E 5 Fall 997 V. Kumar Some Useful Results for Spherial an General Displaements. Spherial Displaements.. Eulers heorem We have seen that a spherial isplaement or a pure rotation is esribe by a 3 3 rotation matrix. oring to Eulers theorem, "ny isplaement of a rigi boy suh that a point on the rigi boy, say, remains fixe, is equivalent to a rotation about a fixe axis through the point." We will start with a general spherial isplaement an show Euler s theorem is vali. Consier a boy fixe frame that is isplae from {F} to {}. he isplaement of any point on the rigi boy an be written as: Rp where p an are the position vetors before an after isplaement respetively, an we are eleting the supersripts an subsripts assoiate with F R. et us fin the eigenvalues an eigenvetors of R by writing: Rp λp he harateristi equation is: R λi axis of rotation u eigenspae spanne by x an x _ φ φ Rv v Rv v
Figure he real eigenvetor, u, an the orthogonal eigenspae for a rotation matrix If the elements of R are enote by R ij, the above equation an be written as: b g 3 λ + λ R + R + R33 λbr R R R g + br R R R g + br R R R gh + R 33 3 3 33 3 3 Beause R is a rotation matrix, R. lso, beause R R R R R R 33 3 3 R R R R R 33 3 3 R R R R R 33 his simplifies the harateristi equation: b g b g λ 3 + λ R + R + R33 λ R + R + R33 + Fatoring the left han sie, we get: b g b 33 g λ λ λ R + R + R + h R, We see that λ is an eigenvalue of R. In other wors, there exists a real vetor, u, suh that all points on the line, p αu remain fixe (invariant) uner the transformation R. hus R is a rotation that leaves this line or axis fixe. Define φ so that b g () R + R + R33 hen the remaining eigenvalues are: iφ iφ λ e + i, λ e i hus the eigenvalues an eigenvetors are: λ λ e i, p x, say φ e i, p x () φ λ 3 3, p u. --
where x is the omplex onjugate of x. Sine R is orthogonal, x an x are perpeniular to u an span the plane perpeniular to the axis of rotation. real basis for this plane an be onstrute using the vetors: v b i x + xg, v bx xg, (3) b g e j iφ iφ Rv λx + λx e x + e x v + v b g e j i i iφ iφ Rv λx λx e x e x v + v s shown in Figure, the effet of the transformation, R, is to rotate vetors in the plane spanne by v an v through an angle φ about u, while vetors along u are invariant. hus the transformation R rotates the rigi boy about u through an angle φ. his proves Eulers theorem. here is a anonial representation of any rotation matrix whih allows us to view it as a rotation through an angle φ about the z axis. If we onsier the tria of orthonormal vetors, {v, v, u}, we an easily show that R an be written in the form: R Λ (4) where, v v u, an Λ. his is essentially a similarity transformation. If we view the rotation from a new frame, {F }, whose orientation is given by the oorinate transformation, F R F, it is lear that in this new frame, the isplaement is a rotation about the z axis through an angle φ. ote that we oul also have hosen to onstrut our anonial representation of a rotation by viewing it as a rotation about the x or the y axis. -3-
.. Representations for finite rotations We have seen that any rotation or spherial isplaement an be esribe by a rotation matrix. However, beause the rotation matrix is orthogonal, its nine elements are not inepenent of eah other. Speifially, they satisfy six inepenent equations: R + R + R 3 R + R + R 3 R + R + R 3 3 33 R R + R R + R R 3 3 R R + R R + R R 3 3 3 33 R R + R R + R R 3 3 33 3 herefore there are only three inepenent parameters that ompletely esribe a given rotation an the rotation matrix is a very reunant representation. In orer to evelop more ompat representations, we pursue two types of representations. he first relies on eomposing a given rotation into three finite suessive rotations an these three rotation angles, alle Euler angles, ompletely esribe the given rotation. he seon is erive from the results of the previous setion an expliitly ientifies the axis of rotation an the angle of rotation. he Euler angles are efine in stanar kinematis an ynamis texts an are not isusse here. Instea, we isuss axis-angle representations in some etail. In this setion we efine Euler parameters whih provie another ompat representation of the axis an angle of rotation. We also evelop a simple result that allow us to obtain the rotation matrix from the axis an angle of rotation. he axis-angle representation Consier the rotation of a rigi boy about u through an angle φ as shown in Figure. Consier the triangle in the figure. We an write: or, p a + a Rp p a a + + -4-
axis of rotation u (a) φ -p p θ (b) () φ +p -p a p a -p Figure Rotation of p about u through φ. he vetor is relate to p by the rotation matrix, R p. Sine the tail of an p are oinient at, we an raw a one whose apex is suh that both vetors are generators of the one emanating from. From the geometry, -5-
p sinθ u p a u p Sine a is perpeniular to u an p, a bu pg. he length an also be expresse as p bp ug u. herefore, b g hb g a p p u u. Substituting a an a in (.6) yiels If we let U b g. u u, this equation yiels: b g (5) Rp p + uu p + u p u u u 3 3 u R I + uu + U his is alle the Rorigues formula, a result that is attribute to Euler, exell, an Rorigues. It esribes the rotation matrix in terms of the angle an axis of the rotation. o obtain the angle, φ, an axis of rotation, u, from the rotation matrix, we an easily erive the following formulas: b g (6) R + R + R33 U R R h (7) here are many solutions for the angle of rotation beause the inverse osine funtion is multivalue. If we fin φ from (6) restriting the range of the inverse osine funtion to the interval [,π], we an fin the axis of rotation from (7) provie φ is not either or π. If R I, φ an the axis of rotation is not well efine. If the trae of the rotation matrix is -, φπ, an from the Euler-exell formula: R uu I -6-
from whih u an be solve. For every rotation of angle φ about the axis u, we an also obtain an equivalent axis-angle representation with a rotation -φ about the axis -u. lso for every solution (u, φ), we have other solutions (u, φ+kπ) for all integer values of k. Euler parameters We now turn to a representation base on half the rotation angle an the axis of rotation. Define the four Euler parameters: φ φ φ φ os, ux sin, uy sin, 3 uz sin (8) ote + + +. his is the equation of a unit sphere (or hypersphere) in R 4, also 3 alle the projetive three sphere. ny point on this sphere orrespons to a unique axis an angle of rotation an therefore a unique rotation matrix. However, there are two points (iametrially opposite) on the sphere that give the same axis an angle. an - enote the same rotation. For example, a rotation of 9 o egrees about the x-axis yiels F H,,, I K an a rotation of 7 o about the same axis yiels F HG,,, although they are really the same rotation. If is partitione as follows: 3 I K J it an be easily seen that (.7) an be written as: i R I + + D (9) -7-
where D 3 3. his is the rotation matrix expresse in terms of Euler parameters.. Spatial isplaements an transformations.. Chasles heorem ne of the most funamental results in spatial kinematis is a theorem that is usually attribute to Chasles (83), although ozzi an Cauhy are reite with earlier results that are similar. he theorem states: "he most general rigi boy isplaement an be proue by a translation along a line followe (or preee) by a rotation about that line." Beause this isplaement is reminisent of the isplaement of a srew, it is alle a srew isplaement, an the line or axis is alle the srew axis. We prove this theorem for the planar ase an then for the spatial ase. For a planar isplaement, one an fin a fixe point of the isplaement. his is a point that is left unhange by the isplaement. Consier a general 3 3 homogeneous transfer matrix: R R, osθ sinθ, sinθ osθ x If R is not the ientity matrix, there is one fixe point on the rigi boy for any isplaement (in ontrast to spatial isplaements) alle the pole (or the instantaneous enter) of the isplaement. If is the position vetor of the pole, R+ whih allows us to fin the pole: I R y b g (). -8-
his point orrespons to the eigenvetor of the matrix for a unit eigenvalue. hus any planar isplaement an be onsiere as either a pure rotation about a point alle the pole (if (R-I) is not singular) or a pure translation in the iretion given by (if RI). Sine pure rotations an pure translations are speial ases of a srew motion, Chasles theorem is prove for the planar ase. If we onsier a general 4 4 homogeneous transfer matrix,, it has four eigenvalues two of whih are equal to. (he other two are omplex onjugate with a magnitue of ). However there are no real eigenvetors orresponing to λ. his implies that a general isplaement has no fixe points. o prove Chasles theorem for spatial isplaements, onsier the following similarity transformation of the matrix : Λ R R R + ow hoose as we i for the anonial representation of the rotation matrix in Equation (4) so that it onsists of the eigenvetors of R: v v u () he 3 3 orthogonal matrix part of Λ reues to the rotation matrix orresponing to a rotation about the z axis: R If we efine, the 3 translation vetor in Λ an be simplifie as follows: -9-
R + R I + h x x y y z + z where. If the top submatrix of ( R-I) is nonsingular, we an always fin suh that x y + x y In other wors, we an always solve the first two equations of ( R-I) - for the first two omponents of as we i in Equation () an let the thir omponent be zero. In this ase, Λ has the form: Λ k where k is given by the thir omponent of the vetor. In this ase, the isplaement an be esribe by a rotation about the z axis through an angle φ an a onurrent translation along the z axis through a istane k. his is a srew isplaement an proves Chasles theorem for a general isplaement. If the top submatrix of ( R-I) is singular, then RI. In this speial ase, Λ is a pure translation given by the vetor. his is a speial ase of the srew isplaement. hus we have prove Chasles theorem for this speial ase as well. he geometri interpretation is shown in Figure 3. We are intereste in the isplaement that takes frame {G } into frame {G }. he matrix G E () (3) --
represents the oorinate transformation from {E } to {G }. In the referene frame {E }, the isplaement is a pure rotation about the z axis through an angle φ, aommpanie by a onurrent translation along the same axis of khφ. hus, E Λ E straightforwar similarity transformation onfirms: k. G G E E G E E G G E Λ G E (4).. Determination of the srew axis from the homogeneous transformation matrix Given the rotation matrix, R, we an fin u, a unit vetor along the srew axis, an φ, the angle of rotation about the srew axis from Equations (6-7). In orer to ompletely speify the srew isplaement, we want to fin at the position vetor of at least one point on the srew axis an the translation of a point on the rigi boy along the srew axis. --
{G } φ {E } hφ {E } {F} {G } u Figure 3 he isplaement of a boy fixe frame from {G } to {G }, the anonial representation of this isplaement from {E } to {E }, an the orresponing srew axis. We proee aoring to the evelopment in the previous subsetion. First, fin an appropriate proper orthogonal matrix suh that its thir olumn is u. ne may hoose to be the matrix efine in Equation () but this requires the omputation of the eigenvetors of R an is unneessary. If we let p enote the projetion of the vetor on a plane perpeniular to u, p - (. u) u (5) then we an hoose v w u (6) where, --
p v w u v, p an p is the magnitue of p or the norm p x os sin y φ φ p b g b g v w. ow Equation () simplifies to: ote that we have use the fat.w. hus, we get the position vetor of a point on the srew axis: p p sinφ, v w os + os b φg φ b g e j (7) he isplaement of a point on the axis is given by the thir omponent of : ku. (8) an the pith of the srew, h, is given by h k φ (9) 3. Referenes [] Bottema,. an Roth, B., heoretial Kinematis. Dover ubliations, 99. [] Bran,., Vetor an ensor nalysis, John Wiley, 947. [3] Hunt, K.H., Kinemati Geometry of ehanisms, Clarenon ress, xfor, 978. [4] Carthy. J.., Introution to heoretial Kinematis,.I.. ress, 99. [5] aul, R., Robot anipulators, athematis, rogramming an Control, he I ress, Cambrige, 98. -3-