Inverse kinematics is a fundamental technique for motion and positioning control in robotics, typically applied to end-effectors. In this paper, we extend the concept of inverse kinematics to guiding vector fields for path following in autonomous mobile robots. The desired path is defined by its implicit equation, i.e., by a collection of points belonging to one or more zero-level sets. These level sets serve as a reference to construct an error signal that drives the guiding vector field toward the desired path, enabling the robot to converge and travel along the path by following such a vector field. We start with the formal exposition on how inverse kinematics can be applied to guiding vector fields for single-integrator robots in an$m$-dimensional Euclidean space. Then, we leverage inverse kinematics to ensure that the level-set error signal behaves as a linear system, facilitating control over the robot's transient motion toward the desired path and allowing for the injection of feed-forward signals to induce precise motion behavior along the path. We then propose solutions to the theoretical and practical challenges of applying this technique to unicycles with constant speeds to follow 2D paths with precise transient control. We finish by validating the predicted theoretical results through real flights with fixed-wing drones.

The autonomous formation flight of fixed-wing drones is hard when the coordination requires the actuation over their speeds since they are critically bounded and aircraft are mostly designed to fly at a nominal airspeed. This paper proposes an algorithm to achieve formation flights of fixed-wing drones without requiring any actuation over their speed. In particular, we guide all the drones to travel over specific paths, e.g., parallel straight lines, and we superpose an oscillatory behavior onto the guiding vector field that drives the drones to the paths. This oscillation enables control over the average velocity along the path, thereby facilitating inter-drone coordination. Each drone adjusts its oscillation amplitude distributively in a closed-loop manner by communicating with neighboring agents in an undirected and connected graph. A novel consensus algorithm is introduced, leveraging a non-negative, asymmetric saturation function. This unconventional saturation is justified since negative amplitudes do not make drones travel backward or have a negative velocity along the path. Rigorous theoretical analysis of the algorithm is complemented by validation through numerical simulations and a real-world formation flight.

This paper focuses on coordinating a robot swarm orbiting a convex path without collisions among the individuals. The individual robots lack braking capabilities and can only adjust their courses while maintaining their constant but different speeds. Instead of controlling the spatial relations between the robots, our formation control algorithm aims to deploy a dense robot swarm that mimics the behavior of tornado schooling fish. To achieve this objective safely, we employ a combination of a scalable overtaking rule, a guiding vector field, and a control barrier function with an adaptive radius to facilitate smooth overtakes. The decision-making process of the robots is distributed, relying only on local information. Practical applications include defensive structures or escorting missions with the added resiliency of a swarm without a centralized command. We provide a rigorous analysis of the proposed strategy and validate its effectiveness through numerical simulations involving a high density of unicycles.