1. More Rigid Transforms, 3D Rotations
ECE 383/ ME 442
More Rigid Transforms, 3D Rotations 1

2. Agenda
Working with transforms
3D rotations and representations

3. Operations on Transforms
Application
Composition
Inversion
Interpolation
Differentiation
Accomplish: changes of coordinate frame
Accomplish: movement

4. Composing 2D transformations
Using the representation q=( 𝑡 ,𝜃) with 𝑡 =( 𝑡 𝑥 , 𝑡 𝑦 )
𝑇 𝑞 𝑥 =𝑅 𝜃 𝑥 + 𝑡
Suppose this is followed by another transformation 𝑇 𝑞 ′ with 𝑞 ′ = 𝑡 ′ , 𝜃 ′ . Then…
𝑇 𝑞 ′ 𝑇 𝑞 𝑥 =𝑅 𝜃 ′ 𝑅 𝜃 𝑥 + 𝑡 + 𝑡 ′
=𝑅 𝜃 ′ 𝑅 𝜃 𝑥 +𝑅 𝜃 ′ 𝑡 + 𝑡 ′
Which is equivalent to the transform 𝑇 𝑞′′ represented by 𝑞 ′′ = 𝑅 𝜃 ′ 𝑡 + 𝑡 ′ ,𝜃+ 𝜃 ′

5. Rigid Transformation in 3D
q = (R,tx,ty,tz) with R a 3x3 rotation matrix
Robot R0R3 given in reference frame T0
What’s the new robot Rq? {Tq(x,y,z) | (x,y,z)  R0}
Define rigid transformation Tq(x,y,z) : R3 R3 R x y z tx ty tz
Tq(x,y,z) = +
3D rotation matrix
Affine translation 5

6. 3-D Rigid Rotations r11 r12 r13 r21 r22 r23 r31 r32 r33 det(A) = +1
Orthogonal rows and columns: ATA=I, AAT=I
||Ax|| = ||x|| for any x, L2 norm ||.||
Product of two rotations is a rotation
(0,1,0)
(r11,r21,r31)
(Right-handed coordinate system)
(r13,r23,r33)
(1,0,0)
(1,0,0)
(r12,r22,r32)
(0,0,1)
(0,0,1) 6

7. Coordinate Frames R R-1 = RT Local frame World frame 𝑟 13 𝑟 23 𝑟 33
𝑟 13 𝑟 23 𝑟 33
(0,1,0) R
𝑥 𝑦 𝑧
𝑟 11 𝑟 21 𝑟 31
(1,0,0)
R 𝑥 𝑦 𝑧
𝑟 12 𝑟 22 𝑟 32
(0,0,1)
R-1
= RT

8. Composing 3D transformations
Using the representation q=(𝑅, 𝑡 ) with 𝑡 =( 𝑡 𝑥 , 𝑡 𝑦 , 𝑡 𝑧 )
𝑇 𝑞 𝑥 =𝑅⋅ 𝑥 + 𝑡
Suppose this is followed by another transformation 𝑇 𝑞 ′ with 𝑞 ′ =(𝑅′, 𝑡 ′). Then…
𝑇 𝑞 ′ 𝑇 𝑞 𝑥 = 𝑅 ′ ⋅ 𝑅⋅ 𝑥 + 𝑡 + 𝑡 ′
= 𝑅 ′ ⋅𝑅 𝑥 + 𝑅 ′ ⋅ 𝑡 + 𝑡 ′
Which is equivalent to the transform 𝑇 𝑞′′ where 𝑞 ′′ = 𝑅 ′ ⋅𝑅, 𝑅 ′ ⋅ 𝑡 + 𝑡 ′

9. Interpolating 2D rotations
Say we want to blend smoothly from 𝑅 𝜃 to 𝑅 𝜃 ′ with a parameter 𝑢∈[0,1] (𝑢=0 gives 𝑅 𝜃 , 𝑢=1 gives 𝑅(𝜃’))
Blending matrices doesn’t work: (1−𝑢)𝑅(𝜃)+𝑢 𝑅(𝜃’) isn’t in general a rotation matrix
Blending angles 𝑅 1−𝑢 𝜃+𝑢 𝜃 ′ … sort of works. What about the boundary at 0 and 2𝜋?
Naïve blend 𝜃
Geodesic blend: pick minimum of CW and CCW displacement 𝜃 𝜃′ 𝜃′

10. Topological spaces Three representations of 2D rotations:
Representation 1: Angle 𝜃∈[0,2𝜋)
Representation 2: Matrix 𝑅= 𝑟 11 𝑟 12 𝑟 21 𝑟 with constraints
𝑅 𝑇 𝑅=𝐼 (in other words, 𝑟 𝑟 12 2 = 𝑟 𝑟 22 2 =1 and 𝑟 11 𝑟 21 + 𝑟 12 𝑟 22 =0)
det(R)=+1
Representation 3: unit vector 𝑥 = 𝑥 1 , 𝑥 2 𝑇 with 𝑥 =1
And a way to convert between them:
𝑅 𝜃 = cos 𝜃 − sin 𝜃 sin 𝜃 cos 𝜃
𝜃 𝑅 =atan2( 𝑟 21 , 𝑟 11 ) (here atan2 gives a range of [0,2𝜋) )
𝑥 𝜃 = cos 𝜃 , sin 𝜃 𝑇 , 𝑥 𝑅 = 𝑟 11 , 𝑟 21 𝑇 , 𝑅 𝑥 = 𝑥 1 − 𝑥 2 𝑥 2 𝑥 1
But rotations don’t “blend” in the way they would on either 0,2𝜋 , 𝑅 2 , or in 𝑀 2,2 …
Fundamentally different topology

11. Some Important Topological Spaces
R: real number line
Rn: N-dimensional Cartesian space
S1: boundary of circle in 2D
S2: surface of sphere in 3D
SO(2), SO(3): set of 2D, 3D orientations (special orthogonal group)
SE(2), SE(3): set of rigid 2D, 3D transformations (special Euclidean group)
Cartesian product A x B, power notation An = A x A … x A
Homeomorphism ~ denotes topological equivalence
Continuous mapping with continuous inverse (bijective)
Cube ~ S2
SO(2) ~ S1
SE(3) ~ SO(3) x R3

12. Topology of SO(3) r11 r12 r13 r21 r22 r23 r31 r32 r33
9 entries, constrained by:
Orthogonal rows and columns: ATA=I, AAT=I
det(A) = +1
(+/- 1 is implied by the first condition)
Six (Row-wise) orthogonality constraints:
1 = 𝑟 𝑟 𝑟 13 2
1 = 𝑟 𝑟 𝑟 23 2
1 = 𝑟 𝑟 𝑟 33 2
0 = 𝑟 11 𝑟 21 + 𝑟 12 𝑟 22 + 𝑟 13 𝑟 23
0 = 𝑟 11 𝑟 31 + 𝑟 12 𝑟 32 + 𝑟 13 𝑟 33
0 = 𝑟 21 𝑟 31 + 𝑟 22 𝑟 32 + 𝑟 23 𝑟 33
9 variables - 6 non-redundant constraints => SO(3) has 3 degrees of freedom 12

13. 4 representations of 3D rotations
3x3 rotation matrix R
Euler angles REuler(f,q,y)
Axis angle RAA(v,q)
Quaternion RQuat(q)
All representations are “equivalent” but may have certain mathematical or computational advantages

14. Axis-aligned rotations
Rotate about:
Z axis
Y axis
X axis
cos θ -sin θ 0
sin θ cos θ 0
cos θ sin θ
-sin θ cos θ
0 cos θ -sin θ
0 sin θ cos θ 14

15. Parameterization of SO(3)
Euler angles: (f,q,y) x y z q x y z x y z f x y z
1  2  3  4 15

16. Euler Angles Which axes to pick, and what order? Convention A,B,C
REuler(f,q,y) = RA(f)RB(q)RC(y)
E.g., ZYZ, ZYX (roll-pitch-yaw), etc 16

17. Euler Angles Which axes to pick, and what order? Disadvantages
Convention A,B,C
REuler(f,q,y) = RA(f)RB(q)RC(y)
E.g., ZYZ, ZYX (roll-pitch-yaw), etc
Disadvantages
Must constrain to range of values
Singularities, e.g. ZYZ when q=0 or p (Gimbal lock)
Interpolation? 17

18. Axis-Angle Representation
Every rotation matrix R can be obtained by rotating the identity matrix by some angle θ about some axis v!
R v = v

19. Axis-Angle Representation
Axis v (||v||=1), angle θ
Rodrigues’ formula: rotate x about v -> x’
x’ = x cos θ + (v x x) sin θ + v (vT x) (1 - cos θ)
Or in matrix form…
RAA(θ,v) = cos θ I + sin θ [v] + (1 - cos θ) v vT
Cross product matrix
0 -vz vy
vz vx
-vy vx 0

20. Recovering Axis and Angle from the Rotation Matrix
θ = Angle(R) = cos-1((r11+r22+r33-1)/2) = cos-1((tr(R)-1)/2)
v = Axis(R) = 1/(2 sin θ)
r32-r23 r13-r31 r21-r12

21. Properties of Axis-Angle
Disadvantages:
Non unique: RAA(θ,v)=RAA(-θ,-v) (can constrain θ to range [0,p])
4 parameters + one unit length constraint ||v||=1; dealing with this constraint is sometimes annoying, for example, in interpolation, optimization, or sampling
Unique encoding: vector w = θv
θ = ||w||, v = w/||w||
“Exponential map” REM(w) = RAA( ||w|| , w/||w|| )

22. Quaternion representation
Generalization of complex numbers
Complex z=z0+i z1, with |z|=1 can represent a 2D rotation. What’s the 3D analogue?
q = q0+q1i + q2j +q3k, where
i2 = j2 = k2 = -1 i = jk = -kj j = ki = -ik k = ij = -ji
Quaternions were a forerunner of vectors and were once mandatory of all students of physics and math!

23. Unit quaternion representation
q = (q0,q1,q2,q3), ||q||=1
Related to axis angle:
q = (cos q/2,vx sin q/2, vy sin q/2, vz sin q/2)
2(q02+q12)-1 2(q1q2-q0q3) 2(q1q3+q0q2)
2(q1q2+q0q3) 2(q02+q22)-1 2(q2q3-q0q1)
2(q1q3-q0q2) 2(q2q3+q0q1) 2(q02+q32)-1
RQ(q) =

24. Properties of Unit Quaternions
4-sphere: 3D manifold in 4D space
Non-unique: Double cover of SO(3)
Advantages (widely used in computer animation):
Quaternion multiplication = rotation composition (slightly faster than matrix *)
Curve interpolation formulas (Shoemake, ‘85)

25. Summary Rotation matrix Euler angles Axis-angle
9 parameters rij in range [-1,1]
Constraint: determinant +1
Advantages: composition, inversion are easy
Euler angles
3 parameters (f,q,y) in range [0,2p)
Axes picked by convention (e.g., Roll-Pitch-Yaw)
Advantages: no constraints, simple interpolation “works”
Disadvantages: multiple representation, singularities, composition & inversion not easy
Axis-angle
4 parameters (q,v), q in [0,p]
Constraint: |v|=1
Advantages: intuitive, inversion is easy
Disadvantages: constraint, composition & interpolation not easy , multiple representation at p
Moment representation (aka exponential map)
3 parameters w (= q*v), ||w||<=p
Advantages: no constraints, inversion easy, relation to angular velocity (as we shall see soon)
Disadvantages: composition & interpolation not easy, multiple representation at p
Quaternion
4 parameters q=(q0,q1,q2,q3)
Constraint: |q|=1
Advantages: composition & interpolation formulas, inversion easy
Disadvantages: constraint, multiple representation R(q) = R(-q)

26. Notion of Distance
θ(R1TR2) = cos-1 ((tr(R1TR2)-1)/2) measures the minimum angle needed to rotate from frame R1 to R2
Makes a good distance metric

27. Actually a geodesic in SO(3)!
Rotation in Motion
Interpolating between two rotation matrices A and B
We want a path X(s) : [0,1] -> SO(3) with R(0) = A and R(1) = B
Let v = Axis(ATB), θ = Angle(ATB)
Verify that X(s) = A RAA(sθ,v) satisfies the desired properties
Actually a geodesic in SO(3)!

28. Differentiating 2D rotations
𝑅 𝜃 = cos 𝜃 − sin 𝜃 sin 𝜃 cos 𝜃
𝑅’ 𝜃 = − sin 𝜃 − cos 𝜃 cos 𝜃 −sin 𝜃
Interpretation:
Suppose the angle moves as a function of time 𝜃 𝑡 , and we wish to observe the movement of a point 𝑦 𝑡 =𝑅 𝜃(𝑡) 𝑥 where the original coordinates 𝑥 are not time-varying
By the chain rule,
𝑦 ′ 𝑡 = 𝜃 ′ (𝑡) 𝑑 𝑑𝜃 𝑅 𝜃 𝑡 ⋅ 𝑥
(Convince yourself by writing out the components of 𝑦(𝑡))
So, the product 𝑅 ′ 𝜃 ⋅ 𝑥 gives the velocity the point 𝑦 would rotate if moved at a constant velocity
𝑅 ′ 𝜃 ⋅ 𝑥 is perpendicular to 𝑦 and of the same magnitude

29. Angular Velocities in 2D
For a rotation trajectory 𝑅 𝜃 𝑡 , we can show:
𝑑 𝑑𝑡 𝑅 𝜃 𝑡 = 𝑑 𝑑𝑡 cos 𝜃 𝑡 − sin 𝜃 𝑡 sin 𝜃 𝑡 cos 𝜃 𝑡 =
− sin 𝜃 𝑡 𝜃′(𝑡) − cos 𝜃 𝑡 𝜃′(𝑡) cos 𝜃 𝑡 𝜃′(𝑡) − sin 𝜃 𝑡 𝜃′(𝑡) =
𝜃 ′ 𝑡 0 − 𝑅 𝜃 𝑡
=> 𝜃 ′ 𝑡 is the angular velocity
For a point in world coordinates 𝑝 𝑡 =𝑅 𝜃 𝑡 𝑝 0 , we have 𝑝 ′ (𝑡)= 𝜃 ′ 𝑡 0 − 𝑝 𝑡

30. Angular Velocities in 3D
For parameterized rotation trajectory REM(wt), we can show:
d/dt REM(wt) = [w] REM(wt)
=> |w| is the speed of rotation
[w] is the cross product matrix
=> w x x is the velocity of some point x, specified in world coordinates, attached to the frame as it rotates
=> w is the angular velocity

31. Recap 5 operations on transforms: Four common representations of SO(3)
Application to points / directions
Composition
Inversion
Interpolation
Differentiation
Four common representations of SO(3)
All four implemented robustly in Python klampt.so3 module, C++ in KrisLibrary/math3d
Notion of distance, geodesic, velocity in SO(3)

32. Next Week
Lab: Intro to Python and Klamp’t