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Research Papers

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

[+] Author and Article Information
Yevgeniy Yesilevskiy

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: yevyes@umich.edu

Zhenyu Gan

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: ganzheny@umich.edu

C. David Remy

Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109
e-mail: cdremy@umich.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received March 16, 2017; final manuscript received February 26, 2018; published online April 5, 2018. Assoc. Editor: Jun Ueda.

J. Mechanisms Robotics 10(3), 031008 (Apr 05, 2018) (11 pages) Paper No: JMR-17-1063; doi: 10.1115/1.4039496 History: Received March 16, 2017; Revised February 26, 2018

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.

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Figures

Grahic Jump Location
Fig. 1

This study investigated energetically optimal hopping motions for a one-dimensional (1D) hopper with either parallel elastic actuation (PEA, shown as schematic in (a) and detailed model in (c)) or series elastic actuation (SEA, (b) and (d)). For both hoppers we simultaneously optimized motion trajectories, actuator inputs, and actuation parameters for three different cost functions: positive mechanical motor work, thermal electrical losses, and positive electrical work. The optimization included main body position y, leg length l, motor position u (only for SEA), motor force after the transmission To, spring stiffness k, and rotary gearbox ratio nr (not shown).

Grahic Jump Location
Fig. 2

The logarithmic regression of the maximum torque conversion efficiency of the rotary gearbox εmax as a function of the rotary gearbox gear ratio nr. The regression was based on 809 gearboxes [32].

Grahic Jump Location
Fig. 3

The optimal positive electrical work values at a hopping height of h = 1.3o. Three cases are shown: a theoretical entirely frictionless transmission, a rotary to linear transmission of nℓ=200 rad/ℓo with a rotary gearbox with friction, and a rotary to linear transmission of nℓ=2 rad/ℓo with a rotary gearbox with friction. SEA is the most energetically economical choice for both the frictionless transmission and nℓ=200 rad/ℓo cases. PEA is better for the nℓ=2 rad/ℓo case. (a) frictionless transmission, (b) rotary gearbox with friction, frictionless nl = 200 rad/o, and (c) rotary gearbox with friction, frictionless nl = 200 rad/o.

Grahic Jump Location
Fig. 4

An energetic breakdown is shown for the three cost functions in the case of a frictionless transmission. For both the positive mechanical motor work and the positive electrical work, SEA is the energetically optimal actuator type. For thermal losses, PEA is the optimal actuator type. (a) Cmech, (b) Ctherm, and (c) Cel.

Grahic Jump Location
Fig. 5

Optimal motions and actuator inputs for the frictionless transmission. Shown are the results for optimizations based on positive mechanical motor work, thermal losses, and electrical work. The leg motion for SEA (dashed line) is very similar for all cost functions. In contrast, for PEA (solid line), the leg motion is drastically different when optimized for positive mechanical motor work as compared to thermal losses. As a result, when positive mechanical motor work and thermal losses are combined into a single cost function, the electrical work, PEA must trade-off between two very different motion strategies.

Grahic Jump Location
Fig. 6

Shown is the energetic breakdown for the three cost functions in the case of a rotary gearbox with friction and nℓ=200rad/ℓo. Similar to the frictionless transmission case, for both Cmech and Cel, SEA was the energetically optimal actuator type. For Ctherm, PEA was the optimal actuator type. For PEA, negative mechanical motor work compensated for a large proportion of the thermal losses (the cross-hatched region, indicating that though both losses occurred, they only were counted once): (a) Cmech, (b) Ctherm, and (c) Cel.

Grahic Jump Location
Fig. 7

Optimal motions and actuator inputs for the rotary gear box with friction and frictionless nℓ=200rad/ℓo case. Shown are the results for positive mechanical motor work (a), thermal losses (b), and electrical work (c). The most notable difference from the frictionless motion was that for PEA electrical work, it was no longer optimal to have the leg oscillate during flight. Instead, PEA and SEA adopted similar strategies during flight, holding their legs near maximum extension.

Grahic Jump Location
Fig. 8

The figure shows the energetic breakdown for the three cost functions in the case of a rotary gearbox with friction with nℓ=2rad/ℓo. PEA was now optimal for all cost functions. The majority of the losses for Cmech arose from frictional and damping losses. For Cel, thermal losses and gear friction dominated the losses. (a) Cmech, (b) Ctherm, and (c) Cel.

Grahic Jump Location
Fig. 9

Optimal motions and actuator inputs for the rotary gear box with friction and frictionless nℓ=2rad/ℓo case. The motion here was very similar for both PEA and SEA for all three cost functions. Both hoppers extended their legs to near maximum extension during flight and held them there until touchdown.

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