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.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
Scale dependence in hydrodynamic regime for jumping on water
Nature Communications Open Access 17 March 2023
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 per month
cancel any time
Subscribe to this journal
Receive 51 print issues and online access
$199.00 per year
only $3.90 per issue
Rent or buy this article
Get just this article for as long as you need it
Prices may be subject to local taxes which are calculated during checkout
MATLAB code for the energy production and utilization models and the state-space model, as well as the spring simulation, are available upon request.
Aristotle. Problemata 3 12–19.
Morowitz, H. J. De motu animalium. Hosp. Pract. 11, 145–149 (1976).
Seifert, H. S. The lunar pogo stick. J. Spacecr. Rockets 4, 941–943 (1967).
Zhao, J. et al. MSU jumper: a single-motor-actuated miniature steerable jumping robot. IEEE Trans. Robot. 29, 602–614 (2013).
Niiyama, R., Nagakubo, A. & Kuniyoshi, Y. In Proc. IEEE Intl Conf. Robotics and Automation 2546–2551 (IEEE, 2007); https://doi.org/10.1109/ROBOT.2007.363848.
Scarfogliero, U., Stefanini, C. & Dario, P. In Proc. IEEE Intl Conf. Robotics and Automation 467–472 (IEEE, 2007); https://doi.org/10.1109/ROBOT.2007.363830.
Li, F. et al. Jumping like an insect: design and dynamic optimization of a jumping mini robot based on bio-mimetic inspiration. Mechatronics 22, 167–176 (2012).
Zhao, J., Xi, N., Gao, B., Mutka, M. W. & Xiao, L. In Proc. IEEE Intl Conf. Robotics and Automation 4614–4619 (IEEE, 2011); https://doi.org/10.1109/ICRA.2011.5980166.
Churaman, W. A., Currano, L. J., Morris, C. J., Rajkowski, J. E. & Bergbreiter, S. The first launch of an autonomous thrust-driven microrobot using nanoporous energetic silicon. J. Microelectromech. Syst. 21, 198–205 (2012).
Armour, R., Paskins, K., Bowyer, A., Vincent, J. & Megill, W. Jumping robots: a biomimetic solution to locomotion across rough terrain. Bioinsp. Biomim. 2, S65–S83 (2007).
Bergbreiter, S. In 2008 IEEE/RSJ Intl Conf. Intelligent Robots and Systems (IROS) 4030–4035 (IEEE, 2008); https://doi.org/10.1109/IROS.2008.4651167.
Koh, J. S. et al. Jumping on water: surface tension-dominated jumping of water striders and robotic insects. Science 349, 517–521 (2015).
Woodward, M. A. & Sitti, M. MultiMo-Bat: a biologically inspired integrated jumping-gliding robot. Int. J. Rob. Res. 33, 1511–1529 (2014).
Haldane, D. W., Plecnik, M. M., Yim, J. K. & Fearing, R. S. Robotic vertical jumping agility via series-elastic power modulation. Sci. Robot. 1, eaag2048 (2016).
Zaitsev, V. et al. Locust-inspired miniature jumping robot. In 2015 IEEE/RSJ Intl Conf. Intelligent Robots and Systems (IROS) 553–558 (IEEE, 2015); https://doi.org/10.1109/IROS.2015.7353426.
Kovač, M., Fuchs, M., Guignard, A., Zufferey, J. C. & Floreano, D. In Proc. IEEE Intl Conf. Robotics and Automation 373–378 (IEEE, 2008); https://doi.org/10.1109/ROBOT.2008.4543236.
Kovač, M., Schlegel, M., Zufferey, J. C. & Floreano, D. Steerable miniature jumping robot. Auton. Robots 28, 295–306 (2010).
Burdick, J. & Fiorini, P. Minimalist jumping robots for celestial exploration. Int. J. Rob. Res. 22, 653–674 (2003).
Alexander, R. M. Leg design and jumping technique for humans, other vertebrates and insects. Phil. Trans. R. Soc. Lond. B. 347, 235–248 (1995).
Alexander, R. M. N. Simple models of human movement. Appl. Mech. Rev. 48, 461–470 (1995).
Scholz, M. N., Bobbert, M. F. & Knoek van Soest, A. J. Scaling and jumping: gravity loses grip on small jumpers. J. Theor. Biol. 240, 554–561 (2006).
Cerquiglini, S., Venerando, A., Wartenweiler, J. & Plagenhoef, S. Biomechanics III. In Medicine & Science in Sports & Exercise (ed. Hoerler, E.) vol. 6 iv (Karger AG, 1974).
Roberts, T. J. & Marsh, R. L. Probing the limits to muscle-powered accelerations: lessons from jumping bullfrogs. J. Exp. Biol. 206, 2567–2580 (2003).
Bobbert, M. F. Effects of isometric scaling on vertical jumping performance. PLoS One 8, e71209 (2013).
Azizi, E. & Roberts, T. J. Muscle performance during frog jumping: influence of elasticity on muscle operating lengths. Proc. R. Soc. B 277, 1523–1530 (2010).
Bennet-Clark, H. C. Scale effects in jumping animals. In Scale Effects in Animal Locomotion (ed. Pedley, T. J.) 185–201 (Academic, 1977).
Sutton, G. P. et al. Why do large animals never actuate their jumps with latch-mediated springs? Because they can jump higher without them. Integr. Comp. Biol. 59, 1609–1618 (2019).
Divi, S. et al. Latch-based control of energy output in spring actuated systems. J. R. Soc. Interface 17, 20200070 (2020).
Longo, S. J. et al. Beyond power amplification: latch-mediated spring actuation is an emerging framework for the study of diverse elastic systems. J. Exp. Biol. 222, jeb197889 (2019).
Bennet-Clark, H. C. & Alder, G. M. The effect of air resistance on the jumping performance of insects. J. Exp. Biol. 82, 105–121 (1979).
Stoeter, S. A. & Papanikolopoulos, N. Kinematic motion model for jumping scout robots. IEEE Trans. Robot. 22, 397–402 (2006).
Ilton, M. et al. The principles of cascading power limits in small, fast biological and engineered systems. Science 360, eaao1082 (2018).
Gabriel, J. M. The effect of animal design on jumping performance. J. Zool. 204, 533–539 (1984).
Roberts, T. J. & Azizi, E. Flexible mechanisms: the diverse roles of biological springs in vertebrate movement. J. Exp. Biol. 214, 353–361 (2011).
Gerratt, A. P. & Bergbreiter, S. Incorporating compliant elastomers for jumping locomotion in microrobots. Smart Mater. Struct. 22, 014010 (2013).
Greenspun, J. & Pister, K. S. J. First leaps of an electrostatic inchworm motor-driven jumping microrobot. In 2018 Solid-State Sensors, Actuators and Microsystems Workshop 159–162 (IEEE, 2018); https://doi.org/10.31438/trf.hh2018.45.
Greenspun, J. & Pister, K. S. J. in Proc. Intl Conf. Manipulation, Automation and Robotics at Small Scales (MARSS) (eds. Haliyo, S. et al.) 258–262 (IEEE, 2017); https://doi.org/10.1109/MARSS.2017.8001944.
Bergbreiter, S. & Pister, K. S. J. In Proc. IEEE Intl Conf. Robotics and Automation 447–453 (IEEE, 2007); https://doi.org/10.1109/ROBOT.2007.363827.
Berg, H. C. The rotary motor of bacterial flagella. Annu. Rev. Biochem. 72, 19–54 (2003).
Ashby, M. Materials Selection in Mechanical Design 4th edn (Elsevier, 2010).
Hall‐Crags, E. C. B. An analysis of the jump of the lesser galago (Galago senegalensis). J. Zool. 147, 20–29 (1965).
Josephson, R. K. Contraction dynamics and power output of skeletal muscle. Annu. Rev. Physiol. 55, 527–546 (1993).
Seok, S., Wang, A., Otten, D. & Kim, S. In IEEE Intl Conf. Intelligent Robots and Systems 1970–1975 (IEEE, 2012); https://doi.org/10.1109/IROS.2012.6386252.
Miao, Z., Mo, J., Li, G., Ning, Y. & Li, B. Wheeled hopping robot with combustion-powered actuator. Int. J. Adv. Robot. Syst. https://doi.org/10.1177/1729881417745608 (2018).
Ackerman, E. Boston dynamics sand flea robot demonstrates astonishing jumping skills. IEEE Spectrum Robotics Blog https://spectrum.ieee.org/boston-dynamics-sand-flea-demonstrates-astonishing-jumping-skills (28 March 2012).
Dowling, K. Power Sources for Small Robots. Technical report no. CMU-RI-TR-97-02 (Carnegie Mellon University, 1997).
Marden, J. H. & Allen, L. R. Molecules, muscles, and machines: universal performance characteristics of motors. Proc. Natl. Acad. Sci. USA 99, 4161–4166 (2002).
Hirt, M. R., Jetz, W., Rall, B. C. & Brose, U. A general scaling law reveals why the largest animals are not the fastest. Nat. Ecol. Evol. 1, 1116–1122 (2017).
Winslow, J., Hrishikeshavan, V. & Chopra, I. Design methodology for small-scale unmanned quadrotors. J. Aircr. 55, 1062–1070 (2018).
Dermitzakis, K., Carbajal, J. P. & Marden, J. H. Scaling laws in robotics. Procedia Comp. Sci. 7, 250–252 (2011).
Mitchell, H. H., Hamilton, T. S., Steggerda, F. R. & Bean, H. W. The chemical composition of the adult human body and its bearing on the biochemistry of growth. J. Biol. Chem. 158, 625–637 (1945).
Hunt, J. F., Zhang, H., Guo, Z. & Fu, F. Cantilever beam static and dynamic response comparison with mid-point bending for thin mdf composite panels. BioResources 8, 115–129 (2013).
Parry, D. A. & Brown, R. H. J. The jumping mechanism of salticid spiders. J. Exp. Biol. 36, 654–664 (1959).
Marsh, R. L. & John-Alder, H. B. Jumping performance of hylid frogs measured with high-speed cine film. J. Exp. Biol. 188, 131–141 (1994).
Evans, M. E. G. The jump of the click beetle (Coleoptera, Elateridae)—a preliminary study. J. Zool. 167, 319–336 (1972).
Brackenbury, J. & Hunt, H. Jumping in springtails: mechanism and dynamics. J. Zool. 229, 217–236 (1993).
Maitland, D. P. Locomotion by jumping in the Mediterranean fruit-fly larva Ceratitis capitata. Nature 355, 159–161 (1992).
Harty, T. H. The Role of the Vertebral Column during Jumping in Quadrupedal Mammals. PhD thesis, Oregon State Univ. (2010).
Schwaner, M. J., Lin, D. C. & McGowan, C. P. Jumping mechanics of desert kangaroo rats. J. Exp. Biol. 221, jeb186700 (2018).
Katz, S. L. & Gosline, J. M. Ontogenetic scaling of jump performance in the african desert locust (Schistocerca gregaria). J. Exp. Biol. 177, 81–111 (1993).
Toro, E., Herrel, A., Vanhooydonck, B. & Irschick, D. J. A biomechanical analysis of intra- and interspecific scaling of jumping and morphology in Caribbean Anolis lizards. J. Exp. Biol. 206, 2641–2652 (2003).
Essner, R. L. Three-dimensional launch kinematics in leaping, parachuting and gliding squirrels. J. Exp. Biol. 205, 2469–2477 (2002).
Gregersen, C. S. & Carrier, D. R. Gear ratios at the limb joints of jumping dogs. J. Biomech. 37, 1011–1018 (2004).
Burrows, M. & Dorosenko, M. Jumping mechanisms and strategies in moths (Lepidoptera). J. Exp. Biol. 218, 1655–1666 (2015).
Bobbert, M. F., Gerritsen, K. G. M., Litjens, M. C. A. & Van Soest, A. J. Why is countermovement jump height greater than squat jump height? Med. Sci. Sports Exerc. 28, 1402–1412 (1996).
Brackenbury, J. & Wang, R. Ballistics and visual targeting in flea-beetles (Alticinae). J. Exp. Biol. 198, 1931–1942 (1995).
Burrows, M. Jumping performance of froghopper insects. J. Exp. Biol. 209, 4607–4621 (2006).
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.
The authors declare no competing interests.
Peer review information
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.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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.
This file contains Supplementary Methods.
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).
Rights and permissions
About this article
Cite this article
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
This article is cited by
Scale dependence in hydrodynamic regime for jumping on water
Nature Communications (2023)
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.