Virtual Constraint for a Quadrotor UAV Enforcing a Body-Axis Pointing Direction

TL;DR

A geometric control framework on SE(3) for quadrotors that enforces pointing-driven missions without completing a full attitude reference is proposed, yielding a constructive control law and, for relevant tasks, closed-form expressions.

Abstract

We propose a geometric control framework on $SE(3)$ for quadrotors that enforces pointing-driven missions without completing a full attitude reference. The mission is encoded through virtual constraints defining a task manifold and an associated set of admissible velocities, and invariance is achieved by a feedback law obtained from a linear system in selected inputs. Under a transversality condition with the effective actuation distribution, the invariance-enforcing input is uniquely defined, yielding a constructive control law and, for relevant tasks, closed-form expressions. We further derive a local off-manifold stabilization extension. As a case study, we lock a body axis to a prescribed line-of-sight direction while maintaining fixed altitude.

Figures (7)

• Figure 1: Illustration of the pointing--altitude task geometry. The red body axis $b_1$ is aligned with the line-of-sight to the target at the origin, while the motion evolves on the prescribed horizontal plane $z=z_0$.
• Figure 2: Pure invariance law ($k_1=k_2=k_3=0$) with on-manifold initialization and no projection: configuration-level task errors $e_{\mathrm{pt}}(t)$ and $e_z(t)$.
• Figure 3: Comparison of the altitude error $e_z(t)=|e_3^\top p(t)-z_0|$ for a structured off-manifold initialization with $\mu_3(0)\neq0$: pure invariance law ($k_3=0$) versus vertical residual damping ($k_3=5$, with $k_1=k_2=0$). Damping $\mu_3$ reduces altitude drift over the simulated interval.
• Figure 4: Comparison of the vertical residual $\mu_3(t)=e_3^\top v(t)$ for the same experiment as in Fig. \ref{['fig:vertical_ez_compare']}, shown on a logarithmic scale.
• Figure 5: Thrust input $f(t)$ for the structured vertical-residual experiment. The input remains bounded in both cases and has comparable magnitude, indicating that the reduction in altitude drift is achieved without excessive control effort.

Theorems & Definitions (7)

• Remark 1
• Proposition 1
• Theorem 1
• proof
• Proposition 2
• proof
• Remark 2