Rotation in the Space

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1 Rotation in the Space (Com S 477/577 Notes) Yan-Bin Jia Sep 6, 2016 The position of a point after some rotation about the origin can simply be obtained by multiplying its coordinates with a matrix One reason for introducing homogeneous coordinates is to be able to describe translation with a matrix so that multiple transformations, whether each is a rotation or a translation, can be concatenated into one described by the product of their respective matrices However, in some applications (such as spaceship tracking), we need only be concerned with rotations of an object, or at least independently from other transformations In such a situation, we often need to extract the rotation axis and angle from a matrix which represents the concatenation of multiple rotations The homogeneous transformation matrix, however, is not well suited for the purpose 1 Euler Angles A rigid body in the space has a coordinate frame attached to itself and located often at the center of mass This frame is referred to as the body frame or local frame The position, orientation, and motion of the body can be described using the body frame relative to a fixed reference frame, called the world frame z B z x B y B Body Frame x World Frame y Figure 1: Body frame (local) vs world frame (global) The rigid body has six degrees of freedom: its position given by the x, y, z coordinates of its center of mass (ie, the origin of the local frame) in the world frame, and its orientation described The material is partially based on Chapters 3 6 of the book [2] 1

2 Roll Yaw Pitch Figure 2: Roll, pitch, yaw of an aircraft by three angles of rotation of its body frame from the world frame To describe this rotation, one choice is to use three angle about the x-, y-, and z-axes A more convenient choice, from the perspective of the rotating body, is to use three angles about the axes of the body frame The latter three angles are called Euler angles For example, an aircraft in flight can perform three independent rotations: roll, about an axis from nose to tail; pitch, nose up or down about an axis from wing to wing; and yaw, nose left or right about an axis from top to bottom There are several conventions for Euler angles, depending on the axes about which the rotations are carried out Here we introduce the Z-Y -X Euler angles The body frame starts in the same orientation as the world frame To achieve its final orientation, the first rotation is by an angle φ, about the body z-axis, the second rotation by an angle θ [0,π] about the body y-axis, and the third rotation by an angle ψ about the body x-axis Here, φ, θ, and ψ correspond to as yaw, pitch, and roll, respectively The three rotations are illustrated in Figure 3, where x B -y B -z B and x B -y B -z B are the two intermediate configurations of the body frame x B-y B -z B z, z B p z B p p z B z B y B φ y B θ y B, y B z B y B x y ψ x B x B x B x B, x B Figure 3: Z-Y -X Euler angles φ (yaw), θ (pitch), ψ (roll) Here, p, p, and p are the same point of coordinates in the frames x B -y B -z B, x B -y B -z B, and x B -y B -z B, respectively Now we derive the homogeneous transformation matrix Consider a point p in the body frame x B -y B -z B Let us determine its coordinates in the original frame x-y-z by considering the four frames backward: x B -y B -z B, x B -y B -z B, x B -y B -z B, and x-y-z Consider a point p with its coordinates given in the body frame Since the frame x B -y B -z B is obtained from the frame x B -y B -z B by a rotation 2

3 about the x B-axis through an angle ψ, the coordinates of p in the latter frame is p = Rot x (ψ)p Next, the frame x B -y B -z B is obtained from the frame x B -y B -z B after a rotation about the y B -axis through an angle θ So the same point has coordinates p = Rot y (θ)p = Rot y (θ)(rot x (ψ)p in the frame x B -y B -z B Similarly, the coordinates of the point in the frame x-y-z is Rot z (φ)p = Rot z (φ)(rot y (θ)(rot x (ψ)p Thus the transformation matrix associated with the Z-Y -Z Euler angles are Rot zyx (φ,θ,ψ) = Rot z (φ) Rot y (θ) Rot x (ψ) cos φ sin φ 0 cos θ 0 sin θ = sin φ cos φ sinθ 0 cos θ cos ψ sin ψ 0 sin ψ cos ψ cos φcos θ cos φsin θ sin ψ sin φcos ψ cos φsin θ cos ψ + sin φsin ψ = sin φcos θ sinφsin θ sin ψ + cos φcos ψ sinφsin θ cos ψ cos φsin ψ sin θ cos θ sin ψ cos θ cos ψ (1) Euler angles are defined in terms of three moving axes Singularities happen when two of these axes coincide Consider the z-x-z angles α, β, γ with β = 0 Here, the first and the third rotations are about the same axis All the Euler angles (α,0,β) with the same α + β value thus describe one rotation Namely, one degree of freedom is lost Linear interpolation from one orientation to another is not well-behaved Imagine, when the latitude and longitude values are interpolated What will happen when longitude goes towards 90 degrees to reach the north pole? All latitude values there make no difference as they end up describing the same point! This is a phenomenon referred to as gimbal lock, drawing its name from certain orientation with three nested moving gimbals in which two of the three axes become collinear restricting the available rotations to only two axes (Watch a nice video tutorial on gimbal lock at The above singularity issue with Euler angles is because they form a 3D box [0,2π] 3, while the rotations constitute a 3D projective space The mapping between the two spaces cannot be a continuous both way (or, strictly speaking, a homeomorphism 1 ) 2 Arbitrary Rotation Axis Let v be a vector that is undergoing a rotation of the amount θ about some axis through the origin with an arbitrary orientation This can be viewed as a rotation about a line that was treated in Appendix A in the notes titled Transformations in Homogeneous Coordinates The resulting vector can be found by first rotating the axis to the z-axis, performing the rotation, and rotating the axis back to its original orientation Now let us present a simpler approach with direct geometric meaning 1 a bijective and continuous function whose inverse is also continuous 3

4 Let ˆk = (k x,k y,k z ) T be the unit vector along the axis To obtain the resulting v from the rotation, we decompose v into two parts: one along ˆk and the other in the plane Π containing the origin and perpendicular to ˆk: v = (v ˆk)ˆk, (2) v = v v = v (v ˆk)ˆk (3) As shown in Figure 4, the component v is not affected by rotation We need only determine the vector v that results from applying the same rotation to v z v w v θ v v ˆk v θ w v v x y Figure 4: The vector v is rotated to v about the axis ˆk The plane Π is spanned by two orthogonal vectors: v and w = ˆk v = ˆk v (4) It is easy to see that v,w, ˆk are the axes of a right-handed system Since ˆk is orthogonal to v, v and w have the same length The vector v lies in the plane Π, and has the form Finally, we have the rotated vector from v: v = v cos θ + w sinθ (5) v = v + v = (v ˆk)ˆk + v cos θ + w sinθ (by (2) and (5)) = (v ˆk)ˆk + (v (v ˆk)ˆk)cos θ + (ˆk v)sin θ (by (3) and (4)) = v cos θ + (ˆk v)sin θ + ˆk(ˆk v)(1 cos θ) (6) Equation (6) is called Rodrigues rotation formula 4

5 Note that v cos θ = ˆk v = cos θ cos θ cos θ 0 k z k y k z 0 k x k y k x 0 v; v; ˆk(ˆk v) = (ˆkˆk T )v k 2 x k x k y k x k z = k y k x ky 2 k y k z v k z k x k z k y kz 2 Substituting the above into (6), we express v as the product of the following 3 3 rotation matrix with v: k x k x (1 cos θ) + cos θ k x k y (1 cos θ) k z sin θ k x k z (1 cos θ) + k y sinθ Rotˆk (θ) = k x k y (1 cos θ) + k z sin θ k y k y (1 cos θ) + cos θ k y k z (1 cos θ) k x sinθ k x k z (1 cos θ) k y sin θ k y k z (1 cos θ) + k x sin θ k z k z (1 cos θ) + cos θ (7) 3 Rotation Matrix We have seen the use of a matrix to represent a rotation Such a matrix is referred to as a rotation matrix In this section we look at the properties of rotation matrix Below let us first review some concepts from linear algebra 31 Eigenvalues An n n matrix A is orthogonal if its columns are unit vectors and orthogonal to each other, namely, if A T A = I n An orthogonal matrix has determinant det(a) = ±1 A complex number λ is called the eigenvalue of A if there exists a vector x 0 such that Ax = λx It tells whether the vector x is stretched or shrunk or reversed or left unchanged when it is multiplied by A The vector x is called an eigenvector of A associated with the eigenvalue λ The set of all eigenvalues is called the spectrum of A The matrix A has exactly n eigenvalues (multiplicities included) They are the roots of the nth degree polynomial det(a λi), called the characteristic polynomial Let λ 1,,λ k be the distinct eigenvalues of this polynomial, then det(a λi) = ( 1) n (λ λ 1 ) δ1 (λ λ k ) δ k, for some δ 1,,δ k > 0 Here δ i, 1 i k, is the algebraic multiplicity of λ i The geometric multiplicity of λ i is n rank(a λ i I) It specifies the maximum number of linearly independent eigenvectors associated with λ i Example 1 The diagonal matrix ai, where I is the n n identity matrix, has the characteristic polynomial (a λ) n So a is the only eigenvalue while every vector x C n, x 0 is an eigenvector Both algebraic and geometric multiplicities of a are n 5

6 Example 2 The nth order matrix A = a a a also has the characteristic polynomial (a λ) n and a as its only eigenvalue The algebraic multiplicity of a is n However, the rank of A ai is n 1; hence the geometric multiplicity of a is n (n 1) = 1 The product of the n eigenvalues equals the determinant of A The sum of the n eigenvalues equals the sum of the n diagonal entries; this number is called the trace of A and denoted Tr(A) The matrix A is invertible if and only if it does not have a zero eigenvalue A symmetric matrix has real eigenvalues A symmetric and positive definite matrix has positive eigenvalues 32 Matrices Describing Rotations All rotation matrices applying to points in the space are 3 3 Theorem 1 A 3 3 matrix describes a rotation about some axis through the origin if and only if it is orthogonal and has determinant 1 Proof ( ) We first show that any rotation matrix A is orthogonal and det(a) = 1 Let v be the corresponding rotation axis and θ the angle of rotation about the axis The inner product of two vectors, say, v 1 and v 2, must be preserved under the rotation; namely, The above implies that (Av 1 ) T (Av 2 ) = v T 1 v 2 v T 1 (A T A I)v 2 = 0, where I is the 3 3 identity matrix Because v 1 and v 2 are arbitrarily chosen, it must hold that A T A I = 0 and thus A T A = I Therefore, A is orthogonal Further more, from the equation det(a T A) = det(a) T det(a) = ( det(a)) 2 = 1 we infer that det(a) must be 1 or 1 But a rotation preserves the right-handedness of a coordinate frame, so we cannot have det(a) = 1 For example, the diagonal matrix with diagonal entries 1, 1, and 1 would describe a reflection about the x-y plane instead of a rotation We have thus shown that det(a) = 1 ( ) Suppose A is a 3 3 orthogonal matrix with determinant 1 We want to show that it represents a rotation First, we show that there exists a vector u such that Au = u This vector 6

7 will be along the rotation axis It suffices to establish that the matrix A has an eigenvalue of 1, which is true if and only if det(a I) = 0 We have det(a I) = det(a T )det(a I) = det(a T A A T ) = det(i A T ) (A orthogonal) = det((i A) T ) = det(i A) = det(a I) Thus det(a I) = 0 Next, we show that the plane containing the origin and orthogonal to u maps to itself under the rotation Consider an arbitrary vector v in this plane, ie, v T u = 0 The vector Av is orthogonal to u because (Av) T u = v T A T Au = v T Iu = v T u = 0 Finally, the inner product of two vectors v 1 and v 2 is invariant under the transformation because A is orthogonal Therefore, the lengths of v 1 and v 2 and the angles between them are preserved under A Hence the transformation represented by A is a rotation Note that a rotation about an axis not through the origin cannot be represented by a 3 3 orthogonal matrix with determinant 1 To see this, let b 0 be a point on the rotation axis Then a point x is rotated into R(x b) + b, where R is a rotation matrix (orthogonal and having determinant 1) However, there does not exist a rotation matrix R such that R x = R(x b) + b unless R = I Two successive rotations about the same axis in R 3 is equivalent to one rotation about this axis through an angle which is the sum of the two rotation angles Even if they are not about the same axis through the origin, their composition is still a rotation through some angle about some axis The reason is that the product of their rotation matrices, each orthogonal with determinant 1, is still orthogonal with determinant 1 Thus, by Theorem 1 the product matrix describes a rotation To state this formally, the composition of any two rotations is equivalent to a rotation Given a rotation matrix, we can extract the Euler angles φ, θ, and ψ based on the form (1) 33 Recovery of Rotation Axis and Angle Consider the following orthogonal matrix with determinant 1: a 11 a 12 a 13 A = a 21 a 22 a 23 a 31 a 32 a 33 Theorem 1 says that it represents a rotation about some axis v in the space How to find this axis of rotation? It follows from the proof of Theorem 1 that v must be an eigenvector of A which 7

8 corresponds to the eigenvalue 1 Namely, (A I)v = 0, from which we can obtain a solution vector by setting one of the non-zero components in v to 1 We makes use of Q = A A T Note that v is also an eigenvector of A T for We have A T v = A T Av = Iv = v where (A A T )v = Qv 0 q 3 q 2 v 1 = q 3 0 q 1 v 2 = q 2 q 1 0 v 3 0 0, (8) 0 q 1 = a 23 a 32, q 2 = a 31 a 13, q 3 = a 12 a 21 The matrix A A T in (8) is not of full rank because det(a A T ) = 0 So the three linear equations are not independent First, we assume v 3 to be non-zero and let v 3 = 1, obtaining q 3 v 2 q 2 = 0, q 3 v 1 + q 1 = 0 Assume q 3 0 first Solving the two equations above gives us the axis of rotation ( q1 v =, q ) T 2,1 q 3 q 3 Equivalently, we have v = (q 1,q 2,q 3 ) T (which also works for q 3 = 0, ie, v 3 = 0) Thus, if a rotation is defined by the 3 3 matrix A = (a ij ), then the axis of rotation is a 23 a 32 v = a 31 a 13 (9) a 12 a 21 Having determined the rotation axis v, we look at how to recover the rotation angle If the rotation happens to be about the z-axis through an angle θ, then we have the rotation matrix cos θ sin θ 0 A = Rot z (θ) sin θ cos θ

9 And the rotation angle θ satisfies cos θ = a 11 We can also utilize the trace and obtain Tr(A) = a 11 + a 22 + a 33 = 1 + 2cos θ cos θ = Tr(A) 1, (10) 2 sin θ = ± 1 sin 2 cos θ 3 Tr(A) = ± 2 + 2Tr(A) (11) 4 The sign of sin θ must agree with that of a 21 In fact, the solution of θ from (10) (11) generalizes to the situation with an arbitrary rotation axis To see this, we use the old trick of applying a sequence of rotations, represented by a matrix Q, to transform the axis v into the z-axis Then we carry out the rotation about the z-axis through the angle θ Finally, we apply a sequence of reverse rotations Q T to transform the z-axis back to v The rotation matrix is represented as A = Q T Rot z (θ)q (12) The trace function is commutative, that is, Tr(CD) = Tr(DC) for any two square matrices of the same dimensions 2 This property allows us to derive the following from (12): ( ) Tr(A) = Tr Q T Rot z (θ)q ( ) ) = Tr( Q T Rot z (θ) Q ( ( ) ) = Tr Q Q T Rot z (θ) ( = Tr( QQ T) ) Rot z (θ) ( ) = Tr I Rot z (θ) ( ) = Tr Rot z (θ) = 1 + 2cos θ Hence the rotation angle θ still satisfies equation (10) when the rotation axis is arbitrary How do we determine the sign in (11)? First, we normalize the rotation axis v from (9) and obtain a unit vector ˆv = (v x,v y,v z ) The rotation about ˆv through an angle θ is described by the following rotation matrix according to (7): Rotˆv (θ) = v x v x (1 cos θ) + cos θ v x v y (1 cos θ) v z sin θ v x v z (1 cos θ) + v y sin θ v x v y (1 cos θ) + v z sin θ v y v y (1 cos θ) + cos θ v y v z (1 cos θ) v x sin θ v x v z (1 cos θ) v y sin θ v y v z (1 cos θ) + v x sin θ v z v z (1 cos θ) + cos θ 2 The proof is directly based on the definition of trace 9

10 to remote object station β (elevation) α (heading) x (north) tangent plane y (east) z earth Figure 5: Tracking an aircraft from a ground station Comparing an off-diagonal element in the matrix A, say a 21, with its corresponding entry in Rotˆv (θ) gives us sin θ = a 21 v x v y (1 cos θ) v z = 2a 21 v x v y (3 Tr(A)) 2v z (13) The angle θ [0,2π) of rotation is thus completely determined from (10) and (13) 4 Application Tracking In this section, we describe an application of rotations in R 3 Consider a remote object, such as an aircraft, which is being tracked from a station on the earth We define a local coordinate system at the station with the xy-plane being the tangent plane to the earth The x- and y-axes point in the directions of the North and East, respectively, and the z-axis points toward the center of the earth The coordinate system is drawn in the figure below The heading α is the angle in the tangent plane between the North and the projection of the direction to the remote object being tracked The elevation is the angle between the tangent plane and the direction to the object The tracking transformation is a rotation about the z-axis through the angle α, followed by a rotation about the new y-axis through the angle β After these two rotations, the x-axis is pointing toward the object 10

11 Let R be the 3 3 matrix to represent the composite rotation; it relates the tracking coordinate system to the station s coordinate system Namely, a point p = (p x,p y,p z ) T in the (re-oriented) tracking frame is Rp in the station s frame Then we have R = Rot z (α) Rot y (β) cos α sinα 0 cos β 0 sin β = sin α cos α sinβ 0 cos β cos α cos β sin α cos α sinβ = sin α cos β cos α sin α sinβ sinβ 0 cos β This composite tracking transformation can be represented as an equivalent rotation through some angle θ about some axis v By (9) we know that the rotation axis is given by v = sin αsin β cos αsin β sinβ sin αcos β sin α sin α 2 sin β 2 cos α 2 sin β 2 sin α 2 cos β 2 (14) Let ˆv = (v x,v y,v z ) be the normalization of v By (10) and (13) the angle of rotation satisfies the following equations: cos θ = sin θ = cos α cos β + cos α + cos β 1, 2 2sin α cos β v xv y (3 Tr(A)) 2v z The domain for the tracking angles is α ( π,π] and β [ π 2, π 2 ] As β increases from π 2 ǫ to π 2 + ǫ, for small ǫ, α instantaneously jumps to α + π For this reason, (α, π 2 ) is a singular point, and the phenomenon is gimbal lock References [1] J Craig Introduction to Robotics: Mechanics and Control 2nd ed, Addison-Wesley, 1989 [2] J B Kuipers Quaternions and Rotation Sequences Princeton University Press, 1999 [3] -in-euler-angles/ 11

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