Chapters 1 – 4: Overview

Home , Gimbal lock

DPRG Chapters 1 – 4: Overview • Photogrammetry: introduction, applications, and tools • GNSS/INS-assisted photogrammetric and LiDAR mapping • LiDAR mapping: principles, applications, mathematical model, and error sources and their impact. • QA/QC of LiDAR mapping

• This chapter will be focusing on an alternative approach for the representation of rotation in 3D space: quaternions – Definition – Properties – Rotation axis and rotation angle representation of a rotation in 3D space

Laser Scanning 1 Ayman F. Habib DPRG

Chapter 5 QUATERNIONS & ROTATION IN 3D SPACE

Laser Scanning 2 Ayman F. Habib DPRG Overview • Quaternions: definition • Quaternion properties • Quaternions and rotation matrices • Quaternion-rotation matrices relationship • Spherical linear interpolation • Concluding remarks

Laser Scanning 3 Ayman F. Habib DPRG Quaternions

Real Part Imaginary Part i i2  j 2  k 2  ijk  1 i  jk  kj j j  ki  ik k k  ij   ji

• The real part for a “Pure Quaternion” is zero.

Laser Scanning 4 Ayman F. Habib DPRG Quaternion Multiplication

;

• Using the rules in the previous slide, we can get the following definition for quaternion multiplication:

Laser Scanning 5 Ayman F. Habib DPRG Quaternion Multiplication

;

& simplify the quaternion multiplication to matrix multiplication – ortho-normal matrices.

Laser Scanning 6 Ayman F. Habib DPRG Quaternion Multiplication

• Unit quaternions: 1 • For unit quaternions:

Laser Scanning 7 Ayman F. Habib DPRG Quaternion Properties

• Quaternion conjugate:

; ∗ ; ∗ ∗ .; ∗ ;

• For unit quaternions: ∗ ;

Laser Scanning 8 Ayman F. Habib DPRG Quaternion Properties

• Quaternion conjugate: ∗ ∗

∗ ∗ ∗ ... . .

Laser Scanning 9 Ayman F. Habib DPRG Quaternions & Rotation Matrices

• Given the following quaternions: ∗ • q is a unit quaternion. is a pure quaternion (real part is zero).

0; ∗ ∗ ; 0; .; ;

Laser Scanning 10 Ayman F. Habib DPRG Quaternions & Rotation Matrices

∗ ; ;

∗ .;

∗ ;

∗ 1 • The product ∗ produces the same vector .

Laser Scanning 11 Ayman F. Habib DPRG Quaternions & Rotation Matrices

0; is perpendicular to . ; 0; .;

; 0; 0;

Laser Scanning 12 Ayman F. Habib DPRG Quaternions & Rotation Matrices

∗ 0; ;

∗ . .;

∗ 0; ∗ 0; 2

Laser Scanning 13 Ayman F. Habib DPRG Quaternions & Rotation Matrices

∗ 0; 2 ∗ ; 2 • From 1 & 2, one can conclude that:

∗ ∗ ;

http://www.euclideanspace.com

Laser Scanning 14 Ayman F. Habib DPRG Quaternions & Rotation Matrices

Plane to the axis 2 ∗

&∗ are pure quaternions & ∗ are the imaginary components of & ∗.

Laser Scanning 15 Ayman F. Habib DPRG Quaternions & Rotation Matrices

; ∗ ;

Plane to the axis 2 ∗

&∗ are pure quaternions & ∗ are the imaginary components of & ∗.

Laser Scanning 16 Ayman F. Habib DPRG Quaternions & Rotation Matrices • Any 3D rotation matrix can be represented by a rotation ( ) around a unit vector ( ). • This rotation can be defined by the following unit quaternion:

cos sin sin sin 2 2 2 2

http://www.euclideanspace.com

Laser Scanning 17 Ayman F. Habib DPRG Quaternions & Rotation Matrices • Rotation maintains the magnitude of a vector:

∗ . ∗

̅∗ . ̅∗

∗ ∗ ̅ ̅ . ∗∗

Laser Scanning 18 Ayman F. Habib DPRG Quaternions & Rotation Matrices • Rotation maintains the angular deviation between two vectors:

∗ ∗ .

̅∗ . ̅ ∗

∗ ∗ ̅ ̅ . ∗∗

Laser Scanning 19 Ayman F. Habib DPRG Quaternions & Rotation Matrices • Rotation maintains the magnitude of a triple product:

,, . • Since: – Quaternion rotation maintains vector magnitude. – Quaternion rotation maintains angular deviation between two vectors. • Then: – Quaternion rotation maintains the magnitude of the triple product. ∗ ∗ ∗ ,, , ,

Laser Scanning 20 Ayman F. Habib DPRG Quaternions & Rotation Matrices • Quaternion/rotation matrix relationship:

∗

∗ ̅∗ ̅∗

Laser Scanning 21 Ayman F. Habib DPRG Quaternions & Rotation Matrices

• Quaternion/rotation matrix relationship:

̅∗

10 00 0 ̅ ∗ 0 0

Laser Scanning 22 Ayman F. Habib DPRG Quaternions & Rotation Matrices

Quaternion to Rotation Transformation

2 2

2 2 & define the same rotation matrix

Laser Scanning 23 Ayman F. Habib DPRG Quaternions & Rotation Matrices

Rotation to Quaternion Transformation (Option # 1)

3 4 1

/ Assumption: 10

Laser Scanning 24 Ayman F. Habib DPRG Quaternions & Rotation Matrices

Rotation to Quaternion Transformation (Option # 2)

3 4 1

4 /

4 / Assumption: 10

Laser Scanning 25 Ayman F. Habib DPRG Quaternions & Rotation Matrices

Rotation to Quaternion Transformation (Option # 3)

3 4 1 1/4

4 /

4 Assumption: 10 /

Laser Scanning 26 Ayman F. Habib DPRG Quaternions & Rotation Matrices

Rotation to Quaternion Transformation (Option # 4)

3 4 4 1 4 1

4 / Assumption: 10

Laser Scanning 27 Ayman F. Habib DPRG Quaternions & Rotation Matrices

Rotation to Quaternion Transformation

• Among the options, choose the one that ensures the highest numerical stability.

• Option # 1: is the largest among ( ).

• Option # 2: is the largest among ( ).

• Option # 3: is the largest among ( ).

• Option # 4: is the largest among ( ).

Laser Scanning 28 Ayman F. Habib DPRG Quaternions & Rotation Matrices

• The product of two quaternions:

; ;

;

cos ; • This product is equivalent to rotation angle ( ) around the axis .

Laser Scanning 29 Ayman F. Habib DPRG Spherical Linear Interpolation • Problem Statement: Given the rotations represented by and , whose angular deviation is , we need to evaluate the interpolated quaternion rotation , whose angular deviations to and are and , respectively.

• As per the figure above: Laser Scanning 30 Ayman F. Habib DPRG Spherical Linear Interpolation

Laser Scanning 31 Ayman F. Habib DPRG Spherical Linear Interpolation

Laser Scanning 32 Ayman F. Habib DPRG Spherical Linear Interpolation

. 1 1

Laser Scanning 33 Ayman F. Habib DPRG Spherical Linear Interpolation • Spherical Linear Interpolation is useful for: – Interpolation of derived rotation matrices from integrated GNSS/INS attitude – This is the case when deriving the rotation matrices at much higher rate than that derived from GNSS/INS unit (LiDAR & Line Camera systems) – Modeling variation of the rotation matrices as time dependent values for Line Camera Systems

Laser Scanning 34 Ayman F. Habib DPRG Quaternions & Rotation Matrices • Quaternions characteristics compared to rotation matrices: – It avoids the gimbal lock problem. • Happens whenever the secondary rotation is 90° • Two rotations take place around the same axis in space. – Quaternion multiplication requires fewer operations compared to multiplication of two rotation matrices. – Quaternion-based rotation requires more operations when compared to traditional rotation of vectors. – Quaternions has one constraint while rotation matrices has 6 orthogonality constraints. – Interpolation of quaternion rotations is much more straight forward than 3D rotation matrices.

Laser Scanning 35 Ayman F. Habib DPRG Gimbal Lock

http://en.wikipedia.org/wiki/Gimbal_lock • A set of three gimbals mounted together to allow three degrees of freedom: roll, pitch and yaw. • When two gimbals rotate around the same axis, the system loses one degree of freedom.

Laser Scanning 36 Ayman F. Habib DPRG Gimbal Lock

Z Y

90°

Laser Scanning 37 Ayman F. Habib DPRG Gimbal Lock

90°

Laser Scanning 38 Ayman F. Habib DPRG Gimbal Lock

Y X

90°

& .

Laser Scanning 39 Ayman F. Habib DPRG Gimbal Lock

Z X Y

X Z

90°, 90°, 90°

Laser Scanning 40 Ayman F. Habib DPRG Gimbal Lock

Z Y

180°

Laser Scanning 41 Ayman F. Habib DPRG Gimbal Lock

Y Z 90°

Laser Scanning 42 Ayman F. Habib DPRG Gimbal Lock

0°

Laser Scanning 43 Ayman F. Habib DPRG Gimbal Lock

Z X Y

X Z

90°, 90°, 90° & 180°, 90°, 0° ‼! Singularity in the derivation of the rotation angles

Laser Scanning 44 Ayman F. Habib