singularity_handling

Singularity Handling: The Four-Layer Conditioning Cascade

Last updated: 2026-06-22

Overview

Singularity handling for the coordinated free-flying controller is a single, scheduled
four-layer cascade built on one scalar, the minimum singular value of the coordinate
transform ${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$ (notation.md).
Because $\mathbf{\Gamma }$ is singular exactly where its lower-right block
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is — the two minimum singular values track with Spearman rank
correlation $\ge 0.9969$ — the same ${\sigma }_G$ schedules every layer, and all four ultimately
diverge through the shared factor $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}$
(dynamic_singularity; current_sota §6).

The four layers, ordered by the floor at which each engages
(singularity_threshold_cascade):

1. Tikhonov $\mathbf{\Gamma }$ regularization — a squared, variable damping on the
   generalized-velocity normal equations.
2. Impedance derate — an emergent slowdown that scales the gains, not the command.
3. Damped ${\mathbf{J}}^{\oplus }$ inverse (three-tier) — undamped → damped-least-squares → hold-last.
4. Hold-last (freeze) — below the honesty floor, the ${\mathbf{J}}^{\oplus }$ and
   $\mathbf{\Gamma }$ inverses are frozen at their last well-conditioned value.

The intellectual spine is the amplification bound. From the SVD
$\mathbf{\Gamma }={\sum }_i{s}_i{\mathbf{u}}_i{\mathbf{v}}_i^{\top }$, reconstructing the
generalized velocity $\mathbf{x}={\mathbf{\Gamma }}^{-1}\mathbf{z}$ amplifies each task
component by $1/s_i$, so for a commanded task speed $\Vert \mathbf{z}\Vert =v$,

\boldsymbol{x}=\sum_i \frac{\boldsymbol{u}_i^\top\boldsymbol{z}}{s_i}\,\boldsymbol{v}_i \;\Rightarrow\; \lVert\boldsymbol{x}\rVert \le \frac{v}{\sigma_{\min}(\boldsymbol{\Gamma})} \;\Leftrightarrow\; \sigma_{\min}(\boldsymbol{\Gamma}) \le \frac{v}{\lVert\boldsymbol{x}\rVert}. \tag{current_sota eq 6.1}

Inverting this bound at the actuator rate limit $x_{\max }$ produces a velocity currency threaded
through three thresholds: ${\sigma }_{\mathrm{crit},1}=v/x_{\max }\simeq 0.143$ (full-speed budget),
${\sigma }_{\mathrm{crit},2}={\sigma }_{\mathrm{crit},1}^2$ (floored-speed budget exhausted → switch to
damping), and ${\sigma }_{\mathrm{crit},3}={\lambda }_J$ (damped-gain peak → freezing is the honest move)
(current_sota eqs 6.2; §6.1). The cascade design is ours; the damped-least-squares machinery
of layers 1 and 3 descends from chiaverini1997singularity
(fixed-base, redundant manipulator — neither free-flying nor free-floating).

Free-flying vs free-floating (load-bearing)

The object that goes singular here is the circumcentroidal Jacobian
${\mathbf{J}}_{{\nu }_e}^{\oplus }$ inside the coordinate transform $\mathbf{\Gamma }$ — a
free-flying construct in which ${\mathbf{\omega }}_b$ is a commanded input. It is not
the free-floating generalized Jacobian (GJM, Umetani 1989) whose rank loss is the classical
dynamic singularity, and the cascade is about kinematic-reconstruction conditioning, not
momentum conservation. Sources that supply singularity maps / avoidance
(calzolari2020singularity, nanos2015avoiding,
papadopoulos1993dynamic) assume a free-floating base; their loci must be re-derived for our
actuated base.

Key Claims

Claim 1 — A single scalar ${\sigma }_G$ schedules all four layers

${\mathbf{\Gamma }}^{-1}$ exists iff ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ (its lower-right block) is
nonsingular, so the two go singular together; empirically
$\mathrm{Spearman}({\sigma }_{\min }(\mathbf{\Gamma }),{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus }))\ge 0.9969$.
A single proximity scalar ${\sigma }_G$ therefore suffices to switch every layer on and off, and the
hazard has one analytic root — the shared $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}$ factor.
Source support: current_sota §6 (final.tex §7); see
singularity_threshold_cascade and
dynamic_singularity. Caveat: the $\ge 0.9969$ link is
empirical, not proven (open).

Claim 2 — Layer 1: Tikhonov $\mathbf{\Gamma }$ regularization (squared variable damping)

The generalized velocity is recovered from the regularized normal equations with a constant
isotropic floor ${\beta }^2=10^{-4}$:

\dot{\boldsymbol{q}}=\big(\boldsymbol{\Gamma}^\top\boldsymbol{\Gamma}+\lambda_\Gamma\boldsymbol{I}\big)^{-1}\boldsymbol{\Gamma}^\top\boldsymbol{y}, \qquad \lambda_\Gamma=\max\!\big(\beta^2,\ \sigma_{\mathrm{crit},1}^2-\sigma_G^2\big). \tag{current_sota eq 6.3}

The damping is squared — added to ${\mathbf{\Gamma }}^{\top }\mathbf{\Gamma }$, whose
eigenvalues are the squared singular values — and switches on once
${\sigma }_G<\sqrt{{\sigma }_{\mathrm{crit},1}^2-{\beta }^2}$ (Chiaverini-style variable damping in the
singular region). Source support: current_sota §6.2 (final.tex eq gamma-solve);
chiaverini1997singularity for the variable-damping form;
topics damped_least_squares,
singularity_robust_inverse.

Claim 3 (KEY) — Layer 2: impedance derate scales the GAINS, not the command

A derate ramp $\gamma ({\sigma }_G)\in [{\sigma }_{\mathrm{crit},1},1]$ slows the system as it nears the
singular set:

\gamma(\sigma_G)= \begin{cases} \sigma_{\mathrm{crit},1} & \sigma_G\le\sigma_{\mathrm{crit},2},\\[4pt] 1 & \sigma_G\ge\sigma_{\mathrm{crit},1},\\[4pt] \sigma_{\mathrm{crit},1}+(1-\sigma_{\mathrm{crit},1})\dfrac{\sigma_G-\sigma_{\mathrm{crit},2}}{\sigma_{\mathrm{crit},1}-\sigma_{\mathrm{crit},2}} & \text{otherwise.} \end{cases} \tag{current_sota eq 6.4}

Crucially $\gamma$ multiplies the control gains and base wrench, not the velocity command:
${\mathbf{K}}_e,{\mathbf{D}}_e\to \gamma {\mathbf{K}}_e,\gamma {\mathbf{D}}_e$ and
${\mathbf{\tau }}_b^{\oplus }\to \gamma {\mathbf{\tau }}_b^{\oplus }$. This is the load-bearing
distinction. Softening $\breve{\mathbf{K}}$ relocates the cruise-lag equilibrium: the
steady-state pose error floor
${\tilde{\mathbf{x}}}_{ss}=-\left({\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}{)}^{-1}\right[\breve{\mathbf{C}}{\breve{\mathbf{v}}}_d+({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c]$
(current_sota eq 4.14) carries a ${\breve{\mathbf{K}}}^{-1}$ factor, so scaling the stiffness by
$\gamma$ scales the lag as ${\tilde{\mathbf{x}}}_{ss}\to \frac{1}{\gamma }{\tilde{\mathbf{x}}}_{ss}$
under a $\gamma$-invariant restoring wrench. The arm therefore goes slack — it tolerates a
larger lag rather than driving harder toward the dying direction — and the slowdown is
emergent: it falls out of the relocated equilibrium, it is never commanded. This holds
${\sigma }_{\min }$ (observed $\min s_{\min ,J}=0.0268$). Source support: current_sota §6.3 (final.tex
§7); steady_state_error_floor;
cartesian_impedance_control.

Claim 4 (KEY) — A literal SPEED derate cannot beat the bound and COLLAPSES the floor

Throttling the command itself, ${\mathbf{\nu }}_e\to \gamma {\mathbf{\nu }}_e$, is a different
operation. It cannot beat the amplification bound (eq 6.1): lowering $v$ does not raise
${\sigma }_{\min }(\mathbf{\Gamma })$, and the throttle’s pose-error growth over-extends the arm
toward the singular direction. The result is a collapsed error floor (observed $1.0\times 10^{-5}$,
versus the impedance derate’s held floor). Hence the terminology is deliberately “impedance
derate,” not “speed derate.” The two are easy to conflate and produce opposite outcomes — a known
failure mode worth flagging in any reimplementation. Source support: current_sota §6.3 (final.tex
§7); A/B comparison and derivation in the private report
generated_reports/math/derate_impedance_vs_speed.md (mirrored target:
math); see the cascade result
singularity_threshold_cascade.

Inconsistency / failure mode

“Impedance derate” (scale $\breve{\mathbf{K}},\breve{\mathbf{D}},{\mathbf{\tau }}_b^{\oplus }$
by $\gamma$) holds ${\sigma }_{\min }$; a “speed derate” (scale the command ${\mathbf{\nu }}_e$ by
$\gamma$) collapses the floor. They share the symbol $\gamma$ but are not interchangeable.

Claim 5 — Layers 3–4: three-tier damped ${\mathbf{J}}^{\oplus }$ inverse, then freeze

With the SVD ${\mathbf{J}}^{\oplus }=\mathbf{U}\mathrm{diag}(s_i){\mathbf{V}}^{\top }$, the
inverse is scheduled in three tiers and frozen below the honesty floor:

(\boldsymbol{J}^\oplus)^+= \begin{cases} \boldsymbol{V}\operatorname{diag}(1/s_i)\boldsymbol{U}^\top, & \sigma_G\ge\sigma_{\mathrm{crit},2}\ \text{(undamped)},\\[6pt] \boldsymbol{V}\operatorname{diag}\!\Big(\dfrac{s_i}{s_i^2+\lambda_J^2}\Big)\boldsymbol{U}^\top, & \sigma_{\mathrm{crit},3}\le\sigma_G<\sigma_{\mathrm{crit},2}\ \text{(damped)},\\[8pt] (\boldsymbol{J}^\oplus)^+_{\mathrm{last}}, & \sigma_G<\sigma_{\mathrm{crit},3}\ \text{(hold-last)}. \end{cases} \tag{current_sota eq 6.5}

The damped tier is the Chiaverini damped-least-squares gain $s/(s^2+{\lambda }_J^2)$, which peaks at
$1/(2{\lambda }_J)$ when $s={\lambda }_J={\sigma }_{\mathrm{crit},3}$; below this, damping drives the dying
direction toward zero, so hold-last (freeze) is the honest action — the ${\mathbf{J}}^{\oplus }$
and $\mathbf{\Gamma }$ inverses are frozen at their last well-conditioned value (the 7-DOF
extension additionally freezes the redundant self-motion / null-space direction below $\sim 10^{-3}$,
where it swings violently). Source support: current_sota §6.4 (final.tex eq jplus-inverse);
singularity_robust_inverse,
damped_least_squares,
chiaverini1997singularity.

Claim 6 — Stability is asserted only on the singularity-free region $\Omega$

The coordinated control law’s Lyapunov stability (current_sota eqs 4.15–4.16) holds on
$\Omega =\{\mathbf{z}:{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus }(\mathbf{q}))>0\}$. The
cascade is precisely the machinery that keeps the trajectory inside (or honestly stalls it at the
boundary of) $\Omega$ where the stability certificate is valid. Source support: current_sota §4.6;
coordinated_control_lyapunov_stability.

Tensions & Open Questions

• Empirical, not proven, equivalence. The $\ge 0.9969$ Spearman link between ${\sigma }_G$ and
  ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ justifies single-scalar scheduling, but an analytic
  bound across the full inertia-parameter envelope is open
  (singularity_threshold_cascade, Caveats).
• Configuration-specific thresholds. ${\sigma }_{\mathrm{crit},1}=0.143$ and
  ${\sigma }_{\mathrm{crit},3}\approx 0.02$ are tuned to the UR3 wrist rate
  ($x_{\max }=2\pi rad/s$) and the $0.90m/s$ cruise; they must be re-derived for
  any other arm or speed.
• Free-flying vs free-floating loci. Classical dynamic-singularity maps and avoidance results
  (calzolari2020singularity, nanos2015avoiding,
  papadopoulos1993dynamic) assume an uncontrolled base. How much does active 6-DOF base control
  shrink or eliminate the singular locus relative to the free-floating $\det [{\mathbf{J}}^{\ast }]=0$ set?
  Open (dynamic_singularity, Open Questions).
• Derate A/B not yet in the public wiki. The impedance-vs-speed A/B comparison lives in the
  private report; the full derivation is deferred to math. Until
  lifted, Claim 4’s $1.0\times 10^{-5}$ / held-floor figures are reported, not re-derived here.
• Layer interaction. Each layer is scheduled by the same ${\sigma }_G$, but their combined
  closed-loop behaviour (Tikhonov damping + impedance derate + damped inverse acting together near
  ${\sigma }_{\mathrm{crit},2}$) has no separate stability proof; only the singularity-free region
  $\Omega$ result is on hand.

Connected Pages

• Result: singularity_threshold_cascade — the
  amplification bound, the three ${\sigma }_{\mathrm{crit}}$ floors, and the four-layer stack stated as
  a result (this page is its narrative/overview companion).
• Results: steady_state_error_floor ·
  coordinated_control_lyapunov_stability ·
  circumcentroidal_decoupling.
• Topics: dynamic_singularity ·
  damped_least_squares ·
  singularity_robust_inverse ·
  cartesian_impedance_control ·
  kinematic_singularity.
• Source: chiaverini1997singularity — fixed-base
  damped-least-squares / variable-damping ancestor of layers 1 and 3.
• Notation: notation.md.