Manipulability Measure

Definition

A manipulability measure is a scalar that quantifies how far a configuration is from a
(Jacobian-rank) singularity — equivalently, how much velocity-transmission ability the mapping
retains in its weakest direction. The classical form is Yoshikawa’s volume measure
$w=\sqrt{\det (\mathbf{J}{\mathbf{J}}^{\top })}$ (used by Ott as a singularity-avoidance
potential), which equals the product of the Jacobian’s singular values. Because $w$ collapses many
singular values into one product and vanishes only at exact rank loss, this project instead uses
the minimum singular value as the proximity scalar — operatively
${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma })$, the smallest singular value of the coordinate
transform $\mathbf{\Gamma }$ — which falls off in direct proportion to the worst-conditioned
direction and is the quantity threaded through the singularity-handling cascade.

Key Equations

All symbols are rendered from notation.md.

Yoshikawa volume measure (the classical manipulability scalar). For a manipulator Jacobian
$\mathbf{J}$,

$$w=\sqrt{\det \left(\mathbf{J}{\mathbf{J}}^{\top }\right)}=\prod _i{s}_i,$$

(Ott eq 3.22, written $m_{\mathrm{kin}}(\mathbf{q})=\sqrt{\det (\mathbf{J}{\mathbf{J}}^{\top })}$)
— the $s_i$ are the singular values of $\mathbf{J}$; $w$ is the volume of the velocity
manipulability ellipsoid and vanishes iff $\mathbf{J}$ loses rank. Ott uses it to build a
singularity-avoidance potential $V_m(\mathbf{q})=k_s(w-m_0{)}^2$ for $w\le m_0$ (else $0$),
added to the impedance torque as $-\partial V_m/\partial \mathbf{q}$ (Ott eqs 3.23, 3.24).

Regime

Ott’s $w=\sqrt{\det (\mathbf{J}{\mathbf{J}}^{\top })}$ is defined on a fixed-base serial
manipulator (neither free-flying nor free-floating; gravity present and compensated). For a
free-flying space manipulator the bare $\mathbf{J}$ must be replaced by the base-coupled map —
here the circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ — before the measure is meaningful.

Operative measure: the minimum singular value. This project monitors proximity to singularity
with the smallest singular value of the coordinate transform $\mathbf{\Gamma }$, not the volume
product:

$${\sigma }_G\equiv {\sigma }_{\min }(\mathbf{\Gamma }),\mathbf{\Gamma }=\sum _i{s}_i{\mathbf{u}}_i{\mathbf{v}}_i^{\top },$$

(current_sota §6 / final.tex §7) — $\mathbf{\Gamma }$ is singular exactly where its lower-right
block ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ is, so ${\sigma }_G$ is a faithful single-scalar proxy for
end-effector-about-CoM conditioning. Empirically the two minimum singular values track almost
perfectly:

$$\mathrm{Spearman}({\sigma }_{\min }(\mathbf{\Gamma }),{\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus }))\ge 0.9969,$$

(current_sota §6 / final.tex §7 / Giordano eq 36).

Why ${\sigma }_{\min }$ and not $w$. The generalized-velocity reconstruction amplifies each task
component by $1/s_i$, so the worst-conditioned direction — set by the minimum singular value, not
the product — bounds the required joint rate:

$$\mathbf{x}=\sum _i\frac{{\mathbf{u}}_i^{\top }\mathbf{z}}{s_i}{\mathbf{v}}_i\Rightarrow \Vert \mathbf{x}\Vert \le \frac{v}{{\sigma }_{\min }(\mathbf{\Gamma })},$$

(current_sota eq 6.1) — a single near-zero $s_i$ blows up $\Vert \mathbf{x}\Vert$ while $w$
may still look healthy, which is why ${\sigma }_G$ is the operative measure for the controller.

Link to the threshold cascade. Inverting the amplification bound at the highest tolerable rate
$x_{\max }$ converts ${\sigma }_G$ into a commanded-speed budget and fixes the first cascade floor:

$${\sigma }_{\mathrm{crit},1}=\frac{v}{x_{\max }}\simeq \frac{0.90}{2\pi }=0.143,{\sigma }_{\mathrm{crit},2}={\sigma }_{\mathrm{crit},1}^2,{\sigma }_{\mathrm{crit},3}={\lambda }_J,$$

(current_sota eqs 6.2, §6.1) — the manipulability scalar ${\sigma }_G$ is the input that schedules the
entire conditioning stack (Tikhonov $\mathbf{\Gamma }$ regularization, impedance derate,
three-tier damped ${\mathbf{J}}^{\oplus }$ inverse, hold-last freeze); see the
singularity threshold cascade.

Source Support

• ott2008cartesian — defines the Yoshikawa measure $w=\sqrt{\det (\mathbf{J}{\mathbf{J}}^{\top })}$ (eq 3.22) and uses it as a singularity-avoidance potential $V_m$ (eqs 3.23, 3.24); restricts analysis to the region ${\sigma }_{\min }(\mathbf{J})>{\sigma }_0>0$ (fixed-base, neither free-flying nor free-floating).
• chiaverini1997singularity — argues for the minimum singular value as the singularity indicator and bases variable damping on it (fixed-base, redundant); the rationale this project adopts in choosing ${\sigma }_G$ over the volume measure.

Related Topics

• singularity_robust_inverse — consumes ${\sigma }_G$ as its scheduling scalar across the undamped / damped / hold-last tiers.
• dynamic_singularity — the configurations ${\sigma }_G$ (via ${\mathbf{J}}_{{\nu }_e}^{\oplus }$) detects in the free-flying regime.
• kinematic_singularity — the inertia-independent rank loss that the classical $w=\sqrt{\det (\mathbf{J}{\mathbf{J}}^{\top })}$ was designed to measure.
• damped_least_squares — the variable-damping rule keyed to the minimum-singular-value measure rather than the volume product.
• generalized_jacobian — the base-coupled Jacobian whose conditioning the measure must be evaluated on for a space manipulator; Yoshikawa’s bare $\mathbf{J}$ does not apply.

Open Questions

• The Spearman $\ge 0.9969$ equivalence between ${\sigma }_G$ and ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ that justifies monitoring $\mathbf{\Gamma }$ is empirical, not proven; whether it holds across the full inertia-parameter envelope is open.
• Whether a Yoshikawa-style volume potential ($V_m$ built on $\sqrt{\det ({\mathbf{J}}_{{\nu }_e}^{\oplus }{\mathbf{J}}_{{\nu }_e}^{\oplus \top })}$) would add any avoidance benefit over ${\sigma }_{\min }$-based scheduling, or merely re-introduce the spurious-local-minimum failure Ott notes (Ott p. 40).
• No analytic relation is recorded between $w$ (volume) and ${\sigma }_{\min }$ for this system; they coincide only when all but one singular value are well-conditioned.