3.2 Synthesis Procedure The synthesis of a US-PM amounts to finding the location of the U and S joints on the base and on the platform respectively, and the lengths of the connecting legs. That is, if for each leg k, we introduce the column array of components p k >p k ; q k ; r k @, with respect to S 1 , of the point P k which is the center of either the U or the S joint on the platform, the column array of components B k >A k ; B k ; C k @, with respect to a frame parallel to S 0 but centered in C, of the point B k which is the center of either the S joint or the U joint on the base, and the length l k = _P k B k _ of the connecting leg between P k and B k , we have to search for N unknown arrays of geometric parameters, i.e. Q k > p k ; B k ; l k @ for k 1, ,N. Use of the method M1 or M2 makes it possible to find the conditions for Q k so that the leg P k B k fits within the desired 2-dof spherical motion between platform and base. Method M1 : Assuming the description and the parameterization of the orientation introduced in Section 3.1, the length of the k-th US-leg is given by PB kk k k k k l RlpB, (2) which, according to Eq. 1 is a function of the motion parameters D and E . However, for the k-th US-leg to comply within the desired motion of the platform, the length l k must not depend on D and E . This means that if one considers the expression 2 2 cscss scccs 2, sc kk k k kk k k k k kk k pq r A lpqrBf qrC DDEDE DDEDE EE DE ªº «» «» «» ¬¼ l , (3) where , (4) 222 2 2 2 cscss scc , cs s c kk kk kk kk kk kk k k kk k k k k k k Ap Aq Ar Bp Bq f Br Cq Cr p q r A B C DDEDE DDE DE E E DE · § ¸ ¨ ¨¸ © ¹ all the coefficients in c D , s D , c E and s E of the polynomial f D , E must vanish, and the length of the k-th leg must become 222222 kkkkkk lpqrABC k . (5) Vanishing of the coefficients of Eq. 4 leads to the following conditions 0 kk Ap , 0 kk Aq , 0 kk Ar , 0 kk Bp , (6.1) 0 kk Bq , 0 kk Br , 0 kk Cq , 0 kk Cr , (6.2) to be satisfied simultaneously for each leg P k B k . R. Vertechy and V. Parenti-Castelli 388 It turns out that the only non trivial solutions, i.e. l k z 0, are obtained for the following two sets of parameters ^ ` 1 222 ;;; 0; 0;0; iiiiiiiiiii A BCp q r l ABC Q , (7) ^ ` 2 22 0; 0; ; ; 0; 0; j jj jjjjjjj A BCpqrlpC Q . (8) Method M2: Since for a general motion of the US-PM the parameters D and E are functions of time, based on the time derivative of the equation cscss scccs sc kk k k kkkkk k k kk k pq r A pq r B qrC DDEDE DDEDE EE ª º « » « » « » ¬ ¼ RlpB , (9) the relative velocity vector v k , of point P k and point B k , is given by scccs sssc cscss cscc 0c kk k k k kkk k k k k pq r q r pq r q r qr DDEDE DEDE DDEDE DEDE EDE s DE ªº ªº «» «» «» «» «» «» ¬¼ ¬¼ v , (10) where D and E are the first time derivatives of the motion parameters D and E , respectively. Since the legs P k B k have constant length l k , it must hold that, during the motion, the components of the velocity v k in the direction of the leg axis must be zero, i.e. ,, kk gh DED DE E 0 vl s , (11) where ,scccscscs kk kk kk kk kk kk gpAqArApBqBrB DDEDEDDEDE DE , (12) ,csccssscc kk kk kk kk kk kk hqBrBqArAqCrCs DE DE DE DE E DE E . (13) Since this condition must hold for every configuration and motion of the platform (that is it should not depend on D , E , D and E ), all the coefficients in c D , s D , c E and s E of the two polynomials g D , E and h D , E must vanish. Vanishing of the coefficients of Eqs. 12-13 leads to the same conditions expressed by Eq. 6 and, therefore, to the non-trivial solutions represented by Eqs. 7-8. That is, legs characterized by the set of parameters 1 i (type-1) and Q 2 j Q (type-2) are the only ones which comply with the desired 2-dof spherical motion. In Section 3.3 we will show that a feasible mechanism requires both types of legs. 389 Synthesis of 2-DOF Spherical Fully Parallel Mechanisms 3.3 Generation of Mechanisms Equations 7-8 provide the geometric conditions for legs of US-type to fit within the desired spherical motion between platform and base. The conditions identify two types of legs. Legs of type-1 have one joint located at point C in the platform and the other joint located anywhere in the base. Legs of type-2 have the joint in the based located on k k 0 axis and the joint in the platform located on i i 1 axis. Generation of mechanisms amounts to combining a proper number I of legs of type-1 and a proper number J of legs of type-2. Of course, the choice of I and J clearly affects the mechanism architecture and its feasibility. In particular, certain conditions on I and J must be satisfied. First, in order for the mechanism to have 2-dof, the axes of the legs in the set ^ 112 2 11 , , , , , I J QQQQ` must belong to a linear variety of lines with rank 4 (Merlet, 1989), usually referred to as linear line congruence. Therefore, at least four legs with linear independent axes are needed, i.e. . Here, the axis of the k-th leg is defined as the line through point P 4IJt k to point B k . Second, since the legs (of type-1) in the set ^ 11 1 , , I QQ` pass through the common point C, i.e. the center of the reference frame 1 , while the legs (of type-2) in the set ^ S 22 1 , , J QQ` lie in the plane defined by the vectors k 0 and i 1 , the axes of the legs within each type form, at most, a linear variety of lines with rank 3 (Merlet, 1989). Indeed, the axes of the legs within the family of type-1 generate at most a bundle of lines, while the axes of the legs within the family of type-2 generate at most a plane of lines. Therefore, in order for the set of geometric parameters ^ 112 2 11 , , , , , I J QQQQ` to define a linear variety with rank 4, at least one leg for each type is needed, i.e. 1 I t and . 1J t That is, in a feasible 2-dof spherical fully parallel mechanism with legs of US-type, the axes of all the legs must define a degenerate congruence, i.e. the variety of lines which lie in the plane defined by unit vectors k 0 and i 1 or pass through the point C of that plane. In practice, depending on the varieties of lines spanned by the axes of the legs within a type, three families of mechanism architectures can be identified: x Family-1 (Fig. 2): The axes of the legs in the set ^ 1 1 1 , , I QQ , , ` define a linear variety with rank 1, i.e. a single line passing through C but with direction different to k 0 , and the axes of the legs in the set ^ 1 22 J QQ , , ` define a linear variety of lines with rank 3, i.e. a plane of lines defined by k 0 and i 1 . x Family-2 (Fig. 3): The axes of the legs in the set ^ 1 1 1 I QQ` define a linear variety with rank 2, i.e. a planar pencil of lines with center in R. Vertechy and V. Parenti-Castelli 390 C but which does not contain the line through k 0 , and the axes of the legs in the set ^ 1 22 , , J QQ , , ` define a linear variety of lines with rank 2, i.e. a planar pencil of lines in the plane defined by k 0 and i 1 . x Family-3 (Fig. 4): The axes of the legs in the set ^ 1 1 1 I QQ , , ` defines a linear variety of lines with rank 3, i.e. a bundle of lines centered in C, and the axes of the legs in the set ^ 1 22 J QQ` define a linear variety of lines with rank 1, i.e. a single line in the plane defined by the unit vectors k 0 and i 1 but which does not pass through C. Figures 2-4 depicted the basic (non-over-constrained) mechanisms which are the generators of the three families. For convenience, in the figures, both U and S joints are represented by circles. Moreover, addition of legs of type - 1 and/or of type - 2 to such basic US- PMs does not alter the mechanism kinematics but renders the systems redundant and with self-motion. Examples of over - constrained US-PM with five and six US - legs are depicted in Fig. 5. For ease of understanding, the redundant legs are drawn in long - dash - dot lines. Note that over - constrained architectures have several advantages with respect to the basic ones. Indeed, the former make it possible to augment the mechanism stiffness - to - weight (a) (b) 391 Figure 2 . Family - 1 . Figure 3 . Family - 2 . Figure 4 . Family - 3. Figure 5 . Over - constrained mechanisms. Synthesis of 2-DOF Spherical Fully Parallel Mechanisms and -to-encumbrance ratios, the mechanism strength-to-weight and -to- encumbrance ratios, allow the mechanism to be preloaded so as to reduce system backlash, and allow the system to be built through simpler elements such as rafters and wireropes. As an example, the mechanism depicted in Fig. 5.a can be made by means of one rafter (leg drawn in long-dash-dot line) and by four wireropes (legs drawn in solid line). Besides, over-constrained architectures further limit the range of motion of the mechanism and render its assemblage more complex. From a kinematic standpoint, note that many of the U and S joints of the mechanisms depicted in Figs. 2-5 may, in practice, be suppressed and/or replaced by simpler pairs. Indeed, joints which are not placed along the axes k 0 and i 1 are idle; the joints which are placed either on k 0 or i 1 work as simpler revolute joint with rotation axis along k 0 or i 1 , respectively; and the joints which are placed on both k 0 and i 1 work as U joints with rotation axes along k 0 and i 1 . However, from the kinetostatic standpoint, suppression of the idle degrees of freedom of the U and S joints makes the legs to bear consistent flexional loads which may cause the mechanism to be oversized. Replacement of the idle pairs with elastic hinges introduces much smaller flectional loads and, therefore, may be the most effective way to implement the mechanism. 4 Actuation The mechanisms presented in Section 3.3 are inherently suited to be actuated in-parallel by motors with linear motion. In practice, the addition of two U P S-legs (P stands for actuated prismatic P joint), each connecting the base and the platform by means of a U joint and an S joint, provides a very simple means to fully control the mechanism throughout the desired range of motion. R. Vertechy and V. Parenti-Castelli 392 Figure 6 . Actuated mechanism of family-3. . decouple the motion about the axes k 0 and i 1 . Indeed, since a force is not able to generate moments about the lines it crosses, it is clear that every U P S-leg whose connecting joint on the base is centered in a point B k , which lies on k 0 , makes it possible to control rotations about i 1 only, while every U P S-leg whose connecting joint on the platform is centered in a point P k , which lays on i 1 , makes it possible to control rotations about k 0 only. A decoupled actuated manipulator obtained from a US-PM of family-3 is represented in Fig. 6 (U P S-legs are drawn as telescopic legs). The actuated U PS-leg, P 5 B 5 , controls the rotation about the axis k 0 only, while the actuated U P S-leg, P 6 B 6 , controls the rotation about the axis i 1 only. Note that the manipulator obtained from the mechanisms of family-3 coincides with the fully parallel spherical wrist with the P actuator on the leg P 4 B 4 locked. 5 Kinematic, Workspace and Singularity Analyses Due to the decoupled actuation of the rotations of the mechanism about the k 0 and i 1 axes, the direct kinematic, workspace and singularity analyses are very straightforward. Indeed, these problems are reduced to the study of two spatial Whitworth’s quick-return mechanisms. Solutions of these problems can be accomplished as in Di Gregorio and Sinatra, 2002, Di Gregorio, 2002, and Carricato and Parenti-Castelli, 2004. 6. Conclusions This paper presented the synthesis of 2-dof spherical fully parallel mechanisms. In particular, by means of two analytical methods derived from the literature, three families of 2-dof spherical mechanisms have analyses have been addressed which show that such mechanisms are very easy to analyze and control. 7. Acknowledgements Support of this work by the Advanced Concept Team (ACT) of The collaboration and discussions with Dr. Carlo Menon of the ACT are acknowledged and appreciated. 393 In practice, proper placements of the actuators make it possible to been synthesized. These families comprise over-constrained mecha - nisms too. Actuation issues and kinematic, workspace and singularity European Space Agency (ESA) under ESTEC/Contract No. 18911/05/ NL/MV is gratefully acknowledged. Synthesis of 2-DOF Spherical Fully Parallel Mechanisms . References Shönflies, A. (1886), Geometrie der Bewegung in Sinthetischer Darstellung , Lipzieg. Bricard, R. (1906), Memoire sur les Displacements a Trajectoires Spheriques, Borel, E. (1908), Memoire sur les Displacements a Trajectories Spheriques, Grassmann Geometry, The Int. J. of Robotic Resear h , vol. 8, pp. 45-56. sc s c st o c Ma, O. and Angeles J. (1992), Architecture Singularities of Parallel Manipulators, Int. 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(1998), Generation of Singular 6-SPS Parallel Manipulators, Proc. of 1998 ASME Design Techni al Conferences , 98DETC/MECH-5952. Dunlop, G.R. and Johnes, T.P. (1999), Position Analysis of a two DOF Parallel Mechanism – the Canterbury Tracker, Mechanism and Machine Theory , vol. 34, pp. 599-614. Wiitala, J.M. and Stanisic, M.M. (2000), Design of an Overconstrained Dextrous Spherical Wrist, Journal of Mechanical Design, vol. 122, pp. 347-353. Bauer, J.R. (2002), Kinematics and Dynamics of a Double-Gimballed Control Moment Gyroscope, Mechanism and Machine Theory, vol. 37, pp. 1513-1529. Husty, M.L. and Karger, A. (2002), Self Motions of the Stewart-Gough-Platforms, an overview, Proceedings of the Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanism and Manipula ors , Quebec, Canada, pp. 131-141. Di Gregorio, R. and Sinatra, R. (2002), Singulariy Curves of a Parallel Pointing System, MECCANICA , vol. 37, pp. 255-268. Di Gregorio, R. (2002), Analytic Determination of Workspace and Singularities in a Parallel Pointing System, Journal of Robotics Systems , vol. 18, pp. 37-43. Wohlhart, K. (2003), Mobile 6-SPS Parallel Manipulators, J urnal of Robotics Systems , vol. 20, pp. 509-516. Carricato, M. and Parenti-Castelli, V. (2004), A Novel Fully Decoupled Two- Degrees-of-Freedom Parallel Wrist, The Int. J. of Robotics Resear h , vol. 23, pp. 661-667. Gogu, G. (2005), Fully-Isotropic Over-Constrained Parallel Wrists with Two Degrees of Freedom, Proc. IEEE Int. Conf. Robotics and Automation, pp. 4025-4030. R. Vertechy and V. Parenti-Castelli 394 Journal de l’Ecole Polytechnique , vol. 11, no. 2, pp. 1-96. Merlet, J.P. (1989), Singular Configurations of Parallel Manipulators and Memoires Presentes par Divers Savants , Paris, vol. 33, no. 2, pp. 1-128. Machine Theory , vol. 28, pp. 553 561. - CONSTRAINT SYNTHESIS FOR PLANAR -R ROBOTS Gim Song Soh gsoh@uci.edu J. Michael McCarthy jmmccart@uci.edu Robotics and Automation Laboratory University of California, Irvine Irvine, CA 92697 Abstract In this paper, we control the joints of a planar nR chain mechanically using RR dyads and obtain a one degree-of-freedom system that guides the end-effector smoothly through five specified task positions. To solve this problem, we specify the nR chain and determine it configurations when its end-effector is positioned in each of the five task positions. This yields a set of RR chain synthesis problems that constrain alternating links in a way that ensures that the relative joint angles required by the task positions are attained. In general, we cannot guaranteed that the resulting assembly will move smoothly between the task positions without jamming, however we present a strategy based on enforcing symmetry of the nR chain that yields useful solutions. Examples of constrained 3R, 4R, 5R and 6R are discussed. The procedure is general and can be applied to arbitrarily long serial chains. 1. Introduction In this paper, we consider the addition of n-1 RR chains to a planar nR serial chain robot in order to mechanically prescribe its movement through five task positions. In the case of a planar 3R robot this is equivalent to the synthesis of a Watt I six-bar linkage. For a planar 4R, 5R and 6R serial chains, we obtain planar eight-bar, 10-bar and 12-bar linkages. In general, our synthesis results transform an n degree- of-freedom nR chain into a single degree-of-freedom 2n-bar linkage that has 3n −2 revolute joints. Our design equations only ensure that the solution linkage can be assembled in each task position, not that it can move smoothly between the positions. Therefore, it can happen that a resulting linkage has task positions reachable from different assemblies. This is known as a circuit defect ?. A fundamental challenge in linkage synthesis is finding © 2006 Springer. Printed in the Netherlands. 395 J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 395–402. N solutions that have the task positions on the same single circuit. In what follows, we present a synthesis strategy that yields successful constrained nR chains that have the five task positions on the same circuit. 2. Literature Review This work is inspired by ?, who derived synthesis equations for planar nR planar serial chains in which the n joints are constrained by a cable drive. They obtained a “single degree-of-freedom coupled serial chain” that they use to design an assistive device. ? formulated and solved the design equations for six-bar linkages, and ? extended this to eight-bar linkages. Our approach is simpler in that we do not attempt to design the entire 2n-bar linkage, rather we assume the nR serial chain is given, and use standard dyad synthesis theory to solve for individual RR constraints, see ?. Once a linkage has been designed, we analyze it to determine its con- figuration for given values of the input crank, in order to simulate its movement. ? presents an analysis methodology for general planar link- ages using complex number loop equations and the Dixon determinant. However, the our synthesis approach yields linkages that are a series of four-bar loops and are easily analyzed individually. 3. Kinematics Equations of a Planar nR Chain Let the configuration of an nR serial chain be defined by the coor- dinates C i =(x i ,y i ), i =1, ,n of each of its revolute joints. The distances a i,i+1 = |C i+1 −C i | defined the lengths of each link. Attach a frame A i to each of these links so so its origin is located at C i and its x axis is directed toward C i+1 . The joints C 1 and C n are the attachments to the base frame F = A 1 and the moving frame M = A n , respectively, and we assume they are the origins of these frames. The joint angles θ i define the relative rotation about the joints C i . Introduce a world frame G and task frame H so the kinematics equa- tions of the nR chain are given by [D]=[G][Z(θ 1 )][X(a 12 )][Z(θ 2 )][X(a 23 )] [X(a n−1,n )[Z(θ n )][H], (1) where [Z(θ i )] and [X(a i,i+1 )] are the 3×3 homogeneous transforms that represent a rotation about the z-axis by θ i , and a translation along the x- axis by a i,i+1 , repspectively. The transformation [G]definestheposition of the base of the chain relative to the world frame, and [H] locates the task frame relative to the end-effector frame. The matrix [D] defines the coordinate transformation from the world frame G to the task frame H. 396 G.S. Soh and J.M. McCarthy (12, 16) (6, 7) (8, 10) (10, 13) 3R 4R 5R 6R nR (2n, 3n-2) 100R (200, 298) Design n-1 RR chains Figure 1. This shows the kinematic structure of mechanically constrained serial chains. The linkage graph is on the left, which has each link as a node and each R joint as an edge. The contracted graph on the right shows only links with three or four edges as nodes. This shows that the structure extends to any length of nR robot. Given five task positions [T j ],j =1, ,5 of the end-effector of this chain, we can solve the equations [D]=[T j ],j=1, ,5, (2) to determine the joint parameter vectors θ j =(θ 1,j ,θ 2,j , ,θ n,j ). Be- cause there are three independent constraints in this equation, we have free parameters when n>3. In what follows, we show how to use these free parameters to facilitate the design of mechanical constraints on the joint movement so the end-effector moves smoothly through the task positions. 397 Constraint Synthesis for Planar N- R Robots- [...]... constrain an nR chain to reach five task positions Figure ?? lists the planar linkages that this procedure allows us to design Notice that in each the chain is a sequence of fourbar linkages extending from the base frame F to the moving frame M Furthermore, while the base and moving frames are binary links having only two revolute joints, the links A1 and An−1 are ternary links, and the remaining links... in space is limited and expensive because of hardware cost and system complexity The objective of this work is to study the system topologies that minimize the number of sensors needed to measure joint torques and reaction jet forces for space robots 413 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 413– 422 © 2006 Springer Printed in the Netherlands 414 P Boning and S Dubowsky Figure... Symposium on Advances in Robot Kinematics A STUDY OF MINIMAL SENSOR TOPOLOGIES FOR SPACE ROBOTS Peggy Boning Department of Mechanical Engineering, MIT Cambridge, MA, USA pboning@mit.edu Steven Dubowsky Department of Mechanical Engineering, MIT Cambridge, MA, USA dubowsky@mit.edu Abstract 1 Sensing in space robotic systems is expensive but necessary to compensate for imprecise actuation, disturbances, and modeling... second four-bar linkage in frame A2 and solve for P2 Next, we move to the thrid four-bar linkage in frame A3 and solve for P3 The procedure continues until we conclude with the n − 1st four-bar linkage in frame An−1 The result is a complete analysis of the mechanically constrained nR serial chain 7 Conclusions Our synthesis of a mechanically constrained nR chain yields a one degree-of-freedom system... space robot needs to perform precise motion and force control using its manipulators and reaction jets [Matsumoto 2002] However, in space actuation can be imprecise and constrained by inherent nonlinearities Sensing is necessary to compensate for torque disturbances due to joint friction, reaction jet variability and bias, thermal warping effects, and modeling errors [Breedveld 1997] Sensing in space... the specification of the nR chain to the location of the base joint C1 in G and the end-effector joint Cn relative to task frame H If the serial chain has three joints, then the inverse kinematics equations are completely prescribed and the synthesis of two RR chains yields the constrained serial chain Figure ?? shows the results of this synthesis In this case, though we obtained a solution that passes... Design, 115(2):21 4-2 22 Krovi, V., Ananthasuresh, G K., and Kumar, V., 2002, “Kinematic and Kinetostatic Synthesis of Planar Coupled Serial Chain Mechanisms,” ASME Journal of Mechanical Design, 124(2):30 1-3 12 Lin, C.-S., and Erdman, A G., 1987, “Dimensional Synthesis of Planar Triads for Six Positions,” Mechanism and Machine Theory, 22:41 1-4 19 Subbian, T., and Flugrad, D R., 1994, “6 and 7 Position Triad... controlled within lower and upper bounds The calculation of a force distribution is theoretically straight forward in the case of a manipulator with m = n + 1 tendons In the case of m > n + 1, optimization is required, which is always expensive in terms of computational time For 6 d.o.f systems, 403 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 403–412 © 2006 Springer Printed in the Netherlands... sequence of four-bar linkage analysis problems We begin the analysis with the results of the inverse kinematics analysis of the nR chain at each of the five task positions Our approach uses the analysis procedure of 4 bar linkage from ? We analyze each of the n-1 4 bar linkages in the order of frame K, A2 , A3 , , An−1 402 G.S Soh and J.M McCarthy Starting from the first four-bar linkage in frame K,... canonical elements are combined to determine the number of sensors needed for the original system Configurations of one- and two-manipulator space robots are examined here and the minimal number of sensors shown Introduction Autonomous robots will be needed for future space missions such as satellite capture and on orbit construction of large space structures including space telescopes and solar power stations . redundant and with self-motion. Examples of over - constrained US-PM with five and six US - legs are depicted in Fig. 5. For ease of understanding, the redundant legs are drawn in long - dash - dot. Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 403–412. 403 a) b) Figure 1. Segesta Testbed and Symbol Definitions for a General Tendon-Based one. n degree- of-freedom nR chain into a single degree-of-freedom 2n-bar linkage that has 3n −2 revolute joints. Our design equations only ensure that the solution linkage can be assembled in each