Null Space Control

Definition

Null-space control exploits the extra degrees of freedom of a redundant manipulator
by injecting joint forces (or motions) that produce zero operational force on the
end-effector, so a secondary objective can be served without disturbing the primary
task. In Khatib’s operational-space formulation, any joint-force command splits
uniquely into an end-effector-producing term plus a term lying in the null space of
the dynamically-consistent generalized inverse $\bar{\mathbf{J}}$; only that
choice of inverse keeps the null-space forces from accelerating the end-effector
(khatib1987unified, §VI–VII). Khatib’s source is a
fixed-base, terrestrial manipulator; for our free-flying space manipulator the
same projector idea carries over, but the relevant “inertia” is the full coupled
base+arm inertia, so dynamic consistency must be taken w.r.t. the coupled system, not
the arm alone.

Key Equations

Symbols per notation.md.

Dynamically-consistent decomposition of the joint force (Khatib eq. 55; $\mathbf{\tau }$
for the physical joint torque, ${\mathbf{\tau }}_0$ an arbitrary secondary joint force):

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

The dynamically-consistent inverse and the task-space (operational-space) inertia
(Khatib eqs. 52, 51 / 50):

$$\bar{\mathbf{J}}={\mathbf{M}}^{-1}{\mathbf{J}}^{\top }\mathbf{\Lambda },\mathbf{\Lambda }=(\mathbf{J}{\mathbf{M}}^{-1}{\mathbf{J}}^{\top }{)}^{-1},\mathbf{N}=\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top }.$$

The second term is a null operational force: ${\bar{\mathbf{J}}}^{\top }\mathbf{N}=0$,
so the projected secondary force $\mathbf{N}{\mathbf{\tau }}_0$ exerts no force on the
end-effector (Khatib eq. 54, $\mathbf{F}={\bar{\mathbf{J}}}^{\top }\mathbf{\tau }$).

Notation flag. Khatib writes the joint-space inertia as $A$ and the generalized
joint force as $\Gamma$. In notation.md, $\mathbf{M}$ is inertia
and the glyph $\mathbf{\Gamma }$ is the load-bearing $12\times (6+n)$ circumcentroidal
coordinate transform — an unrelated object — so Khatib’s $\Gamma$ is rendered here as
the joint torque $\mathbf{\tau }$. The symbols $\mathbf{\Lambda }$ (operational-space
inertia), $\bar{\mathbf{J}}$ (dynamically-consistent inverse), and
$\mathbf{N}=\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top }$ are taken from
the notation.md “Additions” table (Khatib rows). Note $\mathbf{\Lambda }$ here is the
Khatib op-space inertia, not the guidance helix $\mathbf{\Lambda }=\{{\mathbf{\lambda }}_i\}$.

Source Support

• khatib1987unified — primary; derives the
  end-effector equations of motion for a redundant arm (§VI), the dynamically-consistent
  decomposition (eq. 55), and uses null-space joint forces for asymptotic stabilization
  without disturbing the end-effector (§VII, eqs. 63–67), avoiding any explicit
  pseudo-inverse.

Related Topics

• null_space_projection — the projector
  $\mathbf{N}=\mathbf{E}-{\mathbf{J}}^{\top }{\bar{\mathbf{J}}}^{\top }$ this control
  scheme applies; null-space control is its use to drive a secondary objective.
• operational_space_formulation — supplies the
  end-effector dynamics ($\mathbf{\Lambda },\bar{\mathbf{J}}$) on which the
  null-space decomposition rests.
• operational_space_control — the primary-task control
  law; null-space control adds the secondary term that this law leaves free.
• kinematic_redundancy_resolution — the
  velocity/kinematic-level counterpart; null-space control resolves redundancy at the
  force/torque level instead.
• task_priority_control — generalizes the single null-space
  term to a strict hierarchy of tasks, each projected into the predecessors’ null space.

Open Questions

• Khatib’s $\bar{\mathbf{J}}$ uses the fixed-base joint-space inertia $\mathbf{M}$.
  For a free-flying base, is dynamic consistency to be enforced against the full coupled
  base+arm inertia, and does the resulting $\mathbf{N}$ still produce zero base wrench
  (cf. the reaction_null_space of the free-floating regime)?
• §VII proves asymptotic stabilization via dissipative null-space joint forces for a
  fixed base; does the same Lyapunov argument survive the additional base dynamics and
  base-attitude regulation of an actuated free-flying system?