Exploiting redundancy for jerk minimization in robotic manipulators

Exploiting redundancy for jerk minimization in robotic manipulators Giuliano Fabris University of Udine [email protected] Lorenzo Scalera University of Udine [email protected] Paolo Boscariol University of Padova [email protected] Alessandro Gasparetto University of Udine [email protected] Abstract—This paper presents two approaches for minimizing jerk in redundant manipulators, leveraging the redundancy with respect to the desired task. As optimization variables, we consider the position of a selected redundant joint of the robot and the angles that define the end-effector orientation for each considered way point. These strategies are tested on a seven-degree-offreedom robotic system executing a predefined task. The results of numerical simulations highlight the performance of the proposed strategies in reducing the jerk of the robot end-effector. Index Terms—jerk, optimization, trajectory planning, redundancy, robotics I. INTRODUCTION Trajectory planning is a crucial problem in modern robotics. An important objective of trajectory planning for robotic manipulators is the minimization of the jerk, which can reduce mechanical vibrations and stress of the robotic structure, extend the lifetime of the robot, and increase precision [1]. In the literature, several trajectory planning strategies to achieve motion smoothness has been proposed. These include polynomialbased methods [2], the adoption of B-splines [3], radial basis functions [4], and heuristic optimizations [5]. In redundant robots, functionally redundancy can be exploited to further minimize jerk. A robot is defined functionally redundant if it exhibits a number Nof degrees of freedom (DOFs) greater than the number Rof variables needed to perform a desired task (N > R) [6]. Previous studies show that redundancyaware methods can reduce jerk and improve smoothness while preserving the end-effector path [7]. This is especially useful in collaborative robotics and precision manufacturing [8]. Fig. 1. Considered robotic system with 7 DOFs. In this work, we propose two approaches for smooth trajectory planning that exploit different degrees of redundancy. The proposed strategies optimizes the position of a selected redundant joint of the robot and the angles of the end-effector orientation for each considered way point. Both methods are validated on a 7-DOF manipulator performing a pick-andplace task (Fig. 1), showing improvements of up to 94.7% of jerk reduction compared to a reference case. II. PROPOSED APPROACH x′ x y y′ z z′ ϕ x′ y′ y′′ z′ z′′ x′′ θ x′′ y′′ z′′ x′′′ y′′′ ψ Fig. 2. Euler angles ZYZ (ϕ,θ,ψ). We aim at reducing the mechanical vibrations in a redundant robotic manipulator, that has to execute a predefined task, by minimizing the jerk of the robot end-effector ... X. To reach this goal, we propose two strategies (Approach (1), and Approach (2)) that exploit the redundancy of the robot with respect to the considered task. Given a sequence of way points in the Cartesian space that define the task to execute and the time intervals between the way points (with total time T), the proposed approaches optimize the positions q∗of a selected redundant joint of the robot and the orientation of the robot end-effector ϵ∗for each of the considered way points. More in detail, Approach (1) optimizes q∗and two of the angles that define the end-effector orientation (while the third is fixed), whereas Approach (2) optimizes q∗and all these three angles (the selection of the joint q∗to be optimized is beyond the focus of this work). The end-effector orientation is defined using Euler angles ZYZ (ϕ,θ,ψ), defined as in Fig. 2. The optimization problem for both approaches is formulated as: min q∗,ϵ∗ 6 X i=1 ZT 0 ... X2 i(q, ˙q, ¨q, ... q)dt (1) subject to the kinematic constraints of the considered robot. Once q∗and ϵ∗are defined for each way point, the inverse kinematics is computed and the joint-space trajectory (q,˙q,¨q, ... q) of the robot is parameterized over time using 434 2025 I-RIM Conference October 17-19, Rome, Italy ISBN: 9788894580570 10.5281/zenodo.17629640 57 spline curves, based on third and fourth-degree polynomial functions. In Approach (1), ϵ∗includes two of the Euler angles for each way point, whereas in Approach (2) ϵ∗considers all the Euler angles for each way point. Therefore, Approach (1) considers three optimization variables (one joint position and two end-effector orientation angles) for each way point, while Approach (2) takes into account four optimization variables for each way point. III. SIMULATION RESULTS The performance of the proposed approaches are validated on a 7-DOF Panda arm by Franka Emika GmbH (Fig.1). The considered task is defined using 4 way points as represented in Fig. 4. The optimization problem in (1) is implemented in Matlab 2024b using the genetic algorithm ga. Figure 3 shows the percentage of jerk reduction of the two proposed approaches with respect to the reference case ( ... Xrms = 3.01 m/s3), where the orientation of the robot end-effector at the considered way points is fixed, as in [9]. As it can be seen from the figure, the greatest jerk reduction with Approach (1) is achieved when the positions of joint q∗ 7and the orientation angles ϕ∗and θ∗are optimized, obtaining a jerk reduction equal to 94.3% (... Xrms = 2.13 m/s3). Moving to Approach (2), the results of Approach (1) are improved, obtaining a jerk reduction equal to 94.7% when the positions of joint q∗ 7are considered as optimization variables ( ... Xrms = 2.08 m/s3). These results demonstrate the capabilities of the proposed approaches in reducing the jerk (and consequently the mechanical vibrations) of the robot end-effector. Furthermore, Fig. 4 shows the 3D path and the robot configuration at each way point for Approach (2) when q∗ 7is considered. Finally, Fig. 5 reports the end-effector positions, velocities, accelerations, and jerks for the reference and optimal trajectories. Fig. 3. Percentage of jerk reduction with respect to the reference case. ACKNOWLEDGMENT This research was developed within the Interconnected Nord-Est Innovation Ecosystem (iNEST), and partially funded by the European Union NextGeneration EU (PNRR – Mission 4.2, Investment 1.5 – D.D. 1058 23/06/2022, ECS 00000043). REFERENCES [1] Z. Wu, F. Wang, and T. Bao. Optimal Time–Jerk Trajectory Planning for Manipulators Based on a Constrained Multi-Objective Dream Optimization Algorithm. Machines, 13(8):682, 2025. [2] A. Piazzi and A. Visioli. 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