Virtual-Energy Homotopy for Simultaneous Trajectory and Stiffness Co-Design in Flexible-Joint Robotic Arms

Abstract

Flexible-joint arms are attractive for tasks that demand both precision and energy efficiency, such as collaborative assembly. However, enforcing exact terminal constraints in trajectory optimization often leads to illconditioned nonlinear programs. The difficulty grows when joint stiffness is treated as a decision variable alongside the control trajectory, as is needed in electromechanical co-design. We propose Virtual-Energy Homotopy (VEH), which adds a scalar-parameterized damping term directly into the system's Lagrangian. Doing so turns a fragile single NLP into a sequence of well-conditioned ones; the artificial dissipation tends to pull the solver away from poor local minima in our tests. Coupled with an adaptive node-refinement strategy, the framework simultaneously optimizes joint stiffness and motor commands at each homotopy step. On a 4-DOF planar arm, VEH enforced terminal constraints to machine precision in all tests. With simultaneous stiffness optimization, the cost of transport and peak torque both dropped by 42% relative to a fixed-stiffness baseline. Comparative benchmarks also indicate that the proposed energy-based damping achieves better terminal precision while keeping computation time competitive against standard parameter-continuation homotopy. These results support the view that VEH can serve as a practical, reliable algorithmic core for generating agile motions in complex manipulation settings.