An Analytic Solution to the 3D CSC Dubins Path Problem

TL;DR

This work derives an analytic, resultant-based solution to the 3D CSC Dubins path problem by modeling the path as a 5-DOF RRPRR manipulator and performing a careful sequence of eliminations to produce a univariate polynomial in a tangent-half-angle variable. The approach yields all candidate CSC paths, including configurations with up to seven valid solutions, and demonstrates fast computation (on the order of milliseconds per configuration) across a large configuration space. Key innovations include a strategic left-right term arrangement in the matrix model, suppression of the $\theta_4$ variable to avoid degeneracies, and a robust treatment of planar special cases that avoids combinatorial explosion. The results enable exact enumeration of CSC paths for 3D Dubins planning and provide a foundation for selecting shortest or constraint-satisfying paths in practical robotics and drone applications, with future work extending to CCC and helicoidal paths and incorporating joint limits or self-collision considerations.

Abstract

We present an analytic solution to the 3D Dubins path problem for paths composed of an initial circular arc, a straight component, and a final circular arc. These are commonly called CSC paths. By modeling the start and goal configurations of the path as the base frame and final frame of an RRPRR manipulator, we treat this as an inverse kinematics problem. The kinematic features of the 3D Dubins path are built into the constraints of our manipulator model. Furthermore, we show that the number of solutions is not constant, with up to seven valid CSC path solutions even in non-singular regions. An implementation of solution is available at https://github.com/aabecker/dubins3D.

Figures (8)

• Figure 1: 2D slices of the 5-DOF configuration space, showing that the number of valid solutions to the CSC Dubins path ranges from 2 to 7 for the configurations shown. See video overview at https://youtu.be/aWfmgsal0JU.
• Figure 2: Figure showing our RRPRR analogue of the 3D CSC Dubins path. The left shows the RRPRR manipulator that we solve the inverse kinematics for, the right shows the equivalent 3D CSC Dubins path.
• Figure 3: An example of configurations with infinite valid CSC Dubins paths shown with representative paths. The goal orientation $\textbf{v}_g$ and the goal position $\textbf{x}_g$ are colinear with $\textbf{v}_0$ and $\textbf{x}_0$. On the left, $\textbf{v}_g = [0,0,-1]$ and on the right, $\textbf{v}_g = [0,0,1]$.
• Figure 4: Planar paths in 3D occur when $\textbf{x}_g \times \textbf{v}_g$ is only in the $x_0y_0$ plane. The four valid CSC paths in this figure are all on the light blue plane defined by $\textbf{v}_0$, $\textbf{v}_g$, and $\textbf{x}_g$. The blue circles are orthogonal to this plane and show the reachable orientations of the $\psi_1$ and $\psi_2$ arcs.
• Figure 5: Number of valid CSC paths for one million trials, with $\textbf{x}_g$ uniformly sampled in $[-4,4]^3$ and $\textbf{v}_g$ uniformly sampled on the unit sphere.