Singularity Robust Inverse

Definition

A singularity-robust inverse replaces the exact Jacobian (pseudo)inverse with a
conditioning-aware surrogate that stays bounded as the mapping loses rank, trading a small
reconstruction error for finite joint rates near a singularity. In this project the inverse acted
on is the circumcentroidal end-effector Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (motion of the
end-effector $\mathcal{E}$ about the system CoM $\mathcal{C}$; see notation.md),
because ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is the lower-right block of the coordinate transform
$\mathbf{\Gamma }$ and the two go singular together. The implementation is a three-tier
schedule on the proximity scalar ${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$ — undamped
SVD pseudoinverse far from singularity, Chiaverini damped-least-squares in the singular region,
and a frozen hold-last inverse once damping would do more harm than good.

Key Equations

With the SVD ${\mathbf{J}}^{\oplus }=\mathbf{U}\mathrm{diag}(s_i){\mathbf{V}}^{\top }$
(shorthand ${\mathbf{J}}^{\oplus }\equiv {\mathbf{J}}_{{\nu }_e}^{\oplus }$), the applied inverse is
scheduled by ${\sigma }_G$ across the three floors of the
singularity threshold cascade:

$$({\mathbf{J}}^{\oplus }{)}^{+}=\{\begin{array}{ll}\mathbf{V}\mathrm{diag}(1/s_i){\mathbf{U}}^{\top },& {\sigma }_G\ge {\sigma }_{\mathrm{crit},2}\text{(undamped)},\\ \mathbf{V}\mathrm{diag}\left(\frac{s_i}{s_i^2+{\lambda }_J^2}\right){\mathbf{U}}^{\top },& {\sigma }_{\mathrm{crit},3}\le {\sigma }_G<{\sigma }_{\mathrm{crit},2}\text{(damped)},\\ ({\mathbf{J}}^{\oplus }{)}_{\mathrm{last}}^{+},& {\sigma }_G<{\sigma }_{\mathrm{crit},3}\text{(hold-last)}.\end{array}\text{\hspace{1em}(current\_sota eq 6.5)}$$

Tier 1 — undamped pseudoinverse. The Moore–Penrose inverse from the SVD, exact
reconstruction; used wherever the mapping is well-conditioned (${\sigma }_G\ge {\sigma }_{\mathrm{crit},2}={\sigma }_{\mathrm{crit},1}^2$).

Tier 2 — damped-least-squares (DLS). The singular-value gain $1/s_i$ is replaced by the
Chiaverini damped gain $s_i/(s_i^2+{\lambda }_J^2)$, which is bounded by $1/(2{\lambda }_J)$ and rolls
the dying direction off smoothly rather than blowing up. This is the SVD form of the DLS inverse
${\mathbf{J}}^{\ast }={\mathbf{J}}^{\top }(\mathbf{J}{\mathbf{J}}^{\top }+{\lambda }^2\mathbf{I}{)}^{-1}={\sum }_i\frac{s_i}{s_i^2+{\lambda }^2}{\mathbf{v}}_i{\mathbf{u}}_i^{\top }$
(Chiaverini eq 11), used here in the singular region ${\sigma }_{\mathrm{crit},3}\le {\sigma }_G<{\sigma }_{\mathrm{crit},2}$.

Tier 3 — hold-last (freeze). Below ${\sigma }_{\mathrm{crit},3}={\lambda }_J$ the damped gain has
passed its peak ($s_i={\lambda }_J$) and any further damping actively drives the dying direction
toward zero; the honest move is to freeze $({\mathbf{J}}^{\oplus }{)}^{+}$ (and the coupled
$\mathbf{\Gamma }$ inverse) at its last well-conditioned value
$({\mathbf{J}}^{\oplus }{)}_{\mathrm{last}}^{+}$. The redundant ($n=7$) extension additionally freezes
the self-motion (null-space) direction below $\sim 10^{-3}$, where it swings violently.

The damped-gain peak fixes the third threshold: $s/(s^2+{\lambda }_J^2)$ is maximized at $s={\lambda }_J$
with value $1/(2{\lambda }_J)$, so ${\sigma }_{\mathrm{crit},3}={\lambda }_J$ marks the boundary below which
damping is counterproductive (current_sota §6.4; see the cascade derivation in
singularity threshold cascade). All three tiers
diverge through the same $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}$ factor that appears in the
closed-form ${\mathbf{\Gamma }}^{-1}$ (gamma closed-form inverse),
which is why scheduling on the single scalar ${\sigma }_G$ governs the entire conditioning stack.

Regime

The DLS/SVD machinery here is borrowed from fixed-base redundancy resolution
(Chiaverini) and free-floating task control (Sentis–Khatib), but it is applied to the
free-flying circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$. The robustness
mechanism is purely kinematic conditioning and transfers across regimes unchanged; the object
being inverted (an actuated-base circumcentroidal map, not a momentum-folded generalized Jacobian)
is what differs.

Source Support

• holt1992inertial — inertial-space tracking context where the inverse is exercised.
• sze2024obstacle — obstacle-aware planning that relies on a robust inverse near ill-conditioned configurations.
• chiaverini1997singularity — origin of the damped-least-squares inverse ${\mathbf{J}}^{\ast }={\mathbf{J}}^{\top }(\mathbf{J}{\mathbf{J}}^{\top }+{\lambda }^2\mathbf{I}{)}^{-1}$ and the SVD gain $s_i/(s_i^2+{\lambda }^2)$ reused in tier 2 (fixed-base, redundant; neither free-flying nor free-floating).
• sentis2005control — singularity-robust task control via eigen/SVD truncation of the task inertia ${\mathbf{\Lambda }}^{-1}$; the rank-truncated-inverse idea behind the hold-last/freeze tier (free-floating regime; base unactuated).

Related Topics

• damped_least_squares — the tier-2 gain $s_i/(s_i^2+{\lambda }_J^2)$ is exactly the DLS gain; this page schedules it inside the three-tier inverse.
• pseudoinverse_jacobian — tier 1 is the plain Moore–Penrose inverse this technique falls back to when well-conditioned.
• circumcentroidal_motion — defines the object being inverted, ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (EE motion about the CoM).
• dynamic_singularity — the configurations where ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ loses rank and the robust inverse is needed.
• singularity threshold cascade (result) — supplies ${\sigma }_{\mathrm{crit},1},{\sigma }_{\mathrm{crit},2},{\sigma }_{\mathrm{crit},3}$ and the ${\sigma }_G$ scheduling that selects the tier.

Open Questions

• The Spearman $\ge 0.9969$ link between ${\sigma }_G$ and ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ that justifies scheduling the inverse on ${\sigma }_G$ is empirical, not proven.
• ${\lambda }_J$ (hence ${\sigma }_{\mathrm{crit},3}\approx 0.02$) is tuned to the UR3 arm and cruise speed; no analytic rule bounds the tier-2 reconstruction error as a function of ${\lambda }_J$.
• Continuity of the inverse across the tier-2 → tier-3 (freeze) switch, and the transient when it later thaws, is not formally characterized.