# Joints Trajectory Planning of Robot Based on Slime Mould Whale Optimization Algorithm
## 1 Introduction
### Literature Review  
Trajectory planning works to solve the displacement, velocity and acceleration of each joint in real time according to the motion path and trajectory of the manipulator based on its task requirements, so as to generate a motion trajectory. When trajectory planning is carried out in joint space, a time function is generally used to describe joint variables and plan their first-order and second-order time derivatives ([PERSON], 2016). It is therefore also called planning considering differential constraints. In general, the motion path points are first transformed into joint path points using inverse kinematics equations, and then smooth fitting is performed on each joint path point ([PERSON], 2016). Commonly used trajectory planning interpolation algorithms include cubic polynomial, quintic polynomial, cubic B-spline, parabolic transition, mixed interpolation, etc. The cubic polynomial is simple to calculate, but it cannot guarantee continuous acceleration. Due to vibration and impact in the machining process it is a suitable algorithm for simple point-to-point motion. The quintic polynomial increases the acceleration constraint, which reduces the impact and increases the computation. It is suitable for high-speed and high-performance motion from point to point. The multi-segment polynomial interpolation algorithm is suitable for multi-point continuous motion, but the smoothness of the joints is not guaranteed. Spline curves with geometric invariance, local support and other characteristics can make multi-segment trajectory transitions smooth and reduce joint impact. In the Cartesian space, the joint-angles need to be obtained by solving the inverse kinematics equation many times, and angular velocities of the joints are obtained by solving the inverse of the Jacobian matrix ([PERSON], 2016). For the optimal trajectory planning of manipulator joints, the current research focuses on optimization based on time, system energy and the presence of multiple objectives. For the study of time-optimal trajectory planning, [PERSON] et al. (2016) fused kinematic methods and evolutionary strategies for the trajectory planning of optimizers. This is effective for specific articulated robots for which it is difficult to obtain exact dynamics equations. [PERSON] et al. (2016) studied the planning of time-optimal trajectories of space robots under dynamical constraints based on particle swarm algorithms. [PERSON] et al. (2017) interpolated the position sequence of joints using cubic spline. An adaptive genetic algorithm that introduced the penalty function was used to optimize the shortest time trajectory. [PERSON] (2017) proposed a multi-variate time optimization method to optimize the trajectory of each joint based on a smooth trajectory generator. [PERSON] et al. (2017) combined a second-order continuous polynomial interpolation function with the optimization subject of a cosine reduced-weight particle swarm constrained by kinematics and dynamics. A time-optimal (TO) strategy was proposed and the motion time of the manipulator was successfully optimized. [PERSON] et al. (2017) transformed the trajectory of a manipulator into a position and time series in the joint space by inverse kinematics. The quintic B-spline curve was used for interpolation. The objective function was optimized by the non-dominated sorting genetic algorithm (NSGA-II) and a time-optimal, safe, collision-free trajectory planning method was proposed. [PERSON] et al. (2017) designed a time-optimal trajectory planning method for robotic arms, which could search for optimal paths simultaneously. A uniform cubic b-spline interpolation algorithm was proposed to derive the motion curve expression of the manipulator joint with unknown path points. The trajectory was optimized based on the genetic algorithm. [PERSON] et al. [16] introduced chaos strategy and inertia weight to optimize the whale optimization algorithm and applied it to the trajectory planning of a robot arm.