A method for mapping the boundaries of collision-free reachable workspaces

This paper proposes a method for mapping the boundaries of manipulatorsʹ collision-free reachable workspaces. Knowledge of manipulatorsʹ actual available workspaces is important in applications and design, e.g., the pre-surgery planning for medical robots, the design parameter optimization for manipulators, etc. Algorithms for mapping the boundaries of reachable workspaces have been well developed in previous works, but we have not seen any method that takes into account the obstaclesʹ effects on the manipulatorsʹ available workspaces. In this paper, the concept of collision-free reachable workspace is delivered. The obstacle and the end-effecter are modeled as intersections of convex implicit surfaces, which in this paper are defined as convex superquadrics. The minimum distance query between each two superquadrics is collapsed to a convex optimization problem. By employing an interior-point method, the collision-free conditions are obtained and integrated with the continuation method, creating an algorithm which enables the boundary mapping of collision-free reachable workspaces. To demonstrate and validate our approach, a planar three-bar manipulator is tested with the proposed algorithm; the boundaries of the manipulatorʹs collision-free reachable workspaces are calculated and compared with the results of the manipulatorʹs reachable workspaces.