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Asymptotics of motion planning complexity for control-affine systems

Michele Motta, Dario Prandi

TL;DR

This work analyzes the asymptotics of the tubular motion-planning complexity $\mathrm{MC}_\varepsilon(\Gamma,T)$ for control-affine systems satisfying the strong Hörmander condition with corank-one distributions, focusing on step-2 along a transverse curve $\Gamma$. It develops a drift-decomposition framework $X_0 = X_0^\Delta + X_0^\Gamma$, proves a continuity result for $\mathrm{MC}_\varepsilon$, and introduces a reduced metric complexity that collapses the problem to a 1D vertical dynamics. The authors provide explicit asymptotics with computable constants in the Step-2 regime, including a detailed 1D reduction and bang-bang synthesis, and they extend the analysis to 3D with generic Martinet-type singularities, where logarithmic corrections appear at Martinet points. The results yield precise, dimension-dependent formulas for $\mathrm{MC}_\varepsilon(\Gamma,T)$ and give additive contributions from multiple Martinet points, thereby delivering sharp, practically informative estimates for motion-planning complexity in nonholonomic control-affine systems. These insights have potential implications for the design of near-optimal tracking strategies in robotics and related control applications where drift and singularities influence the complexity of motion planning.

Abstract

In this paper, we study the complexity of the approximation of nonadmissible curves for nonlinear control-affine systems satisfying the strong H{ö}rmander condition. Focusing on tubular approximation complexities, we provide asymptotic equivalences, with explicit constants, for all generic situations where the distribution, i.e., the linear part of the control system, is of co-rank one. Namely, we consider curves in step 2 distributions and any dimension. In the 3 dimensional case, we also consider the case of distributions with Martinet-type singularities that are crossed by the curve at isolated points.

Asymptotics of motion planning complexity for control-affine systems

TL;DR

This work analyzes the asymptotics of the tubular motion-planning complexity for control-affine systems satisfying the strong Hörmander condition with corank-one distributions, focusing on step-2 along a transverse curve . It develops a drift-decomposition framework , proves a continuity result for , and introduces a reduced metric complexity that collapses the problem to a 1D vertical dynamics. The authors provide explicit asymptotics with computable constants in the Step-2 regime, including a detailed 1D reduction and bang-bang synthesis, and they extend the analysis to 3D with generic Martinet-type singularities, where logarithmic corrections appear at Martinet points. The results yield precise, dimension-dependent formulas for and give additive contributions from multiple Martinet points, thereby delivering sharp, practically informative estimates for motion-planning complexity in nonholonomic control-affine systems. These insights have potential implications for the design of near-optimal tracking strategies in robotics and related control applications where drift and singularities influence the complexity of motion planning.

Abstract

In this paper, we study the complexity of the approximation of nonadmissible curves for nonlinear control-affine systems satisfying the strong H{ö}rmander condition. Focusing on tubular approximation complexities, we provide asymptotic equivalences, with explicit constants, for all generic situations where the distribution, i.e., the linear part of the control system, is of co-rank one. Namely, we consider curves in step 2 distributions and any dimension. In the 3 dimensional case, we also consider the case of distributions with Martinet-type singularities that are crossed by the curve at isolated points.

Paper Structure

This paper contains 17 sections, 24 theorems, 126 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Preliminaries and normal forms
  4. Continuity of motion complexity
  5. Decomposition of the drift and proof of \ref{['eq:main-thm2']}
  6. Normal forms
  7. Step 2 case: the reduced metric complexity
  8. Proof of Theorem \ref{['thm:reduction']}
  9. Proof of Theorem \ref{['thm:sign-of-control']}
  10. Step 2 case: proof of Theorem \ref{['thm:main']}
  11. Proofs of the auxiliary lemmata
  12. Proof of Lemma \ref{['lem:I^*']}
  13. Proof of Lemma \ref{['lem:I_*']}
  14. Distributions of generic type
  15. Case of a single Martinet point
  16. ...and 2 more sections

Key Result

Theorem 1.2

Assume that $\Delta$ and $\Gamma$ are a distribution and a curve on a smooth manifold $M$ satisfying (H1), (H2) and (H3). Let $\omega$ be a form associated with $(\Delta,\Gamma)$ and define Here, $\mathcal{V}_q$ is any neighbourhood of $q$. Then, recalling the definition of $T_\Gamma$ in eq:T-gamma, we have:

Figures (5)

  • Figure 1: Construction for the proof of Lemma \ref{['lemma:dz-pos']} in case $a_0(z_{\min})=0$.
  • Figure 2: Construction for Lemma \ref{['lem:control_pos']}.
  • Figure 3: Graphical representation of the extremals controls. If the value of $a_0/\alpha$ is above $H$, then $v=0$ and $\dot z=a_0$. When the value of $a_0/\alpha$ is below $H$, then $v=c$. As $c$ grows, the extremal trajectories go faster and faster through the region of the $z$ axis where $a_0/\alpha < H$.
  • Figure 4: Graphical explanation of why $\bar{q} \in \operatorname{int} \Omega$.
  • Figure 5: Graphical representation of the trajectory $z$ of point 1. in Claim \ref{['claim:martinet-opt-synth']}: the blue line is the trajectory $z$, the green circle is the limit cycle of $z$, the red circle represent the distribution $\Delta$, which is transverse to the limit cycle.

Theorems & Definitions (59)

  • Definition 1.1
  • Theorem 1.2: Step 2 case
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 49 more