**Appendix**

*Stability analysis of position control without velocity measurements*

Considering the dynamic equations of translational system (6, Σ1), the dynamics of acceleration error can be rewritten as

$$
\dot{\eta}\_s = \frac{1}{m} \operatorname{Re}\_3 T - \operatorname{ge}\_3 - \dot{\eta}\_d \tag{39}
$$

The closed loop of position system is obtained by substituting the control intermediate vector *U* to the dynamics of acceleration.

$$\begin{cases} S\_{B\bar{\boldsymbol{\eta}}\_{1}} \begin{cases} \ddot{\boldsymbol{\eta}}\_{\boldsymbol{\iota}} = -k\_{\boldsymbol{\rho}} \tanh(\boldsymbol{\eta}\_{\boldsymbol{\iota}}) - k\_{\boldsymbol{d}} \tanh(\boldsymbol{\nu})\\ \dot{\boldsymbol{\nu}} = -A \tanh(\boldsymbol{\nu}) + B\dot{\boldsymbol{\eta}}\_{\boldsymbol{\iota}} \end{cases} \end{cases} \tag{40}$$

Consider the following candidate Lyapunov function

$$\begin{split} V\_{\text{RF}\_1} &= \frac{1}{2} \dot{\eta}\_e^T \dot{\eta}\_e + k\_\rho \log(\cosh(\eta\_e)) \\ &+ k\_\rho B^{-1} \log(\cosh(V)) + \gamma \dot{\eta}\_e (\tanh(\eta\_e) - \tanh(\nu)) \end{split} \tag{41}$$

where *γ* is a small positive parameter to be determined later. Taking the time derivative of Eq. (41) yields

$$\begin{split} \dot{V}\_{\text{BF}\_1} &= \frac{1}{2} \dot{\eta}\_e^{\top} \ddot{\eta}\_e + k\_\rho \tanh(\eta\_e^{\top}) \dot{\eta}\_e \\ &+ k\_d B^{-1} \tanh(\mathbf{v}^{\top}) \dot{\mathbf{v}} + \mathcal{Y} \dot{\eta}\_e (\tanh(\eta\_e) - \tanh(\mathbf{v})) \\ &+ \mathcal{Y} \dot{\eta}\_e^{\top} (\cosh^{-2}(\eta\_e) \dot{\eta}\_e - \cosh^{-2}(\mathbf{v}) \dot{\mathbf{v}}) \end{split} \tag{42}$$

substituting Eqs. (40)–(42) yields

$$\begin{aligned} \dot{\boldsymbol{\nu}}\_{\mathcal{H}\_{\mathcal{H}}} &= -\boldsymbol{k}\_{d} \boldsymbol{B}^{-1} \boldsymbol{A} \tanh(\boldsymbol{\nu}^{\top}) \tanh(\boldsymbol{\nu}) \\ &- \boldsymbol{k}\_{\boldsymbol{\rho}} \boldsymbol{\gamma} \tanh(\boldsymbol{\eta}^{\top}\_{\boldsymbol{\epsilon}}) \tanh(\boldsymbol{\eta}\_{\boldsymbol{\epsilon}}) \\ &+ \boldsymbol{k}\_{d} \boldsymbol{\gamma} \tanh(\boldsymbol{\nu}^{\top}) \tanh(\boldsymbol{\nu}) \\ &+ \boldsymbol{\gamma}(\boldsymbol{k}\_{\boldsymbol{\rho}} - \boldsymbol{k}\_{d}) \tanh(\boldsymbol{\eta}^{\top}\_{\boldsymbol{\epsilon}}) \tanh(\boldsymbol{\nu}) \\ &+ \boldsymbol{\gamma} \dot{\boldsymbol{\eta}}\_{\boldsymbol{\epsilon}}^{\top} \cosh^{-2}(\boldsymbol{\eta}\_{\boldsymbol{\epsilon}}) \dot{\boldsymbol{\eta}}\_{\boldsymbol{\epsilon}} \\ &+ \cosh^{-2}(\boldsymbol{\nu}) \boldsymbol{A} \tanh(\boldsymbol{\nu}) \boldsymbol{\gamma} \boldsymbol{\eta}\_{\boldsymbol{\epsilon}}^{\top} \\ &- \cosh^{-2}(\boldsymbol{\nu}) \dot{\boldsymbol{\eta}}\_{\boldsymbol{\epsilon}} \dot{\boldsymbol{\eta}}\_{\boldsymbol{\epsilon}}^{\top} \boldsymbol{\gamma} \boldsymbol{B} \end{aligned} \tag{43}$$

To achieve boundedness of the system, the last equation must be in the following form:

$$\begin{split} \dot{\mathcal{W}}\_{\rm B1}^{\cdot} &\leq -\frac{k\_{dm}a\_{m}}{b\_{m}} \|\tanh(v)\|\|^{2} \\ &-k\_{pm}\gamma\|\left|\tanh(\eta\_{\epsilon})\right|^{2} \\ &+\gamma\|\left|\dot{\eta}\_{\epsilon}\right|^{2}\|\cosh^{-2}(\eta\_{\epsilon})\| \\ &-b\_{m}\gamma\|\left|\dot{\eta}\_{\epsilon}\right|^{2}\|\cosh^{-2}(\eta\_{\epsilon})\|^{2} \\ &+a\_{M}\gamma\|\left|\dot{\eta}\_{\epsilon}\right|\|\left|\cosh^{-2}(v)\right|\|\left|\tanh(v)\right|\| \\ &+k\_{DM}\gamma\gamma\|\left|\tanh(v)\right|\|^{2} \end{split} \tag{44}$$

Consider the following inequalities:

desired orientation. In the second phase of our control design, barrier Lyapunov function has been employed to design the control torque that ensuring the tracking of the desired attitude derived at first phase of the control design. The stability of the closed loop system has been investigated and the simulation results have been provided to show the good performance of the proposed algorithm. Our next work will be to accomplish real time experiments for the

Considering the dynamic equations of translational system (6, Σ1), the dynamics of acceleration

3 3

The closed loop of position system is obtained by substituting the control intermediate vector

= tanh( ) tanh( )

*k kv*

log(cosh( )) (tanh( ) tanh( ))

 h

h

gh

*kB V v*

2 2

& && & & &

 h h

gh

*kB v v v*

( ( ) ()) cosh cosh


+ +-

&

where *γ* is a small positive parameter to be determined later. Taking the time derivative of Eq.

tanh( ) (tanh( ) tanh( ))

 h

*v v*

h

*e*

 h  h

&& <sup>=</sup> - - && *e d Re T ge <sup>m</sup>* (39)

*v A vB* (40)

(41)

(42)

1

= tanh( )

<sup>1</sup> = log(cosh( )) <sup>2</sup>

<sup>1</sup> <sup>=</sup> tanh( ) <sup>2</sup>

*T T BF e e p e e T d e e*

+

& & && &

+ -

*e e e*

 h h

+

*BF e e p e*

ìï - - <sup>í</sup> ïî - +

& & *e p ed*


+ +-

*p e e*

h

1

1

1


gh

*T*

*V k*

hh

& &

*T*

h h

*V k*

*BF*

*S*

Consider the following candidate Lyapunov function

1

1

substituting Eqs. (40)–(42) yields

h

&&

*Stability analysis of position control without velocity measurements*

proposed algorithm.

152 Recent Progress in Some Aircraft Technologies

error can be rewritten as

*U* to the dynamics of acceleration.

**Appendix**

(41) yields

$$\begin{aligned} \gamma \| |\dot{\eta}\_{\varepsilon}| |^2 \| \cosh^{-2}(\eta\_{\varepsilon}) \| &\le \gamma \| |\dot{\eta}\_{\varepsilon}| \| ^2 \\ \| a\_{\mathcal{M}} \gamma \| |\dot{\eta}\_{\varepsilon}| \| |\cosh^{-2}(v)| \| |\tanh(v)| | \\ &\le a\_{\mathcal{M}} \gamma \| \dot{\eta}\_{\varepsilon} \| \tanh(v) \| \end{aligned} \tag{45}$$

The inequality Eq. (44) can be rewritten in the following form:

$$\begin{split} \mathcal{V}\_{\text{BF}}^{\varepsilon} &\leq -\frac{k\_{dm}a\_{m}}{b\_{m}} \left\lVert \tanh(\upsilon) \right\rVert^{2} \\ &-k\_{pm}\gamma \left\lVert \tanh(\eta\_{\varepsilon}) \right\rVert^{2} + \gamma \left\lVert \eta\_{\varepsilon} \right\rVert^{2} \\ &-b\_{m}\gamma \left\lVert \left\lVert \eta\_{\varepsilon} \right\rVert^{2} \right\rVert \cosh^{-2}(\eta\_{\varepsilon}) \big\lVert \right^{2} \\ &+a\_{M}\gamma \left\lVert \left\lVert \eta\_{\varepsilon} \right\rVert \right\rVert \left\lVert \tanh(\upsilon) \right\rVert \\ &+k\_{DM}\gamma \left\lVert \tanh(\upsilon) \right\rVert^{2} \end{split} \tag{46}$$

where *kdm*,*k pm*,*am*,*bm* are the minimum values of the vectors *kd* ,*kp*,*A*,*B*, respectively, and *kDM* ,*kPM* ,*AM* ,*BM* are the maximum values of these vectors. In order to simplify the stability study of the above equation, we assume that both minimum and maximum values are equal. So, the expression (44) can be rewritten in the following matrix form:

$$\begin{aligned} \text{VBF}\_{\text{BF}} &\leq -\frac{1}{2} \underbrace{\left[\begin{|} \tanh(\eta\_{\epsilon}) \| \right] \right]^{T}}\_{\begin{subarray}{c} \Omega \\ \left[\begin{array}{c} \tau \lambda \left(\tau \right) \| \right] \end{array} \end{subarray}}\_{\begin{subarray}{c} \Omega \\ \left[\begin{array}{c} -\gamma \left(k\_{p} + k\_{d}\right) \\ -\gamma \left(k\_{p} + k\_{d}\right) \end{array} \right]} \xrightarrow{\begin{subarray}{c} \Omega \\ \left[\begin{array}{c} \tau \left(k\_{p} + k\_{d}\right) \\ \end{array} \right]} \frac{2\mu\_{\epsilon}}{3\delta} \text{I} \\ \left[\begin{|} \left[\begin{array}{c} \tanh(\eta\_{\epsilon}) \| \right] \end{array} \right]^{T} & \left[\begin{|} \eta\_{\epsilon} \| \\ \tanh(\upsilon) \| \right] \right]^{T} \\ \end{array} \begin{cases} \frac{\Omega}{2} \\ \frac{2B \cosh^{-2}(\eta)}{3} - \frac{\gamma A}{2} \\ \frac{\gamma}{3B} \end{cases} \end{aligned} \tag{47}$$
 
$$\begin{cases} \frac{2B \cosh^{-2}(\eta)}{3B} \left[ \begin{|aligned} \frac{\left[\begin{|} \eta\_{\epsilon} \| \end{|}}{2} \\ \frac{\left[\begin{|} \eta\_{\epsilon} \| \end{|}}{3B} - \begin{|} \eta\_{\epsilon} \\ \$$

Now, we define the parameters *kd* ,*kp*,*A*,*B* so as to ensure that the terms *Q*1,*Q*2,*β*1,*β*2 are positive definite. To achieve that, the inequality of linear matrix method is applied. For *Q*<sup>1</sup> > 0, this is equivalent to

$$\frac{2k\_dA}{3B} \ge 0 \text{ and } 2\gamma k\_p - \frac{3B\gamma^2 (k\_d + k\_p)}{2k\_dA} \ge 0 \tag{48}$$

For *Q*2 > 0, this is equivalent to

$$\frac{k\_{\bar{d}}A}{3B} \ge 0 \text{ and } \frac{\gamma B \cosh^{-2}(0)\nu}{3} - \frac{3B\gamma^2 A^2}{4k\_{\bar{d}}A} \ge 0\tag{49}$$

Equation (48) gives:

$$\frac{k\_p}{\left(k\_d + k\_p\right)^2} > \frac{3B\gamma}{4k\_dA} \tag{50}$$

A Hierarchical Tracking Controller for Quadrotor Without Linear Velocity Measurements http://dx.doi.org/10.5772/62442 155

and the Eq. (49) gives:

study of the above equation, we assume that both minimum and maximum values are equal.

Now, we define the parameters *kd* ,*kp*,*A*,*B* so as to ensure that the terms *Q*1,*Q*2,*β*1,*β*2 are positive definite. To achieve that, the inequality of linear matrix method is applied. For *Q*<sup>1</sup> > 0, this is

g

*d*

 g

*d*

*<sup>B</sup> k A* (48)

*<sup>B</sup> k A* (49)

*k k kA* (50)

<sup>2</sup> <sup>2</sup> 3( ) >0 2 > 0

<sup>2</sup> 2 2 cosh () 3 > 0 > 0

2

*dp d k B*

3 > ( )4

g


<sup>+</sup> - *<sup>d</sup> d p p*

*k A Bkk and k*

3 2

g

3 34 g

> + *p*

*kA B v B A and*


equivalent to

For *Q*2 > 0, this is equivalent to

Equation (48) gives:

So, the expression (44) can be rewritten in the following matrix form:

154 Recent Progress in Some Aircraft Technologies

$$\frac{\gamma A}{4k\_d} < \frac{1}{9} \tag{51}$$

since cosh−<sup>2</sup> (*v*)≤ −1. For *β*1 > 0, this is equivalent to

$$\frac{k\_d A}{3B} - \gamma k\_d \ge 0\tag{52}$$

This gives:

(47)

$$
\gamma = \frac{A}{3B} \tag{53}
$$

For *β*2 > 0, this is equivalent to

$$\frac{2B}{3} \ge 1\tag{54}$$

Since cosh−<sup>2</sup> (*v*)≤1, one can write

$$B = \frac{3}{2} \tag{55}$$

substituting Eqs. (53)–(55), yields,

$$A = 4.5\text{y}\tag{56}$$

Substituting Eqs. (56) and (55) into (50) yields

$$\frac{k\_p}{(k\_d + k\_p)^2} > \frac{9}{36k\_d} \tag{57}$$

Using Eq. (19), we can define the parameter *kd* as follows:

$$k\_d = \left\|\frac{T}{m}\right\| - k\_\mathcal{g} - kp\tag{58}$$

and

$$(k\_d + k\_p)^2 = (\|\frac{T}{m}\| - k\_\mathcal{g})^2\tag{59}$$

Substituting Eqs. (58) and (59) into Eq. (57), we obtain the second inequality expression

$$\mathfrak{M}k\_{\mathfrak{P}}(\left\|\frac{T}{m}\right\|-k\_{\mathfrak{g}}-k\_{\mathfrak{p}}) > (\left\|\frac{T}{m}\right\|-k\_{\mathfrak{g}})^2\tag{60}$$

By solving this inequality, we obtain the value of parameter *kp*

$$k\_p > \frac{||\frac{T}{m}|| - k\_g}{2} \tag{61}$$

and hence, the value of parameter *kd* can calculated as

$$k\_d = \left\|\frac{T}{m}\right\| - k\_\mathcal{g} - k\mathcal{p} \tag{62}$$

Finally, substituting Eq. (56) into Eq. (51) yields

$$
\gamma \le 0.31 \sqrt{k\_d} \tag{63}
$$

Now, we calculate the total thrust and the desired orientation. As it is presented previous, the control vector *<sup>U</sup>* <sup>=</sup> *<sup>R</sup> <sup>d</sup> <sup>e</sup>*3*<sup>T</sup> <sup>m</sup>* can be written in the standard form as

$$\frac{T^2}{m^2} = U^T U$$

which is equivalent to

$$T = \sqrt{\mu\_x^2 + \mu\_y^2 + \mu\_z^2} \tag{65}$$

Using Eq. (17), the vector *<sup>R</sup> <sup>d</sup> <sup>e</sup>*3 can be calculated as

A Hierarchical Tracking Controller for Quadrotor Without Linear Velocity Measurements http://dx.doi.org/10.5772/62442 157

$$\begin{aligned} R^d e\_i &= \left( \cos(\phi^d) \sin(\theta^d) \cos(\psi^d) + \sin(\phi^d) \sin(\psi^d) \right) \\ \begin{pmatrix} \cos(\phi^d) \sin(\theta^d) \sin(\psi^d) - \sin(\phi^d) \cos(\psi^d) \\ \cos(\phi^d) \cos(\theta^d) \end{pmatrix} \\ \begin{pmatrix} \underline{u}\_s m \\ \overline{T} \\ \underline{u}\_s m \\ \underline{u}\_s m \\ \underline{u}\_s m \\ \underline{u}\_s m \\ \underline{T} \end{pmatrix} \end{aligned} \tag{66}$$

This leads to obtain the values of *φ <sup>d</sup>* ,*θ <sup>d</sup>*

$$\phi^d = \sin^{-1}(\frac{\mu\_l m \sin(\psi^d) - \mu\_\sharp m \cos(\psi^d)}{T}) \tag{67}$$

and

(58)

(59)

(60)

(61)

(62)

and

156 Recent Progress in Some Aircraft Technologies

Substituting Eqs. (58) and (59) into Eq. (57), we obtain the second inequality expression

< 0.31 *<sup>d</sup>*

*<sup>m</sup>* can be written in the standard form as

2

Now, we calculate the total thrust and the desired orientation. As it is presented previous, the

*k* (63)

<sup>2</sup> <sup>=</sup> *<sup>T</sup> <sup>T</sup> U U <sup>m</sup>* (64)

<sup>222</sup> *T uuu* = *xyz* + + (65)

g

By solving this inequality, we obtain the value of parameter *kp*

and hence, the value of parameter *kd* can calculated as

Finally, substituting Eq. (56) into Eq. (51) yields

Using Eq. (17), the vector *<sup>R</sup> <sup>d</sup> <sup>e</sup>*3 can be calculated as

control vector *<sup>U</sup>* <sup>=</sup> *<sup>R</sup> <sup>d</sup> <sup>e</sup>*3*<sup>T</sup>*

which is equivalent to

$$\boldsymbol{\Theta}^d = \tan^{-1}(\frac{\boldsymbol{\mu}\_x \cos(\boldsymbol{\nu}^d) + \boldsymbol{\mu}\_z \sin(\boldsymbol{\nu}^d)}{\boldsymbol{\mu}\_z}) \tag{68}$$

$$U = \begin{bmatrix} u\_x \\ u\_y \\ u\_z \end{bmatrix} \tag{69}$$

#### **Author details**

Yassine Jmili1\*, Nuradeen Fethalla1 , Jawhar Ghomam1,2 and Maarouf Saad1

\*Address all correspondence to: nour32004@yahoo.com

1 Department of Electrical Engineering, Ecole de Technologie Superieure, Montreal, Canada

2 Department of Electrical Engineering, CEM-Lab, National School of Engineering of Sfax, Tunisia
