Liu, Z., Yuan, C., Zhang, Y. et al. A Learning-Based Fault Tolerant Tracking Control of an Unmanned Quadrotor Helicopter. J Intell Robot Syst 84, 145–162 (2016).

文章目录

1 Introduction
2 Description and Dynamics of the Unmanned Quadrotor Helicopter
- 2.1 Nonlinear Model of the Unmanned Quadrotor Helicopter
- 2.2 Linearized Model of the Unmanned Quadrotor Helicopter
- - Assumption 1

1 Introduction

2 Description and Dynamics of the Unmanned Quadrotor Helicopter

如图 1 所示，推力(u1,u2,u3,u4u_1, u_2, u_3, u_4u1,u2,u3,u4)是由四个分别配置在前角、后角、左角和右角的独立电机驱动的螺旋桨产生的。前后电机顺时针旋转，左右电机逆时针旋转。所产生的推力在 zBz _BzB 方向上总是向上的。

因此，1）直接将相同数量的控制信号分配给每台电机即可实现垂直平移； 2）水平平移要求四旋翼直升机提前滚动或俯仰，这样就可以产生向前或横向运动。此外，横滚和俯仰旋转可以通过分配不同数量的控制信号到相反的马达，这可以迫使四旋翼直升机向最慢的马达倾斜 [1]。

2.1 Nonlinear Model of the Unmanned Quadrotor Helicopter

利用 [28] 和 [29] 中的四旋翼直升机模型，常用的四旋翼直升机在地球固定坐标系下的动力学模型可有:
{x¨=(cos⁡ϕsin⁡θcos⁡ψ+sin⁡ϕsin⁡ψ)u1(t)−K1x˙my¨=(cos⁡ϕsin⁡θsin⁡ψ−sin⁡ϕcos⁡ψ)u1(t)−K2y˙mz¨=(cos⁡ϕcos⁡θ)uz(t)−K3z˙m−gϕ¨=u3(t)−K4ϕ˙Ixθ¨=u2(t)−K5θ˙Iyψ¨=u4(t)−K6ψ˙Iz(1)\left\{\begin{aligned} \ddot{x} &= \frac{(\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi)~ u_1(t) - K_1 \dot{x}}{m} \\ \ddot{y} &= \frac{(\cos \phi \sin\theta \sin\psi - \sin\phi \cos\psi)~ u_1(t) - K_2 \dot{y}}{m} \\ \ddot{z} &= \frac{(\cos\phi \cos\theta)~ u_z(t) - K_3 \dot{z}}{m} - g \\ \ddot{\phi} &= \frac{u_3(t) - K_4 \dot{\phi}}{I_x} \\ \ddot{\theta} &= \frac{u_2(t) - K_5 \dot{\theta}}{I_y} \\ \ddot{\psi} &= \frac{u_4(t) - K_6 \dot{\psi}}{I_z} \\ \end{aligned}\right. \tag{1}⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧x¨y¨z¨ϕ¨θ¨ψ¨=m(cosϕsinθcosψ+sinϕsinψ) u1(t)−K1x˙=m(cosϕsinθsinψ−sinϕcosψ) u1(t)−K2y˙=m(cosϕcosθ) uz(t)−K3z˙−g=Ixu3(t)−K4ϕ˙=Iyu2(t)−K5θ˙=Izu4(t)−K6ψ˙(1)

加速度与升力/力矩的关系表示为:
[uz(t)uθ(t)uϕ(t)uψ(t)]=[1111L−L0000L−LCC−C−C][uc1(t)uc2(t)uc3(t)uc4(t)](2)\begin{aligned} \left[\begin{matrix} u_z(t) \\ u_\theta(t) \\ u_\phi(t) \\ u_\psi(t) \\ \end{matrix}\right]&= \left[\begin{matrix} 1 & 1 & 1 & 1 \\ L & -L & 0 & 0 \\ 0 & 0 & L & -L \\ C & C & -C & -C \\ \end{matrix}\right] \left[\begin{matrix} u_{c1}(t) \\ u_{c2}(t) \\ u_{c3}(t) \\ u_{c4}(t) \\ \end{matrix}\right] \end{aligned} \tag{2}⎣⎢⎢⎡uz(t)uθ(t)uϕ(t)uψ(t)⎦⎥⎥⎤=⎣⎢⎢⎡1L0C1−L0C10L−C10−L−C⎦⎥⎥⎤⎣⎢⎢⎡uc1(t)uc2(t)uc3(t)uc4(t)⎦⎥⎥⎤(2)

每个电机由其相应的脉宽调制(PWM)信号控制，其关系定义为:
ui(t)=Kmωms+ωmuci(t)(3)u_i(t) = K_m \frac{\omega_m}{s+\omega_m} u_{ci}(t) \tag{3}ui(t)=Kms+ωmωmuci(t)(3)

2.2 Linearized Model of the Unmanned Quadrotor Helicopter

Assumption 1

假设四旋翼无人直升机在操作期间处于悬停状态 [20]，这表明在竖直方向 uz≈mgu_z \approx mguz≈mg。俯仰角和滚转角的变化幅度也比较小，有 sin⁡ϕ≈ϕ,sin⁡θ≈θ\sin \phi \approx \phi, \sin \theta \approx \thetasinϕ≈ϕ,sinθ≈θ，并且没有航向角变化 ψ≈0\psi \approx 0ψ≈0。另外，当无人机移动速度非常慢时，阻力系数是微不足道的。

那么根据上述假设 1，可以简化公式（1）变为

{x¨=θgy¨=−ϕgz¨=uz(t)/m−gIxθ¨=uθ(t)Iyϕ¨=uϕ(t)Izψ¨=uψ(t)(4)\begin{aligned} \left\{\begin{aligned} &\ddot{x} = \theta g \\ &\ddot{y} = - \phi g \\ &\ddot{z} = u_z(t)/m - g \\ &I_{x} \ddot{\theta} = u_{\theta}(t) \\ &I_{y} \ddot{\phi} = u_{\phi}(t) \\ &I_{z} \ddot{\psi} = u_{\psi}(t) \\ \end{aligned}\right. \end{aligned} \tag{4}⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧x¨=θgy¨=−ϕgz¨=uz(t)/m−gIxθ¨=uθ(t)Iyϕ¨=uϕ(t)Izψ¨=uψ(t)(4)

进一步，当把无人机和它的执行器作为一个整体时，在控制系统设计过程中可以忽略执行器动力学，且不会造成显著的残差，这是由于执行器的时间常数比无人机小得多 [30]。

那么公式（3）就可以简化为 Kmωms+ωm≈KmK_m \frac{\omega_m}{s+\omega_m} \approx K_mKms+ωmωm≈Km，这仍然可以用来描述控制行为的有效性。因此，公式（2）可以写为
[uz(t)uθ(t)uϕ(t)uψ(t)]=[KmKmKmKmKmL−KmL0000KmL−KmLKmCKmC−KmC−KmC][uc1(t)uc2(t)uc3(t)uc4(t)](5)\begin{aligned} \left[\begin{matrix} u_z(t) \\ u_\theta(t) \\ u_\phi(t) \\ u_\psi(t) \\ \end{matrix}\right]&= \left[\begin{matrix} K_m & K_m & K_m & K_m \\ K_mL & -K_mL & 0 & 0 \\ 0 & 0 & K_mL & -K_mL \\ K_mC & K_mC & -K_mC & -K_mC \\ \end{matrix}\right] \left[\begin{matrix} u_{c1}(t) \\ u_{c2}(t) \\ u_{c3}(t) \\ u_{c4}(t) \\ \end{matrix}\right] \end{aligned} \tag{5}⎣⎢⎢⎡uz(t)uθ(t)uϕ(t)uψ(t)⎦⎥⎥⎤=⎣⎢⎢⎡KmKmL0KmCKm−KmL0KmCKm0KmL−KmCKm0−KmL−KmC⎦⎥⎥⎤⎣⎢⎢⎡uc1(t)uc2(t)uc3(t)uc4(t)⎦⎥⎥⎤(5)

这里将欧拉角加速度映射到了螺旋桨转速上。

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【Paper】2016_A Learning-Based Fault Tolerant Tracking Control of an Unmanned Quadrotor Helicopter