OPEN THE MODEL

Every assumption.
Every equation.

Four dimensions become a torque–speed envelope through a small, inspectable model. Here is what comes from established theory, what we derive under assumptions, and what we choose for the demonstration.

01 / WHAT IS SUPPORTED?

The four-slider audit

Assessment of the implemented relationships
Input	How it enters the calculation	Assessment
Air-gap diameter Dg	Changes torque arm, slot space, Kt/Ke and rotor inertia.	The stress–torque identity is established. Fixed flux, turns and waveform factors are assumptions; changing Dg at fixed outside diameter can reduce torque by shrinking the slots.
Stator outside diameter Do	Changes slot and iron areas, current capacity, resistance and stationary mass.	Conditional sizing estimate. Fixed slot fraction, yoke thickness and copper current density do not establish saturation or thermal feasibility.
Stack length L	Scales active torque, Kt/Ke, resistance, mass and rotor inertia.	First-order scaling at fixed cross-section and turns. A constant end-turn factor is a simplification, especially for short stacks.
Gear ratio n	Multiplies torque, divides speed and multiplies reflected rotor inertia by n².	Established ideal-transmission relations. Efficiency is 100% by design; gearbox mass is a separate illustrative estimate.

The motor is an idealized inner-rotor, radial-flux design. Four dimensions alone do not specify a manufacturable motor.

02 / THE CALCULATION CHAIN

From geometry to available torque

Subscript 0 denotes the reference design. Geometry uses millimetres and mm² unless stated otherwise. Electromagnetic torque equations use metres. Angular speed is mechanical rad/s.

Our geometric approximation

1. Reserve space for the yoke, teeth and copper

D_{b} = D_{g} + g

D_{s} = D_{o} - 2 t_{y}

A_{s} = f_{s} \frac{π}{4} ({D_{s}}^{2} - {D_{b}}^{2})

A_{Fe} = \frac{π}{4} ({D_{o}}^{2} - {D_{b}}^{2}) - A_{s}

Dg is the mid-gap diameter; g is the radial gap, so Db is the stator bore diameter. Ds is the slot outer diameter. As is total slot area, not copper area: copper occupies kfill As. The geometric condition Do − Dg ≥ 16 mm reserves space but does not certify magnetic or mechanical feasibility.

The annulus and constant slot-fraction construction is our simplification. Slot-area and winding-space models are discussed in [2], §5.

Established identity + conditional scaling

2. Convert electric loading into torque

T_{motor} = \frac{π}{2} σ {D_{g}}^{2} L

A_{electric} \propto \frac{k_{fill} J A_{s}}{π D_{g}}

\frac{σ}{σ_{0}} = \frac{\frac{A_{s}}{A_{s0}}}{\frac{D_{g}}{D_{g0}}}

\frac{T_{motor}}{T_{0}} = \frac{A_{s}}{A_{s0}} \frac{D_{g}}{D_{g0}} \frac{L}{L_{0}}

The stress identity follows by integrating tangential force over the cylindrical air gap and multiplying by radius. Holding magnetic loading, winding factors, copper fill and current density fixed gives our normalized torque scaling. Waveform and winding factors are absorbed into the reference stress; we do not equate all peak and RMS definitions.

[1], Eq. (3) supports the torque identity; [2], §5 supports the slot-area/current-capacity reasoning. The reference T₀ = 0.9 N·m is our uncalibrated choice. “Rated” here is a nominal model reference, not a verified continuous thermal rating.

Reduced electrical model

3. Keep current and voltage limits separate

T = K_{t} I E = K_{e} ω_{m} V = R I + E

K_{t} = K_{e} = \frac{48}{850} \frac{D_{g}}{76} \frac{L}{14}

R = R_{0} \frac{\frac{L}{L_{0}}}{\frac{A_{s}}{A_{s0}}}

I_{r} = \frac{T_{motor}}{K_{t}} I_{pk} = 2 I_{r}

I_{available} (ω_{m}) = \min (I_{pk}, \max (0, \frac{V - K_{e} | ω_{m} |}{R}))

This is a steady-state, single-port DC equivalent. Its SI voltage and current are defined so that Kt and Ke are numerically equal and EI = Tωm. They are not raw three-phase line, phase, peak, RMS or dq quantities. The effective terminal voltage is assumed to be 48 V; no inverter modulation mapping is made from a physical DC bus.

Fixed flux density, pole count, winding turns and winding factor give Kt, Ke ∝ Dg L. Fixed turns and fill make wire area proportional to As. With a fixed end-turn multiplier, wire length scales with L, hence R ∝ L/As. A larger slot accommodates thicker conductors and higher current; the controller’s absolute current and power ratings are not separately constrained.

Basic motor constants and operating limits: [3]. Winding resistance from conductor length and area: [2], §5.1. R₀ = 0.30 Ω, 850 rad/s and the ×2 current factor are illustrative inputs, not literature fits. This DC reduction omits the inductive voltage limits described by the full PM-machine theory in [2].

Why the old triangular envelope was replaced

The earlier line Tpk(1 − |ω|/ω₀) implicitly made the selected peak current torque equal to voltage-limited stall torque. There was no resistance model to justify that equality. The current implementation clips a voltage-limited line at a separate current limit: a flat low-speed region followed by a decline. Its reference constants still need calibration.

Ideal gearing

4. Map the motor to the joint

ω_{m} = n ω_{joint}

T_{available} (ω_{joint}) = n K_{t} I_{available} (n ω_{joint})

ω_{0,joint} = \frac{V}{K_{e} n}

ω_{base,joint} = \max (0, \frac{V - R I_{pk}}{K_{e} n})

J_{ref} = n^{2} J_{rotor}

At low speed, the current cap sets available torque. Above base speed, the voltage cap dominates. The torque–speed chart and the jumping simulation call the same function. Gearbox efficiency is 100%, so Tjoint ωjoint = Tmotor ωm.

Ideal torque/speed transformation and reflected rotor inertia: [4], §8.9. Gear tooth strength, feasible discrete ratios and transmission losses are not modeled.

Volume calculation + illustrative structural terms

5. Account for mass and inertia

m_{Fe} = ρ_{Fe} A_{Fe} L \cdot 10^{−9}

m_{Cu} = ρ_{Cu} k_{fill} A_{s} L f_{end} \cdot 10^{−9}

m_{rotor} = 0.12 {(\frac{D_{g}}{76})}^{2} \frac{L}{14}

m_{act} = m_{Fe} + m_{Cu} + m_{rotor} + 0.15 + 0.0015 n + 0.085 {(\frac{D_{o}}{100})}^{2}

J_{rotor} = 0.000085 {(\frac{D_{g}}{76})}^{4} \frac{L}{14}

Nominal torque density = \frac{n T_{motor}}{m_{act}}

Areas are mm², lengths mm, densities kg/m³; mass is kg and inertia kg·m². The iron/copper terms use m = ρV. Rotor and ancillary terms are our reference scaling estimates, not a CAD bill of materials or a gearbox sizing equation. The Dg⁴L inertia exponent assumes similar rotor geometry. A wider stator changes moving leg mass without changing rotor inertia at fixed Dg and L.

03 / NUMBERS WE CHOSE

Reference values are not validation data

These values define the demonstration. None has been fitted here to the paper’s Motor-CAD dataset or fabricated prototype. Geometric bounds define the UI search domain, not a certified design domain.

Inputs read directly from the model module
Quantity	Value	Status
Loading parameters…

04 / FOLLOW THE NUMBERS

Your design, evaluated by the same code

The default example is Dg 76 mm, Do 100 mm, L 14 mm, n 18. Open this page from “Inspect these formulas” in the lab to inspect your current hardware.

Intermediate values and the final joint limits
Quantity	Computed value
Loading calculation…

Direct imports: sizing and torque envelope · contact dynamics · integration and flight · bounded design search. Version: loading.

05 / WHAT THE MODEL DOES NOT ESTABLISH

How to interpret the results

Motor accuracy. No finite-element field solution, saturation check, leakage, iron loss, temperature rise, demagnetization, thermal duty cycle or manufacturability check is performed. A nominal current density does not establish a continuous rating.
Electrical accuracy. No phase/dq voltage circle, inductance, field weakening, switching dynamics, bus sag or system power cap. No-load speed is an electrical estimate, not a rotor mechanical speed limit. The braking catch reuses the magnitude envelope symmetrically; regenerative electrical behavior is not solved.
Mechanical accuracy. Thigh and shank are each 250 mm; their mass distributions remain educational approximations. The hip stays on a vertical guide; the foot switches between sticking and Coulomb sliding. Upward release uses a frozen-pose ballistic approximation. Flight joint motion and landing impacts are omitted. Joint/floor limits can end an attempt.
Optimization claims. Search compares candidates within this model and its chosen bounds. It proves neither a global optimum nor superiority of a real actuator. Changing reference assumptions can change the ranking.
Verification versus validation. Automated checks cover geometry constraints, circuit limits, gearing, numerical refinement, friction dissipation and reference contact equations. They do not measure prediction error against physical hardware. The paper’s validation is reported in the paper.

06 / PRIMARY SOURCES

Read the original reasoning

W. L. Soong. Sizing of Electrical Machines. Power Engineering Briefing Note #9, University of Adelaide, 26 September 2008. Eq. (3), first PDF page, and the stator-sizing discussion. Supports the stress/geometry relation, not our reference constants.
James L. Kirtley Jr. Permanent Magnet “Brushless DC” Motors. MIT 6.685 Electric Machines, Course Notes 7, Fall 2013. §5 “Current Rating and Resistance” and §5.1 “Resistance” (printed pp. 22–23). The full machine model is more detailed than this playground.
maxon. Motor data and simulation. Technical documentation, sections on torque constant, speed constant and speed/torque gradient; accessed 10 September 2026. Explains motor-constant conventions and why real iron-core and commutated motors need additional care.
Kevin M. Lynch and Frank C. Park. Modern Robotics: Mechanics, Planning, and Control, Cambridge University Press, 2017. §8.9, Actuation, Gearing, and Friction, official companion material. Supports ideal gearing and reflected inertia.

Every assumption.Every equation.