# The Use of Asymmetric Polynomial Profiles for Planning a Smooth Trajectory
## 1 Introduction
Many papers in the literature discuss the application of polynomials for description of trajectories. However, it should be pointed out that the polynomials of a degree higher than seven were used for the displacement description in only a few papers. [PERSON] and [PERSON] [32] presented the way of planning the trajectory for two robots, taking into account the deformations in the joints. The changes of the displacement of the manipulated object were described by applying 9 th degree polynomials. [PERSON] and [PERSON] [33] used higher-degree polynomials for trajectory planning. They used 5 th, 7 th and 9 th degree polynomials to describe the acceleration profile, with a constraint acceleration. Subsequent papers used polynomials of higher degrees to describe trajectories for various applications [34; 35; 36; 37; 38]. [PERSON] et al. [39] used 6 th, 5 th, 4 th and 3 rd degree polynomials as well as cycloidal formula and elliptical formula to trajectory planning of a surgical micro-robot. [PERSON] and [PERSON] [40] used high degree polynomials for trajectories planning. They used the algorithm CS (Cuckoo Search) for the polynomial coefficients determination. [PERSON] et al. [41] proposed a method for planning the trajectory in which the acceleration was described by means of the 4 th degree polynomial. According to the authors, it is possible to obtain a near time-optimal trajectory by maximizing the constant velocity part under kinematical constraints. [PERSON] and [PERSON] [42] presented planning of the trajectory for a parallel robot. In the end-effector operating space the rectangular transition of the pick-and-place trajectory was rounded with the Lame curve. The algorithm GWO (Grey Wolf Optimizer) was suggested for the energy consumption optimization, whereby the reduction of residual vibration and precision of manipulator work were obtained using 5 th and 6 th degree polynomials. Table 1 presents the exemplary papers in which the most frequent methods of trajectories parameterization were used. The objective of the paper is elaboration of trajectory planning algorithm using asymmetric polynomial profiles. Contrary to the previous papers, the asymmetry k coefficient was introduced to obtain the velocity asymmetric profile. Similar to the previous works [33; 37; 38], the most frequently used description of polynomial trajectory, in the form of a power expansion, was neglected and the root multiplicity was applied for building the polynomial defining the acceleration function. For the given trajectory with kinematic constraints, the problem consists in calculation of one coefficient of the polynomial, asymmetry coefficient as well as the time of realization independent of the polynomial degree. An important aspect of the proposed solution is that the calculated quantities can be considered the dependences describing velocity, position and jerk. This reduces the calculation complexity of parameterization in the time domain. The paper arrangement is as follows. Section 2 presents the way of formulation of acceleration profile using the roots multiplicity of the polynomial and the description of the conditions imposed on the acceleration function. The root multiplicity was collated for the 5 th-9 th degree polynomials and the polynomials for the detailed analysis were chosen. A method of calculating the polynomial coefficient, asymmetry coefficient and motion time at the constraint of single kinematic quantity, i.e., velocity, acceleration and jerk as well as those of a few kinematic quantities at the same time is presented. Section 3 compares the profiles of displacement, velocity, acceleration and jerk for one or a few constraints based on the results of simulation made for the rectilinear trajectory of the end-effector. It also includes the results of simulation for the trajectory with two intermediate points and the comparison of polynomial trajectory with the asymmetric AS-curve. Chapter 4 presents a detailed discussion. The conclusions are given in the last chapter.