Mass budget and proportions

Stage 2 · Mechanics · ~35 min · Before this: 04 Actuators, 05 Leg loads and stairs · numbers from the repository as of 2026-11-02

Every kilogram a humanoid carries has to be held up by its knees, cooled out of its motors, stopped when it falls and paid for in battery. This page shows where Marvin Bot’s 40 kg goes, what would take it back towards the 35 kg target, why making a robot bigger is harder than it looks, and how its body compares with a person of the same height.

You will learn

• Why mass is the number that every other number depends on: joint torque, motor heat, fall energy and runtime.
• The square-cube law: scale a robot’s height by s and the torque its joints need grows roughly as s⁴, while the torque its motors give grows only as s³.
• How a mass budget is kept, item by item, and which savings Marvin Bot has on the table.
• Why mass far from the hip (distal mass) costs more than mass near it when the leg swings.
• How human body proportions (anthropometry) guided the 2026-10-29 redesign from 1.29 m to 1.44 m, and what it cost.

The idea in one picture

Think of climbing stairs with a heavy rucksack: every step costs your thighs more, you tire sooner and you fall harder. A robot is the same, except that its “thighs” are electric motors with a fixed strength, and its “tiredness” is motor heat. A mass budget is simply a list of every part and what it weighs, kept as carefully as a money budget, so the designers know where each kilogram went and which ones they can still win back.

A humanoid standing on one bent leg is a weight on the end of a lever. The knee motor must hold the robot’s weight, m·g, times the horizontal distance from the knee to the line of that weight. So the knee torque grows in direct proportion to the mass: in Marvin Bot’s slow 18 cm stair-step check, the stance knee holds 47 N·m at 41.3 kg, about 1.1 N·m for every kilogram (docs/design/stairs.md). That is already 118% of the knee motor’s continuous (rated) torque: the torque it can hold indefinitely without overheating. Anything over 100% can only be held for a short time. Mass therefore is not a cosmetic number: it is the lever on the robot’s hottest joint.

The same mass shows up everywhere else too:

• Heat. A motor’s copper loss (the heat made by current flowing through its windings) goes with current squared, and current goes with torque, so heat goes with mass squared. 10% more mass means about 21% more heat at the knee.
• Falls. The energy to absorb in a fall is m·g·h (h = how far the centre of mass drops). Twice the mass, twice the energy the structure and shells must take.
• Battery. Walking power grows roughly with weight, so a heavier robot drains the same battery faster; and the battery is itself one of the heaviest items.
• Structure. Landing loads are a multiple of body weight (Marvin Bot designs for 3 × body weight), so every structural part must grow with the mass it carries, which adds mass again.

That last point is the trap: mass begets mass. The playground below is the robot’s real mass budget, so you can see which items are worth chasing. This page builds on the knee and stair loads of 05 Leg loads and stairs; the next page, 08 Structure and code-CAD, shows how the structural parts in this budget are designed and weighed.

Playground 1: the mass budget

Each block is one item of description/mass_budget.yaml, plus the motors (counted from description/robot.yaml and description/actuators.yaml) and the add-on body shells (description/cad_model.yaml), exactly as tools/check_description.py sums them. Block area is proportional to mass (the full panel = 44 kg, so the treemap shrinks as you save). Tick a reduction idea to apply it.

Start here: click the preset buttons one at a time and watch the five readouts under the picture. Names like RS04, RS03, RS10P and RS00 are RobStride motor models, from the largest (RS04, at the hips and knees) to the smallest (RS00, at the wrists); see 04 Actuators.

Presets

Marvin Bot today Repo's two savings Towards 35 kg Add a 2-DOF waist Knee stage

Reduction ideas

Stair consequence

Knee torque at 41.3 kg

The repo's static 18 cm step check: 47 N·m on the stance knee. Scaled in proportion to mass below.

Bare robot (target 35 kg)

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With shells (limit 42 kg)

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Stair knee, % of rated

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Stair hip roll, % of rated

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Knee heat vs today

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How to read this picture: each coloured block is one item, its area proportional to its mass; colours are groups (legend above). The bar underneath stacks the groups into the total, with the 35 kg target (green dashes, bare robot) and the 42 kg hard limit (red dashes, with shells). Readout colours: green = within target or rating, amber = over target or slightly over rating, red = over the limit or well over rating.

Try this

1. Click Repo’s two savings (the carrier board and lighter leg cores, the two ideas listed in mass_budget.yaml). How far does that get you towards 35 kg? Now look at the treemap: which blocks are bigger than either saving?
2. Click Towards 35 kg. It ticks six ideas at once, including the smaller battery that brings runtime down to the 1 h requirement, and the bare robot still weighs about 36.3 kg: 1.3 kg short of the target. What does the knee readout do?
3. Click Add a 2-DOF waist (two extra degrees of freedom, DOF: the waist could also lean sideways and bend forwards, not only twist): waist roll and pitch would help some skills (docs/decisions.md, 2026-10-28), but watch the limit readout and the knee.
4. Tick only 1.5:1 knee stage. The robot gets heavier, yet the knee readout drops a lot. Why? (A reduction stage, an extra set of gears, multiplies the motor’s continuous torque by 1.5 but adds only a little mass.)

From first principles

Why torque scales with mass times length

Torque is force times the perpendicular distance from the joint to the force’s line of action: τ = F · d (τ in N·m, F in newtons, d in metres). For a joint holding up part of the body, F is that part’s weight, m·g (g = 9.81 m/s²), and d is some fraction of a limb length L. So the torque a joint needs to hold a pose is

τ ≈ k · m · g · L

where k is a number between 0 and 1 set by the pose (how bent the leg is). Double the mass and you double the torque; double the limb length at the same pose and you double it again.^[7]

The square-cube law

In everyday words: if you make a robot taller, its weight grows much faster than its height, and the turning force its joints need grows faster still. The motors you can buy do not keep up. That is why a bigger humanoid is not simply a scaled-up smaller one.

Suppose you scale a whole robot up by a factor s in every direction (s = 1.1 means 10% taller, 10% wider, 10% deeper) and keep the same materials. Then:

Quantity	Scales as	Why	s = 1.1
Lengths (limbs, lever arms)	s	by definition	1.10
Cross-section areas (strength of a strut)	s²	width × depth	1.21
Volume, so mass at constant density	s³	width × depth × height	1.33
Joint torque needed, τ ∝ m·L	s⁴	s³ · s	1.46
Torque a scaled-up motor gives	about s³	gap shear stress × gap area × gap radius	1.33

This is Galileo’s and Haldane’s old observation that big animals cannot simply be enlarged small ones: weight grows with volume, strength with area.^[3] The bottom two rows are the ones that matter: the torque needed grows as s⁴ but the torque available only as s³, so the motor’s torque margin shrinks as 1/s. A robot twice the size has half the relative margin.

Go deeper: where the s³ for motors and animals comes from

Biologists measure the same effect across animals from mice to horses, which compensate with straighter limbs that shorten the lever arms.^[4] For electric motors, the torque a motor can make is limited by the magnetic shear stress in its air gap (roughly constant for a given technology) acting over the gap’s area at the gap’s radius, so torque goes as radius² × length, about s³ for a geometrically similar motor.^[5][6] Divide the two and you get the 1/s margin above.

When mass grows less than s³. Real robots are not uniformly scaled. Bought parts keep their size: the computer, the camera, the electronics and often the battery weigh the same whether the robot is 1.2 m or 1.7 m. So the mass grows more slowly than s³ at first, which helps a taller robot. But motors come in steps (RobStride has nothing bigger than the RS04 ^[9]), so once the biggest motor runs out of margin a taller robot cannot simply buy a bigger one. Playground 2 below uses exactly this split: fixed parts plus a part that scales with s³.

Distal mass and leg inertia

Holding a pose is about weight; swinging a leg is about moment of inertia. For a leg rotating about the hip, each piece of mass m_i at distance r_i from the hip axis contributes m_i r_i²:^[7]

I = Σ m_i r_i² (kg·m²), τ = I · α

where α is the angular acceleration (rad/s²). The square means a kilogram at the ankle (0.64 m from the hip) costs about 14 times as much as a kilogram near the hip pitch motor (0.17 m). You feel this yourself: swinging your leg forward in heavy boots is far more tiring than carrying the same weight in a belt pouch. That is why designers move motors up the leg and drive the lower joints through linkages or belts. Marvin Bot’s ankles are an example: their two motors sit on the shin and push the foot through rods (06 The parallel-linkage ankle). Its knee motor, though, sits at the knee; docs/design/concept.md lists “knee motor at the joint or near the hip with a linkage” as an open question. The small playground below lets you test it: drag the sliders and watch the “Hip torque” readout, the torque the hip needs just to swing this leg at the chosen acceleration (lower is better).

Presets

Marvin Bot today Knee motor at the hip Ankle motors high Both moved up

Controls

Knee motor distance from hip

0.325 m = at the knee (today). Less = moved up the thigh, with a 0.2 kg linkage added.

Low ankle motor distance from hip

Today one ankle RS03 sits high under the knee and one low behind the ankle.

Foot mass

Foot plate, ribs and TPU sole: 1.108 kg for two feet (mass_budget.yaml).

Swing acceleration α

How hard the hip swings the leg forward, in radians per second squared (1 rad ≈ 57°); a brisk step is tens of rad/s².

Leg inertia about hip

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Hip torque for α

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Share below the knee

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Swinging mass

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Point-mass estimate of one straight leg below the hip pitch axis (masses from mass_budget.yaml and actuators.yaml, split along the leg by hand). Gravity is ignored here: it adds m·g·r_CoM·sin(angle) on top.

Try this

Move the knee motor to the hip (Knee motor at the hip). The inertia falls by about 12%. Now try Ankle motors high instead: the low ankle motor alone is worth more than the knee motor, because it sits further out. Marvin Bot’s staggered ankle already puts one of its two motors high under the knee (chosen to make the shin narrower, but it helps inertia too), which is part of why moving the knee motor alone is not urgent.

Playground 2: proportions

Human body proportions are well measured. The classic table, by Drillis and Contini and reproduced in Winter’s textbook, gives each segment as a fraction of standing height H: shoulder 0.818 H, elbow 0.630 H, wrist 0.485 H, fingertip 0.377 H, hip (greater trochanter) 0.530 H, knee 0.285 H, ankle 0.039 H, shoulder width 0.259 H.^[1][2] They are averages for adults, good to a few per cent; they are a guide for how a humanoid will look and reach, not a law.

Why care? A humanoid works in rooms built for people: door handles, tables, shelves and stairs. If its hands hang too low or its hips sit too high, it reaches and steps unlike a person, and motions recorded from people (see 17 Motion tracking) fit it badly.

How to read this picture: on the left, Marvin Bot’s front view from description/robot.yaml; on the right, a person of the same height from those fractions. Grey dashed boxes are parts whose size does not change with the robot. Change the height with the slider and compare the readouts: robot / person heights in metres (green = within a few centimetres of a person).

Presets

Marvin Bot today (1.41 m) Before 2026-10-29 (1.29 m) Alternative: legs +90 mm Scaled to 1.20 m Scaled to 1.70 m

Controls

Standing height

Scales every segment of the chosen design by s = H / design height. Fixed parts stay the same size.

Scale motors too

No: keep the RS04 knee (real choice)Yes: imaginary motor that grows as s³

RobStride's largest motor is the RS04, so in practice a taller robot keeps it.

Fingertips: robot / person

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Elbow: robot / person

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Hip axis / height

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Mass (with shells)

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Stair knee, % of rated

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Knee motor Ø / thigh

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Marvin Bot (robot.yaml)Person, Winter / Drillis & ContiniFixed-size parts

Try this

1. Click Before 2026-10-29. The fingertips hang about 0.40 m above the floor in this sketch (0.38 m in the repo’s full model) against a person’s 0.49 m: almost to the knees. Then click Marvin Bot today and see what 100 mm of torso and 50 mm of leg did.
2. Click Alternative: legs +90 mm. The hips reach 0.50 of the height, like a person’s, but the knee readout passes 125%: this is the option rejected on 2026-10-29 because it would need the parked 1.5:1 knee stage.
3. Drag the height down to 1.0 m. The knee becomes comfortable, but look at the grey boxes: the Ø120 mm knee motor and the 124 mm camera no longer fit a body that size. Drag up to 1.8 m and the knee runs out of torque, unless you switch Scale motors too on: even then the margin shrinks as 1/s.

The playground’s knee model: knee % = 118% × (mass / 41.30 kg) × (thigh / 0.325 m). It reproduces the repo’s static stair check for all three designs (109% at 1.29 m, 118% at 1.44 m, 127% with legs +90 mm) to within about 2 points. Mass when scaling: 7.08 kg of fixed parts (computer 0.877, battery 3.4, keep-alive pack 0.4, electronics 1.47, hands 0.8, camera 0.133) plus the rest × s³. Robot landmark heights come from the straight chain of robot.yaml lengths; the repo’s full model puts the fingertips 15 mm lower (0.556 m) and the elbow 6 mm lower (0.934 m) because of mounting offsets this sketch leaves out.

How Marvin Bot does it

In the repo

• One budget, checked automatically. The script tools/check_description.py, run automatically in CI (continuous integration: the tests that run on every change to the repo), sums the motors (from the motor list in description/robot.yaml and the masses in description/actuators.yaml: 20.30 kg) and every other item in description/mass_budget.yaml (19.70 kg): 40.00 kg bare. The add-on body shells, whose masses the CAD writes into description/cad_model.yaml, add 0.98 kg: 40.97 kg. It warns above the 35 kg target (bare) and fails above the 42 kg hard limit (with shells and hands).
• Each item says how it is known: estimate (a guess), cad (computed from the code-CAD solids, see 08 Structure and code-CAD), datasheet or weighed. Today 7 items come from the CAD, the hands from a datasheet, the computer is weighed, and the battery, keep-alive pack, padding, fasteners and limb harness are still estimates.
• The computer was weighed. The Jetson AGX Orin developer kit weighs 0.877 kg without its power supply, against an earlier 1.5 kg estimate (docs/decisions.md, 2026-10-28). The same review added items the CAD did not have: fasteners (0.4 kg, about 4% of the frame) and 7 m of limb harness (0.5 kg). An honest budget went up, not down.
• Motors are more than half the robot. 23 RobStride actuators and 4 servos are 21.8 kg; the 6 RS04 (hip pitch, hip roll, knee) alone are 8.5 kg. The motors were chosen for torque, so their mass is the price of the stair requirement.
• The real gate is knee heat. The 42 kg limit was set because the static stair check put the knee at 118% of its continuous rating; the trained (1.29 m) stair policy ran the knees at 73-75% of rated RMS torque (root mean square, a kind of average over time that tracks heating) (docs/design/stairs.md). Phase 2 of the build plan, a bench test (see 22 Roadmap), measures the RS04’s real continuous rating in a thigh-sized mount.

The 2026-10-29 redesign: 1.29 m to 1.44 m

At 1.29 m the robot’s heights were already close to human fractions at the shoulders and chin (2026-10-26 entry), but the bought hands are long for that size, and with the upper arm hanging 50 mm below the shoulder pitch joint the fingertips reached 0.38 m above the floor against a person’s 0.49 m: the hands hung almost to the knees. The fix (docs/decisions.md, 2026-10-29):

• torso 100 mm longer (shoulders, neck and head rise with it), shoulder roll joint 25 mm below shoulder pitch (was 50);
• legs 50 mm longer: thigh 0.300 → 0.325 m (the extra 25 mm above the hip yaw motor, so the thigh plate still fits the desktop mill), shin 0.290 → 0.315 m;
• result: 1.438 m, fingertips 0.556 m (person 0.542), elbows 0.934 m (person 0.906), hip axis at 0.487 of the height (0.700 / 1.438; the log rounds it to 0.48).

The costs, all from the decision log: +0.64 kg; the static stair knee from 109% to 118% of rated and hip roll from 103% to 105%; the centre of mass 78 mm higher (a taller inverted pendulum, like a broom balanced on a hand, falls more slowly but needs bigger corrections, see 02 Balance); and the policies trained on the old body need retraining. The flat-ground policy has since been retrained twice, on the 1.44 m model (flat v8) and on today’s 1.41 m model with the RS10P arms and the rounded head (flat v9), both with their knees bent more than intended through the stride; the stair policy is still the one trained on the 1.29 m body, which passes all 25 stair runs on today’s model (sim/isaaclab/README.md).

Trade-offs and alternatives

Option	Mass	Trade-off	Status (repo)
Orin module on a compact carrier board instead of the developer kit	−0.4 kg	needs its own heatsink, fan and carrier; the developer kit’s I/O is convenient[8]	listed in mass_budget.yaml (was −1.0 kg when the kit was estimated at 1.5 kg)
Lighter leg cores once real loads are known	−0.5 kg	needs simulated and measured loads first	listed in mass_budget.yaml
13S2P battery instead of 13S3P (13 cells in series, 2 or 3 in parallel; see 09 Power)	≈ −1.1 kg	runtime about 1 h instead of 1.5 h: exactly at the requirement	not taken (2026-10-28)
RS10P at shoulder roll and yaw and the elbow (in place of RS03 and RS06)	−1.52 kg	13 rad/s top speed; holds 9.5 N·m still (1 kg held out sideways for about two minutes)	done 2026-11-02
RS10P at shoulder pitch and the waist too	−1.32 kg	shoulder pitch over its held rating with 1 kg at full reach; the waist’s bearing and hub are built round the RS03	checked and declined (2026-11-02)
7075 leg plates thinned to the same stiffness	−0.3 to −0.5 kg	cost and sourcing (note: same stiffness needs the same thickness, since 7075 is barely stiffer than 6061; the saving comes from strength-limited parts only)	option in stairs.md
Carbon-fibre tube cores for thigh and shin	≈ −0.5 kg	design work, bonded joints	option in stairs.md
Body shells off	−0.98 kg	looks unfinished, less fall protection; the policies already train on the bare robot	shells are an add-on by design
Waist roll and pitch joints (2-DOF waist)	≈ +2.5 kg	more reach and bending; about +$630	to try in simulation first (2026-10-28)
1.5:1 knee reduction stage	≈ +0.2 kg per knee	60 N·m continuous, some backlash, thigh redesign	parked; decided after the bench test
Legs +90 mm instead of +50 mm	small	hips at 0.50 of height (human-like) but knee at 127% of rated	rejected 2026-10-29; would need the knee stage
Shoulders raised 25-40 mm at 1.29 m, or torso +50 mm only	≈ 0	chin hits the shell’s collar looking down; not enough reach	rejected 2026-10-29

Where sources and the repo disagree

• Hip height. The decision log says a person’s hips are “about 0.50” of height. Winter’s table gives the greater trochanter (the bony point on the side of the hip) at 0.530 H;^[1] the hip joint centre sits somewhat below and inside it, so 0.50 for the joint is reasonable. The robot’s hip pitch axis is at 0.487 H. The page compares the robot’s joint axis with the person’s trochanter, so the gap looks larger here than it is.
• Carrier-board saving. Older entries (2026-10-25, 2026-10-26) give −1.0 kg for the Orin module on a carrier board; the budget now says −0.4 kg, because the developer kit was weighed at 0.877 kg rather than the 1.5 kg estimate. The playground uses −0.4 kg.
• Which mass the stair check used. The knee figure (47 N·m) is quoted with “41.30 kg with shells” in stairs.md; the strength check uses the URDF mass (40.3 kg, bare). The playground scales 47 N·m from 41.3 kg, a first-order estimate.

Decisions you can make here

• Which savings to pursue first. The battery, the RS03 set and the leg structure are the biggest blocks. Trade: the 13S2P battery saves about 1.1 kg but cuts runtime from about 1.5 h to the 1 h minimum; RS10P at the shoulder pitch and waist would save 1.3 kg more, but the shoulder pitch could not hold 1 kg at full reach.
• Whether 35 kg is still the right target, or whether the knee’s measured continuous rating (phase 2 bench test) should set the limit instead. Trade: chasing 35 kg costs runtime and margin elsewhere; a measured limit may allow today’s mass, or may demand more cuts.
• Whether to add a 2-DOF waist. Trade: more reach and bending, but about +2.5 kg (over the 42 kg limit unless something else goes) and about +$630.
• Where to put mass on the leg. Trade: the knee motor up the thigh cuts leg inertia by about 12% (an easier, quicker swing), but adds a linkage of about 0.2 kg and design work.
• Whether to move the proportions further towards human (hips at 0.50 H). Trade: more human-like steps and reach, but the stair knee goes to about 127% of rated, so it needs the 1.5:1 knee stage (+0.2 kg per knee, some gear play).

Your decision noteCopy

Built from your settings above. Paste it into an issue, a design discussion or an AI assistant session.

Further reading

• J. B. S. Haldane, On Being the Right Size: a short, very readable essay on why size changes everything.
• A. A. Biewener, Biomechanical consequences of scaling (J. Exp. Biol., 2005): how animals from mice to elephants cope with the square-cube law.
• D. A. Winter, Biomechanics and Motor Control of Human Movement, chapter 4 (anthropometry): segment lengths, masses and inertias of the human body, the standard reference for humanoid proportions.
• P. M. Wensing et al., Proprioceptive actuator design in the MIT Cheetah: how motor size, gear ratio and leg inertia trade off, the reasoning behind quasi-direct drives like the RobStride.
• K. M. Lynch and F. C. Park, Modern Robotics, chapter 8: mass, inertia and the dynamics of open chains.

Sources

1. D. A. Winter, Biomechanics and Motor Control of Human Movement, 4th ed., Wiley, 2009 — used for: segment heights and lengths as fractions of height (Fig. 4.1, after Drillis and Contini)
2. R. Drillis, R. Contini, M. Bluestein, Body segment parameters: a survey of measurement techniques, Artificial Limbs, 1964 — used for: origin of the segment-length fractions and their adult basis
3. J. B. S. Haldane, On Being the Right Size, 1926 — used for: the square-cube argument
4. A. A. Biewener, Biomechanical consequences of scaling, J. Exp. Biol. 208, 2005 — used for: how animals compensate for mass growing faster than strength
5. S. Seok et al., Design principles for energy-efficient legged locomotion and implementation on the MIT Cheetah robot, IEEE/ASME Trans. Mechatronics, 2015 — used for: motor torque scaling with gap radius and size
6. P. M. Wensing, A. Wang, S. Seok, D. Otten, J. Lang, S. Kim, Proprioceptive actuator design in the MIT Cheetah, IEEE T-RO, 2017 — used for: torque density, leg inertia and why distal mass matters
7. K. M. Lynch, F. C. Park, Modern Robotics: Mechanics, Planning, and Control, Cambridge University Press, 2017 — used for: torque, moment of inertia and rigid-body dynamics
8. NVIDIA, Jetson AGX Orin Developer Kit User Guide — used for: the developer kit as a module on a reference carrier board
9. RobStride, Product_Information (manuals and specifications) — used for: RS04 Ø120 mm body, 40 N·m rated and 120 N·m peak; motor masses
10. Orbbec, Gemini 335L product page — used for: camera size 124 × 29 × 27 mm and 133 g