1 HOMOGENEOUS TRANSFORMATIONS

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1 HOMOGENEOUS TRANSFORMATIONS Purpose: The purpose of this chapter is to introduce ou to the Homogeneous Transformation. This simple 4 4 transformation is used in the geometr engines of CAD sstems and in the kinematics model in robot controllers. It is ver useful for eamining rigid-bod position and orientation (pose) of a sequence of robotic links and joint frames. Script Notation: Pre super- and pre sub-scripts are often used to denote frames of reference B C = transformation of frame C relative to frame B C p = vector located in frame C Tsai uses a pre and post script notation Β C = transformation of frame C relative to frame B C p = vector located in frame C The notation in these notes is understood graphicall b the figures and does not alwas use the scripting approach. You should be able to interpret these various notations. H, a 44 matri, will be used to represent a homogeneous transformation. H can represent translation, rotation, stretching or shrinking (scaling), and perspective transformations, and is of the general form H = a b c p a b c p a b c p d d 2 d 3 (.) Thus, given a vector u, its transformation v is represented b v = H u (.2) -

2 v H v u u H (a) Operator Interpretation (b) Frame Interpretation Figure - Interpreting the homogeneous transformation The vector u having components u, u, u must be epanded to a 4 vector of the form: u = u u u (.3) For the case of pure translation the point or vector u is translated b H when a =, a = a = b =, b =, b = c = c =, c = d = d 2 = d 3 = so that H becomes H(p) = p p p and p T = p, p, p represent the translation components along the,, aes, respectivel. -2

3 v p u H(p) p p Figure -2 Translation Eample.: Soln: If u = - 2, determine H(p) required to translate u to point v = -5 v = H(p)u implies that -5 = p p p - 2 = + p p 2+ p Thus, - + p = p = p = 2 + p = -5 p = -7 which gives H(p) = -7-3

4 The scaling transformation H s represents a scaling of vector u when all off-diagonal terms are ero and when a = a, b = b, c = c are not equal to. H s = a b c u H s v u cu u u au bu. Rotation Transformations Figure -3 Scaling transformation The rotation transformation is contained in the 33 submatri of H which we will denote b R H(R,p) = R p (3 3) (3 ) d T ( 3) ( ) (.4) If there is no rotation then R = I = the identit 3 3 matri. If there is rotation onl, then d T = T, and p =. For the special case of rotation about the ais, R assumes the form R (,) = cos -sin sin cos (.5) -4

5 v u, Figure -4 Rotation of a vector We designate the rotated frame b the aes and the original frame b the aes. The effect of rotating u to v is to change its coordinates with respect to the aes but not with respect to the aes. Note that v in the aes has the same coordinates as u in the aes. v α α u The coordinates of v in are Figure -5 Rotation about the ais v = u v = u cos - u sin v = u sin + u cos since the coordinates of v in are same as u in. Thus, v = R(,) u and can be epanded to the homogeneous form v = v v v = R (,) T u u u = H (,) u Similarl, rotations about and aes b give -5

6 R (,) = R (,) = cos sin -sin cos (.6) cos -sin sin cos (.7).2 Multiple Rotations Eample.2: Soln: Rotate u b 9 o about + and 9 o about +, where are the fied base reference aes. What are the final coordinates of the vector u after these two rotations in the base aes? If the rotation order changed, will the final coordinates be the same? Let u T = [ ]. v = R (,9 ) u "rotate u to v" w = R (,9 ) v "rotate v to w" Thus w = R (,9 ) R (,9 ) u w = w = = Rotations are not commutative. Thus R (,9 ) R (,9 ) R (,9 ) R (,9 ).3 Frame Transformation Interpretation -6

7 The columns of H represent the 4 vectors describing a second frame. The first three, when normalied, are the direction cosines of the second frame aes relative to the first reference frame. The last vector locates the secondar frame origin in the reference frame. To illustrate this interpretation, consider the previous eample and Figure -6. In particular note the multiple 9 rotations from to ''' and then to., ', " 9 2 9, ', "," Figure -6 Multiple rotations Considering the previous eample, note that H can be interpreted as two subsequent rotational transformations. H = = R (, 9o ) R (, 9 o ) T The columns of H ultimatel represent (orient and position) a secondar frame """ relative to the original frame: " oriented b, " b, and " b of both frames are the same, the last column is relative to the frame. Since the origins In the more general case of both rotation and displacement the first 3 columns represent the secondar aes orientation with respect to the base aes and the fourth column locates the secondar origin relative to the base origin. Given a rotation onl described b R (rotation onl) and a pure translation described b p, does. -7

8 H(p) H(R)? = H(R) H(p) where and H(R) = R T H(p) = I p T First consider rotation R, and then translation p, to get w = H(p) H(R) u = I p T R T u w = R p T u = H (R, p) u (.8) Net, consider translation first, then rotation: w = H(R) H(p) u = R T I p T u = R Rp T u (.9) The w s are different in (.8) and (.9) and thus H(p) H(R) H(R) H(p) The usual procedures for locating points in translated and rotated reference frames relative to the base aes is to rotate the frame first, then translate the frame origin, i.e., appl (.8)..4 Relative Transformations Since H(R,p) = H(p) H(R) H(R) H(p), what does H(R) H(p) represent? Epanding for w, w = H(R) H(p) u = R Rp T u = R(u + p) (.) where u is understood to be in the form of the epanded vector [u ] T. From (.) we note that the effect of translating first is to cause the rotation R of u relative to the base frame after it has been translated b p. Generaliing: -8

9 If we postmultipl a transformation representing a frame (relative to base aes) b a second transformation (relative to the frame of the first transformation), we make the transformation with respect to the frame aes of the first transformation. Premultipling the frame transformation b the second transformation causes the transformation to be made with respect to the base reference frame. Eample (Paul).3: Given frame C = 2 - and transformation H = - locate frame X = H C and frame Y = C H First, X = HC = = 2 This can be illustrated b Figure -7 where we note that,, and locate the ", ", " aes with respect to the,, and aes, respectivel. -9

10 Z 2 C X " " " Y X Figure -7 X = H C Net, we reverse the order to calculate Y = C H Y = = as illustrated b the following figure: Z C 2 Y X Y Figure -8 Y = C H -

11 Note that postmultipling causes H to be made relative to frame C rather than relative to XYZ base aes..5 Inverse Transformations Given u and the rotational transformation R, the coordinates of u after being rotated b R are defined b v = Ru. The inverse question is given v, what u when rotated b R will give v? This is found b premultipling b R -, the inverse of R, where R - R = I. R - v = R - R u =I u = u (.) Thus, u = R - v. Note that the inverse is onl defined for square matrices with the rank equal to the number of columns (or rows). Now how do we determine R -? Without proof it will be stated that R - = R T (Prove for an case b simpl showing that R T R = R R T = I). Similarl for an displacement matri H (R, p), we can pose a similar question to get u = H - v. What is the inverse of a displacement transformation? Without proof: a a a -p T a H - = b b b -p T b c c c -p T c = RT -R T p T (.2) To prove, show that H - H = I, -R T p T R T R p T = R T R (R T p - R T p T = I T = I Remember the rules for square matrices A, B, C having rank n that (A B C ) T = C T B T A T (.3) (A B C ) - = C - B - A - (.4).6 Homogeneous Transformations Summar A 4 4 homogeneous transformation consists of three components: -

12 rotational, orthogonal 33 sub-matri which is comprised of columns of direction cosines used to orient the aes of one frame relative to another. column vector in 4th column represents the origin of second frame relative to first frame, resolved in the first frame. s in 4th row ecept for in 4,4 position. The homogeneous transformation effectivel merges a frame orientation matri and frame translation vector into one matri. The order of the operation should be viewed as rotation first, then translation. The homogeneous transformation can be viewed as a position/orientation relationship of one frame relative to another frame called the reference frame. The order of multipling frames is important since A B B A. A B can be interpreted as frame A described relative to the first or base frame while frame B is described relative to frame A (usual wa). We can also interpret B in the base frame transformed b A in the base frame. Both interpretations give same result. -2