Gravity compensation and inverse dynamics

Learning outcomes

Derive computed-torque control under explicit assumptions.
Distinguish bias compensation from full inverse dynamics.
Diagnose force accounting with MuJoCo.

Prerequisites: Linear algebra, ODEs, and earlier lectures in this sequence.

Compensation as a model-based feedforward term

Consider a fully actuated robot without contact:

M(q)\dot v+c(q,v)=u+\tau_{\rm passive}.

Here

u

means generalized motor force, which equals ctrl only in our unit-gear direct-drive teaching model. Gravity compensation adds the gravity contribution to a feedback controller. Bias compensation adds all of

c(q,v)

, including velocity-dependent terms.

At zero velocity, c is the gravity bias for a conventional mechanical model. With nonzero velocity, it is generally not just gravity. If damping is already a passive force in the simulator, include it explicitly in any exact cancellation argument.

Computed torque

For hinge coordinates with $e=q-q^\star$ , define

a^\star=\ddot q^\star-K_d(v-v^\star)-K_pe.

Then request

u=M(q)a^\star+c(q,v)-\tau_{\rm passive}.

If the model is exact and the actuator realizes this force without saturation, the error obeys

\ddot e+K_d\dot e+K_pe=0

. The apparent decoupling comes from canceling the coupled dynamics, not from assuming the mass matrix is diagonal.

Underactuated robots cannot realize an arbitrary generalized force. With $u=B(q)\eta$ , solve an allocation problem or redesign the desired acceleration. A floating-base robot with no contact cannot independently command its base acceleration and every joint acceleration.

Worked pendulum

For $I\ddot q+b\dot q+mg\ell\sin q=u$ , the bias is $mg\ell\sin q$ and the passive force is $-b\dot q$ . The computed-torque command becomes

u=I(-k_pe-k_d\dot e)+mg\ell\sin q+b\dot q

for a fixed target. The positive damping-cancellation term is easy to get wrong. Without it, physical damping remains in the closed loop; that may be useful, but the claimed error equation must change.

If the true inertia is twice the model inertia, the cancellation is imperfect. Run the model-mismatch experiment before presenting exact-model behavior as robustness.

MuJoCo force queries

After mj_forward, retrieve c from qfrc_bias and the passive force from qfrc_passive. Use mj_fullM to materialize the dense mass matrix when needed for these small examples.

mj_inverse answers a related query after qacc is set. With constraints and passive forces present, interpreting its result requires the documented decomposition. It is not automatically an actuator command.

Exercise and practice

Practice 2 compares PD, bias-compensated PD, and computed torque on the same two-link arm. Plot error and saturation, then reduce the motor limit.

Check: Why can adding exact gravity compensation worsen an experiment? Possible causes include a wrong sign, wrong frame, double-counted compensation, stale bias data, or actuator saturation. Compensation also consumes some of the available motor force. Diagnose these before changing the feedback gains.