Reactive Robot Navigation Using Quasi-conformal Mappings and Control Barrier Functions

TL;DR

A robot control algorithm suitable for safe reactive navigation tasks in cluttered environments and showcased and analyzed via extensive simulations and experiments performed using different types of robotic systems, including manipulators and mobile robots.

Abstract

This paper presents a robot control algorithm suitable for safe reactive navigation tasks in cluttered environments. The proposed approach consists of transforming the robot workspace into the \emph{ball world}, an artificial representation where all obstacle regions are closed balls. Starting from a polyhedral representation of obstacles in the environment, obtained using exteroceptive sensor readings, a computationally efficient mapping to ball-shaped obstacles is constructed using quasi-conformal mappings and Möbius transformations. The geometry of the ball world is amenable to provably safe navigation tasks achieved via control barrier functions employed to ensure collision-free robot motions with guarantees both on safety and on the absence of deadlocks. The performance of the proposed navigation algorithm is showcased and analyzed via extensive simulations and experiments performed using different types of robotic systems, including manipulators and mobile robots.

Figures (11)

• Figure 1: The approach proposed in this paper to ensure safety of dynamical systems in the presence of multiple non-convex unsafe regions consists of mapping the state space---the polyhedral world---to a ball world, where obstacles are closed balls or the complement of open balls. In the ball world, obstacles are transformed by changing their centers and radii, by means of the inputs $\dot\rho_i^*$ and $\dot q_j^*$. This lets the mapped state $z$, and hence the real state $x$, to remain safe.
• Figure 2: Full qc mapping (using the inverse mapping to transform the states). In each panel, the left box is the polyhedral (real) world, and the right circle is the ball world. Notice the motion of the obstacles in the ball world to prevent the state of the robot mapped into the ball world from colliding with them at each point in time.
• Figure 3: Partial conformal mapping ($\lambda = 10000$). In each panel, the left box is the polyhedral (real) world, and the right circle is the ball world. Notice the motion of the obstacles in the ball world to prevent the state of the robot mapped into the ball world from colliding with them at each point in time.
• Figure 4: Partial conformal mapping for different values of $\lambda$. In each panel, the left box is the polyhedral (real) world, and the right circle is the ball world.
• Figure 5: (a) Computational complexity of Harmonic Map vlantis2018robot and the proposed Full QC for different domain approximations. Note the logarithmic scale on the time axis. (b) Mesh domain approximation with $144$ (left) and $10070$ (right) triangles. (c) Polyhedral worlds of increasing complexity considered in this comparison.