Curvature-Constrained Vector Field for Motion Planning of Nonholonomic Robots

TL;DR

The paper tackles curvature-constrained motion planning for nonholonomic robots by co-designing a curvature-bounded vector field (CVF) with saturated control laws that include a state-dependent dynamic gain. The CVF blends vortex, source, and sink flows to create a stable limit cycle as the target positive limit set, guaranteeing bounded integral-curve curvature and alignment with the desired heading at the target. The authors prove orientation stabilization and almost global convergence under curvature constraints, and validate the approach through extensive simulations and hardware experiments on an Ackermann UGV and a semi-physical fixed-wing UAV. The results highlight improved curvature adherence and robustness compared to existing VF-based methods, with practical real-time implementation demonstrated on real platforms.

Abstract

Vector fields are advantageous in handling nonholonomic motion planning as they provide reference orientation for robots. However, additionally incorporating curvature constraints becomes challenging, due to the interconnection between the design of the curvature-bounded vector field and the tracking controller under underactuation. In this paper, we present a novel framework to co-develop the vector field and the control laws, guiding the nonholonomic robot to the target configuration with curvature-bounded trajectory. First, we formulate the problem by introducing the target positive limit set, which allows the robot to converge to or pass through the target configuration, depending on different dynamics and tasks. Next, we construct a curvature-constrained vector field (CVF) via blending and distributing basic flow fields in workspace and propose the saturated control laws with a dynamic gain, under which the tracking error's magnitude decreases even when saturation occurs. Under the control laws, kinematically constrained nonholonomic robots are guaranteed to track the reference CVF and converge to the target positive limit set with bounded trajectory curvature. Numerical simulations show that the proposed CVF method outperforms other vector-field-based algorithms. Experiments on Ackermann UGVs and semi-physical fixed-wing UAVs demonstrate that the method can be effectively implemented in real-world scenarios.

Figures (16)

• Figure 1: Diagram of the co-design methodology of vector field and control laws for motion planning presented in this paper.
• Figure 2: Three types of elementary flow with singular point at the origin.
• Figure 3: The blended vector field $\mathbf{F}$, where the domains of source, vortex and sink are shaded in red, blue and green. Within the transition region between red and green dotted circles, the blended vector field evolves from source to vortex and then to sink as the distance from the singular point increases.
• Figure 4: The proposed CVF $\mathbf{T}$, where the singular point is the origin.
• Figure 5: Simulation results that verify Theorem \ref{['thm dynamic gain']}. The subfigures \ref{['sub@fig:Lemma2_Exp1']}-\ref{['sub@fig:Lemma2_Exp3']} present the results for Exp 1-3 in the first set. The time intervals during which the robot is moving in the saturation region $\mathcal{S}$ are highlighted with magenta shaded area. The entire simulation results are on the left and the shaded areas are magnified on the right. The results of the Exp 4-7 in the second set are shown by subfigures \ref{['fig:Lemma2_Exp4']}-\ref{['sub@fig:Lemma2_Exp7']}.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Remark 1
• Definition 3
• Definition 4
• Definition 5
• Lemma 1
• Theorem 1
• Remark 2
• Theorem 2