robot interaction safety [5-9]. Pneumatic muscle (PM), as the representative of the flexible actuator in the bionic soft robot, has become an important product of the new actuator due to its lightness, good flexibility, and high power-to-weight ratio [10-12]. As a legged robot that cooperates with people, the use of PM drives can better realise the characteristics of bionics, better adapt to the environment, and have a good application prospects. However, due to the shortcomings of strong non-linearity, hysteresis, and creep of PM, it is difficult to establish an accurate mathematical model for the PM robot to achieve high-precision position tracking [13-15]. Moreover, the coupling of multiple variables, such as the position, contraction force, and stiffness of the PM, is strong, and its application scenarios are often limited.

In order to realise the high-precision control of PM, many researchers have carried out related research. Zhong et al. established a quaternary polynomial phenomenological model of PM as feedforward, and designed an improved Proportional-Integral-Derivative (PID) algorithm to improve the dynamic tracking performance of PM [16]. Ai et al. proposed a model-free adaptive iterative controller to transform the complex dynamics of PM into a dynamic linear model through iterative learning, and achieved a fast convergence rate through a high-order estimation algorithm [17]. Chen et al. considered that long-term work would lead to PM ageing and lead to changes in the model, and used a three-layer neural network to identify and compensate online for unknown non-linear part in the dynamic model of the PM system, ensuring the asymptotic convergence of the tracking error of the PM system, which ensures the stable operation of the PM system [18]. Aiming at the hysteresis of PM, Shakiba et al. designed a feedforward controller based on the Prandtl-Ishlinski model to suppress the hysteresis behaviour of PM to reduce the hysteresis effect in the case of single-frequency lag [19].

Using PM as the driver of the robot means introducing more unknown disturbance terms and unmodelled terms to the robot dynamics model, which increases the difficulty of the controller to track the robot's position. In order to avoid the negative impact of complex accurate modelling and time-varying model parameters on the controller design, researchers have proposed many effective methods. For non-linear systems with time-varying uncertainty, Wang et al. proposed an adaptive model predictive control algorithm, which uses fast adaptation to estimate the uncertainty, so that the controller does not depend on an accurate system model [20]. Tong et al. designed a model-free fuzzy adaptive dynamic surface controller, which approximates the robot model information required in the dynamic surface control algorithm through fuzzy logic, and has good anti-disturbance performance [21]. Wang et al. designed a variable-parameter particle swarm optimisation algorithm to identify the parameters of the dynamic model, which improved the identification accuracy [22]. Neural network has the characteristics of infinite approximation to any non-linear function, and has a fast convergence speed, which is very suitable for the control of robotic system with highly non-linear characteristics [23-25]. Gao et al. used two radial basis function (RBF) neural networks to approximate the non-linear uncertainty of the model and system, and designed aperiodic adaptive laws to improve control efficiency [26]. Considering both the holonomic and non-holonomic constraints, Sun et al. proposed an RBF neural network controller to control the robot's position and approximate unknown dynamic parameters [27].

Considering the uncertainty of the overall dynamic model of the bionic legged robot due to the structure of the crossed four-link knee joint, this paper aims to solve the problem of low control accuracy of the PM robot from the aspects of system modelling and anti-interference design of the control system. First, by analysing the kinematic models of the hip and knee joints, a mapping model between PM contraction force and joint torque is proposed. In addition, the dynamic models of the PMs are obtained through experimental identification, which are used for the design of the PM feedforward controller. Then, according to the characteristics of PM bionic leg, a double closed-loop control of the PM bionic leg based on a neural network is proposed. In the inner control loop, a feedforward neuron PID controller is designed to control the PM contraction force to reduce the influence of the strong non-linearity of the PM on the system. In the control outer loop, three RBF neural networks are used to approximate the dynamic parameters of the bionic leg in real time, and a sliding mode robust controller is designed to control the joint position of the bionic leg. The control outer loop and inner loop work together to further improve the overall anti-interference ability of the system.

2 MODEL OF PM BIONIC LEGGED ROBOT

2.1 Motion model of the PM bionic legged robot

The PM bionic legged robot realises swinging and walking motions by imitating the antagonistic and pulling relationship of human biceps and triceps, and its motion model is shown in Figure 1. The hip joint is a sprocket transmission mechanism, and the knee joint is a crossed four-bar linkage mechanism. The waist muscles above the hip joint of the bionic leg are composed of PM_a and PM_b, and the thigh muscles above the knee joint of the bionic leg are composed of PM_c and PM_d. Two sets of PMs drive the hip and knee joints to rotate by pulling against each other. In Figure 1, F_i (i = a,b,c,d) is the contraction force generated by PM_i, d_i is the terminal displacement of PM_i, r is the radius of the hip joint, $${{\theta }_{\mathrm{j}}}_{k}$$, τ_jk, O_jk (k = 1,2) are the rotation angle, torque, and rotation centre of the joint respectively.

FIGURE 1

Motion model of the bionic legged robot.

The rotation range of the hip joint is defined as $${{\theta }_{\mathrm{j}}}_{1}\in \left[{\theta }_{\mathrm{j}1\text{\_}\mathrm{min}\,},{\theta }_{\mathrm{j}1\text{\_}\mathrm{max}\,}\,\!\!\right]\!\!\,$$, the expressions between $${{\theta }_{\mathrm{j}}}_{1}$$ and the terminal retraction displacements d_a of PM_a and d_b of PM_b, respectively, are as follows:

$$${d}_{\mathrm{a}}=\left({\theta }_{\mathrm{j}1\text{\_}\mathrm{max}\,}+{\theta }_{\mathrm{j}1}\right)\cdot r$$$ (1)

$$${d}_{\mathrm{b}}=\left({\theta }_{\mathrm{j}1\text{\_}\mathrm{min}\,}-{\theta }_{\mathrm{j}1}\right)\cdot r$$$ (2)

The relationship between the contraction force F_a of PM_a and the contraction force F_b of PM_b and the active torque τ_j1 of the hip joint are as follows:

$$${\tau }_{\mathrm{j}1}={\tau }_{\mathrm{a}}-{\tau }_{\mathrm{b}}=\left({F}_{\mathrm{a}}-{F}_{\mathrm{b}}\right)r$$$ (3)

The mapping models between F_a and F_b to the hip joint torque τ_j1 are designed as:

$$${F}_{\mathrm{a}}=\left\{\begin{array}{l}\frac{{\tau }_{\mathrm{j}1}}{r}+{F}_{\text{a0}}\ \ \ \ \ \ {\tau }_{\mathrm{j}1}\ge \text{0}\hfill \\ {F}_{\text{a0}}\ \ \ \ \ \ \ \ \ \ \ \ \ {\tau }_{\mathrm{j}1}< \text{0}\hfill \end{array}\right.$$$ (4)

$$${F}_{\mathrm{b}}=\left\{\begin{array}{l}{F}_{\text{b0}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\tau }_{\mathrm{j}1}\ge \text{0}\hfill \\ \frac{\left\vert {\tau }_{\mathrm{j}1}\right\vert }{r}+{F}_{\text{b0}}\ \ \ \,\ \ \ {\tau }_{\mathrm{j}1}< \text{0}\hfill \end{array}\right.$$$ (5)

where $${F}_{\text{a0}}={F}_{\text{b0}}=\frac{{\tau }_{\mathrm{j}10}}{r}\ge 0$$ are the initial contraction forces obtained from the initial torque τ_j10 of the hip joint, and F_a0 and F_b0 keep the PMs in a contracted state all the time to prevent the PMs from bending during exercise.

The motion model of the knee joint is shown in Figure 2. The knee joint considers the variable force arm and crossed four-link structure of the human knee joint, and the calf is driven through the variable force arm crossed four-link structure ANQT. The intersection point of the link AQ and NT during the movement is the instantaneous rotation centre of the knee joint. The movement mode of the variable force arm can better imitate the movement of the knee joint of the human body, and can realise a large range of rotation. The physical parameters of the crossed four-bar linkage are shown in Table 1.

FIGURE 2

Motion model of knee joint.

TABLE 1. Parameters of the crossed four-bar linkage

Parameter	Length (mm)	Parameter	Length (mm)
LAR	38.7	LNH	40.0
LRQ	75.5	LNT	75.0
LAQ	70.0	LHT	101.0
LAN	35.0	LQT	28.0
LAV	288.0	LNS	306.0
LBV	40.0	LJS	40.0
Lc	337.0	Ld	314.0

For the knee joint model, through geometric calculation analysis, we can get:

$$$\begin{array}{rl}\hfill f\left({d}_{\mathrm{c}}\right)& ={\theta }_{\mathrm{j}2}=\pi +{\vartheta }_{2}-\kappa \hfill \\ \hfill & =\pi +\left({\varepsilon }_{2}-{\chi }_{1}\right)-\left(\pi -{\alpha }_{2}+{\gamma }_{2}-{\beta }_{2}-{\chi }_{2}\right)\hfill \end{array}$$$ (6)

$$$\begin{array}{rl}\hfill f\left({d}_{\mathrm{d}}\right)& ={\theta }_{\mathrm{j}2}=\pi -{\vartheta }_{1}-\varsigma \hfill \\ \hfill & =\pi -\left({\varepsilon }_{1}-{\delta }_{1}\right)-\left({\gamma }_{1}-{\delta }_{2}-{\beta }_{1}+{\alpha }_{2}\right)\hfill \end{array}$$$ (7)

where $${\varepsilon }_{1}=\mathrm{arccos}\left(\frac{{L}_{\text{AV}}^{2}\,+\,{L}_{\text{AB}}^{2}\,-\,{L}_{\text{BV}}^{2}}{2{L}_{\text{AV}}{L}_{\text{AB}}}\right)+\mathrm{arccos}\left(\frac{{L}_{\text{AB}}^{2}+{L}_{\text{AR}}^{2}-\,{L}_{\text{BR}}^{2}}{2{L}_{\text{AB}}{L}_{\text{AR}}}\right)$$, $${\varepsilon }_{2}=\mathrm{arccos}\left(\frac{{L}_{\text{NJ}}^{2}\,+\,{L}_{\text{NS}}^{2}\,-\,{L}_{\text{JS}}^{2}}{2{L}_{\text{NJ}}{L}_{\text{NS}}}\right)+\mathrm{arccos}\left(\frac{{L}_{\text{NJ}}^{2}\,+\,{L}_{\text{NH}}^{2}\,-\,{L}_{\text{JH}}^{2}}{2{L}_{\text{NJ}}{L}_{\text{NH}}}\right)$$, $${\beta }_{1}=\mathrm{arccos}\left(\frac{{L}_{\text{NQ}}^{2}\,+\,{L}_{\text{AQ}}^{2}\,-\,{L}_{\text{AN}}^{2}}{2{L}_{\text{NQ}}{L}_{\text{AQ}}}\right)$$, $${\beta }_{2}=\mathrm{arccos}\left(\frac{{L}_{\text{AT}}^{2}\,+\,{L}_{\text{NT}}^{2}\,-\,{L}_{\text{AN}}^{2}}{2{L}_{\text{AT}}{L}_{\text{NT}}}\right)$$, $${\gamma }_{1}=\mathrm{arccos}\left(\frac{{L}_{\text{QT}}^{2}\,+\,{L}_{\text{NQ}}^{2}\,-\,{L}_{\text{NT}}^{2}}{2{L}_{\text{QT}}{L}_{\text{NQ}}}\right)$$, $${\gamma }_{2}=\mathrm{arccos}\left(\frac{{L}_{\text{QT}}^{2}\,+\,{L}_{\text{AT}}^{2}\,-\,{L}_{\text{AQ}}^{2}}{2{L}_{\text{QT}}{L}_{\text{AT}}}\right)$$, $${L}_{\text{NQ}}=\sqrt{{L}_{\text{AN}}^{2}+{L}_{\text{AQ}}^{2}-2\mathrm{cos}\left(\pi +{\alpha }_{1}-{\varepsilon }_{1}-{\delta }_{3}\right){L}_{\text{AN}}{L}_{\text{AQ}}}$$, $${L}_{\text{AT}}=\sqrt{{L}_{\text{AN}}^{2}+{L}_{\text{NT}}^{2}-2\mathrm{cos}\left(2\pi -{\alpha }_{1}-{\varepsilon }_{2}-{\chi }_{3}\right){L}_{\text{AN}}{L}_{\text{NT}}}$$, L_JH = L_c − d_c, L_BR = L_d − d_d, χ₁ = 0.70 rad, χ₂ = 0.35 rad, χ₃ = 2.09 rad, δ₁ = 1.17 rad, δ₁ = 0.53 rad, δ₁ = 1.44 rad, α₁ = 1.31 rad, α₂ = 0.67 rad.

Then, according to the physical parameters of the crossed four-bar linkage, the expressions of θ_j2 and the terminal retraction displacement d_c and d_d, respectively, are as follows:

$$${d}_{\mathrm{c}}=\text{0}.001{\theta }_{\mathrm{j}2}^{2}-0.429{\theta }_{\mathrm{j}2}+43.15$$$ (8)

$$${d}_{\mathrm{d}}=\text{0}.00065{\theta }_{\mathrm{j}2}^{2}+0.182{\theta }_{\mathrm{j}2}-13.09$$$ (9)

The torques τ_c and τ_d generated by the contraction forces F_c and F_d acting on the endpoints N and A of the crossed four-link can be obtained from the static equilibrium equation:

$$${\tau }_{\mathrm{c}}={F}_{\mathrm{c}}{x}_{\mathrm{c}}={{F}_{\mathrm{c}}}^{\prime }{x}_{\mathrm{c}}^{\prime }$$$ (10)

$$${\tau }_{\mathrm{d}}={F}_{\mathrm{d}}{x}_{\mathrm{d}}={{F}_{\mathrm{d}}}^{\prime }{x}_{\mathrm{d}}^{\prime }$$$ (11)

where x_c and $${x}_{\mathrm{c}}^{\prime }$$ are the force arms of F_c and $${{F}_{\mathrm{c}}}^{\prime }$$ to endpoint N respectively. x_d and $${x}_{\mathrm{d}}^{\prime }$$ are the force arms of F_d and $${{F}_{\mathrm{d}}}^{\prime }$$ to endpoint A respectively. According to the proportional relationship of the two force arms, the proportional coefficient can be obtained:

$$$v=\frac{{\tau }_{\mathrm{d}}}{{\tau }_{\mathrm{c}}}=\frac{{x}_{\mathrm{d}}}{{x}_{\mathrm{c}}}$$$ (12)

In view of Equations (10-11)–(12), the torque at the instantaneous centre of the crossed four-link knee joint is as follows:

$$${\tau }_{\mathrm{j}2}={\tau }_{\mathrm{d}}-{\tau }_{\mathrm{c}}v={F}_{\mathrm{d}}{x}_{\mathrm{d}}-{F}_{\mathrm{c}}{x}_{\mathrm{c}}v$$$ (13)

In the same way as the hip joint, the mapping models between F_c and F_d to the knee joint torque τ_j2 is designed as:

$$${F}_{\mathrm{c}}=\left\{\begin{array}{ll}{F}_{\text{c0}}\hfill & {\tau }_{\mathrm{j}2}\ge \text{0}\hfill \\ \frac{\left\vert {\tau }_{\mathrm{j}2}\right\vert }{v\cdot {x}_{\mathrm{c}}}+{F}_{\text{c0}}\hfill & {\tau }_{\mathrm{j}2}< \text{0}\hfill \end{array}\right.$$$ (14)

$$${F}_{\mathrm{d}}=\left\{\begin{array}{ll}\frac{{\tau }_{\mathrm{j}2}}{{x}_{\mathrm{d}}}+{F}_{\text{d0}}\hfill & {\tau }_{\mathrm{j}2}\ge \text{0}\hfill \\ {F}_{\text{d0}}\hfill & {\tau }_{\mathrm{j}2}< \text{0}\hfill \end{array}\right.$$$ (15)

where $${F}_{\text{c0}}=\frac{{\tau }_{\mathrm{j}20}}{v\cdot {x}_{\mathrm{c}}}\ge 0$$ and $${F}_{\text{d0}}=\frac{{\tau }_{\mathrm{j}20}}{{x}_{\mathrm{d}}}\ge 0$$ are the initial contraction forces obtained from the initial torque τ_j20 of the knee joint.

2.2 Dynamic model of PM

Due to the structural flexibility of PM, this paper selects PM as the actuator of bionic leg, which can enhance the flexibility of joints, reduce the body weight of single-legged robot, and realise safe human–robot interaction.

Considering the strong non-linear characteristics, such as hysteresis and creep of the PM, this section will complete the modelling of the PM drive unit through the experimental identification method based on the three-element method. The three-element modelling method generally simplifies the model into a system of contraction unit, spring unit, and damping unit [28]. However, due to the relationship between the body structure of the PM and the air pressure drive, the PM mostly works at a lower frequency in actual movement, so the damping unit is ignored in the experimental identification. The PM contraction force is simplified to the force generated by the contraction unit H_i (N) and the spring unit K_i (N/mm), respectively, and its mathematical expression is:

$$${F}_{i}={H}_{i}(p)-{K}_{i}(p){d}_{i}$$$ (16)

where p is the air pressure inside the PM.

The isobaric test experiments were carried out on three kinds of PMs used in the bionic single-leg robot, in which the length of PM_a and PM_b are 185 mm, the length of PM_c is 156 mm, and the length of PM_d is 133 mm. The specific steps are to apply different pulling forces to the PM under a fixed air pressure of 0–0.6 bar, and record the displacement and pulling force value of the end of the PM. After data analysis and fitting, the effective force-air pressure and stiffness-air pressure expressions of the 185, 156 and 133 mm PMs can be obtained:

$$$\left\{\begin{array}{ll}{H}_{\text{a,}\mathrm{b}}=231.7{p}_{\text{a,}\mathrm{b}}+125.9\hfill & 0< {p}_{\text{a,}\mathrm{b}}\le 6\hfill \\ {K}_{\text{a,}\mathrm{b}}=\left\{\begin{array}{l}-24.2{p}_{\text{a,}\mathrm{b}}+67.4\hfill \\ 5.10{p}_{\text{a,}\mathrm{b}}+4.5\hfill \end{array}\right.\begin{array}{l}0< {p}_{\text{a,}\mathrm{b}}\le 2.15\hfill \\ 2.15< {p}_{\text{a,}\mathrm{b}}\le 6\hfill \end{array}\hfill & \end{array}\right.$$$ (17)

$$$\left\{\begin{array}{ll}{H}_{\mathrm{c}}=199.3{p}_{\mathrm{c}}+222.6\hfill & 0< {p}_{\mathrm{c}}\le 6\hfill \\ {K}_{\mathrm{c}}=\left\{\begin{array}{l}-21.9{p}_{\mathrm{c}}+53.9\hfill \\ 3.4{p}_{\mathrm{c}}+14.80\hfill \end{array}\right.\begin{array}{l}0< {p}_{\mathrm{c}}\le 1.54\hfill \\ 1.54< {p}_{\mathrm{c}}\le 6\hfill \end{array}\hfill & \end{array}\right.$$$ (18)

$$$\left\{\begin{array}{ll}{H}_{\mathrm{d}}=207.8{p}_{\mathrm{d}}+391.1\hfill & 0< {p}_{\mathrm{d}}\le 6\hfill \\ {K}_{\mathrm{d}}=\left\{\begin{array}{l}-45.5{p}_{\mathrm{d}}+120.5\hfill \\ 6.1{p}_{\mathrm{d}}+11.3\hfill \end{array}\right.\begin{array}{l}0< {p}_{\mathrm{d}}\le 2.12\hfill \\ 2.12< {p}_{\mathrm{d}}\le 6\hfill \end{array}\hfill & \end{array}\right.$$$ (19)

Substituting Equations (17-18)–(19) into Equation (16), respectively, we can obtain the mathematical expressions of the contraction force of PM_a, PM_b, PM_c, and PM_d as follows:

$$${F}_{\text{a,}\mathrm{b}}=\left\{\begin{array}{c}\hfill \left(\begin{array}{l}24.2{p}_{\text{a,}\mathrm{b}}{d}_{\text{a,}\mathrm{b}}+231.7{p}_{\text{a,}\mathrm{b}}\hfill \\ -67.4{d}_{\text{a,}\mathrm{b}}+125.9\hfill \end{array}\right)\,0< {p}_{\text{a,}\mathrm{b}}\le 2.15\hfill \\ \hfill \left(\begin{array}{l}-5.1{p}_{\text{a,}\mathrm{b}}{d}_{\text{a,}\mathrm{b}}+231.7{p}_{\text{a,}\mathrm{b}}\hfill \\ -4.5{d}_{\text{a,}\mathrm{b}}+125.9\hfill \end{array}\right)\,2.15< {p}_{\text{a,}\mathrm{b}}\le 6\hfill \end{array}\right.$$$ (20)

$$${F}_{\mathrm{c}}=\left\{\begin{array}{c}\hfill \left(\begin{array}{l}21.9{p}_{\mathrm{c}}{d}_{\mathrm{c}}+199.3{p}_{\mathrm{c}}\hfill \\ -53.9{d}_{\mathrm{c}}+222.6\hfill \end{array}\right)\,0< {p}_{\mathrm{c}}\le 1.54\hfill \\ \hfill \left(\begin{array}{l}-3.4{p}_{\mathrm{c}}{d}_{\mathrm{c}}+199.3{p}_{\mathrm{c}}\hfill \\ -14.8{d}_{\mathrm{c}}+222.6\hfill \end{array}\right)\,1.54< {p}_{\mathrm{c}}\le 6\hfill \end{array}\right.$$$ (21)

$$${F}_{\mathrm{d}}=\left\{\begin{array}{c}\hfill \left(\begin{array}{l}45.5{p}_{\mathrm{d}}{d}_{\mathrm{d}}+207.8{p}_{\mathrm{d}}\hfill \\ -120.5{d}_{\mathrm{d}}+391.1\hfill \end{array}\right)\,0< {p}_{\mathrm{d}}\le 2.12\hfill \\ \hfill \left(\begin{array}{l}-6.1{p}_{\mathrm{d}}{d}_{\mathrm{d}}+207.8{p}_{\mathrm{d}}\hfill \\ -11.3{d}_{\mathrm{d}}+391.1\hfill \end{array}\right)\,2.12< {p}_{\mathrm{d}}\le 6\hfill \end{array}\right.$$$ (22)

2.3 Dynamic model of the bionic leg

The simplified model of the single-leg mechanism can be regarded as a plane linkage mechanism, and the waist and thigh are fixed. In this paper, the elastic potential energy of PMs is considered, and the dynamic equation of the double-joint bionic leg is established by the Lagrangian modelling method as follows:

$$$\mathbf{I}(\boldsymbol{\uptheta })\ddot{\boldsymbol{\uptheta }}+\mathbf{C}\left(\boldsymbol{\uptheta },\dot{\boldsymbol{\uptheta }}\right)\dot{\boldsymbol{\uptheta }}+\mathbf{G}(\boldsymbol{\uptheta })=\boldsymbol{\uptau }-\mathbf{D}$$$ (23)

where $$\boldsymbol{\uptheta }={\left[{\theta }_{\mathrm{j}1}\quad {\theta }_{\mathrm{j}2}\right]}^{\mathrm{T}}$$ is the joint swing angle matrix, $$\mathbf{I}(\boldsymbol{\uptheta })=\left[\begin{array}{cc}\hfill {I}_{1}\hfill & \hfill {I}_{2}\hfill \\ \hfill {I}_{3}\hfill & \hfill {I}_{4}\hfill \end{array}\right]$$ is the positive definite inertia matrix, $$\mathbf{C}\left(\boldsymbol{\uptheta },\dot{\boldsymbol{\uptheta }}\right)=\left[\begin{array}{cc}\hfill {C}_{1}\hfill & \hfill {C}_{2}\hfill \\ \hfill {C}_{3}\hfill & \hfill {C}_{4}\hfill \end{array}\right]$$ is the Centripetal and Coriolis matrix, $$\mathbf{G}(\boldsymbol{\uptheta })={\left[{G}_{1}\quad {G}_{2}\right]}^{\mathrm{T}}$$ is the gravity and elastic potential energy matrix item, and $$\boldsymbol{\uptau }={\left[{\tau }_{1}\quad {\tau }_{2}\right]}^{\mathrm{T}}$$ is the input torque item, $$\mathbf{D}={\left[{D}_{1}\quad {D}_{2}\right]}^{\mathrm{T}}$$ contains uncertain terms and unknown disturbances. The expressions in the I(θ), $$\mathbf{C}\left(\boldsymbol{\uptheta },\dot{\boldsymbol{\uptheta }}\right)$$ and G(θ) matrices are:

$$$\begin{array}{l}{I}_{1}={m}_{\mathrm{j}1}{l}_{\mathrm{j}1}^{2}+{I}_{\mathrm{j}1}+{I}_{\mathrm{j}2}+{m}_{\mathrm{j}2}\left({L}_{\mathrm{j}1}^{2}+{l}_{\mathrm{j}2}^{2}+2{L}_{\mathrm{j}1}{l}_{\mathrm{j}2}\mathrm{cos}{\theta }_{\mathrm{j}2}\right)\hfill \\ {I}_{2}={m}_{\mathrm{j}2}{l}_{\mathrm{j}2}^{2}+{m}_{\mathrm{j}2}{L}_{\mathrm{j}1}{l}_{\mathrm{j}2}\mathrm{cos}{\theta }_{\mathrm{j}2}+{I}_{\mathrm{j}2}\hfill \\ {I}_{3}={m}_{\mathrm{j}2}{l}_{\mathrm{j}2}^{2}+{m}_{\mathrm{j}2}{L}_{\mathrm{j}1}{l}_{\mathrm{j}2}\mathrm{cos}{\theta }_{\mathrm{j}2}+{I}_{\mathrm{j}2}\hfill \\ {I}_{4}={m}_{\mathrm{j}2}{l}_{\mathrm{j}2}^{2}+{I}_{\mathrm{j}2}\hfill \\ {C}_{1}=-{\dot{\theta }}_{\mathrm{j}2}{m}_{\mathrm{j}2}{L}_{\mathrm{j}1}{l}_{\mathrm{j}2}\mathrm{sin}{\theta }_{\mathrm{j}2}\hfill \\ {C}_{2}=-{\dot{\theta }}_{\mathrm{j}1}{m}_{\mathrm{j}2}{L}_{\mathrm{j}1}{l}_{\mathrm{j}2}\mathrm{sin}{\theta }_{\mathrm{j}2}-{\dot{\theta }}_{\mathrm{j}2}{m}_{\mathrm{j}2}{L}_{\mathrm{j}1}{l}_{\mathrm{j}2}\mathrm{sin}{\theta }_{\mathrm{j}2}\hfill \\ {C}_{3}={\dot{\theta }}_{\mathrm{j}2}{m}_{\mathrm{j}2}{L}_{\mathrm{j}1}{l}_{\mathrm{j}1}\mathrm{sin}{\theta }_{\mathrm{j}2}\hfill \\ {C}_{4}=0\hfill \\ {G}_{1}={m}_{\mathrm{j}1}g{l}_{\mathrm{j}1}\mathrm{sin}{\theta }_{\mathrm{j}1}+{m}_{\mathrm{j}2}g{L}_{\mathrm{j}1}\mathrm{sin}{\theta }_{\mathrm{j}1}+{m}_{\mathrm{j}2}g{l}_{\mathrm{j}2}\mathrm{sin}\left({\theta }_{\mathrm{j}1}+{\theta }_{\mathrm{j}2}\right)\hfill \\ \ \ \ \hspace*{.5em}\ +{K}_{\mathrm{a}}{d}_{\mathrm{a}}r-{K}_{\mathrm{b}}{d}_{\mathrm{b}}r\hfill \\ {G}_{2}={m}_{\mathrm{j}2}g{l}_{\mathrm{j}2}\mathrm{sin}\left({\theta }_{\mathrm{j}1}+{\theta }_{\mathrm{j}2}\right)+{K}_{\mathrm{c}}{d}_{\mathrm{c}}\left(0.002{\theta }_{\mathrm{j}2}-0.429\right)\hfill \\ \ \ \ \ \hspace*{.5em}+{K}_{\mathrm{d}}{d}_{\mathrm{d}}\left(0.0013{\theta }_{\mathrm{j}2}+0.182\right)\hfill \end{array}$$$

where m_j1 and m_j2 are the mass of the thigh and calf, respectively, I_j1 and I_j2 are the moments of inertia of the thigh and calf, respectively, l_j1 is the length from the hip joint to the centre of mass of the thigh, l_j2 is the length from the knee joint to the centre of mass of the calf, L_j1 is the length of the thigh, and g is the acceleration of gravity.

3 CONTROLLER DESIGN

The PM bionic joint system has complex characteristics and strong coupling of multiple variables, such as position, contraction force, and stiffness. It is difficult to establish an accurate mathematical model, which will affect its position tracking performance. Therefore, this paper designs a control method based on a neural network to realise the position tracking control of the PM bionic legged robot. The control block diagram is shown in Figure 3. The outer loop is the joint angle control loop, and the inner loop is the PM contraction force control loop. The PM bionic leg ensures precise tracking of its position through the combined action of the inner and outer rings.

FIGURE 3

Double closed-loop control of pneumatic muscle (PM) bionic legged robot.

3.1 Design of PM contraction force controller

Considering the inherent strong non-linearity and hysteresis of PMs, a feedforward neuron PID controller is designed. In this paper, the inverse model of the PM contraction force models (20-21)–(22) is used as the output of the feedforward control of the contraction force controller. A neuron PID controller is designed to compensate the feedforward output. The neuron adjusts the parameters of the PID controller online by the gradient descent method according to the system state, so as to enhance the adaptability of the PID algorithm to the high non-linearity of PM.

First, the effective force and stiffness expressions (17-18)–(19) can be rewritten as follows:

$$$\left\{\begin{array}{ll}{H}_{i}(k)={q}_{1i}{p}_{i}(k)+{q}_{2i}\hfill & 0< {p}_{i}(k)\le 6\hfill \\ \hfill {K}_{i}(k)=\left\{\begin{array}{cc}\hfill {q}_{3i}{p}_{i}(k)+{q}_{4i}\hfill & \hfill 0< {p}_{i}(k)\le {p}_{mi}\hfill \\ \hfill {q}_{5i}{p}_{i}(k)+{q}_{6i}\hfill & \hfill {p}_{mi}< {p}_{i}(k)\le 6\hfill \end{array}\right.\hfill & \end{array}\right.$$$ (24)

where q_1i and q_2i are constant coefficients in the effective force expression, q_3i, q_4i, q_5i, and q_6i are constant coefficients in the stiffness expression, i = a, b, c, d represent the numbering of the four PMs.

According to Equations (20-21)–(22), the inverse models of PM dynamics can be obtained. Based on these models, an air pressure feedforward controller is designed, and its control output P_fi(k) is:

$$${P}_{\mathrm{f}i}(k)=\left\{\begin{array}{c}\hfill \frac{{F}_{\mathrm{d}i}(k)+{q}_{4i}{d}_{\mathrm{i}}(k)-{q}_{2i}}{{q}_{1i}-{q}_{3i}{d}_{i}(k)}\,0< {p}_{i}\le {p}_{\mathrm{m}i}\hfill \\ \hfill \frac{{F}_{\mathrm{d}i}(k)+{q}_{6i}{d}_{\mathrm{i}}(k)-{q}_{2i}}{{q}_{1i}-{q}_{5i}{d}_{i}(k)}\,{p}_{\mathrm{m}i}< {p}_{i}\le 6\hfill \end{array}\right.$$$ (25)

The PID controller selects the positional PID controller, and its control output P_bi(k) is:

$$$\begin{array}{r}\hfill {P}_{\mathrm{b}i}(k)={K}_{Pi}{e}_{i}(k)+{K}_{Ii}\sum\limits _{n=0}^{k}{e}_{i}(n)\\ +{K}_{Di}\left({e}_{i}(k)-{e}_{i}(k-1)\right)\hfill \end{array}$$$ (26)

where e_i(k) is the difference between the ideal contraction force and the actual contraction force of the PM at time k.

By combining the feedforward control output (25) and the PID feedback control output (26), we can obtain the contraction force control law as:

$$${P}_{i}(k)={P}_{\mathrm{f}i}(k)+{P}_{\mathrm{b}i}(k)$$$ (27)

For the update of K_Pi, K_Ii, K_Di in the control law, the neuron performance index function is set as:

$$${E}_{i}(k)=\frac{1}{2}{\left[{F}_{\mathrm{d}i}(k)-{F}_{i}(k)\right]}^{2}$$$ (28)

The PID parameter adjustment algorithms are as follows:

$$$\left\{\begin{array}{l}{\Delta }{K}_{\mathrm{P}i}(k)=-{r}_{\mathrm{P}i}\frac{\partial E(k)}{\partial {K}_{\mathrm{P}i}}={r}_{\mathrm{P}i}e(k)\frac{\partial {F}_{i}(k)}{\partial {P}_{i}(k-1)}{L}_{\mathrm{P}i}(k-1)\hfill \\ {\Delta }{K}_{\mathrm{I}i}(k)=-{r}_{\mathrm{I}i}\frac{\partial E(k)}{\partial {K}_{\mathrm{I}i}}={r}_{\mathrm{I}i}e(k)\frac{\partial {F}_{i}(k)}{\partial {P}_{i}(k-1)}{L}_{\mathrm{I}i}(k-1)\hfill \\ {\Delta }{K}_{\mathrm{D}i}(k)=-{r}_{\mathrm{D}i}\frac{\partial E(k)}{\partial {K}_{\mathrm{D}i}}={r}_{\mathrm{D}i}e(k)\frac{\partial {F}_{i}(k)}{\partial {P}_{i}(k-1)}{L}_{\mathrm{D}i}(k-1)\hfill \end{array}\right.$$$ (29)

where e(k) = F_di(k) − F_i(k), r_Pi ∈ (0, 1), r_Ii ∈ (0, 1), r_Di ∈ (0, 1) are the learning rates, L_Pi(k) = e_i(k), $${L}_{\mathrm{I}i}(k)={\sum }_{n=0}^{k}{e}_{i}(n)$$, L_Di(k) = e_i(k) − e_i(k − 1). Let $${y}_{\mathrm{n}i}(k-1)=\frac{\partial {F}_{i}(k)}{\partial {P}_{i}(k-1)}$$, and y_ni is replaced approximately by $$\text{sgn}\left(\frac{{F}_{i}(k\,-\,1)-{F}_{i}(k\,-\,2)}{{P}_{i}(k\,-\,2)-{P}_{i}(k\,-\,3)}\right)$$.

In view of Equation (29), the update algorithms for the K_Pi, K_Ii, K_Di can be obtained as follows:

$$$\left\{\begin{array}{l}{K}_{\mathrm{P}i}(k)={K}_{\mathrm{P}i}(k-1)+{r}_{\mathrm{P}i}e(k){y}_{\mathrm{n}i}(k-1){L}_{\mathrm{P}i}(k-1)\hfill \\ {K}_{\mathrm{I}i}(k)={K}_{\mathrm{I}i}(k-1)+{r}_{\mathrm{I}i}e(k){y}_{\mathrm{n}i}(k-1){L}_{\mathrm{I}i}(k-1)\hfill \\ {K}_{\mathrm{D}i}(k)={K}_{\mathrm{D}i}(k-1)+{r}_{\mathrm{D}i}e(k){y}_{\mathrm{n}i}(k-1){L}_{\mathrm{D}i}(k-1)\hfill \end{array}\right.$$$ (30)

3.2 Design of bionic leg position controller

In practical situations, the mathematical model of the robot is usually difficult to obtain accurately or unknown [29, 30]. Due to the crossed four-link knee joint mechanism of the PM bionic leg robot, there are more uncertainties in the overall dynamic model of the bionic legged robot. In order to make the dynamic modelling of the system more accurate and improve the control accuracy of the bionic-legged robot, three RBF neural networks are used in this paper to approximate the dynamic parameters.

The RBF neural network has a simple structure and has consistent approximation to non-linear continuous functions, and its hidden layer activation function is a Gaussian basis function:

$$${\varphi }_{k}=\mathrm{exp}\left(-\frac{{\left\Vert \mathbf{x}-{\boldsymbol{\upmu }}_{k}\right\Vert }^{2}}{2{\sigma }_{k}^{2}}\right),k=1,2,\text{\ldots },m$$$ (31)

where x is the neural network input matrix, μ is the centre node vector matrix, and σ is the node base width.

The dynamic parameters I(θ), $$\mathbf{C}\left(\boldsymbol{\uptheta },{\boldsymbol{\dot{\boldsymbol{\uptheta }}}}\right)$$, G(θ) of the bionic leg are approximated by three RBF neural networks, and their expressions are as follows:

$$$\left\{\begin{array}{l}\mathbf{I}(\boldsymbol{\uptheta })={\mathbf{I}}_{\text{NN}}(\boldsymbol{\uptheta })+{\mathbf{E}}_{\mathbf{I}}^{\prime }\hfill \\ \mathbf{C}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)={\mathbf{C}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)+{\mathbf{E}}_{\mathbf{C}}^{\prime }\hfill \\ \mathbf{G}(\boldsymbol{\uptheta })={\mathbf{G}}_{\text{NN}}(\boldsymbol{\uptheta })+{\mathbf{E}}_{\mathbf{G}}^{\prime }\hfill \end{array}\right.$$$ (32)

where I_NN(θ), $${\mathbf{C}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)$$, G_NN(θ) are the output values of the three RBF neural networks, and $${\mathbf{E}}_{\mathbf{I}}^{\prime }$$, $${\mathbf{E}}_{\mathbf{C}}^{\prime }$$, $${\mathbf{E}}_{\mathbf{G}}^{\prime }$$ are the approximation errors.

Define $${\widehat{\mathbf{I}}}_{\text{NN}}(\boldsymbol{\uptheta })$$, $${\widehat{\mathbf{C}}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)$$, $${\widehat{\mathbf{G}}}_{\text{NN}}(\boldsymbol{\uptheta })$$ as the estimations of I_NN(θ), $${\mathbf{C}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)$$, G_NN(θ), respectively, we have:

$$$\left\{\begin{array}{l}{\widehat{\mathbf{I}}}_{\text{NN}}(\boldsymbol{\uptheta })={\widehat{\mathbf{W}}}_{\mathbf{I}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{I}}(\boldsymbol{\uptheta })\hfill \\ {\widehat{\mathbf{C}}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)={\widehat{\mathbf{W}}}_{\mathbf{C}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{C}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)\hfill \\ {\widehat{\mathbf{G}}}_{\text{NN}}(\boldsymbol{\uptheta })={\widehat{\mathbf{W}}}_{\mathbf{G}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{G}}(\boldsymbol{\uptheta })\hfill \end{array}\right.$$$ (33)

where $${\widehat{\mathbf{W}}}_{\mathbf{I}}$$, $${\widehat{\mathbf{W}}}_{\mathbf{C}}$$, $${\widehat{\mathbf{W}}}_{\mathbf{G}}$$ as the estimations of W_I, W_C, and W_G respectively.

The joint angle tracking error is defined as:

$$$\mathbf{z}={\boldsymbol{\uptheta }}_{\mathrm{d}}-\boldsymbol{\uptheta }$$$ (34)

In view of Equation (34), the sliding surface is defined as:

$$$\mathbf{s}=\dot{\mathbf{z}}+\boldsymbol{\upzeta }\mathbf{z}$$$ (35)

where ζ > 0.

By making use of Equations (34) and (35), we can get:

$$$\dot{\boldsymbol{\uptheta }}=\boldsymbol{\upzeta }\mathbf{z}+{\dot{\boldsymbol{\uptheta }}}_{\mathrm{d}}-\mathbf{s}$$$ (36)

$$$\ddot{\boldsymbol{\uptheta }}=\boldsymbol{\upzeta }\boldsymbol{\dot{\mathbf{z}}}+{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{d}}-\dot{\mathbf{s}}$$$ (37)

In view of Equations (36) and (37), dynamics expression (23) can be rewritten as:

$$$\begin{array}{rl}\hfill \boldsymbol{\uptau }& =\mathbf{I}(\boldsymbol{\uptheta })\left(\boldsymbol{\upzeta }\boldsymbol{\dot{\mathbf{z}}}+{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{d}}-\dot{\mathbf{s}}\right)+\mathbf{C}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)\left(\boldsymbol{\upzeta }\mathbf{z}+{\dot{\boldsymbol{\uptheta }}}_{\mathrm{d}}-\mathbf{s}\right)+\mathbf{G}(\boldsymbol{\uptheta })+\mathbf{D}\hfill \\ \hfill & =\mathbf{I}(\boldsymbol{\uptheta })\left(\boldsymbol{\upzeta }\boldsymbol{\dot{\mathbf{z}}}+{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{d}}\right)+\mathbf{C}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta}}}\right)\left(\boldsymbol{\upzeta }\mathbf{z}+{\dot{\boldsymbol{\uptheta }}}_{\mathrm{d}}\right)+\mathbf{G}(\boldsymbol{\uptheta })\hfill \\ \hfill & -\mathbf{I}(\boldsymbol{\uptheta })\dot{\mathbf{s}}-\mathbf{C}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta}}}\right)\mathbf{s}+\mathbf{D}\hfill \\ \hfill & =\left[{\mathbf{W}}_{\mathbf{I}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{I}}(\boldsymbol{\uptheta })\right]\left(\boldsymbol{\upzeta }\boldsymbol{\dot{\mathbf{z}}}+{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{d}}\right)+\left[{\mathbf{W}}_{\mathbf{C}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{C}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)\right]\left(\boldsymbol{\upzeta }\mathbf{z}+{\dot{\boldsymbol{\uptheta }}}_{\mathrm{d}}\right)\hfill \\ \hfill & +\left[{\mathbf{W}}_{\mathbf{G}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{G}}(\boldsymbol{\uptheta })\right]-\mathbf{I}(\boldsymbol{\uptheta })\dot{\mathbf{s}}-\mathbf{C}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)\mathbf{s}+\mathbf{E}\hfill \end{array}$$$ (38)

where $$\mathbf{E}={\mathbf{E}}_{\mathbf{I}}^{\prime }{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{d}}+{\mathbf{E}}_{\mathbf{C}}^{\prime }{\dot{\boldsymbol{\uptheta }}}_{\mathrm{d}}+{\mathbf{E}}_{\mathbf{G}}^{\prime }+\mathbf{D}$$.

Define the reference variables $${\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}$$ and $${\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}$$ as:

$$${\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}=\boldsymbol{\upzeta }\mathbf{z}+{\dot{\boldsymbol{\uptheta }}}_{\mathrm{d}}=\mathbf{s}+\dot{\boldsymbol{\uptheta }}$$$ (39)

$$${\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}=\boldsymbol{\upzeta }\boldsymbol{\dot{\mathbf{z}}}+{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{d}}=\dot{\mathbf{s}}+\ddot{\boldsymbol{\uptheta }}$$$ (40)

Based on the nominal model of the bionic leg obtained by the three RBF neural networks, the control law of the nominal model is designed as:

$$${\boldsymbol{\uptau }}_{\mathrm{n}}={\widehat{\mathbf{I}}}_{\text{NN}}(\boldsymbol{\uptheta }){\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+{\widehat{\mathbf{C}}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right){\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+{\widehat{\mathbf{G}}}_{\text{NN}}(\boldsymbol{\uptheta })$$$ (41)

In order to suppress the negative influence of unknown disturbance and the approximation error of the neural network, the PI robust control is designed as:

$$${\boldsymbol{\uptau }}_{\mathrm{s}}={\mathbf{Q}}_{\mathrm{P}}\mathbf{s}+{\mathbf{Q}}_{\mathrm{I}}\int \mathbf{s}\mathrm{d}t+{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})$$$ (42)

where Q_P > 0, Q_I > 0, M_ssgn(s) is the robust term, $${\mathbf{M}}_{\mathrm{s}}=\left[\begin{array}{cc}\hfill {M}_{\mathrm{s}1}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {M}_{\mathrm{s}2}\hfill \end{array}\right]$$, $${M}_{\mathrm{s}i}\ge \left\vert {\mathbf{E}}_{i}\right\vert $$, i = 1, 2.

Subsequently, by employing Equations (41) and (42), the control law is designed as follows:

$$$\begin{array}{rl}\hfill \boldsymbol{\uptau }& ={\boldsymbol{\uptau }}_{\mathrm{n}}+{\boldsymbol{\uptau }}_{\mathrm{s}}\hfill \\ \hfill & ={\widehat{\mathbf{I}}}_{\text{NN}}(\boldsymbol{\uptheta }){\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+{\widehat{\mathbf{C}}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right){\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+{\widehat{\mathbf{G}}}_{\text{NN}}(\boldsymbol{\uptheta })\hfill \\ \hfill & +{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})+{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}+{\mathbf{Q}}_{\mathrm{I}}\int \mathbf{s}\mathrm{d}t\hfill \\ \hfill & =\left[{\widehat{\mathbf{W}}}_{\mathbf{I}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{I}}(\boldsymbol{\uptheta })\right]{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+\left[{\widehat{\mathbf{W}}}_{\mathbf{C}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{C}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)\right]{\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}\hfill \\ \hfill & +\left[{\widehat{\mathbf{W}}}_{\mathbf{G}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{G}}(\boldsymbol{\uptheta })\right]+{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})+{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}+{\mathbf{Q}}_{\mathrm{I}}\int \mathbf{s}\mathrm{d}t\hfill \end{array}$$$ (43)

Finally, we design the following update laws:

$$$\left\{\begin{array}{l}{\dot{\widehat{\mathbf{W}}}}_{\mathbf{I}i}={\boldsymbol{\Xi }}_{\mathbf{I}i}{\boldsymbol{\uppsi }}_{\mathbf{I}i}(\boldsymbol{\uptheta }){\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}{\mathbf{s}}_{i}\hfill \\ {\dot{\widehat{\mathbf{W}}}}_{\mathbf{C}i}={\boldsymbol{\Xi }}_{\mathbf{C}i}{\boldsymbol{\uppsi }}_{\mathbf{C}i}\left(\boldsymbol{\uptheta },\dot{\boldsymbol{\uptheta }}\right){\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}{\mathbf{s}}_{i}\hfill \\ {\dot{\widehat{\mathbf{W}}}}_{\mathbf{G}i}={\boldsymbol{\Xi }}_{\mathbf{G}i}{\boldsymbol{\uppsi }}_{\mathbf{G}i}(\boldsymbol{\uptheta }){\mathbf{s}}_{i}\hfill \end{array}\right.$$$ (44)

where $${\dot{\widehat{\mathbf{W}}}}_{\mathbf{I}i}$$, $${\dot{\widehat{\mathbf{W}}}}_{\mathbf{C}i}$$, $${\dot{\widehat{\mathbf{W}}}}_{\mathbf{G}i}$$ are the update laws of $${\widehat{\mathbf{W}}}_{\mathbf{I}i}$$, $${\widehat{\mathbf{W}}}_{\mathbf{C}i}$$, $${\widehat{\mathbf{W}}}_{\mathbf{G}i}$$. Ξ_Ii, Ξ_Ci, Ξ_Gi are positive definite symmetric matrices, ψ_Ii, ψ_Ci, ψ_Gi are the elements in the matrices φ_I(θ), $${\boldsymbol{\upvarphi }}_{\mathbf{C}}\left(\boldsymbol{\uptheta },\dot{\boldsymbol{\uptheta }}\right)$$, φ_G(θ), respectively, i = 1, 2.

To demonstrate the stability of the closed-loop system under the proposed control method, we perform a stability proof.

Proof: The Lyapunov function is chosen as:

$$$\begin{array}{rl}\hfill V& =\frac{1}{2}\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{I}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{I}i}^{-1}{\tilde{\mathbf{W}}}_{\mathbf{I}i}+\frac{1}{2}\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{C}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{C}i}^{-1}{\tilde{\mathbf{W}}}_{\mathbf{C}i}\hfill \\ \hfill & +\frac{1}{2}\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{G}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{G}i}^{-1}{\tilde{\mathbf{W}}}_{\mathbf{G}i}+\frac{1}{2}{\mathbf{s}}^{\mathrm{T}}\mathbf{I}\mathbf{s}\hfill \\ \hfill & +\frac{1}{2}{\left(\int \nolimits_{0}^{t}\mathbf{s}\mathrm{d}\tau \right)}^{\mathrm{T}}{\mathbf{Q}}_{\mathrm{I}}\left(\int \nolimits_{0}^{t}\mathbf{s}\mathrm{d}\tau \right)\hfill \end{array}$$$ (45)

The derivative of V is:

$$$\begin{array}{rl}\hfill \dot{V}& =\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{I}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{I}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{I}i}+\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{C}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{C}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{C}i}\hfill \\ \hfill & +\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{G}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{G}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{G}i}+{\mathbf{s}}^{\mathrm{T}}\left[\frac{1}{2}\dot{\mathbf{I}}s+\mathbf{I}\boldsymbol{\dot{\mathbf{s}}}+{\mathbf{Q}}_{\mathrm{I}}\left(\int \nolimits_{0}^{t}\mathbf{s}\mathrm{d}\tau \right)\right]\hfill \end{array}$$$ (46)

Since I and C satisfy the following equation:

$$${\mathbf{s}}^{\mathrm{T}}\left(\dot{\mathbf{I}}-2\mathbf{C}\right)\mathbf{s}=0$$$ (47)

Thus, Equation (46) can be rewritten as:

$$$\begin{array}{r}\hfill \dot{V}=\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{I}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{I}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{I}i}+\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{C}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{C}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{C}i}\\ +\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{G}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{G}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{G}i}+{\mathbf{s}}^{\mathrm{T}}\left[\mathbf{C}\mathbf{s}+\mathbf{I}\boldsymbol{\dot{\mathbf{s}}}+{\mathbf{Q}}_{\mathrm{I}}\left(\int \nolimits_{0}^{t}\mathbf{s}\mathrm{d}\tau \right)\right]\hfill \end{array}$$$ (48)

According to Equations (38) and (43), the following equations can be obtained:

$$$\begin{array}{rl}\hfill & \mathbf{I}\dot{\mathbf{s}}+\mathbf{C}\mathbf{s}+{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}+{\mathbf{Q}}_{\mathrm{I}}\int \mathbf{s}\mathrm{d}t+{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})\hfill \\ \hfill & =\left[\left({\mathbf{W}}_{\mathbf{I}}^{\mathrm{T}}-{\widehat{\mathbf{W}}}_{\mathbf{I}}^{\mathrm{T}}\right)\cdot {\boldsymbol{\upvarphi }}_{\mathbf{I}}(\boldsymbol{\uptheta })\right]{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+\left[\left({\mathbf{W}}_{\mathbf{C}}^{\mathrm{T}}-{\widehat{\mathbf{W}}}_{\mathbf{C}}^{\mathrm{T}}\right)\cdot {\boldsymbol{\upvarphi }}_{\mathbf{C}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)\right]{\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}\hfill \\ \hfill & +\left[\left({\mathbf{W}}_{\mathbf{G}}^{\mathrm{T}}-{\widehat{\mathbf{W}}}_{\mathbf{G}}^{\mathrm{T}}\right)\cdot {\boldsymbol{\upvarphi }}_{\mathbf{G}}(\boldsymbol{\uptheta })\right]+\mathbf{E}\hfill \end{array}$$$ (49)

Let $${\tilde{\mathbf{W}}}_{\mathbf{I}}={\mathbf{W}}_{\mathbf{I}}-{\widehat{\mathbf{W}}}_{\mathbf{I}},{\tilde{\mathbf{W}}}_{\mathbf{C}}={\mathbf{W}}_{\mathbf{C}}-{\widehat{\mathbf{W}}}_{\mathbf{C}},{\tilde{\mathbf{W}}}_{\mathbf{G}}={\mathbf{W}}_{\mathbf{G}}-{\widehat{\mathbf{W}}}_{\mathbf{G}}$$, Equation (49) can be rewritten as:

$$$\begin{array}{rl}\hfill & \mathbf{I}\dot{\mathbf{s}}+\mathbf{C}\mathbf{s}+{\mathbf{Q}}_{\mathrm{I}}\int \mathbf{s}\mathrm{d}t\hfill \\ \hfill & =\left[{\tilde{\mathbf{W}}}_{\mathbf{I}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{I}}(\boldsymbol{\uptheta })\right]{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+\left[{\tilde{\mathbf{W}}}_{\mathbf{C}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{C}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta }}}\right)\right]{\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}\hfill \\ \hfill & +\left[{\tilde{\mathbf{W}}}_{\mathbf{G}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{G}}(\boldsymbol{\uptheta })\right]+\mathbf{E}-{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}-{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})\hfill \end{array}$$$ (50)

Substituting Equation (50) into Equation (48), we get:

$$$\begin{array}{rl}\hfill & \dot{V}=\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{I}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{I}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{I}i}+\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{C}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{C}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{C}i}\hfill \\ \hfill & +\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{G}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{G}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{G}i}+{\mathbf{s}}^{\mathrm{T}}\left\{\left[{\tilde{\mathbf{W}}}_{\mathbf{I}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{I}}(\boldsymbol{\uptheta })\right]{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}\right.\hfill \\ \hfill & \left.+\left[{\tilde{\mathbf{W}}}_{\mathbf{C}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{C}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)\right]{\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+\left[{\tilde{\mathbf{W}}}_{\mathbf{G}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{G}}(\boldsymbol{\uptheta })\right]\right.\hfill \\ \hfill &\left. +\mathbf{E}-{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}-{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})\right\}\hfill \end{array}$$$ (51)

Furthermore,

$$$\begin{array}{rl}\hfill {\mathbf{s}}^{\mathrm{T}}{\tilde{\mathbf{W}}}_{\mathbf{I}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{I}}(\boldsymbol{\uptheta }){\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}& ={\left[\begin{array}{c}\hfill {\mathbf{s}}_{1}\hfill \\ \hfill {\mathbf{s}}_{2}\hfill \end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}\hfill {\tilde{\mathbf{W}}}_{\mathbf{I}1}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{I}1}{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}\hfill \\ \hfill {\tilde{\mathbf{W}}}_{\mathbf{I}2}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{I}2}{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}\hfill \end{array}\right]\hfill \\ \hfill & =\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{I}}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{I}i}{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}{\mathbf{s}}_{i}\hfill \end{array}$$$ (52)

$$$\begin{array}{rl}\hfill {\mathbf{s}}^{\mathrm{T}}{\tilde{\mathbf{W}}}_{\mathbf{C}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{C}}(\boldsymbol{\uptheta }){\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}& ={\left[\begin{array}{c}\hfill {\mathbf{s}}_{1}\hfill \\ \hfill {\mathbf{s}}_{2}\hfill \end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}\hfill {\tilde{\mathbf{W}}}_{\mathbf{C}1}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{C}1}{\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}\hfill \\ \hfill {\tilde{\mathbf{W}}}_{\mathbf{C}2}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{C}2}{\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}\hfill \end{array}\right]\hfill \\ \hfill & =\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{C}}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{C}i}{\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}{\mathbf{s}}_{i}\hfill \end{array}$$$ (53)

$$$\begin{array}{rl}\hfill {\mathbf{s}}^{\mathrm{T}}{\tilde{\mathbf{W}}}_{\mathbf{G}}^{\mathrm{T}}\cdot {\boldsymbol{\upvarphi }}_{\mathbf{G}}(\boldsymbol{\uptheta })& ={\left[\begin{array}{c}\hfill {\mathbf{s}}_{1}\hfill \\ \hfill {\mathbf{s}}_{2}\hfill \end{array}\right]}^{\mathrm{T}}\left[\begin{array}{c}\hfill {\tilde{\mathbf{W}}}_{\mathbf{G}1}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{G}1}\hfill \\ \hfill {\tilde{\mathbf{W}}}_{\mathbf{G}2}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{G}2}\hfill \end{array}\right]\hfill \\ \hfill & =\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{G}}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{G}i}{\mathbf{s}}_{i}\hfill \end{array}$$$ (54)

Equation (51) can be arranged in the form:

$$$\begin{array}{rl}\hfill \dot{V}& =\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{I}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{I}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{I}i}+\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{C}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{C}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{C}i}\hfill \\ \hfill & +\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{G}i}^{\mathrm{T}}{\boldsymbol{\Xi }}_{\mathbf{G}i}^{-1}{\dot{\tilde{\mathbf{W}}}}_{\mathbf{G}i}+\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{I}}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{I}i}{\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}{\mathbf{s}}_{i}\hfill \\ \hfill & +\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{C}}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{C}i}{\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}{\mathbf{s}}_{i}+\sum\limits _{i=1}^{2}{\tilde{\mathbf{W}}}_{\mathbf{G}}^{\mathrm{T}}{\boldsymbol{\uppsi }}_{\mathbf{G}i}{\mathbf{s}}_{i}\hfill \\ \hfill & +{\mathbf{s}}^{\mathrm{T}}\mathbf{E}-{\mathbf{s}}^{\mathrm{T}}{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}-{\mathbf{s}}^{\mathrm{T}}{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})\hfill \end{array}$$$ (55)

Since $${\dot{\widehat{\mathbf{W}}}}_{\mathbf{I}i}=-{\dot{\tilde{\mathbf{W}}}}_{\mathbf{I}i}$$, $${\dot{\widehat{\mathbf{W}}}}_{\mathbf{C}i}=-{\dot{\tilde{\mathbf{W}}}}_{\mathbf{C}i}$$, $${\dot{\widehat{\mathbf{W}}}}_{\mathbf{G}i}=-{\dot{\tilde{\mathbf{W}}}}_{\mathbf{G}i}$$, $${M}_{\mathrm{s}i}\ge \left\vert {\mathbf{E}}_{i}\right\vert $$, combined with Equation (44), we can get:

$$$\dot{V}={\mathbf{s}}^{\mathrm{T}}\mathbf{E}-{\mathbf{s}}^{\mathrm{T}}{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}-{\mathbf{s}}^{\mathrm{T}}{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})\le -{\mathbf{s}}^{\mathrm{T}}{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}\le 0$$$ (56)

From Equations (45) and (56), we can get that both $${\tilde{\mathbf{W}}}_{\mathbf{I}}$$, $${\tilde{\mathbf{W}}}_{\mathbf{C}}$$, $${\tilde{\mathbf{W}}}_{\mathbf{G}}$$ and s are bounded. According to the LaSalle invariance principle, since when $$\dot{V}\equiv 0$$, s = 0, the system is asymptotically stable.

4 SIMULATION STUDY

In this section, the effectiveness of the control strategy of the PM bionic legged robot is verified by simulation.

First, the simulation model of the PM bionic leg was built in the MATLAB/Simulink simulation environment, and its physical parameters are shown in Table 2. Based on multiple debugging, the optimal position controller parameters are obtained. Let $$\mathbf{E}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$$, set the controller parameters Q_P = 2.5E, Q_I = 0.5E, ζ = 10.0E, Ξ_Ii = 5.7E, Ξ_Ci = 1.5E, Ξ_Gi = 1500.0E. For the three RBF neural networks used to approximate the bionic leg dynamics parameter, set their parameters μ_i = [−1.5 −0.75 0 0.75 1.5], σ_i = 1.185, the initial weights W_i(0) are all 0.

TABLE 2. Physical parameters of the bionic leg

Parameter	Value	Parameter	Value
mj1	2.97 kg	mj2	0.54 kg
lj1	12.40 mm	lj2	14.60 mm
Lj1	40.00 mm	Lj1	35.50 mm

Meanwhile, after the feedforward neuron PID controller has been updated for many times, the PID parameters may become too large, which will cause the system to oscillate. Therefore, in order to avoid system instability due to large PID parameters, we normalise the PID parameters to obtain:

$$$\left\{\begin{array}{l}{{K}_{\mathrm{P}}}^{\prime }=\frac{{K}_{\mathrm{P}}}{\vert {K}_{\mathrm{P}}\vert +\vert {K}_{\mathrm{I}}\vert +\vert {K}_{\mathrm{D}}\vert }\hfill \\ {{K}_{\mathrm{I}}}^{\prime }=\frac{{K}_{\mathrm{I}}}{\vert {K}_{\mathrm{P}}\vert +\vert {K}_{\mathrm{I}}\vert +\vert {K}_{\mathrm{D}}\vert }\hfill \\ {{K}_{\mathrm{D}}}^{\prime }=\frac{{K}_{\mathrm{D}}}{\vert {K}_{\mathrm{P}}\vert +\vert {K}_{\mathrm{I}}\vert +\vert {K}_{\mathrm{D}}\vert }\hfill \end{array}\right.$$$ (57)

The initial values of the parameters of the feedforward neuron PID controller corresponding to the four PMs are all set to K_Pi = 10, K_Ii = 10, K_Di = 5. The hip joint radius r = 0.05 m. In the contraction force mapping model, set the initial torques τ_j10 = τ_j20 = 0.15 N m. In addition, in order to show the advantages of the designed method, rewrite controller (43) as:

$$$\begin{array}{rl}\hfill \boldsymbol{\uptau }& ={\mathbf{I}}_{0}(\boldsymbol{\uptheta }){\ddot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+{\mathbf{C}}_{0}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta}}}\right){\dot{\boldsymbol{\uptheta }}}_{\mathrm{s}}+{\mathbf{G}}_{0}(\boldsymbol{\uptheta })\hfill \\ \hfill & +{\mathbf{M}}_{\mathrm{s}}\text{sgn}(\mathbf{s})+{\mathbf{Q}}_{\mathrm{P}}\mathbf{s}+{\mathbf{Q}}_{\mathrm{I}}\int \mathbf{s}\mathrm{d}t\hfill \end{array}$$$ (58)

where I₀, C₀ and G₀ are defined as nominal models, which are directly obtained from the simplified double-joint plane mechanism. We conduct simulation experiments with this controller as a comparison controller, and the controller parameters are consistent with the controller parameters in this paper.

Set the desired position curve of the hip joint as a sine curve of −60° to −30°, and the desired position curve of the knee joint as a sine curve of 60°–90°, and the period of both is 5 s. The angle and angle error response curves of the knee and hip joints are shown in Figures 4 and 5 respectively. Under the control method in this paper, the system reaches the steady state in about 2 s, while the control method with the nominal model reaches the steady state in a shorter time. Under the control method in this paper, the tracking error ranges of the hip and knee joints are −0.120° to 0.121° and −0.031° to 0.035° respectively. Under the control method with the nominal model, the tracking error ranges of the hip and knee joints are −0.095° to 0.125° and −0.639° to −0.108° respectively. It can be seen that for a simple fixed-axis mechanism such as the hip joint, the control effect of the method in this paper is similar to that of the method with a nominal model. For a complex non-fixed-axis mechanism such as the knee joint, since the parameters of the nominal model are obtained directly from the simplified double-joint mechanism, the crossed four-bar linkage is not considered. This paper uses three neural networks to approximate the system model. Therefore, the control method with a nominal model is far less effective in the control of the knee joint than the control method in this paper.

FIGURE 4

Angle tracking curves of hip joint: (a) Angle tracking curves. (b) Angle tracking error curves.

FIGURE 5

Angle tracking curves of knee joint: (a) Angle tracking curves. (b) Angle tracking error curves.

In order to quantitatively analyse the steady-state performance of robust controllers approximated by neural network local models. According to the analysis of angle tracking error z, Table 3 gives the values of maximum error (E_max), average error (E_ave) and standard deviation error (E_sd) after stable tracking, expressed as:

$$$\begin{array}{l}{E}_{\mathrm{max}}=\mathrm{max}\,\vert z(k)\vert \hfill \\ {E}_{\text{ave}}=\frac{1}{k}\sum\limits _{k=1}^{25000}\vert z(k)\vert \hfill \\ {E}_{\text{sd}}=\sqrt{\frac{1}{k}{{\sum }_{k=1}^{25000}\left(z(k)-\frac{1}{k}{\sum }_{1}^{25000}z(k)\right)}^{2}}\hfill \end{array}$$$ (59)

where k ∈ [1, 25,000] is the sampling point, and the sampling point time interval is 0.001 s, so there are 25,000 sampling points in 0–25 s.

TABLE 3. Steady-state performance of the pneumatic muscle (PM) bionic legged robot under two algorithms

Joint	RBF approximation		Nominal model
	Hip	Knee	Hip	Knee
Emax (°)	0.121	0.035	0.125	0.639
Eave (°)	0.075	0.013	0.049	0.323
Esd (°)	0.082	0.016	0.058	0.118

In order to more intuitively display the approximation effect of the three RBF neural networks, this paper compares the two-norm values of the matrices $${\widehat{\mathbf{I}}}_{\text{NN}}(\boldsymbol{\uptheta })$$, $${\widehat{\mathbf{C}}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)$$ and $${\widehat{\mathbf{G}}}_{\text{NN}}(\boldsymbol{\uptheta })$$ estimated by the neural network with the two-norm values of the matrices I(θ), $$\mathbf{C}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)$$ and G(θ) output by the bionic leg dynamics model in the simulation. Figure 6 shows the approximation effect of three RBF neural networks. The actual values are the two-norm values of the matrices I(θ), $$\mathbf{C}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)$$ and G(θ) in the dynamic model of the bionic leg during the simulation run. It can be seen from the figure that the three RBF neural networks have a good approximation effect on the dynamic parameters of the bionic leg. On the basis of the estimated values $${\widehat{\mathbf{I}}}_{\text{NN}}(\boldsymbol{\uptheta })$$, $${\widehat{\mathbf{C}}}_{\text{NN}}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)$$ and $${\widehat{\mathbf{G}}}_{\text{NN}}(\boldsymbol{\uptheta })$$, precise dynamic control of the PM bionic leg is carried out through sliding mode robust control.

FIGURE 6

Approximation curves of two-norm values of I(θ), $$\mathbf{C}\left(\boldsymbol{\uptheta },\boldsymbol{\dot{\boldsymbol{\uptheta} }}\right)$$ and G(θ): (a) Estimated curves. (b) Estimated error curves.

The torque response curves of the hip and knee joints are shown in Figure 7, and the position controller outputs the ideal torque control values τ_j1d and τ_j2d according to the system state. According to the obtained actual contraction forces F_a, F_b, F_c and F_d, the actual torque control values τ_j1 and τ_j2 are calculated from the actual contraction forces by the PM contraction force mapping models (4), (5), (14) and (15).

FIGURE 7

Torque tracking curves of the hip and knee joints: (a) Torque tracking curves. (b) Torque tracking error curves.

The response curves of the contraction force of the four PMs in the control inner ring are shown in Figure 8 and 9. According to the ideal joint torques τ_j1d and τ_j2d, the ideal contraction force F_ad, F_bd, F_cd, and F_dd of the four PMs are obtained through the PM contraction force mapping models (4), (5), (14) and (15).

FIGURE 8

Contraction force curves of pneumatic muscle (PM) - PM_a and PM_b: (a) Contraction force tracking curves. (b) Contraction force tracking error curves.

FIGURE 9

Contraction force curves of pneumatic muscle (PM) - PM_c and PM_d: (a) Contraction force tracking curves. (b) Contraction force tracking error curves.

For the hip joint, the contraction force provided by PM_a has been maintained at around the set initial contraction force value of 3 N. The change trend of the contraction force provided by PM_b is opposite to the change trend of the hip joint angle, and the contraction force generated by PM_b is mainly used to overcome the influence of gravity during the joint rotation process. For the knee joint, the contraction force provided by the PM_c varies between 4–6 N, with the maximum contractile force being provided when the knee joint angle reaches around 90°. The shank mainly relies on the contraction force generated by PM_d to drive through the crossed four-bar linkage. The results show that the designed feedforward neuron PID controller has good control performance for the control of PM contraction force.

The parameter variation curves of the neuron PID contraction force controllers is shown in Figure 10. The controller parameters K_Pi, K_Ii, and K_Di of the four PMs are adjusted in real time by the gradient descent method according to the performance index formula (28) to enhance the stability and anti-disturbance of the inner loop of the system control. The dynamic performance of the PM bionic leg is further improved through the combined action of the PM contraction force control inner loop and the bionic leg position control outer loop.

FIGURE 10

Proportional-Integral-Derivative (PID) parameter curves: (a) PID parameter curves of pneumatic muscle (PM) - PM_a and PM_b, (b) PID parameter curves of PM_c and PM_d.

In order to verify that the designed control strategy has better anti-interference ability, a larger disturbance term D is added in the simulation experiment.

Set the disturbance term as follows:

$$$\mathbf{D}=\left[\begin{array}{c}\hfill 5{\dot{\theta }}_{\mathrm{j}1}+\text{sgn}\left({\dot{\theta }}_{\mathrm{j}1}\right)+0.5\mathrm{sin}(3\mathrm{t})+0.3\mathrm{cos}(\mathrm{t})\hfill \\ \hfill 5{\dot{\theta }}_{\mathrm{j}2}+\text{sgn}\left({\dot{\theta }}_{\mathrm{j}2}\right)+0.5\mathrm{sin}(3\mathrm{t})+0.3\mathrm{cos}(\mathrm{t})\hfill \end{array}\right]$$$ (60)

The angle and angle error response curves of the hip and knee joints are shown in Figures 11 and 12 respectively. Figure 13 shows the torque response curves of hip and knee joints. It can be seen from the figure that in the case of large disturbance, after the system is stabilised, the hip joint angle tracking error is within the range of −1.011° to 1.039°, and the knee joint angle tracking error is within the range of −0.875° to 1.067°. In addition, Table 4 provides specific data on the steady-state performance of the system.

FIGURE 11

Angle tracking curves of hip joint. (a) Angle tracking curves. (b) Angle tracking error curve.

FIGURE 12

Angle tracking curves of knee joint: (a) Angle tracking curves. (b) Angle tracking error curve.

FIGURE 13

Torque tracking curves of the hip and knee joints: (a) Torque tracking curves. (b) Torque tracking error curves.

TABLE 4. Steady-state performance of the PM bionic legged robot under disturbance conditions

Joint	Hip	Knee
Emax (°)	1.039	1.067
Eave (°)	0.310	0.248
Esd (°)	0.376	0.310

• Abbreviation: PM, pneumatic muscles.

Figures 14 and 15 are the tracking curves of the contraction forces of the four PMs after adding the disturbance. Compared with the undisturbed contraction force response curve, in order to counteract the negative effects of disturbances on the system, the four PMs all generated larger contraction force, and the tracking accuracy remained within a certain range. The results show that the algorithm still has good anti-interference ability.

FIGURE 14

Contraction force curves of pneumatic muscle (PM) - PM_a and PM_b: (a) Contraction force tracking curves. (b) Contraction force tracking error curves.

FIGURE 15

Contraction force curves of PM_c and PM_d: (a) Contraction force tracking curves. (b) Contraction force tracking error curves.

It can be seen from the simulation results that in the presence of large disturbances, the overall control accuracy of the PM bionic legged robot is kept within a reasonable range, which can ensure a good control effect, indicating that the designed control method has strong robustness.

In order to verify the generality of the method designed in this paper, the simulation experiments are carried out with the period T set to 2.5 s and 7.5 s respectively. Figures 16 and 17 show the simulation results, from which it can be seen that the method has better performance under different periods. Table 5 provides an analysis of the tracking error for the two periods.

FIGURE 16

Angle tracking curves at T = 2.5 s: (a) Angle tracking curves. (b) Angle tracking error curves.

FIGURE 17

Angle tracking curves at T = 7.5 s: (a) Angle tracking curves. (b) Angle tracking error curves.

TABLE 5. Steady-state performance of the pneumatic muscle (PM) bionic legged robot under different periods

Joint	T = 2.5 s		T = 7.5 s
	Hip	Knee	Hip	Knee
Emax (°)	0.212	0.297	0.084	0.048
Eave (°)	0.121	0.110	0.051	0.010
Esd (°)	0.137	0.128	0.057	0.014

5 EXPERIMENTAL STUDY

The designed control method is applied on the physical platform to further verify the effectiveness of the method.

The physical platform of the PM bionic legged robot is shown in Figure 18. The platform uses four PMs as drivers, and provides air supply to the PMs through the air compressor. The Beckhoff controller is used to control the opening of the four pressure regulating proportional valves. Two angle encoders are used to measure the real-time angle of the hip and knee joints, and four tensile force sensors are used to measure the real-time contraction force of the PMs.

FIGURE 18

The physical platform of the pneumatic muscle (PM) bionic legged robot.

In practical situations, due to the strong non-linearity of the PM, its control bandwidth has research significance. In this paper, the performance of the pressure regulating proportional valve used is first tested. The experimental results of the frequency f being 4 Hz are shown in Figure 19. According to the definition of bandwidth, the bandwidth of the pressure regulating proportional valve is about 4 Hz.

FIGURE 19

Sinusoidal response curve at f = 4 Hz.

Then, the proportional valve was connected with the PM, and its performance was tested at the frequency f of 0.5, 1, 2 and 4 Hz. The experimental results are shown in Figures 20 and 21. At 4 Hz, the phase lag is close to half a cycle and the amplitude is attenuated by about 50%.

FIGURE 20

Sinusoidal response curves at f = 0.5 Hz and f = 1 Hz: (a) f = 0.5 Hz, (b) f = 1 Hz.

FIGURE 21

Sinusoidal response curves at f = 2 Hz and f = 4 Hz: (a) f = 2 Hz, (b) f = 4 Hz.

In order to study the hopping ability of the bionic leg, we carried out the corresponding step experiment. The device is installed on the vertical slide rail and can move up and down. In the initial stage, the initial air pressure of the four PMs is given respectively, and the bionic leg is made to stand still on the ground under the action of the auxiliary device. As shown in Figure 18, the sole of the foot is equipped with a pressure sensor. After the system is stabilised, increase the air pressure of PM_a and PM_c to 3.5 bar, and the joint drives the calf to kick to the ground. When the plantar pressure is equal to the static plantar pressure, PM_a and PM_c return to the initial air pressure, and the pressure of PM_b and PM_d increases to 3.5 bar, so that the calf lifts up and the bionic leg lifts off the ground. When the soles of the feet land on the ground again, the air pressures of the four PMs all return to their initial values, completing a hop. Figure 22 shows the actual air pressure change through the proportional valve outlets and the value change of the plantar pressure sensor during the process from standing to continuous hopping. It can be seen that the bottom of the foot is off the ground about 35% of the time in a jump cycle, showing its good hopping performance and power-to-weight ratio performance.

FIGURE 22

The experimental curves of hopping: (a) Air pressure curves. (b) Plantar pressure curve.

Fasten the bionic leg device on the vertical slide rail. Similar to the simulation, the desired position curve of the hip joint is set as a sine curve of −60° to −30°, and the desired position curve of the knee joint is set as a sine curve of 60°–90°, both with a period of 5 s. The experimental results of the angle response curves of the hip and knee joints are shown in Figures 23 and 24. The response curves of torques are shown in Figure 25. It can be seen from the figure that in the actual situation, the whole bionic legged robot is in a vertical downward state under the action of gravity in the initial stage. The actual angle of the position encoder has a large deviation from the expected angle given at the beginning, resulting in a large change in both joint angles in the initial stage. After the system is stabilised, the hip joint angle tracking error is within the range of −1.177° to 1.425°, and the knee joint angle tracking error is within the range of −1.508° to 3.806°. Table 6 provides an analysis of the tracking error.

FIGURE 23

Angle tracking curves of hip joint: (a) Angle tracking curves. (b) Angle tracking error curve.

FIGURE 24

Angle tracking curves of knee joint: (a) Angle tracking curves. (b) Angle tracking error curve.

FIGURE 25

Torque tracking curves of the hip and knee joints: (a) Torque tracking curves. (b) Torque tracking error curves.

TABLE 6. Steady-state performance of the pneumatic muscle (PM) bionic legged robot

Joint	Hip	Knee
Emax (°)	1.425	3.806
Eave (°)	0.409	1.109
Esd (°)	0.561	1.461

The response curves of the contraction force of the four PMs are shown in Figures 26 and 27. After the system is stabilised, F_a remains around 120 N, F_b varies within 140–205 N, F_c varies within 80–180 N, and F_d varies within 115–155 N. The control accuracy of the PM contraction force is generally maintained within a reasonable range.

FIGURE 26

Contraction force curves of pneumatic muscle (PM) - PM_a and PM_b: (a) Contraction force tracking curves. (b) Contraction force tracking error curves.

FIGURE 27

Contraction force curves of pneumatic muscle (PM) - PM_c and PM_d: (a) Contraction force tracking curves. (b) Contraction force tracking error curves.

Compared with the simulation situation, the tracking error of the physical experiment is larger. The main reasons can be found through the analysis as follows: the simulation environment is ideal, and many uncertainties in the actual experiment cannot be accurately simulated in the simulation, such as hysteresis, creep, and elastic deformation of PMs. In addition, due to the complexity of the cross-four-link knee joint mechanism, there is a multi-axis rotational friction term, which also leads to the fact that the rotation error of the knee joint is larger than the rotation error of the hip joint. Overall, it can be seen from the simulation verification and physical experiment verification that the designed control method is effective and reliable.

6 CONCLUSIONS

In this paper, aiming at the non-linear modelling of PM actuator and the motion control of PM bionic leg, a double closed-loop control method of PM bionic leg based on neural network is proposed. The theoretical calculation, simulation results and physical experimental results showed that the control method has good transient and steady-state performance, and it has good robustness to load disturbances. The main contributions are given as follows:

• (1)

  A double closed-loop control strategy of PM bionic legs is designed, which takes the contraction force control of PMs as the control inner loop and the angle control of the bionic joints as the control outer loop. Through the contraction force mapping model, the inner and outer loops of the control work together to improve the anti-disturbance performance of the system.

• (2)

  Based on the three-element model of PM, a feedforward neuron PID controller is designed. The PM contraction force model is used as the output of the feedforward control, and the PID controller makes up for the output of the feedforward controller. The neuron adjusts the PID parameters in real time according to the motion state of the PM, which effectively suppresses the negative control effect caused by the non-linearity of the PM.

• (3)

  A sliding mode robust controller with local model approximation is designed, which approximates the real-time dynamic parameters of the bionic leg through three RBF neural networks to improve the modelling accuracy. The stability of the closed-loop system is proved by Lyapunov function, and its effectiveness is verified by simulation and physical experiments.

ACKNOWLEDGEMENTS

This work was supported by the Zhejiang Province Key Research and Development Program Project under Grant No. 2021C01069.

CONFLICT OF INTEREST

The authors declare no conflicts of interest.

Open Research

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.