Task Priority Redundancy

Definition

Task-priority redundancy resolution exploits a kinematically redundant manipulator’s extra
degrees of freedom by ordering its tasks: a higher-priority task is solved exactly via the
Jacobian pseudoinverse, and lower-priority tasks are projected onto the null space of the
higher-priority task Jacobian so they cannot perturb it. Caccavale & Siciliano (2001) apply this
within a closed-loop inverse kinematics (CLIK) scheme for a space manipulator, treating the
end-effector pose as the primary task and the spacecraft attitude (or a joint-limit / obstacle
constraint) as the secondary “constraint” task. The cited source assumes a free-floating
base: manipulator reaction on the uncontrolled spacecraft is folded in through the
generalized Jacobian ${\mathbf{J}}_g$, so all redundancy is measured
against the augmented spacecraft-plus-end-effector task. Redundancy exists only when the joint
count $n$ exceeds the total task dimension $m_s+m_e$ (spacecraft + end-effector).

Key Equations

Symbols per notation.md.

Task-priority CLIK solution with the end-effector task at top priority (Caccavale & Siciliano
2001, Eq. 21), in canonical notation (${\mathbf{J}}_g$ = generalized Jacobian, $\mathbf{E}$ =
identity, $(\cdot {)}^{+}$ = pseudoinverse):

$$\dot{\mathbf{q}}={\mathbf{J}}_g^{+}\left({\mathbf{\nu }}_d+{\mathbf{K}}_P{\tilde{\mathbf{x}}}_e\right)+\underset{\text{null-space projector}}{\underbrace{\left(\mathbf{E}-{\mathbf{J}}_g^{+}{\mathbf{J}}_g\right)}}{\mathbf{J}}_C^{\top }{\mathbf{K}}_C{\tilde{\mathbf{x}}}_C$$

The projector $\mathbf{E}-{\mathbf{J}}_g^{+}{\mathbf{J}}_g$ confines the secondary
(constraint) contribution to the null space of ${\mathbf{J}}_g$, so the constraint task cannot
interfere with the prioritized end-effector task. The secondary term uses the transpose
${\mathbf{J}}_C^{\top }$ (not the pseudoinverse), with a feedback correction ${\mathbf{K}}_C{\tilde{\mathbf{x}}}_C$ ensuring convergence — this avoids a second pseudoinversion and is more
robust to algorithmic singularities from conflicting tasks. Swapping subscripts $G\leftrightarrow C$
re-prioritizes the constraint task above the end-effector (source Eq. 24).

Note: ${\mathbf{K}}_P,{\mathbf{K}}_C$ are the primary/secondary positive-definite CLIK feedback
gains and ${\mathbf{J}}_C,{\tilde{\mathbf{x}}}_C$ the constraint-task Jacobian / error; these are
not in notation.md and are introduced source-faithfully here.

Source Support

• caccavale2001kinematic — primary; derives the task-priority CLIK algorithm (Eq. 21) for a redundant free-floating space manipulator, using the generalized Jacobian to absorb base reaction and the unit quaternion for both spacecraft and end-effector orientation error. Introduces the “projected-transpose” secondary task to dodge a second pseudoinverse.

Related Topics

• task_priority_redundancy_resolution — the general method this page instantiates for the space-manipulator case.
• redundancy_resolution — the broader family of techniques for using extra DOF; task priority is one ordering strategy within it.
• kinematic_redundancy — the precondition ($n>m_s+m_e$) that makes a null-space projection nontrivial.
• null_space_projection — the operator $\mathbf{E}-{\mathbf{J}}_g^{+}{\mathbf{J}}_g$ that isolates the lower-priority task.
• task_prioritization — the ordering principle (which task yields to which) realized here by projection.
• generalized_jacobian — ${\mathbf{J}}_g$, the free-floating construct that replaces the fixed-base Jacobian in every term above.

Open Questions

• The source assumes a free-floating base and resolves redundancy against the generalized Jacobian ${\mathbf{J}}_g$. For our free-flying (fully-actuated 6-DOF base) system, base attitude is an actuated task rather than a passive constraint absorbed by ${\mathbf{J}}_g$ — does the same priority ordering carry over, or should the base become a coordinated primary task using the circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ instead?
• Task-priority projection inherits algorithmic singularities when the primary and secondary tasks conflict; how does this interact with the kinematic/dynamic singularities of ${\mathbf{J}}_g$ (and, in our regime, $\mathbf{\Gamma }$) near rank loss?