Almost Global Trajectory Tracking for Quadrotors Using Thrust Direction Control on $\mathcal{S}^2$

TL;DR

The paper tackles almost global trajectory tracking for quadrotors with nonlinear manifold dynamics by moving beyond cascade ISS analyses. It develops a backstepping-like control design that extends a translational double-integrator controller to attitude control on the unit sphere $\mathcal{S}^2$, using a composite Lyapunov function $V = V_\xi(x_1,x_2) + \frac{1 - \zeta^T x_3}{2 k_2(1 + \zeta^T x_3)}$ to certify stability. The main contribution is a globally robust controller with an explicit region of attraction $\mathcal{D}$ (excluding the singular attitude $x_3 = -\zeta$) and demonstrated improvements in transient and steady-state performance through both simulations and real-world experiments. This approach offers a practical, geometrically grounded trajectory-tracking strategy that leverages thrust-direction control on $\mathcal{S}^2$ to achieve almost global stability. The results indicate significant potential for robust, aggressive quadrotor maneuvers with simpler tuning compared to prior global solutions.

Abstract

Many of the existing works on quadrotor control address the trajectory tracking problem by employing a cascade design in which the translational and rotational dynamics are stabilized by two separate controllers. The stability of the cascade is often proved by employing trajectory-based arguments, most notably, integral input-to-state stability. In this paper, we follow a different route and present a control law ensuring that a composite function constructed from the translational and rotational tracking errors is a Lyapunov function for the closed-loop cascade. In particular, starting from a generic control law for the double integrator, we develop a suitable attitude control extension, by leveraging a backstepping-like procedure. Using this construction, we provide an almost global stability certificate. The proposed design employs the unit sphere $\mathcal{S}^2$ to describe the rotational degrees of freedom required for position control. This enables a simpler controller tuning and an improved tracking performance with respect to previous global solutions. The new design is demonstrated via numerical simulations and on real-world experiments.

Figures (10)

• Figure 1: Illustration of the proposed approach on a trajectory tracking maneuver in which the quadrtotor starts from an inverted flight configuration. The capability to perform such type of maneuvers is achieved by controlling the thrust direction vector directly on its configuration manifold $\mathcal{S}^2$.
• Figure 2: Position trajectories followed by the quadrotor over 100 simulation runs (thin gray lines), and reference position trajectory $p_r$ (red line). The initial conditions are marked with a circle.
• Figure 3: Initial transient of the position errors $x_1=[x_{1,x}\,x_{1,y}\,x_{1,z}]^T$ and of the trust direction error $\eta$, corresponding to the trajectories in Fig. \ref{['dronetraj']}.
• Figure 4: Initial transient of the position errors $x_1=[x_{1,x}\,x_{1,y}\,x_{1,z}]^T$ and of the trust direction error $\eta$ for the proposed control law (black solid line) and the control law in kooijman2019trajectory (blue dash-dotted line).
• Figure 5: Control inputs $f$, $\omega_x$ and $\omega_y$ corresponding to the trajectories in Fig. \ref{['trackbsl']}; $\omega_x$ and $\omega_y$ denote the first two entries of $\omega$.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Theorem 1
• proof
• Remark 3
• Corollary 1
• proof