Structure and motion

Mass loop, convergence condition, protection, reach and payload, peak power

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Structure, actuation and energy are one coupled loop. They cannot be specified separately, and cutting the loop anywhere leaves the other ends meaningless.

Every figure in this chapter is reproducible: python scripts/gems_budget.py --check recomputes them and confirms each one still appears here. See scripts/.

2.1 The mass loop

Body mass divides into two kinds:

• Scaling mass — structure, actuators, battery. The heavier the body, the more of each it needs, because they exist to carry and move that very mass.
• Non-scaling mass — armour, compute, sensors, hands, skin, harness. These are placed on the body from outside; their size is set by the mission, not by the body's mass.

With m the total mass and m_ext the non-scaling part:

m  =  f_str·m  +  f_act·m  +  f_bat·m  +  m_ext

     f_bat = p·t / e
     p = specific power in motion (W/kg)
     t = required free-running endurance (hours)
     e = battery energy density (Wh/kg)

Solving:

m  =  m_ext / (1 − Σf)      where  Σf = f_str + f_act + p·t/e
γ  =  1 / (1 − Σf)          — the growth factor

Convergence condition: Σf < 1.

This is a structural statement, not a design point. If the three scaling terms sum to the whole body, nothing is left for armour, sensors or hands — and no mass satisfies the system, at any price. Such a design is not heavy; it does not exist.

γ reads directly as the mass price of each kilogram of function: adding 1 kg of armour costs not 1 kg but γ kg of body.

2.2 The four coefficients

f_str is left open because it depends on material choices not yet made; its range is narrow enough that no conclusion below changes sign.

f_act is no longer an assumption. It follows from the declared joint count and the declared torque density, and 2.6 shows the derivation. It is listed as a coefficient because the loop consumes it as one, not because it is free.

It has two parts, and the loop treats them differently. Leg and trunk torque scales with body mass, so their actuators are a true fraction of the body: f_act_s = 0.226 at 80 Nm/kg, and this is the part that enters Σf. Arm, wrist and neck torque is fixed by payload and geometry, not by body mass, so their actuators are non-scaling mass — 9.2 kg at 80 Nm/kg — and enter m_ext beside armour and sensors, where γ multiplies them. The f_act of the table is the sum of both over the body, 0.32 at the 2-hour point. Treating the whole of it as scaling, as this section did before 2026-09-26, overstated γ and made a lighter body look worse than it is.

2.3 The endurance ceiling

Setting Σf = 1 and solving for t:

t_max  =  (1 − f_str − f_act) · e / p

Here f_act is the scaling part of 2.2, f_act_s = 0.226 at 80 Nm/kg; the fixed part sits in m_ext and does not bound t. At f_str = 0.30:

And γ inflates sharply on approach (e=450):

Read this correctly. It does not say the body runs for 10 hours. It says that at p=20 W/kg on a durable-grade pack, no free-running body beyond 10.7 hours exists — and that from about 8 hours the price is already meaningless, since γ=8.5 means each kilogram of armour costs 8.5 kg of body.

Endurance remains ⟦CTRL⟧. What this ceiling adds is a bound the controller cannot choose past, because on the far side there is no expensive design — there is no design.

2.4 Protection

The group 5 standard is one material level covering handgun rounds, knives and everyday impact together. Bullets and blades defeat armour by different mechanisms — a blade tip passes between soft fibres where a bullet does not — so a single body facing both needs a multi-layer package.

Sourced. Against a skin area of ~1.6 m² for a 1.65 m body (1.8 m² at the 1.75 m declared until 2026-09-26, scaled by the square of height):

Armour is non-scaling mass, so it enters m_ext and is multiplied by γ. At γ=2.6, the reference point, choosing 80% coverage over 50% adds ~3 kg of armour — and ~7 kg of body. Coverage and level are under discussion (D-8): the reference design keeps 65% until that is settled. This is where the trade bites hardest, and it is ⟦CTRL⟧.

Rifle-rated protection (NIJ III/IV) is achievable but adds substantially more mass. The anchor level proposed here is handgun plus blade plus everyday impact; anything above is ⟦CTRL⟧.

2.5 Mass envelope

m = γ · m_ext, with m_ext = armour (5–13 kg) plus fixed mass — compute, sensors, hands, skin, harness: 15–25 kg, ⟦IMPL⟧ — plus the arm, wrist and neck actuators of 2.2, 9.2 kg at 80 Nm/kg, which do not scale with the body.

Independent cross-check. A real 1.2 m, 35 kg, 25-DOF humanoid scaled by mass ratio (1.65/1.2)³ = 2.60 gives ~91 kg for a 1.65 m body of the same architecture, before armour. That figure lands inside the 76–123 kg band at the 2-hour mark, beside the reference design's 96 kg. Two independent derivations — one from the coupled loop, one from geometric scaling of a machine that exists — agree to an order of magnitude.

2.6 Actuation and manipulation

The basis matters as much as the number

Torque density is quoted on two incompatible bases, differing by a factor of four. Peak torque over actuator module mass — motor, gearbox and housing — is what module datasheets give, and what this specification declares. Total integrated actuator mass, counting cooling and wiring, yields 18–22 Nm/kg for the state of the art. A figure carried across from one basis to the other will size a body that cannot be built.

This specification declares peak-over-module, at the reduction ratio of the modules it was derived from — about 16:1, ~100 rpm no-load. Torque density is not a ceiling of the motor: the same motor sold at 48:1 claims 159 Nm/kg at about half the speed (electrical §6b). What the ratio trades is output speed, so a density figure without its ratio, like one without its basis, is a number waiting to be misread. The cost of standing at 80–90 is still real: part availability narrows to a handful of modules, and the joints that need more torque than those modules give must take a higher ratio and lose speed, which the sizing here does not yet check.

Why 80 is the floor

The declared kinematics need 2475 Nm summed across 31 joints at the 96 kg reference point (joint-by-joint sizing); 1737 Nm of it scales with body mass and 738 Nm does not. The three trunk axes are sized from measured human trunk strength per kilogram (electrical §6a), the arms from the declared payload below (electrical §6b). Divide:

The declared band and the f_act band of 2.2 are the same constraint seen twice. 80–90 Nm/kg maps onto f_act 0.29–0.32. The arithmetic alone would allow a floor near 74 Nm/kg, where f_act touches 0.35; the declared floor was kept at 80 when the arm payload came down on 2026-09-26, as margin — for decision D-1 on the knee, for a dynamic trunk figure above the isometric one, and for the continuous-torque question electrical §6b leaves open. Below the floor the actuators eat the mass budget and the loop stops converging. The two figures cannot drift apart, because each is derivable from the other and scripts/gems_budget.py --check holds them together.

Arm torque

Declared group 2 capability: daily manipulation and moderate precision, with 15 kg in one hand, arm straight and horizontal, load at the grip centre — the posture of maximum gravitational moment. Shoulder roll carries the same posture abducted, at 11 kg. Decided 2026-09-26; the reasoning is at electrical §6b.

The lever is to the grip centre, not the fingertip: 0.61 m from the shoulder and 0.31 m from the elbow (hardware/kinematics.md). The arm's own weight is added — its joint modules weighed at 80 Nm/kg from their own torque, plus structure and hand allowances that are ⟦IMPL⟧ until CAD exists:

For the lever alone, τ = M·g·L, with the corresponding actuator mass at 80 Nm/kg:

Load cases

Decided 2026-09-26: the declared loads are static objects. Lifting, carrying or dragging a person is outside group 2 — it needs dynamic factors no source gives and a responsibility no paper design can carry. Safe contact with people (group 4 of 01) is not a load case and stays core: touching, steadying and walking beside are in; lifting is out.

Each case is a mass, how many hands hold it, the horizontal lever from the shoulder and elbow axes to its centre of mass, and a status. A load held close to the body is carried by the legs and trunk, so its mass is added to every per-kilogram row of theirs — the rule the sizing lacked until this date. Arm self-weight is included. At the 96 kg reference point:

Two things the table says. The arm cases are cheap: the horizontal arm sets the shoulder and nothing else comes near it. The leg cases are where the mass goes: 20 kg held close costs the hip 35 Nm, whatever the hands do. The bag hanging straight down costs the shoulder nothing and is a bearing case — the AKH70-48 output bearing is rated 8680 N static, 18 times that bag.

Carry capacity follows: the leg joint with the largest per-kilogram demand (hip pitch and knee, 1.77 Nm/kg) reaches a module's torque at (body + load) = torque / 1.77. Against the AKH70-48's 222 Nm peak, the 2-hour body at 96 kg carries 29 kg in one stand-up or step, while a 4-hour body at 125 kg carries 0 kg — itself and nothing else. Every kilogram off the body is a kilogram onto the hands, which is why D-8 and the load cases are one decision, and why the reference design moved to 2 hours. At the module's 74 Nm rated torque no body in 2.5 climbs stairs with a load for long; how long its peak may be held is the open question of electrical §6b. scripts/gems_budget.py prints both figures for any point.

A single joint is not the binding constraint, but the arms are the lever on f_act. The legs and trunk carry 1737 Nm of the 2475 and scale with the body, so they set the absolute torque each module must deliver and the peak power of 2.7, not the fraction. The arms, wrists and neck are the fixed 738 Nm, and the fixed part is what moves f_act when the body gets lighter — halving the payload from the 30 kg this table once declared took f_act at 80 Nm/kg from 0.34 to 0.30 at the old 130 kg point, and the move to the 96 kg reference point took it to 0.32: a lighter body, and a larger share of it in the arms.

2.7 Peak power is limited by the source, not the actuators

Actuators and pack both scale with the body, so both operating points of 2.5 have to be priced separately:

The source falls short in every cell — by between 1.1× and 13×. The actuators' 3–5 kW/kg ceiling is not reachable from the battery at any point on this table, and high-energy-density chemistries generally trade away C-rate, so choosing the ceiling-grade pack widens the gap rather than closing it.

Only the most favourable corner — the larger pack at 10C against the low end of the actuator band — comes close, at 1.1×. Everywhere else the margin is wide.

Supercapacitors are listed as an option in 03. These figures say it more precisely: without them, the body's peak power is set by the battery, not by the actuators, and every claim about fast reflex, catching and jumping falls in the source-limited region. A supercapacitor does not change total energy; it changes instantaneous power available. Fitting one remains ⟦CTRL⟧, but the trade is now quantified.

2.8 Hard constraints

Three axes draw on one budget and cannot all be maximised:

Choose two; pay on the third. A body is not free to weigh whatever it likes: the coupled loop converges only in a narrow region, and the energy measures of 03 are what widen that region, by lowering effective p.

Operating point: ⟦CTRL⟧.