A weld or connect in a robot's MJCF is a stiff spring, not a rigid joint: limits of the stiff rule¶
What is wrong. A <weld> or <connect> equality in a robot's MJCF that sets no solref / solimp was
MuJoCo's soft spring: under load the two bodies came apart 2.3 mm per kN on a 9 kg pair. Since the default change
(alloy.build(scene, welds=...) left out is "stiff"), such rows take the stiff rule of scene attachments and hold
3-30 kN to hundredths of a millimetre. What is left: a weld is still a spring whose force stiffness falls with the
lighter body, it trails by a step-size error in motion, and welds="mujoco" (or a row written by hand) keeps
whatever the file says.
The number. Rig: a free 20 kg body welded to a 5 kg body on a slide, pulled sideways (the audit's ERR-11 rig;
tests/precision/test_weld_stiff_law.py), 1.25 ms rigid step, CPU, MuJoCo C and MuJoCo Warp alike:
| pull | welds="mujoco" (MuJoCo's row; the default before the change) |
default (welds="stiff") |
|---|---|---|
| 3 kN | 7.0 mm, JOINT_FAIL | 0.0015 mm, OK |
| 10 kN | 23.3 mm, JOINT_FAIL | 0.0051 mm, OK |
| 30 kN | 70.0 mm, JOINT_FAIL | 0.0152 mm, OK |
Both columns are the row's own compliance to within 2 % (the law): a row carrying a force F rests at
r = F W (1 - d) / (d² k), where W is the two bodies' inverse weights summed (0.117 per kg on the rig), d the
impedance (solimp) and k the stiffness from solref. MuJoCo's defaults (0.02 s, 0.9-0.95) give 2.0e-5 s² x W per
newton; the stiff rule gives 4.3e-9 s² x W at the 1.25 ms step, about 4,600 times less. The stiff rule's stretch grows
with the square of the rigid step (sim.routes() names the step).
In motion a weld also trails by a step-size error that stiffness does not remove. Over one step MuJoCo holds the
row's direction fixed, so a body carried around a curve at acceleration a keeps a small row velocity, and the row's
damping holds it off by B h a / (2 K d): about a h τ for a time constant τ, and (5/3) a h² under the stiff rule
(2.6 µm per m/s² at 1.25 ms). Measured on a 1 kg tool welded 0.4 m out on a link spinning at 2 / 4 / 8 rad/s (the
same law file): default 0.38 / 0.39 / 0.44 mm off (mostly the sag under its own weight), stiff 0.004 / 0.017 /
0.066 mm. A position servo whose target is ramped by sim.run(targets=...) gets a new target once per control
period. A stiff servo then kicks its link every period, and the weld follows those kicks with a larger error: 4.4
mrad and 1.8 mm on that rig at 4 rad/s, the same in plain MuJoCo with the same controls. A shorter control period
(alloy.Scene(dt=...)) makes the kicks smaller. On ordinary arms: the shipped
Franka Panda carrying a 0.5 kg tool welded to its hand peaks at 0.01 mm with stiff welds through slow, fast and
pick ramps, against 0.42-1.49 mm with the default rows, which already read JOINT_FAIL holding still (2.6 mm sag). A
light two-link arm on a very stiff servo (kp 20,000) does read JOINT_FAIL with stiff welds on fast ramps at
dt = 0.01 (2.0-2.2 mm; default 19.6-43.1 mm); at dt = 0.002 it peaks at 0.55 mm.
The stiff rule is stiff in acceleration, so in force a weld is as stiff as its most mobile body lets it be: about
ω² / ((1 - d) W), which is 0.36 / (h² (1 - d) W), or 2.3e8 / W N/m at 1.25 ms with W in 1/kg. The 9 kg pair
above holds 2e9 N/m. A 1 kg body welded to a 1 g carrier (W = 334 per kg) holds 6.9e5 N/m: 0.02 mm under 10 N at
1.25 ms, 0.33 mm at a 5 ms step. A 1 t body hung from that 1 g carrier is outside what a weld row can do: it sags
about 0.1 m and swings, against 5-7 m with MuJoCo's default row. For such a pair, make the heavy body a child of the
carrier (Workaround). The stiff rule costs nothing measurable: 1,024 and 4,096 worlds on one L4 step equally fast (within
0.5 %, three alternating runs), and the solver takes the same single iteration per step.
Where the weld point is matters. In MuJoCo a weld's anchor is a point of its second body, so
<weld body1="b"/> (second body: the world) anchors at the world origin, and a load on b far from that point is
held by the weld's rotation rows: a 20 kg body 0.6 m from the origin moves 0.58 mm under 3 kN with stiff rows (the
same in plain MuJoCo; MuJoCo's default row goes unstable there), against 0.0007 mm with anchor at the body
(<weld body1="b" anchor="0.1 0 0.6"/>) or written <weld body1="world" body2="b"/>.
A weld's rotation rows follow the same rule and are just as limited by the bodies' rotational inverse weights, so a moment shows as an angle first: five 1 kg boxes welded in a 1 m cantilever bend 0.4 mrad under their own weight and 1.7 mrad with 50 N at the tip (JOINT_FAIL from about 25 N), while their weld points move 0.001 mm; released, they are back to 0.0004 mm. MuJoCo's own rows let the same cantilever fall 0.78 m.
How health reads it: JOINT_FAIL (DEGRADED) above 1 mm or 1 mrad for an MJCF weld, which health judges both as a loop joint at that flat floor and as a row at its own limit. On a ramped arm the angle can trip first: a tool welded 0.3 m out on a kp 2,000 link ramped 1.5 rad in 0.5 s peaks at 0.29 mm and about 1 mrad with stiff rows. The stiff rows read far below both. JOINT_FAIL counts here are from a tree with the flag's hold between checks (since e21f29f28).
Why. It is MuJoCo's soft-constraint model working as specified, at defaults chosen for a 2 ms step and light
loads; nothing is wrong in the step or the solver (plain MuJoCo C gives the same numbers). Alloy's stiff rule for
attachments (alloy/scene/weld_impedance.py: stiffness set from the rigid step, impedance 0.999) was never applied
to equalities that come in through a robot file.
Workaround.
- Nothing to do for the common case: leaving welds out gives the stiff rule. Rows that write their own solref or
solimp stay as written (a row written out as exactly solref="0.02 1" solimp="0.9 0.95 0.001 0.5 2" counts as
a default); welds="mujoco" keeps every row as written. alloy.load(path), the command line and the Gym
environments build with the default.
- For a heavy body on a light carrier, or wherever even microns matter: do without the equality. Make the second
body a child of the first in the MJCF, with no joint between them; it then moves exactly with it
(docs/guide/robots.md, Grasping).
- Anchor a weld to the world at the body: <weld body1="world" body2="b"/> or anchor= at the body's position.
Planned fix. None for the rows: the remaining limits are those of a soft-constraint step (smaller steps, or a tree joint instead of an equality, remove them). A public call that attaches or welds an object while a simulation runs is separate and not built (grasp-no-attach-or-weld.md).