Industrial Robotics Theory Modelling and Control Part 5 ppt

Cable-based Robot Manipulators with Translational Degrees of Freedom 229 5.1 Tensionability of DishBot For the tensionability of DishBot, Theorem 1 given in Section 3 is not sufficient. The reason is that the spine force only affects the tension of drive cables and has no influence on the passive ones. As a result, it cannot leverage all the ten- sions. However, it should be noted that each passive cable has a pre-tensioning spring to maintain the tension. Therefore, the tensionability can be still proved based on its definition in Section 2. For the proof of tensionability, we use the idea of Theorem 1 (not the theorem itself), i.e. tension can be generated in the cables to any extent while the static equilibrium is satisfied. The free body diagram of the end-effector is shown in Fig. 8. The passive ca- bles are in parallel with the spine and their tensions are shown by a super- script p while for the tension of drive cables, a superscript d is used. The static equilibrium equations are found to be: () 0wuuuF =−−−+++= ¦ ˆˆˆˆ 321332211 ddd s ddd f ττττττ (34) ()() () 0wruwrwrM =×−×−×−= ¦ ˆˆˆˆ 1312211 ppp τττ (35) where i d i u ˆ τ 's are drive cable forces (i=1,2,3), w ˆ p j τ 's are passive cable forces, r j 's are the position vectors of the anchor points of the passive cables (j=1,2,3) and w ˆ s f is the spine force. Figure 8. Free body diagram of DishBot’s end-effector 230 Industrial Robotics: Theory, Modelling and Control A quick inspection of the above equations shows that Eq. (35), which is the equilibrium of the moments, is a set of homogenous equations independent from the spine force which, in general, results in zero tension for passive ca- bles. The tension of the drive cables are found from Eq. (34). Note that the drive cables form a cone which contains the spine and hence, using Lemma 4, the drive cable tensions are positive as long as the equivalent spine force, ppp s f 321 τττ −−− , is positive (compressive). As a conclusion, tension in the drive cables can be generated to any desired level by choosing a large enough spine force but it does not affect the tension in the passive cables. In order for DishBot to be tensionable, we also need to show that the tension in the passive cables can be increased to any desired level. For this purpose, note that, in Fig. 8, if O ′ is the geometrical center of the three anchor points of the passive cables (P 1 , P 2 and P 3 ) then Eq. (35) has non-zero solutions for passive cable tensions. In this case, the solution would be pppp ττττ === 321 where p τ is an arbitrary real value and thus can be positive. It is known that such a geometrical center coincides with the centroid of triangle ¨P 1 P 2 P 3 . As a result, if O ′ is the centroid of triangle ¨P 1 P 2 P 3 , positive equal tensions in passive ca- bles are determined only by the pre-tensioning springs. As a conclusion, DishBot is tensionable as long as the following conditions are met: 1. O ′ is the centroid of ¨P 1 P 2 P 3 , 2. The pretension of the pre-tensioning springs are equal ( p τ ) and positive (tensile), 3. The spine lies inside the cone of the drive cables. 4. The spine force satisfies 0 321 >−−− ppp s f τττ which means p s f τ 3> . 6. Planar cable-based manipulators with translational motion Planar manipulators with translational motion (in XY plane) are sufficient for many industrial pick-and-place applications such as packaging and material handling. Simplicity of these manipulators compared to spatial ones further facilitates their applications where a two axis motion is sufficient (Chan, 2005). Two new designs of planar cables-based manipulators with translational mo- tion are studied here that are tensionable everywhere in their workspace. Schematic diagrams of these manipulators are shown in Fig. 9. The spine is connected to the base and end-effector by revolute joints. The end-effector is constrained by three cables. Two of the cables form a parallelogram which eliminates the rotation of the end-effector as long as the cables are taut. As a result, the end-effector can only move in X and Y directions. Cable-based Robot Manipulators with Translational Degrees of Freedom 231 Figure 9. Planar cable-based manipulators with pure translational degrees of freedom In the first design (Fig. 9a), the parallelogram is maintained by two winches with a common shaft which makes them move simultaneously and hence, keep the cable lengths equal. Similar to BetaBot and DishBot, the workspace of this manipulator is only limited by the minimum and maximum lengths of the 232 Industrial Robotics: Theory, Modelling and Control spine and hence it can theoretically span a half circle above the base. In the second design (Fig. 9b), a pair of synchronous rotating arms preserves the par- allelogram without changing the length of the cables and therefore, possess a smaller workspace. The synchronization can be obtained by a pair of pulleys and a timing belt or a simple 4-bar parallelogram as seen in Fig. 9b. The kinematics of these manipulators consist of a single planar cone and hence easy to formulate for both direct and inverse solutions. In this paper, however, our main focus is on the their tensionability and rigidity which is presented in the following. 6.1 Tensionability of Planar Manipulators The planar manipulators of Fig. 9 are both tensionable everywhere in their workspaces. This can be proved using an approach similar to the one that was used for BetaBot. There are two geometrical conditions that should be met for the tensionability of these two manipulators. As depicted in Fig. 10a, these two conditions are as follows: Figure 10. a) The configuration of the cables and spine in planar manipulators, b) The free body diagram of the end-effector Condition 1. Cable 1 (Fid. 10a) is always on the right and Cable 3 is al- ways on the left side of the spine. This is obtained if the spine is hinged to the base at a proper point between the two sets of cables. Condition 2. On the end-effector, the spine and Cable 3 are concurrent at point E which is located somewhere between Cables 1 and 2. Cable-based Robot Manipulators with Translational Degrees of Freedom 233 To prove the tensionability, we show that a compressive spine force can be balanced by positive tensions in the cables. The proof is quite similar to the one of Theorem 2 and is briefly explained here. We first consider the force equilibrium of the end-effector subject to a com- pressive spine force. According to the free body diagram shown in Fig. 10b, we have: wuu0wuu ˆˆˆˆˆˆ 23112311 ss ff −=+=++ τ σ τ σ (36) Due to Condition 1, the direction of the spine, w ˆ is located between 1 ˆ u and 2 ˆ u (cable directions). Therefore, the projection of w ˆ s f− on 1 ˆ u and 2 ˆ u will be posi- tive and hence 31 , τ σ >0. Now, let: 1 21 2 21 21 1 1 and στστ PP EP PP EP == (37) It is clear that 121 σ τ τ =+ . Since 0 1 > σ and due to the distribution given in Eq. (37), the moment of 11 ˆ u τ about E cancels the one of 12 ˆ u τ and hence, these two forces can be replaced by 11 ˆ u σ without violating the static equilibrium. Finally, since all three forces on the end-effector, 11 ˆ u σ , 11 ˆ u τ and 12 ˆ u τ , are concurrent at E, the equilibrium of the moments is also met which completes the proof. 7. Conclusion In this paper, several new cable-based manipulators with pure translational motion were introduced and their rigidity where thoroughly studied. The sig- nificance of these new designs can be summarized in two major advantages over the other cable-based manipulators: 1. Cables are utilized to provide kinematic constraints to eliminate rotational motion of the end-effector. In many industrial applications, reduced DoF manipulators are sufficient to do the job at a lower cost (less number of axes). 2. These manipulators can be rigidified everywhere in their workspace using a sufficiently large pretension in the cables. In order to study the rigidity of these manipulators, the concept of tensionabil- ity was used and a theorem was given to provide a sufficient condition for ten- sionability. Using this theorem, tensionability of each manipulator was proved 234 Industrial Robotics: Theory, Modelling and Control using line geometry and static equilibrium in vector form. For each of these manipulators, it was shown that as long as certain conditions are met by the geometry of the manipulator, the tensionable workspace in which the manipu- lator can be rigidified, is identical to the geometrical workspace found from the kinematic analysis. BetaBot and the planar manipulators are tensionable everywhere and can be rigidified only by a sufficiently large spine force. In DishBot, on top of the geometrical conditions, a relation between the spine force and pre-tensioning springs of passive cables should be also satisfied to maintain the rigidity of the manipulator. 8. References Barrette G.; Gosselin C. (2005), Determination of the dynamic workspace of ca- ble- driven planar parallel mechanisms, Journal of Mechanical Design, Vol. 127, No. 3, pp. 242-248 Behzadipour S. (2005), High-speed Cable-based Robots with Translational Motion, PhD Thesis, University of Waterloo, Waterloo, ON, Canada Behzadipour S.; Khajepour A., (2006), Stiffness of Cable-based Parallel Ma- nipulators with Application to the Stability Analysis, ASME Journal of Mechanical Design, Vol. 128, No. 1, 303-310 Chan E. (2005), Design and Implementation of a High Speed Cable-Based Planar Parallel Manipulator, MASc Thesis, University of Waterloo, Waterloo, ON, Canada. Clavel R., (1991), Conception d'un robot parallele rapide a 4 degres deliberate, PhD Thesis, EPFL, Lausanne Dekker R. & Khajepour A. & Behzadipour S. (2006), Design and Testing of an Ultra High-Speed Cable Robot”, Journal of Robotics and Automation, Vol. 21, No. 1, pp. 25-34 Ferraresi C.; Paoloni M.; Pastorelli S.; Pescarmona F, (2004), A new 6-DOF par- allel robotic structure actuated by wires: the WiRo-6.3, Journal of Robot- ics Systems, Vol. 21, No. 11, pp. 581-595 Gallina P; Rosati G., (2002), Manipulability of a planar wire driven haptic de- vice, Mechanism and Machine Theory, Vol. 37, pp. 215-228 Gouttefarde M.; Gosselin C. (2006), Analysis of the wrench-closure workspace of planar parallel cable-driven mechanisms, IEEE Transactions on Robotics, Vol. 22, No. 3, pp. 434-445 Kawamura S.; Choe W.; Tanak S. Pandian S.R., (1995),Development of an ul- trahigh speed robot FALCON usign wire driven systems, Proceedings of IEEE International Conference on Robotics and Automation, pp. 215-220, IEEE, 1995 Cable-based Robot Manipulators with Translational Degrees of Freedom 235 Khajepour A. & Behzadipour S. & Dekker R. & Chan E. (2003), Light Weight Parallel Manipulators Using Active/Passive Cables, US patent provisional file No. 10/615,595 Landsberger S.E.; Sheridan T.B., (1985), A new design for parallel link manipu- lators, Proceedings of the International Conference on Cybernetics and Society, pp. 81-88, Tuscon AZ, 1985 Landsberger S.E.; Shanmugasundram P. (1992), Workspace of a Parallel Link Crane, Proceedings of IMACS/SICE International Symposium on Robotics, Mechatron- ics and Manufacturing Systems, pp. 479-486, 1992 Ming A.; Higuchi T. (1994a), Study on multiple degree-of-freedom positioning mechanism using wires (Part1), International Journal of Japan Society of Precision Engineering, Vol. 28, No. 2, pp. 131-138 Ming A.; Higuchi T. (1994b), Study on multiple degree-of-freedom positioning mechanism using wires (Part2), International Journal of Japan Society of Precision Engineering, Vol. 28, No. 3, pp. 235-242 Ogahara Y.; Kawato Y.; Takemura K.; Naeno T., (2003), A wire-driven minia- ture five fingered robot hand using elastic elements as joints, Pro- ceedings of IEEE/RSJ International Conference on Intelligent Robots and Sys- tems, pp. 2672-2677, Las Vegas, Nevada, 2003 Oh s.; Makala K. K.; Agrawal S., (2005a) Dynamic modeling and robust con- troller design of a two-stage parallel cable robot, Multibody System Dy- namics, Vol. 13, No. 4, pp. 385-399 Oh s.; Makala K. K.; Agrawal S., Albus J. S. (2005b), A dual-stage planar cable robot: Dynamic modeling and design of a robust controller with posi- tive inputs, ASME Journal of Mechanical Design, Vol. 127, No. 4, pp. 612- 620 Pusey j.; Fattah A.; Agrawal S.; Messina E., (2004), Design and workspace analysis of a 6-6 cable-suspended parallel robot, Mechanisms and Ma- chine Theory, Vol. 39, No. 7, pp. 761-778 Robers G.R; Graham T.; Lippitt T. (1998), On the inverse kinematics, statics, and fault tolerance of cable-suspended robots, Journal of Robotic Sys- tems, Vol. 15, No. 10, pp. 581-597 Stump E.; Kumar V., (2006), Workspace of cable-actuated parallel manipula- tors, ASME Journal of Mechanical Design, Vol. 128, No. 1, pp. 159-167 Tadokoro S.; Nishioka S.; Kimura T. (1996), On fundamental design of wire configurations of wire-driven parallel manipulators with redundancy, ASME Proceeding of Japan/USA Symposium on Flexible Automation, pp. 151-158 Tsai L-W., (1996), Kinematics of A Three DOF Platform With Three Extensible Limbs, In Recent Advances in Robot Kinematics, Lenarcic J. and Parenti- Castelli V., pp. 401-410, Kluwer Academic, Netherlands 236 Industrial Robotics: Theory, Modelling and Control Verhoeven R.; Hiller M.; Tadokoro S. (1998), Workspace, stiffness, singularities and classification of tendon-driven stewart platforms, In Advances in Robot Kinematics Analysis and Control, Lenarcic J. and Husty L., pp. 105- 114, Kluwer Academic, Netherlands Verhoeven R.; Hiller M. (2000), Estimating the controllable workspace of ten- don- based Stewart platforms, In Advances in Robot Kinematics, Lenarcic J. and Stanisic M., pp. 277-284, Kluwer Academic, Netherlands Yamaguchi F., (2002), A Totally Four-dimensional Approach / Computer-Aided Geometric Design, Springer-Verlag, Tokyo 237 8 A Complete Family of Kinematically-Simple Joint Layouts: Layout Models, Associated Displacement Problem Solutions and Applications Scott Nokleby and Ron Podhorodeski 1. Introduction Podhorodeski and Pittens (1992, 1994) and Podhorodeski (1992) defined a ki- nematically-simple (KS) layout as a manipulator layout that incorporates a spherical group of joints at the wrist with a main-arm comprised of success- fully parallel or perpendicular joints with no unnecessary offsets or link lengths between joints. Having a spherical group of joints within the layouts ensures, as demonstrated by Pieper (1968), that a closed-form solution for the inverse displacement problem exists. Using the notation of possible joint axes directions shown in Figure 1 and ar- guments of kinematic equivalency and mobility of the layouts, Podhorodeski and Pittens (1992, 1994) showed that there are only five unique, revolute-only, main-arm joint layouts representative of all layouts belonging to the KS family. These layouts have joint directions CBE, CAE, BCE, BEF, and AEF and are de- noted KS 1 to 5 in Figure 2. Figure 1. Possible Joint Directions for the KS Family of Layouts 238 Industrial Robotics: Theory, Modelling and Control KS 1 - CBE KS 2 - CAE KS 3 - BCE KS 4 - BEF KS 5 - AEF KS 6 - CCE KS 7 - BBE KS 8 - CED KS 9 - ACE KS 10 - ACF KS 11 - CFD KS 12 - BCF KS 13 - CED Figure 2. KS Family of Joint Layouts Podhorodeski (1992) extended the work of Podhorodeski and Pittens (1992, 1994) to include prismatic joints in the layouts. Podhorodeski (1992) con- cluded that there are 17 layouts belonging to the KS family: five layouts com- prised of three revolute joints; nine layouts comprised of two revolute joints and one prismatic joint; two layouts comprised of one revolute joint and two prismatic joints; and one layout comprised of three prismatic joints. However, four of the layouts comprised of two revolute joints and one prismatic joint (layouts he denotes AAE, AAF, ABF, and BAE) are not kinematically simple, by the definition set out in this chapter, due to an unnecessary offset existing between the second and third joints. [...]... becomes: c4 c5c6 − s4 s6 s5c6 s4c5c6 + c4 s6 − c4c5 s6 − s4c6 − s5 s6 − s4 c5 s6 + c4c6 c4 s5 r11 − c5 = r21 s4 s5 r31 r12 r22 r32 r13 r23 r33 (17) Using element (2, 3) of equation (17) allows θ 5 to be solved as: ( ) 2 θ 5 = atan2 ± 1 − c5 , c5 , where c5 = −r23 (18) Using elements (1, 3) and (3, 3) of equation (17) allows θ 4 to be solved as: r33 r & c4 = 13 s5 s5 Using elements (3, 1) and (3, 2) of... values and thus the right-hand-side of equation (14) is known, i.e., rij , i = 1 to 3 and j = 1 to 3, are known values 3 Substituting the elements of the rotation matrix 6 R = 3 sph R sph R into equation 6 (14) yields: c4c5c6 − s4 s6 3 6 R= 3 sph R R= sph 6 3 sph − c4c5 s6 − s4c6 c4 s5 r12 r13 R s4c5c6 + c4 s6 − s4c5 s6 + c4c6 s4 s5 = r21 r22 r23 − s5c6 s5 s6 c5 r11 r31 r32 r33 ( 15) 258 Industrial Robotics: ... ee where )( ) 6 0 6 T 45T 56 T eeT= sphT sph6TeeT sph 4 (10) 6 ee T is the homogeneous transformation describing the end-effector frame Fee with respect to frame F6 and would be dependent on the type of tool attached, 0 sph T is defined in equation (3), and c 4 c5 c 6 − s 4 s 6 s 4 c5 c 6 + c 4 s 6 sph 6T = − s 5 c6 0 − c4 c5 s6 − s4 c6 − s4 c5 s6 + c4 c6 s5 s 6 c 4 s5 s 4 s5 c5 0 0 0 0 0 1 sph 6 T is:... pz − d1 ) g 13 d3 = px − g d2 = − py d1 = p z Table 6 Inverse Displacement Solutions for KS 6 to 13 253 254 Industrial Robotics: Theory, Modelling and Control Referring to Tables 5 and 6, for KS 1 to 6, 8, 9, and 10, up to four possible solutions exist to the inverse displacement problem For KS 7, 11, and 12, up to two possible solutions exist for the inverse displacement problem For KS 13 there is only... s4c5r32 2 2 2 − s4 c 5 − c 4 c6 = −s4c5r31 − c 4r32 2 2 2 − s4 c 5 − c 4 A Complete Family of Kinematically-Simple Joint Layouts: Layout Models… 259 Note that if s5 = 0 , joint axes 4 and 6 are collinear and the solutions for θ 4 and θ 6 are not unique In this case, θ 4 can be chosen arbitrarily and θ 6 can be solved for Similar solutions can be found for the cases where α 3 equals 0 and − π 2 5. 3.2... forward displacement problem: 0 sph R 248 Industrial Robotics: Theory, Modelling and Control is the change in orientation due to the first three joints and 0 p o0 →osph is the location of the spherical wrist centre The homogeneous transformations 0 sph T for the KS family of layouts can be found in Tables 3 and 4 Note that in Tables 3 and 4, ci and si denote cos(θ i ) and sin (θ i ) , respectively KS F j... frame F j with respect to frame F j −1 and frame F j −1 to the origin of frame F j j −1 p o j−1 →o j is a vector from the origin of A Complete Family of Kinematically-Simple Joint Layouts: Layout Models… KS 1 KS 2 KS 3 2 45 KS 4 KS 5 Figure 5 Zero-Displacement Diagrams for Layouts with Three Revolute Joints (KS 1 to 5) 246 Industrial Robotics: Theory, Modelling and Control KS 6 KS 7 KS 8 KS 9 KS 10 Figure... Uncertainty and Redundancy Related Concerns, Proceedings of the 3rd Workshop on Advances in Robot Kinematics, September 7-9, 1992, Ferrara, Italy, pp 150 - 156 264 Industrial Robotics: Theory, Modelling and Control Podhorodeski, R P & Nokleby, S B (2000) Reconfigurable Main-Arm for Assembly of All Revolute-Only Kinematically Simple Branches Journal of Robotic Systems, Vol 17, No 7, pp 3 65- 373 Podhorodeski,... spatial motion 240 Industrial Robotics: Theory, Modelling and Control (a) Layout CBF (b) Layout CBD Figure 3 Examples of the Two Types of Degenerate Revolute-Revolute-Revolute Layouts 2.2 Layouts with Two Revolute Joints and One Prismatic Joint Layouts consisting of two revolute joints and one prismatic joint can take on three forms: prismatic-revolute-revolute; revolute-revolute-prismatic; and revolute-prismatic-revolute... c3 gpz f 2 + g 2 + 2s3 fg A similar procedure can be followed for the other KS layouts Inverse displacement solutions for all 13 of the KS layouts are summarized in Tables 5 and 6  252 KS 1 Industrial Robotics: Theory, Modelling and Control Inverse Displacement Solutions θ1 = atan2( p y , px ) or atan2(− p y , − p x ) ) ( 2 2 p x + p y + p z2 − f 2 − g 2 2 θ 3 = atan2 s3 , ± 1 − s3 , where s3 = 2 fg . CAE, BCE, BEF, and AEF and are de- noted KS 1 to 5 in Figure 2. Figure 1. Possible Joint Directions for the KS Family of Layouts 238 Industrial Robotics: Theory, Modelling and Control KS 1 -. KS 1 KS 2 KS 3 KS 4 KS 5 Figure 5. Zero-Displacement Diagrams for Layouts with Three Revolute Joints (KS 1 to 5) 246 Industrial Robotics: Theory, Modelling and Control KS 6 KS 7 KS 8. Kluwer Academic, Netherlands 236 Industrial Robotics: Theory, Modelling and Control Verhoeven R.; Hiller M.; Tadokoro S. (1998), Workspace, stiffness, singularities and classification of tendon-driven