roceedings of the World Congress on Engineering 212 Vol III WCE 212, July 4-6, 212, London, U.K. Singularity Analysis of a arallel Manipulator with Cylindrical Joints Zhumadil Zh. Baigunchekov, and Myrzabay B. Izmambetov Abstract In this paper we are presenting some results of the singularity analysis of the new parallel manipulator with cylindrical oints. We have derived the differential kinematic relations between two vectors: mobile-platform velocity and the active-oint rates. hese relations comprise two matrices, the forward - and the inverse - kinematics Jacobians. he analytical approach to the research of these relations, based only on linear algebra, has yielded interesting results on identification of the singular configurations of the parallel manipulator. Index erms Jacobian matrix, mobile platform, parallel manipulator, singularity analysis. I. INRODUCION arallel manipulators in comparison with serial manipulators have big carrying capacity, high velocities and positioning accuracy [1]. However, parallel manipulators have many singularity configurations in which they cannot carry out their functional tasks [2]. Gosselin and Angeles [3], sai [4] describe three types of singularity, based on the properties of the Jacobian matrices. he Jacobian of a parallel manipulator can be divided into two matrix: one matrix J x associated with the direct kinematics and the other matrix J q associated with the inverse kinematics. In this paper some results for identification of the singularity configuration of a parallel manipulator with cylindrical oints (M 3CCC based on research of degeneration of Jacobian matrices are presented. II. GEOMERY OF A M 3CCC A M 3CCC is formed by connection of a mobile platform 3 with a base by three spatial dyads АВС, DEF, GHI of kind СCС, where C - cylindrical oint (Fig. 1. A spatial kinematic chain with two links and three kinematic pairs is called a spatial dyad which has zero degree-of-freedom. As a spatial dyad has zero degree-of-freedom it does not impose restriction on a mobile platform 3 and six degree-of-freedom of a mobile platform is remained. Each cylindrical oint has two degree-of-freedom: one rotation and one translation. Six movements of input cylindrical oints A, F, I (three rotations and three translations are generalized coordinates of a parallel manipulator with cylindrical oints M 3CCC. A M 3CCC is intended for reproduction of movement of a mobile platform 3 or the frame X Y Z attached to it with respect to base frame U o V o W o U U q(, V V q(, W W q( q(, q(, q(, where q(=[ 7 (, s 7 (, 8 (, s 8 (, 9 (, s 9 (,] - a vector of the input generalized coordinates;, и - the components of relative orientation of coordinate systems X Y Z and OU o V o W o. W O U B C 2 6 D 7 s 7 V A Fig. 1. arallel manipulator with cylindrical oints M 3CCC For definition of constant and variable parameters of parallel manipulator we fixed the right Cartesian coordinate systems UVW and XYZ to each element of each oint. he axes W and Z of coordinate systems UVW and XYZ are directed on a axis of rotation and translation of cylindrical oints. ransformation matrix Z 3 Р E 1 7 8 k between coordinate systems UVW and X kyk Z fixed on the ends of binary link k has a k view (Fig. 2 F 4 s 8 Y X 5 G I s 9 9 H (1 Manuscript received March 22, 212; revised April 13, 212. Zh. Zh. Baigunchekov is with the Department of Science, Kazakh-British echnical University, Almaty, Kazakhstan (phone: +7(727-2-727-572; fax: +7(727-2-72-489; e-mail: bzh47@mail.ru. M. B. Izmambetov is with the Research Laboratory of Mechatronics and Robotics, Kazakh-British echnical University, Almaty, Kazakhstan (e-mail: m.izmambetov@kbtu.kz. k k ( a k, b k, c k, k, k, k ISBN: 978-988-19252-2- ISSN: 278-958 (rin; ISSN: 278-966 (Online WCE 212
roceedings of the World Congress on Engineering 212 Vol III WCE 212, July 4-6, 212, London, U.K. W Z k V // W Y k О U k а k О k k Х k ( t k l Fig. 2. Binary link k of kind CC t = t t t 11 21 31 41 t12 t22 t32 t42 t13 t23 t33 t43 t14 t 24 1 = t34, (2 τk Rk t44 OO7 io7io1 io1 io2io2io3i O can be written as r ro s i i a i s i a i s i g 7 7 71 1 12 2 i, i 1, 2, 3, (3 where t 11 1, t 12 t13 t14, t21 a k cos k b k sin k sin k, t cos cos sin cos sin, 22 k k k k k t23 cos k sin k sin k cos k cos k, t24 sin k sin k, t31 a k sin k b k cos k sin k, t32 sin k cos k cos k cos k sin k, t33 cos k cos k cos k sin k sin k, t34 cos k sin k, t41 c k b k cos k, t42 sin k sin k, t43 sin k cos k, t44 cos k. he following six parameters define the relative positions of the two coordinate systems U V W and X kyk Z : a k k - a distance from axis W to axis Zk which is measured along the direction of tk; tk a common perpendicular between axes W and Zk; k - an angle between positive directions of axes W and Zk which is measured counter clockwise relatively to positive direction of tk; bk - a distance from direction of tk to direction of the axis Xk which is measured along positive direction of an axis Zk; k - an angle between positive directions of tk and axis Xk which is measured counter clockwise relatively tо positive direction of axis Zk; ck- a distance from direction of an axis U to direction of tk which is measured along positive direction of an axis W; k - an angle between positive directions of axis U and tk which is measured counter clockwise relatively to positive direction of an axis W. where Fig. 3. Geometry of the i-th leg of a M 3CCC III. GENERAION OF JACOBIAN MARICES 3MCCC has same three legs of kind CCC. Let s consider the i-th leg which has a kinematic chain OO7 io7io1i O1 io2io2io3i (Fig. 3. A loop-closure equation for the kinematic chain 1 1 1 1 a cos r r a sin c (4 ISBN: 978-988-19252-2- ISSN: 278-958 (rin; ISSN: 278-966 (Online WCE 212
roceedings of the World Congress on Engineering 212 Vol III WCE 212, July 4-6, 212, London, U.K. 1 1 g i O2 i c 2i a 2i g i gi 1 1 = a2i cos 2i 2i (5 a2i sin 2i c2i Differentiating (3 with respect to time, we obtain r s 7ie7i θ 7i r7i s 1i e1i θ 1i a12i s 2ie2i ω fi, i 1, 2, 3 (6,,, V - a vector U W of velocity of a point Р; s 7i s 7i e7i, θ 7i 7i e7i - vectors of velocity of the active oints; r7i a71i s1i a12i ; s 1i e1i, θ 1i 1i e1 i, s 2i e2 i - vectors of velocity of the passive oints; where r U V, W U V W,, - a vector of angular velocity of a mobile platform; fi O2i g i s 2i. Dot-multiplying both sides of (6 by a 12i leads to IV. SINGULARIY ANALYSIS OF HE M WIH 6 DOF For a M 3CCC the first type of singularity corresponds to the case when the matrix J q has rank deficiency, i.e. when at least the elements one of the rows of this matrix are zero. hen shall be fulfilled the following conditions a 12 i e7i, (9 e 7 i ( r7i a12i. (1 From the first condition (9 a 12i e7i is followed i.e. the vector a 12i determining the shortest distance between the axes of passive cylindrical oints is perpendicular to the axis of the input cylindrical oint. It is easy to see that this is possible in cases а if 71i or, in always, (11 b if, in only 71i or. (12 1i he case (11 defines the third type of singularity configuration when a finite change of the coordinate s 7i has no effect on the movement of the platform (Fig. 4. a12 i r s 7 i ( a12 i e 7i θ 7 i e7 i ( r7i a12i ( fi a12i, i 1, 2, 3 (7 s 2i Equation (7 can be presented in the matrix form a 12i J xx J qq (8 U 7 by using the notation x r,, q s71, 71, s72, 72, s73, 73, a 71i Z 7i, W 7i where the Jacobian matrices J x and J q are defined as follows Z 1i, W 1i 7i J x a a a 12,1 12,2 12,3 ( f a ( f ( f 1 2 3 a a 12,1 12,2 12,3, s 7i Fig. 4. he third type of singularity of the M 3CCC J q a12,1 e71 e71( r71 a12,1 a12,2 e72 e72 ( r72 a12,2 a12,3 e73 e73 ( r73 a12,3 he case (12 implies that a 12 i a 71i. At the same time the condition (1 implies that the three vectors e 7 i, r7i, a12i must lie either in parallel planes or in the same plane. his arrangement of these vectors, subect to (12 holds only when the point O 2 i defining the end of the vector r 7i lies in the plane O7 iu 7iW7i (this plane is denoted by i (Fig. 5. ISBN: 978-988-19252-2- ISSN: 278-958 (rin; ISSN: 278-966 (Online WCE 212
roceedings of the World Congress on Engineering 212 Vol III WCE 212, July 4-6, 212, London, U.K. s 2i U 7 the last three columns - the second group of components of the vectors n f a relative to an inertial coordinate system, i.e. i i 12i i Z 1i, W 1i, a 12i 7i a 71i s 7i Fig. 5. he first type of singularity of the M 3CCC Since the vectors a 12 i, a 71i determine the shortest distance between the cylindrical oints, in this type of configuration the i-th leg is fully stretched or folded position. herefore the set of Cartesian velocities of platform which corresponds to the velocities of point O 2 i parallel to the vectors a 12 i, a 71i may not be reproduced. his set of the Cartesian velocities is given by the set of rotation of platform around an arbitrary line in the plane containing the point O 2 i and orthogonal to the vectors a 12 i, a 71 i. In addition, any force applied to the platform and line of action which lies in the i, and a pair of forces applied parallel to this plane do not affect the rotational component of the i-th cylindrical actuator. his is explained by the fact that the moments of force and a pair of forces with respect to the axis O 7 iz 7 i, i.e. relative to the axis of rotation of the input oint are zero. he second type of singularity occurs when matrix J x lacks the range, that is when rows or columns of this matrix are linearly dependent. his type of singularities in contrast to the first is located inside the workspace. For the corresponding configuration of the M3CCC the nonzero output Cartesian velocities x of the platform are displayed in the zero vector by conversion J x. hese speeds of platform are possible even in still actuators of input oints. Let s rewrite the matrix J x in the form a12,1x a12,1y a12,1z n1x n1y n1z J x a12,2x a12,2y a12,2z n2x n2y n2z, (13 a12,3x a12,3y a12,3z n3x n3y n3z where the elements of the first three columns are the components of one group of vectors a, and elements of 12i Z 7i, W 7i ISBN: 978-988-19252-2- ISSN: 278-958 (rin; ISSN: 278-966 (Online 12i [ 12iX, a12iy, a12iz i fi a12i [ ix, niy, niz a a ], n n ], i 1, 2, 3., (14 For definiteness assume that the vectors g i lie in a plane perpendicular to the axes of the platform and kinematic cylindrical oints of platform. Consider the case where the three columns of vectors are linearly dependent. his case means that all the vectors of this group are parallel to each other and parallel to a common plane. Let s assume without proof the the following theorem. heorem. wo vectors laying in different and mutually intersecting planes are parallel if and only if each of these vectors are parallel to the line along which their planes are intersected. he statement 1. he parallelism of all vectors n i, i 1,2,3 is possible only if each of these vectors are parallel to the axis Z of the coordinate system X Y Z associated with the mobile platform. his statement means that in this case the vectors f i and a 12i, i 1,2, 3 according to the definition of the vector product of two vectors must lie in the plane РХ Р У Р. hen all cylindrical oints axis of the platform must also lie in the plane РХ Р У Р (this plane is denoted by Ω (Fig. 6. hus we find that the last z-components of vectors a 12i, i 1,2, 3 with respect to the local coordinate system X Y Z should be zero, i.e. a 12iZ, i 1,2, 3. We can get the form (8 with respect to the coordinate system X Y Z associated with the platform. For this case the third, fourth and fifth columns of the matrix J x will be zero. hen the zero space of the matrix J x is generated by the vector [,, 1, 1, 1, ]. his zero space corresponds to the set of local rotations of the platform relative to an arbitrary axis of plane X Y and local displacement along the axis z with fixed actuators. In addition, the force acting along the axis z and a pair of forces applied to the platform in a parallel axis z of plane do not affect to actuators, i.e. manipulator is not able to resist these loads. he statement 2. All vectors n i, i 1,2, 3 can be simultaneously parallel to one arbitrary plane if all axis of cylindrical oints of platform are orthogonal to this plane. Assume that all axis of cylindrical oints are orthogonal to the plane X Y of the platform (Fig.7. hen the vectors a12i and n i i 1,2, 3 lie in planes parallel to the plane X Y and therefore a and n, i 1,2, 3. hat is 12iZ the third and fourth columns of the matrix J x, compiled on a system of coordinates X Y Z, will be zero. In this case the zero space of matrix J x is generated by vector [,, 1, 1,, ]. his zero space corresponds to a local rotation of the platform relative to the axis Z and its finite movement along the axis Z with fixed actuators. In addition, the iz WCE 212
roceedings of the World Congress on Engineering 212 Vol III WCE 212, July 4-6, 212, London, U.K. Z а 12,2 Y X а 12,1 g 2 g 1 g 3 а 12,3 Ω Fig. 6. he second type of singularity of the M 3CCC Z а 12,2 а 12,1 X Y g 2 а 12,3 Ω g 1 g 3 Fig 7. he third type of singularity of the M 3CCC force acting along the axis Z and a pair of forces applied to the platform in a parallel axis Z of a plane do not affect to the actuators, i.e. parallel manipulator is not able to resist these loads. hus, we found another case that led to the singularity of the third kind when the axis of the cylindrical oints of the platform are perpendicular to its plane, which leads to the finite no controlled movements of platform in the orthogonal direction to the plane of platform in the fixed actuators. V. CONCLUSION On the basis of the equations of closed kinematic chains of the legs of a M3CCC the velocity vectors are formed. On the basis of the Jacobi matrix the conditions for the existence of singular configurations are defined and their geometrical and physical interpretations are given. hese results can be used in designing of construction of such manipulators. Besides the obtained conditions of configuration singularities can be expressed through constants and variable parameters of the parallel manipulator that is important for the control of such configurations. REFERENCES [1] J.. Merlet, arallel Robots, Kluwer Academic ublishers, London, 2. [2] Zh. Baigunchekov, and M.Izmambetov, "Singularity Analysis of the New arallel Manipulator with 6 Degree-of-Freedom", World Congress on Engineering WCE21, roceedings, Vol. II, 21, pp. 1472-1477. [3] C. Gosselin, and J. Angeles, "Singularity analysis of closed-loop kinematic chains," IEEE rans. Robotics & Autom., Vol. 6, No. 3, 199, pp. 281-29. [4] L. W. sai, Robot analysis: the mechanics of serial and parallel manipulators, John Wiley & Sons, Inc., New York, 1999. ISBN: 978-988-19252-2- ISSN: 278-958 (rin; ISSN: 278-966 (Online WCE 212