Robust collision-free guidance and control for fully actuated multirotor aerial vehicles

Abstract

This paper is concerned with the robust guidance and control of fully actuated multirotor aerial vehicles in the presence of moving obstacles, linear velocity constraints, and matched model uncertainties and disturbances. We address this problem by adopting a hierarchical flight control architecture consisting of a supervisory outer-loop guidance module and an inner-loop stabilizing control one. The position and attitude control laws are designed using a proportional–derivative approach combined with a high-order sliding mode disturbance observer. The resulting inner-loop control strategy is arbitrarily smooth and robust (in the sliding mode sense) with respect to model disturbances and uncertainties. On the other hand, we propose a robust collision-free guidance strategy that extends the continuous-control-obstacles method to drive the vehicle to a target pose under velocity constraints, disturbances, and uncertainties, in an environment containing moving obstacles. The overall method has been numerically evaluated and shown to be effective in providing satisfactory tracking performance, collision-free guidance, satisfaction of linear velocity constraints, and computational viability. Furthermore, it is shown to outperform an analogous scheme based on the original continuous-control-obstacles method and conventional sliding mode inner-loop control laws.

Proof of Lemma 3

The solution of the position reference filter differential equation (32) is given by

$$\begin{aligned} {{\textbf {y}}}(t) = e^{{\textbf {A}}\delta t}{{\textbf {y}}}(t_0) + \int _{t_0}^{t}e^{{\textbf {A}}(t-\tau )}{} {\textbf {B}}{{{\textbf {v}}}}_r^* \text {d} \tau . \end{aligned}$$

(49)

where $\delta t\triangleq t - t_0$.

The integral of (49) cannot be directly calculated since matrix ${\textbf {A}}$ is singular. To analytically calculate (49), we consider ${{{\textbf {v}}}}_r^*$ as a constant input and define an augmented vector ${\textbf {w}} \triangleq ({\textbf {y}}, {{{\textbf {v}}}}_r^*)\in {\mathbb {R}}^{(3p+6)}$. Then, using (32) we can write the following dynamic model

$$\begin{aligned} \dot{{\textbf {w}}} = \bar{{\textbf {A}}}{{\textbf {w}}}, \end{aligned}$$

(50)

where

$$\begin{aligned} \bar{{\textbf {A}}} \triangleq \left[ \begin{array}{cc} {\textbf {A}}&{} {\textbf {B}} \\ {\textbf {0}}_{3\times (3p+3)} &{} {\textbf {0}}_{3\times 3} \end{array} \right] \in {\mathbb {R}}^{(3p+6) \times (3p+6)}. \end{aligned}$$

The solution of (50) is easily calculated and is given by ${\textbf {w}}(t) = e^{\bar{{\textbf {A}}}(\delta t)}{} {\textbf {w}}(0)$. Using the power series definition of matrix exponential, $e^{\bar{{\textbf {A}}}(\delta t)}$ can be calculated by

$$\begin{aligned} e^{\bar{{\textbf {A}}}(\delta t)} = \left[ \begin{array}{cc} e^{{\textbf {A}} \delta t} &{} \int _{t_0}^{t}e^{{\textbf {A}}(t-\tau )}{} {\textbf {B}} \text {d} \tau \\ {\textbf {0}}_{3\times (3p+3)} &{} {\textbf {I}}_{3} \end{array}\right] . \end{aligned}$$

(51)

Then, using (51), equation (49) can be rewritten as ${{\textbf {y}}}(t) = e^{{\textbf {A}}\delta t}{{\textbf {y}}}(t_0) + {\textbf {G}}(\delta t) {{{\textbf {v}}}}_r^*,$ where

$$\begin{aligned} {\textbf {G}}(\delta t) \triangleq \left[ {\textbf {I}}_{(3p+3)}, {\textbf {0}}_{(3p+3) \times 3}\right] e^{\bar{{\textbf {A}}}(\delta t)} \left[ \begin{array}{c} {\textbf {0}}_{(3p+3) \times 3} \\ {\textbf {I}}_3 \end{array} \right] , \end{aligned}$$

thus completing the proof. $\square$