Path Planning and Optimization for Cuspidal 6R Manipulators

TL;DR

The paper addresses the challenging issue of path planning for cuspidal manipulators, which can switch IK solutions without encountering singularities. It introduces an efficient identification method using IK-Geo to prove cuspidality (including ABB GoFa) and develops a graph-based path-planning framework that enumerates all IK solutions along a task-space path and solves a shortest-path in a weighted DAG to obtain the optimal joint-space trajectory. A path-optimization scheme then offsets the workpiece/base pose to minimize a chosen metric while preserving feasibility, demonstrated with significant reductions in RMS joint velocity for both 3R and 6R cases. The results highlight that cuspidal robots, while challenging, can be effectively utilized with appropriate IK solvers and planning tools, enabling practical applications in welding, grinding, and additive/subtractive processes. The work also provides open-source code and lays groundwork for future enhancements in efficiency, collision handling, and extension to higher-DOF systems. $IK$-Geo and the graph-based approach offer a principled way to harness multiple IK solutions for robust, efficient motion of cuspidal manipulators.

Abstract

A cuspidal robot can move from one inverse kinematics (IK) solution to another without crossing a singularity. Multiple industrial robots are cuspidal. They tend to have a beautiful mechanical design, but they pose path planning challenges. A task-space path may have a valid IK solution for each point along the path, but a continuous joint-space path may depend on the choice of the IK solution or even be infeasible. This paper presents new analysis, path planning, and optimization methods to enhance the utility of cuspidal robots. We first demonstrate an efficient method to identify cuspidal robots and show, for the first time, that the ABB GoFa and certain robots with three parallel joint axes are cuspidal. We then propose a new path planning method for cuspidal robots by finding all IK solutions for each point along a task-space path and constructing a graph to connect each vertex corresponding to an IK solution. Graph edges have a weight based on the optimization metric, such as minimizing joint velocity. The optimal feasible path is the shortest path in the graph. This method can find non-singular paths as well as smooth paths which pass through singularities. Finally, we incorporate this path planning method into a path optimization algorithm. Given a fixed workspace toolpath, we optimize the offset of the toolpath in the robot base frame while ensuring continuous joint motion. Code examples are available in a publicly accessible repository.

Figures (11)

• Figure 1: Industrial cuspidal robots. (a) FANUC CRX-10iA/L. (b) Kinova Link 6. (c) ABB GoFa CRB 15000 5 kg.
• Figure 2: Cuspidal 3R manipulator with an infeasible path (straight line in task space) and a nonsingular change of solution (straight line in joint space). (a) 3D view using cylindrical coordinates $(\rho, \phi, z)$. Points with singularities are marked for the $\phi=0$, $\rho>0$ half-plane. (b) All solutions for joint angle 1 over the infeasible path. Line width corresponds to which aspect the IK solution is in. (c) One aspect in joint space bounded by singularities. Nonsingular change of solution is shown. (d) Forward kinematics mapping of one aspect to task space. Also shown are the infeasible path and the nonsingular change of solution.
• Figure 3: MoveL path for the ABB GoFa which is feasible depending on the initial pose. In this case, there are eight initial and ten final IK solutions but only six feasible paths.
• Figure 4: MoveL path for the FANUC CRX-10iA/L with eight initial and final IK solutions, up to twelve intermediate solutions, and only two feasible paths.
• Figure 5: Path for the noncuspidal ABB IRB 6640 which is entirely in the workspace but is infeasible. The blue annulus centered at $\rho<0$ shows the task space reachable by shoulder back configurations. The black annulus centered at $\rho>0$ shows the task space reachable by shoulder forward configurations. The path shown passes through the hole in the black annulus and also leaves the outer boundary of the blue annulus. Therefore, neither configuration can achieve the entirety of the desired path.