Null Space Projection

Definition

For a kinematically redundant manipulator ($n$ joints, $m<n$ task DOF), the null space of the
task Jacobian $\mathbf{J}$ is the set of joint motions that produce no end-effector (task) motion —
i.e. self-motion of the structure. A null-space projector is an idempotent operator that maps
an arbitrary joint vector onto that null space, so a secondary objective (joint-limit avoidance,
manipulability, a constraint task) can be served without disturbing the primary task. Siciliano (1990)
gives the kinematic (velocity-level) projector $\mathbf{E}-{\mathbf{J}}^{†}\mathbf{J}$ built from the
Moore–Penrose pseudoinverse; Khatib (1995) gives the dynamic (torque-level) projector
$\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top }$ built from the dynamically consistent inverse $\bar{\mathbf{J}}$,
whose null-space torques excite no end-effector acceleration. Both derivations are carried out for a
fixed-base terrestrial manipulator (Siciliano’s survey, and Khatib’s macro/mini and multiarm
structures). Khatib does not stop there, however: he states the fixed-base results “directly extend
to holonomic mobile manipulator systems” and reports implementation on a free-flying space robot
(Russakow & Khatib 1992), with the free-flying base treated as the macro structure — so the dynamic
projector is not regime-agnostic-by-omission, it is explicitly claimed to carry over to an actuated base.

Key Equations

Symbols per notation.md.

Kinematic null-space projection (Siciliano 1990, Eq. 6). The general redundancy-resolution velocity
solution is the minimum-norm particular solution plus a homogeneous term that lives in the null space of
$\mathbf{J}$:

$\dot{\mathbf{q}}={\mathbf{J}}^{†}\dot{\mathbf{x}}+(\mathbf{E}-{\mathbf{J}}^{†}\mathbf{J}){\dot{\mathbf{q}}}_0,$

where ${\mathbf{J}}^{†}={\mathbf{J}}^{\top }(\mathbf{J}{\mathbf{J}}^{\top }{)}^{-1}$, $\mathbf{E}$ is the identity, and ${\dot{\mathbf{q}}}_0$ is arbitrary.
The projector $(\mathbf{E}-{\mathbf{J}}^{†}\mathbf{J})$ is Hermitian and idempotent, so ${\dot{\mathbf{q}}}_0$ yields pure
self-motion ($\mathbf{J}\dot{\mathbf{q}}=\dot{\mathbf{x}}$ regardless of ${\dot{\mathbf{q}}}_0$). The gradient-projection choice
${\dot{\mathbf{q}}}_0=\partial h/\partial \mathbf{q}$ steers the redundancy toward optimizing a cost $h(\mathbf{q})$.

Dynamic (dynamically consistent) null-space projection (Khatib 1995). At torque level, the projector
uses the dynamically consistent inverse $\bar{\mathbf{J}}={\mathbf{M}}^{-1}{\mathbf{J}}^{\top }\mathbf{\Lambda }$ with task inertia
$\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1}$:

$\mathbf{N}=\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top },\mathbf{\tau }={\mathbf{J}}^{\top }\mathbf{F}+\mathbf{N}{\mathbf{\tau }}_0.$

$\mathbf{N}{\mathbf{\tau }}_0$ is the unique generalized inverse for which arbitrary null-space torques ${\mathbf{\tau }}_0$ produce
no end-effector acceleration (dynamic consistency), so the end-effector and self-motion controllers
decouple.

Source Support

• siciliano1990tutorial — primary for the kinematic projector: derives the velocity-level form $\dot{\mathbf{q}}={\mathbf{J}}^{†}\dot{\mathbf{x}}+(\mathbf{E}-{\mathbf{J}}^{†}\mathbf{J}){\dot{\mathbf{q}}}_0$ and the gradient-projection method; fixed-base survey.
• khatib1995inertial — the dynamic projector: shows null-space joint torques produce no end-effector acceleration, and that the dynamically consistent inverse is the unique one making this hold (Theorem 1); derived for fixed-base macro/mini and multiarm structures, but Khatib explicitly extends the result to holonomic mobile / free-flying bases (Russakow & Khatib 1992).

Related Topics

• null_space_control — the control task placed inside the projected null space (the secondary objective ${\dot{\mathbf{q}}}_0$ or ${\mathbf{\tau }}_0$ acted on here).
• dynamically_consistent_inverse — the generalized inverse $\bar{\mathbf{J}}$ that defines the dynamic projector $\mathbf{N}$ (Khatib).
• kinematic_redundancy_resolution — the broader problem; null-space projection is the homogeneous-solution mechanism it relies on.
• task_priority_redundancy_resolution — stacks projectors so lower-priority tasks live in the higher-priority task’s null space.
• pseudoinverse_jacobian — ${\mathbf{J}}^{†}$ supplies the particular (minimum-norm) solution that the null-space term augments.
• reaction_null_space — the space analogue: instead of $\mathbf{J}$, the projector kernels the base–arm coupling inertia so arm motion produces zero base reaction.

Open Questions

• Both derivations are fixed-base. Khatib asserts the result extends to holonomic mobile / free-flying bases (treating the base as a macro structure), but does not work the actuated-base dynamics explicitly. For our free-flying system the relevant operator is the coupled inertia $\mathbf{M}$ over $(6+n)$ DOF and the circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$; does Khatib’s dynamically consistent projector carry over verbatim when the “base” rows are actuated rather than fixed — and does the momentum coupling change the null-space structure relative to his macro/mini case?
• Khatib’s $\mathbf{N}$ and the reaction_null_space projector use different matrices (task Jacobian vs. base–arm coupling block ${\mathbf{M}}_{bm}$): when, if ever, do they coincide, and which is the right one for redundancy resolution that simultaneously regulates EE pose and base attitude?
• The kinematic projector inherits the pseudoinverse’s ill-conditioning near a kinematic_singularity; how does this interact with the ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ threshold cascade used for our singularity handling?