**2. Dynamic model**

In this section, Euler angles are used to represent the quadrotor dynamical model. External disturbances and model parameter uncertainties change are considered as well. An IBS controller is derived and tested in simulation. The stability analysis is obtained via a selected Lyapunov function. The full quadrotor dynamic model including the gyroscope effects *G*ð Þ *ω* is

$$\begin{cases} \dot{\mathbf{p}} = \mathbf{v} \\ \dot{\mathbf{v}} = -\mathbf{g}\mathbf{e} + \frac{f}{m}R\_{\theta}\mathbf{e} \\ \dot{R}\_{\theta} = R\_{\theta}\mathbf{S}(\boldsymbol{\omega}) \\ J\dot{\boldsymbol{\omega}} = -\mathbf{S}(\boldsymbol{\omega})J\boldsymbol{\omega} - \mathbf{G}(\boldsymbol{\omega}) + \boldsymbol{\tau}\_{E} \end{cases} \tag{1}$$

and the rotational matrix *R<sup>θ</sup>* from the inertial frame to the body frame is

$$R\_{\theta} = \begin{bmatrix} c\psi c\theta & c\psi s\theta s\rho - s\psi c\rho & c\psi s\theta c\rho + s\psi s\rho \\ s\psi c\theta & s\psi s\theta s\rho + c\psi c\rho & s\psi s\theta c\rho - c\psi s\rho \\ -s\theta & c\theta s\rho & c\theta c\rho \end{bmatrix} . \tag{2}$$

where *m* is the quadrotor mass, *ω* ¼ *ωx*, *ωy*, *ω<sup>z</sup>* � �*<sup>T</sup>* is the angular velocity in the body frame, *J* is the 3 � 3 diagonal matrix representing three inertial moments in the body frame, *τ<sup>E</sup>* is the torque vector applied on the quadrotor, **v** ¼ *vx*, *vy*, *vz* � �*<sup>T</sup>* is the linear velocity, **p** ¼ ½ � *x*, *y*, *z <sup>T</sup>* is the position vector, *S* is the skew-symmetric cross product matrix, and the vector **<sup>e</sup>** <sup>¼</sup> ½ � 0, 0, 1 *<sup>T</sup>*.

Assuming that *φ*, *θ*, *ωx*, *ω<sup>y</sup>* and *ω<sup>z</sup>* are very small, *ζ* ¼ ½ � *φ*, *θ*, *ψ <sup>T</sup>*, *<sup>η</sup>* <sup>¼</sup> \_ *ζ* ¼ *φ*\_ , \_ *<sup>θ</sup>*, *<sup>ψ</sup>*\_ � �*<sup>T</sup>* <sup>¼</sup> *<sup>ω</sup>x*,*ωy*,*ω<sup>z</sup>* � �*<sup>T</sup>* and *<sup>η</sup>*\_ <sup>¼</sup> *<sup>φ</sup>*€, €*θ*, *<sup>ψ</sup>*€ � �*<sup>T</sup>* <sup>¼</sup> *<sup>ω</sup>*\_ *<sup>x</sup>*, *<sup>ω</sup>*\_ *<sup>y</sup>*, *<sup>ω</sup>*\_ *<sup>z</sup>* � �*<sup>T</sup>* , then the attitude control part of Eq. (1) can be written as:

$$\begin{cases} \begin{aligned} \label{1.1} f &= mg \\ \ddot{\boldsymbol{\rho}} &= \dot{\boldsymbol{\theta}} \dot{\boldsymbol{\nu}} \frac{J\_{\boldsymbol{\gamma}} - J\_{x}}{J\_{x}} + \frac{J\_{r}}{J\_{x}} \dot{\boldsymbol{\theta}} \boldsymbol{\Omega} + \frac{\boldsymbol{\tau}\_{\boldsymbol{\Psi}}}{J\_{x}} \\ \ddot{\boldsymbol{\theta}} &= \dot{\boldsymbol{\rho}} \dot{\boldsymbol{\nu}} \frac{J\_{x} - J\_{x}}{J\_{\boldsymbol{\gamma}}} - \frac{J\_{r}}{J\_{\boldsymbol{\gamma}}} \dot{\boldsymbol{\rho}} \boldsymbol{\Omega} + \frac{\boldsymbol{\tau}\_{\boldsymbol{\theta}}}{J\_{\boldsymbol{\gamma}}} \\ \end{aligned} \end{cases} \tag{3}$$
 
$$\begin{aligned} \begin{aligned} \ddot{\boldsymbol{\nu}} &= \dot{\boldsymbol{\rho}} \dot{\boldsymbol{\theta}} \frac{J\_{x} - J\_{\boldsymbol{\gamma}}}{J\_{x}} + \frac{\boldsymbol{\tau}\_{\boldsymbol{\theta}}}{J\_{x}} \end{aligned} \tag{3}$$

### **3. Quadrotors formation problem**

The full dynamic model based on Euler angles (1) of a quadrotor can be written as:

$$\begin{cases} \dot{\mathbf{p}}\_i = \mathbf{v}\_i \\ \dot{\mathbf{v}}\_i = -\mathbf{g}\mathbf{e} + \frac{f\_i}{m\_i} R\_{i\theta} \mathbf{e} \\ \dot{\zeta}\_i = \eta\_i \\ J\_i \dot{\eta}\_i = S(\eta\_i) f\_i \eta\_i + G(\eta\_i) - \tau\_{iE} \end{cases} \tag{4}$$

where *i* is *L* for the leader and *F* for the follower.

The leader-follower formation control problem to be solved in this chapter is a distributed control scheme of one leader and one follower. The leader control problem is formulated as a trajectory tracking, and the follower control problem is also formulated as a tracking problem, but with a different tracking target.

The follower keeps its yaw angle the same as the leader when it maintains the formation pattern. It moves to a desired position **p***Fd*, which is determined by a desired distance *d*, a desired incidence angle *ρ*, and a desired bearing angle *σ*. A new frame *F*<sup>0</sup> is defined by the translation of the leader frame *L* to the frame with the desired follower position **p***Fd* as the origin. As shown in **Figure 1**, the desired incidence angle is measured between the desired distance *d* and the *x* � *y* plane in the new frame *F*<sup>0</sup> , and the desired bearing angle is measured between the *x* axis and the projection of the *d* in *x* � *y* plane in the new frame *F*<sup>0</sup> . The desired position **p***Fd* is

**Figure 1.** *Body frames in formation.*

*Integral Backstepping Controller for UAVs Team Formation DOI: http://dx.doi.org/10.5772/intechopen.93731*

$$\mathbf{p}\_{Fd} = \mathbf{p}\_L - R\_{Lq}^T d \begin{bmatrix} \cos \rho \cos \sigma\\ \cos \rho \sin \sigma\\ \sin \rho \end{bmatrix}. \tag{5}$$

Assume both the leader and the follower are able to obtain their own pose information and the follower is able to obtain the leader's pose information via wireless communication. The design goal of the controllers is to find the state feedback control law for the thrust and torque inputs for both the leader and the follower. The leader-follower formation control problem is solved if both conditions (6) and (7) are satisfied.

$$\begin{cases} \lim\_{t \to \infty} (\mathbf{p}\_{Fd} - \mathbf{p}\_F) = \mathbf{0} \\ \lim\_{t \to \infty} (\boldsymbol{\mu}\_L - \boldsymbol{\mu}\_F) = \mathbf{0} \end{cases} \tag{6}$$

and

$$\begin{cases} \lim\_{t \to \infty} (\mathbf{p}\_{Ld} - \mathbf{p}\_L) = \mathbf{0} \\ \lim\_{t \to \infty} (\boldsymbol{\mu}\_{Ld} - \boldsymbol{\mu}\_L) = \mathbf{0} \end{cases} \tag{7}$$

The communication among the robots is assumed to be available. The position **p***L*, yaw angle *ψ<sup>L</sup>* of the leader *L* and its first and second derivatives *ψ*\_ *<sup>L</sup>* and *ψ*€*<sup>L</sup>* are assumed to be available and measurable. The linear velocity of the leader *L* and its derivatives **v***<sup>L</sup>* and **v**\_ *<sup>L</sup>* are assumed bounded and available for the follower.

#### **4. Formation IBS controllers**

Integral backstepping control is one of the popular control approaches for both individual and multiple quadrotors. In this section, the integral backstepping control is applied for the individual quadrotor path tracking and leader-follower formation problems. This nonlinear control technique divide the control into two loops, the inner loop is for the attitude stabilisation and the outer loop is for the position control as shown in **Figure 2**.

In this case, the leader and the follower desired roll and pitch angles are assumed to be *θLd* ¼ *θFd* ¼ 0 and *φLd* ¼ *φFd*. The dynamic model of a quadrotor is

**Figure 2.** *Two-loop control block diagram.*

**Figure 3.** *Initial system.*

represented based on Euler angles representation and includes some modelled aerodynamical effects as a nonlinear part. The IBS controller is designed for the translational part to track the desired trajectory. Stability analysis is achieved via a suitable Lyapunov function. The external disturbance and model parameters uncertainty are considered in the simulation tests in all circumstances.

#### **4.1 Backstepping control concept**

Backstepping is a recursive design mechanism to asymptotically stabilise a controller for the following system [16]:

$$\begin{cases} \dot{\mathbf{x}} = f(\mathbf{x}) + \mathbf{g}(\mathbf{x})\Gamma\\ \dot{\Gamma} = \mathbf{u} \end{cases} \tag{8}$$

This system is described as an initial system in **Figure 3**, where **x**∈ *<sup>n</sup>* and Γ∈ are the system state and **<sup>u</sup>**<sup>∈</sup> is the control input. *<sup>f</sup>*, *<sup>g</sup>* : D ! *<sup>n</sup>* are assumed to be smooth and *f*ð Þ¼ 0 0. A stabilising state feedback control law Γ ¼ Φð Þ **x** , assuming Φð Þ¼ 0 0, exists, in addition to a Lyapunov function *V*<sup>1</sup> : D ! <sup>þ</sup> such that

$$\dot{V}\_1(\mathbf{x}) = \frac{\partial V\_1}{\partial \mathbf{x}} [f(\mathbf{x}) + \mathbf{g}(\mathbf{x})\Phi(\mathbf{x})] \le -V\_\varepsilon(\mathbf{x}), \forall \mathbf{x} \in \mathcal{D} \tag{9}$$

where *Vε*ð Þ **x** : D ! <sup>þ</sup> is a positive semidefinite function. Now, the following algebraic manipulation is required: by adding and subtracting the term *g*ð Þ **x** Φð Þ **x** to/from the subsystem (8) we can have the following system:

$$
\dot{\mathbf{x}} = f(\mathbf{x}) + \mathbf{g}(\mathbf{x})\Phi(\mathbf{x}) + \mathbf{g}(\mathbf{x})s \tag{10}
$$

where *s* ¼ Γ � Φð Þ **x** , by this construction, when *s* ! 0, **x**\_ ¼ *f*ð Þþ **x** *g*ð Þ **x** Φð Þ **x** which is asymptotically stable. The derivative of *s* is

$$
\dot{s} = \dot{\Gamma} - \dot{\Phi}(\mathbf{x}) = \mathbf{u} - \dot{\Phi}(\mathbf{x}) = \nu \tag{11}
$$

which is the backstepping, since Φð Þ **x** is stepped back by differentiation as described in **Figure 4**. So we have

$$\begin{aligned} \dot{\mathbf{x}} &= f(\mathbf{x}) + \mathbf{g}(\mathbf{x})\Phi(\mathbf{x}) + \mathbf{g}(\mathbf{x})s \\ \dot{s} &= \nu \end{aligned} \tag{12}$$

This system is equivalent to the initial system (8), where <sup>Φ</sup>\_ <sup>¼</sup> <sup>∂</sup><sup>Φ</sup> *<sup>∂</sup>***<sup>x</sup> x**\_ ¼ ∂Φ *<sup>∂</sup>***<sup>x</sup>** ½ � *f*ð Þþ **x** *g*ð Þ **x** Γ . The next step is to stabilise the system (12), and the following Lyapunov function is considered:

*Integral Backstepping Controller for UAVs Team Formation DOI: http://dx.doi.org/10.5772/intechopen.93731*

$$V(\mathbf{x}, \mathbf{s}) = V\_1(\mathbf{x}) + \frac{1}{2}\mathbf{s}^2. \tag{13}$$

Then

$$\begin{split} \dot{V} &= \frac{\partial V\_1}{\partial \mathbf{x}} [f(\mathbf{x}) + \mathbf{g}(\mathbf{x})\Phi(\mathbf{x}) + \mathbf{g}(\mathbf{x})\mathbf{s}] + s\nu \\ &\leq -V\_\varepsilon(\mathbf{x}) + \left[\frac{\partial V\_1}{\partial \mathbf{x}} \mathbf{g}(\mathbf{x}) + \nu \right] \varsigma. \end{split} \tag{14}$$

Let

$$\begin{cases} \nu = -\frac{\partial V\_1}{\partial \mathbf{x}} \mathbf{g}(\mathbf{x}) - \epsilon \mathbf{s} \\ \epsilon > 0 \end{cases} \tag{15}$$

Then

$$
\dot{V} \le -V\_\varepsilon(\mathbf{x}) - \varepsilon \mathbf{s}^2 < 0. \tag{16}
$$

This signifies that the origin ð Þ **x** ¼ 0, *s* ¼ 0 is asymptotically stable. Since , then the origin **x** ¼ 0 and Γ ¼ 0 is also asymptotically stable. In the next step an integral part is added to the BS controller to eliminate the steady state error which occurred in the simulation results and is called IBS.

#### **4.2 Follower integral backstepping controller**

The IBS controller for the follower is to track the leader and maintain a desired distance between them with desired incidence and bearing angles.

In this subsection, we derive the IBS controller for the follower. Let us use the follower translational part (17):

$$
\ddot{\mathbf{p}}\_F = f\left(\mathbf{p}\_F\right) + \mathbf{g}\left(\mathbf{p}\_F\right) f\_F \tag{17}
$$

where

$$f\begin{pmatrix} \mathbf{p}\_F \end{pmatrix} = \begin{bmatrix} \mathbf{0} & \mathbf{0} & -\mathbf{g} \end{bmatrix}^T \tag{18}$$

$$\mathbf{g}\left(\mathbf{p}\_F\right) = \begin{bmatrix} \boldsymbol{u}\_{Fx}/\boldsymbol{m}\_F & \boldsymbol{u}\_{Fy}/\boldsymbol{m}\_F & \boldsymbol{c}\boldsymbol{\theta}\_F\boldsymbol{c}\boldsymbol{q}\_F/\boldsymbol{m}\_F \end{bmatrix}^T \tag{19}$$

where

$$\mathfrak{u}\_{F\mathfrak{x}} = \left(\mathfrak{c}\mathfrak{y}\_F \mathfrak{s} \theta\_F \mathfrak{c}\mathfrak{q}\_F + \mathfrak{s}\mathfrak{y}\_F \mathfrak{s} \mathfrak{q}\_F\right) \tag{20}$$

$$
\mu\_{F\circ} = (\mathfrak{s}\mu\_F \mathfrak{s}\theta\_F \mathfrak{c}\varphi\_F - \mathfrak{c}\mathfrak{y}\_F \mathfrak{s}\varphi\_F). \tag{21}
$$

**Figure 4.** *Backstepping system.*

Then the position tracking error between the leader and the follower can be calculated as:

$$\ddot{\mathbf{p}}\_F = \mathbf{p}\_{Fd} - \mathbf{p}\_F = \mathbf{p}\_L - \mathbf{R}\_L^T d \begin{bmatrix} \cos \rho \cos \sigma\\ \cos \rho \sin \sigma\\ \sin \rho \end{bmatrix} - \mathbf{p}\_F \tag{22}$$

and its derivative is

$$
\dot{\mathbf{p}}\_F = \dot{\mathbf{p}}\_{Fd} - \dot{\mathbf{p}}\_F = \dot{\mathbf{p}}\_{Fd} - \mathbf{v}\_F \tag{23}
$$

where **v***<sup>F</sup>* is a virtual control, and its desirable value can be described as:

$$\mathbf{v}\_F^d = \dot{\mathbf{p}}\_{Fd} + b\_F \tilde{\mathbf{p}}\_F + k\_F \overline{\mathbf{p}}\_F \tag{24}$$

where the integration of the follower position error is added to minimise the steady-state error.

Now, consider the linear velocity error between the leader and the follower as:

$$
\tilde{\mathbf{v}}\_F = \mathbf{v}\_F^d - \dot{\mathbf{p}}\_F. \tag{25}
$$

By substituting (24) into (25) we obtain

$$
\tilde{\mathbf{v}}\_F = \dot{\mathbf{p}}\_{Fd} + b\_F \tilde{\mathbf{p}}\_F + k\_F \overline{\mathbf{p}}\_F - \dot{\mathbf{p}}\_F \tag{26}
$$

and its time derivative becomes

$$
\dot{\tilde{\mathbf{v}}}\_{F} = \ddot{\mathbf{p}}\_{Fd} + b\_{F}\dot{\tilde{\mathbf{p}}}\_{F} + k\_{F}\tilde{\mathbf{p}}\_{F} - \ddot{\mathbf{p}}\_{F}.\tag{27}
$$

Then from (24) and (25) we can rewrite (23) in terms of the linear velocity error as:

$$
\dot{\tilde{\mathbf{p}}}\_F = \tilde{\mathbf{v}}\_F - b\_F \tilde{\mathbf{p}}\_F - k\_F \overline{\mathbf{p}}\_F. \tag{28}
$$

By substituting (17) and (28) into (27), the time derivative of the linear velocity error can be rewritten as:

$$
\dot{\tilde{\mathbf{v}}}\_{F} = \ddot{\mathbf{p}}\_{Fd} + b\_F \tilde{\mathbf{v}}\_{F} - b\_F^2 \dot{\tilde{\mathbf{p}}}\_{F} - b\_F k\_F \overline{\mathbf{p}}\_{F} + k\_F \tilde{\mathbf{p}}\_{F} - f\left(\mathbf{p}\_F\right) - \mathbf{g}\left(\mathbf{p}\_F\right) f\_F. \tag{29}
$$

The desirable time derivative of the linear velocity error is supposed to be

$$
\dot{\tilde{\mathbf{v}}}\_F = -\boldsymbol{\sigma}\_F \tilde{\mathbf{v}}\_F - \tilde{\mathbf{p}}\_F. \tag{30}
$$

Now, the total thrust *f <sup>F</sup>*, the longitudinal *uFx* and the lateral *uFy* motion control can be found by subtracting (29) from (30) as follows:

$$f\_F = (\mathbf{g} + \dot{\nu}\_{Lx} + (\mathbf{1} - b\_{Fx}^2 + k\_{Fx})\tilde{\mathbf{z}}\_F + (b\_{Fx} + c\_{Fx})\tilde{\nu}\_{Fx} - b\_{Fx}k\_{Fx}\tilde{\mathbf{z}}\_F$$

$$-d(R\_{\theta31}\cos\rho\cos\sigma + R\_{\theta32}\cos\rho\sin\sigma + R\_{\theta33}\sin\rho))\frac{m\_F}{c\theta\_F c\rho\_F} \tag{31}$$

$$\mathbf{u}\_{\rm Fx} = (\dot{\nu}\_{\rm Lx} + (1 - b\_{\rm Fx}^2 + k\_{\rm Fx})\tilde{\mathbf{x}}\_{\rm F} + (b\_{\rm Fx} + c\_{\rm Fx})\tilde{\nu}\_{\rm Fx} - b\_{\rm Fx}k\_{\rm Fx}\overline{\mathbf{x}}\_{\rm F}$$

$$-d(R\_{\theta11}\cos\rho\cos\sigma + R\_{\theta12}\cos\rho\sin\sigma + R\_{\theta13}\sin\rho))\frac{m\_{\rm F}}{f\_{\rm F}}\tag{32}$$

*Integral Backstepping Controller for UAVs Team Formation DOI: http://dx.doi.org/10.5772/intechopen.93731*

$$\begin{split} u\_{F\text{y}} &= \left(\dot{v}\_{L\text{y}} + \left(\mathbf{1} - b\_{F\text{y}}^{2} + k\_{F\text{y}}\right)\ddot{\mathbf{y}}\_{F} + \left(b\_{F\text{y}} + c\_{F\text{y}}\right)\ddot{v}\_{F\text{y}} - b\_{F\text{y}}k\_{F\text{y}}\overline{\mathbf{y}}\_{F} \\ &- d\left(R\_{\theta21}\cos\rho\cos\sigma + R\_{\theta22}\cos\rho\sin\sigma + R\_{\theta23}\sin\rho\right)\right) \frac{m\_{F}}{f\_{F}}.\end{split} \tag{33}$$

For the attitude stability, the following nonlinear *PD*<sup>2</sup> controller (34) proposed in [17] was implemented and tested in simulation for both the leader and the follower:

$$
\tau\_{\rm E} = \boldsymbol{\alpha} \times \boldsymbol{J} \boldsymbol{\alpha} + \mathbf{G}(\tilde{\boldsymbol{\alpha}}) - (\mu\_3 + \mu\_2 \mu\_1) \tilde{\mathbf{q}} - \mu\_3 \dot{\mathbf{J}} \dot{\tilde{\mathbf{q}}} - \mu\_2 \tilde{\boldsymbol{\alpha}}.\tag{34}
$$

where *μ*1, *μ*<sup>2</sup> and *μ*<sup>3</sup> are constants.

Next, we show the stability of the follower's translational part.

#### **4.3 Follower controller stability analysis**

The following candidate Lyapunov function is chosen for the stability analysis for the follower's translational part with the IBS controller:

$$V = \frac{1}{2} \left( \tilde{\mathbf{p}}\_F^T \tilde{\mathbf{p}}\_F + \tilde{\mathbf{v}}\_F^T \tilde{\mathbf{v}}\_F + k\_F \overline{\mathbf{p}}\_F^T \overline{\mathbf{p}}\_F \right) \tag{35}$$

and its time derivative is

$$
\dot{V} = \tilde{\mathbf{p}}\_F^T \dot{\tilde{\mathbf{p}}}\_F + \tilde{\mathbf{v}}\_F^T \dot{\tilde{\mathbf{v}}}\_F + k\_F \overline{\mathbf{p}}\_F^T \dot{\tilde{\mathbf{p}}}\_F. \tag{36}
$$

By substituting **p**\_ *<sup>F</sup>* ¼ **p**~*<sup>F</sup>* and Eqs. (28) and (30) into (36), Eq. (36) becomes

$$\dot{V} = -b\_F \tilde{\mathbf{p}}\_F^T \tilde{\mathbf{p}}\_F - c\_F \tilde{\mathbf{v}}\_F^T \tilde{\mathbf{v}}\_F \le \mathbf{0}.\tag{37}$$

Finally, (37) is less than zero provided *bF* and *cF* are positive diagonal matrices, i.e. *<sup>V</sup>*\_ <sup>&</sup>lt;0, <sup>∀</sup> **<sup>p</sup>**~*F*, **<sup>v</sup>**~*<sup>F</sup>* 6¼ 0 and *<sup>V</sup>*\_ ð Þ¼ <sup>0</sup> 0. It can be concluded from the positive definition of *V* and applying LaSalle theorem that a global asymptotic stability is guaranteed. This leads us to conclude that lim *<sup>t</sup>*!<sup>∞</sup> **p**~*<sup>F</sup>* ¼ 0 and lim *<sup>t</sup>*!<sup>∞</sup> **v**~*<sup>F</sup>* ¼ 0, which meets the position condition of (6).

#### **4.4 Leader IBS controller**

The leader is to track a desired trajectory **p***Ld*. Its IBS controller is developed by following the procedure described for the follower quadrotor.

The result is that the total force and horizontal position control laws *f <sup>L</sup>*, *uLx* and *uLy* can be written using Euler angles dynamic model representation as:

$$f\_L f\_L = \left(\ddot{\mathbf{z}}\_{Ld} + \mathbf{g} + \left(\mathbf{1} - b\_{Lx}\,^2 + k\_{Lx}\right)\ddot{\mathbf{z}}\_L + (b\_{Lx} + c\_{Lx})\ddot{v}\_{Lx} - b\_{Lx}k\_{Lx}\overline{\mathbf{z}}\_L\right) \frac{m\_L}{c\_{\theta L}\mathbf{c}\_{\phi L}}\tag{38}$$

$$\boldsymbol{\mu}\_{L\mathbf{x}} = \left(\ddot{\mathbf{x}}\_{Ld} + \left(\mathbf{1} - \boldsymbol{b}\_{L\mathbf{x}}\,^2 + \boldsymbol{k}\_{L\mathbf{x}}\right)\ddot{\mathbf{x}}\_{L} + \left(\boldsymbol{b}\_{L\mathbf{x}} + \boldsymbol{c}\_{L\mathbf{x}}\right)\ddot{\boldsymbol{v}}\_{L\mathbf{x}} - \boldsymbol{b}\_{L\mathbf{x}}\boldsymbol{k}\_{L\mathbf{x}}\overline{\mathbf{x}}\_{L}\right)\frac{\boldsymbol{m}\_{L}}{\boldsymbol{f}\_{L}}\tag{39}$$

$$
\mu\_{L\eta} = \left(\ddot{\mathbf{y}}\_{Ld} + \left(\mathbf{1} - \mathbf{b}\_{L\eta}\,^2 + \mathbf{k}\_{L\eta}\right)\ddot{\mathbf{y}}\_{L} + \left(\mathbf{b}\_{L\eta} + \mathbf{c}\_{L\eta}\right)\ddot{\mathbf{y}}\_{L\eta} - \mathbf{b}\_{L\eta}\mathbf{k}\_{L\eta}\overline{\mathbf{y}}\_{L}\right)\frac{m\_L}{f\_L}.\tag{40}
$$

The torque vector applied to the leader quadrotor *τLE* ∈ <sup>3</sup> is a nonlinear *PD*<sup>2</sup> controller (34). These leader controllers are used for path tracking tests.
