When there are two unknown forces (such as in Fig. 7(c)) the forces and torques cannot be calculated directly. This situation can often be solved by starting at several points in the chain and propagating the known forces and moments to a common point. In other cases additional information (such as provided by an additional force/torque sensor) is needed to permit a solution. When all of the links in the system have been visited it is possible to determine if the given set of sensors is sufficient or if additional sensors are required. 2.3 Analysis of Canonical Elements The above analysis can be applied case by case to the canonical elements in Fig. 3. First consider the chain with known loads (Fig. 3(a)). Starting with the link on the far left, it is possible to find the actuator torques on the first joint. Continuing with the links from left to right it is possible to find forces and torques on all joints in this system. Hence there are enough sensors to completely identify all actuation efforts for this case. The canonical element chain with one unknown load also has enough sensors, but it is necessary to work inward from both ends of the chain simultaneously so that the single unknown load at the middle link can be determined. However, for any chain that has more than one unknown load (as in Fig. 3(c)) all actuator efforts cannot be determined without adding more sensors. Loops can be resolved into two chains joined by two branching links. Loops are analysed by starting with a link that has only known applied loads and propagating the loads in both directions around the loop until the chain rejoins. It can be shown that there are not enough sensors to determine all actuation efforts for any of the three canonical elements with loops. However, inserting a sensor in a loop converts this problem into the case of the chain with known loads (Fig. 3(a)). To summarize, for all the canonical elements, only a chain with known loads (Fig. 3(a)) and a chain with one unknown load (Fig. 3(b)) have enough sensory information to determine all actuation efforts. Using the analysis of the canonical elements, it is straightforward to apply the results to the original system to determine sensor placement. For any given robot configuration with multiple manipulators, links, branches, etc., it is possible to enumerate potential sensor placements, divide the system into subsystems at the sensors, classify each subsystem by its canonical element, eliminate the layouts where there is not enough sensory information, and find the minimal number and placement of sensors for the system. P. Boning and S. Dubowsky 418 3. System Topologies Systems such as Fig. 2 were studied to determine the torques at each joint and the reaction jet forces. The parameters varied are number of manipulators (p = 1, 2), number of links per manipulator (n = 1, 2, many), reaction jets or not (free-flying or free-floating), and payload or not. The primary locations for sensor placement are the manipulator wrist and the manipulator base where it joins the spacecraft. For most cases it is not necessary to enumerate all the cases where the sensor is placed at any joint in between since they are often equivalent to the cases where there are sensors at the ends of manipulator. 3.1 Minimum Sensor Configurations The sensor placement method presented above was applied to space robots with one and two manipulators. A collection of single manipulator cases with and without thrusters is summarized in Fig. 8. In all cases with a single manipulator there was adequate sensing. Fig . 9-12 summarize the results for space robots with two manipulators. The cases in Fig. 9 do not have (or are not firing their) thrusters. The first row shows possible sensor placements when one sensor is available. It can be placed between the manipulator and the spacecraft, at the end effector, between both manipulators and the spacecraft, or at the end of both end- effectors. The second row shows placement of two sensors, the third three sensors, and the last row the only configuration with four sensors. All cases reduce to the canonical chain elements with at most one unknown load, except for the two loop cases which are crossed out. The crossed out cases do not have enough sensing to determine all actuation efforts. From the remaining cases which do have enough information it is easy to determine the minimal sensors (one) and their potential locations. These cases are outlined in bold. Fig. 10 shows the same cases as Fig. 9, except that the space robots now have thrusters. The addition of the unknown thruster loads does not change the results. There are still only two cases which do not have enough sensing, and a single force torque sensor is enough to determine actuation. Fig. 11 shows robots which have no thrusters but carry a payload grasped by both manipulators creating a closed loop. Most of the loops are broken by a sensor so actuation can be determined but there are only two places to put a single sensor to determine actuation. Fig. 12 shows the robots from Fig. 11 with thrusters. Once again addition of unknown thruster forces does not significantly change the results. A Study of Minimal Sensor Topologies 419 s Figure 8. Space robot configurations for a single manipulator, no thrusters Figure 9. Space robot configurations for two manipulators and no thrusters Figure 10. Space robot configurations for two manipulators and thrusters Figure 11. Space robot configurations for two manipulators and payload Figure 12. Space robot configurations for two manipulators, payload, thrusters P. Boning and S. Dubowsky 420 . . . . . 4. Illustrative Examples To demonstrate the validity of the basic method it is applied to the system shown in Fig. 2. One sensor could be used; however two sensors very well even though the manipulator is unable to follow the torque command due to very large friction. In this case, a torque loop is not closed to compensate for friction. With a valid torque estimate this could be accomplished but control algorithms are beyond the scope of this paper. Fig. 14 shows the results of the same sensor measurements used to estimate the applied spacecraft reaction jet forces. The actual reaction jet forces do not match the commanded due to nonlinear effects. However the method is still able to estimate these forces. 0 2.25 4.5 −2000 −1000 0 1000 2000 time [sec] Commanded Torque Manipulator 1 Joint 1 Estimated Torque (dashed line) and Applied Torque (solid line) virtually the same Joint 1 Torque [N−m] Figure 13. Manipulator 1 torques for the satellite capture task 0 2.25 4.5 −1000 −500 0 500 1000 time [sec] F x [N] Commanded Spacecraft Force Estimated Force (dashed line) Applied Force (solid line) Maximum Error <5% Figure 14. Continuous commanded net thruster forces, satellite capture task 5. Conclusions In space robots, actuator effort sensing is required for precise control. However, such sensing adds complexity, weight and cost. Hence it is important to minimize the number of sensors used. Here it is shown that there are minimum sensor configurations that are able to determine system actuation precisely. It was found that a base force torque sensor are included here to provide sensor redundancy. With failure of one sensor this system could still maintain precise control. While the above results are valid for 3D systems, for clarity 2D cases of a satellite capture task were simulated. The simulations were done in Matlab for the free-flying space robot, firing its thrusters at the same time as the manipulator end-effectors were tracking the grasp points on the satellite [Boning 2006]. Fig. 13 shows joint torques for the first manipulator. The mani- pulators have very high friction, 50% of their maximum capable tor- que. The method’s torque estimate and actual applied torque agree . . A Study of Minimal Sensor Topologies 421 for each manipulator can provide an estimate for friction in the joints and applied reaction jets. A wrist force torque sensor for each manipulator can also be used to estimate joint friction and applied reaction jet forces. However, additional sensors are needed for cases when there are closed kinematic loop configurations. The methods shown here can be applied to other situations such as unknown contact forces at the end-effector, unknown payload mass, or payload gripped with pin joints (rather than rigidly grasped). Systems with reaction wheels can be considered with this methodology. The approach is useful to study redundant sensor configurations and accommodate sensor failure. Current studies are underway to consider the effects of higher order dynamics and sensor noise. An experimental validation of the method is expected to be completed shortly (in time for the ARK conference). Acknowledgment The support of the Japan Aerospace Exploration Agency (JAXA) is acknowledged by the authors. The authors would like to thank Prof. Ohkami, Mr. Ueno, and Mr. Ishijima for their useful comments. References Boning, P. and Dubowsky, S. (2006), Identification of Actuation Efforts using Limited Sensory Information for Space Robots, Proc. of the IEEE International Conference on Robotics and Automation, (to be published). Breedveld, P., Diepenbroek, A., van Lunteren, T. (1997), Real-time Simulation of Friction in a Flexible Space Manipulator, Proceedings of the 8th International Conference on Advanced Robotics, Monterey, CA. Satellite Capturing Strategy using Agile Orbital Servicing Vehicle, Hyper- OSV, Proc. of the IEEE Int Conf. on Robotics and Automation, Wash. DC. Morel, G., Iagnemma, K., and Dubowsky, S. (2000), The Precise Control of Manipulators with High Joint-Friction Using Base Force/Torque Sensing, Automatica: The Journal of the International Federation of Automatic Control, Staritz, P.J., Skaff, S., Urmson, C., and Whittaker, W. (2001), Skyworker: A Robot for Assembly, Inspection and Maintenance of Large Scale Orbital Facilities, Proc. IEEE Int. Conf. on Robotics and Automation, Seoul, Korea. Ueno, H., Nishimaki, T., Oda, M., and Inaba, N. (2003), Autonomous Cooperative Robots for Space Structure Assembly and Maintenance, Proc. 7th Int. Symp. on Artificial Intelligence, Robotics, and Automation in Space, NARA, Japan. Vischer D. and Khatib O. (1995), Design and Development of High-Performance Wilson, E., Lages, C., and Mah, R. (2002), Gyro-based maximum-likelihood thruster fault detection and identification, Proc. of the American Control Conference, Anchorage, AK. P. Boning and S. Dubowsky 422 Matsumoto, S., Ohkami, Y., Wakabayashi, Y., Oda, M., and Ueno, H. (2002), No. 7, vol. 36, pp. 931-941. Torque-Controlled Joints, IEEE Trans. Robotics & Automation, No. 4, vol. 11. KINEMATICS AND OPTIMIZATION TRANSLATING 3-CCR/3-RCC PARALLEL MECHANISMS Massimo Callegari and Matteo-Claudio Palpacelli Polytechnic University of Marche, Department of Mechanics Ancona, Italy {m.callegari, m.palpacelli}@univpm.it Abstract The paper presents the kinematics of the 3-RCC/3-CCR translating parallel mechanisms and several machines of such family are described in detail taking into account different possible kinds of actuations. They all share good kinematic properties as for instance simple closed-form relations and convex workspace, but differ for overall dimensions of the mobile platform and dynamic behaviour: therefore the concepts have been optimized and compared against common performance indices, to determine the best solutions for selected classes of applications. Based on such results, a prototype robot has been finally built. Keywords: Parallel robots, Translating Parallel Machines, Kinematic analysis, 1. Introduction The kinematics of parallel mechanisms with full spatial mobility is usually very complex and sometimes cannot even be solved in closed form, as for example in the well-known case of the general Gough- Stewart platform (Liu and Fitzgerald, 2003). For this reason it is recently growing the interest of researchers and industry towards minor mobility mechanisms, able to perform elementary motions like pure rotations, pure translations or planar displacements: in this way the complexity of the analytical problem is greatly reduced while the advantages of closed- loop actuation are still preserved, eventually by having two parallel mechanisms mounted in series or cooperating in parallel for the accomplishment of a given task. Many new translating parallel machines (TPMs) have been studied in the last years, including the 3-RPC mechanism by (Callegari and Tarantini, 2003) that showed good kinematic performances, e.g. simple equations, high stiffness, convex workspace, and so on. However the aforementioned concept presented also some clear weak points, such as a cumbersome moving platform, poor dynamic performances and an © 2006 Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 423–432. 423 Conceptual design, Optimization OF THE overconstrained structure: therefore the study has been enlarged to the whole family of machines with 3-RCC/3-CCR kinematics (Callegari and Marzetti, 2003) to assess whether other mechanisms of this set show better features than the one previously identified. The paper briefly presents three mechanisms selected among the most optimization and compares the resulting performances. 2. A whole family of mechanisms can be defined by the functional design schematically represented in Fig. 1a: a mobile platform is connected to the fixed base by three identical limbs, that are composed by two members coupled by a cylindrical pair; the lower link of each leg is connected to the frame by a revolute joint while the upper one is connected to the platform by a cylindrical pair. Such kind of mechanisms are conventionally called 3-RCC to indicate the sequence of the joints in the three (identical) limbs, starting from the fixed frame and moving towards the platform; the 3-CCR architecture, shown in Fig. 1b, is simply obtained by kinematic inversion. It can be easily seen that the described architecture is characterized by 3 d.o.f.’s in space where in the general case spatial translations are coupled with changes of orientation of the platform. Nevertheless for particular geometrical configurations such mechanisms can provide motions of pure translation, i.e. iff (i) the axes of the revolute and outer cylindrical joint of i th limb are parallel to the same unit vector i u ˆ and (ii) ji uu ˆˆ ≠ for i j (i,j = 1,2,3). (a) (b) It must be pointed out that in these cases the two links of each leg do not turn around the cylindrical pair, which could well be substituted by a prismatic joint, giving rise to the already mentioned 3-RPC (or 3-CPR) overconstrained mechanism. 424 M. Callegari and M C. Palpacelli Description of the Family of Mechanisms Figure 1. The 3-RCC (a) and 3-CCR (b) parallel mechanisms. to the possible different actuations and finally performs a kinematic interesting elements of this class, develops their kinematics according 3. In this section three different mechanisms are considered. The 3-RCC architecture shown in Fig. 1a can be actuated either by driving the base revolute joints or by controlling legs’ extensions: the mobile platform of such concepts turns out to be rather bulky because of the (unavoidable) length of the stroke of the cylindrical pairs mounted on the platform itself. This drawback can be possibly avoided by inverting the kinematic structure of the mechanism, Fig. 1b: in this case the legs can be actuated by directly driving the sliders running along the fixed slideways. Unfortunately a mere inversion of the mechanism is not possible because the resulting kinematics shown in Fig. 1b is singular in its entire workspace; therefore two more concepts are presented, slightly different from first mechanism but both characterized by a 3-CCR architecture. 3.1 The configuration of the mechanism is symmetric, Fig. 1a, with the axes of the three revolute pairs forming an equilateral triangle on the fixed base; in the same manner, another equilateral triangle is formed on the moving platform by the axes of the cylindrical pairs; moreover, the legs are perpendicular to the joints connecting them with the two bases. In case the machine is driven by controlling legs’ lengths, IPK is characterized by a single configuration of the mechanism while DPK presents 8 different solutions that can be simply evaluated in closed-form (Callegari and Tarantini, 2003). When the unit vectors of the three limbs become linearly dependent, the manipulator gets stuck in a singular configuration: therefore the locus of all the singular points is given by a right cylinder, that however can be moved outside the workspace with a mechanism can be easily driven by means of rotary motors lumped at the end of the limbs and ball screws, see Fig. 2: such design yields very good static properties (e.g. a high thrust can be delivered since the legs are loaded only by normal forces) but the resulting dynamic behaviour is poor due to the relevant mass of the mobile platform and the high inertia Better dynamic performances can be obtained by directly powering the base revolute joints, even if the limbs are loaded by bending moments in this case. (Callegari and Marzetti, 2003) developed the complete kinematics of such mechanism: it is shown that a (very simple) unique solution exists for both direct and inverse position problem and no translation or rotation singularities are possible at all. 425 Kinematics and Optimization Kinematics of Some Mechanisms 3-RCC with Triangular Configuration proper dimensioning of the machine. No constraint (rotation) singu- larities exist apart from the surface of the mentioned cylinder. This of the spinning or tilting masses (Callegari et al., 2006). 3.2 The mere kinematic inversion of the 3-RCC mechanism just described is shown in Fig. 1b: (Callegari and Marzetti, 2003) proved that such As a matter of fact, in order to give full mobility to the inverted mechanism it is necessary that the direction of the axes of the three pairs connecting the limbs to the ground are linearly independent. (a) (b) th For instance, in the tetrahedral configuration shown in Fig. 3a such axes stem from the origin of the fixed frame () zyxO ˆ , ˆ , ˆ and are tilted by α radians with respect to the horizontal plane. If a frame () wvuP ˆ , ˆ , ˆ , parallel to () zyxO ˆ , ˆ , ˆ , is attached to the mobile platform at the intersection of the 426 M. Callegari and M C. Palpacelli Figure 2. Sketch of the 3-RCC translational platform (actuation on legs’ length). 3-CCR with Tetrahedral Configuration limb. Figure 3 . Sketch of the tetrahedral 3-CCR architecture and loop-closure for i mechanism is useless since all the points of its workspace are singular. three revolute joints, the vector OP=p can be taken to specify the position of the moving platform. In order to simplify the study of the mechanism, other frames can be easily defined for each link as shown in Fig. 3b: e.g. the frame ) ˆ , ˆ , ˆ ( iiii zyxA , attached at of i th limb, is obtained by starting at the global frame () zyxO ˆ , ˆ , ˆ , then it is rotated by i ϕ around the (current) z i axis and then by α around the (current) y i axis, finally a translation a i along the direction of the (current) x i axis is performed, to allow for the variable sliding of the cylindrical pair. Moreover, the articular coordinates of the 3 joints are defined as follows: a i and i ϑ for the first cylindrical pair, d i and β i for the second one and γ i for the revolute joint. It is also noted that the configuration of Fig. 3 is characterized by maximum symmetry, therefore °=°=°= 240,120,0 321 ϕϕϕ and tt i = for i = 1,2,3. The loop-closure equation of i th limb is given by: iii tdap ++= (1) that can be expressed in the local i A frame as: » » » ¼ º « « « ¬ ª − » » » ¼ º « « « ¬ ª − » » » ¼ º « « « ¬ ª = » » » ¼ º « « « ¬ ª » » » ¼ º « « « ¬ ª − − i ii ii ii i i i z y x ii ii ii c sd ss sc c t a p p p csssc cs scscc ϑ ϑ βϑ βϑ β ααϕαϕ ϕϕ ααϕαϕ 0 0 00 (2) The mobility analysis developed in the Appendix shows that unfortunately such architecture is liable of constraints singularities inside its entire workspace; therefore, in order to prevent it rotating, it is necessary to turn to an overconstrained architecture by replacing limbs’ internal cylindrical pairs with a prismatic pairs, resulting with a 3-CPR kinematics. The following analysis will be performed with reference to this case, where Eq. 2 still holds with β i = 0. Inverse position kinematics relations provide the actuated variables as functions of platform s position p: iziyixi tspcspccpa +++= ααϕαϕ (3) The same Eq. 3, written for i=1,2,3, is used to solve the direct position kinematics and find the values of zyx ppp ,, as functions of i a . The direct derivation of Eq. 1 is the base for the velocity analysis: iiiiiip da dȦdav ×+⋅+⋅= ˆ ˆ (4) By dot-multiplying both sides of Eq. 4 by the unit vector i a ˆ and collecting the 3 relations for i=1,2,3 in a single expression in matrix form, av = p J , the expressions of the Jacobian matrix is obtained: 427 Kinematics and Optimization ’ [...]... lies in a plane as shown in Figure 7 Figure 6 A HHH chain with a local infinitesimal singularity Figure 7 Infinitesimal singularity of RRR generator of planar motion C.-C Lee and J.M Herv é 442 5 Infinitesimal Singularity in Chains with One P Let us consider a pseudo-planar chain including a P pair generating {T (s)} An infinitesimal displacement belonging to the subgroup {T (s)} is s the point transformation... more, with linear Delta that might be used for designing machinetools Most of recent researches in that field have proposed different designs to obtain Scara motions either for serving as pick -and- place robots, or for 445 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 445–454 © 2006 Springer Printed in the Netherlands 446 S Krut et al being a part of a more complex machine-tool; some of... discriminate two types of singularity, namely permanent singularity and local singularity The permanent singularity yields an inadequate chain that does never generate the desired Y motion Local 435 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 435– 444 © 2006 Springer Printed in the Netherlands C.-C Lee and J.M Hervé 436 singularities are specific of particular poses of the chain that... Zamora-Esquivel, E Bayro-Corrochano Kinematics and grasping using conformal geometric algebra 473 R Subramanian, K Kazerounian Application of kinematics tools in the study of internal mobility of protein molecules 481 O Altuzarra, C Pinto, V Petuya, A Hernandez Motion pattern singularity in lower mobility parallel manipulators 489 PSEUDO-PLANAR MOTION GENERATORS Chung-Ching Lee Dept Tool & Die-Making... link maintains the parallelism and the chain is a wrong generator of Y-motion In a PHP array, the angle between the two P pairs can change and the chain generates Y-motion when the P pairs are not parallel and may become locally singular in a possible posture with transitory parallel P pairs Both cases are shown in Figure 3 (a) permanent finite singularity (b) locally finite singularity Figure 3 Finite... under way (a) Figure 6 (b) Concept (a) and prototype (b) of the 3-CPU translation robot References Callegari, M., Marzetti, P (2003), Kinematics of a Family of Parallel Translating Mechanisms, Proc Intl Workshop on Robotics in Alpe-Adria-Danube Region, Cassino (Italy), May 7-1 0 Callegari, M., Palpacelli, M.-C., Principi, M (2006), Dynamics Modelling and Control of the 3-RCC Translational Platform, submitted... Kinematics and Optimization 5 431 Conclusions The paper has presented the kinematic properties of the translating platforms characterized by the 3-RCC architecture: it has been shown that also its 3-CCR kinematic inversion can be considered but, due to rotation singularities, the rotation of the inner cylindrical joint must be avoided by turning to the 3-CPR overconstrained architecture After having... Figure 5 Finite local singularity of Y-motion generators with two coaxial H pairs 440 4 C.-C Lee and J.M Hervé Infinitesimal Singularity of HHH Generators Through an infinitesimal displacement belonging to the {H (N, w, p)} subgroup, a generic point M of the Euclidean affine 3D space is transformed into another point M' = M + dM, that is, M M M' = N +( p/2 ) u+ exp( w ) (NM) N (4) where is an infinitesimal... cylindrical joints (3-CPR architecture) Motion Synthesis and Mobility C.-C Lee, J.M Herv´ e Pseudo-planar motion generators 435 S Krut, F Pierrot, O Company On PKM with articulated travelling-plate and large tilting angles 445 C.R Diez-Mart´nez, J.M Rico, J.J Cervantes-S´nchez, ı a J Gallardo Mobility and connectivity in multiloop linkages 455 K Tcho´, J Jakubiak n Jacobian inverse kinematics algorithms... Generators Figure 10 Infinitesimal singularity of RPR G-motion generator 443 Figu re 11 Infinitesimal local singularity of PRR G-motion generator To sum up, a HPH, PHH or HHP open chain aiming to be a generator of Y-motion has a local infinitesimal singularity iff the P pair is perpendicular to the plane of the two parallel H axes What is more, based on the above findings, the possible local singular postures . weak points, such as a cumbersome moving platform, poor dynamic performances and an © 2006 Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, . Springer. Printed in the Netherlands. 435 three-dimensional (3D) Lie group. The possible architectures of the - “ ” J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 435–444. Pseudo-planar. aligning the ground cylindrical pairs along a Cartesian frame. 3.3 The setting of the joints shown in Fig. 4 gives rise to a full Cartesian kinematics; in fact inverse and direct position kinematics