Vectors in Three Dimensions and Transformations

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1 Vectors in Three Dimensions and Transformations University of Pennsylvania 1

2 Scalar and Vector Functions φ(q 1, q 2,...,q n ) is a scalar function of n variables φ(q 1, q 2,...,q n ) is independent of reference frames scalar invariant a 3 a 2 a 1 e 3 e 2 e 1 NB: No assumptions of orthogonality! v(q 1, q 2,...,q n ) is a vector function v of n variables. In any reference frame {A}, we can find three linearly independent (LI) vectors a 1, a 2, and a 3 that are basis vectors. The vector function v(q 1, q 2,...,q n ) can be expressed as a linear combination of the three vectors: v(q 1, q 2,...,q n ) = v 1 (q 1, q 2,...,q n ) a 1 + v 2 (q 1, q 2,...,q n ) a 2 + v 3 (q 1, q 2,...,q n ) a 3 The three coefficients are the three scalar functions v 1, v 2, and v 3. They are called components and these three functions are unique once the vectors a 1, a 2, and a 3 are specified. University of Pennsylvania 2

3 Reference Frames e 3 e 2 e 1 a 3 a 2 a 1 Components of vectors depend on the reference triad A robotic arm is a system of rigid bodies (reference frames) A, B, and C. D is the inertial or the laboratory reference frame that is considered fixed. University of Pennsylvania 3

4 Position, Velocity and Acceleration Vectors p is a position vector of P in A Emanates from a point fixed to A Ends up at P A v P is the velocity of P in A p P O p' A a P is the acceleration of P in A A O' A a P = A d( A v P ) dt What if a different position vector were chosen? University of Pennsylvania 4

5 Standard Reference Triad Three LI vectors rigidly attached to a reference frame satisfying the equations below constitute a standard reference triad or simply a reference triad. a 3 e 3 e 2 e 1 a 2 a 1 Projection rule Composition rule University of Pennsylvania 5

6 Transformation of Vectors a e 3 3 a e 2 2 a e 1 1 Components in {A} and in {E} are related by: E ( ) A p [ p] i = e i a j j [ ] j E [ p] i e i = i j = j A A [ p] j = e i a j a j ( ) [ p] j a j e i ( ) i, j i A = e i a j i j [ p ] j e i ( ) A p [ ] j e i e i E [ p] = [ E R A ] A [ p] e 1 a 1 e 1 a 2 e 1 a 3 = e 2 a 1 e 2 a 2 e 2 a 3 e 3 a 1 e 3 a 2 e 3 a 3 A [ p] Rotation matrix that transforms components in {A} into components in {E} University of Pennsylvania 6

7 Rotation Matrices University of Pennsylvania 7

8 Orthogonal MEAM 535 Properties of Rotation Matrices Matrix times its transpose equals 1 Special orthogonal Determinant is +1 ξ 3 Closed under multiplication ξ 2 Composition A R C = A R B B R C The inverse of a rotation matrix is also a rotation matrix Composition and inverse operations are continuous functions ξ 1 SO(3) The set of all rotations is a Lie Group, SO(3) SO(3) can be parameterized by 3 coordinates ξ 1 ξ 2 ξ 3 University of Pennsylvania 8

9 Example: Simple Rotations Rotation about the x-axis through θ a 3 e 3 a 2 θ e 2 z θ e 1 y x University of Pennsylvania 9

10 Example: Rotation Rotation about the y-axis through θ Rotation about the z-axis through θ a 1 e 1 θ a 3 e 3 a 1 e 1 θ e 3 e 2 e 2 a 2 University of Pennsylvania 10

11 Composition of Three Rotations A R D = A R B B R C C R D A R D = Rot(x, ψ) Rot(y, φ) Rot(z, θ) University of Pennsylvania 11

12 Euler Angles Any rotation can be described by three rotations about linearly independent axes. ξ 3 Rotations can be parameterized by 3 coordinates (angles) ξ rotation matrix ξ 1 Almost 1-1 transformation University of Pennsylvania 12

13 Euler Angles: Parameterization of Rotation Matrices Sequence of three rotations about bodyfixed axes Rot(z, φ) Rot(y, θ) Rot(z, ψ) Are these LI? Three Euler Angles φ, θ, and ψ Parameterize rotations R = Rot(z, φ) Rot(y, θ) Rot(z, ψ) Note θ= 0 is a special (singular) case University of Pennsylvania 13

14 Determination of Euler Angles R = Rot(z, φ) Rot(y, θ) Rot(z, ψ) University of Pennsylvania 14

15 Determination of Euler Angles If R 33 < 1, If R 33 = 1, If R 33 = -1, R = Rot(z, φ) Rot(y, θ) Rot(z, ψ) Two sets of Euler angles for every R for almost all R s! University of Pennsylvania 15

16 Euler Angles b 3 See notes a 3 c 3 b 2 a 1 a 2 c 1 α c 2 d 3 β γ d 2 b 1 d 1 University of Pennsylvania 16

17 Rotations MEAM 535 Euler s Theorem Any displacement of a rigid body such that a point on the rigid body, say O, remains fixed, is equivalent to a rotation about a fixed axis through the point O. Chasles Theorem for General Displacements (later) The most general rigid body displacement can be produced by a translation along a line followed (or preceded) by a rotation about that line. University of Pennsylvania 17

18 Proof of Euler s Theorem Displacement from {F} to {M} P = Rp Solve the eigenvalue problem (find the vector that is unaffected by R): Rp = λ p Let Three eigenvalues and eigenvectors are: University of Pennsylvania 18

19 Axis and Angle of Rotation Rotation Matrix to Axis and Angle 1. Solve Ru = 1 u for unit vector along axis 2. Find angle of rotation: Axis and Angle to Rotation Matrix? University of Pennsylvania 19

20 The Axis/Angle for a Rotation Matrix v u w = u v Select v to be orthogonal to u Let w = u v The other two eigenvectors are perpendicular to u Q University of Pennsylvania 20 Q Λ

21 p u φ v (p.u)u vcosφ + u vsinφ p-(p.u)u φ ( p ( p.u)u )cosφ + u ( p ( p.u)u )sinφ (p.u)u pcosφ + uu T ( 1 cosφ)p + u psinφ w = u v University of Pennsylvania 21

22 Axis/Angle to Rotation Matrix Rotation about u through φ Rodrigues formula p u φ Rp where, Notes: 1. Map from R to (u, φ) is one to many. - restrict φ to the interval [0,π] 2. Singular R = I trace(r) = -1 Extracting the axis and the angle from the rotation matrix 1. Find the eigenvector corresponding to λ=1. 2. From Rodrigues formula: See notes - University of Pennsylvania 22

23 The 3 1 vector a and its 3 3 skew symmetric matrix counterpart A a A For any vector b a b = A b University of Pennsylvania 23