Task Priority Control

Definition

Task-priority control assigns a strict hierarchy to competing tasks so that a
lower-priority task is executed only in the null space of the higher-priority one, and
therefore can never corrupt it. In bruschi2025singularity
the primary task is end-effector pose tracking (the capture objective) and the secondary
task is base pose tracking (safe standoff and attitude); the secondary command is projected
through ${\mathbf{N}}_N=\mathbf{I}-{\mathbf{J}}_N^{†}{\mathbf{J}}_N$ so it acts only
on the residual redundancy. This regime is free-flying (the design “leverages the full or
over actuation capabilities” of a fully-actuated base): a redundant $6+N$ actuated DOF set
(${\mathbf{J}}_N$ full row rank $6$ by construction) is what guarantees the end-effector task
remains solvable — and hence singularity-free — even when the manipulator Jacobian ${\mathbf{J}}_m(q)$
loses rank at a kinematic singularity. The hierarchy induces a cascade in the closed-loop error
system and yields almost-global tracking of any smooth end-effector trajectory; the secondary
task converges only when the two trajectories are compatible (share a common generalized
velocity after finite time).

Key Equations

Symbols per notation.md.

Source-specific symbols not in notation.md (used here as in the source): ${\mathbf{J}}_N=[{\mathbf{J}}_b{\mathbf{J}}_m]$
is the whole-system end-effector Jacobian (base + arm columns) — distinct from the fixed-base
${\mathbf{J}}_{{\nu }_e}$ ($\equiv {\mathbf{J}}_m$) of notation.md; ${\mathbf{J}}_0=[{\mathbf{I}}_{6}0]$ is the
base Jacobian; ${\mathrm{v}}^c$ is the outer-loop virtual velocity command; $w({\tilde{X}}_k,t)$ is the
per-task velocity controller (feedback ${\gamma }^x$ + feedforward).

Double-task (priority) velocity command — primary in the range of ${\mathbf{J}}_N^{†}$,
secondary in its null space:

$${\mathrm{v}}^c={\mathbf{J}}_N(q{)}^{†}w({\tilde{X}}_N,t)+\underset{\text{null-space projector}}{\underbrace{{\mathbf{N}}_N(q)}}{\mathbf{J}}_0^{\top }w({\tilde{X}}_0,t),{\mathbf{N}}_N(q):={\mathbf{I}}_{6+N}-{\mathbf{J}}_N(q{)}^{†}{\mathbf{J}}_N(q).$$

Because ${\mathbf{J}}_N$ is full row rank for all $q$, the primary error obeys
${\mathbf{J}}_N{\mathrm{v}}^c=w({\tilde{X}}_N,t)$ (the null-space term is annihilated,
${\mathbf{J}}_N{\mathbf{N}}_N=0$), so the end-effector task is unaffected by
the secondary command and by manipulator singularities.

Source Support

• bruschi2025singularity — primary source: derives the
  hierarchical inner–outer-loop task-priority law (Eq. 12) for a fully-actuated space robot,
  proves the cascade-structured closed-loop stability, and establishes singularity-free almost-global
  end-effector tracking plus the compatibility condition for the secondary base task.

Related Topics

• task_prioritization — the general principle (operational-space task
  hierarchies); this page is its trajectory-tracking instantiation for a space robot.
• task_priority_redundancy_resolution — the kinematic
  redundancy-resolution machinery (successive null-space projection) that task-priority control uses
  to rank tasks.
• singularity_robust_inverse — Bruschi extends this singularity-robust
  approach to space robots; the priority hierarchy avoids needing to invert the rank-deficient
  ${\mathbf{J}}_m$ directly.
• null_space_projection — supplies the projector ${\mathbf{N}}_N$ that
  confines the secondary task to the redundant DOF.
• trajectory_tracking — the outer-loop objective each prioritized task
  realizes (pose-error dynamics driving ${\tilde{X}}_k\to \mathbf{I}$).

Open Questions

• The source’s ${\mathbf{J}}_N$ is full row rank $6$ by construction precisely because the base
  is fully actuated; how does this hierarchy degrade for a free-floating base, where momentum
  conservation couples the columns and the effective Jacobian (GJM) can lose rank dynamically?
• Secondary (base) tracking is guaranteed only for compatible trajectories; for our inspection
  guidance, when do a desired end-effector helix and a desired base standoff fail compatibility, and
  what is the resulting steady-state base-pose error?
• The priority law is derived at the kinematic level and wrapped in an inner loop; does the
  singularity-free guarantee survive the inner-loop velocity-tracking error $\tilde{\mathrm{v}}$ that
  the kinematic design neglects?