Robust tracking control of unknown models for space in-cabin robots with a pneumatic continuum arm

Abstract

The service robots of space station in-cabin have attracted more and more attention. The space in-cabin robot with a pneumatic continuum arm is studied in this paper. It could be safer, more efficient and more flexible than the space rigid robot. However, the coupling motion of the moving base and the pneumatic continuum continuous arm brings a new challenge for controlling the end-effector to track the desired path. In this paper, a new control method based on the zeroing neural network (ZNN) is developed to solve the high-precision kinematics trajectory tracking control problem of unknown models. The real-time Jacobian matrix of the in-cabin robots with a pneumatic continuum arm is estimated by the input–output information when the parameter and the structure of the kinematic model are unknown. Moreover, this paper also employs a modified activation function power-sigmoid activation function (PSAF) to improve the robustness. In addition, it is proved through the Lyapunov stability theory that the proposed control approach is convergent and stable. Finally, the simulation results are given to show the effectiveness and robustness of the proposed control method for space in-cabin robots with a pneumatic continuum arm.

Introduction

In recent years, China has also made a breakthrough in space station technology with the continuous development of space technology. China has successfully launched the Tianhe Core Module and created a large-scale, long-term manned national space laboratory to develop China's Space Station (CSS) between 2021 and 2022 [1]. At present, the long-term care of the Space Station and the implementation of complex scientific studies primarily rely on human astronauts. Astronauts need to do many microgravity experiments in biology, physics, astronomy, and other fields in the Space Station. So it is important to help astronauts perform their work efficiently [2]. In International Space Station (ISS), several in-cabin assistant robots have been proposed and flown on orbit, which includes Personal Satellite Assistant (PSA) [3], Astrobee [4, 5] Synchronized Position Hold Engage and Reorient Experimental Satellite (SPHERES) [6] Int-Ball [7] et al.

However, only Astrobee is fitted with a small rigid manipulator in the above space station robots. They could not help the astronauts carry things or make some simple operations in the space station. Therefore, we propose a new space station robot’s scheme that installs a pneumatic continuum arm under the micro-nano robot, as shown in Fig. 1. This pneumatic continuum arm is a special kind of robotic manipulator, including a base and two sections pneumatic manipulator consisting of three bellows, which exhibits biological behaviors similar to trunks by the folded structure composed of elastic elements and flexible materials [8]. Compared with traditional manipulators, pneumatic continuum manipulators are safer and more flexible [9]. In the limited space station, it can not only improve the work efficiency of astronauts but also ensure the safety of astronauts. So the in-cabin robots with a pneumatic continuum arm can help astronauts carry cargo and perform routine inspections of instruments. Furthermore, it can work collaboratively and interactively with the astronauts [11].

Fig. 1

The model of space in-cabin robots with a pneumatic continuum arm and some partial schematic diagrams

Based on the above analysis, pneumatic continuum manipulators have great prospects in the future, especially in aerospace applications. However, there are few applications of continuum arms in aerospace [12]. There are two main reasons: there is no scholar who studied the kinematic model of the space continuum arm of the moving base, and the lack of effective and practical control algorithms.

Unlike traditional rigid robots, pneumatic continuum manipulators generally have no fixed joints and rods but an infinite degree of freedom theoretically. Therefore, it is difficult to model the kinematics of continuum robots without strong assumptions. The analytical models mainly include the Pseudo Rigid Body Model and the Constant Curvature Model, which are very fast to compute and can be derived easily for control law proofs [13]. This paper adopts the Constant Curvature Model to solve the Jacobian of soft continuum robots. The following assumptions are made the bending always keeps the shape of constant curvature and there is no torsion produced [15]. However, the continuum robots in the space station are weightless, so the bending case can be treated completely as constant curvature.

The tracking control of the space continuum robots is an important problem in a wide range of applications, which moves the end-effector to track the desired trajectory in the inertial space [16]. Generally, the kinematic control of the end-effector can be implemented using the inverse kinematics calculation method when the desired path is known. In addition, the infinite DOF of continuum robots make inverse kinematics multi-solutions. The coupled motion of the moving base and the continuous arm also increases the difficulty of the solution. Therefore, the acquisition of efficient solutions is challenging. Many researchers have proposed different methods to solve the problem. Falkenhahn et al. [17] have been used to generate a time-optimal trajectory by minimizing pose error based on Quadratic Programming (QP). Salvador et al. [18] developed a novel kinematic model that can control the end-effector of a continuum manipulator in narrow environments combining inverse kinematic and forward kinematic. The above control methods need that the model of the continuum arm has been known in advance.

However, the structural complexity and the particularity of space application make it difficult to build and compute an accurate model for space in-cabin robots with a pneumatic continuum arm. Therefore, the control methods of unknown model can play a crucial part. Jin et al. [19] developed a model-less feedback controller by estimating the Jacobian matrix of soft manipulators derived from the structural theory of the model. Li et al. [20] researched a path-tracking control of unknown model mainly based on an adaptive Kalman filter, which is used to find the Jacobian matrix of the continuum manipulator using only pressures and tip position. Tan et al. [21] adopted the zeroing neurodynamic theory to present a model-less trajectory tracking control method for continuum manipulators. The control of continuum robots has made great progress.

However, current research on motion control of continuum robots is based on fixed bases. In practical engineering applications, continuum robots are often mounted on moving platforms such as smart cars and flying machines. In this paper, the continuum arm is mounted under the space service robot base. The coupling motion between the continuum robots and the space-moving base has to be considered. Current model-free kinematic tracking control methods mainly consider optimality and hardly pay attention to the robustness of the control methods. As a result, they will no longer be applicable to scenarios with complex coupled motion. Hence, it is important to investigate a more effective, higher precision, and more robust control method for space in-cabin robots with a pneumatic continuum arm.

This paper researches the trajectory tracking control of unknown models for space in-cabin robots with a pneumatic continuum arm based on the zeroing neural network. In this method, the high precision tracking control can be achieved only by the actuator input, sensory output, and the attitude of base information, when the coupling kinematic model is unknown. Moreover, PSAF is employed to overcome the influence of base coupling motion. The main contributions in this paper are listed as the following facts:

• The scheme of the micro-nano robot with a pneumatic continuum arm in space station is proposed for the first time, and the coupling kinematic model between the pneumatic continuum arm and the moving base is established.

• A tracking control method of unknown models based on modified-function-activated ZNN is proposed for solving the coupling motion problem of the moving base and the pneumatic continuum arm.

• The convergence, stability and simulation results are proved and presented. The accuracy and the robustness of trajectory tracking are greatly improved by employing the modified activation function PSAF.

The rest of this paper is organized in four sections. In Modeling the space continuum robots, the motion model of space in-cabin robots with a pneumatic continuum arm is systematically studied. In Trajectory tracking controller of unknown model design, a robust tracking control method of unknown model based on zeroing neural network is presented. The next section presents the Simulation results. Finally, Conclusion generalizes the work of this paper.

Modeling the space continuum robots

Scenario description

The target trajectory motion of space in-cabin robots with a pneumatic continuum arm is researched in this paper. This motion can represent a typical application scenario for the space station in-cabin robots. Specifically, the continuum robot of the space station in-cabin has a pneumatic continuum soft arm mounted on a micro-nano robot base, and the arm has two sections consisting of three bellows, as shown in Fig. 2.

Fig. 2

The model of space in-cabin robots with a pneumatic continuum arm

To demonstrate the kinematics of space in-cabin robots with a pneumatic continuum arm, the following series of coordinate systems are defined [22]. $\sum I$ is the inertial coordinate system, $\sum B$ denotes the base coordinate system, $\sum G$ is the center of space robot system mass coordinate [23], $\sum 1, \, \sum 2, \, \sum 3$ are some local coordinate systems of each section continuum arm, in Fig. 2. For convenience, the mathematical notations and abbreviations are defined as follows in this paper.

$m_{0} ,m_{1} ,m_{2} \in R^{1}$ the mass of the space robot base and each section continuum arm.

${\varvec{P}}_{e} \in {\mathbb{R}}^{3}$ the position vector of the end-effector in $\sum I$.

${\varvec{r}}_{0} \in {\mathbb{R}}^{3}$ the position vector for the center of mass (CM) of the base in $\sum I$.

${\varvec{r}}_{i} \in {\mathbb{R}}^{3} , \, i \in 1,2$ the position vector for the CM of the ith section in $\sum I$.

${\varvec{r}}_{g} \in {\mathbb{R}}^{3}$ the position vector for the CM of the space robot system in $\sum I$.

${\varvec{b}}_{0} \in {\mathbb{R}}^{3}$ the position vector from the CM of the base in $\sum I$.

${\varvec{q}} \in {\mathbb{R}}^{6}$ the state vector of actuators. $q_{ij}$ represents the length of the jth bellow in the ith section.

$\alpha ,\beta ,\theta \in {\mathbb{R}}$ the attitude angle of the base.

$\varphi_{i} \in {\mathbb{R}}^{1}$ the deformation orientation of the ith section.

$\kappa_{i} \in {\mathbb{R}}^{1}$ the curvature of the ith section.

$l_{i} \in {\mathbb{R}}^{1}$ the backbone length of the ith section.

${\varvec{n}}_{e} \in {\mathbb{R}}^{3}$ the unit orientation vector of the end-effector.

$\alpha_{1} , \, \alpha_{2} \in {\mathbb{R}}^{1}$ the orientation angle of the end-effector.

${\varvec{T}}_{i}^{i + 1} \in {\mathbb{R}}^{4 \times 4}$ the transformation matrix from $\sum i$ to $\sum i + 1$.

${\varvec{R}}_{i}^{i + 1} \in {\mathbb{R}}^{3 \times 3}$ the rotation matrix from $\sum i$ to $\sum i + 1$.

${\varvec{O}}_{i}^{i + 1} \in {\mathbb{R}}^{3 \times 1}$ the translation matrix from $\sum i$ to $\sum i + 1$.

$\gamma$ the iteration parameter of ZNN.

${\mathbb{F}}( \cdot )$ the activation functions of ZNN.

$f( \cdot )$ the nonlinear mapping describing the kinematic of robots.

${\varvec{q}}(t), \, \dot{\user2{q}}(t), \, \user2{\ddot{q}}(t) \in {\mathbb{R}}^{6}$ the length of actuators at time t, the derivative, and the second derivative.

${\varvec{P}}_{a} (t) \in {\mathbb{R}}^{5}$ including the actual position vector and the actual orientation vector of the end-effector.

${\varvec{P}}_{d} (t) \in {\mathbb{R}}^{5}$ including the desired position vector and the desired orientation vector of the end-effector.

${\varvec{J}}({\varvec{u}}(t)), \, {\varvec{J}}(t) \in {\mathbb{R}}^{5 \times 6}$ the Jacobian matrix of robots.

ZNN Zeroing neural network.

LAF Linear activation functions.

PSAF Power-sigmoid activation function.

RMSE Root mean squared error

Kinematic modeling

The kinematic model of space in-cabin robots with a pneumatic continuum arm is illuminated. The research uses a bellows-driven soft continuum arm mounted on a micro robot base in this paper. The kinematics model can be represented by two mappings based on the theory of constant curvature [24], as shown in Fig. 3. The first mapping denotes the transformation from the actuator state to the configuration state, and the second mapping denotes the transformation from the configuration state to the end-effector’s state. The task space here can be viewed as an expression of an inertial coordinate system.

Fig. 3

The mapping relation in the kinematics of the bellows-driven soft continuum arm based on constant curvature

In actuator space, ${\varvec{q}} \in R^{6}$ represents the state of actuators consisting of six bellows. Each section consists of three bellows. Two sections in series are connected as a pneumatic continuum arm. Therefore $\user2{q = }[q_{11} ,q_{12} ,q_{13} ,q_{21} ,q_{22} ,q_{23} ]^{{\text{T}}}$ denotes the state of actuators, and $q_{ij}$ denotes the state of the jth bellow in the ith section. The configuration space is denoted by a triplet $(\varphi_{i} , \, \kappa_{i} , \, l_{i} )$. The items of $(\varphi_{i} , \, \kappa_{i} , \, l_{i} )$ denote the orientation of the movement, bending curvature, and center point length of the ith section respectively.

The end-effector’s position vector can be defined as ${\varvec{P}}_{e} { = [}x, \, y, \, z{]}$ in the inertial coordinate system. Besides, the end-effector’s orientation can be represented by a unit vector ${\varvec{n}}_{e} { = [}n_{x} , \, n_{y} , \, n_{z} {]}$. Since continuum robots do not twist, the end-effector’s orientation movement can be expressed using two azimuth angles $\alpha_{1} , \, \alpha_{2}$, as shown in Fig. 4.

$$ \begin{gathered} \alpha_{1} = \arctan (n_{y} /n_{x} ) \hfill \\ \alpha_{2} = \arctan (n_{z} /\sqrt {n_{x}^{2} + n_{y}^{2} } ) \hfill \\ \end{gathered} $$

(1)

Fig. 4

The orientation of the end-effector

Since the end-effector’s position and orientation are denoted as a vector ${\varvec{P}}_{a} = [x,y,z,\alpha_{1} ,\alpha_{2} ]$.

The mapping between the actuator state and the configuration state of the pneumatic continuum arm’s ith section can be denoted as follows:

$$ \begin{gathered} \varphi_{i} = \arctan 2\left( {3(q_{i2} - q_{i3} ),\sqrt 3 (q_{i2} + q_{i3} - 2q_{i1} )} \right) \hfill \\ \kappa_{i} = \frac{{2\sqrt {q_{i1}^{2} + q_{i2}^{2} + q_{i3}^{2} - q_{i1} q_{i2} - q_{i2} q_{i3} - q_{i3} q_{i1} } }}{{d(q_{i1} + q_{i2} + q_{i3} )}} \hfill \\ l_{i} = \frac{{q_{i1} + q_{i2} + q_{i3} }}{3} \hfill \\ \end{gathered} $$

(2)

where d is the radius of the pneumatic continuum arm as shown in Figs. 5, 6.

Fig. 5

The configuration space of the continuum arm’s ith section

Fig. 6

The sectional view of the continuum arm’s ith section

The mapping relationship between the configuration space and task space is represented by coordinate transformation as illustrated in Fig. 2. The relative pose and position between coordinate frames $\sum i$ and $\sum i + 1$ can be described by several homogeneous transformation matrices ${\varvec{T}}_{i}^{i + 1} \in {\mathbb{R}}^{4 \times 4}$ including a rotation matrix ${\varvec{R}}_{i}^{i + 1}$ and a translation vector ${\varvec{O}}_{i}^{i + 1}$, as follows:

$$ {\varvec{T}}_{i}^{i + 1} = \left[ {\begin{array}{*{20}c} {{\varvec{R}}_{i}^{i + 1} } & {{\varvec{O}}_{i}^{i + 1} } \\ {0_{1 \times 3} } & 1 \\ \end{array} } \right] $$

(3)

where $R_{i}^{i + 1}$ is the rotation matrix of the continuum arm. For the sake of convenience, c and s denote cosine and sine functions respectively in this paper.

$$ {\varvec{R}}_{i}^{i + 1} = \left[ {\begin{array}{*{20}c} {c^{2} \varphi_{i} (c(\kappa_{i} l_{i} ) - 1) + 1} & {s\varphi_{i} c\varphi_{i} (c(\kappa_{i} l_{i} ) - 1)} & {c\varphi_{i} s(\kappa_{i} l_{i} )} \\ {s\varphi_{i} c\varphi_{i} ({\text{c}} (\kappa_{i} l_{i} ) - 1)} & {c^{2} \varphi_{i} (1 - c(\kappa_{i} l_{i} )) + c(\kappa_{i} l_{i} )} & {s\varphi_{i} s(\kappa_{i} l_{i} )} \\ { - c_{i} \varphi_{i} s(\kappa_{i} l_{i} )} & { - s\varphi_{i} s(\kappa_{i} l_{i} )} & {c(\kappa_{i} l_{i} )} \\ \end{array} } \right] $$

(4)

where ${\varvec{O}}_{i}^{i + 1}$ is translation vector as follows:

$$ {\varvec{O}}_{i}^{i + 1} = \left[ {\begin{array}{*{20}c} {\frac{{c\varphi_{i} (1 - c(\kappa_{i} l_{i} ))}}{{\kappa_{i} }}} & {\frac{{s\varphi_{i} (1 - c(\kappa_{i} l_{i} ))}}{{\kappa_{i} }}} & {\frac{{s(\kappa_{i} l_{i} )}}{{\kappa_{i} }}} \\ \end{array} } \right] $$

(5)

Considering the moving base, ${\varvec{R}}_{I}^{B}$ is the rotation matrix of the micro-nano robot base as follows:

$$ {\varvec{R}}_{I}^{B} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {c\alpha } & { - s\alpha } \\ 0 & {s\alpha } & {c\alpha } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {c\beta } & 0 & {s\beta } \\ 0 & 1 & 0 \\ { - s\beta } & 0 & {c\beta } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {c\theta } & { - s\theta } & 0 \\ {s\theta } & {c\theta } & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

(6)

where $\alpha , \, \beta , \, \theta$ denote the attitude angle of the base which are controlled and known. The translation vector ${\varvec{O}}_{I}^{B}$ is defined as

$$ {\varvec{O}}_{I}^{B} = {\varvec{r}}_{0} $$

(7)

where ${\varvec{r}}_{0}$ is the position vector of CM of the base in $\sum I$. Hence the transformation matrix ${\varvec{T}}_{I}^{B}$ is defined as

$$ {\varvec{T}}_{I}^{B} { = }\left[ {\begin{array}{*{20}c} {{\varvec{R}}_{I}^{B} } & {{\varvec{O}}_{I}^{B} } \\ {0_{1 \times 3} } & 1 \\ \end{array} } \right] $$

(8)

Then the transformation matrix ${\varvec{T}}_{B}^{1}$ as illustrated in Fig. 2. can be described as follows

$$ {\varvec{T}}_{B}^{1} = \left[ {\begin{array}{*{20}c} {I_{3 \times 3} } & {{\varvec{b}}_{0} } \\ {0_{1 \times 3} } & 1 \\ \end{array} } \right] $$

(9)

Considering the major characteristic of space continuum robots, the kinematic of the space continuum robots is coupled to the base in the space station microgravity environment. Since there is no external force applied to the entire system, ${\varvec{r}}_{g}$ remains constant [25]. Then, the geometrical definition of $\sum G$ is introduced [26], and thus an additional equation arises as

$$ \begin{gathered} {\varvec{r}}_{1} = {\varvec{r}}_{0} + {\varvec{R}}_{I}^{B} {\varvec{b}}_{0} + {\varvec{R}}_{I}^{B} {\varvec{R}}_{B}^{1} {\varvec{O}}_{1}^{{m_{1} }} \hfill \\ {\varvec{r}}_{2} = {\varvec{r}}_{0} + {\varvec{R}}_{I}^{B} {\varvec{b}}_{0} + {\varvec{R}}_{I}^{B} {\varvec{R}}_{B}^{1} {\varvec{O}}_{1}^{2} \user2{ + R}_{I}^{B} {\varvec{R}}_{B}^{1} {\varvec{R}}_{1}^{2} {\varvec{O}}_{2}^{{m_{2} }} \hfill \\ \end{gathered} $$

(10)

To simplify the model, the ${\varvec{r}}_{g}$ holds zero in $\sum I$. According to the geometric features of the system as illustrated in Fig. 2, ${\varvec{r}}_{1} , \, {\varvec{r}}_{2}$ can be described as follows

$$ m_{0} {\varvec{r}}_{0} + m_{1} {\varvec{r}}_{1} + m_{2} {\varvec{r}}_{2} = {\varvec{r}}_{g} (m_{0} + m_{1} + m_{2} ) $$

(11)

where regard the position of ${{l_{i} } \mathord{\left/ {\vphantom {{l_{i} } 2}} \right. \kern-0pt} 2}$ as the CM of the ith section. Therefore, ${\varvec{O}}_{i}^{{m_{i} }} , \, i = 1,2$ can be presented as follows:

$$ {\varvec{O}}_{i}^{{m_{i} }} = \left[ {\begin{array}{*{20}c} {\frac{{c\varphi_{i} (1 - c({{\kappa_{i} l_{i} } \mathord{\left/ {\vphantom {{\kappa_{i} l_{i} } 2}} \right. \kern-0pt} 2}))}}{{\kappa_{i} }}} & {\frac{{s\varphi_{i} (1 - c({{\kappa_{i} l_{i} } \mathord{\left/ {\vphantom {{\kappa_{i} l_{i} } 2}} \right. \kern-0pt} 2}))}}{{\kappa_{i} }}} & {\frac{{s({{\kappa_{i} l_{i} } \mathord{\left/ {\vphantom {{\kappa_{i} l_{i} } 2}} \right. \kern-0pt} 2})}}{{\kappa_{i} }}} \\ \end{array} } \right] $$

(12)

Substituting ${\varvec{r}}_{g} \equiv 0$, Eq. (11) and Eq. (12) into Eq. (10), ${\varvec{r}}_{0}$ can be inferred

$$ {\varvec{r}}_{0} = - \frac{{m_{1} ({\varvec{R}}_{I}^{B} {\varvec{b}}_{0} + {\varvec{R}}_{I}^{B} {\varvec{R}}_{B}^{1} {\varvec{O}}_{1}^{{m_{1} }} )}}{{m_{0} + m_{1} + m_{2} }} - \frac{{m_{2} \user2{(R}_{I}^{B} {\varvec{b}}_{0} + {\varvec{R}}_{I}^{B} {\varvec{R}}_{B}^{1} {\varvec{O}}_{1}^{2} \user2{ + R}_{I}^{B} {\varvec{R}}_{B}^{1} {\varvec{R}}_{1}^{2} {\varvec{O}}_{2}^{{m_{2} }} )}}{{m_{0} + m_{1} + m_{2} }} $$

(13)

Therefore, the end-effector’s position vector of in $\sum I$ can be expressed as follows:

$$ \left[ {\begin{array}{*{20}c} {{\varvec{P}}_{e} } \\ 1 \\ \end{array} } \right] = {\varvec{T}}_{I}^{B} {\varvec{T}}_{B}^{1} {\varvec{T}}_{1}^{2} {\varvec{T}}_{2}^{3} \left[ {\begin{array}{*{20}c} {0_{3 \times 1} } \\ 1 \\ \end{array} } \right] $$

(14)

The direction vector of the end-effector ${\varvec{n}}_{e} = {[}n_{x} , \, n_{y} , \, n_{z} {]}$ can be calculated as follows:

$$ {\varvec{n}}_{e} = {\varvec{R}}_{I}^{B} {\varvec{R}}_{B}^{1} {\varvec{R}}_{1}^{2} {\varvec{R}}_{2}^{3} \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 1 \\ \end{array} } \right] $$

(15)

The orientation angle $\alpha_{1} , \, \alpha_{2}$ can be calculated according to Eq. (1) and ${\varvec{n}}_{e}$.

In this paper, a tracking control method of unknown model is investigated. But it is worth noting that we need to build the kinematic model of space in-cabin robots with a pneumatic continuum arm when need to do some simulations. In the actual space in-cabin robot system, it only needs to acquire the actuator input, sensory output, and the attitude of base information.

Trajectory tracking controller of unknown model design

In this section, a tracking control method of unknown models is introduced. This method based on ZNN is developed to solve the high precision kinematics trajectory tracking control problem of unknown models for space in-cabin robots with a pneumatic continuum arm.

Controller design

Generally, we only consider the control of bellow actuators, when the attitude of the base is controlled and invariant. The kinematic model of space in-cabin robots with a pneumatic continuum arm can be expressed as follows:

$$ {\varvec{P}}_{a} (t) = f({\varvec{q}}(t)) $$

(16)

where ${\varvec{P}}_{a} (t) \in {\mathbb{R}}^{5}$ represents the actual trajectory of the end-effector, ${\varvec{q}}(t) \in {\mathbb{R}}^{6}$ represents the length of the bellow actuators, and $f( \cdot )$:${\mathbb{R}}^{6}$$\to$${\mathbb{R}}^{5}$ is a kinematic mapping. By differentiating Eq. (16), we can obtain

$$ \dot{\user2{P}}_{a} (t) = {\varvec{J}}(t)\dot{\user2{q}}(t) $$

(17)

where ${\varvec{J}}(t) = {{\partial f({\varvec{q}}(t))} \mathord{\left/ {\vphantom {{\partial f({\varvec{q}}(t))} {\partial {\varvec{q}}(t)}}} \right. \kern-0pt} {\partial {\varvec{q}}(t)}} \in {\mathbb{R}}^{n \times m}$ is the unknown Jacobian matrix of the continuum arm, $\dot{\user2{P}}_{a} (t)$ denotes the actual trajectory velocity of the end-effector, $\dot{\user2{q}}(t)$ represents the change rate of the actuators.

To realize the tracking control problem solving of the unknown model, ${\varvec{q}}(t)$ is generated to drive the continuum arm along the desired path and the desired orientation. For the first error function, we define

$$ {\varvec{e}}(t) = {\varvec{P}}_{d} (t) - {\varvec{P}}_{a} (t) \in {\mathbb{R}}^{5} $$

(18)

Then, using the ZNN $\dot{\user2{e}}(t) = - \gamma {\mathbb{F}}({\varvec{e}}(t))$ can make the error function converge to zero [27], we can obtain

$$ \dot{\user2{P}}_{d} (t) - \dot{\user2{P}}_{a} (t) = - \gamma {\mathbb{F}}({\varvec{P}}_{d} (t) - {\varvec{P}}_{a} (t)) $$

(19)

where $\dot{\user2{P}}_{d} (t)$ is the velocity of the desired trajectory, $\gamma \in {\mathbb{R}}^{ + }$ is a constant parameter to determine the rate of convergence, and ${\mathbb{F}}( \cdot )$ denotes the activation function [32]. Then, substituting Eq. (17) into Eq. (19), we have

$$ \dot{\user2{P}}_{d} (t) - \dot{\user2{P}}_{a} (t) = - \gamma {\mathbb{F}}({\varvec{P}}_{d} (t) - {\varvec{P}}_{a} (t)) $$

(20)

the solution of Eq. (20) is rewritten as

$$ \dot{\user2{q}}(t) = {\varvec{J}}^{\dag } (t)(\dot{\user2{P}}_{d} (t) + \gamma {\mathbb{F}}({\varvec{P}}_{d} (t) - {\varvec{P}}_{a} (t))) $$

(21)

where ${\varvec{J}}^{\dag } (t)$ is the pseudo-inverse of ${\varvec{J}}(t)$. In this paper, we employ two different activation functions including a linear activation function and a modified activation function.

• (1) Linear activation function (LAF)

The linear activation function is presented as follows:

$$ {\mathbb{F}}(x) = x $$

(22)

• (2) Power-sigmoid activation function (PSAF) [29]

The power-sigmoid activation function is nonlinear, as follows:

$$ {\mathbb{F}}(x) = \left\{ {\begin{array}{*{20}ll} x^{p} \, & \quad \left| x \right| \ge 1 \\ \frac{1 + \exp ( - \xi )}{{1 - \exp ( - \xi )}} \cdot \frac{1 - \exp ( - \xi x)}{{1 + \exp ( - \xi x)}}& \quad \left| x \right| < 1 \\ \end{array} } \right. $$

(23)

where $p \ge 3$ is an odd number and $\xi \ge 2$.

The pseudo-inverse of ${\varvec{J}}(t)$ should be calculated firstly if we want to solve the differential Eq. (21). ${\varvec{P}}_{d} (t)$ and $\dot{\user2{P}}_{d} (t)$ are the desired trajectory and the velocity of the desired trajectory, respectively. And ${\varvec{P}}_{a} (t)$ can be acquired by sensor metrical data. Since the kinematics of the continuum arm is unknown, we must estimate the real-time Jacobian matrix ${\varvec{J}}(t)$.

The second error function is proposed to estimate the Jacobian matrix, it is:

$$ {\varvec{\varepsilon}}(t) = \dot{\user2{P}}_{a} (t) - {\varvec{J}}(t)\dot{\user2{q}}(t) \in {\mathbb{R}}^{5} $$

(24)

Utilize the ZNN design formula $\dot{\user2{\varepsilon }}(t) = - \gamma F(\varepsilon (t))$ again and we obtain

$$ \user2{\ddot{P}}_{a} (t) - \dot{\user2{J}}(t)\dot{\user2{q}}(t) - {\varvec{J}}(t)\user2{\ddot{q}}(t) = - \gamma (\dot{\user2{P}}_{a} (t) - {\varvec{J}}(t)\dot{\user2{q}}(t)) $$

(25)

The above equation can be rewritten as

$$ \dot{\user2{J}}(t) = (\user2{\ddot{P}}_{a} (t) - {\varvec{J}}(t)\user2{\ddot{q}}(t) + \gamma (\dot{\user2{P}}_{a} (t) - {\varvec{J}}(t)\dot{\user2{q}}(t)))\dot{\user2{q}}^{\dag } (t) $$

(26)

where $\dot{\user2{q}}^{\dag } (t)$ is the pseudo-inverse of $\dot{\user2{q}}(t)$, $\dot{\user2{P}}_{a} (t)$ and $\user2{\ddot{P}}_{a} (t)$ can be acquired by sensor measurement information, $\user2{\ddot{q}}(t)$ is calculated based calculus of differences, as $\user2{\ddot{q}}(t) = (\dot{\user2{q}}(t) - \dot{\user2{q}}(t - \Delta t))/\Delta t$. Then, ${\varvec{J}}(t)$ and ${\varvec{J}}^{\dag } (t)$ can be calculated by solving numerical differential Eq. (26).

Consider that there is always some noise in practical engineering. Therefore, add the noise term $\zeta (t) \in {\mathbb{R}}^{5}$ to the first ZNN design formula as follows:

$$ \dot{\user2{e}}(t) = - \gamma {\mathbb{F}}({\varvec{e}}(t)) + \zeta (t) $$

(27)

Then, Eq. (21) will be written as follows:

$$ \dot{\user2{q}}(t) = {\varvec{J}}^{\dag } (t)(\dot{\user2{P}}_{d} (t) + \gamma {\mathbb{F}}({\varvec{P}}_{d} (t) - {\varvec{P}}_{a} (t)) + \zeta (t)) $$

(28)

By solving Eqs. (26) and (28), we can obtain the ${\varvec{q}}(t)$ and ${\varvec{J}}(t)$. The pneumatic continuum arm is actuated by controlling ${\varvec{q}}(t)$. The complete control architecture of the tracking control system based on ZNN is shown in Fig. 7.

Fig. 7

The complete control architecture of the tracking control system based on ZNN

Theoretical analysis

In this part, the convergence and the stability of the proposed control mothed of unknown model based on ZNN are analyzed by the following theorems.

(a) Convergence analysis

The convergence of the ZNN method will be proved by theorem 1 and theorem 2.

Theorem 1.

To realize the trajectory tracking control of unknown model for the in-cabin robots with a pneumatic continuum arm as Eq. (16), the error of the end-effector’s actual trajectory ${\varvec{P}}_{a} (t)$ and the desired trajectory ${\varvec{P}}_{d} (t)$ will converge to zero when using the LAF Eq. (22).

Proof.

First, define a Lyapunov function candidate as follows.

$$ V_{1} = \frac{{\left\| {{\varvec{e}}(t)} \right\|^{2} }}{2} = \frac{{{\varvec{e}}(t)^{{\text{T}}} {\varvec{e}}(t)}}{2} $$

(29)

where ${\varvec{e}}(t) = {\varvec{P}}_{d} (t) - {\varvec{P}}_{a} (t)$, $\left\| \cdot \right\|_{2}$ represents the Euclidean norm of the vector. We can easily obtain $V_{1} \ge 0$. Taking the time derivative of $V_{1} (t)$ and substituting Eq. (22) LAF, one can obtain

$$ \dot{V}_{1} = - \gamma {\varvec{e}}^{{\text{T}}} (t){\mathbb{F}}({\varvec{e}}(t)) = - \gamma \sum\limits_{j = 1}^{n} {|e_{j} (t)|^{2} } $$

(30)

where $e_{j} (t)$ represents the jth element of ${\varvec{e}}(t)$ and $j = 1,2...,n$. In addition $\gamma > 0$, $V_{1}$ is obvious negative-definite and ${\varvec{e}}(t)$ can converge to zero. Considering the ZNN formula of LAF $\dot{\user2{e}}(t) = - \gamma {\varvec{e}}(t)$, we can get

$$ {\varvec{e}}(t) = {\varvec{e}}(0)\exp ( - \gamma t) $$

(31)

Obviously, we have

$$ \mathop {\lim }\limits_{t \to \infty } {\varvec{e}}(t) = \mathop {\lim }\limits_{t \to \infty } {\varvec{e}}(0)\exp ( - \gamma t) = 0 $$

(32)

By substituting Eq. (18) into Eq. (32), we can obtain

$$ \mathop {\lim }\limits_{t \to \infty } ({\varvec{P}}_{d} (t) - {\varvec{P}}_{a} (t)) = 0 $$

(33)

Therefore, the above Eq. (33) can be represented as follows

$$ {\varvec{P}}_{d} (t) \to {\varvec{P}}_{a} (t), \, as \, t \to + \infty $$

(34)

Therefore, the actual trajectory ${\varvec{P}}_{a} (t)$ will be convergent to the desired trajectory ${\varvec{P}}_{d} (t)$. Thus, the proof of the theorem is completed.

Theorem 2.

To realize the trajectory tracking control of unknown model for the in-cabin robots with a pneumatic continuum arm as Eq. (16), the error of the end-effector’s actual trajectory ${\varvec{P}}_{a} (t)$ and the desired trajectory ${\varvec{P}}_{d} (t)$ will achieve the better convergent result when using the PSAF as Eq. (23), comparing with using LAF Eq. (22).

Proof.

Similarly, the Lyapunov function candidate $V_{1}$ can be used. For the PSAF as Eq. (23), the ZNN formula becomes $\dot{\user2{e}}(t) = - \gamma {\mathbb{F}}({\varvec{e}}(t))$, and we can obtain the time derivative of $V_{1} (t)$ as follow two cases [31].

(1) In the case of $\left| {e_{j} (t)} \right| \ge 1$, where $e_{j} (t)$ represents the jth element of ${\varvec{e}}(t)$ and $j = 1,2...,n$. The time derivative of $V_{1} (t)$ can be obtained as follows

$$ \dot{V}_{1} = - \gamma {\varvec{e}}^{{\text{T}}} (t){\mathbb{F}}({\varvec{e}}(t)) = - \gamma \sum\limits_{j = 1}^{n} {|e_{j} (t)|^{p + 1} } $$

(35)

where $p \ge 3$, $|e_{j} (t)|^{p + 1} \gg |e_{j} (t)|^{2}$. Hence PSAF has much faster convergence rate, when $\left| {e_{j} (t)} \right| \ge 1$. And the solution of $\dot{\user2{e}}(t) = - \gamma {\mathbb{F}}({\varvec{e}}(t))$ can be given as follows

$$ {\varvec{e}}(t) = {\varvec{e}}(0)\left\{ {(p - 1){\varvec{e}}^{p - 1} (0)\gamma t + 1} \right\}^{{ - \frac{1}{p - 1}}} $$

(36)

when $p = 3$, one can obtain

$$ {\varvec{e}}(t) = {\varvec{e}}(0)/\sqrt {2{\varvec{e}}^{2} (0)\gamma t + 1} $$

(37)

Obviously,

$$ \mathop {\lim }\limits_{t \to \infty } {\varvec{e}}(t) = 0 $$

(38)

Therefore, we can obtain

$$ {\varvec{P}}_{d} (t) \to {\varvec{P}}_{a} (t), \, as \, t \to + \infty $$

(39)

(2) In the case of $\left| {e_{j} (t)} \right| < 1$, the solution of $\dot{\user2{e}}(t) = - \gamma {\mathbb{F}}({\varvec{e}}(t))$ is given as follows

$$ e_{j} (t) = - \ln (1 + Z(t))/\xi $$

(40)

where

$$ Z(t) = \frac{\eta \exp ( - \xi \gamma t)}{2} - {\text{sgn}} (e_{j} (0))\sqrt {(1 + \frac{\eta \exp ( - \xi \gamma t)}{2}) - 1} $$

(41)

where $\eta$ is a constant $\eta = (\exp ( - \xi e_{j} (0)) - 1)^{2} /\exp ( - \xi e_{j} (0))$. Using the Taylor series expansion Eq. (40), we can obtain

$$ |e_{j} (t)| \le \left| - \frac{Z}{\xi } + \frac{{Z^{2} }}{(2\xi )} - \frac{{Z^{3} }}{(3\xi )} + \cdot \cdot \cdot \right| $$

(42)

when $|Z| < 1$, there exists $Z + Z^{2} + Z^{3} + \cdot \cdot \cdot = Z/(1 - Z)$. Substituting into Eq. (40) yields

$$ |e_{j} (t)| \le \left|\frac{{|Z| + |Z|^{2} + |Z|^{3} + \cdot \cdot \cdot }}{\xi }\right| = \frac{|Z|}{{(\xi (1 - |Z|))}} $$

(43)

Due to $|Z| < 1$, the error $e_{j} (t) > - \ln 2/\xi$. Equation (41) satisfies the inequality as follows

$$ |Z(t)| \le \frac{\eta \exp ( - \xi \gamma t)}{2} + \sqrt {\frac{{\eta^{2} \exp ( - 2\xi \gamma t) + 4\eta \exp ( - \xi \gamma t)}}{4}} \, \le \frac{{\eta + \sqrt {\eta^{2} + 4\eta } }}{2}\exp \left( - \frac{\xi \gamma t}{2}\right) \, $$

(44)

there exists $z_{0} = |Z(t)| < 1$, we have

$$ |e_{j} (t)| \le \frac{1}{{\xi (1 - z_{0} )}}|z| \le \frac{{\eta + \sqrt {\eta^{2} + 4\eta } }}{{2\xi (1 - z_{0} )}}\exp \left( - \frac{\xi \gamma t}{2}\right) \, $$

(45)

when $\xi \ge 2$, PSAF possesses much faster convergence rate for $e_{j} (t) > - \ln 2/\xi$, comparing with using LAF.

According to the above two cases, we can draw a conclusion that PSAF can achieve much faster convergence rate, as compared with using LAF. Thus, the proof of the theorem is completed.

(b) Stability analysis

Theorem 3.

Considering the trajectory tracking control of unknown model for the in-cabin robots with a pneumatic continuum arm, ${\varvec{e}}(t)$ and ${\varvec{\varepsilon}}(t)$ can converge to zero, hence the control method described in (21) and (26) is stable in the sense of Lyapunov.

Proof.

For the tracking control of the in-cabin robots with a pneumatic continuum arm described by (16), define a Lyapunov function candidate [32] as follows.

$$ V_{2} (t) = \frac{{\left\| {{\varvec{e}}(t)} \right\|_{2}^{2} + \left\| {{\varvec{\varepsilon}}(t)} \right\|_{2}^{2} }}{2} = \frac{{{\varvec{e}}^{{\text{T}}} (t){\varvec{e}}(t) + {\varvec{\varepsilon}}^{{\text{\rm T}}} (t){\varvec{\varepsilon}}(t)}}{2} $$

(46)

where ${\varvec{e}}(t) = {\varvec{P}}_{d} (t) - {\varvec{P}}_{a} (t)$ and ${\varvec{\varepsilon}}(t) = \dot{\user2{P}}_{a} (t) - {\varvec{J}}(t)\dot{\user2{q}}(t)$. $V(t) = 0$ only if ${\varvec{e}}(t) = 0$ and ${\varvec{\varepsilon}}(t) = 0$, $V(t) > 0$ in other cases. It is easily checked that the Lyapunov function Eq. (41) is positive-definite. Taking the time derivative of $V(t)$, one can obtain

$$ \dot{V}_{2} (t) = {\varvec{e}}^{{\text{T}}} (t)\frac{{{\varvec{e}}(t)}}{dt} + {\varvec{\varepsilon}}^{{\text{\rm T}}} (t)\frac{{{\varvec{\varepsilon}}(t)}}{dt} = - \gamma {\varvec{e}}^{{\text{T}}} (t){\mathbb{F}}({\varvec{e}}(t)) - \gamma {\varvec{\varepsilon}}^{{\text{\rm T}}} (t){\mathbb{F}}({\varvec{\varepsilon}}(t)) $$

(47)

Both LAF and PSAF are odd functions and satisfy as follows

$$ {\mathbb{F}}(x)\left\{ {\begin{array}{*{20}c} { \ge 0, \, x \ge 0} \\ { < 0, \, x < 0} \\ \end{array} } \right. $$

(48)

In addition $\gamma > 0$, $\dot{V}(t)$ is obvious negative-definite. Therefore, the proposed tracking control method of the known model can be proved to be stable. Thus, the proof of this theorem is completed.

Remark 1.

It is worth noting that the work of this paper is based on simulations which means $\dot{\user2{P}}_{a} (t)$ and $\user2{\ddot{P}}_{a} (t)$ are easily obtained. However, in the actual applications, the parts are difficult to obtain directly. Therefore, when the method is applied in practice, we need to consider the ability and efficiency of the sensor to acquire velocity and acceleration.

Simulation results

In this section, the numerical simulations are conducted for two different reference trajectories to verify the effectiveness and robustness of the unknown models tracking control method. Besides, the effect of parameter variation on tracking error is analyzed by the simulations. Finally, the performance of two different activation functions is compared including LAF and PSAF. Some parameters of LAF and PSAF are given in Table 1. The simulation environment is performed in Windows 11 using Matlab2021a. The CPU of the computer is 11th Gen Intel(R) Core(TM) i5-11,400.

Table 1 Activation functions parameter

In the simulation, the radius of the continuum arm is set as $d = 0.05{\text{m}}$. The mass of the space robot base and each section's continuous arm are chosen $m_{0} = 5kg$, $m1 = m2 = 0.5kg$. The attitude angles of the base are controlled and invariant, which are assumed as $\alpha ,\beta ,\theta = 0$. The initial value of the Jacobian matrix ${\varvec{J}}(0)$ can be computed as ${\varvec{J}}_{i} (0) = \Delta {\varvec{P}}_{ai} /\Delta q$ by driving each actuator independently. The state of the bellows is chosen ${\varvec{q}}(0) = [200,210,190,200,210,190][200,210,190,200,210,190]$ at the beginning time.

Without loss of generality, we define a circular path and a clover path in the three-dimensional space. Then we take the circular trajectory of the 3D space as followed:

$$ {\varvec{P}}_{dX} (t) = k_{1} \cos \left( {2\pi \sin^{2} (0.5\pi t/Td)} \right) - k + 0.0875 $$

$$ {\varvec{P}}_{dY} (t) = k_{1} \cos (\pi /6)\sin \left( {2\pi \sin^{2} (0.5\pi t/T_{d} )} \right) $$

$$ {\varvec{P}}_{dZ} (t) = k_{1} \sin (\pi /6)\sin \left( {2\pi \sin^{2} (0.5\pi t/T_{d} )} \right) + 0.4362 $$

where $k_{2} = 0.05$ m is the radius of the space's circular path. Then the desired orientation angle is set as ${\varvec{P}}_{{d\alpha_{1} }} (t) = 0$ rad,${\varvec{P}}_{{d\alpha_{2} }} (t) = 1.11$ rad. The task duration of the circular trajectory is $T_{d} = 20s$. Another reference trajectory of the clover shape is adopted as followed:

$$ {\varvec{P}}_{dX} (t) = k_{2} \sin (3\pi t/T_{d} )\cos (\pi t/T_{d} ) + 0.0875 $$

$$ {\varvec{P}}_{dY} (t) = k_{2} \sin (3\pi t/T_{d} )\sin (\pi t/T_{d} ) $$

$$ {\varvec{P}}_{dZ} (t) = 0.4362 $$

where $k_{1} = 0.02$ m. Then the desired orientation angle is set as ${\varvec{P}}_{{d\alpha_{1} }} (t) = 0$ rad,${\varvec{P}}_{{d\alpha_{2} }} (t) = 1.11$ rad. The task duration of the circular trajectory is also $T_{d} = 20s$.

The simulation results are shown in Fig. 8. It shows the trajectory tracking results of the circular path without noise when employing LAF. As we can see, Fig. 8a illustrates the circular trajectory tracking control in the three-dimensional space. The changes of the bellows' length are shown in Fig. 8b. Figure 8c illustrates the displacement of the base in the trajectory tracking process. The continuum robots bend motion, which causes the base's center of mass to change. Figure 8d, e depict the position error and the orientation error of the continuum arm end-effector respectively. It can be seen from Fig. 8d that the position error does not exceed 1 mm. In addition, Fig. 8e shows that the orientation error does not exceed $2 \times 10^{ - 3} {\text{rad}}$, which can be negligible.

Fig. 8

The simulations result of the proposed method for tracking the circular path without noise when employing LAF

The simulation results are shown in Fig. 9. It shows the trajectory tracking results of the clover path without noise when employing LAF. As we can see, Fig. 9a illustrates the circular trajectory tracking control in the three-dimensional space. The changes of the bellows' length are shown in Fig. 9b. Figure 9c illustrates the displacement of the base in the trajectory tracking process. The continuum robots bend motion, which causes the base's center of mass to change. Figure 9d, e depict the position error and the orientation error of the continuum arm end-effector respectively. It can be seen from Fig. 9d that the position error does not exceed 0.5 mm. In addition, Fig. 9e shows that the orientation error does not exceed $4 \times 10^{ - 4} {\text{rad}}$, which can be negligible. All bellows vary within a reachable range of lengths in different reference trajectories. Through the above analysis, the feasibility and accuracy of the tracking control method can be verified.

Fig. 9

The simulations result of the proposed method for tracking the clover path without noise when employing LAF

Here the effect of parameter variations is explored. We choose different γ values for the simulation including $\gamma = 1,\gamma = 3,\gamma = 5,\gamma = 10$ and make a comparison employing LAF and PSAF. Here the uniform is tracking the circle trajectory. As shown in Fig. 10, it shows that $\left\| {e_{p} (t)} \right\|_{2}$ and $\left\| {e_{o} (t)} \right\|_{2}$ decreases as γ increases where $\left\| {e_{p} (t)} \right\|_{2}$ and $\left\| {e_{o} (t)} \right\|_{2}$ denote the norm of the position error vector and the norm of the orientation error vector respectively [30]. In Table 2, we can obtain the same conclusion according to the RMSE results. And they are equally applicable to LAF and PSAF. However, as γ becomes larger, it increases the number of iterations and the amount of computation of the algorithm. In terms of this feature, γ should be selected appropriately for the simulative purpose. Therefore, the parameters γ are selected as Table 1 by default in the later simulations.

Fig. 10

The comparison of the tracking error for different parameters when tracking the circular path

Table 2 Position and orientation RMSE for different parameters when tracking the circular path

From Table 2, we can also obtain that the RMSE of the position and the orientation employing PSAF is much smaller compared to LAF. Then the trajectory tracking result employing two activation functions is compared without noise in Fig. 11. Similarly, Fig. 11a illustrates the desired trajectory, the actual trajectory based LAF and the actual trajectory based PSAF. It can be seen that the actual trajectory based PSAF is closer to the desired trajectory than the actual trajectory based LAF. Figure 11b, d show the tracking $\left\| {e_{p} (t)} \right\|_{2}$ and $\left\| {e_{o} (t)} \right\|_{2}$ with two activation functions respectively. Figure 11c, e illustrate the boxplot of the trajectory tracking error with different activation functions of LAF and PSAF. By comparison, it can be found that PSAF is more accurate to track the desired trajectory than LAF both in terms of position and orientation.

Fig. 11

The comparison of the path trajectory tracking result without noise employing LAF and PSAF

Finally, the robustness of the method after adding different types of noise is verified by some simulations. Robustness analysis is discussed by comparing the performance of different activation functions under the same disturbance of noises. Firstly, Fig. 12 shows the trajectory tracking result with the sinusoidal noise $\xi = 0.005\sin (5\pi t/T_{d} )$ m employing LAF and PSAF. From Fig. 12a–e that the PSAF can achieve better robustness performance compared with the LAF in the case of the sinusoidal noise. And Fig. 13 illustrates the trajectory tracking result with temporal abrupt noise $\xi = 0.04$ m employing LAF and PSAF, which is similar in that the robustness of PSAF is superior. It is obvious that the position tracking result is more visible employing PSAF compared to the orientation because the orientation itself is smaller in magnitude. It shows that the RMSE of the tracking error is smaller employing PSAF for the same noise in Table 3. Therefore, the accuracy and the robustness of trajectory tracking are greatly improved by the modified activation function PSAF.

Fig. 12

The comparison of the path trajectory tracking result with sinusoidal noise employing LAF and PSAF

Fig. 13

The comparison of the path trajectory tracking result with temporal abrupt noise employing LAF and PSAF

Table 3 Position and orientation RMSE for different noises when tracking the circular path

Conclusion

In this paper, the robust tracking control method of unknown model is developed to solve the kinematic control problem for space in-cabin robots with a pneumatic continuum arm. By using the zeroing neural network, the robust tracking control method is proposed, which can estimate the time-varying Jacobian matrix with the unknown coupling kinematic model between the continuum arm and the space moving base. Then the modified activation function PSAF is employed to improve the tracking accuracy and the robustness of the controller. In addition, the theoretical analyses of the convergence and the stability are given for the proposed control approach. Finally, the simulation results are given to verify the effectiveness of the proposed control method for space in-cabin robots with a pneumatic continuum arm. Moreover, PSAF is indicated to be superior to LAF in tracking accuracy and robustness by comparison. In the future, we will build a real physical experiment platform to verify the effectiveness of the method in this paper.