International Journal of Aerospace Engineering

Volume 2016, Article ID 7375685, 13 pages

http://dx.doi.org/10.1155/2016/7375685

## Kane Method Based Dynamics Modeling and Control Study for Space Manipulator Capturing a Space Target

College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 29 January 2016; Revised 3 June 2016; Accepted 15 June 2016

Academic Editor: Christopher J. Damaren

Copyright © 2016 Yanhua Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Dynamics modeling and control problem of a two-link manipulator mounted on a spacecraft (so-called carrier) freely flying around a space target on earth’s circular orbit is studied in the paper. The influence of the carrier’s relative movement on its manipulator is considered in dynamics modeling; nevertheless, that of the manipulator on its carrier is neglected with the assumption that the mass and inertia moment of the manipulator is far less than that of the carrier. Meanwhile, we suppose that the attitude control system of the carrier guarantees its side on which the manipulator is mounted points accurately always the space target during approaching operation. The ideal constraint forces can be out of consideration in dynamics modeling as Kane method is used. The path functions of the manipulator’s end-effector approaching the space target as well as the manipulator’s joints control torque functions are programmed to meet the soft touch requirement that the end-effector’s relative velocity to the space target is zero at touch moment. Numerical simulation validation is conducted finally.

#### 1. Introduction

Tethered space robot and space manipulator are two means for on-orbit operation such as satellite maintenance, refueling, retrieval, or space debris removal. The main advantage of the former over the latter is its long operation distance (from tens of meters to dozens of kilometers), while the disadvantage of the former is obvious such as inherent instability in retrieval procedure and huge difficulties in control design for the procedure [1]. Compared to tethered space robot, space manipulator can work efficiently within the scope of tens of meters or smaller [2, 3], such as SRMS (Shuttle Remote Manipulator System) [2].

Generally, the system dynamics of space manipulator is more complex than that on the ground [4, 5]. In the field of space manipulator dynamics, [6] presents the singularity-free dynamic equations of the dynamically equivalent manipulator (DEM) of spacecraft-manipulator systems, which is computationally efficient and meanwhile possesses the same dynamic properties as the corresponding free-floating spacecraft-manipulator system. Reference [7] outlines a procedure for modeling three-dimensional flexible multibody manipulators during maneuvering and payload capture. The manipulators are capable of performing large rigid-body translations and rotations, as well as experiencing deformation as a result of their inherent flexibility. This motion is highly nonlinear, so a nonlinear corotational finite element method is used. Reference [8] presents a study to understand the true dynamics of the tested space manipulator aiming at the problem encountered in ground physical simulation that the air-bearing support system changes the dynamics characteristics of the manipulator such as the natural frequencies, stiffness, and damping. References [9–11] study the configuration, dynamics, and control problem of a tendon-actuated lightweight space manipulator.

Trajectory optimization of manipulator with redundant degree of freedom for capturing space target is researched in [12, 13] with the influence of manipulator’s movement on its carrier taken into account. An improved manipulator trajectory optimization method for such system with nonconserved momentum and angular momentum is presented in [14]. The method allows minimization of a quadratic norm connected with the power use of motors in manipulator joints. The method differs from its previous version in including the possibility of constraint final velocity of the end-effector and by considering period of the capture maneuver as a parameter that is optimized.

The closed loop control law is designed for manipulator when approaching and capturing space target in [15, 16]. In [15], the flexibility of manipulator’s joints is considered, and then the notion of artificial potential is introduced to design the control law so as to guarantee the stability of the closed loop, while linear quadratic control theory is adopted in [16]. In [17], contact mechanics between the end-effector of manipulator and space target is taken into account, and the dynamics for the whole procedure is modeled; then the control law is designed. Reference [18] has discussed the control strategies for the autonomous target capture of a free-flying space robot. The highlight is made on the reactionless manipulation and the generalized Jacobian matrix. The reactionless manipulation is obtained from the reaction null-space (RNS) of the coupling inertia matrix that describes the dynamic interaction between the manipulator and the base attitude motions. The obtained manipulator motion yields zero attitude disturbance on the base satellite. Reference [19] presents a method for control of micro-macro manipulators. The method splits the control problem into two parts. The first part is to devise a controller which moves the macro manipulator close to the desired path, and the second part is to design a controller to enforce the tracking error of the end-effector to zero. The main advantage of the controller is that it does not try to prevent fast motion of macro manipulator to avoid excitation of higher modes. In [20], fuzzy logic is applied to control a novel two-link robotic arm. The control system has three levels. A conventional controller consisting of the feedback linearization technique combined with proportional-derivative control is used in the bottom level of the hierarchical system to control the servomotors of the robot. A second layer consists of a servo-expert that preprocesses the high-resolution information coming from joint encoders and extracts the status of the system. A third, intelligent layer is added at the top of the hierarchy to complete the control structure. The main purpose of the top level is to tune the parameters of the conventional controller to improve the response of the system.

For tethered space robot dynamics and control problem, the involved orbit dynamics is taken into account adequately [21–30]. In [21–25], the main satellite or shuttle (from which the tethered space robot is released out) is on circular orbit around the earth, while in [26–30], the influence of the eccentricity of the main satellite’s ellipse orbit on the dynamics of the tethered system is studied. Compared to the tethered space robot, almost all the dynamics modeling for space manipulator presented in documented literatures to date is conducted with respect to inertial reference system; that is to say, the influence of the relative orbit dynamics between the carrier and the target on the system dynamics modeling is not taken into account. The paper attempts to derive the dynamics model of two-link manipulator when its end-effector approaches target based on Kane method with respect to the carrier body-fixed noninertial coordinate system; that is to say, the relative orbit dynamics of the carrier to the space target is considered in system dynamics modeling. Then the end-effector’s path function and the manipulator’s corresponding joints control torque function are programmed to meet the requirement of soft touch, which means that the relative velocity of the end-effector to the target is zero at touch moment. Finally, numerical simulation will be conducted to validate the work done in the paper.

#### 2. System Dynamics Modeling

##### 2.1. Preliminary Work

A space target is supposed to fly on a circular orbit around the earth. An operation spacecraft (called carrier uniformly afterwards in the paper) flies freely around the target governed by the well-known homogenous C-W equations [31], as shown in Figure 1. In the figure, the origin of coordinate system denotes the space target. Axis points the direction of the velocity of the space target relative to the earth’s center, and axis lies on the extended line from the earth’s center to the space target with the positive direction deviating from the earth. The ellipse denotes the relative trajectory of the carrier to the target and is supposed to be coplanar with the space target’s circular orbit around the earth. The direction of the carrier flying around the target in Figure 1 should be anticlockwise according to C-W equations. Two links of the manipulator are identified as I and II. Their mass, moment of inertia with respect to their centers of mass, length, and angle of rotation are ( corresponding to I, II), respectively, and both of the two links are assumed to be homogenous in mass. Two motors are mounted on the joints identified as I and II, and their driving torques are and , respectively. The distance between joint I and the center of mass of the carrier is . The assumption that the mass and inertia moment of the carrier is far bigger than that of the manipulator is adopted here, and then the influence of the manipulator’s movement to the carrier can be ignored. We also suppose that the attitude control system of the carrier guarantees its side on which the manipulator is mounted points accurately always the space target during approaching operation.