The measurement #
The same flexion was applied to all five motors, the model simulated for 3 s from rest, and the steady-state joint angles recorded — 41 flexion values from 0 to 1.
| Flexion | Force | Index MCP / PIP / DIP | Thumb CMC / MP / IP |
|---|---|---|---|
| 0.000 | +50.0 N | −3.3° / −3.2° / −3.3° | −2.9° / −2.7° / −2.7° |
| 0.250 | +25.0 N | −1.7° / −1.6° / −1.7° | −1.5° / −1.3° / −1.4° |
| 0.500 | 0.0 N | 0.0° / 0.0° / 0.0° | 0.0° / 0.0° / 0.0° |
| 0.625 | −12.5 N | 91.1° / 91.0° / 91.0° | 90.6° / 53.9° / 45.6° |
| 1.000 | −50.0 N | 94.0° / 93.8° / 93.9° | 92.3° / 54.2° / 46.0° |
Joint limits are 90°; values just above it are MuJoCo’s soft limit being pressed.
Zooming into the wall (index MCP, steady state from rest):
| Tendon force | 0 N | −0.1 N | −0.5 N | −1.0 N | −2.5 N |
|---|---|---|---|---|---|
| Index MCP bend | 0.0° | 1.0° | 62.1° | 90.1° | 90.2° |
The finger’s entire proportional range fits between 0 and −1 N — 1% of the ±50 N command range, a flexion band of 0.50 to 0.51.
| Open (+50 N) | Closed (−50 N) |
|---|---|
 |
 |
Timing and hysteresis #
| Test | Result |
|---|---|
| Close from open at −50 N (MCP reaches 85°) | 48 ms |
| Re-open from closed at +50 N (MCP below 10°) | 44 ms |
| From closed, command 0 N for 2 s | still 69.6° — barely drifts |
| From closed, command +1 N for 2 s | fully open (−0.1°) |
A finger does not return to open at flexion 0.5; it opens only once the command crosses to the push side. Around 0.5 the finger simply stays where it last was.
Why: nothing pushes back #
A proportional position needs a restoring force that grows with displacement — a spring. At equilibrium, per joint:
$$\underbrace{r\,F_\text{tendon}}_{\text{tendon torque}} \;=\; \underbrace{k_\text{joint}\,q}_{\text{joint spring}} \;+\; \underbrace{\tau_\text{springs}}_{\text{elastic tendons}} \;+\; \text{friction} + \text{gravity}$$| Restoring element | In robot.xml |
Contribution |
|---|---|---|
| Joint stiffness \(k_\text{joint}\) | 0 — not set | none |
| Joint damping | 0.01 | slows motion; no equilibrium force |
| Joint friction loss | 0.001 | a tiny dead-band |
| Tendon stiffness (flexor + extensor) | 15 N/m | a full curl stretches the extensor ~12 mm → ≈ 0.2 N |
| Gravity on phalanges | a few grams each | tiny, orientation-dependent |
Nothing grows fast enough to balance even 1 N of pull, so any net pull accelerates the finger into its joint limit, and the limit becomes the equilibrium. Force control against a near-zero spring is a switch.
The thumb’s MP and IP joints stopping near 54° and 46° even at −50 N hasn’t been isolated; the likely explanation is the thumb tendon’s via-point geometry once the CMC hits its stop.
Is the project wrong, then? #
No — it’s a binary grasp twin, which is exactly how the live demo looks: a crisp open ↔ fist mirror. The fingers travel in ~45 ms, faster than one vision frame, so it looks instantaneous. The smooth flexion signal isn’t wasted either; it’s on ROS 2 for any consumer. But partial poses can’t be mirrored, and calibration mostly moves when the switch flips.
Three fixes, tested #
Every option below stayed numerically stable over a 3 s settle. Numbers are index MCP / PIP / DIP bend.
Keep force control; give each finger joint a return spring (jnt_stiffness on the 15 finger joints):
| Stiffness (N·m/rad) | −5 N | −12.5 N | −25 N | −50 N |
|---|---|---|---|---|
| 0 — current | 90 / 90 / 90 | 91 / 91 / 91 | 92 / 92 / 92 | 94 / 94 / 94 |
| 0.03 | 86 / 75 / 75 | 91 / 90 / 90 | 92 / 91 / 92 | 94 / 93 / 93 |
| 0.1 | 24 / 21 / 20 | 72 / 62 / 62 | 91 / 90 / 90 | 92 / 92 / 92 |
At 0.1 the finger is proportional over the first ~25 N of pull (flexion 0.5 → 0.75); go stiffer (start near 0.2) to spread the curl across the full half.
<!-- default class of robot.xml — finger joints, not the servo_* hinges -->
<joint frictionloss="0.001" armature="0.0001" damping="0.01" stiffness="0.1"/>Force control still only curls on the pull half. To map all of 0 → 1 onto curl, set FORCE_OPEN = 0
and let the springs open the hand.
Raise stiffness on the passive tendon_ext_* tendons (currently 15 N/m) so the extensor becomes the
return spring — the rubber-band approach many 3D-printed hands use. Same principle as option A, applied
along the tendon path rather than per joint.
Untested here — sweep it the same way before relying on it.
Real SG90 servos are position devices: a horn angle winds the string to a length. Model that directly — a position actuator on each flexor tendon — and interpolate flexion onto length. From the model, the index flexor is 0.1294 m open and 0.0870 m closed.
| Gain | t = 0.25 | t = 0.5 | t = 0.75 | t = 1.0 |
|---|---|---|---|---|
| kp = 200 | 33 / 22 / 18 | 61 / 43 / 36 | 85 / 61 / 55 | 90 / 85 / 82 |
| kp = 1000 | 30 / 24 / 22 | 58 / 46 / 42 | 81 / 65 / 63 | 90 / 90 / 90 |
kv = 2·√(kp·0.001): ≈ 0.9 for kp 200, 2 for kp 1000.
<position name="pull_index" tendon="tendon_flex_index" kp="1000" kv="2" ctrlrange="0.0870 0.1294"/>target_length = L_open[finger] + flexion * (L_closed[finger] - L_open[finger])
self.data.ctrl[self.motors[finger]] = target_lengthA smooth, monotonic curl over the whole range — and the same command a hardware servo driver needs.
Option C is tracked for implementation in issue #1.
Reproduce it #
env -u PYTHONPATH MUJOCO_GL=glfw venv/bin/python docs/tools/make_figures.py allIt rewrites docs/data/flexion_sweep.json and regenerates every chart and render from the model and
the constants in standalone/main.py.
