Flight Dynamics & Control Equations of Motion of 6 dof Rigid Aircraft-Kinematics

Transcription

1 Flight Dynamic & Control Equation of Motion of 6 dof Rigid Aircraft-Kinematic Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity

2 Outline Rotation Matrix Angular Velocity Euler Angle Kinematic equation

3 Rigid Body kinematic Conider two reference frame: a pace frame () and a ody frame () with common origin (R 0 0). Let { x, y, z } e an orthonormal ai for the pace frame. Let { x, y, z } e an orthonormal ai for the ody frame. he orientation of the ody frame i pecified relative to the pace frame if the ai vector { x, y, z } are pecified in the coordinate of the pace frame. x z R R 0 z y Rotation Matrix xx xy xz L: yx yy yz zx zy zz x xx x xy y xz z y yx x yy y yz z z zx x zy y zz z r x y

4 Propertie of the Rotation Matrix 1) L convert ody coordinate to pace coordinate. o ee thi, uppoe a vector r ha coordinate r, r, r in the ody frame and r, r, r in the pace frame: x y z r r + r + r r + r x x y y z z x x y multiply ucceively y,, to prove x y z y xy yy zy y r z xz yz zz r z + r y z z rx xx yx zx rx r r r Lr 2) Orthonormal unit vector LL I, L L 3) Right hand coordinate ytem det L 1 4) ranlation plu rotation R R + Lr, R R + Lr + Lr 0 0 x y z 1

5 Succeive Rotation Suppoe a ucceion of rotation are made, ay L, then L, then L, then the total rotation i defined y L LLL Example: coψ inψ 0 1) Rotation of angle ψ aout z-axi L1 inψ coψ z coθ 0 inθ 2) Rotation of angle θ aout y-axi L x θ in θ 0 coθ z ) Rotation of angle φ aout x-axi L3 0 coφ inφ φ 0 inφ coφ y

6 Euler Angle Conider a reference frame fixed in the ody, with origin located y the poition vector R and angular orientation denoted y L, oth relative to a fixed inertial (pace) frame. L can e parameterized y the Euler angle ψθφ,, (yaw, pitch, roll) repreenting equential rotation aout the axe z, y, x, repectively: L coθcoψ coθinψ inθ ψ, θ, φ inφinθcoψ coφinψ inφinθinψ + coφcoψ inφcoθ co φ inθcoψ + inφinψ coφinθinψ inφcoψ coφcoθ ( ) z 0 ψ Standard coordinate frame employ 3,2,1 or z,y,x convention for defining Euler angle. φ θ y x

7 Angular Velocity ~ 1 Conider a rotation L L+ L L Definition : L lim t 0 t Definition : A quare matrix A i called anti-ymmetric if A A. Example: A 3 3 anti-ymmetric matrix ha the general form 0 -c c 0 a a 0 Notice that there are only 3 independent element ac,,. In thi ene every 3 3 matrix i equal to a 3-vector. We ue the notation a 0 -c v and v c 0 a c a 0 Definition : he cro product of two 3-vector u, v i u v uv

8 Angular Velocity ~ 2 Propoition : LL Proof: ut ( ) ( ) and LL ( ) ( ) L L I, L+ L L+ L I and antiymmetric matrice. ( ) L L+ LL+ L L 0 LL + LL 0 LL LL Definition : Define the antiymmetric matrice ω L+ L L+ L L L+ L L+ L L+ L L L L, ω LL. ω ~ ω, ω ~ ω are the angular velocity in pace and ody coordinate, repectively. Note: ω L ωl, ω L ωl, L ωl, L ωl R R + Lr + Lr R + ω Lr + Lr R + ω Lr + Lr 0 0 0

9 Velocity in Body and Space Coordinate Conider a ody frame () and a pace frame () with common origin and the only relative motion i rotation. If r i the poition vector of any point fixed in the ody, then ( ) ( ), ( t) r ( t) L ( t) r L ( t) Lr ( t) ω ( t) r ( t) ω ( t) r ( t) r t L t r r contant v Similarly, ( ) ( ) ( ) ( ) ω ( ) ( ) ( ) ω ( ) ( ) ( ) ω ( ) ( ) ω ( ) ( ) ranlation + Rotation Inertial velocity in pace coordinate V R R + Lr R + ω Lr R + ω L r v t Ltv t Lt tr t Lt tl tr t tr t t r t R R + r R R + r R R + r Inertial velocity in ody coordinate V LR LR + LL r LR + L ω L r LR + ω r LR + ω r

10 Euler Angle Kinematic Recall the fundamental kinematic relationhip: Lt φ define the coordinate vector q : θ, ψ ( ) ω ( tlt ) ( ) q Γ ( q) ω, 1 inφtanθ coφtanθ 1 0 inφ Γ ( q) 0 coφ inφ 1, ( q ) 0 coφ coθinφ Γ 0 inφecθ coφecθ 0 inφ coθcoφ

11 Kinematic Equation Summary inertial pace location x coθcoψ coψ inθinφ coφinψ coφcoψ inθ + inφinψ u y coθinψ coφcoψ inθinφinψ coψ inφ coφinθinψ v + + z inθ coθinφ coθcoφ w angular orientation φ 1 inφtanθ coφtanθ p θ 0 coφ inφ q ψ 0 inφecθ coφecθ r