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Abstract

The paper focuses on two automatic systems for the attitude and position’s control of the microaerial vehicles—insect type by using a nonlinear dynamic model, which describes the motion of microaerial vehicles with respect to the Earth tied frame. The attitude control is adaptive type, with the estimation of the inertia moments’ matrix and of the dynamic damping coefficients’ matrix in two variants: by means of the attitude vector or by using the quaternion vector. The new resulting control architectures use a vector for the control of the microaerial vehicles’ attitude, a proportional-derivative linear dynamic compensator, an error vector (whose elements are the estimated deviations of the inertia moments and dynamic damping coefficients with respect to the real ones), and the Lyapunov theory. In the two variants of the adaptive control, the control law is represented by the command aerodynamic moments and the wing rotation’s command vector, respectively; the control law for the microaerial vehicle position’s control is deduced in the same way. The two obtained control systems are validated by complex numerical simulations.

Keywords

Microaerial vehicle adaptive control attitude quaternion wing

Introduction

The microaerial vehicles (MAVs) are very interesting aerial vehicles for researchers from the aerospace domain or from other related areas: mechatronics, automation, electronics, and so on. In the last years, the studies and the research have evolved in several directions: aerodynamics, flight dynamics and automatic control, navigation, sensors, transducers, actuators, state observers.

In terms of research on aerodynamics and flight dynamics, studies by Deng et al.,^1,2 Schenato,³ and Paranjape et al.⁴ can be mentioned. Considerable progress has been made relative to the MAVs’ modeling, attitude’s stabilization and flight’s control, as well as in the area of improvement and adaptation to MAVs’ flight of the control algorithms (optimal or adaptive algorithms) used in the flight control of a large aerospace vehicles’ class in terms of external or internal disturbances.^2–13

Progress has been also made towards: (1) the modeling and the manufacturing of the miniaturized highly performing servo-actuators (thorax of the insect-type MAVs) to achieve the wing’s beat motion in the case of flying mini-robots and to command the MAVs’ motion through the modification of the wing’s beat and attack angles^14–16; (2) the manufacturing of miniaturized sensors and transducers, which merge the signals provided by the accelerometers, gyro, optical, or magnetic sensors; (3) the design and software implementation of the linear or nonlinear state observers, which serve the navigation and flight control systems.^4,7,17–21

The MAVs can be regarded sometimes as the physical models of the insects. Such mechanisms generally consist of three subsystems: a command subsystem (electrical engine or piezoelectrical actuator), a wing actuation equipment (the cinematic mechanism), and a controller. The recent studies and researches have culminated in the development of efficient flying robots, but, from our information, none of them have adaptive controllers based on the attitude or the quaternion vectors, Lyapunov theory, proportional–derivative (PD) dynamic compensators, and on the estimation of the inertia moments’ matrices and of the dynamic damping coefficients; this is achieved in this paper, and it is interesting to see if such adaptive controllers can guarantee the control of MAVs’ attitude and position. Thus, our aim is to design new adaptive systems for the control of the MAVs by using the attitude vector or the quaternion one.

The paper is organized as follows: the dynamics of MAVs is presented in the second section; the design of the automatic systems for the control of MAVs’ attitude and position, by using the attitude vector or the quaternion vector, is presented in the third section; in the fourth section of the paper, complex simulations to validate the proposed automatic control systems are performed and the obtained graphical characteristics are analyzed; finally, some conclusions are shared in the last section.

Dynamics of the microaerial vehicles

The dynamics of MAVs is generally nonlinear while in the case of the gliding flight, the dynamics is linear. The uncertainties caused by the MAV dynamics’ incomplete knowing lead to the necessity of a robust controller. In these circumstances, the Lyapunov direct method is used.

Let us consider $Θ = [ϕ θ ψ] T$ as the attitude vector (the angular position of MAV) and $ω p = Θ \cdot = ⌊ ϕ \cdot θ \cdot ψ \cdot ⌋ T$ as the MAV angular rates’ vector. We also consider oxyz – the MAV tied frame and OXYZ – the Earth tied frame (Darboux frame) having the axis OX tangent to the parallel and East oriented, OY – tangent to the meridian and North oriented, and OZ – the locus vertical line (Zenith oriented); the axes of the frame oxyz are oriented as follows: ox – the longitudinal axis (pointing the flight direction), oy – the axis oriented towards the right wing, and oz – perpendicular to the plane oxy and downward oriented. $ϕ, θ, ψ$ are respectively the roll, pitch, and yaw (direction) angles, i.e. the angles of the MAV tied frame with respect to the Earth tied frame. Also, let us denote p = [XYZ]^T as the position vector (the vector of linear coordinates) with respect to the Earth tied frame and $p \cdot = V p = [V X V Y V Z] T = ⌊ X \cdot Y \cdot Z \cdot ⌋ T$ as the vector of linear velocities, oriented after the axes of the same frame. Let us consider $P b = [xyz] T$ as the vector of the MAVs’ linear coordinates with respect to the oxyz frame, $V b = [V x V y V z] T$ as the vector of linear velocities with respect to the MAV tied frame (oxyz), and $ω b = [ω x ω y ω z] T$ as the vector of MAVs’ angular rates with respect to the axes of the oxyz frame.

The transformation equations of the position vectors and angular rates’ vector between the two frames (oxyz and OXYZ) are, respectively

p = R p b, p \cdot = V p = R V b

(1) with R being the rotation matrix (the successive rotations

R ψ, R θ, R ϕ

of MAV with the angles

ψ, θ, ϕ

with respect to the OXYZ frame)¹⁸;

R = (R ψ R θ R ϕ) - 1

with

R ψ = [\cos ψ - \sin ψ 0 \sin ψ \cos ψ 0001], R θ = [1000 \cos θ \sin θ 0 - \sin θ \cos θ], R ϕ = [010 \cos ϕ 0 - \sin ϕ - \sin ϕ 0 - \cos ϕ]

(2)

Similarly, for the rotation motion,¹⁸ we have

ω p = Θ \cdot = W - 1 ω b, ω b = W ω p = W Θ \cdot

(3) with the matrix

W = [10 - \sin θ 0 \cos ϕ \sin ϕ \cos θ 0 - \sin ϕ \cos ϕ \cos θ]

(4)

The resultants of the forces and moments acting upon the MAVs’ body are $F \to b$ and $M \to b$

F \to b = {F \to}_{b}^{a} + {F \to}_{b}^{g} + {F \to}_{b}^{v}, M \to b = {M \to}_{b}^{a} + {M \to}_{b}^{g} + {M \to}_{b}^{v}

(5) where the first component is the aerodynamic one, the second is the damping component, while the third one is associated with the weight; the aerodynamic component of the resultant force and moment is induced by the left wing

{F \to}_{b}^{al}

and by the right wing

{F \to}_{b}^{ar}

{F \to}_{b}^{a} = {F \to}_{b}^{al} + {F \to}_{b}^{ar}, {M \to}_{b}^{a} = r \to l \times {F \to}_{b}^{al} + r \to r \times {F \to}_{b}^{ar}

(6) where

r \to l

and

r \to r

are the position vectors of the left and right wings’ pressure centers with respect to the MAVs’ mass center (point o); the aerodynamic force and moment produced by the two wings are¹

F_{a}^{a} = [F_{d}^{l} \cos φ L + F_{d}^{r} \cos φ R F_{d}^{l} \cos φ L - F_{d}^{r} \cos φ R F_{l}^{l} + F_{l}^{r}], M_{a}^{a} = r a [- F_{d}^{l} \cos φ L - F_{d}^{r} \cos φ R - F_{d}^{l} \sin φ L - F_{d}^{r} \sin φ R F_{d}^{l} + F_{d}^{r}]

(7) where r_a is the position vector of the resultant aerodynamic forces

{F \to}_{a}^{a}, {F \to}_{l}^{l}, {F \to}_{l}^{r}

, the lift forces

{F \to}_{d}^{l}, {F \to}_{d}^{r}

are the wings’ drag forces,

φ L

and

φ R

are the angles of the wings in the beat plane (“l”, “L” – left, “r”, “R” – right); denoting

M

_φ as the transformation (rotation) matrix of the wing in the beat plane, we obtain

F_{b}^{a} = M φ F_{a}^{a}, M_{b}^{a} = - M φ r ab F_{a}^{a} + M φ M_{a}^{a}

(8) where r_ab is the position vector of the force

{F \to}_{a}^{a}

with respect to the MAVs’ mass center.

The modules of the dynamic damping components from equation (5) are

F_{b}^{v} = - f v V b, M_{b}^{v} = 0

(9) where f_v is the dynamic damping coefficient; the components due to the gravity are

F_{b}^{g} = - m 0 R - 1 g = - m 0 R - 1 [00 g] T, M_{b}^{g} = 0

(10) where g is the gravitational acceleration vector, while m₀ is the MAVs’ mass.

The block diagram of the ensemble wing i.e. MAVs’ body is presented in Figure 1.

Figure 1.

Block diagram of the ensemble wing – MAVs’ body.

The MAVs’ dynamics is described by the forces and moments’ equilibrium equations¹⁸

m 0 (\cdot \to V b + ω \to b \times V \to b) = F \to b

(11)

\frac{\partial K \to b}{\partial t} + ω \to b \times K \to b = M \to b

(12) where K _b = J_bω _b is the rotation kinetic moment and J_b is the inertia moment matrix with respect to the axes of the MAVs’ tied frame:

J b = [J xx 0 - J xz 0 J yy 0 - J xz 0 J zz]

Neglecting the second term in equation (11) and taking into account equations (1) and (5) with equations (9) and (10), equation (11) becomes

p ·· = - \frac{f v}{m 0} p \cdot - g + \frac{1}{m 0} R F_{b}^{a}

(13)

Equation (12) may also be written in the form

K \cdot b + ω_{b}^{\times} K b = M_{b}^{a}

(14)

M _a = M_b^a (see equations (5), (9), and (10)) and $ω_{b}^{\times}$ has the form⁵

ω_{b}^{\times} = [0 - ω z - ω y ω z 0 - ω x - ω y - ω x 0] = (3) W Θ \cdot \times

(15)

K b = J b ω b = (3) J b W Θ \cdot, K \cdot b = (J b W) Θ ·· + (J b W \cdot) Θ \cdot

(16)

With these, equation (14) gets the form: $(J b W) Θ ·· + (J b W \cdot) Θ \cdot + W Θ \cdot \times (J b W) Θ \cdot = M_{b}^{a}$ or, by using the notations: $m (Θ) = J b W, n (Θ, Θ \cdot) = J b W \cdot, σ (Θ, Θ \cdot) = W Θ \cdot \times m (Θ) Θ \cdot$ , it becomes

m (Θ) Θ ·· + n (Θ, Θ \cdot) Θ \cdot = M_{b}^{a} - σ (Θ, Θ \cdot)

(17)

σ (Θ, Θ \cdot)

is a vector whose components are disturbing moments. The matrix

W \cdot = W \cdot (Θ, Θ \cdot)

is obtained by derivation of the matrix W having the form (4)

W \cdot = [00 - Δ \cdot ¯ θ \cos (Δ θ ¯) 0 - Δ \cdot ¯ ϕ \sin (Δ ϕ ¯) Δ \cdot ¯ ϕ \cos (Δ ϕ ¯) cos (Δ θ ¯) - Δ \cdot ¯ θ \sin (Δ ϕ ¯) sin (Δ θ ¯) 0 - Δ \cdot ¯ ϕ \cos (Δ ϕ ¯) - Δ \cdot ¯ ϕ \sin (Δ ϕ ¯) cos (Δ θ ¯) - Δ \cdot ¯ θ \cos (Δ ϕ ¯) sin (Δ θ ¯)]

(18)

“Δ” means the variation of the angles or angular rates; the line above the variables will be explained later in this paper.

Automatic control of the MAVs’ attitude and position

Control of the MAVs’ attitude and position using the attitude vector

In Figure 2, we present the control system’s block diagram. The control vector is $Δ u = [φ R α R φ L α L] T$ ;

u = u 0 + Δ u, u 0 = [g φ g α g φ g α] T

(19) where u ₀ is the permanent component of the command vector u ,

u, g φ = φ 0 \cos ω t, g α = α 0 \cos ω t,

φ 0

is the amplitude of the beat angle φ,

α 0

is the amplitude of the wing’s attack angle α, ω = 2πf = 2π/T, f is the frequency, T is the period of the wing’s beat; on the component u ₀ we superpose the component³

Δ u = G Δ v, v = [v_{1}^{r} v_{2}^{r} v_{3}^{r} v_{1}^{l} v_{2}^{l} v_{3}^{l}] T

(20) where v is the vector of cinematic parameters (the vector of virtual commands); we have chosen three cinematic parameters for each wing. The matrix G (4 × 6) has the form²

G = [g 1 0 g 3 0000 g 2 0000000 g 1 0 g 3 0000 g 2 0], g 1 = g 2 = \frac{π}{15} \sin 3 ω t, g 3 = \frac{π}{15}

(21)

Figure 2.

The general block diagram of the MAVs’ control system.

The component Δ u = [ΔU₁ ΔU₂]^T is the vector of the voltages generated by the servo-actuators (SA) for the rotation of the wings with the supplementary angles Δφ and Δα.

In Figure 2, we denoted with x – the state vector consisting of the MAVs’ linear and angular displacements and velocities; h( x ) is the control law. Considering that all the physical variables are of harmonic type (period T), we will work with their mean values, calculated in a time equal to period T; considering a variable B, we use the notation $B ¯$ for the mean value of B and $Δ B ¯$ for the variation of $B ¯$ with respect to B₀ (the reference value of B). The variation $Δ A ¯$ (the variation of the mean aerodynamic vector) has, as components, the mean values of the aerodynamic forces’ vector $(Δ {F ¯}_{b}^{a})$ and of the aerodynamics moments’ vector $(Δ {M ¯}_{b}^{a})$ , i.e. $Δ A = ¯ [Δ {F ¯}_{b}^{a} Δ {M ¯}_{b}^{a}] T$ .

The linearization of the function, which expresses the dependency between $A = ¯ [{F ¯}_{b}^{a} {M ¯}_{b}^{a}] T$ and v is difficult to be obtained analytically

A ¯ = A ¯ 0 + Δ A ¯ ≅ A ¯ 0 + A ¯ 1 Δ v, A ¯ 1 = [A F A M] T

(22)

The matrices $A ¯ 0$ and $A ¯ 1$ can be deduced by selecting random vectors v = v ₀ + Δ v = Δ v as inputs for an aerodynamic model of subsystem: SA = Wing.^1,3 Thus, in Schenato³ for a MAV, insect type, we have

A ¯ 0 = [00 G 0 000] T A F = G 0 [00 0.1 0.1 00000000 0.15 0.15 0000] A M = G 0 l [0.05 - 0.05 000000 0.01 - 0.01 0.05 0.05 00 0.06 - 0.06 00]

(23) where G₀ = m₀g is the weight and l is the length of MAV.

From equations (20) and (22) one successively obtains where ${A ¯}_{1}^{+}$ is the pseudo-inverse of the matrix $A ¯ 1$ ; the same notations are used for the matrices A_F and A_M; equation (24) is equivalent to one of the following (for the translation and rotation motions)

Δ {F ¯}_{b}^{a} = (A F G +) Δ u ¯, Δ {M ¯}_{b}^{a} = (A M G +) Δ u ¯

(25)

In Figure 3, we present the block diagram of adaptive system for the control of the MAVs’ attitude and position. The inner loop achieves the control of the attitude, while the external one achieves the control of the MAVs’ position.

Figure 3.

Block diagram of the system for the control of the MAVs’ attitude and position.

Taking into account that $W = W (Θ), W \cdot = W \cdot (Θ, Θ \cdot)$ and, consequently, $m = m (Θ), n = n (Θ, Θ \cdot),$ we design an adaptive control law for the control of the attitude. Thus, using the methodology from the sliding control approach,⁴ we define a vector-type variable h _a consisting of the angular rate error vector $e \cdot a$ and the attitude error e _a = Θ _d − Θ, with Θ _d being the desired attitude; $h a = e \cdot a + λ a e a$ , with λ _a being the diagonal matrix whose elements (positive defined) are the gain coefficients; h _a is canceled in the same time with e _a and $e \cdot a$ (in the same time with the canceling of the angular rate and with the attitude’s stabilization).

We denote with a, the vector whose components are the elements of the matrices $m (Θ)$ and $n (Θ, Θ \cdot)$

a = [m ij n ij] T, i, j = 1, 3 ¯

(26) The estimate of this vector is

a \land = [m \land ij n \land ij] T, i, j = 1, 3 ¯

(27) with

a ~ = a \land - a

and the composed vector-type variable h _a, we use the Lyapunov function

V a = \frac{1}{2} h_{a}^{T} m (Θ) h a + \frac{1}{2} a ~ T Γ - 1 a ~

(28) where Γ is a diagonal matrix having the gain coefficients as elements. By time derivation of equation (28), we get:

V \cdot a = h_{a}^{T} m (Θ) h \cdot a + \frac{1}{2} h_{a}^{T} \frac{d m (Θ)}{d t} h a + ~ \cdot a T Γ - 1 a ~

. Replacing into this equation

e ·· a = Θ ·· d - Θ ·· = - Θ ·· and \frac{d m (Θ)}{d t}

with its expanded expression:

\frac{d m (Θ)}{d t} = J b W \cdot = n (Θ, Θ \cdot), h \cdot a = e ·· a + λ a e \cdot a

, the following equation is resulted

Or, taking into account equation (17), we get

V \cdot a = - h_{a}^{T} [M_{b}^{a} - σ (Θ, Θ \cdot) - n (Θ, Θ \cdot) (Θ \cdot d - e \cdot a) - m (Θ) λ a e \cdot a - \frac{1}{2} n (Θ, Θ \cdot) h a] + ~ \cdot a T Γ - 1 a ~ = - h_{a}^{T} [M_{b}^{a} - σ (Θ, Θ \cdot) - n (Θ, Θ \cdot) Θ \cdot d - (m (Θ) λ a - n (Θ, Θ \cdot)) e \cdot a - \frac{1}{2} n (Θ, Θ \cdot) h a] + ~ \cdot a T Γ - 1 a ~

(29)

Above, we have taken into account the equation: $Θ \cdot d = Θ ·· d = 0$ because $Θ d$ is constant. Now, we neglect $σ (Θ, Θ \cdot)$ and we choose $M_{b}^{a}$ in the following form

M_{b}^{a} = k a h a + n \land (Θ, Θ \cdot) Θ \cdot d + [m \land (Θ)] λ a - n \land (Θ, Θ \cdot) e \cdot a + \frac{1}{2} n \land (Θ, Θ \cdot) h a = k a h a + h ad

(30) where

m \land (Θ)

and

n \land (Θ, Θ \cdot)

are the estimated matrices (with the coefficients from equation (27)), while k_a is the diagonal matrix whose elements (all positive) are the gain coefficients; h _ad is the adaptive component of the control law. Replacing equation (30) into (29), we obtain

V \cdot a = - h_{a}^{T} [k a h a + n ~ (Θ, Θ \cdot) Θ \cdot d + [m ~ (Θ) λ a - n ~ (Θ, Θ \cdot)] e \cdot a + \frac{1}{2} n ~ (Θ, Θ \cdot) h a] + ~ \cdot a T Γ - 1 a ~

(31) where

m ~ (Θ) = m ̂ (Θ) - m (Θ), n ~ (Θ, Θ \cdot) = n ̂ (Θ, Θ \cdot) - n (Θ, Θ \cdot)

With the parameterization matrix Y verifying the equation

Y a ~ = n ~ (Θ, Θ \cdot) Θ \cdot d + [m ~ (Θ) λ a - n ~ (Θ, Θ \cdot)] e \cdot a + \frac{1}{2} n ~ (Θ, Θ \cdot) h a

(32) equation (31) becomes

V \cdot a = h_{a}^{T} k a h a - h_{a}^{T} Y a ~ + ~ \cdot a T Γ - 1 a ~

(33)

Now, we impose

h_{a}^{T} Y = ~ \cdot a T Γ - 1

(34) which, by right multiplication with Γ, becomes:

h_{a}^{T} Y Γ = ~ \cdot a T

or, because Γ^T= Γ

~ \cdot a = Γ Y T h a

(35)

Taking into account equation (34), equation (33) takes the form: $V \cdot a = h_{a}^{T} k a h a < 0$ . The closed-loop system is asymptotic stable. Thus, $h a \to 0$ and $a ~ \to 0 (a \land \to a),$ i.e. $m \land (Θ) \to m (Θ)$ $n \land (Θ, Θ \cdot) \to n (Θ, Θ \cdot)$ . Therefore, $e a \to 0 (Θ, Θ d, Δ Θ \to Δ Θ d)$ and $e \cdot a \to 0 (Θ \cdot \to Θ \cdot d, Δ Θ \cdot \to Δ Θ \cdot d), M_{b}^{a} \to n (Θ, Θ \cdot) Θ \cdot d = h ad$ .

The block diagram of the adaptive system for the control of MAVs’ attitude is presented in Figure 4, where the variables for which we previously omitted the sign “Δ” (variation) have now this sign. The variables are the mean ones, i.e. with a line above them. The component $Δ u ¯ = [Δ φ ¯ R Δ α ¯ R Δ φ ¯ L Δ α ¯ L] T$ is calculated with the second equation (25).

Figure 4.

Block diagram of the automatic system for the control of the MAVs’ attitude.

To design the system for the adaptive control of the MAVs’ position (described by equation (13)) we choose the Lyapunov function: $V p = \frac{1}{2} h_{p}^{T} h p, h p = e \cdot p + λ p e p,$ with λ _p being the diagonal matrix whose positive elements are the gain coefficients, $e p = p d - p, e \cdot p = p \cdot d - p \cdot, e ·· p = p ·· d - p ··$ ( p _d , $p \cdot d$ , and $p ·· d$ are the desired vectors of the coordinates, velocities, and accelerations, respectively). By time derivation of V_p and taking into account equation (13), we obtain: $V \cdot p = h_{p}^{T} h \cdot p = h_{p}^{T} (e ·· p + λ p e \cdot p) = h_{p}^{T} (p ·· p - p ·· + λ p e \cdot p) = - h_{p}^{T} \frac{1}{m 0} R F_{b}^{a} + h_{p}^{T} (p ·· d + λ p e \cdot p + \frac{f v}{m 0} p \cdot + g)$ .

Choosing vector $F_{b}^{a}$ such that

\frac{1}{m 0} R F_{b}^{a} = k p h p (p ·· d + λ p e \cdot p + g + \frac{f v}{m 0} p \cdot)

(36) with k_p being the diagonal matrix with positive elements (the gain coefficients), it results

V \cdot p = h_{p}^{T} k p h p < 0

(37)

As a consequence, the closed-loop system is asymptotically stable. Replacing equation (36) into equation (13), we obtain the following

e ·· p + λ p e \cdot p + k p h p = 0

(38)

The condition (37) ensures the convergence of the variable h _p , i.e.

h p = e \cdot p + λ p e p \to 0

(39)

Because λ_p is a diagonal and positive-defined matrix, from equation (39) it yields that h_p, e_p, and $e \cdot p$ converge to zero simultaneously $(p \to p d)$ and $(p \cdot \to p \cdot d)$ . Therefore, according to equation (38), one obtains the convergence $e \cdot\cdot p \to 0$ in the same time with the convergence of the other variables $(p \cdot\cdot \to p \cdot\cdot d)$ .

In Figure 5, we present the block diagram of the automatic system for the control of the MAVs’ position; I₃ is the (4 × 6) identity matrix. As in Figure 4, the variables represent the mean variations. The command vector $Δ u ¯ = [Δ φ ¯ R Δ α ¯ R Δ φ ¯ L Δ α ¯ L] T$ is calculated with the first equation (25).

Figure 5.

Block diagram of the automatic system for the control of the MAVs’ position.

Control of the MAVs’ attitude and position using the quaternion vector

In this section, we present an adaptive algorithm for the control of the MAVs’ attitude by using the quaternion vector.

The Euler’s theory shows that a rigid body’s attitude may be modified through the rotation of the rigid solid with respect to a proper axis (Euler axis). This axis, fixed with respect to the body of the rigid solid, is stationary in the inertial space. We consider the unitary vector $ε = [ɛ 1 ɛ 2 ɛ 3] T$ along the Euler axis, with ɛ₁, ɛ₂, ɛ₃ as the directory cosines of this vector relative to the inertial frame; the following equation is verified: $ɛ_{1}^{2} + ɛ_{2}^{2} + ɛ_{3}^{2} = 1$ . We define the components of the quaternion vector⁵

q 1 = ɛ 1 \sin \frac{γ}{2}, q 2 = ɛ 2 \sin \frac{γ}{2}, q 3 = ɛ 3 \sin \frac{γ}{2}, q 4 = ɛ 4 \cos \frac{γ}{2}

(40) where γ is the rigid solid’s rotation angle relative to the Euler’s axis. The quaternions verify the condition:

q_{1}^{2} + q_{2}^{2} + q_{3}^{2} + q_{4}^{2} = 1

. The first three components (q₁, q₂, q₃) form the quaternion:

q = [q 1 q 2 q 3] T = ε \sin \frac{γ}{2}

. This verifies the equation

q \cdot = ω X q; ω X = \underset{ω X}{\underset{︸}{[0 - ω 3 ω 2 ω 3 0 - ω 1 - ω 2 ω 1 0]}}, ω P = Θ \cdot = [ω 1 ω 2 ω 3] = [ϕ \cdot θ \cdot ψ \cdot]

(41) We considered the Earth tied frame (OXYZ) as the inertial frame.

The rotation matrix of the MAV tied frame (oxyz) relative to the Earth tied frame can be expressed by means of equation⁷: $R = R (q, q 4) = (q_{4}^{2} - q T q) I 3 + 2 qq T - 2 q 4 q X$ .

A form of the differential equation of the quaternion q is⁶

q \cdot = F (q) ω

(42) with F( q ) in the following form²¹

F (q) = \frac{1}{2} {I 3 + q X qq T - \frac{1}{2} [1 - q T q] I 3}, q X = [0 - q 3 q 2 q 3 0 - q 1 - q 2 q 1 0]

(43)

The MAVs’ dynamics (equation (17)) is equivalent to the following one

m (Θ) ω \cdot + n (Θ, Θ \cdot) ω = M_{b}^{a} - ω_{b}^{X} K b

(44)

By time derivation of equation (42), we obtain: $q \cdot\cdot = F (q) ω \cdot + F \cdot (q, q \cdot) F - 1 (q, q \cdot) ω$ , or, replacing $ω$ from equation (42) we get

q ·· = F (q) ω \cdot + F \cdot (q, q \cdot) F - 1 (q) q \cdot

(45) where the expression of

F \cdot (q, q \cdot)

is obtained by time derivation of equation (43), i.e.

F \cdot (q, q \cdot) = \frac{1}{2} [q \cdot X + 2 qq \cdot T - q \cdot T q I 3]

(46)

By left multiplying equation (45) with $m (Θ) F - 1 (q)$ and eliminating $m (Θ) ω \cdot$ between equation (44) and the new form of equation (45), we obtain

M (q) q ·· + N (q, q \cdot) q \cdot = M_{b}^{a} - ω_{b}^{a} K B

(47) where

M (q) = m (Θ) F - 1 (q), N (q, q \cdot) = [n (Θ \cdot, Θ \cdot) - M (q) F \cdot (q, q \cdot)] F - 1 (q)

(48)

By using vectors (26) and (27), we form the vectors ã = â–a and $q ~ = q d - q,$ with q _d being the desired quaternion; $q ~$ should not be interpreted as the quaternion error but the deviation of the vector q with respect to the vector q _d . We define the vector-type variable as

h a = q ~ \cdot + λ a q ~ = q \cdot d - q \cdot + λ a q ~ = q \cdot r - q \cdot, q \cdot r = q \cdot d + λ a q ~

(49) The significance of the matrix λ _a has been presented above.

We define the Lyapunov function⁸

V a = \frac{1}{2} h_{a}^{T} M (q) h a + a ~ T Γ - 1 a ~

(50) where Γ is the diagonal matrix whose elements (positive defined) are the gain coefficients; ã has the above presented form, with ã being the estimation of the vector a, its components being the elements of the matrices M( q ) and

N (q, q \cdot)

By time derivation of equation (50), we obtain

V \cdot a = h_{a}^{T} M (q) h \cdot a + h_{a}^{T} \frac{d M (q)}{d t} h a + ~ \cdot a T Γ - 1 a ~ = h_{a}^{T} [M (q) q ·· r - M (q) q ··] + N (q, q \cdot) h a + ~ \cdot a T Γ - 1 a ~ = h_{a}^{T} [M (q) q ·· r + N (q, q \cdot) q \cdot - (M_{b}^{a} - ω_{b}^{X} K b)] + N (q, q \cdot) h a + ~ \cdot a T Γ - 1 a ~ \Leftrightarrow V \cdot a = h_{a}^{T} [M (q) q ·· r + N (q, q \cdot) (q \cdot r - h a) - (M_{b}^{a} - ω_{b}^{X} K b)] + N (q, q \cdot) h a + ~ \cdot a T Γ - 1 a ~ = - h_{a}^{T} [- M (q) q ·· r - N (q, q \cdot) \cdot q \cdot r {+ (M}_{b}^{a} - ω_{b}^{X} K b)] + ~ \cdot a T Γ - 1 a ~

(51)

Choosing

M_{b}^{a} = ω_{b}^{X} K \land b + M \land (q) q ·· r + N \land (q, q \cdot) q \cdot r + k a h a

(52) where

M \land (q), N \land (q, q \cdot)

, and

ω_{b}^{X} K \land b

(negligible term) represent the estimates of

M (q), N (q, q \cdot)

, and

ω_{b}^{X} K b, k a

is the diagonal matrix having constant and positive elements; the expression of

V \cdot a

becomes

V \cdot a = - h_{a}^{T} [M ~ (q) q ·· r + N ~ (q, q \cdot) q r + k a h a] + ~ \cdot a T Γ - 1 a ~

(53) where

M ~ (q) = M \land (q) - M (q), N ~ (q, q \cdot) = N \land (q, q \cdot) - N (q, q \cdot)

Writing

M ~ (q) q ·· r + N ~ (q, q \cdot) q \cdot r = Y a ~

(54) equation (53) takes the form

V \cdot a = - h_{a}^{T} k a h a - h_{a}^{T} Y a ~ + ~ \cdot a T Γ - 1 a ~

(55)

Now, we impose the fulfillment of the condition

h_{a}^{T} Y = ~ \cdot a T Γ - 1

(56) which, by right multiplying with Γ, becomes:

h_{a}^{T} Y Γ = ~ \cdot a T

~ \cdot a = Γ Y T h a

(57) because

Γ = Γ T

Taking into account equation (56), equation (55) becomes: $V \cdot a = - h_{a}^{T} k a h a < 0$ . As a consequence, the closed-loop system is asymptotically stable.

In Figure 6, we present the structure of the adaptive system for the control of the MAVs’ attitude using the quaternion vector and the estimation of the inertia moments’ matrix $m (Θ)$ and of the dynamic damping coefficients’ matrix $n (Θ, Θ \cdot)$ ; we mention that equation (52) is used under the form

M_{b}^{a} = M \land (q) q ·· r + N \land (q, q \cdot) q \cdot r + k a h a

(58)

Figure 6.

Block diagram of the automatic system for the adaptive control of the MAV by using the quaternion vector.

In Figure 6, we denoted with Δ x the variation of the angular variable x and with $Δ x ¯$ the mean value of this variable. The vector $Δ u ¯ = [Δ φ ¯ R Δ α ¯ R Δ φ ¯ L Δ α ¯ L] T$ is the command vector, which produces the command aerodynamic moment $Δ {M ¯}_{b}^{a}$ and this is calculated with equation (25) where $A_{M}^{+}$ is the pseudo-inverse of matrix A_M.

Numerical simulation results

We choose the following numerical values for a MAV – insect type²: m₀ = 100 mg, f_v = 0.01 kg/s, l = 10 mm, f = 150 Hz, T = 1/f = 0.0066 s, J_xx = 2 × 10 ^− 7kg m², J_xz = 6 × 10 ^− 7kg m², J_yy = 618 × 10 ^− 7kg m², J_zz =8 × 10 ^− 7kg m². The states’ initial values are: $Δ Θ ¯ (0) = [Δ ϕ ¯ (0) Δ θ ¯ (0) Δ ψ ¯ (0)] T = [- 0.5 - 0.90.8] T$ deg, $Δ Θ \cdot = [Δ \cdot ¯ ϕ (0) Δ \cdot ¯ θ (0) Δ \cdot ¯ ψ (0)] T = [2.54.50.9] T$ deg/s, $Δ p ¯ (0) = [Δ X ¯ (0) Δ Y ¯ (0) Δ Z ¯ (0)] T = [0.180.350.09] T m, Δ \cdot ¯ p [Δ V ¯ X (0) Δ V ¯ Y (0) Δ V ¯ Z (0)] = [Δ \cdot ¯ X (0) Δ \cdot ¯ Y (0) Δ \cdot ¯ Z (0)] T = [000] T$ m/s. The following matrices have been used: λ_a = λ_b = 5I₃ and k_a = k_a = 10I₃ Matrix G has the form (21), while matrices A_F and A_M have the forms (23).

The architectures in Figures 4 and 5 were software implemented in Matlab/Simulink and the characteristics in Figures 7 and 8 were obtained. The components $Δ F ¯ x, Δ F ¯ y, Δ F ¯ z,$ are nondimensional; $Δ F ¯ x = Δ {F ¯}_{bx}^{a} / G 0$ , $Δ F ¯ y = Δ {F ¯}_{by}^{a} / G 0$ , $Δ F ¯ z = Δ {F ¯}_{bz}^{a} / G 0$ with G₀ = m₀g being the weight of the MAV – insect type and $Δ {F ¯}_{bx}^{a}, Δ {F ¯}_{by}^{a}, Δ {F ¯}_{bz}^{a} -$ the components of the vector $Δ {F ¯}_{b}^{a}$ . The control system in Figure 6 were also software implemented in Matlab/Simulink and the characteristics in Figure 9 were obtained.

Figure 7.

Dynamic characteristics of the system for the adaptive control of the MAVs’ attitude – Figure 4.

Figure 8.

Dynamic characteristics of the system for the adaptive control of the MAVs’ position – Figure 5.

Figure 9.

Dynamic characteristics of the automatic system for the adaptive control of the MAV by using the quaternion vector – Figure 6.

By comparative analysis of the dynamic characteristics in Figures 7 and 9, we can notice that in the first case (Figure 7), the variables have a nonperiodical behavior, while in the second case (Figure 9) the variables have an oscillatory behavior, with overshoot. Also, the transient regime period is shorter in the first case.

Conclusions

In this paper, we presented two types of automatic systems for the adaptive control of the MAVs’ attitude: using the attitude vector or the quaternion vector. Also, we designed an automatic system for the control of MAVs’ position relative to the Earth tied frame. The dynamic model of the MAV is nonlinear; the adaptive control laws are based on the estimation of the inertia moments’ matrices and the dynamic damping coefficients.

To design the adaptive control laws, we have chosen Lyapunov functions having components, which depend on the error vector ã and vector h _a (provided by a PD dynamic compensator). From the condition that the closed-loop system is asymptotic stable, the differential equation of the deviation ã is deduced with respect to h _a and the parameterization matrix Y; also, we deduced the adaptive control laws for the attitude (for both types of systems), these being expressed with respect to the command aerodynamic moments’ vector and the inherent command vector (having as components the beat and the attack angle of the MAVs’ wing). Similarly, we designed the automatic control law of the MAVs’ position by using another Lyapunov function (with respect to ã and the signal provided by a linear dynamic compensator of PD type). The validation of the theoretical results has been achieved by means of complex numerical simulations; the results are quite good, which are being analyzed in the fourth section of the paper.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the grant no. 89/1.10.2015 (Modern architectures for the control of aircraft landing) of the Romanian National Authority for Scientific Research and Innovation, CNCS – UEFISCDI, project code PN-II-RU-TE-2014-4-0849.

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Automatic control of the microaerial vehicles’ attitude and position

Abstract

Keywords

Introduction

Dynamics of the microaerial vehicles

Automatic control of the MAVs’ attitude and position

Control of the MAVs’ attitude and position using the attitude vector

Control of the MAVs’ attitude and position using the quaternion vector

Numerical simulation results

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References