Autonomous quadrotor collision avoidance and destination seeking in a GPS-denied environment

Abstract

We present a new integrated guidance and control method for autonomous collision avoidance and navigation in an unmapped GPS-denied environment that contains unknown obstacles. The algorithm is implemented on an experimental custom quadrotor that uses onboard vision sensing (i.e., an Intel RealSense R200) to detect the positions of obstacles. We demonstrate autonomous collision avoidance and destination seeking in experiments, where the quadrotor navigates unknown GPS-denied environments. All feedback measurements are obtained from onboard sensors. The new guidance and control algorithm uses a nonlinear inner-loop attitude controller; a nonlinear middle-loop velocity controller; and an ellipsoidal-potential-field outer-loop guidance algorithm for collision avoidance and destination seeking. The main analytic result regarding the inner-loop control shows that every quadrotor attitude with pitch between $\pm 90^{\circ }$ is a locally exponentially stable equilibrium of the closed-loop attitude dynamics, and we quantify the region of attraction for each attitude equilibrium.

Appendix A. Proof of Theorem 1

Proof

It follows from (5) and (11) that

$$\begin{aligned} \dot{e}_\varPhi = A(e_\phi ,e_\theta ) e_\omega - K_\varPhi e_\varPhi , \end{aligned}$$

where

$$\begin{aligned} A(e_\phi ,e_\theta ) \triangleq \begin{bmatrix} 1 &{}\quad \mathrm{s}_{e_\phi +\phi _{\mathrm{e}}} \mathrm{t}_{e_\theta +\theta _{\mathrm{e}}} &{}\quad \mathrm{c}_{e_\phi +\phi _{\mathrm{e}}} \mathrm{t}_{e_\theta +\theta _{\mathrm{e}}} \\ 0 &{}\quad \mathrm{c}_{e_\phi +\phi _{\mathrm{e}}} &{}\quad -\mathrm{s}_{e_\phi +\phi _{\mathrm{e}}} \\ 0 &{}\quad \mathrm{s}_{e_\phi +\phi _{\mathrm{e}}} / \mathrm{c}_{e_\theta +\theta _{\mathrm{e}}} &{}\quad \mathrm{c}_{e_\phi +\phi _{\mathrm{e}}} / \mathrm{c}_{e_\theta +\theta _{\mathrm{e}}} \end{bmatrix}. \end{aligned}$$

Substituting (12) into (10) yields

$$\begin{aligned} \dot{e}_\omega = -K_\omega e_\omega - A^{\mathrm{T}}(e_\phi ,e_\theta ) e_\varPhi . \end{aligned}$$

Next, define $D \triangleq \{ (e_\varPhi , e_\omega ) \in \mathbb {R}^6: | e_2^{\mathrm{T}} e_\varPhi +\theta _{\mathrm{e}} | < \frac{\pi }{2} \}$, and it follows that for all $(e_\varPhi ,e_\omega ) \in D$, the derivative of V along the trajectories of (5), (6), and (10)–(12) is

$$\begin{aligned} \dot{V}(e_\varPhi ,e_\omega )&\triangleq \dfrac{\partial V}{\partial e_\varPhi } \dot{e}_\varPhi + \dfrac{\partial V}{\partial e_\omega } \dot{e}_\omega \\&= - 2(e_\varPhi ^{\mathrm{T}} K_\varPhi e_\varPhi + e_\omega ^{\mathrm{T}} K_\omega e_\omega ) \\&\le - c_1 V(e_\varPhi ,e_\omega ), \end{aligned}$$

where $c_1 \triangleq 2 \, \mathrm{min} \, \{\lambda _\mathrm{min}(K_\varPhi ),\lambda _{\mathrm{min}}(K_\omega )\}$ is positive. Thus, $(\varPhi ,\omega ) \equiv (\varPhi _{\mathrm{e}},0)$ is a locally exponentially stable equilibrium of (5), (6), and (10)–(12).

To show the final statement of the result, define $R_{\mathrm{A}}' \triangleq \{ (e_\varPhi ,e_\omega ) \in \mathbb {R}^6: V(e_\varPhi ,e_\omega ) < (\frac{\pi }{2} - | {\theta _{\mathrm{e}}} | )^2 \}$. Since $R_{\mathrm{A}}' \subseteq D$ and for all $(e_\varPhi ,e_\omega ) \in D {\setminus } \{0\}$, $\dot{V}(e_\varPhi ,e_\omega ) < 0$, it follows that $R_{\mathrm{A}}'$ is invariant with respect to (5), (6), and (10)–(12).

Let $(\varPhi (0),\omega (0)) \in R_{\mathrm{A}}$, and it follows that $(e_\varPhi (0),e_\omega (0)) \in R_{\mathrm{A}}'$. Since $R_{\mathrm{A}}'$ is invariant with respect to (5), (6), and (10)–(12), it follows that for all $t \ge 0$, $(e_\varPhi (t),e_\omega (t)) \in R_{\mathrm{A}}' \subseteq D$. Therefore,

$$\begin{aligned} 0&\le \int _{0}^{\infty } V(e_\varPhi (t),e_\omega (t)) \, \mathrm{d}t \\&\le - \dfrac{1}{c_1}\int _{0}^{\infty } \dot{V}(e_\varPhi (t),e_\omega (t)) \, \mathrm{d}t \\&= \dfrac{1}{c_1}\bigg [V(e_\varPhi (0),e_\omega (0)) - \lim _{t \rightarrow \infty } V(e_\varPhi (t),e_\omega (t))\bigg ] \\&\le \dfrac{1}{c_1}V(e_\varPhi (0),e_\omega (0)), \end{aligned}$$

which implies that $\int _{0}^{\infty } V(e_\varPhi (t),e_\omega (t)) \, \mathrm{d}t$ exists. Since, in addition, $\dot{V}(e_\varPhi (t),e_\omega (t))$ is bounded, it follows from Barbalat’s lemma that $\lim _{t \rightarrow \infty } V(e_\varPhi (t),e_\omega (t))=0$, which confirms the last statement of the result. $\square$