Figure 5. Object relative position desired finger without the need of solving any kind of inverse kinematics equations C. Canudas, G. Bastin, B. Siciliano. Given the the differential kinematics equation ˙ X 3 = 1 125 X 3 · L 2 + 1 375 X 3 · L 3 − 2 35 X 3 · L 1 ˙q 1 ˙q 4 , (27) If we want to reach the point H(s 1 ,t 1 ), we require that the suitable velocity at the very end of the finger should be proportional to the error at each instance V i = −0.7(X 3 − H(s 1 ,t 1 )). This velocity is mapped into the phase space by means of using the Jacobian inverse. Here we use simply the pseudo-inverse. with j 1 = 1 125 X 3 · L 2 + 1 375 X 3 · L 3 and j 2 = − 2 35 X 3 · L 1 ∆q 1 ∆q 4 =(j 1 ∧ j 2 ) −1 · V i ∧ j 2 j 1 ∧ V i (28) Applying this control rule, one can move any of the fingers at a desired position above an object, so that an adequate grasp is accomplish. 5. Results In this section we present the experimental results of our grasping algorithm. In Figure 6, the inferior images correspond to the simulated scenario and the other ones are real. In this experiment the object was suspended manually above the grasping hand, simply to check whether the has been opened correctly or not. We can see that for each object the algorithm manages to find the singular grasp points, so that the object is hold properly and in equilibrium. Note that the found points correspond to the expected grasping points. Kinematics and Grasping 4 7 9 or it is possible to implement a control law which will allow to move the Figure 6. 6. Conclusion Using conformal geometric algebra we show that it is possible to find three grasping points for each kind of object, based on the intrinsic information of the object. The hand s kinematic and the object structure can be easily related to each other in order to manage a natural and feasible grasping where force equilibrium is always guaranteed. References computational geometry”. G. Somer, editor, Geometric Computing with Clifford Algebras, pages 27-52. Springer-Verlag Heidelberg. the kinematics of robot manipulators. Journal of Robotics Systems, 17(9):495-516. Carlos Canudas de Wit, Georges Bastin, Bruno Siciliano. (1996) Theory of Robot Control, Springer. Arbitrary 3D Objects. ICRA, pages 1890-1896. Detroit, Michigan. Andrew T. Miller, Steffen Knoop, Peter K. Allen, Henrik I. Christensen. (2003) ” Au- tomatic Grasp Planning Using Shape Primitives,” In Proceedings of the IEEE International Conference on Robotics and Automation, pp. 1824-1829. J. Zamora-Esquivel and E. Bayro-Corrochano480 Grasping some objects. Li, H. Hestenes, D. Rockwood A. (2001).“Generalized Homogeneous coordinates for Bayro-Corrochano, E. and K¨ahler, D. (2000). Motor Algebra Approach for Computing Ch Borst, Fischer M. and Hirzinger, G. (1999) A Fast and Robust Grasp Planner for, , ,, . . . Raghavendran Subramanian Graduate Student Dept. of Mechanical Engineering University of Connecticut Storrs, CT – 06268 raghavendran@engr.uconn.edu Kazem Kazerounian Professor Dept. of Mechanical Engineering University of Connecticut Storrs, CT – 06268 kazem@engr.uconn.edu Abstract In this paper, we present a new methodology to identify the rigid domains in a protein molecule. This procedure also identifies the flexible domains as well as their degree of flexibility. Identification of rigid domains significantly simplifies the motion modeling procedures (such as molecular dynamics) that use geometric features of a protein as variables. Keywords: 1. Introduction Proteins are the building blocks that play an essential role in a variety of basic biological functions such as signal transduction, ligand binding, catalysis, regulation of activity, transport of metabolites, formation of larger assemblies and cellular locomotion. Its internal motions results in conformational transitions and often relate structure to its function. Hence, comprehending the protein internal motion is the key to the understanding of the structural relationship of these natural machines to their function. Protein molecules have always been observed with rigid domains connected by flexible portions as shown in the figure 1. Kinematics serial chain model of proteins has been established and justified in few of our previous works (Kazerounian 2004; Kazerounian, Kazerounian June 2002). As the long snake type serial linkage folds, new bonds are created between atoms of the residues that are not © 2006 Springer. Printed in the Netherlands. 481 J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 481– 488. APPLICATION OF KINEMATICS TOOLS OF PROTEIN MOLECULES IN THE STUDY OF INTERNAL MOBILITY inematics, mobilities, functions, graph theory, nano machines, closed loops Latif et al., 2005; Kazerounian, Latif et al., 2005; Subramanian 2005; K neighbors. These bonds transform the open loop linkage to a linkage with some relatively large links (rigid body domains) and closed kinematic loops. These bonds are generally categorized as follows: 1) Hydrogen Bonds (main chain to main chain, main chain to Side chain and side chain to side chain), and 2) Disulphide bonds. To gain insight into a protein function, we must understand the are five different computational methods reported in literature to identify rigid domains of the protein. Two of the methodsalso attempt to 1995) involves comparison of two conformations of a protein to identify the rigid domains in a molecule. The second method (Wriggers and Schulten 1997) also compares two different conformations of the same dynamics of a protein molecule. The procedure creates an equivalent elastic network model with atoms as masses serially connected one after the This mathematical treatment yields vibrational frequency modes of all the atoms. An atom for which all frequency modes are computed to be zero, will be considered as a part of a rigid domain. This method is computationally, a very expensive procedure even with a network of just C D atoms. Fourth method is a variant of the third method. In this method normal mode analysis Carlo simulation is used to form the trajectory of all the atoms. This requires only one conformation to identify the rigid domains in a protein molecule. It uses the distance constraints between atoms due to the a rigidity matrix which on further manipulation based on the set of rules defined under the rigidity theory, one can find rigid domains. This method disregards the presence of disulphide bonds in a protein molecule which also reduces the mobility of atoms in proteins. The methodology presented in this paper treats the protein molecule as a kinematic chain that has open as well as closed loops. In the recent 482 R. Subramanian and K. Kazerounian kinematics and the mobility of the internal motion of the protein. T here establish the mobility of the chain. F irst method (Nichols, Rose et al. protein. It uses a least square technique to best fit the two con- formations. Third method (Levitt, Sander et al., 1985) is based on the other by springs . in its open and closed conformation. coupled with molecular dynamics (Doruker, Bahar et al., 2002) and Monte method too requires unreasonably excessive computation. Fifth method presence of covalent and the hydrogen bonds between them. I t develops (Jacobs, Rader et al., 2001) is based on the graph theory. This method Figure 1. Ribbon view of a peptide chain , past, many kinematicians (Crossley 1965; Woo 1967; Manolescu 1973; type synthesis of mechanisms especially to identify non-isomorphic mechanisms and to enumerate mechanisms. This method also uses graph theory based on the primary (linear) structure of the protein, and uses the atom coordinates to detect the hydrogen and disulphide bonds. The resulting graph maintains the information on the connectivity of links in the protein mechanism and thereby identifies all the loops formed by hydrogen and disulphide bonds. The loops that are kinematically over- constrained form rigid structures. This is an iterative process that 2. Identification of the Hydrogen Bonds Hydrogen bonds occur when two electronegative atoms interact with the same hydrogen. The hydrogen atom is covalently attached to one atom (commonly called donor), and interacts electrostatically with the other atom (commonly called acceptor). This (hydrogen). Hydrogen bond possesses some degree of orientational preference and has the characteristics of a covalent bond (although it is weak). Several fine works in the literature have focused on this directional behavior of hydrogen bonds(Baker and Hubbard 1984; Eswar and Ramakrishnan 2000). These works have established generalized geometric characteristics for identification of the hydrogen bonds when the positional coordinates of the electronegative atoms and the hydrogen configuration. A shortfall of these data files is that the hydrogen atom positions are usually not recorded. C O C N C H 483 Application of Kinematics Tools results in the identification of all the rigid domains. Figure 2. Location of Hydrogen atom with respect to the neighboring atoms. atoms and the proton interaction is due to ween the electronegative the dipole effect bet- atoms are known. Protein Data Bank (PDB) (Berman, Westbrook et al., Mruthyunjaya and Raghavan 1979) have extensively used graph theory for 2000) offers the coordinate position of all the atoms in a protein 2.1 Hydrogen Atom Position Calculation The chemical (directional) nature of the covalent bonds leads to a unique relative position of a hydrogen atom with respect to the positional coordinate of its neighboring atoms. Hence the coordinates of a hydrogen atom can be established theoretically using coordinates of its neighbor atoms (figure 2). The detailed procedure and formulation based on figure 2 is included in Rigid body assumption in proteins has been established and justified in few of our previous works (Subramanian 2005). 2.2 Criteria to Establish Hydrogen Bonds There are predominantly three types of hydrogen bonds observed in the protein structures. They are main chain to main chain, main chain to side chain and side chain to side chain hydrogen bonds. The majority of the main chain to main chain hydrogen bonds are local in nature involving less than six consecutive residues in the primary sequence of a protein. As mentioned earlier, the directional nature of the hydrogen bond results in a set of geometric criteria to be established to identify the presence of hydrogen bonds. These geometric criteria solely depend on the coordinate positions of two electronegative atoms and a hydrogen atom. These geometric criteria are different for different sets of the electronegative atoms and the geometric conditions for identification of the hydrogen bonds are quite extensive. Reference (Subramanian 2005) and the exhaustive conditions for selecting each one of the three possible hydrogen bonds, as developed by the authors. 3. Identification of the Disulphide Bonds A disulphide bridge is formed between two cysteine residues by the oxidation of their sulfur atoms to form a double bond. Thus two cysteine residues connect through their sulphur atoms and form loops in the open chain. In proteins disulphide bridges contribute significantly to the stability of proteins. Two parameters have been established (Sowdhamini, Srinivasan et al. 1989) to identify the presence of disulphide bonds between two cysteine residues in a protein molecule. They are based on the geometric features that exist between the two test residues. The distance parameters include the distance between the two alpha carbon atoms and the distance between the two beta carbon atoms of two cysteine residues in the primary sequence. The two geometric conditions are that the first distance mentioned lies within 3.8Å to 6.5Å and the second distance lies within 3.4Å to 4.5Å. These criteria are checked for all the possible 484 R. Subramanian and K. Kazerounian , combinations of any two cysteines in a protein molecule. Those combinations that meet the above requirements are assumed to form the disulphide bonds. 4. Application of the Graph Theory to Loop The internal mobility of a protein chain is a function of how various links in the open chain model connect by means of hydrogen and disulphide bonds. These bonds transform the open loop linkage to more complex multi closed loop system. The size of the protein molecule and the large number of such bonds demands a sophisticated method of accounting for connections within the molecule. Graph theory is an ideal tool for this purpose The equivalent linkage mechanisms to protein chains can be described as a graph with links as edges and joints as vertices and is a very useful tool to represent the connectivity between links. A two dimensional matrix (commonly called as connectivity matrix) mathematically represent the connectivity between all the links. Prior to the detection of the over-constrained loops from the given connectivity matrix, all the side chain links which do not participate in the loop formation will be eliminated from the connectivity matrix. This will reduce the computational complexity of the problem of detecting the over- constrained closed loops. As a first step, we will eliminate all those side chain links which do not participate in the loop formation. This process starts from the end link of all the side chains. If the end link has only one joint, then it does not form a loop. Consequently the link preceding the removed end link becomes the end link itself. This procedure iteratively eliminates all the links of the side chains that do not form closed loops except the first link of the side chains (that is connected to the two main chain links). The graph after the previous step will have only one side chain link for all the side chains which are not involved in the loop formation. As mentioned earlier, these side chains will be connected only to two main chain links and no side chain links. In the second step the procedure eliminates all such first links of the side chains. This requires that the side chain links be differentiated from main chain links by their index numbers. This can be done by storing the index numbers of all the main chain links and all the side chain links in two different vectors. This information is readily available from Protein Data Bank (PDB) files. Note that in the above two stages the size and the values in the connectivity matrix changed while eliminating all the side chain links that were not part of any closed loops. This will leave the graph with 485 Application of Kinematics Tools Detection in a Protein Chain none of the open ended side chain branches off the main chain. Thus the complexity involved with maintaining the information of the un- influencing side chain links is avoided. This also reduces the computational needs for solving the problem of detecting all the over constrained loops. The procedure we have developed to detect all the over-constrained loops involves finding all the closed loops with two links, three links, four links, five links and six links respectively, until all the over-constrained loops are detected. The steps to detect loops with m links (m = 2 to 6) are 2005). The search starts with any link and corresponding to that link in the connectivity matrix, we follow the trail of the links connected until we arrive back on the link we started the search with. This indicates that the loop is closed and a counter keeps track of the number of links in that closed loop. In the repeated local search for all the over-constrained loops, we change the connectivity matrix every time an over-constrained loop is found. The changes are as follow: all the links of an over-constrained loop is replaced by a single new link, thus the rows and columns corresponding to these links are dropped from the connectivity matrix and a new row and column is appended to the connectivity matrix to represent this new link. All the connections to all the links of this over-constrained loop will now be the connections to this new link. 5. Results and Discussion numerous protein molecules to identify their rigid domains and flexible portions. One such numerical experiment was on the protein BPTI (PDB 486 R. Subramanian and K. Kazerounian briefly explained as follows. The detailed algorithm for this detec- tion process is included in reference(Subramanian of a “1” in the row through the detection Figure 3. Kinematic Sketch of the protein BPTI (1K6U) with its rigid and flexible domains. The methodology developed in this work was succesfully applied to - ID: 1K6U). Bovine Pancreatic Trypsin Inhibitor (PDB ID: 1K6U) is a 58 residue long protein. We identified a total of 26 hydrogen bonds in the protein molecule of which 19 were main chain to main chain hydrogen bonds and the rest of the hydrogen bonds were between main chain and side chains. This protein molecule was identified with 3 disulphide bonds. 4 rigid domains (R1 to R4) and 5 flexible portions (F1 to F5). The kinematic sketch for this protein is shown in figure 3 (kinematic arrangement) and a 3-D illustration in Figure 4. The alpha helices and beta sheets as expected formed rigid domains or part of rigid domains. Among all the flexible chains, 3 of them were closed loops. The degrees of freedom for such constrained closed loops are also reported. These are as follow: 7 for F2, 1 for F3 and 5 for F4. These results were compared visually with the motion of the protein molecules available in the website: The results were observed to be consistent with these motion pictures for each of the three protein molecules. 6. Conclusion (and contact). The coordinate value of all the atoms in the protein is used only to establish the location of hydrogen and disulphide bonds. It also finds all the flexible portions of a protein molecule and calculates its degrees of freedom, a numerical value as a flexibility measure, for each of these flexible portions. This methodology has been successfully tested on several proteins from PDB. 7. References Baker, E. N. and R. E. Hubbard (1984). Hydrogen bonding in globular proteins. Prog Biophys Mol Biol 44(2): 97-179. orange and blue) represent the different 487 Application of Kinematics Tools “ ” Figure 4. Color code based distinction between the rigid domains and flexible portions of the protein, BPTI (PDB ID: 1K6U). Red portions are the flexible regions and other colors (pink, green, rigid domains in the protein. The protein molecule had http://molmovdb.mbb.yale.edu/molmovdb/ (Echols, Milburn et al., 2003). tify all the rigid domains in a protein identified in a PDB type format W e have developed a computationally efficient methodology to iden- Berman, H. M., J. Westbrook, et al. (2000). The Protein Data Bank. Nucleic Acids Res 28(1): 235-42. Crossley, F. R. E. (1965). The permutations of Kinematic Chains of Eight Members or Less from Graph Theoretic Viewpoint. Developments in Theoretical and Applied Mechanics 2: 467-487. Doruker, P., I. Bahar, et al. (2002). Collective deformations in proteins determined by a mode analysis of molecular dynamics trajectories. POLYMER 43(2): 431-439. Echols, N., D. Milburn, et al. (2003). MolMovDB: analysis and visualization of conformational change and structural flexibility. Nucleic Acids Research 31(1): 478-482. Eswar, N. and C. Ramakrishnan (2000). Deterministic features of side-chain main-chain hydrogen bonds in globular protein structures. Protein Eng 13(4): 227-38. Jacobs, D. J., A. J. Rader, et al. (2001). Protein flexibility predictions using graph theory. Proteins 44(2): 150-65. Kazerounian, K. (2004). From mechanisms and robotics to protein conformation and drug design. Journal of Mechanical Design 126(1): 40-45. Kazerounian, K. (June 2002). Is Design of New Drugs a Challenge for Kinematics? Proceedings of the 8th Int. Conf. on Advance Robot Kinematics - ARK, Caldes de Malavalla, Spain. Kazerounian, K., K. Latif, et al. (2005). Protofold: A successive kinetostatic compliance method for protein conformation prediction. Journal of Mechanical Design 127(4): 712-717. Kazerounian, K., K. Latif, et al. (2005). Nano-kinematics for analysis of protein molecules. Journal of Mechanical Design 127(4): 699-711. Levitt, M., C. Sander, et al. (1985). Protein normal-mode dynamics: trypsin inhibitor, crambin, ribonuclease and lysozyme. J Mol Biol 181(3): 423-47. Manolescu, N. I. (1973). A Method based on Barnov Trusses, and using Graph Theory to find the set of Planar Jointed Kinematic Chains and Mechanisms. mechanism and machine theory 8(1): 3-22. Mruthyunjaya, T. S. and M. R. Raghavan (1979). Structural Analysis of Kinematic Chains and Mechanisms based on Matrix Representation. ASME Journal of Mechanical Design 101: 488-494. Nichols, W. L., G. D. Rose, et al. (1995). Rigid Domains in Proteins - an Algorithmic Approach to Their Identification. Proteins-Structure Function and Genetics 23(1): 38-48. Sowdhamini, R., N. Srinivasan, et al. (1989). Stereochemical modeling of disulfide bridges. Criteria for introduction into proteins by site-directed mutagenesis. Protein Eng 3(2): 95-103. Subramanian, R. (2005). Calibration of Structural Variables and Mobility Analysis of Protein molecules. Mechanical Engineering Department. Storrs, University of Connecticut. MS. Woo, L. S. (1967). Type Synthesis of Plane Linkages. ASME Journal of Engineering for Industry: 159-172. Wriggers, W. and K. Schulten (1997). Protein domain movements: Detection of rigid domains and visualization of hinges in comparisons of atomic coordinates. Proteins-Structure Function and Genetics 29(1): 1-14. 488 R. Subramanian and K. Kazerounian “ ” “ ” “ ” “ ” “ ” ” ” “ ” “ ” ” “ ” “ ” “ ” “ ” “ ” “ ” “ “ “ [...]... Sugimoto K., Duffy J and Hunt K.H (1982), Special configurations of spatial mechanisms and robot arms, Mechanism and Machine Theory, vol 17, no 2, pp 11 9-1 32 Merlet J -P (1989), Singular configurations of parallel manipulators and Grassmann geometry, Int J of Robotics Res., vol 8, no 5, pp 4 5-5 6 Gosselin C and Angeles J (1990), Singularity analysis of closed-loop kinematic chains, IEEE Trans Robot Autom.,... 28 1-2 90 Zlatanov D., Fenton R.G and Benhabib B (1994), Singularity analysis of mechanisms and robots via velocity-equation model of the instantaneous kinematics, IEEE Int Conf Rob Autom., vol 2, pp 98 6-9 91 Park F.C and Kim J.W (1999), Singularity analysis of closed kinematic chains, ASME Journal of Mechanical Design, vol 121, pp 3 2-3 8 Zlatanov D.S., Bonev I.A and Gosselin C M (2002), Constraint singularities... variation of the translational and rotational directions with the pose This does not imply a singularity, 489 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 489–496 © 2006 Springer Printed in the Netherlands 490 O Altuzarra et al but it does condition the kinematic characteristics of the manipulator such as its manipulability The nature of the DOF of the moving platform is usually unaltered... Int Conf Rob Autom., Washington DC, USA Ma O and Angeles J (1992), Architecture singularities of parallel manipulators, Int J Robot Automt., vol 7, no 1, pp 2 3-2 9 Company O., Krut S and Pierrot F (2006), Internal singularity analysis of a class of lower mobility parallel manipulators with articulated traveling plate, IEEE Tran on Robotics, vol 22, no 1, pp 1-1 1 Wenger P (2004), Uniqueness Domains and. .. IEEE Tran on Robotics, vol 20, no 4, pp 74 5-7 50 Thomas F., Ottaviano E., Ros Ll and Ceccarelli M (2005), Performance Analysis of a 3-2 -1 Pose Estimation Device, IEEE Tran on Robotics, vol 21, no 3, pp 28 8-2 97 Huang Z and Cao Y (2005), Property Identification of the Singularity Loci of a Class of Gough-Stewart Manipulators, The Int J Robotics Research, vol 24, no 8, pp 67 5-6 85 Liu X -J.,Wang J and Pritschow... references as Freudenstein, 1962, Hunt, 1978, Sugimoto et al., 1982, Merlet, 1989, Gosselin and Angeles, 1990, Zlatanov et al., 1994, and Park and Kim, 1999 These works and some others have stated fundamental concepts as direct kinematic singularity, inverse kinematic singularity, or increased mobility configuration Some more specific concepts have also been issued, such as constraint singularity Zlatanov... defined with origin at point O and a moving frame is attached to the platform with origin at point P The loopclosure position equation is stated for every limb relating vector p that positions point P with: vectors ai that locate the fixed S joints Ai of the linear actuators, vectors that go from Ai to Bi (being si a unit vector in the direction of actuators and ρi the length of the actuator), and vectors... PATTERN SINGULARITY IN LOWER MOBILITY PARALLEL MANIPULATORS Oscar Altuzarra, Charles Pinto, Victor Petuya, Alfonso Hernandez Department of Mechanical Engineering University of the Basque Country, Alameda de Urquijo s/n 48013 Bilbao, Spain [oscar.altuzarra, charles.pinto, victor.petuya, a.hernandez]@ehu.es Abstract Many procedures to detect singularities in manipulators have been described in the literature... the platform to the inputs of the manipulator A rank deficiency in the first Jacobian in Eq 15, called Jx , imply a direct kinematic singularity A rank deficiency of the second Jacobian, called Jq , is not possible in this manipulator, and hence no singularity in the inverse problem exists The Jacobian Jx must be inverted and postmultiplied by the second Jacobian to get the following expression of the... (2005), Kinematics, singularity and workspace of planar 5R symmetrical parallel mechanisms, Mechanism and Machine Theory, In press Author Index Degani, A., 229 Dhanik, A., 201 Di Gregorio, R., 167 Diez-Martínez, C.R., 455 Donelan, P., 41 Dubowsky, S., 413 Dunlop, R., 285 Altuzarra, O., 489 Ambike, S., 177 Andrade-Cetto, J., 3 Angeles, J., 359 Babi , J., 147 Bajd, T., 185 Bamberger, H., 75 Bayro-Corrochano, . observed in the protein structures. They are main chain to main chain, main chain to side chain and side chain to side chain hydrogen bonds. The majority of the main chain to main chain hydrogen. of Kinematics Tools Detection in a Protein Chain none of the open ended side chain branches off the main chain. Thus the complexity involved with maintaining the information of the un- influencing. Bonds (main chain to main chain, main chain to Side chain and side chain to side chain), and 2) Disulphide bonds. To gain insight into a protein function, we must understand the are five different