Comparison of cost functions for electrically driven running robots

Abstract

Abstract-In this work we apply optimal control to create running gaits for the model of an electrically driven one legged hopper, and compare the results obtained for five different objective functions. By using high compliant series elastic actuators, the motions of joint and motor are decoupled, which allows the exploitation of natural dynamics.Depending on the cost function, this exploitation varies. Energy is injected at different points of time, the amplitude of actuator action changes significantly, and the optimal gear ratios differ by a factor of two. Variations are, however, comparable over a wide range of hopping heights and running velocities. Purely force-based cost functions prove to be ill-suited for such non-conservative systems, and it is shown that thermal electrical losses, in contrast to common belief, do not dominate energy expenditure. The numerical results are corroborated by detailed analytical considerations which give general insights into optimal excitation with electric actuators.

A comparison of series and parallel elasticity in a monoped hopper

“…Minimizing the positive actuator work is equivalent to minimizing the sum of the collision losses, the negative actuator work, and the damping losses [14]. A variety of strategies can be used in an optimal motion to minimize these losses (Fig 3).…”

“…Most importantly, a suitable adaptation of input and motion trajectories allows for the exploitation of the properties of each actuation concept and could thus have a huge impact on the overall power consumption. The actuator inertia in an SEA, for example, can be used to temporarily store kinetic energy [14], and the need for leg retraction with PEA can be reduced by a smart choice of motion trajectories. Due to their differing strengths and weaknesses, a SEA driven robot will hence look and move differently than a PEA driven robot.…”

Excitation and stabilization of passive dynamics in locomotion using hierarchical operational space control

“…Hence, as task with lowest priority we choose either to optimize for efficiency or for robustness against slippage. Since the energy consumption of electric motors is often quantified by the integral over current square [33], efficiency can be optimized by torque minimization (W τ = I, τ des = 0). (Risk of) slippage on the other side is mainly depending on the local tangential forces and can hence be lowered by a force minimization task that selects the local tangential forces by W F .…”

Bio-inspired swing leg control for spring-mass robots running on ground with unexpected height disturbance

“…We use this integral as the criteria for the actuator electric work. The mathematical formula for this criteria is defined as the following function which has been used as the cost function for optimization problems in other studies [30,31,32]:…”