2. Preliminaries. x 2 + y 2 + z 2 = a 2 ( 1 )

Transcription

1 x 2 + y 2 + z 2 = a 2 ( 1 ) V. Kumar 2. Preliminaries 2.1 Homogeneous coordinates When writing algebraic equations for such geometric objects as planes and circles, we are used to writing equations that are not homogenous in all the coordinates. For example, consider a coordinate system O-x-y-z in three-dimensional Euclidean space, and the equation of a sphere of radius a with the center at O: The degree of the first three terms in the equation in x, y, and z is equal to 2, while the degree of the last term is equal to 0. Similarly, consider a plane with the normal s ti + uj + vk and let its distance from the origin, O, be plane in this space is: t 2 +u 2. The equation of the 2 + v tx + uy + vz + s = 0 ( 2 ) Note that the first three terms are of degree 1 and the fourth of degree 0. In both cases, the equations are not homogeneous. There are two difficulties with this [KHH 78]. First, the nonhomogeneity can cause inconsistencies in derivations and yield spurious results especially when problems of intersections are solved. Second, x, y, and z are non homogeneous coordinates and they do not offer a good representation of geometric objects that are located very far away from the origin. We introduce the idea of homogeneous coordinates to overcome these two difficulties, although, as we shall see, the main advantage for us is that we will be able to develop a compact representation for transforming points and planes from one coordinate system to another. The difference between homogeneous and nonhomogeneous coordinates is best illustrated by the simple example of the plane in Equation ( 2 ). The equation can be made homogeneous if we describe the point in the projective three-space denoted by P 3.

2 Essentially we use four real coordinates (x, y, z, w) to represent a point in P 3, with the following two conditions [SR 49]: (a) Not all coordinates can be zero we exclude the point (0, 0, 0, 0); and (b) There is an equivalence relationship defined by: (x, y, z, w) ~ (x, y, z, w ) if there is a nonzero real number α such that x = αx, y = αy, z = αz, w = αw. Thus, a point (x, y, z) can be represented by a vector r = [ xyzw,,, ] T in R 4. The absolute values of the four coordinates are not significant. Instead, it is the three ratios (for example, x/w, y/w, and z/w) that are important because (x, y, z, w) ~ (x/w, y/w, z/w, 1) provided (1/w) =/ 0. Thus we can think of the standard Euclidean space as being the plane w = 1 in the projective three-space. Now, consider the plane in Equation ( 2 ) given by (t, u, v, s). If we represent it by a vector, π=[, tuvs,,] T, the equation of the plane is given by: x π T y r = [ t u v s] = 0 z w ( 3 ) This equation represents the incidence of the point r on the plane π. It is also worth noting that the vector π is the representation of the plane in homogeneous coordinates. y multiplying the vector by any non zero real scalar, we get the same plane - απ (α=/ 0) and π are the same plane. In fact this parallel between points and planes extends further. Equation ( 3 ) represents all points on the plane π and is homogeneous in (x,y,z,w). It also represents all planes through a given point r, in which case, it is homogeneous in (t,u,v,s). The plane w=0 gives us all the points at infinity. Similarly, the plane s=0 gives us all the planes containing the origin. The coordinates (x, -2-12/31/01

3 y, z) yield the direction cosines of a line from the origin to the point (x, y, z, w) while (t,u,v) are the direction ratios of a line perpendicular to the plane (t, u, v, s). Finally, the distance of a point (x, y, z, w) from the origin is x + y + z w 2, while the distance of a plane (t, u, v, s) from the origin is s t + u + v. In conclusion it is worth noting there is an entire branch of geometry called projective geometry that deals with such geometric relationships as incidence and colinearity. In fact, projective geometry is more basic than Euclidean geometry in the sense that it does not require the definition of a metric, and therefore is not based on such concepts as lengths and angles [WM 86]. Homogeneous coordinates are fundamental to the study of projective geometry. 2.2 Transformations and displacements Coordinate transformations Consider the frame {} attached to a body, and a reference frame {} that is attached to the body as shown in Figure 1. We are interested in relating the position vector of any point in {} to the position vector of the same point in {}. Thus we want to define a coordinate transformation that transforms the components of a position vector or coordinates measured in {} to those measured in {}. Since such a transformation must transform points into points and lines into lines, and further, preserve parallelism, it must be affine. In other words, if we consider a generic point P with position vectors r P and r P in reference frames {} and {} respectively, the transformation must have the form [R 90]: r = X r+ y ( 4 ) -3-12/31/01

4 r P and r P are n 1 vectors of coordinates of the same generic point P in reference frames {} and {}, X is a n n matrix, and y is a n 1 vector. This works for planar and spatial transformations - n = 2 for planar kinematics and n = 3 for spatial kinematics. The leading superscript against a vector indicates the reference frame in which it is measured. When necessary, a trailing superscript is used with a position vector to explicitly indicate the point that it refers to. z' z {} r P P y' r P O' r y O x' x Figure 1 The two rigid bodies, and, and the two body fixed reference frames, {} and {}. y letting the origin of frame {}, O, be the generic point, it is easy to see that the vector y must be the position vector of the orgin O of {}in the reference frame {}. In other words, y = r O. ( 5 ) y considering the special case in which the origins of the two frames coincide, it is clear that X models the orientation of the reference frame {} with respect to the reference frame {}. ccordingly, we call it a rotation matrix with the notation R. The leading superscript and trailing subscript in R show that the n n matrix rotates position vectors in {} into position vectors in {} which when added to r O yields the correct result. Thus, Equation ( 4 ) can be written in the form: -4-12/31/01

5 P P O r = R r + r ( 6 ) This convention with superscripts and subscripts for matrices is extensively used, sometimes with minor variations, in used in multibody dynamics, computer graphics and robotics [RP 81]. This rule for the transformation of points attached to a rigid body induces a rule for transformation of vectors attached to a rigid body. Consider a vector u attached to a rigid body and two points P and Q so that the vector going from P to Q is the vector u. Let the points P and Q have position vectors: p = r P and q = r Q in frame {} and p = r P and q = r Q in frame {} respectively. The vector can be measured in frame {} or in frame {} either as u or u. Clearly, u = q- p and u = q- p. ccording to Equation ( 6 ), we can write O p = R p+ r O q= R q+ r Subtracting one equation from the other, it is clear that the vector u satisfies the relationship: ( ) ( ) u = p q = R p q = R u In other words, the components of the vector u can be transformed from u to u by the transformation law for vectors: u = R u ( 7 ) -5-12/31/01

6 There are two properties of the transformation that restrict the form of R. Since T must preserve the rigidity of the body, the distance between any two points p and q (or p and q) must be the same regardless of the coordinate system in which it is measured. The distance between p and q is given by: p q = p q ( R p O r ) ( R q O r ) R ( p q) = + + = Expanding both sides using the standard Euclidean norm for vectors, we get: ( ) T ( ) ( ) T T p q p q = p q [ R ] [ R ]( p q) The equality works for any pair of points if and only if T [ R ] = [ R ] 1 ( 8 ) Such a matrix is called an orthogonal (or an orthonormal) matrix. The determinant of an orthogonal matrix is ±1, since T [ ] [ ] 2 R = R R = 1. nother property of a coordinate transformation is that it preserves the cross product of any two vectors in the following sense. If u and v are vectors attached to the rigid body so that, u = R u, v = R v we want the cross product of u and v to transform into the cross product of u and v: u v = R ( u v) It can be shown that this property in ( 9 ) can only be satisfied by a proper orthogonal matrix. proper orthogonal matrix is an orthogonal matrix R, with R = 1. In fact, an ( 9 ) -6-12/31/01

7 orthogonal matrix with a determinant of +1 is called a rotation matrix. If R = 1, it is called a reflection matrix. Rotation matrices are used to define the orientation of a rigid body, and as we will show later, the set of all rotation matrices form a group Displacements We have seen that Equation ( 6 ) is a coordinate transformation which transforms the position vector of any point in frame {} to the position vector in frame {}. We now show that there is a different interpretation for the same equation. Let us consider two positions a rigid body from the vantage point of a reference frame {}. Let us choose {}, so that the body fixed reference frame coincides with {} at the first instance. s shown in Figure 2, let the body fixed reference frame be coincident with {} at the second position. The point P on the body is displaced to P. ased on Equation ( 6 ), we can write ut since P is fixed to the body, we can write: P P O r = R r + r P P r = r P P O r = R r + r ( 10 ) In other words, the rotation matrix R and the vector r O, describe an operator that displaces a generic point P and therefore any point attached to the body, from its original position to its new position /31/01

8 z' {} z {} r P r P P y' r P P r O O' O y x' x Figure 2 The rigid body displacement of a rigid body from an initial position and orientation to a final position and orientation. The body fixed reference frame is coincident with {} in the initial position and orientation, and with {} in its final position and orientation. The point P attached to the rigid body moves from P to P. This dual interpretation of Equation ( 6 ) suggests that we can think of any transformation from frame {} to frame {}, as a displacement of a reference frame attached to a rigid body relative to the frame {}, from a position and orientation that is initially coincident with the frame {} to the position and orientation of the reference frame {}. nd conversely, given any displacement frame {} to frame {} as shown in Figure 2,, we can immediately associate a coordinate transformation from frame {} to frame {} Homogeneous transformation matrices So far we have used 3 1 position vectors in Equations ( 6 ) and ( 10 ) to represent points attached to rigid bodies. If we use homogeneous coordinates to describe points in the projective space, P 3, the coordinate transformation and the displacement can be represented by a 4 4 matrix: -8-12/31/01

9 P O P r R r w = r w ( 11 ) The 4 4 matrix = O' R 0 r is called a homogeneous transformation matrix because the generic point is described in frame {} and frame {} in homogeneous coordinates. Note that in order for the matrix to have the form shown above, the fourth coordinate w must be the same on both sides. In fact, it is convenient to take w to equal 1. Similarly, the displacement operator that describes the displacement of a generic point attached to a rigid body, in reference frame {}, is given by a 4 4 homogeneous transformation matrix. From Equation ( 10 ), it is evident we can write: P O P r R r w = r w ( 12 ) For all practical purposes, there is no difference between Equations ( 6 ) and ( 11 ), or Equations ( 10 ) and ( 12 ). The homogeneous transformation matrix has the advantage of being more compact, and as we shall see, lends itself to the derivation of properties for the group of rigid body displacements. However, for most symbolic calculations such as those encountered in robot direct and inverse kinematics, it may be convenient to use the form of ( 6 ) where the rotation matrix and the translation vector are distinct. There are two special cases that are of interest. pure rotation of rigid body that leaves the origin of {} fixed is described by the matrix, = R Similarly, a pure translation of the rigid body is given by: -9-12/31/01

10 0 r = O. 2.3 Transformations and displacements with multiple reference frames Composition Consider any three rigid bodies,,, and C, with reference frames {}, {}, and {C} attached to them respectively. If the coordinate transformation from {} to {} is described by the homogeneous transformation matrix,, and the coordinate transformation from {C} to {} is described by the homogeneous transformation matrix, C, then the homogeneous transformation matrix describing the coordinate transformation from {C} to {} is given by the composition of the two transformations: where C = C ( 13 ) = O' R 0 r 1 3 1, O R C r C = and C = O C R 0 r /31/01

11 z' z {} {} y' O' O r O y r O {C} z' x' O C x x' y' Figure 3 Three rigid bodies,,, and C, with reference frames {}, {}, and {C} attached to them. z' z {} POSITION 2 POSITION 1 {} O' y' O y {C} z'' x' O'' x x'' POSITION 3 y'' Figure 4 rigid body is displaced from position 1 to position 2, and then from position 2 to position 3. The body fixed reference reference frame is coincident with {} in position 1, with {} in position 2, and with {C} in position /31/01

12 Similarly, if we consider two consecutive displacements, from position 1 to position 2, and then from position 2 to position 3, the composite displacement (from position 1 to position 3) can be described by multiplying the transformation matrices describing the two displacements. This is shown in Figure 4. The body fixed reference reference frame is coincident with {} in position 1, with {} in position 2, and with {C} in position 3. The first displacement is described by: = O' R 0 r 1 3 1, in frame {}. The second displacement, from position 2 to position 3, is described by: O R C r C = in frame {}. The composite displacement, from position 1 to position 3, is described by: O R C r C = in frame {}. The composition rule in Equation ( 13 ) remains valid. However, in order for this formula to work, notice that the second displacement must be described in the intermediate reference frame, frame {}, while the first displacement and the composite displacement must be described in the first reference frame (frame {}). We have implicitly assumed but it is easy to verify that the product of two homogeneous transformation matrices is also a homogeneous transformation matrix. In Equation ( 13 ), the matrix multiplication of and C yields:,, /31/01

13 O R C r C = 0 1 O O R r R C r = O O R RC R r + r = 0 1 where the refers to the standard multiplication operation between matrices (and vectors). It is clear that, R C = R R C, the product of two proper orthogonal matrices, is another proper orthogonal matrix and therefore another rotation matrix, while, r O = R r O + r O is a 3 1 vector. Thus the composition of and transformation matrix: C yields another homogeneous O O O R C r R RC R r r C = 0 = Using the above formula, it is easy to obtain the formula for the inverse of the displacement. Letting {C} be the same as {}, we find: T 1 [ ] [ ] [ ] = T O R R r 0 Note that the composition operation is not commutative. In general, C is not the same as C. This is quite obvious from the well-known fact that matrix multiplication is not commutative. However, translations are commutative, as are rotations about the same fixed axis /31/01

14 2.4 Similarity Transformations Up to this point in our discussions of coordinate transformations (and displacements) we have considered two frames and described the coordinate transformation (or displacement) in the first frame. In other words, when we discuss the coordinate transformation from {} to {}, we mean the position and orientation of {} in the reference frame {}. In the context of displacements, we are referring to the displacement of a reference frame attached to the object from an initial position (and orientation) coincident with {}, to a final position (and orientation) coincident with {}. Thus, in Figure 5, D is a displacement from frame {} to frame {}, we would describe the displacement using a homogeneous transformation matrix consisting of the rotation matrix describing the orientation of {} in {}, and the position vector of the origin of {} in {}. However, what if we wanted to describe the same physical displacement in another reference frame, say {F}? We would have to find a body fixed reference frame that coincides with frame {F} in position 1, and look at the position and orientation of this reference frame (denoted by {G} in Figure 5). Thus, the same physical displacement can be modeled as two different displacements. It can be described in reference frame {} by, and in reference frame {F} by F G. nd there is no reason to believe that the two homogeneous matrices are the same. There is a natural relationship between and F G in Figure 5. There are two bodyfixed reference frames. The first is displaced from {} to {}, while at the same times, the second from {F}to {G}. Thus, there must be a rigid body transformation that relates {} to {F}, and the same transformation must relate {} to {G}. This transformation is shown by the block arrow T in the figure and is given by the homogeneous transformation matrix, F = G Using the composition rule for matrices, it is clear that: F G = F G Since, /31/01

15 F = G = ( G ) -1 we have the result: F G = F ( F ) -1 ( 14 ) This equation allows us to relate two different descriptions of the same physical displacement. In other words, it transforms the displacement, the description in {}, into the displacement F G, the description in {F}. More generally, such a transformation is called a similarity transformation. It is the transformation rule for any second order tensor from one coordinate system {} to another coordinate system {F}. D z' {} POSITION 2 {} z O' T y' Z {G} POSITION 1 O x' y D Y X x T Z X {F} Y Figure 5 rigid body is displaced from position 1 to position 2 (denoted by the block arrow D). This displacement can be described in reference frame {} by, and in reference frame {F} by F G /31/01

16 We now consider a direct application of ( 14 ) to derive a useful result for the composition of displacements. If we consider three successive positions of the same rigid body as shown in Figure 3, the composition rule given by Equation ( 13 ) gives us the composite displacement from position 1 to position 3. ut as noted earlier, the first displacement (between position 1 and 2) must be described in frame {}, while the second displacement (between position 2 and 3) must be described in a different frame, frame {}. What happens when we want to study two consecutive displacements of a rigid body from a third frame, say an absolute frame, {F}? In order to answer this question, we transform each of the three displacements, C,, and C, to the new frame {F}. s {} is displaced to {}, let {F} be displaced to {G}. Similarly, as {} is displaced to {C}, let {F} be displaced to {H}. F G = F ( F ) -1 F H = F C ( F ) -1 If we consider the composite displacement, {} to {C}, the body fixed frame coincident with {F} at the initial position will go to a position and orientation different from {H}. Let us call this position and orientation {I}. F I = F C ( F ) -1 We can manipulate the three equations to obtain a relationship between the composite displacement F I, and the two successive displacements F G and F H. F I = F C ( F ) -1 = F C ( F ) -1 = F C ( F ) -1 = F ( F ) -1 F C ( F ) -1 F ( F ) -1 = F F H ( F ) -1 = F H F G In other words, /31/01

17 F I = F H F G ( 15 ) We have the surprising result that the order of matrix multiplication required for composing displacements, all described in the same reference frame, is different from the order in Equation ( 13 ). 2.5 Lines z (x,0, z ) 1 1 (0, y, z ) 2 2 y Figure 6 x The four coordinates for a line intersecting the x-z and the y-z planes /31/01

18 z u l ρ n O ρ y Figure 7 x line can be described in terms of a unit vector, u, along the line and the position vector, ρ, of any point on the line. In previous sections, we have discussed the description of points and how we can transform position vectors describing points from one coordinate system to another. In this section, we will focus on a different geometric entity, the line. We will first develop an elegant way of mathematically representing a line, and then discuss the transformation rules for lines. line can be uniquely specified by four independent parameters, i.e., a line has four degrees of freedom. To see this, consider a line that intersects the x-z and the y-z planes. If it does not, we can pick a coordinate system in which it does intersect these planes. The line can be described by its intersection with the two coordinate planes as shown in Figure 6. In this case, ( x 1,z 1,y 2,z 2 ) completely specify the line. y varying any one of the four coordinates, we get a different line, and in this way, we can describe any line in space. There is an alternative description of a line which is more elegant. s shown in Figure 7, let us define u, a vector parallel to the line, l, and ρ, the position vector of any point on l. The two vectors, u and ρ completely specify the line. While u can be specified in terms of three scalar components, note that we can let u be a unit vector so that /31/01

19 u u = 1. Further, we can decompose ρ into ρ = ρ n +αu so that ρ n. u = 0 and α is an appropriate scalar. We can take α to be zero so that ρ = ρ n is the position vector that is perpendicular to the line. The two vectors, u and ρ n, can be used to describe the line and the six scalar components describing these vectors are not independent: u u = 1 ρn. u = 0 ased on this observation, Julius Plücker developed a set of homogeneous coordinates to describe lines. Consider a system of two vectors, the first being a vector parallel to the line, u = [ L, M, N] T, and the second being the moment of the line about the origin, O, given by O [ PQR] u = ρ u = ρ u =,, n Note that u need not be normalized. The set of six numbers [L, M, N, P, Q, R] are called Plücker's coordinates. They are a system of homogeneous coordinates: [L, M, N, P, Q, R] and [α L, α M, α N, α P, α Q, α R] are the same line as long as α =/ 0.. The space of such six-dimensional vectors is the projective five-space, P 5. For every vector in this space, it is the five ratios of the six coordinates (for example, M/L, N/L, P/L, Q/L, and R/L) and not the actual values of the coordinates that are significant. Further, those vectors in P 5 that satisfy O u u = 0 represent a line. In other words, T LP + MQ + NR = 0 ( 16 ) This is called the quadratic identity. y insisting that the Plücker coordinates be normalized so that u = [ L, M, N] T is a unit vector we get a second condition: /31/01

20 L 2 + M 2 + N 2 = 1 ( 17 ) The set of all coordinates that satisfy Equations ( 16 ) and ( 17 ) form a quadric surface in a six-dimensional space. This quadric in P 5 is called the Klein quadric. Each point on the Klein quadric corresponds to a unique line. Similarly, the normalized Plücker coordinates for a line are unique, except for a sign two sets of normalized Plücker coordinates, [L, M, N, P, Q, R] and [-L, -M, -N, -P, -Q, -R] represent the same line. From this point on we will use the term line vector to represent a vector of six coordinates that satisfy Equation ( 16 ), and a unit line vector to mean a line vector that also satisfies Equation ( 17 ). We can arrive at the Plücker coordinates for a line, or the line vector, by knowing the coordinates of any two points on the line. If (x 1, y 1, z 1, w 1 ) and (x 2, y 2, z 2, w 2 ) are two points on the line, the determinants of the 2 2 submatrices of the so-called Grassmanian matrix, w 1 x 1 y 1 z 1 w 2 x 2 y 2 z 2 yield the Plücker line coordinates: w x L = 1 1 w2 x2 M = w1 y1 w2 y2 x y R = 1 1 x2 y2 Q = z1 x1 z2 x2 N = w 1 z 1 w2 z2 y z P = 1 1 y2 z2 ( 18 ) It is instructive to verify this for the special case w 1 = w 2 = w. It is easy to see that /31/01

21 ( LMN,, ) x2 x1 T = wy2 y1 z2 z1 1 w x 1i y 1j z 1k w x 2 x 1 i y 2 y 1 j z 2 z 1 k [ ] T ( PQR,,,) = ( + + ) ( ) + ( ) + ( ) = ( yz 1 2 zy 1 2) ( zx 1 2 z2x1) ( xy x y ) So far we have formed the 6 1 line vector of Plücker line coordinates using the ordering above: [L, M, N; P, Q, R] T. In contrast to this, we will later find it useful to use a different ordering scheme: [P, Q, R; L, M, N,] T. Coordinates ordered this way are often called axis coordinates. The coordinates [L, M, N; P, Q, R] T are called ray coordinates. In both, a semi-colon is often used to separate the first three elements from the last three. If p R 6 represents a line vector using ray coordinates, we can generate p, the line in axis coordinates, using the transformation: p = p ( 19 ) and conversely, p = p where = is a symmetric and idempotent operator, 2 = = I and T = /31/01

22 However, we will avoid possible confusion by staying with the ray coordinate representation for unit line vectors. In the system of homogenous coordinates (x, y, z, w), w=0 with the other three coordinates not all equal to zero describes the set of points at infinity. Similarly, if the Plücker coordinates L=M=N=0, and P, Q, R are not all equal to zero, we get the lines at infinity. In order to see this, consider a line at infinity along a unit vector u. Since the line is infinitely far from the origin, ρ n must be infinitely large and therefore, u O = ρ n u must be infinitely larger than u. If we multiply both vectors by a very small scalar α, in the limit, L, M, and N tend to zero, while P, Q, and R remain finite. This is evident from Equation ( 18 ), If we pick two points on the line at infinity, w 1 =w 2 =w = 0, while the other coordinates, x 1, y 1, z 1, x 2, y 2, z 2 are finite. Therefore the coordinates L=M=N=0, while P, Q, R are finite. line at infinity is characterized by three components and thus behaves like a free vector 1. The coordinates can be normalized so that P 2 +Q 2 +R 2 =1. ( 20 ) Thus, a unit line vector for a line at infinity, is a line vector with L=M=N=0, and P, Q, R satisfying Equation ( 20 ) instead of Equation ( 17 ). However, the direction cosines associated with P, Q and R do not give us the direction of the line. Instead [P, Q, R] represents the moment vector, a vector that is perpendicular to the direction of the line. Thus, a line at that is perpendicular to the x-axis (parallel to the y-z plane), after normalization, has the coordinates [0,0,0;1,0,0] T. Similarly a line at that is perpendicular to the y-axis has all components equal to zero except for Q. We conclude this preliminary discussion on Plücker coordinates with a comment regarding Plücker's views on mechanics. pure force behaves like a line a proper description requires, in addition to its magnitude, its direction cosines and its moment 1 free vector is a vector in R 3 and its components are invariant under changes in the origin. Of course, its components change if the coordinate system is rotated. This is in contrast to a bound vector that is bound to /31/01

23 about a reference point, say the origin O. Similarly, a pure rotational velocity is described by its axis (i.e., a line) in addition to its magnitude. Thus pure forces and pure angular velocities are called line vectors. Such vectors, also called bound vectors, are bound to lines in space a translation of a line vector results in a different line vector. Pure couples and pure translational velocities behave like lines at. They are invariant with respect to changes in origin. In contrast a line vector is not invariant with respect to changes in origin. In summary, forces and angular velocities are line vectors which behave like lines (that are at a finite distance from the origin), while couples and translational velocities are free vectors whose characteristics are similar to lines at. z' z {} r P P y' r P r O O' y l O x' x Figure 8 The two rigid bodies, and, and the two body fixed reference frames, {} and {}. The line l is attached to the rigid body. Our discussion of lines and their representations is not complete without exploring the transformation laws for line vectors. Consider the two reference frames and the line shown in Figure 8. In the reference frame {}, the line vector is given by: a line. Since the Plücker coordinates of a line are not invariant under changes in origin, a bound vector is not like a vector in R /31/01

24 p = u u O P, u = r where u is a vector parallel to the line and P is any point on the line. On the other hand, the line vector in reference frame {}, is given by: p = u O u u P = r It is clear that the vectors in the two frames are related by the transformation law in Equation ( 7 ). Therefore, u = R u u u. u O is the moment of u about the origin of {} and is given by: u O = r P u = ( r O + R r P ) R u = r O R u + R ( r P u) = r O R u + R u O. We introduce a 3 3 matrix operator that allows us to compute the cross product of a vector. For any 3 1 vector a, we define the matrix $a, given by: 0 a3 a2 $a = a3 0 a1, a 0 2 a1 where a = [a 1, a 2, a 3 ] T. It is easy to verify that for any vector b, the vector cross product a b is equivalent to multiplying $a by the vector b: a b = $a b. rmed with this skew symmetric matrix operator, we can write, O O O u = r$ R u+ R u. Therefore, the two 6 1 line vectors are related by the 6 6 transformation matrix, Γ, given by: p = Γ p, /31/01

25 where, Γ O = R 0 r$ R R ( 21 ) where O $r and R are 3 3 matrices and 0 is a 3 3 zero matrix. This is the transformation law for 6 1 Plücker coordinate line vectors. 2.6 References [1] all, R. S., Treatise on the Theory of Screws, Cambridge University Press, [2] oothby, W. M., n Introduction to differentiable manifolds and Riemannian Geometry, cademic Press, [3] ottema, O. and Roth,., Theoretical Kinematics. Dover Publications, [4] rand, L., Vector and Tensor nalysis, John Wiley, [5] Hunt, K.H., Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, [6] McCarthy. J.M., Introduction to Theoretical Kinematics, M.I.T. Press, [7] Paul, R., Robot Manipulators, Mathematics, Programming and Control, The MIT Press, Cambridge, [8] Plücker, J. On a new geometry of space. Phil Trans. 155, , [9] Semple, J.G. and Roth, L. Introduction to lgebraic Geometry, Clarendon Press, Oxford [10] Sommerville, D.M.Y., nalytical geometry of three dimensions. Cambridge University Press /31/01