# Trajectory Planning and Optimization for a Par4 Parallel Robot Based on Energy Consumption
## 1 Introduction  
In recent years, parallel robots have been widely developed for their optimal dynamic properties [1, 2], especially parallel robots with lower mobility such as Delta, H4, X4, and Par4. [3, 4]. At present, parallel robots are widely used in high-precision fields [5, 6, 7] such as aviation manufacturing, electronic packaging, and 3D printing. A reasonable trajectory planning of a robot can help it not only to perform the desired tasks with satisfactory performance [8], but it is also conducive to improving efficiency [9], reducing energy consumption, and extending the operating life [10, 11]. At present, there are many studies on robot trajectory planning, such as in [12] (point-to-point dynamic trajectory planning was carried out for six degree-of-freedom (six-DOF) parallel robots, attaining good effects), [13] (point-to-point dynamic trajectory planning was carried out for three degree-of-freedom (three-DOF) parallel robots, also attaining good effects), and [14] (a new \(s-\tilde{s}\) plane path method was adopted to design a dynamic feasible point-to-point path and periodic trajectories, based on the forces of geometric constraints). In general, robot trajectory planning can be either on the end-effector or on the joint level [15]. If it is done on the joint level, the end-effector trajectory of the robot should be mapped to the joint motion trajectory based on inverse kinematics [16], then only the joint motion needs to be controlled, and the result of the motion planning is only evaluated at each joint [17]. If done on the end-effectorlevel, the motion planning is directly on the end-effector, and the evaluation and control are all applied on the end-effector [18]. In [19], trajectory planning for a Delta parallel robot was carried out on the end-effector and on the joint level, respectively. Through comparative analysis, it was found that the trajectory planning on the joint level could effectively reduce the maximum torque of the motor and the vibration amplitude of the robot end in the movement process. In view of this, from the lower rated torque of the motor and the lower vibration amplitude of the robot end of view, in this paper the trajectory planning of a Par4 parallel robot will be carried out on the joint level, i.e., first, we will design the shape of movement track and the motion law on the end-effector level, then, the motion feature of the end-effector will be mapped to the joint space, and the follow-up evaluation and control will all be conducted at each joint. Trajectory planning and trajectory optimization are often carried out simultaneously, and the common optimization objectives are minimum time [20], minimum energy [21], minimum pulsation [22], and maximum load-carrying capacity [23]. The minimum time optimization, directly associated with the production efficiency of the robot, was the first of these objectives to be considered, having been widely applied. For instance, in [24], considering three different types of constraints, smooth joint trajectories were constructed with the cubic polynomial function, a genetic algorithm procedure with parallel-populations was applied for the purpose of minimizing time, and finally, the validity of the method was verified; in [25], a time-optimal trajectory planning method was proposed, based on quintic Pythagorean Hodograph (PH) curves, in order to realize a smooth and high-speed operation of a Delta parallel robot. If we blindly pursue the minimum execution time at actual operation, it will easily lead to the discontinuity of torque and acceleration. Energy consumption is also one of the most concerning indexes of parallel robots, and especially when the energy source of the robot system is limited, it is more meaningful to study the minimum energy consumption than other optimization objectives. At present, some researchers have done trajectory planning for robots with minimal energy consumption.