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Abstract #

Humanoid spec sheets list peak joint torque and battery life. Neither tells a buyer how long the robot can stand still holding a box. We computed static joint torques for five open humanoid models from MuJoCo Menagerie: Unitree G1, Unitree H1, Booster T1, Apptronik Apollo and Fourier N1. Each held 0 to 20 kg while standing, squatting or balancing on one leg. We passed the torque on the most loaded joint through a two-node motor thermal model. That model reproduces published peak-current durations of a quasi-direct-drive actuator to within 12%. For the G1, we also built a battery budget from measured copper-loss coefficients and the measured rest power. The weakest link is never the legs when the robot holds a box in front of it. On the G1 it is the 5 N·m wrist pitch joint. The largest box the G1 can hold indefinitely weighs 2.23 kg, close to Unitree’s own “about 2 kg” arm load. Across the five models, that payload ranges from 0 kg (H1, limited by the ankle in our reference stance) to 16.47 kg (Apollo). Holding 5 kg, the G1’s wrist winding reaches its limit after 16.2 s in the model, or 64.8 s with four times the thermal capacitance. The battery would last 1.42 h at the same 296.9 W draw. When a humanoid holds a box, motor heat runs out long before the battery does.

1. Introduction #

Our earlier analysis of humanoid thermal management showed that actuator heat sets the operating envelope of a humanoid as much as its torque does [1]^[2]. A related article on actuation set out how torque budgets are drawn up per joint [2]^[3]. This article puts the two together on a task that sounds trivial: stand still and hold a box.

Vendors publish two kinds of numbers. Unitree lists the G1 with a maximum knee torque of 90 N·m (120 N·m for the EDU version), a 9000 mAh battery, “about 2 h” of battery life and an arm load of “about 2 kg”, adding that the arm load “varies greatly under different arm extension postures” [3]^[4]. For the H1 it lists an 864 Wh battery and joint torques of up to 360 N·m. For the heavier H1-2 it gives a rated arm payload of about 7 kg and a peak of about 21 kg [4]^[5]. Peak torque alone does not say whether a joint can deliver torque at a given posture for a given time [5]^[6], and battery life alone does not say what limits a static hold. Holding a load does no mechanical work, yet every motor that resists gravity turns current into heat.

The question is timely. Loco-manipulation policies are being trained to carry payloads on these exact robots, and thermal-aware control is starting to appear in legged robots. However, published papers rarely give a simple per-robot table of what can be held and for how long. We provide one, built from open models, and we state each assumption.

RQ1: Which joint limits how much a humanoid can hold, and how much torque headroom do five open humanoid models have when they stand, squat or balance on one leg with a 0-20 kg box? RQ2: How long can the bottleneck motor sustain each hold before its winding reaches the thermal limit, and what payload can be held indefinitely? RQ3: For the Unitree G1, which runs out first during a static hold, the battery or the motors’ thermal margin?

2. Existing Approaches (2025-2026 state of the art) #

2.1 Learned whole-body loco-manipulation under load #

Most current payload work trains reinforcement-learning (RL) policies. FALCON splits whole-body control into a lower-body policy for balance and an upper-body policy for end-effector tracking. It trains them with a force curriculum “while respecting torque limits”, and on real robots it transports payloads with 0-20 N of end-effector force [6]^[7]. HULK combines model predictive control with RL for heavy objects, one teacher for wrist forces and one for holding large objects against the body. It is evaluated in simulation and on a Unitree G1, at 10 kg per arm in simulation [7]^[8]. HEFT teleoperates a 65 kg full-size humanoid through squats and walking under payloads of up to 24 kg [8]^[9]. HALO identifies the robot model and the unknown payload with differentiable MuJoCo before training, so that heavy-load policies transfer to hardware [9]^[10]. HOTICE trains separate upper- and lower-body policies that share observations to move objects through clutter [10]^[11]. All of these respect instantaneous torque limits. None of them reports how long a pose can be held before a motor overheats.

2.2 Thermal-aware control #

A second line of work puts motor temperature into the controller. On a Unitree A1 quadruped carrying 3 kg, a baseline policy triggered overheat protection after about 7 minutes. A policy with motor temperatures in its observations and thermal-constraint rewards ran for over 27 minutes [11]^[12]. A follow-up adds a residual thermal policy on top of a nominal one. With 3 kg, it walked for over 13 minutes, while the nominal policy overheated in about 5 [12]^[13]. The Olaf character robot feeds actuator temperatures to its policies because small neck actuators driving a large head risk overheating [13]^[14]. The closest precedent for static poses is NASA’s Valkyrie. A first-order, effort-based thermal model predicted motor temperatures from joint torque alone, and balancing on one leg heated the stance-leg actuators to a 75 °C warning level in less than two minutes [14]^[15].

2.3 Actuator characterisation #

Quasi-direct-drive (QDD) actuators, a motor with a low-ratio gearbox, now dominate legged robots. They need high currents for torque and therefore heat up [15]^[16]. Design studies now optimise single-stage planetary gearboxes for these actuators systematically [16]^[17]. Vendors state continuous and peak ratings. A typical commercial unit, the CubeMars AK80-64, lists a rated torque of 48 N·m and a peak of 120 N·m [17]^[18]. Urs et al. measured a full two-node thermal model of an open-source QDD actuator. They also published how long the actuator can sustain a given current: 9.2 A for 20 s and 22 A for 2 s [15]^[16]. The Human-Level Actuation framework argues that thermal sustainability belongs next to torque and power in any joint comparison [5]^[6].

2.4 Energy and battery models #

Deniz et al. identified a physics-based electrical power model for the 7-degree-of-freedom (DoF) left arm of a real Unitree G1. Copper losses dominate the shoulder yaw and elbow, and the whole robot draws 135 W at rest [18]^[19]. The same group then used that model to train energy-efficient reaching policies [19]^[20]. On the locomotion side, passive-dynamics-inspired training cut the hardware cost of transport of a G1 by 16.3% with a motion prior [20]^[21]. These studies treat energy per motion; they do not compare battery runtime with thermal runtime during a static hold.

flowchart TD
    Q[How much can a humanoid hold, and for how long?] --> A[Learned loco-manipulation]
    Q --> B[Thermal-aware control]
    Q --> C[Actuator characterisation]
    Q --> D[Energy and battery models]
    A --> A1[Respects peak torque limits, no time dimension]
    B --> B1[Temperature in the policy, mostly quadrupeds]
    C --> C1[Continuous vs peak ratings, measured thermal models]
    D --> D1[Energy per motion, measured copper losses]
    A1 --> G[Gap: per-robot hold time from torque, heat and battery together]
    B1 --> G
    C1 --> G
    D1 --> G

3. Method and Data #

3.1 Robot models #

We used the MJCF (MuJoCo XML) models of five humanoids from MuJoCo Menagerie, commit 98a70c569c4c1b6da467a20012f766ee1f8dc521 [21]^[22], in MuJoCo 3.17.0. Every body in these models has explicit inertia, so we removed visual meshes without changing any mass property. The models’ total masses are 33.34 kg (G1), 51.44 kg (H1), 31.61 kg (T1), 80.9 kg (Apollo) and 39.73 kg (N1). Unitree states about 35 kg for the G1 and about 47 kg for the H1 [3]^[4][4]^[5]. Torque limits come from the MJCF itself: the actuator forcerange for four robots, and ctrlrange times gear for the H1’s torque motors. These are the limits a simulation enforces, and we treat them as peak torque. They do not always match the vendor’s numbers. The G1 model allows 139 N·m at the knee against the stated 90 or 120 N·m [3]^[4]. The H1 model allows 300 N·m at the knee and 40 N·m at the ankle, against about 360 and 59 N·m on the vendor page [4]^[5].

3.2 Postures and statics #

Each posture fixes hip and knee flexion: standing (10° and 20°), half squat (45° and 90°) and deep squat (70° and 110°). For one-leg stance, the swing leg is raised to 30° and 60°. The code solves the ankle pitch and trunk pitch so that each foot is flat and the whole-body centre of mass, robot plus box, sits over the centre of the support area. In one-leg stance it also solves the stance hip roll and ankle roll (base roll for the H1, which has no ankle roll). A variant places the centre of mass over the ankle joints instead. Inverse kinematics puts the grip points in front of the shoulders in two arm poses. In “carry”, the hands are 0.60 L forward and 0.45 L below the shoulder, where L is the arm’s reach. In “reach”, they are 0.95 L forward and 0.05 L below. The box weight is split equally between the two hands.

Joint torques follow from static equilibrium: gravity torques minus the foot contact wrenches minus the box load, with the foot wrenches chosen as the minimum-norm set that balances the floating base. Utilisation is |τ|/τ_limit for every actuated joint, and the bottleneck is the joint with the highest utilisation. Postures that need a joint outside its range, or where the hands miss the target by more than 5 cm, are flagged infeasible and left out of the charts.

3.3 Motor heat #

Each bottleneck joint gets the two-node thermal model of Urs et al. [15]^[16], with fans on: winding-to-housing resistance 0.828 K/W, housing-to-ambient 1.44 K/W, winding capacitance 15.9 J/K and housing capacitance 146 J/K. That gives time constants of 13.2 s and 210.2 s. Copper loss scales with torque squared. We normalise it so that holding the continuous torque leaves the winding exactly at its limit in steady state. Time to limit then depends only on the ratio of held torque to continuous torque and on the time constants.

We set continuous torque to one third of the MJCF limit. This matches the H1-2’s rated-to-peak arm payload of 7 kg to 21 kg [4]^[5]. Other anchors are higher: 0.4 for the AK80-64 [17]^[18] and 0.4059 for the MIT Mini-Cheetah actuator (6.9 N·m continuous against 17 N·m peak, from the table in [15]^[16]). We therefore sweep the ratio over 0.25, 0.333 and 0.4, and the thermal capacitance from ×0.5 to ×4, because humanoid joint motors are larger than the 0.5 kg actuator that was measured. Every hold starts cold.

As a check, the model must reproduce the published sustainable peaks of the same actuator. Its continuous current is 5.587 A: 4.4 N·m continuous torque, divided by the 7.5:1 ratio and by a torque constant of 0.105 N·m/A.

3.4 Battery budget for the G1 #

Power during a hold is the measured 135 W rest power [18]^[19] plus the copper loss b·τ² of every joint. For the arm joints, b is the coefficient measured on a G1 (for example 0.4184 W/(N·m)² for shoulder pitch and 0.3942 for the elbow) [18]^[19]. For the other joints we assume equal copper loss at each joint’s torque limit, matching the elbow’s 246.4 W at 25 N·m (our assumption). Battery energy is our estimate: 9 Ah × 13 cells × 3.6 V nominal = 421.2 Wh, from Unitree’s 9000 mAh, 13-string specification [3]^[4]. The vendor’s “about 2 h” battery life implies an average draw of 210.6 W.

flowchart LR
    M[Menagerie MJCF, 5 robots] --> P[Posture solver: flat feet, CoM over support, arm IK]
    P --> S[Static equilibrium: joint torques with box load]
    S --> U[Utilisation per joint vs MJCF limit]
    U --> B[Bottleneck joint]
    B --> T[Two-node thermal model, validated on published peaks]
    T --> H[Time to thermal limit and continuous payload]
    S --> E[G1 copper loss plus 135 W rest power]
    E --> R[Battery hours at 421.2 Wh]
    H --> C[Which limit binds]
    R --> C

The full run takes about two minutes on one CPU core. Code, models and outputs: stabilarity/hub/research/how-long-can-a-humanoid-robot-hold-a-box-joint-torque-motor-heat-and-battery-limits-on-five-open-robot-models.

4. Results #

4.1 RQ1: The bottleneck is in the arms, and on the G1 it is the wrist #

Chart 1 plots the utilisation of the most loaded joint against box mass. With empty hands, every robot standing in the carry pose uses less than half of any joint’s limit: 9.4% on the G1, 41.3% on the H1, 8.8% on the T1, 11.1% on Apollo and 19.7% on the N1. Adding a box moves the bottleneck from the legs or waist to the arms on every robot.

Robot	MJCF mass (kg)	Bottleneck at 0 kg	Bottleneck at 10 kg	Utilisation at 10 kg (%)	Peak payload, carry (kg)	Continuous payload, carry (kg)	Continuous payload, reach (kg)
Unitree G1	33.34	waist (4.72 of 50 N·m)	wrist pitch (6.07 of 5 N·m)	121.4	8.11	2.23	1.21
Unitree H1	51.44	ankle (16.53 of 40 N·m)	elbow (14.7 of 18 N·m)	81.7	12.43	0 (ankle); 3.65 with CoM over ankle	0 (ankle)
Booster T1	31.61	shoulder pitch (1.58 of 18 N·m)	shoulder pitch (13.32 of 18 N·m)	74.0	13.85	3.86	2.26
Apptronik Apollo	80.9	ankle (16.64 of 150 N·m)	shoulder pitch (25.0 of 114 N·m)	21.9	52.8	16.47	8.2
Fourier N1	39.73	ankle (5.9 of 30 N·m)	elbow (10.91 of 30 N·m)	36.4	29.5	9.08	6.48

Source: our MuJoCo statics on Menagerie models. Standing, carry pose. “Peak payload” is the box mass at which the bottleneck reaches 100% of its MJCF limit; “continuous payload” is the mass at which it reaches one third.

The G1 result has the clearest real-world counterpart. Its wrist pitch joint has a 5 N·m limit in the model. The grip point lies a few centimetres in front of that axis, so each kilogram in the hands loads the wrist directly. The wrist saturates at 8.11 kg, and its continuous payload is 2.23 kg. Unitree’s own figure is “about 2 kg” for the G1 and about 3 kg for the EDU [3]^[4]. With arms extended, the G1’s shoulder pitch becomes the bottleneck first, and the continuous payload falls to 1.21 kg.

Leg posture barely matters for the arms. On the G1 at 5 kg, wrist utilisation is 64.7% standing, 64.7% in a half squat and 64.4% in a deep squat. Without a box, the deep squat raises the G1 knee to 28.98 N·m, 20.9% of its 139 N·m limit. Squats are hard for other reasons: our deep squat needs the H1 ankle at -55.6° against a stop at -0.87 rad (-49.8°), so that pose is infeasible on the H1 model.

Balancing on one leg is the real leg test. On the N1, the stance hip roll reaches 46.29 of 54 N·m (85.7%) with empty hands. On Apollo, the stance hip yaw reaches 42.0% and is the bottleneck. On the T1 and the H1, the posture we prescribed needs hip roll beyond the joint range (-13.7° against a stop at -0.2 rad on the T1, 59.8° against 0.43 rad on the H1). Those robots would have to stand on one leg differently, and we do not report their numbers as feasible.

The H1 row shows how much the stance matters. Our reference posture puts the centre of mass over the middle of the foot, 0.067 m in front of the H1’s ankle. That costs 16.53 N·m at each ankle, 41.3% of the model’s 40 N·m limit, which is above our continuous threshold before any box is added. Moving the centre of mass over the ankles drops the empty-handed bottleneck to 11.7% (hip roll). The continuous payload then becomes 3.65 kg, limited by the elbow. The vendor’s ankle figure of about 59 N·m [4]^[5] would also remove the problem.

4.2 RQ2: Above the continuous line, holds last seconds to minutes #

First, the thermal model passes its check (Chart 3). With the fan-on parameters it predicts 10.27 A for 20 s and 24.63 A for 2 s, against the published 9.2 A and 22 A [15]^[16]. Those errors are +11.6% and +12.0%. The ratio between the two peaks is 2.398 against a published 2.391, an error of 0.3%. The paper’s own table lists slightly higher peak torques for the same durations (7.5 N·m for 20 s and 19.1 N·m for 2 s), which our predictions match more closely. The model is therefore slightly optimistic: it allows about 12% more current than the real actuator for the same duration. The fan-off parameter set predicts 18.58 A and 48.42 A, far from the published figures, so we use fan-on.

Chart 2 and the table below apply the model to the bottleneck joints. Below the continuous line, a hold never reaches the limit. Above it, time drops steeply.

Robot, standing carry	5 kg	10 kg	20 kg	5 kg, capacitance ×4
Unitree G1 (wrist pitch)	16.2 s	3.0 s (above peak)	0.8 s (above peak)	64.8 s
Unitree H1 (ankle, then elbow)	100.7 s	8.0 s	1.7 s (above peak)	402.7 s
Booster T1 (shoulder pitch)	177.4 s	10.5 s	2.2 s (above peak)	709.8 s
Apptronik Apollo (shoulder pitch)	no limit	no limit	206.6 s	no limit
Fourier N1 (elbow)	no limit	349.1 s	13.0 s	no limit

Source: our two-node thermal model, cold start, continuous torque = 1/3 of the MJCF limit. “Above peak” marks holds that also exceed the MJCF torque limit, which the robot could not hold at all.

At the 5 kg point, the G1 hold lasts 6.9 s if continuous torque is a quarter of the limit and 36.8 s if it is 0.4. It lasts 64.8 s if the motor has four times the measured thermal mass. That spread is wide, but every value is about a minute or less. The one-leg results show the same pattern without a box. The N1’s stance hip roll, at 85.7% of its limit, reaches the thermal limit in 7.0 s, and Apollo’s stance hip yaw in 152.5 s. This agrees with Valkyrie, whose stance leg reached its warning temperature in less than two minutes [14]^[15].

4.3 RQ3: For the G1, heat binds long before the battery #

G1 posture and box	Arm copper loss (W)	Other joints (W)	Total power (W)	Battery hold (h)	Thermal hold
Stand, carry, 0 kg	6.5	6.68	148.2	2.84	no limit
Stand, carry, 2 kg	35.62	14.74	185.4	2.27	no limit
Stand, carry, 5 kg	125.08	36.79	296.9	1.42	16.2 s
Stand, arms extended, 2 kg	103.13	43.86	282.0	1.49	206.7 s
Stand, arms extended, 5 kg	280.56	101.0	516.6	0.82	17.4 s
Deep squat, carry, 5 kg	107.34	77.86	320.2	1.32	16.5 s
One leg, carry, 5 kg	127.82	80.18	343.0	1.23	16.2 s

Source: our estimate. Rest power and arm copper-loss coefficients measured on a G1 [18]^[19]; battery energy 421.2 Wh estimated from the vendor’s 9 Ah, 13-string pack [3]^[4].

A G1 standing empty-handed draws 148.2 W, and its 421.2 Wh pack would last 2.84 h. That is consistent with the vendor’s “about 2 h” for normal use, which implies 210.6 W on average. Holding 2 kg raises the draw to 185.4 W and still leaves 2.27 h. At 5 kg the draw roughly doubles to 296.9 W, and the pack still lasts 1.42 h. The wrist winding, however, reaches its limit in 16.2 s. Chart 4 sets the two limits side by side on a log scale. Where a thermal limit exists, it arrives more than ten times sooner than the battery limit, and more than a hundred times sooner with a 5 kg box.

The copper-loss coefficients also give a plausibility check on the one-third assumption. At one third of its 25 N·m limit, the G1 elbow would dissipate 27.37 W. Through the measured actuator’s 2.268 K/W total thermal resistance, that would be a steady winding rise of 62.1 K. This is a consistency check only: if the real allowed rise of the G1 elbow were much smaller, its continuous ratio would be lower than one third.

flowchart TD
    S[Static hold requested] --> U{Bottleneck torque above continuous rating?}
    U -->|No| N[Indefinite hold, battery sets the limit: hours]
    U -->|Yes| P{Above the peak limit?}
    P -->|Yes| X[Cannot hold at all]
    P -->|No| T[Thermal limit: seconds to minutes]
    T --> A[Change posture: hug the box, centre of mass over ankles]
    T --> B[Share load: lower the box, swap arms, rest cycles]
    T --> C[Bigger or better-cooled arm and wrist motors]

5. Discussion #

What practitioners should take from this. Battery life is the wrong number for holding tasks. On the G1, any box that pushes the wrist or shoulder above about one third of its limit turns a battery limit of hours into a thermal limit of seconds. A spec sheet that listed continuous torque per joint, or a rated payload in a stated arm pose, would answer the holding question directly. Unitree already does this for the H1-2 (7 kg rated, 21 kg peak) [4]^[5], and the actuation framework of [5]^[6] argues for it.

The bottleneck sits where the lever arm is long and the motor is small: wrists, elbows and shoulders, not knees. The grip matters as much as the box. HULK’s teacher that holds large objects against the body [7]^[8] moves the load path away from the wrist, which our results suggest is thermally as well as mechanically sensible. The centre-of-mass target matters too. Moving it from mid-foot to over the ankles turned the H1’s continuous payload from 0 kg into 3.65 kg without any hardware change. Thermal-aware policies on quadrupeds already trade such choices against task performance [11]^[12][12]^[13]. Our numbers suggest humanoid manipulation policies need the same temperature input, as Olaf’s controller already has [13]^[14].

Limitations and threats to validity.

• Torque limits. MJCF torque limits are what simulators enforce, not certified actuator ratings, and the three we could compare differ from vendor figures by 16% to 54%. A different limit rescales every utilisation in proportion.
• Thermal parameters. We apply one measured thermal parameter set, from a 0.5 kg research actuator with fans, to every joint. We bracket the size effect with a ×0.5-×4 capacitance sweep but do not model cooling paths, temperature-dependent winding resistance or heat flow between neighbouring joints. The model is about 12% optimistic on its validation points.
• Continuous-to-peak ratio. The one-third ratio rests on a payload ratio, which approximates a torque ratio only when the arm’s own weight is small. The anchors span 0.33 to 0.41.
• Postures. Our postures are static and symmetric, with a minimum-norm split of foot forces. Real controllers move, shift weight and add feedback torques, all of which add heat.
• Battery. The battery budget leaves out gearbox friction, which dominates some G1 arm joints [18]^[19], and uses a nominal cell voltage of 3.6 V that we assumed.
• Scope. No robot was measured on hardware for this article. Every time in Section 4 is a model output.

What would change the conclusion. Three findings would change it. A G1 wrist that sustains 5 kg for minutes on hardware would mean either a larger thermal mass or a continuous rating above one third. Published continuous-torque tables for humanoid joints would replace our ratio sweep. Measured winding temperatures during a timed hold, the protocol that [5]^[6] proposes, would test the model directly. The ordering, however, is robust: in every G1 scenario where a thermal limit exists, it arrives more than ten times sooner than the battery limit.

6. Conclusion #

RQ1 Finding: The arms, not the legs, limit how much a standing humanoid can hold, and on the G1 the 5 N·m wrist pitch joint is the weakest link. Measured by the box mass at which the bottleneck joint reaches one third of its MJCF limit, the continuous payload is 2.23 kg for the G1 (Unitree states about 2 kg), 3.86 kg for the T1, 9.08 kg for the N1 and 16.47 kg for Apollo. For the H1 it is 0 kg in our mid-foot stance and 3.65 kg with the centre of mass over the ankles. Peak payloads range from 8.11 kg (G1) to 52.8 kg (Apollo). This matters for the series because it identifies small distal actuators, not leg motors, as the component that sets the payload of today’s humanoids.

RQ2 Finding: Above the continuous line, holds are measured in seconds to minutes. Measured by time to the winding limit in a two-node model that reproduces published actuator peaks to within 12%, the G1 holding 5 kg reaches its limit in 16.2 s (6.9-64.8 s across the parameter sweep), the T1 in 177.4 s and the H1 in 100.7 s, while Apollo holds 10 kg with no limit. This matters for the series because hold time, not peak torque, is the figure a buyer of a humanoid for warehouse or service work actually needs.

RQ3 Finding: For the G1, motor heat runs out long before the battery. Measured by holding time at a 5 kg box, the battery would last 1.42 h at 296.9 W, while the wrist reaches its thermal limit in 16.2 s; at 2 kg the battery still lasts 2.27 h and there is no thermal limit. This matters for the series because battery life is the wrong runtime metric for manipulation, and runtime claims should state the payload and arm pose.

The next question is empirical. A timed hold on real hardware, with winding temperatures logged, would show how far these open-model estimates are from the robots they describe.