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Engineered jumpers overcome biological limits via work multiplication

Nature volume 604, pages 657–661 (2022)Cite this article

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Abstract

For centuries, scientists have explored the limits of biological jump height1,2, and for decades, engineers have designed jumping machines3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18 that often mimicked or took inspiration from biological jumpers. Despite these efforts, general analyses are missing that compare the energetics of biological and engineered jumpers across scale. Here we show how biological and engineered jumpers have key differences in their jump energetics. The jump height of a biological jumper is limited by the work its linear motor (muscle) can produce in a single stroke. By contrast, the jump height of an engineered device can be far greater because its ratcheted or rotary motor can ‘multiply work’ during repeated strokes or rotations. As a consequence of these differences in energy production, biological and engineered jumpers should have divergent designs for maximizing jump height. Following these insights, we created a device that can jump over 30 metres high, to our knowledge far higher than previous engineered jumpers and over an order of magnitude higher than the best biological jumpers. Our work advances the understanding of jumping, shows a new level of performance, and underscores the importance of considering the differences between engineered and biological systems.

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Fig. 1: Graphical overview of the modelling framework.
Fig. 2: Trends of specific-energy upper bounds for biological versus engineered, and direct-drive versus spring-actuated jumpers.
Fig. 3: From insights to an engineered jumper that exceeds 30 m.

Data availability

All data are available in Extended Data Tables 1–3.

Code availability

MATLAB code for the energy production and utilization models and the state-space model, as well as the spring simulation, are available upon request.

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Acknowledgements

We thank W. Heap for assistance with jumper design and testing, K. Chen for assistance testing jumpers, F. Porter for assistance with modelling, H. Bluestone for editorial suggestions, G. Hawkes for early discussions on the limits of jumping, S. Rufeisen and K. Park for help filming, A. Sauret for sharing high-speed videography equipment, and K. Fields for technical support. This work was partially supported by an Early Career Faculty grant from NASA’s Space Technology Research Grants Program.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, CA, USA

    Elliot W. Hawkes, Charles Xiao, Christopher Keeley & Matthew R. Begley

  2. Disney Research, Pasadena, CA, USA

    Richard-Alexandre Peloquin & Morgan T. Pope

  3. Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA, USA

    Günter Niemeyer

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  1. Elliot W. Hawkes
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  2. Charles Xiao
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  3. Richard-Alexandre Peloquin
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Contributions

E.W.H., M.T.P. and G.N. designed the research; E.W.H., R.-A.P. and C.K. performed the experiments; E.W.H., M.T.P., G.N., C.K., C.X. and R.-A.P. analysed the data; C.X., M.R.B. and G.N. performed modelling and simulations; C.K., R.-A.P. and E.W.H. built the jumpers; E.W.H., G.N. and C.X. wrote the paper; E.W.H., M.T.P. and G.N. supervised the project.

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Correspondence to Elliot W. Hawkes.

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Nature thanks Sawyer Fuller, Gregory Sutton and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended Data Fig. 1 Work multiplication in more detail.

a, Similar to a ratcheted motor in Fig. 1, a rotary motor can accomplish work multiplication through multiple rotations instead of multiple strokes. b, The output work of a biological jumper is determined by fixed parameters (motor stroke, leg stroke and motor force), but work multiplication overcomes this for engineered jumpers: For biological jumpers, motor stroke and leg stroke determine an effective gear ratio, if the entire stroke of both is to be used (in animals, the gear ratio varies around this value slightly throughout the jump)19. With this determined gear ratio and a fixed motor force (assuming a size of motor), the leg force is determined. Finally, with the fixed leg stroke and determined leg force, the output work is determined. By contrast, for engineered jumpers, although the leg stroke is roughly fixed (assuming a size of jumper), the motor can make multiple strokes or rotations, allowing the gear ratio to be designed (higher gear ratio will result in more strokes, at the cost of more time). With this designed gear ratio and a fixed motor force (assuming a size of motor), the leg force is also multiplied with respect to the leg force in the single-stroke case. Finally, with the fixed leg stroke and the multiplied leg force, the output work is also multiplied.

Extended Data Fig. 2 Biological mechanism specific-energy data.

The model (Fig. 2a–c) predicts an upper limit to specific energy for all biological jumping mechanisms, regardless of transmission type, at approximately 200 J kg−1 (dash-dot green). Across scales found in nature, this limit holds. Note that the energy utilization was estimated at 15%, similar to previous biological work26,27. However, variation likely occurs, with jumpers with higher take-off velocities likely having more mass dedicated to jumping muscles, and thus having a higher energy utilization efficiency. A higher utilization efficiency, for example, 30%, would result in a lower mechanism specific energy than shown here. The model also predicts a limit due to motor specific power. Direct-drive jumpers fall on or below this limit (dashed blue). Non-latched spring-actuated jumpers can exceed this limit, and latched spring-actuated jumpers can exceed it by even greater amounts (distance from blue dashed line). However, all still fall below. See Extended Data Table 1 for data.

Extended Data Fig. 3 Direct-actuated jumper simulations.

a, The produced energy specific to the jumper mass. b, The centre-of-mass kinetic energy, specific to the jumper mass. c, The acceleration time. d, The optimal fixed reduction, G, producing the highest acceleration velocity for each jumper scale. The simulations are performed (i) for biological jumpers with fixed reductions of 0.01, 0.1, and 1 (dotted lines), and (ii) for biological jumpers (blue solid) and engineered jumpers (red solid: no linkage; red dotted: with linkage) using variable reduction to operate at maximum power. Each fixed reduction is only possible up to a limiting scale, where the motor force balances the body weight. Biological jumpers operating at full power are also limited in scale, as the motor runs out of stroke. Consequently, biological energy production is always limited by the motor energy (black dashed line). Finally, when operating at the optimal fixed or full-power variable reduction, the acceleration time scales with a 2/3 power of size, reflected in the same scaling in energy and gear reduction.

Extended Data Fig. 4 Spring-actuated engineered jumper simulations.

a, The produced energy specific to the jumper mass. b, The centre-of-mass kinetic energy, specific to the jumper mass. c, The acceleration time. The simulations are performed for spring mass ratios of ranging from 0.001–10. A lower mass ratio lowers the produced energy specific to the total mass and also imposes an upper bound on size, as smaller springs cannot match larger weight forces. The acceleration time scales nearly linearly with the size, and bigger springs create faster jumps.

Extended Data Fig. 5 Jumper design details.

a, Ashby plot of materials with the largest material factor, or the square of yield strength over density. At low elastic moduli are elastomers, but these require a passive linkage to load in tension. At high elastic moduli are fibre-reinforced composites, which can act as stand-alone compression bow springs, but have lower specific energies than elastomers in tension. We therefore design a hybrid spring with elastomer in tension and carbon fibre in bending, replacing the passive linkage. b, Force–displacement plot of our hybrid linkage-spring, with total area under the curve (energy) shown (24.2 J). c, Schematic and pictures of the minimalistic release mechanism for unlatching. During winding of the string, the motor shaft turns, pulling the string over a shaft supported by bearings in the arm and compressing the hybrid spring-linkage. With further winding, a lever on the string eventually hits the latch, prying it open. The arm swings open, allowing the string to unspool from the shaft. d, Components of the jumper before assembly. e, Self-righting mechanism. Without a self-righting mechanism, the top-heavy jumper will roll nose-down during compression of the bow springs, given its mass distribution. However, if tapered and split bows are added between each pair of the main, non-tapered bow springs, the behaviour can be reversed. The taper in the bow near the nose creates a high radius of curvature during compression, contacting the ground and forcing the nose to roll upward. The split section continues this as the jumper nears completion of the righting behaviour.

Extended Data Fig. 6 Simulating the presented jumper across the spring–motor mass ratio and scale.

Using the state-space model modified with the specifics of the presented jumper, we simulated jump height. We included both energy production and energy utilization. When the spring–motor mass ratio is increased to infinite, we see only a 17% increase in jump height (from 32.9 to 38.6 m). When the scale is increased by 10×, we find an increase of only 19% in jump height (from 32.9 to 39.1 m). The star denotes the presented jumper (0.3 m scale, 32.9 m jump height).

Extended Data Fig. 7 Schematic of simplified jumper.

a, Schematic of jumper used in Fig. 1a. b–d, Free-body diagrams of the body, top linkage, and bottom linkage, respectively.

Extended Data Table 1 Biological jump data
Full size table
Extended Data Table 2 Engineered jumper data
Full size table
Extended Data Table 3 Jumper specifications
Full size table

Supplementary information

Supplementary Information

This file contains Supplementary Methods.

Peer Review File

Supplementary Video 1

Slow Motion Take-off. This video shows a jumper taking off in slow motion (1/10, 1/100, and 1/500× real time). Take-off occurs in 9 ms after latch release. Stills from this video are shown in Fig. 3d. (4 MB, MP4).

Supplementary Video 2

26 m Jump. This video shows a device jumping over 30 m high in an outdoor test. The person is 1.83 m tall. (8.6 MB, MP4).

Supplementary Video 3

Self-righting Indoors. This video shows a jumper self-righting on a flat surface (2× real time). The device uses an additional set of four tapered legs between the main power legs to perform the righting behaviour. Stills from this video are shown in Fig. 3e. (17.6 Mb, MP4).

Supplementary Video 4

Self-righting Outdoors. This video shows a jumper launching up a bluff outdoors. After landing, the device self-rights and jumps a second time. (28.5 Mb, MP4).

Supplementary Video 5

On-board Camera. This video shows the perspective from a small onboard camera during a jump. The video is slowed down (speeds shown on relevant frames) and paused at the apex. (25.3 MB, MP4).

Supplementary Video 6

Balcony Fly-by. This video shows a device, tuned to jump three stories, next to a building. Cameras are placed on each balcony, visualizing the flight path. (34.3 MB, MP4).

Supplementary Video 7

View from Below. This video shows the jumper taking off from a perspective below the jumper looking skyward. (13.3 MB, MP4).

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Hawkes, E.W., Xiao, C., Peloquin, RA. et al. Engineered jumpers overcome biological limits via work multiplication. Nature 604, 657–661 (2022). https://doi.org/10.1038/s41586-022-04606-3

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