Unmanned Aerial Vehicles

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3 Unmanned Aerial Vehicles

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5 Unmanned Aerial Vehicles Embedded Control Edited by Rogelio Lozano

6 First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Objets volants miniatures : Modélisation et commande embarquée published 2007 in France by Hermes Science/Lavoisier LAVOISIER 2007 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc St George s Road 111 River Street London SW19 4EU Hoboken, NJ UK USA ISTE Ltd The rights of Rogelio Lozano to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act Library of Congress Cataloging-in-Publication Data Unmanned aerial vehicles : embedded control / edited by Rogelio Lozano. p. cm. "Adapted from Objets volants miniatures : Modélisation et commande embarquée published 2007." Includes bibliographical references and index. ISBN Drone aircraft--automatic control. 2. Embedded computer systems. I. Lozano, R. (Rogelio), TL589.5.U '6--dc British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.

7 Table of Contents Chapter 1. Aerodynamic Configurations and Dynamic Models... 1 Pedro CASTILLO and Alejandro DZUL 1.1. Aerodynamic configurations Dynamicmodels Newton-Eulerapproach Euler-Lagrangeapproach Quaternionapproach Example: dynamic model of a quad-rotor rotorcraft Bibliography Chapter 2. Nested Saturation Control for Stabilizing the PVTOL Aircraft 21 Isabelle FANTONI and Amparo PALOMINO 2.1. Introduction Bibliographicalstudy ThePVTOLaircraftmodel Controlstrategy Control of the vertical displacement y Control of the roll angle θ and the horizontal displacement x Boundedness of θ Boundedness of θ Boundedness of x Boundedness of x Convergence of θ, θ, x and x tozero Other control strategies for the stabilization of the PVTOL aircraft Experimentalresults Conclusions Bibliography v

8 vi Unmanned Aerial Vehicles Chapter 3. Two-Rotor VTOL Mini UAV: Design, Modeling and Control. 41 Juan ESCARENO, SergioSALAZAR andeduardorondon 3.1. Introduction Dynamicmodel Kinematics Dynamics Forcesactingonthevehicle Torquesactingonthevehicle Modelforcontrolanalysis Controlstrategy Altitude control Horizontalmotioncontrol Attitude control Experimentalsetup Onboard flight system (OFS) Outboardvisualsystem Position Opticalflow Experimentalresults Concludingremarks Bibliography Chapter 4. Autonomous Hovering of a Two-Rotor UAV Anand SANCHEZ, JuanESCARENO and Octavio GARCIA 4.1. Introduction Two-rotorUAV Description Dynamicmodel Translationalmotion Rotationalmotion Reducedmodel Controlalgorithmdesign Experimentalplatform Real-timePC-controlsystem(PCCS) Sensors and communication hardware Experimentalresults Conclusion Bibliography Chapter 5. Modeling and Control of a Convertible Plane UAV Octavio GARCIA, JuanESCARENO and Victor ROSAS 5.1. Introduction

9 Contents vii 5.2.ConvertibleplaneUAV Verticalmode Transition maneuver Horizontalmode Mathematicalmodel Translationofthevehicle Orientationofthevehicle Eulerangles Aerodynamic axes Torques Equationsofmotion Controllerdesign Hovercontrol Axialsystem Longitudinalsystem Lateralsystem Simulation and experimental results Transition maneuver control Horizontalflightcontrol Embeddedsystem Experimentalplatform Microcontroller Inertialmeasurementunit(IMU) Sensorfusion Conclusionsandfutureworks Conclusions Futureworks Bibliography Chapter 6. Control of Different UAVs with Tilting Rotors Juan ESCARENO, Anand SANCHEZ and Octavio GARCIA 6.1. Introduction DynamicmodelofaflyingVTOLvehicle Kinematics Dynamics Attitude control of a flying VTOL vehicle Triple tilting rotor rotorcraft: Delta Kinetics of Delta Torques acting on the Delta Experimentalsetup Avionics Sensor module (SM) On-board microcontroller (OBM)

10 viii Unmanned Aerial Vehicles Data acquisition module (DAQ) Experimentalresults Single tilting rotor rotorcraft: T-Plane Forces and torques acting on the vehicle Experimentalresults Experimentalplatform Experimentaltest Concludingremarks Bibliography Chapter 7. Improving Attitude Stabilization of a Quad-Rotor Using Motor Current Feedback Anand SANCHEZ, LuisGARCIA-CARRILLO, Eduardo RONDON and Octavio GARCIA 7.1. Introduction Brushless DC motor and speed controller Quad-rotor Dynamicmodel Controlstrategy Attitude control Armaturecurrentcontrol Systemconfiguration Aerialvehicle Ground station Visionsystem Experimentalresults Concludingremarks Bibliography Chapter 8. Robust Control Design Techniques Applied to Mini-Rotorcraft UAV: Simulation and Experimental Results José Alfredo GUERRERO, Gerardo ROMERO, Rogelio LOZANO and Efraín ALCORTA 8.1. Introduction Dynamicmodel Problemstatement Robustcontroldesign Simulationandexperimentalresults Simulations Experimentalplatform Conclusions Bibliography

11 Contents ix Chapter 9. Hover Stabilization of a Quad-Rotor Using a Single Camera. 167 Hugo ROMERO and Sergio SALAZAR 9.1. Introduction Visualservoing Directvisualservoing Indirectvisualservoing Position based visual servoing Image-basedvisualservoing Position-imagevisualservoing Cameracalibration Two-planecalibrationapproach Homogenous transformation approach Poseestimation Perspective of n-pointsapproach Plane-pose-basedapproach Dynamicmodelandcontrolstrategy Platformarchitecture Experimentalresults Cameracalibrationresults Testingphase Real-timeresults Discussionandconclusions Bibliography Chapter 10. Vision-Based Position Control of a Two-Rotor VTOL Mini UAV Eduardo RONDON, SergioSALAZAR, JuanESCARENO and Rogelio LOZANO Introduction Position and velocity estimation Inertialsensors Visualsensors Position Opticalflow(OF) Kalman-basedsensorfusion Dynamicmodel Controlstrategy Frontal subsystem (S camy ) Lateral subsystem (S camx ) Heading subsystem (S ψ ) Experimentaltestbedandresults Experimentalresults Concludingremarks

12 x Unmanned Aerial Vehicles 10.7.Bibliography Chapter 11. Optic Flow-Based Vision System for Autonomous 3D Localization and Control of Small Aerial Vehicles Farid KENDOUL, Isabelle FANTONI and Kenzo NONAMI Introduction Related work and the proposed 3NKF framework Opticflowcomputation Structurefrommotionproblem Bioinspired vision-based aerial navigation Brief description of the proposed framework Prediction-based algorithm with adaptive patch for accurate and efficientopticflowcalculation Searchcenterprediction Combined block-matching and differential algorithm Nominal OF computation using a block-matching algorithm(bma) Subpixel OF computation using a differential algorithm(da) Optic flow interpretation for UAV 3D motion estimation and obstacles detection(sfmproblem) Imagingmodel Fusion of OF and angular rate data EKF-based algorithm for motion and structure estimation Aerial platform description and real-time implementation Quadrotor-basedaerialplatform Real-timesoftware D flight tests and experimental results Experimental methodology and safety procedures Opticflow-basedvelocitycontrol Optic flow-based position control Fully autonomous indoor flight using optic flow Conclusionandfuturework Bibliography Chapter 12. Real-Time Stabilization of an Eight-Rotor UAV Using Stereo Vision and Optical Flow Hugo ROMERO, SergioSALAZAR and José GÓMEZ 12.1.Stereovision Dreconstruction Keypoints matching algorithm Opticalflow-basedcontrol Lucas-Kanadeapproach

13 Contents xi 12.5.Eight-rotorUAV Dynamicmodel Translational subsystem model Rotational subsystem model Controlstrategy Attitude control Horizontal displacements and altitude control Systemconcept Real-timeexperiments Bibliography Chapter 13. Three-Dimensional Localization Juan Gerardo CASTREJON-LOZANO and Alejandro DZUL 13.1.Kalmanfilters LinearKalmanfilter ExtendedKalmanfilter UnscentedKalmanfilter UKFalgorithm AdditiveUKFalgorithm Square-root UKF algorithm Additive square-root UKF algorithm Spherical simplex sigma-point Kalman filters Spherical simplex sigma-point approach Spherical simplex UKF algorithm AdditiveSS-UKFAlgorithm Square-root SS-UKF algorithm Square-root additive SS-UKF algorithm Robot localization Typesoflocalization Dead reckoning (navigation systems) Apriorimap-basedlocalization Simultaneous localization and mapping (SLAM) Inertial navigation theoretical framework Navigation equations in the navigation frame Simulations Quad-rotorhelicopter Inertialnavigationsimulations Conclusions Bibliography Chapter 14. Updated Flight Plan for an Autonomous Aircraft in a Windy Environment Yasmina BESTAOUI and Fouzia LAKHLEF Introduction

14 xii Unmanned Aerial Vehicles 14.2.Modeling Down-draftmodeling Translational dynamics Updatedflightplanning Basicproblemstatement Hierarchical planning structure Updates of the reference trajectories: time optimal problem AnalysisofthefirstsetofsolutionsS Conclusions Bibliography List of Authors Index

15 Chapter 1 Aerodynamic Configurations and Dynamic Models 1.1. Aerodynamic configurations In this chapter, we present the aerodynamic configurations commonly used for UAV (unmanned aerial vehicles) control design. Our presentation is focused on mini-vehicles like the airplane (fixed wing models), the flapping wing UAV aircrafts, and the rotorcrafts (rotary wing models). The rotorcrafts will also be classified according to the number of rotors they are equipped with: 1, 2, 3 or 4. A UAV, also called drone, is a self-descriptive term commonly used to describe military and civil applications of the latest generations of pilotless aircraft. UAVs are defined as aircrafts without the onboard presence of human pilots, used to perform intelligence, surveillance, and reconnaissance missions. The technological objective of UAVs is to serve across the full range of missions cited previously. UAVs present several basic advantages compared to manned systems that include better maneuvrability, lower cost, smaller radar signatures, longer endurance, and minor risk to crew. Usually, people, and also we ourselves, tend to use the terms airplane and aircraft as synonymous. However, dictionary defines an aircraft as any craft that flies through the air, whether it be an airplane, a helicopter, a missile, a glider, a balloon, a blimp, or any other vehicle that uses the air to generate lift for flight. On the other hand, the Chapter written by Pedro CASTILLO and Alejandro DZUL. 1

16 2 Unmanned Aerial Vehicles term airplane is more specific and refers only to a powered vehicle with fixed wings to generate lift. Each type of mini-aerial vehicle presents advantages and disadvantages but scenarios are used to represent different types of UAV. For instance, fixed-wing UAVs can easily achieve high efficiency and long flight times compared to other UAVs, consequently they are well suited to operating during required extended loitering times. Nevertheless they are usually unable to enter buildings since they cannot hover or make the tight turns required. In opposition to fixed-wing UAVs, rotary-wing UAVs (like vertical take-off and landing aircrafts VTOL or short take-off and landing aircrafts STOL) can easily hover and move in any direction during a shorter flight time [HIR 97]. The last flapping-wing configuration offers the best potential in terms of miniaturization and maneuvrability compared to fixed- and rotary-wing UAVs, but are usually very inferior to fixed- and rotary-wing MAVs (micro air vehicles (MAVs)). Single rotor configuration This type of aerodynamic configuration is composed of a single rotor and ailerons to compensate the rotor torque (yaw control input). Since the rotor does not hold swashplate, it has extra ailerons to produce pitch and roll torques. This type of flying machine is particularly difficult to control even for experienced pilots. The single rotor configuration is mechanically simpler than standard helicopters but it does not have as much control authority. In both cases, a significant amount of energy is used in the anti-torque, i.e. to stop the fuselage turning around the vertical axis. However, due to its mechanical simplicity, this configuration seems more suitable for micro-aircraft than the other configurations. In this type of configuration, we firstly find the planes called 3D or STOL, see Figure 1.1. Figure 1.1. The 3D plane

17 Dynamic Models 3 Twin rotor configuration In this type of configuration, we can distinguish those that use one or two swashplates (i.e. collective pitch) and those that use fixed pitch. Among the configurations that use cyclic plates, we can quote the following: the classic helicopter, the tandem helicopter and the coaxial helicopter. The aircraft configuration with two rotors without swashplate: we find in this category the twin rotors aircraft with ailerons, i.e. two rotors placed on different axes (or in coaxial rotor configuration) and the ailerons orientated in direction of the rotor air flow in order to obtain the required torques to control the vehicle in 3D. Let us note that the rotors can turn in opposite direction or in the same direction. In order to better describe this configuration, we can mention for example the T-Wing of the University of Sydney, see Figure 1.2 [STO 01]. It is also possible to get two counter-rotating rotors on the same axis and ailerons in the air flow direction of the rotors. This last configuration is very compact but difficult to control. Finally, we can have two rotors which tilt on two axes (bi-rotor with counter-rotating propellers in tandem). In this configuration, the propellers do not have a swashplate and the rotors can tilt in two different directions to generate the pitch and the roll torque. The roll torque is obtained by the speed difference between the two rotors. Figure 1.2. The classic helicopter and the T-Wing aircraft Multi-rotors In this category, we find the three-rotor rotorcrafts, the four-rotor rotorcrafts and the rotorcrafts with more than four rotors. The four-rotor rotorcraft or quad-rotor is the most popular multi-rotor rotorcraft (Figure 1.3). With this type of rotorcraft we can attempt to achieve stable hovering and precise flight by balancing the forces produced by the four rotors. One of the

18 4 Unmanned Aerial Vehicles advantages of using a multi-rotor helicopter is the increased payload capacity. Therefore, with higher lift heavy weights can be carried. Quad-rotors are highly maneuverable and enable vertical take-off and landing, as well as flying into tough conditions to reach specified areas. The main disadvantages are the heavy weight of the aircraft and the high consumption of energy due to extra motors. The quadrotor is superior to the others rotor configurations, from the control authority point of view. Controlled hover and low-speed flight has been successfully demonstrated. However, further improvements are required before demonstrating sustained controlled forward-flight. When internal-combustion engines are used, multiple-rotor configurations have disadvantages compared to single-rotor configurations because of the complexity of the transmission gear. Figure 1.3. Helicopter with four rotors Airship An airship or dirigible is a lighter-than-airaircraft that can be steered and propelled through the air using rudders and propellers or other thrust. In opposition to other aerodynamic aircrafts such as fixed-wing aircrafts and helicopters that produce lift by moving a wing or airfoil through the air, the aerostatic aircrafts (airships, hot air balloons, etc.) stay aloft by filling a large cavity like a balloon with a lifting gas. Major types of airship are non-rigid (or blimps), semi-rigid and rigid. Blimps are small airships without internal skeletons but semi-rigid airships are a bit larger and have any forms of internal support such as a fixed keel. Airplane An aeroplane, or airplane, is a kind of aircraft which uses wings in order to generate lift. The body of the plane is called the fuselage. It is usually a long tube shape. The wing surfaces are smooth and their shape helps to push the air over the top of the wing more rapidly than the air travel, as it approaches the wing. As the wing moves, the air flowing over the top has further to go and moves faster than the

19 Dynamic Models 5 Figure 1.4. The LSC s airship air underneath the wing. So the pressure of the air above the wing is lower creating a depression which produces the upward lift. The design of the wings determines how fast and high the plane can fly. The wings are called airfoils. The hinged control surfaces are used to steer and control the airplane. The flaps and ailerons are connected to the backside of the wings. The flaps may move backward and forward modifying the surface of the wing area, but also may tilt downward increasing the curve of the wing. There are also slats, located at the top of the wing, which move out to create a larger wing space. It helps to increase a lifting force to the wing at slower speeds like take-off and landing. The ailerons are hinged on the wings and move downward to push the air down and make the wing tilt up. This moves the plane to the side and helps it turn during flight. After landing, the spoilers are used like air brakes to reduce any remaining lift and slow down the airplane. The tail at the rear of the plane provides stability and the fin is the vertical part of the tail. The rudder at the back of the plane moves left and right to control the left or right movement of the plane. The elevators are placed at the rear of the plane and they can be raised or lowered to change the direction of the plane s nose. The plane will go up or down, depending on the direction toward which the elevators are moved. Flapping-wing UAV A new trend in the UAV community is to take inspiration from flying insects or birds to achieve unprecedented flight capabilities. Biological systems are not only interesting for their smart way of relying on unsteady aerodynamics using flapping wings, they are increasingly inspiring engineers for other aspects such as distributed sensing and acting, sensor fusion and information processing. Birds demonstrate that flapping-wing flight (FWF) is a versatile flight mode, compatible with hovering, forward flight and gliding to save energy. However, design is challenging because

20 6 Unmanned Aerial Vehicles Figure 1.5. The plane configuration aerodynamic efficiency is conditioned by complex movements of the wings and because many interactions exist between morphological (wing area, aspect ratio) and kinematic parameters (flapping frequency, stroke amplitude, wing unfolding) [RAK 04]. Figure 1.6. The dragonfly 1.2. Dynamic models The dynamic representation of a flying object is of course one of the main goals to be solved before the control strategy development. In this chapter, three approaches to modeling a flying object will be presented (Newtonian, Lagrangian and quaternion approaches). The flying object is considered to be a solid object moving in a 3D environment, submitted to forces and torques applied to the body depending on the type of flying object considered [CAS 05, LOZ 00]. The dynamic model is then used to express and represent the behavior of the system over time. At the end of this chapter, we will present the dynamic model of a helicopter with four rotors.

21 Dynamic Models 7 Figure 1.7. Geometrical representation of a rigid body Newton-Euler approach A rigid body is a system of particles in which the distances between the particles do not vary. We find in literature [GOL 80] different ways of presenting rigid body dynamics moving in a 3D space. Newton-Euler and Euler-Lagrange approaches are the most prominent. The Newton-Euler approach is used, firstly, to develop the dynamics body and to represent it in the body frame and then in the inertial frame [KOO 98]. After these manipulations, we express these dynamics using the Euler-Lagrange approach. Consider Figure 1.7. The rigid body is marked by the letter C.LetI = E x,e y,e z denote a right-hand inertial frame stationary with respect to the earth and such that E z denotes the vertical direction downwards into the earth [MUR 94]. The vector ξ = (x, y, z) denotes the position of the centre of mass of the rigid body relative to the frame I. LetC = E 1,E 2,E 3 be a (right-hand) body fixed frame for C. The orientation of the helicopter is given by a rotation R : C Iwhere R SO(3) is an orthogonal rotation matrix. The orientation of the rigid body is given by the three Euler angles η =(ψ, θ, φ), which are the classic yaw, pitch and roll Euler angles commonly used in aerodynamic applications. The rotation matrix R(ψ, θ, φ) SO(3) representing the orientation of the airframe C relative to a fixed inertial frame c θ c ψ s ψ s θ s θ R = c ψ s θ s φ s ψ c φ s ψ s θ s φ + c ψ c φ c θ s φ, c ψ s θ c φ + s ψ s φ s ψ s θ c φ c ψ s φ c θ c φ

22 8 Unmanned Aerial Vehicles where c α (respectively s α ) denote cos α (respectively sin α). Let F and τ = (τ 1,τ 2,τ 3 ) be the external thrust and torques applied to center of mass of C relative to the frame C. Then, the dynamic model of a rigid object evolving in SE(3) and using Newton s classic equations of motion is ξ = RV, (1.1) m V = Ω mv + F, (1.2) Ṙ = RˆΩ, (1.3) I Ω = Ω IΩ+τ, (1.4) where the vector F represents the gravitational force and all other forces applied to the body relative to the frame C. For the helicopter, these forces are produced by the rotation of the rotors. The vector V = R T ξ R 3 represents the speed of the center of mass of the rigid body relative to the frame C, m represents the total mass of the body and g is the constant of gravity. Ω R 3 describes the angular velocity and matrix I R 3 3 represents the inertia body, both relative to the frame C. The matrix ˆΩ represents the skew-symmetric matrix of Ω and is given by 0 Ω 3 Ω 2 ˆΩ = Ω 3 0 Ω 1. Ω 2 Ω 1 0 An angular velocity in the body fixed frame C is related to the generalized velocities η = ( ψ, θ, φ) (in the region where the Euler angles are valid) via the standard kinematic relationship φ ψs θ Ω= θc φ + ψc θ s φ. ψc θ c φ θs φ Defining s θ 0 1 W η = c θ s φ c φ 0, c θ c φ s φ 0 then η = W 1 η Ω.

23 Dynamic Models 9 To represent the dynamic model of the rigid body C in the inertial frame I, itis necessary to specify the F coordinates in I. Thus, we use f = RF. (1.5) Define υ = ξ R 3 as the body speed relative to frame I. Therefore, the complete model dynamics of a rigid body relative to the inertial frame is given by the following equations: ξ = υ, (1.6) m υ = f, (1.7) Ṙ = RˆΩ, (1.8) I Ω = Ω IΩ+τ. (1.9) We expressed the rotation dynamic of the model in the frame C because the rotation velocity measurements are always obtained in this frame Euler-Lagrange approach Another way to represent a dynamic model is by using Euler-Lagrange equations of motion. Define the generalized coordinates of the helicopter as q =(ξ,η) T =(x, y, z, ψ, θ, φ) T R 6, (1.10) where ξ and η represent the position and orientation of the helicopter with respect to the inertial-fixed frame respectively (see Figure 1.7). The translational and rotational kinetic energy of the helicopter are where J = W T η IW η. T trans = m 2 ξ T m ξ (ẋ2 = +ẏ 2 +ż 2), (1.11) 2 T rot = 1 2 ΩT IΩ= 1 2 ηt J η, (1.12) We consider that the potential energy of the body consists of the gravitational potential energy U = mgz (1.13)

24 10 Unmanned Aerial Vehicles and thus the Lagrangian function is defined as L = T trans + T rot U = m 2 (ẋ2 +ẏ 2 +ż 2) ΩT IΩ+mgz, (1.14) which satisfies the Euler-Lagrange equation ( ) d L L dt q q = F L, (1.15) where F L represents the forces and torques applied to the fuselage. After some algebraic steps, we can obtain the following standards equations: M(q) q + C(q, q) q + G(q) =F L, (1.16) where M(q) R 6 6 is the symmetric positive definite inertia matrix, C(q, q) R 6 6 is the matrix of centrifugal and Coriolis forces. Finally, G(q) R 6 is the gravity force vector. Moreover, the matrices M and C verify the passivity property necessary if Ṁ 2C = P,whereP denotes an antisymmetric matrix Quaternion approach The quaternions represent another way of describing the dynamics of a mobile vehicle. This type of representation is used as an alternative to model the attitude dynamics in order to avoid the singularities given by a classic 3D representation (Euler angles or Rodrigues parameters) [CHO 92]. Quaternion is based on a four-parameter representation that gives a more global parametrization; however, it fails for a 180º rotation about some axis [FJE 94]. Then, the minimal number of parameters avoiding any singularity are five. A unit quaternion is composed of four real numbers (q 0, q 1, q 2, q 3 ) giving a rotation representation with the constraint: 3 qi 2 i=0 =1, (1.17) where the parameters belong to a complex number: Q = [ q0 q ] (1.18)

25 Dynamic Models 11 from which q 0 represents the scalar part and q = q 1 q 2 q 3 (1.19) denotes a vector part. In the context of a dynamic representation for an object s orientation, in a 3D space, all unit quaternion can form a matrix [ISI 03] 1 2q2 2 2q2 3 2q 1 q 2 2q 0 q 3 2q 1 q 3 +2q 0 q 2 R(Q) = 2q 1 q 2 +2q 0 q 3 1 2q1 2 2q3 2 2q 2 q 3 2q 0 q 1 (1.20) 2q 1 q 3 2q 0 q 2 2q 2 q 3 +2q 0 q 1 1 2q 2 1 2q 2 2 that satisfies R(Q)R T (Q) = I and det (R(Q)) = 1, as a consequence R(Q) SO(3),thatisR(Q) can be considered as a rotation matrix. Thus, if we have a rotation matrix R SO(3), then a unit quaternion Q can be created such that R = R(Q). (1.21) In [CHO 92, ISI 03], we can find the next relations: ( ) θ Q = ϑ + ϱ =cos +sin 2 where with 0 θ<π. ( θ 2 ) λ, (1.22) ( ) θ q 0 = ϑ =cos, 2 (1.23) ( ) θ sin = q 2 q, (1.24) λ 1 λ = λ 2 = q qt q R3 (1.25) λ 3 Using simple calculus, it is easy to show that eˆλθ = R ( q(θ, λ) ), (1.26) where 0 λ 3 λ 2 ˆλ = λ 3 0 λ 1. (1.27) λ 2 λ 1 0

26 12 Unmanned Aerial Vehicles Another existing relation between a rotation matrix R = {r ij } (with i, j =1, 2, 3) and the unit quaternion is given by [SCI 00] q 0 = 1 1+r11 + r 22 + r 33, (1.28) 2 q = 1 r 32 r 23 r 13 r 31 = 1 sgn ( ) r 32 r 23 r11 r 22 r q 0 2 sgn ( ) r 13 r 31 r22 r 33 r r 21 r 12 sgn ( ), (1.29) r 21 r 12 r33 r 11 r which is well defined for all positive values of 1 +r 11 + r 22 + r 33. If this value is negative, then we can use some algorithms which solve this problem [PAI 92, SHE 78]. One goal of the matrix rotation using quaternion is linked to the possibility of expressing the solution of the equation Ṙ = RˆΩ (1.30) in terms of the solution of an associated differential equation (1.28)-(1.29), which is defined on the set of unit quaternions [ISI 03]. In order to obtain this equation, we start with the quaternion propagation rule Q = 1 [ ] 0 2 Q (1.31) Ω and using quaternion algebra [CHO 92], we obtain Q = 1 2 E(Q)Ω = 1 D(Ω)Q, (1.32) 2 where [ ] q T E(Q) =, (1.33) q 0 I +ˆq [ 0 ] Ω T D(Ω) = Ω ˆΩ. (1.34) The inverse of (1.32) is given by Ω=2E T (Q) Q. (1.35)

27 Dynamic Models 13 Thus, the dynamic model evolving in a 3D space, related with an inertial frame, can be represented by ξ = υ, (1.36) m υ = f, (1.37) Q = 1 E(Q)Ω, (1.38) 2 I Ω = Ω IΩ+τ. (1.39) Example: dynamic model of a quad-rotor rotorcraft In this section we derive a dynamic model of the quad-rotor helicopter. This model is obtained by representing the aircraft as a solid body evolving in a 3D space and subject to the main thrust and three torques (see Figure 1.8). Figure 1.8. Helicopter with four-rotor scheme Define the Lagrangian L = T trans + T rot U,

28 14 Unmanned Aerial Vehicles where T trans = m 2 (ẋ2 +ẏ 2 +ż 2 ) is the translational kinetic energy, T rot = 1 2 ΩT IΩ is the rotational kinetic energy,u = mgz is the potential energy of the aircraft, z is the rotorcraft altitude, m denotes the mass of the quad-rotor,ω is the vector of the angular velocity, I is the inertia matrix, and g is the acceleration due to gravity. where Define J = J(η) =W T η IW η, (1.40) I xx 0 0 I = 0 I yy I zz The model of the full rotorcraft dynamics is obtained from Euler-Lagrange s equations with external generalized forces [ ] d Ltrans dt ξ L trans 0 ξ [ ] f [ ] d Lrot 0 L rot =, (1.41) τ dt η η where f = RF L is the translational force applied to the rotorcraft due to main thrust and τ represents the yaw, pitch, and roll moments. We ignore the small body forces because they are generally of a much smaller magnitude than the principal control inputs f and τ. From Figure 1.8 or 1.9, it follows that 0 F L = 0, (1.42) u where u = f 1 + f 2 + f 3 + f 4 and for i =1,...,4, f i is the force produced by motor M i, as shown in Figure 1.9. Typically, f i = k i wi 2 where k i > 0 is a constant depending on the density of air, the radius, shape of the blade and other factors and ω i is the angular speed of the ith motor. The generalized torques are thus 4 τ ψ τ Mi τ τ θ = i=1 ( ) τ f2 f 4 l, (1.43) φ ( ) f3 f 1 l

29 Dynamic Models 15 where l is the distance between the motors and the center of gravity, and τ Mi moment produced by motor M i. is the Since the Lagrangian contains no cross terms in the kinetic energy combining ξ with η, the Euler-Lagrange equation can be partitioned into dynamics for ξ coordinates and the η coordinates. Figure 1.9. Forces scheme The Euler-Lagrange equations for the translation motion are d dt L trans ξ = m ξ, ] ξ = m ξ, 0 L trans = 0. ξ mg [ Ltrans Finally, we obtain mẍ f = mÿ. (1.44) m z mg

30 16 Unmanned Aerial Vehicles or For the η coordinates, we have d dt [ Lrot η [ d η T J η ] dt η ] L rot η = τ 1 ( η T J η ) = τ. 2 η Thus, we get J η + J η 1 ( η T J η ) = τ. 2 η Define the Coriolis/centripetal vector as we may write V (η, η) = J η 1 ( η T J η ), 2 η J η + V (η, η) =τ, (1.45) but we can rewrite V (η, η) as ( V (η, η) = J 1 ( η T J )) η 2 η = C(η, η) η, (1.46) where C(η, η) is referred to as the Coriolis term and contains the gyroscopic and centrifugal terms associated with the η dependence on J. From (1.44), (1.45) and (1.46), we can obtain the dynamic model of the helicopter with four rotors mẍ f = mÿ, m z mg (1.47) τ = J η + C(η, η) η. Note that the η-dynamic can be written in the general form as M(η) η + C(η, η) η = τ, (1.48)