Issue 042 - Robotics - Sprint power and battery energy
How much battery energy does a humanoid robot need to sprint 100 meters?
Reuters reported that China's Tiangong Ultra humanoid robot ran 100 meters in 8.86 seconds at the World Humanoid Robot Games in Beijing, improving on the 9.39-second robot record it had set only days earlier.
The problem
Estimate how much battery energy Tiangong Ultra would need to complete one all-out 100-meter sprint.
Then estimate the peak or average power the robot's battery and motors must deliver during the run. Use your result to answer: is the hard part battery capacity, peak power delivery, motor control, or mechanical durability?
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I used KE = 1/2mv^2 to find the kinetic energy maintained, on average, over the sprint. Then, to make the math easy, I assumed roughly that amount of energy would need to be generated by the robot per second to accelerate to that speed and maintain it throughout the sprint.
I also assumed the robot was about 33% energy efficient, meaning it would need to carry about 3x the energy required by the physical effort. I assumed a chemical battery with energy density and overall density similar to a AA battery, and I estimated a typical AA battery as about 0.05 kg carrying about 20 kJ.
kinetic energy ~= 1/2 x 40 kg x (10 m/s)^2
~= 2,000 JI treated that as about 2,000 J per second over a 10-second sprint:
mechanical-ish sprint energy ~= 2,000 J/s x 10 s
~= 2 x 10^4 J
~= 20 kJ
battery energy with 33% efficiency
~= 3 x 20 kJ
~= 60 kJFor perspective, if a human eats 2,500 calories daily, that is about 10 million J:
daily food energy ~= 2,500 kcal x 4,000 J/kcal
~= 1 x 10^7 J
average human power ~= 1 x 10^7 J / 86,000 s
~= 110 WDuring strenuous exercise, I guessed a human might reach about 800 J/s, so a 10-second sprint might consume about 8,000 J. That would put the robot's sprint energy rate at several times a human's strenuous output.
For the battery, if a AA battery carries about 20 kJ, then 60 kJ would require about 3 to 4 AA batteries, or 0.15 to 0.2 kg. That is fine if the sprint is all that needs to be accomplished. But if it had to maintain this effort for 10 minutes, it would need about 60x more battery mass, or 9 to 12 kg. For 30 minutes, it would need 27 to 36 kg; for an hour, 54 to 72 kg, which would more than double the mass of the robot.
Calibration Score
Matt's Calibration Score: 65 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 10/30. Mass and speed pegs were good, but sprint power and battery-power pegs were missing.
Model: 15/30. Forward kinetic energy was only a partial model for running energy.
Math: 10/10. The arithmetic was clean.
Result: 30/30. Despite the flawed model, the final energy estimate landed close to the corrected order.
Grounding facts
A 100 kJ sprint is tiny compared with daily food energy: a 2,500-kcal day is about 10 MJ, or 100 times larger. But power tells a different story. Releasing 100 kJ in about 10 seconds is about 10 kW, far above a human's full-day average power.
That is the useful robot lesson: endurance asks for energy capacity, while explosive movement asks for power density, motor torque, heat rejection, and control.
After checking sources
Check and recalibrate
Matt's mass and speed pegs were close enough. The reported mass is about 55 kg, and the 8.86-second 100 m gives an average speed of about 11.3 m/s.
speed ~= 100 m / 8.86 s
~= 11 m/s
KE floor ~= 1/2 x 55 kg x (11 m/s)^2
~= 3.3 x 10^3 JThat 3 kJ is only the energy in the robot's forward motion. It does not count the repeated leg swing, impact losses, vertical motion, balance correction, heat in motors and electronics, or stopping at the end.
A useful physiology-inspired peg is that all-out sprinting can involve average metabolic power around 50 to 100 W/kg, with peak values above 100 W/kg. A humanoid robot is not a human body, but this gives a good Fermi-scale demand for violent biped locomotion:
sprint power ~= robot mass x power per kg
~= 55 kg x (50 to 150 W/kg)
~= 2.8 to 8.3 kWOver roughly 9 seconds:
sprint energy ~= power x time
~= (2.8 to 8.3 kW) x 9 s
~= 25 to 75 kJNow add motor, battery, control, gearbox, and heat losses. A reasonable battery-side estimate is roughly:
battery energy ~= 50 to 150 kJ
~= order 1 x 10^5 JThat means Matt's 60 kJ answer is within the right order of magnitude, even though the method leaned too hard on forward kinetic energy.
The battery-capacity mass is surprisingly small if you only care about one sprint. Modern lithium-ion batteries often store around 200 Wh/kg, and:
200 Wh/kg ~= 200 x 3,600 J/kg
~= 7.2 x 10^5 J/kg
~= order 10^6 J/kg
mass for 100 kJ ~= 1 x 10^5 / 7 x 10^5
~= 0.14 kgSo pure energy capacity for one 100 m run is not the limiting factor. The harder battery problem is power: delivering several kilowatts in short bursts without voltage sag, overheating, or damaging the pack. A 6 kW load from a 0.2 kg battery would require an extreme discharge rate; a larger pack makes the power delivery more manageable.
The context answer is therefore: a single sprint does not require much stored energy, but it demands high peak power, precise motor control, impact tolerance, balance recovery, and thermal management. That is why the race is a real robotics milestone even if the energy total is small.
Post-check reflection
Matt's reflection
I have made this error before, using the kinetic energy of a moving object to estimate the total energy necessary to maintain a speed. I still think it was sort of fair, since sprinting is falling, catching yourself, and propelling yourself forward again. At the speed of a sprinter, I bet that is a dominant energy use, almost like having to accelerate to sprint speed repeatedly. If not, I am way off base for calculating energy use for this activity. Looks like I came reasonably close to the right energy estimate despite the flawed approach.
I was low on estimating energy consumption. The 100 to 150 W/kg peak sprint-power peg is important to keep track of. Another useful peg is modern lithium batteries holding roughly 10^6 J/kg.
I also did not consider power capability at all, not just capacity. The batteries need to be able to produce something like 6 kW. One more peg might be about 200 Wh/kg.
Ultimately I think my estimates were within an order of magnitude, and this problem really helps emphasize the challenge of powering dynamic robots.
Recommended memory peg
Remember KE = 1/2mv^2 for the lower bound, but use energy = power x time for the full sprint. Useful pegs: elite sprinting can involve 50 to 100 W/kg average and more than 100 W/kg peak; modern lithium-ion batteries are around 200 Wh/kg, or roughly 10^6 J/kg.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.