Pluck¨ er Basis Vectors

Roy Featherstone Department of Information Engineering The Australian National University Canberra, ACT 0200, Australia Email: [email protected]

Abstract— 6-D vectors are routinely expressed in Pluck¨ er Pluck¨ er coordinate vector and the 6-D vector it represents; it coordinates; yet there is almost no mention in the literature of explains the relationship between a 6-D vector and the two the basis vectors that give rise to these coordinates. This paper 3-D vectors that define it; and it shows how to differentiate identifies the Pluck¨ er basis vectors, and uses them to explain the following: the relationship between a 6-D vector and its Pluck¨ er a 6-D vector in a moving Pluck¨ er coordinate system, using coordinates, the relationship between a 6-D vector and the pair of acceleration as an example. For the sake of a concrete expo- 3-D vectors used to define it, and the correct way to differentiate sition, this paper uses the notation and terminology of spatial a 6-D vector in a moving coordinate system. vectors; but the results apply generally to any 6-D vector that is expressed using Pluck¨ er coordinates. I. INTRODUCTION The rest of this paper is organized as follows. First, the 6-D vectors are used to describe the motions of rigid bodies Pluck¨ er basis vectors are described. Then the topic of dual and the forces acting upon them. They are therefore useful for coordinate systems is discussed. This is relevant to those 6-D describing the kinematics and dynamics of rigid-body systems vector formalisms in which force vectors are deemed to occupy in general, and robot mechanisms in particular. 6-D vectors a different vector space to motion vectors. Next, a convenient come in various forms, such as twists, wrenches, motors, spa- operator notation is introduced, that allows us to express and tial vectors, Lie-algebra elements, and simple concatenations manipulate the mappings between coordinate vectors and the of pairs of 3-D vectors. For examples, see [1], [2], [3], [4], [6], vectors they represent, as determined by the basis vectors. The [8], [9], [10], [12], [16]. Nearly all such vectors are expressed paper then proceeds to examine the nature of the relationship using Pluck¨ er coordinates—a system of coordinates invented between spatial vectors and the pairs of Euclidean vectors that in the 1860s by J. Pluck¨ er [11]. are used to define them. Finally, the method of differentiation It is a basic tenet of linear algebra that a coordinate system in a moving Pluck¨ er coordinate system is explained, and the on a vector space is defined by a basis. It therefore follows that results used to illuminate the relationship between competing Pluck¨ er coordinates must be defined by a basis. Yet, despite definitions of 6-D acceleration vectors. the widespread use of Pluck¨ er coordinates, there appears to be almost no mention of the basis vectors that give rise to them. II. PLU¨ CKER BASIS VECTORS The only example the author could find is the standard basis Different kinds of vector belong to different vector spaces. described in [4]. We therefore begin by defining the vector spaces En and Rn for The role of a basis is to define the relationship between the n-dimensional Euclidean and coordinate vectors, respectively. coordinates and the vectors they represent. Without an explicit Elements of En have properties of magnitude and direction, definition of the Pluck¨ er basis vectors, there is not a clear while elements of Rn are n-tuples of real numbers. description of the relationship between Pluck¨ er coordinates Spatial vectors are not Euclidean, and therefore do not and the 6-D vectors they represent. This has occasionally led belong in E6. Furthermore, it is useful to maintain a distinction to confusion over the true nature of 6-D vectors, as evidenced between those vectors that describe the motions of rigid by the recent debate on the definition of the 6-D acceleration bodies and those that describe the forces acting upon them. vector [5], [7], [13], [14], [15]. There is a tendency to regard We therefore define two vector spaces, M6 and F6, one for 6-D vectors as being ordered pairs of 3-D vectors, but this spatial motion vectors and one for spatial forces. Elements model is inaccurate, as it does not properly take into account of M6 describe velocities, accelerations, directions of motion the role of the reference point. freedom, and so on. Elements of F6 describe forces, momenta, This paper makes the following contributions: it defines the impulses, and so on. Pluck¨ er basis vectors; it explains the relationship between a Spatial vectors are usually constructed from pairs of 3D Euclidean vectors. Let us now examine how this is done. In 0 c 2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional particular, let us examine the construction of a velocity vector purposes or for creating new collective works for resale or redistribution to and a force vector. servers or lists, or to reuse any copyrighted component of this work in other Referring to Figure 1, the velocity of a rigid body can be works must be obtained from the IEEE. E3 This paper appeared in Proc. IEEE Int. Conf. Robotics & Automation, specified by a pair of vectors, ω, vO ∈ , where ω is the Orlando, FL, May 15–19 2006, pp. 1892–1897. angular velocity of the body as a whole, and vO is the linear ^ dOz v = αdOz + rαdx vO dz ω vP α dx z d ω OP O Oy rαdx r vO O P d y Ox dy x (a) (b)

Fig. 1. Rigid body velocity Fig. 2. Pluck¨ er motion basis (a), and example (b) velocity of the body-fixed point that is passing through an represents the same rigid-body velocity as the two 3D vectors arbitrary given point, O, in the underlying physical space. ω and vO. To establish the relationship between vˆ and its 6 The rigid-body velocity described by these two vectors is coordinates, we define the following basis on M : the sum of a pure rotation, in which the body rotates with 6 D = {d , d , d , d , d , d } ⊂ M , (5) angular velocity ω about an axis passing through O, and a O Ox Oy Oz x y z pure translation given by vO. This can be seen in the formula in which dOx, dOy and dOz are unit rotations about the −−→ directed lines Ox, Oy and Oz (which pass through O in vP = vO + ω × OP , (1) the x, y and z directions, respectively), and dx, dy and which gives the velocity of the body-fixed point at P . The dz are unit translations in the x, y and z directions (see right-hand side is the sum of a component due to the transla- Figure 2(a)). Thus, if the body were rotating about Ox with tion of the whole rigid body by vO, and a component due to an angular velocity of magnitude α, then its spatial velocity its rotation by ω about an axis passing through O. would be α dOx. Likewise, if the body were translating with Observe how the meanings change as the two vectors are a linear velocity of vO, then its spatial velocity would be combined: ω on its own is a disembodied angular velocity that vOx dx + vOy dy + vOz dz. Note the difference between this is the same for any choice of O, and vO on its own refers expression and the one in (4): the former is a spatial vector 6 specifically to the one point in the body that coincides with O (i.e., an element of M ), and the latter a Euclidean vector (an 3 at the current instant; but when we combine the two, the rigid- element of E ). A third example is shown in Figure 2(b). This body velocity they describe is the sum of a rotation specifically example shows a rotation of magnitude α about an axis that about an axis passing through O, and a pure translation in is parallel to the z axis and passes through the point (0, r, 0). which every point in the body travels with velocity vO. This motion is represented by the spatial vector αdOz +rαdx. Let us introduce a Cartesian coordinate frame, Oxyz, with Observe that the translational component equals the velocity its origin at O. This frame defines three mutually perpendicular of a particle at the origin that is rotating about (0, r, 0) with directions, x, y and z. These directions allow us to define an an angular velocity of α. orthonormal basis, It can be seen, by inspection, that the spatial vector 3 C = {i, j, k} ⊂ E , (2) vˆ = ωx dOx + ωy dOy + ωz dOz + vOx dx + vOy dy + vOz dz (6) in which the unit vectors i, j and k point in the x, y and represents the same rigid-body velocity as that described by z directions, respectively. This basis gives rise to a Cartesian the two vectors ω and vO above. We may therefore conclude E3 coordinate system on , such that ω and vO can be expressed that DO is the basis that gives rise to the Pluck¨ er coordinate in terms of their Cartesian coordinates: system in M6 associated with Oxyz, and that the coordinate ω i j k vector = ωx + ωy + ωz (3) T R6 vˆO = [ ωx ωy ωz vOx vOy vOz ] ∈ (7) and represents the spatial vector vˆ ∈ M6 in the Pluck¨ er coordinate vO = vOx i + vOy j + vOz k . (4) system defined by the basis DO. Equation (7) is often written We can now say that the Euclidean vectors ω and vO are in the form 1 ω T ω represented by the coordinate vectors = [ ωx ωy ωz ] ∈ vˆO = , (8) R3 T R3 vO and vO = [ vOx vOy vOz ] ∈ , respectively, in the coordinate system defined by the basis C. in which the right-hand side is the concatenation of the two It is well known that the six numbers ωx, . . . , vOz are coordinate vectors ω and vO. 6 the Pluck¨ er coordinates of a spatial vector,2 vˆ ∈ M , that Observe the pattern of subscripts in (6). Each quantity that depends on the location of O contains an O in its subscript. 1 Coordinate vectors are underlined to distinguish them from the vectors Note, however, that the subscript in d refers to the line they represent. Ox 2Spatial vectors other than basis vectors are marked with a hat. Basis vectors Ox, whereas the subscript in vOx is really two subscripts are left unmarked. run together, since vOx is the x coordinate of vO. Although 6 individual terms may vary, it can be shown that the complete Let D = {d1, . . . , d6} be an arbitrary basis on M . For any expression on the right-hand side of (6) is invariant with choice of D, there exists a unique basis E = {e1, . . . , e6} on respect to both the position and orientation of Oxyz. Thus, vˆ F6 with the property that is a genuinely invariant representation of rigid-body velocity. 1 if i = j A spatial force vector is constructed in a similar manner. di · ej = (15)  0 otherwise. Any system of applied forces acting on a single rigid body is equivalent to a single resultant force vector, f, together E is the dual (or reciprocal) basis to D, and vice versa. The with a moment vector, nO, giving the moment of the force pair (D, E) can be called a dual basis pair, or simply a dual system about an arbitrary given point, O. Although f itself is basis. It defines a dual coordinate system encompassing both M6 F6 M6 independent of O, the quantity it represents is a force acting and , in which elements of are expressed via D F6 on the rigid body along a line passing through O. and elements of via E. A dual coordinate system is the Introducing the coordinate frame Oxyz, and the basis C, we spatial-vector equivalent of a Cartesian coordinate system in can express f and nO in terms of their Cartesian coordinates: a Euclidean vector space. In any dual coordinate system, the following equations hold: f = f i + f j + f k (9) x y z mˆ · fˆ = mˆ T fˆ, (16) and ei · mˆ = mi (17) nO = nOx i + nOy j + nOz k . (10) and As before, the six numbers nOx, . . . , fz are the Pluck¨ er di · fˆ = fi , (18) coordinates of a spatial force vector, fˆ ∈ F6, representing the where mˆ and fˆ are the coordinate vectors representing mˆ ∈ same force system as f and n . To establish the relationship O M6 ˆ F6 between fˆ and its coordinates, we define the following basis and f ∈ in bases D and E, respectively, and mi and on F6: fi are the individual coordinates. These results follow directly from (15). If A and B are any two dual coordinate systems, 6 B M EO = {ex, ey, ez, eOx, eOy, eOz} ⊂ F , (11) and XA is the coordinate transformation matrix from A to B coordinates for motion vectors, then the corresponding in which ex, ey and ez are unit couples in the x, y and z transformation matrix for force vectors is e e e directions, and Ox, Oy and Oz are unit forces along the B F B M −T lines Ox, Oy and Oz. Again, it can be seen, by inspection, XA = ( XA ) . (19) that the spatial vector This equation follows from the invariance property of the scalar product, which can be expressed as fˆ = nOx ex + nOy ey + nOz ez + fx eOx + fy eOy + fz eOz (12) mˆ T fˆ mˆ T fˆ A A = B B (20) represents the same force system as the two vectors f and nO. m fˆ We may therefore conclude that EO is the basis that gives rise for all ˆ , , A and B. to the Pluck¨ er coordinate system in F6 associated with Oxyz, The Pluck¨ er bases in (5) and (11) satisfy (15), so the basis M6 F6 and that the coordinate vector pair (DO, EO) defines a dual coordinate system on and . It is possible to write the elements of DO in a different order, fˆ T R6 O = [ nOx nOy nOz fx fy fz ] ∈ (13) provided one does the same to EO. The result is a reordering of the cordinates in the coordinate vectors. For example, we ˆ F6 represents the spatial vector f ∈ in the Pluck¨ er coordinate could rewrite DO and EO as system defined by the basis EO. Equation (13) can be written in the form DO = {dx, dy, dz, dOx, dOy, dOz} n fˆ = O , (14) and O f   EO = {eOx, eOy, eOz, ex, ey, ez} , in which the right-hand side is the concatenation of the two in which case the coordinate vectors representing vˆ and fˆ coordinate vectors nO and f. would be T vˆO = [ vOx vOy vOz ωx ωy ωz ] III. DUAL COORDINATE SYSTEMS and Neither M6 nor F6 defines an inner product on its elements. fˆ = [ f f f n n n ]T . Instead, there is a scalar product that takes one argument from O x y z Ox Oy Oz each space and produces a real number representing work. Some authors prefer this linear-before-angular ordering to the Thus, if mˆ ∈ M6 and fˆ ∈ F6, then fˆ · mˆ is the work done angular-before-linear ordering in (7) and (13). With the aid of by a force fˆ acting on a rigid body moving with motion mˆ . Pluck¨ er bases, it is immediately obvious that the difference For convenience, we define mˆ · fˆ to mean the same as fˆ· mˆ , between these two orderings is purely cosmetic—they both but the expressions fˆ· fˆ and mˆ · mˆ are not defined. represent the same spatial vectors. IV. BASIS MAPPINGS use, consider the task of formulating the transformation matrix between two coordinate systems, A and B. Let u and u Suppose B = {b1, . . . , bn} is a basis on a vector space U. A B Given B, we can express any vector u ∈ U in the form be the coordinate vectors representing the vector u ∈ U in A BX n and B coordinates. If A is the coordinate transformation matrix from A to B, then we have u = bi ui , =1 B Xi uB = XA uA . u u B where i are the coordinates of in . This equation can also However, if BA and BB are the basis maps for A and B, then be written in the form we also have u1 −1 −1 u = B u = B B u , . B B B A A u = b1 · · · bn  .  = B u , (21) . so   un B B−1 B   XA = B A . B where is the operator that maps coordinate vectors to the Expanding this equation gives vectors they represent in the basis B. We therefore call B the B B−1 basis mapping associated with B. Formally, B is a mapping XA = B bA1 · · · bAn Rn B−1 B−1 from to U defined as follows: = B bA1 · · · B bAn ; u1 B−1   n but B bAi is just the coordinate vector representing bAi in B B Rn  .  B : 7→ U : . 7→ bi ui . (22) coordinates, so we may conclude that XA is a square matrix Xi=1 un whose columns are the coordinates of the old basis vectors   in the new coordinate system. (This is a standard result. The B If is written as a 1 × n array of basis vectors, as shown point is simply the speed with which it can be obtained.) in (21), then the action of B on u can be understood as the result of a formal matrix multiplication between the two. V. RELATIONSHIP BETWEEN SPATIAL AND EUCLIDEAN B is a 1 : 1 mapping, and is therefore invertible; so there VECTORS −1 −1 must exist an inverse mapping, B , that satisfies u = B u. Let us examine the relationship between a spatial vector and −1 A formal expression for B can be stated as the pair of Euclidean vectors that are used to define it. If we rot ∗· partition DO into two sub-bases, DO = {dOx, dOy, dOz} and b1 lin DO = {dx, dy, dz}, then (6) can be written as follows: B−1  .  = . , (23) rot lin ∗ vˆ = D ω + D v b · O O O  n  Drot C−1 ω Dlin C−1 v ∗ ∗ ∗ = O + O O b b b · M M where { 1, . . . , n} is the dual basis to B, and i is the = Rot ω + Lin v , ∗ · O O (27) operator that maps any vector u ∈ U to the scalar bi u. −1 Expanding B B gives where ∗ ∗ ∗ M Drot C−1 · · · b1· b1 · b1 · · · b1 · bn RotO = O = dOx i + dOy j + dOz k (28) B−1 B  .   . . .  = . b1 · · · bn = . .. . and ∗ ∗ ∗ M Dlin C−1 · · · b ·   b · b1 · · · b · b  Lin = O = dx i + dy j + dz k . (29)  n   n n n Expressions like dOx i· are dyads that map Euclidean vectors which equates to the identity matrix because of the reciprocity 3 to spatial motion vectors. dOx i· maps any vector v ∈ E condition (15). Special cases of interest are: 6 to dOx (i · v) = dOx vx ∈ M , and so on. The operators i· M M −1 RotO and Lin are therefore both dyadics (sums of dyads). C = j· , (24) M RotO maps Euclidean vectors to the set of pure rotations k· M   about axes passing through O; and Lin maps Euclidean · vectors to the set of pure translations. It can be shown that ex M M −1 both Rot and Lin are independent of the orientation of D =  .  O O . (25) the coordinate frame, Oxyz, that gave rise to the bases C and · M eOz  DO. Furthermore, Lin is also independent of the position   M M and of O. Therefore, Lin is an invariant tensor, while RotO dOx· depends only on the position of O. E−1  .  A similar analysis can be performed for force vectors: O = . . (26) ˆ E rot E lin  d ·  f = O nO + O f  z  E rot C−1 E lin C−1 Basis mappings provide a simple but powerful tool for = O nO + O f F F expressing the relationships between vectors. To illustrate their = Rot nO + LinO f , (30) where that is moving with a velocity of vˆ (coordinate vector vˆO = F T T T Rot = ex i· + ey j· + ez k· (31) [ ω vO ] ), then −1 C C˙ = ω× , (36) and F · · · −1 ω 0 LinO = eOx i + eOy j + eOz k . (32) D D˙ × O O = vˆO× = (37) vO× ω× It is not accurate to describe ω, vO, nO and f as being the angular and linear components of vˆ and fˆ. However, it is and −1 ∗ ω× vO× possible to regard them as the vector-valued coordinates of vˆ E E˙ O = vˆ × = , (38) O O  0 ω×  and fˆ in the coordinate systems defined by the basis tensors M M F F RotO , Lin , Rot and LinO. where ω× is the 3×3 matrix that maps a 3D coordinate vector, E3 E3 M6 ∗ In summary, the mapping from × to either or u, to the cross product ω × u, and vˆO× and vˆO× are the F6 is accomplished by a pair of dyadic tensors, one of which analogous spatial-vector operators. vˆO× maps a motion vector ∗ is invariant, while the other is a function of the location of to a motion vector, while vˆO× maps a force to a force. ω× the reference point, O, that was used when specifying the two is given by the formula Euclidean vectors. ωx 0 −ωz ωy VI. DIFFERENTIATION ωy× =  ωz 0 −ωx . (39) ωz −ωy ωx 0 Let u(t) be a vector-valued, differentiable function of a     real variable t. The derivative of u with respect to t is itself For the three special cases above, (35) becomes a vector, and is given by −1 C v˙ = v˙ + ω × v , (40) d u(t + δt) − u(t) u(t) = lim . (33) D−1 ˙ ˙ dt δt→0 δt O mˆ = mˆ + vˆO × mˆ (41) This equation is valid for any variable t; but we will assume and E−1 ˆ˙ ˆ˙ ∗ ˆ below that t denotes time, and we will use the standard dot O f = f + vˆO × f , (42) notation for time derivatives (du/dt = u˙ , etc.). 3 6 Equation (33) applies to all vectors, including coordinate where v, mˆ and fˆ denote general elements of E , M and 6 vectors. It therefore follows that the derivative of a coordinate F , respectively. Except for the use of basis-mapping notation, vector is its component-wise derivative: (40) is a standard result that can be found in many textbooks. Observe that we have been able to treat this subject using only u1(t) u1(t + δt) − u1(t) u˙ 1 one kind of differential operator, instead of the usual two. This d . 1 . .  .  = lim  .  =  .  . is because the basis-mapping notation gives us a symbol for dt . δt→0 δt . . un(t) un(t + δt) − un(t) u˙ n “the coordinate vector that represents. . . ”.      (34) Let B be a basis on U, and let u be the coordinate vector that VII. ACCELERATION represents the vector u ∈ U in B coordinates. The derivatives Nothing has yet been said about the velocity of O. The of u and u are u˙ and u˙ , respectively, but the coordinate vector role of O is to specify a point where something is measured, −1 that represents u˙ in B is B u˙ . The relationship between u˙ or a point through which something passes. Thus, quantities −1 and B u˙ is given by like vO, nO, dOx, etc., all depend on the position of O, but not its velocity. It is therefore possible to assign any −1 −1 d B u˙ = B ( (B u)) desired velocity to O without affecting the definitions of dt − spatial vectors. However, if the velocity of O is nonzero, then B 1 B u B˙ u = ( ˙ + ) D E −1 O and O are moving bases, and this must be taken into = u˙ + B B˙ u . (35) account when differentiating a spatial vector. Referring back to Figure 1, suppose we make O track a This is the general formula for differentiation in a moving point in the moving body, so that the two points coincide coordinate system. If the coordinate system is stationary then −1 permanently. The body’s velocity would still be characterized B˙ = 0 and B u˙ = u˙ . −1 by the two Euclidean vectors ω and v ; it would still have a Expanding the term B B˙ gives O spatial velocity of vˆ, as defined in (6); and vˆ would still be −1 −1 B B˙ = B b˙ 1 · · · b˙ n represented in DO coordinates by its coordinate vector, vˆO, −1 −1 as defined in (7) and (8). However, O itself would now have = B b˙ 1 · · · B b˙ n , a velocity of v , and so would Oxyz.   O which is a square matrix whose columns are the coordinates Let vˆOxyz denote the spatial velocity of the coordinate of the derivatives of the basis vectors. Three special cases are frame, and let vˆOxyz be the coordinate vector representing of particular interest. If C, DO and EO are the orthonormal vˆOxyz in DO coordinates. If we set the angular velocity of 0T T T and Pluck¨ er bases derived from a coordinate frame Oxyz the coordinate frame to zero, then vˆOxyz = [ vO ] . The acceleration of a rigid body is simply the time- then DO is a time-varying coordinate system, and this must derivative of its velocity. The spatial acceleration of the be taken into account when performing the differentiation. body in Figure 1 is therefore vˆ˙ , and the coordinate vector D−1 ˙ VIII. CONCLUSION representing its acceleration in DO coordinates is O vˆ. We D−1 ˙ This paper has introduced the concept of Pluck¨ er bases, and can obtain an expression for O vˆ directly from (41) as follows: an operator notation to express explicitly how a basis maps a coordinate vector to the vector it represents. Using these D−1 ˙ ˙ O vˆ = vˆO + vˆOxyz × vˆO tools, the paper explains the following: the precise relationship ω˙ 0 ω between spatial vectors and the pairs of 3-D vectors that = + × define them; the correct way to differentiate a spatial vector v˙ O vO vO in a moving Pluck¨ er coordinate system; and why the classical ω˙ 0 = + . (43) description of rigid-body acceleration is not the derivative of v˙ v × ω  O  O  spatial velocity. In the classical textbook treatment, the acceleration of a Although this paper has been written in the language of spa- rigid body is usually defined by an angular acceleration vector, tial vectors, the results reported here are applicable generally ω˙ , and the linear acceleration, r¨, of a chosen body-fixed point to any 6-D vector that is represented in Pluck¨ er coordinates. whose position is given by r. If we define r to be the position REFERENCES of O relative to some fixed datum, then v = r˙ and v˙ = r¨, O O [1] E. J. Baker and K. Wohlhart. Motor Calculus: A New Theoretical Device and (43) becomes for Mechanics. Graz, Austria: Institute for Mechanics, TU Graz, 1996. (English translation of [8] and [9].) −1 ω˙ 0 A Treatise on the Theory of Screws D vˆ˙ = + . (44) [2] R. S. Ball. . London: Cambridge O r¨ r˙ × ω University Press, 1900. Republished 1998. [3] L. Brand. Vector and Tensor Analysis, 4th ed. New York/London: Wi- ley/Chapman and Hall, 1953. 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