Redundancy Resolution

Definition

Redundancy resolution is the choice of one joint-space command among the continuum admitted by
a kinematically redundant manipulator (joints $n$ exceeding task dimension $m$, see
kinematic_redundancy): the inverse-kinematics map is ill-posed, so a
resolution scheme must single out a unique $\dot{\mathbf{q}}$. The classical resolution operates
at velocity level — the pseudoinverse_jacobian returns the minimum-norm
solution and the residual freedom is steered through the Jacobian
null_space_projection to serve secondary objectives. The page is regime-general:
the resolution math is identical for fixed-base and space arms, but what counts as the task and how
base motion enters differs by regime — note the assumption of each source. dambrosio2024redundant
treats a free-flying chaser (6-DOF actively controlled base + 7-DOF arm, one degree of redundancy
for a 6-DOF end-effector pose task) and resolves the redundancy with a learned policy rather than a
pseudoinverse, sidestepping the ill-posed IK at run time.

Key Equations

Symbols per notation.md.

Velocity-level (resolved-motion-rate) redundancy resolution for a desired end-effector twist
${\mathbf{\nu }}_{ed}$, with minimum-norm primary term plus a null-space secondary command
${\dot{\mathbf{q}}}_0$:

$$\dot{\mathbf{q}}={\mathbf{J}}_{{\nu }_e}^{+}{\mathbf{\nu }}_{ed}+\mathbf{N}{\dot{\mathbf{q}}}_0,\mathbf{N}=\mathbf{E}-{\mathbf{J}}_{{\nu }_e}^{+}{\mathbf{J}}_{{\nu }_e}.$$

The redundancy is the non-trivial kernel: $\dim \ker {\mathbf{J}}_{{\nu }_e}=n-\mathrm{rank}{\mathbf{J}}_{{\nu }_e}>0$,
which is exactly what $\mathbf{N}$ projects onto.

The learning-based alternative used by dambrosio2024redundant
replaces the analytic inverse with a trained PPO policy ${\pi }_{\theta }$ mapping the observed pose error to
the desired joint rates (notation ${\pi }_{\theta },\mathbf{o}$ source-faithful, not canonicalized in notation.md):

$${\dot{\mathbf{q}}}_m^{\ast }={\pi }_{\theta }(\mathbf{o}),\mathbf{o}=[{\mathbf{q}}_m,{\dot{\mathbf{q}}}_m,\tilde{\mathbf{r}},\widetilde{\mathrm{DCM}},\tilde{\mathbf{v}},\tilde{\mathbf{\omega }}{]}^{\top },$$

where the policy output is the desired joint rate ${\dot{\mathbf{q}}}_m^{\ast }$ (source Eq. 10) — distinct from
the observed current rate ${\dot{\mathbf{q}}}_m$ in $\mathbf{o}$ — and its time-integral ${\mathbf{q}}_m^{\ast }$,
together with ${\dot{\mathbf{q}}}_m^{\ast }$, is passed to a feedback-linearizing PD law
$\mathbf{\tau }=\mathbf{M}\mathrm{PD}({\mathbf{q}}^{\ast }-\mathbf{q})+\mathbf{C}\dot{\mathbf{q}}$
(source $H,C$ = our $\mathbf{M},\mathbf{C}$).

Source Support

• dambrosio2024redundant — primary; explicitly a free-flying
  7-DOF redundant arm on a fully-actuated 6-DOF base. Motivates redundancy resolution by the “ill-posed
  inverse kinematics” of the redundant arm, then resolves it with a PPO/DRL guidance policy that outputs
  the 7 joint rates directly (no pseudoinverse / null-space projection), avoiding real-time optimization.

Related Topics

• kinematic_redundancy — the condition ($n>m$) that creates the solution
  continuum this technique must resolve.
• pseudoinverse_jacobian — supplies the minimum-norm primary term
  ${\mathbf{J}}_{{\nu }_e}^{+}{\mathbf{\nu }}_{ed}$ of the classical resolution.
• null_space_projection — the projector $\mathbf{N}$ that routes the
  redundant freedom to secondary objectives without disturbing the end-effector task.
• task_priority_redundancy — the priority-ordered generalization, where
  lower-priority tasks live in the null space of higher-priority ones.
• resolved_motion_rate_control — the velocity-level IK framework in
  which the resolution equation above is posed.

Open Questions

• D’Ambrosio’s policy resolves redundancy implicitly (no closed-form null-space command): can a
  secondary objective — base-reaction minimization, manipulability, joint limits — be guaranteed, or
  only encouraged through the reward shaping?
• The source is free-flying with the base held at a synchronized state by a separate PD loop; classical
  free-floating resolution folds momentum conservation into a generalized Jacobian
  (generalized_jacobian). For our actively-actuated base, is the plain
  kinematic ${\mathbf{J}}_{{\nu }_e}^{+}$ resolution sufficient, or should base reaction still shape the
  null-space command?
• Learned resolution gives no a-priori guarantee of avoiding kinematic singularities of
  ${\mathbf{J}}_{{\nu }_e}$; how does the policy behave as ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e})\to 0$?