Elements of Coordinate System Transformations

Transcription

1 B Elements of Coordinate System Transformations Coordinate system transformation is a powerfl tool for solving many geometrical and kinematic problems that pertain to the design of gear ctting tools and the kinematics of gear machining processes. Conseqent coordinate system transformations can easily be described analytically with the implementation of matrices. The se of matrices for coordinate system transformation can be traced back to the late 1940s [41], early 1950s [11, 42 and others]. The implementation of coordinate system transformations is necessary for representation in a common coordinate system a) of the gear ctting tool, and b) of its motion relative to the tooth flank of the work-gear. At every instant of time, the configration position and orientation) of the gear ctting tool relative to the work-gear can be described analytically with the help of a homogeneos transformation matrix corresponding to the displacement of the ctting tool from its crrent location to a certain consective location. B.1 Coordinate System Transformation In this text, coordinate system transformations are discssed briefly from the standpoint of their implementation for the prposes of gear ctting tool design. The interested reader may wish to go to [66, 70, 73, 74, 80] and other advanced sorces for details. B.1.1 Introdction Homogenos coordinates tilize a mathematical trick to embed three-dimensional coordinates and transformations into a for-dimensional matrix format. As a reslt, inversions or combinations of linear transformations are simplified to inversion or mltiplication of the corresponding matrices. Geometry of Srfaces: A Practical Gide for Mechanical Engineers, First Edition. Stephen P. Radzevich John Wiley & Sons, Ltd. Pblished 2013 by John Wiley & Sons, Ltd.

2 212 Appendix B: Elements of Coordinate System Transformations Homogenos coordinate vectors. Instead of representing each point rx, y, z) in threedimensional space with a single three-dimensional vector x r = y z B.1) homogenos coordinates allow each point rx, y, z) to be represented by any of an infinite nmber of for-dimensional vectors T x r = T y T z T B.2) The three-dimensional vector corresponding to any for-dimensional vector can be compted by dividing the first three elements by the forth, and a for-dimensional vector corresponding to any three-dimensional vector can be created by simply adding a forth element and setting it eqal to one. Homogenos coordinate transformation matrices of the dimension 4 4. Homogenos coordinate transformation matrices operate on for-dimensional homogenos vector representations of traditional three-dimensional coordinate locations. Any three-dimensional linear transformation translation, rotation, etc.) can be represented by a 4 4 homogenos coordinate transformation matrix. In fact, becase of the redndant representation of three-space in a homogenos coordinate system, an infinite nmber of different 4 4 homogenos coordinate transformation matrices is available to perform any given linear transformation. This redndancy can be eliminated to provide a niqe representation by dividing all elements of a 4 4 homogenos transformation matrix by the last element which will become eqal to one). This means that a 4 4 homogenos transformation matrix can incorporate as many as 15 independent parameters. The generic format representation of a homogenos transformation eqation for mapping the three-dimensional coordinate x 1, y 1, z 1 ) to the three-dimensional coordinate x 2, y 2, z 2 )is T x 2 T a T b T c T d T x 2 T y 2 T z 2 = T e T f T g T h T i T j T k T m T y 2 T z 2 T T n T p T q T T B.3) If any two matrices or vectors of this eqation are known, the third matrix or vector) can be compted and then the redndant T element in the soltion can be eliminated by dividing all elements of the matrix by the last element. Varios transformation models can be sed to constrain the form of the matrix to transformations with fewer degrees of freedom.

3 Appendix B: Elements of Coordinate System Transformations 213 a x a 2 a z X1 X2 a) b) c) Figre B.1 Analytical description of the operators of translation Tr a x, X), Tr a y, Y ), Tr a z, Z) along the coordinate axes. B.1.2 Translations Translation of a coordinate system is one of the major linear transformations sed for the prposes of gear ctting tool design. Translations of the coordinate system along the axes of the coordinate system are illstrated in Fig. B.1. The translations can be described analytically by the homogenos transformation matrices of dimension 4 4. For the analytical description of translation along coordinate axes, the operators of translation Tr a x, X), Tr a y, Y ) and Tr a z, Z) are sed. The operators yield matrix representation in the form a x Tr a x, X) = B.4) Tr a y, Y ) = a y B.5) Tr a z, Z) = a z B.6) Here, a x, a y, a z are signed vales that denote distances of translations along corresponding axes. Consider the two coordinate systems and shifted along the axis on a x [Fig. B.1a)]. Let a point m in the coordinate system be given by the position vector r 2 m). In the coordinate system that same point m can be specified by the position vector r 1 m). Then, the position vector r 1 m) can be expressed in terms of the position vector

4 214 Appendix B: Elements of Coordinate System Transformations ϕ x ϕ y ϕ z a) b) c) Figre B.2 Analytical description of the operators of rotation Rt ϕ x, X), Rt ϕ y, Y ), Rt ϕ z, Z) abot the coordinate axes. r 2 m) by the eqation r 1 m) = Tr a x, X) r 2 m). Eqations similar to that above are valid for other operators Tr a y, Y ) and Tr a z, Z) of the coordinate system transformation [Fig. B.1b, c)]. Any coordinate system transformation that doesn t change the orientation of a geometrical object is an orientation-preserving transformation, or a direct transformation. Therefore, the transformation of translation is an example of a direct transformation. B.1.3 Rotation abot a coordinate axis Rotation of a coordinate system abot a coordinate axis is another kind of major linear transformation sed for the prpose of gear ctting tool design. Rotation of the coordinate system abot the axis of the coordinate system is illstrated in Fig. B.2. For an analytical description of rotation abot coordinate axes, the operators of rotation Rt ϕ x, X), Rt ϕ y, Y ) and Rt ϕ z, Z) are sed. The operators yield representation in the form of homogenos matrices Rt ϕ x, X) = 0 cos ϕ x sin ϕ x 0 0 sin ϕ x cos ϕ x 0 cos ϕ y 0 sin ϕ y 0 Rt ϕ y, Y ) = sin ϕ y 0 cos ϕ y 0 cos ϕ z sin ϕ z 0 0 Rt ϕ z, Z) = sin ϕ z cos ϕ z B.7) B.8) B.9)

5 Appendix B: Elements of Coordinate System Transformations 215 Here, ϕ x, ϕ y, ϕ z are signed vales that denote angles of rotation abot the corresponding axis: ϕ x is rotation arond the X-axis pitch); ϕ y is rotation arond the Y -axis roll), and ϕ z is rotation arond the Z-axis yaw). Consider two coordinate systems and trned abot the -axis throgh the angle ϕ x [Fig. B.2a)]. In the coordinate system, a certain point m is given by the position vector r 2 m). In the coordinate system, that same point m can be specified by the position vector r 1 m). Then, the position vector r 1 m) can be expressed in terms of the position vector r 2 m) by the eqation r 1 m) = Rt ϕ x, X) r 2 m). Eqations similar to that above are valid for other operators Rt ϕ y, Y ) and Rt ϕ z, Z) of the coordinate system transformation [Fig. B.2b, c)]. B.1.4 Resltant coordinate system transformation The operators of translation Tr a x, X), Tr a y, Y ), Tr a z, Z) together with the operators of rotation Rt ϕ x, X), Rt ϕ y, Y ), Rt ϕ z, Z) are sed to compose the operator Rs 1 2) of the resltant coordinate system transformation. The operator Rs 1 2) of the resltant coordinate system transformation describes analytically the transition from the initial coordinate system to a certain coordinate system. Consider three conseqent translations along the coordinate axes, and. Sppose that a point P on a rigid body goes throgh a translation describing a straight path from P 1 to P 2 with a change of coordinates a x, a y, a z ). This motion can be described with an operator of the resltant coordinate system transformation Rs 1 2), which can be expressed in terms of the operators Tr a x, X), Tr a y, Y ), Tr a z, Z) of elementary coordinate system transformations. The operator Rs 1 2) is eqal to a x Rs 1 2) = Tr a z, Z) Tr a y, Y ) Tr a x, X) = a y a z B.10) In this particlar case, the operator of the resltant coordinate system transformation Rs 1 2) can be interpreted as the operator Tr a, A) of translation along an axis A [Rs 1 2) = Tr a, A)]. Evidently, the axis A is always an axis throgh the origin. Similarly, three conseqent rotations abot coordinate axes can be described with another operator of the resltant coordinate system transformation Rs 1 2) Rs 1 2) = Rt ϕ z, Z) Rt ϕ y, Y ) Rt ϕ x, X) B.11) In this particlar case, the operator of the resltant coordinate system transformation Rs 1 2) can be interpreted as the operator Rt ϕ,a) of rotation abot an axis A [Rs 1 2) = Rt ϕ,a)]. Evidently, the axis A is always an axis throgh the origin. In practice, it is often necessary to perform coordinate system transformations comprising translations along and rotations abot coordinate axes. For example, the expression Rs 1 5) = Tr a x, X) Rt ϕ z, Z) Rt ϕ x, X) Tr a y, Y ) B.12)

6 216 Appendix B: Elements of Coordinate System Transformations a x1 Z 4 Z 5 a y4 Z 3 X1, X2 Y 5 Y 4 Z 3 X 5 X 4 Z 4 Y 5 a x1 r 1 M ) M a y4 Z 5 r 5 M ) X 5 ϕ y3 Y 3 ϕ z2 Y 3 X 3 X 3 Y 4 X 4 Figre B.3 Example of the resltant coordinate system transformation. indicates that the transition from the coordinate system to the coordinate system X 5 Y 5 Z 5 Fig. B.3) is performed in the following for steps: a) translation Tr a y, Y ), followed by b) rotation Rt ϕ x, X), followed by c) second rotation Rt ϕ z, Z),followedbyd) translation Tr a x, X). Ultimately, the eqality r 1 M) = Rt 5 1) r 5 M) is observed. When the operator Rs 1 t) of a resltant coordinate system transformation is known, then the transition in the opposite direction can be performed by means of the operator Rs t 1) of the inverse coordinate system transformation. The eqality Rs t 1) = Rs 1 1 t) B.13) is valid for the operator Rs t 1) of the inverse coordinate system transformation. B.1.5 Screw motion abot a coordinate axis Operators for analytical description of screw motions abot an axis of the Cartesian coordinate system are a particlar case of operators of the resltant coordinate system transformation. By definition Fig. B.4), the operator Sc x ϕ x, p x ) of a screw motion abot the X-axis of the Cartesian coordinate system XYZ is eqal to Sc x ϕ x, p x ) = Rt ϕ x, X) Tr a x, X) B.14)

7 Appendix B: Elements of Coordinate System Transformations 217 ax = px ϕx X1 X2 ϕ x Figre B.4 Analytical description of the operator of screw motion Sc x ϕ x, p x ). After sbstitting the operator of translation Tr a x, X) [Eq. B.4)] and the operator of rotation Rt ϕ x, X) [Eq. B.7)], Eq. B.14) casts into the expression p x ϕ x Sc x ϕ x, p x ) = 0 cos ϕ x sin ϕ x 0 0 sin ϕ x cos ϕ x 0 B.15) for comptation of the operator of the screw motion Sc x ϕ x, p x ) abot the X-axis. The operators of screw motion Sc y ϕ y, p y ) and Sc z ϕ z, p z ) abot the Y - and Z-axis correspondingly are defined in a way similar to that above. For the operator of screw motion Sc x ϕ x, p x ): Sc y ϕ y, p y ) = Rt ϕ y, Y ) Tr a y, Y ) Sc z ϕ z, p z ) = Rt ϕ z, Z) Tr a z, Z) B.16) B.17) Using Eqs B.5) and B.6) together with Eqs B.8) and B.9), one can come p with the expressions cos ϕ y 0 sin ϕ y 0 Sc y ϕ y, p y ) = p y ϕ y sin ϕ y 0 cos ϕ y 0 cos ϕ z sin ϕ z 0 0 Sc z ϕ z, p z ) = sin ϕ z cos ϕ z p z ϕ z B.18) B.19) for comptation of the operators of screw motion Sc y ϕ y, p y ) and Sc z ϕ z, p z ) abot the Y - and Z-axis.

8 218 Appendix B: Elements of Coordinate System Transformations ax = Rw ϕy R w V ϕ y ω Figre B.5 Illstration of the transformation of rolling Rl x ϕ y, Y ) of a coordinate system. Screw motions abot a coordinate axis, as well as screw srfaces, are common in the design of gear ctting tools. This makes practical se of the operators of screw motion Sc x ϕ x, p x ), Sc y ϕ y, p y ) and Sc z ϕ z, p z ) when designing gear ctting tools. If necessary, an operator of the screw motion abot an arbitrary axis either throgh the origin of the coordinate system or not throgh the origin of the coordinate system can be derived in a similar way as the operators Sc x ϕ x, p x ), Sc y ϕ y, p y ) and Sc z ϕ z, p z ). B.1.6 Rolling motion of a coordinate system One more practical combination of a rotation and a translation is often sed when designing a gear ctting tool. Consider a Cartesian coordinate system Fig. B.5). The coordinate system is traveling in the direction of the -axis. The speed of translation is denoted V. The coordinate system is rotating abot the -axis simltaneosly with the translation. The speed of rotation is denoted ω. Assme that the ratio V / ω is constant. Under sch a scenario, the resltant motion of the reference system to its arbitrary position allows interpretation in the form of rolling with no sliding of a cylinder of radis R w over the plane. The plane is parallel to the coordinate -plane, and remote from it at a distance R w. For comptation of the radis of the rolling cylinder, the expression R w = V / ω can be sed. Owing to the rolling of the cylinder of radis R w over the plane being performed with no sliding, a certain correspondence between the translation and rotation of the coordinate system is established. When the coordinate system trns throgh a certain angle ϕ y, the translation of the origin of the coordinate system along the -axis is eqal to a x = ϕ r R w. The transition from the coordinate system to the coordinate system can be described analytically by the operator of the resltant coordinate system transformation Rs 1 2), where Rs 1 2) = Rt ϕ y, ) Tr a x, ) B.20) Here, Tr a x, ) designates the operator of translation along the -axis, and Rt ϕ y, ) is the operator of rotation abot the -axis. Operators of the reslting coordinate system transformation of this kind are referred to as operators of rolling motion over a plane.

9 Appendix B: Elements of Coordinate System Transformations 219 When the translation is performed along the -axis, and the rotation is performed abot the -axis, the operator of rolling is denoted Rl x ϕ y, Y ). In this particlar case, the eqality Rl x ϕ y, Y ) = Rs 1 2) [see Eq. B.20)] is valid. Based on this eqality, the operator of rolling over a plane Rl x ϕ y, Y ) can be compted from the eqation cos ϕ y 0 sin ϕ y a x cos ϕ y Rl x ϕ y, Y ) = sin ϕ y 0 cos ϕ y a x sin ϕ y B.21) While rotation remains abot the -axis, translation can be performed not along the -axis bt along the -axis instead. For rolling of this kind, the operator of rolling is eqal to cos ϕ y 0 sin ϕ y a z sin ϕ y Rl z ϕ y, Y ) = sin ϕ y 0 cos ϕ y a z cos ϕ y B.22) For cases when the rotation is performed abot the -axis, the corresponding operators of rolling are Rl y ϕ x, X) = 0 cos ϕ x sin ϕ x a y cos ϕ x 0 sin ϕ x cos ϕ x a y sin ϕ x B.23) for the case of rolling along the -axis and Rl z ϕ x, X) = 0 cos ϕ x sin ϕ x a z sin ϕ x 0 sin ϕ x cos ϕ x a z cos ϕ x for the case of rolling along the -axis. Similar expressions can be derived for the case of rotation abot the -axis cos ϕ z sin ϕ z 0 a x cos ϕ z Rl x ϕ z, Z) = sin ϕ z cos ϕ z 0 a x sin ϕ z cos ϕ z sin ϕ z 0 a y sin ϕ z Rl y ϕ z, Z) = sin ϕ z cos ϕ z 0 a y cos ϕ z B.24) B.25) B.26)

10 220 Appendix B: Elements of Coordinate System Transformations ϕ 1 * * ϕ 2 R 1 * R 2 * O 1 P O 2 ϕ 2 ω 1 ω 2 C Figre B.6 Derivation of the operator of rolling Rr ϕ 1, ) of two coordinate systems. Use of the operators of rolling [Eqs B.21) throgh B.26)] significantly simplifies analytical description of the coordinate system transformations. B.1.7 Rolling of two coordinate systems When designing a gear ctting tool, combinations of two rotations abot parallel axes are of particlar interest. As an example, consider two Cartesian coordinate systems and as shown in Fig. B.6. The coordinate systems and are rotated abot their axes and. The axes of rotation are parallel to each other ). Rotations ω 1 and ω 2 of the coordinate systems can be interpreted so that a circle of a certain radis R 1 associated with the coordinate system is rolling with no sliding over a circle of corresponding radis R 2 associated with the coordinate system. When the centerdistance C is known, the radii R 1 and R 2 of the circles can be expressed in terms of the center-distance C and the given rotations ω 1 and ω 2. For the comptations, the formlae 1 R 1 = C 1 + R 2 = C 1 + B.27) B.28) can be sed. Here, the ratio ω 1 /ω 2 is denoted. In the initial configration, the - and -axes align with each other. The - and - axes are parallel to each other. In Fig. B.6, the initial configration of the coordinate systems and is labeled X 1 Y 1 Z 1 and X 2 Y 2 Z 2. When the coordinate system trns throgh a certain angle ϕ 1, the coordinate system trns throgh the corresponding angle ϕ 2. When the angle ϕ 1 is known, the corresponding angle ϕ 2 is eqal to ϕ 2 = ϕ 1 /.

11 Appendix B: Elements of Coordinate System Transformations 221 The transition from the coordinate system to the coordinate system can be described analytically by the operator of the resltant coordinate system transformation Rs 1 2). In the case nder consideration, the operator Rs 1 2) can be expressed in terms of the operators of the elementary coordinate system transformations Rs 1 2) = Rt ϕ 1, ) Rt ϕ 1 /, ) Tr C, ) B.29) Other eqivalent combinations of the operators of elementary coordinate system transformations can reslt in the same operator Rs 1 2) of the resltant coordinate system transformation. The interested reader may wish to derive the eqivalent expressions for the operator Rs 1 2). The operator of the resltant coordinate system transformation [see Eq. B.29)] is referred to as an operator of rolling motion over a cylinder. When rotations are performed arond the - and -axes, the operator of rolling motion over a cylinder is denoted Rr ϕ 1, ). In this particlar case, the eqality Rr ϕ 1, ) = Rs 1 2) [see Eq. B.29)] is valid. Based on this eqality, the operator of rolling Rr ϕ 1, ) over a cylinder can be compted from the eqation Rr ϕ 1, ) = sin cos ϕ ) ϕ ) sin ϕ ) cos ϕ ) 0 C B.30) For the inverse transformation, the inverse operator of rolling of two coordinate systems Rr ϕ 2, ) can be sed. It is eqal to Rr ϕ 2, ) = Rr 1 ϕ 1, ). In terms of operators of the elementary coordinate system transformations, the operator Rr ϕ 2, ) can be expressed as Rr ϕ 2, ) = Rt ϕ 1 /, ) Rt ϕ 1, ) Tr C, ) B.31) Other eqivalent combinations of the operators of elementary coordinate system transformations can reslt in the same operator Rr ϕ 2, ) of the resltant coordinate system transformation. The interested reader may wish to derive the eqivalent expressions for the operator Rr ϕ 2, ). For the comptation of the operator of rolling of two coordinate systems Rr ϕ 2, ), the eqation can be sed. cos Rr ϕ 2, ) = ϕ sin ϕ ) ) sin ϕ ) cos ϕ ) 0 C B.32)

12 222 Appendix B: Elements of Coordinate System Transformations Similar to the expression [see Eq. B.30)] derived for the comptation of the operator of rolling Rr ϕ 1, ) arond the - and -axes, corresponding formlae can be derived for the comptation of the operators of rolling Rr ϕ 1, ) and Rr ϕ 1, ) abot parallel axes and, as well as abot parallel axes and. Use of the operators of rolling abot two axes Rr ϕ 1, ), Rr ϕ 1, ) and Rr ϕ 1, ) sbstantially simplifies analytical description of the coordinate system transformations. B.2 Conversion of the Coordinate System Orientation Application of the matrix method of coordinate system transformation presmes that both coordinate systems i and i ± 1) are of the same hand. This means that it is assmed from the very beginning that both of them are either right-hand oriented or left-hand oriented Cartesian reference systems. In the event the coordinate systems i and i ± 1) are of opposite hand, one of the coordinate systems needs to be converted into an oppositely oriented Cartesian coordinate system. For conversion of a left-hand oriented Cartesian coordinate system into a right-hand oriented coordinate system and vice versa, operators of reflection are sed. In order to change the direction of the X i -axis of the initial coordinate system i to the opposite direction in this case, in the new coordinate system i ± 1) the eqalities X i±1 = X i, Y i±1 Y i and Z i±1 Z i are observed), the operator of reflection Rf x Y i Z i ) can be applied. The operator of reflection yields representation in matrix form as Rf x Y i Z i ) = B.33) Similarly, implementation of the operators of reflection Rf y X i Z i ) and Rf z X i Y i ) reslts in the directions of the Y i - and Z i -axes being reversed. The operators of reflection Rf y X i Z i ) and Rf z X i Y i ) in this case can be expressed analytically in matrix form as Rf y X i Z i ) = Rf z X i Y i ) = B.34) B.35) A linear transformation that reverses the direction of the coordinate axis is an opposite transformation. The transformation of reflection is an example of an orientation-reversing transformation.

13 Appendix B: Elements of Coordinate System Transformations 223 B.3 Transformation of Srface Fndamental Forms Every coordinate system transformation reslts in a corresponding change to the eqation of the gear tooth srface G and/or the generating srface T of the gear ctting tool. Becase of this, it is necessary to recalclate the coefficients of the first 1.g and of the second 2.g fndamental of the srfaces G as many times as the coordinate system transformation is performed. This rotine and time-consming operation can be eliminated if the operators of coordinate system transformations are sed directly on the fndamental forms 1.g and 2.g. After being compted in an initial coordinate system, the fndamental magnitdes E g, F g, G g, L g, M g and N g of the forms 1.g and 2.g can be determined in any new coordinate system sing for this prpose the operators of translation, rotation and reslting coordinate system transformation. Sch transformations of the fndamental magnitdes 1.g and 2.g become possible de to implementation of the formlae below. Consider a gear tooth srface G that is given by the eqation r g = r g U g, V g ), where U g, V g ) G. For convenience, the first fndamental form 1.g of the gear tooth srface G is represented in matrix form [66, 74] as E g F g 0 0 du g [ ] [ 1.g = dug dv g 0 0 ] F g G g dv g 0 0 B.36) Similarly, the eqation of the second fndamental form 2.g of the srface G can be given by the eqation L g M g 0 0 du g [ ] [ 2.g = dug dv g 0 0 ] M g N g dv g 0 0 B.37) The coordinate system transformation with the operator of the resltant linear transformation Rs 1 2) transfers the eqation r g = r g U g, V g ) of the gear tooth srface G, initially given in, to the eqation r g = r g U g, V g ) of that same srface P in a new coordinate system. It is clear that the position vector of a point of the tooth flank point G in the first reference system differs from the position vector of that same point in the second reference system i.e. r g r g ). The operator of the reslting linear transformation Rs 1 2) of the srface P having first 1.g and second 2.g fndamental forms from the initial coordinate system to the new coordinate system reslts in new fndamental forms expressed in the form [66, 73, 74] [ 1.g ] = Rs T 1 2) [ 1.g ] Rs 1 2) B.38) [ 2.g ] = Rs T 1 2) [ 2.g ] Rs 1 2) B.39) Eqations B.38) and B.39) reveal that after the coordinate system transformation is complete, the first 1.g and second 2.g fndamental forms of the srface G in the coordinate system

14 224 Appendix B: Elements of Coordinate System Transformations are expressed in terms of the first 1.g and second 2.g fndamental forms initially represented in the coordinate system. In order to convert the fndamental forms 1.g and 2.g to the new coordinate system, either the corresponding fndamental form 1.g or 2.g needs to be pre-mltiplied by Rs 1 2) and after that post-mltiplied by Rs T 1 2). Implementation of Eqs B.38) and B.39) significantly simplifies the formlae of transformations.eqations [ 1.c ] = Rs T 1 2) [ 1.c ] Rs 1 2) B.40) [ 2.c ] = Rs T 1 2) [ 2.c ] Rs 1 2) B.41) similar to those above are valid for the generating srface T of the gear ctting tool. Implementation of the elements of screw calcls for the prpose of coordinate system transformation is a possible way to enhance the approach. In this case, screw operators of jst one kind can be applied for all kinds of coordinate system transformations.