Hybrid Tensegrity Humanoid

MPC-in-the-loop design evaluation of a full-size humanoid built from tensegrity structure. MuJoCo models, an MJPC (iLQG) walking controller, and the experiment suite behind the paper MPC-in-the-Loop Design Evaluation of a Hybrid Tensegrity Humanoid (under review; PDF in paper/).

The question

Which joints of a tensegrity humanoid must remain conventional bearings for the robot to be controllable, and what does each bearing cost in mass and buy in control? Three simulated machines of identical anthropometry are compared under receding-horizon iLQG:

machine	joints	outcome
pure tensegrity	none; 210 free DoF, 306 cables	stands when anchored; no planner attempted (tractability, not controllability: every counter-wound joint has a strict self-stress, t₆ > 0)
fully hinged (27 DoF, tendon-driven)	bearing at every joint	walks (7/10 at 0.21 m/s), but its cable network raises passive impact loads (+14 % at 0.1 m) and its cable transmission is short of torque at 13 of 27 DoF mid-stride
hybrid (this design)	bearings at hips and knees; tensegrity joints at ankles and waist	walks 10/10 at 0.20 m/s at the specified series-elastic cable stiffness, 20.1 kg at 1.68 m, 0.8 kg of articulation hardware

The allocation rule that follows: bearings where the gait's torque requirements concentrate, series-elastic cable interfaces where impacts enter the structure.

Key results

Mass. 20.1 kg at 1.68 m (11.9 kg/m), against 30–80 kg for built humanoids of similar stature. Every member is sized from simulated load traces: 60 CFRP tubes at the minimum manufacturable wall (manufacturability-, not load-limited), 12-strand UHMWPE tendons, AK70-class hip/knee drives through ~3:1 tendon loops, XM540-class tension drives at ankles and waist.

The bearing-free joint. The foot's three-strut mast overlaps 0.2 H into the shank cage; twelve cables in four families join them. Overlap gives force closure, but only a counter-wound pair of cross families gives the strict self-stress (t₆ > 0) that tension-only actuators need to reject disturbances. At the specified 40 kN/m series-elastic stiffness the joint attenuates drop impacts by 27–32 % against a mass-matched pin joint; at 400 kN/m (bare rope) the effect inverts.

Walking across the specified stiffness range (same controller, no retuning):

interface cables	planner / plant step	walks	upright 16 s	speed	real time
2.4 kN/m (development configuration)	8 / 4 ms	10/10	10/10	0.19 m/s	0.29×
20 kN/m	2 / 1 ms	9/10	8/10	0.19 m/s	0.062×
40 kN/m (design point)	1 / 0.5 ms	10/10	10/10	0.20 m/s	0.034×
168 kN/m	0.82 / 0.41 ms	5/5	5/5	0.12 m/s	0.030×

The 2.4 kN/m development configuration was chosen for the planner, not the hardware: it is the stiffest cable the 8 ms planner step integrates stably and it plans ~8× faster, which made the tuning runs of the design iteration tractable. Stiffer cables need proportionally smaller steps (MJPC's 512-knot horizon cap sets the 0.82 ms floor at 168 kN/m).

Robustness and drives (40 kN/m design point unless noted). Ten Monte Carlo trials at 20 kN/m with cable stiffness ±30 %, prestress ±20 %, body masses ±15 % and ground friction μ ∈ [0.3, 0.9] walk 10/10 (lowest μ walked: 0.32). Over 120 s of logged walking, leg-torque commands exceed the 74 N·m drive peak in two isolated episodes of ≤ 1.5 ms; RMS demand is 5.3 N·m motor-side, 67 % of the AK70-10 rating.

Capability envelope (development configuration; 40 kN/m results in the paper):

Reliable speed band 0.30–0.45 m/s command (0.10–0.19 m/s achieved; 0.23 m/s after retuning stride and clearance); trunk payload 10/10 at 2 kg, 0/10 at 12 kg on the tensegrity waist (a 4 cm forward placement restores 10 kg at 10/10); single-arm carry 2 kg at soft cables, 4 kg at 40 kN/m, where the hybrid becomes motor-limited like the hinged machine.

Joint-count study. Rigidizing the hybrid's tensegrity joints raises speed monotonically (0.19 → 0.21 → 0.23 m/s) and removes compliance; the gait tuned for the compliant waist fails outright (0/5) when that joint is rigidized until one parameter is retuned. Removing the bearings removes planner tractability.

Repository layout

• mujoco/ — parametric model generators (generate_hybrid_humanoid.py builds the hybrid from one node-geometry description; the pure and hinged variants have their own generators) and compiled XML models.
• experiments/ — every experiment in the paper, one script each:
  • hybrid_baseline_verify.py — development-configuration baseline and controls
  • e27_walker_stiffness.py — the walker at 20/40/168 kN/m (design point)
  • e28_stiff_sweeps.py — payload and Monte Carlo robustness at stiff cables
  • e10/e19 — bearing-free joint testbed: closure, drops, stiffness sweep
  • e15, e26 — cable-family search, counter-wound chain
  • e17, e20 — joint-count study and rigid-waist retune
  • e1–e9, e11 — the fully hinged E-suite (impact, stiffness, degradation, proximal mass, tendon realisability, authority audit)
  • make_numbers.py — regenerates paper/numbers.tex (every number in the paper) from the results JSONs; plot_hybrid_sweeps.py regenerates Fig. 4. Outputs land in experiments/results/ (gitignored, ~4 GB, regenerable).
• paper/ — LaTeX source, figures, video attachment.
• design/, references/ — design notes and bibliography.
• HYBRID_WALKER.md — working notes on the hybrid walker.

Reproducing

python -m venv .venv && .venv/bin/pip install mujoco numpy scipy matplotlib
.venv/bin/python mujoco/generate_hybrid_humanoid.py        # model
.venv/bin/python experiments/e27_walker_stiffness.py run 40 10   # design-point walker, n=10
.venv/bin/python experiments/make_numbers.py               # paper/numbers.tex
cd paper && latexmk -pdf codesign_paper.tex

Walking experiments drive MJPC's headless testspeed runner and need the mujoco_mpc build below. At the design point one 16 s trial takes about nine minutes of wall time (0.034× real time).

Rebuilding mujoco_mpc

The mujoco_mpc/ directory (a clone of google-deepmind/mujoco_mpc with the tensegrity tasks) is not committed. To reconstruct it:

git clone https://github.com/google-deepmind/mujoco_mpc
cd mujoco_mpc
git checkout ff572a21e7c2bf9fda62e1862a758da7e9a8719b
git apply ../mujoco_mpc_tensegrity.patch

The patch adds mjpc/tasks/tensegrity/ (task code, XML models, STL meshes) and registers the tasks in mjpc/tasks/tasks.cc, mjpc/agent.cc, mjpc/app.cc, mjpc/testspeed.cc, and mjpc/CMakeLists.txt. Build with CMake and launch mjpc --task="Hybrid Walk" for the interactive view.

Acknowledgment

AI (Claude Fable 5.1, Anthropic, via Claude Code) was used to support all aspects of this work: the model generator, MJPC task, experiment and analysis scripts, figures, and manuscript drafting, all to the authors' specification and under their direction and review. Gemini Flash 3.6 (Google) was used to produce mock reviews of manuscript drafts. The paper's Acknowledgment gives the full disclosure.