Cost of transport

Cost of transport (COT), also called specific cost of transport, specific resistance, or the dimensionless cost of transport, is the standard measure of how much energy a moving body spends to travel. It is defined as the energy used to carry a unit of weight over a unit of distance, and in its usual form it is a pure number with no units. A lower cost of transport means more efficient travel. The metric is used across biology and engineering, which is what makes it powerful: it lets a swimming fish, a walking human, a cargo ship, and a legged robot be placed on the same axis. In robotics it is the quantity most often cited when comparing the energy efficiency of different locomotion strategies, and it sits at the center of the long-running debate over whether wheels or legs are the better way to move a machine. ^[1][3]

The formula

The cost of transport is most simply written in an energy form:

COT = E / (m g d)

where E is the energy used, m is the mass of the moving body, g is the acceleration due to gravity (about 9.81 m/s^2), and d is the distance travelled. The product m g is the body's weight, and m g d is the mechanical work it would take to lift that weight straight up through a height equal to the distance travelled. So the cost of transport can be read as a ratio: the energy actually spent to go a distance d, divided by the energy it would take to hoist the whole vehicle by that same distance. ^[7]

An equivalent power form is used when energy is measured as a rate:

COT = P / (m g v)

where P is the power consumption and v is the travel speed. The two forms describe the same quantity. Average power is energy divided by time, P = E / t, and average speed is distance divided by time, v = d / t. Substituting these into the power form gives P / (m g v) = (E / t) / (m g d / t) = E / (m g d), the energy form. ^[1]

Because energy divided by (weight times distance) is joules divided by (newtons times metres), and a newton-metre is a joule, the ratio is dimensionless. It has the same numerical value in any consistent system of units, which is exactly why it travels so well between fields. ^[1] Gabrielli and von Karman, who introduced the vehicle version of the metric in 1950, described the same number as "a kind of global friction coefficient for the vehicle": the cost of transport times the weight is an effective resistance force that, acting over the distance travelled, would consume all the energy used, so dividing by weight yields a drag-like coefficient comparable to a friction coefficient. ^[2]

Two properties follow from the definition. First, lower is better: a body with a cost of transport of 0.1 spends a tenth of the lifting energy to travel a given distance, and is more efficient than one at 1.0. Second, cost of transport depends on speed. It is infinite when a body stands still and burns energy going nowhere, it falls as speed increases, and for most animals and vehicles it passes through a minimum at an intermediate, energy-optimal speed before rising again. ^[1] Minimal biped models optimised purely for energy reproduce walking at low speeds and running at high speeds, which is one reason the metric is treated as a principled objective rather than an arbitrary score. ^[18]

Mechanical, metabolic, and total electrical cost of transport

The single hardest thing about the cost of transport is that the "energy used" in the numerator can be counted in several different ways, and the resulting numbers differ by large factors. Three accountings are common.

The mechanical cost of transport counts only the positive mechanical work done at the joints or by the actuators. It ignores every conversion loss: the inefficiency of muscle, the friction and heat in motors and gearboxes, the energy spent by electronics, and the cost of simply holding posture against gravity. ^[3]

The metabolic cost of transport, used for animals, counts the chemical energy consumed, normally measured through oxygen uptake. It includes the large losses of the muscles themselves. It comes in two sub-flavours that are frequently confused. The gross metabolic cost includes the resting or basal metabolism the animal would burn even standing still; the net metabolic cost subtracts that resting rate to isolate the energy attributable to moving. ^[1][10]

The total electrical cost of transport, used for robots and battery-powered vehicles, counts the total energy drawn from the battery or wall plug: motors, gearboxes, sensors, onboard computers, cooling, and the power spent just standing. This is sometimes called the specific energetic cost of transport. ^[3][7]

These accountings can differ by a factor of several for the same machine performing the same motion. The Cornell passive-dynamic biped consumes about 11 watts total but only about 3 watts of mechanical work, roughly a factor of four. ^[3] For a walking human the mechanical cost of transport is near 0.05, the net metabolic cost is near 0.2, and the gross metabolic cost of a brisk walk is about 0.38, so a single act of walking can be quoted with three different cost-of-transport numbers depending only on the accounting convention. ^[1][3]

The practical consequence is a warning. Comparing a mechanical cost of transport to a metabolic or total-electrical one is a category error that flatters the mechanical number, because the mechanical figure has quietly discarded most of the energy actually consumed. A machine whose "cost of transport" of 0.05 is mechanical is not competitive with an animal whose 0.2 is metabolic; the honest comparison is mechanical-to-mechanical or total-to-total. The careful studies in this field state which one they mean and compare like with like. ^[3]

History

The metric has two independent origins, one in engineering and one in biology.

In 1950 Giuseppe Gabrielli, director of engineering at Fiat, and Theodore von Karman, the aerodynamicist, delivered the Thurston Lecture published as "What Price Speed? Specific Power Required for Propulsion of Vehicles." ^[2] They collected installed power, gross weight, and maximum speed for a wide range of vehicles, from merchant ships and submarines to trucks, railcars, cars, airships, and aircraft, and plotted a dimensionless quantity they called the specific tractive force or specific resistance, epsilon = P / (W V), the ratio of installed power to the product of gross weight and maximum speed. This is the cost of transport, computed at the design point of full installed power and top speed. Their central result is a frontier. When all vehicles are thrown together, the most efficient of each class trace a lower edge below which no vehicle falls: ships have the lowest specific resistance at low speed, terrestrial vehicles are best at medium speed, and aircraft are best at high speed, so no single mode is efficient everywhere. ^[2] Later summaries of the diagram describe the frontier as a limit line in which specific resistance rises roughly in proportion to speed, with the best large ships approaching a slope on the order of 4 x 10^-4 seconds per metre. ^[19] Gabrielli and von Karman also showed why speed becomes expensive: for merchant ships near their speed limit, a 1 percent increase in speed demanded about 6 percent more power per ton, because wave resistance climbs steeply as the ship's speed approaches the wave-propagation speed set by its Froude number. ^[2]

The biological thread runs through the 1970s. Vance A. Tucker measured the energetic cost of moving in animals and machines and defined the dimensionless cost of transport as metabolic power input divided by weight times speed. ^[1][6] Knut Schmidt-Nielsen's 1972 Science paper "Locomotion: Energy Cost of Swimming, Flying, and Running" assembled the comparative picture that became the metric's most famous biological result. ^[5] Tucker's 1975 American Scientist article carried the memorable subtitle that walking and running are extremely inefficient forms of locomotion, and that much greater efficiency is achieved by birds, fish, and bicyclists. ^[1]

Cost of transport in animals

Tucker's and Schmidt-Nielsen's data produced a clean ordering that holds across the animal kingdom regardless of taxonomy: per unit distance, swimming is cheaper than flying, which is cheaper than running. ^[1][5][6] On Tucker's chart the swimmers form the lowest band, the fliers a middle band, and the runners the highest, and the separation is large. Tucker put the gap between a flier and a runner of the same size at roughly a factor of 27 for a small animal of about 0.01 kg. ^[6] A cyclist, a human using a machine to convert leg motion into rolling, has about one-quarter the cost of transport of a walker and holds the lowest cost of transport ever measured for an animal, though Tucker noted that large fish and whales, sitting on the swimmers' line below 0.1, may do better still. ^[1]

Cost of transport also falls with body size. Across roughly twelve orders of magnitude of body mass, from a fruit fly to a horse, the minimum cost of transport for a given mode declines as animals get larger, so a large animal moves its kilograms more cheaply than a small one. ^[1] Tucker observed that large birds and mammals end up with costs of transport about the same as automobiles and small propeller aircraft. ^[6] More recent allometric work has questioned how far the "bigger is always cheaper" reading can be pushed once body size is properly accounted for, but the broad downward trend is not in dispute. ^[17]

For concrete human numbers, Tucker measured a 70 kg person reaching a minimum cost of transport at a fast walk of 1.75 m/s, where the metabolic rate was 452 watts and the gross metabolic cost of transport was 0.376; jogging at 3.5 m/s raised the metabolic rate to 1,122 watts and the cost of transport to 0.467. ^[1] The reason running is so costly is muscular. The mean efficiency of a flying bird's muscles is about 0.2, but the muscles of a runner are repeatedly stretched while active, and their effective efficiency often falls below 0.05, so most of the metabolic energy is lost as heat rather than turned into useful work. The same human muscles reach an efficiency near 0.25 when driving a bicycle, which is why cycling is so much cheaper than running. ^[1]

Cost of transport in vehicles and robots

The metric became a fixture of legged robotics as a way to expose how wasteful early walking machines were. In a 2005 Science paper, Steve Collins, Andy Ruina, Russ Tedrake, and Martijn Wisse compared powered walking robots built on passive-dynamic principles against both humans and a conventional humanoid. ^[3] Passive-dynamic walkers descend a shallow slope with no motors or control at all, driven only by gravity, yet produce strikingly human-like bipedal gaits; the tradition traces to Tad McGeer's 1990 work and Collins's earlier kneed 3D walker. ^[8][9] Collins and colleagues added small power sources so their machines could walk on level ground while keeping that efficiency. Their 13 kg Cornell biped, walking at 0.4 m/s, achieved a specific mechanical cost of transport of about 0.055 and a total electrical cost of transport of about 0.2, essentially matching a human, whose mechanical cost is about 0.05 and whose metabolic cost is about 0.2. The Delft biped reached a mechanical cost of about 0.08 and the MIT learning biped about 0.02 or above. ^[3]

The contrast case is Honda's ASIMO, then the best-known joint-angle-controlled humanoid. From ASIMO's published specifications, a 510 newton robot that walked at up to 1.6 km/h and drained a 38.4 volt, 10 amp-hour battery in about half an hour, Collins and colleagues estimated a total electrical cost of transport of about 3.2, and, assuming a 50 percent drivetrain efficiency, a mechanical cost of transport of about 1.6. ^[3][11] On a like-for-like total basis, ASIMO's 3.2 was roughly sixteen times a human's 0.2, and Collins concluded it used at least ten times the energy of a typical human. ^[3] It is worth being precise about the 1.6 figure, because it is easy to misreport: 1.6 is ASIMO's estimated mechanical cost of transport, not its total. Its total electrical cost of transport is the larger 3.2. A separate compilation, using a different ASIMO specification of 1.8 kilowatts at 1.5 m/s for a 54 kg robot, puts ASIMO's total cost of transport nearer 2, so estimates for the machine's total figure span roughly 2 to 3 depending on the version and assumptions. ^[7][15] Either way, the passive-dynamic result stands: careful mechanical design brought a robot's efficiency to human levels, an order of magnitude better than the actively controlled humanoid.

The cleanest illustration of wheels against legs comes from putting both on the same machine. Marko Bjelonic and colleagues at ETH Zurich equipped ANYmal, a torque-controlled quadruped robot developed there and commercialised by ANYbotics, with non-steerable powered wheels at the end of each leg, making it a wheeled-legged robot that can both walk and drive. ^[4][14] On flat terrain at 2 m/s, driving on the wheels, the robot achieved a mechanical cost of transport of 0.1 at a mechanical power consumption of 63.64 watts. The paper reports that this was 83 percent lower than the same robot trotting on its legs and 17 percent lower than skating with passive wheels; the 83 percent reduction implies a trotting cost of transport of about 0.59, and the skating value about 0.12, all mechanical and all on flat ground. ^[4][12] Driving also let the wheeled ANYmal reach 4 m/s, well above the 1.5 m/s top speed previously recorded for the same platform under a learned legged controller, so on flat ground the wheels won on efficiency and speed at once. ^[4][13] In other words, on smooth terrain and holding the actuators, mass, and electronics fixed, switching from legs to wheels cut the cost of transport nearly sixfold. That is the quantitative core of the wheels-versus-legs efficiency argument.

Other robots fill in the range. A compilation by Kashiri and colleagues lists the bipedal robot Cassie at a total cost of transport of about 0.7 (a 30 kg robot walking at 1.0 m/s on about 200 watts), the efficiency-focused humanoid DURUS near a target of 1, and the hydraulically actuated quadruped BigDog, from Boston Dynamics, at a total cost of transport around 15, an order of magnitude worse than electric walkers because hydraulic power delivery is lossy. ^[7][16] Cassie's maker, Agility Robotics, later built the delivery robot Digit on similar efficiency-minded principles.

Representative values

The table gathers sourced figures. The energy-basis column is essential: numbers on different bases are not directly comparable, and the vehicle figures from Gabrielli and von Karman are design-point values (installed power at top speed) rather than cruise-efficiency figures.

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Why it matters for robot design

For a battery-powered machine the total-electrical cost of transport is close to a direct statement of range and endurance. Range scales as the usable battery energy divided by the product of weight, gravity, and cost of transport, so halving the cost of transport doubles how far the robot can go on a charge, or halves the battery mass needed for a fixed range. Because a bigger battery adds weight that must itself be carried, cost of transport enters the sizing of nearly every mobile robot. ^[3][7]

It also drives actuator and drivetrain choices. The gap between a robot's mechanical and total cost of transport is dominated by conversion losses in motors and gearboxes and by the cost of holding posture, which is why high-gear-ratio geared limbs, backdrivable transmissions, series elastic elements, and passive-dynamic designs that let gravity and inertia do part of the work all show up as cost-of-transport improvements. ASIMO's high figure came largely from stiff, precise, energy-hungry joint-angle control; the passive-dynamic walkers won by letting the mechanics carry the gait. ^[3]

Most visibly, cost of transport is the anchor of the robot form factor debate over wheels versus legs. The ANYmal experiment quantifies the intuition that wheels are far cheaper on flat, hard ground: nearly a sixfold reduction in cost of transport when the same machine rolls instead of steps. ^[4] Legs earn their much higher cost of transport by crossing terrain wheels cannot, which is why wheeled-legged robots exist at all, and why cost of transport alone never settles the form-factor question. On a warehouse floor a wheeled mobile robot wins on energy by a wide margin; over rubble or stairs the comparison inverts, because a wheel that cannot proceed has an effectively infinite cost of transport.

Limitations of the metric

Cost of transport is a narrow measure, and its narrowness is often misread as a verdict on overall capability.

It is speed-dependent, so any single number is really a number at a stated speed; quoting a cost of transport without its speed, and ideally without noting whether it is the speed-minimised value, is incomplete. ^[1]

It is defined for steady, level travel and says nothing about task, terrain, or payload. A robot optimised for the lowest cost of transport on a treadmill may be useless on gravel, and a machine that can climb stairs will always look inefficient beside one that only ever rolls on tile. Cost of transport measures the price of moving, not the value of where the machine can go. ^[4]

The mechanical-versus-total ambiguity discussed above is the metric's most common source of error, and it is not academic: mechanical and total figures for the same walk can differ by a factor of several, so a comparison that mixes the two can be wrong by an order of magnitude. ^[3] For animals, the gross-versus-net metabolic choice adds a further factor.

Finally, cost of transport treats the transported mass as a given. A heavier machine can post a lower cost of transport while consuming more absolute energy per trip, because the weight sits in the denominator, so a low cost of transport does not by itself mean low energy use for a fixed task. The metric answers one clean question, how cheaply weight is moved, and should be read as that and nothing more.

ELI5

Imagine you have to drag a heavy backpack across a field. Cost of transport is a fairness score for how good you are at it: it compares the energy you actually burn to the energy it would take to just lift the backpack straight up as high as the field is long. A small score means you barely waste anything, like a fish gliding through water. A big score means you are working hard and mostly making heat, like a stiff robot that clomps along using lots of battery. Because it is only a comparison of energy to weight-times-distance, it comes out as a plain number with no units, so you can use the very same score for a fish, a person on a bike, a truck, and a robot dog, and see at a glance which one moves its weight most cheaply. The catch is that the score only measures moving cheaply on flat ground; it does not measure whether the robot can climb the stairs at the edge of the field.

See also

• Robot form factor
• Robot locomotion
• Bipedal locomotion
• Quadruped robot
• Wheeled-legged robot
• ANYbotics
• Boston Dynamics
• Actuator

References

1. Tucker, V. A. (1975). "The Energetic Cost of Moving About." American Scientist, 63(4), 413-419. https://www.jstor.org/stable/27845517 ↩
2. Gabrielli, G., and von Karman, T. (1950). "What Price Speed? Specific Power Required for Propulsion of Vehicles." Mechanical Engineering, 72(10), 775-781. https://gwern.net/doc/technology/1950-gabrielli.pdf ↩
3. Collins, S., Ruina, A., Tedrake, R., and Wisse, M. (2005). "Efficient Bipedal Robots Based on Passive-Dynamic Walkers." Science, 307(5712), 1082-1085. https://doi.org/10.1126/science.1107799 ↩
4. Bjelonic, M., Bellicoso, C. D., de Viragh, Y., Sako, D., Tresoldi, F. D., Jenelten, F., and Hutter, M. (2019). "Keep Rollin': Whole-Body Motion Control and Planning for Wheeled Quadrupedal Robots." IEEE Robotics and Automation Letters, 4(2), 2116-2123. https://arxiv.org/abs/1809.03557 ↩
5. Schmidt-Nielsen, K. (1972). "Locomotion: Energy Cost of Swimming, Flying, and Running." Science, 177(4045), 222-228. https://doi.org/10.1126/science.177.4045.222 ↩
6. Tucker, V. A. (1970). "Energetic Cost of Locomotion in Animals." Comparative Biochemistry and Physiology, 34(4), 841-846. https://doi.org/10.1016/0010-406X(70)91006-6 ↩
7. Kashiri, N., et al. (2018). "An Overview on Principles for Energy Efficient Robot Locomotion." Frontiers in Robotics and AI, 5, 129. https://doi.org/10.3389/frobt.2018.00129 ↩
8. McGeer, T. (1990). "Passive Dynamic Walking." International Journal of Robotics Research, 9(2), 62-82. https://doi.org/10.1177/027836499000900206 ↩
9. Collins, S. H., Wisse, M., and Ruina, A. (2001). "A Three-Dimensional Passive-Dynamic Walking Robot with Two Legs and Knees." International Journal of Robotics Research, 20(7), 607-615. https://doi.org/10.1177/02783640122067561 ↩
10. Donelan, J. M., Kram, R., and Kuo, A. D. (2002). "Mechanical Work for Step-to-Step Transitions Is a Major Determinant of the Metabolic Cost of Human Walking." Journal of Experimental Biology, 205(23), 3717-3727. https://doi.org/10.1242/jeb.205.23.3717 ↩
11. Honda Motor Co. "ASIMO Specifications and Technical Information." https://asimo.honda.com/ ↩
12. Bjelonic, M., Bellicoso, C. D., Tiryaki, M. E., and Hutter, M. (2018). "Skating with a Force Controlled Quadrupedal Robot." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 7555-7561. https://doi.org/10.1109/IROS.2018.8593829 ↩
13. Hwangbo, J., Lee, J., Dosovitskiy, A., Bellicoso, D., Tsounis, V., Koltun, V., and Hutter, M. (2019). "Learning Agile and Dynamic Motor Skills for Legged Robots." Science Robotics, 4(26), eaau5872. https://doi.org/10.1126/scirobotics.aau5872 ↩
14. Hutter, M., et al. (2016). "ANYmal: A Highly Mobile and Dynamic Quadrupedal Robot." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 38-44. https://doi.org/10.1109/IROS.2016.7758092 ↩
15. Sakagami, Y., Watanabe, R., Aoyama, C., Matsunaga, S., Higaki, N., and Fujimura, K. (2002). "The Intelligent ASIMO: System Overview and Integration." IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2478-2483. https://doi.org/10.1109/IRDS.2002.1041641 ↩
16. Reher, J., Cousineau, E. A., Hereid, A., Hubicki, C. M., and Ames, A. D. (2016). "Realizing Dynamic and Efficient Bipedal Locomotion on the Humanoid Robot DURUS." IEEE International Conference on Robotics and Automation (ICRA), 1794-1801. https://doi.org/10.1109/ICRA.2016.7487325 ↩
17. Alexander, R. McN. (2003). Principles of Animal Locomotion. Princeton University Press. https://press.princeton.edu/books/paperback/9780691126340/principles-of-animal-locomotion ↩
18. Srinivasan, M., and Ruina, A. (2006). "Computer Optimization of a Minimal Biped Model Discovers Walking and Running." Nature, 439(7072), 72-75. https://doi.org/10.1038/nature04113 ↩
19. "Von Karman-Gabrielli Diagram." Wikipedia. https://en.wikipedia.org/wiki/Von_K%C3%A1rm%C3%A1n%E2%80%93Gabrielli_diagram ↩