Task Prioritization

Definition

Task prioritization organizes a collection of $N$ task objectives $\{x_k(q)\mid k=1,\dots ,N\}$ — where the index $k$ is the priority rank — into a strict control hierarchy: each lower-priority objective is executed only in the null space of all higher-priority ones, so it can never disturb them. In the operational-space (Khatib) formulation this is realized at the torque level by nesting null-space projectors, giving the recursive law of eq. (26)–(28) below; rank loss of a projected (extended) Jacobian flags the corresponding task as locally infeasible (a task/algorithmic singularity). sentis2005control builds this hierarchy for free-floating humanoids by folding momentum conservation into a Generalized Jacobian, so the base’s reactive motion is absorbed into each task’s kinematics rather than commanded. The construction is regime-agnostic in form, but the specific Jacobian it prioritizes is the free-floating one; an actuated (free-flying) base would prioritize a different system Jacobian (see Open Questions).

Key Equations

Symbols per notation.md.

Notation conflict (flagged). sentis2005control writes the control torque as $\Gamma$. In notation.md the glyph $\mathbf{\Gamma }$ is the load-bearing $12\times (6+n)$ coordinate transform. To avoid clobbering it, the torque is written $\mathbf{\tau }$ here (our canonical joint-torque symbol) and the null-space projector $N_{\mathrm{task}(k)}$ keeps the source’s $N$ (consistent with notation.md’s null-space projector $\mathbf{N}$). $A$ in the source is the full inertia $\mathbf{M}$; $\bar{\mathbf{J}}$ is the dynamically-consistent inverse $\bar{\mathbf{J}}={\mathbf{M}}^{-1}{\mathbf{J}}^{\top }\mathbf{\Lambda }$.

Prioritized torque hierarchy — each task projected through the null spaces of all higher-priority tasks (source eq. 26):

$$\mathbf{\tau }={\mathbf{\tau }}_{\mathrm{task}(1)}+{\mathbf{N}}_{\mathrm{task}(1)}^{\top }\left({\mathbf{\tau }}_{\mathrm{task}(2)}+{\mathbf{N}}_{\mathrm{task}(2)}^{\top }\left({\mathbf{\tau }}_{\mathrm{task}(3)}+\cdots \right)\right)$$

Cumulative (“previous”) null-space projector — introduced as the nested product of all prior null spaces (source eq. 27) and given here in its closed-form unique solution (source eq. 33) — and the Extended Generalized Jacobian of task $k$ (source eq. 30):

$${\mathbf{N}}_{\mathrm{prev}(k)}=\mathbf{E}-\sum _{i=1}^{k-1}{\bar{\mathbf{J}}}_{i\mid \mathrm{prev}(i)}{\mathbf{J}}_{i\mid \mathrm{prev}(i)},{\mathbf{J}}_{k\mid \mathrm{prev}(k)}={\mathbf{J}}_k{\mathbf{N}}_{\mathrm{prev}(k)}$$

Task feasibility via the extended operational-space inertia: task $k$ is infeasible exactly when ${\mathbf{J}}_{k\mid \mathrm{prev}(k)}$ drops rank, i.e. when ${\mathbf{\Lambda }}_{k\mid \mathrm{prev}(k)}^{-1}={\mathbf{J}}_{k\mid \mathrm{prev}(k)}{\mathbf{M}}^{-1}{\mathbf{J}}_{k\mid \mathrm{prev}(k)}^{\top }$ acquires a zero eigenvalue (source eqs. 31, 37).

Source Support

• sentis2005control — primary. Derives the prioritized torque hierarchy (eq. 26–28), the Extended Generalized Jacobian ${\mathbf{J}}_{k\mid \mathrm{prev}(k)}$ (eq. 30) and its unique cumulative null-space projector (eq. 33) for free-floating humanoids, plus an eigen-decomposition test for per-task feasibility under singularity (eq. 37). Note: “free-floating” here is the source’s own term for an uncontrolled/reactive base, applied to a terrestrial humanoid using the space-robotics Generalized-Jacobian formalism — not our free-flying regime.

Related Topics

• task_priority_control — the broader control scheme this page’s hierarchy instantiates; prioritization is its organizing principle.
• task_priority_redundancy — prioritization is only meaningful when the system has surplus DOF; this links the hierarchy to the redundancy it consumes.
• null_space_projection — supplies the projector $\mathbf{N}=\mathbf{E}-\bar{\mathbf{J}}\mathbf{J}$ that nests one priority level inside another.
• operational_space_control — the Khatib dynamics ($\mathbf{\Lambda }$, $\bar{\mathbf{J}}$) in which this prioritization is posed at the torque level.
• redundancy_resolution — task prioritization is one strategy for resolving redundancy: a strict hierarchy rather than a weighted/optimized blend.

Open Questions

• The source’s hierarchy prioritizes the free-floating Generalized Jacobian (momentum-conserving, base uncontrolled). For our free-flying system the base is fully actuated and momentum is not conserved by the same constraint — does the same nested-null-space ordering apply if the base 6-DOF become commanded tasks rather than reactive states, and which system Jacobian (${\mathbf{J}}_{{\nu }_e}^{\oplus }$?) replaces ${\mathbf{J}}_k$?
• Feasibility is judged by rank of the projected Jacobian ${\mathbf{J}}_{k\mid \mathrm{prev}(k)}$ (an algorithmic singularity that can arise even when each task alone is non-singular). How does this interact with our ${\sigma }_G={\sigma }_{\min }(\mathbf{\Gamma })$ singularity diagnostic and threshold cascade — are they measuring the same conditioning, or complementary failures?