Closed Loop Inverse Kinematics

Definition

Closed-loop inverse kinematics (CLIK) solves the differential inverse-kinematics problem
by integrating joint velocities while feeding back the task-space pose error between the
desired and the forward-kinematics-computed end-effector trajectory, so that discrete-time
numerical drift is suppressed rather than accumulated (caccavale2001kinematic).
It is the inverse-kinematics layer of kinematic control: the manipulator kinematics is
resolved outside the motion loop, separating singularity/redundancy handling from the
joint-space tracking controller. Caccavale & Siciliano pose it for a free-floating
space manipulator (uncontrolled, reactive base), so the resolving map is the generalized
Jacobian, which folds in the base reaction — this regime assumption matters for our
free-flying (fully-actuated base) system, where the resolving Jacobian is a different object.

Key Equations

Symbols per notation.md.

Quaternion-based CLIK at the velocity level (eq. 19), using the geometric/generalized
Jacobian ${\mathbf{J}}_g$ so no representation singularity arises:

$$\dot{\mathbf{q}}={\mathbf{J}}_g^{-1}\left({\mathbf{v}}_d+{\mathbf{K}}_P{\mathbf{e}}_{de}\right),{\mathbf{e}}_{de}=[{\tilde{\mathbf{x}}}_r^{\top }{\mathbf{ϵ}}_{de}^{\top }{]}^{\top },$$

with ${\tilde{\mathbf{x}}}_r$ the EE position error, ${\mathbf{ϵ}}_{de}$ the
vector part of the orientation-error quaternion (the registry’s $\mathbf{ϵ}$),
and ${\mathbf{v}}_d$ the desired EE twist feedforward. Note: ${\mathbf{K}}_P=\mathrm{diag}\{k_{Pp}\mathbf{E},k_{Po}\mathbf{E}\}\succ 0$
is the CLIK feedback gain; this symbol is not in notation.md (distinct from the
outer-loop stiffness $\breve{\mathbf{K}}$ and from $\mathbf{\Gamma }$) — flagged for central addition.

Substituting into the differential kinematics gives the closed-loop error dynamics
— the defining CLIK property. The position error is governed by a homogeneous,
exponentially stable linear equation (eq. 16):

$${\dot{\tilde{\mathbf{x}}}}_r+k_{Pp}{\tilde{\mathbf{x}}}_r=0.$$

The quaternion orientation error (eq. 20) is non-homogeneous: it contains the
end-effector angular-velocity error $\Delta {\mathbf{\omega }}_{de}={\mathbf{\omega }}_d-{\mathbf{\omega }}_e$
rather than the time derivative of ${\mathbf{ϵ}}_{de}$, so its convergence is established by a
Lyapunov argument rather than read off as a linear-ODE eigenvalue:

$$\Delta {\mathbf{\omega }}_{de}+k_{Po}{\mathbf{ϵ}}_{de}=0.$$

Redundancy resolution with task priority (eq. 21): the higher-priority EE task uses the
pseudoinverse ${\mathbf{J}}_g^{†}$ and the constraint task (e.g. spacecraft attitude)
is projected onto its null space:

$$\dot{\mathbf{q}}={\mathbf{J}}_g^{†}\left({\mathbf{v}}_d+{\mathbf{K}}_P{\mathbf{e}}_{de}\right)+(\mathbb{I}-{\mathbf{J}}_g^{†}{\mathbf{J}}_g){\mathbf{J}}_C^{\top }{\mathbf{K}}_C{\mathbf{e}}_C.$$

Source Support

• caccavale2001kinematic — primary: derives the Euler-angle and unit-quaternion CLIK algorithms for a redundant free-floating space manipulator, proves exponential convergence of the position error (and the Euler-angle error away from representation singularities) while establishing the quaternion-error convergence by a Lyapunov argument, and adds Jacobian-transpose and null-space (task-priority) redundancy variants.

Related Topics

• resolved_motion_rate_control — CLIK is RMRC plus a pose-error feedback correction term; without the feedback it reduces to open-loop resolved-rate control, which drifts under discrete-time integration.
• task_space_error_dynamics — the closed-loop error equations (16, 20) are the task-space error dynamics CLIK is designed to drive to zero: the position error (16) decays exponentially, while the quaternion-orientation error (20) is shown convergent by a Lyapunov argument.
• pseudoinverse_jacobian — the Jacobian inverse/pseudoinverse ${\mathbf{J}}_g^{†}$ is the resolving map that converts the feedback-corrected task velocity into joint rates.
• kinematic_redundancy_resolution — the null-space projector $(\mathbb{I}-{\mathbf{J}}_g^{†}{\mathbf{J}}_g)$ in eq. (21) carries a secondary (constraint) task without disturbing the prioritized EE task.
• damped_least_squares — CLIK still inverts ${\mathbf{J}}_g$, so near a kinematic_singularity the plain inverse blows up; damped least-squares replaces it to bound joint rates.

Open Questions

• The source assumes a free-floating base (resolving via the generalized Jacobian ${\mathbf{J}}_g$). For our free-flying fully-actuated base the resolving Jacobian is the circumcentroidal ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, not ${\mathbf{J}}_g$ — does the exponential-stability argument for eqs. (16)/(20) carry over unchanged, or does base actuation change the error coupling?
• Eqs. (16)/(20) are pure kinematic error dynamics that ignore base/arm dynamic_coupling; when CLIK only feeds a joint-space tracking controller, does the realized closed loop preserve exponential convergence under the coupled inertia $\mathbf{M}$?
• The task-priority variant (eq. 21) assumes ${\mathbf{J}}_g$ non-singular for the pseudoinverse term; how should the singularity-robust path (damped/hold-last) interact with the null-space constraint task near a singularity?