**1. Introduction**

Mapping the path of an end-effector onto a configuration trajectory for the robot to accomplish a desired collision-free task is a well-known problem in robotics [1]. The consideration of redundancy, where the actuated degrees of freedom of the manipulator exceed the end-effector variables defining its functionality in the task space, adds an interesting dimension to the planning problem. It effectively facilitates a scheme where additional objectives can also be incorporated along the way. Beyond obstacle avoidance, constraints such as minimal energy, jerk-free paths, anthropomorphism and so forth can thus be considered. A particularly attractive scenario in motion planning is the avoidance of undesirable singularities in joint space [2], which limits the ability to move in certain task space directions. Increasing the manipulability of the robotic system at each time step is regarded as an effective means of moving away from the neigborhood of such configurations [3], thus reducing the hazardous condition whereby small task space movements may translate to large joint velocities. Avoiding near-singular regions is also a particularly concerning situation when the manipulator might be operating in close proximity to human operators. Moreover, operating away from singularity regions also relaxes the effect of undesirable dynamics that otherwise impose additional perturbances to the robot controllers, hence permitting superior end-effector precision whilst executing the desired task.

This brief proposes a stochastic method that exploits the particular kinematics of closed-chain mechanisms with redundant actuation and a well-known manipulability measure [4] to track the desired end-effector task-space motion in an efficient manner. The approach departs from global solutions with high computational costs, or optimal formulations that can only be solved numerically, without any guarantee of success except for simple obstacle-free problems. The approach has been tested through simulation on

**Citation:** Gil Aparicio, A.; Valls Miro, J. An Efficient Stochastic Constrained Path Planner for Redundant Manipulators. *Appl. Sci.* **2021**, *11*, 10636. https://doi.org/ 10.3390/app112210636

Academic Editors: Luis Gracia and Carlos Perez-Vidal

Received: 30 September 2021 Accepted: 4 November 2021 Published: 11 November 2021

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a number of redundant multibody topologies, and via experimental deployment on the Rethink Robotics 7R Sawyer arm.

The rest of the paper is organised as follows: Section 2 provides broad coverage of techniques in relation with this motion planning for redundant manipulators in the presence of obstacles. Next, Section 3 describes the kinematics of redundant manipulators and the exploitation of the null space. The stochastic algorithm to generate collision-free trajectories is described in Section 4. Section 5 presents a set of experiments carried out both in simulation and with a real platform. Finally, the main conclusions are described in Section 6.
