Introduction to quaternions. Mathematics. Operations
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1 Introduction to quaternions Topics: Definition Mathematics Operations Euler Angles (optional) intro to quaternions 1 noel.h.hughes@gmail.com
2 Euler's Theorem y y Angle! rotation follows right hand rule x Euler Rotation Axis e = e 1 e 2 e 3 e 2 x e 3 z e 1 z intro to quaternions 2 Euler's Theorem: (paraphrased) The rotational relationship between any two coordinate frames can be described by a unit vector about which rotation takes place and a total rotation angle.
3 Quaternion Definition Discovered by William Rowan Hamilton in 1843 while walking with Lady Hamilton, crossing the Broom Bridge on the way to Dublin. He scratched the fundamental formulation i 2 = j 2 = k 2 = ijk = 1 into a stone on the side of the bridge. Q = q 1 q 2 q 3 q 4 = e sin(θ/2) cos(θ/2) = (e 1 e 2 e 3 )sin(θ/2) cos(θ/2) = q 1 i q 2 j q 3 k q 4 e 1 e 2 e 3 must be unit vector q 1 q 2 q 3 called the "vector elements" q 4 called the "scalar element" intro to quaternions 3 There is not an accepted standard on the order. In some instances, the scalar element is first, sometimes denoted q 0
4 Example 90 degree rotation about z axis e = Q = 0.0*sin(45), 0.0*sin(45), 1.0*sin(45), cos(45) intro to quaternions 4 =
5 Coordinate Frames Earth Centered Inertial ECI ECI has: Z axis along Earth rotation axis, X axis along the Vernal equinox (intersection of Equatorial and Ecliptic planes) Fixed in inertial space. Sometimes ECI is defined as aligned with ECR at a particular time intro to quaternions 5 Earth Centered Relative ECR or ECEF (Earth Centered, Earth Fixed) Mars Centered Inertial ECR has: Z axis along Earth rotation axis, X axis through intersection of Prime Meridian and Equator Rotates with Earth.
6 More Coordinate Frames Body Fixed Sensor Boresight An infinite number of coordinate frames can be defined Bottom Line: Always know what coordinate frame(s) you are working in! intro to quaternions 6
7 What Do We Do with Quaternions? ECR Describe vehicle orientation: e.g. where, rotationally, is the vehicle relative to a given reference frame? Body Fixed Transform vectors: e.g. I know where a target is in ECR, where is it relative to the body (in the body fixed frame)? intro to quaternions 7 Rotate vectors: e.g. I know a vector in my vehicle coordinate frame, where is it pointing in ECR?
8 Quaternion Mathematics Basic Functions: Normalization Multiplication Inverse, Identity Equality Successive Rotations Two Coordinate frames Vector cross and dot products Operations with Quaternions: Vector Rotation, Transformation Attitude/Orientation Propagation Vehicle maneuver/re-orientation Attitude/orientation knowledge update Appendage Pointing intro to quaternions 8
9 Quaternion Normalization To normalize quaternion, make it's magnitude = 1.0 Q n = q 1 q 2 q 3 q 4 q 1 2 q 2 2 q 3 2 q 4 2 Any quaternion that represents a rotation MUST be normal. Normalize before and after all operations. intro to quaternions 9
10 Quaternion Multiplication i i k j k j multiplication rules i 2 = j 2 = k 2 = ijk = -1 ij = k jk = i ki = j ji = -k ik = -j kj = -i Q A Q B = (q a1 i q a2 j q a3 k q a4 ) (q b1 i q b2 j q b3 k q b4 ) intro to quaternions 10 multiplication performed term by term
11 Commutivity, Associativity Q A Q B Q B Q A Not Commutive (Q A Q B ) Q C = Q A (Q B Q C ) Associative intro to quaternions 11
12 Quaternion Inverse, Identity inverse: Q -1 = Q* = -q 1 i -q 2 j -q 3 k q 4 (also called conjugate) identity: Q I = (a zero angle rotation) Multiplication by conjugate Q* Q = Q Q* = Q I = Multiplication by identity Q Q I = Q I Q = Q These are the only multiplicative operations that are commutative intro to quaternions 12 Q A Q B Q B Q A in general
13 Quaternion Equality Reversing signs of all four elements of a quaternion yields the same quaternion, mathematically Interpretation: Negative rotation about the opposite rotation axis Warning: Be very careful about software that may encounter an equivalent quaternion with reversed signs. A 1 o error in one direction may be interpreted as a 359 o error in the opposite direction IT HAS HAPPENED!!!!!!!!!!!!!!!! intro to quaternions 13
14 Successive Rotations Given: Q 1 Q 2 Q 3... Q n = sequence of rotations Q total = Q 1 Q 2 Q 3... Q n successive rotations are described by successive multiplications of the quaternions intro to quaternions 14
15 Two Coordinate Frames Given: Two coordinate frames, described by Q A Q B both referenced to the same coordinate frame To determine the quaternion going from A to B: start with: Q B = Q A Q AB multiply both sides by inverse of Q A Q A * Q B = Q A * Q A Q AB or Q AB = Q A * Q B intro to quaternions 15
16 Vector Cross, Dot Products V C V 2 V 1 Angle! Cross Product V C = V 1 X V 2 Dot Product P = V 1 V 2 = V 1 V 2 cos(θ) V C perpendicular to both V 1 and V 2 V C = V 1 V 2 sin(θ) intro to quaternions 16 If V1 and V2 are unit vectors, Θ = cos -1 (P)
17 Vector Transformation y y frame A frame B V A Q AB x z z Vector, V A, is known in coordinate frame A What is it in frame B? - V B x intro to quaternions 17 V B = Q* AB V A Q AB a zero (0) is appended to V to make it compatible with quaternion multiplication
18 Vector Transformation, Example: Vector from Star Catalog J2000 Earth Centered Inertial Star Vector, referenced to ECI V ECI = What is star vector in Spacecraft coordinate frame? V SC Spacecraft Coordinate Frame Q sc = Relative to ECI intro to quaternions 18
19 Vector Transformation, Example: Vector from Star Catalog (in ECI) to Spacecraft Coordinate Frame V SC = Q * SC V ECI Q SC from two charts back Q SC = V ECI = V SC = intro to quaternions 19
20 Vector Rotation y y frame A frame B Q AB V B x z z Vector, V B, is known in coordinate frame B What is it in frame A? - V A x intro to quaternions 20 V A = Q AB V B Q* AB a zero (0) is appended to V to make it compatible with quaternion multiplication
21 Vector Rotation Example: Star Camera Boresight J2000 Earth Centered Inertial Star Camera Boresight Star Vector Camera in boresight Body Frame In body frame V STb = V sb = What is star camera boresight in J2000 coordinate frame? V 2000 intro to quaternions 21
22 Vector Rotation Example: Star Camera Boresight from SC Body Frame to ECI V A = Q AB V B Q* AB from two charts back Q sc = V sb = V 2000 = intro to quaternions 22
23 Another View V 2000 = intro to quaternions 23
24 Attitude/Orientation Propagation We know orientation at time t 1 Q(t 1 ) and rotational rates ω What is orientation at time t 2? t 2 = t 1 + Δt Q(t 2 ) ω 2 ω = ω 1 ω 2 ω 3 0 = body rotation vector with zero (0) appended ω 3 ω 1 Calculate quaternion derivative Integrate forward in time intro to quaternions 24 Q = ½ Qω Q(t 2 ) = t 2!. Q dt + Q(t 1 ) t 1 (do not normalize)
25 Attitude/Orientation Knowledge Update Attitude/Orientation Propagation Process Inertial Measurement Unit Body Rotational Rates [ω 0.0] ½ Qω Q est Q est is "estimated" attitude Errors accumulate over time, requires update from external reference Star Camera, Sun sensor, planetary horizon sensor are common external reference Updating using quaternions: point sensor at known star compare observed with anticipated star position intro to quaternions 25 generate delta quaternion update Q est
26 Updating Attitude V o - Observed star vector V a - Anticipated star vector, based on initial Q est sensor V d = V a X V o delta rotation axis (normalize) Θ d = cos -1 (V a V o ) delta rotation angle Q d = (V d )sin(θ d /2) cos(θ d /2) delta quaternion Update Q est Q upd = Q est Q d updated attitude estimate intro to quaternions 26
27 Attitude Change/Reorientation V a V w Where we are commanded attitude = Q a Where we want to be commanded attitude = Q w rotation axis V R = V a X V w (normalize) rotation angle Θ R = cos -1 (V a V w ) rotation quaternion Q R = V R sin(θ R /2) cos(θ R /2) intro to quaternions 27 final attitude Q w = Q a Q R
28 Appendage Pointing Azimuth Azimuth Axis Gimbal Base Frame intro to quaternions 28
29 Appendage Pointing Azimuth - Elevation Azimuth Frame (after -30 deg rotation) Gimbal Base Frame Boresight 30 deg intro to quaternions 29
30 intro to quaternions 30 Appendage Pointing Example Newest Top Secret Satellite
31 Appendage Pointing Example Azimuth and Elevation Gimbal Vehicle frame Z Y Antenna Boresight is Z axis El = 30 deg Relative to Az Frame Azimuth Frame Az = -30 deg Relative to ABF X Antenna Base Frame (ABF) Pitch 90, yaw 180 Relative to vehicle frame Azimuth axis fixed in Vehicle Frame Elevation axis rotates with azimuth and Is fixed to the antenna intro to quaternions 31
32 Appendage Pointing Example Quaternions What is the orientation of the antenna relative to the universe (reference - ECI)? Vehicle attitude Q v relative to ECI Vehicle to ABF Q P90 = 0.0 sin(45) 0.0 cos(45) Q Y180 = Q VABF = Q P90 Q Y180 Azimuth Q az = sin(θ AZ /2) cos(θ AZ /2) Elevation Q El = 0.0 sin(θ EL /2) 0.0 cos(θ EL /2) Antenna boresight relative to ECI Q ANT = Q v Q VABF Q az Q El intro to quaternions 32
33 Appendage Pointing: The Problem Ø We have: Target Vector o Offboard Sensor In ECI, ECR, MCI, etc. o Onboard sensor In sensor frame, vehicle frame, etc Gimbal frame o Relative to Vehicle Many geometries Ø What we need to find Gimbal Angles To Point Payload At Target intro to quaternions 33
34 Gimbal Geometries Azimuth, Elevation Roll, Pitch Pitch, Roll Many Payload boresight may be offset Gimbal Axis Vehicle may be one or two axes of pointing Nadir, yaw for azimuth Spinner with spin axis orbit normal: despun platform, into, out of spin for azimuth elevation gimbal Single axis gimbal relative to nadir for push broom Single axis scanning gimbal Each Has Unique Math to Point intro to quaternions 34
35 Example: Find Azimuth, Elevation Gimbal Angles to Point at Target Vector Vehicle coordinate frame Z Gimbal Frame: Antenna Base Frame Az, 0.0, 0.0 Azimuth Gimbal Axis Fixed to Vehicle Target Vector Offboard sensor, => Given in ECI X Y Azimuth Gimbal Axis In ABF: V AZ = [1 0 0] Elevation Gimbal Axis Rotates with Azimuth intro to quaternions 35
36 Step 1: Transform Target Vector into Gimbal Frame (from earlier slide) Vehicle attitude Q v relative to reference (ECI) Vehicle to ABF Q P90 = 0.0 sin(45) 0.0 cos(45) Q Y180 = Q VABF = Q P90 Q Y180 ECI to ABF Q E2ABF = Q v Q VABF Target Vector in ECI = V ECI Target Vector in ABF V TABF = Q * E2ABF V ECI Q E2ABF intro to quaternions 36
37 Step 2: Find Azimuth Angle Cross Product of Target Vector with Azimuth Axis V 1 = V TABF X V AZ (normalize) Azimuth Axis V AZ = [1 0 0 ] V 1 is in ABF Y-Z Plane, Perpendicular to Target Vector V 1 V TABF intro to quaternions 37 Elevation Axis
38 Step 2: Find Azimuth Angle (cont d) Cross Product Az axis with V1 V 2 = V AZ X V 1 (normalize) V AZ V 1 is in ABF Y-Z Plane, Projection of V TABF On Y-Z plane V TABF V 1 V 2 intro to quaternions 38
39 Step 2: Find Azimuth Angle (more) Azimuth Magnitude is Angle between ABF boresight vector (Z ABF ) and V 2 Θ AZ = cos -1 (V 2 Z ABF ) (dot product) Right Hand Rule Determines Sign Of Azimuth Angle V AZ V TABF Θ AZ V 1 Z ABF = [0 0 1] V 2 intro to quaternions 39
40 Step 3: Find Elevation Angle Elevation is Angle between V TABF and V 2 Θ EL = cos -1 (V 2 V TABF ) Right Hand Rule Determines Sign Of Elevation Angle V AZ Elevation axis Rotated by Az angle Θ EL V 1 Θ AZ V 2 Z ABF intro to quaternions 40
41 Conclusion Quaternions simple, elegant to use unambiguous non-singular straightforward operations Questions? intro to quaternions 41
42 Euler Angles Three successive rotations about independent axes. Sometimes called "yaw, pitch, roll" angles or "heading, attitude and bank". Describe rotational relationship between two coordinate frames Quaternions and Euler angles contain the same information. Conversion from one to the other: one-to-one correspondence between a quaternion and a set of Euler angles intro to quaternions 42
43 Advantages, Disadvantages Euler Angles Advantages: Familiarity - nose up/down, left/right, roll left/right Disadvantages: Ambiguous - 12 rotation sequences - ypr (321), rpy (123), ypy (323),... Vector tranformations and rotations require calculating corresponding matrix or quaternion Singularities - e.g. 90 pitch causes instantaneous 180 deg change in euler angles Equations of motion are horrible intro to quaternions 43
44 Advantages, Disadvantages (cont'd) Quaternions: Advantages: No ambiguity No singularities Transformations: two quaternion multiplications Equations of motion simple Successive rotations: successive multiplications Disadvantages: Unfamiliar intro to quaternions 44 let's look at some maneuvers
45 1) pick correct Euler sequence 2) calculate matrix 3) perform vector/matrix multiplication Transformation with Euler Angles intro to quaternions 45
46 intro to quaternions 46 Propagation with Euler Angles
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