# The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory
## 1 Introduction  
Trajectory planning, e.g., of industrial manipulators, mobile robots, toolheads of treatment machines or optical disk drives, etc., is an important issue in the field of robotics, machines construction, automatic and control engineering being a subject of investigations for many researchers. Broader elaboration of the issues associated with planning trajectories of machines and robots executory systems can be found in the book by [PERSON] and [PERSON] [1]. The papers in scientific journals can be divided into several groups. Some of them focus on planning smooth trajectories with constraints of selected kinematic quantities. In papers [2; 3], algebraic splines, B-splines and trigonometric splines were used for trajectory planning. The method of trajectory planning technique described by B-splines was presented in [4]. The method of trajectory planning with jerk and acceleration constraints that uses algebraic splines to ensure a smooth trajectory is presented in [5]. [PERSON] et al. [6] used Catmull-Rom splines for trajectory planning. The S-curve and AS-curve for trajectory planning are described in many papers [7; 8; 9; 10; 11; 12; 13; 14; 15]. Some papers were on planning trajectory and its optimization applying various functions of purpose and various constraints. Planning of minimal-time trajectory by means of algebraic splines with kinematic and dynamic constraints is discussed in [16; 17]. The S-curve was used for a planning trajectory in paper [18]. The author used constant values of jerk at the transition between the constant acceleration and deceleration periods. In this way, the minimal time of transition from the actual to assumed velocity on the motion sections with imposed jerk and acceleration constraints was obtained. [PERSON] et al. [19] used an S-curve with fifteen segment motion profile for trajectory planning. It is shown that these trajectories are time-optimal in the most relevant cases. [PERSON] et al. [20] usedalgebraic splines for trajectory description and optimized them by means of GA (Genetic Algorithms). The aim of optimization was obtaining a minimal jerk, taking into account the additional constraints. The papers by [PERSON] and [PERSON] present the way of planning smooth trajectories where B-splines and algebraic splines were applied for the description. The objective function takes into account the execution time as well as the integral squared jerk [21; 22; 23]. [PERSON] et al. [24] used the general model of the S-curve for planning minimal-time trajectory. They presented also a strategy based on the trigonometric models of this curve. [PERSON] and [PERSON] [25; 26] used the S-curve employing the filters FIR (Finite Impulse Response) for planning the minimum-time trajectories with kinematic and frequency constraints. Paper [27] applied the S-curve with the constraints of snap, jerk, acceleration and velocity as well as the optimization algorithm for reducing vibrations and developing the quality of machines CNC processing. [PERSON] et al. [28] employed B-splines for planning trajectories. They applied the algorithm SQP (Sequential Quadratic Programming) for the optimization of motion time taking into account the limits on the velocities and accelerations for each joint of the robot. [PERSON] et al. [29] used the S-curve for planning the trajectory of the end-effector of the manipulator placed on a mobile platform. To obtain the time-optimal trajectory they made use of the MASA (Multi-Axis Synchronization Algorithm). For planning the trajectory, [PERSON] et al. [30] proposed B-splines which were optimized by means of the hybrid algorithm WOA-GA (Whale Optimization Algorithm and Genetic Algorithm). As the objective function, the integral of the squared jerk along the entire trajectory and the total execution time were taken. [PERSON] et al. [31] proposed the algorithm of planning the PTP (Point-To-Point) trajectory in the joint space. It is based on a locally improved asymmetrical jerk motion profile to obtain time-optimal and smooth joint trajectories.