My final semesters at RPI consisted of research in three areas:

Cuspidal robots can switch inverse kinematics (IK) solutions without crossing singularities, so classical path planners may fail. I identified new cuspidal robots, proposed planning and optimization methods.

Redundant robots have an extra degree of freedom in joint space. I applied cuspidality analysis to 7-DOF redundant robot arms for the first time. I also analyzed the ABB YuMi kinematics, providing the first publicly available complete forward and inverse kinematics using the ABB's redundancy parameterization.

Parallel robots have loops in their kinematic structure. I demonstrated how to use my IK solver IK-Geo to solve the forward and inverse kinematics of several parallel robots. This method finds all forward and inverse kinematics solutions, including singular solutions.

Cuspidal Robots

Cuspidal manipulators can switch between IK solutions without passing through a singularity. This leads to surprising behavior that makes classical path planners fail. In our paper Path Planning and Optimization for Cuspidal 6R Manipulators, we propose several new insights and algorithms for this increasingly common class of manipulators.

We are the first to show the ABB GoFa and some 3-parallel-axis robots are cuspidal. We also propose a graph-based planner to find optimal joint paths for a given end effector path and an optimizer to adjust the workpiece placement.

One reason why classical path planners may struggle with cuspidal robots is because they have task-space paths with valid IK solutions at every point along the path but no continuous joint-space path. In the video below, we show such an infeasible path for a 3R cuspidal manipulator. This path can be extended to form a closed loop in task space, demonstrating a nonrepeatable path. Although the manipulator can follow the desired path for one loop, the nonsingular change of IK solution means it encounters a singularity before it can follow the loop again.

Then, we show results from our path planner and optimizer, which find the best workpiece placement to minimize joint movement. Examples include a 3R cuspidal robot and the 6R FANUC CRX 10iA-L.

I presented the findings from our paper at ICRA 2025 Late Breaking Results Poster Session, shown below.

Redundant Robots

7-DOF robots have an extra redundant degree of freedom that allows the robot to move while keeping the end effector still, which is called self-motion. However, redundancy parameterizations create new algorithmic singularities. In our previous paper, Redundancy parameterization and inverse kinematics of 7-DOF revolute manipulators, we found efficient IK solutions for 7-DOF robot arms using IK-Geo, and we found a new redundancy parameterization called the stereographic SEW angle that reduces the presence of algorithmic singularities in the workspace. The conventional SEW angle and the new stereographic SEW angle fit into a new framework we call the general SEW angle.

In our recent paper Redundancy Parameterization of the ABB YuMi Robot Arm, we provided the first complete and validated definition of the SEW angle used for the YuMi. This definition is compatible with the general SEW angle framework. We found complete numerical conditions for all algorithmic singularities. We also demonstrated using IK-Geo and 2D search to find all IK solutions with the ABB SEW angle parameterization. This demonstrates the highly general capabilities of the subproblem decomposition method as the ABB YuMi has no consecutive intersecting or parallel joint axes. This is the first IK solver that can find all IK solutions for the YuMi in the literature.

In my doctoral dissertation, Efficient Singularity-Robust Inverse Kinematics and Redundancy Management for Robotic Systems, we apply cuspidality analysis to redundant arms for the first time. While redundant arms can usually travel between self-motion manifolds without encountering a singularity, certain 7-DOF arms may or may not be cuspidal once the redundancy is parameterized. We find the ABB YuMi is cuspidal after parameterization, while the KUKA iiwa is not.

Parallel Mechanisms

Parallel robots are a class of robots fundamentally different from serial arms: Instead of a serial kinematic chain, there are loops in the kinematic structure. Additionally, only some joints are actuated, while the rest are passive and usually have no angle sensors. Parallel manipulators often have superior speed and stiffness compared to serial manipulators. This makes them good for applications like pick and place or machining.

In my doctoral dissertation, I found efficient inverse kinematics and forward kinematics solutions for parallel manipulators using subproblem decomposition. These solutions also demonstrate applications of the subproblem decomposition method for robots with prismatic joints. Solutions were found for planar and spatial five-bar linkages, the Eclipse robot, delta robots, and several variations of Stewart platforms. As is the case for serial arm inverse kinematics, these solutions are robust to singular poses. For parallel robots, this includes both unmanipulable singularities and unstable singularities.