## 3 Optimal Time-Jerk Problem Description  
A high-speed robot is expected to move quickly while performing pick-and-place tasks. However, increasing the frequency of pick and place unavoidably increases acceleration and jerk, thereby affecting the stability and position accuracy of the robot motion. Trajectory optimization is an effective scheme for reducing the robot jerk. An optimization process can be reviewed according to the following two aspects: firstly, finding the shortest movement time, and secondly, the motion stability during operation. Since the motion stability is mainly reflected in the acceleration, it is necessary to minimize the peak acceleration of the robot during the process. Jerk is defined as the first derivative of acceleration with respect to time, which reflects the change rate of acceleration and is an important indicator for evaluating the smoothness of the robot motion. More specifically, the greater the jerkiness, the faster the acceleration change. According to [PERSON]'s second law, acceleration affects the applied force on the robot driving joint. Accordingly, when the acceleration changes frequently, the motor that drives the joint should constantly change the output torque to control the robot movement, thereby leading to the chattering phenomenon in the motor and affecting the motion accuracy of the robot. Therefore, it is essential to consider the variations in acceleration on the dynamic performance of the robot. This is especially the case for high-speed pick-and-place tasks. The larger the jerk, the lower the smoothness of the robot motion. In this paper, the average cumulative effect of trajectory mutation is defined as an index to evaluate the smoothness of the trajectory. To consider the total running time and the robot trajectory, the optimization objective can be mathematically expressed as follows:  
\[\left\{ \begin{aligned} \kappa_{1}=\sum_{i=1}^{n-1}h_{i}\\ \kappa_{2}=\sum_{j=1}^{N}\sqrt{\frac{1}{t_{f}}\int_{0}^{t_{f}} \left(\vec{p}_{j}(t)\right)^{2}}dt\end{aligned} \right. \tag{2}\]  
Figure 1: Delta parallel robot. (**a**) Prototype of Delta parallel robot; (**b**) Sketch of Delta parallel robot. where \(\kappa_{1}\) and \(\kappa_{2}\) are the objective functions for the motion time and the robot stability, respectively. Moreover, \(t_{f}\) and \(p_{j}\) denote the total movement time and the angular displacement of the driver arm, respectively. Equation (2) is subject to the following kinematic constraints:  
\[\left|\dot{p}_{j}\right|\leq v_{\max}\]  
\[\left|\dot{p}_{j}\right|\leq acc_{\max} \tag{3}\]  
\[\left|\dot{\overline{p}}_{j}\right|\leq jerk_{\max}\]  
where \(v_{\max}\), \(acc_{\max}\), and \(jerk_{\max}\) are the maximum values of the velocity, acceleration, and jerk, respectively. The NURBS curve lies within the merging set of individual convex packages formed by the control vertices [29]. Consequently, this curve can be applied to convert the kinematic constraints to constraints on individual control vertices. The converted constraints can be expressed in the form below:  
\[\left\{ \begin{array}{l}\left|d_{j}^{1}\right|\leq v_{\max}\\ \\ \left|d_{j}^{2}\right|\leq acc_{\max}\\ \\ \left|d_{j}^{3}\right|\leq jerk_{\max}\end{array} \right. \tag{4}\]  
However, the time and jerk are a set of conflicting objectives. On the on hand, as the running time reduces, the corresponding jerk fluctuations increase, which is not conducive to smoothing the trajectory. On the other hand, as the motion jerk reduces, the total motion time increases, which affects the efficiency of the operation. Accordingly, weighting factors are usually used in multi-objective optimization problems to optimize the running time and meet the acceleration requirements. According to the weight selection of the two optimization objectives, the optimization objective function is defined as follows:  
\[f=\frac{\sum_{i=1}^{n}\left(w\left[ \begin{array}{cc}\kappa_{1}&\kappa_{2} \end{array}