A new 6-DOF parallel robot with simple kinematic model

Transcription

1 A new -DOF parallel robot with simple kinematic model Nicholas Seward, Ilian A. Bonev, Senior Member, IEEE Abstract his paper presents a novel si-legged parallel robot, the kinematic model of which is simpler than that of the simplest si-ais serial robot. he new robot is the -DOF etension of the Cartesian parallel robot. It consists of three pairs of base-mounted prismatic actuators, the directions in each pair parallel to one of the aes of a Cartesian coordinate sstem. In each of the si legs, there are also two passive revolute joints, the aes of which are parallel to the direction of the prismatic joint. Finall, each leg is attached to the mobile platform via a spherical joint. he direct kinematics of the novel parallel robot can be solved easil b partitioning the orientation and the position of the mobile platform. here are eight distinct solutions, which can be found directl b solving a linear sstem and alternating the signs of three radicals. his parallel robot has a large workspace and is suitable for machining or rapid prototping, as detailed in this paper. Kewords: Cartesian parallel robot, direct kinematics, singularities, rapid prototping. I. INRODUCION he invention of the Cartesian parallel robot [], also known as the ripteron (Fig. ), in 00 (see [] for more details), was a ke turning point in the theor of parallel mechanisms. Different designs of parallel robots had been known for decades, but none as simple as the Cartesian parallel robot, the kinematic model of which resumes to,, z, where,, and are the active-joint variables, and,, and z are the position coordinates of the mobile platform. Later, the concept was etended to DOFs (degrees of freedom) with the four-legged Quadrupteron robot [], but the kinematic model of the latter is no more linear. Its direct kinematics problem requires the solution of a quadratic equation. Later, the ripteron concept was etended to DOFs with the five-legged Pentapteron [,]. Its forward kinematics, however, boils down to solving a polnomial of degree 80, and allows at least 08 real solutions [7]. Etending the concept of the simplest parallel robot to DOFs ielded one of the most comple parallel robots. We present in this article the -DOF missing member of the so-called Multipteron famil. Contrar to epectations, the kinematic model of the Heapteron, as we call it, is not comple. In fact, it is simpler than the kinematic model of the simplest -DOF serial robot. Furthermore, the Heapteron can have a fairl large workspace, if properl designed. Figure. he ripteron Cartesian parallel robot []. In the net section, we present the Heapteron s geometric design and solve its inverse and direct kinematics. hen, in Section III, we find its singularit loci and discuss its workspace. Section IV presents the mechanical design of a D printing device based on the Heapteron architecture. Finall, we present our conclusions in Section V. II. KINEMAIC MODELING OF HE HEXAPERON he Heapteron consists of si legs, each composed of a clindrical joint, a revolute joint, and a spherical joint, arranged orthogonall as shown in Fig., where O-z is the base reference frame and C- is the mobile reference frame. In each leg, the aes of the two passive revolute joints are parallel to the direction of the actuated prismatic joint. z O ρ ρ B B B B B C B ρ ρ N. Seward is computer science instructor at the Arkansas School for Mathematics, Sciences and the Arts, Hot Springs, Arkansas 790, USA ( nicholas.seward@gmail.com). I. Bonev is professor at the École de technologie supérieure, Montreal, Quebec HC K, Canada ( ilian.bonev@etsmtl.ca). ρ ρ Figure. Schematic of the Heapteron.

2 he coordinates of the centers of the spherical joints, i.e., of points B i (i =,,, ), epressed in the mobile frame, are given with the following vectors: 0 0 rb, r, 0, 0, B r B r B rb, r. B 0 0 here are other arrangements of the platform joints that lead to mechanisms with kinematic models of similar simplicit, but this arrangement seems to lead to larger workspace. Also, note that at zero orientation (when the orientations of the mobile frame and the base frame are the same), each pair of adjacent legs would interfere, unless the are designed with an offset or curved, as illustrated in Fig.. his offset, however, does not change the kinematics of the mechanism, since each active joint variable ρ i will be defined as the distance between B i and the plane passing through O and normal to the direction of actuator i. hus, the kinematic model of the Heapteron is represented b the following si simple equations: B B B B B B where p,, z i p Rr, i p Rr, j p Rr, j p Rr, k p Rr, k p Rr, is the position vector of point C with respect to the base frame, R is the orthogonal rotation matri representing the orientation of the mobile frame with respect to the base frame, and i, j, and k are the unit vectors along, respectivel, the,, and z ais of the base frame. Equations ( 8) represent the solution of the inverse kinematics of the Heapteron. Obviousl, there is onl one solution per leg, and serial singularities occur whenever a leg is full stretched or full folded. In order to solve the direct kinematics, we need to manipulate ( 8), but first we need to choose a particular orientation representation. As usual, some orientation representations make the direct kinematics more difficult to solve, while others lead to elegant solutions. For eample, among all Euler angles conventions, onl the ZYX fied angle set leads to relativel compact direct kinematic equations. However, we will use Euler parameters, since the resulting direct kinematic equations are even more compact and smmetric. Based on Euler parameters, the rotation matri is epressed as follows: qq qq q0q qq q0q R qq qq 0 q q qq qq 0, qq qq 0 qq qq 0 qq where the four Euler parameters satisf the following constraint equation q q q q 0. Net, we will get rid of the position coordinates b simpl subtracting () from (), () from (), and (7) from (8), and then dividing each new equation b, which ields: where qq qq, qq qq, qq qq,,,. here are different avenues for solving (0 ), including the use of Gröbner basis, but the following approach is b far the most elegant one. We obtain four new equations b subtracting () and () from the sum of (0) and (); subtracting () and () from the sum of (0) and (); subtracting () and () from the sum of (0) and (); adding (0), (), () and (). Factoring the left sides of these four new equations ields: q q q q, 0 q q q q, 0 q q q q, 0 q q q q. 0 Of course, the direct kinematics of the Heapteron will have a solution onl if the right hand side of each of the above four equations is nonnegative. In order to solve ( 7), we take the square roots on each side, which ields the following sstem of four linear equations in the Euler parameters: Q q q q q, 0 Q q q q q, 0 Q q q q q, 0 Q q q q q, 0 where,,,, and,, etc.

3 Finall, solving the set of four linear equations (8 ) ields q 0, q, q, q. However, since the quaternion q0, q, q, q is equivalent to the quaternion q0, q, q, q, we can eliminate δ (i.e., set it equal to ), which means that we have onl eight distinct solutions for the orientation of the mobile platform. Now that we know the orientation of the mobile platform, we can obtain its position b adding () to (), () to () and (7) to (8) and rearranging: 0, qq q q 0, q q q q 0. z qq q q Fig. shows a top view of the eight poses of the mobile platform for a numerical eample where.7,.,.,.,.8, and.. For clarit, the legs are not shown (though the first four legs are hinted b dashed lines), but one can easil imagine that not all poses are feasible on a real prototpe. Note that Fig. shows the first solution, which is feasible. In addition to Fig., able presents the corresponding position coordinates and Euler angles, according to the ZYX fied angle set, where R(,, ) R ( ) R ( ) R ( ). z ABLE I. AN EXAMPLE OF HE EIGH SOLUIONS O HE DIREC KINEMAIC PROBLEM OF HE HEXAPERON Solution Pose of the mobile frame # z Figure. op view of the eight solutions to the direct kinematic problem presented in able (legs not shown). It is important to note that the direct kinematics of the Heapteron is a special case of the si-points-on-si-planes problem studied in [8] and, to a certain etent, in [9]. Indeed, the author of [8] essentiall proved that the direct kinematics of a Heapteron in which the spatial arrangement of the si platform joints and the directions of the actuated prismatic joints are arbitrar is equivalent to solving a univariate polnomial of degree 8. hus, what is particularl interesting about our Heapteron is that each of its eight direct kinematic solutions can be obtained directl, without solving an polnomial. What is even more important is that we can select which assembl mode we are interested in and calculate the corresponding unique direct kinematic solution, as done in [0] for the Agile Ee, b specifing δ, δ, and δ.

4 Of course, the Heapteron is not the onl -DOF parallel robot with trivial direct kinematics. However, virtuall all other such parallel mechanisms are either three legged (e.g., [ ]) or asmmetrical (e.g., [ ]) or require double or triple spherical joints (e.g., []). III. SINGULARIIES AND WORKSPACE OF HE HEXAPERON As alread mentioned, the Heapteron is at a serial singularit whenever one of its si legs is full stretched or full folded (i.e., when in a leg, the center of the spherical joint is coplanar with the aes of the two passive revolute joints). In order to find the parallel singularities, we should obtain the Jacobian matri and eamine the conditions that lead to zeroing of its determinant. Skipping the derivations, the Jacobian matri of the Heapteron is RrB i i RrB i i RrB j j J RrB j j RrB k k RrB k k he determinant of this matri is. QQQQ det J 8, which means that the Heapteron is at a parallel singularit whenever Q j = 0, i.e., whenever j = 0 (j =,,, ). Fig. shows that the four planes j = 0 form a pramid in the,, active-joint space. Note that the interior of that pramid is the allowable space, where j > 0 (j =,,, ), i.e., where the Heapteron has eight assembl modes. Figure. Singularit surfaces of the Heapteron in its active-joint space. Finall, note that if we use the ZYX fied angle set, then the determinant of the Jacobian matri is det J 8cos cos cos. Now, even though the Heapteron is a -DOF mechanism, we are mostl interested in its -ais capabilities, or more specificall in the maimum tilt angles that it can achieve. It is eas to show that the angle between the ais of the mobile frame and the z ais and the base frame is cos cos cos Clearl, θ should alwas remain in the open range ( 90, 90 ) to avoid parallel singularities. Similarl, it is possible to show that should also alwas remain in the open range ( 90, 90 ) to avoid parallel singularities. herefore, in theor, the maimum tilt angle of the mobile platform can be anthing below 90. As for the position workspace, as shown in [], if the legs are sufficientl long, the onl theoretical limits are the strokes of the actuators. In practice, however, the main limits (particularl for the orientation workspace) are the interferences between the legs. Since these legs must have comple curved shapes in order to reduce interferences, it is ver difficult to compute the workspace of the real Heapteron. IV. MECHANICAL DESIGN OF A HEXAPERON FOR D PRINING Fused deposition modeling (FDM) of plastics tpicall limits the overhang of individual laers to less than. Models requiring overhangs greater than this need printed support structures to prevent unintended part deformation. Yet, D printers have been shown to print upside down and in zero-gravit indicating that the limit is from the normal of the laer instead of from the gravit vector. he vast majorit of D printers are -DOF robots (Cartesian serial robots or Delta parallel robots []) and tpicall print in planar laers perpendicular to gravit. he additional DOFs of the Heapteron allow for unconventional printing of curved and tilted laers. Software will need to be written that will produce laers that intersect the surface of a model at ± from its normal. his will allow FDM to produce models with negative overhangs that previousl required techniques such as selective laser sintering. Additionall, D printers have been required to print on planar surfaces. he possibilit for curved and tilted laers will allow for printing directl on non-planar object. For instance, it would be possible to print a plastic handhold around the small bar on a dumbbell. Previousl, this would have been impossible without disassembl of the dumbbell. he mechanical design of the Heapteron (Fig. ) took advantage of the sstem s smmetr. Most parts are made in triple, including laser cut plates from aluminum, and the even- and odd-numbered legs, also machined from aluminum. he etremities of each leg, which provide the offsets necessar to reduce collisions, are made of steel (Fig. ). All revolute joints are mounted on ball bearings. Finall, each of the si platform joints is made of a Cardan joint with two revolute joints on each side. his redundanc increases significantl the range of the joints as shown in Fig., while avoiding a gimbal lock.

5 Initial simulations show the mobile platform can tilt up to about in an direction (Fig. ). However, after analzing the first two prototpes that are currentl being machined, the mechanical design will be optimized in order to allow greater tilt angles. Figure. Mechanical design of the Heapteron under construction. V. CONCLUSIONS AND FUURE WORK Among all parallel robots with si identical legs, the Heapteron is probabl the one with simplest kinematic model. Naturall, this first paper focused mainl on solving the direct kinematics of the Heapteron and studing its singularities. Net, its mechanical design will be optimized to allow the largest possible tilt angle in an direction. he redundant rotational DOF about the should, however, be taken into account. Finall, the potential for using the Heapteron design for building a machine tool will also be eplored. Although we have not et performed detailed workspace analses for an particular actual prototpe, there is no doubt that the Heapteron has a larger workspace than an other heapod sstem. VI. ACKNOWLEDGEMENS he authors would like to thank Jonathan Coulombe for the mechanical design of the Heapteron. Figure. A closeup of the mobile platform at an etreme orientation. For actuation, si NEMA step motors with maimal torque of.9 Nm are used. Movement to the carriages (machined from aluminum) is transmitted through three 00 mm 0 mm linear rails and si mm ball screws with 0 mm pitch. he overall dimensions of the machine are about 70 mm 70 mm 70 mm. Finall, the mobile platform, also machined from aluminum, is equipped with a hot end that heats up to about 8 C as.7 mm filament is pressed into it. he nozzle of the hot end is of diameter 0. mm. Not pictured in Figs. and is an FEP eflon tube (i.e., a Bowden tube) that runs from the top of the mobile platform to a remote filament drive, also not pictured. Eventuall, a gripper can be incorporated into the mobile platform to allow for automated assembl of printed parts and part removal. REFERENCES [] X. Kong and C. M. Gosselin, Kinematics and singularit analsis of a novel tpe of -CRR -DOF translational parallel manipulator, he International Journal of Robotics Research, Vol., No. 9, September, 00, pp [] C. M. Gosselin, X. Kong, S. Foucault, and I. A. Bonev, I.A., A full decoupled -DOF translational parallel mechanism, 00 Parallel Kinematic Machines International Conference, pp. 9 0, Chemnitz, German, April 0, 00. [] I.A. Bonev, he true origins of parallel robots, available at [] P.-L. Richard, C.M. Gosselin and X. Kong, Kinematic analsis and prototping of a partiall decoupled -DOF R parallel manipulator, Proceedings of the 00 ASME Design Engineering echnical Conferences & Computers and Information in Engineering Conference, Philadelphia, USA, September 0, 00. [] X. Kong and C.M. Gosselin, pe snthesis of -DOF parallel manipulators based on screw theor, Journal of Robotic Sstems, Vol., No. 0, 00, pp. 7. [] C. Gosselin, M.. Masouleh, V. Duchaine, P. L. Richard, S. Foucault, and X. Kong, Parallel mechanisms of the Multipteron famil: Kinematic architectures and benchmarking, IEEE International Conference on Robotics and Automation, Roma, Ital, 0 April, pp. 0 (007). [7] M.. Masouleh, M. Hust, and C. Gosselin, Forward kinematic problem of -PRUR parallel mechanisms using stud parameters, Advances in Robot Kinematics: Motion in Man and Machine, Springer, pp., 00. [8] C.W. Wampler, On a rigid bod subject to point-plane constraints, Journal of Mechanical Design, Vol. 8, No., pp. 8, 00. [9] X.S. Gao, D. Lei, Q. Liao, and G.-F. Zhang, Generalized Stewart- Gough platforms and their direct kinematics, IEEE ransactions on Robotics and Automation, Vol., No., pp., 00. [0] X. Kong and C.M. Gosselin, A formula that produces a unique solution to the forward displacement analsis of a quadratic spherical parallel manipulator: the Agile Ee, Journal of Mechanisms and Robotics, Vol., No., 00. [] Y.K. Bun and H.-S. Cho, Analsis of a novel -DOF, -PPSP parallel manipulator, International Journal of Robotics Research, Vol., No., pp , 997.

6 [] C. Chen,. Garal, S. Caro, S. Abewardena, D. Chablat and G. Moroz, A si degree of freedom epicclic-parallel manipulator, Journal of Mechanisms and Robotics, Vol., No., 0. [] SmarPods b SmarAct, consulted Februar, 0. [] C.-D. Zhang, and S.M. Song, Forward kinematics of a class of parallel (Stewart) platforms with closed-form solutions, IEEE International Conference on Robotics and Automation, pp. 7 8, Sacramento, April, 99. [] P. Nanua, Direct kinematics of parallel mechanisms, Master s hesis, Ohio State Universit, USA, 988. [] I.A. Bonev, Affordable D printers Parallel robots catch on, available at 0.