Interval Arithmetic

Definition

Interval Arithmetic (IA) is a model for self-validated numerical analysis (Moore 1966): each scalar
is replaced by an interval $[a,b]=\{x\in \mathbb{R}\mid a\le x\le b\}$ enclosing its possible values, and
arithmetic operations are redefined so that an evaluated expression always returns an interval containing
all true results (the correctness property). Applied over an interval box in configuration space, IA
yields guaranteed bounds rather than sampled point estimates — the property calzolari2020singularity
exploits to certify that a parametric matrix (the generalized Jacobian) stays nonsingular across a whole
region, not merely at tested points. The pure-IA evaluation is fast but conservative (the “wrapping effect”:
repeated variables and inter-element dependencies inflate the enclosure); the paper therefore pairs it with
3rd-order Taylor models (polynomial + interval remainder) to tighten the bound. Regime: the cited
application is to a free-floating space manipulator (uncontrolled base, generalized Jacobian ${\mathbf{J}}_m^{\ast }$);
IA itself is regime-agnostic — it is a numerical certification tool, not a dynamics model.

Key Equations

Symbols per notation.md.
The interval superscript $(\cdot {)}^I$, the $mid(\cdot )/rad(\cdot )$ (midpoint / radius) operators, and the
spectral radius $\rho (\cdot )$ are source-faithful (Calzolari 2020) and are not in notation.md.
${\mathbf{J}}_m^{\ast }$ is the free-floating generalized Jacobian (notation.md’s ${\mathbf{J}}^{\ast }$, Papadopoulos).

Interval of a scalar:
$[a,b]=\{x\in \mathbb{R}\mid a\le x\le b\}.$

Interval evaluation of the generalized Jacobian over a configuration box ${\mathbf{q}}^I$ (Calzolari Eq. 14):
${\mathbf{J}}_m^{\ast I}={\mathbf{J}}_m^I({\mathbf{q}}^I)-{\mathbf{J}}_b^I({\mathbf{q}}^I)[{\mathbf{M}}_b^I({\mathbf{q}}^I){]}^{-1}{\mathbf{M}}_{bm}^I({\mathbf{q}}^I).$

Strong-regularity (nonsingularity) sufficient condition over the box — Rex & Rohn, with $mid({\mathbf{A}}^I)$ nonsingular (Calzolari Eq. 15):
$\rho \left(|mid({\mathbf{A}}^I{)}^{-1}|rad({\mathbf{A}}^I)\right)<1\Rightarrow {\mathbf{A}}^I\text{regular (singularity-free)}.$

Source Support

• calzolari2020singularity — primary and sole source: introduces IA (Eq. 14 for ${\mathbf{J}}_m^{\ast I}$, Eq. 15 regularity test), contrasts it with 3rd-order Taylor models (tighter, slower), and embeds both in a branch-and-bound search that returns singularity maps (S-Maps) of a 6-DOF free-floating space robot; computations use the CORA MATLAB toolbox.

Related Topics

• singularity_robust_inverse — IA certifies where the (generalized) Jacobian is invertible; the singularity-robust inverse handles the configurations where it is not. Complementary: offline guarantee vs. online conditioning.
• manipulability_measure — the paper’s alternative to formal certification is locally maximizing manipulability at viapoints; IA instead gives a hard pass/fail over a region rather than a scalar nearness score.
• motion_planning — S-Maps feed trajectory planning by providing the guaranteed distance to the nearest singularity set, supporting singularity-avoidant path feasibility in an NLP context.
• damped_least_squares — the runtime mitigation our system uses near rank loss; IA is the offline, configuration-space counterpart that maps where damping is needed.
• ffsm_dynamics — IA is applied to the manipulator’s generalized Jacobian / coupling inertia (${\mathbf{M}}_{bm}$, ${\mathbf{M}}_{em}$), the dynamic objects whose rank loss defines a dynamic_singularity.
• See also generalized_jacobian — the matrix ${\mathbf{J}}_m^{\ast }$ whose regularity IA certifies.

Open Questions

• The cited application certifies the free-floating generalized Jacobian ${\mathbf{J}}_m^{\ast }$. For our free-flying (fully-actuated base) system the relevant map is $\mathbf{\Gamma }$ / the circumcentroidal Jacobian ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ — does the same IA + Taylor-model regularity machinery apply to ${\sigma }_{\min }(\mathbf{\Gamma })$, and is it less conservative since base reaction is not folded in?
• IA’s wrapping effect made pure IA prove regularity only for “tiny boxes,” forcing Taylor models for efficiency. What box granularity / precision $ϵ$ would make S-Maps tractable for our redundant (7+ DOF) arm, where the configuration space is higher-dimensional?
• Could the guaranteed distance-to-singularity from an S-Map serve as a precomputed, certified constraint in a risk-aware planner, replacing online ${\sigma }_{\min }$ thresholding?