  Compared to rigid actuators, Series Elastic Actuators (SEAs) offer a
potential reduction of motor energy consumption and peak power, though these
benefits are highly dependent on the design of the torque-elongation profile of
the elastic element. In the case of linear springs, natural dynamics is a
traditional method for this design, but it has two major limitations: arbitrary
load trajectories are difficult or impossible to analyze and it does not
consider actuator constraints. Parametric optimization is also a popular design
method that addresses these limitations, but solutions are only optimal within
the space of the parameters. To overcome these limitations, we propose a
non-parametric convex optimization program for the design of the nonlinear
elastic element that minimizes energy consumption and peak power for an
arbitrary periodic reference trajectory. To obtain convexity, we introduce a
convex approximation to the expression of peak power; energy consumption is
shown to be convex without approximation. The combination of peak power and
energy consumption in the cost function leads to a multiobjective convex
optimization framework that comprises the main contribution of this paper. As a
case study, we recover the elongation-torque profile of a cubic spring, given
its natural oscillation as the reference load. We then design nonlinear SEAs
for an ankle prosthesis that minimize energy consumption and peak power for
different trajectories and extend the range of achievable tasks when subject to
actuator constraints.
