has a constant direction with respect to a fixed frame, the second axis is orthogonal to the first one. A detailed kinematic analysis is carried out and leads to geometrical conditions to be verified by the mechanism for proper functioning. Then a kinematic modelling illustrates the mechanism simplicity and provides a first evaluation of the machine’s workspace. Finally, preliminary information is given regarding practical implementation of this new architecture. The proposed machine is a 6-actuator / 5-dof parallel mechanism. In Fig. 1, a joint-and-loop graph is depicted: grey boxes represent actuated joints; white boxes passive joints and circles express a kinematic coupling between two joints. R coupling coupling Moving Platform Base P P P S S S S P S S S S P P S S S S P S S S S P As for Delta and H4 architectures, the actuators are fixed on the base. Actuators may be rotational or linear; one S-joint per chain may be replaced by U-joint (to get rid of internal motions). One must notice the machine’s symmetrical architecture: the machine’s upper and lower parts are identically made of a “spatial-parallelogram” and two single rods. Each single rod is connected to the “spatial parallelogram”. Note that, in a general matter, the “spatial-parallelogram” chains 2 parallelogram” made of PR(RR) 2 R chains (as done on the Orthoglide, see Chablat et al., 2002) adds two constraints on the mechanism (3 translations and 1 rotation remain feasible). 449 chains) only add one constraint on a mechanism (3 translations and 2 rotations remain feasible) while a “spatial- (that is, the P(SS) Figure 1. Joint-and-loop graph. Figure 2. Kinematics scheme. On PKM with Articulated Travelling-plate 3.1 General Concept The travelling plate is the one introduced in Krut et al., 2003, with the I4L robot: while the two sub-parts shift one relatively to the other, a mechanical device transforms this motion into a rotation. Two types of travelling plates exist (see Fig. 3): Type 1 is made of two prismatic joints and two kinematically coupled rack-and-pinion systems. It has a symmetrical design. (a) Type 1 (b) Type 2 3.2 Singularities analysis is often based on the analysis of the standard Jacobean matrices x J and q J representing the input-output velocity relationship: qx J qJx , (1) where q and x are respectively the joint velocity vector and the operational velocity vector. But other kind of singularities can occur (Zlatanov et al., 1998). To reveal them, a deeper analysis is required. At first, we will recall the fact that “spatial parallelograms” can be seen in two different ways. The realistic case where spherical joints are modelled as 3-DoF joints and not as 2-DoF joints is considered here. Then, two types of modelling will be given: one suggesting that the linear guide is a cylindrical joint (isostatic modelling), and another assuming that it is a prismatic joint (over- constrained modelling). In both cases, geometrical constraints, which must be fulfilled to get rid of internal singularities, will be derived. According to Hervé’s notations (see Hervé, 1999) about displacement subgroups, ^` T stands for the subgroup of spatial translations and ^` ()X u stands for the subgroup of Schoenflies displacements (or Scara motions), where u is a unitary vector collinear to the rotation axis. If a closed loop mechanism is composed of two chains producing Schoenflies displacements with zvu, then: ^`^`^` () () X XT uv (2) 450 S. Krut et al. Figure 3. Different possibilities of travelling plate. the parts. Type 2 is made up with one part less, but loses Type 1’s symmetrical design, which is good for balancing the loads among all A Remark on Singularity Analysis 2 easily handled with such a technique since those chains correspond to Schoenflies subgroup. (a) RR(RR) 2 R (b) R(SS) 2 The case of machines with R(SS) 2 chains (Fig. 4-b) is more complex: each chain provides 5 DoF, 3T-2R, and does not correspond to a group. Indeed, it is possible that the union ( ) of two 3T-2R chains generates a 3T-3R motion. This implies the recourse to a more complex analysis when dealing with mechanisms based on R(SS) 2 chains: for lack of space, this is recalled here, but the reader may find relevant information in Company et al., 2006. In this section, the focus is given to a particular design, where the six linear motors are all collinear with vector z workspace in this particular direction. The selected geometrical parameters are as follows 0.45 mH , 0.08 mI and 0.4 mJ 1. The values of geometrical parameters are: 0.05 mD , 0.06 mE and 1 0.05 m/radk . Note that the amplification ratio 1 k is chosen equal to D in order to have same rotation capabilities for T and M (+/-90 degrees for this design). Lengths of rods are: 0.9 m i l , {1, , 6}i ! . Actuators limits are: 01.3m i qdd . Fig 5 presents the domain where the condition number of the normalized Jacobean matrix is smaller than 8 (note that by the actuators’ stroke). 451 e : this guarantees a large along the Z direction corresponding to z , the workspace is only limited (See Fig. 2 for geometrical parameters explanations). The travelling plate is of type Figure 4. Two ways to model “spatial parallelograms”. that is to say that such a mechanism will produce only three trans- lations. The case of machines with RR(RR) R chains (Fig. 4-a) is On PKM with Articulated Travelling-plate u re 3.3 Workspace Analysis O (m ) z (m)y (m)x 1 O 4 O 2 O 3 O 5 O 6 O Figure 5. Workspace for cond( -1 -1 qxx JJW) < 8 Figure 6. CAD View of the Eureka It could be interesting, for simplicity purposes, to connect the “single rods” directly to the travelling plate; however, such a practical design faces too many self-collisions. The machine depicted in Fig. 7 (left) shows such a practical design. Another architecture avoiding self-collisions is shown in Fig. 7 (right). It involves curved shapes of the single rods in order to avoid self-collisions. A prototype is about to be built. The practical design is extremely simple thanks to Linear motors (Fig. 6). Dimensions are the ones introduced for computing the workspace. Rods and travelling plate are S . Krut et a l . 4 52 prototype. Figure 7. Self-collision-free design #1 and #2. 3.4 Practical Design Considerations made of aluminium. Instead of using rack-and-pinion systems, the mobile platform has been equipped with cable-pulley devices. This kinematics provides the same displacements as those of the Tricept robot. This design is well suited for the manipulation of light objects, but other applications are still possible. A design of a haptic master arm based on this kinematics is proposed in Fig. 8. It uses revolute actuators instead of prismatic ones, so the footprint is reduced. DD motors are used in order to reduce friction. The required range for angular displacements is +/ 45 degrees. This allows the use of an articulated travelling plate based on a planar parallelogram provides three Translations plus two Rotations (3T-2R). The missing rotation (to get the complete master arm) is obtained using a carried revolute axis, located directly on the ending stick. This is similar to the design of classical master arms, such as the PHANToM (SensAble Technologies). In this paper, several techniques for reaching high tilting angles have been presented, with a focus on solutions related to articulated travelling plates. Even though such results are still at an early stage of development, they show that it might be possible to use (i) on the one hand, travelling plates embedding passive joints which allows local motion amplification, and (ii) on the other hand, actuation redundancy as 453 to provide the desired rotation. The translation to rotation trans- formation is then suppressed and friction reduced. The Eureka base Figure 8. CAD view of the Eureka haptic arm. On PKM with Articulated Travelling-plate – 4. Conclusion a way to overcome some singular positions that usually limit the range of motion. References Angeles, J., Morozov, A., Navarro, O., A novel manipulator architecture for the production of SCARA motions, Proceddings of IEEE International Conference on Robotics and Automation, San Francisco, April 24-28, 2000, pp. 2370-2375. Chablat, D., and Wenger, P., Design of a Three-Axis Isotropic Parallel Workshop on Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators, October 3-4, Québec, Québec, Canada, 2002. Clavel, R., Une nouvelle structure de manipulateur parallèle pour la robotique légère, APII, 23(6), 1985, pp. 371-386. Company, O., Krut, S., and Pierrot, F., Internal Singularity Analysis of a Class of Lower Mobility Parallel Manipulators with Articulated Travelling Plate, IEEE Transactions on Robotics, 2006 (to appear). Koevermans, W.P., Design and performance of the four dof motion system of the NLR research flight simulator, Proceedings of AGARD Conference, La Haye, 20-23 October 1975, pp. 17-1/17-11. Krut, S., Company, O., Benoit, M., Ota, H., and Pierrot, F., I4: A new parallel mechanism for Scara motions, Proceedings of IEEE International Conference on Robotics and Automation, Taipei, Taiwan, September 14-19, 2003. Pierrot, F., and Company, O., H4: a new family of 4-dof parallel robots, Proceedings of IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Atlanta, Georgia, USA, September 19-22, 1999, pp. 508-513. Reboulet, C., Rapport d’avancement projet VAP, thème 7, phase 3. Rapport de Recherche 7743, CNES/DERA, January 1991. Rolland L., The Manta and the Kanuk: Novel 4 dof parallel mechanism for industrial handling, Proceedings of ASME Dynamic Systems and Control Ryu, S.J., Kim, J.W., Hwang, J.C., Park C., Ho, H.S., Lee, K., Lee, Y., Cornel U., Park, F.C., and Kim, J., ECLIPSE: An Overactuated Parallel Mechanism for Rapid Machining, Proceedings of ASME International Mechanical Engineering Congress and Exposition, Vol. 8, USA, 1998, pp. 681-689. Zlatanov, D., Fenton, R.G., and Benhabib, B., Identification and classification of the singular configurations of mechanisms, Mechanism and Machine Theory, Vol. 33, No. 6, pp. 743-760, August 1998. 454 S. Krut et al. pp. 831-844. Division IMECE’99 Conference, Nashville, November 14-19, 1999, vol. 67, Manipulator for Machining Applications: The Orthoglide, Proceedings of 719-730. Hervé, J.M., The Lie group of rigid body displacements, a fundamental tool for mechanism design, Mechanism and Machine Theory, Vol. 34, 1999, pp. MOBILITY AND CONNECTIVITY Abstract This contribution provides new definitions of infinitesimal mobility and connectivity of kinematic chains. These definitions are straightforwardly connected with accepted definitions of finite mobility and connectivity. Further, screw theory is applied to the determination of the infinitesimal mobility and connectivity of multi-loop linkages. These results pro vide a guide for the determination of the finite mobility and conne ctivit y of general topology of multiloop linkages, one of the important remaining problems in mobility theory. Keywords: 1. Introduction The last three years have seen a flurry of studies about mobility and connectivity of kinematic chains. For the most part, these analyses have been focused on single-loop kinematic chains and parallel platforms. Most of the few studies about mobility and connectivity of general mul- tiloop linkages deal with the mobility and connectivity determined from their velocity analysis. It is well known that the information gathered via velocity analysis of any class of kinematic chains does not provide conclusive information about their mobility and connectivity. In this © 2006 Springer. Printed in the Netherlands. 455 J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 455–464. IN MULTILOOP LINKAGES contribution, it is shown that higher analyses, in particularorder Mobility, connectivity, multiloop linkages, screw theory C´esar Real Diez-Mart´ınez creal@iteso.edu Jos´e M. Rico, J. Jes´us Cervantes-S´anchez jrico, jecer@salamanca.ugto.mx Jaime Gallardo a gjaime@itc.mx Instituto Tecnol´ogico y de Estudios Superiores de Occidente Guadalajara, Jal. M´exico Facultad de Ingenier a Meca´ nica El ´ectrica y Electr´onica Universidad de Guanajuato, Salamanca, Gto. M ´exico Departamento de Ingenier´a Mecanic´ Instituto Tecnol´ogico de Celaya, Celaya, Gto. M ´exico i noted that Wohlhart, 1999 and Wohlhart, 2000, employed higher order analyses to shed light into the characteristics of singular positions of fully parallel platforms. In contrast, in this contribution the authors are interested in general topology multiloop linkages. Further, the higher order analyses are employed, in addition, as an important aid in the determination of their mobility, a problem that remain unsolved in this general case. 2. Mobility and Connectivity In this section, a review of the concepts of mobility and connectivity as well as new definitions of infinitesimal mobility and connectivity are presented. Definition 1. Consider a single-loop or multiple loop kinematic chain. The finite mobility of the chain, denoted by M F ,inagiven configuration is the number, minimum and necessary, of scalar variables required to determine the pose, with respect to a link regarded as refer- ence, of all the remaining links of the kinematic chain. In our approach, the finite mobility depends not only on the kinematic chain, but also of the configuration of the kinematic chain to be analyzed, and it becomes a property of the configuration of the kinematic chain and its neighborhood. This definition is motivated by the presence of kinematotropic chains, Galletti and Fanghella, 2001, and differs from the definition, usually presented in undergraduate and graduate textbooks, and adopted by Gogu, 2005. Consider now the velocity analysis equation of the kinematic chain, in a given configuration, which can be written as follows Jω = 0, (1) where, the Jacobian matrix, J, is a matrix with as many columns as screws associated with the kinematic pairs of the chain and as many rows as fundamental circuits and loops of the kinematic chain, multiplied by 6. Additionally, ω is the vector of joint rates, translational or angular, associated with the screws of the chain, and the vector 0 has the same number of rows as the Jacobian matrix. Definition 2. Consider a kinematic chain whose velocity analysis equation is given by Eq. 1. Then, the kinematic chain has first order infinitesimal mobility if there exists a vector ω 1 = 0 that satisfies the Eq. 1. Moreover, the number of independent components of the vector ω 1 determines the number of first order degrees of freedom, or first acceleration analysis, can be successfully employed in shedding light on the mobility and connectivity of general multiloop linkages. It should be 456 C.R. Diez- Martínez et al. order infinitesimal mobility, denoted by M 1 , of the chain in a given configuration. Consider now the acceleration analysis equation of the kinematic chain in the same given configuration, which can be written as follows J ˙ ω = −$ L , (2) where, $ L is the Lie screw that contains terms of the form [ω i $ i ω j $ j ], where the bracket represents the dual motor product or Lie product, Unlike the velocity analysis equation, the accel- eration analysis equation is non-homogeneous. Further, the solution of a non-homogeneous linear system is given by the sum of the subspace solution of the associated homogeneous system and a particular solution associated homogeneous linear system is given by J ˙ ω = 0. (3) Therefore, a vector ˙ ω whose components are numerically equal to those of any of the vectors ω 1 , solution of the Eq. 1, is also a solution of Eq. 3. Furthermore, a necessary and sufficient condition for the Eq. 2 to have a particular solution is given, Bentley and Cooke, 1971, by Rank(J)=Rank(J, $ L ). (4) If Rank(J) is less than the number of matrix rows, Eq. 4 frequently im- poses additional conditions over the components of the vector ω 1 .These additional conditions require that one or more of the independent com- ponents of ω 1 satisfy additional equations, frequently, these conditions require that one or more of the independent components of ω 1 be zero. Let ω 2 be the solution of both, the velocity and the acceleration analy- ses equations. This result, provides the rationale for defining the second order infinitesimal mobility of the chain. Definition 3. Consider a kinematic chain in a given configuration, such that the velocity and acceleration analyses equations are given by Eqs. 1, 2. The chain has a second order infinitesimal mobility if there is a non-zero vector ω 2 that satisfies both equations. Furthermore, the number of independent components of the vector ω 2 determines the num- ber of second order degrees of freedom, or second order infinitesimal mobility, denoted by M 2 , of the chain in a given configuration. Similarly, it is possible to define similar concepts regarding the con- nectivity. Consider a kinematic chain in a given configuration. The finite connectivity between a pair of links (i, j), in the kinematic 457 see Rico et al., 1999. Mobility and Connectivity of the non-homogeneous system, Bentley and Cooke, 1971. Thus, the Definition 4. chain is the minimum and necessary number of joint – scalar – variables that determine the pose of one link with respect to the other, and it if denoted as C F (i, j). Consider an arbitrary kinematic chain and assume that (i, j)isan arbitrary pair of links of the chain. Further, assume that the velocity analysis of the chain has been solved and that the unique velocity state, i V j 1 , of link j with respect to link i, following all possible paths between links i and j has been determined. The velocity state i V j 1 depends on independent variables that solve the velocity analysis solution, contained in ω 1 . These elements are linear or angular velocities associated with the kinematic pairs of the chain. Further, i V j 1 is a vector space. Then it is possible to define the first order infinitesimal connectivity. Consider a kinematic chain in a given configuration tivity between links (i, j), denoted by C 1 (i, j), is defined as C 1 (i, j)=dim( i V j 1 ). (5) Consider a kinematic chain in a given configuration and let (i, j) be a pair of links. Further, assume that the velocity and acceleration analyses of the chain has been solved. The number of in- dependent variables of the vector ω 2 might be less than the number of independent variables of the vector ω 1 . Assume that the unique velocity state of link j with respect to link i, using the solution of the velocity and acceleration analyses ω 2 , denoted by i V 2 j , is also known. Then, their second order infinitesimal connectivity, denoted by C 2 (i, j), is defined by C 2 (i, j)=dim( i V 2 j ). (6) Higher order mobilities and connectivities can be defined accordingly . 3. Mobility and Connectivity in Multiloop Linkages Consider the multiloop spatial kinematic chain shown in Fig. 1, pro- posed by Fayet, 1995 and used by Wohlhart, 2004. The chain has tw o spherical pairs, four cylindrical pairs, a planar pair and three revolut e pairs. 458 Definition 6. Definition 5. C.R. Diez- Martínez et al. and let (i, j) be a pair of links. The first order infinitesimal connec - [...]... outputs The end-point map of this system determines the kinematics of the mobile manipulator Tcho´ and Jakubiak, 2003 The Jacobian inverse kinematics algorithm n for mobile manipulators are conveniently devised using the continua465 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 465–472 © 2006 Springer Printed in the Netherlands 466 h K Tchon and J Jakubiak tion method Chitour and Sussmann,... Architectural Mobility of Platforms, On Advances in Robot Kinematics, Lenarˇiˇ, J and Staniˇi´, eds Dordrecht: cc sc Kluwer Academic Publishers, pp 36 5-3 74 Wohlhart, K (2004), Screw Spaces and Connectivities in Multiloop Linkages, On Advances in Robot Kinematics, Lenarˇiˇ, J and Galletti, C eds Dordrecht: Kluwer cc Academic Publishers, pp 9 7-1 04 JACOBIAN INVERSE KINEMATICS ALGORITHMS WITH VARIABLE STEPLENGTH... Sastry and H J Sussmann, Springer-Verlag, New York, pp 9 1-1 25 Deuflhard, P (2004), Newton Methods for Nonlinear Problems, Springer-Verlag, Berlin, 2004 Divelbiss, A., Seereeram, S., and Wen, J.T (1998), Kinematic path planning for robots with holonomic and nonholonomic constraints In: Essays on Mathematical Robotics, ed by J Baillieul, S S Sastry and H J Sussmann, Springer-Verlag, New York, pp 12 7-1 50... Mobility and Relative Freedoms in Multiloop Linkages and Structures, Mechanism and Machine Theory, vol 16, no 6, pp 58 3-5 97 Bentley, D L., and Cooke, K L (1971), Linear Algebra with Differential Equations, New York: Holt, Rinehart and Winston Fayet, M (1995), M´canismes Multi-Boucles I D´termination des Espaces de Torseurs e e Cin´matiques Dans un M´canisme Multi-Boucles Quelconque, Mechanism and Mae e chine... manipulators: a derivation and performance assessment of Jacobian inverse kinematics algorithms, Int J Control, vol 76, no 14, pp 138 7-1 419 Tcho´, K., and Jakubiak, J (2005), A repeatable inverse kinematics algorithm with n linear invariant subspaces for mobile manipulators, IEEE Trans Syst., Man, Cybernet., – Part B Cybernetics, vol 35, no 5, pp 105 1-1 057 KINEMATICS AND GRASPING USING CONFORMAL GEOMETRIC... conformal geometry in a very elegant way In this geometry an Euclidean vector space R3 473 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 473–480 © 2006 Springer Printed in the Netherlands 474 J Zamora-Esquivel and E Bayro-Corrochano is represented in R4,1 This space has orthonormal vector basis given by {ei } and eij = ei ∧ ej are bivectorial basis where e23 , e31 and e12 correspond... approximation defines a finite-dimensional, band-limited endoge˜ = nous configuration space X ∼ R11 , and yields the band-limited kine˜ q ,T (λ, x) and the band-limited analytic Jacobian matics K 0 ˜ ˜ Jq0 ,T (λ, x) = C(T, x) T ˜ ˜ ˜ Φ(T, s)B(s)P (s)ds, D(T, x) 0 In consequence, a discrete band-limited Jacobian pseudoinverse inverse kinematics algorithm (6) will take the following form λk+1 xk+1 = λk xk... ALGEBRA Julio Zamora-Esquivel and Eduardo Bayro-Corrochano Center of research and advanced studies of IPN, Unit Guadalajara Geovis Laboratory Jalisco, Mexico jzamora,edb@gdl.cinvestav.mx Abstract 1 In this paper we introduce the conformal geometric algebra in the field of robot grasping It help us to tackle problems of object modelling, hand kinematics and vision system using a unifying geometric language... 20 1-2 17 Galletti, C., and Fanghella, P (2001), Single-Loop Kinematotropic Mechanisms,Mechanism and Machine Theory, vol 36, no 6, pp 74 3-7 61 Gogu, G (2005), Mobility of Mechanisms: a Critical Review, Mechanism and Machine Theory, vol 40, no 9, pp 106 8-1 097 464 C.R Diez-Martínez et al Rico, J M., Gallardo, J., and Duffy, J (1999), Screw Theory and Higher Order Analyses of Open Serial and Closed Chains,... The remaining degree of freedom is related to the translational motion, along the same axis y, and produced by the screws 1b $1c , located in point C, and 7 $7a , located in point J, while the revolute joints located between them remain inactive This degree of freedom is passive when the fixed and moving platforms are links 1 and 5 The conclusion is that the finite mobility of the multiloop linkage is . chains does not provide conclusive information about their mobility and connectivity. In this © 2006 Springer. Printed in the Netherlands. 455 J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, . suggesting that the linear guide is a cylindrical joint (isostatic modelling), and another assuming that it is a prismatic joint (over- constrained modelling). In both cases, geometrical constraints,. 6-actuator / 5-dof parallel mechanism. In Fig. 1, a joint -and- loop graph is depicted: grey boxes represent actuated joints; white boxes passive joints and circles express a kinematic coupling