01 · Question
Why can identical joint motion move feet differently?
C-1N motivated the question: synchronized joint commands can still produce different torso- or world-frame foot motion. The bench isolates that geometry in small MuJoCo legs.
02 · Prediction
Rotate the same motion; rotate its shared-frame component.
For the same positive first-joint perturbation, the predicted world-X response was large at 0°, near zero at 90°, and equally large with the opposite sign at 180°. The foot should still move at 90°—mostly along world Z.
03 · Experiment · inspect the geometry
Hold a foot force fixed; change only the pose.
This is an illustrative geometric map, not a browser physics simulation. It shows why the same desired Cartesian foot force can require different joint torques at different poses.
04 · Observation
Both numerical checks matched.
At the initial one-link pose, MuJoCo’s X Jacobian and a centered finite difference agreed: −1.000000 at 0°, approximately 0 at 90°, and +1.000000 at 180°.
In a separate frozen two-joint test, the same q̇ = [0.70, −0.35] rad/s produced different world X-Z foot velocities in bent and open poses. In both cases, J(q)q̇ matched MuJoCo’s reported foot velocity to the printed numerical precision.
05 · Model update
The Jacobian is a local physical map.
J(q) maps joint velocity to foot velocity at the current pose. It is not a coordinate-system conversion, and it does not solve stability. For a chosen foot force, J(q)ᵀf maps that force into the joint torques implied by the current lever geometry.
Polar coordinates only rename a point. Torque tells us how strongly a force turns a joint.
06 · Free-body context
Whole-robot forces and leg torques are different views.
The full free-body diagram separates gravity and ground-reaction forces on the robot from actuator torques on one leg. Choose a view, then inspect the question it answers.
Whole robot: Which external ground forces balance gravity and keep the body from rotating?
07 · Stop boundary
Stop when the physical map is clear.
This is where the work stopped: the same joint velocity can move a foot differently because the leg pose changes the local geometry, and the same desired foot force can require different motor torques for the same reason. The next question is whole-body standing—not a longer proof of this one-leg map.
Evidence