Damped Least Squares

Definition

Damped least squares (DLS), a.k.a. the Levenberg–Marquardt or singularity-robust inverse, replaces the exact Jacobian inverse with the solution of a Tikhonov-regularized least-squares problem, trading exact task-velocity reconstruction for bounded joint velocities near a singularity. For a map $\mathbf{J}$ with SVD singular values $s_i$, the damped inverse weights each direction by $s_i/(s_i^2+{\lambda }^2)$ instead of $1/s_i$, so the gain stays finite (peaking at $1/(2\lambda )$) as $s_i\to 0$. In this project DLS appears twice in the singularity stack: as Tikhonov regularization of the coordinate transform $\mathbf{\Gamma }$ (damping ${\lambda }_{\Gamma }$, notation.md) and as the damped circumcentroidal-Jacobian inverse $({\mathbf{J}}^{\oplus }{)}^{+}$ (damping ${\lambda }_J$), with the damping switched on variably (Chiaverini) only inside a singular region.

Key Equations

Tikhonov regularization of $\mathbf{\Gamma }$ (Layer 1). The generalized velocity is recovered from the regularized normal equations, with a constant isotropic floor ${\beta }^2=10^{-4}$ and variable damping that switches on once ${\sigma }_G<\sqrt{{\sigma }_{\mathrm{crit},1}^2-{\beta }^2}$:

\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 here is squared: it is added to ${\mathbf{\Gamma }}^{\top }\mathbf{\Gamma }$, whose eigenvalues are the squared singular values, so ${\lambda }_{\Gamma }$ acts in ${\sigma }^2$-units against ${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$.

Damped circumcentroidal-Jacobian inverse (Layer 3, middle tier). With the SVD ${\mathbf{J}}^{\oplus }=\mathbf{U}\mathrm{diag}(s_i){\mathbf{V}}^{\top }$, the damped pseudoinverse used in the singular band ${\sigma }_{\mathrm{crit},3}\le {\sigma }_G<{\sigma }_{\mathrm{crit},2}$ is

(\boldsymbol{J}^\oplus)^+=\boldsymbol{V}\operatorname{diag}\!\Big(\frac{s_i}{s_i^2+\lambda_J^2}\Big)\boldsymbol{U}^\top. \tag{current_sota eq 6.5}

Each per-direction gain $s_i/(s_i^2+{\lambda }_J^2)$ peaks at $1/(2{\lambda }_J)$ exactly when $s_i={\lambda }_J$ — this peak fixes the third threshold ${\sigma }_{\mathrm{crit},3}={\lambda }_J$ in the singularity threshold cascade, below which damping would actively drive the dying direction toward zero and freezing (hold-last) is the honest move.

Fixed-base ancestor (Chiaverini). Both project forms specialize Chiaverini’s singularity-robust inverse and its variable damping, for a fixed-base end-effector Jacobian ${\mathbf{J}}_{{\nu }_e}$ with minimum singular value ${\sigma }_m$ and activation threshold $\epsilon$:

$${\mathbf{J}}_{{\nu }_e}^{\ast }={\mathbf{J}}_{{\nu }_e}^{\top }({\mathbf{J}}_{{\nu }_e}{\mathbf{J}}_{{\nu }_e}^{\top }+{\lambda }^2\mathbf{I}{)}^{-1}=\sum _i\frac{s_i}{s_i^2+{\lambda }^2}{\mathbf{v}}_i{\mathbf{u}}_i^{\top },{\lambda }^2=\{\begin{array}{ll}0,& {\sigma }_m\ge \epsilon ,\\ (1-({\sigma }_m/\epsilon {)}^2){\lambda }_{\max }^2,& {\sigma }_m<\epsilon .\end{array}\text{\hspace{1em}(Chiaverini eqs 11, 15)}$$

Confining ${\lambda }^2$ to the region ${\sigma }_m<\epsilon$ keeps the inverse as accurate as the pseudoinverse away from singularities while staying continuous and well-conditioned through the transition; multiple simultaneous singularities add an isotropic floor ${\beta }^2\ll {\lambda }^2$ (Chiaverini eq 20) — the same role ${\beta }^2$ plays in eq 6.3 above.

Regime

Chiaverini (fixed-base, redundant arm) and Ott (fixed-base rigid manipulator) are neither free-flying nor free-floating. The transferable content is the SVD/variable-damping machinery; this project applies it to the free-flying circumcentroidal transform $\mathbf{\Gamma }$ and its block ${\mathbf{J}}_{{\nu }_e}^{\oplus }$.

Source Support

• chiaverini1997singularity — origin of the DLS inverse ${\mathbf{J}}^{\ast }={\mathbf{J}}^{\top }(\mathbf{J}{\mathbf{J}}^{\top }+{\lambda }^2\mathbf{I}{)}^{-1}$, the $s_i/(s_i^2+{\lambda }^2)$ damped-gain form, variable damping in a singular region, and the isotropic floor ${\beta }^2$. Fixed-base, redundant; neither flying nor floating.
• ott2008cartesian — operational-space inertia $\mathbf{\Lambda }={\mathbf{J}}^{-T}\mathbf{M}{\mathbf{J}}^{-1}$ and the manipulability-potential singularity treatment; the damped/regularized inverse is the kinematic counterpart of its impedance-distortion-near-singularity analysis. Fixed-base rigid manipulator.
• holt1992inertial — (carried from the original stub) inertial-frame singularity-robust inverse usage.
• Code provenance (ours). The project implements this as gamma_regularization (g_reg) — the source never says “Tikhonov”; the $({\mathbf{\Gamma }}^{\top }\mathbf{\Gamma }+\lambda \mathbf{I}{)}^{-1}{\mathbf{\Gamma }}^{\top }$ shape is what identifies it (a structural, not lexical, join — ast-grep matches it, grep tikhonov does not). sims_gnc/breve_controller.py:484–490 is eq 6.3 verbatim: onset = g_reg.floor (the ${\sigma }_{\mathrm{crit},1}$ role; $=0.020$ in the rejected mission_tikhonov_020_jun19 A/B), constant floor = g_reg.damping ($={\beta }^2$). Source-verified 2026-06-23.

Related Topics

• singularity_robust_inverse — DLS is the canonical realization of the singularity-robust (Levenberg–Marquardt) inverse.
• singularity_threshold_cascade — schedules where DLS damping switches on (${\sigma }_{\mathrm{crit},2}$) and where it yields to hold-last (${\sigma }_{\mathrm{crit},3}={\lambda }_J$).
• pseudoinverse_jacobian — DLS reduces to the Moore–Penrose pseudoinverse as $\lambda \to 0$ (the undamped tier).
• kinematic_singularity — the rank-loss condition DLS is built to ride through.
• circumcentroidal_motion — supplies $\mathbf{\Gamma }$ and ${\mathbf{J}}_{{\nu }_e}^{\oplus }$, the objects this project damps.

Open Questions

• The activation threshold $\epsilon$ (Chiaverini) and the project floors ${\sigma }_{\mathrm{crit},1\dots 3}$ are tuned, not derived from a tracking-accuracy bound; what is the optimal $\lambda$ that minimizes reconstruction error for a given singularity proximity?
• Damping the squared transform (${\mathbf{\Gamma }}^{\top }\mathbf{\Gamma }+{\lambda }_{\Gamma }\mathbf{I}$) vs. damping the Jacobian SVD directly ($s_i/(s_i^2+{\lambda }_J^2)$) are reconciled only empirically via the Spearman $\ge 0.9969$ link between ${\sigma }_G$ and ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$; an analytic equivalence is open.
• Subsystem reach ([base, ee]). This regularization damps the coupled $9\times 9$ circumcentroidal block ($\mathbf{\Gamma }$, ${\mathbf{J}}_{{\nu }_e}^{\oplus }$), not the decoupled $3\times 3$ CoM loop. Whether near-singular spikes in the CoM channel ${\tilde{\mathbf{x}}}_c$ are this block’s forward-Euler artifact (cured in code by implicit base-damping integration) or a separate CoM-loop effect is unresolved — decide via the $dt$-independent lag-1 autocorrelation test on the logged ${\dot{\tilde{\mathbf{x}}}}_c$ (subsystem_decomposition, evidence_map).