# A Time-Optimal Continuous Jerk Trajectory Planning Algorithm for Manipulators
## 1 Introduction
No matter what kind of planning method is used, the kinematics and dynamics constraints of the robot must be considered to make the first two derivatives of the robot's motion continuous, and a continuous, smooth trajectory without large impact and reversal can be given within the range required by the performance of each robot component [19; 20; 21]. More and more practical applications show that trajectory planning can not only focus on the motion effect and task requirements, but also consider the efficiency, energy consumption, stationarity and other factors in the practical application process. In [22], an ethical trajectory planning algorithm is proposed that makes a variety of ethically significant decisions aimed at equitably distributing risk among road users. It is of more practical significance for engineering to find the optimal trajectory planning scheme under each working condition to guide the robot's operation [11; 23; 24]. To avoid the vibration of the manipulator, prolong the service life of the joints and improve the trajectory tracking accuracy, minimal jerk and jerk continuous trajectory planning have received close attention [25]. Time-optimal trajectory planning aims to plan the shortest execution time trajectory according to the given points while the boundary constraints are satisfied, that is, constructing a trajectory \(p_{m}=f_{m}(t)\) satisfying kinematic constraints, while making \(p_{mi}=f_{m}(t_{i})\) hold and \(T\) minimized, where \(m=1,2,...N,i=0,1,...,n;N\) is the number of joints of the manipulator; and \(n\) is the number of joint positions that the joint trajectory must pass through. When \(f_{m}(t)\) satisfies at least \(C^{3}\) continuity, the jerk trajectory is continuous. In view of the adverse effects of excessive impact on robots, a large number of scholars have discussed such problems as vibration, overshoot, mechanical wear and reduced life caused by the jerk [26; 27; 28]. In order to eliminate the adverse jitter effect and obtain a higher level of continuity and regularity, some works try to construct trajectories by increasing the number of polynomials. In robot picking operations and high-precision positioning systems, seventh-order polynomial interpolation is usually used to satisfy the pumping boundary conditions [29]. Ref. [30] combined cubic splines and seventh-order polynomials to ensure zero jitter at the start and end points of joint motion. The particle swarm optimization (PSO) algorithm is used to generate the time-optimal smooth trajectory. Ref. [31] described a smooth fifth-order B-spline trajectory planning method, whose objective functions were defined as movement time and integral squared acceleration. The human arm modeled by the robot arm also has a minimum jerk setting in terms of movement, which is conducive to humans smoothly grasping and lifting objects [32]. The optimal trajectory planning for the impact of industrial robots aims to find the trajectory that minimizes the impact of the robot. On the one hand, the trajectory tracking error can be reduced to a large extent by reducing the impact of the robot during movement. On the other hand, the defects such as resonance, jitter, mechanical wear and service life reduction caused by excessive impact can be greatly reduced so that the robot can run stably and smoothly. At present, for time-optimal trajectory planning, various algorithms have their own unique problems, and so far, there is no general optimization algorithm to determine the time-optimal trajectory. Thus, in this paper, we propose a new optimal trajectory planning method for the manipulator to optimize its operating efficiency and ensure the smoothness of the motion process. First, the 7 th-degree B-spline with C6 continuity is used to interpolate the joint position sequence, and the continuous jerk trajectory is obtained, whose joint start-stop velocity, acceleration and pulsation can be set separately.