Euler angles(2) o Concept used in rotational e about yi Zi(parallel to r) kinematics to describe body O about X orientation w.r.t. inertial frame v about Zb 小Yaw o Sequence of three angles and Roll Bod prescription for rotating one Xi CM reference frame into another (parallel to v o Can be defined as a transformation Pitch matrix body/inertial as shown: TB/ (rxy direction) Yi o Euler angles are non-unique and r exact sequence is critical nadir Goal: Describe kinematics of body-fixed Note frame with respect to rotating local vertical B/Ⅰ-1M/B-1B/I (Pitch, Roll, Y aw)=(0, o, y)-+ Euler Angles Transformation cos y siny 0 ocos 0 - 0 from body to B/l=J-siny cosy 0l 0 coso sing 0 Inertial” frame: 0 -sino coso sin 0 cos 0 YAW ROLL PITCH
Euler Angles (2) Angles (2) Concept used in rotational kinematics to describe body orientation w.r.t. inertial frame Sequence of three angles and prescription for rotating one reference frame into another Can be defined as a transformation matrix body/inertial as shown: TB/I Euler angles are non-unique and exact sequence is critical Zi (parallel to r ) Yaw Pitch Roll Xi (parallel to v ) (r x v direction) Body CM Goal: Describe kinematics of body-fixed frame with respect to rotating local vertical Yi nadir r / YAW ROLL PITCH cos sin 0 1 0 0 cos 0 -sin -sin cos 0 0 cos sin 0 1 0 0 0 1 0 -sin cos sin 0 cos TB I ψ ψ θθ ψψ φφ φ φθ θ = ⋅⋅ Note: about Yi about X’ about Zb θ φ ψ 1 // / T TTT BI IB BI − = = Transformation from Body to “Inertial” frame: (Pitch, Roll, Yaw) = () Euler Angles
Qu quaternions o Main problem computationally is q=A vector describes the the existence of a singularity axis of rotation o Problem can be avoided by an 0/% 14=A scalar describes the 43[q4 application of Euler's theorem amount of rotation EULERS THEOREM The Orientation of a body is uniquely B specified by a vector giving the direction of a body axis and a scalar specifying a rotation angle about the axis. B o Definition introduces a redundant A: Inertial fourth element which eliminates sin B: Body the singularity o This is the“ quaternion” concept 12=ky si o Quaternions have no intuitively interpretable meaning to the human 93=kosin mind, but are computationally convenient 94=CoS
Quaternions Quaternions Main problem computationally is the existence of a singularity Problem can be avoided by an application of Euler’s theorem: The Orientation of a body is uniquely specified by a vector giving the direction of a body axis and a scalar specifying a rotation angle about the axis. EULER’S THEOREM Definition introduces a redundant fourth element, which eliminates the singularity. This is the “quaternion ” concept Quaternions have no intuitively interpretable meaning to the human mind, but are computationally convenient = = 4 4 3 2 1 q q q q q q Q Jefferson Memorial ZA ˆ X A ˆ YA ˆ X B ˆ YB ˆ ZB ˆ θ K A ˆ = z y x A k k k Kˆ = = = = 2 cos 2 sin 2 sin 2 sin 4 3 2 1 θ θ θ θ q q k q k q k z y x axis of rotation. q = A vector describesthe amount of rotation. = A scalar describesthe q 4 A: Inertial B: Body
Quaternion Demo MATLaB) 4 Quaternion Demonstration [区 Euler Angles Ya 1127614 D 180 Ptch:26836 Deg 90 Rdk32395 D 180 lorth Quaternion Representation Q=[01702208053] Azimuth -128.2378 Deg 1608244 D Beta「2434972 eg Fast render Reset C Dynamic Static Help Close
Quaternion Demo (MATLAB) Quaternion Demo (MATLAB)
Comparison of attitude descriptions Method Euler Direction Angular Quaternions ngles Cosines Velocity o Pluses If given o, y, 0 Orientation Vector Computational then a unique defines a properties robust orientation is unique dir-cos commutes w.r.t Ideal for digital defined matrix R addition control implement Minuses Given orient 6 constraints Integration w.r.t Not Intuitive then euler must be met. time does not Need transforms non-unique non -intuitive give orientation Singularity Needs transform Must store Best for Best for initial condition digital control analytical and Implementation acS design work
Comparison of Attitude Descriptions Comparison of Attitude Descriptions Method Euler Angles Direction Cosines Angular Velocity Quaternions Pluses If given φ,ψ,θ then a unique orientation is defined Orientation defines a unique dir-cos matrix R Vector properties, commutes w.r.t addition Computationally robust Ideal for digital control implement Minuses Given orient then Euler non-unique Singularity 6 constraints must be met, non-intuitive Integration w.r.t time does not give orientation Needs transform Not Intuitive Need transforms Best for Best for analytical and analytical and ACS design work ACS design work Best for Best for digital control digital control implementation implementation Must store initial condition
Rigid body kinematics Z Body Time derivatives CM (non-inertial K Rotating @ Angular velocity of body frame Inertial R Body frame frame Y A pplied to BASIC RULES INERTIAL BODY +oxp position vector r=R+p Position Expressed in 广=R+p × p Rate the inertial frame BODY =R+ p+2×p BODy+×D+0×(×P) Acceleration Inertial relative accel angular cor centripetal accel of Cm w.r.t. CM
Rigid Body Kinematics Rigid Body Kinematics Inertial Inertial Frame Time Derivatives: (non-inertial) X Y Z Body CM Rotating Rotating i Body Frame Body Frame J K ^ ^ ^ ^ ^ ^ k j I r R = Angular velocity of Body Frame BASIC RULE: INERTIAL BODY Applied to ρ ρ ωρ = +× position vector r: ( ) BODY BODY BODY 2 r R r R r R ρ ρ ωρ ρ ωρ ωρω ωρ = + =+ +× =+ + × +×+× × Position Rate Acceleration Inertial accel of CM relative accel w.r.t. CM centripetal coriolis angular accel Expressed in the Inertial Frame