# Trajectory Planning and Optimization for a Par4 Parallel Robot Based on Energy Consumption
## 1 Introduction
For example, in [26], a method of trajectory planning with minimum energy damping was proposed for a flexible-base robot, considering the point-to-point motion task. The maximum residual vibration amplitude and operating energy were taken as objective functions, and the radial basis function networks (RBFNs) were optimized using non-dominated sorting genetic algorithms-II (NSGA-II). In [27], a design method based on the concept of universal redundancy was proposed to improve the dynamic performance of the mechanism. After the servo motor's motion path was divided into multiple segments, the trajectory planning of each segment was designed, respectively, applying the law of polynomial motion. Some polynomial parameters, as well as the speed and acceleration of the working point, should be selected with the minimal optimization for energy and vibration force, based on the genetic algorithm. The validity of the method was verified by experiments, but since there were many parameters to be optimized, it was easy to fall into the local optimum. An underactuated self-reconfigurable robot with active and passive joints was designed in [28], and reconfiguration planning and scheduling were carried out with the aim of minimizing energy consumption based on the genetic algorithm. In [29], three trajectories were designed based on a model of an underwater robot, based on minimum energy, minimum turbulence, and minimum torque change rate, respectively. Finally, the fourth trajectory with minimal energy based on the genetic algorithm was planned, in which a polynomial approximates the time histories of the trajectory in joint space; the running time and the coordinates of an intermediate point were taken as the object to be optimized, and the quartic polynomial was selected as its motion law. The advantage of this method was that it was not reliant on an accurate mathematical model; however, since the intermediate point was obtained by random search, the shape of the trajectory could not be well controlled. At present, to the best of our knowledge, no trajectory planning involving energy consumption has yet been elaborated for a Par4 parallel robot. In this paper, in order to study the trajectory planning and energy consumption of the Par4 parallel robot, the minimum energy consumption was taken as the optimized object. High-speed pick-and-place operation is one of the main working modes of parallel robots [30; 31], and a gate-shaped trajectory is usually adopted, as shown in Figure 1. In the figure, A and H are the pick-and-place endpoints, B, C, E, F and G the intermediate points on the pick-and-place track, \(H_{a}\) is the vertical displacement, also known as pick-up height, and \(B_{a}\) is the horizontal displacement, also known as pick-up span. The trajectory is symmetric with respect to the dot line MN. A polynomial of third order or more, a modified trapezoid and Bang-Bang algorithms are usually applied as the law of motion [32; 33]. Due to the right-angle connection of the conventional gate trajectory (as shown by the solid line in Figure 1), it is easy to cause the discontinuity of the motor's speed and acceleration in the process of movement, which leads to severe mechanical vibration in the robot. Therefore, a smooth transition is often applied instead of the right-angle connection, as shown by the dotted line in Figure 1. The ideal transition curve should meet the criterion of continuous curvature. In [34], taking the five-bar mechanism as the object, a variety of motion trajectories were compared and analyzed, and the results show that the Lame curve is better than the others in terms of residual vibration and the precision of the final position. However, there was no optimization research about the trajectory with the Lame curve in [34].