Fundamentals of Robotic Mechanical Systems Chapter 2: Mathematical Background

Transcription

1 Fundamentals of Robotic Mechanical Systems Chapter 2: Mathematical Background Jorge Angeles Department of Mechanical Engineering & Centre for Intelligent Machines McGill University 2009 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

2 Outline 1 Preambule 2 Linear Transformations 3 Rigid-Body Rotations The Cross-Product Matrix The Rotation Matrix The Linear Invariants of a 3 3 Matrix Linear Invariants of a Rotation Examples Euler-Rodrigues Parameters 4 Composition of Reflections and Rotations 5 Coordinate Transformations and Homogeneous Coordinates Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

3 Preamble Links are regarded as rigid bodies General motion : rotation and translation Invariant concepts : items that do not change upon a change of coordinate frame Examples : distances between points and angles between lines Vectors vs. components Tensors vs. matrices We will use matrix whenever we mean tensor. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

4 Linear Transformations Vector space : Set of objects called vectors that follow certain algebraic rules The physical 3-dimensional space is a particular case of a vector space vectors : in boldface lower case tensors and matrices : in boldface upper case Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

5 Algebraic Rules for Vectors Let v, v 1, v 2, v 3 and w be elements of a vector space V over the real field while α and β are real numbers : (i) v 1 + v 2 = v 2 + v 1 (ii) v + 0 = v where 0 : the zero vector (iii) v 1 + (v 2 + v 3 ) = (v 1 + v 2 ) + v 3 (iv) v + w = 0 w = v (v) α(βv) = (αβ)v (vi) α = 1 αv = v (vii) (α + β)v = αv + βv α(v 1 + v 2 ) = αv 1 + αv 2 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

6 Linear Transformations (Cont d) L : linear transformation operator over the real field (i) homogeneity : v = Lu Properties of linear transformations : L(αu) = αv (ii) additivity : L(u 1 + u 2 ) = v 1 + v 2 Range of L : Kernel of L : v = Lu for every u in U u N of U mapped by L into 0 V Examples of transformations in 3-dimensional Euclidean space E 3 : projections, reflections, rotations, affine transformations Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

7 Linear Transformations (Cont d) Orthogonal projection P of E 3 onto itself : linear transformation of E 3 onto a plane Π passing through the origin with unit normal n. Properties : P 2 = P : idempotent; Pn = 0 : singular (2.1a) The projection p of p onto a plane Π of unit vector n : FIG.: Projection. p = p n(n T p) = (1 nn T )p = Pp (2.1b) P = 1 nn T (2.4) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

8 Linear Transformations (Cont d) Reflection R of E 3 onto a plane Π passing through the origin with a unit normal n : FIG.: reflection. p = p 2nn T p (1 2nn T )p = Rp R = 1 2nn T (2.5) Properties of reflections : R 2 = 1, R 1 = R Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

9 Linear Transformations (Cont d) Orthogonal decomposition v = v + v axial component ( v-par ) : v ee T v normal component ( v-perp ) v v v (1 ee T )v Vector space (2.6a) (2.6b) Basis of a vector space V : set of linearly independent vectors {v i } n 1 where n is the dimension of V v = α 1 v 1 + α 2 v α n v n (2.7) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

10 Linear Transformations (Cont d) A linear transformation of U into V with bases B U = {u j } m 1 B V = {v i } n 1 : Luj = l1jv1 + l2jv2 + + lnjvn and j = 1,..., m l 11 l 12 l 1m [L] B V l 21 l 22 l 2m BU (2.10) l n1 l n2 l nm Definition The jth column of L with respect to B U and B V is composed of the n real coefficients l ij of the representation of the image of the jth vector of B U in terms of B V. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

11 Linear Transformations (Cont d) If Le = λe, then e : eigenvector of L, λ : eigenvalue of L Characteristic polynomial of L : det(λ1 L) = 0 (2.11) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

12 Example Example What is the representation of the reflection R of E 3 into itself, with respect to the x-y plane, in terms of unit vectors parallel to the X, Y, Z axes that form the coordinate frame F? Ri = i, Rj = j, Rk = k [Ri] F = 1 0 0, [Rj] F = 0 1, [Rk] F = [R] F = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

13 Rigid-Body Rotations Linear isomorphism : one-to-one linear transformation mapping space V onto itself Isometry : distance between any two points is preserved : d (u v) T (u v) (2.12) where u and v are position vectors of two points. Volume of tetrahedron formed by triad {u, v, w} : V 1 6 u v w = 1 ([ ]) det u v w (2.13) 6 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

14 Rigid-Body Rotations (Cont d) Magnitudes of vectors {u, v, w} : u u T u, v v T v, w w T w (2.14) Under isotropy mapping {u, v, w} into {u, v, w } : u = u, v = v, w = w (2.15a) det[u v w ] = ± det[u v w] (2.15b) If +, then rotation ; otherwise, reflection Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

15 Rigid-Body Rotations (Cont d) A position vector p of a point in E 3 undergoes a rotation Q : p = Qp, p T p = p T p Q T Q = 1 and Q is an orthogonal matrix. Moreover, for T [u v w] and T [u v w ], T = QT (2.20) Q is a proper orthogonal matrix. det(t) = det(t ) (2.21a) det(q) = +1 (2.21b) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

16 Rigid-Body Rotations (Cont d) Theorem The eigenvalues of a proper orthogonal matrix Q lie on the unit circle centered at the origin of the complex plane. Proof : Qe = λe, λ : eigenvalue of Q, λ : complex conjugate of λ, e Q = λe e : eigenvector of Q e : complex conjugate of e For real Q : Q = Q T, e = e T e T Q Qe = λλe T e e T Q T Qe = λλe T e (2.25) Q is orthogonal e T e = λ 2 e T e λ 2 = 1 (2.27) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

17 Rigid-Body Rotations (Cont d) Corollary A proper orthogonal 3 3 matrix has at least one eigenvalue that is +1. Qe = e (2.28) Theorem (Euler, 1776) A rigid-body motion about a point O leaves fixed a set of points lying on a line L that passes through O and is parallel to eigenvector e of Q associated with the eigenvalue +1. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

18 Rigid-Body Rotations (Cont d) Theorem (Cayley-Hamilton) If P (λ) is the characteristic polynomial of n n matrix A, i.e., then, P (λ) = det(λ1 A) = λ n + a n 1 λ n a 1 λ + a 0 (2.29) A n + a n 1 A n a 1 A + a 0 1 = O (2.30) O : n n zero matrix Any power p n of the n n matrix A can be expressed as linear combination of the first n powers of A : 1, A,, A n 1 where 1 = A 0 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

19 The Cross-Product Matrix If u = u(t) U and v = v(u(t)) V are m- and n-dimensional vectors, respectively, while t is a real variable, and f = f(u, v) is a real-valued function, then u(t) = u 1 (t) u 2 (t). u m (t), v(t) = v 1 (t) v 2 (t). v n (t) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

20 Cross-Product Matrix (Cont d) f u f/ u 1 f/ u 2. f/ u m, f v f/ v 1 f/ v 2. f/ v n (2.31) v u v 1 / u 1 v 1 / u 2 v 1 / u m v 2 / u 1 v 2 / u 2 v 2 / u m (2.32) v n / u 1 v n / u 2 v n / u m df du = f u + ( ) T v f u v (2.33) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

21 Cross-Product Matrix (Cont d) If f = f(u, v, t) and v = v(u, t), then df dt = f ( f t + u + ( f v ) T du dt + ) T v du u dt ( ) f T v v t (2.34) and dv dt = v t + v du u dt (2.35) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

22 Example Example Let the components of v and x in certain reference frame F be given as v 1 [v] F = v 2 v 3, [x] F = what are the components of [v x] and Solution : [v x] F = [ ] (v x) x F x 1 x 2 x 3 [ ] (v x) x in the same frame F? v 2 x 3 v 3 x 2 v 3 x 1 v 1 x 3 v 1 x 2 v 2 x 1 = 0 v 3 v 2 v 3 0 v 1 v 2 v 1 0 (2.36a) (2.36b) (2.36c) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

23 Cross-Product Matrix (Cont d) v x = Vx (2.37) Theorem The cross-product matrix A of any 3-dimensional vector a is skew-symmetric, i.e., A T = A a (a b) = A 2 b (2.38) with A 2 = a aa T (2.39) and being the Euclidean norm of the vector. The cross-product matrix (CPM) A of any 3-dimensional vector a is uniquely defined. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

24 The Rotation Matrix FIG.: Rotation of a rigid body about a line. OQ : the axial component of p along e QP : normal component of p with respect to e p = OQ + QP (2.40) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

25 Rotation Matrix (Cont d) OQ= ee T p (2.41) QP = (cos φ) QP +(sin φ) QP (2.42) QP = (1 ee T )p, QP = e p Ep QP = cos φ(1 ee T )p + sin φep (2.45) p = [ee T + cos φ(1 ee T ) + sin φe]p (2.47) p is a linear transformation of p with a rotation matrix e, φ : The natural invariants of Q Q = ee T + cos φ(1 ee T ) + sin φe (2.48) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

26 Rotation Matrix (Cont d) Special cases : leading to a symmetric Q. φ = π Q = 1 + 2ee T (2.49) φ = 0 Q = 1 Under no circumstance does the rotation matrix become skew-symmetric, for a 3 3 skew-symmetric matrix is by necessity singular. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

27 Rotation Matrix (Cont d) The series expansion of Q for a fixed value of e : Q(φ) = Q(0) + Q (0)φ + 1 2! Q (0)φ k! Q(k) (0)φ k + where (k) is the kth derivative of Q with respect to φ. Since Q (k) (0) = E k, E (2k+1) = ( 1) k E, E 2k = ( 1) k (1 ee T ), Q(φ) = 1 + Eφ + 1 2! E2 φ k! Ek φ k + Q(φ) = e Eφ (2.53) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

28 Rotation Matrix (Cont d) Q(φ) = 1+ [ 1 2! φ ! φ4 + 1 (2k)! ( 1)k φ 2k + ](1 ee T ) }{{} cos φ 1 +[φ 1 1 3! φ3 + + (2k + 1)! ( 1)k φ 2k+1 + ]E }{{} sin φ Q = 1 + sin φe + (1 cos φ)e 2 (2.54) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

29 Rotation Matrix (Cont d) Canonical Forms of the Rotation Matrix If X axis is parallel to the axis of rotation, i.e., to [e] X = [ ] T, then [E] X = 0 0 1, [E 2 ] X = [Q] X = 0 cos φ sin φ 0 sin φ cos φ cos φ 0 sin φ [Q] Y = sin φ 0 cos φ cos φ sin φ 0 [ Q ] Z = sin φ cos φ (2.55a) (2.55b) (2.55c) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

30 The Linear Invariants of a 3 3 Matrix For any 3 3 matrix A : Cartesian decomposition A S 1 2 (A + AT ), A SS 1 2 (A AT ) (2.56) A S : symmetric part, A SS : skew-symmetric part. The axial vector is a vector a with the property a v A SS v (2.57) for any vector v. The trace of A is the sum of the eigenvalues of A S. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

31 Linear Invariants of a 3 3 Matrix (Cont d) Let entries of matrix A be a ij, for i, j = 1, 2, 3. Then, a 32 a 23 vect(a) a 1 2 a 13 a 31 a 21 a 12 tr(a) a 11 + a 22 + a 33 (2.58) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

32 Linear Invariants of a 3 3 Matrix (Cont d) Theorem The vector of a 3 3 matrix vanishes if and only if it is symmetric, whereas the trace of an n n matrix vanishes if the matrix is skew symmetric. For 3-dimensional vectors a and b : vect(ab T ) = 1 2 a b (2.59) tr(ab T ) = a T b (2.60) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

33 Linear Invariants of a 3 3 Matrix (Cont d) Proof of relation (2.59) : Let w denote vect(ab T ) w v = Wv (2.61) W : skew-symmetric component of ab T W 1 2 (abt ba T ) (2.62) Wv = w v = 1 2 [(bt v)a (a T v)b] (2.63) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

34 Linear Invariants of a 3 3 Matrix (Cont d) Compare relation (2.63) with the double-cross product : (b a) v = (b T v)a (a T v)b (2.64) w = 1 2 b a (2.65) The trace of an n n matrix can vanish without the matrix necessarily being skew-symmetric, while the trace of a skew-symmetric matrix necessarily vanishes. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

35 Linear Invariants of a Rotation Matrices 1, ee T and cos φ(1 ee T ) are symmetric ; matrix sin φe is skew-symmetric. Hence, vect(q) = vect(sin φ E) = sin φ e (2.66) tr(q) = tr[ee T + cos φ(1 ee T )] e T e + cos φ(3 e T e) = cos φ (2.67) cos φ = tr(q) 1 2 (2.68) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

36 Linear Invariants of a Rotation (Cont d) in a given frame. vect(q) q = q 1 q 2 q 3 Introducing q 0 cos φ as a linear invariant of the rotation matrix, the rotation matrix can be fully defined by four scalar parameters : These components of λ are not independent since λ [q 1, q 2, q 3, q 0 ] T (2.69) q 2 + q 2 0 sin 2 φ + cos 2 φ = 1 (2.70) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

37 Linear Invariants of a Rotation (Cont d) λ 2 q q q q 2 0 = 1 (2.71) sin φ = ± q, e = q/ sin φ (2.73) ) Q = qqt q 2 + q 0 (1 qqt q 2 + Q (2.74a) Q (q x) x Q = q Q + qqt 1 + q 0 (2.74b) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

38 Linear Invariants of a Rotation Linear invariants are not suitable to represent a rotation when φ is either π or close to it. Changing the sign of e does not change the rotation matrix, provided that the sign of φ is also changed. Choose 0 φ π sin φ = q, e = q q Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

39 Examples Example If [e] F = [ 3/3, 3/3, 3/3 ] T in a given coordinate frame F and φ = 120, what is Q in F? Solution : From the data, [ee T ]F = 1 3 cos φ = 1 2, sin φ = [ ] = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

40 Examples (Cont d) [ 1 ee T ] F = , [ E ] F [Q] F = [Q] F = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

41 Examples (Cont d) Example Is the linear transformation below a rotation? [ Q ] F = If so, find its natural invariants. Solution : Test for orthogonality : [ Q ] F [ Q T ] F = = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

42 Examples (Cont d) det( Q ) = +1 a rotation Find e and φ : vect(q) = (sin φ)e [ q ] F sin φ [e] F = 1 2 [ ] T 3 sin φ q = φ = 60 or [e] F = [q] F 3 q = [ ] T cos φ tr(q) = 0, cos φ = 1 2 φ = 120 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

43 Examples (Cont d) Example A coordinate frame X 1, Y 1, Z 1 is rotated into a configuration X 2, Y 2, Z 2 in such a way that X 2 = Y 1, Y 2 = Z 1, Z 2 = X 1 Find the matrix representation of the rotation in X 1, Y 1, Z 1 coordinates. Compute the direction of the axis and the angle of rotation. Solution : Let i 1, j 1, k 1 be unit vectors parallel to X 1, Y 1, Z 1, respectively, i 2, j 2, k 2 being defined correspondingly. i 2 = j 1, j 2 = k 1, k 2 = i 1 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

44 Examples (Cont d) [ Q ] 1 = [q] 1 [ vect(q) ] 1 = sin φ [ e ] 1 = 1 2 cos φ = 1 2 [ tr(q) 1 ] = 1 2 Assume sin φ 0 sin φ = q = [ e ] 1 = [q] 1 3 sin φ = φ = 120 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

45 Examples (Cont d) Example Show that the matrix P in eq. (2.4) satisfies (2.1a). Solution : Is P idempotent? Moreover, P 2 = (1 nn T )(1 nn T ) = 1 2nn T + nn T nn T = 1 nn T = P Pn = (1 nn T )n = n nn T n = n n = 0 n is eigenvector of P with eigenvalue 0 and, hence, spans the nullspace of P Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

46 Examples (Cont d) Example The representations of three linear transformations in a given coordinate frame F : [A ] F = , [ B ] F = , [ C ] F = Identify the orthogonal projection, the reflection and the rotation and find its invariants. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

47 Examples (Cont d) Solution : B and C are symmetric. Rotation matrix is symmetric if and only if sin φ = 0 i.e., φ = 0 or π, with trace 3 or 1. But tr(b) = 2, tr(c) = 1 B and C are not rotations [ AA T ] F = , det(a) = A is a rotation Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

48 Examples (Cont d) Natural invariants of A : 1 sin φ [e] F = [vect(a)] F = cos φ = 1 2 [tr(a) 1] = 1 2 (2 1) = 1 2 For sin φ 0 sin φ = vect(a) = 3 2 φ = 60 [e] F = [vect(a)] 1 F 3 vect(a) = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

49 Examples (Cont d) Test B and C for orthogonality : [ BB T ] F = = [ B 2 ] F = [ B ] F B is not orthogonal but idempotent [ CC T ] F = C is orthogonal Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

50 Examples (Cont d) det(c) = 1 C is a reflection det(b) = 0 B is singular : a projection The unit vector [ n ] F = [ n 1, n 2, n 3 ] T spanning the nullspace of B can be determined from : Bn = 0 nn T = 1 B Produces two solutions, one the negative of the other. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

51 Examples (Cont d) n 2 nn T 1 n 1 n 2 n 1 n 3 = n 1 n 2 n 2 2 n 2 n 3 = n 1 n 3 n 2 n 3 n n = ± From C = 1 2nn T follows the same solution for n Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

52 Examples (Cont d) Example The vector and the trace of a rotation matrix Q in a reference frame F are [vect(q)] F = 1 1 1, tr(q) = Find the matrix representation of Q in the given coordinate frame and in a frame with its Z-axis parallel to vect(q). Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

53 Examples (Cont d) Solution : Use eq.(2.74a) to determine rotation matrix Q [ qq T ] F = , q 2 = [ Q ] F = [ Q ] F = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

54 Examples (Cont d) Denote by Z a coordinate frame with Z axis parallel to q 0 3 [ q ] Z = 0, [ qq T ] Z = /2 3 3/2 0 [ Q ] Z = [ Q ] Z = 3/2 1/ Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

55 Examples (Cont d) Example A procedure for trajectory planning produced a matrix z [ Q ] = x y representing a rotation for a certain pick-and-place operation where x, y, and z are entries that are unrecognizable due to failures in the printing hardware. Find the missing entries. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

56 Examples (Cont d) Solution : Product Q T Q must be 1 : z (x z 1) 0.5( y + z) Q T Q = x (x + y) y 2 Upon equating the diagonal elements to unity : x = ±0.250, y = 0, z = ±0.750 while the vanishing of the off-diagonal entries leads to : x = 0.250, y = 0, z = det(q) = +1 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

57 Euler-Rodrigues Parameters r sin ( ) φ 2 e, r 0 = cos ( ) φ 2 (2.75) Q = (r 0 2 r r)1 + 2rr T + 2r 0 R (2.76) where, for arbitrary x, (r x) R x φ π : 1 + q0 r 0 = ±, r = q 2 2r 0 (2.77) φ = π : r 0 = 0, r = e (2.78) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

58 The Matrix Square Root The square root for a 3 3 matrix is a linear combination of 1 (the identity matrix), the matrix itself and its square, the coefficients being found using the eigenvalues of the matrix (from Cayley-Hamil- ton s Theorem). Square root of a rotation matrix : a rotation through angle φ/2 about axis parallel to e. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

59 The Matrix Square Root (Cont d) A diagonalizable n n matrix admits as many square roots as there are combinations of the two possible roots of individual eigenvalues. Example : the eight square roots of the 3 3 identity matrix : , , , , 0 0 1, , Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

60 The Matrix Square Root (Cont d) Not all 2 3 = 8 square roots of a 3 3 proper orthogonal matrix are proper orthogonal but there is one and only one that represents a rotation of φ/2 denoted Q : r = vect( Q), r 0 = tr( Q) 1 (2.79) 2 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

61 Euler-Rodrigues Parameters (Cont d) Linear invariants can be derived from the rotation matrix by simple additions and subtractions, but fail to produce information on the axis of rotation when φ = π. Euler-Rodrigues parameters require square roots and entail sign ambiguities, but produce information on the axis of rotation for any value of the rotation angle φ. Note : Not to be confused with Euler angles, which are not invariant and admit multiple definitions. No single set of Euler angles exists for a given rotation ; set depends on how the rotation is decomposed into three simpler rotations. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

62 Example Example Find the Euler-Rodrigues parameters of the proper orthogonal matrix Q = Solution : Q is symmetric φ = π. Impossible to determine the direction of the axis of rotation from the linear invariants. However, note that ee T = 1 (1 + Q) 2 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

63 Example (Cont d) e 2 1 e 1 e 2 e 1 e 3 e 1 e 2 e 2 2 e 2 e 3 = e 1 e 3 e 2 e 3 e [ ] T e = ± ( ) 1 φ 3 r = e sin = ±, r 0 = cos ( ) φ = 0 2 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

64 Composition of Reflections and Rotations Let R be a pure reflection & Q a rotation T = QR and T = RQ are neither symmetric nor self-inverse, yet both are reflections! An improper orthogonal transformation (not symmetric) can always be decomposed into the product of a rotation and a pure reflection. This decomposition is not unique. For arbitrary n : R 1 2nn T Q = TR 1 TR = T 2(Tn)n T Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

65 Example Example Place both palms in the diving position. Hold a sheet of paper between them. The sheet defines a hand plane in each hand, its unit normal, hand normal, pointing outside of the hand, n R and n L for the right and left hand, respectively. o R and o L : unit vectors pointing in the direction of the finger axes of each of the hands n L = n R, o L = o R Now, without moving your right hand, let the left hand attain a position whereby the left-hand normal lies at right angles with the right-hand normal, the palm pointing downwards and the finger axes of the two hands remaining parallel. Find the representation of the transformation carrying the right hand to the final configuration of the left hand, in terms of the unit vectors n R and o R. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

66 Example (Cont d) Solution : Desired transformation : T = QR Reflection mapping the right hand into the left hand : R = 1 2n R n T R Left hand rotates from diver position about an axis o R through an angle of 90 cw. From eq.(2.48) and above information, Q = o R o T R + O R where O R is the cross-product matrix of o R T = o R o T R + 2O R 2(o R n R )n T R Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

67 Coordinate Transformations and Homogeneous Coordinates Coordinate Transformations Between Frames with a Common Origin A = {X, Y, Z}, B = {X, Y, Z}, Q: A B (2.80) [ p ] A = [ x y z ] T (2.81) Find [ p ] B in terms of [ p ] A and Q Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

68 Coordinate Transformations Between Frames with a Common Origin FIG.: Coordinate transformation : (a) coordinates of point P in the A-frame ; and (b) relative orientation of frame B with respect to A. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

69 Coordinate Transformations Between Frames with a Common Origin (Cont d) Point P is attached to frame A, while frame A undergoes a rotation Q about its origin and is carried to frame B : π = Qp (2.82) where π is the position vector of point Π : rotated position of P [ π ] B = [ x y z ] T (2.83) [ π ] A = [ ξ η ζ ] T (2.84) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

70 Coordinate Transformations Between Frames with a Common Origin (Cont d) Theorem The representations of the position vector π of any point in two frames A and B, denoted by [ π ] A and [ π ] B respectively, are related by [ π ] A = [ Q ] A [ π ] B (2.85) Proof : [ π ] A = [ Q ] A [ p ] A (2.86) [ π ] B = [ p ] A (2.87) [ π ] A = [ Q ] A [ π ] B (2.88) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

71 Coordinate Transformations Between Frames with a Common Origin (Cont d) Theorem The representations of Q carrying A into B in these two frames are identical, i.e., [ Q ] A = [ Q ] B (2.89) Proof : [ Qp ] A = [ Q ] A [ Qp ] B, [ Q ] A [ p ] A = [ Q ] A [ Qp ] B [ p ] A = [ Q ] B [ p ] B (2.90) [ p ] A = [ Q ] A [ p ] B (2.91) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

72 Coordinate Transformations Between Frames with a Common Origin (Cont d) Theorem The inverse relation of Theorem 1 is given by [ π ] B = [ Q T ] B [ π ] A (2.92) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

73 Example Example FIG.: Coordinate frames A and B with a common origin. Find the representations of Q rotating A into B in these two frames and show that they are identical. Moreover, if [ p ] A = [ 1, 1, 1 ] T, find [ p ] B. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

74 Example (Cont d) Solution : Let i, j, and k be unit vectors in the directions of the X-, Y -, and Z-axes, respectively, unit vectors ι, γ, and κ being defined likewise, as parallel to the X -, Y-, and Z-axes of Fig. 2.2 Q i ι = k, Q j γ = i, Q k κ = j [ Q ] A = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

75 Example (Cont d) Likewise, From Definition Qι = = 1 = j = κ Qγ = ι, Qκ = γ [ Q ] B = = [ Q ] A Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

76 Example (Cont d) If we represent this matrix in a frame whose X-axis is directed along the axis of rotation of Q, [ Q ] X = 0 cos φ sin φ with φ = sin φ cos φ [ Q ] X = 0 1/2 3/2 0 3/2 1/ [ p ] B = = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

77 Coordinate Transformation with Origin Shift Theorem The representations of the position vector p of a point P of the Euclidean 3-dimensional space in two frames A and B are related by [ p ] A = [ b ] A + [ Q ] A [ π ] B (2.93a) [ π ] B = [ Q T ] B ([ b ] A + [ p ] A ) (2.93b) with b defined as the vector directed from the origin of A to that of B, and π the vector directed from the origin of B to P, as depicted in Fig Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

78 Coordinate Transformation with Origin Shift (Cont d) FIG.: Coordinate frames with different origins. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

79 Coordinate Transformation with Origin Shift (Cont d) Proof : p = b + π (2.94) [ p ] A = [ b ] A + [ π ] A π is assumed to be readily available in B [ p ] A = [ b ] A + [ Q ] A [ π ] B To prove eq.(2.93b), we simply solve eq.(2.94) for π and apply eq.(2.92) to the equation thus resulting. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

80 Example Example If [ b ] A = [ 1, 1, 1 ] T and A & B have the relative orientations as in Example 4, find the position vector, in B, of a point P of position vector [ p ] A given as in the same example. Solution : Find [ π ] B : [ b ] A + [ p ] A = [ ] T By virtue of Theorem 2, Q ] B = [ Q T ] A : [ Q ] B = [ π ] B = = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

81 Homogeneous Coordinates General coordinate transformation involving a shift of the origin is not linear, e.g., the first terms of the RHS of eq.(2.93a) is nonhomogeneous. It can be represented in homogeneous form by introducing homogeneous coordinates : {p} M [ [ p ] T M 1] T (2.95) Then, the affine transformation of eq.(2.93a) becomes {p} A = {T} A {π} B with : { T } A [ ] [ Q ]A [ b ] A 0 T, 1 (2.97) { T 1 } B = [ [ Q T ] B ] [ b ] B 0 T 1 (2.98) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

82 Homogeneous Coordinates (Cont d) F k, for k = i 1, i, i + 1 : coordinate frames with origins at O k Q i 1 : the rotation of F i 1 into F i Q i : the rotation of F i into F i+1 If three origins coincide : with inverse relation [ p ] i = [ Q T i 1 ] i 1 [ p ] i 1 [ p ] i+1 = [ Q T i ] i [ p ] i = [ Q T i ] i [ Q T i 1 ] i 1 [ p ] i 1 [ p ] i 1 = [ Q i 1 ] i 1 [ Q i ] i [ p ] i+1 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

83 Homogeneous Coordinates (Cont d) If origins do not coincide, a i 1 = O i 1 O i, a i = O i O i+1 [ [ Qi 1 ] i 1 [ a i 1 ] i 1 Inverse transformation : {T i 1 } i 1 = 0 [ T 1 ] [ Qi ] {T i } i = i [ a i ] i 0 T 1 {T i 1 } 1 i = {T i } 1 i+1 = ], [ [ Q T i 1 ] i [ Q T i 1 ] ] i[ a i 1 ] i 1 0 T 1 [ [ Q T i ] i+1 [Q T i ] ] i+1[ a i ] i 0 T 1 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

84 Homogeneous Coordinates (Cont d) Therefore, the coordinate transformations are {p i 1 } i 1 = {T i 1 } i 1 {p i } i (2.105) {p i 1 } i 1 = {T i 1 } i 1 {T i } i {p i+1 } i+1 (2.106) with the corresponding inverse transformations { p i } i = {T i 1 } 1 i 1 { p i 1 } i 1 (2.107) { p i+1 } i+1 = {T i } 1 i { p i } i = {T i } 1 i {T i 1 } 1 i 1 { p i 1 } i 1 (2.108) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

85 Homogeneous Coordinates (Cont d) If P lies at infinity : lim {p} M = p {p} M p [ ] [ e ]M 1/ p ( ) ( lim p p lim p [ ]) [ e ]M 1/ p = ( ) [ ] [ e lim p ]M p 0 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

86 Homogeneous Coordinates (Cont d) We define the homogeneous coordinates of a point P lying at infinity as [ ] [ e ]M { p } M (2.109) 0 Let Q = [ ] e 1 e 2 e 3 and triad { ek } 3 1 be orthonormal {T} A of eq.(2.97) becomes { T } A = [ e1 e 2 e 3 ] b (2.110) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

87 Example Example Find the equation of an ellipsoid in A with semiaxes of a = 1, b = 2, c = 3 and with its centre O B of position [ b ] A = [ 1, 2, 3 ] T, its axes X, Y, Z defining a coordinate frame B. The direction cosines of X are (0.933, 0.067, 0.354), whereas Y b and Y u while u X -axis. u, v, w are unit vectors parallel to the X -, Y-, and Z-axes, respectively : [ u ] A = 0.067, v = u b u b, w = u v Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

88 Example (Cont d) [ v ] A = 0.843, [ w ] A = [ Q ] A = [ u, v, w ] A = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

89 Example (Cont d) If [ p ] A = [ p 1, p 2, p 3 ] T and [ π ] B = [ π 1, π 2, π 3 ] T, B : π π π = [ Q T ] B = [ Q T ] A = p 1 [ π ] B = p p 3 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

90 Example (Cont d) Therefore, π 1 = 0.933p p p π 2 = 0.243p p p 3 π 3 = 0.266p p p Upon substitution of the foregoing relations into the ellipsoid equation in B, A: p p p p p p p 1 p p 2 p p 1 p = 0 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

91 Similarity Transformations v = α 1 a 1 + α 2 a α n a n v = β 1 b 1 + β 2 b β n b n α 1 α 2 α n [ v ] A =., [ v ] B = b j = a 1j a 1 + a 2j a a nj a n, β 1 β 2. β n j = 1,..., n (2.113) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

92 Similarity Transformations (Cont d) v = β 1 (a 11 a 1 + a 21 a a n1 a n ) + β 2 (a 12 a 1 + a 22 a a n2 a n ). + β n (a 1n a 1 + a 2n a a nn a n ) v = (a 11 β 1 + a 12 β a 1n β n )a 1 + (a 21 β 1 + a 22 β a 2n β n )a 2. + (a n1 β 1 + a n2 β a nn β n )a n Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

93 Similarity Transformations (Cont d) [ v ] A = [ A ] A [ v ] B (2.117) where Inverse relationship : a 11 a 12 a 1n a 21 a 22 a 2n [ A ] A..... (2.118). a n1 a n2 a nn [ v ] B = [ A 1 ] A [ v ] A (2.119) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

94 Similarity Transformations (Cont d) l 11 l 12 l 1n l 21 l 22 l 2n [ L ] A = l n1 l n2 l nn (2.120) We want to find relation between [L] A and [L] B. To this end, let Lv = w (2.121) whose representations in A and B are [ L ] A [ v ] A = [ w ] A (2.122) [ L ] B [ v ] B = [ w ] B (2.123) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

95 Similarity Transformations (Cont d) We assume that the image vector w is identical to vector v in the range of L (not always the case). [ w ] A = [ A ] A [ w ] B (2.124) Substitution of eq.(2.124) into eq.(2.122) yields [ A ] A [ w ] B = [ L ] A [ A ] A [ v ] B (2.125) Hence, [ w ] B = [ A 1 ] A [ L ] A [ A ] A [ v ] B (2.126) [ L ] B = [ A 1 ] A [ L ] A [ A ] A (2.127) [ L ] A = [ A] A [ L ] B [ A 1 ] A (2.128) Equations (2.117), (2.119), (2.127) & (2.128) are the similarity transformations sought. They preserve invariant quantities Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

96 Similarity Transformations (Cont d) Theorem The characteristic polynomial of a given n n matrix remains unchanged under a similarity transformation. Moreover, the eigenvalues of two matrix representations of the same n n linear transformation are identical, and if [ e ] B is an eigenvector of [ L ] B, then under the similarity transformation (2.128), the corresponding eigenvector of [ L ] A is [ e ] A = [ A ] A [ e ] B. Proof : The characteristic polynomial of [ L ] B is P (λ) = det(λ[ 1 ] B [ L ] B ) (2.129) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

97 Similarity Transformations (Cont d) P (λ) det(λ[ A 1 ] A [ 1 ] A [ A ] A [ A 1 ] A [ L ] A [ A ] A ) = det([ A 1 ] A (λ[ 1 ] A [ L ] A )[ A ] A ) = det([ A 1 ] A )det(λ[ 1 ] A [ L ] A )det([ A ] A ) det([ A 1 ] A )det([ A ] A ) = 1 The characteristic polynomials of [ L ] A and [ L ] B are identical [ L ] A & [ L ] B have same eigenvalues. Let [ L ] B [ e ] B = λ[ e ] B : [ A 1 ] A [ L ] A [ A ] A [ e ] B = λ[ e ] B [ L ] A [ A ] A [ e ] B = λ[ A ] A [ e ] B (2.130) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

98 Similarity Transformations (Cont d) Theorem If [ L ] A and [ L ] B are related by the similarity transformation (2.127), then [ L k ] B = [ A 1 ] A [ L k ] A [ A ] A (2.131) for any integer k. Proof : For k = 2 For k = n [ L 2 ] B [ A 1 ] A [ L ] A [ A ] A [ A 1 ] A [ L ] A [ A ] A = [ A 1 ] A [ L 2 ] A [ A ] A [ L n+1 ] B [ A 1 ] A [ L n ] A [ A ] A [ A 1 ] A [ L ] A [ A ] A = [ A 1 ] A [ L n+1 ] A [ A ] A Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

99 Similarity Transformations (Cont d) Theorem The trace of an n n matrix does not change under a similarity transformation. Proof : Let [ A ], [ B ] and [ C ] be three different n n matrix arrays, in a given reference frame Let a ij, b ij, and c ij be the components of the said arrays, with indices ranging from 1 to n tr([ A ] [ B ] [ C ]) a ij b jk c ki = b jk c ki a ij tr([ B ] [ C ] [ A ]) tr([ L ] B ) = tr([ A 1 ] A [ L ] A [ A ] A ) = tr([ A ] A [ A 1 ] A [ L ] A ) = tr([ L ] A ) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

100 Example Example We consider the equilateral triangle Fig. 2.4 of side 2, with vertices P 1, P 2, P 3, and coordinate frames A(X, Y) and B(X, Y ), both with origin at the centroid of the triangle. Let P be a 2 2 matrix : P = [ p 1 p 2 ] with p i being the position vector of P i in a given coordinate frame. Show that matrix P does not obey a similarity transformation upon a change of frame, and compute its trace in frames A and B to make it apparent that this matrix does not comply with the conditions of Theorem 3. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

101 Example (Cont d) FIG.: Two coordinate frames used to represent the position vectors of the corners of an equilateral triangle. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

102 [ [ P ] A = 1 0 3/3 2 3/3 Example (Cont d) ] [, [ P ] B = /3 tr([ P ] A ) = tr([ P ] B ) = 3 3 [ p i ] A = [ Q ] A [ p i ] B [ P ] A = [ Q ] A [ P ] B Let R PP T then [ 1 ] 3/3 [ R ] A =, [ R ] B = 3/3 5/3 3/3 ] [ ] 1 3/3 3/3 5/3 tr([ R ] B ) = 8 3 [ R ] A = [ PP T ] A = [ Q ] A [ P ] B ([ Q ] A [ P ] B ) T = [ Q ] A [ PP ] T B[ Q T ] A which is a similarity transformation Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

103 Invariance Concepts The scalar function f( ) of the position vector p is invariant if : f([p] B ) = f([p] A ) vector f is invariant if : [f] A = [Q] A [f] B matrix F is invariant if : [F] A = [Q] A [F] B [Q T ] A Moreover, [ a ] T A [ b ] A = [ a ]T B [ b ] B [ a b ] A = [ Q ] A [ a b ] B Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

104 The kth moment of an n n matrix T : Invariance Concepts (Cont d) with I 0 = tr(1) = n. I k tr(t k ), k = 0, 1,... Theorem The moments of a n n matrix are invariant under a similarity transformation. Proof : [ T k ] B = [ A 1 ] A [ T k ] A [ A ] A (2.140) [ I k ] B = tr( [ A 1 ] [ A T k ] A [ A ] A ) tr([ A ] [ A A 1 ] [ A T k ] A ) = tr( [ T k ] ) [ I A k ] A Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

105 Invariance Concepts (Cont d) Theorem An n n matrix has only n linearly independent moments. Proof : Let the characteristic polynomial of T be P (λ) = a 0 + a 1 λ + + a n 1 λ n 1 + λ n = 0 Applying the Cayley-Hamilton Theorem : a a 1 T + + a n 1 T n 1 + T n = 0 Take the trace of both sides of the above equation : a 0 I 0 + a 1 I a n 1 I n 1 + I n = 0 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

106 Invariance Concepts (Cont d) The vector invariants of an n n matrix are its eigenvectors. The eigenvectors of symmetric matrices are real and mutually orthogonal, its eigenvalues being real as well. This is not so for skew-symmetric matrices but for 3 3 skew-symmetric matrices, one eigenvalue is real, namely, 0, and its associated vector is the axial vector of this matrix. Two n n matrices related by a similarity transformation have the same set of moments. However, if two n n symmetric matrices share their first n moments { I k } n 1 0, they are not necessarily related by a similarity transformation. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

107 Invariance Concepts (Cont d) For example : A = [ ] 1 0, B = 0 1 [ ] Both matrices share the two moments : I 0 = 2 & I 1 = 2. The second moments are different : [ ] tr(a 2 ) = 2, tr(b ) = tr = To test whether two different matrices represent the same linear transformation, we must verify that they share the same set of n moments { I k } n 1 since all n n matrices share the same I 0 = n. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

108 Invariance Concepts (Cont d) The foregoing discussion does not apply, in general, to nonsymmetric matrices : they are not fully characterized by their eigenvalues. For example : [ ] 1 1 A = 0 1 Moments I 0 = 2, I 1 = tr(a) = 2 are equal to those of the 2 2 identity matrix but the transformations are different. If two symmetric matrices, A and B, represent the same transformation, they are related by a similarity transformation : B = T 1 AT Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

109 Examples Example Find out whether two symmetric matrices A = 0 1 0, B = are related by a similarity transformation. tr(a) = tr(b) = 4. What about I 2 and I 3? Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

110 Examples (Cont d) A 2 = 0 1 0, B 2 = tr(a 2 ) = tr(b 2 ) = A 3 = 0 1 0, B 3 = tr(a 3 ) = tr(b 3 ) = 19 Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

111 Examples (Cont d) Example Same as the previous example, for : A = 0 1 0, B = tr(a) = tr(b) = 2, but tr(a 2 ) = 10 while tr(b 2 ) = 6 Therefore, A and B are not related by a similarity transformation Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

112 Applications to Redundant Sensing If redundant sensors are introduced, and we attach frames A and B to each of these, then each sensor can be used to determine the orientation of the end-effect- or with respect to a reference configuration. For this task it is needed to measure rotation R that each frame underwent from the reference configuration. Assume that the measurements produce the orthogonal matrices A and B, representing R in A and B, respectively. Determine the relative orientation Q of frame B with respect to frame A (instrument calibration). Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

113 Applications to Redundant Sensing (Cont d) We have : A [ R ] A and B [ R ] B Need to determine [ Q ] A or [ Q ] B, which can be obtained from A = [ Q ] A B[ Q T ] A or A [ Q ] A = [ Q ] A B This problem can be solved with three invariant vectors associated with A and B. Since A and B are orthogonal, they admit one real invariant vector : their axial vector. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

114 Applications to Redundant Sensing (Cont d) To determine Q, we need two more invariant vectors, represented in A and B. To this end, take two measurements of the orientation of A 0 and B 0 with respect to A and B. Let the matrices representing these orientations be given by A i and B i, with a i and b i being the corresponding axial vectors, for i = 1, 2. There are two possibilities : (i) Neither of a 1 and a 2 and, consequently, neither of b 1 and b 2, is zero ; (ii) at least one of a 1 and a 2, and consequently, the corresponding vector of the { b 1, b 2 } pair, vanishes. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

115 Applications to Redundant Sensing (Cont d) In the first case compute a 3rd vector for each set a 3 = a 1 a 2, b 3 = b 1 b 2 (2.143) In the second case, two more possibilities : angle of rotation of orthogonal matrix, A 1 or A 2, whose axial vector vanishes, is either 0 or π. If angle is 0, then A, as well as B, underwent a pure translation from A 0 and B 0, correspondingly. Then, a new measurement is needed, involving a rotation. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

116 Applications to Redundant Sensing (Cont d) If the angle is π, then associated rotation is symmetric and the unit vector e parallel to its axis can be determined from eq.(2.49) in both A and B. Then, we end up with two pairs of nonzero vectors {a i } 2 1 and {b i } 2 1. a i = [ Q ] A b i, for i = 1, 2, 3 (2.144) E = [ Q ] A F (2.145) E [ a 1 a 2 a 3 ], F [ b1 b 2 b 3 ] (2.146) [ Q ] A = EF 1 (2.147) Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

117 Applications to Redundant Sensing (Cont d) Therefore, F 1 = 1 (b 2 b 3 ) T (b 3 b 1 ) T, b 1 b 2 b 3 (2.148) (b 1 b 2 ) T [ Q ] A = 1 [a 1(b 2 b 3 ) T + a 2 (b 3 b 1 ) T + a 3 (b 1 b 2 ) T ] Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

118 Example Example (Hand-Eye Calibration) Determine the relative orientation of a frame B attached to a camera mounted on a robot end-effector, with respect to a frame A fixed to the latter, as shown in Fig It is assumed that two measurements of the orientation of the two frames with respect to frames A 0 and B 0 in the reference configuration of the end-effector are available. These measurements produce the orientation matrices A i of the frame fixed to the camera and B i of the frame fixed to the end-effector, for i = 1, 2. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

119 Example (Cont d) The numerical data : A 1 = A 2 = B 1 = B 2 = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

120 Example (Cont d) FIG.: Measuring the orientation of a camera-fixed coordinate frame with respect to a frame fixed to a robotic end-effector. Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

121 Example (Cont d) a 1 = a 2 = a 3 = , b 1 = , , b 2 = , b 3 = = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122

122 Example (Cont d) a 1 (b 2 b 3 ) T = a 2 (b 3 b 1 ) T = a 3 (b 1 b 2 ) T = [ Q ] A = Jorge Angeles (McGill University ) Chapter 2: Mathematical Background / 122