## 2 Point-to-Point Energy Optimization  
The proposed method demonstrates the use of particle swarm optimization (PSO) in the robotic field for PTP trajectory energy optimization. There are several papers that are devoted to PTP optimization, each targeting different goals--many of them aiming at minimal task operation time. However, time is not always an available objective to optimize, especially in specific cases where the arm waits for the completion of a task performed by another agent, gluing, or welding. We focus this manuscript on the energy optimization of possible fixed time trajectories having specific start and end positions. This kind of task represents a typical arm movement during a pick and place job. We only limit the scope of the following research to energy optimization for a robotic arm with six degrees of freedom (DOF) in PTP tasks. Various authors search for energy-efficient path planning and energy-saving methods for robotic arms. There are detailed and depth reviews [3; 7] on various optimization aspects in the field of robotic manipulators. Using the calculus of variations, one can solve the PTP energy optimization as shown in [8; 9]. A simple example of such a problem is to find the curve giving the shortest length for the PTP connection. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. The topology of PTP connection by arm is more complex, and direct computation of geodesics is very difficult [10; 11]. In [12], the authors solve the energy optimization of a manipulator with the use of a genetic algorithm (GA), where the chromosome represents the arm trajectory, as \(n\) points in the joint space. Research by [PERSON] et al. [13] solves the energy optimization for straight, square, and combined motion using robot simulation software, while taking into consideration velocity, acceleration, and trajectory. The optimization module then iteratively evaluates all of the solutions and chooses the one with the lowest energy demand. [PERSON] [2] measures the total robot energy consumption of the whole manipulator, the control unit, and robotic arm, while using an external measurement unit. The simulations and optimization rely on dynamic behavior modeling in Catia software. The optimization key elements are robot accuracy, productivity (execution time), and power consumption reduction. [PERSON] et al. [14] compare the energy optimization results from GA and Constraint Immune multiobjective optimization algorithm. They present the trajectories as eight-point cubic B-Spline curves. [PERSON] and others [15] optimize the energy and time of the trajectories of robotic arms from ABB Corporation, but the producer's policy restricts the use of technical data. The authors of [15] optimize a path with sequential quadratic programming while using a Matlab model that they directly obtain from proprietary software from ABB. Further information on the topic of PTP trajectory energy optimization can be found in review papers [16] in Chapter 4, [3] in Chapter 3, or [17]. All of the papers dealing with PTP energy optimization share similar features. First, there are inverse and direct approaches to optimization. For the inverse approach previously discussed in [18; 19; 20], the researchers develop a mathematical model for the theoretical prediction of power consumption during the execution of particular trajectories. Subsequently, they integrate the data to obtain the theoretical energy consumption estimation over time. Even for a simple movement, there are numerous possible trajectories for an exact specification of the movement. The analytical models vary in complexity and need to handle various aspects of robotic arm movement for all six DOF.