Differential Inverse Kinematics

For manipulators with many joints, inverse kinematics is usually solved as a local differential problem. Rather than solving for joint angles in closed form, we update them iteratively using the Jacobian of the forward kinematics map.

Local linearization

If $x = f(q)$ maps joint configuration $q$ to task-space coordinates $x$, then a small update satisfies

$$\Delta x \approx J(q)\,\Delta q,$$

where $J(q)$ is the manipulator Jacobian. The inverse problem is to find a joint increment $\Delta q$ that reduces a task error $e = x_d - x$.

Damped least squares

Near singularities the pseudoinverse can become numerically unstable. Damped least squares regularizes the problem:

$$\Delta q = J^\top (J J^\top + \lambda^2 I)^{-1} e.$$

The damping term $\lambda$ trades exactness for robustness. In practice this is often the better choice for real manipulators, especially when sensor noise and model mismatch are unavoidable.

Redundancy resolution

Redundant manipulators allow secondary objectives in the Jacobian null space, for example posture regularization, joint-limit avoidance, or obstacle biasing.

dq_primary = J.T @ torch.linalg.solve(J @ J.T + lam**2 * I_task, error)
dq_null = (I_joint - J_pinv @ J) @ dq_secondary
dq = dq_primary + dq_null

This makes inverse kinematics relevant well beyond textbook derivations. It sits directly inside practical robot stacks, often beneath planners or high-level policies.