Vector fields are advantageous in handling non holonomic motion planning, as they provide the robot with reference orientation across the workspace. However, additionally incorporating curvature constraints presents challenges due to the interconnection between the design of the curvature-bounded vector field and the tracking controller under limited actuation. In this paper, we present a novel framework to co-develop the vector field and the control law, 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 either converge to or pass through the target configuration, depending on its dynamics and the specific tasks. Next, we construct a curvature-constrained vector field (CVF) via blending and embedding elementary flows in the workspace. To track such CVF, a saturated control law with dynamic gains is proposed, under which the tracking error's magnitude decreases even when saturation occurs. Under the control law, the kinematically constrained nonholonomic robot is 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.

This article investigates the roto-translation invariant (RTI) formation of multiple underactuated planar rigid bodies, which are established under the framework of matrix Lie groups. The main contribution is that we define the RTI and pseudo RTI (P-RTI) formation of planar rigid bodies. Different from the common formation given in the earth-fixed frame, the RTI formation is defined in the body-fixed frame so that it possesses a rigid-body motion obtained by composing rotation and translation simultaneously. Moreover, regarding fully actuated planar rigid bodies, we propose the velocity and force requirements to maintain the RTI formation, which are derived based on the kinematic and dynamic model, respectively. Another contribution of this article is that the RTI formation feasibility is investigated for underactuated planar rigid bodies subject to nonholonomic constraints on velocities and accelerations. To be more specific, we study the occasions when wheeled mobile robots and underactuated surface vessels can maintain the RTI or P-RTI formation. Finally, the results of the simulation and experiment are presented so as to exhibit the RTI and P-RTI formation intuitively.