A GENERAL FRAMEWORK FOR MINIMIZING ENERGY CONSUMPTION OF SERIES ELASTIC ACTUATORS WITH REGENERATION

Abstract

The use of actuators with inherent compliance, such as series elastic actuators (SEAs), has become traditional for robotic systems working in close contact with humans. SEAs can reduce the energy consumption for a given task compared to rigid actuators, but this reduction is highly dependent on the design of the SEA's elastic element. This design is often based on natural dynamics or a parameterized optimization, but both approaches have limitations. The natural dynamics approach cannot consider actuator constraints or arbitrary reference trajectories, and a parameterized elastic element can only be optimized within the given parameter space. In this work, we propose a solution to these limitations by formulating the design of the SEA's elastic element as a non-parametric convex optimization problem, which yields a globally optimal conservative elastic element while respecting actuator constraints. Convexity is proven for the case of an arbitrary periodic reference trajectory with a SEA capable of energy regeneration. We discuss the optimization results for the tasks defined by the human ankle motion during level-ground walking and the natural motion of a single mass-spring system with a nonlinear spring. For all these tasks, the designed SEA reduces energy consumption and satisfies the actuator's constraints.

FIGURE 1

Energy flow of an SEA: Dashed lines indicate that the energy path may or may not exist depending on the construction of the device. For instance, energy flowing from the drive to the battery requires drivers capable of regeneration. Energy flowing from the transmission to the electric motor requires the motor-transmission system to be backdrivable.

FIGURE 2

Diagram of a SEA. Eqns. 3–4 illustrate the system’s equations of motion.

FIGURE 3

(a) Single mass-spring system. The elastic element describes the nonlinear spring with ${\tau }_e=k{q}_l^3$. (b) Double mass-spring system. The equilibrium position of the elastic element is q_l = q_m/r, elongation is defined as δ = q_l − q_m/r. Motor and transmission are considered to be backdrivable.

FIGURE 4

The reference trajectory of the load is defined by the natural oscillation of the single mass-spring system in Fig. 3-(a) with α = 40 Nm/rad³, I_l = 125 gm², and q_l(0) = π/2rad.

FIGURE 5

Optimization results considering natural oscillation of a nonlinear spring as the reference motion. The solid line corresponds to the elastic element that minimizes the energy consumption due to viscous friction (VF). It matches τ_ela = αδ³, the nonlinear spring used in the single mass-spring system. The dotted line describes the elastic element that minimizes winding losses (WL) due to Joule heating. The dashed line describes the elastic element that minimizes both winding losses and viscous friction, i.e., total energy (TE). The corresponding energy expenditure is shown in Table 2.

FIGURE 6

Motion of the human ankle during level ground walking as shown in [26]. Slow, normal, and fast walking speeds are equivalent to cadences of 87, 105, and 123 steps per minute. The gait cycle begins with heel contact of one foot and finishes with the subsequent occurrence of the same foot [26]. In average, the ankle of a 75 kg subject walking at normal speed provides about 17 J during a single gait cycle. In the lower figure, translucent regions denote the minimum and maximum external torques corresponding to 65 kg and 85 kg subjects.

FIGURE 7

Optimization results considering motion of the ankle as the reference trajectory. Dotted, solid, and dashed lines indicate results for slow, normal, and fast level-ground walking speeds, respectively. Translucent regions denote upper and lower bounds corresponding to 85 kg and 65 kg subjects.

FIGURE 8

Energy savings for the ankle reference trajectory. Results for slow, normal, and fast level-ground walking for three different subject’s weights.