Coordinated Trajectory Planning of Discrete-Serpentine Heterogeneous Multi-Arm Space Robot for Capturing Tumbling Targets Using Manipulability Optimization

Abstract

The discrete-serpentine heterogeneous multi-arm space robot (DSHMASR) has more advantages than single discrete space robots or single serpentine space robots in complex tasks of on-orbit servicing. However, the mechanical structure complexity of the DSHMASR poses challenges for modeling and motion planning. In this paper, a coupled kinematic model and a coordinated trajectory planning method for the DSHMASR were proposed to address these issues. Firstly, an uncontrolled satellite and the DSHMASR were modeled based on the momentum conservation law. The generalized Jacobian matrix J_g of the space robotic system was derived. Secondly, the manipulation capability of the DSHMASR was analyzed based on the null-space of J_g. Furthermore, the cooperative capturing-monitoring trajectory planning method for DSHMASR was presented through the manipulability optimization. The expected trajectory of each arm’s tip can be obtained by pose deviations and velocity deviations between the tip and the target point. Additionally, the optimized joint velocities of each arm were calculated by combining differential kinematics and manipulability optimization. Therefore, the manipulability of DSHMASR in the direction of the capture operation was enhanced simultaneously as it approached the target satellite. Finally, the proposed algorithm was demonstrated by establishing the Adams–Simulink co-simulation model. Comparisons with traditional approaches further confirm the outperformance of the proposed method in terms of manipulation capability.

1. Introduction

Capturing and removing space debris or malfunctional satellites pose critical challenges, especially under non-cooperative and tumbling conditions [1,2,3,4]. In this case, the space robotic system serves as one of the most useful ways for handling such uncontrolled space objects and protecting the Earth’s orbital environment [5]. Robinson [6] summarized existing robotic arms and classified them as discrete robots, serpentine robots and continuum robots, respectively. Thereinto, discrete manipulators are constructed by a series of rigid links, each of which is connected by a single degree of freedom joint. Serpentine robots also utilize discrete joints, but with much shorter rigid links connected by a large density of joints. Different from the above two types of robots, continuum robots have no rigid links and identifiable rotational joints. In fact, compared with continuum robots, discrete robots and serpentine robots are more widely used for the on-orbit service (OOS) including maintenance, reuse or other operations on uncontrolled targets [7,8]. Therefore, the research on discrete arms and serpentine arms is more attractive to scholars.

The stiff connection grasp using space manipulators acts as an essential prerequisite for the OOS. In this way, scholars have made certain contributions to relevant issues of capturing uncontrolled space objects with discrete space manipulators. Papadopoulos and Dubowsky [9,10] presented the kinematic and dynamics modeling method for free-floating discrete arms (FFDAs) to analyze the differences between FFDAs and fixed-base discrete arms (FBDAs). Yoshida et al. [11] proposed a bias momentum method for FFDAs approaching a tumbling satellite with the minimization attitude deviation of the base, before FFDAs made contacts with the target. To grasp a non-cooperative satellite with unknown dynamic parameters, Aghili [12] introduced an optimal estimation and guidance scheme to acquire the trajectory and rendezvous point at the time of grasping. Zhang [13] proposed a configuration optimization method to decrease the maximum contact force of the FFDA, aiming to capture a tumbling target. In addition to a single discrete manipulator, the discrete dual-arm or discrete multi-arm have also received extensive attention due to their preferable reliability and stability in the aspect of uncontrolled satellite capture. Peng et al. [14] designed a hybrid Kalman filter that combined an extended Kalman filter and an unscented Kalman filter to achieve efficient parameters estimation for target satellites, which was applied in the coordinated trajectory planning method for the dual-arm to capture tumbling targets. Based on the “Area-Oriented Capture” concept, Xu et al. [15] proposed the dual-arm coordinated capture method with larger capture tolerance, shorter capture time and higher efficiency compared with fixed-point oriented capture methods. To minimize the base disturbance, Yan et al. [16] transformed the target satellite capture problem into a configuration optimization of the dual-arm, so as to generate the expected trajectory of end-effector. Recent studies mainly focus on the modeling, planning and control of discrete multi-arm space robots [17,18,19]. However, discrete robotic arms have two disadvantages. On the one hand, the large mass and inertia of manipulators can trigger strong coupling effects between the base and the manipulator. On the other hand, discrete arms fail to carry out precision operations such as precise monitoring and narrow space detection due to limited degrees of freedom and large structural dimensions.

Compared with discrete arms, cable-driven serpentine manipulators have excellent characteristics of electromechanical separation, high dexterity, light weight and small size. Therefore, they can not only improve the operational accuracy of space tasks that are not easily implemented by discrete arms, but also reduce the risk of collision between the arm and the target satellite [20]. Recently, some research has been conducted in aspects of design [21], trajectory planning [22] and control [23] of serpentine arms. Mu [24] discussed typical applications, key technologies, and grand challenges of hyper-redundant manipulators with multiple degrees of freedom and slender links, demonstrating superior dexterity and excellent operability for OOS. Xidias [25] proposed a trajectory planning method for a hyper-redundant manipulator that is based on optimal time theory, aiming to move the manipulator from the initial configuration to the final configuration in a complex 3D workspace. To perform monitoring and maintenance tasks in a confined space, Hu et al. [26] presented a pose-configuration simultaneous planning method based on an equivalent kinematics model of the space serpentine arm. In reference [27], an efficient dynamic model of serpentine manipulators through multi-body dynamics was established based on spatial operator algebra. Reference [28] further established a general framework for studying the stiffness modeling of cable-driven serpentine manipulators. The control strategy for serpentine robotic arms to accurately track a desired trajectory has also been investigated [29,30,31]. However, the aforementioned research mainly assumed that the base of serpentine manipulators remained in a fixed state. The scholars of CMU Biorobotics Lab [32] designed a serpentine robot and defined a body frame that is determined by the averaged position of the robot to describe the robot’s pose. Further, the serpentine robot [33] was used to assist with search and rescue efforts in the wake of the Mexico City earthquake in September 2017. In addition, there is little research on serpentine manipulators under free-floating conditions.

Heterogeneous robotic arms have been rarely studied up till now. Marangoz et al. [34] computed decentralized collision-aware inverse kinematics for heterogeneous multi-arm systems composed of a single 6-DOF arm and two 7-DOF arms using a method in which the trust region based on non-linear optimization was used to follow end-effector trajectories while avoiding collisions. Reference [35] proposed a new type of hybrid rigid-continuum dual-arm space robot (HRCDASR). A coordinated motion planning method was established based on the task priority for the HRCDASR to complete target operation. Wang et al. [36] presented a robotic arm that was capable of switching between a rigid arm and a continuum arm, and the experimental testing was carried out to validate the design, the kinematic model, and the motion performance of this arm. However, the above research is only applicable to ground scenes where the arm’s base is fixed. The desired trajectory tracking and manipulability optimization of heterogeneous robotic arms are rarely addressed.

Previously, discrete robotic arms ensured reliable capture operations with strong load capacity and superior stiffness, whereas serpentine robotic arms carried out complex tasks with excellent maneuverability and superior dexterity. However, most of the space robotic systems previously used for target satellite manipulation either focused solely on discrete arms or dedicated serpentine arms. Despite some research on the types of space robots with heterogeneous robotic arms, the analysis of manipulation capability as the core index of task operation is not considered in these methods.

Therefore, in this paper, a discrete-serpentine heterogeneous multi-arm space robot (DSHMASR) composed of both discrete arms and a serpentine arm was designed, aiming to deal with problems of cooperative capturing and monitoring of tumbling target satellites based on manipulability optimization of the DSHMASR. Discrete arms with the same mechanical structure were used as mission manipulators, enabling high-tolerance capturing and high-stiff locking for the target satellite, while the serpentine arm was regarded as an auxiliary arm to provide monitoring for the discrete dual-arm arms. Due to the involvement of two different kinds of arms, the new space robotic system is more complex than the traditional space robot in the kinematics modeling and coordinated planning. To overcome such challenges, the kinematics model of the DSHMASR was firstly established to obtain the generalized Jacobian matrix. The manipulation ability of the DSHMASR was then analyzed based on the null-space of the Jacobian matrix. Furthermore, the desired monitoring position of the serpentine arm was determined by the two capture points that are expected grasp positions for both discrete arms. Finally, the cooperative capturing-monitoring trajectory was generated for the DSHMASR by integrating differential kinematics with manipulability optimization.

The key contributions of this paper are as follows. Firstly, a new type of heterogeneous multi-arm space robot with a discrete dual-arm and a serpentine arm is proposed, which has many advantages in complex tasks of on-orbit servicing. Secondly, the coupled kinematics model of the multi-arm space robot is established. Based on this, its generalized Jacobian matrix is obtained. Moreover, the manipulation ability of the heterogeneous multi-arm space robot in the mission direction is analyzed by the above Jacobian matrix. The coordinated trajectory planning method based on manipulability optimization is proposed for grasping non-cooperative satellites. The proposed method not only ensures the successful capture of non-cooperative satellites, but also improves the manipulability in the direction of capture operation.

The remainder of this paper is structured as follows: Section 2 establishes the kinematics models of a tumbling target satellite and the DSHMASR. In Section 3, the manipulability measure of the DSHMASR is defined, and the manipulability analysis is conducted to improve the manipulation capability of mission arms. Based on manipulability optimization, Section 4 proposes the coordinated capturing-monitoring trajectory planning method for the DSHMASR. In Section 5, a co-simulation system is developed to verify the proposed method. The autonomous capturing-monitoring case using the DSHMASR is simulated. Based on the analysis of previous sections, Section 6 summarizes the work of this study.

2. Modeling of the Target Satellite and Space Robotic System

2.1. Dynamics of an Uncontrolled Target Satellite

The simplified model of an uncontrolled satellite is shown in Figure 1a. Here, {X_b, Y_b, Z_b} is the fixed coordinate system, and {X_m, Y_m, Z_m} represents the inertia principal axes system of the target satellite. For the sake of description, both are marked as Σ_b and Σ_m. Their origins are all set on the center (point C) of the target satellite and denoted by O_b. Assuming that the point C in the target satellite is the reference point, the angular momentum of the target satellite is expressed by

$$h^{\mathrm{C}}={\displaystyle {\int }_{\mathrm{B}}r\times \left(\omega \times r\right) \mathrm{d}m}=\left(A^{\mathrm{T}}\cdot {}^{\mathrm{b}}I_{\mathrm{t}}\cdot A\right)\cdot \omega$$

(1)

where A is the transformation matrix from Σ_b to the inertial coordinate frame. dm is the infinitesimal mass. Its angular velocity is denoted by ω. r is the position vector between the point C and the point where the infinitesimal mass is.

${}^{\mathrm{b}}I_{\mathrm{t}}$ is the inertial matrix of the target satellite expressed in the Σ_b, and it is written as follows:

$${}^{\mathrm{b}}I_{\mathrm{t}}=\left[\begin{array}{ccc}{I}_{xx}& -I_{xy}& -I_{xz}\\ -I_{yx}& I_{yy}& -I_{yz}\\ -I_{zx}& -I_{zy}& I_{zz}\end{array}\right]$$

(2)

I_xx, I_yy and I_zz are moments of inertia about the x-axis, y-axis and z-axis, respectively. I_xy, I_xz and I_yz are products of inertia. Differentiating both sides of Equation (1), the attitude dynamics model of the target satellite can be obtained, i.e.:

$$\left[\begin{array}{c}{M}_x\\ M_y\\ M_z\end{array}\right]=\left(A\cdot {}^{\mathrm{b}}I_{\mathrm{t}}\cdot A^{\mathrm{T}}\right)\left[\begin{array}{c}{\dot{\omega }}_x\\ {\dot{\omega }}_y\\ {\dot{\omega }}_z\end{array}\right] +\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]\times \left(A\cdot {}^{\mathrm{b}}I_{\mathrm{t}}\cdot A^{\mathrm{T}}\right)\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]$$

(3)

ω_x, ω_y and ω_z are angular velocity about the x-axis, y-axis and z-axis. Considering that ${}^{\mathrm{b}}I_{\mathrm{t}}$ is a diagonal matrix, there exists a matrix that can make its non-diagonal elements be zeros.

$$A_{\mathrm{b}\to \mathrm{m}}\left[\begin{array}{ccc}{I}_{xx}& -I_{xy}& -I_{xz}\\ -I_{xy}& I_{yy}& -I_{yz}\\ -I_{xz}& -I_{yz}& I_{zz}\end{array}\right]A_{\mathrm{b}\to \mathrm{m}}^{\mathrm{T}}=\left[\begin{array}{ccc}{I}_{mx}& 0& 0\\ 0& I_{my}& 0\\ 0& 0& I_{mz}\end{array}\right]={}^{\mathrm{m}}I_{\mathrm{t}}$$

(4)

I_mx, I_my and I_mz are moments of inertia about the x-axis, y-axis and z-axis of Σ_m. Assuming no external forces and torques acting upon the target satellite, its momentum is conserved, and mass center position is unchanged. Moreover, due to M_x = M_y = M_z = 0, the attitude dynamics equation of the uncontrolled spacecraft can be written as

$$\left\{\begin{array}{l}{I}_{mx}{\dot{\omega }}_x+{\omega }_y{\omega }_z(I_{mz}-I_{my})=0\\ I_{my}{\dot{\omega }}_y+{\omega }_x{\omega }_z(I_{mx}-I_{mz})=0\\ I_{mz}{\dot{\omega }}_z+{\omega }_x{\omega }_y(I_{my}-I_{mx})=0\end{array}\right.$$

(5)

${\dot{\omega }}_x$, ${\dot{\omega }}_y$ and ${\dot{\omega }}_z$ are angular acceleration about the x-axis, y-axis and z-axis. As can be seen from Equation (5), the movements of each axis are mutually coupled when the spacecraft is tumbling. In fact, uncontrolled satellites are often subjected to environmental forces or torques. Therefore, the direction and amplitude of angular momentum h^C will change. Moreover, due to friction, liquid sloshing, structural flexibility, and other energy dissipation factors, the spinning axis of the target satellite will also undergo changes. As a result, a practical uncontrolled spacecraft will move with three types of motion, i.e., rotation, precession and nutation (see Figure 1b).

2.2. Kinematics of the Discrete-Serpentine Heterogeneous Multi-Arm Space Robot

As shown in Figure 2, the discrete-serpentine heterogeneous multi-arm space robot (DSHMASR) is composed of a discrete dual-arm (named by Arm-a and Arm-b), a serpentine arm (denoted by Arm-c) and a spacecraft platform.

Both discrete arms are driven by joint motors and share identical mechanical structures. Each joint of the discrete arm has only one rotational axis. Different from discrete arms, the serpentine arm is driven by cables, and each joint of the arm is a universal joint that has two degrees of freedom (DOFs). Here, the serpentine arm refers to a manipulator arm that has multiple degrees of rotation. It can move in three dimensional space with 6-DOF movement (three-dimensional translation and three-dimensional rotation in three directions). The numbers of their joints are n_a, n_b and n_c, respectively. Here, {x₀, y₀, z₀} represents the base frame, and {x_I, y_I, z_I} is the inertial frame. The definitions of common variables are listed in

Table 1. Definition of symbols (k = a, b or c; i = 1, 2,…, n_k).

Variable	Definition
B0	The base of the DSHMASR
$B_i^k$	The ith link of Arm-k
$J_i^k$	The ith joint of Arm-k
$a_i^k$	$\mathrm{The} \mathrm{vector} \mathrm{from} J_i^k$$\mathrm{to} \mathrm{the} \mathrm{centroid} \mathrm{of} B_i^k$
$b_0^k$	$\mathrm{The} \mathrm{vector} \mathrm{from} \mathrm{base}’\mathrm{s} \mathrm{centroid} \mathrm{to} J_1^k$
$b_i^k$	$\mathrm{The} \mathrm{vector} \mathrm{from} \mathrm{the} \mathrm{centroid} \mathrm{of} B_i^k$$\mathrm{to} J_{i+1}^k$
$r_0^{\phantom{­}}$	The position vector of the base B0
$r_i^k$	$\mathrm{The} \mathrm{position} \mathrm{vector} \mathrm{of} \mathrm{the} \mathrm{centroid} \mathrm{of} B_i^k$
$p_i^k$	$\mathrm{The} \mathrm{position} \mathrm{vector} \mathrm{of} J_i^k$
$p_{\mathrm{e}}^k$	The position vector of the tip of Arm-k
Ek	The tip of Arm-k
$v_0$$, {\omega }_0$	The base’s linear velocity and angular velocity
$v_{\mathrm{e}}^k$$, {\omega }_{\mathrm{e}}^k$	The linear velocity and angular velocity of Arm-k tip
Om	The m-dimensional zero matrix
Em	The m-dimensional identity matrix

. Unless otherwise stated, all the variables are described in the inertial frame.

For the kth arm (k = a, b or c), the position vectors of the centroid of $B_i^k$ are derived as follows.

$$r_i^k=r_0+b_0^k+{\displaystyle \sum _{j=1}^i\left(a_j^k+b_j^k\right)}+a_i^k$$

(6)

On basis of position relationship between the centroid of $B_{n_k}^k$ and the tip of Arm-k, the position vector of Arm-k tip can be expressed as:

$$p_{\mathrm{e}}^k=r_0+b_0^k+{\displaystyle \sum _{j=1}^{n_k}\left(a_j^k+b_j^k\right)}$$

(7)

Differentiating Equation (7), the velocities of Arm-k tip are obtained. According to the preceding discussion, only one rotational axis exists at the each joint of Arm-a and Arm-b. However, for Arm-c, there are two rotational axes for each joint because it is a universal joint. Therefore, the linear velocities of Arm-a, Arm-b and Arm-c tips can be derived as follows.

$$\left\{\begin{array}{l}{v}_{\mathrm{e}}^k=v_0+{\omega }_0\times \left(p_{\mathrm{e}}^k-r_0\right)+{\displaystyle \sum _{i=1}^{n_k}\left[k_i^k\times \left(p_{\mathrm{e}}^k-p_i^k\right)\right]{\dot{\theta }}_i^k},\hspace{1em}k=\mathrm{a},\mathrm{b}\\ v_{\mathrm{e}}^{\mathrm{c}}=v_0+{\omega }_0\times \left(p_{\mathrm{e}}^{\mathrm{c}}-r_0\right)+{\displaystyle \sum _{h=1}^{n_{\mathrm{c}}}\left[\left(k_{h,1}^{\mathrm{c}}{\dot{\theta }}_{h,1}^{\mathrm{c}}+k_{h,2}^{\mathrm{c}}{\dot{\theta }}_{h,2}^{\mathrm{c}}\right)\times \left(p_{\mathrm{e}}^{\mathrm{c}}-p_h^{\mathrm{c}}\right)\right]}\end{array}\right.$$

(8)

where $k_i^k$ and ${\dot{\theta }}_i^k$ are the rotation axis vector and angular velocity of the jth joint for Arm-k (k = a, b). $k_{h,m}^{\mathrm{c}}$ and ${\dot{\theta }}_{h,m}^{\mathrm{c}}$ are the mth rotation axis vector and angular velocity of the hth joint for Arm-c (m = 1, 2).

Based on relative motion, the angular velocity of each arm’s end-effector can be obtained as follows.

$$\left\{\begin{array}{l}{\omega }_{\mathrm{e}}^k={\omega }_0+{\displaystyle \sum _{i=1}^{n_k}{k}_i^k}{\dot{\theta }}_i^k,\hspace{1em}k=\mathrm{a},\mathrm{b}\\ {\omega }_{\mathrm{e}}^{\mathrm{c}}={\omega }_0+{\displaystyle \sum _{h=1}^{n_{\mathrm{c}}}\left(k_{h,1}^{\mathrm{c}}{\dot{\theta }}_{h,1}^{\mathrm{c}}+k_{h,2}^{\mathrm{c}}{\dot{\theta }}_{h,2}^{\mathrm{c}}\right)}\end{array}\right.$$

(9)

Thus, the generalized velocity of Arm-k end-effector can be expressed as

$${\dot{x}}_{\mathrm{e}}^k=\left[\begin{array}{l}{v}_{\mathrm{e}}^k\\ {\omega }_{\mathrm{e}}^k\end{array}\right]=J_{\mathrm{b}}^k\left[\begin{array}{l}{v}_0\\ {\omega }_0\end{array}\right]+J_{\mathrm{m}}^k{\dot{\Theta }}^k, k=\mathrm{a},\mathrm{b},\mathrm{c}$$

(10)

where ${\dot{\Theta }}^k$ is the joint angular velocity of Arm-k (k = a, b or c). $J_{\mathrm{b}}^k$ and $J_{\mathrm{m}}^k$ are Jacobian matrices related to the base motion and the Arm-k motion, respectively, which can be obtained as follows.

$$J_{\mathrm{b}}^k=\left[\begin{array}{cc}{E}_3& -{\tilde{p}}_{0\mathrm{e}}^k\\ O_3& E_3\end{array}\right]\in {\mathbf{R}}^{6\times 6}, k=\mathrm{a},\mathrm{b},\mathrm{c}$$

(11)

$$J_{\mathrm{m}}^k=\left[\begin{array}{ccc}{k}_1^k\times \left(p_{\mathrm{e}}^k-p_1^k\right)& \dots & k_n^k\times \left(p_{\mathrm{e}}^k-p_{n_k}^k\right)\\ k_1^k& \dots & k_{n_k}^k\end{array}\right]\in {\mathbf{R}}^{6\times n_k},\hspace{1em}k=\mathrm{a},\mathrm{b}$$

(12)

$$J_{\mathrm{m}}^{\mathrm{c}}=\left[\begin{array}{ccccc}{k}_{1,1}^{\mathrm{c}}\times \delta p_{\mathrm{e},1}^{\mathrm{c}}& k_{1,2}^{\mathrm{c}}\times \delta p_{\mathrm{e},1}^{\mathrm{c}}& \dots & k_{n_{\mathrm{c}},1}^{\mathrm{c}}\times \delta p_{\mathrm{e},n_{\mathrm{c}}}^{\mathrm{c}}& k_{n_{\mathrm{c}},2}^{\mathrm{c}}\times \delta p_{\mathrm{e},n_{\mathrm{c}}}^{\mathrm{c}}\\ k_{1,1}^{\mathrm{c}}& k_{1,2}^{\mathrm{c}}& \dots & k_{n_{\mathrm{c}},1}^{\mathrm{c}}& k_{n_{\mathrm{c}},2}^{\mathrm{c}}\end{array}\right]\in {\mathbf{R}}^{6\times \left(2n_{\mathrm{c}}\right)}$$

(13)

$\delta p_{\mathrm{e},i}^{\mathrm{c}}$ is the vector from the ith joint to the Arm-c tip. $p_{0\mathrm{e}}^k$ is the vector from the base’s centroid to the tip of Arm-k, i.e., $p_{0\mathrm{e}}^k=p_{\mathrm{e}}^k-p_0^k$. ${\tilde{p}}_{0\mathrm{e}}^k$ is the skew symmetric matrix of $p_{0\mathrm{e}}^k$, and it can be obtained by

$${\tilde{p}}_{0\mathrm{e}}^k=\left[\begin{array}{ccc}0& -z& y\\ z& 0& x\\ -y& x& 0\end{array}\right], \mathrm{if} p_{0\mathrm{e}}^k={\left[\begin{array}{ccc}x& y& z\end{array}\right]}^{\mathrm{T}}, k=\mathrm{a},\mathrm{b},\mathrm{c}$$

(14)

Since the discrete-serpentine heterogeneous multi-arm space robot is in the micro gravity environment, it is not subject to external forces or torques. Assuming the linear and angular momentum of this space robot are both zero in the initial moment, its momentum always remains zero based on the momentum conservation laws, that is:

$$\left[\begin{array}{c}{P}_0\\ L_0\end{array}\right]=\left[\begin{array}{cc}M{E}_3& M{\left({\tilde{r}}_{0\mathrm{g}}^{\phantom{­}}\right)}^{\mathrm{T}}\\ M{\tilde{r}}_{\mathrm{g}}& I_{\mathrm{w}}\end{array}\right]\left[\begin{array}{l}{v}_0\\ {\omega }_0\end{array}\right]+\left[\begin{array}{c}{J}_{\mathrm{T}w}^{\mathrm{a}}\\ I_{\varphi }^{\mathrm{a}}\end{array}\right]{\dot{\Theta }}^{\mathrm{a}}+\left[\begin{array}{c}{J}_{\mathrm{T}w}^{\mathrm{b}}\\ I_{\varphi }^{\mathrm{b}}\end{array}\right]{\dot{\Theta }}^{\mathrm{b}}+\left[\begin{array}{c}{J}_{\mathrm{T}w}^{\mathrm{c}}\\ I_{\varphi }^{\mathrm{c}}\end{array}\right]{\dot{\Theta }}^{\mathrm{c}}=\left[\begin{array}{c}\mathbf{0}\\ \mathbf{0}\end{array}\right]$$

(15)

where $P_0$ and $L_0$ are initial linear and angular momentum. M is the total mass of the DSHMASR. $r_{\mathrm{g}}$ is the position vector of the center of mass for the DSHMASR.

$$\left\{\begin{array}{l}{J}_{\mathrm{T}w}^k={\displaystyle \sum _{i=1}^{n_k}\left(m_i^k{J}_{\mathrm{T}i}^k\right)\in {\mathbf{R}}^{3\times n_k}},\hspace{1em}k=\mathrm{a},\mathrm{b}\\ J_{\mathrm{T}w}^{\mathrm{c}}={\displaystyle \sum _{i=1}^{n_{\mathrm{c}}}\left(m_i^{\mathrm{c}}{J}_{\mathrm{T}i}^{\mathrm{c}}\right)}\in {\mathbf{R}}^{3\times \left(2n_{\mathrm{c}}\right)}\end{array}\right.$$

(16)

$$\left\{\begin{array}{l}{J}_{\mathrm{T}i}^k=\left[k_1^k\times \left(r_i^k-p_1^k\right),\dots ,k_i^k\times \left(r_i^k-p_i^k\right),0,\dots ,0\right]\in {\mathbf{R}}^{3\times n_k},\hspace{1em}k=\mathrm{a},\mathrm{b}\\ J_{\mathrm{T}i}^{\mathrm{c}}=\left[k_{1,1}^{\mathrm{c}}\times \left(r_i^{\mathrm{c}}-p_1^{\mathrm{c}}\right),k_{1,2}^{\mathrm{c}}\times \left(r_i^{\mathrm{c}}-p_1^{\mathrm{c}}\right),\dots ,k_{i,1}^{\mathrm{c}}\times \left(r_i^{\mathrm{c}}-p_i^{\mathrm{c}}\right),k_{i,2}^{\mathrm{c}}\times \left(r_i^{\mathrm{c}}-p_i^{\mathrm{c}}\right),0,\dots ,0\right]\in {\mathbf{R}}^{3\times \left(2n_{\mathrm{c}}\right)}\end{array}\right.$$

(17)

$$\left\{\begin{array}{l}{I}_{\varphi }^k={\displaystyle \sum _{i=1}^{n_k}\left(I_i^k{J}_{\mathrm{R}i}^k+m_i^k{\tilde{r}}_i^k{J}_{\mathrm{T}i}^k\right)}\in {\mathbf{R}}^{3\times n_k},\hspace{1em}k=\mathrm{a},\mathrm{b}\\ I_{\varphi }^{\mathrm{c}}={\displaystyle \sum _{i=1}^{n_{\mathrm{c}}}\left(I_i^{\mathrm{c}}{J}_{\mathrm{R}i}^{\mathrm{c}}+m_i^{\mathrm{c}}{\tilde{r}}_i^{\mathrm{c}}{J}_{\mathrm{T}i}^{\mathrm{c}}\right)}\in {\mathbf{R}}^{3\times \left(2n_{\mathrm{c}}\right)}\end{array}\right.$$

(18)

$$\left\{\begin{array}{l}{J}_{\mathrm{R}i}^k=\left[k_1^k,\dots ,k_i^k,0,\dots ,0\right]\in {\mathbf{R}}^{3\times n_k},\hspace{1em}k=\mathrm{a},\mathrm{b}\\ J_{\mathrm{R}i}^{\mathrm{c}}=\left[k_{1,1}^{\mathrm{c}},k_{1,2}^{\mathrm{c}},\dots ,k_{i,1}^{\mathrm{c}},k_{i,2}^{\mathrm{c}},0,\dots ,0\right]\in {\mathbf{R}}^{3\times \left(2n_{\mathrm{c}}\right)}\end{array}\right.$$

(19)

$$I_{\mathrm{w}}=I_0+{\displaystyle \sum _{i=1}^{n_k}\left[I_i^k-m_i^k{\tilde{r}}_i^k{\tilde{r}}_{0i}^k\right]},\hspace{1em}k=\mathrm{a},\mathrm{b},\mathrm{c}$$

(20)

$m_i^k$ and $I_i^k$ are the mass and inertia matrix of $B_i^k$. $I_0$ is the inertia matrix of the base. According to Equation (15), the base’s linear velocity and angular velocity are calculated by the following equations.

$$v_0=-\left({\displaystyle \frac{J_{\mathrm{T}w}^{\mathrm{a}}}{M}}+{\tilde{r}}_{0\mathrm{g}}^{\phantom{­}}{I}_{\mathrm{s}}^{-1}{I}_{\Theta }^{\mathrm{a}}\right){\dot{\Theta }}^{\mathrm{a}}-\left({\displaystyle \frac{J_{\mathrm{T}w}^{\mathrm{b}}}{M}}+{\tilde{r}}_{0\mathrm{g}}^{\phantom{­}}{I}_{\mathrm{s}}^{-1}{I}_{\Theta }^{\mathrm{b}}\right){\dot{\Theta }}^{\mathrm{b}}-\left({\displaystyle \frac{J_{\mathrm{T}w}^{\mathrm{c}}}{M}}+{\tilde{r}}_{0\mathrm{g}}^{\phantom{­}}{I}_{\mathrm{s}}^{-1}{I}_{\Theta }^{\mathrm{c}}\right){\dot{\Theta }}^{\mathrm{c}}$$

(21)

$${\omega }_0=-I_{\mathrm{s}}^{-1}\left(I_{\Theta }^{\mathrm{a}}{\dot{\Theta }}^{\mathrm{a}}+I_{\Theta }^{\mathrm{b}}{\dot{\Theta }}^{\mathrm{b}}+I_{\Theta }^{\mathrm{c}}{\dot{\Theta }}^{\mathrm{c}}\right)$$

(22)

where $I_{\mathrm{s}}=M{\tilde{r}}_{\mathrm{g}}{\tilde{r}}_{0\mathrm{g}}^{\phantom{­}}+I_{\mathrm{w}}$, $I_{\Theta }^k=I_{\varphi }^k-{\tilde{r}}_{\mathrm{g}}{J}_{\mathrm{T}w}^k, k=\mathrm{a},\mathrm{b},\mathrm{c}$.

Synthesizing Equations (21) and (22), the generalized velocity of the base is yielded.

$$\left[\begin{array}{l}{v}_0\\ {\omega }_0\end{array}\right]=J_{\mathrm{bm}}^{\mathrm{a}}{\dot{\Theta }}^{\mathrm{a}}+J_{\mathrm{bm}}^{\mathrm{b}}{\dot{\Theta }}^{\mathrm{b}}+J_{\mathrm{bm}}^{\mathrm{c}}{\dot{\Theta }}^{\mathrm{c}}=\left[\begin{array}{ccc}{J}_{\mathrm{bm}}^{\mathrm{a}}& J_{\mathrm{bm}}^{\mathrm{b}}& J_{\mathrm{bm}}^{\mathrm{c}}\end{array}\right]\left[\begin{array}{c}{\dot{\Theta }}^{\mathrm{a}}\\ {\dot{\Theta }}^{\mathrm{b}}\\ {\dot{\Theta }}^{\mathrm{c}}\end{array}\right]$$

(23)

where $J_{\mathrm{bm}}^k$ is the Jacobian matrix relating the base to arm-k. It can be obtained as follows.

$$J_{\mathrm{bm}}^k=\left[\begin{array}{c}-{\tilde{r}}_{0\mathrm{g}}^{\phantom{­}}{I}_{\mathrm{s}}^{-1}{I}_{\Theta }^k-{\displaystyle \frac{J_{\mathrm{T}w}^k}{M}}\\ -I_{\mathrm{s}}^{-1}{I}_{\Theta }^k\end{array}\right]$$

(24)

Substituting Equation (23) into Equation (10), the mapping relationship of end-effector velocity and joint angular velocity for the DSHMASR can be expressed as follows.

$$\left[\begin{array}{c}{\dot{x}}_{\mathrm{e}}^{\mathrm{a}}\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{b}}\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{c}}\end{array}\right] =\left[\begin{array}{ccc}{J}_{\mathrm{aa}}& J_{\mathrm{ab}}& J_{\mathrm{ac}}\\ J_{\mathrm{ba}}& J_{\mathrm{bb}}& J_{\mathrm{bc}}\\ J_{\mathrm{ca}}& J_{\mathrm{cb}}& J_{\mathrm{cb}}\end{array}\right]\left[\begin{array}{c}{\dot{\Theta }}^{\mathrm{a}}\\ {\dot{\Theta }}^{\mathrm{b}}\\ {\dot{\Theta }}^{\mathrm{c}}\end{array}\right]=J_{\mathrm{g}}\dot{\Theta }$$

(25)

where

$$\left\{\begin{array}{l}{J}_{\mathrm{aa}}=J_{\mathrm{b}}^{\mathrm{a}}{J}_{\mathrm{bm}}^{\mathrm{a}}+J_{\mathrm{m}}^{\mathrm{a}}\\ J_{\mathrm{ab}}=J_{\mathrm{b}}^{\mathrm{a}}{J}_{\mathrm{bm}}^{\mathrm{b}}\\ J_{\mathrm{ac}}=J_{\mathrm{b}}^{\mathrm{a}}{J}_{\mathrm{bm}}^{\mathrm{c}}\end{array}\right.,\hspace{1em}\left\{\begin{array}{l}{J}_{\mathrm{ba}}=J_{\mathrm{b}}^{\mathrm{b}}{J}_{\mathrm{bm}}^{\mathrm{a}}\\ J_{\mathrm{ab}}=J_{\mathrm{b}}^{\mathrm{b}}{J}_{\mathrm{bm}}^{\mathrm{b}}+J_{\mathrm{m}}^{\mathrm{b}}\\ J_{\mathrm{ac}}=J_{\mathrm{b}}^{\mathrm{b}}{J}_{\mathrm{bm}}^{\mathrm{c}}\end{array}\right.,\hspace{1em}\left\{\begin{array}{l}{J}_{\mathrm{ca}}=J_{\mathrm{b}}^{\mathrm{c}}{J}_{\mathrm{bm}}^{\mathrm{a}}\\ J_{\mathrm{cb}}=J_{\mathrm{b}}^{\mathrm{c}}{J}_{\mathrm{bm}}^{\mathrm{b}}\\ J_{\mathrm{cc}}=J_{\mathrm{b}}^{\mathrm{c}}{J}_{\mathrm{bm}}^{\mathrm{c}}+J_{\mathrm{m}}^{\mathrm{c}}\end{array}\right.$$

(26)

J_g represents the generalized Jacobian matrix of the DSHMASR. $\dot{\Theta }$ means the generalized joint angular velocity, i.e., $\dot{\Theta }={[{({\dot{\Theta }}^{\mathrm{a}})}^{\mathrm{T}},{({\dot{\Theta }}^{\mathrm{b}})}^{\mathrm{T}},{({\dot{\Theta }}^{\mathrm{c}})}^{\mathrm{T}}]}^{\mathrm{T}}\in {\mathbf{R}}^{\left(n_{\mathrm{a}}+n_{\mathrm{b}}+2n_{\mathrm{c}}\right)\times 1}$.

3. Manipulability Analysis

The manipulation capability of manipulators is generally reflected by manipulability. And a larger operational velocity or force can be realized by manipulability optimization. During the process of cooperative capturing-monitoring for a tumbling target satellite, the discrete dual-arm acts as the mission arm to capture the target satellite. Meanwhile, the serpentine manipulator serves as the auxiliary arm, enabling monitoring for the discrete dual-arm capturing the target. In this section, a manipulability measure for mission arms is defined, and the manipulability optimization is studied for improving the manipulation capability in the mission direction.

3.1. Definition of Manipulability Measure for Mission Arms

According to Equation (25), the end-effector velocity of each discrete arm can be obtained by

$$\left\{\begin{array}{l}{\dot{x}}_{\mathrm{e}}^{\mathrm{a}}=\left[\begin{array}{ccc}{J}_{\mathrm{aa}}& J_{\mathrm{ab}}& J_{\mathrm{ac}}\end{array}\right]\dot{\Theta }=J_{\mathrm{g}}^{\mathrm{a}}\dot{\Theta }\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{b}}=\left[\begin{array}{ccc}{J}_{\mathrm{ba}}& J_{\mathrm{bb}}& J_{\mathrm{bc}}\end{array}\right]\dot{\Theta }=J_{\mathrm{g}}^{\mathrm{b}}\dot{\Theta }\end{array}\right.$$

(27)

Assuming each joint of the discrete arms (Arm-a and Arm-b) has the same maximum angular velocity, the unit sphere of joint velocity, i.e., ${\dot{\Theta }}^{\mathrm{T}}\dot{\Theta }=1$, can be mapped into the ellipsoid of end-effector velocity by $J_{\mathrm{g}}^{\mathrm{a}}$ and $J_{\mathrm{g}}^{\mathrm{b}}$.

$${({\dot{x}}_{\mathrm{e}}^k)}^{\mathrm{T}}\cdot {(J_{\mathrm{g}}^k{(J_{\mathrm{g}}^k)}^{\mathrm{T}})}^{-1}\cdot {\dot{x}}_{\mathrm{e}}^k=1,\hspace{1em}k=\mathrm{a},\mathrm{b}$$

(28)

Letting ${\dot{x}}_{\mathrm{e}}^k={\lambda }^k{e}^k$, ${\lambda }^k$ and $e^k$ are the magnitude and unit direction vector of ${\dot{x}}_{\mathrm{e}}^k$, respectively. While mission arms capture the target satellite, it would be better if the end-effector’s velocity is as high as possible. Therefore, the manipulability measure of each mission arm in the direction $e^k$ is calculated as follows:

$$M_{\mathrm{m}}^k={\left({(e^k)}^{\mathrm{T}}\cdot {(J_{\mathrm{g}}^k{(J_{\mathrm{g}}^k)}^{\mathrm{T}})}^{-1}\cdot e^k\right)}^{-1},\hspace{1em}k=\mathrm{a},\mathrm{b}$$

(29)

In the paper, the comprehensive manipulation capability of the discrete dual-arm (mission arms) is considered, and its manipulability measure is defined as follows:

$$M_{\mathrm{m}}=w_1{M}_{\mathrm{m}}^{\mathrm{a}}+w_2{M}_{\mathrm{m}}^{\mathrm{b}}$$

(30)

where $w_1$ and $w_2$ are weight coefficients of the manipulability measure for Arm-a and Arm-b, respectively. The coefficients $w_1$ and $w_2$ can influence the trajectory planning of the end-effector. In practice, the $w_1$ and $w_2$ are greater than or equal to zero, and their sum is equal to 1. Therefore, if $w_1$ is greater than 0.5, it means the manipulability of Arm-a gains more consideration. Otherwise, the manipulability of Arm-b will obtain more attention than that of Arm-a.

It should be noted that the definition of manipulability measurement refers to the operational velocity ability of both discrete arms’ end-effectors in the mission direction. In other words, the manipulability measurement is used to optimize the configuration of mission arms such that the end-effectors’ velocities are as high as possible under the same joint angular velocity.

3.2. Manipulability Optimization of Mission Arms

Letting $s=\partial M_{\mathrm{m}}/\partial \Theta$, the derivative of the manipulability measure can be calculated by

$${\dot{M}}_{\mathrm{m}}=s^{\mathrm{T}}\cdot \dot{\Theta }$$

(31)

Based on Equation (25), the generalized joint angular velocity of the DSHMASR is resolved as follows.

$$\dot{\Theta }=J_{\mathrm{g}}{\phantom{­}}^{\#}\left[\begin{array}{c}{\dot{x}}_{\mathrm{e}}^{\mathrm{a}}\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{b}}\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{c}}\end{array}\right]+J_{\mathrm{g}\_\mathrm{null}}\phi$$

(32)

where (•)^# represents the pseudo-inverse operation of a matrix. J_{g_null} is the null-space of the Jacobian matrix J_g, and $J_{\mathrm{g}\_\mathrm{null}}=E_{46}-J_{\mathrm{g}}{\phantom{­}}^{\#}{J}_{\mathrm{g}}$. $\phi$ is an arbitrary vector.

Substituting Equation (32) into Equation (31), the following equation can be obtained:

$${\dot{M}}_{\mathrm{m}}=s^{\mathrm{T}}{J}_{\mathrm{g}}{\phantom{­}}^{\#}\left[\begin{array}{c}{\dot{x}}_{\mathrm{e}}^{\mathrm{a}}\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{b}}\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{c}}\end{array}\right]+s^{\mathrm{T}}{J}_{\mathrm{g}\_\mathrm{null}}\phi$$

(33)

Because $\phi$ is an arbitrary vector, it can be set as follows:

$$\phi =\rho s$$

(34)

where $\rho$ is a positive number. Substituting Equation (34) into Equation (33), ${\dot{M}}_{\mathrm{m}}^k$ can be calculated as follows:

$${\dot{M}}_{\mathrm{m}}=s^{\mathrm{T}}{J}_{\mathrm{g}}{\phantom{­}}^{\#}\left[\begin{array}{c}{\dot{x}}_{\mathrm{e}}^{\mathrm{a}}\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{b}}\\ {\dot{x}}_{\mathrm{e}}^{\mathrm{c}}\end{array}\right]+\rho s^{\mathrm{T}}{J}_{\mathrm{g}\_\mathrm{null}}s$$

(35)

From the above formula, it can be seen that $\rho s^{\mathrm{T}}{J}_{\mathrm{g}\_\mathrm{null}}s$ is always greater than zero when $s$ is not a zero matrix, as J_{g_null} is a positive definite matrix. Therefore, ${\dot{M}}_{\mathrm{m}}$ will increase if $\phi$ is set by Equation (34), and $M_{\mathrm{m}}$ will become larger while ${\dot{M}}_{\mathrm{m}}$ is increased. It reveals that the manipulation capability is improved as long as $\phi$ is chosen as $\rho s$.

4. Coordinated Capturing-Monitoring Trajectory Planning Based on Manipulability Optimization

4.1. Cooperative Motion Planning Strategy

The motion planning strategy for the discrete-serpentine heterogeneous multi-arm space robot (DSHMASR) for cooperatively capturing and monitoring a tumbling target is depicted in Figure 3. During the process, the docking ring is taken as the object to be grasped, owing to its large structural dimensions and high connection stiffness.

Two discrete arms (Arm-a and Arm-b) serve as primary operational manipulators and offer high-tolerance capturing and high-stiff locking for the uncontrolled target satellite. Simultaneously, the serpentine arm (Arm-c), acting as an auxiliary arm, is employed to provide monitoring for both discrete arms. As shown in Figure 3, $p_{\mathrm{t}}^{\mathrm{a}}$ and $p_{\mathrm{t}}^{\mathrm{b}}$ are capture points for Arm-a and Arm-b, respectively, and they are both on the docking ring. $k_{\mathrm{t}}$ is the normal vector of the docking ring plane. To achieve the best monitoring perspective, the direction of the Arm-c tip should be perpendicular to the plane where the docking ring is. Additionally, to ensure that Arm-c offers the same monitoring conditions for capture points of Arm-a and Arm-b, the theoretical monitoring position should be at the midpoint between $p_{\mathrm{t}}^{\mathrm{a}}$ and $p_{\mathrm{t}}^{\mathrm{b}}$, i.e., the point ${\stackrel{⌢}{p}}_{\mathrm{t}}^{\mathrm{c}}$ satisfying $‖\overrightarrow{p_{\mathrm{t}}^{\mathrm{a}}{\stackrel{⌢}{p}}_{\mathrm{t}}^{\mathrm{c}}}‖=‖\overrightarrow{{\stackrel{⌢}{p}}_{\mathrm{t}}^{\mathrm{c}}{p}_{\mathrm{t}}^{\mathrm{b}}}‖$. But the tip of Arm-c must not reach the point ${\stackrel{⌢}{p}}_{\mathrm{t}}^{\mathrm{c}}$. Otherwise, the monitoring field of vision will be lost for Arm-c. Hence, a certain distance between the Arm-c tip and theoretical position (${\stackrel{⌢}{p}}_{\mathrm{t}}^{\mathrm{c}}$) must be maintained to provide a clear viewing angle for monitoring capture points. Assuming the distance is $d_{\mathrm{c}}$, the desired position of the Arm-c tip is the point $p_{\mathrm{t}}^{\mathrm{c}}$, and $\overrightarrow{p_{\mathrm{t}}^{\mathrm{c}}{\stackrel{⌢}{p}}_{\mathrm{t}}^{\mathrm{c}}}$ is perpendicular to the plane of the docking ring. The vector $k_{\mathrm{t}}$ is also regarded as the desired direction for Arm-a and Arm-b. The flowchart of the coordinated trajectory planning algorithm is shown in Figure 4.

Firstly, the kinematics model of DSHMASR is established, and the movement state of the docking ring can be obtained through visual measurement, thereby acquiring the position, velocity and normal vector of the docking ring center. The expected positions and expected orientations of Arm-k (k = a, b, c) can be figured out. The position deviation $\Delta {\mathit{P}}_{\mathrm{e}}^k$ and orientation $\Delta O_{\mathrm{e}}^k$ are calculated based on the above data. To ensure that the end-effector of Arm-c is the first to reach the desired monitoring position, the trajectory planning of Arm-c is implemented, until $‖\Delta {\mathit{P}}_{\mathrm{e}}^{\mathrm{c}}‖$ and $‖\Delta {\mathit{O}}_{\mathrm{e}}^{\mathrm{c}}‖$ do not exceed the threshold values (i.e., ${\epsilon }_{\mathrm{p}}^{\mathrm{c}}$ and ${\epsilon }_{\mathrm{o}}^{\mathrm{c}}$, which are determined by the monitoring tolerance). Furthermore, a judgment was made to determine whether the magnitudes of position and orientation errors of Arm-a and Arm-b are less than or equal to the threshold values (i.e., ${\epsilon }_{\mathrm{p}}^{\mathrm{a}}$, ${\epsilon }_{\mathrm{o}}^{\mathrm{b}}$ ${\epsilon }_{\mathrm{p}}^{\mathrm{b}}$, and ${\epsilon }_{\mathrm{o}}^{\mathrm{b}}$, which are set to denote the capturing tolerance). If this is true, the grippers of these two arms will be closed, and the target satellite will be captured. Otherwise, the coordinated trajectory of Arm-a, Arm-b and Arm-c will be planned based on the pose deviation and movement estimation of the target satellite. The joint motion of each arm is determined by the velocity-level kinematics based on manipulability optimization. The above joint data are used as the desired values for the servo controllers. Finally, Arm-a and Arm-b are actuated to track and approach the desired capture points until their end-effectors enter the capturing box, while the end-effector of Arm-c remains at the desired monitoring position.

4.2. Trajectory Planning for the End-Effector

As shown in Figure 3, based on the movement data of the target satellite, the position and attitude deviations between the Arm-k tip and the target point can be calculated by

$$\left\{\begin{array}{l}\Delta {\mathit{P}}_{\mathrm{e}}^k=p_{\mathrm{t}}^k-p_{\mathrm{e}}^k\\ \Delta O_{\mathrm{e}}^k={\kappa }^k{\theta }^k\end{array}\right.$$

(36)

where ${\kappa }^k$ and ${\theta }^k$ are the desired rotation direction and desired rotation angle of Arm-k tip. To grasp the docking ring securely, the expected direction (denoted by $k_{\mathrm{e}}^k$) of the end-effector is required to be perpendicular to the plane of the docking ring. To make the vector $k_{\mathrm{e}}^k$ collinear with the vector $k_{\mathrm{t}}$, the end-effector should rotate ${\theta }^k$ about ${\kappa }^k$. Therefore, ${\kappa }^k$ and ${\theta }^k$ can be determined as follows.

$$\left\{\begin{array}{l}{\kappa }^k=\left(k_{\mathrm{e}}^k\times k_{\mathrm{t}}\right)/\left(‖k_{\mathrm{e}}^k\times k_{\mathrm{t}}‖\right)\\ {\theta }^k=\mathrm{acos}\left(\left(k_{\mathrm{e}}^k\cdot k_{\mathrm{t}}\right)/\left(‖k_{\mathrm{e}}^k‖‖k_{\mathrm{t}}‖\right)\right)\end{array}\right.$$

(37)

On the basis of pose deviations and velocity deviations between end-effectors and target points, the expected tip velocity of Arm-k can be planned as follows:

$$\left[\begin{array}{c}{v}_{\mathrm{ed}}^k\\ {\omega }_{\mathrm{ed}}^k\end{array}\right]=\left[\begin{array}{cc}{K}_{\mathrm{p}}^k& O_3\\ O_3& K_{\mathrm{o}}^k\end{array}\right]\left[\begin{array}{c}\Delta {\mathit{P}}_{\mathrm{e}}^k\\ \Delta O_{\mathrm{e}}^k\end{array}\right]+\left[\begin{array}{c}{\widehat{v}}_{\mathrm{t}}^k\\ {\widehat{\omega }}_{\mathrm{t}}^k\end{array}\right]$$

(38)

where $v_{\mathrm{ed}}^k$ and ${\omega }_{\mathrm{ed}}^k$ are the expected linear and angular velocities of the Arm-k tip, respectively. $K_{\mathrm{p}}^k$ and $K_{\mathrm{o}}^k$ are proportional gain matrices corresponding to the position and orientation deviations. ${\widehat{v}}_{\mathrm{t}}^k$ and ${\widehat{\omega }}_{\mathrm{t}}^k$ are the linear velocity and angular velocity of the target point $p_{\mathrm{t}}^k$, and they can be obtained as follows:

$$\left[\begin{array}{c}{\widehat{v}}_{\mathrm{t}}^k\\ {\widehat{\omega }}_{\mathrm{t}}^k\end{array}\right]=\left[\begin{array}{cc}{E}_3& {\tilde{p}}_{{\mathrm{o}}_{\mathrm{DR}}\mathrm{t}}^k\\ O_3& E_3\end{array}\right]\left[\begin{array}{c}{\widehat{v}}_{{\mathrm{o}}_{\mathrm{DR}}}^k\\ {\widehat{\omega }}_{{\mathrm{o}}_{\mathrm{DR}}}^k\end{array}\right]$$

(39)

where ${\widehat{v}}_{{\mathrm{o}}_{\mathrm{DR}}}^k$ and ${\widehat{\omega }}_{{\mathrm{o}}_{\mathrm{DR}}}^k$ are estimated linear and angular velocities of the docking ring center. $p_{{\mathrm{o}}_{\mathrm{DR}}\mathrm{t}}^k$ is the vector from the docking ring center to the target point, i.e., $p_{{\mathrm{o}}_{\mathrm{DR}}\mathrm{t}}^k=p_{{\mathrm{o}}_{\mathrm{DR}}}-p_{\mathrm{t}}^k$. ${\tilde{p}}_{{\mathrm{o}}_{\mathrm{DR}}\mathrm{t}}^k$ can be obtained on the basis of Equation (14).

Practically, the end-effector velocity of Arm-k is in a limited range. Therefore, the desired velocities must also be confined to the range. Furthermore, it is necessary to ensure a smooth startup of each arm at the initial time. Thus, the desired linear and angular velocities of the Arm-k tip are adjusted as follows:

$$v_{\mathrm{ed}}^k=\left\{\begin{array}{ll}\left(v_{\mathrm{em}}^k/‖v_{\mathrm{ed}}^k‖\right)\cdot {\left(t/t_{\mathrm{s}}\right)}^2\cdot v_{\mathrm{ed}}^k, & \mathrm{if} 0\le t\le t_{\mathrm{s}}\\ \left(v_{\mathrm{em}}^k/‖v_{\mathrm{ed}}^k‖\right)\cdot v_{\mathrm{ed}}^k, & \mathrm{else} \mathrm{if} v_{\mathrm{ed}}^k\ge v_{\mathrm{em}}^k\\ v_{\mathrm{ed}}^k, & \mathrm{else} \end{array}\right.$$

(40)

$${\mathit{\omega }}_{\mathrm{ed}}^k=\left\{\begin{array}{ll}\left({\omega }_{\mathrm{em}}^k/‖{\mathit{\omega }}_{\mathrm{ed}}^k‖\right)\cdot {\left(t/t_{\mathrm{s}}\right)}^2\cdot {\mathit{\omega }}_{\mathrm{ed}}^k, & \mathrm{if} 0\le t\le t_{\mathrm{s}}\\ \left({\omega }_{\mathrm{em}}^k/‖{\mathit{\omega }}_{\mathrm{ed}}^k‖\right)\cdot {\mathit{\omega }}_{\mathrm{ed}}^k, & \mathrm{else} \mathrm{if} {\omega }_{\mathrm{ed}}^k\ge {\omega }_{\mathrm{em}}^k\\ {\mathit{\omega }}_{\mathrm{ed}}^k, & \mathrm{else} \end{array}\right.$$

(41)

$$\left\{\begin{array}{l}‖v_{\mathrm{ed}}^k‖=\sqrt{{\left(v_{\mathrm{e}x}^k\right)}^2+{\left(v_{\mathrm{e}y}^k\right)}^2+{\left(v_{\mathrm{e}z}^k\right)}^2}\\ ‖{\mathit{\omega }}_{\mathrm{ed}}^k‖=\sqrt{{\left({\omega }_{\mathrm{e}x}^k\right)}^2+{\left({\omega }_{\mathrm{e}y}^k\right)}^2+{\left({\omega }_{\mathrm{e}z}^k\right)}^2}\end{array}\right.$$

(42)

where t_s denotes the startup time, which is set to guarantee that Arm-k starts smoothly to achieve the maximum values of the linear and angular velocities, i.e., $v_{\mathrm{em}}^k$ and ${\omega }_{\mathrm{em}}^k$.

4.3. Joint Motion Optimization

The generalized joint angular velocity of the discrete-serpentine heterogeneous multi-arm space robot system in differential form is given as Equation (32). To simultaneously capture and monitor the target satellite, the desired end-effector velocity of Arm-k is planned according to Equation (38). Substituting Equation (38) into Equation (32), the joint angular velocity is calculated as follows:

$$\dot{\Theta }=J_{\mathrm{g}}{\phantom{­}}^{\#}\left[\begin{array}{c}{K}_{\mathrm{p}}^{\mathrm{a}}\Delta {\mathit{P}}_{\mathrm{e}}^{\mathrm{a}}+{\widehat{v}}_{\mathrm{t}}^{\mathrm{a}}\\ K_{\mathrm{o}}^{\mathrm{a}}\Delta O_{\mathrm{e}}^{\mathrm{a}}+{\widehat{\omega }}_{\mathrm{t}}^{\mathrm{a}}\\ K_{\mathrm{p}}^{\mathrm{b}}\Delta {\mathit{P}}_{\mathrm{e}}^{\mathrm{b}}+{\widehat{v}}_{\mathrm{t}}^{\mathrm{b}}\\ K_{\mathrm{o}}^{\mathrm{b}}\Delta O_{\mathrm{e}}^{\mathrm{b}}+{\widehat{\omega }}_{\mathrm{t}}^{\mathrm{b}}\\ K_{\mathrm{p}}^{\mathrm{c}}\Delta {\mathit{P}}_{\mathrm{e}}^{\mathrm{c}}+{\widehat{v}}_{\mathrm{t}}^{\mathrm{c}}\\ K_{\mathrm{o}}^{\mathrm{c}}\Delta O_{\mathrm{e}}^{\mathrm{c}}+{\widehat{\omega }}_{\mathrm{t}}^{\mathrm{c}}\end{array}\right]+J_{\mathrm{g}\_\mathrm{null}}\phi$$

(43)

During the whole process of approaching-monitoring tumbling targets, it is expected that the discrete dual-arm possesses superior manipulation capability in the direction of mission operation, especially in the velocity orientation of the capture points. Therefore, based on Equation (30), the comprehensive manipulation ability of the discrete dual-arm is obtained as follows.

$$M_{\mathrm{m}}=w_1{\left({(x_{\mathrm{t}}^{\mathrm{a}})}^{\mathrm{T}}\cdot {(J_{\mathrm{g}}^{\mathrm{a}}{(J_{\mathrm{g}}^{\mathrm{a}})}^{\mathrm{T}})}^{-1}\cdot x_{\mathrm{t}}^{\mathrm{a}}\right)}^{-1}+w_2{\left({(x_{\mathrm{t}}^{\mathrm{b}})}^{\mathrm{T}}\cdot {(J_{\mathrm{g}}^{\mathrm{b}}{(J_{\mathrm{g}}^{\mathrm{b}})}^{\mathrm{T}})}^{-1}\cdot x_{\mathrm{t}}^{\mathrm{b}}\right)}^{-1}$$

(44)

As discussed in Section 3.2, when $\phi$ is set by Equation (34), $M_{\mathrm{m}}$ will become larger as ${\dot{M}}_{\mathrm{m}}$ is increased. Substituting Equation (34) into Equation (43), the desired joint angular velocity of the discrete-serpentine heterogeneous multi-arm space robot is derived as follows.

$$\dot{\Theta }=J_{\mathrm{g}}{\phantom{­}}^{\#}\left[\begin{array}{c}{K}_{\mathrm{p}}^{\mathrm{a}}\Delta {\mathit{P}}_{\mathrm{e}}^{\mathrm{a}}+{\widehat{v}}_{\mathrm{t}}^{\mathrm{a}}\\ K_{\mathrm{o}}^{\mathrm{a}}\Delta O_{\mathrm{e}}^{\mathrm{a}}+{\widehat{\omega }}_{\mathrm{t}}^{\mathrm{a}}\\ K_{\mathrm{p}}^{\mathrm{b}}\Delta {\mathit{P}}_{\mathrm{e}}^{\mathrm{b}}+{\widehat{v}}_{\mathrm{t}}^{\mathrm{b}}\\ K_{\mathrm{o}}^{\mathrm{b}}\Delta O_{\mathrm{e}}^{\mathrm{b}}+{\widehat{\omega }}_{\mathrm{t}}^{\mathrm{b}}\\ K_{\mathrm{p}}^{\mathrm{c}}\Delta {\mathit{P}}_{\mathrm{e}}^{\mathrm{c}}+{\widehat{v}}_{\mathrm{t}}^{\mathrm{c}}\\ K_{\mathrm{o}}^{\mathrm{c}}\Delta O_{\mathrm{e}}^{\mathrm{c}}+{\widehat{\omega }}_{\mathrm{t}}^{\mathrm{c}}\end{array}\right]+J_{\mathrm{g}\_\mathrm{null}}\left[\begin{array}{c}{\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\Theta }^{\mathrm{a}}}}\\ {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\Theta }^{\mathrm{b}}}}\\ {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\Theta }^{\mathrm{c}}}}\end{array}\right]$$

(45)

where $J_{\mathrm{g}}{\phantom{­}}^{\#}$ and $J_{\mathrm{g}\_\mathrm{null}}$ are the pseudo-inverse matrix and the null-space matrix of the Jacobian matrix $J_{\mathrm{g}}$. $\Delta {\mathit{P}}_{\mathrm{e}}^k$ and $\Delta O_{\mathrm{e}}^k$ (k = a, b, c) are position deviations and attitude deviations between the Arm-k tip and its desired point. $K_{\mathrm{p}}^k$ and $K_{\mathrm{o}}^k$ (k = a, b, c) are proportional coefficient matrices regarding position and attitude deviations. ${\widehat{v}}_{\mathrm{t}}^k$ and ${\widehat{\omega }}_{\mathrm{t}}^k$ are linear velocity and angular velocity of the desired point for the Arm-k tip.

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\Theta }^{\mathrm{a}}}}={\left[\begin{array}{cccc}{\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_1^{\mathrm{a}}}}& {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_2^{\mathrm{a}}}}& \dots & {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_{n_{\mathrm{a}}}^{\mathrm{a}}}}\end{array}\right]}^{\mathrm{T}}\\ {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\Theta }^{\mathrm{b}}}}={\left[\begin{array}{cccc}{\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_1^{\mathrm{b}}}}& {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_2^{\mathrm{b}}}}& \dots & {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_{n_{\mathrm{b}}}^{\mathrm{b}}}}\end{array}\right]}^{\mathrm{T}}\\ {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\Theta }^{\mathrm{c}}}}={\left[\begin{array}{ccccccc}{\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_{1,1}^{\mathrm{c}}}}& {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_{1,2}^{\mathrm{c}}}}& {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_{2,1}^{\mathrm{c}}}}& {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_{2,2}^{\mathrm{c}}}}& \dots & {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_{n_{\mathrm{c}},1}^{\mathrm{c}}}}& {\displaystyle \frac{\partial M_{\mathrm{m}}}{\partial {\theta }_{n_{\mathrm{c}},2}^{\mathrm{c}}}}\end{array}\right]}^{\mathrm{T}}\end{array}\right.$$

(46)

Equation (45) will be utilized as a motion actuation command for the discrete-serpentine heterogeneous multi-arm space robot, so as to cooperatively capture and monitor the tumbling target until the target satellite is grasped.

5. Simulation Study

5.1. Parameters of the Simulation Model

A discrete-serpentine heterogeneous multi-arm space robot system for the cooperative capturing-monitoring of an uncontrolled satellite is taken as an example for the simulation study. The robotic system is composed of a base, two discrete manipulators (Arm-a and Arm-b) with the same structures and a serpentine arm (Arm-c), as shown in Figure 5.

Here, {x₀, y₀, z₀} and {x_I, y_I, z_I} are the base-fixed frame and the inertial frame, respectively. The classical Denavit–Hartenberg (D-H) method is employed to describe the kinematic model of each arm. Therefore, the D-H frames of each discrete manipulator are established as shown in Figure 6, and its D-H parameters are listed in

Table 2. D-H parameters of each discrete manipulator.

Link i	θi (°)	αi (°)	ai (mm)	di (mm)
1	−90	90	0	d1 = 100
2	90	90	0	d2 = 190
3	−90	0	a3 = 1810	0
4	0	0	a4 = 1810	0
5	−90	90	0	d5 = 570
6	90	90	0	d6 = 190
7	−90	0	a7 = 150	d7 = 190

.

The serpentine arm contains 16 links with an identical mechanical structure. Any two adjacent links are connected through a universal joint. Each universal joint possesses a Pitch axis and a Yaw axis. Thus, the serpentine arm has a total of 32 DOFs. The D-H frames and parameters of the serpentine arm are given in Figure 7 and

Table 3. D-H parameters of the serpentine manipulator.

Link i	θi (°)	αi (°)	ai (mm)	di (mm)
1	180	90	0	0
2	180	90	0	222
3	180	90	0	0
4	180	90	0	222
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
29	180	90	0	0
30	180	90	0	222
31	180	90	0	0
32	180	90	0	222

, respectively.

The mass properties of Arm-a are listed in

Table 4. The mass properties of Arm-a.

Variates		B0	${\mathit{B}}_1^{\mathbf{a}}$	${\mathit{B}}_2^{\mathbf{a}}$	${\mathit{B}}_3^{\mathbf{a}}$	${\mathit{B}}_4^{\mathbf{a}}$	${\mathit{B}}_5^{\mathbf{a}}$	${\mathit{B}}_6^{\mathbf{a}}$	${\mathit{B}}_7^{\mathbf{a}}$
Mass (kg)		2000	2.5	2.5	5.6	5.6	2.5	2.5	2.5
${}^{i}a_i^{\mathrm{a}}$(m)		/	0	0	0	0.905	0	0	0
		/	0	0	−0.905	0	0	0	0
		/	−0.1	−0.095	−0.095	−0.095	−0.095	−0.095	−0.095
${}^{i}b_i^{\mathrm{a}}$(m)		−0.9231	0.095	−0.095	0	0.905	0.095	−0.095	0
		1.9195	0	0	−0.905	0	0	0	−0.095
		−0.7729	0	0	−0.095	−0.095	0	0	0
${}^{i}I_i^{\mathrm{a}}$(kg·m2)	Ixx	436	0.01	0.01	1.60	0.325	0.01	0.01	0.01
	Iyy	105	0.01	0.01	0.325	1.60	0.01	0.01	0.01
	Izz	436	0.0086	0.0086	1.60	1.60	0.0086	0.0086	0.0086

. For each link, it is assumed the products of inertia are zeros, i.e., I_xy = I_xz = I_yz = 0. Arm-b has the same mass properties as Arm-a with the exception of the mounting location (${}^{0}b_0^{\mathrm{b}}= {[1.6823,1.5998,-0.7555]}^{\mathrm{T}}\mathrm{m}$).

For the serpentine arm, the mass properties of each link are given as follows.

$$\left\{\begin{array}{l}{m}_i^{\mathrm{c}}=0.15 \mathrm{kg}\\ {}^{i}a_i^{\mathrm{c}}={}^{i}b_i^{\mathrm{c}}={\left[\begin{array}{ccc}0.111,& 0,& 0\end{array}\right]}^{\mathrm{T}}\mathrm{m}\\ {}^{0}b_0^{\mathrm{c}}= \left[\begin{array}{ccc}0.4052,& 1.9134,& -1.2755\end{array}\right]{\phantom{­}}^{\mathrm{T}}\mathrm{m}\end{array}\right.,\hspace{1em}i=1,2,\dots ,16$$

(47)

$${}^{i}I_i^{\mathrm{c}}=\left[\begin{array}{ccc}0.001& 0& 0\\ 0& 0.021& 0\\ 0& 0& 0.021\end{array}\right]\mathrm{kg}\cdot {\mathrm{m}}^2$$

(48)

The mass of the target satellite is $m_{\mathrm{t}}=300 \mathrm{kg}$ and its inertia matrix is as follows.

$${}^{\mathrm{m}}I_{\mathrm{t}}=\left[\begin{array}{ccc}60& 0& 0\\ 0& 60& 0\\ 0& 0& 50\end{array}\right]\mathrm{kg}\cdot {\mathrm{m}}^2$$

(49)

In the simulation, the tumbling motion of the target satellite is considered. Its equivalent three-axis angular velocities relative to the fixed coordinate system of the target satellite are ${}^{\mathrm{b}}\omega _{\mathrm{t}}={[\begin{array}{ccc}0.515{\phantom{­}}^{\mathrm{o}}/\mathrm{s},& 0.091{\phantom{­}}^{\mathrm{o}}/\mathrm{s},& 4.973{\phantom{­}}^{\mathrm{o}}/\mathrm{s}\end{array}]}^{\mathrm{T}}$.

To verify the proposed method, a co-simulation system based on Adams2014 and Matlab2014/Simulink software is established. The dynamics models of the discrete-serpentine heterogeneous multi-arm space robot and the target satellite are constructed based on Adams2014. The planner and controller are achieved in Matlab2014/Simulink.

5.2. Target Satellite Capturing-Monitoring Simulation

5.2.1. Parameter Initialization

In the simulation, the maximum linear velocity and angular velocity of the Arm-a tip and Arm-b tip are set as $v_{\mathrm{em}}^{\mathrm{a}}=v_{\mathrm{em}}^{\mathrm{b}}=125 \mathrm{mm}/\mathrm{s}$ and ${\omega }_{\mathrm{em}}^{\mathrm{a}}={\omega }_{\mathrm{em}}^{\mathrm{b}}= 8{\phantom{­}}^{\mathrm{o}}/\mathrm{s}$, respectively. For Arm-c, they are $v_{\mathrm{em}}^{\mathrm{c}}=200 \mathrm{mm}/\mathrm{s}$ and ${\omega }_{\mathrm{em}}^{\mathrm{c}}= 8{\phantom{­}}^{\mathrm{o}}/\mathrm{s}$ The position and attitude threshold values for capturing the target satellite are set as ${\epsilon }_{\mathrm{p}}^{\mathrm{a}}={\epsilon }_{\mathrm{p}}^{\mathrm{b}}=30 \mathrm{mm}$ and ${\epsilon }_{\mathrm{o}}^{\mathrm{a}}={\epsilon }_{\mathrm{o}}^{\mathrm{b}}= 5{\phantom{­}}^{\mathrm{o}}$. This means that the end-effectors of both arms can be considered to reach their capture points when the position and attitude deviations are less than or equal to 30 mm and 5°. Similarly, the position and attitude threshold values for monitoring the target satellite are set as ${\epsilon }_{\mathrm{p}}^{\mathrm{c}}=10 \mathrm{mm}$ and ${\epsilon }_{\mathrm{o}}^{\mathrm{c}}=0.1{\phantom{­}}^{\mathrm{o}}$. This implies the end-effector of Arm-c is also deemed to reach the desired monitoring point as long as the position and attitude deviations are less than or equal to 10 mm and 0.1°, respectively.

The initial joint angles of each arm (Arm-a, Arm-b and Arm-c) are set as follows.

$${\Theta }_0^{\mathrm{a}}={\left[-{180}^{\mathrm{o}}, {160}^{\mathrm{o}}, {45}^{\mathrm{o}}, -{110}^{\mathrm{o}}, {80}^{\mathrm{o}}, {190}^{\mathrm{o}}, 30^{\mathrm{o}}\right]}^{\mathrm{T}}$$

(50)

$${\Theta }_0^{\mathrm{b}}={\left[{180}^{\mathrm{o}}, {20}^{\mathrm{o}}, -{45}^{\mathrm{o}}, {110}^{\mathrm{o}}, -{80}^{\mathrm{o}},{ 10}^{\mathrm{o}}, -{35}^{\mathrm{o}}\right]}^{\mathrm{T}}$$

(51)

$${\Theta }_0^{\mathrm{c}}={\left[\begin{array}{cccccccc}{19}^{\mathrm{o}},& -2^{\mathrm{o}},& {19}^{\mathrm{o}},& -1^{\mathrm{o}},& {19}^{\mathrm{o}},& 0^{\mathrm{o}},& {19}^{\mathrm{o}},& 1^{\mathrm{o}},\dots \dots \\ -{18}^{\mathrm{o}},& 1^{\mathrm{o}},& -{18}^{\mathrm{o}},& 1^{\mathrm{o}},& -{18}^{\mathrm{o}},& 1^{\mathrm{o}},& -{18}^{\mathrm{o}},& 1^{\mathrm{o}},\dots \dots \\ -{18}^{\mathrm{o}},& 1^{\mathrm{o}},& -{18}^{\mathrm{o}},& 1^{\mathrm{o}},& -{18}^{\mathrm{o}},& 1^{\mathrm{o}},& -{18}^{\mathrm{o}},& 1^{\mathrm{o}},\dots \dots \\ {16}^{\mathrm{o}},& 2^{\mathrm{o}},& {16}^{\mathrm{o}},& -2^{\mathrm{o}},& {16}^{\mathrm{o}},& -1^{\mathrm{o}},& {16}^{\mathrm{o}},& -1^{\mathrm{o}}\end{array}\right]}^{\mathrm{T}}$$

(52)

For the base, the initial configuration including its attitude (X-Y-Z Euler angles) and position with respect to the inertial coordinate system is as follows:

$$\left\{\begin{array}{l}{\psi }_{\mathrm{b}0}={\left[\begin{array}{ccc}{2.9952}^{\mathrm{o}},& {0.2586}^{\mathrm{o}},& {6.9932}^{\mathrm{o}}\end{array}\right]}^{\mathrm{T}}\\ r_{\mathrm{b}0}={\left[-163.25\mathrm{mm}, 1250.13\mathrm{mm}, 85.64\mathrm{mm}\right]}^{\mathrm{T}} \end{array}\right.$$

(53)

The initial attitudes (Z-X-Z Euler angles) and positions of the capture points for Arm-a and Arm-b in the inertial coordinate frame are as follows:

$$\left\{\begin{array}{l}{\psi }_{\mathrm{t}0}^{\mathrm{a}}={\left[180{.0}^{\mathrm{o}}, 90{.0}^{\mathrm{o}}, {130.0}^{\mathrm{o}}\right]}^{\mathrm{T}}\\ p_{\mathrm{t}0}^{\mathrm{a}}={\left[-292.09\mathrm{mm}, 6839.76\mathrm{mm}, -1401.08\mathrm{mm}\right]}^{\mathrm{T}} \end{array}\right.$$

(54)

$$\left\{\begin{array}{l}{\psi }_{\mathrm{t}0}^{\mathrm{b}}={\left[180{.0}^{\mathrm{o}}, 90{.0}^{\mathrm{o}}, {130.0}^{\mathrm{o}}\right]}^{\mathrm{T}}\\ p_{\mathrm{t}0}^{\mathrm{b}}={\left[412.32\mathrm{mm}, 6839.76\mathrm{mm},-696.65\mathrm{mm}\right]}^{\mathrm{T}} \end{array}\right.$$

(55)

As previously stated, the theoretical monitoring position of Arm-c should be at the midpoint between two capture points. But the actual monitoring position must be maintained at a certain distance between the Arm-c tip and the theoretical monitoring position. At the initial moment, the attitude (Z-X-Z Euler angles) and position of the theoretical monitoring position are as follows:

$$\left\{\begin{array}{l}{\psi }_{\mathrm{t}0}^{\mathrm{c}}={\psi }_{\mathrm{t}0}^{\mathrm{a}}={\psi }_{\mathrm{t}0}^{\mathrm{b}}={\left[180{.0}^{\mathrm{o}}, 90{.0}^{\mathrm{o}}, {130.0}^{\mathrm{o}}\right]}^{\mathrm{T}}\\ {\stackrel{⌢}{p}}_{\mathrm{t}0}^{\mathrm{c}}={\left[60.12\mathrm{mm}, 6839.76\mathrm{mm}, -1048.87\mathrm{mm}\right]}^{\mathrm{T}} \end{array}\right.$$

(56)

Setting the distance to 1.0 m, i.e., $d_{\mathrm{c}}=1000\mathrm{mm}$, the actual position of the monitoring position is as follows:

$$p_{\mathrm{t}0}^{\mathrm{c}}=\left[60.12\mathrm{mm}, 5839.76\mathrm{mm}, -1048.87\mathrm{mm}\right]{\phantom{­}}^{\mathrm{T}}$$

(57)

5.2.2. Simulation Results

Based on manipulability optimization, the simulated analysis of the cooperative capturing-monitoring planning method was carried out using the co-simulation system. The model parameters and initial conditions remained the same as in the previous section. Through the simulation results, the curves of pose (position and attitude) deviations between the end-effector and its corresponding capture point for Arm-a and Arm-b are respectively shown in Figure 8 and Figure 9. The relative pose curves between the end-effector and the monitoring point for Arm-c are depicted in Figure 10.

As shown in Figure 10, when the simulation time is at 3.6 s, the position and orientation deviations for Arm-c are [0.0052 mm, −0.0951 mm, 0.0055 mm] and 0.0933°, which are both less than the corresponding threshold values given in Section 5.2.1. This indicates the Arm-c tip has reached the actual monitoring position. After t = 3.6 s, its position and orientation deviations are also less than the threshold values. Therefore, the Arm-c tip is always kept in the actual monitoring position. Until t = 15.7 s, the pose deviations for Arm-a are [21.37 mm, −19.79 mm, 4.04 mm] and 0.018°, and they are [12.65 mm, −8.26 mm, −7.42 mm] and 0.016° for Arm-b, which are all less than the respective threshold values given in Section 5.2.1. The results indicate that both end-effectors enter the capturing box and the target satellite can be grasped.

The variation of the joint angles for Arm-a, Arm-b and Arm-c are respectively illustrated in Figure 11, Figure 12, Figure 13 and Figure 14. Through Figure 11 and Figure 12, the joint angles of Arm-a and Arm-b are unchanged until the time is 3.6 s (i.e., t₁ = 3.6 s). This means that both arms are always under the motionless state if the Arm-c tip has not reached the actual monitoring position.

In addition, for mission arms (Arm-a and Arm-b), the comprehensive manipulability of two methods is compared to demonstrate that the proposed method with manipulability optimization is better than the traditional method. The comprehensive manipulability variation of mission arms is as shown in Figure 15. The reason for only improving the manipulability as much as possible can be divided into two aspects. On the one hand, it is the real-time variation of the manipulability optimization direction. On the other hand, the main task of the space robot remains the capture of the target, which can significantly affect the trajectory planning of the end-effector. As a result, the manipulability first increases and then decreases, as shown in Figure 15.

The concrete data are listed in

Table 5. The comparison of comprehensive manipulability for two methods.

Manipulability	Proportional Distribution	The Traditional Method	The Proposed Method	Improved By
The maximum value	w1 = 0.4, w2 = 0.6	7.5333	9.0891	20.65%
	w1 = 0.5, w2 = 0.5	7.5333	10.2659	36.27%
	w1 = 0.6, w2 = 0.4	7.5333	11.4807	52.40%
The mean value	w1 = 0.4, w2 = 0.6	6.0004	7.0155	16.92%
	w1 = 0.5, w2 = 0.5	6.0004	7.4172	23.61%
	w1 = 0.6, w2 = 0.4	6.0004	7.8196	30.23%
The final moment value	w1 = 0.4, w2 = 0.6	3.4634	5.5083	59.04%
	w1 = 0.5, w2 = 0.5	3.4634	6.3485	83.30%
	w1 = 0.6, w2 = 0.4	3.4634	7.1885	107.56%

. Here, the maximum value refers to the peak value of the manipulability curves shown in Figure 15. The mean value is the average value of the manipulability curves. The final moment value is the value of the manipulability curves at the end of the simulation. In order to further verify the advantages of the proposed method, more simulation analyses were implemented, i.e., w₁ = 0.4, w₂ = 0.6; w₁ = 0.5, w₂ = 0.5; and w₁ = 0.6, w₂ = 0.4. According to the results, the maximum value, the mean value and the final moment value are 7.5333, 6.0004 and 3.4634 for the traditional method. In the proposed method, the maximum values are respectively increased by 20.65%, 36.27% and 52.40% for the three simulation cases. They are 16.92%, 23.61% and 30.23% for the mean value and 59.04%, 83.30% and 107.56% for the final moment value. Based on the above results, the manipulation ability of the mission arms will be enhanced significantly when the proposed method is adopted to capture tumbling targets.

The variations of the base centroid position and attitude (X-Y-Z Euler angles) are shown in Figure 16. The amounts of change with respect to the initial values are [3.2 mm, −13.64 mm, −4.43 mm] for position and [2.21°, −1.28°, 6.64°] for attitude. The 3D models during the simulation are depicted in Figure 17, corresponding to simulation milestones at t = 0.0 s, 3.6 s, 6.0 s, 9.0 s, 12.0 s, and 15.7 s, respectively.

6. Conclusions

The autonomous grasp of non-cooperative target satellites, one of the most critical technologies for on-orbit services, faces great challenges due to the complex motion characteristics and insufficient pose information interaction. A single discrete arm or serpentine arm fails to accomplish the above challenging task. Therefore, a discrete-serpentine heterogeneous multi-arm space robot (DSHMASR) with both 7-DOF discrete arms and a 32-DOF serpentine arm is designed to satisfy both the requirements of stabilization capture and dexterous manipulation for uncontrolled satellites. The dynamic model of a non-cooperative target satellite and the kinematics of the DSHMASR are established for the first time. Subsequently, a coordinated trajectory planning method for DSHMASR based on manipulability optimization is proposed for grasping non-cooperative targets. During the process, two discrete arms are used as the primary operational manipulators and offer high-tolerance capturing and high-stiff locking for targets, while the serpentine arm is employed to be an auxiliary arm and provide monitoring for the discrete arms. The desired end-effector trajectory of each arm is generated on the basis of the pose deviations between the arm’s tip and its corresponding target point. Further, the joint angular velocities are resolved by the velocity-level kinematics equations based on manipulability optimization. Finally, a co-simulation system is built to verify the proposed method. The simulation results illustrate that the proposed method can enable each discrete arm’s tip to reach the capture point accurately while the end-effector of the serpentine arm always maintains the actual monitoring position. Compared with the traditional method, the manipulation ability of the mission arms will be enhanced significantly. The methodological framework and simulation findings in this paper may serve as an essential reference for further studies of DSHMASR.

The current research only focuses on the dynamics and trajectory planning of the DSHMASR between the joint space and Cartesian space of the end-effector. The mapping relation between the joint space and the cable space is not considered for the serpentine arm of the DSHMASR, which will be studied in future modeling and planning efforts. Moreover, we will strive to conduct further research on dynamic coupling analysis and adaptive compliant control issues for the DSHMASR.