Stability Preserving Safe Control of a Bicopter

TL;DR

This work tackles the challenge of maintaining safety constraints for a multicopter while preserving stability by transforming a constrained problem into an unconstrained one through an atanh-based state transformation. A backstepping control design with a log-cosh Lyapunov function is developed, yielding an explicit control law that enforces a forward-invariant safety set without solving constrained optimization online. The bicopter in a vertical plane serves as the testbed, with rigorous stability analysis based on Barbashin–Krasovskii–LaSalle confirming asymptotic convergence to a desired waypoint alongside safety guarantees. Numerical simulations demonstrate that position and velocity remain within predefined bounds while tracking a polytopic trajectory, underscoring the practical applicability for real-time safety-critical deployment and enabling future extensions to asymmetric bounds and adaptation.

Abstract

This paper presents a control law for stabilization and trajectory tracking of a multicopter subject to safety constraints. The proposed approach guarantees forward invariance of a prescribed safety set while ensuring smooth tracking performance. Unlike conventional control barrier function methods, the constrained control problem is transformed into an unconstrained one using state-dependent mappings together with carefully constructed Lyapunov functions. This approach enables explicit synthesis of the control law, instead of requiring a solution of constrained optimization at each step. The transformation also enables the controller to enforce safety without sacrificing stability or performance. Simulation results for a polytopic reference trajectory confined within a designated safe region demonstrate the effectiveness of the proposed method.

Figures (7)

• Figure 1: Bicopter configuration considered in this paper. The bicopter is constrained to the $\hat{\imath} _{\rm A}- \hat{\jmath} _{\rm A}$ plane and rotates about the $\hat{k}_{\rm A}$ (out of the page) axis of the inertial frame $\rm F_A.$$\rm F_B$ is fixed to the bicopter such that the forces $\overset{\rightharpoonup}{f}_1$ and $\overset{\rightharpoonup}{f}_2$ are along $\hat{\jmath}_{\rm B}.$ Note that $\overset{\rightharpoonup}{r}_{{\rm c}/w} = r_1 \hat{\imath}_{\rm A} + r_2 \hat{\jmath}_{\rm A}.$
• Figure 2: Nonlinear map $\mathop{\mathrm{atanh}}\nolimits{x}$ used to map the constraint set to an unconstrained set.
• Figure 3: Block diagram illustrating the MCBC controller implementation.
• Figure 4: Constrained trajectory tracking response. Closed-loop position tracking response of the bicopter. The desired position is shown in dashed black, the position response is shown in solid blue, and the safe set is shown in yellow. Note that the position response is always contained within the desired safe set for positions ${\mathcal{S}}.$
• Figure 5: Constrained trajectory tracking response. Closed-loop velocity response of the bicopter. The velocity response is shown in solid blue, and the safe set is shown in pink. Note that the velocity response is always contained within the desired safe set for velocities ${\mathcal{S}}.$

Theorems & Definitions (14)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Proposition III.1
• proof
• Proposition III.2
• proof
• Proposition III.3
• proof