Rotation

Rotation

Wanho Choi (wanochoi.com)

Orthonormal Matrix

A_n×n = x| |1 x2 ⋯ x|n

| | |

x_i : the i-th column vector of A (i : 1,2,⋯, n) x_i ⋅ x_{j = {}0 if i ≠ j_{1 if i = j} AAT _{= (A}T _A)T _{= I}T _{= I} AT_{A =} − x₁ − − x₂ − ⋮ − x_n − | | | x₁ x₂ ⋯ x_n | | | = x₁ ⋅ x₁ x₁⋅ x₂ ⋯ x₁⋅ x_n x₂ ⋅ x₁ x₂⋅ x₂ ⋯ x₂⋅ x_n ⋮ ⋮ ⋱ ⋮ x_n ⋅ x₁ x_n⋅ x₂ ⋯ x_n⋅ x_n = 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ 1 = I

Inverse & Determinant

AA

T

= A

T

A = I

∵ A

T

= IA

T

= (A

−1

A)A

T

= A

−1

(AA

T

) = A

−1

I = A

−1

∴ A

−1

= A

T

∵ 1 = det(I) = det(AA

T

) = det(A)det(A

T

) = det(A)

2

Rotation Matrix

• (rotation matrix) ⟺ (orthonormal matrix w/ det=1)

R = r₁₁ r₁₂ r₁₃ r₂₁ r₂₂ r₂₃ r₃₁ r₃₂ r₃₃ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ RRT = RTR = I RT = R−1 det(R) = 1 r_i ⋅r_j = r_iTr_j = 0 i

(

≠ j & i, j = 0,1,2

)

1 i

(

= j & i, j = 0,1,2

)

⎧ ⎨ ⎪ ⎩⎪ R = | | | r₀ r₁ r₂ | | | ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥

Rotation Matrix

• (rotation matrix) ⟺ (orthonormal matrix w/ det=1)

• proof) Ax 2 = Ax( )T( )Ax = xTATAx = xT

( )

ATA x = xTIx = xTx = x 2 x+ y 2 = x + y( )T(x+ y)= xTx+ 2xTy+ yTy = x 2 + 2xTy+ y 2 similarly, A x( + y) 2 = Ax + Ay 2 = Ax + Ay( )T(Ax + Ay) = Ax 2 + 2 Ax( )T( )Ay + Ay 2 = x 2 + 2 Ax( )T Ay ( )+ y ∴ Ax 2 = x 2 ∴ Ax( )T Ay ( ) = xT y : angle preservation : length preservation

Rotation Matrix

• (rotation matrix) ⟺ (orthonormal matrix w/ det=1)

• example)

det

([cosθ −sinθsinθ cosθ ]) = cos2θ + sin2θ = 1

det

• Special Orthogonal group in 3D Euclidean space • _Special: • _Orthogonal:

SO(3)

det(R) = 1 RRT = RTR = I

SO(3)

= R ∈!

{

3×3

| RR

T

= R

T

R

= I, det(R) = 1

}

1. R₁, R₂ ∈SO(3) ⇒ R₁R₂ ∈SO(3) R₁R₂

(

R₁R₂

)

T = R₁R₂R₂TR₁T = R₁R₁T = I

det R

(

₁R₂

)

= det R

( )

₁ ⋅det R

( )

₂ = 1 2. R ∈SO(3) ⇒ RT ∈SO(3)

3. R

(

₁R₂

)

R₃ = R₁

(

R₂R₃

)

det(R) = −1: reflection

The Column Vectors

• (the i-th column vector): the i-th rotated axis

r₁₁ r₁₂ r₁₃ r₂₁ r₂₂ r₂₃ r₃₁ r₃₂ r₃₃ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 1 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = r₁₁ r₂₁ r₃₁ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ r₁₁ r₁₂ r₁₃ r₂₁ r₂₂ r₂₃ r₃₁ r₃₂ r₃₃ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 0 1 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = r₁₂ r₂₂ r₃₂ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ r₁₁ r₁₂ r₁₃ r₂₁ r₂₂ r₂₃ r₃₁ r₃₂ r₃₃ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = r₁₃ r₂₃ r₃₃ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ R = r₁₁ r₁₂ r₁₃ r₂₁ r₂₂ r₂₃ r₃₁ r₃₂ r₃₃ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ rotated z-axis rotated y-axis rotated x-axis

Coordinate Rotation

• coordinate {A} ↔ coordinate {B}

(

x_A, y_A, z_A

)

↔ x

(

_B, y_B, z_B

)

x_A y_A z_A x_B y_B z_B u v w x_A 1 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ + yA 0 1 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ + zA 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = xB u_x u_y u_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ + yB v_x v_y v_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ + zB w_x w_y w_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⇔ 1 0 0 0 1 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ x_A y_A z_A ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = u_x v_x w_x u_y v_y w_y u_z v_z w_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ x_B y_B z_B ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ x_A y_A z_A ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = u_x v_x w_x u_y v_y w_y u_z v_z w_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ x_B y_B z_B ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ x_B y_B z_B ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = u_x u_y u_z v_x v_y v_z w_x w_y w_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ x_A y_A z_A ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥

Euler’s Rotation Theorem

• Any 3D rotation can be expressed as no more than

three rotations around coordinate axis, which is

equivalent to a single rotation about some axis that runs through the fixed point.

Euler Angles

• The representation of a rotation as three consecutive axis rotations, three angles

R_z(α)= cosα −sinα 0 0 sinα cosα 0 0 0 0 1 0 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ Rx

( )

α ⋅ Ry

( )

β ⋅ Rz

( )

γ R_x

( )

α ⋅ R_z

( )

γ ⋅ R_y

( )

β R_y

( )

β ⋅ R_x

( )

α ⋅ R_z

( )

γ R_y

( )

β ⋅ R_z

( )

γ ⋅ R_x

( )

α R_z

( )

γ ⋅ R_x

( )

α ⋅ R_y

( )

β R_z

( )

γ ⋅ R_y

( )

β ⋅ R_x

( )

α R_x(β)= 1 0 0 0 0 cosβ −sinβ 0 0 sinβ cosβ 0 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ R_y(γ )= cosγ 0 sinγ 0 0 1 0 0 −sinγ 0 cosγ 0 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥

Euler Angles: Interpolation

• A rotation matrix cannot be component-wisely

interpolated! 1 2 0 0 1 0 1 0 −1 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ + 1 2 0 0 −1 0 1 0 1 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 0 0 0 0 1 0 0 0 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ rotation matrix (O) rotation matrix (O) rotation matrix (X)

Euler Angles: Interpolation

• It is can be solved by interpolating angles individually.

• _{But, the interpolation between two Euler angles is not}

Axis Angle (Rotational Axis)

• A rotation can be defined by a unit axis and an angle.

q = p₀ + w = p₀ + ucosθ + vsinθ

= p₀ + p − p

(

₀

)

cosθ + n × p

(

)

sinθ

= n⋅p

(

)

n + p − n⋅p

(

(

)

n

)

cosθ + n × p

(

)

sinθ = pcosθ + n⋅p

(

)

n 1

(

− cosθ

)

+ n × p

(

)

sinθ

θ p _q o v u θ u w= Ru v r n p₀ w p₀ w= r cosθ r sinθ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = r cosθ 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ +⎡ _{r sin}0 _θ ⎣ ⎢ ⎤ ⎦ ⎥ = r 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥⋅cosθ + 0⎡ _r ⎣ ⎢ ⎤ ⎦ ⎥⋅sinθ = ux u_y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥⋅cosθ + v_x v_y ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥⋅sinθ = u⋅cosθ + v ⋅sinθ

Axis Angle: Interpolation

a = p − o n = a × b a × b b = q − o p q o θ θ = cos−1 a a ⋅ b b ⎛ ⎝⎜ ⎞ ⎠⎟ n rotation axis: rotation angle: r

q = pcosθ + n⋅p

(

)

n 1

(

− cosθ

)

+ n × p

(

)

sinθ

Rodriguez’s Formula

q = pcosθ + n⋅p

(

)

n 1

(

− cosθ

)

+ n × p

(

)

sinθ

N≡ [n] = 0 −n_z n_y n_z 0 −n_x −ny nx 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⇔ n × p = Np N2= −ny 2_{− n} z 2 n_xn_y n_zn_x n_xn_y −n_z2− n x 2 _n ynz n_zn_x n_yn_z −n_x2− n y 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = n_x2−1 n_xn_y n_zn_x n_xn_y n_y2−1 n ynz n_zn_x n_yn_z n_z2−1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = n_x2 n_xn_y n_zn_x n_xn_y n_y2 _n ynz n_zn_x n_yn_z n_z2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ − 1 0 0 0 1 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = nnT− I ∵nx 2+ n y 2+ n z 2 = 1 ( ) n⋅p ( )n= n_xp_x+ n_yp_y+ n_zp_z ( )n_x nxpx+ nypy+ nzpz ( )ny n_xp_x+ n_yp_y+ n_zp_z ( )n_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = n_x2_p x+ nxnypy+ nznxpz nxnypx+ ny 2 py+ nynzpz n_zn_xp_x+ n_yn_zp_y+ n_z2p_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = n_x2 _n xny nznx nxny ny 2 nynz n_zn_x n_yn_z n_z2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ p_x py p_z ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = nn( )T p= N( 2+ I)p

∴q = pcosθ + (N2 + I)p 1− cos

(

θ

)

+ Np

( )

sinθ = Icos⎡⎣ θ + (N2 + I) 1− cos

(

θ

)

+ Nsinθ ⎤⎦p = I + Nsin⎡⎣ θ + N2

(

1− cosθ

)

⎤⎦p

= Rp R :rotation matrix

(

)

R = I + Nsinθ + N2

(

1− cosθ

)

Quaternion

• 4D complex number

• : unit vectors along three perpendicular Cartesian axes in 4D space

q = w + xˆi + yˆj + z ˆk (w, x, y, z) ∈!4

ˆi, ˆj, ˆk

ˆiˆj = ˆk = − ˆjˆi, ˆjˆk = ˆi = − ˆkˆj, ˆkˆi = ˆj = −ˆiˆk ˆi2 = ˆj2 = ˆk2 = −1, ˆiˆjˆk = −1

s = w: real part

v = (x, y,z): imaginary part

Quaternion Operations

• Addition and subtraction

• Multiplication q₁q₂ = (s₁ + v₁)(s₂ + v₂) = w

(

₁ + x₁ˆi + y₁ˆj + z₁ˆk

)

(

w₂ + x₂ˆi + y₂ ˆj + z₂ ˆk

)

= w( ₁w₂ − x₁x₂ − y₁y₂ − z₁z₂)+ w₁

(

w₂ + x₂ˆi + y₂ ˆj + z₂ ˆk

)

+ w₂

(

w₁ + x₁ˆi + y₁ˆj + z₁ˆk

)

+ y( ₁z₂ − z₁y₂)ˆi + z( ₁x₂ − x₁z₂) ˆj + x( ₁y₂ − y₁x₂) ˆk = s( ₁s₂ − v₁⋅v₂)+ s( ₁v₂ + s₂v₁ + v₁ × v₂) q₁ ± q₂ = (s₁ + v₁)± (s₂ + v₂) = w( ₁ ± w₂)+ x( ₁ ± x₂)ˆi + y( ₁ ± y₂) ˆj + z( ₁ ± z₂) ˆk = (s₁ ± s₂)+ (v₁ + v₂) q₁q₂ ≠ q₂q₁ ( )

Quaternion Operations

• Conjugate • _{Norm (magnitude)} • Inverse q = s − v = w − xˆi − yˆj − z ˆk q = qq = s2 + v ⋅v = w2 + x2 + y2 + z2 q−1 = q q ∵ qq −1 ₌ qq q = 1 ⎛ ⎝⎜ ⎞ ⎠⎟

Unit Quaternion for Rotation

• Unit quaternion represents 3D rotation.

q = cos(θ / 2)+ usin(θ / 2) u = u₁ˆi + u₂ˆj + u₃ ˆk u = u( 1,u2,u3) u = 1

q 2 = cos2(θ / 2)+ usin(θ / 2)⋅usin(θ / 2) = cos2(θ / 2)+ sin2(θ / 2) u 2 = 1

rotation angle: θ unit rotation axis: u

The same result as the rotation angle expression proof

Example

qpq−1 = cosθ 2 + 0ˆi + 0 ˆj + sin θ 2 ˆk ⎛

⎝⎜ ⎞⎠⎟(0+ xˆi + yˆj + z ˆk) cosθ2 − 0ˆi − 0 ˆj − sin

θ 2 ˆk ⎛ ⎝⎜ ⎞⎠⎟ = cosθ 2 + 0ˆi + 0 ˆj + sin θ 2 ˆk ⎛ ⎝⎜ ⎞⎠⎟ z sinθ2 + (x cos θ 2 − ysin θ 2)ˆi+ (xsin θ 2 + ycos θ 2) ˆj+ zcos θ 2 ˆk ⎛ ⎝⎜ ⎞⎠⎟ = 0 + x(cos2θ 2 − ysin 2θ 2)− 2ycos θ 2sin θ 2 ⎛

⎝⎜ ⎞⎠⎟ˆi + 2xcosθ2sin

θ 2 + y(cos 2θ 2 − ysin 2θ 2) ⎛ ⎝⎜ ⎞⎠⎟ ˆj + z cos2θ2 + sin 2θ 2 ⎛ ⎝⎜ ⎞⎠⎟ ˆk = 0 + x cos( θ− ysinθ)ˆi + xsin( θ+ ycosθ)ˆj + zˆk

= 0, x cos( θ − ysinθ, x sinθ + ycosθ, z)

cosθ −sinθ 0 0 sinθ cosθ 0 0 0 0 1 0 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ x y z 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = x cosθ − ysinθ x sinθ + ycosθ 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥

•

Concatenating Quaternions

q₂

(

q₁pq₁−1

)

q−1₂ = q

(

₂q₁

)

p q

(

₁−1q₂−1

)

= q

(

₂q₁

)

p q

(

₂q₁

)

−1

The rotation using q₁ and then using q₂ = the rotation using q₂q₁

SLERP

(Spherical Linear Interpolation)

• A unit quaternion is a point on a 4D unit sphere.

• _{Therefore, every point on a 4D unit sphere}

represents a 3D rotation.

• _{Interpolating orientations means moving on a 4D unit}

SLERP

(Spherical Linear Interpolation)

SLERP(q_a, q_b, t) = sin ((1 − t)θ)

sinθ qa + sin (tθ)sinθ qb

cosθ = q_a ⋅ q_b

t ∈ [0,1] {q_qa_b; , t = 0_{; , t = 1}

LERP(q_a, q_b, t) = (1 − t)q_a + tq_b

θ = cos−1 _(q

SLERP

(Spherical Linear Interpolation)

θ 1 − tθ tθ a b q x y q = cos(tθ)x + sin(tθ)y

y = Normalize (b − (b ⋅ x)x) = Normalize (b − (b ⋅ a)a) = Normalize (b − cosθa) = b − cosθa

b − cosθa =

b − cosθa

(b − cosθa) ⋅ (b − cosθa)

= b − cosθa

b ⋅ b − 2(a ⋅ b)cosθ + (a ⋅ a)cos2_θ

= b − cosθa

1 − 2cos2_{θ + cos}2_θ = b − cosθa_{1 − cos}2_θ = b − cosθasinθ

= cos(tθ)a + sin(tθ) b − cosθa_{sinθ = (cos(tθ) −} sin(tθ)cosθ_{sinθ ) a +} sin(tθ)_sinθ b = sinθcos(tθ) − cosθsin(tθ)_sinθ a + sin(tθ)_sinθ b = sin(θ − tθ)_sinθ a + sin(tθ)_sinθ b

= sin ((1 − t)θ)

sinθ a + sin(tθ)sinθ b

∵ sin (α + β) = sinα cosβ + cosα sinβ

SLERP

(Spherical Linear Interpolation)

SLERP(q_a, q_b, t) = sin ((1 − t)θ)_sinθ q_a + sin (tθ)_sinθ q_b

= q_a(q_a−1q_b)t = q_b(q_b−1q_a)1−t = (q_bq_a−1)tq_a = (q_aq_b−1)1−tq_b

Conversion