Small unmanned aircraft theory and practice pdf download

The ii -axis points north, the ji -axis points east, and the ki -axis points into the earth. In this section, we will define and describe the following coordinate frames: the inertial frame, the vehicle frame, the vehicle-1 frame, the vehicle-2 frame, the body frame, the stability frame, and the wind frame.

The inertial and vehicle frames are related by a translation, while the remaining frames are related by rotations. The angles defining the relative orientations of the vehicle, vehicle1, vehicle-2, and body frames are the roll, pitch, and yaw angles that describe the attitude of the aircraft. These angles are commonly known as Euler angles.

The rotation angles that define the relative orientation of the body, stability, and wind coordinate frames are the angle of attack and sideslip angles. Throughout the book we assume a flat, non-rotating earth—a valid assumption for MAVs. As shown in figure 2. This coordinate system is sometimes referred to as a north-east-down NED reference frame. It is common for north to be referred to as the inertial x direction, east to be referred to as the inertial y direction, and down to be referred to as the inertial z direction.

However, the axes of F v are aligned with the axis of the inertial frame 13 Coordinate Frames Figure 2. The iv -axis points north, the jv -axis points east, and the kv -axis points into the earth. Figure 2. The iv1 -axis points out the nose of the aircraft, the jv1 -axis points out the right wing, and the kv1 -axis points into the earth. In other words, the unit vector iv points north, jv points east, and kv points toward the center of the earth, as shown in figure 2.

In the absence of additional rotations, iv1 points out the nose of the airframe, jv1 points out the right wing, and kv1 is aligned with kv and points into the earth. The vehicle-1 frame is shown in figure 2. The iv2 -axis points out the nose of the aircraft, the jv2 -axis points out the right wing, and the kv2 -axis points out the belly. The unit vector iv2 points out the nose of the aircraft, jv2 points out the right wing, and kv2 points out the belly, as shown in figure 2.

Therefore, the origin is the center of mass, ib points out the nose of the airframe, jb points out the right wing, and kb points out the belly. The body frame is shown 15 Coordinate Frames Figure 2.

The ib-axis points out the nose of the airframe, the jb-axis points out the right wing, and the kb-axis points out the belly. The directions indicated by the ib , jb , and kb unit vectors are sometimes referred to as the body x, the body y, and the body z directions, respectively. Euler angles are commonly used because they provide an intuitive means for representing the orientation of a body in three dimensions. The physical interpretation of Euler angles is clear and this contributes to their widespread use.

Euler angle representations, 16 Chapter 2 Figure 2. The is -axis points along the projection of the airspeed vector onto the ib-kb plane of the body frame, the js -axis is identical to the jb-axis of the body frame, and the ks -axis is constructed to make a right-handed coordinate system.

Note that the angle of attack is defined as a left-handed rotation about the body jb-axis. This singularity is commonly referred to as gimbal lock. A common alternative to Euler angles is the quaternion. While the quaternion attitude representation lacks the intuitive appeal of Euler angles, they are free of mathematical singularities and are computationally more efficient.

Quaternion attitude representations are discussed in appendix B. We refer to the velocity of the aircraft relative to the surrounding air as the airspeed vector, denoted Va. The magnitude of the airspeed vector is simply referred to as the airspeed, Va. To generate lift, the wings of the airframe must fly at a positive angle with respect to the airspeed vector.

The need for a lefthanded rotation is caused by the definition of positive angle of attack, which is positive for a right-handed rotation from the stability frame is axis to the body frame ib axis. The iw -axis points along the airspeed vector. The unit vector iw is aligned with the airspeed vector Va. The aerodynamic forces, however, depend on the velocity of the airframe relative to the surrounding air. When wind is not present, these velocities are the same.

However, wind is almost always present with MAVs and we must carefully distinguish between airspeed, represented by the velocity with respect to the surrounding air Va , and the ground speed, represented by the velocity with respect to the inertial frame Vg. The bodyframe velocity components u, v, and w are states of the MAV system and are readily available from the solution of the equations of motion.

The wind velocity components uw , vw , and ww typically come from a wind model as inputs to the equations of motion. We define u, v, and w as the body-frame components of the ground speed vector and ur , vr , and wr as the body-frame components of the airspeed vector to clearly distinguish between the two. The significant effect of wind on MAVs is important to understand, more so than for larger conventional aircraft, where the airspeed is typically much greater than the wind speed.

Having introduced the concepts of reference frames, airframe velocity, wind velocity, and the airspeed vector, we can discuss some important definitions relating to the navigation of MAVs. The direction of the ground speed vector relative to an inertial frame is specified using two angles.

The relationship between the groundspeed vector, the airspeed vector, and the wind vector, which is given by equation 2. A more detailed depiction of the wind triangle is given in the horizontal plane in figure 2.

The north direction is indicated by the ii vector, and the direction that the vehicle is pointed is shown by the ib vector, which is fixed in the direction of the body x-axis. The direction the vehicle is traveling with respect to the surrounding air mass is given by the airspeed vector Va. The crab angle is the difference between course and heading. In the absence of wind, the crab angle is zero.

The direction the vehicle is traveling with respect to the ground is shown by the velocity vector Vg. If there is a constant ambient wind, the aircraft will need to crab into the wind in order to follow a ground track that is not aligned with the wind.

Taking the squared norm of both sides of equation 2. When solving the quadratic equation for Vg , the positive root is taken since Vg must be positive.

With both Va and Vg known, the third row of equation 2. Because wind typically has a significant impact on the flight behavior of small unmanned aircraft, we have tried to carefully account for it throughout the text. If wind effects are negligible, however, some important simplifications result.

Suppose that we are given two coordinate frames, F i and F b , as shown in figure 2. Suppose that the vector p is moving in F b and that F b is rotating but not translating with respect to F i.

Our objective is to find the time derivative of p as seen from frame F i. The first three terms on the right-hand side of equation 2. Thus, the differentiation is carried out in the moving frame. We denote this local derivative term by. Given that 25 Coordinate Frames ib , jb , and kb are fixed in the F b frame, their derivatives can be calculated as shown in [9] as. We can rewrite the last three terms of equation 2.

We will use this relation as we derive equations of motion for the MAV in chapter 3. We have described how rotation matrices can be used to transform coordinates in one frame of reference to coordinates in another frame of reference. An understanding of these orientations is essential to the derivation of equations of motion and the modeling of aerodynamic forces involved in MAV flight.

We have introduced the wind triangle and have made the relationships between airspeed, ground speed, wind speed, heading, course, flight-path angle, and air-mass-referenced flight-path angle explicit. We have also derived an expression for the differentiation of a vector in a rotating reference frame.

Notes and References There are many references on coordinate frames and rotations matrices. A particularly good overview of rotation matrices is [10]. Overviews of attitude representations are included in [8, 11]. The definition of the different aircraft frames can be found in [4, 1, 7, 12]. A particularly good explanation is given in [13].

Vector differentiation is discussed in most textbooks on mechanics, including [14, 15, 16, 9]. Coordinate Frames 27 2. Creating animations in Simulink is described in appendix C and example files are contained on the textbook website. Read appendix C and study carefully the spacecraft animation using vertices and faces given at the textbook website.

Create an animation drawing of the aircraft shown in figure 2. Using a Simulink model like the one given on the website, verify that the aircraft is correctly rotated and translated in the animation. In the animation file, switch the order of rotation and translation so that the aircraft is first translated and then rotated, and observe the effect. Deriving the nonlinear equations of motion for a MAV is the focus of chapters 3 and 4.

In chapter 5, we linearize the equations of motion to create transfer-function and state-space models appropriate for control design. In this chapter, we derive the expressions for the kinematics and the dynamics of a rigid body.

In this chapter, we will focus on defining the relations between positions and velocities the kinematics and relations between forces and moments and the momentum dynamics.

In chapter 4, we will concentrate on the definition of the forces and moments involved, particularly the aerodynamic forces and moments. In chapter 5, we will combine these relations to form the complete nonlinear equations of motion. While the expressions derived in this chapter are general to any rigid body, we will use notation and coordinate frames that are typical in the aeronautics literature. In particular, in section 3.

In section 3. There are three position states and three velocity states associated with the translational motion of the MAV. Similarly, there are three angular position and three angular velocity states associated with the rotational motion.

The state variables are listed in table 3. The state variables are shown schematically in figure 3. The northeast-down positions of the MAV pn, pe , pd are defined relative to the inertial frame. The linear velocities u, v, w and the angular velocities p, q, r of the MAV are defined with respect to the body frame.

The 29 Kinematics and Dynamics Table 3. Generally, the angular rates p,. The remainder of this chapter is devoted to formulating the equations of motion corresponding to each of the states listed in table 3. The components u, v, and w correspond to the inertial velocity of the vehicle projected onto the ib , jb , and kb axes, respectively.

On the other hand, the translational position of the MAV is usually measured and expressed in an inertial reference frame. This is a kinematic relation in that it relates the derivative of position to velocity: forces or accelerations are not considered.

The angular rates are defined in the body frame F b. We will assume a flat earth model, which is appropriate for small and miniature air vehicles.

Even though motion is referenced to a fixed frame, it can be expressed using vector components associated with other frames, such as the body frame. Vbg is the velocity of the MAV with respect to the ground as expressed in the body frame. The external forces include gravity, aerodynamic forces, and propulsion forces. The derivative of velocity taken in the inertial frame can be written in terms of the derivative in the body frame and the angular velocity according to equation 2.

Combining 3. This expression is true provided that moments are summed about the center of mass of the MAV. The derivative of angular momentum taken in the inertial frame can be expanded using equation 2.

The larger J x is in value, the more the aircraft opposes angular acceleration about the x axis. This line of thinking, of course, applies to the moments of inertia J y and J z as well. In practice, the inertia matrix is not calculated using equation 3. Instead, it is numerically calculated from mass properties using CAD models or it is measured experimentally using equipment such as a bifilar pendulum [17, 18].

Because the integrals in equation 3. Taking derivatives and substituting from the body frame, hence dt b into equation 3. They are not complete in that the externally applied forces and moments are not yet defined.

Models for forces and moments due to gravity, 37 Kinematics and Dynamics aerodynamics, and propulsion will be derived in chapter 4. In appendix B, an alternative formulation to these equations that uses quaternions to represent the MAV attitude is given.

This model will be the basis for analysis, simulation, and control design that will be discussed in forthcoming chapters. Notes and References The material in this chapter is standard, and similar discussions can be found in textbooks on mechanics [14, 15, 19], space dynamics [20, 21], flight dynamics [1, 2, 5, 7, 12, 22] and robotics [10, 23]. Equations 3. In equations 3. To correctly represent the motion of the aircraft in the inertial frame using ur , vr , and wr as states, the effect of wind speed and wind acceleration must be taken into account.

Read appendix D on building s-functions in Simulink, and also the Matlab documentation on s-functions. Implement the MAV equations of motion given in equations 3. Assume that the inputs to the block are the forces and moments applied to the MAV in the body frame. Block parameters should include the mass, the moments and products of inertia, and the initial conditions for each state.

Use the parameters given in appendix E. Simulink templates are provided on the website. Connect the equations of motion to the animation block developed in the previous chapter. Verify that the equations of motion are correct by individually setting the forces and moments along each axis to a nonzero value and convincing yourself that the motion is appropriate. Since J xz is non-zero, there is gyroscopic coupling between roll and yaw.

To test your simulation, set J xz to zero and place nonzero moments on l and n and verify that there is no coupling between the roll and yaw axes. Verify that when J xz is not zero, there is coupling between the roll and yaw axes. Following [5], we will assume that the forces and moments are primarily due to three sources, namely, gravity, aerodynamics, and propulsion.

In this chapter, we derive expressions for each of the forces and moments. Gravitational forces are discussed in section 4. Atmospheric disturbances, described in section 4. This force acts in the ki direction and is proportional to the mass of the MAV by the gravitational constant g.

Therefore, we must transform the 40 Chapter 4 Figure 4. The strength and distribution of the pressure acting on the MAV is a function of the airspeed, the air density, and the shape and attitude of the MAV. Instead of attempting to characterize the pressure distribution around the wing, the common approach is to capture the effect of the pressure with a combination of forces and a moment.

For example, if we consider the longitudinal ib -kb plane, the effect of the pressure acting on the MAV body can be modeled using a lift force, a drag force, and a moment. As shown in figure 4. For airfoils generally, the lift, drag, and pitching moment coefficients are significantly influenced by the airfoil shape, Reynolds number, Mach number, and the angle of attack.

For the range of airspeeds flown by small and miniature aircraft, the Reynolds number and Mach number effects are approximately constant. It is common to decompose the aerodynamic forces and moments into two groups: longitudinal and lateral. The longitudinal forces and moments act in the ib -kb plane, also called the pitch plane.

They include the forces in the ib and kb directions caused by lift and drag and the moment about the jb axis. The lateral forces and moments include the force in the jb direction and the moments about the ib and kb axes. The control surfaces 42 Chapter 4 Figure 4.

For standard aircraft configurations, the control surfaces include the elevator, the aileron, and the rudder. Other surfaces, including spoilers, flaps, and canards, will not be discussed in this book but are modeled similarly.

Figure 4. The positive direction of a control surface deflection can be determined by applying the righthand rule to the hinge axis of the control surface. For example, the hinge axis of the elevator is aligned with the body jb axis. Applying the right-hand rule about the jb axis implies that a positive deflection for the elevator is trailing edge down.

Similarly, positive deflection for the rudder is trailing edge left. Finally, positive aileron deflection is trailing edge down on each aileron. For small aircraft, there are two other standard configurations. The first is the v-tail configuration as shown in figure 4. The control surfaces for a v-tail are called ruddervators. Driving the ruddervators differentially has the same effect as a rudder, producing a torque about kb.

Driving the ruddervators together has the same effect as an elevator, producing 43 Forces and Moments Figure 4. The ruddervators replace the rudder and the elevator. Driving the ruddervators together has the same effect as an elevator, and driving them differentially has the same effect as a rudder. The other standard configuration for small aircraft is the flying wing depicted in figure 4.

The control surfaces for a flying wing are called elevons. Driving the elevons differentially has the same effect as ailerons, producing a torque about ib , while driving the elevons together has the same effect as an elevator, causing a torque about jb. They are the aerodynamic forces and moment with which we are perhaps most 44 Chapter 4 Figure 4.

The elevons replace the aileron and the elevator. Driving the elevons together has the same effect as an elevator, and driving them differentially has the same effect as ailerons.

By definition, the lift and drag forces are aligned with the axes of the stability frame, as shown in figure 4. When represented as a vector, the pitching moment also aligns with the js axis of the stability frame. The lift and drag forces and the pitching moment are heavily influenced by the angle of attack.

For small angles of attack, however, the flow over the wing will remain laminar and attached. Under these conditions, the lift, drag, and pitching moment can be modeled with acceptable accuracy using linear approximations. It is common to nondimensionalize the partial derivatives of this linear approximation.

We can then rewrite equation 4. Under typical, low-angle-of-attack flight conditions, they are a sufficiently accurate representation of the forces and moments produced.

The flow over the aircraft body is laminar and attached and the flow field over the aircraft is termed quasisteady, meaning it only changes slowly with respect to time. The shape of the flow field is predictable and changes in response to changes in the angle of attack, pitch rate, and elevator deflection.

The quasi-steady behavior of the flow field results in longitudinal aerodynamic forces and torques that are predictable and fairly straightforward to model, as shown above.

In contrast to the quasi-steady aerodynamics typically experienced by aircraft, unsteady aerodynamics are challenging to model and predict. Unsteady aerodynamics are characterized by nonlinear, 46 Chapter 4 Figure 4. Designed for advanced undergraduate or graduate students in engineering or the sciences, this book offers a bridge to the aerodynamics and control of UAV flight. Timothy W. Most of the mathematics involved is very straightforward and the results are presented in a very clear manner.

This is a text that should be very useful to those working on unmanned aerial vehicles and may even be of interest to those working on unmanned land or marine vehicles. Despite their importance, no previous textbook has accessibly introduced UAVs to students in the engineering, computer, and science disciplines--until now. Small Unmanned Aircraft provides a concise but comprehensive description of the key concepts and technologies underlying the dynamics, control, and guidance of fixed-wing unmanned aircraft, and enables all students with an introductory-level background in controls or robotics to enter this exciting and important area.

The authors explore the essential underlying physics and sensors of UAV problems, including low-level autopilot for stability and higher-level autopilot functions of path planning. The textbook leads the student from rigid-body dynamics through aerodynamics, stability augmentation, and state estimation using onboard sensors, to maneuvering through obstacles. Students begin by modeling rigid-body dynamics, then add aerodynamics and sensor models. They develop low-level autopilot code, extended Kalman filters for state estimation, path-following routines, and high-level path-planning algorithms.

The final chapter of the book focuses on UAV guidance using machine vision. Designed for advanced undergraduate or graduate students in engineering or the sciences, this book offers a bridge to the aerodynamics and control of UAV flight. It Recasts like six red months saw used at this development. This pdf Developments in Tissue Converting and Packaging means running a child file to direct itself from Individual springs.

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