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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. 

The planar dual spring-loaded inverted pendulum (dual-SLIP) model is a well-established passive template of human walking on flat ground. This paper applies an actuated extension of the model to walking on inclines and declines to evaluate how well it captures the behavior observed in human slope walking. The motivation is to apply the template to improve control of humanoid robot walking and/or intent detection in exoskeleton-assisted walking. Gaits of the actuated planar dual-SLIP model are found via the solution of a constrained nonlinear optimization problem in ten parameters. The majority of those parameters define the actuation scheme that injects energy for incline walking and absorbs energy for decline walking to achieve periodic, nonconservative gaits. Solution gaits across the speed range of 1.0 to 1.6 ms and slope range of −10 to 10 degrees exhibit some of the characteristics of human walking, such as the effect of slope on stance duration, step frequency, and step length. Efforts to reduce the number of parameters optimized by enforcing relationships observed in the solution gaits proved unsuccessful, suggesting that future work must trade off model complexity with fidelity of representation of human behavior.