Optimal Control, Guidance and Estimation Lecture 17 Overview of Flight Dynamics III Prof. adhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Six DOF Model Prof. adhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
eference Frames and Dynamic ariables Body frame dm ds d Q Inertial frame U P W 3 Dynamic (Force and Moment) Equations ef: oskam J., Airplane Flight Dynamics and Automatic Controls, 1995 1 Uɺ WQ g sin Θ + ( + ) m 1 ɺ WP U + g sin cos Θ + Y + Y m 1 Wɺ UQ P + g cos cos Θ + Z + Z m ( ) ( ) ( ) ( ) Pɺ c Q + c PQ + c L + L + c + 1 2 3 4 2 2 ( ) ( ) Qɺ c P c P + c M + M 5 6 7 ( ) ( ) ɺ c PQ c Q + c L + L + c + 8 2 4 9 Q U P W 4
Kinematic Equations Prof. adhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Orientation of Airplane wrt. Inertial Frame: Euler Angles ranslate the inertial frame and make it coincide with the CG Make the sequential transformation of this frame so as to make it parallel to the body frame. Common sequence: Ψ, Θ, Y Z P Y Z 6
Euler Angles ef: oskam J., Airplane Flight Dynamics and Automatic Controls, 1995 ranslate Y Z parallel to itself until its center coincides with the YZ system. ename Y Z as 1 Y 1 Z 1 for convenience. Y 1 Z Y 1 P Z 1 Y otate the system 1 Y 1 Z 1 about Z 1 axis over an angle ψ his yields the coordinate system 2 Y 2 Z 2 Z 7 Euler Angles ef: oskam J., Airplane Flight Dynamics and Automatic Controls, 1995 otate the system 2 Y 2 Z 2 about Y 2 axis over an angle Θ his yields the coordinate system 3 Y 3 Z 3 otate the system 3 Y 3 Z 3 about Y 3 axis over an angle his yields the coordinate system YZ OPIMAL ADACED COOL, GUIDACE SYSEM AD DESIG ESIMAIO Dr. Prof. adhakant adhakant Padhi, Padhi, AE Dept., AE Dept., IISc-Bangalore 8
Flight Path elative to Earth Fixed Coordinates (Inertial Frame) ef: oskam J., Airplane Flight Dynamics and Automatic Controls, 1995 Y Z xɺ U1 yɺ 1 zɺ W 1 Y Z Y Z 1 1 1 Y Z 1 1 1 2 2 2 U1 cos Ψ sin Ψ U 2 1 sin cos Ψ Ψ 2 W 1 1 W 2 1 Y Z P Z 1 Y 1 Y Z 9 Flight Path elative to Earth Fixed Coordinates (Inertial Frame) ef: oskam J., Airplane Flight Dynamics and Automatic Controls, 1995 Y Z Y Z 2 2 2 3 3 3 U 2 cosθ sin Θ U 3 2 1 3 W 2 sin Θ cos Θ W 3 3 Y3 Z3 Y Z U 3 1 U 3 cos sin W 3 sin cos W 1
Flight Path elative to Earth Fixed Coordinates (Inertial Frame) xɺ I xɺ U1 yɺ y I 1 zɺ I zɺ W 1 cos Ψ sin Ψ U 2 sin cos Ψ Ψ 2 1 W 2 cos Ψ sin Ψ cosθ sin Θ U 3 sin Ψ cos Ψ 1 3 1 sin Θ cos Θ W 3 cos Ψ sin Ψ cosθ sin Θ 1 U sin Ψ cos 1 cos sin Ψ 1 sin Θ cos Θ sin cos W 11 elationship Between Ψ, Θ, P, Q, and ω ip + jq + k Ψ ɺ + Θ ɺ + ɺ Y P However, ɺ Ψ k1ψ ɺ k2ψɺ (otation is about Z1) ɺ Θ j2θ ɺ j3θɺ (otation is about Y2 ) ɺ i ɺ iɺ (otation is about ) Y 3 3 Y Z 12
elationship Between Ψ, Θ, P, Q, 2 3 3 { isin cos ( j sin k cos )} + ( j cos k sin ) Θ ɺ + iɺ and Using co-ordinate transformation rules, we can write: k i sin Θ + k cos Θ j3 cos sin j k 3 sin cos k Using these relationships, we can write: ω Θ + Θ + Ψɺ ip + jq + k 13 elationship Between Ψ, Θ, P, Q, Equating the coefficients, P ɺ Ψɺ sin Θ Q Θɺ cos + Ψɺ cos Θsin Ψɺ cos Θcos Θɺ sin In matrix form, P 1 sin Θ ɺ Q cos cos Θsin Θɺ sin cos Θcos Ψɺ and 14
elationship Between Ψ, Θ, P, Q, aking the inverse transformation, 1 ɺ 1 sin Θ P Θ ɺ cos cos sin Q Θ Ψɺ sin cos Θcos P + Qsin tan Θ + cos tan Θ Q cos sin ( Q sin + cos ) secθ and 15 Components of Gravitational Force in Body Co-ordinates k g k g ig + jg + kg 1 Y Z It is desirable to express g g, g, Y Z in terms of g and Euler angles Θ and. Y Z CG Y ote that Ψ does not affect the component of gravity because k k. ote that: i i i j k 3 k 3 (,, : Body frame) k cos + j sin 1 2 g Z 16
Components of Gravitational Force k i sin Θ + k cos Θ (already derived) 2 3 3 3 1 2 ( ) i sin Θ + k cos + j sin cos Θ But k, we get: ( ) k1g g i sin Θ + k cos + j sin cos Θ ig + jg + kg Y Z Comparing the coefficients of i, j, k one can write: g g g Y Z k g sin Θ g sin cos Θ g cos cosθ Y Z CG g Y Z 17 Kinematic Equations ate of Change of Euler Angles: ɺ P + Qsin tan Θ + cos tan Θ Θɺ Q cos sin Ψɺ sin + cos secθ ( Q ) Flight Path in the Inertial Frame: xɺ I cos Ψ sin Ψ cos Θ sin Θ 1 U yɺ I sin cos 1 cos sin Ψ Ψ zɺ I 1 sin Θ cos Θ sin cos W 18
Complete Six-DOF Model Prof. adhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Airplane Dynamics: Six Degree-of-Freedom Model ef: oskam J., Airplane Flight Dynamics and Automatic Controls, 1995 1 Uɺ WQ g sin Θ + ( + ) m 1 ɺ WP U + g sin cos Θ + ( Y + Y ) m 1 Wɺ UQ P + g cos cos Θ + ( Z + Z ) m Pɺ c Q + c PQ+c L + L +c + ( ) ( ) 1 2 3 4 2 2 ( ) ( ) Qɺ c P c P + c M + M 5 6 7 ( ) ( ) ɺ c PQ c Q + c L + L + c + 8 2 4 9 ɺ P + Q sin tan Θ + cos tan Θ Θɺ Q cos sin xɺ I cosψ sinψ cosθ sinθ 1 U yɺ I sinψ cosψ 1 cosφ sinφ Ψɺ ( Qsin + cos ) sec Θ zɺ I 1 sinθ cosθ sinφ cosφ W h ɺ z ɺ U sin Θ cos Θsin W cos Θ cos I OPIMAL ADACED COOL, GUIDACE SYSEM AD DESIG ESIMAIO Dr. Prof. adhakant adhakant Padhi, Padhi, AE Dept., AE Dept., IISc-Bangalore 2
Airplane Dynamics: Six Degree-of-Freedom Model ef: oskam J., Airplane Flight Dynamics and Automatic Controls, 1995 where c1 Izz( Iyy Izz) Ixz c2 Ixz Izz + Ixx Iyy ( ) c3 1 Izz 2 c4 ( IxxIzz Ixz ) Iyz 2 c 8 Ixx Ixx Iyy I xz ( ) + c I 9 xx 2 c I I ) / I 5 c I / I c 6 7 1/ I zz xx yy xz yy yy OPIMAL ADACED COOL, GUIDACE SYSEM AD DESIG ESIMAIO Dr. Prof. adhakant adhakant Padhi, Padhi, AE Dept., AE Dept., IISc-Bangalore 21 Airplane Dynamics: Six Degree-of-Freedom Model ef: oskam J., Airplane Flight Dynamics and Automatic Controls, 1995 cos cos Ψ i i i i 1 Y cos sin Ψ i i i i 1 Z sin i i i 1 ( cos sin ) ( sin ) L Ψ z y i i i i i i i i i 1 1 ( cos cos ) ( sin ) M Ψ z + x i i i i i i i i i 1 1 ( cos cos ) ( cos sin ) Ψ y + Ψ x i i i i i i i i i i 1 1 C 1 D C O D C D C s i ( ) ( )( ) h α Dδ α α qs α E E Z δ Z + s CL C O L C L C α i h Lδ i E h L Ls Cl Cl Cl β δ A δ δ A ( ) ( ) qsb α α β + s Cn Cn C β δ n δ A δ δ A Y qs CY qs CY β + CY C β δ Y A δ δ [ 1 α ] M qsc C qsc C C C i + C m mo m mi h h m δ α δ E E ( α ) cosα sinα σ i imax i sinα cosα OPIMAL ADACED COOL, GUIDACE SYSEM AD DESIG ESIMAIO Dr. Prof. adhakant adhakant Padhi, Padhi, AE Dept., AE Dept., IISc-Bangalore 22
Six Degree-of-Freedom Model: Important Observations he Six-DOF model consists of 12 equations, out of which 3 equations (namely x, y and ψ) are decoupled with the non-rotating and flat earth assumptions. Hence, for flight control design, usually 9 equations are good enough. One still need to integrate the x, y and ψ equations, however, to know the trajectory of the vehicle in the inertial frame. he 9 equations are further reduced to 8 equations (assuming constant height and neglecting the z equation) and then decoupled into 4 longitudinal equations and 4 lateral equations in the linearized (small disturbance) theory. 23 Six Degree-of-Freedom Model: Important Observations An airplane is symmetric about its Z-plane. Hence: I xy I Missiles and launch vehicles are typically symmetric about both Z-plane as well as Y-plane. Hence: yz I xy I yz I zx 24
Other eference Frames used in Flight Dynamics Stability Frame Body frame is rotated by α about the Y-axis S Y, Ys α P Wind Frame Stability frame is further rotated by β about stability Z s -axis Z Z S ote: Body, Stability and Wind frame variables are related through rotational transformations. 25 Small-Disturbance Flight Dynamics eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. Prof. adhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
Linearization using Small Perturbation heory Perturbation in the variables: U U + U + W W + W P P + P Q Q + Q + + Y Y + Y Z Z + Z + Y Y + Y Z Z + Z M M + M + L L + L + φ Θ Θ + θ Ψ Ψ + ψ δ δ + δ δ δ + δ δ δ + δ A A A E E E 27 rim Condition for Straight and Level Flight Assume: Select: Enforce: Solve for: erify: P Q Y Z, z ( i. e. h ) I ypically rue t Uɺ ɺ Wɺ Pɺ Qɺ ɺ ɺ Θ ɺ zɺ I U, W,, Y, Z, L, M,, Θ Y L M 28
Linearization using Small Perturbation heory eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. U + W + δ E + δ U W δ δ Y Y Y Y Y + P + + δ P δ E Z Z Z Z Z Z Z U + W + Wɺ + Q + δ E + δ U W Wɺ Q δ δ L L L L L L + P + + δ + δ A P δ δ M M M M M M M U + W + Wɺ + Q + δ E + δ U W Wɺ Q δ δ + P + + δ + δ A P δ δ A A E E 29 State ariable epresentation of Longitudinal Dynamics eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. State space form: ɺ A + BU c A U W g Z Z U U W MU + M ZU MW M ZW M Q M U Wɺ + Wɺ + Wɺ 1 δ E δ Zδ Z E δ B Mδ + M Z M E W δe δ M Z ɺ + Wɺ δ U W Q θ U δ E c δ 1 1 U, W etc. m U m W 3
Phugoid Mode eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. Lightly damped Changes in pitch attitude, altitude, velocity Constant angle of attack 31 Phugoid Mode eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. W α U α W State Equations: Frequency: ω n p Damping ratio: ζ Uɺ Z ɺ θ U P ZU g U U 2ω np U U g U θ 32
Short Period Mode eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. Heavily damped Short time period Constant velocity 33 Short Period Mode eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. U Wɺ ZW U W State Equations: Q MW + M Z W W M Q + M U W Q ɺ ɺ ɺ Zα M Q Frequency: ωn SP M α U Z M Q + M ɺ α + U Damping atio: ζ SP 2ω nsp α 34
State ariable epresentation of Lateral Dynamics State space form: ɺ A + BU c Y YP ( U Y ) g cosθ * I Z I Z I Z L + LP + P L + I I I A I Z I Z I Z + L P + LP + L IZ IZ IZ 1 1 Yδ * I Z I Z Lδ + A δ L A δ + δ I I B I Z I Z δ + L L A δ A δ + δ IZ I Z P φ U δ A c δ 35 State ariable epresentation of Lateral-directional Dynamics I, then ote : If Z Y ( YP u Y ) g cosθ Yδ L LP L Lδ L A δ A B P δ A δ 1 Aircraft esponses Spiral Mode: Slowly convergent or divergent motion olling Mode: Highly convergent motion Dutch oll Mode: Lightly damped oscillatory motion having low frequency 36
Lateral Dynamic Instabilities eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. Directional divergence Do not possess directional stability end towards ever-increasing angle of sideslip Largely controlled by rudder Spiral divergence Spiral divergence tends to gradual spiraling motion & leads to high speed spiral dive on oscillatory divergent motion Largely controlled by ailrons 37 Lateral Dynamic Instabilities eference:. C. elson, Flight Stability and Automatic Control, McGraw-Hill, 1989. Dutch roll oscillation Coupled directional-spiral oscillation Combination of rolling and yawing oscillation of same frequency but out of phase each other ime period can be of 3 to 15 sec Yaw damper is used for improving the system damping and used to improves both spiral and dutch roll characteristics 38
Attitude epresentation eference: H. Schaub and J. L. Junkins, Analytical Mechanics of Space Systems, AIAA, 23. Prof. adhakant Padhi Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Attitude epresentation Definition: Attitude coordinates are set of parameters that completely describes the orientation of a rigid body relative to some reference frame. arious Possibilities: Euler Angles Direction Cosines Quaternions (Euler parameters) Broad Class: hree parameter representation Four parameter representation (quaternions) 4
Direction cosine matrix Let the two refrence frames and B each be defined through sets of { } { ˆ} { ˆ} orthonormal right-handed sets of vectors n and b where we use the shorthand vectrix notation cosα11 cosα12 cosα13 ˆb cosα21 cosα22 cosα23 cosα31 cosα32 cosα 33 { nˆ } { b ˆ} [ C]{ nˆ }, C is called the "direction cosine matrix", C cos( bˆ, nˆ ) bˆ nˆ ij i j i j 41 Properties of DCM Direction cosine matrix [C] is orthogonal, [C][C] [I 3x3 ] cosα cosα cosα 11 21 31 { } { ˆ } [ ] { ˆ} { ˆ ˆn cosα12 cosα 22 cos α 32 b C b, b} [ C][ C] { bˆ } cosα13 cosα 23 cosα 33 Inverse of [C] is the transpose of [C], [C] -1 [C] Determinant of DCM is ± 1, det(cc ) det([i 3x3 ])1 Direction cosine matrix is the most fundamental, but highly redundant, method of describing a relative orientation. Minimum 3 parameters are used to describe a reference frame orientation, it has 9 entries, hence 6 extra parameters are redundant through orthogonality condition. 42
Euler angles Standard set of Euler angles used in aircraft and missile is (3-2-1)i.e. (ψ,θ,). his sequence is called asymmetric set. Other is (3-1-3) set of Euler angles, to define the orientation of orbit planes of the planets relative to the earth orbit s plane. It is called symmetric set. DCM can be parameterized in terms of the Euler angles. Each Euler angle defines a successive rotation about one of the body axes 43 Euler angles he direction cosine matrix in terms of the (3-2-1)Euler angles is cθ 2cθ 1 cθ 2sθ 1 sθ 2 C sθ 3sθ 2cθ 1 cθ 3sθ1 sθ 3sθ 2sθ 1 cθ 3cθ 1 sθ 3cθ + 2, where cθ cos θ, sθ sinθ cθ 3sθ 2cθ 1 + sθ 3sθ 1 cθ 3sθ 2sθ 1 sθ 3cθ 1 cθ 3cθ 2 1 C 12 1 1 C 23 ψ θ1 tan, θ θ1 sin ( C13 ), φ θ3 tan C11 C33 Euler angles provide a compact,3 parameter attitude description whose coordinates are easy to visualize 44
Quaternion Another popular set of attitude coordinates are the four Euler parameters (quaternions-4d vector space). hey provide a redundant, non-singular attitude description and are well suited to describe arbitrary, large rotations. q q + q, q + iq + jq + kq 1 2 3 k ij ji, i jk kj, j ki ik 2 2 2 i j k ijk 1 Equality: p q p q, p q, p q, p q 1 1 2 2 3 3 ( ) ( ) ( ) ( ) Addition: p+ q p + q + i p + q + j p + q + k p + q 1 1 2 2 3 3 Multiplication: cq cq + cq i + cq j + cq k 1 2 3 45 Quaternion Conjugate of quaternion : q q iq1 jq2 kq 3 orm of quaternion : q q q, q q q, q q + q + q + q 2 2 2 2 2 2 1 2 3 Unit quaternion (normalized quaternion) : A unit quaternion, q, is a quaternion such that q 1. Inverse of quaternion: 1, 1 1 1 1 q q qq q qq q qq q q 1 q q 2 46
Quaternion: Principal rotation vector heorem: A rigid body or coordinate reference frame can be brought from an arbitrary initial orientation to an arbitrary final orientation by a single rigid rotation through a principal angle about the principal axis. he Euler parameter vector is defined in terms of the principal rotation elements as ( ) ( ) ( ) ( ) q cos, q1 e1 sin, q2 e2 sin, q3 e3 sin 2 2 2 2 47 Quaternion he constraint equation in quaternion algebra (a holonomic constraint) geometrically describes a four-dimensional unit sphere. Any rotation described through the Euler parameters has a trajectory on the surface of this constraint sphere. Euler angles to Quaternion: ψ ( ) ( θ ) ( ψ ) ( ) ( θ ) ( 2 2 2 2 2 2) ψ ( ) ( θ ) ( ψ ) ( ) ( θ ) ( 2 2 2 2 2 2) ψ ( ) ( θ ) ( ψ 2 2 2) ( 2 ) ( θ 2) ( 2) ψ ( 2 ) ( θ 2) ( ψ 2) ( 2 ) ( θ 2) ( 2) q cos cos cos + sin sin sin q1 cos cos sin sin sin cos q2 cos sin cos + sin cos sin q3 sin cos cos cos sin sin Quaternions to DCM: 2 2 2 2 q + q1 q2 q3 2( q1q 2 + q q3) 2( q1q3 q q2) 2 2 2 2 C 2( q1q2 q q3) q q1 + q2 q3 2( q2q3 + q q1 ) 2 2 2 2 2( q1q3 q q2) 2( q2q3 q q1 ) q q1 q2 q + + 3 [ ] 48
OPIMAL COOL, GUIDACE AD ESIMAIO Prof. adhakant Padhi, AE 49