# 基于极限环的四旋翼空间圆形编队控制Finite-Time Spatial Circle Formation Control of Quadrotor Systems Based on Limit Cycle

DOI: 10.12677/DSC.2020.93013, PDF, HTML, XML, 下载: 98  浏览: 574  国家自然科学基金支持

Abstract: In this paper, a quadrotor formation system with limited sensing ability is capable of moving unanimously to form the desired circular formation only using the position information of the target center of the circle and the neighboring quadrotors. By employing a method of decoupling control based on limit-cycle, the spatial circle formation is decoupled into three sub-problems: height control, circular convergence, and hovering together with the spacing distribution. Then a distributed controller composed of height control, circular convergence control, and spacing distribution control is designed to drive the quadrotors to form a circle in the space. This controller has the collision-avoidance among the quadrotors under consideration as well. It can constitute the desired circular distribution of relative position at the designated height and hold the desired formation when the quadrotors fly around the target. Lastly, the effectiveness of the control law is verified by simulations and experiments.

1. 引言

2. 四旋翼建模

2.1. 四旋翼数学模型

Figure 1. Schematic diagram of type X quadrotor

Table 1. Parameters and their description

$\left\{\begin{array}{l}{\tau }_{x}=\sqrt{2}{C}_{T}l\left(-{\omega }_{1}^{2}+{\omega }_{2}^{2}+{\omega }_{3}^{2}-{\omega }_{4}^{2}\right)/2\hfill \\ {\tau }_{y}=\sqrt{2}{C}_{T}l\left({\omega }_{1}^{2}-{\omega }_{2}^{2}+{\omega }_{3}^{2}-{\omega }_{4}^{2}\right)/2\hfill \\ {\tau }_{z}={C}_{D}\left(+{\omega }_{1}^{2}+{\omega }_{2}^{2}-{\omega }_{3}^{2}-{\omega }_{4}^{2}\right)\hfill \end{array}$ (1)

$\left\{\begin{array}{l}{\stackrel{¨}{x}}_{\text{e}}={F}_{\text{sum}}\left(\mathrm{cos}\phi \mathrm{sin}\theta \mathrm{cos}\psi +\mathrm{sin}\phi \mathrm{sin}\psi \right)/m\hfill \\ {\stackrel{¨}{y}}_{\text{e}}={F}_{\text{sum}}\left(\mathrm{cos}\phi \mathrm{sin}\theta \mathrm{sin}\psi +\mathrm{sin}\phi \mathrm{cos}\psi \right)/m\hfill \\ {\stackrel{¨}{z}}_{\text{e}}={F}_{\text{sum}}\mathrm{cos}\phi \mathrm{cos}\theta /m-g\hfill \\ \stackrel{¨}{\phi }=\left({I}_{y}-{I}_{z}\right)\stackrel{˙}{\theta }\stackrel{˙}{\psi }/{I}_{x}+{\tau }_{x}/{I}_{x}\hfill \\ \stackrel{¨}{\theta }=\left({I}_{z}-{I}_{x}\right)\stackrel{˙}{\phi }\stackrel{˙}{\psi }/{I}_{y}+{\tau }_{y}/{I}_{y}\hfill \\ \stackrel{¨}{\psi }=\left({I}_{x}-{I}_{y}\right)\stackrel{˙}{\phi }\stackrel{˙}{\theta }/{I}_{z}+{\tau }_{z}/{I}_{z}\hfill \end{array}$ (2)

$\left[\begin{array}{c}{\stackrel{¨}{x}}_{\text{b}}\\ {\stackrel{¨}{y}}_{\text{b}}\end{array}\right]=\left[\begin{array}{cc}\mathrm{cos}\psi & -\mathrm{sin}\psi \\ \mathrm{sin}\psi & \mathrm{cos}\psi \end{array}\right]\left[\begin{array}{c}{\stackrel{¨}{x}}_{\text{e}}\\ {\stackrel{¨}{y}}_{\text{e}}\end{array}\right]$ (3)

$\left\{\begin{array}{l}\phi =\mathrm{arctan}\left(-\left({\stackrel{¨}{x}}_{\text{e}}\mathrm{cos}\psi +{\stackrel{¨}{y}}_{\text{e}}\mathrm{sin}\psi \right)/g\right)\\ \theta =\mathrm{arctan}\left(\left(-{\stackrel{¨}{x}}_{\text{e}}\mathrm{sin}\psi +{\stackrel{¨}{y}}_{\text{e}}\mathrm{cos}\psi \right)\mathrm{cos}\phi /g\right)\end{array}$ (4)

2.2. 系统感知拓扑结构及队形

Figure 2. (a) Minimum safety distance; (b) Collision avoidance distance

${\stackrel{¯}{p}}_{i}={p}_{i}-{p}_{0}={\left[\begin{array}{ccc}{\stackrel{¯}{x}}_{i}& {\stackrel{¯}{y}}_{i}& {\stackrel{¯}{z}}_{i}\end{array}\right]}^{\text{T}},\text{}i\in {I}_{n}$ (5)

${\stackrel{¯}{p}}_{i}=\left[\begin{array}{c}{\rho }_{i}\mathrm{cos}{\alpha }_{i}\\ \begin{array}{c}{\rho }_{i}\mathrm{sin}{\alpha }_{i}\\ {\stackrel{¯}{z}}_{i}\end{array}\end{array}\right],\text{}i\in {I}_{n}$ (6)

Figure 3. Top view of the quadrotor space system number

$i+1=\left\{\begin{array}{l}i+1,\text{}i=1,2,\cdots ,N-1\hfill \\ 1,\text{}i=N\hfill \end{array}$

$i-1=\left\{\begin{array}{l}i-1,\text{}i=2,3,\cdots ,N\hfill \\ N,\text{}i=1\hfill \end{array}$ (7)

$\left\{\begin{array}{l}{\stackrel{^}{p}}_{i}={p}_{i+1}-{p}_{i}\hfill \\ {\stackrel{^}{p}}_{i-1}={p}_{i}-{p}_{i-1}\hfill \end{array}\text{,}i\in {I}_{n}$ (8)

$d={\left[{d}_{1},{d}_{2},\cdots ,{d}_{N}\right]}^{\text{T}}\text{,}{\sum }_{i=1}^{N}{d}_{i}=2\pi$ (9)

Figure 4. Top view of topological structure information perceptible by each quadrotor

Figure 5. (a) Space formation trajectory; (b) Top view of quadrotor space trajectory and expected formation description

${\stackrel{˙}{\stackrel{¯}{p}}}_{i}=\left[\begin{array}{c}{\stackrel{˙}{\stackrel{¯}{x}}}_{\text{e}i}\\ {\stackrel{˙}{\stackrel{¯}{y}}}_{\text{e}i}\\ {\stackrel{˙}{\stackrel{¯}{z}}}_{\text{e}i}\end{array}\right]=\left[\begin{array}{c}{u}_{xi}\\ {u}_{yi}\\ {u}_{zi}\end{array}\right],\text{}i\in {I}_{n}$ (10)

$\left\{\begin{array}{l}\begin{array}{l}{z}_{i}^{*}=h\text{}\left(高度控制\right)\\ {\rho }_{i}^{*}=r\text{}\left(目标盘旋\right)\end{array}\hfill \\ {\stackrel{^}{\alpha }}_{i}^{*}={d}_{i}\text{}\left(间距调整\right)\hfill \end{array},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\in {I}_{n}$ (11)

3. 控制器设计

Figure 6. Block diagram of the quadrotors space circular formation controller

3.1. 高度控制

${u}_{i}^{\text{height}}={K}_{\text{P}}\left({e}_{zi}+\frac{1}{{K}_{\text{I}}}\int {e}_{zi}\text{d}t+\frac{{K}_{\text{D}}\text{d}{e}_{zi}}{\text{d}t}\right),\text{}i\in {I}_{n}$ (12)

3.2. 圆形汇聚控制

${u}_{i}^{\text{converge}}=\lambda \left[\begin{array}{cc}\gamma \left({r}^{2}-{‖{\stackrel{¯}{p}}_{i}‖}^{2}\right)& -\text{1}\\ \text{1}& \gamma \left({r}^{2}-{‖{\stackrel{¯}{p}}_{i}‖}^{2}\right)\end{array}\right]\left[\begin{array}{c}{\stackrel{¯}{x}}_{\text{e}i}\\ {\stackrel{¯}{y}}_{\text{e}i}\end{array}\right],\text{}i\in {I}_{n}$ (13)

3.3. 间距布局控制

${u}_{i}^{\text{distribution}}={\left(\frac{{d}_{i-1}}{{d}_{i}+{d}_{i-1}}{\stackrel{^}{\alpha }}_{i}-\frac{{d}_{i}}{{d}_{i}+{d}_{i-1}}{\stackrel{^}{\alpha }}_{i-1}\right)}^{\alpha },\text{}i\in {I}_{n}$ (14)

${u}_{i}^{\text{layout}}={c}_{1}+\frac{{c}_{2}}{2\pi }{u}_{i}^{\text{distribution}},\text{}i\in {I}_{n}$ (15)

3.4. 避免碰撞控制

${P}_{ij}\left({p}_{i},{p}_{j}\right)={\left(\mathrm{min}\left\{0,\frac{{‖{p}_{i}-{p}_{j}‖}^{2}-{R}_{2}^{2}}{{‖{p}_{i}-{p}_{j}‖}^{2}-{R}_{1}^{2}}\right\}\right)}^{2}\text{},\text{}i,j\in {I}_{n}$ (16)

$\frac{\partial {P}_{ij}}{\partial {p}_{i}}=\left\{\begin{array}{l}0\text{,}‖{p}_{i}-{p}_{j}‖\ge {R}_{2}\hfill \\ \frac{4\left({R}_{2}^{2}-{R}_{1}^{2}\right)\left({‖{p}_{i}-{p}_{j}‖}^{2}-{R}_{2}^{2}\right)\left({p}_{i}-{p}_{j}\right)}{{\left({‖{p}_{i}-{p}_{j}‖}^{2}-{R}_{1}^{2}\right)}^{3}},\text{}{R}_{1}<‖{p}_{i}-{p}_{j}‖<{R}_{2}\hfill \end{array}$ (17)

${P}_{i}={P}_{i,i-1}+{P}_{i,i+1},\text{}i\in {I}_{n}$ (18)

${u}_{i}^{\text{collisionavoidance}}=-\left(\frac{\partial {P}_{i,i-1}}{\partial {p}_{i}}+\frac{\partial {P}_{i,i+1}}{\partial {p}_{i}}\right),\text{}i\in {I}_{n}$ (19)

3.5. 空间圆形编队控制器融合

${\stackrel{˙}{\stackrel{¯}{p}}}_{i}=\left[\begin{array}{l}{u}_{xi}\hfill \\ {u}_{yi}\hfill \\ {u}_{zi}\hfill \end{array}\right]=\left[\begin{array}{c}{u}_{i}^{\text{converge}}×{u}_{i}^{\text{layout}}+\beta {u}_{i}^{\text{collisionavoidance}}{}_{\left(2×1\right)}\\ {u}_{i}^{\text{height}}\end{array}\right],\text{}i\in {I}_{n}$ (20)

$\left[\begin{array}{c}{\stackrel{¨}{x}}_{\text{e}}\\ {\stackrel{¨}{y}}_{\text{e}}\end{array}\right]=\left[\begin{array}{c}{\stackrel{˙}{u}}_{xi}\\ {\stackrel{˙}{u}}_{yi}\end{array}\right],\text{}i\in {I}_{n}$ (21)

4. 仿真与实验

Table 2. Space circular formation parameters

4.1. 仿真结果

Figure 7. For parameter $\alpha =1,\beta =0$, the simulation diagram is formed by the asymptotic distribution of circular relative positions without avoiding collision

Figure 8. For parameter $\alpha =1,\beta =200$, the simulation diagram is formed by the asymptotic distribution of circular relative positions and avoiding collision

Figure 9. For parameter $\alpha =0.1,\beta =200$, the simulation diagram is formed by circular relative positions and avoiding collision in limited time

4.2. 实验验证

Figure 10. Actual flight rendering

Figure 11. The actual flight data record graph formed by the asymptotic distribution of circular relative position

Figure 12. The actual flight data record graph formed in a finite time by the circular relative position distribution

5. 总结与展望

NOTES

*通讯作者。

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