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AER1216 Lecture 6: Fixed-Wing Dynamics and Control AER1216 Lecture 6: Fixed-Wing Dynamics and Control Prof. Hugh H.T. Liu Feb. 26, 2016 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 Outline Vehicle Kinematics Fixed-Wing UAV Dynamics: Equations of Motion Aerodynamic Forces and Derivatives Linearization: Small Perturbation Theory Dynamic Modes and Performance Criteria UAV Autopilot 2 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 1. Vehicle KinematicsI 1.1 Aircraft Reference Frames I An inertial Earth Surface Frame is defined with its origin somewhere on the surface and near the vehicle as much as possible (so that a flat surface might be assumed in simplifying dynamics analysis); axis zE directed vertically ~ down; xE points north, and yE east. ~ ~ I The air-trajectory reference frame is also called the wind axes system. It has origin fixed to the vehicle (usually at the center of mass), and xW is directed along the velocity vector v of ~ ~ the vehicle relative to the atmosphere. The axis zW lies in the ~ plane of symmetry of the vehicle. If the atmosphere is at rest, then OW would trace out the trajectory of the vehicle relative to the Earth, and xW would be always tangent to it. ~ 3 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 1. Vehicle KinematicsII 1.1 Aircraft Reference Frames I Any set of axes fixed in a rigid-body is a body-fixed reference frame (and the angular velocity of the frame is the same as that of the body). In flight vehicles, the origin is usually chosen at the center of mass, the axes x; z are chosen in the ~ ~ plane of vertical symmetry, where z is directed downwards. ~ 4 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 1.2 The Vehicle Euler AnglesI In flight dynamics, the Euler angles used are those which rotate the earth-surface frame FE into coincidence with the relevant axis system (frame), usually either FB or FW , denoted by ( ; θ; φ) and (µ, γ; σ) respectively. Figure 6.1 shows the sequence of rotations. 5 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 1.2 The Vehicle Euler AnglesII Figure 6.1: Airplane Orientation 6 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 1. a rotation about zE = z1, is called the yaw angle: ~ ~ −π ≤ < π or 0 ≤ < 2π, fxE ; yE ; zE g = fx1; y1; z1g =) x2; y2; z2 ~ ~ ~ ~ ~ ~ ~ ~ ~ 2. a rotation θ about y2, θ is called the pitch angle: ~ −π=2 ≤ θ ≤ π=2, fx2; y2; z2g =) fx3; y3; z3g ~ ~ ~ ~ ~ ~ 3. a rotation φ about x3 = x, φ is called the roll angle: ~ ~ −π ≤ φ < π or 0 ≤ φ ≤ 2π, fx3; y3; z3g =) fx; y; zg ~ ~ ~ ~ ~ ~ 7 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 The rotation matrix from FE to FB is given by FB = C BE FE (6.1) ~ ¯ ~ where C BE = C 1(φ)C 2(θ)C 3( ) ¯ ¯ ¯ ¯ 2 cos θ cos cos θ sin − sin θ 3 4 sin φ sin θ cos − cos φ sin sin φ sin θ sin + cos φ cos sin φ cos θ 5 cos φ sin θ cos + sin φ sin cos φ sin θ sin − sin φ cos cos φ cos θ (6.2) and C EB ¯ 2 cos θ cos sin φ sin θ cos − cos φ sin cos φ sin θ cos + sin φ sin 3 4 cos θ sin sin φ sin θ sin + cos φ cos cos φ sin θ sin − sin φ cos 5 − sin θ sin φ cos θ cos φ cos θ (6.3) 8 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 Similarly, the rotation matrix from FE to FW is given by FW = C WE FE (6.4) ~ ¯ ~ where C WE = C 1(µ)C 2(γ)C 3(σ). ¯ ¯ ¯ ¯ 1. a rotation σ about zE = z1, σ is called the heading angle, ~ ~ fxE ; yE ; zE g = fx1; y1; z1g =) x2; y2; z2 ~ ~ ~ ~ ~ ~ ~ ~ ~ 2. a rotation γ about y2, γ is called the flight path angle, ~ fx2; y2; z2g =) fx3; y3; z3g ~ ~ ~ ~ ~ ~ 3. a rotation µ about x3 = x, µ is called the bank angle, ~ ~ fx3; y3; z3g =) fx; y; zg ~ ~ ~ ~ ~ ~ 9 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 The rotation matrix from FE to FB is given by FW = C WE FE (6.5) ~ ¯ ~ where C WE = C 1(µ)C 2(γ)C 3(σ) ¯ ¯ ¯ ¯ 2 cos γ cos σ cos γ sin σ − sin γ 3 4 sin µ sin γ cos σ − cos µ sin σ sin µ sin γ sin σ + cos µ cos σ sin µ cos γ 5 cos µ sin γ cos σ + sin µ sin σ cos µ sin γ sin σ − sin µ cos σ cos µ cos γ (6.6) and C EW ¯ 2 cos γ cos σ sin µ sin γ cos σ − cos µ sin σ cos µ sin γ cos σ + sin µ sin σ 3 4 cos γ sin σ sin µ sin θ sin σ + cos µ cos σ cos µ sin θ sin σ − sin µ cos σ 5 − sin θ sin µ cos θ cos µ cos θ (6.7) 10 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 Consider the velocity vector expression in FE and FB respectively, 2 3 2 3 x_E u v E = 4 y_E 5 = C EB v B = C EB 4 v 5 (6.8) ¯ ¯ ¯ ¯ z_E w 11 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 The angular velocity of FB relative to the inertial frame FE is T denoted by !B and is expressed by !B = [p q r] . The angular ~ ¯ velocity vector following the Euler rotations are ! = _z1 + θ_y2 + φ_x3 (6.9) ~ ~ ~ ~ 2 φ_ 3 _ = [x3 y2 z1] 4 θ 5 (6.10) ~ ~ ~ _ Transform axes z1; y2; x3 to body frame axes x; y; z, one obtains ~ ~ ~ ~ ~ ~ 2 1 3 x3 = x = [x y z] 4 0 5 (6.11) ~ ~ ~ ~ ~ 0 2 0 3 y2 = y3 = cos φy − sin φz = [x y z] 4 cos φ 5 (6.12) ~ ~ ~ ~ ~ ~ ~ − sin φ 12 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 z1 = z2 = − sin θx3 + cos θz3 (6.13) ~ ~ ~ ~ = − sin θx + cos θ(sin φy + cos φz) (6.14) ~ ~ ~ 2 − sin θ 3 = [x y z] 4 cos θ sin φ 5 (6.15) ~ ~ ~ cos θ cos φ Therefore !B = S B Θ_ B ¯ ¯ ¯ 2 p 3 2 1 0 − sin θ 3 2 φ_ 3 2 φ_ 3 _ ∆ _ 4 q 5 = 4 0 cos φ sin φ cos θ 5 4 θ 5 = S B 4 θ 5 r 0 − sin φ cos φ cos θ _ ¯ _ (6.16) 13 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 1.3 The Aerodynamic AnglesI The velocity of the vehicle relative to the atmosphere v can either ~ be given by its three coordinates (u; v; w) in a body-fixed frame FB , or alternatively by the magnitude V and two suitable defined angles. These angles, which are of fundamental importance in determining the aerodynamic forces that act on the vehicle, are defined as the angle of attack and sideslip angle respectively. Angle of Attack: w α = tan−1 − π ≤ α ≤ π (6.17) u Sideslip Angle: v β = sin−1 − π ≤ β ≤ π (6.18) V 14 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 1.3 The Aerodynamic AnglesII It follows that the velocity components in the body-fixed frame are represented by u = V cos β cos α (6.19) v = V sin β (6.20) w = V cos β sin α (6.21) This relationship reveals the rotation matrix from FW to FB by the following Euler rotation sequence: (−β; α; 0), i.e. FB = C BW FW (6.22) ~ ¯ ~ 15 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 1.3 The Aerodynamic Angles III where C BW = C 1(0)C 2(α)C 3(−β) ¯ ¯ ¯ ¯ 2 cos α 0 − sin α 3 2 cos β − sin β 0 3 = 4 0 1 0 5 4 sin β cos β 0 5 sin α 0 cos αx 0 0 1 2 cos α cos β − cos α sin β − sin α 3 = 4 sin β cos β 0 5 (6.23) sin α cos β − sin α sin β cos α Obviously, the velocity of the vehicle relative to the atmosphere v ~ T expression under FW is v W = [V 0 0] , and the formulas 6.19 - ¯ 6.21 can be obtained by v B = C BW v W . ¯ ¯ ¯ 16 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 Kinematics Diagram Figure 6.2: Vehicle Kinematics Diagram 17 AER1216 - FUNDAMENTALS OF UAVS Lect. 6: Fixed-Wing Dyn. & Cntl. H.H.T. Liu c 2015-16 2. Fixed-Wing UAV Dynamics: Equations of MotionI Given the expressions under body-fixed frame FB : fx; y; zg: ~ ~ ~ T f B = [XYZ] (6.24) ¯ T !B = [p q r] (6.25) ¯ T v B = [u v w] (6.26) ¯ one gets the following force equation expressions: × f B = m(v_ B + ! v B ) (6.27) ¯ ¯ ¯B ¯ leading to X − mg sin θ = m(_u + qw − rv) (6.28) Y + mg cos θ sin φ = m(_v + ru − pw) (6.29) Z + mg cos θ cos φ = m(_w + pv − qu) (6.30) 18 AER1216 - FUNDAMENTALS OF UAVS Lect.
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