Motion · actuation energetics
Joules per Punch
Where the energy of a humanoid strike goes, why every segmented spine tested loses to one waist joint, and what a spine is actually for.
The Charlot Lab · Institute for Physical AI @ Bailey Military Institute
Takes the sum E_task = E_compute + E_actuation from the total-task-energy bench to a strike. A single-file bench, eight printed gates, no dependencies.
1. What question does this report answer?
The robot boxing matches of 2025 and 2026 put a humanoid's whole-body coordination on public display, and the display reads as flailing: fast hands, still torso, feet that exist to not fall over 7. The standing explanation blames the policy. A better explanation starts from the body. Filimonov's force decomposition on master boxers assigns roughly three-quarters of straight-punch force to leg drive and trunk rotation together 1, and the base G1 offers a single waist yaw joint on that entire power path, with the EDU variant adding roll and pitch 6. A policy cannot express a motor program its body has no channel for.
From there the engineering proposal writes itself: give the robot a spine. Replace the single waist joint with a stack of vertebral modules and let the trunk deliver its 37 percent. This report asks whether that proposal survives contact with an actuator, in the only currency a battery understands. It prices the same strike on four architectures and reports where every joule goes.
2. How is a punch priced so that bodies can be compared?
Fist kinetic energy will not serve as the task, because an arm-only punch can be fast and still weak. What a target receives is impulse, and impulse is set by the effective mass the body presents at the contact point along the punch direction. Khatib's operational-space formulation gives that quantity exactly 2:
| Quantity | Definition | Value used |
|---|---|---|
| Effective mass | m_eff = 1 / (uᵀ J M(q)⁻¹ Jᵀ u), the inertia presented at the fist along direction u | floor of 2.4 kg, 7 percent of the 35 kg body |
| Delivered momentum | P = m_eff · v_fist | P* = 9.0 kg·m/s, identical for every architecture |
| Fist speed band | a strike, not a shove and not a slap | 2.8 to 3.8 m/s at contact, 280 ms from guard |
Each constraint closes a loophole the optimiser otherwise takes. Momentum alone lets a body trade effective mass against speed one for one, and the search returns either a fast, nearly massless arm or a slow shove with the whole torso behind it; the speed band excludes both. The effective-mass floor is what a coach means by putting your weight behind it, and the sport-science literature grounds the choice: trained strikers present roughly twice the effective mass of novices at similar hand speeds 9. Every architecture must deliver the same strike; whichever spends fewer joules wins.
Every architecture is also granted a perfect energy-optimal controller, found by a multi-start constrained search over strike profiles, so the comparison is the floor the body imposes rather than any policy's skill. The electrical cost per actuator follows the identified G1 arm power model: copper loss as torque squared over the motor constant, viscous and Coulomb friction, positive mechanical work, and idle 3. Loss magnitudes are calibrated so a nominal reach lands near the 71.5 joules measured on the physical G1 4, and the sibling bench carries that calibration 11. Absolute joules are indicative; the comparisons between architectures are the result.
3. What does the baseline strike cost, and where does it go?
Architecture A is the G1 as built: a rigid trunk on one waist joint driven by one 1.8 kg motor. Its energy-optimal strike delivers the target momentum at 3.19 kg of effective mass and 2.83 m/s, and the bill decomposes as follows.
| Term | Joules | Share | What it is |
|---|---|---|---|
| Positive mechanical work | 106.4 | 39% | the kinetic energy of the strike itself, plus work against coupling |
| Copper loss | 83.0 | 31% | torque squared over the motor constant, of which the trunk motor pays 7.3 J |
| Bearing friction | 52.4 | 19% | viscous and Coulomb, per joint |
| Motor viscous | 23.3 | 9% | speed-squared loss referred to each actuator's output |
| Idle | 5.6 | 2% | drive electronics and holding current over the strike window |
| Total actuation | 270.7 | 100% | the number every other architecture is measured against |
Two features of this decomposition drive everything that follows. Copper is the second-largest term and the only one that depends on how torque is produced rather than on the motion itself. And the trunk's share of it is small in the baseline, which is the first hint of Section 4's result: even a perfect controller barely uses the one trunk joint it has.
4. What happens when momentum is all that is scored?
Drop the effective-mass floor, the way a reward function scores a hit sensor or a momentum estimate, and re-run the search. The optimum barely changes: 3.22 kg at 2.80 m/s, 268.3 joules, and the trunk motor pays 6.9 of the 82.0 joules of copper, 8 percent of the copper bill for the joint that anatomy says should carry 37 percent of the force.
Driving twelve kilograms of torso is not worth the joules when a lighter, faster arm meets the same score. The flailing in the ring is therefore close to optimal play for the objective in use, and this is a result about morphology and scoring before it is a result about any policy. Training longer against the same reward converges to the same arm punch. The correction belongs in the objective: score the effective mass presented at contact, a quantity computable from the model the controller already carries 2, and the trunk becomes worth paying for.
5. What does adding vertebrae cost?
Architecture B is the naive spine: N trunk joints, each driven by its own rotary module, the arrangement every modular-actuator catalogue suggests. The actuator mass budget matters, so both conventions are run. B splits the baseline's 1.8 kg of trunk motor across the N modules; B′ gives every module the full 1.8 kg and lets trunk mass grow.
| Trunk joints | B: fixed total motor mass | vs A | B′: full-size modules | vs A |
|---|---|---|---|---|
| 1 (baseline A) | 270.7 J | 270.7 J | ||
| 2 | 306.1 J | +13% | 277.2 J | +2% |
| 3 | 394.2 J | +46% | 286.5 J | +6% |
| 5 | 749.8 J | +177% | 303.8 J | +12% |
| 8 | 1,941.8 J | +617% | 317.3 J | +17% |
| 12 | 5,325.8 J | +1,867% | 359.1 J | +33% |
| 17 | 13,101.9 J | +4,740% | 405.0 J | +50% |
Seventeen is the human thoracolumbar count, and at seventeen the split-budget spine pays 48 times the baseline for the identical strike. The full-size-module column avoids the catastrophe by spending 28.8 kg of additional motor mass, 30.6 kg of trunk motor in all, to reach a 50 percent penalty. There is no crossover to find: the segmented trunk never dips below the single joint at any count, under either convention.
6. Is the motor scaling law doing the work?
The result rests on two pieces of physics, so each is stated and then stressed.
Serial joints share motion, not torque. In a serial chain every joint transmits the full torque of everything distal to it. Splitting one waist joint into N vertebrae does not divide the load; it replicates it, so N modules pay N copper terms of the same size where the baseline paid one. Bearing friction moves the other way, since the same total rotation split N ways costs 1/N in a speed-squared term, and the bench shows exactly that in its bearing column. Copper dominates.
Smaller motors are worse at torque. For geometrically similar machines the motor constant squared, torque squared per watt of copper, scales as actuator mass to the five-thirds 8. Under a fixed mass budget each of N modules is 1/N the size, so each pays a further factor of N to the five-thirds. The two effects compound: the trunk copper multiplies as N to the one plus the exponent, roughly N to the eight-thirds at the geometric-similarity value.
The exponent is the load-bearing assumption, so it is swept across its physical range. An exponent of 1.00 says copper splits perfectly with mass, crediting the segmented design with everything geometry refuses it.
| Trunk joints | exponent 1.00 | 1.33 | 1.67 (geometric similarity) | 2.00 |
|---|---|---|---|---|
| 3, split budget | +30% | +36% | +43% | +52% |
| 8, split budget | +219% | +365% | +604% | +997% |
The geometric-similarity column re-prices the architecture inside the sensitivity harness with its own multi-start search, so its +604 percent for eight joints sits a few percent under the main table's +617 for the same body: that spread is the search scatter, printed rather than reconciled. At the most generous exponent physically available, an eight-segment spine still pays more than three times the baseline. The N-fold replication of the torque bill carries the result on its own, and that term is set by the topology of a serial chain rather than by any motor technology. Better magnets do not repeal it.
7. What is a spine actually for?
Architecture C is the biological arrangement read as an engineering specification: many kinematic degrees of freedom driven by few actuators that each span multiple joints, with elastic elements in series. A tendon crossing eight vertebrae is one tension, one copper bill, applied along the whole span; the price is that one actuator can only command the block-mean torque of the joints it crosses, and the bench charges that residual against a 35 percent feasibility cap.
| Configuration | Strike energy | vs A | Where the joules moved |
|---|---|---|---|
| 8 joints, 2 actuators | 284.1 J | +5% | copper back near baseline: 8.5 J on the trunk |
| 12 joints, 3 actuators | 295.5 J | +9% | a third actuator adds its own copper and idle |
| 17 joints, 3 actuators | 291.9 J | +8% | five more vertebrae than the row above, for a rounding error |
| 17 joints, 4 actuators | 338.8 J | +25% | the fourth actuator, not the seventeen joints, moves the bill |
Series elasticity is the other lever, and it is priced by a quantity that is easy to get wrong. A linear element storing energy E through a winding travel of theta carries a peak torque of 2E over theta, so the copper cost of loading it falls quadratically with the travel available and does not fall with winding time: winding slowly at the same torque holds the same torque-squared loss for longer. Priced by travel, elastic pre-load on the baseline body reaches a 1 percent saving at a 75 percent elastic share, real but small at this strike energy. Priced by duration, the same model returns large fictitious savings, which is the version of this proposal that does not survive its own arithmetic. Segmentation earns its keep exactly where it buys winding travel for such an element, which is a second, independent reason the useful spine is tendon-driven.
8. Where does compute sit in a strike?
The sum this series prices is E_task = E_compute + E_actuation, and the compute term of a punch is small in a specific, quantifiable way. A 23-degree-of-freedom locomotion-class policy at 50 Hz executes about 175,000 multiply-accumulates per step. Over the 280 ms strike:
| Platform duty | Platform energy | Arithmetic energy | Share of E_task | A 10x smaller policy saves |
|---|---|---|---|---|
| Orin NX class, policy only, 15 W | 4.20 J | 2.5 × 10⁻⁶ J | 1.5% | under a millionth of a percent |
| with perception, 25 W | 7.00 J | 2.5 × 10⁻⁶ J | 2.5% | under a millionth of a percent |
The arithmetic sits six orders of magnitude below the platform's own idle, and the platform itself is 1.5 to 2.5 percent of the strike. Compressing the policy therefore optimises a term that is not there, which reproduces on a strike what the embodied-efficiency literature reports for manipulation 5 and what the sibling bench measures per completed reach 11. The compute lever that does exist is duty cycle: platform watts times seconds is the whole compute bill, so gating the platform moves it and shrinking the network does not. TR-2026-40 states the same bound from the other side, as the reciprocal of the non-compute share 13.
9. What did the bench have to survive?
A dynamics model that is wrong produces confident nonsense, so the bench refuses to print results until eight gates pass, and the run output opens with them. Two of the eight exist because an earlier revision of the bench produced a finding that was fake, and both failures generalise beyond this bench.
| Gate | What it pins | Result |
|---|---|---|
| Energy conservation | free rotation about a vertical axis holds kinetic energy over 80 ms | drift 3.2 × 10⁻⁶ |
| Inverse dynamics identity | recursive Newton-Euler equals M(q) times acceleration at zero velocity | 2.4 × 10⁻¹⁶ |
| Work-energy along the strike | net mechanical work equals the kinetic energy created, on the actual profile | 6.4 × 10⁻⁴ |
| Analytic effective mass | a single link presents I over r squared, tangentially | exact |
| Locked-joint control | a 5-joint trunk with 4 joints frozen reproduces the 1-joint chain | 1.8 × 10⁻¹⁶ |
| Degenerate-direction guard | a near-motionless fist is rejected as inadmissible | rejects at 2.8 × 10⁻⁴ m/s |
| Scaling law wiring | halving module mass costs 3.17x in copper, as the exponent demands | exact |
| Optimiser soundness | tightening a constraint never lowers the reported optimum | slack −0.6% |
The degenerate-direction guard. The first revision reported a 98 percent energy saving for a three-segment spine. Near a singular punch direction the quadratic form under Khatib's effective mass goes to zero, so the effective mass diverges, and the momentum target is met by a fist moving at a third of a millimetre per second. The optimiser found that attractor immediately, and the row it produced looked exactly like a result. The general rule: a physically unbounded quantity in an objective is an optimiser attractor, and it must be bounded explicitly rather than trusted to be unreachable.
Optimiser soundness. A later revision returned a strictly more constrained run 7 percent cheaper than a less constrained one, which no true optimum permits: tightening a constraint shrinks the feasible set. The cause was a redundant radial dimension in the search parameterisation that the scale solver already handled, so the descent wandered among local minima and the wandering read as a morphology difference. The gate that now enforces monotonicity is a one-line check any constrained search can carry, and it converts a silent convergence failure into a loud one.
10. What is established, and what would extend it?
Established, within the model. That a fixed strike admits a clean architecture comparison once effective mass, momentum and speed are all pinned (Section 2). That the segmented, per-joint-actuated trunk costs more than a single waist joint at every count tested, under both mass conventions, and that the penalty survives the scaling exponent swept to its most generous value (Sections 5 and 6). That the copper bill follows actuator count once actuators span joints, making joint count free (Section 7). That elastic storage is priced by winding travel rather than winding time (Section 7). That the compute share of a strike is 1.5 to 2.5 percent with the arithmetic six orders below platform idle (Section 8).
Open. Whether a tendon-driven trunk's routing friction and residual-torque cost, charged here as a block-mean constraint with a 35 percent cap, hold at those values on hardware. Whether the strike profile a learned whole-body policy discovers against an effective-mass-aware reward matches the optimiser's floor, which is a training experiment the sibling bench's recipe supports. And the anchor itself: this review did not locate a published joules-per-strike measurement for any humanoid, so the absolute scale here rests on the reach measurement of the identified power model's platform 4. An instrumented strike, pack current against a force plate through contact, would replace a calibrated scale with a measured one and is within reach of any lab that owns the robot these matches made famous.
Prior art and position. Operational-space effective mass, serial-chain dynamics, motor scaling and series-elastic actuation are established fields. The punch force decomposition is forty years old 1, and tendon-driven musculoskeletal humanoids are a mature research line whose energetic rationale this result supports from the pricing side. What this report contributes is the joining: the strike stated as an effective-mass task, the segmented trunk priced per joule against its single-joint baseline under an identified loss model, the actuator-count reading of the multiarticular alternative, and the two optimiser failures converted into reusable gates. This review did not locate these stated together in the literature it read.
11. Conclusions
A punch prices a body. Asked to deliver a fixed strike, a humanoid with one waist joint spends 271 joules, and every segmented spine bolted onto the same torso spends more, because a serial chain replicates torque down its whole length and torque squared is what copper charges for. The number that governs the bill is not how many joints the trunk has but how many actuators drive it, and the arrangement that makes vertebrae free, few tension elements spanning many joints with elastic series compliance, is the one the biological spine has used all along. The robot in the ring is not failing to use a spine it lacks; scored on momentum, it is correctly declining to pay for the torso it has. The cheapest improvements to the spectacle are therefore an objective that scores effective mass at contact, and a trunk actuated like a tendon rather than like a column of servos. The joules were never in the network. On a strike, as on a reach, they are in the windings, and the windings answer to topology.
12. The forcing function
| What is bounded | The physics that sets it | The engineering change that moves it | What becomes possible |
|---|---|---|---|
| A segmented, per-joint-driven trunk pays 13 to 4,740 percent over one waist joint | Serial joints each transmit the full distal torque, and copper charges torque squared per actuator, worsened by mass to the five-thirds under a split budget | Multiarticular actuation: one tension element spanning many joints, so the copper bill scales with actuator count | Vertebral count becomes free; segments are added for travel, compliance and impact tolerance at no energy cost |
| Elastic pre-load saves 1 percent on a rigid trunk | Storage torque is 2E over the winding travel, so cost falls with travel and never with winding time | Long-travel, low-stiffness series elements wound through a segmented trunk's full range | A genuine stretch-shorten cycle, the one mechanism here that removes peak actuator power rather than relocating it |
| Momentum-scored strikes leave the torso still, at 8 percent of the copper bill | Effective mass and fist speed trade one for one in the score, and the arm is the cheaper channel | Score the effective mass presented at contact, computable from the controller's own model | Whole-body strikes become worth learning; until the objective names m_eff, more training data converges to the same arm punch |
| Compute is 1.5 to 2.5 percent of a strike, its arithmetic a millionth of that | Platform watts times seconds is the whole compute bill; the multiply-accumulates are noise within it | Duty-cycle the platform between control-relevant windows instead of shrinking the network | The efficiency budget moves to the terms that exist: trajectory shape, actuation topology, winding losses |
| The absolute scale rests on a calibrated reach, 71.5 J, rather than a measured strike | This review did not locate a strike-energy measurement on humanoid hardware; the model's scale is calibrated on a reach | Pack-current metering through an instrumented strike against a force plate, on the platform already in the ring | Joules per punch becomes a reported figure rather than a modelled one, and this table's first row becomes testable on hardware |
A humanoid trunk that pays for itself in a strike was out of reach while the only actuation on offer was one rotary module per joint, and this report prices exactly why. It is demonstrably approachable today: the multiarticular rows already sit within 5 percent of the single-joint baseline while carrying seventeen vertebrae of travel and compliance, and the tendon-driven humanoid literature has built such trunks for a decade. It becomes ordinary when trunk actuation is specified by actuator count and winding travel rather than joint count, which is a design-rule change, the most tractable kind of change there is.
References
- Filimonov, V.I., Koptsev, K.N., Husyanov, Z.M. and Nazarov, S.S. Boxing: Means of increasing strength of the punch. National Strength and Conditioning Association Journal 7(6), 65 (1985). Master-class decomposition: 38.5 percent leg drive, 37.3 percent trunk rotation, 24.1 percent arm extension. reported · via the sport-science literature
- Khatib, O. A Unified Approach for Motion and Force Control of Robot Manipulators: The Operational Space Formulation. IEEE Journal on Robotics and Automation 3(1), 43 (1987). The effective-mass definition used throughout. read · framing and definition
- Deniz et al. Physics-based electrical power model of the Unitree G1 7-DOF arm. arXiv:2606.15915 (2026). Copper, viscous, Coulomb and baseline-torque structure; R² 0.965 on unseen speeds. The loss form and per-joint dominance pattern this bench reproduces. measured · third party
- Energy-aware reinforcement learning on the same arm, arXiv:2606.15918 (2026). Mean 71.5 ± 48.3 J per reach, measured on the physical G1. The calibration anchor for absolute scale. measured · third party
- Li, Yang, Chen, Chen, Lin and Li. From Inference Efficiency to Embodied Efficiency. arXiv:2603.19131 (2026). Methods that reduce computation under conventional metrics often increase end-to-end execution cost on robotic platforms. read · abstract, via the Institute record
- Unitree Robotics, G1 published specification: about 35 kg, 23 degrees of freedom in the base configuration, waist degrees of freedom by variant. vendor specification
- CES 2026 humanoid boxing demonstrations, press record: Interesting Engineering, 6 January 2026, and the eWeek and Rest of World coverage of the 2025 Hangzhou tournament and the San Francisco exhibition bouts. reported · press
- Seok, Wang, Otten and Kim. Actuator design for high force proprioceptive control in fast legged locomotion. IROS 2012. Motor-constant scaling for geometrically similar electric machines, the exponent Section 6 sweeps from 1.00 to 2.00. read · scaling argument
- Neto, O.P., Magini, M. and Saba, M.M.F. The Role of Effective Mass and Hand Speed in the Performance of Kung Fu Athletes Compared with Nonpractitioners. Journal of Applied Biomechanics 23(2), 139 (2007). Trained strikers present roughly twice the effective mass of novices at similar hand speeds; the basis for the 7 percent floor. reported · abstract
- Institute for Physical AI @ BMI, Charlot Lab.
punch_energetics.rs, the joules-per-strike bench behind every table above: exact planar recursive Newton-Euler dynamics, the identified loss form of reference 3, a multi-start constrained search granted to every architecture, and the eight gates of Section 9 printed at the head of the run output alongside the full sweep. modelled · ours, eight gates printed - Institute for Physical AI @ BMI, Charlot Lab.
total_task_energy.rs, the sibling bench: the same sum per completed reach, with the compression ablation and the calibration against reference 4. modelled · ours - Institute for Physical AI @ BMI, Energy Observability, Technical Report TR-2026-36. The measurement doctrine this bench's gates apply. ours · prior report
- Institute for Physical AI @ BMI, The Physical AI Hardware Lottery, Technical Report TR-2026-40. States the whole-robot bound on any compute improvement as the reciprocal of the non-compute share, 1.03 to 2.44 times on the bodies measured so far; Section 8 here is the strike-side instance. ours · prior report