  Design of robotic systems that safely and efficiently operate in uncertain
operational conditions, such as rehabilitation and physical assistance robots,
remains an important challenge in the field. Current methods for the design of
energy efficient series elastic actuators use an optimization formulation that
typically assumes known operational conditions. This approach could lead to
actuators that cannot perform in uncertain environments because elongation,
speed, or torque requirements may be beyond actuator specifications when the
operation deviates from its nominal conditions. Addressing this gap, we propose
a convex optimization formulation to design the stiffness of series elastic
actuators to minimize energy consumption and satisfy actuator constraints
despite uncertainty due to manufacturing of the spring, unmodeled dynamics,
efficiency of the transmission, and the kinematics and kinetics of the load. In
our formulation, we express energy consumption as a scalar convex-quadratic
function of compliance. In the unconstrained case, this quadratic equation
provides an analytical solution to the optimal value of stiffness that
minimizes energy consumption for arbitrary periodic reference trajectories. As
actuator constraints, we consider peak motor torque, peak motor velocity,
limitations due to the speed-torque relationship of DC motors, and peak
elongation of the spring. As a simulation case study, we apply our formulation
to the robust design of a series elastic actuator for a powered prosthetic
ankle. Our simulation results indicate that a small trade-off between energy
efficiency and robustness is justified to design actuators that can operate
with uncertainty.
