Modeling and Control of a Co-Axial Helicopter. Farid Zare Seisan

Transcription

1 Modeling and Control of a Co-Axial Helicopter by Farid Zare Seisan A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto Copyright c 2012 by Farid Zare Seisan

2 Abstract Modeling and Control of a Co-Axial Helicopter Farid Zare Seisan Master of Applied Science Graduate Department of Electrical and Computer Engineering University of Toronto 2012 This thesis lays the foundations for the development of a small autonomous coaxial helicopter. This is an helicopter with two propellers mounted on the same axis and revolving in opposite directions. To steer the helicopter, this thesis proposes a mechanism that moves the helicopter s centre of mass. Although such a mechanism has already been investigated experimentally in the literature, it has never been rigorously modeled, and a theoretical analysis has never been performed. This thesis, for the first time, presents an accurate mathematical model of the coaxial helicopter which takes into account the gyroscopic effects of the rotors, the reaction forces and torques exerted by the moving mass actuator on the helicopter body, and the fact that the inertia of the helicopter is time-varying. A nonlinear controller is rigorously derived which makes the helicopter hover at desired positions in three-space. A number of physical prototypes are discussed. None of them is capable of autonomous flight yet, but the experimental and simulation results provide reassurances that the proposed methodology is viable. ii

3 Dedication In loving memory of my father, Salman, and for my kind and loving mother, Sirosiyeh. iii

4 Acknowledgements I would like to thank my thesis supervisor Dr. Manfredi Maggiore for his support and guidance througout my graduate studies. His enthusiasm and great knowledge for control has been inspirational even through the hardest of times. His interest and clarity during his lectures were motivation for me to pursue a career in the field of control systems. I would also like to thank Dr. Edward J. Davison for sharing his astonishing life experiences in the control system field that inspired me to pursue my education in this field early in my studies. Furthermore, I would like to thank Don Gardner for designing and helping me build the physical prototypes, Cameron Fulford and the rest of the Quanser team for their continued help and support. iv

5 Contents List of Figures vii List of Tables ix 1 Introduction Configurations and steering mechanisms Literature review Thesis contributions Modeling Setup of the coaxial helicopter and coordinate frames Rotational motion Rotational kinematics Internal forces Rotational dynamics Dynamics of the moving mass mechanism Translational motion of helicopter Aerodynamic thrust generated by the two rotors Complete model Preliminary Helicopter Prototypes and Attitude Stabilization Helicopter prototypes v

6 3.2 Simplified rotational dynamics and attitude control Discussion of experimental results Position Control Design Simplified model for control design Position control design Step 1: design for translational subsystem Step 2: design for rotational subsystem Step 3: Asymptotic stability of χ Simulation Results Conclusion 64 Appendix 66 A Derivative of the Inertia Tensor 66 Bibliography 68 vi

7 List of Figures 1.1 A rendering of a coaxial helicopter with moving mass mechanism Torque generation with moving mass mechanism Coaxial helicopter s main components Illustration of coordinate frames used in modelling the helicopter Side views of the first prototype Detail of the moving mass actuator Side views of the second prototype Top views of the second prototype Side view of the third prototype with a small battery Top view of the third prototype with a small battery Side view of the third prototype with a large battery Simulation result of the attitude controller for the Euler angles Control inputs of the attitude controller for the rotors torque Control inputs of the attitude controller, position of the COG of the body Result of robustness test of the attitude controller, Euler angles Attitude controller input with parameter perturbations, rotors torque Attitude controller input with PP, position of the COG of the body First prototype mounted on the platform used for yaw control tests Block diagram of proposed position controller vii

8 4.2 Test 1: components of the helicopter displacement vector rio. i Test 1: Euler angles Θ Test 1: angular speeds of the rotors Test 1: position of the COG of the helicopter w.r.t. the body frame Test 2: components of the helicopter displacement vector rio i Test 2: Euler angles Θ Test 2: angular speeds of the rotors Test 2: position of the COG of the helicopter w.r.t. the body frame Test 3: components of the helicopter displacement vector rio i Test 3: Euler angles Θ Test 3: angular speeds of the rotors Test 3: rotor torques Test 3: position of the COG of the helicopter w.r.t. the body frame viii

9 List of Tables 2.1 Definition of various special points Initial condition Simulation parameters Perturbed parameters for robustness test Controller parameters Reference position and yaw angle Actual plant parameters v.s. parameters used in the controller for test 2 59 ix

10 Chapter 1 Introduction Unmanned aerial vehicles (UAVs) are aircrafts that are controlled remotely or flown automatically without any human on board. The history of UAVs goes back to as far as 1916, soon after the human fascination with flying conquered the sky. As embedded microprocessors have become faster and battery technology has improved, so the development of UAVs has progressed. Nowadays, UAVs are employed for surveillance, security, and rescue missions. They can also be used for educational purposes in research laboratories because of their compact sizes. UAVs can be broadly classified as either fixed-wing (airplane-line) or rotary-wing (helicopter-like). Rotary-wing aircrafts have the advantage of being able to hover, and for this reason they are used in surveillance applications. In recent years, most UAV research on rotary-wing aircrafts has focused on quadrotor helicopters. These are aircrafts with four rotors rotating around four distinct parallel axes. The popularity of quadrotors may be due to their maneuverability and ease of control. A number of companies (in Canada, we mention Draganfly, Aeryon Labs, and Quanser) are commercializing quadrotors in the civilian and educational markets. In contrast to quadrotors, coaxial helicopters are aircrafts with two rotors rotating in opposite directions around the same axis. Compared to quadrotors, coaxial helicopters consume less power for the same payload, and they can 1

11 Chapter 1. Introduction 2 Figure 1.1: A rendering of a coaxial helicopter with moving mass mechanism. be made much smaller. However, coaxial helicopters are more difficult to control. This is perhaps the reason that coaxial helicopters have received significantly less attention than quadrotors, and their commercialization is limited (to the best of our knowledge, the only company commercializing autonomous coaxial helicopters is Skybotix Technologies in Switzerland). The focus of this thesis is on laying out the foundations for the development of an educational UAV in the form of an autonomous coaxial helicopter. An illustration of one of the prototypes presented in this thesis is found in Figure 1. Unlike a conventional helicopter in which one main rotor creates lift and a secondary rotor provides yaw control, a coaxial helicopter is built using a pair of rotors rotating in opposite directions around the same axis. Coaxial helicopters have a number of advantages over other configurations. Some of these advantages are compactness, symmetry, power efficiency, and high payload. These are described later in this chapter in more detail. To steer the helicopter, in this thesis we propose to use a mechanism that moves the helicopter s centre of mass. Even though the moving mass mechanism further simplifies the rotor system, it increases the complexity of the dynamical equations of motion due to a time varying inertia and the reaction forces it exerts on the helicopter body. The advantages of using a moving centre

12 Chapter 1. Introduction 3 of mass steering mechanism over the standard swash plate commonly used in conventional helicopters are explained in the following. The design of the helicopter starts with developing an accurate mathematical model of the system. A number of physical prototypes are presented that represent successive improvements in the design. A nonlinear position controller is designed and tested in simulation. Although none of the physical prototypes is capable of autonomous flight, our experimental results and theoretical analysis strongly indicate that the moving mass paradigm is viable and indeed promising. 1.1 Configurations and steering mechanisms The main design choices in developing a rotary wing aircraft are the number of rotors (at least two), their configuration, and the steering mechanism. Conventional helicopters use one main rotor to produce lift and a secondary tail rotor to control yaw, i.e., to prevent the helicopter from spinning around its vertical axis. The main rotor is mounted on a clever mechanism called a swash plate which varies the pitch of the blades over the course of each revolution in such a way that the thrust vector can be controlled by the pilot. A quadrotor helicopter utilizes four rotors rotating around four distinct parallel axes. To prevent the aircraft from spinning around its vertical axis, two rotors revolve clockwise, and the remaining two revolve counter-clockwise. By varying the rotor speeds, torques can be generated to steer the helicopter. The advantages of this configuration over the conventional one is that all rotors produce lift. Moreover, the torque generation mechanism is elementary and it does not require a swash plate. The disadvantage is that the configuration of rotors takes space, and it poses a fundamental limit to the miniaturization of the aircraft. A coaxial helicopter features two rotors mounted on the same axis and revolving in opposite directions. As in quadrotor helicopters, both rotors generate lift. The yaw

13 Chapter 1. Introduction 4 angle of the helicopter is controlled by the differential rotor velocity. Since the rotors are mounted on the same axis, the coaxial configuration is compact, and it can be miniaturized. While quadrotor helicopters have redundant actuation, coaxial helicopters have the minimum number of rotors needed for stable flight, and as a result they are more power efficient. By that it is meant that for the same payload, coaxial helicopters consume less power than quadrotors. T COM r τ = r T Figure 1.2: A moving mass actuator steers the helicopter by shifting the centre of mass of the helicopter on a plane perpendicular to the thrust vector. If the centre of mass of the helicopter is moved off the thrust axis through a displacement vector r, the thrust vector T produces a torque τ = r T. By controlling r, we can produce torques around any axis on the shaded plane. There are two ways to generate steering torques in coaxial helicopters. One may either use a swash plate or move the centre of mass of the helicopter using a suitable mechanism. In this thesis we will adopt this latter approach. Figure 1.2 provides a schematic representation of how torques are generated by a moving mass mechanism. Such a mechanism avoids the complexity and reliability problems inherent with swash plates (miniature helicopters are prone to crashes or abrupt landings). There are, however, challenges with centre of mass steering. The main challenge is the design of a fast and accurate moving mass actuator. A secondary challenge is the fact that the reaction forces and torques caused by the moving mass actuator on the helicopter body can potentially have negative effects on its performance. Finally, it must be noted that a

14 Chapter 1. Introduction 5 moving mass actuator requires some power expenditure [2] so it decreases somewhat the theoretical power efficiency of the aircraft. 1.2 Literature review The literature on UAVs is vast and it covers all aspects of implementation and flight control. Significant work has been done on the attitude and position control problems. The reader can refer to, for instance, [18, 20, 11, 7, 12, 4, 6, 3, 21, 16, 1, 2]. The papers in [2, 3] present a coaxial helicopter with centre of mass steering attained by moving the battery. The battery is mounted on a spherical pendulum at the bottom of the aircraft. The paper does not present the mathematical model of the system. A PD controller is used to control the attitude of the helicopter and the distance of the helicopter from the ground. We surmise that a pilot must manually control the position of the helicopter to prevent drifts. As mentioned earlier, the battery is used to move the centre of mass of the helicopter. This is the same idea adopted in this thesis. However, rather than moving the battery along the surface of a sphere, in our setup the battery moves on the horizontal plane. This has the advantage of giving a linear relationship between the movement of the battery and that of the centre of mass. Another difference between the approach proposed in this thesis and the one in [2, 3] is the fact that in our setup the moving battery is placed near the centre of mass of the helicopter. This provides greater sensitivity of the rotational motion of the helicopter with respect to the control input, and therefore greater maneuverability. The work in[21] proposes a coaxial helicopter, but the implementation details provided are insufficient. For instance, the steering mechanism is not presented. While a clear mathematical model is not presented in [21], some arguments are made that refer to a mathematical model. The coaxial helicopter configuration is the only conspicuous similarity between the work done in [21] and the work done in this thesis.

15 Chapter 1. Introduction 6 In [16] the authors propose a deformable steering structure for a coaxial helicopter, but the mathematical model and experimental results are not presented. 1.3 Thesis contributions This thesis makes three contributions: It presents, for the first time, an accurate mathematical model of a coaxial helicopter steered by a moving mass actuator. This is done in Chapter 2. The model takes into account the gyroscopic effects of the rotors, the reaction forces and torques exerted by the actuator on the helicopter body, and the fact that the inertia of the helicopter is time-varying. It presents a number of physical helicopter prototypes. This is done in Chapter 3. None of the prototypes is capable of autonomous flight yet, but experimental as well as simulation results indicate that the proposed moving mass steering mechanism is viable. It develops, in Chapter 4, a nonlinear position controller with a rigorous stability analysis and extensive simulation results.

16 Chapter 2 Modeling In this chapter we develop a detailed mathematical model of a coaxial helicopter with a moving mass steering mechanism. This model is used for simulation purposes. A simplified version of the model in which the reaction forces caused by the moving mass mechanism are ignored is used in Chapter 4 for position control design. This chapter is organized as follows. We begin, in Section 2.1, by presenting the structure of the helicopter and its main components. In Section 2.2 we model the rotational dynamics. In Section 2.3 we model the dynamics of the moving mass mechanism. In Section 2.4 we model the translational dynamics of the helicopter. In Section 2.5 we determine an expression for the thrust produced by the two propellers. Finally, in Section 2.6 we present the complete helicopter model. 2.1 Setup of the coaxial helicopter and coordinate frames A sketch of the coaxial helicopter investigated in this thesis is presented in Figure 2.1. The helicopter is divided into two sections: the rotor system and the fuselage. The rotor system is composed of the main shaft andthe two propellers, or rotors. The 7

17 Chapter 2. Modeling 8 Figure 2.1: The main components of the coaxial helicopter. propellers, mounted on top of each other on a coaxial shaft, rotate in opposite directions, and they both generate lift. The counter-rotation is used to prevent the helicopter from spinning around the rotation axis of the propellers. The fuselage, otherwise known as the main body of the helicopter, supports the moving mass actuator, the motors, the battery, the processor, the sensors, the data acquisition board (DA), and the legs of the helicopter. The battery is mounted on top of a moving mass actuator. This actuator is built using two servomotors (rotary or linear, depending on the implementation), and is used to accurately control the displacement of thebattery onthe x o -y o plane (refer tofigure2.1). The effect ofthis actuationisto move the center of gravity (COG) of the helicopter and, in so doing, generate torques around the x o and y o axes. More specifically, assuming that the total thrust of the helicopter is parallel to the z o axis, a displacement of the COG along the x o axis generates a torque

18 Chapter 2. Modeling 9 around the y o axis, while a displacement along the y o axis generates a torque around the x o axis. Various implementation details are presented in the next chapter. Figure 2.2: Illustration of coordinate frames used in modelling the helicopter. Our model relies on a number of coordinate frames, depicted in Figure 2.2, and several special points listed in Table 2.1, and described next. We fix an inertial frame with origin i, and axes x i y i z i. We place a body frame rigidly on the helicopter, with origin o and axes x o y o z o. When the helicopter is hovering, axes x o y o are parallel to the ground, and axis z o points downward. The origin o is chosen to be the position of the COG of the helicopter when this latter is hovering. The instantaneous COG of the helicopter is denoted by c. Due to the moving mass in the fuselage, generally when the helicopter is not hovering, c o. We place two coordinate frames at the COG of the top and bottom rotors. Their origins are denoted t and b, respectively. These frames move rigidly with the rotors, but

19 Chapter 2. Modeling 10 point meaning i o origin of inertial frame origin of the body frame, position of the COG of the helicopter during hovering c t b j s f instantaneous position of the COG of the helicopter COG of the top rotor COG of the bottom rotor revolute joint where the coaxial shaft of the rotors joins the fuselage position of the COG of the moving mass during hovering instantaneous position of the COG of the moving mass Table 2.1: Definition of various special points axes z t and z b are parallel to z o. We will let j denote the point where the coaxial shaft of the rotors joins the fuselage. Finally, in order to specify the displacement of the moving mass, we will denote by s the position of the COG of the moving mass when the helicopter is hovering (this is also referred to as the trimmed or rest position of the moving mass), and we will denote by f the instantaneous position of the COG of the moving mass. Clearly, s = f when the helicopter is hovering. Throughout this chapter we will use the following notation. If a and b are two points, we will denote by r ab the geometric vector b a, and by d ab the length of this vector. If v is a geometric vector, and x o y o z o is an orthonormal coordinate frame, we will denote by v o the representation of v in the coordinates of x o y o z o. The three components of such a representation will be labelled using subscripts x, y, z. To illustrate our conventions, the vector roc o is the displacement of the instantaneous COG of the helicopter, c, from its

20 Chapter 2. Modeling 11 hovering position, o, expressed in the body frame x o y o z o. Its components are roc o = r o oc x r o oc y r o oc z. 2.2 Rotational motion In this section we model the rotational dynamics of the coaxial helicopter. We begin by writing the kinematic equations. We then determine the internal forces, and use them to write the dynamic equations Rotational kinematics We denote by R o the rotation matrix between inertial and body frames so that, if v is a geometric vector, the relationship between its coordinate representations in the inertial and body frames is given by v i = R o v o. As is customary in the modeling of aircrafts, we parametrize R o using yaw (ψ), pitch (θ), and roll (φ) angles. These angles are also called the ZYX Euler angles associated with R o, and have the following geometric meaning. To get the body frame, one must rotate the inertial frame around its z axis by angle ψ, then rotate the resulting frame around its y axis by angle θ, and finally rotate this third frame around its x axis by angle φ. Thus, cosψ sinψ 0 cosθ 0 sinθ R o = sinψ cosψ cosφ sinφ sinθ 0 cosθ 0 sinφ cosφ Using the short-hand notation c θ,s θ for cos(θ) and sin(θ), the rotation matrix becomes, c θ c ψ s φ s θ c ψ c φ s ψ c φ s θ c ψ +s φ s ψ R o = c θ s ψ s φ s θ s ψ +c φ c ψ c φ s θ s ψ s φ c ψ. s θ s φ c θ c φ c θ

21 Chapter 2. Modeling 12 This parametrization of R o is singular at θ = ± π, but the helicopter is not expected to 2 operate at this attitude (when the thrust vector is horizontal). In what follows, we will denote Θ = [φ θ ψ]. Letting ω denote the angular velocity of the helicopter, and using the fact that [9], for any geometric vector v, Ṙ o v o = R o (ω o v o ), we have φ φsinφ 1 0 sinθ ω o = θcosφ+ ψcosθsinφ θsinφ+ ψcosθcosφ = 0 cosφ cosθsinφ 0 sinφ cosθcosφ Θ. Letting 1 0 sinθ Y(Θ) = 0 cosφ cosθsinφ, 0 sinφ cosθcosφ the time derivative of the vector of Euler angels is given by Θ = Y 1 (Θ)ω o, (2.1) where Y 1 (Θ) is the inverse of Y(Θ) and is given by 1 sinφtanθ cosφtanθ Y 1 (Θ) = 0 cosφ sinφ. 0 sinφsecθ cosφsecθ Equation (2.1) represents the rotational kinematics of the helicopter. Note that Y 1 (Θ) is undefined when θ = ±π/2. This is a consequence of the singularity in the Euler angle parametrization discussed earlier Internal forces In order to write the rotational dynamics of the helicopter, we need to determine the internal force arising from the revolute joint j connecting the coaxial shaft of the rotors

22 Chapter 2. Modeling 13 to the fuselage. We will denote this force F I, and we will let F t, F b be the contributions to F I due to the top and bottom rotors, respectively, so that F I = F t +F b. Furthermore, we will let Ω t and Ω b denote the angular velocities of the top and bottom propellers, respectively, and we will let τ t and τ t be the z-components of the torques exerted by the motors on the top and bottom rotors. Finally, we will let I t t and Ib b denote the inertia matrices of the two rotors in the respective rotor frames, and assume that these are diagonal matrices. Before proceeding further, we need to understand the relationship between the rotor inertia in the rotor frames and its representation in the body frame. We will assume that the two rotors are identical, so we have It t = It b = diag(i r xx,i ryy,i rzz ). Letting R t, R b betherotationmatrices between thebodyframeandthetwo rotorframes, we have It o = R tit t R t, Io b = R bib b R b. The frame of the top rotor is obtained by rotating the body frame around its z axis by some angle ψ t, so that after matrix multiplication, c 2 ψ t I rxx +s 2 ψ t I ryy c ψt s ψt I rxx c ψt s ψt I ryy 0 It o = c ψt s ψt I rxx c ψt s ψt I ryy s 2 ψ t I rxx +c 2 ψ t I ryy I t r zz Exploiting the symmetry of the rotor, we have I rxx = I ryy, and so we conclude that I o t = diag(i rxx,i rxx,i rzz ). By following an analogous reasoning one can see that the same expression holds for Ib o, and so I o t = Io b = diag(i r xx,i rxx,i rzz ). Next, we write the equations of motion of the two rotors in the body frame, It o Ω [ o t +Ωo t Io t Ωo t = ro jt Fo t τt o b( ) ] Ω o 2 t z Ib o Ω o b +Ωo b Io b Ωo b = ro jb Fo b [0 + 0 (τ ob b( ) )] 2, (2.2) Ω obz

23 Chapter 2. Modeling 14 where b ( Ω o t z ) 2 and b ( Ω o b z ) 2 are the torques due to air resistance, and b > 0 is a physical parameter dependent on the air density, the propeller geometry, and other factors (see [4], [18], and [12] for more details). Note the sign difference in the torques τt o b ( ) Ω o 2 t z and (τ ob b( ) ) 2 Ω obz. This is due to the fact that the engines apply torques in opposite directions and the propellers are counter-rotating. Since the points j, t, b lie on the axis z o, we have r o jt = [0 0 d jt] and r o jb = [0 0 d jb]. Using the expressions for I o t, Io b found earlier, the components of the dynamics of the top rotor (first equation in (2.2)) read as I rxx Ωo tx +Ω o t y Ω o t z (I rzz I ryy ) = d jt F ty (2.3) I ryy Ωo ty +Ω o t x Ω o t z (I rxx I rzz ) = d jt F tx (2.4) I rzz Ωo tz = τ o t b(ω o t z ) 2. (2.5) The same equation holds for the bottom rotor by replacing the subscript t with b and changing the sign of the right-hand side of (2.5). The revolute joint imposes the constraint that the components of the rotor angular velocities in the x o y o plane coincide with analogous components of the body angular velocity, i.e., ω o x = Ω o t x = Ω o b x, ω o y = Ω o t y = Ω o b y. Substituting these identities in the rotor dynamics (2.3), (2.4), and solving for F tx, F ty, F bx, and F by we obtain F tx = I r yy ω o y +ωo x Ωo t z (I rxx I rzz ) d jt F ty = I r xx ω o x +ω o yω o t z (I rzz I ryy ) d jt F bx = I r yy ω o y +ωo x Ωo b z (I rxx I rzz ) d jb F ty = I r xx ω o x +ω o yω o b z (I rzz I ryy ) d jb The x o and y o -components of the total force F o I are given by F o I x = F tx + F bx, F o I y =

24 Chapter 2. Modeling 15 F ty +F by, and so F o I x = I r yy ω o y(d jt +d jb )+ω o x(d jb Ω o t z +d jt Ω o b z )(I rxx I rzz ) d jt d jb F o I y = I r xx ω o x (d jt +d jb )+ω o y (d jbω o t z +d jt Ω o b z )(I rzz I ryy ) d jt d jb. The equation above can be simplified if we assume that d jb = d jt. This approximation is reasonable because d jt d jb min{ d jt, d jb }. With this approximation we obtain F o I x = 2I r yy ω o y +ωo x (Ωo t z +Ω o b z )(I rxx I rzz ) d jt F o I y = 2I r xx ω o x +ω o y(ω o t z +Ω o b z )(I rzz I ryy ) d jt. (2.6) The expressions above will be used in the next section to determine the rotational dynamics of the helicopter body Rotational dynamics Euler s equation describing the rotational motion the helicopter in body coordinates is (see [5]) dl o c dt +ωo L o = τ o c, wherel o c isthebody-referenced angularmomentumofthehelicopter fuselageatitscog, I o c is the body-referenced inertia matrix of the fuselage at its COG, and τo c is the sum of all the torques about the COG of the helicopter body. The angular momentum is given by L o c = I o cω o. In computing the time derivative of L o c, we must take into account the fact that, due to the moving mass, the inertia matrix is time-dependent. Accordingly, the rotational dynamics of the helicopter in body coordinates are given by I o c ωo +ω o I o c ωo + I o c ωo = τ o c, (2.7) There are four torques acting on the helicopter. Torque due to internal force. As we have seen in the previous section, at the revolute joint j there is an internal force F I exerted on the rotors, and a reaction force

25 Chapter 2. Modeling 16 F I exerted on the helicopter body. This latter force produces a torque 0 d jo FI o y τf o I = 0 ( Fo ) = d jo FI o x d jo 0 = d 2I rxx ω x o +ωo y (Ωo t z +Ω o b z )(I rzz I ryy ) jo d jt 2I ryy ω y o +ωx(ω o o t z +Ω o b z )(I rxx I rzz ). 0 Torque due to the engines. The two engines produce torques with magnitude τ o t and τ o b on the two rotors, and two reaction torques in the opposite direction on the body, whose net effect is the torque 0 τe o = 0. τt o +τb o Torque due to the moving mass actuator. The moving mass actuator imparts a force F A on the moving mass. This force vector is horizontal in body coordinates, i.e., it lies on the x o y o plane. The reaction force, F A, produces a torque at the COG of the fuselage, as follows τf o A = rcf o ( Fo A ). We will express the displacement r o cf in terms of ro oc, the displacement of the helicopter s COG from the origin of the body frame which, as we will see, is part of our state vector. Using the identities r of = r oc +r cf, and r of = r os +r sf, we have r o cf = ro os +ro sf ro oc. Recall that r o sf is the displacement of the moving mass relative to its rest position. This displacement is responsible for the displacement of the centre of mass of the helicopter from its equilibrium position, roc o. Specifically, the relationship between these two dis-

26 Chapter 2. Modeling 17 placements is rsf o = (M/m)ro oc. Substituting in the expression for rcf o, we get r o cf = ro os + M m m ro oc. Since both vectors F o A and ro oc lie in the xo y o plane (i.e., they are horizontal in body coordinates), the torque produced by the moving mass actuator is FA o y ros o z τf o A = FA o x ros o z. FA o x ros o y FA o y ros o x + M m m (Fo A x roc o y FA o y roc o x ) Torque due to the aerodynamic thrust. The two rotors produce a thrust vector F T at the point o, pointing upward along the negative z o axis. In body coordinates, the thrust vector is given by F o T = [0 0 T], where T depends on the rotors angular velocities, and it will be determined later. At the COG of the helicopter, the torque produced by the thrust vector is 0 0 Troc o y τf o T = rco o 0 = ro oc 0 = Troc o x. T T 0 Going back to the dynamics in (2.7), the total torque at the COG of the helicopter body is τc o = τf o I + τe o + τo F A + τf o T. Substituting in the expressions for these torques found above we obtain the final expression for the rotational dynamics of the fuselage in body coordinates (Ic o +Jo r ) ωo +[(Ω o t z +Ω o b z )L o r + I c o ]ωo +ω o Ic o ωo FA o y ros o z +Troc o y = FA o x ros o z Troc o x, FA o x ros o y FA o y ros o x + M m m (Fo A x roc o y FA o y roc o x ) τt o +τb o (2.8)

27 Chapter 2. Modeling 18 where Jr o = 2d jo diag(i rxx,i ryy,0) d jt L o r = d 0 I rzz I ryy 0 jo d jt I rxx I rzz The term [(Ω o t z +Ω o b z )L o r]ω o in the left-hand side of equation (2.8) models the impact of the gyroscopic effect of the two propellers on the rotational dynamics of the helicopter. Since the propellers are counter-rotating, at near-hovering conditions the sum of the angular speeds (Ω o t z +Ω o b z ) is approximately zero, and therefore the gyroscopic effect of the two propellers is negligible. This is an important advantage of co-axial helicopters over traditional helicopters. 2.3 Dynamics of the moving mass mechanism In this section we model the translational dynamics of the moving mass mechanism. Let M be the total mass of the helicopter, and m be the moving mass, so that the mass of the helicopter without the moving apparatus is M m. The moving mass motor imparts a force F A on the mass m, and a reaction force F A on the mass M m. Both masses are affected by the gravity force. Moreover, mass M m is subjected to the aerodynamic thrust F T produced by the rotors. Finally, the constraint that mass m slides on top of mass M m is modeled by a constraint force F C acting on mass M m, with the corresponding reaction force F C acting on mass m. Since the moving mass slides in the x o y o plane in body coordinates, the constraint force is directed along the z o axis, and it representation in body coordinates is F C = [0 0 C]. Referring to Table 2.1, recall that f is the instantaneous position of the COG of the moving mass, and o, the origin of the body frame, is the position of the COG of the helicopter during hovering. It is reasonable and realistic to assume that o coincides with

28 Chapter 2. Modeling 19 the COG of the helicopter without the moving mass. In conclusion, the displacement of the COG of mass m is r if, while that of the COG of mass M m is r io. Newton s equations for masses M m and m in inertial coordinates are: 0 0 (M m) r io i = R o Fi A, T +C (M m)g 0 0 m r if i = R o Fi A. C mg Our objective in this section is to write the dynamics of moving mass m in body coordinates. Recall that the displacement of the moving mass from the origin of the body frame is the vector r of. In light of the fact that the displacement r oc of the helicopter s COG with respect to its rest position is related to r of by roc o = m M ro sf = m M (ro of ro os ), (2.9) where r o os is a constant vector, we will equivalently represent the motion of the moving mass in terms of r o oc. This is useful because the first two components of ro oc affect the rotational dynamics in (2.8). Since r of = r if r io, using the two Newton equations above we get 0 r of i = R o 0 + T MC m(m m) M m M m(m m) Fi A. (2.10) Next, we need to rewrite the above equation in body coordinates. To this end, we use the fact that rof i = R orof o. Taking the time derivative of both sides of this equation, and using the identity Ṙov o = R o (ω o v o ), we get ṙ i of = R oṙ o of +Ṙor o of = R o(ṙ o of +ωo r o of ).

29 Chapter 2. Modeling 20 Taking one more time derivative we obtain r i of = R o( r o of + ωo r o of +ωo ṙ o of )+R o(ω o (ṙ o of +ωo r o of )) = R o ( r of o +2ωo ṙof o + ωo rof o +ωo (ω o rof o )). (2.11) Substituting this expression in (2.10) and multiplying from the left by Ro we obtain 0 r of o +2ωo ṙof o + ωo rof o +ωo (ω o rof o ) = 0 + M m(m m) Fo A. (2.12) T MC m(m m) M m Next, we need to determine the intensity of the constraint force, C. To this end, we use the constrain that mass m is only allowed to slide horizontally in body coordinates, so that [0 0 1] r o of = 0, or e 3 r o of = 0, where e 3 = [0 0 1]. Left-multiplying bothsides of (2.12) by e 3 and using the constraint above we obtain T m(m m) MC M m = e 3(2ω o ṙof o + ω o rof o +ω o (ω o rof)) o M m(m m) e 3FA. o The actuator force F o A lies in the xo y o plane, and therefore, e 3 Fo A = 0. Using this fact and substituting the expression above into (2.12) we obtain r o of +2ωo ṙ o of + ωo r o of +ωo (ω o r o of ) = e 3e 3 (2ωo ṙ o of + ωo r o of +ωo (ω o r o of )) + M m(m m) Fo A. Collecting terms, we have r o of +(I e 3 e 3) [ 2ω o ṙ o of + ω o r o of +ω o (ω o r o of) ] = M m(m m) Fo A. (2.13) We are only interested in the first two components of r o of because the moving mass slides horizontally in body coordinates, and because FA o is horizontal as well. To this end, let Π = 1 0 0, 0 1 0

30 Chapter 2. Modeling 21 and for a vector v o, denote v o = vo x vy o = Πv o. From (2.13) we have Π [ r o of +2ω o ṙ o of + ω o r o of +ω o (ω o r o of) ] = M m(m m) F o A. (2.14) The expression above will be used in the next section to derive the translational dynamics of the helicopter. Now we derive the dynamics of the vector r o oc = Πro oc. Using (2.9), we have rof o = ro os + M m ro oc. Moreover, ro of = ro os +ro sf, and since points s and f lie on the same horizontal line in body coordinates, the third component of the vector r o of is ro os z, so that r o of = col( ro of,ro os z ). Therefore, r o of = r o os + M m col( ro oc,0). (2.15) The vector r o os is independent of the displacement of the moving mass, and therefore its time derivative is zero, implying that ṙof o = M m col( r oc o,0), ro of = M m col( r oc o,0). (2.16) Substituting these expressions into (2.14) we get r oc o = Π m M ( ωo ros o +ωo (ω o ros o )) +2ω o r o oc + ω o 0 ro oc 0 +ω o ω o ro oc 0 + F o A M m. (2.17) This second-order differential equation for r o oc expresses the motion of the helicopter s COG in body coordinates. The constant vector ros o, representing the displacement between the origin of the body coordinates and the rest position of the moving mass, can be determined experimentally.

31 Chapter 2. Modeling Translational motion of helicopter We are now ready to determine the translational motion of the helicopter. In particular, we want to characterize the dynamics of rio o, the displacement of the origin of the body frame from the origin of the inertial frame expressed in body coordinates. Newton s equation for the helicopter s COG is M r i ic = Fi, where F i is the sum of all the external forces acting on the helicopter. Using the fact that r ic = r io +r oc, we have M r i io +M ri oc = Fi. (2.18) Following the same steps leading to the expression in (2.11), we get r oc i = R o( r oc o +2ωo ṙoc o + ωo roc o +ωo (ω o roc o )). (2.19) Recall the relationship in (2.9) between the vector r oc, representing the displacement of the COG of the helicopter with respect to its rest position, and the vector r sf, r o oc = (m/m)(rof o ro os ). Recall that the vector ro os, representing the displacement between the origin of the body coordinates and the rest position of the moving mass, is constant. Substituting (2.9) and (2.19) into (2.18) and using the fact that ṙ o os = r o os = 0, we obtain [ M r io i +mr o ω o ros o ωo (ω o ros o ) ] + r of o +2ωo ṙof o + ωo rof o +ωo (ω o rof o ) = F i. (2.20) Now we focus on the second line of equation (2.20), and we bring in the moving mass dynamics of equation (2.14). Noticing that the identity matrix can be partitioned as I = Π, e 3

32 Chapter 2. Modeling 23 premultiplying the second line of (2.20) by the above expression, and using equation(2.14) we get r of o +2ωo ṙof o + ωo rof o +ωo (ω o rof o ) = e 3 M F A o m(m m) (, 2ω o ṙof o + ωo rof o +ωo (ω o rof o )) where we have used the fact that e 3 ro of = 0. Using the identities (2.15)-(2.16) and letting ( µ( r oc o, r oc o,ωo, ω o ) = e m 3 M ( ωo ros o +ωo (ω o ros o )) (2.21) +2ω o r oc o + ω o ro oc +ω o ω o ro oc, we have r o of +2ωo ṙ o of + ωo r o of +ωo (ω o r o of ) = M m F o A M m. µ( r oc o, r oc o,ωo, ω o ) Substituting this expression in the second line of (2.20) we get M r i io +mr o ω o r o os ωo (ω o r o os )+ M m F o A M m = F i. µ( r oc o, r oc o,ωo, ω o ) We are left with the characterization of the total external force F i. This is simply the sum of the force due to gravity and the aerodynamic thrust generated by the rotors, 0 0 F i = 0 +R o 0. Mg T In conclusion, the translational dynamics of the helicopter are given by r io i = R o m F o 0 A M ( ωo ros o +ωo (ω o ros o )) M m + µ( r oc o, 0 r oc o,ωo, ω o )+ T. M g (2.22)

33 Chapter 2. Modeling Aerodynamic thrust generated by the two rotors The thrust generated by a propeller is modeled in great detail in [9]. In this thesis we adopt a simplified aerodynamic model in which the thrust produced by each rotor is proportional to its squared angular speed, so that the vertical component of the total thrust produced by the two rotors is given by T = Γ[ ( Ω o t z ) 2 + ( Ω o bz ) 2], (2.23) where Γ > 0 is a parameter that depends on the physical properties of the propellers and other factors (please see [18], [12], and [14] for more details). The parameter Γ can be experimentally estimated using a torque-force sensor. 2.6 Complete model In this section we summarize the mathematical model of the helicopter. The model contains the following components: The propeller dynamics in (2.5) whose states are the angular speeds Ω o t z, Ω o b z. The translational dynamics in (2.22) whose state is (r i io,ṙi io ). The rotational kinematics in (2.1) whose state is the vector of Euler angles Θ, and the rotational dynamics in (2.8) whose state is the angular velocity ω o. The moving mass dynamics in (2.17) with states the horizontal displacement vector r o oc and the velocity vector r o oc. The overall state of the system is (Ω o t z,ω o b z,rio i,ṙi io,θ,ωo, r oc o, r oc o ). The control inputs are the engine torques τ o t, τo b, and the actuator force F o A = col(fo A x,f o A y ). The complete

34 Chapter 2. Modeling 25 helicopter dynamics are as follows: I rzz Ωo tx = τ o t b(ω o t z ) 2 (2.24) I rzz Ωo bz = ( τb o b(ωo b z ) 2) (2.25) r io i = R o m F o 0 A M ( ωo ros o +ωo (ω o ros o )) M m + µ( r oc o, 0 r oc o,ωo, ω o )+ T (2.26) M g Θ = Y 1 (Θ)ω o (Ic o +Jr) ω o o +[(Ω o t z +Ω o b z )L o r + I c]ω o o +ω o Icω o o FA o y ros o z +Troc o y = FA o x ros o z Troc o x FA o x ros o y FA o y ros o x + M m m (Fo A x roc o y FA o y roc o x ) τt o +τb o r oc o = Π m M ( ωo ros o +ωo (ω o ros o )) (2.27) (2.28) +2ω o r o oc + ω o ro oc +ω o 0 0 ω o ro oc 0 + F o A M m. (2.29)

35 Chapter 2. Modeling 26 In the above, we have µ( r o oc, r o oc,ωo, ω o ) = e 3 +2ω o r o oc + ω o 0 ( m M ( ωo ros o +ωo (ω o ros o )) ro oc 0 +ω o T = Γ[ ( ) Ω o 2 ( ) t z + Ω o 2] bz 1 sinφtanθ cosφtanθ Y 1 (Θ) = 0 cosφ sinφ 0 sinφsecθ cosφsecθ Jr o = 2d jo diag(i rxx,i ryy,0) d jt L o r = d 0 I rzz I ryy 0 jo d jt I rxx I rzz Π = ω o ro oc 0 The matrix I c o in (2.28) can be computed analytically as a function of (ro oc,ṙo oc ). We refer the interested reader to Appendix A..

36 Chapter 3 Preliminary Helicopter Prototypes and Attitude Stabilization This chapter presents three helicopter prototypes built in collaboration with Quanser, and a basic PD attitude stabilizer tested both in simulation and experimentally. By attitude stabilization one refers to the problem of stabilizing the rotation matrix of the body frame of the helicopter to a desired rotation matrix. In terms of the vector of Euler angles Θ used in this thesis, the attitude control problem corresponds to making Θ Θ d, where Θ d is the vector of desired Euler angles. This chapter is organized as follows. In Section 3.1 we present three physical prototypes, pointing out the challenges that were faced in their implementation. In Section 3.2 we present a simplified mathematical model of the rotational dynamics that conforms to the characteristics of the moving mass actuator. Then, we present a simple PD attitude stabilizer and the associated simulation results. Finally, in Section 3.3 we discuss the experiment results of the attitude stabilizer. 27

37 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization Helicopter prototypes Before building a physical prototype, we performed parameter sensitivity analysis by simulating the mathematical model presented in Section 2.6. The sensitivity analysis is required to find the optimal parts and optimal position for the vital components of a helicopter within the physical prototype. For example, the size of the motors and propellers, weight of the moving mass and its maximum displacement, and the position of the moving mass were among the decisions made based on the simulation results of the sensitivity analysis for variety of possible designs. The most important result of the sensitivity analysis was the optimal location of the moving mass mechanism. The simulation results verified that the moving mass mechanism is more effective if it is placed as close as possible to the centre of gravity of the helicopter. Placing the moving mass mechanism near the centre of gravity of the helicopter decreases the helicopter s inertia and subsequently increases its maneuverability. Also, some components of the reaction torques caused by the moving mass actuators are reduced when the moving mass mechanism is placed near the centre of gravity of the helicopter. The sensitivity analysis also verified that the moving mass mechanism should not be placed above the centre of gravity of the helicopter (on the negative z axis), because the reaction torques from the moving mass actuator would be acting on the body in the opposition direction of the desired torque from the thrust. Three prototypes were built during the course of this thesis research, and a fourth prototype is being built by Quanser at the time of writing this document. The moving mass mechanisms and the prototypes were designed by Don Gardner of Quanser. We now describe the three prototypes and the challenges we faced with each of them. Prototype 1. The first prototype, seen in Figure 3.1, was built using an AEO- RC CR23L coaxial motor to drive the propellers, two Turnigy BLS980 Digital Brushless Heli Rudder Servos as the moving mass mechanism (one for each direction) and four

38 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization29 Figure 3.1: Side views of the first prototype. aluminum legs. The moving mass mechanism is illustrated in Figure 3.2. Sensors, wireless connection, and the processor on board were provided by HiQ, the microcontroller board developed by Quanser. The body of the helicopter was designed by Quanser engineers using SolidWorks and built using a 3D printer. The helicopter was powered by a single 3-cell, 2500mAh, 11.1V Lithium-ion polymer battery. Two counter-rotating propellers of the same size, 10 5in, were used for this model. The total weight of the prototype was about 700g. This prototype had several problems. The coaxial motor was not sufficiently powerful and proved to be poorly manufactured. The square shape of the body resulted in a problematic air flow for the rotors. The aluminum legs were too short and heavy. The helicopter was very close totheground, which resulted inasignificant groundeffect. 1 Prototype 2. The second prototype, seen in Figures 3.3 and 3.4, used the same battery and servo motors as prototype 1, but a different coaxial motor, the AEO-RC CR28M. This time the helicopter body was designed to improve the air flow of the 1 The ground effect is described in [10] as follows: Because the ground must be a streamline to the flow, the rotor slipstream tends to rapidly expand as it approaches the surface. This alters the slipstream velocity, the induced velocity in the plane of the rotor, and, therefore, the rotor thrust and power. Similar effects are obtained both in hover and forward flight, but the effects are strongest in the hovering state. When the hovering rotor is operating in ground effect, the rotor thrust is found to be increased for a given power.

39 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization30 Figure 3.2: Detail of the moving mass actuator. Two servomotors, mounted perpendicular to each other, pull two linkages pinned on the helicopter body. The servomotors are mounted on linear guides. By pulling the linkage, the servomotor slides on the linear guide. propellers. The legs were made of carbon fiber, and were longer and lighter than the aluminum legs used in prototype 1. Two counter-rotating propellers, in, were used. The improvements afforded by this prototype were better airflow for the rotors, greater lift, and a reduced ground effect. However, the motor proved to be unreliable. Prototype 3. The third prototype was made in three different configurations using the Himax CR coaxial motor. All three configurations used the same servo motors as the first and second prototypes. The batteries and the weight of the moving masses separate the three configurations from one another. The third prototype was the heaviest among the three, between 1kg and 1.3kg. The first configuration used a 3cell, 2500mAh, 11.1V Lithium-ion polymer battery. With a heavier helicopter body, the moving mass actuator of Figure 3.2 did not displace sufficient mass in order to move the COG of the helicopter by a significant amount. Additionally, the battery only provided

40 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization31 Figure 3.3: Side views of the second prototype. Figure 3.4: Top views of the second prototype. enough power for about 4 minutes of flying, which was not long enough for testing. The second configuration, shown in Figure 3.7, used a higher capacity 3cell, 5000mAh, 11.1V Lithium-ion polymer battery. We increased the moving mass by attaching brass plates to each of the servomotors. The larger battery provided higher current and lasted longer, giving more flight time, but the battery increased the total weight making the moving mass insufficient to move the COG of the helicopter by a significant amount. At this point of our development it became clear that the main implementation challenge

41 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization32 Figure 3.5: Side view of the third prototype with a small battery. Figure 3.6: Top view of the third prototype with a small battery. was the design of the moving mass actuator. The actuator should displace a large enough mass to significantly affect the displacement of the COG of the helicopter. At the same time, the actuator response should be sufficiently fast to guarantee closed-loop stability. In an attempt to address the limitations of the two configurations just described, the third configuration used two 3cell, 2500mAh, 11.1V Lithium-ion polymer batteries connected in parallel to each other. This time, each battery was attached to one of the servos, moving in the x and y directions. In this configuration, the moving mass was large enough that the actuatorwas ableto displace thecog of thehelicopter by a significant amount. This

42 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization33 Figure 3.7: Side view of the third prototype with a large battery. prototype showed promise, and allowed us to successfully test a PD attitude controller presented in what follows. Unfortunately, the moving mass mechanism proved to be unreliable, and its performance started to deteriorate after a few tests. The actuator mechanism showed in Figure 3.2 exerts a torsional force at the juncture between each servomotor and the respective linear guide. The result is that the torque produced by the servomotor places excessive stress on the shaft of the motor as well as the juncture, causing friction and a progressive degradation of the motor. Additionally, with the added mass of the batteries the servomotors cannot reach their rated speed of 10 cm/s, and hence the performance of the helicopter is affected. The fourth prototype being developed uses a completely different actuation mechanism which should overcome the problems described above.

43 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization Simplified rotational dynamics and attitude control Recall the rotational dynamics of the helicopter presented in Section 2.6: I rzz Ωo tz = τ o t b(ωo t z ) 2 I rzz Ωo bz = ( τ o b b(ωo b z ) 2) Θ = Y 1 (Θ)ω o (Ic o +Jr) ω o o +[(Ω o t z +Ω o b z )L o r + I c]ω o o +ω o Icω o o FA o y ros o z +Troc o y = FA o x ros o z Troc o x FA o x ros o y FA o y ros o x + M m m (Fo A x roc o y FA o y roc o x ) τt o +τo b r oc o = Π m M ( ωo ros o +ω o (ω o ros)) o (3.1) +2ω o r o oc + ω o 0 ro oc 0 +ω o ω o ro oc 0 + F o A M m. The control inputs in the model above are the engine torques τt, o τb o, and the actuator force F o A = col(fo A x,f o A y ). The moving mass actuator depicted in Figure 3.2 relies on two servomotors. Servomotors accept a reference position in input, rather than a force, and it is customary to assume that the actuator dynamics are fast enough that the reference position is attained instantaneously. This fact creates a discrepancy between the mathematical model of the moving mass and the actuator implementation: in the mathematical model, the displacement of the moving mass and its derivative are essentially states 2 ; in the actuator, the displacement of the actuator is the control input. To resolve this inconsistency, we need to simplify the mathematical model above by neglecting the 2 Actually, the displacement of the COG of the helicopter, r o oc, and its derivative ṙo oc, are states. These are related to the displacement of the moving mass and its derivative through (2.15)-(2.16).

44 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization35 actuator dynamics, and considering the planar displacement of the helicopter s COG, r oc o = col(ro oc x,roc o y ), as the new control input. The difficulty in neglecting the actuator dynamics arises fromthe fact that theactuator force F A o affects the torque vector in(3.1). However, assuming that the components of the vector ros o are small, the terms in (3.1) involving F o A can be neglected. This is justified provided that the moving mass is designed so that its rest position is very close to the COG of the helicopter. With the assumption that d os 1, the rotational dynamics model becomes I rzz Ωo tz = τ o t b(ωo t z ) 2 I rzz Ωo bz = ( τ o b b(ω o b z ) 2) Θ = Y 1 (Θ)ω o (I o c +Jo r ) ωo +[(Ω o t z +Ω o b z )L o r + I o c ]ωo +ω o I o c ωo = Tr o oc y Troc o x. τt o +τo b As discussed earlier, the control inputs are (τ o t,τo b,ro oc x,r o oc y ). We will assume that the Euler angles in Θ, and the angular velocity in body coordinates, ω o, are available for feedback. In practice, rate gyroscopes provide an estimate of ω o. Sensing Θ, on the other hand, is far more difficult. In a laboratory setting, camera feedback can be used for this purpose. LetΘ d = [φ d θ d ψ d ] beavectorofdesiredeulerangles. Theattitudeofthehelicopter can be controlled using decentralized PD controllers. τ o t = K P1 (ψ ψ d ) K D1 ω o z + τ o t τ o b = K P 1 (ψ ψ d )+K D1 ω o z + τo b where K Pi > 0,K Di > 0, for i = 1,2,3. r o oc x = K P2 (θ θ d )+K D2 ω o y r o oc x = K P3 (φ φ d ) K D3 ω o x,

45 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization36 The r o oc x and r o oc y feedbacks control the roll and pitch. Note that τ o t τo b = 2K P 1 (ψ ψ d ) 2K D1 ω o z. This is a yaw controller. The bias terms τ t o, τ b o are chosen so as to assign the total thrust generated by the two rotors. In steady-state, if the torques are constant then the two rotor speeds are given by (Ω o t z ) 2 = τ t o/b, (Ωo b z ) 2 = τ b o /b. The total thrust is, therefore, T = Γ( τ t o/b + τo b /b). Since only the attitude of the helicopter is of importance at this point, we would like to keep the thrust around the hovering point, which is T = Mg. Therefore, we set τ t o = bmg 2Γ, τ b o = bmg 2Γ. The control gains K Pi and K Di can be found using any conventional control theory methods. We will not present any stability analysis since in the next chapter we will present a detailed control design for a position controller. For now, it suffices to say that the control structure we have presented above is standard. The PD controller above is tested via simulation using the parameters given in Table 3.2 and the reference attitude Θ d = [0 0 0], with the initial conditions given in Table 3.1. Figures 3.8 to 3.10 are the results of the PD attitude controller simulation test. The robustness of the PD attitude controller is tested later on in this section. State Value Unit φ 0.2 rad θ 0.3 rad ψ 0.1 rad ωx o 2 rad/s ωy o 1 rad/s ωz o 3 rad/s Table 3.1: Initial condition.

46 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization37 Parameter Value Unit M Kg g 9.8 m/s 2 m 0.5 Kg b ros o x 8 mm ros o y 6 mm ros o z 25 mm Ic o xx Kg.m 2 Ic o xy Kg.m 2 Ic o xz Kg.m 2 Ic o yy Kg.m 2 Ic o yz Kg.m 2 Ic o zz Kg.m 2 I rxx Kg.m 2 I ryy Kg.m 2 I czz Kg.m 2 Table 3.2: Simulation parameters. To test the robustness of the PD controller some perturbation in the system parameters have been introduced. Table 3.3 compares the actual values to the perturbed values of the parameters used by the controller. The simulations are done for the same set of initial conditions as before, Table 3.1, and the results are given in Figures 3.11 to 3.13.

47 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization Roll Pitch Yaw Radians Seconds Figure 3.8: Simulation result of the PD attitude controller for the Euler angles: Θ. 3.3 Discussion of experimental results The attitude controller presented in the previous section was tested on all three helicopter prototypes, and movies were recorded. The experiments were divided into two steps. First, we tested the yaw controller to avoid spinning of the helicopter around its vertical axis. To do that, the physical prototypes were mounted on the platform shown in Figure The platform was used to constraint the motion of the helicopter so that it could only spin about the z o axis. In this setting, the yaw controller worked well. The second step of the experiment was the test and tuning of the full attitude controller. Due to the limitations of the moving mass actuator described in the previous section, the full attitude controller did not perform well and it was not able to achieve any stability for the first two prototypes and for the first two configurations of the third prototype. The full controller seemed to work well and it was able to stabilize the last configuration of the third prototype when the helicopter was tethered to the ground. Unfortunately, the moving mass mechanism performance started to deteriorate after a few tests; however,

48 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization Magnitude of the input torque for the top rotor Magnitude of the input torque for the bottom rotor Newton meter Seconds Figure 3.9: Control inputs of the attitude controller for the top and bottom rotors torque: τ o t and τ o b. the result of the PD attitude stabilizer for the last prototype was promising before the failure of the moving mass mechanism.

49 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization x Position of the COG w.r.t. the body frame in x direction Position of the COG w.r.t. the body frame in y direction Meters Seconds Figure 3.10: Control inputs of the attitude controller for the position of the COG of the body with respect to the origin of the body frame in x and y directions: r o oc x andr o oc y. Parameter Actual Value Perturbed Value Unit M Kg m Kg Ic o xx Kg.m 2 Ic o xy Kg.m 2 Ic o xz Kg.m 2 Ic o yy Kg.m 2 Ic o yz Kg.m 2 Ic o zz Kg.m 2 I rxx Kg.m 2 I ryy Kg.m 2 I czz Kg.m 2 Table 3.3: Perturbed parameters for robustness test.

50 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization Roll Pitch Yaw 0.3 Radians Seconds Figure 3.11: Simulation result of the attitude controller for the Euler angles in the presence of parameter perturbations Magnitude of the input torque for the top rotor Magnitude of the input torque for the bottom rotor Newton meter Seconds Figure 3.12: Control inputs of the attitude controller for the top and bottom rotors torque in the presence of parameter perturbations.

51 Chapter 3. Preliminary Helicopter Prototypes and Attitude Stabilization42 5 x Position of the COG w.r.t. the body frame in x direction Position of the COG w.r.t. the body frame in y direction 3 Meters Seconds Figure 3.13: Control inputs of the attitude controller for the position of the COG of the body with respect to the origin of the body frame in x and y directions in the presence of parameter perturbations. Figure 3.14: First prototype mounted on the platform used for yaw control tests.