Robotics & Automation. Lecture 03. Representation of SO(3) John T. Wen. September 3, 2008

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1 Robotics & Automation Lecture 03 Representation of SO(3) John T. Wen September 3, 2008

2 Last Time Transformation of vectors: v a = R ab v b Transformation of linear transforms: L a = R ab L b R ba R SO(3) has the interpretation of representation of one frame in another, direction cosine matrix, coordinate transformation matrix, and rotation of frames. Any rotation matrix R can be written as (Euler-Rodrigues Formula): R = I 3 + sinθˆk + (1 cosθ)ˆk 2. September 3, 2008Copyrighted by John T. Wen Page 1

3 Exponential Formula Consider eˆkθ. Eigenvalues of eˆkθ are 1, e ± jθ (since ˆk has eigenvalues {0,± j}). From Cayley-Hamilton Theorem, we have eˆkθ = a 0 I + a 1ˆk + a 2ˆk 2. Substituting the eigenvalues for ˆk, we have 1 = a 0 e jθ = a 0 + a 1 j a 2 e jθ = a 0 a 1 j a 2. Therefore, a 0 = 1, a 1 = sinθ, a 2 = 1 cosθ, which results in the same expression as the Euler-Rodrigues formula. Also, the eigenvalues of eˆkθ are { 1,e ± jθ} with k as the eigenvector corresponding to the eigenvalue 1. Euler s (Rotation) Theorem: Given R SO(3), there exists k (k R 3 and k = 1) and θ R such that R = eˆkθ. In words, every rotation corresponds to a single rotation about a unit vector. k is called the equivalent axis and θ the equivalent angle. September 3, 2008Copyrighted by John T. Wen Page 2

4 Representation of SO(3) Given (k,θ), we can find a corresponding R SO(3). How do we find (k,θ) for a given R SO(3)? If so (i.e., Euler s theorem), is it unique? Given R SO(3), find (k, θ). Exercise: show that sinθ = ± R R T /2, cosθ = (trr 1)/2. Therefore, θ may be recovered from θ = atan2(sinθ,cosθ) (with sign ambiguity). Equivalent axis, k, may be solved from (if sinθ > 0): What if R = R T = I? k = (R R T ) /(2sinθ). September 3, 2008Copyrighted by John T. Wen Page 3

5 Example Let s consider a rotation about the z axis, then R = I + sinθẑ + (1 cosθ)ẑẑ. Verify it is the same as before. In general, use MATLAB expm function to verify exponential formula September 3, 2008Copyrighted by John T. Wen Page 4

6 Other Representations: Unit Quaternion Unit Quaternion (Euler Parameters) (q 0,q v ) (scalar and vector quaternion) with q q T v q v = 1. R(q) = I + 2q 0 ˆq v + 2 ˆq v ˆq v, q 0 = cos(θ/2),q v = sin(θ/2)k. Can you write (q 0,q v ) in terms of R directly (without computing (k,θ) first)? Quaternion is a generalization of complex number: q 0 + q 1 i + q 2 j + q 3 k with i 2 = j 2 = k 2 = 1, i j = k, jk = i, ki = j. Quaternion multiplication, q a q b, is given by (q a,0 I qt a,v q a,v ˆq a,v ) q b,0 q b,v Nice property of quaternion: R(q a )R(q b ) = R(q a q b ). Also, we shall see that quaternion is a nonsingular representation.. September 3, 2008Copyrighted by John T. Wen Page 5

7 Vector Quaternion and Gibbs Vector Vector Quaternion: q v = sin(θ/2)k (replace q 0 by + 1 q v 2 ). Gibbs Vector or Euler-Rodrigues parameters: s = tan(θ/2)k. (Write R in terms of s.) Note that in contrast to Euler angles, R is either a polynomial or rational functions of the representation parameters no trigonometric functions! September 3, 2008Copyrighted by John T. Wen Page 6

8 Other Representations: Euler Angles Euler angles: three consecutive principal axes rotations, e.g., 313 Euler angles: R 03 = R 01 (β 1 )R 12 (β 2 )R 23 (β 3 ). where R i 1,i (β i ) = eâi 1β i, a i 1 is one of the coordinate vectors of the i 1th frame. September 3, 2008Copyrighted by John T. Wen Page 7

9 Euler Angle (Cont.) R 03 = eẑβ 1 e ˆxβ 2 eẑβ 3 c 1 s c 3 s 3 0 = s 1 c c 2 s 2 s 3 c s 2 c c 1 c 3 s 1 c 2 s 3 c 1 s 3 s 1 c 2 c 3 s 1 s 2 = s 1 c 3 + c 1 c 2 s 3 s 1 s 3 + c 1 c 2 c 3 c 1 s 2 s 2 s 3 s 2 c 3 c 2 Solving (β 1,β 2,β 3 ) from R: β 2 = cos 1 R 33 [0,π] or cos 1 R 33 β 1 = atan2(r 13, R 23 ) if s 2 0 or atan2(r 13, R 23 ) + π otherwise β 3 = atan2(r 31,R 32 ) if s 2 0 or atan2(r 13, R 23 ) + π otherwise. There are two solutions in general and they merge when β 2 = 0 or π. We shall see that this corresponds to the singularity in the representation. September 3, 2008Copyrighted by John T. Wen Page 8

10 Euler Angle (Cont.) There could be redefinition of the axis as well, e.g., the tool frame representation for the original Unimate controller (for PUMA robots): R 03 = eẑβ eŷβ 2 eẑβ In general: if p is a representation of R, then R(p) is sometimes called the forward kinematics, and p(r) is called the inverse kinematics. September 3, 2008Copyrighted by John T. Wen Page 9