Analytical expression of motion profiles with elliptic jerk

Abstract

The paper discusses the analytical expressions of a motion profile characterized by elliptic jerk. This motion profile is obtained through a kinematic approach, defining the jerk profile and then obtaining acceleration, velocity, and displacement laws by successive integrations. A dimensionless formulation is adopted for the sake of generality. The main characteristics of the profile are analyzed, outlining the relationships between the profile parameters. A kinematic comparison with other motion laws is carried out: trapezoidal velocity, trapezoidal acceleration, cycloidal, sinusoidal jerk, and modified sinusoidal jerk. Then, the features of these motion profiles are evaluated in a dynamic case study, assessing the vibrations induced to a second-order linear system with different levels of damping. The results show that the proposed motion law provides a good compromise between different performance indexes (settling time, maximum absolute values of velocity and acceleration).

Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot

“…The number of independent dimensionless profile parameters can be further reduced by adding the following two hypotheses: t ad,papj = t ad,panj , and t ad,nanj = t ad,napj . It is possible to demonstrate that these two further assumptions lead to symmetric peak profiles both in acceleration and in deceleration (j ad1 = j ad3 and j ad5 = j ad7 ) [10]. It can be understood graphically considering that the integral of the jerk is the acceleration; consequently, if the semi-elliptical areas delimited by the jerk profiles of phases #1 and #3 are equal, the acceleration is null at the beginning of phase #4 (constant velocity); similarly, if the semielliptical areas delimited by the jerk profiles of phases #5 and #7 are equal, the acceleration is null at the end of the rest-to-rest motion.…”

“…Figure 3 shows the elliptic jerk motion profile [10] in dimensionless formulation, adopted for the sake of generality. For any motion displacement d and motion duration T, it is possible to define: The elliptic jerk profile is divided into seven phases (#1-#7, Figure 3).…”

“…A profile fulfilling these conditions is named a symmetric elliptic jerk profile (Figure 4). For a symmetric elliptic jerk profile in dimensionless formulation, the number of independent parameters is four (t ad,pa , t ad,papj = t ad,panj , t ad,na , t ad,nanj = t ad,napj ) [10].…”

“…A profile fulfilling these conditions is named a symmetric elliptic jerk profile (Figure 4). For a symmetric elliptic jerk profile in dimensionless formulation, the number of independent parameters is four (tad,pa, tad,papj = tad,panj, tad,na, tad,nanj = tad,napj) [10]. In [10] the elliptic jerk motion profile (EJ) has been compared to other motion laws discussed in the scientific literature: the trapezoidal velocity profile (TV, [5]), the trapezoidal acceleration profile or S-curve (TA, [5]), the cycloidal profile (CY, [5]), the sinusoidal jerk profile (SJ, [20]), and the modified sinusoidal jerk profile (MSJ, [21]).…”

“…In this paper, a recently proposed motion profile with elliptic jerk [9] is compared to other profiles discussed in the scientific literature with reference to the position control of robots. In [9,10] the elliptic jerk profile has been discussed with application to single-input single-output (SISO) systems. In this work, the investigation is extended to a multi-input multi-output (MIMO) system, a serial robot with Cartesian space position control.…”