Development of a Novel Convex Optimization Framework for Mono- and Bi-Articular Parallel Elastic Elements in Robotic Arms

Abstract As robotic systems become more prevalent, the need for energy-efficient designs continues to grow. Parallel elastic actuation (PEA) and Biarticular PEA (BPEA) offer a promising approach by reducing actuator torque through passive load support. However, existing methods for configuring these elastic elements often rely on nonconvex optimization and are tailored to a single predefined task, limiting their generality and increasing computational cost. This article introduces a novel convex optimization framework for the design of PEA and BPEA in robotic arms, aiming to reduce actuator torque requirements by minimizing root-mean-square (RMS) actuator torque in varied tasks. For the PEA-only case, spring parameter tuning is reformulated as a convex quadratic program and therefore admits a global optimum for the stated model and constraints. For the BPEA and combined configurations, fixing the pulley ratios yields convex quadratic subproblems, while an outer discretized search identifies the best solution on the prescribed pulley-ratio mesh. A case study shows that the framework provides comparable or improved torque reduction while optimizing much more quickly. Even when the robot performs a wide range of tasks, a single fixed spring configuration can still substantially reduce actuator torque requirements. This framework provides an efficient optimization tool for the robotic-arm configuration considered and establishes a basis for future extensions to higher-degree-of-freedom systems, supporting sustainable and cost-effective automation.