Kinematic Redundancy Resolution
Definition
Kinematic redundancy resolution is the procedure for selecting a unique joint-space solution to the
inverse-kinematics problem when a manipulator is kinematically redundant with respect to its task
(joint count exceeds task dimension , so has
infinitely many solutions). Following Siciliano’s tutorial, resolution is almost always performed
locally, at the velocity level, by inverting the differential map and using the surplus degrees of freedom to meet secondary
objectives — avoiding kinematic singularities, joint limits, or obstacles,
or optimizing a manipulability measure. The four canonical families are:
the Jacobian pseudoinverse (minimum-norm), gradient projection into the
null space, task-space augmentation with
task priority, and inverse-kinematic functions. The source is
fixed-base terrestrial robotics — it carries no base-reaction or momentum-conservation terms, so for
our free-flying space manipulator these schemes apply only to the arm Jacobian and must be extended
before they describe the coupled base+arm system.
Key Equations
Symbols per notation.md.
Differential kinematics of the redundant () task and the minimum-norm (pseudoinverse) solution:
\dot{\boldsymbol q}=\boldsymbol J^{\dagger}\dot{\boldsymbol x},\quad \boldsymbol J^{\dagger}=\boldsymbol J^{\top}\!\left(\boldsymbol J\boldsymbol J^{\top}\right)^{-1}.$$ General resolution = minimum-norm term plus a homogeneous self-motion projected into the Jacobian null space (gradient-projection method, with $\dot{\boldsymbol q}_0=(\partial h/\partial\boldsymbol q)^{\top}$ for a scalar cost $h(\boldsymbol q)$): $$\dot{\boldsymbol q}=\boldsymbol J^{\dagger}\dot{\boldsymbol x} +\left(\boldsymbol E-\boldsymbol J^{\dagger}\boldsymbol J\right)\dot{\boldsymbol q}_0 .$$ > Notation note: the kinematic null-space projector here is $\boldsymbol E-\boldsymbol J^{\dagger}\boldsymbol J$ > (right pseudoinverse), **distinct from** [notation.md](../notation.md)'s > $\boldsymbol N=\boldsymbol E-\boldsymbol J^{\top}\bar{\boldsymbol J}^{\top}$, which is Khatib's > *dynamically-consistent* projector using the generalized inverse $\bar{\boldsymbol J}$. Same role > (a self-motion that leaves the task unaffected), different inverse. $\dot{\boldsymbol q}_0$ is the > arbitrary secondary-task joint velocity (symbol absent from notation.md; named here, not in conflict). ## Source Support - [siciliano1990tutorial](../sources/siciliano1990tutorial.md) — primary; the tutorial that organizes redundancy resolution into its four canonical families (pseudoinverse, gradient projection, augmented/task-priority, inverse-kinematic functions), and flags pseudoinverse instability near singularities (Baillieul et al. 1984), the damped-least-squares fix (Wampler; Nakamura & Hanafusa 1986), non-repeatability of the minimum-norm solution, and algorithmic singularities of the augmented Jacobian. Fixed-base terrestrial regime. ## Related Topics - [kinematic_redundancy](kinematic_redundancy.md) — the underlying phenomenon ($n>m$) that *creates* the freedom this technique resolves. - [redundancy_resolution](redundancy_resolution.md) — the broader (incl. dynamic / momentum-aware) resolution concept; this page is its purely-kinematic, velocity-level specialization. - [pseudoinverse_jacobian](pseudoinverse_jacobian.md) — supplies the minimum-norm particular solution $\boldsymbol J^{\dagger}\dot{\boldsymbol x}$ and the variable-damping remedy near singularities. - [null_space_projection](null_space_projection.md) — the projector $(\boldsymbol E-\boldsymbol J^{\dagger}\boldsymbol J)$ that injects secondary objectives as task-invariant self-motion. - [task_priority_redundancy_resolution](task_priority_redundancy_resolution.md) — the augmented-task family with strict priority ordering, the source's recommended scheme for handling multiple tasks. ## Open Questions - Siciliano's schemes resolve a single (arm) Jacobian on a **fixed base**. For our free-flying manipulator, does the redundant DOF resolve against the arm Jacobian alone, or against the coupled base+arm / [circumcentroidal Jacobian](circumcentroidal_motion.md) $\boldsymbol J_{\nu_e}^{\oplus}$ — and does the actuated base supply extra DOF that change the "redundancy" bookkeeping ($n>m$)? - The source warns that the minimum-norm pseudoinverse solution is **not repeatable** for cyclic tasks. Does any of our cyclic inspection guidance rely on cyclic joint behaviour, and if so does this non-repeatability matter, or does the augmented/extended-Jacobian (repeatable) variant become necessary? - Self-motion swings violently as the singular value collapses (see the notation/terminology note on freezing self-motion below $\sim10^{-3}$). How should the null-space term be regularized so secondary objectives degrade gracefully rather than amplifying near a singularity?