Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 12 pps

Figure 2. Example of fully-isotropic hexapod: Isoglide6-E1(a) and its associated Figure 3. Example of fully-isotropic hexapod: Isoglide6-E2 (a) and its 327Fully-Isotropic Hexapods graph (b). associated graph (b). joints or by introducing some new joints with idle mobilities to obtain non overconstrained (isostatic) solutions. Due to space limitations, we have reduced our presentation in this paper to fully-isotropic overconstrained solutions without idle mobilities integrating just revolute and prismatic pairs in the legs A, B and C. B and C of type PɠRɠRɠR (Fig. 2) and complex legs A, B and C of type PɠRɠRb 1 ɠR (Fig. 3). The workspace of these solutions must be correlated with the angular capability of the homokinetic joints and translational capability of the telescopic shafts. 3. Conclusions of 2197 fully-isotropic hexapods with six degrees of mobility called Isoglide6-E. Special legs were conceived to achieve fully-isotropic conditions. The Jacobian matrix mapping the joint and the operational vector spaces of the fully-isotropic hexapods presented in this paper is realize a one-to-one mapping between the actuated joint velocity space and the operational velocity space. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission. Moreover, the solutions of fully-isotropic hexapods presented in this paper have all actuators mounted directly on the fixed base. As far as we are aware, this paper presents for the first time fully-isotropic parallel manipulators with six degrees of freedom and a method for their structural synthesis. 4. Acknowledgement This work was sustained by CNRS (The French National Council of Scientific Research) in the frame of the projects ROBEA-MAX (2002- 2003) and ROBEA-MP2 (2004-2006). References Angeles, J. (1987), Fundamentals of Robotic Mechanical Systems : Theory, Methods, and Algorithms, New York, Springer. Angeles, J. (2004), The qualitative synthesis of parallel manipulators, Trans. ASME Journal Mech. Design, vol. 126, pp. 617-624. 328 the 6 × 6 identity matrix throughout the entire workspace. These solutions G. Gogu An approach has been proposed for structural synthesis of a family The two examples presented in Figs. 2 and 3 have elementary legs A, Carricato, M. and Parenti-Castelli, V. (2002), Singularity-free fully-isotropic translational parallel mechanisms, Int. Journal of Robotics Research, no. 2, vol. 22, pp. 161-174. Caricato, M. (2005), Fully-isotropic four degrees-of-freedom parallel mechanisms for Schoenflies motion, Int. Journal of Robotics Research, no. 5, vol. 24, pp. 397-414. articulate, Braúov, Ed. Orientul Latin. Braúov, Ed. Trisedes Press. Fang, Y., and Tsai, L W. 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(2004b), Structural synthesis of fully-isotropic translational parallel robots via theory of linear transformations, European Journal of Mechanics– A/Solids, no. 6, vol. 23, pp. 1021-1039. Gogu, G. (2004c), Fully-isotropic over-constrained planar parallel manipulators, Proceedings of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems (IROS 2004), Sendai, pp. 3519-3520. Gogu, G. (2005a), Singularity-free fully-isotropic parallel manipulators with Schönflies motions, Proceedings of 12th International Conference on Advanced Robotics (ICAR 2005), Seattle, pp. 194-201. Gogu, G. (2005b), Fully-isotropic parallel robots with four degrees of freedom T2R2- type, Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2005), Edmonton, pp. 1190-1195. Gogu, G. (2005c), Fully-isotropic T1R2-type parallel manipulators with three degrees of freedom, Proceedings of International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE 2005), Long Beach, Paper DETC2005-84313. Gogu, G. (2005d), Fully-isotropic over-constrained parallel wrists with two degrees of freedom, Proceedings of IEEE International Conference on Robotics and Automation (ICRA 2005), Barcelona, pp. 4025-4030. Gogu, G. (2005e), Mobility and spatiality of parallel robots revisited via theory of linear transformations, European Journal of Mechanics / A –Solids, vol. 24, pp. 690-711. Proceedings of the 9th World Congress on the Theory of Machines and Mechanisms, Milan, pp. 2079-2082. 329 Fully-Isotropic Hexapods DudiĦӽ, F., Diaconescu, D., Jaliu, C., Bârsan, A. and Neagoe, M. (2001a), Cuplaje mobile DudiĦӽ, F., Diaconescu, D., Lateú, M. and Neagoe, M. (2001b), Cuplaje mobile podomorfe, Fattah, A. and Hasan Ghasemi A.M. 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(1993), Synthesis of parallel manipulators using Hervé, J.M. (2004), New translational parallel manipulators with extensible Huang, Z. and Li, Q.C. (2002), General methodology for type synthesis of Huang, Z. and Li, Q.C. (2003), Type synthesis of symmetrical lower-mobility Hunt, K.H. (1978), Kinematic Geometry of Mechanisms, Oxford University Press. Hunt, K.H. (1983), Structural kinematics of in-parallel-actuated robot arms, Kim, H.S. and Tsai, L W. (2002), Evaluation of a Cartesian parallel Kong, X., Gosselin, C.M. (2001), Generation of parallel manipulators with three Kong, X. and Gosselin, C.M. (2002), Type synthesis of linear translational Kong, X. and Gosselin, C.M. (2004a). Type sysnthesis of 3T1R parallel Kong, X. and Gosselin, C.M. (2004b), Type synthesis of analytic translational Kong, X. and Gosselin, C.M. (2004c), Type synthesis of 3-dof translational Li, Q., Huang, Z. and Hervé, J. M. (2004), Type synthesis of 3R2T 5-DOF parallel , 2 édition, Paris, Hermès. Merlet, J.P. (1997), Les robots parallèls Merlet, J.P. (2000), Parallel Robots, Dordrecht, Kluwer Academic Publishers. ANEWCALIBRATIONSTATEGYFOR A CLASS OF PARALLEL MECHANISMS Philipp Last Institute of Machine Tools and Production Technology Langer Kamp 19b,38106 Braunschweig, Germany p.last@tu-bs.de J¨urgen Hesselbach Institute of Machine Tools and Production Technology Langer Kamp 19b,38106 Braunschweig, Germany j.hesselbach@tu-bs.de Abstract Geometric calibration has been proven to be an efficient way to en- hance absolute accuracy of robotic systems. The idea is to identify the geometric parameters of the kinematic model matching the real robots’ geometry. Basically calibration is performed by analyzing the differ- ence between conflicting information gained by the kinematic model and corresponding redundant information. In all existing robot calibra- tion approaches required redundancy is achieved either by extra sensors or by special constraint devices. This paper for the first timeproposes a calibration method that does not rely on any extra device, thus being very economical. The presented technique which only holds for parallel robots is based on a method that allows passing singularities of type two. By means of simulation studies using a FiveBar-structure as an example the approach is verified. Keywords: Parallel Kinematics, Calibration, Singularities 1. Introduction Although automated robot programming is a well engineered technol- ogy, most robotic systems are still programmed by using the teach-in approach. This is due to an insufficient absolute accuracy offered by most industrial robots. Positioning errors are mainly caused by a devia- tion between the controller model and the real robots’ geometry effected by thermal influences, manufacturing and assembly tolerances [Mooring et al., 2005]. Geometric calibration has been shown to be a suitable method to overcome that drawback. It is a process by which the pa- rameters of the kinematic model are estimated in a way that best fits the real robot. Parametric calibration requires redundant measurement information that is usually obtained by additional internal or external © 2006 Springer. Printed in the Netherlands. 331 J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 331–338. measurement systems such as lasertracker-devices, theodolites, camera systems or passive joint sensors. Alternatively the robots’ degree of freedom (dof) may be restrained by passive devices. In that case the actuator encoders of the system deliver enough information allowing for parameter identification. Various calibration technique of both cate- gories are compared in [Hollerbach and Wampler, 1996]. This contribution for the first time presents a calibration strategy which does not require any calibration equipment. Due to the aban- donment of external measurement systems or constraint devices the proposed calibration approach is very inexpensive compared to other techniques. Furthermore it belongs to the class of self-calibration meth- ods [Maurine et al., 2005] and can thus be completely automated and repeated whenever necessary. Redundancy is achieved by special knowl- edge about singular configurations of type two which need to be passed in order to identify the kinematic parameters of a parallel kinematic ma- nipulator. Our approach is therefore based on a technique that allows to savely guide a parallel mechanism through singularities of type two, introduced in [Helm, 2003]. Without a loss of generality the new cali- bration approach will be explained and validated by means of a simple 2-dof planar parallel structure, the FiveBar-robot [Sachau et al., 2002]. 2. As mentioned in the previous section our new calibration approach re- lies on passing singularities of type two. Because these constitute struc- ture configurations where two solutions of the direct kinematic problem (DKP) coincide, they are also called direct kinematic singularities. It is well known that a robot-structure is uncontrollable in this kind of configurations [Hesselbach et al., 2005] and hence particular strategies need to be applied to savely guide a manipulator through singularities of type two. With the intention of workspace enlargement Helm pre- sented a technique to pass direct kinematic singularities, which has been approach has been extended to spatial parallel structures in tuate the robot system during passing the singular configuration and to use an additional driving force to guide the structure through the direct kinematic singularity. By means of the planar FiveBar-structure the configuration (a) the structure is underactuated by releasing one actua- tor (b). While the second actuator is kept at a constant motor-position P. Last and J. Hesselbach332 approach is exemplarily summarized in Fig. 1. In a pose near the singular experimentally proven at a planar robot-structure [Helm, 2003]. The et al., 2005]. Both methods rely on the basic idea to temporarily underac- [Budde Idea of the Calibration cheme the endeffector-point C passes the singularity (c) driven by gravity S g q 1 q 2 g q 1 q 1 q 2 q 2 a) c ) d) q 1 q 2 g b) x y A 1 A 2 C B 1 B 2 active joint passive joint L 11 L 12 L 22 L 21 endeffector movement movement of the released actuator released joint Figure 1. FiveBar-structure: Kinematic design (a) and basic steps while passing leased actuator can be activated again. Instead of exploiting gravity as the driving force which has been also done in [Budde et al., 2005], struc- ture inertia may be used to pass the singularity as described in [Helm, 2003]. Mathematically singular configurations of type two can be detected by det(J)=0 (1) with the Jacobian matrix J = ∂q ∂X relating endeffector velocities ˙ X and actuator velocities ˙ q [Gosseling and Angeles, 1990] ˙ q = J ˙ X . (2) Following the above mentioned strategy to pass the singularity one actu- ator needs to be released while all others remain at a constant position. In that case the systems’ dof is one and J reduces to a scalar expression J = ˙q ˙ X . (3) Since for scalar values det(J)=J , (4) 333 A New Calibration Stategy a singular configuration in (b)-(d). influence until it reaches a nonsingular configuration (d) in which the re- a direct kinematic singularity is according to Eq. 1 reached under the condition ˙q =0∧ ˙ X =0 . (5) Thus, in a singular structure configuration the velocity of the released actuator ˙q released is zero while the endeffector is still in motion. This cor- responds to the structure in Fig. 1 where the released actuator changes its direction of movement exactly in the singular configuration (indicated by the dashed line in a,b,c). Consequently by observing the movement of the released actuator by its own encoder it is possible to identify and save the actuator coordinate ˆq sing released that corresponds to a singular config- uration. Furthermore, since particular geometric conditions need to be fulfilled at a singular configuration of type two, it is possible to compute the actuator coordinate q sing released (k)from the kinematic model including the kinematic parameters k.Comparing both information leads to a residual r(k)=ˆq sing released − q sing released (k). (6) Passing the direct kinematic singularity at different locations allows for aformulation of different residual functions. These may be assembled real robot-structure, then r(k)=0. Since we assumeparameter errors, mathematical optimization methods may be applied to find k such that r(k)isminimized. 3. In order to validate the presented approach the FiveBar-structure from Fig. 1 will be calibrated. Its kinematic model is defined by only five parameters. These are (Fig. 1a): 1parameter L 0 defining the distance A 1 A 2 between the two actu- ator base points. 1parameter L i1 for each kinematic chain i =1, 2 describing the length of the crank A i B i 1parameter L i2 for each kinematic chain i =1, 2 specifying the rod length B i C While typical kinematic problems are concernd with relating endef- fector and actuator coordinates the calibration approach presented here requires to determine the actuator coordinate q sing released from arbitrary given fixed actuator coordinates q fixed and a vector of kinematic pa- rameters k q sing released = f SKP (k, q fixed ). (7) in a vector r(k). Ideally, if the kinematic model exactly matches the FiveBar- obot inematics R K 334 P. Last and J. Hesselbach We refer to this problem as the Singular Kinematic Problem (SKP). The FiveBar-structures’ SKP can be solved analytically. For brevity index f is introduced for parameters of the chain with the fixed actuator and index r for parameters of the chain whose actuator is released. We assumethatr A f =[x A f ,y A f ] T and r A r =[x A r ,y A r ] T pointing from the base coordinate system to point A f and A r respectively, are given with the restriction that |x A r − x A f | = L 0 and consequently y A r = y A f .By known vector r B r =[x B r ,y B r ] T , q released canbesolvedto q released = q r = atan 2(y B r − y A r ,x B r − x A r ). (8) The structure is in a singular configuration under the geometric con- dition that the two rods of the robot build a common line. Hence r B r can be computed by intersecting a circle K I with its center in B f and radius R I = L 12 + L 22 and a second circle K II around A r with radius R II = L r1 . With the substitutes S 1 = x 2 A r +y 2 A r −(x A f +cos q f L f 1 ) 2 −(y A f +sin q f L f 1 ) 2 +R 2 I −R 2 II 2y A r −2y A f −2sinq f L f 1 (9) S 2 = x A f +cos q f L f 1 −x A r y A r −y A f −sin q f L f 1 (10) S 3 = S 1 S 2 −S 2 (y A f +sin q f L f 1 )−(x A f +cos q f L f 1 ) S 2 2 +1 (11) S 4 = (S 1 −y A f −sin q f L f 1 ) 2 +(x A f +cos q f L f 1 ) 2 −R 2 I S 2 2 +1 (12) two solutions x B r 1,2 = − S 3 ±  S 2 3 − S 4 ; y B r 1,2 = S 1 + S 2 x B r 1,2 (13) can be derived for x B r and y B r , leading to two solutions of the FiveBar- structures’ SKP with Eq. 8. 4. Requirements and imitations The technique presented above is a very promising strategy to cali- brate parallel mechanisms and thus to enhance their absolute accuracy. There are however several limitations: As there is a risk of damaging a robot-structure it is usually avoided to approach direct kinematic singularities. Due to this several par- allel structures are dimensionedanddesignedinawaythatnosin- gular configuration of type two exist in their workspace. Obviously these manipulators cannot be calibrated by means of the proposed calibration scheme. L 335 A New Calibration Stategy Each time a singularity is passed one redundant information can be gathered. In order to identify n parameters by the calibration process at least n independent informations need to be determined. This requires that different singular configurations exist, which is for example not the case for the Paraplacer-structure presented in [Helm, 2003]. If only angular measurements are used for parameter calibration no unique parameter-set can be identified as each scaled version of the robot defines a possible solution to the calibration problem. Consequently one metric parameter needs to be known in advance and serves as a reference-dimension during calibration, meaning problem is the same for all calibration strategies). Since direct kinematic singularities for the FiveBar-robot occur under the geometric condition of the two rods building one line, the calibration process cannot differ between a parameter deviation of L 12 and one in L 22 . This however results from the particular design feature that both kinematic chains are directly connected to each other in one joint. Most parallel structures, especially those with adof> 2 are designed in a way that the chains are not directly connected to each other but to an additonal passive platform link. In that case this problem does not occur. The last two itemslimit parameter identification for the FiveBar- robot under consideration. In order to circumvent the scaling problem, it is assumed that L 0 is exactly known and will serve as the reference parameter during calibration. Furthermore, as it cannot be differed between parameter-deviations in L 12 and L 22 , L 12 + L 22 will be handled as one parameter of the calibration procedure. In summary only three of the five parameters describing the system can be identified. 5. Simulation tudies In order to validate the proposed calibration approach various sim- tor k real containing the actual robot geometry-parameters is generated which adds random values in the range [±1mm] to the nominal values of the three kinematic parameters L 11 , L 21 , L 12 + L 22 that are supposed to be identified by the calibration process. Nominal parameters k nom as well as typical real robot parameters k real Gathering of redundant information is simulated by application of ˆq sing 2,j = f SKP (k real , q 1,j ), where j indicates a specific configuration. This that this parameter remains constant during calibration. (This S ulation studies have been performed. For simulation purposes a vec- 336 P. Last and J. Hesselbach are given in Table 1. [...]... Advances in Robot Kinematics, 339–348 © 2006 Springer Printed in the Netherlands 340 M Krefft and J Hesselbach the kinematics is defined This is a typical approach because the tools for kinematic and dynamic analysis are often not the same and the dynamic analysis is usually very difficult to perform In this paper we will discuss new performance criteria By integrating the kinematics and dynamics in the... and particularly because of the unique complication of nonassembly during the numerical solution of the associated optimization problem It is of interest to note that the problem of non-assembly in the optimal synthesis of linkages has also received little attention [Minnaar et al., 2001] 349 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 349 –356 © 2006 Springer Printed in the Netherlands... real values within a small number of iteration steps indicating that the algorithm works successful deviation [mm] 1e - 0 1e-2 1e-4 1e-6 1e-8 1e-10 1e -1 2 1 2 3 iteration step L12+L 22 4 5 L22 6 L11 parameter Figure 2 Simulation result corresponding to Table 1 showing the effectiveness of the presented calibration approach 338 6 P Last and J Hesselbach Conclusion For the first time a robot calibration... Parallel Robot, Proc of the IEEE International Conference on Robotics and Automation, ICRA, Barcelona, Spain, pp 350 7-3 512 Hollerbach, J M., Wampler, C W (1999), The Calibration Index and Taxonomy for Robot Kinematic Calibration Methods, International Journal of Robotics Research, vol 15 Maurine, P., Liu, D M., Uchiyama, M (1998), Self Calibration of a New Hexa Parallel Robot, Proc of the 4th Japan-France... Kinematically Driven System and Non-Assembly In general a kinematically driven system is described by a combination of n kinematic and driving constraint equations of the form [4]: Φ(q, t) = 0 (1) 352 J.A Snyman where q is a n-vector of generalized coordinates and t denotes the instant in time It is assumed here that equation (1) explicitly or implicitly specifies the position vector r(t) of the end-effector... metal forming, which was designed at the IWF in 2002 (Fig 3) The advantage of the hydraulic actuators is the high power to weight ratio For the metal forming process the platform has to rotate around the pitch and roll axes with a sin-characteristics respectively, resulting in a tumbling motion with a maximum angle of 15◦ and 1 Hz The process forces are 50kN in z-direction and 5kN in the xy-plane Beside... warning that a lockup configuration and nonassembly is being approached In the following section a simple example is presented that illustrates such a situation and shows how the behavior of the manipulator may be monitored in practice 4 Simple E xample of Lock-Up and Non-Assembly Consider a two-link revolute manipulator (l1 = l2 = 1) with the configuration specified by link angles φ1 and φ2 as shown in. .. design is obtained that results in lock-up and resultant non-assembly further along the path (between the initial and end point), then obviously the evaluation of the objective function (4) is not possible The behavior of the accelerations near lock-up, as discussed in the previous sections, now points to a practical strategy for obtaining a meaningful value for the objective function even if non-assembly... pp 111 124 Erdman, A.G and Sandor, G.N (1997), Mechanism design: analysis and synthesis, Upper Saddle River, New Jersey, Prentice-Hall Haug, E.J (1989), Computer-aided kinematics and dynamics of mechanical systems, Boston, Allyn and Bacon Minnaar, R.J., Tortorelli, D.A., and Snyman, J.A (2001), On nonassembly in the optimal dimensional synthesis of planar mechanisms, Structural and Multidisciplinary... general method for solving this particular problem Often the nature of the system being optimized dictates the method to be adopted in overcoming this problem Here the previous problem-specific methodology of Snyman and Berner, for overcoming difficulty (iv), is reviewed and justified in terms of general kinematic constraint equations 2 Problem Formulation In the optimal design of a serially linked manipulator, . Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 339–348. 339 Mathias Krefft m.krefft@tu-bs.de Juergen Hesselbach Institute of Machine Tools and Production. additional internal or external © 2006 Springer. Printed in the Netherlands. 331 J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 331–338. measurement systems such as lasertracker-devices,. , Lie-groups: Y-STAR and H -ROBOT, Proceedings of IEEE Intl. Workshop on Advanced Robotics, Tsukuba, pp. 7 5-8 0. parallelogram, Proceedings of the 11 th World Congress in Mechanism and Machine