Energy-Optimal Hopping in Parallel and Series Elastic One-Dimensional Monopeds

Abstract

In this paper, we examine the question of whether parallel elastic actuation or series elastic actuation is better suited for hopping robots. To this end, we compare and contrast the two actuation concepts in energy optimal hopping motions. To enable a fair comparison, we employ optimal control to identify motion trajectories, actuator inputs, and system parameters that are optimally suited for each actuator concept. In other words, we compare the best possible hopper with parallel elastic actuation to the best possible hopper with series elastic actuation. The optimizations are conducted for three different cost functions: positive mechanical motor work, thermal electrical losses, and positive electrical work. Furthermore, we look at three representative cases for converting rotary motor motion to linear leg motion in a legged robot. Our model featured an electric DC-motor model, a gearbox with friction, damping in the leg spring, and contact collisions. We find that the optimal actuator choice depends both on the cost function and conversion of motor motion to leg motion. When considering only thermal electrical losses, parallel elastic actuation always performs better. In terms of positive mechanical motor work and positive electrical work, series elastic actuation is better when there is little friction in the gear-train. For higher gear-train friction parallel elastic actuation is more economical for these cost functions as well.1 This value is twice the maximum continuous torque the motor can provide. It is the maximum value suggested through personal communication with Maxon Motor.

An Overview on Principles for Energy Efficient Robot Locomotion

“…As animal structure and motion have simultaneously evolved to be specifically designed to perform desired tasks efficiently and effectively, it is essential to account for both morphology and motion in robot design and control. Yesilevskiy et al (2015Yesilevskiy et al ( , 2018a discussed the effect of morphological variations in legged robots, and showed for a one-dimensional monoped hopper that is driven by a geared DC motor, with the correct choice of the transmission parameter, a hopper with Series Elastic Actuators (SEA) is more energetically efficient than one with Parallel Elastic Actuator (PEA). It is mostly due to the fact that, for a hopper with PEA, the motor inertia contributes to energetic losses due to the ground contact collisions.…”

Exploring the Limits of a Redundant Actuation System Through Co-Design

“…The key aspect making this problem one of co-design is the additional decision variables in ρ, which contains the design parameters: motor masses, spring stiffnesses and gear ratios. The motor constant and inertia of each motor were related to the motor mass by the relationships presented in [15] and reported in the technical report [20]. Detailed path and boundary constraints (12c) and (12d) are provided in [20] as well.…”

“…The frequency range in the analysis goes from 0.1 to 10 Hz, with 0.1 Hz resolution. The set of values for the spring stiffness is [30, 50, 100] Nm/rad, while for the gear ratio of the SEA it is [5,10,15,20,30,50,100,150,200].…”

“…Motor specifications include: peak torque τ max = 7.13 Nm, rotor inertia I se = 10 −5 kg • m 2 and motor constant K m = 0.64 Nm/ √ Watt. The gearbox efficiency was considered to scale exponentially with the gear ratio as suggested in [15].…”

Towards the Exploitation of Physical Compliance in Segmented and Electrically Actuated Robotic Legs: A Review Focused on Elastic Mechanisms

“…Joint parallel compliance can be viewed from the standpoint of biologically inspired design, mimicking the compliant property within muscles and ligaments, as well as joint capsules. Theoretical studies have revealed that the addition of physical parallel compliance is beneficial to reduce peaks of motor torque and power, and even energetic cost, provided that the parallel compliant mechanism is designed and tuned properly [56][57][58][59][60][61].…”