DETERMINING THE 3 DETERMINING THE 3DETERMINING THE 3 DETERMINING THE 3 × ×× × 3 ROTATION MATRICES 3 ROTATION MATRICES 3 ROTATION MATRICES 3 ROTATION MATRICES THAT SATISFY THREE L THAT SATISFY THREE LTHAT SATISFY THREE L THAT SATISFY THREE LINEAR EQUATIONS IN INEAR EQUATIONS IN INEAR EQUATIONS IN INEAR EQUATIONS IN THE DI THE DITHE DI THE DIRECTION COSINES RECTION COSINESRECTION COSINES RECTION COSINES Carlo Innocenti DIMeC – University of Modena and Reggio Emilia – Italy carlo.innocenti@unimore.it Davide Paganelli DIEM – University of Bologna – Italy davide.paganelli@mail.ing.unibo.it Abstract AbstractAbstract Abstract The paper presents a solution to all the spatial kinematics problems that - sines satisfy three linear equations. After having expressed the direction co - sines in terms of the Rodrigues parameters, a classical elimination method to solve three quadratic equations in three unknowns is here extended to in - pa Keywords : KeywordsKeywords Keywords Rotation matrix, direction cosines, Rodrigues parameters 1. 1. 1. 1. Introduction IntroductionIntroduction Introduction A whole class of problems of spatial kinematics can be solved by de- three given linear equations. Owing to the orthogonality constraints among the direction cosines, these problems are equivalent to solving a set of nine equations: three linear and six quadratic. Rather than tackling right away the solution of such an equation set, it is computationally more efficient to replace, in each equation, all un - known direction cosines by their expressions in terms of the Rodrigues parameters. In doing so, all orthogonality constraints are implicitly ful - filled, whereas the former linear equations in the direction cosines turn into second - order equations in the Rodrigues parameters. Unfortunately, the known algebraic elimination algorithms that solve a set of three quadratic equations – such as the Sylvester method – are 23 © 2006 Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 23 32. – require determination of the 3 × 3 rotation matrices whose nine direction co clude all solutions at infinity. Therefore no admissible 3 × 3 rotation matrix is rametrization of orientation. A case study exemplifies the new method. neglected even though it corresponds to a singularity of the Rodrigues termining all 3 × 3 rotation matrices whose nine direction cosines obey : : unable to find real solutions at infinity, which are here of interest too because infinite real Rodrigues parameters are associated to finite real exist, these algorithms might fail to determine even the finite solutions. After exemplifying the recurrence in kinematics of the addressed three - equation set in the direction cosines, this paper presents an origi - nal procedure to find all real solutions of the equation set. The proposed procedure – based on the Rodrigues parametrization of orientation and presented with reference to the Sylvester algebraic elimination algorithm – is able to identify all real solutions in terms of Rodrigues parameters, both finite and at infinity. Therefore its adoption guarantees that no real neglected. A numerical example shows application of the proposed computational 2. 2. 2. 2. A linear three - equation set in nine direction cosines is the unifying factor behind a number of seemingly different kinematics problems, such as those epitomized in Fig. 1. Although these problems have already been solved in the literature by ad - hoc algorithms, they could be also worked - ditions in the direction cosines. In this respect, the procedure proposed in this paper is a viable alternative to already-known solving methods. Figure 1 . a) Fully-parallel spherical wrist; b) rigid body supported at six points by six planes. not always suitable to the case at hand. The reason is twofold: i) they are 24 3 × 3 rotation matrices, and ii) in case one or more solutions at infinity 3 × 3 rotation matrix compatible with the original three linear equations is procedure to a case study. TThhe e Relelevevaanncce e tto o Kiinnememaattiiccss out by determining all 3 × 3 rotation matrices satisfying three linear con tics aims at determining all possible orientations of the moving platform Figure 1a shows a fully parallel spherical wrist, whose direct kinema C. Innocenti and D. Paganelli for a given set of actuator lengths (Innocenti and Parenti-Castelli, 1993). If v vv v i and w ww w i are the coordinate vectors of points Q i and P i relative to the fixed ( S ) and movable ( S’ ) reference frames respectively, and R RR R is the rotation matrix for transformation of coordinates from S’ to S , then – by applying Carnot’s theorem to triangle OQ i P i – the compatibility equa - tions can be written as +− = = 2 2 ( 1, ,3) TT T ii i ii i i Livv ww vRw (1) These equations are linear in the (unknown) elements of matrix R RR R. Figure 1b refers to another kinematics problem, which consists in find - ing any possible positions of a rigid body C supported at six given points P i ( i=1, ,6) by six fixed planes (Innocenti, 1994; Wampler, 2006). The co - ordinate vector w ww w i of each point P i is known with respect to a reference frame S’ attached to C . Each supporting plane is defined with respect to the fixed frame S by the coordinate vector v vv v i of a point Q i lying on the plane, together with the components in S of a unit vector n nn n i orthogonal to the plane. The unknown position of C with respect to S is parametrized through the coordinate vector s s s s of the origin of S’ with respect to S , to - gether with the rotation matrix R RR R for transformation of coordinates from S’ to S . The compatibility equations can be written as : ( ) [ ] 0 ( 1, ,6) T iii i+−= =nsRw v (2) They are linear in both the elements of R RR R and the components of s ss s. If there exist three supporting planes not parallel to the same line, three of these equations can be linearly solved for the components of vector s ss s, and their expressions inserted into the remaining three equations. Therefore a linear three-equation set that has the nine direction cosines of matrix R RR R as only unknowns is obtained once more. Other kinematics problems susceptible of being reduced to the same linear formulation as the one just exemplified are traceable in Gosselin et al., 1994, Husain and Waldron, 1994, Wohlhart, 1994, Callegari et al. 2004. 3. 3. 3. 3. If ij r ( i,j =1,2,3) is the ij th element (direction cosine) of a rotation matrix RR ij k k equations that has to be solved for ij r ( i, j =1,2,3) is 25 The Equations to be Solved The Equations to be Solved 3 × 3 Determining the Rotation Matrics , RR and a , , b ( i, j, k =1, ,3) are known quantities, the set of three linear = == ∑ , , 1, ,3 ( 1, ,3) ij k ij k ij ar b k (3) The expressions of r ij in terms of the vector of Rodrigues parameters p pp p = ( p 1 , p 2 , p 3 ) T are concisely given by (Bottema and Roth, 1979) −++ = +  (1 ) 2 2 1 TT T ppI p pp R pp (4) where p  is the skew - symmetric matrix associated with vector p pp p, i.e., =×pe p e  for any three - component vector e ee e. As is known, the vector p pp p of Rodrigues parameters corresponds to a finite rotation of amplitude 1 2tanθ − = p about the axis defined by unit vector upp/= . Unfortunately, the Rodrigues parametrization of orientation is singu - lar for any half - a - turn rotation ( θ = π rad) about any line because, in this instance, at least one of the components of p pp p approaches infinity. By considering Eq. (4), Eq. (3) can be re - written as : () ,, 222 , 1, ,3; 1, ,3 123 1 0 1, ,3 1 ij k i j i k i k ij i j i App BpC k ppp =≤ = ⎛⎞ ⎟ ⎜ ⎟ ++== ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ +++ ⎝⎠ ∑∑ (5) where quantities A ij,k , B i,k , and C k (i,j,k = 1, ,3; i ≤ j) are known because dependent on the given quantities a ij , k and b k only. Because the denominator of Eq. (5) does not vanish for any real vector p pp p, if p pp p does not approach infinity Eq. (5) can be simplified as follows () ,, , 1, ,3; 1, ,3 0 1, ,3 ij k i j i k i k ij i j i App BpC k =≤ = ++== ∑∑ (6) Conversely, in case the denominator of Eq. (5) approaches infinity, so does at least one of the components of p pp p. If both the numerator and the denominator of Eq. (5) are homogenized by replacing p i with expression x i /x 0 (i = 1, ,3), and subsequently multiplied by x 0 2 , the resulting denomi - nator is definitely different from zero (the real quantities x 0 , x 1 , x 2 , and x 3 cannot vanish simultaneously). Finally, for x 0 = 0 (which means that at least one Rodrigues parameter approaches infinity), Eq. (5) becomes () , , 1, ,3; 0 1, ,3 ij k i j ij i j Axx k =≤ == ∑ (7) 26 C. Innocenti and D. Paganelli This is a set of three homogeneous quadratic equations in three un - knowns, namely, the components of vector x xx x = (x 1 , x 2 , x 3 ) T . If the set of the non - vanishing vectors that satisfy Eq. (7) is parti - tioned into equivalence classes so that two solution vectors parallel one to the other belong to the same class, then each class corresponds to a vector p pp p of Rodrigues parameters which satisfies Eq. (5) and has infinite magnitude. Finding all real solutions of Eq. (5) – both finite and at infinity – has been thus reduced to determining all real finite solutions of Eq. (6), to - gether with all equivalence classes of real solutions of Eq. (7). This im - plies that all real solutions of Eq. (6) – including those at infinity – need to be computed. Bezout’s theorem (Semple and Roth, 1949) ensures that the maximum number of these solutions is eight. 4. 4. 4. 4. As will be proven further on, the existence of solutions at infinity might affect the search for the finite solutions. It is therefore convenient to compute the solutions at infinity first. The Appendix at the end of the paper briefly summarizes the mathe - matical tools that will be taken advantage of in this section. 4.1 4.1 4.1 4.1 s The solutions at infinity, if existent, can be found by identifying Eq. (7) with Eq. (1 - A) of the Appendix. For the case at hand, Eq. (3-A) becomes ( ) = 222 123121323 T xxxxxxxxx M0 (8) where M MM M is a 6 × 6 matrix that depends on coefficients A ij,k of Eq. (7) only. In case the determinant of M MM M is different from zero, there is only the trivial solution for Eq. (7), and no solution at infinity exists for Eq. (6). Conversely, if the determinant of M MM M vanishes, Eq. (7) has non - vanishing solutions. The number of equivalence classes of these solutions matches the number of solutions at infinity for Eq. (6). Determination of all solutions of Eq. (7) poses no hurdles and will not be detailed in this paper. Suffices it to say that, in the worst possible scenario, the classes of equivalence for the solutions of Eq. (7) can be found by solving a set of two quadratic equations in two unknowns. 27 The Solving Procedure The Solving Procedure Solutions at Infinity s Solutions at Infinity 3 × 3 Determining the Rotation Matrics 4.2 4.2 4.2 4.2 In most cases, the finite solutions of Eq. (6) can be determined through the procedure described by Roth, 1993, and here briefly summarized. If ( α , β , γ ) is a permutation of indices (1,2,3), two of the three unknowns, say p α and p β , are first replaced in Eq. (6) by quantities y α / y 0 and y β / y 0 . Fol - lowing multiplication by y 0 2 , the ensuing equation set is obtained: () () ( ) () ,min,max,,,0 ,; 22 ,, 0 0 1, ,3 ij k i j i i k i k i ij or i j i or kkk Ayy A p B yy ApBpCy k γγγ αβ αβ γγ γ γ γ =≤ = ⎡⎤ ++ ⎣⎦ +++== ∑∑ (9) which is homogeneous with respect to unknowns y 0 , y α , and y β . If a triplet of values for p α , p β , and p γ fulfils Eq. (6), Eq. (9) must be satisfied by the same value of p γ together with a non-vanishing triplet of values for y 0 , y α , and y β . By also taking into account the dependence on p γ of the coefficients of the homogeneous system in Eq. (9), the solvability condition for Eq. (9) that corresponds to Eq. (3-A) turns into ( ) 222 000 () T p y y y yy yy yy γαβαβαβ =N0 (10) The solution of this linear set is meaningful only if the triplet ( y 0 , y α , y β ) does not vanish, i.e., if the following condition is satisfied (see Eq. (4 - A)) γ =det ( ) 0 pN (11) This univariate polynomial equation in p γ has degree not greater than eight (Roth, 1993). It is the outcome of elimination of unknowns p α and p β from Eq. (6). For every root of Eq. (11), the corresponding values of p α and p β can be easily found by Eq. (10) through linear determination of a non-vanishing triplet ( y 0 , y α , y β ). Thus far is the outline of the procedure that has been presented – without investigating its singularities – in Roth, 1993. It is worth noting that Eq. (11) is unable to yield solutions at infinity. Things keep manageable if an infinite p γ satisfies Eq. (5) for some values of p α and p β , as Eq. (11) has a degree lower than eight and its roots con - vey information on finite solutions only. Regrettably, should an infinite solution to Eq. (5) exist for a finite p γ (i.e., only p α or p β or both approach infinity) then Eq. (11) vanishes and the described elimination method becomes pointless. 28 Finite Solutionss Finite Solutionss C. Innocenti and D. Paganelli This latter drawback can be explained by noticing that – for p α or p β approaching infinity – Eq. (10) should hold for y 0 = 0 and for some (not simultaneously vanishing) values of y α and y β , irrespective of the value of p γ (the left-hand side of Eq. (9) does not depend on p γ when y 0 = 0). Conse - quently, the determinant of 6 × 6 matrix N NN N( p γ ) should vanish for any finite p γ , which also means that Eq. (11) collapses into a useless identity. If it is not possible to choose index γ so as to circumvent the just men tioned inconvenience, the classical elimination method is definitely un able to find any finite solution to Eq. (6). Even a different set of Rodri - gues parameters consequent on a randomly - chosen offset rotation does not guarantee removal of the inconvenience. 4.3 4.3 4.3 4.3 Adding robustness Adding robustnessAdding robustness Adding robustness To overcome the drawback outlined at the end of the previous subsec - tion, once the solutions at infinity of Eq. (6) have been computed (see subsection 4.1), and prior of attempting determination of the finite solu - tions, the vector p pp p of Rodrigues parameters is replaced by vector q qq q= (q 1 , q 2 , q 3 ) T , related to the former by the ensuing relation =qLp (12) where L LL L i s a 3 × 3 non-singular constant matrix whose third row is not orthogonal to each non-vanishing vector (x 1 , x 2 , x 3 ) T that solves Eq. (7). By selecting γ = 3 and replacing q 1 and q 2 with quantities z 1 / z 0 and z 2 / z 0 , Eq. (9) turns into ( ) ( ) () ,3,3,0 , 1 ,2; 1,2 22 33, 3 3, 3 0 0 1, ,3 ij k i j i k i k i ij i j i kkk Azz A q B zz AqBqCz k =≤ = ′′′ ++ ′′′ +++== ∑∑ (13) where coefficients A ij,k , B i,k , and C k , depend on the coefficients of Eq. (6) and on the chosen matrix L LL L. By applying the elimination procedure de - scribed in the previous subsection, the correspondent of Eq. (11) is ′ = 3 det ( ) 0 qN (14) Differently from Eq. (11), Eq. (14) does not lose trace of the finite solu - tions of Eq. (6), because any solution at infinity in terms of p pp p involves a vector q qq q whose third component, q 3 , approaches infinity too. 29 ′ ′ ′ which is a univariate polynomial equation in the unknown . q 3 3 × 3 Determining the Rotation Matrics 5. 5. 5. 5. The ensuing linear equation set in the direction cosines is considered : 21 22 23 31 32 33 11 12 21 22 33 rrr10 rrr10 rrr3rr10 ⎧ +++= ⎪ ⎪ ⎪ +++= ⎨ ⎪ ⎪ +++ −+= ⎪ ⎩ In terms of homogenized Rodrigues parameters ( x 1 , x 2 , x 3 , these equa - tions have three solutions at infinity, i.e., (1, −1,0), (0,1, −1), and (1,0,0). Since each Rodrigues parameter is finite for at least one solution at infin - ity, the change of variable in Eq. (12) is crucial. The third row of LL is ex pressly chosen not normal to each of the three solutions at infinity. A possible expression for L L L is 1 0 0 0 1 0 1 1 1 ⎛ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ − − ⎝ ⎠ L Following the change of variables in Eq. (12), Eq. (14) yields 54 3 2 33 3 3 3 9 54 126 57 9 0qq q q q−+ − + −= The only real root of this equation is q 3 = 3. Back-substitution of this root into the analogous of Eq. (10) completes determination of vector q qq q =( −1,1,3) T . Next, Eq. (12) results into p pp p=(−1,1, −1) T . The rotation matri - ces corresponding to the four real solutions − three at infinity in terms of Rodrigues parameters, and the other finite − are respectively (see Eq. 4): 010 100 100 001 1 0 0 , 0 0 1 , 0 1 0 , 1 0 0 . 001 0 10 001 0 10 − − − − −− − − − − 6. 6. 6. 6. Conclusions ConclusionsConclusions Conclusions matrices satisfying three linear equations in the direction cosines. The proposed procedure is based on the Rodrigues parametrization of orienta - tion and takes advantage of a classical algebraic elimination method in order to solve a set of three quadratic equations in three unknowns. To avoid neglecting any possible 3 × 3 rotation matrix, the classical 30 ) L Numerical Example Numerical Example This paper has presented a new procedure to find all 3 × 3 real rotation C. Innocenti and D. Paganelli ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ tion method has been extended in the paper so that it keeps effective even in case one or more Rodrigues parameters approach infinity. A numerical example has shown application of the proposed procedure to a case study. References ReferencesReferences References Bottema, O., and Roth, B. (1979), Theoretical Kinematics, North-Holland Pub - lishing Co., Amsterdam, NL. Callegari M., Marzetti P., and Olivieri B. (2004), Kinematics of a Parallel Mecha - nism for the Generation of Spherical Motions, On Advances in Robot Kine mat - ics (J. Lenarčič and C. Galletti (eds.)), Kluwer Academic Publishers, the Neth erlands, pp. 449-458. Gosselin, C.M., Sefrioui J., and Richard, M.J. (1994), On the Direct Kinematics of Spherical Three - Degree - of - Freedom Parallel Manipulators of General Archi - tecture, ASME Journal of Mechanical Design, vol. 116, no. 2, pp. 594-598. Husain, M., and Waldron, K.J. (1994), Direct Position Kinematics of the 3-1-1-1 Stewart Platforms, ASME Journal of Mech. Design, vol. 116, no. 4, pp. 1102- 1107. Innocenti, C. (1994), Direct Position Analysis in Analytical Form of the Parallel Manipulator That Features a Planar Platform Supported at Six Points by Six Planes, Proc. of the 1994 Engineering Systems Design and Analysis Confer- ence, July 4-7, London, U.K., PD-Vol. 64-8.3, ASME, N.Y., pp. 803-808. Innocenti, C., and Parenti-Castelli, V. (1993), Echelon Form Solution of Direct Kinematics for the General Fully-Parallel Spherical Wrist, Mechanism and Machine Theory vol. 28, no. 4, pp. 553-561. Roth, B. (1993), Computations in Kinematics, in Computational Kinematics , Kluwer Academic Publisher, the Netherlands, pp. 3-14. Salmon, G. (1885), Modern Higher Algebra, Hodges, Figgis, and Co., Dublin. Semple, J.G., and Roth, L. (1949), Introduction to Algebraic Geometry, Oxford University Press, London, UK. Wampler, C.W. (2006), On a Rigid Body Subject to Point-Plane Constraints, ASME Journal of Mechanical Design, vol. 128, no. 1, pp. 151-158. Wohlhart, K. (1994), Displacement Analysis of the General Spherical Stewart Platform, Mechanism and Machine Theory, vol. 29, no. 4, pp. 581-589. Appendix AppendixAppendix Appendix Let f ff f(g gg g) be an n - dimensional vector function that depends on an n-dimensional vector g gg g. If all components of f ff f are homogeneous functions of the same degree in the components of g gg g, for any non-vanishing solution of the following homogenous system 31 elimina , 3 × 3 Determining the Rotation Matrics the ensuing condition holds (Salmon, 1885) D ∇=0 (2-A) where D is the determinant of the Jacobian matrix of f ff f. Sylvester (Salmon, 1885) has suggested the following procedure in or - der to assess whether a set of three second-order homogeneous equations in three unknowns has non - vanishing solutions : i) compute the determinant D (which is a third-order homogeneous polynomial in the components g i , i = 1, ,3, of vector g gg g); ii) determine the gradient of D (its components are quadratic homo- geneous polynomials in g i , i = 1, ,3); iii) consider Eqs. (1-A)-(2-A) as a set of six equations that are linear and homogeneous in the six monomials g i g j (i,j = 1, ,3, i ≤ j) ( ) = 222 1 23121323 T ggggggggg H0 (3-A) where H HH H is a 6 × 6 matrix whose elements are functions of the coef- ficients of Eq. (1-A). The original set of three homogeneous quadratic equations has non- vanishing solutions if and only if the ensuing condition is satisfied =det 0 H (4-A) 32 =()fg 0 (1-A) C. Innocenti and D. Paganelli [...]... selecting the robot joints and the driver locations are discussed In this paper it is assumed that the readers have a basic knowledge of the displacement groups in kinematics, as can be found, for instance, in Hervé, 1978 and in Section 5.3 of Angeles, 1988 2 Single-loop Kinematotropic Chains Figure 1 shows three single-loop kinematotropic chains presented by Galletti and Fanghella, 20 01 They are shown in. .. 20 00 Merlet J -P Singular configurations of parallel manipulators and Grassmann geometry Int J of Robotics Research, 8(5):45–56, October 1989 Merlet J -P and Daney D A formal-numerical approach to determine the presence of singularity within the workspace of a parallel robot In F.C Park C.C Iurascu, editor, Computational Kinematics, pages 167–176 EJCK, Seoul, May, 2 0 -2 2, 20 01 Monsarrat B and Gosselin... displacement group G1, and the subchain from body a2 to b2 generates a group G2 The kinematic pairs KPa and KPb provide displacements between bodies a1 and a2, and b1 and b2 respectively: the meaning of G1, G2, KPa , and KP b, are given in Table 1 51 Parallel Robots that Change their Group of Motion Let G be the intersection group between G1 and G2 By moving KPa and KPb the group G changes and, as a consequence,... mechanisms in which variations in the position variables can 49 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 49–56 © 20 06 Springer Printed in the Netherlands 50 P Fanghella, C Galletti and E Giannotti result in changes in the permanent finite mobility of the mechanisms Wohlhart, 1996, called this very peculiar mobility property kinematotropy In this paper, some results concerning kinematotropic... d.o.f., with drivers P1, P2 and P3 Starting again from the singular position, by moving driver P4 the plane formed by pairs P1-P11-P 12, and the planes formed by P2-P7-P8 and P3-P9-P10 become not parallel, so their intersection group gives a prismatic constraint with axis parallel to the common axis of pairs P4, P5, and P6 Therefore, the robot acts as “cylindrical” platform, with 2 d.o.f., with two drivers:... Theoretical Kinematics, MIT Press Ravani, B., and Roth, B., (1983), Motion synthesis using kinematic mappings, ASME Journal of Mechanisms, Transmissions, and Automation in Design, vol 105, pp 46 0-4 67 Schilling, R.J., and Lee, H., (1988), Engineering Analysis- a Vector Space Approach, Wiley & Sons Shoemake, K., and Duff, T., (19 92) , Matrix animation and polar decomposition, Proc of Graphics Interface ’ 92, pp 25 8 -2 64... Oulu, Finland Larochelle, P., Murray, A., and Angeles, J., (20 04), SVD and PD Based Projection Metrics on SE(n), in Lenarˇiˇ, J and Galletti, C (editors), On Advances in Robot cc Kinematics, Kluwer Academic Publishers, pp 1 3 -2 2, 20 04 Larochelle, P., and McCarthy, J.M., (1995), Planar motion synthesis using an approximate bi-invariant metric, ASME Journal of Mechanical Design, vol 117, no 4, pp 64 6-6 51... Printed in the Netherlands 34 P M Larochelle referred to as being bi-invariant, is desirable Interestingly, for the specific case of orienting a finite rigid body in SO(n) bi-invariant metrics do exist Larochelle and McCarthy, 1995 presented an algorithm for approximating displacements in SE (2) with spherical orientations in SO(3) By utilizing the bi-invariant metric of Ravani and Roth, 1983 they arrived... be achieved by all chains, 3 legs are introduced to connect the platform and the robot frame The general scheme of the kinematic chain of the robots is shown in Fig 3 The kinematic chains of the 3 robots obtained in this way are reported in Fig 4 Each chain is shown in the singular position that separates the two branches of positions belonging to different groups (E, S, Y and C) In order to exemplify... They are shown in the singular position that separates the two branches of positions in which the displacements between bodies a1 and b2 belong to different displacement groups b1 KPb GR1 KPb b2 b1 GR2 a) a2 a1 GR1 b2 KPa a1 GR1 GR2 a2 b) b1 a1 KPa KPb c) b2 a2 KPa Figure 1 GR2 Single-loop kinematotropic chains All these chains can be represented by the scheme of Fig 2 The subchain from body a1 to b1 . are 23 © 20 06 Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 23 32. – require determination of the 3 × 3 rotation matrices whose nine direction. in the unknown . q 3 3 × 3 Determining the Rotation Matrics 5. 5. 5. 5. The ensuing linear equation set in the direction cosines is considered : 21 22 23 31 32 33 11 12 21 22 . matrix that involves the full twist of the end-effector. Indeed for a robot with n d.o.f. © 20 06 Springer. Printed in the Netherlands. 41 J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics,