Inverse Kinematics on Guiding Vector Fields for Robot Path Following

TL;DR

This work extends inverse kinematics to guiding vector fields for autonomous path following when the path is described implicitly by level-set equations. By decomposing the GVF into a tangential component and a converging component projected via $v_C=J_\phi^T(J_\phi J_\phi^T)^{-1}u_\phi$, the error dynamics $\dot{\phi}=u_\phi$ become explicitly controllable, enabling linear-like behavior and the injection of feed-forward or time-varying targets $\gamma(t)$. The approach is developed for single-integrator robots in $\mathbb{R}^m$ and adapted to 2D unicycles and fixed-wing UAVs with constant speed, including smoothness and speed-boundedness considerations; stability follows from the chosen $u_\phi$ and full-rank Jacobians. Flight experiments with a fixed-wing UAV validate controlled convergence to circular paths and demonstrate telemetry improvements via behavior-driven IK-GVF, with open-source Paparazzi implementations provided. Overall, IK-GVF offers analytical, reactive guarantees for precise path convergence and programmable behavior without requiring heavy solvers.

Abstract

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.

Figures (4)

• Figure 1: 0.9m wingspan fixed-wing drone used during the experiments.
• Figure 2: Telemetry data of the X-Y position received in the ground station at a frequency of 1Hz from the fixed-wing drone, which tracks two different IK-GVFs to follow a circular path of radius $r = 100$ meters. On the left (red dots), the desired kinematic of the error signal follows $\dot \phi = - k_\phi \phi(t)$, while on the right (blue dots) $\dot \phi = \dot \gamma(t) - k_\phi (\phi(t) - \gamma(t))$, where $\gamma(t) = A \sin(\omega_\gamma t)$, with $A = 0.03$ and $\omega_\gamma = 1.3$ rad/s, drives the desired behavior. For both vector fields, the design parameters are $k_\phi = 0.25$ and $k_\theta = 0.8$. Transmission quality improves (resulting in more uniformly distributed data) when the controlled oscillations are induced.
• Figure 3: The fixed-wing drone in Figure \ref{['fig: sonic']}, following the IK-GVF \ref{['eq:IK_VF']}, converges to a circular path with radius $r=100$ meters. The ground speed of the drone is between 14 and 17 m/s, depending on the wind (around 2 m/s). The design parameters are $k_\phi = 0.25$ and $k_\theta = 0.6$. On the right, from top to bottom, the actual kinematic of $\phi(t)$ in comparison with the predicted one, given by $\dot \phi_p(t) = - k_\phi \phi_d(t)$, starting from $t_p = 8$ s (vertical dashed black line). Prior to this, the drone aligns with $f$ and proceeds directly toward the path. The drone tracks the designed transitory satisfactorily as predicted. Next is the control input $\omega$ commanded by the controller (as defined in \ref{['eq:omega']}), along with $\dot \theta_c = \frac{1}{v^2}f^\top E^\top \dot f$. Finally, the norms of $v_T$ and $v_C$, which oscillate due to the wind, are displayed.
• Figure 4: The fixed-wing drone in Figure \ref{['fig: sonic']} converges to a circular path of radius $r = 100$ by following a behavior-driven IK-GVF with $\gamma(t) = A \sin(\omega_\gamma t)$, where $A = 0.1$, and $\omega_\gamma = 0.55$ rad/s. The ground speed of the drone is between 14 and 17 m/s, depending on the wind (around 2m/s). The design parameters are $k_\phi = 0.25$ and $k_\theta = 0.8$. On the right, from top to bottom, the actual kinematic of $\phi(t)$ in comparison with the predicted one, given by $\dot \phi_p = \dot \gamma(t) - k_\phi (\phi(t) - \gamma(t))$, where $\gamma(t) = A \sin(\omega_\gamma t)$ drives the desired behavior. Regarding the initial condition, $\phi_p(0) = \phi(0) + \gamma(t_p)$, with $t_p = 6.55$ s. The phase between the red and blue signals have been adjusted by hand since we are inducing $\dot\gamma$ and not $\gamma$. The error between the actual and desired behavior is around 5 meters. Next is the control input $\omega$, commanded by the controller (as defined in \ref{['eq:omega']}), along with $\dot \theta_c = \frac{1}{v^2}f^\top E^\top \dot f$. Finally, the norms of $v_T$ and $v_C$, which oscillate due to the wind, are displayed.

Theorems & Definitions (9)

• Definition 1
• Proposition 1
• proof
• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof