singularity_threshold_cascade

Singularity Threshold Cascade and the Four-Layer Conditioning Stack

Statement

Proximity to singularity for the coordinated free-flying controller is measured by 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 — empirically the two minimum singular values track with
Spearman rank correlation $\ge 0.9969$ — a single scalar ${\sigma }_G$ suffices to schedule the
whole conditioning stack.

Amplification bound. From the SVD $\mathbf{\Gamma }={\sum }_i{s}_i{\mathbf{u}}_i{\mathbf{v}}_i^{\top }$,
the generalized-velocity reconstruction $\mathbf{x}={\mathbf{\Gamma }}^{-1}\mathbf{z}$
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}

The cascade (ours). Threading one velocity currency through three floors:

• First floor — full-speed budget. Setting $\Vert \mathbf{x}\Vert$ to the highest
  tolerable rate $x_{\max }$ turns the bound into a commanded-speed budget, \sigma_{\mathrm{crit},1}=\frac{v}{x_{\max}}\simeq\frac{0.90}{2\pi}=0.143, \tag{current_sota eq 6.2} with $v_{\mathrm{ref}}=0.90m/s$ and the wrist rate $x_{\max }=2\pi rad/s$.
  Above ${\sigma }_{\mathrm{crit},1}$ no derating is needed.
• Second floor — floored-speed budget. ${\sigma }_{\mathrm{crit},2}={\sigma }_{\mathrm{crit},1}^2$.
  Once the impedance derate has bottomed out, the speed budget is exhausted and the controller
  switches from regularization to explicit damping.
• Third floor — damped-inverse honesty floor. ${\sigma }_{\mathrm{crit},3}={\lambda }_J$. The
  damped-inverse gain $s/(s^2+{\lambda }_J^2)$ peaks at $1/(2{\lambda }_J)$ when $s={\lambda }_J$; below
  this, damping actively drives the dying direction toward zero, so the honest action is to freeze
  (hold-last). This reproduces the code’s $0.02$.

The four-layer conditioning stack (ours). Built on ${\sigma }_G$:

1. Tikhonov $\mathbf{\Gamma }$ regularization — recover the generalized velocity 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 because it is added to ${\mathbf{\Gamma }}^{\top }\mathbf{\Gamma }$,
   whose eigenvalues are the squared singular values; it switches on once
   ${\sigma }_G<\sqrt{{\sigma }_{\mathrm{crit},1}^2-{\beta }^2}$.
2. Impedance derate — an emergent slowdown via the gain ramp
   $\gamma ({\sigma }_G)\in [{\sigma }_{\mathrm{crit},1},1]$: \gamma(\sigma_G)= \begin{cases} \sigma_{\mathrm{crit},1} & \sigma_G\le\sigma_{\mathrm{crit},2},\\[2pt] 1 & \sigma_G\ge\sigma_{\mathrm{crit},1},\\[2pt] \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} applied to the arm gains ${\mathbf{K}}_e,{\mathbf{D}}_e\to \gamma {\mathbf{K}}_e,\gamma {\mathbf{D}}_e$
   and the base wrench ${\mathbf{\tau }}_b^{\oplus }\to \gamma {\mathbf{\tau }}_b^{\oplus }$. Crucially
   $\gamma$ scales the gains, not the command: softening $\breve{\mathbf{K}}$ relocates the
   cruise-lag equilibrium to a larger lag (the arm goes slack rather than driving into the singular
   direction), which holds ${\sigma }_{\min }$. A literal speed derate ($\gamma {\mathbf{\nu }}_e$)
   is a different operation that cannot beat the amplification bound and collapses the error floor.
3. Damped ${\mathbf{J}}^{\oplus }$ inverse (three-tier) — with the SVD
   ${\mathbf{J}}^{\oplus }=\mathbf{U}\mathrm{diag}(s_i){\mathbf{V}}^{\top }$: (\boldsymbol{J}^\oplus)^+= \begin{cases} \boldsymbol{V}\operatorname{diag}(1/s_i)\boldsymbol{U}^\top, & \sigma_G\ge\sigma_{\mathrm{crit},2}\ \text{(undamped)},\\[4pt] \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)},\\[6pt] (\boldsymbol{J}^\oplus)^+_{\mathrm{last}}, & \sigma_G<\sigma_{\mathrm{crit},3}\ \text{(hold-last)}. \end{cases} \tag{current_sota eq 6.5}
4. Hold-last (freeze) — below ${\sigma }_{\mathrm{crit},3}$ 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.)

All four layers diverge through the shared $({\mathbf{J}}_{{\nu }_e}^{\oplus }{)}^{-1}$ factor — the
analytic root of the $\mathbf{\Gamma }$ conditioning.

Assumptions

• The circumcentroidal map $\mathbf{\Gamma }$ and its lower-right block
  ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ go singular together (see
  circumcentroidal_decoupling); ${\sigma }_G$ is a faithful proxy
  for end-effector-about-CoM conditioning (Spearman $\ge 0.9969$ with
  ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$).
• Free-flying regime (fully-actuated 6-DOF base). The amplification bound and cascade are about
  conditioning of the kinematic reconstruction, not about momentum conservation — they do not
  assume the free-floating GJM machinery.
• Numerical values are configuration-specific: $v_{\mathrm{ref}}=0.90m/s$,
  $x_{\max }=2\pi rad/s$ (UR3 wrist rate), ${\beta }^2=10^{-4}$, ${\lambda }_J$ giving
  ${\sigma }_{\mathrm{crit},3}\approx 0.02$.
• The nonredundant ($n=6$) case is assumed for the three-tier inverse; the redundant ($n=7$) case
  layers an additional null-space freeze.

Proof sketch

The four-layer threshold sizing is derived in
threshold_derivation.md; the determinant root of the
elbow singularity in singularity_geometry.md; and the
impedance-vs-speed-derate A/B in
derate_impedance_vs_speed.md. Aligning
threshold_derivation’s detailed ${\sigma }_6$-floor audit ($0.005/0.02/0.025/0.049$) with the
${\sigma }_{\mathrm{crit}}$ cascade below is a Phase-D reconciliation check.

1. SVD $\mathbf{\Gamma }={\sum }_i{s}_i{\mathbf{u}}_i{\mathbf{v}}_i^{\top }$; invert to get
   $\mathbf{x}={\sum }_i({\mathbf{u}}_i^{\top }\mathbf{z}/s_i){\mathbf{v}}_i$. The dominant
   $1/{\sigma }_{\min }$ term gives the operator-norm bound
   $\Vert \mathbf{x}\Vert \le v/{\sigma }_{\min }(\mathbf{\Gamma })$ (eq 6.1).
2. Invert the bound at the velocity limit $x_{\max }$ to read off the first floor
   ${\sigma }_{\mathrm{crit},1}=v/x_{\max }$ (eq 6.2); square it for the second floor; identify
   the damped-gain peak $s={\lambda }_J$ as the third floor.
3. Show each layer’s switch-on condition aligns with a floor: Tikhonov damping
   ${\lambda }_{\Gamma }=\max ({\beta }^2,{\sigma }_{\mathrm{crit},1}^2-{\sigma }_G^2)$ activates at
   ${\sigma }_{\mathrm{crit},1}$; the derate ramp $\gamma$ saturates between
   ${\sigma }_{\mathrm{crit},2}$ and ${\sigma }_{\mathrm{crit},1}$; the three-tier inverse switches
   undamped→damped at ${\sigma }_{\mathrm{crit},2}$ and damped→freeze at ${\sigma }_{\mathrm{crit},3}$.
4. Argue (impedance vs. speed derate) that scaling gains relocates the cruise-lag equilibrium
   (eq 4.14, circumcentroidal_decoupling) rather than throttling
   the command, hence holds ${\sigma }_{\min }$ instead of collapsing the error floor. (A/B comparison
   to be lifted from the private derate report into generated/math/.)

The chiaverini1997singularity SVD/variable-damping machinery (the DLS inverse
${\mathbf{J}}^{\ast }={\mathbf{J}}^{\top }(\mathbf{J}{\mathbf{J}}^{\top }+{\lambda }^2\mathbf{I}{)}^{-1}$
and the singular-region variable damping ${\lambda }^2=(1-({\sigma }_m/\epsilon {)}^2){\lambda }_{\max }^2$)
is the fixed-base ancestor of layers 1 and 3; the cascade design — the three coupled thresholds and
the impedance-vs-speed derate distinction — is ours.

Source / provenance

• Literature: chiaverini1997singularity — the
  damped-least-squares inverse, numerical filtering, and variable damping in a singular region
  (fixed-base, redundant manipulator; neither free-flying nor free-floating). Source of the
  $s/(s^2+{\lambda }^2)$ damped-gain form reused in layers 1 and 3.
• Ours: the threshold cascade (${\sigma }_{\mathrm{crit},1},{\sigma }_{\mathrm{crit},2},{\sigma }_{\mathrm{crit},3}$),
  the single-${\sigma }_G$ scheduling across all four layers, the impedance-derate-vs-speed-derate
  distinction, and the application to the circumcentroidal coordinate transform $\mathbf{\Gamma }$.
• Private master: my_writing/equations/current_sota.md §6.1–6.4 (eqs 6.1–6.5), derived from
  final.tex §7.

Caveats

• ${\sigma }_{\mathrm{crit},1}=0.143$ and ${\sigma }_{\mathrm{crit},3}\approx 0.02$ are tuned to the UR3
  arm and the $0.90m/s$ cruise; they must be re-derived for any other arm/speed.
• The Spearman $\ge 0.9969$ link between ${\sigma }_G$ and ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$
  is empirical, not proven; an analytic bound is open.
• A literal speed derate collapses the tracking-error floor (~$10^{-5}$) where the impedance
  derate holds it; conflating the two is a known failure mode — keep them distinct.
• The hat/breve distinction is load-bearing for the underlying $\hat{\mathbf{M}},\breve{\mathbf{M}}$
  blocks; see circumcentroidal_decoupling. This page lives on the
  reduced/circumcentroidal side (${\mathbf{J}}_{{\nu }_e}^{\oplus }$, the $\breve{(\cdot )}$ subsystem).
• Subsystem attribution ([base, ee]). All four layers — Tikhonov $\mathbf{\Gamma }$
  regularization included — act on the coupled $9\times 9$ circumcentroidal block, not on the
  decoupled $3\times 3$ CoM loop. So “Tikhonov damps the CoM” is a category error: whether near-singular
  spikes in the CoM channel ${\tilde{\mathbf{x}}}_c$ share this block’s forward-Euler mechanism is an
  open question — see subsystem_decomposition,
  error_floor.md §4, evidence in
  evidence_map.
• The 7-DOF redundant self-motion freeze ($\sim 10^{-3}$) is a separate mechanism, not part of the
  three-floor cascade above.

Related

• Sibling result: circumcentroidal_decoupling — supplies
  $\mathbf{\Gamma }$, the hat/breve distinction, and the cruise-lag equilibrium (eq 4.14).
• Sibling result: singularity_geometry — the determinant root (why the
  system sits near singular in the first place).
• Sibling result: posture_oracle_gap — bounds how much of the oracle’s
  wall-clear route survives live dynamics.
• Sibling result: discretization_stability_omega_b — the
  forward-Euler ripple this cascade is exonerated of (driven by the same near-singular
  ${\mathbf{\Gamma }}^{-1}$ blow-up, but upstream of ${\sigma }_G$ scheduling).
• Topics: damped_least_squares ·
  singularity_robust_inverse ·
  dynamic_singularity ·
  circumcentroidal_motion ·
  cartesian_impedance_control.
• Source: chiaverini1997singularity.
• Notation: notation.md.

Implementation (sims wiki)

External — into the code wiki via the sims_wiki/ symlink (resolves in Obsidian, not GitHub).

• breve_controller — the four-layer regularization / derate / damped-inverse / hold-last stack scheduled on ${\sigma }_G$.