# A Time-Optimal Continuous Jerk Trajectory Planning Algorithm for Manipulators
## 1 Introduction  
With the continuous development of industrial automation, many manual operations in the field of industrial production have been replaced by industrial robots. With its accuracy, flexibility and high efficiency, industrial robots have greatly improved labor productivity, reduced the burden on workers, and can even perform many complex tasks that cannot be performed by humans [1, 2]. Motion control technology guarantees the stable operation of robots, and the motion control algorithm is the core of robot control. In order to improve the working efficiency of robots, avoid the harm caused by joint vibration and prolong the service life of robots, trajectory planning for the robots has received close attention. The trajectory planning scheme determines the motion mode and operation performance of robots. Different planning schemes are required for different applications, which greatly promotes the development of trajectory planning research of modern industrial robots [3]. In modern industrial automation applications, such as welding and spraying, the robot end effector must move according to certain operating requirements, and there are strict requirements for its displacement, speed and even acceleration, so specific trajectory planning must be carried out. On the other hand, the quality of the trajectory planning scheme directly determines the movement mode, operation accuracy and service life of the robot. A good trajectory planning scheme can not only enable the robot to complete the task accurately, but also ensure good motion stability and small mechanical wear, as well as have a good performance in time utilization and energy consumption [4, 5, 6]. Therefore, trajectory optimization has become one of the research hotspots in the field of robotics [7, 8]. Traditional time-optimal motion algorithms with jump acceleration mostly focus on maximizing efficiency from the perspective of time, which will cause flutter, which may damage the actuator and transmission components, resulting in large tracking errors, making it difficult to realize these laws on the machine in reality [9]. Therefore, from the point of view of practical implementation, non-smooth trajectories should be avoided as much as possible. The non-smooth trajectory may cause sudden changes in joint torque, which the actuator cannot provide. Improved motion control can be achieved through accurate dynamic modeling of the robot [10], or by forcing the programming module to generate trajectories with sufficiently high continuity, preferably at least to a level of acceleration where the jerk is finite [11]. In addition, the planned trajectory should also meet specific constraints on position, speed and acceleration according to the mechanical characteristics of the drive device to ensure the dynamic performance of the system [12]. [PERSON] [13] used sequential quadratic programming to improve the payload capacity of a robotic arm by optimizing the cubic spline and Bernstein polynomial trajectories. A polynomial model combined with minimum disturbance theory was used to create a smooth human-like motion for the auxiliary robot [14]. However, motion constraints were not considered. Ref. [15] presented a constrained control scheme to steer a UAV to the desired position while ensuring constraint satisfaction at all times. The authors of [16] utilized the input shaping algorithm instead of the jerk constraint in the trajectory optimization model to achieve a smooth trajectory. In [17], an optimal trajectory planning method based on a Pythagorean curve for delta robot pick-and-place operation time was proposed. The authors of [18] expressed the equation of motion with a fifth-degree polynomial function and obtained the minimum kinematically constrained jump trajectory.