# Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot
###### Abstract  
The paper discusses the application of a motion profile with an elliptic jerk to Cartesian space position control of serial robots. This motion profile is obtained by means of a kinematic approach, starting from the jerk profile and then calculating acceleration, velocity and position by successive integrations. Until now, this profile has been compared to other motion laws (trapezoidal velocity, trapezoidal acceleration, cycloidal, sinusoidal jerk, modified sinusoidal jerk) considering single-input single-output systems. In this work, the comparison is extended to nonlinear multi-input multi-output systems, investigating the application to Cartesian space position control of serial robots. As case study, a 4-DOF SCARA-like architecture with elastic balancing is considered; both an integer-order and a fractional-order controller are applied. Multibody simulation results show that, independently of the controller, the behavior of the robot using the elliptic jerk profile is similar to the case of adopting the sinusoidal jerk and modified sinusoidal jerk laws, but with a slight reduction in the position error (\(-3.8\%\) with respect to the sinusoidal jerk law and \(-0.8\%\) with respect to the modified sinusoidal jerk law in terms of Integral Square Error) and of the control effort (\(-8.2\%\) with respect to the sinusoidal jerk law and \(-1.3\%\) with respect to the modified sinusoidal jerk law in terms of Integral Control Effort). elliptic jerk; motion profile; Cartesian space control; SCARA robot; Schoenflies' motion; multibody simulation Article Application of Elliptic Jerk Motion Profile to Cartesian Space Position Control of a Serial Robot. 2023135601. [[https://doi.org/10.3390/app13095601](https://doi.org/10.3390/app13095601)]([https://doi.org/10.3390/app13095601](https://doi.org/10.3390/app13095601))  
21 March 202317 April 202317 April 202330 April  
Besides their simplicity, the scientific interest about kinematic methods has another motivation: in the case of repetitive tasks the robot motion can be optimized offline to maximize a specific performance index by dynamic methods. On the other hand, in some cases, it is not possible to optimize the robot motion offline, since the trajectory must be replanned depending on external events, not predictable a priori. An example, with growing industrial importance, is collaborative robotics, in which the robot must modify its trajectory when a human operator enters its workspace [8]. In these cases, the robot motion must be quickly replanned online without time-expensive optimization techniques, but adopting predefined motion profiles. Commonly, to avoid collisions with obstacles, the robot trajectory must be controlled in the Cartesian space of the end-effector external coordinates. 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. As a case study, the SCARA-like RRFbR robot (Revolute, Revolute, Four-bar, Revolute) with elastic balancing is considered [11]. As regards the algorithm for Cartesian space position control, both an integer-order (IO) and a fractional-order (FO) scheme have been taken into account. Classical Cartesian space controllers (KD) are IO, with a stiffness matrix K that defines the end-effector stiffness in the external coordinates, and a damping matrix D which expresses the end-effector damping, related to the first-order derivatives of the end-effector coordinate errors. Consequently, the KD Cartesian space position control represents a MIMO extension of the PD control for SISO systems, in which there is a term P proportional to the error and a term D proportional to the first-order derivative of the error.