🤖 AI Summary
In multi-task scenarios involving parallel elastic actuators, simultaneous optimization of spring stiffness and preload remains challenging due to inherent trade-offs between energy consumption and actuation force. This paper proposes a geometric convex optimization framework to address this issue. We formulate multi-task elastic performance guarantees as a convex optimization problem with elliptical constraints—establishing, for the first time, that the objective function (source-side force or energy consumption) is a convex quadratic function of stiffness and preload, thereby enabling rigorous geometric interpretation and analytical verifiability in parameter space. The method integrates musculo-motor hybrid dynamics modeling with convex geometric constraint design, and is experimentally validated on both a knee exoskeleton and an electric ankle prosthesis platform. Open-source code supports automated parameter selection. Results demonstrate that the proposed approach systematically reduces both actuation energy consumption and peak force across arbitrary task sets.
📝 Abstract
A spring in parallel with an effort source (e.g., electric motor or human muscle) can reduce its energy consumption and effort (i.e., torque or force) depending on the spring stiffness, spring preload, and actuation task. However, selecting the spring stiffness and preload that guarantees effort or energy reduction for an arbitrary set of tasks is a design challenge. This work formulates a convex optimization problem to guarantee that a parallel spring reduces the root-mean-square source effort or energy consumption for multiple tasks. Specifically, we guarantee the benefits across multiple tasks by enforcing a set of convex quadratic constraints in our optimization variables, the parallel spring stiffness and preload. These quadratic constraints are equivalent to ellipses in the stiffness and preload plane; any combination of stiffness and preload inside the ellipse represents a parallel spring that minimizes effort source or energy consumption with respect to an actuator without a spring. This geometric interpretation intuitively guides the stiffness and preload selection process. We analytically and experimentally prove the convex quadratic function of the spring stiffness and preload. As applications, we analyze the stiffness and preload selection of a parallel spring for a knee exoskeleton using human muscle as the effort source and a prosthetic ankle powered by electric motors. The source code associated with our framework is available as supplemental open-source software.