#### 期刊菜单

Fault Cooperative Tolerant Control for Multiple Quadrotor Systems with Sensor Faults
DOI: 10.12677/JAST.2021.94014, PDF, HTML, XML, 下载: 140  浏览: 243  科研立项经费支持

Abstract: Due to the local communication among individuals in multiple quadrotor systems, once some quadrotors in the formation have sensor faults, the faulty quadrotors and the ones which can receive the information from them may deviate from the desired path such that the predesigned formation shape cannot be maintained. Thus, a distributed control scheme based on the dynamic surface control is proposed for multiple quadrotor systems with unknown senor faults to maintain a leader-follower formation. Moreover, the leader can always follow the external reference signals under the designed controllers. By doing simulation studies on Qball-X4 quadrotors, the results show the effectiveness of the proposed method.

1. 引言

Figure 1. Schematic diagram for the topology of the considered leader- follower formation

2. 基础知识

$\mathcal{G}\triangleq \left(\mathcal{V},\mathcal{E},\mathcal{A}\right)$，它由n个节点和m条有向弧组成，所有节点和有向弧分别组成节点集 $\mathcal{V}=\left\{1,2,\cdots ,n\right\}$ 和有向弧集 $\mathcal{E}=\left\{{\mathcal{E}}^{1},\cdots ,{\mathcal{E}}^{m}\right\}$。其中每条有向弧都代表着2个节点之间的有向连接，例如有向弧 $\left(i,j\right)\in \mathcal{E}$ 表明节

${\mathcal{N}}_{i}\triangleq \left\{j\in \mathcal{V}|\left(j,i\right)\in \mathcal{E}\right\}$

${R}_{i}\triangleq \left\{j\in \mathcal{V}|\left(i,j\right)\in \mathcal{E}\right\}$

${\mathcal{A}}_{ij}=\left\{\begin{array}{l}1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }若\left(j,i\right)\in \mathcal{E}\hfill \\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}否则\hfill \end{array}$

$L\triangleq D-\mathcal{A}$

3. 问题描述

1) 每个节点i ( $i\in \mathcal{V}$ )需跟随其父节点，而主节点则需跟踪参考信号；

2) 每个节点i ( $i\in \mathcal{V}$ )需与其邻集中的每个节点j ( $j\in {\mathcal{N}}_{i}$ )保持预设的相对距离，该相对距离分解到惯性坐标系后，定义为 ${d}_{ij}\triangleq {\left[{d}_{ijx},{d}_{ijy},{d}_{ijz}\right]}^{\text{T}}$，如图2所示。

Figure 2. The relative distance between node i and node j in the inertial frame

$\left\{\begin{array}{l}{\stackrel{¨}{x}}_{i}={u}_{i1}\left(\mathrm{cos}{\varphi }_{i}\mathrm{sin}{\theta }_{i}cos{\psi }_{i}+\mathrm{sin}{\varphi }_{i}\mathrm{sin}{\psi }_{i}\right)\\ {\stackrel{¨}{y}}_{i}={u}_{i1}\left(\mathrm{cos}{\varphi }_{i}\mathrm{sin}{\theta }_{i}\mathrm{sin}{\psi }_{i}-\mathrm{sin}{\varphi }_{i}\mathrm{cos}{\psi }_{i}\right)\\ {\stackrel{¨}{z}}_{i}={u}_{i1}\mathrm{cos}{\varphi }_{i}\mathrm{cos}{\theta }_{i}-g\\ {\stackrel{¨}{\theta }}_{i}={u}_{i2}+\frac{{J}_{\varphi }-{J}_{\psi }}{{J}_{\theta }}{\stackrel{˙}{\psi }}_{i}{\stackrel{˙}{\varphi }}_{i}\\ {\stackrel{¨}{\varphi }}_{i}={u}_{i3}+\frac{{J}_{\psi }-{J}_{\theta }}{{J}_{\varphi }}{\stackrel{˙}{\psi }}_{i}{\stackrel{˙}{\theta }}_{i}\\ {\stackrel{¨}{\psi }}_{i}={u}_{i4}+\frac{{J}_{\theta }-{J}_{\varphi }}{{J}_{\psi }}{\stackrel{˙}{\varphi }}_{i}{\stackrel{˙}{\theta }}_{i}\end{array}$ (1)

$\left\{\begin{array}{l}{\stackrel{˙}{w}}_{i1}={w}_{i2}\\ {\stackrel{˙}{w}}_{i2}={u}_{i1}{g}_{1}\left({w}_{i5}\right)\tau \left({w}_{i3}\right)\\ {\stackrel{˙}{w}}_{i3}={w}_{i4}\\ {\stackrel{˙}{w}}_{i4}={v}_{i1}+{f}_{i1}\end{array}$ (2)

$\left\{\begin{array}{l}{\stackrel{˙}{w}}_{i5}={w}_{i6}\\ {\stackrel{˙}{w}}_{i6}={g}_{2}\left({w}_{i3}\right){v}_{i2}+{f}_{i2}\end{array}$ (3)

${g}_{1}=\frac{1}{m}\left[\begin{array}{cc}\mathrm{sin}{\psi }_{i}& \mathrm{cos}{\psi }_{i}\\ -\mathrm{cos}{\psi }_{i}& \mathrm{sin}{\psi }_{i}\end{array}\right],\text{\hspace{0.17em}}{g}_{2}=\left[\begin{array}{cc}\frac{\mathrm{cos}{\varphi }_{i}\mathrm{cos}{\theta }_{i}}{m}& 0\\ 0& 1\end{array}\right]$

${f}_{i1}=\left[\begin{array}{c}\frac{{J}_{\varphi }-{J}_{\psi }}{{J}_{\theta }}{\stackrel{˙}{\psi }}_{i}{\stackrel{˙}{\varphi }}_{i}\\ \frac{{J}_{\psi }-{J}_{\theta }}{{J}_{\varphi }}{\stackrel{˙}{\psi }}_{i}{\stackrel{˙}{\theta }}_{i}\end{array}\right],\text{\hspace{0.17em}}{f}_{i2}=\left[\begin{array}{c}-g\\ \frac{{J}_{\theta }-{J}_{\varphi }}{{J}_{\psi }}{\stackrel{˙}{\varphi }}_{i}{\stackrel{˙}{\theta }}_{i}\end{array}\right]$

$\tau \left({w}_{i3}\right)=\left[\begin{array}{c}\mathrm{sin}{\varphi }_{i}\\ \mathrm{cos}{\varphi }_{i}\mathrm{sin}{\theta }_{i}\end{array}\right]$

$\left\{\begin{array}{l}{y}_{oi_1}={x}_{i1}\\ {y}_{oi_2}={x}_{i2}\end{array}$

$r\triangleq {\left[{x}_{r},{y}_{r},{z}_{r},{\psi }_{r}\right]}^{\text{T}}$，其余所有僚机都无法直接获知参考信号r的信息，因此如果系统中缺少协同控制，则

$\left\{\begin{array}{l}{y}_{oi_1}={x}_{i1}\\ {y}_{oi_2}={x}_{i2}+{f}_{si}\end{array}$ (4)

${b}_{i}\left(t\right)=\left\{\begin{array}{l}{b}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\stackrel{˙}{b}}_{i}\left(t\right)\equiv 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}偏移故障\hfill \\ {b}_{i}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}|{b}_{i}\left(t\right)|={\lambda }_{i}t,\text{\hspace{0.17em}}\text{\hspace{0.17em}}漂移故障\hfill \\ {b}_{i}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}|{b}_{i}\left(t\right)|<{\stackrel{¯}{b}}_{i},\text{\hspace{0.17em}}{\stackrel{˙}{b}}_{i}\left(t\right)\to 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}精度损失\hfill \end{array}$

$\left\{\begin{array}{l}{\stackrel{˙}{w}}_{i1}={w}_{i2}-{f}_{si1}\\ {\stackrel{˙}{w}}_{i2}={u}_{i1}{g}_{1}\left({w}_{i5}\right)\tau \left({w}_{i3}\right)\\ {\stackrel{˙}{w}}_{i3}={w}_{i4}-{f}_{si2}\\ {\stackrel{˙}{w}}_{i4}={v}_{i1}+{f}_{i1}+\delta {f}_{i1}\end{array}$ (5)

$\left\{\begin{array}{l}{\stackrel{˙}{w}}_{i5}={w}_{i6}-{f}_{si3}\\ {\stackrel{˙}{w}}_{i6}={g}_{2}\left({w}_{i3}\right){v}_{i2}+{f}_{i2}+\delta {f}_{i2}\end{array}$ (6)

${\left[{x}_{1},{y}_{1},{z}_{1},{\psi }_{1}\right]}^{\text{T}}\to r$${\left[{x}_{i}-{x}_{j},{y}_{i}-{y}_{j},{z}_{i}-{z}_{j}\right]}^{\text{T}}\to {d}_{ij}$${\psi }_{i}-{\psi }_{j}\to 0$，其中 $j\in {\mathcal{N}}_{i}$

4. 协同容错控制设计

${\sigma }_{i1}\triangleq {w}_{i2}-{b}_{i}{w}_{r2}+{b}_{i}\left({w}_{i1}-{w}_{r1}\right)+\underset{j\in {\mathcal{N}}_{i}}{\sum }{a}_{ij}\left({w}_{i1}-{w}_{j1}-{d}_{ijxy}\right)$ (7)

${\sigma }_{i2}\triangleq {w}_{i3}-{w}_{i3d}$ (8)

${\sigma }_{i3}\triangleq {w}_{i4}-{w}_{i4d}$ (9)

${\sigma }_{i4}\triangleq {w}_{i6}-{b}_{i}{w}_{r6}+{b}_{i}\left({w}_{i5}-{w}_{r5}\right)+\underset{j\in {\mathcal{N}}_{i}}{\sum }{a}_{ij}\left({w}_{i5}-{w}_{j5}-{h}_{ij}\right)$ (10)

${b}_{i}=\left\{\begin{array}{l}1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}即\text{\hspace{0.17em}}i\text{\hspace{0.17em}}为主机\hfill \\ 0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\in \mathcal{V}-\left\{1\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}即\text{\hspace{0.17em}}i\text{\hspace{0.17em}}为僚机\hfill \end{array}$ (11)

$\begin{array}{c}{\stackrel{˙}{w}}_{i3d}=\frac{{J}^{-1}\left({w}_{i3d}\right)}{{\epsilon }_{1}}\left\{\frac{{g}_{1}^{-1}}{{u}_{i1}}×\left[-{k}_{1}{\sigma }_{i1}-{b}_{i}\left({w}_{i2}-{\stackrel{^}{f}}_{si1}-{w}_{r2}\right)-\underset{j\in {\mathcal{N}}_{i}}{\sum }{a}_{ij}\left({w}_{i2}-{\stackrel{^}{f}}_{si1}-{w}_{j2}\right)\\ \text{\hspace{0.17em}}+{b}_{i}{\stackrel{˙}{w}}_{r2}-{\epsilon }_{1}\frac{\text{d}{u}_{i1}{g}_{1}}{\text{d}t}\tau \left({w}_{i3d}\right)\right]-\tau \left({w}_{i3d}\right)\right\}\end{array}$ (12)

${\stackrel{˙}{w}}_{i4d}=\frac{1}{{\epsilon }_{2}}\left(-{k}_{2}{\sigma }_{i3}+{\stackrel{˙}{w}}_{i3d}+{\stackrel{^}{f}}_{si2}-{w}_{i4d}\right)$ (13)

${v}_{i1},\text{\hspace{0.17em}}{v}_{i2}$ 设计成如下形式：

${v}_{i1}=-{k}_{3}{\sigma }_{i3}+{\stackrel{˙}{w}}_{i4d}-{f}_{i1}-\delta {\stackrel{^}{f}}_{i1}$ (14)

${v}_{i2}={g}_{2}^{-1}\left[-{k}_{4}{\sigma }_{i4}-{f}_{i2}-\delta {\stackrel{^}{f}}_{i2}-\underset{j\in {\mathcal{N}}_{i}}{\sum }{a}_{ij}\left({w}_{i6}-{\stackrel{^}{f}}_{si3}-{w}_{j6}\right)-{b}_{i}\left({w}_{i6}-{\stackrel{^}{f}}_{si3}-{w}_{r6}\right)+{b}_{i}{\stackrel{˙}{w}}_{r6}\right]$ (15)

${\stackrel{˙}{\stackrel{^}{f}}}_{si1}=-{k}_{1}\left[\left(\underset{j\in {\mathcal{N}}_{i}}{\sum }{a}_{ij}+{b}_{i}\right){e}_{i1}-\underset{k\in ch\left(i\right)}{\sum }{a}_{ki}{e}_{k1}\right]+{\beta }_{1}{\stackrel{˜}{f}}_{si1}$ (16)

${\stackrel{˙}{\stackrel{^}{f}}}_{si2}=-{\sigma }_{i2}+{\beta }_{2}{\stackrel{˜}{f}}_{si2}$ (17)

${\stackrel{˙}{\stackrel{^}{f}}}_{si3}=-\left(\underset{j\in {\mathcal{N}}_{i}}{\sum }{a}_{ij}+{b}_{i}\right){\sigma }_{i4}+\underset{k\in ch\left(i\right)}{\sum }{a}_{ki}{\sigma }_{k4}+{\beta }_{3}{\stackrel{˜}{f}}_{si3}$ (18)

$\delta {\stackrel{˙}{\overline{)\stackrel{^}{f}}}}_{i1}={\sigma }_{i3}+{\beta }_{4}\delta {\stackrel{˜}{f}}_{i1}$ (19)

$\delta {\stackrel{˙}{\overline{)\stackrel{^}{f}}}}_{i2}={\sigma }_{i4}+{\beta }_{5}\delta {\stackrel{˜}{f}}_{i2}$ (20)

$\begin{array}{c}{\stackrel{˙}{w}}_{j3d}=\frac{{J}^{-1}\left({w}_{j3d}\right)}{{\epsilon }_{1}}\left\{\frac{{g}_{1}^{-1}}{{u}_{j1}}×\left[-{k}_{1}{\sigma }_{j1}-{b}_{j}\left({w}_{j2}-{w}_{r2}\right)-\underset{k\in {\mathcal{N}}_{j}}{\sum }{a}_{jk}\left({w}_{j2}-{w}_{k2}\right)\\ \text{\hspace{0.17em}}+{b}_{j}{\stackrel{˙}{w}}_{r2}-{a}_{ji}{\stackrel{^}{f}}_{si1}-{\epsilon }_{1}\frac{\text{d}{u}_{j1}{g}_{1}}{\text{d}t}\tau \left({w}_{j3d}\right)\right]-\tau \left({w}_{j3d}\right)\right\}\end{array}$ (21)

${\stackrel{˙}{w}}_{j4d}=\frac{1}{{\epsilon }_{2}}\left(-{k}_{2}{\sigma }_{j3}+{\stackrel{˙}{w}}_{j3d}-{w}_{j4d}\right)$ (22)

${\stackrel{˙}{v}}_{j1}=-{k}_{3}{\sigma }_{j3}+{\stackrel{˙}{w}}_{j4d}-{f}_{j1}$ (23)

${v}_{j2}={g}_{2}^{-1}\left[-{k}_{4}{\sigma }_{j4}-{f}_{j2}-\underset{k\in {\mathcal{N}}_{j}}{\sum }{a}_{jk}\left({w}_{j6}-{w}_{j6}\right)-{a}_{ji}{\stackrel{^}{f}}_{si3}-{b}_{j}\left({w}_{j6}-{w}_{r6}\right)+{b}_{j}{\stackrel{˙}{w}}_{r6}\right]$ (24)

$\begin{array}{c}{\stackrel{˙}{w}}_{k3d}=\frac{{J}^{-1}\left({w}_{k3d}\right)}{{\epsilon }_{1}}\left\{\frac{{g}_{1}^{-1}}{{u}_{k1}}×\left[-{k}_{1}{\sigma }_{k1}-{b}_{k}\left({w}_{k2}-{w}_{r2}\right)-\underset{i\in {\mathcal{N}}_{k}}{\sum }{a}_{ki}\left({w}_{k2}-{w}_{i2}\right)\\ \text{\hspace{0.17em}}+{b}_{k}{\stackrel{˙}{w}}_{r2}-{\epsilon }_{1}\frac{\text{d}{u}_{k1}{g}_{1}}{\text{d}t}\tau \left({w}_{k3d}\right)\right]-\tau \left({w}_{k3d}\right)\right\}\end{array}$ (25)

${\stackrel{˙}{w}}_{k4d}=\frac{1}{{\epsilon }_{2}}\left(-{k}_{2}{\sigma }_{k3}+{\stackrel{˙}{w}}_{k3d}-{w}_{k4d}\right)$ (26)

${\stackrel{˙}{v}}_{k1}=-{k}_{3}{\sigma }_{k3}+{\stackrel{˙}{w}}_{k4d}-{f}_{k1}$ (27)

${v}_{k2}={g}_{2}^{-1}\left[-{k}_{4}{\sigma }_{k4}-{f}_{k2}-\underset{i\in {\mathcal{N}}_{k}}{\sum }{a}_{ki}\left({w}_{k6}-{w}_{i6}\right)-{b}_{k}\left({w}_{k6}-{w}_{r6}\right)+{b}_{k}{\stackrel{˙}{w}}_{r6}\right]$ (28)

5. 稳定性分析

${\sigma }_{1}\triangleq \left[{w}_{2}-\left(B\otimes {I}_{2}\right)\left({w}_{r2}\otimes I\right)\right]+\left[\left(L+B\right)\otimes {I}_{2}\right]\left({w}_{1}-{w}_{r1}\otimes I\right)-\left(\mathcal{A}\otimes {I}_{2}\right)×C$

${\sigma }_{2}\triangleq {w}_{3}-{w}_{3d},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{3}\triangleq {w}_{4}-{w}_{4d}$

${\sigma }_{4}\triangleq \left[{w}_{6}-\left(B\otimes {I}_{2}\right)\left({w}_{r6}\otimes I\right)\right]+\left[\left(L+B\right)\otimes {I}_{2}\right]\left({w}_{5}-{w}_{r5}\otimes I\right)-\left(\mathcal{A}\otimes {I}_{2}\right)×H$

${\sigma }_{1}$ 的导数，得到：

${\stackrel{˙}{\sigma }}_{1}=\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\tau \left({w}_{3}\right)-\left(B\otimes {I}_{2}\right)\left({\stackrel{˙}{w}}_{r2}\otimes I\right)+\left[\left(L+B\right)\otimes {I}_{2}\right]\left({w}_{2}-{f}_{s1}-{w}_{r2}\otimes I\right)$ (29)

$\tau \left({\stackrel{¯}{w}}_{3}\right)={\left[\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\right]}^{-1}\left\{-{k}_{1}{\sigma }_{1}-\left[\left(L+B\right)\otimes {I}_{2}\right]\left({w}_{2}-{\stackrel{^}{f}}_{s1}-{w}_{r2}\otimes I\right)+\left(B\otimes {I}_{2}\right)\left({\stackrel{˙}{w}}_{r2}\otimes I\right)\right\}$ (30)

$\begin{array}{c}\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\tau \left({\stackrel{¯}{w}}_{3}\right)\triangleq {\epsilon }_{1}\frac{\text{d}\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\tau \left({w}_{3d}\right)}{\text{d}t}+\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\tau \left({w}_{3d}\right)\\ =\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\left[{\epsilon }_{1}J\left({w}_{3d}\right){\stackrel{˙}{w}}_{3d}+\tau \left({w}_{3d}\right)\right]+{\epsilon }_{1}\frac{\text{d}\left({U}_{1}\otimes {I}_{2}\right)G}{\text{d}t}\tau \left({w}_{3d}\right)\end{array}$ (31)

$\begin{array}{c}{\stackrel{˙}{w}}_{3d}=\frac{{J}^{-1}\left({w}_{3d}\right)}{{\epsilon }_{1}}{\left[\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\right]}^{-1}\left\{-{k}_{1}{\sigma }_{1}-\left[\left(L+B\right)\otimes {I}_{2}\right]\left({w}_{2}-{\stackrel{^}{f}}_{s1}-{w}_{r2}\otimes I\right)\begin{array}{c}\text{ }\\ \text{ }\end{array}\\ \text{\hspace{0.17em}}+\left(B\otimes {I}_{2}\right)\left({\stackrel{˙}{w}}_{r2}\otimes I\right)-{\epsilon }_{1}\frac{\text{d}\left({U}_{1}\otimes {I}_{2}\right)G}{\text{d}t}\tau \left({w}_{3d}\right)-\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\tau \left({w}_{3d}\right)\right\}\end{array}$ (32)

${\stackrel{¯}{w}}_{4}=-{k}_{2}{\sigma }_{2}+{\stackrel{˙}{w}}_{3d}+{\stackrel{^}{f}}_{s2}$ (33)

${\stackrel{¯}{w}}_{4}\triangleq {\epsilon }_{2}{\stackrel{˙}{w}}_{4d}+{w}_{4d}$ (34)

${\stackrel{˙}{w}}_{4d}=\frac{1}{{\epsilon }_{2}}\left(-{k}_{2}{\sigma }_{2}+{\stackrel{˙}{w}}_{3d}+{\stackrel{^}{f}}_{s2}-{w}_{4d}\right)$ (35)

${v}_{1}=-{k}_{3}{\sigma }_{3}+{\stackrel{˙}{w}}_{4d}-{f}_{1}-\delta {\stackrel{^}{f}}_{1}$ (36)

${v}_{2}={G}_{2}^{-1}\left\{-{k}_{4}{\sigma }_{4}-{f}_{2}-\delta {\stackrel{^}{f}}_{2}-\left[\left(L+B\right)\otimes {I}_{2}\right]\left({w}_{6}-{\stackrel{^}{f}}_{s3}-{w}_{r6}\otimes I\right)+\left(B\otimes {I}_{2}\right)\left({\stackrel{˙}{w}}_{r6}\otimes I\right)\right\}$ (37)

${e}_{1}\triangleq \left({U}_{1}\otimes {I}_{2}\right){G}_{1}\left(\tau \left({w}_{3d}\right)-\tau \left({\stackrel{¯}{w}}_{3}\right)\right)$ (38)

${e}_{2}\triangleq {w}_{4d}-{\stackrel{¯}{w}}_{4}$ (39)

${\stackrel{˜}{f}}_{si}={f}_{si}-{\stackrel{^}{f}}_{si},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,3$ (40)

$\delta {\stackrel{˜}{f}}_{i}=\delta {f}_{i}-\delta {\stackrel{^}{f}}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2$ (41)

${\stackrel{˙}{\sigma }}_{1}={e}_{1}-{k}_{1}{\sigma }_{1}-\left[\left(L+B\right)\otimes {I}_{2}\right]{\stackrel{˜}{f}}_{s1}+\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\left[\tau \left({w}_{3}\right)-\tau \left({w}_{3d}\right)\right]$

${\stackrel{˙}{\sigma }}_{2}\text{=}{\sigma }_{3}+{e}_{2}-{k}_{2}{\sigma }_{2}-{\stackrel{˜}{f}}_{s2}$

${\stackrel{˙}{\sigma }}_{3}=-{k}_{3}{\sigma }_{3}+\delta {\stackrel{˜}{f}}_{1}$

${\stackrel{˙}{\sigma }}_{4}=-{k}_{4}{\sigma }_{4}+\delta {\stackrel{˜}{f}}_{2}-\left[\left(L+B\right)\otimes {I}_{2}\right]{\stackrel{˜}{f}}_{s3}$

$\begin{array}{c}{\stackrel{˙}{e}}_{1}=-\frac{{e}_{1}}{{\epsilon }_{1}}+{k}_{1}{e}_{1}-{k}_{1}^{2}{\sigma }_{1}+{k}_{1}\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\left[\tau \left({w}_{3}\right)-\tau \left({w}_{3d}\right)\right]\\ \text{\hspace{0.17em}}\text{ }\text{ }+\left[\left(L+B\right)\otimes {I}_{2}\right]\left[\left({U}_{1}\otimes {I}_{2}\right){G}_{1}\tau \left({w}_{3}\right)-{\stackrel{˙}{\stackrel{^}{f}}}_{s1}-{\stackrel{˙}{w}}_{r2}\otimes {I}_{2}-{k}_{1}{\stackrel{˜}{f}}_{s1}\right]\\ \text{\hspace{0.17em}}\text{ }\text{ }-\left(B\otimes {I}_{2}\right)\left({\stackrel{¨}{w}}_{r2}\otimes {I}_{2}\right)\end{array}$

${\stackrel{˙}{e}}_{2}=-\frac{{e}_{2}}{{\epsilon }_{2}}+{k}_{2}\left({\sigma }_{3}-{k}_{2}{\sigma }_{2}+{e}_{2}-{\stackrel{˜}{f}}_{s2}\right)-{\stackrel{¨}{w}}_{3d}-{\stackrel{˙}{\stackrel{^}{f}}}_{s2}$

$V=\frac{1}{2}\left(\underset{i=1}{\overset{4}{\sum }}{\sigma }_{i}^{\text{T}}{\sigma }_{i}+\underset{i=1}{\overset{2}{\sum }}{e}_{i}^{\text{T}}{e}_{i}+\underset{i=1}{\overset{2}{\sum }}{\left(\delta {\stackrel{˜}{f}}_{i}\right)}^{\text{T}}\delta {\stackrel{˜}{f}}_{i}+\underset{i=1}{\overset{3}{\sum }}{\stackrel{˜}{f}}_{si}^{\text{T}}{\stackrel{˜}{f}}_{si}\right)$

$\begin{array}{c}\stackrel{˙}{V}\le -{k}_{4}{‖{\sigma }_{4}‖}^{2}+\frac{1}{2}\left(1+{\epsilon }_{3}|1-{k}_{1}^{2}|-2{k}_{1}\right){‖{\sigma }_{1}‖}^{2}\text{+}\frac{1}{2}\left(1-2{k}_{2}+{\epsilon }_{4}|1-{k}_{2}^{2}|\right){‖{\sigma }_{2}‖}^{2}\\ \text{\hspace{0.17em}}\text{ }\text{ }+\frac{1}{2}\left(-2{k}_{3}+{k}_{2}+1\right){‖{\sigma }_{3}‖}^{2}+\left(-\frac{1}{{\epsilon }_{1}}+{k}_{1}+\frac{|1-{k}_{1}^{2}|}{2{\epsilon }_{3}}+\frac{{k}_{1}+1}{2}\right){‖{e}_{1}‖}^{2}\\ \text{\hspace{0.17em}}\text{ }\text{ }+\left(-\frac{1}{{\epsilon }_{2}}+\frac{|1-{k}_{2}^{2}|}{2{\epsilon }_{4}}+\frac{3{k}_{2}+1}{2}\right){‖{e}_{2}‖}^{2}+\frac{{k}_{2}+1}{2}{a}_{1}^{2}+\frac{1}{2}{a}_{2}^{2}+\frac{1}{2}{a}_{3}^{2}-\underset{i=1}{\overset{3}{\sum }}{\beta }_{i}{‖{f}_{si}‖}^{2}\\ \text{\hspace{0.17em}}\text{ }\text{ }-{\sigma }_{1}^{\text{T}}\left[\left(L+B\right)\otimes {I}_{2}\right]{\stackrel{˜}{f}}_{s1}-{e}_{1}^{\text{T}}\left[\left(L+B\right)\otimes {I}_{2}\right]{\stackrel{˙}{\stackrel{^}{f}}}_{s1}+{\stackrel{˜}{f}}_{s1}^{\text{T}}{\stackrel{˙}{f}}_{s1}\\ \text{\hspace{0.17em}}\text{ }\text{ }-{e}_{2}^{\text{T}}{\stackrel{˙}{\stackrel{^}{f}}}_{s2}+{\stackrel{˜}{f}}_{s2}^{\text{T}}{\stackrel{˙}{f}}_{s2}-{\sigma }_{4}^{\text{T}}\left[\left(L+B\right)\otimes {I}_{2}\right]{\stackrel{˜}{f}}_{s3}+{\stackrel{˜}{f}}_{s3}^{\text{T}}{\stackrel{˙}{f}}_{s3}\end{array}$

$\begin{array}{c}\stackrel{˙}{V}\le \left(-{k}_{4}+\frac{{a}_{4}}{2}\right){‖{\sigma }_{4}‖}^{2}+\frac{1}{2}\left(1+{a}_{4}+{\epsilon }_{3}|1-{k}_{1}^{2}|-2{k}_{1}\right){‖{\sigma }_{1}‖}^{2}\\ \text{\hspace{0.17em}}+\frac{1}{2}\left(1-2{k}_{2}+{\epsilon }_{4}|1-{k}_{2}^{2}|\right){‖{\sigma }_{2}‖}^{2}+\frac{1}{2}\left(-2{k}_{3}+{k}_{2}+1\right){‖{\sigma }_{3}‖}^{2}\\ \text{\hspace{0.17em}}+\frac{1}{2}\left(-\frac{2}{{\epsilon }_{1}}+2{k}_{1}+\frac{|1-{k}_{1}^{2}|}{{\epsilon }_{3}}+{k}_{1}+1+{a}_{4}\right){‖{e}_{1}‖}^{2}\\ \text{\hspace{0.17em}}+\frac{1}{2}\left(-\frac{2}{{\epsilon }_{2}}+\frac{|1-{k}_{2}^{2}|}{{\epsilon }_{4}}+3{k}_{2}+2\right){‖{e}_{2}‖}^{2}+\left(\frac{{a}_{4}+1}{2}-{\beta }_{1}\right){‖{\stackrel{˜}{f}}_{s1}‖}^{2}\\ \text{\hspace{0.17em}}+\left(\frac{{a}_{4}+1}{2}-{\beta }_{3}\right){‖{\stackrel{˜}{f}}_{s3}‖}^{2}+\left(\frac{1}{2}-{\beta }_{2}\right){‖{\stackrel{˜}{f}}_{s2}‖}^{2}+\eta \end{array}$

$1+{a}_{4}+{\epsilon }_{3}|1-{k}_{1}^{2}|-2{k}_{1}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}1-2{k}_{2}+{\epsilon }_{4}|1-{k}_{2}^{2}|<0$

$-2{k}_{3}+{k}_{2}+1<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{2}{{\epsilon }_{1}}+2{k}_{1}+\frac{|1-{k}_{1}^{2}|}{{\epsilon }_{3}}+{k}_{1}+1+{a}_{4}<0$

$-{k}_{4}+\frac{{a}_{4}}{2}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{2}{{\epsilon }_{2}}+\frac{|1-{k}_{2}^{2}|}{{\epsilon }_{4}}+3{k}_{2}+2<0$

$\frac{{a}_{4}+1}{2}-{\beta }_{1}<0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{2}<{\beta }_{2}$

6. 仿真实例

Figure 3. Qball-X4

$\begin{array}{l}{J}_{\theta }=0.03\text{\hspace{0.17em}}\text{kg}\cdot {\text{m}}^{2},\text{\hspace{0.17em}}{J}_{\varphi }=0.03\text{\hspace{0.17em}}\text{kg}\cdot {\text{m}}^{2},\\ {J}_{\psi }=0.04\text{\hspace{0.17em}}\text{kg}\cdot {\text{m}}^{2},\text{\hspace{0.17em}}{J}_{r}=0.06\text{\hspace{0.17em}}\text{kg}\cdot {\text{m}}^{2},\\ m=1.4\text{\hspace{0.17em}}\text{kg},\text{\hspace{0.17em}}l=0.2\text{\hspace{0.17em}}\text{m}\end{array}$

Figure 4. The graph topology of the considered multi-quadrotor systems in simulation

${x}_{1}-{x}_{2}=1\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}{y}_{1}-{y}_{2}=1\text{\hspace{0.17em}}\text{m}$

${x}_{1}-{x}_{3}=-1\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}{y}_{1}-{y}_{3}=1\text{\hspace{0.17em}}\text{m}$

${x}_{2}-{x}_{4}=-1\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}{y}_{2}-{y}_{4}=1\text{\hspace{0.17em}}\text{m}$

${\left[{x}_{1}\left(0\right),\text{\hspace{0.17em}}{y}_{1}\left(0\right),\text{\hspace{0.17em}}{z}_{1}\left(0\right)\right]}^{\text{T}}={\left[0\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}0.8\text{\hspace{0.17em}}\text{m}\right]}^{\text{T}}$

${\left[{x}_{2}\left(0\right),\text{\hspace{0.17em}}{y}_{2}\left(0\right),\text{\hspace{0.17em}}{z}_{2}\left(0\right)\right]}^{\text{T}}={\left[0\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}0.6\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{m}\right]}^{\text{T}}$

${\left[{x}_{3}\left(0\right),\text{\hspace{0.17em}}{y}_{3}\left(0\right),\text{\hspace{0.17em}}{z}_{3}\left(0\right)\right]}^{\text{T}}={\left[0.4\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}-0.8\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}1.2\text{\hspace{0.17em}}\text{m}\right]}^{\text{T}}$

${\left[{x}_{4}\left(0\right),\text{\hspace{0.17em}}{y}_{4}\left(0\right),\text{\hspace{0.17em}}{z}_{4}\left(0\right)\right]}^{\text{T}}={\left[-1.1\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}-1\text{\hspace{0.17em}}\text{m},\text{\hspace{0.17em}}0.9\text{\hspace{0.17em}}\text{m}\right]}^{\text{T}}$

7. 结论

Figure 5. Position trajectory of each quadrotor in x-y-z space

Figure 6. Yaw angle trajectory of each quadrotor

NOTES

*通讯作者。

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