**1. Introduction**

Space robotics are now increasingly employed in outer space for various tasks, such as the assembly and maintenance of the International Space Station (ISS) [1], and Lunar Base construction [2]. Examples of space robotics include the Japanese Experiment Module Remote Manipulator System (JEMRMS) of the Japan Aerospace Exploration Agency (JAXA) [3] and the Canadarm 2 of MacDonald Dettwiler and Associates Ltd. (MDA) [4], while examples of performing tasks include the Experimental Test Satellite VII (EST-VII) [5] and the Robot Technology Experiment [6]. The free-flying space robot (FFSR1) and the free-floating space robot (FFSR2) are two main types of space robotics. The base attitude of FFSR1 is controlled, contributing to establishing a good connection between the ground and the base spacecraft. This is because the attitude of the base spacecraft should be well-managed when sending signals to, or receiving signals from, the ground. The base spacecraft of FFSR2 is in free-floating mode, contributing to saving energy, which is part of the superiority of FFSR2 since energy is precious in outer-space environments.

The trajectory planning for space robotics, aiming at generating the time histories of joints (or end effector) and contributing to the desired motion of robots, attracts extensive attention from both scientists and practitioners. The study of trajectory planning is essential for robots before physical manipulation and has been well developed. A primary trajectory-planning approach is based on the inverse kinematics of space robotics. Concepts such as the generalized Jacobian matrix [7], the enhanced disturbance map [8], the path independent workspace [9] and the reaction null-space [10] have been successively proposed for trajectory planning. The disadvantage of this approach is that it leads to kinematic singularity in some cases. Aiming at avoiding this kinematic singularity, an approach based on the direct kinematics of space robotics has been well-received. To begin with, mathematical functions such as the polynomial, trigonometric [11], or the Bézier function [12] are employed to depict the joint trajectories. Then, according to predefined conditions, mathematical functions are depicted with a set of unknown coefficients. In the aforementioned mathematical functions, the polynomial function denotes the combination of constants and variables with limited addition and multiplication calculations, the trigonometric function is an elementary function with variables of angles, and the Bézier function contributes to the derivation of a smooth curve based on four random points. Next, the trajectory-planning problem is converted into a parameter-optimization problem, with the objective function of minimum maneuver time or maximum manipulability of the robotic system, or minimum attitude disturbance acting on the free-floating base spacecraft during the robotic maneuver. Finally, the unknown coefficients are optimized by the optimization algorithms, including the basic heuristic algorithms such as the particle-swarm optimization algorithm (PSO) [13] and genetic algorithm (GA) [14,15], and the improved optimization algorithms such as hybrid PSO [16] and the constrained differential evolution algorithm (DE) [17]. In the aforementioned algorithms, the PSO is a random search algorithm based on group collaboration, more specifically, simulating the foraging behavior of a flock of birds. The DE, based on population, is a self-adaptive optimization algorithm with global search capability. To the authors' knowledge, the approaches mentioned above aim at solving the point-to-point trajectory-planning problem, meaning that the space robot executes one task in each travel. If the robot executes two or more tasks in each travel, it could save much energy. Maneuvering time would also be reduced, which is a significant advantage in an emergency. Therefore, it is meaningful to study the multitask-based trajectory-planning problem (MTTP) for space robotics.

As a matter of fact, the MTTP for industrial robotics has been widely studied for its high productivity and low cost [18,19], and two categories of approaches were developed. For simplicity, the location of each task is usually considered to be a waypoint, which is also followed. One category of approaches aims at directly solving the MTTP. In [20], the MTTP is studied using the branch-and-bound method, where multiple inverse kinematic solutions of the robotic system are not considered. The branch-and-bound method aims at searching for solutions to solve optimization problems with constraints, where the feasible solution space is finite.

In [21], the MTTP is transformed into a mixed-integer nonlinear programming problem, where a solver is developed to improve calculation speed. The approach developed in [21] works well with a relatively small number of waypoints. In [22], an improved genetic algorithm (IGA) was developed where each chromosome consists of two parts. The first part represents the waypoint sequence, while the second part represents the sequence of the joint configurations at the waypoints, which corresponds to the first part. In [23], each chromosome consists of three parts, the waypoint sequence, the sequence of joint configurations, and the robot placement. The technique of dividing each chromosome into several parts was also employed in [24–26]. A common point of IGAs in [22–26] is that the parameter denoting the joint angular velocity is predefined as a constant. The other category of approaches aims at dividing MTTP into several subproblems which are then solved successively. In [27,28], MTTP is divided into two subproblems, the problem of the waypoint sequence, and the problem of joint trajectories. In [27], Tabu search was employed to optimize the waypoint sequence, after which joint trajectories were derived according to the inverse kinematics of industrial robotics. The Tabu search algorithm, a meta-heuristic algorithm, searches for the global optimal solution by constructing a Tabu table with the functions of cycling and memory. In [28], an improved lazy algorithm, according to the direct kinematics of industrial robotics, was developed to optimize the joint trajectories. Among MTTP documents for industrial robotics, manipulators are nonredundant, meaning that a point in the task space corresponds with finite points in the joint space. However, the redundant robot [29] provides the manipulator with high dexterity, contributing to the solution of problems such as obstacle avoidance, singularity avoidance, and joint limits. Unlike ground environments [30,31], there is microgravity in outer-space environments. Moreover, for FFSR2, strong coupling between manipulator and free-floating base is usually generated during a maneuver. Therefore, the aforementioned approaches, which were developed to solve MTTP for nonredundant industrial robotics, cannot be directly employed to solve MTTP for redundant space robotics.

In some cases, an urgent task should be performed in ISS, while the functional completeness of ISS should be guaranteed in advance. Component maintenance, assembly, and refueling of ISS should be finished successively, contributing to the functional completeness of ISS before performing the urgent task. Finishing a series of tasks successively also contributes to saving energy in outer space. Therefore, The MTTP for redundant space robotics is studied. In the MTTP, the location of each task is simplified as a task point, called waypoint. The position and attitude of each waypoint are predefined in the Cartesian space. First, the end effector is required to visit the waypoints with minimum time, thus the sequential order of visiting the waypoints should be optimized. Second, the joint configuration corresponding to each waypoint should be optimized, thus the feasible joint movements between adjacent waypoints can be guaranteed. Third, the joint movements should meet the predefined constraints. Piecewise continuous-sine functions with cubic polynomial arguments are employed to depict the joint trajectories along the waypoints, where each piece of sine function depicts one joint trajectory between adjacent waypoints. With predefined conditions, each piece of sine function is depicted with one unknown parameter which should be optimized. The MTTP is converted into a parameter-optimization problem. An IGA is developed to optimize the unknown parameters. In the IGA, each chromosome consists of three parts. The first part denotes the waypoint sequence, the second part denotes the sequence of joint configurations corresponding with the first part, and the third part denotes a special value corresponding with the depiction of the joint trajectories. Since the system is redundant, each waypoint corresponds with infinite joint configurations. Moreover, considering the sequence of joint configurations directly leads to combination explosion. An approach based on the concept of the dual-arm angle [32] was employed to formulate joint configurations. At each waypoint, eight joint configurations were derived with an assignment of the arm angle.

The rest is organized as follows. Preliminaries and adopted notation for space robotics are presented in Section 2. In the same section, the MTTP, the objective functions, the approach to depicting the joint configurations at each waypoint, and the approach of formulating the joint trajectories are explained. The IGA encoding and updating mechanisms, together with the IGA optimization process, are introduced in Section 3. In Section 4, numerical simulations are carried out to validate IGA, including comparisons with two other approaches. Section 5 concludes the work and gives an outlook for further research.
