Robust Control of Space Robots Considering Friction Characteristics

Abstract

In this chapter, a planar two-bar space robot is taken as the research object, and the dynamics equations of fully rigid and joint flexible space robot systems are derived by using momentum conservation and Lagrange equation of the second kind. Aiming at flexible joint space robot system, the system is decomposed into fast and slow subsystems based on singular perturbation. Considering the friction characteristics of joints, a robust controller based on the friction upper bound was designed to offset the influence of friction torque and improve the flat roof phenomenon in joint trajectory tracking control. For flexible joint space robot system, a fast variable subsystem controller is designed using moment differential to actively suppress flexible vibration. The controller of slow subsystem is designed by robust method and a bounded friction compensation term is introduced to offset the influence of joint friction torque. The effectiveness of the proposed manipulator control scheme is verified by numerical simulation experiments.

1. Introduction

Space robot systems play an irreplaceable role in the construction of space laboratories, in-orbit maintenance of spacecraft, and recycling of space garbage. Space robots have the ability to perform tasks in the space environment that poses a threat to human life, reducing the risk of debris and strong radiation to astronauts. Therefore, the research on space robots has been widely valued by scholars at home and abroad [1, 2, 3, 4, 5, 6, 7, 8, 9].

In low Earth orbit environment, there are various disturbances acting on the space robot system, such as thin atmosphere, joint friction torque, liquid propellant slosh of spacecraft and so on. Therefore, when designing the controller of space robot, it is necessary to consider the influence of system uncertainty and external disturbance, as well as the influence of joint friction torque on the response.

2. Kinematics and dynamics modeling of floating-based space robots

The space robot system is in free-floating state, and its base does not form rigid connection with the inertial system object, so there is a strong coupling effect between the base of the space robot system and each rod of the robot arm.

In order to establish the relationship between trajectory point kinematics and control force, the robot dynamics model should be built. The commonly used multi-body dynamic modeling methods for robot dynamics include Newton-Euler method [1], Kane method [2], Lagrange method of the second kind [3], virtual robotic arm method [4] and Roberson-Weittenberg method [5]. The above modeling method can be extended to rigid space robot system modeling after proper transformation.

In this chapter, the second Lagrange equation and momentum conservation relationship are used to establish a dynamic model of space robot with uncontrolled carrier position and controlled carrier attitude, and the dynamic equation of flexible joint floating space robot system is established based on the simplified linear torsion spring joint model of flexible joint. The work in this chapter lays the foundation for the space robot dynamic control scheme in the following paper.

2.1 Dynamics model of space robot system with attitude controlled but carrier position not controlled

As shown in Figure 1, the model is a planar motion fully rigid two-bar space robot consisting of a free-floating carrier $B0$ and two rigid arms $B1$, $B2$.

Figure 1.

Free-floating space manipulator system.

The symbols involved in formula derivation in Figure 1 and this section are defined as follows:

$\mathit{OXY}$: inertial coordinate system;

$O_{\mathit{0}}{X}_{\mathit{0}}{Y}_{\mathit{0}}$: space robot carrier coordinate system;

$O_i{X}_i{Y}_i$: coordinate system of each manipulator $\left(i=1,2\right)$;

$C$: total center of mass of space robot system;

$\mathit{Oci}$: center of mass of each part $\mathit{Bi}\left(i=0,1,2\right)$;

$O\mathit{0}$: the origin of the carrier coordinate system coinciding with the center of mass$O\mathrm{c}0$;

$O\mathit{1}$, $O\mathit{2}$_: respectively are the rotary joint axis of the arm rod $B1$, $B2$;

$x_{\mathit{0}}$: line direction from the origin $O\mathit{0}$ to $O\mathit{1}$ of the carrier coordinate system;

$x_i$: mechanical arm $\mathit{Bi}$ along the bar direction;

${\mathbf{e}}_i$: a unit vector along the axis $x_i\left(i=0,1,2\right)$;

${\mathbf{z}}_i$: unit vector perpendicular to the axis of each joint of the plane $\mathit{OXY}$;

${\mathbf{r}}_i$: position vector of centroid of each part of inertial system $\mathit{OXY}$, where $\left(i=0,1,2\right)$;

${\mathbf{r}}_c$: vector pointing from the origin $O$ of inertial system to the total center of mass $C$ of the system;

$l_{\mathit{0}}$: the distance along the axis $O\mathit{0}$ to $O\mathit{1}$;

$l_i$: length of each arm $\mathit{Bi}$ along the axis $x_i\left(i=1,2\right)$;

$a_{\mathit{1}}$: the distance between the center of mass $\mathit{Oc}1$ of rod 1 and the rotating hinge joint$O\mathit{1}$;

$a_{\mathit{2}}$: the distance between the center of mass $\mathit{Oc}2$ of rod 2 and the rotating hinge joint$O\mathit{2}$;

$m_i$: the mass of each part $\mathit{Bi}$$\left(i=0,1,2,3\right)$, where $m_3$ represents the end-load mass;

$M$: total mass of space robot system, where $M={\sum }_{i=\mathit{0}}^{\mathit{3}}{m}_i$;

$\mathit{Ii}$: the central inertia tensor of each part $\left(i=0,1,2,3\right)$, where $I_3$ represents the end-load mass;

$\mathit{Iim}$: the moment of inertia of the motor at the ith joint;

$q_{\mathit{0}}$: attitude Angle of the carrier;

$q_i$: relative joint Angle of the $\mathrm{i}$th arm $\left(i=1,2\right)$;

$\mathit{\omega i}$: the rotation angular speed of the carrier and each arm bar $\left(i=0,1,2\right)$;

By analyzing the geometric position relation of the system in Figure 2, the position vector relation of the center of mass $\mathit{Oci}$ of each part of the floating based rigid space robot in the inertial system $\mathit{OXY}$ can be expressed as follows:

Figure 2.

Friction characteristics of joints.

$${\mathbf{r}}_1={\mathbf{r}}_0+l_{\mathit{0}}{\mathbf{e}}_0+a_{\mathit{1}}{\mathbf{e}}_1$$E1

$${\mathbf{r}}_2={\mathbf{r}}_0+l_{\mathit{0}}{\mathbf{e}}_0+l_1{\mathbf{e}}_1+a_2{\mathbf{e}}_2$$E2

where, the position vector ${\mathbf{r}}_0=\left[x_0,y_0\right]$ of the carrier’s centroid, the unit vector ${\mathbf{e}}_i$along the axis $x_i$, the attitude Angle of the carrier $q_{\mathit{0}}$ and the Angle of each arm joint $q_i\left(i=1,2\right)$ are expressed as follows:

$${\mathbf{e}}_0={\left[sin{q}_{\mathit{0}}cos{q}_{\mathit{0}}\right]}^{\mathrm{T}}$$E3

$${\mathbf{e}}_1={\left[sin\left(q_{\mathit{0}}+q_{\mathit{1}}\right)cos\left(q_{\mathit{0}}+q_{\mathit{1}}\right)\right]}^{\mathrm{T}}$$E4

$${\mathbf{e}}_2={\left[sin\left(q_{\mathit{0}}+q_{\mathit{1}}+q_{\mathit{2}}\right)cos\left(q_{\mathit{0}}+q_{\mathit{1}}+q_{\mathit{2}}\right)\right]}^{\mathrm{T}}$$E5

The velocity vector relationship between the centroid $\mathit{Oci}\left(i=0,1,2\right)$ of each component and the terminal P point can be obtained:

$${\dot{\mathbf{r}}}_1={\dot{\mathbf{r}}}_0+l_{\mathit{0}}{\dot{\mathbf{e}}}_0+a_{\mathit{1}}{\dot{\mathbf{e}}}_1$$E6

$${\dot{\mathbf{r}}}_2={\dot{\mathbf{r}}}_0+l_{\mathit{0}}{\dot{\mathbf{e}}}_0+l_1{\dot{\mathbf{e}}}_1+a_2{\dot{\mathbf{e}}}_2$$E7

$${\dot{\mathbf{r}}}_p={\dot{\mathbf{r}}}_0+l_{\mathit{0}}{\dot{\mathbf{e}}}_0+l_1{\dot{\mathbf{e}}}_1+l_2{\dot{\mathbf{e}}}_2$$E8

${\mathbf{r}}_p$: the position vector at the end P of the manipulator in the inertial frame.

Where ${\dot{\mathbf{e}}}_0={\dot{q}}_{\mathit{0}}{\left[cos{q}_{\mathit{0}}-sin{q}_{\mathit{0}}\right]}^{\mathrm{T}}$, ${\dot{\mathbf{e}}}_1=\left({\dot{q}}_{\mathit{0}}+{\dot{q}}_{\mathit{1}}\right){\left[cos\left(q_{\mathit{0}}+q_{\mathit{1}}\right)-sin\left(q_{\mathit{0}}+q_{\mathit{1}}\right)\right]}^{\mathrm{T}}$, ${\dot{\mathbf{e}}}_2=\left({\dot{q}}_{\mathit{0}}+{\dot{q}}_{\mathit{1}}+{\dot{q}}_{\mathit{2}}\right){\left[cos\left(q_{\mathit{0}}+q_{\mathit{1}}+q_{\mathit{2}}\right)-sin\left(q_{\mathit{0}}+q_{\mathit{1}}+q_{\mathit{2}}\right)\right]}^{\mathrm{T}}$.

The relationship between the velocity of P point at the end of the deduced arm and the generalized velocity ${\dot{\mathbf{q}}}_{p0}$ in the inertial system is as follows:

$${\dot{\mathbf{r}}}_p=\left[\begin{array}{c}{\dot{x}}_p\\ {\dot{y}}_p\end{array}\right]=\left[\begin{array}{c}{J}_{011}{J}_{012}\begin{array}{ccc}{J}_{013}& J_{014}& J_{015}\end{array}\\ \begin{array}{ccc}{J}_{021}& J_{022}& \begin{array}{ccc}{J}_{023}& J_{024}& J_{025}\end{array}\end{array}\end{array}\right]\left[\begin{array}{c}{\dot{x}}_0\\ \begin{array}{c}{\dot{y}}_0\\ {\dot{q}}_0\\ {\dot{q}}_1\end{array}\\ {\dot{q}}_2\end{array}\right]={\mathbf{J}}_0{\dot{\mathbf{q}}}_{p0}$$E9

The generalized velocity ${\dot{\mathbf{q}}}_{p0}={\left[{\dot{x}}_0,{\dot{y}}_0,{\dot{q}}_0,{\dot{q}}_1,{\dot{q}}_2\right]}^{\mathrm{T}}$, ${\mathbf{J}}_0$ is the Jacobian velocity matrix of the floating space robot system with fully controllable position and attitude.

Considering that the space robot studied in this chapter is based on the assumption that the attitude of the carrier is controlled and its position is not controlled; that is, the attitude of the carrier is controlled by the reaction wheel driven by the motor, and the system itself is not controlled by external forces. The free-floating space robot system follows the momentum conservation relationship without the weak gravity effect. From the definition of the total centroid of the system, we can see the following relation:

$$M{\mathbf{r}}_c=\sum _{i=0}^2{m}_i{\mathbf{r}}_i$$E10

${\mathbf{r}}_c=0$, the centroid position of the initial state space robot system is selected as the origin $O$ of the inertial system. Therefore, by substituting Eq. (31) into Eq. (35), the vector expression of the carrier centroid position can be obtained:

$${\mathbf{r}}_0=L_{00}{\mathbf{e}}_0+L_{01}{\mathbf{e}}_1+L_{02}{\mathbf{e}}_2$$E11

Where $L_{00}=-\left(M-m_0\right)l_0/M$, $L_{01}=-\left(m_1{a}_1+m_2{l}_1+m_3{l}_1\right)/M$, $L_{02}=-\left(m_2{a}_2+m_3{l}_2\right)/M$.

Suppose that the initial momentum of the system is zero, that is ${\dot{\mathbf{r}}}_c=0$, from the conservation of momentum:

$$M{\dot{\mathbf{r}}}_c=\sum _{i=0}^3{m}_i{\dot{\mathbf{r}}}_i=0$$E12

The relationship between the velocity of point P at the end of the space robot arm in the inertial system and the generalized velocity $\dot{\mathbf{q}}$ is deduced as follows:

$${\mathbf{r}}_p=\left[\begin{array}{c}{\dot{x}}_p\\ {\dot{y}}_p\end{array}\right]=\left[\begin{array}{c}\begin{array}{ccc}{J}_{11}& J_{12}& J_{13}\end{array}\\ \begin{array}{ccc}{J}_{21}& J_{22}& J_{23}\end{array}\end{array}\right]\left[\begin{array}{c}{\dot{q}}_0\\ {\dot{q}}_1\\ {\dot{q}}_2\end{array}\right]=\mathbf{J}\dot{\mathbf{q}}$$E13

where $\dot{\mathbf{q}}={\left(\begin{array}{ccc}{\dot{q}}_0& {\dot{q}}_1& {\dot{q}}_2\end{array}\right)}^{\mathrm{T}}$, $\mathbf{J}$ is the Jacobian matrix of system velocity controlling carrier attitude. The items are as follows:

$J_{11}=L_{30}cos{q}_0+L_{31}cos\left(q_0+q_1\right)+L_{32}cos\left(q_0+q_1+q_2\right)$,

$J_{12}=L_{31}cos\left(q_0+q_1\right)+L_{32}cos\left(q_0+q_1+q_2\right)$, $J_{13}=L_{32}cos\left(q_0+q_1+q_2\right)$;

$J_{21}=-L_{30}sin{q}_0-L_{31}sin\left(q_0+q_1\right)-L_{32}sin\left(q_0+q_1+q_2\right)$,

$J_{22}=-L_{31}sin\left(q_0+q_1\right)-L_{32}sin\left(q_0+q_1+q_2\right)$, $J_{23}=-L_{32}sin\left(q_0+q_1+q_2\right)$

where $L_{10}=L_{00}+l_0$, $L_{11}=L_{01}+a_1$, $L_{12}=L_{02}$; $L_{20}=L_{00}+a_0$, $L_{21}=L_{01}+l_1$, $L_{22}=L_{02}+a_2$; $L_{30}=L_{00}+a_0$, $L_{31}=L_{01}+l_1$, $L_{32}=L_{02}+l_2$.

In this section, the deduced velocity Jacobian matrix $\mathbf{J}$ of the space robot system with no carrier position control and attitude control establishes the basis for the discussion of tracking the inertial space trajectory of the manipulator end of the space robot system in the following chapters.

The structure diagram of a fully rigid space robot system that does not control the position of the carrier but controls the attitude of the carrier is shown in Figure 1. Based on the velocity-vector relationship and the kinematics Jacobian matrix derived in Section 1.1, the dynamics equation of the system will be established in this section by combining the Lagrange equation of the second kind and the momentum conservation relationship.

According to the multi-body dynamics theory, according to aforesaid Eq. (13), the kinetic energy of each part of the space robot can be expressed as follows:

$$T_i=\frac{1}{2}{m}_i{{\dot{\mathbf{r}}}_i}^2+\frac{1}{2}{I}_i{{\omega }_i}^2$$E14

Where $i=0,1,2,3$, ${\omega }_0={\dot{q}}_0{\mathbf{z}}_0$, ${\omega }_1=\left({\dot{q}}_0+{\dot{q}}_1\right){\mathbf{z}}_1$, ${\omega }_2=\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right){\mathbf{z}}_2$.

The kinetic energy of the system is expressed as follows:

$$\begin{array}{rcl}T& =& T_0+T_1+T_2+T_3=\frac{1}{2}\sum _{i=0}^3\left(m_i{{\dot{\mathbf{r}}}_i}^2+I_i{{\omega }_i}^2\right)\\ & =& \frac{1}{2}{\varphi }_1{W}_1\left(\mathbf{q},\dot{\mathbf{q}}\right)+\frac{1}{2}{\varphi }_2{W}_2\left(\mathbf{q},\dot{\mathbf{q}}\right)+\frac{1}{2}{\varphi }_3{W}_3\left(\mathbf{q},\dot{\mathbf{q}}\right)+{\varphi }_4{W}_4\left(\mathbf{q},\dot{\mathbf{q}}\right)+{\varphi }_5{W}_5\left(\mathbf{q},\dot{\mathbf{q}}\right)\\ & & +{\varphi }_6{W}_6\left(\mathbf{q},\dot{\mathbf{q}}\right)\end{array}$$E15

Where $W_1={{\dot{q}}_0}^2$, $W_2={\left({\dot{q}}_0+{\dot{q}}_1\right)}^2$, $W_3={\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_3\right)}^2$, $W_4={\dot{q}}_0\left({\dot{q}}_0+{\dot{q}}_1\right)cos{q}_1$, $W_5={\dot{q}}_0\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)cos\left(q_1+q_2\right)$, $W_6=\left({\dot{q}}_0+{\dot{q}}_1\right)\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)cos{q}_2$, ${\varphi }_i\left(i=1-6\right)$ is the inertial parameter term, the specific items are as follows:

$${\varphi }_1=m_0{L_{00}}^2+m_1{L_{10}}^2+m_2{L_{20}}^2+m_3{L_{30}}^2+I_0$$

$${\varphi }_2=m_0{L_{01}}^2+m_1{L_{11}}^2+m_2{L_{21}}^2+m_3{L_{31}}^2+I_1$$

$${\varphi }_3=m_0{L_{02}}^2+m_1{L_{12}}^2+m_2{L_{22}}^2+m_3{L_{32}}^2+I_2+I_3$$

$${\varphi }_4=m_0{L}_{00}{L}_{01}+m_1{L}_{10}{L}_{11}+m_2{L}_{20}{L}_{21}+m_3{L}_{30}{L}_{31}$$

$${\varphi }_5=m_0{L}_{00}{L}_{02}+m_1{L}_{10}{L}_{12}+m_2{L}_{20}{L}_{22}+m_3{L}_{30}{L}_{32}$$

$${\varphi }_6=m_0{L}_{01}{L}_{02}+m_1{L}_{11}{L}_{12}+m_2{L}_{21}{L}_{22}+m_3{L}_{31}{L}_{32}$$

If the weak gravitational factor is ignored and the system’s gravitational potential energy $V=0$ is ignored, the system’s electromotive force is $L=T-V=T$. The space robot system takes the generalized coordinates $\mathbf{q}={\left(\begin{array}{ccc}{q}_0& q_1& q_2\end{array}\right)}^{\mathrm{T}}$ and puts the Lagrange function of the system’s dynamic force into the second Lagrange equation $L$ as shown below:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{\mathbf{q}}}\right)-\frac{\partial L}{\partial \mathbf{q}}={\mathbf{Q}}_0$$E16

Where ${\mathbf{Q}}_0$ is the generalized force vector. As shown in Figure 1, the dynamics equation of the space robot system is as follows:

$$\mathbf{D}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}=\mathbf{\tau }$$E17

Where $\mathbf{D}\left(\mathbf{q}\right)\in {\mathbf{R}}^{3\times 3}$ is a symmetric, positive definite mass matrix, $\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}\in {\mathbf{R}}^{3\times 1}$ is a vector containing Coriolis force and centrifugal force. $\mathbf{q}={\left[\begin{array}{ccc}{q}_0& q_1& q_2\end{array}\right]}^{\mathrm{T}}$ is the column vector of the generalized coordinates of the system. $q_0$ is the carrier attitude Angle, $q_1,q_2$ is the joint Angle of the boom, $\mathbf{\tau }\in {\mathbf{R}}^{3\times 1}$ is the vector formed by the carrier attitude control moment and joint control moment.

The terms of the matrix $\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)$ are not unique, which can be properly selected $\mathbf{D}\left(\mathbf{q}\right)$ and $\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)$ distinguished as follows:

$D_{11}={\varphi }_1+{\varphi }_2+{\varphi }_3+2{\varphi }_{4}cos{q}_1+2{\varphi }_{5}cos\left(q_1+q_2\right)+2{\varphi }_{6}cos{q}_2$,

$D_{12}={\varphi }_2+{\varphi }_3+{\varphi }_{4}cos{q}_1+{\varphi }_{5}cos\left(q_1+q_2\right)+2{\varphi }_{6}cos{q}_2$, $D_{13}={\varphi }_3+{\varphi }_{5}cos\left(q_1+q_2\right)+{\varphi }_{6}cos{q}_2$

$D_{21}=D_{12}$, $D_{22}={\varphi }_2+{\varphi }_3+2{\varphi }_{6}cos{q}_2$, $D_{23}={\varphi }_3+{\varphi }_{6}cos{q}_2$, $D_{31}=D_{13}$, $D_{32}=D_{23}$, $D_{33}={\varphi }_3$.

$H_{11}=-{\varphi }_4{\dot{q}}_{1}sin{q}_1-{\varphi }_5\left({\dot{q}}_1+{\dot{q}}_2\right)sin\left(q_1+q_2\right)-{\varphi }_6{\dot{q}}_{2}sin{q}_2$,

$H_{12}=-{\varphi }_4\left({\dot{q}}_1+{\dot{q}}_2\right)sin{q}_1-{\varphi }_5\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)sin\left(q_1+q_2\right)-{\varphi }_6{\dot{q}}_{2}sin{q}_2$,

$H_{13}=-{\varphi }_5\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)sin\left(q_1+q_2\right)-{\varphi }_6\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)sin{q}_2$,

$H_{21}={\varphi }_4{\dot{q}}_{0}sin{q}_1+{\varphi }_5{\dot{q}}_{0}sin\left(q_1+q_2\right)-{\varphi }_6{\dot{q}}_{2}sin{q}_2$, $H_{22}=-{\varphi }_6{\dot{q}}_{2}sin{q}_2{H}_{23}=-{\varphi }_6\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)sin{q}_2$, $H_{31}={\varphi }_5{\dot{q}}_{0}sin\left(q_1+q_2\right)+{\varphi }_6\left({\dot{q}}_0+{\dot{q}}_1\right)sin{q}_2$,

$H_{32}={\varphi }_6\left({\dot{q}}_0+{\dot{q}}_1\right)sin{q}_2$, $H_{33}=0$.

Then, $\mathbf{D}\left(\mathbf{q}\right)$ and $\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)$ matrix satisfies the following relation:

$${\mathbf{z}}^{\mathrm{T}}\mathbf{Hz}=\frac{1}{2}{\mathbf{z}}^{\mathrm{T}}\dot{\mathbf{D}}\mathbf{z}$$E18

Where $\mathbf{z}\in {\mathtt{R}}^{3\times 1}$is any vector.

2.2 Kinematic modeling of flexible joint floating-based space robot system

As shown in Figure 3, the research object is a plane-moving flexible joint space robot. The space robot system consists of a floating carrier $B0$, two flexible joints and rigid arms $B1$, $B2$. The dynamics model of space robot system established in this section is aimed at the situation that the position of the carrier is not controlled but the attitude of the carrier is controlled.

Figure 3.

Flexible-joint space robot.

The symbol definitions involved in formula derivation in Figure 3 and this section is supplemented as follows:

$\mathbf{K}$: represents the diagonal stiffness matrix of the joint of the manipulator;

$q_i$: actual joint Angle of the ith arm $\left(i=1,2\right)$;

$q_{\mathit{mi}}$: respectively refers to the Angle of the motor of the ith joint $\left(i=1,2\right)$;

$\mathit{\omega i}$: the rotation angular speed of the carrier and each arm $\left(i=0,1,2\right)$.

As shown in Figure 4, the joint of mechanical arm is generally composed of servo motor, harmonic reducer and arm rod. According to the flexible joint manipulator model studied by Spong [6], the joint flexibility can be simplified into a subsystem composed of joint actuator, linear torsional spring with constant inertia coefficient and arm.

Figure 4.

Simplified model of flexible joint.

Where $K$ is the equivalent joint stiffness.

As shown in Figure 4, the floating space robot system satisfies the law of conservation of momentum without loss of generality. Assuming that the initial momentum of the system acting without external force is 0, the following formula can be obtained:

$$M{\dot{\mathbf{r}}}_c=\sum _{i=0}^3{m}_i{\dot{\mathbf{r}}}_i=0$$E19

By taking the derivative of time t and combining with the total centroid relation (19), the expression of centroid velocity vector of each part can be obtained:

$${\dot{\mathbf{r}}}_0=\left[\begin{array}{c}-{\dot{q}}_0{L}_{00}sin{q}_0-\left({\dot{q}}_0+{\dot{q}}_1\right)L_{01}sin\left(q_0+q_1\right)-\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)L_{02}sin\left(q_0+q_1+q_2\right)\\ {\dot{q}}_0{L}_{00}cos{q}_0+\left({\dot{q}}_0+{\dot{q}}_1\right)L_{01}cos\left(q_0+q_1\right)+\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)L_{02}cos\left(q_0+q_1+q_2\right)\end{array}\right]$$E20

$${\dot{\mathbf{r}}}_1=\left[\begin{array}{c}-{\dot{q}}_0{L}_{10}sin{q}_0-\left({\dot{q}}_0+{\dot{q}}_1\right)L_{11}sin\left(q_0+q_1\right)-\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)L_{12}sin\left(q_0+q_1+q_2\right)\\ {\dot{q}}_0{L}_{10}cos{q}_0+\left({\dot{q}}_0+{\dot{q}}_1\right)L_{11}cos\left(q_0+q_1\right)+\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)L_{12}cos\left(q_0+q_1+q_2\right)\end{array}\right]$$E21

$${\dot{\mathbf{r}}}_2=\left[\begin{array}{c}-{\dot{q}}_0{L}_{20}sin{q}_0-\left({\dot{q}}_0+{\dot{q}}_1\right)L_{21}sin\left(q_0+q_1\right)-\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)L_{22}sin\left(q_0+q_1+q_2\right)\\ {\dot{q}}_0{L}_{20}cos{q}_0+\left({\dot{q}}_0+{\dot{q}}_1\right)L_{21}cos\left(q_0+q_1\right)+\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)L_{22}cos\left(q_0+q_1+q_2\right)\end{array}\right]$$E22

$${\dot{\mathbf{r}}}_p=\left[\begin{array}{c}-{\dot{q}}_0{L}_{30}sin{q}_0-\left({\dot{q}}_0+{\dot{q}}_1\right)L_{31}sin\left(q_0+q_1\right)-\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)L_{32}sin\left(q_0+q_1+q_2\right)\\ {\dot{q}}_0{L}_{30}cos{q}_0+\left({\dot{q}}_0+{\dot{q}}_1\right)L_{31}cos\left(q_0+q_1\right)+\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right)L_{32}cos\left(q_0+q_1+q_2\right)\end{array}\right]$$E23

The kinetic energy of each part of the system carrier and the boom is expressed as follows:

$$T_i=\frac{1}{2}{m}_i{{\dot{\mathbf{r}}}_i}^2+\frac{1}{2}{I}_i{{\omega }_i}^2$$E24

Where $i=0,1,2,3$, ${\omega }_0={\dot{q}}_0{\mathbf{z}}_0$, ${\omega }_1=\left({\dot{q}}_0+{\dot{q}}_1\right){\mathbf{z}}_1$, ${\omega }_2=\left({\dot{q}}_0+{\dot{q}}_1+{\dot{q}}_2\right){\mathbf{z}}_2$.

Kinetic energy of motor rotor component at the ith joint:

$$T_{\mathrm{mi}}=\frac{1}{2}{I}_{\mathit{im}}{{\dot{q}}_{\mathit{mi}}}^2,\left(i=1,2\right)$$E25

Potential energy accumulated by elastic deformation at the ith joint:

$$V_{\mathrm{mi}}=\frac{1}{2}{k}_{\mathit{im}}{\left({\dot{q}}_{\mathit{mi}}-{\dot{q}}_i\right)}^2,\left(i=1,2\right)$$E26

The Lagrange emotive force of the system can be expressed as follows:

$$\begin{array}{l}L=T-V\\ =\frac{1}{2}\sum _{i=0}^3\left(m_i{{\dot{\mathbf{r}}}_i}^2+I_i{{\omega }_i}^2\right)+\sum _{i=1}^2{T}_{\mathrm{mi}}-\sum _{i=1}^2{V}_{\mathrm{mi}}\end{array}$$E27

According to the hypothesis (20)-(23), the space robot system satisfies the law of conservation of momentum. From the second Lagrange equation, the system dynamics equation of this kind of space robot can be obtained, which controls the attitude of the carrier but not the position of the carrier, without external force and without considering the weak gravity factor:

$$\mathbf{D}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}=[\begin{array}{cc}{\mathbf{\tau }}_{\mathbf{0}}& \mathbf{\tau }]{}^{\mathrm{T}}\end{array}$$E28

$$\mathbf{I}{\ddot{\mathbf{q}}}_{\mathbf{m}}+\mathbf{\tau }={\mathbf{\tau }}_{\mathbf{m}}$$E29

Where ${\ddot{\mathbf{q}}}_{\mathbf{m}}={\left(\begin{array}{cc}{\ddot{q}}_{m1}& {\ddot{q}}_{m2}\end{array}\right)}^{\mathrm{T}}$, $\mathbf{\theta }=[{\begin{array}{cc}{q}_1& q_2]\end{array}}^{\mathrm{T}}$ are the actual response Angle of the two boom bars, the diagonal stiffness matrix $\mathbf{K}=diag\left(\begin{array}{cc}{k}_1& k_2\end{array}\right)\in {\mathrm{R}}^{2\times 2}$, and ${\mathbf{\tau }}_{\mathbf{m}}\in {\mathtt{R}}^{2\times 1}$ is the control torque vector actually generated by the servo motor at the joint. And terms$\mathbf{D}\left(\mathbf{q}\right)$ and $\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)$ are the same as in Eq. (18).

Similarly, the easy proof formulae (28) and (29) always satisfy the following formula:

$${\mathbf{\zeta }}^T\mathit{H}\mathbf{\zeta }=\frac{1}{2}{\mathbf{\zeta }}^{\mathrm{T}}\dot{\mathit{D}}\mathbf{\zeta }$$

Where $\mathbf{\zeta }\in {\mathtt{R}}^{3\times 1}$ is any vector.

For the purpose of controller design, Eqs. (28) and (29) is decomposed as follows:

$$\mathbf{D}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}=\mathbf{W\Phi }$$E30

Where $\mathbf{\Phi }={\left(\begin{array}{cc}\begin{array}{ccc}{\varphi }_1& {\varphi }_2& {\varphi }_3\end{array}& \begin{array}{ccc}{\varphi }_4& {\varphi }_5& {\varphi }_6\end{array}\end{array}\right)}^{\mathrm{T}}$ is the combination vector of inertial parameters, $\mathbf{W}$ is the function matrix containing $\mathbf{q}$, $\dot{\mathbf{q}}$, and $\ddot{\mathbf{q}}$, all items do not contain inertial parameters, and the specific items are shown as follows:

$$\mathbf{W}=\left[\begin{array}{cc}\begin{array}{ccc}\begin{array}{c}{W}_{11}\\ W_{21}\\ W_{31}\end{array}& \begin{array}{c}{W}_{12}\\ W_{22}\\ W_{32}\end{array}& \begin{array}{c}{W}_{13}\\ W_{23}\\ W_{33}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}{W}_{14}\\ W_{24}\\ W_{34}\end{array}& \begin{array}{c}{W}_{15}\\ W_{25}\\ W_{35}\end{array}& \begin{array}{c}{W}_{16}\\ W_{26}\\ W_{36}\end{array}\end{array}\end{array}\right]$$

Where $W_{11}={\ddot{q}}_0$, $W_{12}={\ddot{q}}_0+{\ddot{q}}_1$, $W_{13}={\ddot{q}}_0+{\ddot{q}}_1+{\ddot{q}}_2$, $W_{14}=\left(2{\ddot{q}}_0+{\ddot{q}}_1\right)cos{q}_1-\left(2{\dot{q}}_0{\dot{q}}_1+{{\dot{q}}_1}^2\right)sin{q}_1$,

$$W_{15}=\left(2{\ddot{q}}_0+{\ddot{q}}_1+{\ddot{q}}_2\right)cos\left(q_1+q_2\right)-\left(2{\dot{q}}_0{\dot{q}}_1+2{\dot{q}}_0{\dot{q}}_2+2{\dot{q}}_1{\dot{q}}_2+{{\dot{q}}_1}^2+{{\dot{q}}_2}^2\right)sin\left(q_1+q_2\right),$$

$W_{16}=\left(2{\ddot{q}}_0+2{\ddot{q}}_1+{\ddot{q}}_2\right)cos{q}_2-\left(2{\dot{q}}_0{\dot{q}}_2+2{\dot{q}}_1{\dot{q}}_2+{{\dot{q}}_2}^2\right)sin{q}_2$, $W_{21}=0$, $W_{22}=W_{12}$, $W_{23}=W_{13}$,

$W_{24}={\ddot{q}}_{0}cos{q}_1+{{\dot{q}}_0}^{2}sin{q}_1$, $W_{25}={\ddot{q}}_{0}cos\left(q_1+q_2\right)+{{\dot{q}}_0}^{2}sin\left(q_1+q_2\right)$, $W_{26}=W_{16}$,

$W_{31}=0$, $W_{32}=0$, $W_{33}=W_{13}$, $W_{34}=0$, $W_{35}=W_{25}$, $W_{36}=\left({\ddot{q}}_0+{\ddot{q}}_1\right)cos{q}_2-{\left({\dot{q}}_0+{\dot{q}}_1\right)}^{2}sin{q}_2$.

3. Robust adaptive control of space robot considering friction characteristics

The joint friction of space robot is complicated and nonlinear. The friction torque is the internal force of the system, and its coupling effect will affect the steady-state tracking error of the system. Conventional PID or robust method for trajectory tracking control has large steady-state error or limit cycle oscillation, and crawling occurs at low speed, which is difficult to achieve the expected real-time trajectory tracking effect [7, 8, 10]. At present, most control schemes considering frgziction use observer or neural network, which requires a large amount of computation and is not conducive to real-time online control of space robots [11].

In this chapter, a robust adaptive control strategy based on friction compensation is proposed by combining robust control theory and adaptive thought. The Lyapunov function of the system was constructed, and the tracking error convergence was proved and verified by numerical simulation. The proposed control strategy can effectively offset the influence of friction torque, improve the flat roof phenomenon in real-time tracking control, and improve the real-time trajectory tracking performance of the space robot.

As shown in Figure 1, based on the assumption that the attitude of the carrier is controlled and its position is not controlled, the dynamics equation of the fully rigid two-bar space robot system can be expressed as follows by reference in Eq. (17):

$$\mathbf{D}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}=\mathbf{\tau }$$E31

According to the analysis in Section 1, Eq. (30) shows that the dynamics equation of space robot can be expressed in the following form:

$$\mathbf{D}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}=\mathbf{W\Phi }$$E32

Where $\mathbf{\Phi }$ is the combination vector of the inertial parameters, and $\mathbf{W}$ is the function matrix containing $\mathbf{q}$, $\dot{\mathbf{q}}$ and $\ddot{\mathbf{q}}$.

To represent the nominal value ${\mathbf{\Phi }}_0$ of the inertia parameter vector of the system and the error ${\mathbf{\Phi }}_e$ between the nominal value and the actual value of the inertia parameter vector, then:

$$\mathbf{\Phi }={\mathbf{\Phi }}_0+{\mathbf{\Phi }}_e$$E33

The space robot dynamics equation of the nominal model can be expressed as follows:

$${\mathbf{D}}_0\left(\mathbf{q}\right)\ddot{\mathbf{q}}+{\mathbf{H}}_0\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}={\mathbf{W\Phi }}_0$$E34

So the dynamics equation of the actual space robot system can be expressed as follows:

$${\mathbf{D}}_0\left(\mathbf{q}\right)\ddot{\mathbf{q}}+{\mathbf{H}}_0\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}+\mathrm{\Delta }\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)=\mathbf{\tau }$$E35

Where $\mathrm{\Delta }\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)$ represents the uncertainty caused by the inaccuracy of system structure parameters.

Considering the influence of joint friction factors on the system, the joint friction has strong nonlinear characteristics. When the joint rotation speed approaches zero, the friction torque at the joint is large. When the joint velocity is not zero, the friction torque is small, and the friction characteristics of the joint can be approximated by the nonlinear function shown in Figure 2.

It is difficult to accurately establish the specific function form $f\left(\dot{q}\right)$ of friction torque, but its upper bound $\widehat{f}\left(\dot{q}\right)$ is known and can be expressed in the following form:

$$\begin{array}{c}\widehat{f}\left(\dot{q}\right)={\mu }_{\mathrm{s}}\left(\dot{q}\right)\lambda \left(\dot{q}\right)+{\mu }_{\mathrm{m}}\left(\dot{q}\right)\left(1-\lambda \left(\dot{q}\right)\right)\\ \left|{\mu }_{\mathrm{s}}\left(\dot{q}\right)\right|\le {\gamma }_{\mathrm{s}},\left|{\mu }_{\mathrm{m}}\left(\dot{q}\right)\right|\le {\gamma }_{\mathrm{f}}\end{array}$$E36

Where ${\mu }_{\mathrm{m}}$, ${\mu }_{\mathrm{s}}$ represents the unknown static friction force and dynamic friction force, and ${\gamma }_{\mathrm{f}}$, ${\gamma }_{\mathrm{s}}$represents the upper bound of static friction and dynamic friction, respectively, and the switching function $\lambda \left(\dot{q}\right)$ is expressed as follows:

$$\lambda \left({\dot{q}}_i\right)=\left\{\begin{array}{c}\begin{array}{cc}1,& \left|{\dot{q}}_i\right|>\mathit{\epsilon }\end{array}\\ \begin{array}{cc}0& \left|{\dot{q}}_i\right|\le \mathit{\epsilon }\end{array}\end{array}\right.,i=1,2$$E37

Where $\mathit{\epsilon }\mathit{>}\mathit{0}$ and sufficiently small.

The dynamics equation of the actual space robot model considering friction characteristics can be expressed as follows:

$${\mathbf{D}}_0\left(\mathbf{q}\right)\ddot{\mathbf{q}}+{\mathbf{H}}_0\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}+\mathrm{\Delta }\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)=\mathbf{\tau }-\mathbf{f}$$E38

Where $f$ represents the friction torque, and the rotation of the carrier relative to the inertial system is not affected by the friction torque, $f\left(1\right)=0$.

Let ${\mathbf{q}}_{\mathrm{d}}$ be the expected trajectory of the system, the design control law is as follows:

$$\mathbf{\tau }=\mathbf{u}+{\mathbf{D}}_0\left(\mathbf{q}\right){\ddot{\mathbf{q}}}_d+{\mathbf{H}}_0\left(\mathbf{q},\dot{\mathbf{q}}\right){\dot{\mathbf{q}}}_d$$E39

Joint tracking error $\mathbf{e}=\mathbf{q}-{\mathbf{q}}_{\mathrm{d}}$ is defined. The actual space robot system error equation can be expressed as follows:

$${\mathbf{D}}_0\left(\mathbf{q}\right)\ddot{e}+{\mathbf{H}}_0\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{e}+\mathbf{\Delta H}=u-f$$E40

Define state variables as follows:

$$\left\{\begin{array}{c}{\mathbf{x}}_1=\mathbf{e}\\ {\mathbf{x}}_2=\dot{\mathbf{e}}+\mathbf{e}\end{array}\right.$$E41

The state equation of system error can be obtained from Eqs. (38) and (41) as follows:

$$\left\{\begin{array}{c}{\dot{\mathbf{x}}}_1={\mathbf{x}}_2-{\mathbf{x}}_1\\ {\mathbf{D}}_0{\dot{\mathbf{x}}}_2={\mathbf{D}}_0\left({\mathbf{x}}_2-{\mathbf{x}}_1\right)-{\mathbf{H}}_0\left({\mathbf{x}}_2-{\mathbf{x}}_1\right)+\mathbf{u}-\mathbf{f}-\mathbf{\Delta H}\end{array}\right.$$E42

The design control law $\mathbf{u}$ is as follows:

$$\mathbf{u}={\mathbf{u}}_{\mathrm{r}}+{\mathbf{u}}_{\mathrm{c}}+{\mathbf{u}}_{\mathrm{f}}$$E43

Where ${\mathbf{u}}_{\mathrm{r}}$ is the error compensation term, ${\mathbf{u}}_{\mathrm{c}}$ represents the structural uncertainty fitting term, and ${\mathbf{u}}_{\mathrm{f}}$ is the friction compensation term. The specific form is as follows:

$$\left\{\begin{array}{c}{\mathbf{u}}_{\mathrm{r}}=-{\mathbf{D}}_0\left({\mathbf{x}}_2-{\mathbf{x}}_1\right)-{\mathbf{H}}_0{\mathbf{x}}_1-{\mathbf{x}}_1-p_{\mathbf{2}}{\mathbf{x}}_2+\frac{1}{4{\gamma }^2}{\mathbf{x}}_2\\ {\mathbf{u}}_c=W{\widehat{\mathbf{\Phi }}}_{\mathrm{e}}\\ {\mathbf{u}}_f=-\mathbf{\lambda }\left(\dot{\mathbf{q}}\right)sgn\left({\mathbf{x}}_2\right){\gamma }_{\mathit{s}}-\left(1-\mathbf{\lambda }\left(\dot{\mathbf{q}}\right)sgn\left({\mathbf{x}}_2\right){\gamma }_{\mathit{f}}\right)\end{array}\right.$$E44

Where ${\widehat{\mathbf{\Phi }}}_{\mathrm{e}}$ represents the estimated error of the nominal value ${\mathbf{\Phi }}_e$ and the actual value of the inertial parameter vector, which $P_2$ is greater than zero and is a positive constant.

The uncertainty caused by the inaccuracy of system structure parameters $\mathrm{\Delta }\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)$ is compensated by the fitting merge ${\mathbf{u}}_{\mathrm{c}}$:

$$\mathrm{\Delta }\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)=\mathbf{W}{\widehat{\mathbf{\Phi }}}_{\mathrm{e}}+{\mathbf{\epsilon }}_{\mathbf{f}}$$E45

Where ${\mathbf{\epsilon }}_{\mathbf{f}}$ is the fitting error of structural uncertainty term.

The law of design adaptation is as follows:

$${\dot{\widehat{\mathbf{\Phi }}}}_{\mathrm{e}}=-\lambda {\mathbf{W}}^{\mathrm{T}}{\mathbf{x}}_2$$E46

Where $\lambda >0$ is a constant. Defined estimation error ${\tilde{\mathbf{\Phi }}}_e={\widehat{\mathbf{\Phi }}}_{\mathrm{e}}-{\mathbf{\Phi }}_e$.

The evaluation signal $\mathbf{Z}\mathit{=}{P}_1{\mathbf{x}}_1$ is defined, and the fitting error ${\mathbf{\epsilon }}_{\mathbf{f}}$ of the uncertain term is regarded as the perturbation term of the closed-loop system. To prove that the closed-loop system satisfies L2 gain condition, the Lyapunov-like function of the system is constructed, which makes the dissipation inequality as shown below valid:

$$\dot{V}\le \left\{{\gamma }^2{‖{\mathbf{\epsilon }}_{\mathbf{f}}‖}^2-{‖\mathbf{z}\left(t\right)‖}^2\right\}$$E47

Where $\gamma >0$ represents the interference suppression level.

Construct the Lyapunov function as follows:

$$V=\frac{1}{2}{{\mathbf{x}}_1}^{\mathit{T}}{\mathbf{x}}_1+\frac{1}{2}{{\mathbf{x}}_2}^{\mathit{T}}{\mathbf{D}}_0{\mathbf{x}}_2+\frac{1}{2\lambda }\mathtt{tr}\left({{\tilde{\mathbf{\Phi }}}_e}^{\mathtt{T}}{\tilde{\mathbf{\Phi }}}_e\right)$$E48

By taking the derivative of the above equation with respect to time, we can obtain from the dissipation inequality (47):

$$\begin{array}{l}H=\dot{V}-\left({\gamma }^2{‖{\mathbf{\epsilon }}_{\mathbf{f}}‖}^2-{‖\mathbf{z}\left(t\right)‖}^2\right)\\ ={{\mathbf{x}}_1}^{\mathit{T}}{\dot{\mathbf{x}}}_1+{{\mathbf{x}}_2}^{\mathit{T}}\left({\mathbf{D}}_0{\dot{\mathbf{x}}}_2+\frac{1}{2}{\dot{\mathbf{D}}}_0{\mathbf{x}}_{\mathit{2}}\right)-{\mathbf{W}}^{\mathit{T}}{\widehat{\mathbf{\Phi }}}_{\mathrm{e}}{\mathbf{x}}_{\mathit{2}}-\left({\gamma }^2{‖{\mathbf{\epsilon }}_{\mathbf{f}}‖}^2-{‖\mathbf{z}\left(t\right)‖}^2\right)\\ =\left(1-{{\mathtt{P}}_{\mathtt{1}}}^2\right){‖{\mathbf{x}}_1‖}^2-{{\mathbf{x}}_2}^{\mathtt{T}}{\mathbf{\epsilon }}_{\mathbf{f}}-\frac{1}{\lambda }\mathtt{tr}\left({{\dot{\tilde{\mathbf{\Phi }}}}_e}^{\mathtt{T}}{\tilde{\mathbf{\Phi }}}_e\right)-{\gamma }^2{‖{\mathbf{\epsilon }}_{\mathbf{f}}‖}^2\\ +{{\mathbf{x}}_2}^{\mathit{T}}\left(-P_2{\mathbf{x}}_2+{\mathbf{W}}^{\mathit{T}}{\widehat{\mathbf{\Phi }}}_{\mathrm{e}}+\frac{1}{4{\gamma }^2}{\mathbf{x}}_2+{\mathbf{u}}_{\mathrm{f}}-\mathbf{f}\right)\\ \le -\eta {‖{\mathbf{x}}_1‖}^2-P_2{‖{\mathbf{x}}_2‖}^2+{{\mathbf{x}}_2}^{\mathit{T}}\left({\mathbf{u}}_{\mathrm{f}}-\mathbf{f}\right)\end{array}$$E49

Where $\eta <1-{{\mathtt{P}}_{\mathtt{1}}}^2$ is a positive constant.

By analyzing the friction compensation term ${\mathbf{u}}_{\mathrm{f}}$, the following inequality relation is established:

$$\begin{array}{l}{\mathbf{x}}_2>0,\left\{\begin{array}{c}-{\gamma }_{\mathit{s}}-{\mu }_{\mathit{s}}\left(\dot{q}\right)<0,\left|\dot{q}\right|>\mathit{\epsilon }\\ -{\gamma }_{\mathit{f}}-{\mu }_{\mathit{m}}\left(\dot{q}\right)<0,\left|\dot{q}\right|\le \mathit{\epsilon }\end{array}\right.\\ {\mathbf{x}}_2<0,\left\{\begin{array}{c}{\gamma }_{\mathit{s}}-{\mu }_{\mathit{s}}\left(\dot{q}\right)>0,\left|\dot{q}\right|>\mathit{\epsilon }\\ {\gamma }_{\mathit{f}}-{\mu }_{\mathit{m}}\left(\dot{q}\right)>0,\left|\dot{q}\right|\le \mathit{\epsilon }\end{array}\right.\end{array}$$

Therefore, ${{\mathbf{x}}_2}^{\mathit{T}}\left({\mathbf{u}}_{\mathrm{f}}-\mathbf{f}\right)<0$ is true, $H<0$ is always true, and the closed-loop system meets L2 gain condition $J<\gamma$. According to Lyapunov’s stability theorem, the closed-loop system satisfies the asymptotic stability condition, that is, $\underset{t\to \infty }{lim\mathbf{e}}\left(t\right)=0$,$\underset{t\to \infty }{lim\dot{\mathbf{e}}}\left(t\right)=0$.

The research object of value simulation experiment is the planar two-bar space robot as shown in Figure 1.

The inertia parameters of the system are selected as follows:

Quality of each component $m_0=40\mathrm{kg}$, $m_1=2\mathrm{kg}$, $m_2=1\mathrm{kg}$; the moment of inertia are respectively $J_0=34.17\mathrm{kg}\cdot {\mathrm{m}}^2$, $J_1=1.5\mathrm{kg}\cdot {\mathrm{m}}^2$, $J_2=0.75\mathrm{kg}\cdot {\mathrm{m}}^2$; the mechanical arm is set as a uniform rod, and the length of the rod is respectively $l_1=3\mathrm{m}$, $l_2=3\mathrm{m}$, where the distance along the axis $x_{\mathit{0}}$ is $l_0=1.5\mathrm{m}$ from point $O\mathit{0}$ to $O\mathit{1}$. The unknown load at the end of the nominal arm $m_{\mathrm{p}}=2\mathrm{kg}$, from the end load to the moment of inertia $O_{\mathrm{c}2}$ is $J_3=1.5\mathrm{kg}\cdot {\mathrm{m}}^2$. The actual load acting on the end of the manipulator is $m_{\mathrm{pe}}=4\mathrm{kg}$, other parameters are the same as the nominal model. Initial configuration of system simulation $q_0\left(0\right)=0.1\mathrm{rad}$, $q_1\left(0\right)=0.22\mathrm{rad}$,$q_2\left(0\right)=0.79\mathrm{rad}$ and expected trajectory are as follows:

$$\left\{\begin{array}{c}{q}_{\mathrm{d}0}=\mathrm{\pi }/6\mathrm{rad}\\ q_{\mathrm{d}1}=\mathrm{\pi }/8‐sin\left(\mathrm{\pi }t/3\right)\mathrm{rad}\\ q_{\mathrm{d}2}=cos\left(\mathrm{\pi }t/3\right)\mathrm{rad}\end{array}\right.$$

The continuous smooth static friction law proposed by Feeny was adopted to represent the friction characteristics of joints [12]:

$$\mathbf{f}\left(\dot{\mathbf{q}}\right)={\mathbf{f}}_{\mathrm{v}}\dot{\mathbf{q}}+\left({\mathbf{f}}_{\mathrm{c}}+{\mathbf{f}}_{\mathrm{cs}}sech\left(\beta \dot{\mathbf{q}}\right)tanh\left(\alpha \dot{\mathbf{q}}\right)\right)$$

The viscous friction factor $f_{\mathrm{v}}=0.00986$; $f_{\mathrm{c}}=0.743$ is coulomb friction factor; $f_{\mathrm{cs}}=3.99$ and $\alpha =3.24$ structure Stribeck friction; $\beta =0.799$ ensure the continuity of the equation when the relative sliding velocity is zero, $f\left(1\right)=0$.

The simulation results are as follows:

As can be seen in Figure 5, when friction compensation control is not enabled, the coupling effect of joint friction torque will cause interference to the attitude of the carrier, making it difficult to maintain the desired attitude Angle stably. Figures 6 and 7 show the tracking response of the system to the command trajectory. Due to the existence of friction factors, the joint Angle tracking has obvious response hysteresis, and the actual trajectory lags behind the expected trajectory, that is, $\left|\dot{\mathbf{q}}\right|\le \left|{\dot{\mathbf{q}}}_{\mathrm{d}}\right|$. With the friction torque compensation control turned on, the simulation results show that the friction compensation term can effectively offset the coupling effect of joint friction torque on the carrier, and can maintain the desired attitude of the carrier stably. After friction compensation control, the hysteresis phenomenon of joint Angle tracking is effectively improved, and the real-time trajectory tracking performance is improved.

Figure 5.

Carrier attitude angle.

Figure 6.

Joint angle of arm 1.

Figure 7.

Joint angle of arm 2.

3.1 Summary

1. This section discusses the trajectory tracking control problem of a fully rigid space robot with uncertain structural parameters under the influence of joint friction torque, and designs a robust adaptive control strategy based on friction compensation.

2. The Lyapunov function of the system was constructed, and the convergence of tracking errors was proved and verified by numerical simulation. The design scheme can effectively improve the response hysteria caused by friction factor, and effectively reduce the influence of joint coupling factors on the carrier attitude control.

3. The designed controller has the advantages of simple structure and small amount of calculation, and can be applied to real-time trajectory tracking control of three-dimensional space robots by extension.

4. Robust control and active suppression of elastic vibration of flexible joint space robot considering friction characteristics

In practical engineering applications, the space manipulator mostly uses harmonic reducer as the joint transmission mechanism. The application of harmonic reducer makes the joint of the manipulator have inevitable flexibility, which brings the coupling between the servo motor Angle and the joint Angle, and easily causes the manipulator jitter and the system response hysteretic problems [12, 13, 14]. In the undamped space environment, the vibration attenuation is slow, which brings great difficulty to the control of space robot. At present, there are few control schemes for space robots that take joint friction into consideration, and most studies take friction into consideration the disturbance term [7, 15, 16, 17]. In most control schemes, friction compensation is based on on-line observer, and the friction torque is observed and compensated as the disturbance term. Moreover, the structure of most controllers is complex or dependent on neural network, which is not conducive to real-time joint control [18, 19, 20].

In this section, a flexible joint space robot with uncontrolled carrier position and attitude is taken as the research object, and the control problem of space robot under the influence of strong nonlinear friction torque is explored. Based on the joint flexibility compensation idea and singular perturbation theory, a L2 gain robust controller based on the upper bound of friction force was designed, and the Lyapunov function of the system was constructed. The convergence of tracking errors was proved and verified by co-simulation.

As shown in Figure 3, the object of study is a space robot system with flexible joints that moves in a plane. The system whose attitude is controlled by the reaction wheel satisfies the law of conservation of momentum. Let the initial momentum be 0. Based on the assumption that the attitude of the carrier is controlled without controlling its position, the system dynamics model of the flexible joint space robot is derived from Eqs. (28) and (29):

$$\mathbf{D}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}=[\begin{array}{cc}{\mathbf{\tau }}_{\mathbf{0}}& \mathbf{\tau }]{}^{\mathrm{T}}\end{array}$$E50

$$\mathbf{J}{\ddot{\mathbf{\theta }}}_{\mathbf{m}}+\mathbf{\tau }={\mathbf{\tau }}_{\mathbf{m}}$$E51

$$\mathbf{\tau }=\mathbf{K}\left({\mathbf{\theta }}_{\mathbf{m}}-\mathbf{\theta }\right)$$E52

The rotational inertia $\mathbf{J}\in {\mathrm{R}}^{2\times 2}$ of the motor is the diagonal and constant matrix, and$\mathbf{K}\in {\mathrm{R}}^{2\times 2}$ is the diagonal stiffness matrix of the joint. $\mathbf{q}={\left[\begin{array}{ccc}{q}_0& q_1& q_2\end{array}\right]}^{\mathrm{T}}$,$\mathbf{\theta }={\left[\begin{array}{cc}{q}_1& q_2\end{array}\right]}^{\mathrm{T}}$ is the actual rotation Angle of the boom, ${\mathbf{\theta }}_{\mathbf{m}}={\left[\begin{array}{cc}{q}_{\mathrm{m}1}& q_{\mathrm{m}2}\end{array}\right]}^{\mathrm{T}}$ is the motor rotation Angle vector, $\mathbf{\tau }\in {\mathrm{R}}^{2\times 1}$ is the control torque vector acting on the mechanical arm, ${\mathbf{\tau }}_{\mathrm{m}}\in {\mathrm{R}}^{2\times 1}$ represents the actual control torque vector of the servo motor.

The dynamic equation of torque can be obtained from Eqs. (51) and (52):

$${\mathbf{JK}}^{-1}\ddot{\mathbf{\tau }}+\mathbf{\tau }={\mathbf{\tau }}_{\mathrm{m}}-\mathbf{J}\ddot{\mathbf{\theta }}$$E53

Let the motor control input be as follows:

$${\mathbf{\tau }}_{\mathrm{m}}={\mathbf{K}}_{\mathrm{r}}{\mathbf{\tau }}_{\mathrm{r}}+{\mathbf{u}}_{\mathrm{fs}}$$E54

Where ${\mathbf{K}}_{\mathrm{r}}=\mathrm{I}+{\mathbf{K}}_{\mathrm{b}}$, ${\mathbf{K}}_{\mathrm{b}}\in {\mathrm{R}}^{2\times 2}$ is the diagonal, positive invariant matrix, $\mathrm{I}$ is the identity matrix; ${\mathbf{\tau }}_{\mathrm{r}}\in {\mathrm{R}}^{2\times 1}$ is the control quantity to be designed; Joint flexibility compensation controller ${\mathbf{u}}_{\mathrm{fs}}=-{\mathbf{K}}_{\mathrm{b}}\mathbf{\tau }$. According to Eqs. (54) and (53), it can be obtained that:

$${{\mathbf{JK}}_{\mathrm{e}}}^{-1}\ddot{\mathbf{\tau }}+\mathbf{\tau }={\mathbf{\tau }}_{\mathrm{r}}-{{\mathbf{K}}_{\mathrm{r}}}^{-1}\mathbf{J}\ddot{\mathbf{\theta }}$$E55

Where ${\mathbf{K}}_{\mathrm{e}}={\mathbf{KK}}_{\mathrm{r}}$ is the equivalent stiffness matrix after flexibility compensation.

The control input ${\mathbf{\tau }}_{\mathrm{r}}$ based on singular perturbation is designed as follows:

$${\mathbf{\tau }}_{\mathrm{r}}={\mathbf{\tau }}_{\mathrm{rl}}+{\mathbf{\tau }}_{\mathrm{rf}}$$E56

Where ${\mathbf{\tau }}_{\mathrm{rl}}\in {\mathrm{R}}^{2\times 1}$ is the control input of slow variable subsystem and ${\mathbf{\tau }}_{\mathrm{rf}}\in {\mathrm{R}}^{2\times 1}$ is the control input of fast variable subsystem.

Define sufficiently small positive numbers $\mu >0$, let ${\mathbf{K}}_{\mathrm{e}}={\mathbf{K}}_{\mathrm{a}1}/{\mu }^{\mathit{2}}$; ${\mathbf{K}}_{\mathrm{a}1}\in {\mathrm{R}}^{2\times 2}$ is of a similar order of magnitude to the variable in the slow subsystem. The fast subsystem equation is derived from Eq. (55) as follows:

$${\mu }^2\mathbf{J}\ddot{\mathbf{\tau }}+{\mathbf{K}}_{\mathrm{a}1}\mathbf{\tau }={\mathbf{K}}_{\mathrm{a}1}\left({\mathbf{\tau }}_{\mathrm{rl}}-{{\mathbf{K}}_{\mathrm{r}}}^{-1}\mathbf{J}\ddot{\mathbf{\theta }}\right)+{\mathbf{K}}_{\mathrm{a}1}{\mathbf{\tau }}_{\mathrm{rf}}$$E57

The torque differential controller is designed as follows:

$${\mathbf{\tau }}_{\mathrm{rf}}=-\mu {\mathbf{K}}_{\mathrm{a}2}\mathbf{K}\left({\dot{\mathbf{\theta }}}_{\mathrm{m}}-\dot{\mathbf{\theta }}\right)$$E58

Let $\mu =0$, the dynamic equation of the slow subsystem is as follows:

$$\overline{\mathbf{D}}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}={\left[\begin{array}{cc}{\tau }_0& {\mathbf{\tau }}_{\mathbf{rl}}\end{array}\right]}^{\mathrm{T}}$$E59

Among them $\overline{\mathbf{D}}\left(\mathbf{q}\right)=\left[\begin{array}{cc}{\overline{\mathbf{D}}}_{11}& {\overline{\mathbf{D}}}_{12}\\ {\overline{\mathbf{D}}}_{21}& {\overline{\mathbf{D}}}_{22}\end{array}\right]$, ${\overline{\mathbf{D}}}_{11}={\mathbf{D}}_{11}$, ${\overline{\mathbf{D}}}_{12}={\mathbf{D}}_{12}$, ${\overline{\mathbf{D}}}_{21}={\mathbf{D}}_{21}$, ${\overline{\mathbf{D}}}_{22}={\mathbf{D}}_{22}+{{\mathbf{K}}_{\mathrm{r}}}^{-1}\mathbf{J}$; where ${{\mathbf{K}}_{\mathrm{r}}}^{-1}\mathbf{J}$ is a constant matrix, the first derivative is always zero, and Eq. (17) can be proved similarly. For any column vector $\mathbf{z}\in {\mathrm{R}}^{3\times 1}$, it is always:

$${\mathbf{z}}^{\mathrm{T}}\mathbf{Hz}=\frac{1}{2}{\mathbf{z}}^{\mathrm{T}}\dot{\overline{\mathbf{D}}}\mathbf{z}$$E60

The dynamic equation of slow subsystem considering the influence of joint friction can be expressed as follows:

$$\overline{\mathbf{D}}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}={\left[\begin{array}{cc}{\tau }_0& {\mathbf{\tau }}_{\mathbf{rl}}\end{array}\right]}^{\mathrm{T}}-f$$E61

Where $f$ represents the friction torque, $f\left(1\right)=0$.

The robot joint friction shows strong nonlinear characteristics at low speed, as shown in Figure 8. When the joint rotation speed approaches zero, the friction torque at the joint is large. When the joint velocity is not zero, the friction torque is small. Many scholars have studied the friction characteristics of joint, and the precise friction model is complicated, which makes the calculation amount surge. Consider using the nonlinear function shown in Figure 8 to approximate the friction characteristics of the joint.

Figure 8.

Friction characteristics of joints.

In this section, it is studied that the form of joint friction characteristic function is difficult to accurately establish, but its upper bound $\widehat{f}\left(\dot{q}\right)$ is known, which is expressed as follows:

$$\begin{array}{c}\widehat{f}\left(\dot{q}\right)={\mu }_{\mathrm{s}}\left(\dot{q}\right)\lambda \left(\dot{q}\right)+{\mu }_{\mathrm{m}}\left(\dot{q}\right)\left(1-\lambda \left(\dot{q}\right)\right)\\ \left|{\mu }_{\mathrm{s}}\left(\dot{q}\right)\right|\le {\gamma }_{\mathrm{s}},\left|{\mu }_{\mathrm{m}}\left(\dot{q}\right)\right|\le {\gamma }_{\mathrm{f}}\end{array}$$E62

Where ${\mu }_{\mathrm{f}}$, ${\mu }_{\mathrm{s}}$ represent the unknown static friction force and dynamic friction force, ${\gamma }_{\mathrm{f}}$ and ${\gamma }_{\mathrm{s}}$ are the upper bound of static friction and dynamic friction, respectively, $\lambda \left(\dot{q}\right)$ is the switching function, expressed as follows:

$$\lambda \left({\dot{q}}_i\right)=\left\{\begin{array}{c}\begin{array}{cc}1,& \left|{\dot{q}}_i\right|>\mathit{\epsilon }\end{array}\\ \begin{array}{cc}0& \left|{\dot{q}}_i\right|\le \mathit{\epsilon }\end{array}\end{array}\right.,i=1,2$$E63

Where $\mathit{\epsilon }\mathit{>}\mathit{0}$ and sufficiently small.

Set ${\mathbf{q}}_{\mathrm{d}}$ as the expected output of the system, and define the joint Angle tracking error as follows:

$$\mathbf{e}=\mathbf{q}-{\mathbf{q}}_{\mathrm{d}}$$E64

By introducing the control quantity $\mathbf{u}$ to be designed, the space robot tracking problem is transformed into an asymptotic stability control problem:

$$\overline{\mathbf{D}}\left(\mathbf{q}\right){\ddot{\mathbf{q}}}_{\mathrm{d}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right){\dot{\mathbf{q}}}_{\mathrm{d}}+\mathbf{u}={\left[\begin{array}{cc}{\tau }_0& {\mathbf{\tau }}_{\mathrm{rl}}\end{array}\right]}^{\mathit{T}}$$E65

Considering the external interference torque $\mathbf{d}\in {\mathrm{R}}^{3\times 1}$ acting on the system, the dynamics model of the slow subsystem is as follows:

$$\overline{\mathbf{D}}\left(\mathbf{q}\right)\ddot{\mathbf{q}}+\mathbf{H}\left(\mathbf{q},\dot{\mathbf{q}}\right)\dot{\mathbf{q}}={\left[\begin{array}{cc}{\tau }_0& {\mathbf{\tau }}_{\mathrm{rl}}\end{array}\right]}^{\mathit{T}}+\mathbf{d}-f$$E66

Let ${\mathbf{x}}_1=\mathbf{e}$, ${\mathbf{x}}_2=\dot{\mathbf{e}}+\mathbf{e}$ and the state-space equation of the dynamics model of space robot system considering friction characteristics and external interference torque is as follows:

$$\left\{\begin{array}{c}{\dot{\mathbf{x}}}_1={\mathbf{x}}_2-{\mathbf{x}}_1\\ {\overline{\mathbf{D}}\dot{\mathbf{x}}}_2=\overline{\mathbf{D}}\left({\mathbf{x}}_2-{\mathbf{x}}_1\right)-\mathbf{H}\left({\mathbf{x}}_2-{\mathbf{x}}_1\right)+\mathbf{u}+\mathbf{d}-\mathbf{f}\end{array}\right.$$E67

The controller $\mathbf{u}$ is designed as follows:

$$\mathbf{u}={\mathbf{u}}_{\mathrm{r}}+{\mathbf{u}}_{\mathrm{d}}+{\mathbf{u}}_{\mathrm{f}}$$E68

Where ${\mathbf{u}}_{\mathrm{r}}$ is the tracking error compensation term, ${\mathbf{u}}_{\mathrm{d}}$ is the disturbance compensation term and ${\mathbf{u}}_{\mathrm{f}}$ is the friction compensation term. The evaluation signal $\mathbf{Z}\mathit{=}{\left[\begin{array}{cc}{P}_{\mathit{1}}{\mathbf{e}}_1& P_2{\mathbf{e}}_2\end{array}\right]}^{\mathit{T}}$, $P_1\ge 0$, $P_2\ge 0$ is defined as the given weighting coefficient, which is expressed as follows:

$$\left\{\begin{array}{c}{\mathbf{u}}_{\mathrm{r}}=-\overline{\mathbf{D}}\left({\mathbf{x}}_2-{\mathbf{x}}_1\right)-{\mathbf{Hx}}_1-{\mathbf{x}}_1\\ {\mathbf{u}}_{\mathrm{d}}=2{P_2}^2{\mathbf{x}}_1-{\mathbf{K}}_{\mathrm{d}}{\mathbf{x}}_2\\ {\mathbf{u}}_{\mathrm{fs}}=-\left(\mathbf{\lambda }\left(\dot{\mathbf{q}}\right)sgn\left({\mathbf{x}}_2\right){\gamma }_{\mathit{s}}+\left(1-\mathbf{\lambda }\left(\dot{\mathbf{q}}\right)sgn\left({\mathbf{x}}_2\right){\gamma }_{\mathit{f}}\right)\right)\end{array}\right.$$E69

Construct the energy function $V$. If the system satisfies L2 gain condition, then the following dissipation inequality holds:

$$\dot{V}\le \left\{{\gamma }^2{‖\mathbf{d}\left(t\right)‖}^2-{‖\mathbf{z}\left(t\right)‖}^2\right\}$$E70

Where $\gamma >0$ represents the suppression level of system interference, and the value $\gamma$ reflects the influence of external disturbance on system tracking error.

The Lyapunov function of the closed-loop system is constructed as follows:

$$V=\frac{1}{2}{{\mathbf{x}}_1}^{\mathit{T}}{\mathbf{x}}_1+\frac{1}{2}{{\mathbf{x}}_2}^{\mathit{T}}\overline{\mathbf{D}}{\mathbf{x}}_2$$E71

The derivative of $V$ with respect to time T is:

$$\begin{array}{l}H=\dot{V}-\left({\gamma }^2{‖\mathbf{d}\left(t\right)‖}^2-{‖\mathbf{z}\left(t\right)‖}^2\right)\\ ={{\mathbf{x}}_1}^{\mathit{T}}{\mathbf{x}}_1+{P_{\mathit{1}}}^2{‖{\mathbf{x}}_1‖}^2+{P_2}^2{‖{\mathbf{x}}_2-{\mathbf{x}}_1‖}^2-{\gamma }^2{‖\mathbf{d}\left(t\right)‖}^2\\ +{{\mathbf{x}}_2}^{\mathit{T}}\left[\overline{\mathbf{D}}\left({\mathbf{x}}_2-{\mathbf{x}}_1\right)+{\mathbf{Hx}}_1-{\mathbf{x}}_1+\mathbf{u}+\mathbf{d}-\mathbf{f}\right]\\ \le -{\delta }_1{‖{\mathbf{x}}_1‖}^2-{\delta }_2{‖{\mathbf{x}}_2‖}^2+{{\mathbf{x}}_2}^{\mathrm{T}}\left({\mathbf{u}}_{\mathrm{fs}}-\mathbf{f}\right)\end{array}$$E72

Where ${\delta }_1$ and ${\delta }_2$ are sufficiently small but greater than 0, and satisfies the following formula:

$$\left\{\begin{array}{c}1-{P_1}^2-{P_2}^2\ge {\delta }_1\\ ‖{\mathbf{K}}_{\mathrm{d}}‖-{P_2}^2-\frac{1}{4{\gamma }^2}\ge {\delta }_2\end{array}\right.$$E73

From Eq. (69), we can see that the following inequality relationship is valid:

$$\begin{array}{l}{\mathbf{x}}_2>0,\left\{\begin{array}{c}-{\gamma }_{\mathit{s}}-{\mu }_{\mathit{s}}\left(\dot{q}\right)<0,\left|\dot{q}\right|>\epsilon \\ -{\gamma }_{\mathit{f}}-{\mu }_{\mathit{m}}\left(\dot{q}\right)<0,\left|\dot{q}\right|\le \epsilon \end{array}\right.\\ {\mathbf{x}}_2<0,\left\{\begin{array}{c}{\gamma }_{\mathit{s}}-{\mu }_{\mathit{s}}\left(\dot{q}\right)>0,\left|\dot{q}\right|>\epsilon \\ {\gamma }_{\mathit{f}}-{\mu }_{\mathit{m}}\left(\dot{q}\right)>0,\left|\dot{q}\right|\le \epsilon \end{array}\right.\end{array}$$

Therefore, $H<0$ is always true, and the closed-loop system meets L2 gain condition$J<\gamma$. According to Lyapunov’s stability theorem, the closed-loop system satisfies the asymptotic stability condition, that is, $\underset{t\to \infty }{lim\mathbf{e}}\left(t\right)=0$, $\underset{t\to \infty }{lim\dot{\mathbf{e}}}\left(t\right)=0$.

The inertia parameters of the system are selected as follows:

Carrier parameters are $m_0=40\mathrm{kg}$, $I_0=34.17\mathrm{kg}.{\mathrm{m}}^2$, $l_0=1.5\mathrm{m}$; the inertia parameter of boom rod 1 is $m_1=2\mathrm{kg}$, $I_1=1.5\mathrm{kg}.{\mathrm{m}}^2$, $l_1=3\mathrm{m}$, and the inertia parameter of boom rod 2 is $m_2=1\mathrm{kg}$, $I_2=0.8\mathrm{kg}.{\mathrm{m}}^2$, $l_2=3\mathrm{m}$. The stiffness of the flexible joint is set $\mathbf{K}=diag\left(k_1,k_2\right)$, where $k_1=100\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$,$k_2=100\mathrm{N}\cdot \mathrm{m}/\mathrm{rad}$; the moment of inertia of the motor $\mathbf{J}=diag\left(J_1,J_2\right)$, where $J_1=0.08\mathrm{kg}\cdot {\mathrm{m}}^2$, $J_2=0.08\mathrm{kg}\cdot {\mathrm{m}}^2$. Select the controller parameter $P_1=0.5$, $P_2=0.5$, ${\mathbf{K}}_{\mathrm{d}}=\mathit{diag}\left(\mathit{50},\mathit{50},\mathit{50}\right)$, ${\mathbf{K}}_{\mathrm{b}}\mathit{=}\mathit{diag}\left(65,65\right)$, $\gamma =1$, ${\gamma }_{\mathrm{s}}=2$, ${\gamma }_{\mathrm{f}}=4$, ${\mathbf{K}}_{\mathrm{a}1}\mathit{=}\mathit{diag}\left(\mathit{0.05},\mathit{0.05}\right)$, ${\mathbf{K}}_{\mathrm{a}2}\mathit{=}\mathit{diag}\left(\mathit{25},\mathit{25}\right)$, $\mathit{\epsilon }=0.01$. Initial system configuration $q_0\left(0\right)=0\mathrm{rad}$, $q_1\left(0\right)=q_{\mathrm{m}1}\left(0\right)=0.22\mathrm{rad}$, $q_2\left(0\right)=q_{\mathrm{m}2}\left(0\right)=0.79\mathrm{rad}$. The expected trajectory of the system is as follows:

$$\left\{\begin{array}{c}{q}_{\mathrm{d}0}=\mathrm{\pi }/6\mathrm{rad}\\ q_{\mathrm{d}1}=\mathrm{\pi }/8‐sin\left(\mathrm{\pi }t/3\right)\mathrm{rad}\\ q_{\mathrm{d}2}=cos\left(\mathrm{\pi }t/3\right)\mathrm{rad}\end{array}\right.$$

The continuous smooth static friction law proposed by Feeny was adopted to represent the friction characteristics of joints [21]:

$$\mathbf{f}\left(\dot{\mathbf{q}}\right)={\mathbf{f}}_{\mathrm{v}}\dot{\mathbf{q}}+\left({\mathbf{f}}_{\mathrm{c}}+{\mathbf{f}}_{\mathrm{cs}}sech\left(\beta \dot{\mathbf{q}}\right)tanh\left(\alpha \dot{\mathbf{q}}\right)\right)$$

The viscous friction factor $f_{\mathrm{v}}=0.00986$; $f_{\mathrm{c}}=0.743$ is coulomb friction factor; $f_{\mathrm{cs}}=3.99$ and $\alpha =3.24$ structure Stribeck friction; $\beta =0.799$ ensure the continuity of the equation when the relative sliding velocity is zero, $f\left(1\right)=0$ (Figure 9).

Figure 9.

Carrier attitude angle.

The simulation results are as follows:

Figure 10 shows the tracking control of the carrier attitude Angle. It can be seen that without friction compensation control, the coupling effect of joint friction torque will cause interference to the carrier attitude, making it difficult to maintain the desired attitude Angle stably. Under the action of friction compensation control, the friction compensation term can effectively offset the coupling effect of joint friction torque on the carrier, and the carrier can maintain the desired attitude stably after 5 s. Figure 10 and Figure 11 show the joint angle tracking control situation. Due to the existence of friction factors, the joint angle tracking has obvious response hysteresis, the actual trajectory lags behind the expected trajectory, $\left|\dot{\mathbf{q}}\right|\le \left|{\dot{\mathbf{q}}}_{\mathrm{d}}\right|$, and the real-time tracking error is large. After friction compensation control, the hysteresis phenomenon of joint angle tracking is effectively improved, and the real-time trajectory tracking performance is improved.

Figure 10.

Joint angle of arm 1.

Figure 11.

Joint angle of arm 2.

In Figures 12 and 13, ${\tau }_0$, ${\tau }_1$ and ${\tau }_2$ represent the carrier attitude, joint 1 and joint 2 control input, respectively. By comparison, it can be seen that the controller designed in this chapter can effectively track the expected trajectory stably, the torque differential controller can effectively suppress the vibration caused by flexible joints, and the L2 gain robust controller based on friction compensation for the slow subsystem can effectively improve the response hysteria caused by joint friction torque and reduce the interference of friction torque coupling on the carrier attitude.

Figure 12.

Control input with friction compensation turn off.

Figure 13.

Control input with friction compensation turn on.

Figure 14 shows the joint Angle tracking curve and tracking error of the space robot system when the torque differential controller ${\mathbf{\tau }}_{\mathrm{rf}}$ and friction compensation control ${\mathbf{u}}_{\mathrm{f}}$ of the fast variable subsystem are turned off. By comparing Figures 9-11 and 13, it can be seen that after closing the torque differential controller ${\mathbf{\tau }}_{\mathrm{rf}}$, the vibration caused by joint flexibility causes the instability of the carrier and joint motion, and the system oscillates and diverges, and the joint angle tracking control cannot be effectively implemented. It can be proved that the torque differential controller given by Eq. (58) can effectively suppress the vibration caused by joint flexibility.

Figure 14.

Joint angle and error without vibration suppression. (a) Carrier attitude angle, (b) joint 1 angle, (c) joint 2 angle, (d) tracking error.

4.1 Summary

In this section, the trajectory tracking control problem of flexible joint space robot system influenced by joint flexibility and joint friction torque is discussed. Based on joint compensation and singular perturbation method, the torque differential controller for the fast subsystem is designed. For the slow subsystem, the L2 gain controller based on the upper bound of friction is robust. The scheme can restrain the influence of friction factors well. Through the co-simulation, it can be seen that the space robot with floating base is different from the ground robot with fixed base. Its base attitude is affected by the coupling effect of joint friction torque. The controller designed can reduce the influence of joint coupling effect on the carrier attitude angle and effectively improve the response hysteretic problem caused by the friction factor.

The Lyapunov equation of the system is constructed, the convergence of the control law is proved by the stability theory, and the simulation proves that the controller designed in this chapter can effectively improve the response hysteretic problem caused by friction, and effectively reduce the influence of joint coupling factors on the attitude control of the carrier.