1 Coordinate Transformation

Transcription

1 1 Coordinate Transformation 1.1 HOMOGENEOUS COORDINATES A position vector in a three-dimensional space (Fig ) may be represented (i) in vector form as r m = O m M = x m i m + y m j m + z m k m (1.1.1) where (i m, j m, k m ) are the unit vectors of coordinate axes, and (ii) by the column matrix x m r m = y m. (1.1.2) z m The subscript m indicates that the position vector is represented in coordinate system S m (x m, y m, z m ). To save space while designating a vector, we will also represent the position vector by the row matrix, r m = [x m y m z m ] T. (1.1.3) The superscript T means that r T m is a transpose matrix with respect to r m. A point the end of the position vector is determined in Cartesian coordinates with three numbers: x, y, z. Generally, coordinate transformation in matrix operations needs mixed matrix operations where both multiplication and addition of matrices must be used. However, only multiplication of matrices is needed if position vectors are represented with homogeneous coordinates. Application of such coordinates for coordinate transformation in theory of mechanisms has been proposed by Denavit & Hartenberg [1955] and by Litvin [1955]. Homogeneous coordinates of a point in a threedimensional space are determined by four numbers (x, y, z, t ) which are not equal to zero simultaneously and of which only three are independent. Assuming that t 0, ordinary coordinates and homogeneous coordinates may be related as follows: x = x t y = y t z = z t. (1.1.4) 1

2 2 Coordinate Transformation Figure 1.1.1: Position vector in Cartesian coordinate system. With t = 1, a point may be specified by homogeneous coordinates such as (x, y, z, 1), and a position vector may be represented by x m y m r m = z m or 1 r m = [x m y m z m 1] T. 1.2 COORDINATE TRANSFORMATION IN MATRIX REPRESENTATION Consider two coordinate systems S m (x m, y m, z m ) and S n (x n, y n, z n ) (Fig ). Point M is represented in coordinate system S m by the position vector r m = [x m y m z m 1] T. (1.2.1) The same point M can be determined in coordinate system S n by the position vector r n = [x n y n z n 1] T (1.2.2) with the matrix equation r n = M nm r m. (1.2.3)

3 1.2 Coordinate Transformation in Matrix Representation 3 Figure 1.2.1: Derivation of coordinate transformation. Matrix M nm is represented by a 11 a 12 a 13 a 14 a M nm = 21 a 22 a 23 a 24 a 31 a 32 a 33 a (i n i m ) (i n j m ) (i n k m ) (O n O m i n ) (j = n i m ) (j n j m ) (j n k m ) (O n O m j n ) (k n i m ) (k n j m ) (k n k m ) (O n O m k n ) = cos( x n, x m ) cos( y n, x m ) cos( z n, x m ) cos( x n, y m ) cos( y n, y m ) cos( z n, y m ) cos( x n, z m ) x (O m) n cos( y n, z m ) y (O m) n cos( z n, z m ) z (O m). (1.2.4) Here, (i n, j n, k n ) are the unit vectors of the axes of the new coordinate system; (i m, j m, k m ) are the unit vectors of the axes of the old coordinate system; O n and O m are the origins of the new and old coordinate systems; subscript nm in the designation M nm indicates that the coordinate transformation is performed from S m to n

4 4 Coordinate Transformation S n. The determination of elements a lk (k = 1, 2, 3; l = 1, 2, 3) of matrix M nm is based on the following rules: (i) Elements of the 3 3 submatrix a 11 a 12 a 13 L nm = a 21 a 22 a 23 (1.2.5) a 31 a 32 a 33 represent the direction cosines of the old unit vectors (i m, j m, k m ) in the new coordinate systems S n. For instance, a 21 = cos( y n, x m ), a 32 = cos( z n, y m ), and so on. The subscripts of elements a kl in matrix (1.2.5) indicate the number l of the old coordinate axis and the number k of the new coordinate axis. Axes x, y, z are given numbers 1, 2, and 3, respectively. (ii) Elements a 14, a 24, and a 34 represent the new coordinates x (O m) n, y (O m) n, z (O m) n of the old origin O m. Recall that nine elements of matrix L nm are related by six equations that express the following: (1) Elements of each row (or column) are direction cosines of a unit vector. Thus, a a a 13 2 = 1, a a a 31 2 = 1,. (1.2.6) (2) Due to orthogonality of unit vectors of coordinate axes, we have [a 11 a 12 a 13 ][a 21 a 22 a 23 ] T = 0 [a 11 a 21 a 31 ][a 12 a 22 a 32 ] T = 0. (1.2.7) An element of matrix L nm can be represented by a respective determinant of the second order [Strang, 1988]. For instance, a 11 = a 22 a 23 a 32 a 33, a 23 = ( 1) a 11 a 12 a 31 a 32. (1.2.8) To determine the new coordinates (x n, y n, z n, 1) of point M, we have to use the rule of multiplication of a square matrix (4 4) and a column matrix (4 1). (The number of rows in the column matrix is equal to the number of columns in matrix M nm.) Equation (1.2.3) yields x n = a 11 x m + a 12 y m + a 13 z m + a 14 y n = a 21 x m + a 22 y m + a 23 z m + a 24 z n = a 31 x m + a 32 y m + a 33 z m + a 34. (1.2.9) The purpose of the inverse coordinate transformation is to determine the coordinates (x m, y m, z m ), taking as given coordinates (x n, y n, z n ). The inverse coordinate transformation is represented by r m = M mn r n. (1.2.10) The inverse matrix M mn indeed exists if the determinant of matrix M nm differs from zero.

5 1.2 Coordinate Transformation in Matrix Representation 5 There is a simple rule that allows the elements of the inverse matrix to be determined in terms of elements of the direct matrix. Consider that matrix M nm is given by a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 M nm = a 31 a 32 a 33 a 34. (1.2.11) It is necessary to determine the elements of matrix M mn represented by b 11 b 12 b 13 b 14 M mn = b 21 b 22 b 23 b 24 b 31 b 32 b 33 b 34. (1.2.12) Here, M mn = M 1 nm, M mnm nm = I where I is the identity matrix. The submatrix L mn of the order (3 3) is determined as follows: b 11 b 12 b 13 a 11 a 21 a 31 L mn = b 21 b 22 b 23 = a 12 a 22 a 32 = L T nm. (1.2.13) b 31 b 32 b 33 a 13 a 23 a 33 The remaining elements (b 14, b 24, and b 34 ) are determined with the following equations: : a 11 : a 12 a 13 : a 14 : : a 21 : a 22 a 23 : a 24 : b 14 = (a 11 a 14 + a 21 a 24 + a 31 a 34 ) : a 31 : a 32 a 33 : a 34 : : 0 : 0 0 : 1 : a 11 : a 12 : a 13 : a 14 : a 21 : a 22 : a 23 : a 24 : b 24 = (a 12 a 14 + a 22 a 24 + a 32 a 34 ) a 31 : a 32 : a 33 : a 34 : 0 : 0 : 0 : 1 : a 11 a 12 : a 13 : : a 14 : b 34 = (a 13 a 14 + a 23 a 24 + a 33 a 34 ) a 21 a 22 : a 23 : : a 24 : a 31 a 32 : a 33 : : a 34 :. (1.2.14) 0 0 :0: :1: The columns to be multiplied are marked. To perform successive coordinate transformation, we need only to follow the product rule of matrix algebra. For instance, the matrix equation r p = M p(p 1) M (p 1)(p 2) M 32 M 21 r 1 (1.2.15) represents successive coordinate transformation from S 1 to S 2, from S 2 to S 3,...,from S p 1 to S p.

6 6 Coordinate Transformation To perform transformation of components of free vectors, we need only to apply 3 3 submatrices L, which may be obtained by eliminating the last row and the last column of the corresponding matrix M. This results from the fact that the free-vector components (projections on coordinate axes) do not depend on the location of the origin of the coordinate system. The transformation of vector components of a free vector A from system S m to S n is represented by the matrix equation A n = L nm A m (1.2.16) where A n = A xn A yn a 11 a 12 a 13, L nm = a 21 a 22 a 23, A m = A xm A ym. (1.2.17) A zn a 31 a 32 a 33 A zm A normal to the gear tooth surface is a sliding vector because it may be translated along its line of action. However, we may transform the surface normal as a free vector if the surface point where the surface normal is considered will be transferred simultaneously. 1.3 ROTATION ABOUT AN AXIS Two Main Problems We consider a general case in which the rotation is performed about an axis that does not coincide with any axis of the employed coordinate system. We designate the unit vector of the axis of rotation by c (Fig ) and assume that the rotation about c may be performed either counterclockwise or clockwise. Henceforth we consider two coordinate systems: (i) the fixed one, S a ; and (ii) the movable one, S b. There are two typical problems related to rotation about c. The first one can be formulated as follows. Consider that a position vector is rigidly connected to the movable body. The initial position of the position vector is designated by OA = ρ (Fig ). After rotation through an angle φ about c, vector ρ will take a new position designated by OA = ρ. Both vectors, ρ and ρ (Fig ), are considered to be in the same coordinate system, say S a. Our goal is to develop an equation that relates components of vectors ρ a and ρ a. (The subscript a indicates that the two vectors are represented in the same coordinate system S a.) Matrix equation ρ a = L aρ a (1.3.1) describes the relation between the components of vectors ρ and ρ that are represented in the same coordinate system S a. The other problem concerns representation of the same position vector in different coordinate systems. Our goal is to derive matrix L ba in matrix equation ρ b = L ba ρ a. (1.3.2)

7 1.3 Rotation About an Axis 7 Figure 1.3.1: Rigid body rotation. The designations ρ a and ρ b indicate that the same position vector ρ is represented in coordinate systems S a and S b, respectively. Although the same position vector is considered, the components of ρ in coordinate systems S a and S b are different and we designate them by and ρ a = a 1 i a + a 2 j a + a 3 k a (1.3.3) ρ b = b 1 i b + b 2 j b + b 3 k b. (1.3.4) Matrix L ba is an operator that transforms the components [a 1 a 2 a 3 ] T into [b 1 b 2 b 3 ] T. It will be shown below that operators L a and L ba are related. Problem 1. Relations between components of vectors ρ a and ρ a. Recall that ρ a and ρa are two position vectors that are represented in the same coordinate system S a. Vector ρ represents the initial position of the position vector, before rotation, and ρ represents the position vector after rotation about c. The following derivations are based on the assumption that rotation about c is performed counterclockwise. The procedure of derivations (see also Suh & Radcliffe, 1978, Shabana, 1989, and others) is as follows. Step 1: We represent ρa by the equation (Fig ) ρ a = OM + MN + NA (1.3.5)

8 8 Coordinate Transformation where OM = (c a ρ a )c a = (c a ρ a )c a (1.3.6) and c a is the unit vector of the axis of rotation that is represented in S a. Step 2: Vector ρ a is represented by the equation that yields ρ a = OM + MA = (c a ρ a )c a + MA (1.3.7) MA = ρ a (c a ρ a )c a. (1.3.8) We emphasize that a vector being rotated about c generates a cone with an apex angle α. Thus, both vectors, ρ and ρ, are the generatrices of the same cone, as shown in Fig Step 3: Vector MN has the same direction as MA and this yields MN = MA cos φ = MA cos φ = ρ sin α cos φ (1.3.9) where α is the apex angle of the generated cone, MA =ρ sin α, and ρ is the magnitude of ρ. Equations (1.3.8) and (1.3.9) yield MN = MN MA MA = [ρ a (c a ρ a )c a ] cos φ. (1.3.10) Step 4: Vector NA has the same direction as (c a ρ a ) and may be represented by Here, NA = c a ρ a c a ρ a NA =sin φ(c a ρ a ). (1.3.11) NA = MA sin φ = ρ sin α sin φ, c a ρ a = ρ sin α. Step 5: Equations (1.3.5), (1.3.6), (1.3.10), and (1.3.11) yield ρ a = ρ a cos φ + (1 cos φ)(c a ρ a )c a + sin φ(c a ρ a ). (1.3.12) Step 6: It is easy to prove that because (c a ρ a )c a = c a (c a ρ a ) + ρ a (1.3.13) c a (c a ρ a ) = (c a ρ a )c a ρ a (c a c a ). Step 7: Equations (1.3.12) and (1.3.13) yield ρ a = ρ a + (1 cos φ)[c a (c a ρ a )] + sin φ(c a ρ a ). (1.3.14) Equation (1.3.14) is known as the Rodrigues formula. According to the investigation by Cheng & Gupta [1989], this equation deserves to be called the Euler Rodrigues, formula.

9 1.3 Rotation About an Axis 9 Step 8: Additional derivations are directed at representation of the Euler Rodrigues formula in matrix form. The cross product (c a ρ a ) may be represented in matrix form by c a ρ a = C s ρ a (1.3.15) where C s is the skew-symmetric matrix represented by 0 c 3 c 2 C s = c 3 0 c 1. (1.3.16) c 2 c 1 0 Vector c a is represented by c a = c 1 i a + c 2 j a + c 3 k a. (1.3.17) Step 9: Equations (1.3.14), (1.3.15), and (1.3.16) yield the following matrix representation of the Euler Rodrigues formula: ρ a = [ I + (1 cos φ)(c s ) 2 + sin φc s ] ρ a = L a ρ a (1.3.18) where I is the 3 3 identity matrix. While deriving Eqs. (1.3.14) and (1.3.18), we assumed that the rotation is performed counterclockwise. For the case of clockwise rotation, it is necessary to change the sign preceding sin φ to its opposite. The expression for matrix L a that will cover two directions of rotation is L a = I + (1 cos φ)(c s ) 2 ± sin φc s. (1.3.19) The upper sign preceding sin φ corresponds to counterclockwise rotation and the lower sign corresponds to rotation in a clockwise direction. In both cases the unit vector c must be expressed by the same Eq. (1.3.17) that determines the orientation of c but not the direction of rotation. The direction of rotation is identified with the proper sign preceding sin φ in Eq. (1.3.19). Problem 2. Recall that our goal is to derive the operator L ba in matrix equation (1.3.2) that transforms components of the same vector (see Eqs. (1.3.3) and (1.3.4)). It will be shown below that the sought-for operator is represented as L ba = L T a = I + (1 cos φ)(cs ) 2 sin φc s. (1.3.20) Operator L ba can be obtained from operator L a given by Eq. (1.3.19) by changing the sign of the angle of rotation, φ. The upper and lower signs preceding sin φ in Eq. (1.3.20) correspond to the cases where S a will coincide with S b by rotation counterclockwise and clockwise, respectively. The proof is based on the determination of components of the same vector, say vector OA shown in Fig , in coordinate systems S a and S b. Step 1: We consider initially that vector OA is represented in S a as ρ a = [a 1 a 2 a 3 ] T. (1.3.21) Step 2: To determine components of vector OA in S b we consider first that coordinate system S b and the previously mentioned position vector are rotated as one rigid body

10 10 Coordinate Transformation about c. After rotation through angle φ, position vector OA will take the position OA and can be represented in S b as OA = a 1 i b + a 2 j b + a 3 k b. (1.3.22) It is obvious that vector OA has in S b the same components as vector OA has in S a. Step 3: We consider now in S b two vectors OA and OA. Vector OA will coincide with OA after clockwise rotation about c. The components of vectors OA and OA in S b are related by an equation that is similar to Eq. (1.3.19). The difference is that we now have to consider that the rotation from OA to OA is performed clockwise. Then we obtain (OA) b = L b (OA ) b = [ I + (1 cos φ)(c s ) 2 sin φc s ] (OA ) b. (1.3.23) Designating components of (OA) b by [b 1 b 2 b 3 ] T, we receive [b 1 b 2 b 3 ] T = [I + (1 cos φ)(c s ) 2 sin φc s ][a 1 a 2 a 3 ] T. (1.3.24) Step 4: We have now obtained components of the same vector OA in coordinate systems S a and S b, respectively. The matrix equation that describes transformation of components of OA is (OA) b = L ba (OA) a. (1.3.25) For the case in which rotation from S a to S b is performed counterclockwise we have obtained that L ba = I + (1 cos φ)(c s ) 2 sin φc s. (1.3.26) Similarly, for the case in which rotation from S a to S b is performed clockwise, we obtain L ba = I + (1 cos φ)(c s ) 2 + sin φc s. (1.3.27) The general description of operator L ba and the respective coordinate transformation are as follows: ρ b = L ba ρ a = [ I + (1 cos φ)(c s ) 2 sin φc s ] ρ a. (1.3.28) The upper and lower signs preceding sin φ correspond to the cases in which rotation from S a to S b is performed counterclockwise and clockwise, respectively. In our identification of coordinate systems S a and S b we do not use the terms fixed and movable. We just consider that S a is the previous coordinate system and S b is the new one, and we take into account how the rotation from S a to S b is performed: counterclockwise or clockwise. Matrix L ba Using Eqs. (1.3.26) and (1.3.27), we may represent elements of matrix L ba in terms of components of unit vector c of the axis of rotation and the angle of rotation φ. Thus,