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A New Approach to Motion Planning in 3D for a Dubins Vehicle: Special Case on a Sphere

Deepak Prakash Kumar, Swaroop Darbha, Satyanarayana Gupta Manyam, David Casbeer

TL;DR

The work tackles 3D motion planning for curvature-constrained (Dubins-like) vehicles by solving the sphere-constrained problem on a unit sphere and connecting configurations under pitch and yaw-rate bounds. Using a Sabban-frame representation and PMP on the Lie group $SO(3)$, it distinguishes abnormal ($\lambda=0$) and normal ($\lambda=1$) control regimes and derives a phase-portrait-based characterization of candidate optimal paths, revealing precise path-type families such as $CGC$, $CCC$, $CCCC$, and $CC_\pi C$, with generalizations up to $r\le\frac{\sqrt{3}}{2}$. The main contributions include a complete set of candidate normal/abnormal paths for $r$ in $(\tfrac{1}{2},\tfrac{\sqrt{3}}{2}]$, bounds on the number of $C$-segments (e.g., at most two for $r\le\tfrac{1}{\sqrt{2}}$ and at most three for $r\le\tfrac{\sqrt{3}}{2}$), and analytic construction of all candidates alongside open-source code. The results demonstrate how the turning-radius parameter influences optimal path types, with practical implications for 3D path planning of high-speed aerial vehicles over spherical surfaces, and the paper provides a public repository for path construction and numerical verification.

Abstract

In this article, a new approach for 3D motion planning, applicable to aerial vehicles, is proposed to connect an initial and final configuration subject to pitch rate and yaw rate constraints. The motion planning problem for a curvature-constrained vehicle over the surface of a sphere is identified as an intermediary problem to be solved, and it is the focus of this paper. In this article, the optimal path candidates for a vehicle with a minimum turning radius $r$ moving over a unit sphere are derived using a phase portrait approach. We show that the optimal path is $CGC$ or concatenations of $C$ segments through simple proofs, where $C = L, R$ denotes a turn of radius $r$ and $G$ denotes a great circular arc. We generalize the previous result of optimal paths being $CGC$ and $CCC$ paths for $r \in \left(0, \frac{1}{2} \right]\bigcup\{\frac{1}{\sqrt{2}}\}$ to $r \leq \frac{\sqrt{3}}{2}$ to account for vehicles with a larger $r$. We show that the optimal path is $CGC, CCCC,$ for $r \leq \frac{1}{\sqrt{2}},$ and $CGC, CC_πC, CCCCC$ for $r \leq \frac{\sqrt{3}}{2}.$ Additionally, we analytically construct all candidate paths and provide the code in a publicly accessible repository.

A New Approach to Motion Planning in 3D for a Dubins Vehicle: Special Case on a Sphere

TL;DR

The work tackles 3D motion planning for curvature-constrained (Dubins-like) vehicles by solving the sphere-constrained problem on a unit sphere and connecting configurations under pitch and yaw-rate bounds. Using a Sabban-frame representation and PMP on the Lie group , it distinguishes abnormal () and normal () control regimes and derives a phase-portrait-based characterization of candidate optimal paths, revealing precise path-type families such as , , , and , with generalizations up to . The main contributions include a complete set of candidate normal/abnormal paths for in , bounds on the number of -segments (e.g., at most two for and at most three for ), and analytic construction of all candidates alongside open-source code. The results demonstrate how the turning-radius parameter influences optimal path types, with practical implications for 3D path planning of high-speed aerial vehicles over spherical surfaces, and the paper provides a public repository for path construction and numerical verification.

Abstract

In this article, a new approach for 3D motion planning, applicable to aerial vehicles, is proposed to connect an initial and final configuration subject to pitch rate and yaw rate constraints. The motion planning problem for a curvature-constrained vehicle over the surface of a sphere is identified as an intermediary problem to be solved, and it is the focus of this paper. In this article, the optimal path candidates for a vehicle with a minimum turning radius moving over a unit sphere are derived using a phase portrait approach. We show that the optimal path is or concatenations of segments through simple proofs, where denotes a turn of radius and denotes a great circular arc. We generalize the previous result of optimal paths being and paths for to to account for vehicles with a larger . We show that the optimal path is for and for Additionally, we analytically construct all candidate paths and provide the code in a publicly accessible repository.

Paper Structure

This paper contains 14 sections, 11 theorems, 106 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Planning in 3D
  3. Proposed model and connection to motion planning on sphere
  4. Planning on a sphere
  5. Problem Formulation and Hamiltonian Construction
  6. Characterization of Optimal Path
  7. Case 1: $\lambda = 0$
  8. Case 2.1: $\lambda = 1,$ $\lambda_{H_{12}} < \frac{U_{max}}{1 + U_{max}^2}$
  9. Case 2.2: $\lambda = 1,$ $\lambda_{H_{12}} = \frac{U_{max}}{1 + U_{max}^2}$
  10. Case 2.3: $\lambda = 1,$ $\lambda_{H_{12}} > \frac{U_{max}}{1 + U_{max}^2}$
  11. Path Construction and Results
  12. Conclusion
  13. Computations for non-optimality of $RLRLR$ path for $r \leq \frac{1}{\sqrt{2}}$ in Lemma \ref{['lemma: non-optimality_CCCCC']}
  14. Computations for non-optimality of $RLRLRL$ path for $r \leq \frac{\sqrt{3}}{2}$ in Lemma \ref{['lemma: claim2']}

Key Result

Lemma 1

If $H_{12} \equiv 0,$ then $\lambda$ cannot be zero; further, for $\lambda = 1,$$\kappa \equiv 0.$

Figures (15)

  • Figure 1: Depiction of osculating spheres at initial and final configuration corresponding to the pitch rate constraints
  • Figure 2: Visualization of segments
  • Figure 3: Initial and final configuration on sphere
  • Figure 4: Turns on a sphere generalization
  • Figure 5: Overview of cases and results. (In this figure, whenever we refer to the optimal path being a particular type, the optimal path can be a degenerate path of the same as well. For example, for $\lambda = 1$ and $r \leq \frac{1}{\sqrt{2}},$ the optimal path can be a degenerate path of $CC_{\pi + \beta} C_{\pi + \beta} C,$ which includes $C, CC,$$CC_{\pi + \beta}C$.)
  • ...and 10 more figures

Theorems & Definitions (36)

  • Lemma 1
  • Remark
  • Remark
  • Proposition 2
  • proof
  • Lemma 3
  • proof
  • Remark
  • Remark
  • Lemma 4
  • ...and 26 more