Analysis and control of a parallel lower limb based on pneumatic artificial muscles

Abstract

Most robots that are actuated by antagonistic pneumatic artificial muscles are controlled by various control algorithms that cannot adequately imitate the actual muscle distribution of human limbs. Other robots in which the distribution of pneumatic artificial muscle is similar to that of human limbs can only analyze the position of the robot using perceptual data instead of rational knowledge. In order to better imitate the movement of a human limb, the article proposes a humanoid lower limb in the form of a parallel mechanism where muscle is unevenly distributed. Next, the kinematic and dynamic movements of bionic hip joint are analyzed, where the joint movement is controlled by an observer-based fuzzy adaptive control algorithm as a whole rather than each individual pneumatic artificial muscle and parameters that are optimized by a neural network. Finally, experimental results are provided to confirm the effectiveness of the proposed method. We also document the role of muscle in trajectory tracking for the piriformis and musculi obturator internus in isobaric processes.

Introduction

Pneumatic artificial muscle (PAM) comprises an internal rubber hose, external fiber, and a metallic sheath fixed at both ends. A PAM allows for radial expansion while inflating and output longitudinal force. When it is deflating, a joint driven by antagonistic PAM can move freely with passive characteristics, especially when both PAM are inflated. Joints can rotate with the changes in air pressure. A PAM is a high power-to-weight ratio actuator with excellent compliance, similar to human muscle and has been widely used in robots. Almost all past research studies have been restricted to a one-dimensional robot actuated by antagonistically PAM. For example, Sayama et al. ¹ and Narioka et al. ² designed robot legs actuated by monoarticular and biarticular PAM, respectively. Hosoda et al. ³ proposed biarticular and monoarticular muscles that play different roles in jumping. Choi et al. ⁴ expanded on this research and presented a two-dimensional leg. Other researchers who studied on control algorithms have used a sliding mode control based on a nonlinear disturbance observer, ⁵ an adaptive impedance controller based on an evolutionary dynamic fuzzy neural network, ⁶ and fractional fuzzy adaptive sliding mode control. ⁷ Others have used adaptive self-organizing fuzzy sliding mode control, ⁸ gain scheduling neural multiple-input multiple-output (MIMO) dynamic neural proportional-integral-derivative (DNN PID) control, ⁹ hybrid feed-forward inverse nonlinear autoregressive with exogenous input (NARX) fuzzy model-PID control, ¹⁰ proxy-based sliding mode control, ¹¹ and twisting and super twisting sliding control. ¹²

Of course, other researchers have designed robots with PAM and the distribution of PAM is similar to that of human muscle, they only analyzed robot with perceptual instead of rational knowledge, not to mention accurate trajectory control.^13–16 Parallel mechanisms actuated by PAM have been designed of which individual PAM has been controlled using fuzzy control, ¹⁷ adaptive robust control, ¹⁸ or sliding mode control with adaptive fuzzy cerebellar model articulation controller (CMAC). ¹⁹ However, no attention has been paid to controlling the mechanism as a whole.

In this article, after analyzing the biological muscle structure of the human lower limb, we propose a humanoid lower limb actuated by PAM in the form of a parallel mechanism where the muscle is unevenly distributed instead of employing antagonistic muscles. Then, the system is simplified and a fuzzy adaptive control algorithm is adopted to control a hip joint as a whole rather than to control every PAM individually, and neural network is applied for optimizing fuzzy basis function parameters.

In this article, section “System analysis” addresses the humanoid lower limb kinematic and dynamic analysis, section “Observer-based fuzzy adaptive control” introduces observer-based fuzzy adaptive control, section “Parameter optimization and stability analysis” uses neural network to optimize the fuzzy parameters, and section “Experimental results” describes how the experiment was conducted and analyzes the properties of a muscle and hip joint.

System analysis

Based on the analysis of the human musculoskeletal system, we designed a humanoid lower limb with PAM (Figure 1) ²⁰ . First, the pelvis was fixed to the frame, and the femoris was connected to the pelvis and hip joint platform by a thread. Moreover, the iliopsoas, musculi obturator internus, and musculi piriformis are distributed between the pelvis and hip joint platform. It can be simplified as parallel manipulator (Figure 2(a)).

Figure 1.

Humanoid lower limb: (a) photo of the prototype of the humanoid lower limb and (b) diagram of the thigh with various muscles (1, pelvis; 2, hamstrings; 3, musculi piriformis; 4, femur; 5, biceps flexor cruris; 6, musculi obturator internus; 7, iliacus; 8, hip platform; 9, gracilis; 10, rectus femoris; 11, knee platform).

Figure 2.

Schematic diagram of hip joint: (a) the principle structure of the hip joint, (b) point coordinates in the pelvis, and (c) point coordinates in the hip platform.

In the hip joint, comparing to $A_x{A}_y{A}_z$ transformed matrix of $A_x^{\prime }{A}_y^{\prime }{A}_z^{\prime }$ is

$$R_h=\left(\begin{array}{ccc}c{\beta }_{1}c{\gamma }_1& -c{\beta }_{1}s{\gamma }_1& s{\beta }_1\\ s{\alpha }_{1}s{\beta }_{1}c{\gamma }_1+c{\alpha }_{1}s{\gamma }_1& -s{\alpha }_{1}s{\beta }_{1}s{\gamma }_1+c{\alpha }_{1}c{\gamma }_1& -s{\alpha }_{1}c{\beta }_1\\ -c{\alpha }_{1}s{\beta }_{1}c{\gamma }_1+s{\alpha }_{1}s{\gamma }_1& c{\alpha }_{1}s{\beta }_{1}s{\gamma }_1+s{\alpha }_{1}c{\gamma }_1& c{\alpha }_{1}c{\beta }_1\end{array}\right)$$

(1)

where $c=\cos$ and $s=\sin$ ; ${\alpha }_1$ , ${\beta }_1$ , and ${\gamma }_1$ are flexion-extension, adduction-abduction, and medial-lateral rotation angles in the hip joint.

The fixed point of muscles coordinated in the pelvic area are $A_1$ , $A_2$ , and $A_3$ , and their coordination in $A_x{A}_y{A}_z$ yields (Figure 2(b))

$$A_1^a=\left(\begin{array}{c}{r}_{A1}\cos (180-\eta )\hfill \\ \multicolumn{1}{c}{r_{A1}\sin (180-\eta )}\\ \multicolumn{1}{c}{0}\end{array}\right),A_2^a=\left(\begin{array}{c}{r}_{A2}\cos \eta \hfill \\ \multicolumn{1}{c}{r_{A2}\sin \eta }\\ \multicolumn{1}{c}{0}\end{array}\right),A_3^a=\left(\begin{array}{c}{r}_{A3}\cos (270+\xi )\hfill \\ \multicolumn{1}{c}{r_{A3}\sin (270+\xi )}\\ \multicolumn{1}{c}{z_{A3}}\end{array}\right)$$

where ξ and η are the azimuth angle of points; $r_{A1}$ , $r_{A2}$ , $r_{A3}$ are the points of $A_1$ , $A_2$ , $A_3$ circle radius in the cylindrical-coordinate system; and $r_{A1}=r_{A2}$ , while $z_{A3}$ is value in the vertical direction.

Coordinates of point $B_1$ and $B_2$ can be expressed as (Figure 2(c))

$$B_i^b={\left(\begin{array}{ccc}{r}_b\cos \phi \hfill & r_b\sin \phi \hfill & 0\hfill \end{array}\right)}^{T}i=1,\dots ,2\phi =i\times 180$$

(2)

where $r_b$ is circle radius of $B_1$ and $B_2$ and ϕ is azimuth angle.

The coordinates of $B_0$ in $A_x{A}_y{A}_z$ are

$$p_h=(\begin{array}{ccc}0\hfill & -y_{B0}^a\hfill & -z_{B0}^a\hfill \end{array}{)}^T$$

(3)

where $y_{B0}^a$ and $z_{B0}^a$ are the coordinates of point $B_0$ in the y and z directions.

$B_1$ and $B_2$ in $A_x{A}_y{A}_z$ can be written as

$$B_i^a=R_h{B}_i^b+p_h$$

(4)

Muscle length in hip joint can be expressed as

$$l_h=((x_{\mathrm{Ai}}-x_{\mathrm{bj}}{)}^2+(y_{\mathrm{Ai}}-y_{\mathrm{bj}}{)}^2+(z_{\mathrm{Ai}}-z_{\mathrm{bj}}{)}^2{)}^{1/2}i=1,2,3j=1,2$$

(5)

where $l_h$ is length of muscle in hip joint; $l_1$ , $l_2$ , and $l_3$ are length of piriformis, musculi obturator internus, and iliopsoas; and $x_{\mathrm{Ai}}$ , $y_{\mathrm{Ai}}$ , $z_{\mathrm{Ai}}$ , $x_{\mathrm{Bj}}$ , $y_{\mathrm{Bj}}$ , and $z_{\mathrm{Bj}}$ are point coordinates that fix a PAM in pelvis and hip joint platform.

In the hip joint, the input and output are muscle length changes and angle changes; by taking the derivation of time, following relationship can be obtained

$${\stackrel{·}{l}}_h=J_h{\stackrel{·}{\theta }}_h$$

(6)

where ${\stackrel{·}{l}}_h$ are the velocities of the piriformis, musculi obturator internus, and iliopsoas, and ${\stackrel{·}{\theta }}_h$ is pitch, yaw, and roll angular velocities. $J_h$ is used to depict the relationship between input and output and is a matrix, so we call it structure matrix.

Equation (6) can also be expressed as

$$\left(\begin{array}{c}{\stackrel{·}{l}}_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{l}}_2}\\ \multicolumn{1}{c}{{\stackrel{·}{l}}_3}\end{array}\right)=\left(\begin{array}{ccc}{J}_{11}\hfill & J_{12}\hfill & J_{13}\hfill \\ \multicolumn{1}{c}{J_{21}}& J_{22}\hfill & J_{23}\hfill \\ \multicolumn{1}{c}{J_{31}}& J_{32}\hfill & J_{33}\hfill \end{array}\right)\left(\begin{array}{c}{\stackrel{·}{\alpha }}_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\beta }}_1}\\ \multicolumn{1}{c}{{\stackrel{·}{\gamma }}_1}\end{array}\right)$$

(7)

$$\begin{array}{c}\multicolumn{1}{c}{J_{11}={\stackrel{·}{l}}_1/{\stackrel{·}{\alpha }}_1=(c{\alpha }_{1}s{\beta }_{1}c{\gamma }_1-s{\alpha }_{1}s{\gamma }_1)r_b}\\ \multicolumn{1}{c}{+(s{\alpha }_{1}s{\beta }_{1}c{\gamma }_1+c{\alpha }_{1}s{\gamma }_1)r_b}\\ \multicolumn{1}{c}{J_{12}={\stackrel{·}{l}}_1/{\stackrel{·}{\beta }}_1=-s{\beta }_{1}c{\gamma }_1{r}_b+s{\alpha }_{1}c{\beta }_{1}c{\gamma }_1{r}_b-c{\alpha }_{1}c{\beta }_{1}c{\gamma }_1{r}_b}\\ \multicolumn{1}{c}{J_{23}={\stackrel{·}{l}}_1/{\stackrel{·}{\gamma }}_1=-c{\beta }_{1}s{\gamma }_1{r}_b+(-s{\alpha }_{1}s{\beta }_{1}s{\gamma }_1+c{\alpha }_{1}c{\gamma }_1)r_b}\\ \multicolumn{1}{c}{+(c{\alpha }_{1}s{\beta }_{1}s{\gamma }_1+s{\alpha }_{1}c{\gamma }_1)r_b}\\ \multicolumn{1}{c}{J_{21}={\stackrel{·}{l}}_2/{\stackrel{·}{\alpha }}_1=(-c{\alpha }_{1}s{\beta }_{1}c{\gamma }_1+s{\alpha }_{1}s{\gamma }_1)r_b}\\ \multicolumn{1}{c}{-(s{\alpha }_{1}s{\beta }_{1}c{\gamma }_1+c{\alpha }_{1}c{\gamma }_1)r_b}\\ \multicolumn{1}{c}{J_{22}={\stackrel{·}{l}}_2/{\stackrel{·}{\beta }}_1=s{\beta }_{1}c{\gamma }_1{r}_b-s{\alpha }_{1}c{\beta }_{1}c{\gamma }_1{r}_b+c{\alpha }_{1}c{\beta }_{1}c{\gamma }_1{r}_b}\\ \multicolumn{1}{c}{J_{23}={\stackrel{·}{l}}_2/{\stackrel{·}{\gamma }}_1=c{\beta }_{1}s{\gamma }_1{r}_b+(s{\alpha }_{1}s{\beta }_{1}s{\gamma }_1-c{\alpha }_{1}c{\gamma }_1)r_b}\\ \multicolumn{1}{c}{-(c{\alpha }_{1}s{\beta }_{1}s{\gamma }_1+s{\alpha }_{1}c{\gamma }_1)r_b}\\ \multicolumn{1}{c}{J_{31}={\stackrel{·}{l}}_3/{\stackrel{·}{\alpha }}_1=(-c{\alpha }_{1}s{\beta }_{1}c{\gamma }_1+s{\alpha }_{1}s{\gamma }_1)r_b}\\ \multicolumn{1}{c}{-(s{\alpha }_{1}s{\beta }_{1}c{\gamma }_1+c{\alpha }_{1}s{\gamma }_1)r_b}\\ \multicolumn{1}{c}{J_{32}={\stackrel{·}{l}}_3/{\stackrel{·}{\beta }}_1=s{\beta }_{1}c{\gamma }_1{r}_b-s{\alpha }_{1}c{\beta }_{1}c{\gamma }_1{r}_b+c{\alpha }_{1}c{\beta }_{1}c{\gamma }_1{r}_b}\\ \multicolumn{1}{c}{J_{33}={\stackrel{·}{l}}_3/{\stackrel{·}{\gamma }}_1=c{\beta }_{1}s{\gamma }_1{r}_b+(s{\alpha }_{1}s{\beta }_{1}s{\gamma }_1-c{\alpha }_{1}c{\gamma }_1)r_b}\\ \multicolumn{1}{c}{-(c{\alpha }_{1}s{\beta }_{1}c{\gamma }_1+s{\alpha }_{1}c{\gamma }_1)r_b}\end{array}$$

where $J_{11}$ , $J_{12}$ , $J_{13}$ , $J_{21}$ , $J_{22}$ , $J_{23}$ , $J_{31}$ , $J_{32}$ , and $J_{33}$ are the subitems of the structure matrix; ${\stackrel{·}{l}}_1$ , ${\stackrel{·}{l}}_2$ , and ${\stackrel{·}{l}}_3$ are the contraction velocities of piriformis, musculi obturator internus, and iliopsoas; ${\stackrel{·}{\alpha }}_1$ , ${\stackrel{·}{\beta }}_1$ , and ${\stackrel{·}{\gamma }}_1$ are the angular velocities of pitch, yaw, and roll of the hip joint.

Based on the principle of virtual work, ²¹ the external and inertia forces of the femur and PAM received are zero, so the dynamics equations for hip joint are

$$\delta l_h^T{f}_b+\delta {\stackrel{.}{x}}_h^T{F}_B+{\displaystyle \sum _{i=1}^3\delta x_{hi}^T{F}_{hi}}=0$$

(8)

where $\delta l_h$ and $f_b$ are the muscle contraction velocities and force in the general coordinates; $\delta x_{\mathrm{hi}}$ and $F_{\mathrm{hi}}$ are contraction velocities and force of iliopsoas, musculi obturator internus, and musculi piriformis in the generalized independent coordinates; and $F_B$ are the external and inertia forces that act on muscle.

Equation (8) can be further expressed as

$${\tau }_b=J_b^{-T}{I}_B{\stackrel{·}{\theta }}_h+J_b^{-T}{\theta }_h\times \left(I_B{\theta }_h\right)-{\tau }_d$$

(9)

where ${\tau }_b$ is torque in the hip joint, $J_b$ is transfer matrix that PAM displacement are transferred to generalized coordinates, $I_B$ is lower limb inertia, and ${\tau }_d$ is external disturbance.

It can also be represented with the general dynamics model of the robot

$$M_h(q_1){\stackrel{··}{q}}_1+C_h(\begin{array}{cc}{\stackrel{·}{q}}_1\hfill & q_1\hfill \end{array})={\tau }_b+{\tau }_d$$

(10)

where ${\stackrel{··}{q}}_1={\stackrel{·}{\theta }}_h$ is angular accelerations, and the inertial matrix and Coriolis force are $M_h(q_1)=J_b^{-T}{I}_B$ , $C_h(\begin{array}{cc}{\stackrel{·}{q}}_1\hfill & q_1\hfill \end{array})=J_b^{-T}[{\theta }_h\times (I_B{\theta }_h)]$ , respectively.

Lower limb dynamic equation can be transferred into nonlinear MIMO system as

$$\{\begin{array}{c}\left[\begin{array}{c}{\stackrel{\cdot }{x}}_1\\ {\stackrel{\cdot }{x}}_2\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ 1\end{array}\right]\left(\left[\begin{array}{c}0\\ f_m\left(x\right)\end{array}\right]+\left[\begin{array}{c}0\\ g_{mn}\left(x\right)u\end{array}\right]\right)\\ y=\left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\end{array}$$

(11)

where $x_1={\theta }_h=(\begin{array}{ccc}{\alpha }_1\hfill & {\beta }_1\hfill & {\gamma }_1\hfill \end{array})$ and $x_2={\stackrel{·}{\theta }}_h=(\begin{array}{ccc}{\stackrel{·}{\alpha }}_1\hfill & {\stackrel{·}{\beta }}_1\hfill & {\stackrel{·}{\gamma }}_1\hfill \end{array})$ are the angle and angular velocities, $f_m(x)=-M^{-1}(q)C(\begin{array}{cc}\stackrel{·}{q}\hfill & q\hfill \end{array})=\left[\begin{array}{c}{f}_1(x)\hfill \\ \multicolumn{1}{c}{f_2(x)}\\ \multicolumn{1}{c}{f_3(x)}\end{array}\right]$ and $g_{\mathrm{mn}}(x)=-M^{-1}(q)=\left[\begin{array}{ccc}{g}_{11}(x)\hfill & g_{12}(x)\hfill & g_{13}(x)\hfill \\ \multicolumn{1}{c}{g_{21}(x)}& g_{22}(x)\hfill & g_{23}(x)\hfill \\ \multicolumn{1}{c}{g_{31}(x)}& g_{32}(x)\hfill & g_{33}(x)\hfill \end{array}\right]$ are nonlinear functions, while $M(q)$ , $C(\begin{array}{cc}\stackrel{·}{q}\hfill & q\hfill \end{array})$ , and $G(q)$ are Inertia, Centrifugal and Coriolis forces, and Gravity of robots dynamic model, respectively. Moreover, $u_i=[\begin{array}{ccc}{u}_1\hfill & u_2\hfill & u_3\hfill \end{array}{]}^T$ is control law while $d_i=M^{-1}(q){\tau }_d$ is deemed as external interference.

The energy equilibrium equation in PAM can be given by

$$\stackrel{·}{W}+\stackrel{·}{U}={\stackrel{·}{q}}_{\mathrm{in}}-{\stackrel{·}{q}}_{\mathrm{out}}+k{C}_v({\stackrel{·}{m}}_{\mathrm{in}}{T}_{\mathrm{in}}-{\stackrel{·}{m}}_{\mathrm{out}}{T}_{\mathrm{out}})$$

(12)

where W is muscle contractibility power in inflation and deflation, U is energy change inside PAM, while $q_{\mathrm{in}}$ , $q_{\mathrm{out}}$ , $m_{\mathrm{in}}$ , $m_{\mathrm{out}}$ , $T_{\mathrm{in}}$ , and $T_{\mathrm{out}}$ are the heat, air mass, and temperature during charge and discharge.

Pressure change inside PAM becomes

$$\mathrm{\Delta }{P}_i=k_1\frac{\mathrm{RT}}{V_i}({\stackrel{·}{m}}_{\mathrm{in}}-{\stackrel{·}{m}}_{\mathrm{out}})-k_2\frac{P_i}{V_i}\mathrm{\Delta }{V}_i$$

(13)

where $\mathrm{\Delta }{P}_i$ and $\mathrm{\Delta }{V}_i$ are pressure and volume change rate, $k_1$ and $k_2$ are coefficient, and in an adiabatic process and isothermal process, they are 1.4 and 1, respectively.

It can also be expressed as

$$\mathrm{\Delta }{P}_i=\frac{\mathrm{RT}}{V_i}({\stackrel{·}{m}}_{\mathrm{in}}-{\stackrel{·}{m}}_{\mathrm{out}})-\frac{P_i}{V_i}\mathrm{\Delta }{V}_i$$

(14)

Valve port effective area for one pulse-width modulation (PWM) cycle is

$$A_e=A_{\mathrm{emax}}·\frac{T_{\mathrm{on}}}{T_{\mathrm{off}}}=A_{\mathrm{emax}}·d$$

(15)

where $A_e$ is an effective area of nozzle, $A_{\mathrm{emax}}$ is the maximum effective area of valve port, and d is duty ratio.

Air pressure change inside PAM in non-pressure control mode as

$$\mathrm{\Delta }{P}_i=\{\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{RT}{A}_{\mathrm{emax}}f(P_d,P_u)}{V_i}d+\frac{\mathrm{RT}{A}_{\mathrm{emax}}f(P_d,P_u)}{V_i}{d}_{\min }-\frac{P_i}{V_i}\mathrm{\Delta }{V}_i{P}_d/P_u>0.528}\\ \multicolumn{1}{c}{\frac{\mathrm{RT}{A}_{\mathrm{emax}}f(P,P_u)}{V_i}d+\frac{\mathrm{RT}{A}_{\mathrm{emax}}f(P,P_u)}{V_i}{d}_{\min }-\frac{P_i}{V_i}\mathrm{\Delta }{V}_i{P}_d/P_u\le 0.528}\end{array}\begin{array}{c}{d}_{\min }\le d\le d_{\max }\hfill \end{array}$$

(16)

or

$$\mathrm{\Delta }{P}_i=\{\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{RT}{A}_{\mathrm{emax}}f(P_d,P_u)}{V_i}d+\frac{\mathrm{RT}{A}_{\mathrm{emax}}f(P_d,P_u)}{V_i}{d}_{\min }-\frac{P_i}{V_i}\mathrm{\Delta }{V}_i{P}_d/P_u>0.528}\\ \multicolumn{1}{c}{\frac{\mathrm{RT}{A}_{\mathrm{emax}}f(P,P_u)}{V_i}d+\frac{\mathrm{RT}{A}_{\mathrm{emax}}f(P,P_u)}{V_i}{d}_{\min }-\frac{P_i}{V_i}\mathrm{\Delta }{V}_i{P}_d/P_u\le 0.528}\end{array}-d_{\max }\le d\le -d_{\min }$$

(17)

where $d_{\min }$ and $d_{\max }$ are the minimum and maximum duty ratio of high speed on-off valve.

PAM output force model can be fitted with least-square and be calculated experimentally

$$f_i(l_i,P_i)=\pi r_0^2·P_i[a_i{(1-k_i\mathrm{\Delta }{l}_i)}^2+b_i]$$

(18)

where $r_0$ is initial radius of PAM; $P_i$ and $\mathrm{\Delta }{l}_i$ are air pressure and contraction length of PAM, while $a_i$ , $k_i$ , and $b_i$ are the undetermined coefficients and $a_i=3/{\tan }^2{\upsilon }_0$ , $b_i=1/{\sin }^2{\upsilon }_0$ , and ${\upsilon }_0$ is initial angle between central axis of PAM and its wire netting.

Observer-based fuzzy adaptive control

If $f_m(x)$ and $g_{\mathrm{mn}}(x)$ are known and $d_h=[\begin{array}{ccc}{d}_1\hfill & d_2\hfill & d_3\hfill \end{array}{]}^T=0^{22-30}$ , the control law of the hip joint can be written in the following form

$$\left[\begin{array}{c}{u}_1\hfill \\ \multicolumn{1}{c}{u_2}\\ \multicolumn{1}{c}{u_3}\end{array}\right]={\left[\begin{array}{ccc}{g}_{11}\left(x\right)\hfill & g_{12}\left(x\right)\hfill & g_{13}\left(x\right)\hfill \\ \multicolumn{1}{c}{g_{21}\left(x\right)}& g_{22}\left(x\right)\hfill & g_{23}\left(x\right)\hfill \\ \multicolumn{1}{c}{g_{31}\left(x\right)}& g_{32}\left(x\right)\hfill & g_{33}\left(x\right)\hfill \end{array}\right]}^{-1}(\left[\begin{array}{c}{f}_1\left(x\right)\hfill \\ \multicolumn{1}{c}{f_2\left(x\right)}\\ \multicolumn{1}{c}{f_3\left(x\right)}\end{array}\right]-\left[\begin{array}{c}\stackrel{··}{{\alpha }_1}\hfill \\ \multicolumn{1}{c}{\stackrel{··}{{\beta }_1}}\\ \multicolumn{1}{c}{\stackrel{··}{{\gamma }_1}}\end{array}\right])$$

(19)

Choosing the right $K_c=[\begin{array}{ccc}{k}_0\hfill & k_1\hfill & k_2\hfill \end{array}{]}^T$ , the polynomial $k_2\stackrel{··}{e}+k_1\stackrel{·}{e}+k_{0}e$ has all its roots in the left half plane, and the control law can be defined as

$$\left[\begin{array}{c}{u}_1\hfill \\ \multicolumn{1}{c}{u_2}\\ \multicolumn{1}{c}{u_3}\end{array}\right]={\left[\begin{array}{ccc}{g}_{11}\left(x\right)\hfill & g_{12}\left(x\right)\hfill & g_{13}\left(x\right)\hfill \\ \multicolumn{1}{c}{g_{21}\left(x\right)}& g_{22}\left(x\right)\hfill & g_{23}\left(x\right)\hfill \\ \multicolumn{1}{c}{g_{31}\left(x\right)}& g_{32}\left(x\right)\hfill & g_{33}\left(x\right)\hfill \end{array}\right]}^{-1}(-\left[\begin{array}{c}{f}_1\left(x\right)\hfill \\ \multicolumn{1}{c}{f_2\left(x\right)}\\ \multicolumn{1}{c}{f_3\left(x\right)}\end{array}\right]+\left[\begin{array}{c}\stackrel{··}{{\alpha }_1}\hfill \\ \multicolumn{1}{c}{\stackrel{··}{{\beta }_1}}\\ \multicolumn{1}{c}{\stackrel{··}{{\gamma }_1}}\end{array}\right]+\left[\begin{array}{c}{k}_0\hfill \\ \multicolumn{1}{c}{k_1}\\ \multicolumn{1}{c}{k_2}\end{array}\right]{\left[\begin{array}{ccc}e\hfill & \stackrel{·}{e}\hfill & \stackrel{··}{e}\hfill \end{array}\right]}^T)$$

(20)

Then, following error equation can be derived $k_2\stackrel{··}{e}+k_1\stackrel{·}{e}+k_{0}e=0$ ; obviously this cannot be realized.

Based on the fuzzy control with universal approximation property, let the nonlinear function $f_m(x)$ and $g_{\mathrm{mn}}(x)$ be approximated by fuzzy systems ${\hat{f}}_m(x)$ and ${\hat{g}}_{\mathrm{mn}}(x)$ .

${\stackrel{·}{x}}_1$ and ${\stackrel{·}{x}}_2$ are the state variables which are not measured directly; we can take angle and angular velocities as a whole $x=(\begin{array}{cc}{x}_1\hfill & x_2\hfill \end{array})$ , so that equation (11) can be further expressed as

$$\{\begin{array}{c}\stackrel{·}{x}=\mathrm{Ax}+B(f_m(x)+g_{\mathrm{mn}}(x)u+d)\hfill \\ \multicolumn{1}{c}{y=C^{T}x}\end{array}$$

(21)

where $A=\left[\begin{array}{cc}0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$ , $B=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]$ , and $C=\left[\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]$ .

$\hat{x}$ and $\hat{y}$ are introduced to estimate x and y, fuzzy adaptive observer as follows

$$\{\begin{array}{c}\stackrel{·}{\hat{x}}=A\hat{x}+B[{\hat{f}}_m(\hat{x})+{\hat{g}}_{\mathrm{mn}}(\hat{x})u+\hat{d}]+K_0[y-C^T\hat{x}]\hfill \\ \multicolumn{1}{c}{\hat{y}=C^T\hat{x}}\end{array}$$

(22)

Define observation error $\tilde{e}=x-\hat{x}$ and $\tilde{y}=y-\hat{y}$ and combine equation (21) with equation (22)

$$\{\begin{array}{l}\stackrel{\cdot }{\tilde{e}}=A\tilde{e}-K_0\tilde{y}+B{u}_{er}+B\left[(f_m\left(x\right)-{\hat{f}}_m(\hat{x}))+(g_{mn}\left(x\right)-{\hat{g}}_{mn}(\hat{x}))u\right]\\ \tilde{y}=C^T\tilde{e}\end{array}$$

(23)

then

$$\{\begin{array}{l}\stackrel{\cdot }{\tilde{e}}=(A-K_0{C}^T)+B{u}_{er}+B\left[(f_m(x)-{\hat{f}}_m(\hat{x}))+(g_{mn}(x)-{\hat{g}}_{mn}(\hat{x}))u\right]\\ \tilde{y}=C^T\tilde{e}\end{array}$$

(24)

The control law in equation (20) becomes

$$\left[\begin{array}{c}{u}_1\\ u_2\\ u_3\end{array}\right]={\left[\begin{array}{ccc}{\hat{g}}_{11}(\hat{x})& {\hat{g}}_{12}(\hat{x})& {\hat{g}}_{13}(\hat{x})\\ {\hat{g}}_{21}(\hat{x})& {\hat{g}}_{22}(\hat{x})& {\hat{g}}_{23}(\hat{x})\\ {\hat{g}}_{31}(\hat{x})& {\hat{g}}_{32}(\hat{x})& {\hat{g}}_{33}(\hat{x})\end{array}\right]}^{-1}\left[-\left[\begin{array}{c}{\hat{f}}_1(\hat{x})\\ {\hat{f}}_2(\hat{x})\\ {\hat{f}}_3(\hat{x})\end{array}\right]+\left[\begin{array}{c}\stackrel{\cdot \cdot }{\alpha }\\ \stackrel{\cdot \cdot }{\beta }\\ \stackrel{\cdot \cdot }{\gamma }\end{array}\right]+\left[\begin{array}{c}{k}_0\\ k_1\\ k_2\end{array}\right]{\left[\begin{array}{ccc}\tilde{e}& \stackrel{\cdot }{\tilde{e}}& \stackrel{\cdot \cdot }{\tilde{e}}\end{array}\right]}^T-\left[\begin{array}{c}{u}_r\\ u_r\\ u_r\end{array}\right]\right]$$

(25)

where adaptive control can be calculated by

$${\hat{f}}_m(\hat{x})={\left[\begin{array}{c}{\hat{f}}_1(\hat{x})\hfill \\ \multicolumn{1}{c}{{\hat{f}}_2(\hat{x})}\\ \multicolumn{1}{c}{{\hat{f}}_3(\hat{x})}\end{array}\right]}^{\mathrm{T}}={\mathrm{\Theta }}_f^T{\Phi }_f(\hat{x})=\left[\begin{array}{c}{\theta }_{f1}^T\hfill \\ \multicolumn{1}{c}{{\theta }_{f2}^T}\\ \multicolumn{1}{c}{{\theta }_{f3}^T}\end{array}\right]·\left[\begin{array}{c}{\xi }_1(\hat{x})\hfill \\ \multicolumn{1}{c}{{\xi }_2(\hat{x})}\\ \multicolumn{1}{c}{{\xi }_3(\hat{x})}\end{array}\right]=\left[\begin{array}{c}{\theta }_{f1}^T{\xi }_1(\hat{x})\hfill \\ \multicolumn{1}{c}{{\theta }_{f2}^T{\xi }_2(\hat{x})}\\ \multicolumn{1}{c}{{\theta }_{f3}^T{\xi }_3(\hat{x})}\end{array}\right]$$

(26)

$$\begin{array}{l}{\mathrm{\Theta }}_f={\displaystyle \int {\stackrel{\cdot }{\mathrm{\Theta }}}_f}={\displaystyle \int {\stackrel{\cdot }{\tilde{\mathrm{\Theta }}}}_f}={\displaystyle \int -{\gamma }_1{\tilde{e}}^{T}PB{\Phi }_f(\hat{x})}\\ \begin{array}{c}={\displaystyle \int \left[\begin{array}{cc}-{\gamma }_1\tilde{e}& -{\gamma }_1\stackrel{\cdot }{\tilde{e}}\end{array}\right]\left[\begin{array}{cc}{P}_{11}& P_{12}\\ P_{21}& P_{22}\end{array}\right]\left[\begin{array}{c}0\\ 1\end{array}\right]\left[\begin{array}{c}{\xi }_1(\hat{x})\\ {\xi }_2(\hat{x})\\ {\xi }_3(\hat{x})\end{array}\right]}=\end{array}\left[\begin{array}{c}{\theta }_{f1}^T\\ {\theta }_{f2}^T\\ {\theta }_{f3}^T\end{array}\right]\end{array}$$

(27)

$$\begin{array}{c}{\hat{g}}_{\mathrm{mn}}(\hat{x})=\left[\begin{array}{ccc}{\hat{g}}_1(\hat{x})\hfill & {\hat{g}}_2(\hat{x})\hfill & {\hat{g}}_3(\hat{x})\hfill \end{array}\right]={\mathrm{\Theta }}_g^T{\Phi }_g(\hat{x})=\left[\begin{array}{c}{\mathrm{\Theta }}_{g11}\hfill \\ \multicolumn{1}{c}{{\mathrm{\Theta }}_{g21}}\\ \multicolumn{1}{c}{{\mathrm{\Theta }}_{g31}}\end{array}\right]·\mathrm{diag}\left[\begin{array}{ccc}{\Phi }_{g11}(\hat{x})\hfill & {\Phi }_{g22}(\hat{x})\hfill & {\Phi }_{g33}(\hat{x})\hfill \end{array}\right]\hfill \\ \begin{array}{c}=\hfill \end{array}\left[\begin{array}{ccc}{\theta }_{g11}^T\hfill & {\theta }_{g12}^T\hfill & {\theta }_{g13}^T\hfill \\ \multicolumn{1}{c}{{\theta }_{g21}^T}& {\theta }_{g22}^T\hfill & {\theta }_{g23}^T\hfill \\ \multicolumn{1}{c}{{\theta }_{g31}^T}& {\theta }_{g32}^T\hfill & {\theta }_{g33}^T\hfill \end{array}\right]·\left[\begin{array}{c}{\xi }_1(\hat{x})\hfill \\ \multicolumn{1}{c}{{\xi }_2(\hat{x})}\\ \multicolumn{1}{c}{{\xi }_3(\hat{x})}\end{array}\right]=\left[\begin{array}{ccc}{\theta }_{g11}^T{\xi }_1(\hat{x})\hfill & {\theta }_{g12}^T{\xi }_1(\hat{x})\hfill & {\theta }_{g13}^T{\xi }_1(\hat{x})\hfill \\ \multicolumn{1}{c}{{\theta }_{g21}^T{\xi }_2(\hat{x})}& {\theta }_{g22}^T{\xi }_2(\hat{x})\hfill & {\theta }_{g23}^T{\xi }_2(\hat{x})\hfill \\ \multicolumn{1}{c}{{\theta }_{g31}^T{\xi }_3(\hat{x})}& {\theta }_{g32}^T{\xi }_3(\hat{x})\hfill & {\theta }_{g33}^T{\xi }_3(\hat{x})\hfill \end{array}\right]\hfill \end{array}$$

(28)

$$\begin{array}{c}{\mathrm{\Theta }}_{\mathrm{gi}}=\int {\stackrel{·}{\mathrm{\Theta }}}_{\mathrm{gi}}=\int {\stackrel{·}{\mathrm{\Theta }}}_{\mathrm{gi}}=\int -{\gamma }_2{\tilde{e}}^T\mathrm{PB}{\Phi }_g(\hat{x})u\hfill \\ \begin{array}{c}=\int \left[\begin{array}{cc}-{\gamma }_1\tilde{e}\hfill & -{\gamma }_1\stackrel{·}{\stackrel{~}{e}}\hfill \end{array}\right]\left[\begin{array}{cc}{P}_{11}\hfill & P_{12}\hfill \\ \multicolumn{1}{c}{P_{21}}& P_{22}\hfill \end{array}\right]\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]\left[\begin{array}{ccc}\xi (\hat{x})\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \xi (\hat{x})\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \xi (\hat{x})\hfill \end{array}\right]\left[\begin{array}{c}{u}_1\hfill \\ \multicolumn{1}{c}{u_2}\\ \multicolumn{1}{c}{u_3}\end{array}\right]=\hfill \end{array}\left[\begin{array}{c}{\theta }_{g1i}^T\hfill \\ \multicolumn{1}{c}{{\theta }_{g2i}^T}\\ \multicolumn{1}{c}{{\theta }_{g3i}^T}\end{array}\right]\hfill \end{array}$$

(29)

where ${\hat{f}}_1(\hat{x})$ , ${\hat{f}}_2(\hat{x})$ , ${\hat{f}}_3(\hat{x})$ and ${\hat{g}}_1(\hat{x})$ , ${\hat{g}}_2(\hat{x})$ , ${\hat{g}}_3(\hat{x})$ are subentries of ${\hat{f}}_m(\hat{x})$ and ${\hat{g}}_{\mathrm{mn}}(\hat{x})$ ; ${\theta }_{f1}$ , ${\theta }_{f2}$ , ${\theta }_{f3}$ and ${\theta }_{g11}$ , ${\theta }_{g21}$ , ${\theta }_{g31}$ , ${\theta }_{g12}$ , ${\theta }_{g22}$ , ${\theta }_{g32}$ , ${\theta }_{g13}$ , ${\theta }_{g23}$ , ${\theta }_{g33}$ are corresponding adjustable parameters of ${\hat{f}}_m(\hat{x})$ and ${\hat{g}}_{\mathrm{mn}}(\hat{x})$ . ${\mathrm{\Theta }}_f$ and ${\mathrm{\Theta }}_g$ are compact adjustable parameters of ${\hat{f}}_m(\hat{x})$ and ${\hat{g}}_{\mathrm{mn}}(\hat{x})$ , ${\Phi }_f(\hat{x})$ and ${\Phi }_g(\hat{x})$ are fuzzy basis functions of ${\hat{f}}_m(\hat{x})$ and ${\hat{g}}_{\mathrm{mn}}(\hat{x})$ , while

$$\xi (\hat{x})={\left[\begin{array}{ccc}{\xi }_1(\hat{x})\hfill & {\xi }_2(\hat{x})\hfill & {\xi }_3(\hat{x})\hfill \end{array}\right]}^T$$

$${\xi }_1(\hat{x})=\frac{{\mu }_{F_1^1}({\hat{x}}_1)·{\mu }_{F_2^1}({\hat{x}}_2)·{\mu }_{F_3^1}({\hat{x}}_3)}{{\mu }_{F_1^1}({\hat{x}}_1)·{\mu }_{F_2^1}({\hat{x}}_2)·{\mu }_{F_3^1}({\hat{x}}_3)+{\mu }_{F_1^2}({\hat{x}}_1)·{\mu }_{F_2^2}({\hat{x}}_2)·{\mu }_{F_3^2}({\hat{x}}_3)+{\mu }_{F_1^3}({\hat{x}}_1)·{\mu }_{F_2^3}({\hat{x}}_2)·{\mu }_{F_3^3}({\hat{x}}_3)}$$

$${\xi }_2(\hat{x})=\frac{{\mu }_{F_1^2}({\hat{x}}_1)·{\mu }_{F_2^2}({\hat{x}}_2)·{\mu }_{F_3^2}({\hat{x}}_3)}{{\mu }_{F_1^1}({\hat{x}}_1)·{\mu }_{F_2^1}({\hat{x}}_2)·{\mu }_{F_3^1}({\hat{x}}_3)+{\mu }_{F_1^2}({\hat{x}}_1)·{\mu }_{F_2^2}({\hat{x}}_2)·{\mu }_{F_3^2}({\hat{x}}_3)+{\mu }_{F_1^3}({\hat{x}}_1)·{\mu }_{F_2^3}({\hat{x}}_2)·{\mu }_{F_3^3}({\hat{x}}_3)}$$

$${\xi }_3(\hat{x})=\frac{{\mu }_{F_1^3}({\hat{x}}_1)·{\mu }_{F_2^3}({\hat{x}}_2)·{\mu }_{F_3^3}({\hat{x}}_3)}{{\mu }_{F_1^1}({\hat{x}}_1)·{\mu }_{F_2^1}({\hat{x}}_2)·{\mu }_{F_3^1}({\hat{x}}_3)+{\mu }_{F_1^2}({\hat{x}}_1)·{\mu }_{F_2^2}({\hat{x}}_2)·{\mu }_{F_3^2}({\hat{x}}_3)+{\mu }_{F_1^3}({\hat{x}}_1)·{\mu }_{F_2^3}({\hat{x}}_2)·{\mu }_{F_3^3}({\hat{x}}_3)}$$

Parameter optimization and stability analysis

The essential of Gauss basis neural network function uses error back-propagation, and the weights of each layer are adjusted to obtain the minimal value of mean square error between the actual and reference outputs (Figure 3). The input and output are air pressure in the PAM and hip joint angles.

Figure 3.

Gauss basis neural network parameter optimization structure.

In the input layer, the node input and the node output are

$${\mathrm{net}}_i^{\left(1\right)}(N)=x_i^{\left(1\right)}(N)i=1,2,3$$

(30)

$$y_i^{\left(1\right)}(N)=f_i^{\left(1\right)}({\mathrm{net}}_i^{\left(1\right)}(N))={\mathrm{net}}_i^{\left(1\right)}(N)$$

(31)

Inputs are delivered directly to the next layer, so input variables are equal to output variables.

In the membership layer, Gauss basis neural network function is adopted as membership function, for node of j, input and the outputs are

$${\mathrm{net}}_j^{\left(2\right)}(N)=-{\left(\frac{x_i^{\left(2\right)}-{\overline{x}}_j^l}{{\sigma }_j^l}\right)}^2\begin{array}{c}j=1,2,3\hfill \end{array}$$

(32)

$$y_j^{\left(2\right)}(N)=f_i^{\left(2\right)}({\mathrm{net}}_j^{\left(2\right)}(N))=\exp ({\mathrm{net}}_j^{\left(2\right)}(N))$$

(33)

where ${\overline{x}}_j^l$ and ${\sigma }_j^l$ are the mean and standard derivation of the jth node based on the Gauss basis neural network function.

In the rule layer, the main math computation involves multiplication; for node of k, input and the output are

$${\mathrm{net}}_k^{\left(3\right)}(N)=\underset{j=1}{\overset{3}{\Pi }}{\omega }_{\mathrm{jk}}^{\left(3\right)}{x}_j^{\left(3\right)}(N)k=1,2,3$$

(34)

$$y_k^{\left(3\right)}(N)=f_k^{\left(3\right)}({\mathrm{net}}_k^{\left(3\right)}(N))={\mathrm{net}}_k^{\left(3\right)}(N)$$

(35)

where ${\omega }_{\mathrm{jk}}^{(3)}=1$ .

In the output layer, node of k, input and the output are

$${\mathrm{net}}_o^{\left(4\right)}(N)=\sum _{k=1}^3{\omega }_{\mathrm{ko}}^{\left(4\right)}{x}_k^{\left(4\right)}(N)o=1,2,3$$

(36)

$$y_o^{\left(4\right)}(N)=f_o^{\left(4\right)}({\mathrm{net}}_o^{\left(4\right)}(N))={\mathrm{net}}_o^{\left(4\right)}(N)$$

(37)

The neural network reference output angle and actual mean square error can be defined as

$$E=\frac{1}{2}[{({\alpha }_h-{\alpha }_1)}^2+{({\beta }_h-{\beta }_1)}^2+{({\gamma }_h-{\gamma }_1)}^2]=\frac{1}{2}(e_{\alpha }^2+e_{\beta }^2+e_{\gamma }^2)$$

(38)

where $e_{\alpha }$ , $e_{\beta }$ , and $e_{\gamma }$ are pitch, yaw, and roll angular errors in the hip joint.

Back-propagation neural network is shown as follows.

In the output layer, the error is given by

$$\begin{array}{l}{\delta }_o^{\left(4\right)}=-\frac{\partial E}{\partial ne{t}_o^{\left(4\right)}}=-\left[\frac{\partial E}{\partial e_{\alpha }}\frac{\partial e_{\alpha }}{\partial {\alpha }_1}\frac{\partial {\alpha }_1}{\partial y_o^{\left(4\right)}}\frac{\partial y_o^{\left(4\right)}}{\partial ne{t}_o^{\left(4\right)}}+\frac{\partial E}{\partial e_{\beta }}\frac{\partial e_{\beta }}{\partial {\beta }_1}\frac{\partial {\beta }_1}{\partial y_o^{\left(4\right)}}\frac{\partial y_o^{\left(4\right)}}{\partial ne{t}_o^{\left(4\right)}}+\frac{\partial E}{\partial e_{\gamma }}\frac{\partial e_{\gamma }}{\partial {\gamma }_1}\frac{\partial {\gamma }_1}{\partial y_o^{\left(4\right)}}\frac{\partial y_o^{\left(4\right)}}{\partial ne{t}_o^{\left(4\right)}}\right]\\ \begin{array}{c}=\end{array}{e}_{\alpha }\frac{\partial {\alpha }_1}{\partial y_o^{\left(4\right)}}+e_{\beta }\frac{\partial {\beta }_1}{\partial y_o^{\left(4\right)}}+e_{\gamma }\frac{\partial {\gamma }_1}{\partial y_o^{\left(4\right)}}\end{array}$$

(39)

and connective weight is updated as follows

$$\begin{array}{c}\mathrm{\Delta }{\omega }_{\mathrm{ko}}^{\left(4\right)}=-{\eta }_{\omega }\frac{\partial E}{\partial {\omega }_{\mathrm{ko}}^{\left(4\right)}}\hfill \\ \begin{array}{c}=[-{\eta }_{\omega }\frac{\partial E}{\partial y_o^{\left(4\right)}}\frac{\partial y_o^{\left(4\right)}}{\partial {\mathrm{net}}_o^{\left(4\right)}}]\hfill \end{array}\left[\frac{\partial {\mathrm{net}}_o^{\left(4\right)}}{\partial {\omega }_{\mathrm{ko}}^{\left(4\right)}}\right]={\eta }_{\omega }{\delta }_o^{\left(4\right)}{x}_k^{\left(4\right)}\hfill \end{array}$$

(40)

$$-\frac{\partial E}{\partial x_k^{\left(4\right)}}=[-\frac{\partial E}{\partial y_o^{\left(4\right)}}]\left[\frac{\partial y_o^{\left(4\right)}}{\partial x_k^{\left(4\right)}}\right]=\{\begin{array}{c}\begin{array}{cc}{\omega }_{\mathrm{ko}}^{\left(4\right)}{\delta }_o^{\left(4\right)}\hfill & \partial x_k^{\left(4\right)}\ne 0\hfill \end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{cc}0\hfill & \partial x_k^{\left(4\right)}=0\hfill \end{array}}\end{array}$$

(41)

$${\omega }_{\mathrm{ko}}^{\left(4\right)}(N+1)={\omega }_{\mathrm{ko}}^{\left(4\right)}(N)+\mathrm{\Delta }{\omega }_{\mathrm{ko}}^{\left(4\right)}$$

(42)

where ${\eta }_{\omega }$ is learning-rate factor.

Error coefficients in rule layer are computed as

$${\delta }_k^{\left(3\right)}=-\frac{\partial E}{\partial {\mathrm{net}}_k^{\left(3\right)}}=[-\frac{\partial E}{\partial y_o^{\left(4\right)}}\frac{\partial y_o^{\left(4\right)}}{\partial {\mathrm{net}}_o^{\left(4\right)}}]\left[\frac{\partial {\mathrm{net}}_o^{\left(4\right)}}{\partial y_k^{\left(3\right)}}\frac{\partial y_k^{\left(3\right)}}{\partial {\mathrm{net}}_k^{\left(3\right)}}\right]={\delta }_o^{\left(4\right)}{\omega }_{\mathrm{ko}}^{\left(4\right)}$$

(43)

where ${\delta }_o^{(4)}=e_{\alpha }+\mathrm{\Delta }{e}_{\alpha }+e_{\beta }+\mathrm{\Delta }{e}_{\beta }+e_{\gamma }+\mathrm{\Delta }{e}_{\gamma }$ ; $\mathrm{\Delta }{e}_{\alpha }$ , $\mathrm{\Delta }{e}_{\beta }$ , and $\mathrm{\Delta }{e}_{\gamma }$ are pitch, yaw, and roll angular velocities error, respectively.

Error coefficients in membership layer can be calculated as

$$\begin{array}{l}{\delta }_j^{\left(2\right)}=-\frac{\partial E}{\partial ne{t}_j^{\left(2\right)}}\\ \begin{array}{c}=\left[-\frac{\partial E}{\partial y_o^{\left(4\right)}}\frac{\partial y_o^{\left(4\right)}}{\partial ne{t}_o^{\left(4\right)}}\frac{\partial ne{t}_o^{\left(4\right)}}{\partial y_k^{\left(3\right)}}\frac{\partial y_k^{\left(3\right)}}{\partial ne{t}_k^{\left(3\right)}}\right]\left[\frac{\partial ne{t}_k^{\left(3\right)}}{\partial y_j^{\left(2\right)}}\frac{\partial y_j^{\left(2\right)}}{\partial ne{t}_j^{\left(2\right)}}\right]\end{array}\\ \begin{array}{c}={\displaystyle \sum _{k=1}^3{\delta }_k^{\left(3\right)}{y}_k^{\left(3\right)}}\end{array}\end{array}$$

(44)

$$\begin{array}{c}\mathrm{\Delta }{\overline{x}}_j^l=-{\eta }_m\frac{\partial E}{\partial {\overline{x}}_j^l}\hfill \\ \begin{array}{c}=[-{\eta }_m\frac{\partial E}{\partial y_j^{\left(2\right)}}\frac{\partial y_j^{\left(2\right)}}{\partial {\mathrm{net}}_j^{\left(2\right)}}\frac{\partial {\mathrm{net}}_j^{\left(2\right)}}{\partial {\overline{x}}_j^l}]\hfill \end{array}\hfill \\ \begin{array}{c}={\eta }_m{\delta }_j^{\left(2\right)}\hfill \end{array}\frac{2(x_i^{\left(2\right)}-m_{\mathrm{ij}})}{{({\sigma }_j^l)}^3}\hfill \end{array}$$

(45)

$$\begin{array}{l}\mathrm{\Delta }{\sigma }_j^l=-{\eta }_{\sigma }\frac{\partial E}{\partial {\sigma }_{ij}}\\ \begin{array}{c}=\left[-{\eta }_{\sigma }\frac{\partial E}{\partial y_j^2}\frac{\partial y_j^{\left(2\right)}}{\partial ne{t}_j^2}\frac{\partial ne{t}_j^{\left(2\right)}}{\partial {\sigma }_j^l}\right]\end{array}\\ \begin{array}{c}={\eta }_{\sigma }{\delta }_j^{\left(2\right)}\end{array}\frac{2{(x_i^{\left(2\right)}-m_{ij})}^2}{{({\sigma }_j^l)}^3}\end{array}$$

(46)

In the input layer, relative parameters can be deduced by the following recursive formula

$${\overline{x}}_j^l(N+1)={\overline{x}}_j^l(N)+\mathrm{\Delta }{\overline{x}}_j^l$$

(47)

$${\sigma }_j^l(N+1)={\sigma }_j^l(N)+\mathrm{\Delta }{\sigma }_j^l$$

(48)

The dynamic observer error is

$$\begin{array}{l}\stackrel{\cdot }{\tilde{e}}=(A-K_0{C}^T)\tilde{e}+B{u}_r+B[({\hat{f}}_m(\hat{x}|{\mathrm{\Theta }}_f^{*})-{\hat{f}}_m(\hat{x}))+({\hat{g}}_{mn}(\hat{x}|{\mathrm{\Theta }}_g^{*})\\ -{\hat{g}}_{mn}(\hat{x}))u]+B\left[(f_m\left(x\right)-{\hat{f}}_m(\hat{x}|{\mathrm{\Theta }}_f^{*}))+(g_{mn}\left(x\right)-{\hat{g}}_{mn}(\hat{x}|{\mathrm{\Theta }}_g^{*}))u\right]\\ =(A-K_0{C}^T)\tilde{e}+B{u}_{er}+B{\tilde{\mathrm{\Theta }}}_f\xi (\hat{x})+B{\tilde{\mathrm{\Theta }}}_g\xi (\hat{x})u+Bw\end{array}$$

(49)

where

$$\{\begin{array}{c}{\tilde{\mathrm{\Theta }}}_f={\mathrm{\Theta }}_f^{*}-{\mathrm{\Theta }}_f\hfill \\ \multicolumn{1}{c}{{\tilde{\mathrm{\Theta }}}_g={\mathrm{\Theta }}_g^{*}-{\mathrm{\Theta }}_g}\end{array}$$

(50)

and ${\stackrel{·}{\stackrel{~}{\mathrm{\Theta }}}}_f={\stackrel{·}{\mathrm{\Theta }}}_f$ , ${\stackrel{·}{\stackrel{~}{\mathrm{\Theta }}}}_g={\stackrel{·}{\mathrm{\Theta }}}_g$ .

Consider Lyapunov function candidate as

$$V=\frac{1}{2}{\tilde{e}}^{T}P\tilde{e}+\frac{1}{2{\gamma }_1}{\tilde{\mathrm{\Theta }}}_f^T{\tilde{\mathrm{\Theta }}}_f+\frac{1}{2{\gamma }_2}{\tilde{\mathrm{\Theta }}}_g^T{\tilde{\mathrm{\Theta }}}_g$$

(51)

where ${\gamma }_1$ and ${\gamma }_2$ are constant positives.

The time derivative of equation (51) is

$$\stackrel{·}{V}=\frac{1}{2}{\stackrel{·}{\stackrel{~}{e}}}^{T}P\tilde{e}+\frac{1}{2}{\tilde{e}}^{T}P\stackrel{·}{\stackrel{~}{e}}+\frac{1}{{\gamma }_1}{\stackrel{·}{\stackrel{~}{\mathrm{\Theta }}}}_f^T{\tilde{\mathrm{\Theta }}}_f+\frac{1}{{\gamma }_2}{\stackrel{·}{\stackrel{~}{\mathrm{\Theta }}}}_g^T{\tilde{\mathrm{\Theta }}}_g$$

(52)

We can rewrite equation (52) as follows

$$\begin{array}{l}\stackrel{\cdot }{V}=\frac{1}{2}[{\tilde{e}}^T{(A-K_0{C}^T)}^{T}P\tilde{e}+u_{er}{B}^{T}P\tilde{e}+{\xi }^T(\hat{x}){\tilde{\mathrm{\Theta }}}_f^T{B}^{T}P\tilde{e}+{\xi }^T(\hat{x}){\tilde{\mathrm{\Theta }}}_g^T{B}^{T}P\tilde{e}u\\ +w^T{B}^{T}P\tilde{e}+{\tilde{e}}^{T}P(A-K_0{C}^T)\tilde{e}+{\tilde{e}}^{T}PB{u}_{er}+{\tilde{e}}^{T}PB{\tilde{\mathrm{\Theta }}}_f\xi (\hat{x})+{\tilde{e}}^{T}PB{\tilde{\mathrm{\Theta }}}_g\xi (\hat{x})u+{\tilde{e}}^{T}PBw]\\ +\frac{1}{{\gamma }_1}\stackrel{\cdot }{{\tilde{\mathrm{\Theta }}}_f^T}{\tilde{\mathrm{\Theta }}}_f+\frac{1}{{\gamma }_2}\stackrel{\cdot }{{\tilde{\mathrm{\Theta }}}_g^T}{\tilde{\mathrm{\Theta }}}_g\end{array}$$

(53)

$u_{\mathrm{er}}$ is a robust compensator, which can be defined as

$$u_{\mathrm{er}}=-\frac{1}{r}{B}^{T}P\tilde{e}$$

(54)

$$\begin{array}{c}{u}_{\mathrm{ri}}=-\frac{1}{r_i}{B}^{T}P\tilde{e}=-\frac{1}{r_i}·{\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{1}\end{array}\right]}^T\left[\begin{array}{cc}{P}_{11}\hfill & P_{12}\hfill \\ \multicolumn{1}{c}{P_{21}}& P_{22}\hfill \end{array}\right]\left[\begin{array}{c}\tilde{e}\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\stackrel{~}{e}}}\end{array}\right]\hfill \\ \begin{array}{c}=\hfill \end{array}-\frac{1}{r_i}(P_{21}\tilde{e}+P_{22}\stackrel{·}{\stackrel{~}{e}})\hfill \end{array}$$

(55)

$$\begin{array}{c}{u}_r=\left[\begin{array}{c}{u}_{r1}\hfill \\ \multicolumn{1}{c}{u_{r2}}\\ \multicolumn{1}{c}{u_{r3}}\end{array}\right]=-\frac{1}{r_i}(P_{21}\left[\begin{array}{c}{\tilde{e}}_{\alpha }\hfill \\ \multicolumn{1}{c}{{\tilde{e}}_{\beta }}\\ \multicolumn{1}{c}{{\tilde{e}}_{\gamma }}\end{array}\right]+P_{22}\left[\begin{array}{c}{\stackrel{·}{\stackrel{~}{e}}}_{\alpha }\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\stackrel{~}{e}}}_{\beta }}\\ \multicolumn{1}{c}{{\stackrel{·}{\stackrel{~}{e}}}_{\gamma }}\end{array}\right])\hfill \\ =\left[\begin{array}{c}-\frac{1}{r_i}(P_{21}{\tilde{e}}_{\alpha }+P_{22}{\stackrel{·}{\stackrel{~}{e}}}_{\alpha })\hfill \\ \multicolumn{1}{c}{-\frac{1}{r_i}(P_{21}{\tilde{e}}_{\beta }+P_{22}{\stackrel{·}{\stackrel{~}{e}}}_{\beta })}\\ \multicolumn{1}{c}{-\frac{1}{r_i}(P_{21}{\tilde{e}}_{\gamma }+P_{22}{\stackrel{·}{\stackrel{~}{e}}}_{\gamma })}\end{array}\right]\hfill \end{array}$$

(56)

Equation (53) can be written as

$$\begin{array}{c}\stackrel{·}{V}=\frac{1}{2}{\tilde{e}}^T[P(A-K_0{C}^T)+{(A-K_0{C}^T)}^{T}P-\frac{2}{r_i}\mathrm{PB}{B}^{T}P]\tilde{e}\hfill \\ \multicolumn{1}{c}{+\frac{1}{2}[w^T{B}^{T}P\tilde{e}+{\tilde{e}}^T\mathrm{PBw}]+\frac{1}{{\gamma }_1}[{\gamma }_1{\tilde{e}}^T\mathrm{PB}{\xi }^T(\hat{x})+{\stackrel{·}{\stackrel{~}{\mathrm{\Theta }}}}_f^T]{\tilde{\mathrm{\Theta }}}_f}\\ \multicolumn{1}{c}{+\frac{1}{{\gamma }_2}[{\gamma }_2{\tilde{e}}^T\mathrm{PB}{\xi }^T(\hat{x})u+{\stackrel{·}{\stackrel{~}{\mathrm{\Theta }}}}_g^T]{\tilde{\mathrm{\Theta }}}_g}\end{array}$$

(57)

The adaptive laws are given by

$$\{\begin{array}{c}{\stackrel{·}{\stackrel{~}{\mathrm{\Theta }}}}_f=-{\gamma }_1{\tilde{e}}^T\mathrm{PB}\xi (\hat{x})\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\stackrel{~}{\mathrm{\Theta }}}}_g=-{\gamma }_2{\tilde{e}}^T\mathrm{PB}\xi (\hat{x})u}\end{array}$$

(58)

Equation (57) can be simplified as

$$\begin{array}{c}\stackrel{·}{V}\le \frac{1}{2}{\tilde{e}}^T{Q}_1\tilde{e}-\frac{1}{2{\rho }^2}{\tilde{e}}^T\mathrm{PB}{B}^{T}P\tilde{e}+\frac{1}{2}[w^T{B}^{T}P\tilde{e}+{\tilde{e}}^T\mathrm{PBw}]\hfill \\ \begin{array}{c}=\hfill \end{array}\frac{1}{2}{\tilde{e}}^T{Q}_1\tilde{e}+\frac{1}{2}{\rho }^2{w}_1^2-\frac{1}{2}{[\frac{1}{\rho }{B}_s^T{P}_2\tilde{e}-\rho w_1]}^T\hfill \\ \multicolumn{1}{c}{[\frac{1}{\rho }{B}_s^T{P}_2\tilde{e}-\rho w_1]\le \frac{1}{2}{\tilde{e}}^T{Q}_1\tilde{e}+\frac{1}{2}{\rho }^2{w}^2}\end{array}$$

(59)

Let $Q=\mathrm{diag}[Q_1]$ , and $E^T={\left[\begin{array}{cc}\tilde{e}\hfill & \stackrel{·}{\stackrel{~}{e}}\hfill \end{array}\right]}^T$ ; then, equation (59) is equivalent to

$$\stackrel{·}{V}\le \frac{1}{2}{E}^T\mathrm{QE}+\frac{1}{2}{\rho }^2{w}^2$$

(60)

Integrating equation (60) from 0 to T, we have

$$V(T)-V(0)\le -\frac{1}{2}\underset{0}{\overset{T}{\int }}{E}^T\mathrm{QE}+\frac{1}{2}{\rho }^2\underset{0}{\overset{T}{\int }}{w}^2\mathrm{dt}$$

(61)

Since $V(T)\ge 0$ , equation (61) can be written as

$$\frac{1}{2}\underset{0}{\overset{T}{\int }}{E}^T\mathrm{QE}\le \frac{1}{2}{E}^T(0)\mathrm{PE}(0)+\frac{1}{2{\gamma }_1}{\tilde{\mathrm{\Theta }}}_f^T(0){\tilde{\mathrm{\Theta }}}_f^T(0)+\frac{1}{2{\gamma }_2}{\tilde{\mathrm{\Theta }}}_g^T(0){\tilde{\mathrm{\Theta }}}_g^T(0)+\frac{1}{2}{\rho }^2\underset{0}{\overset{T}{\int }}{w}^2\mathrm{dt}$$

(62)

Using Barbalat’s lemma, we can conclude that the tracking errors converge asymptotically to zero.

Experimental results

Figures 4–9 show the trajectories tracking and tracking errors of pitch (1.18°), yaw (4.54°), and roll angles (2.03°). The tracking errors are relatively large in the early stages because of a dead zone in the joint and PAM. Another larger tracking error occurs around 5 s simultaneously; the tracking errors of pitch, yaw, and roll angles were 2.3°, 1.2°, and 2.21° and are caused by resonance. Other reasons for tracking errors may include the nonlinear properties of PAM, a dead zone of the high speed on-off valve, and hysteresis influenced by the trachea. Overall, the tracking errors fluctuate around zero which further proved the stability of the control algorithm in this experiment.

Figure 4.

The tracking curve of pitch angle over time.

Figure 5.

The tracking error of pitch angle over time.

Figure 6.

The tracking curve of yaw angle over time.

Figure 7.

The tracking error of yaw angle over time.

Figure 8.

The tracking curve of roll angle over time.

Figure 9.

The tracking error of roll angle over time.

Pressure change (Figure 10) remained constant in the piriformis and musculi obturator internus. They are in an isobaric process which plays a role of consolidation. Meanwhile, pressure of iliopsoas changes all the time; in the first half the main trend is inflation, while in the latter half it is just the opposite. It is generally believed that the iliopsoas is mainly responsible for the movement of the hip joint.

Figure 10.

Air pressure change over time.

Figure 11 shows a change in the control signal input, because the non-pressure control mode has many advantages, and the valve works in this mode. We test the control parameters of the valve and learned that when the control signal input change is smaller than 20%, they work in the full closed positions, and when the control signal input change is greater than 72.9%, they work in the full open positions, respectively.

Figure 11.

Control input of valve over time.

Figure 12 shows change in muscle length. When combined with Figure 10 during isobaric expansion, although the air pressure remained constant in the piriformis and musculi obturator internus, their lengths change during the whole process. The tension and contraction in the iliopsoas work together to realize trajectories tracking. It also further proved that the movement of the limb is realized by different muscles that are compatible with each other, which is natural evolution.

Figure 12.

Muscle length change over time.

Conclusion

In conclusion, we design a spatial humanoid lower limb driven by PAM, which can be simplified as a parallel mechanism. Then, it was simplified into a nonlinear MIMO system through deducing the dynamics of the hip joint and using angular velocities and angular accelerations as input. However, two variables were not measured directly. Therefore, a fuzzy adaptive observer was designed to estimate and control variables, while through the combination of Gauss basis neural network function, a novel Gauss basis fuzzy adaptive control algorithm was constructed. In addition, proposed control algorithm was verified by the experiment.