End-effector pose estimation of the Monash Epicyclic-Parallel Manipulator through the visual observation of its legs

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1 End-effetor pose estimation of the Monash Epiyli-Parallel Manipulator through the visual observation of its legs Alessia Vignolo,Sébastien Briot, Philippe Martinet,, Chao Chen 3 Abstrat Past researh works have shown that it was possible to evaluate the end-effetor pose of parallel robots by vision. First of all, it was proposed to diretly observed the end-effetor. However, this observation may be not possible (e.g. in the ase of a hapti devie for whih the end-effetor may be hidden by the user hand). Therefore, it has been proposed another type of end-effetor pose estimation based on the observation of the diretions of the legs. Eventeresting results were obtained, this method is not suitable for some partiular parallel robot families (e.g. the Monash Epiyli-Parallel Manipulator, MEPaM). This paper proposes a new approah for the estimation of the endeffetor pose: by observing the mehanism legs, it is possible to extrat the Plüker oordinates of their lines and determine the end-effetor pose. The new end-effetor pose estimation method is applied to the MEPaM. All results are validated on a MEPaM simulator reated using ADAMS/Controls and interfaed with Matlab/Simulink. I. INTRODUCTION If ompared to serial robots, parallel robots have several advantages : higher speeds, higher dynami performanes and larger payload-to-weight ratio []. One of the disadvantages is that their ontrol is triky beause of the high kinemati oupling, learane effets in joints and manufaturing errors whih deteriorate stability and auray. In order to redue this stability and auray degradation, whih is due to the omplex kinemati struture of the robot, sensor-based ontrol may be used. Vision-based ontrol has showts ability for estimating and ontrolling the end-effetor pose, e.g. by the diret observation of the end-effetor pose [], [3], [4]. However, in some ases the diret observation of the end-effetor may be impossible or diffiult (for instane, with a hapti devie where the user ould prevent the observability of the endeffetor). A good alternative for parallel robots is to observe the robot legs. In this ase, by a diret measure of anternal property aquired through an external sensor, i.e. the robot leg onfiguration observed by vision, this tehnique allows to estimate the end-effetor pose [5] whih is an external property. This method has been applied in [6] for the ontrol of a Gough-Stewart (GS) platform [7]: the visual primitives were This work was supported by the Frenh Federation AtlanSTIC A. Vignolo, S. Briot, and P. Martinet are with the Institut de Reherhe en Communiations et Cybernétique de Nantes (IRCCyN), UMR CNRS 6597, Nantes, Frane P. Martinet is also with the Eole Centrale de Nantes, Nantes, Frane 3 C. Ches with Monash University, Clayton, Australia /5/$3. 5 IEEE represented by the leg diretions and the end-effetor pose estimation was obtained from their reonstrution from the image. The same method was then applied to the Adept Quattro and other robots of the same family [8], [9]. However, as proven [6], the approah annot be applied to any type of robot family: for instane it was not possible to estimate the pose of a partiular family of parallel robots for whih the first joints of the legs are prismati joints whose diretions are all parallel [9] that. For example, in the ase of the PRRRP robot with parallel P joints (Fig. ), the endeffetor pose an not be reonstruted by the observation of leg diretions u i only as, for the same values of u and u,infinite possible end-effetor poses an be found. In order to overome this problem, a solution would be to use more information ontained in the image, suh as the Plüker oordinates of the lines passing through the legs. It will be shown this paper that by observing the leg projetion the amera frame, it is possible to extrat not only the leg diretion but also the Plüker oordinates of the leg. Using the Plüker oordinates of the lines passing through the legs for the visual servoing is equivalent to use the leg diretion plus their distane and position with respet to the amera frame. Thus, the line passing through the legs are fully defined. Estimating the end-effetor pose in the ase of the PRRRP robot of Fig. is similar to finding the intersetion point of the lines L and L passing through the legs. In this paper the new end-effetor pose estimation approah based on the use of the Plüker oordinates of the lines passing through the legs is introdued and it is applied to the MEPaM [], a hapti devie based on a six-degrees-of-freedom (dof ) parallel kinemati mahine. II. MEPAM A. Arhiteture desription MEPaM has annovative design (Fig. ). The atuators are mounted on the base instead of the moving limbs, reduing their mass and inertia. It has three legs and an preserve six-dof thanks to planetary-belt systems, whih transmit power to the moving legs. There are three planetary-belt systems, and eah one provides two-dof in a plane and is driven by two motors: a lower motor drives the lever arm A via a short stiff belt, and a upper motor drives the sun pulley via a long stiff belt. The planetary-belt system ends with the lever arm B, whose end is In the following of the paper, R and P stand for passive revolute and prismati joints, respetively, while R and P stand for ative revolute and prismati joints, respetively. In the paper, u i stands for unit vetor.

2 Some other robot onfig. for the same leg diretions u u C (x,y) L L leads to some geometri onditions for MEPaM s singularities, whih are shown Fig. 3. Fig.. The PRRRP robot Fig.. Virtual model of MEPaM (one leg is hidden for larity) attahed to a ylindrial joint perpendiular to the driving plane of the system. Then, one universal joint links every ylindrial joint to a vertex of the triangular end-effetor. B. Forward kinematis of MEPaM based on the atuator angles and singularities The forward kinematis model of MEPaM based on the atuator angles and its singularities have been already analyzed in []. The serial and parallel singularities an be found from the analysis of the rank defiieny of the serial and parallel kinemati Jaobian matries, defined through the relation J S Θ = J P t () where J S and J P are the serial and parallel kinemati Jaobian matries, respetively, Θ is the vetor of ative joints rates and t is the twist of the moving platform. Matrix J S is rank defiient when there is serial singularity: in suh a ase for the MEPaM, the arms are fully extended or folded, and, in suh a onfiguration, the triangular end-effetor is not able to move in the arms diretion, and it loses one-dof. Conerning the parallel singularities whih appear when the matrix J P is rank defiient, in [] they were determined from geometri onditions derived by Grassmann-Cayley algebra (GCA) []. The proedure, omitted for reason of brevity, Fig. 3. a) A singular onfiguration of MEPaM where its moving platform is vertial; b) A singular onfiguration of MEPaM where lines L and L interset Anteresting feature of the manipulator is that the parallel singularity is not dependent on the end-effetor orientation. Moreover, the design of the devie prevents the end-effetor from reahing the parallel singularities thanks to the physial limits of the universal joints. C. Fore ontroller for a hapti devie The fore ontroller [] was applied to the MEPaM, as shown Fig. 4. The symbols in the figure are: virtual environment impedane Z env, penetration of the avatar through the virtual environment X env, desired fore F d, admittane Z h, inverse Jaobian JT (depending on the robot pose) whih maps F d into desired torque ommands τ d, whih are applied to the hapti devie. The torques to resist a given fore F are dependent on the robot onfiguration. Suh type of ontroller requires the estimation of the endeffetor pose X. Usually, it is estimated using the information of the joints (suh as in Fig. 4), but, in order to overpass the

3 X env - + F Z d τ env J T d - Θ (X,Θ) Z h X Diret Kin. Fig. 4. The open-loop fore ontrol. model errors and improve the auray, it is better to use vision and to measure the position of some robot elements as lose as possible to the end-effetor. The most effiient method would be to observe diretly the end-effetor. However, for an hapti devie, the user hand may prevent this observation. To avoid this problem, we propose to observe the ylindrial links whih are diretly onneted to the end-effetor. In the next setion, we dislose about the end-effetor pose estimation based on leg observation. Fig. 5. u i A ylinder projeted in the image u i R i III. A. Line modeling END-EFFECTOR POSE ESTIMATION USING LEG OBSERVATION A line L in 3D may be parameterized by its Binormalized Plüker oordinates [3]: h i θ i h i L ( u, n, n) () where u is a unit vetor parameterizing the diretion of the line 3, s a unit vetor orthogonal to the line interpretation plane L (defined as the plane lying on the line L and passing through the amera frame origin) and s a salar superior or equal to zero. n and n are linked through the relation n n = P u where P is the loation of a point P on the line with respet to the amera frame. The harateristi equation of suh a line projeted in the image plane is given the amera frame by [3]: n T P = (3) With the matrix K formed by intrinsi parameters of the amera, one an obtain the line equation pixel oordinates p n from: In pixel oordinates, the line equatios given by: p n Tp p = (4) Indeed, replaing p p with K p in this expression yields: By omparing (3) and (5), one obtains p n = p n T K p = (5) K T n K T n, n = KTp n K Tp n 3 In the following of the paper, the supersript before the vetor denotes the frame in whih the vetor is expressed ( b for the base frame, for the amera frame and p for the prixel frame). If there is no supersript, the vetor an be written any frame. (6) Fig. 6. Edges of a ylinder observed by vision B. Cylindrial leg observation The legs of parallel robots have usually ylindrial rosssetions []. The edges of ylindrial leg i are given, in the amera frame, by [4] (Figs. 5 and 6): n i = os θ i h i sin θ i u i h i (7) n i =+osθ i h i sin θ i u i h i (8) where os θ i = / h i R i h i, sin θ i = R i / h i and ( u i, h i, h i ) are the Binormalized Plüker oordinates of the ylinder axis and R i is the ylinder radius. From [4], the leg orientation an be defined in the amera frame by n i u i = n i n i n i (9) It is known that eah ylinder edge is a spatial line parameterized by the Plüker oordinates ( u i, n j i, n j i ) (Fig 5). Moreover, any point A i (of oordinates A i in the amera frame) loated on the axis of the ylinder is at the distane R i from the edge of the ylinder. Consequently, any edge of a ylinder edge is totally defined by the following relations, valid in any frame: n jt i A i = R i () n jt i n j i = () u T i n j i = ()

4 A iz z i A i q ib wiz q ia θ ib θ ia x i w ix A ix Fig. 8. The end-effetor of MEPaM onneted to A, A and A 3 Fig. 7. The belt-pulley transmission of MEPaM for the leg i The vetor h i = h i h i an be omputed using the edges of the ylindrial leg i too, and it is given by h i = D i u i (3) where D i is the position of the point D i in the amera frame, whih is the losest point of the axis of the i-th leg to the amera. It is given by D i = R i sin(θ i ) n i + n i n i + n i (4) C. MEPaM kinematis based on Plüker oordinates The objetive of the work is to use the information of the leg Plüker oordinates instead of the joint angles to estimate the MEPaM end-effetor pose, therefore the forward kinemati problem has to be solved using them. The Figs. 7 and 8 are useful to introdue the kinematis of MEPaM. Figure 7 shows the kinemati representation of the MEPaM legs and Fig. 8 shows the frame assignment for the driving planes: three fixed frames, F (O, X, Y, Z ), F (O, X, Y, Z )andf 3 (O 3, X 3, Y 3, Z 3 ), are attahed to the base in an equilateral triangle formation, while a moving frame F 4 (B, X 4, Y 4, Z 4 ) is attahed to the triangular end-effetor of side length d 3. A i and B i are the respetive Cartesian oordinate vetors of the endpoints of the i-th leg A i and B i. The upper-left index indiates the frame in whih the vetor is expressed. Sine eah ylindrial leg attahed to the triangular platform is orthogonal to the plane of the orresponding leg, we have A x A 3 x A 3x B =, B =, 3 B 3 = (5) l A z l A z l 3 3 A 3z where l i is the length of the i-th ylindrial leg and i A ix and i A iz are the x and z oordinates of the point A i in the loal frame F i (Fig. 7). In order to estimate MEPaM end-effetor pose, we first need to estimate through vision the position of the points A i (Figs. 7 and 8) whih are neessary in the relation (5). We have [ i A i ] = i T [ A i ] (6) The values of i T are given by the robot design, and are equal to: T = r (7) C( 7 6 π) S( 7 6 π) r C( 7 6 π) 3 r S( 7 6 π) S( T = 6 π) C( 7 6 π) r S( 7 6 π) 3 r C( 7 6 π) C( 6 π) S( 6 π) (8) r 3 C( 6 π)+ 3 r3 S( 6 π) 3 T = S( 6 π) C( 6 π) r3 S( 6 π)+ 3 r3 C( 6 π) (9) where r, r and r 3 are the distanes between the base frame with respet to the origin of the frames F, F and F 3 respetively, while A i will be estimated through vision as the intersetion between the line passing through i-th leg and its orresponding perpendiular plane shown Fig. 8. Note that in these equations, S(q) stands for sin q and C(q) stands for os q. To ompute the positions of points A i, we use the following equations. Given a line expressed in Plüker oordinates as [u, h, h] and a plane expressed in Plüker oordinates too

5 as [N n] 4, the general formula whih gives the point of intersetion between them is [5] Â i =( (hh) N nu un) () where Âi are the Plüker oordinates of the point A i (equivalent to the homogeneous oordinates of the point A i ). From the robot design parameters, the transformation matries between frames F, F and F 3 are given by os(π/3) sin(π/3) d T = 3T = 3 sin(π/3) os(π/3) T = () Fig. 9. A pinhole amera model The position of all the verties of the platform an be transformed into F, B i = i T i B i () for i =, 3. The geometri onstraints on the devie are given by B i B j = d 3 (3) for i =,, 3 and j = i+ (mod 3), where d 3 is the (onstant) side length of the platform. The onstraint equations, Eqs. (3), ontain only three variables l i,i=,, 3, and an be written the form D l + D l + D = (4) E l 3 + E l 3 + E = (5) F l 3 + F l 3 + F = (6) where D j,e j,f j (j =,, ), whih are not developed here for brevity, are funtions of l,l and l 3, respetively, as well as d 3, A ix and A iz (i =,, 3). By means of dialyti elimination, the system of Eqs. (4), (5), (6) an be redued to a univariate polynomial of order four in the variable l G 4 l 4 + G 3 l 4 + G l + G l + G = (7) One Eq. (7) has been solved for l, substitution of the result into Eqs. (4), (5), (6) will allow for determination of l and l 3. Solutions of l i for i =,, 3 areusedtoestimatethe end-effetor pose. The position of the end-effetor is simply the origin of F 4,givenby B =[A x,l,a z ] T. The endeffetor orientatios parameterized by 4Q =[i, j, k] where i = B B B B j = ( B 3 B ) ii T ( B 3 B ) ( B 3 B ) ii T ( B 3 B ) (8) (9) k = i j (3) 4 Given the Cartesian equation of a plane ax + by + z + d =, its Plüker oordinates are [N n] where N =(a, b, ) and n = d D. Singularities of the new model It an be shown that, from [], the standard forward kinemati problem (based on the enoder values Θ) havethe same inputs as the forward kinemati problem based on the leg pose visual estimation shown subsetion III-C, whih are the positions of points A i. Evef the positions of points A i are estimated through different manners (through the enoder positions and the leg geometri parameters in [] while it is through vision the urrent work), the struture of the forward kinemati model remains the same. Therefore, the singularities of the new model based on the Plüker oordinates are the same of the ones shown [] (the parallel singularities dislosed in Setion II-B). IV. SIMULATION RESULTS In this setion, some examples of simulation results are presented to show the effiieny of the end-effetor pose through the leg observation. These results have been got with a model in Adams of MEPaM with the following nominal parameters (in meters): r i =.65; d =[.375,.375,.375]; d =[.375,.375,.375]; w x =, w z =.. Moreover, the radius of the observed ylinder is equal to m. We simulated a motion of MEPaM. The initial position and orientation of the platform is: X d =[,,.8, pi/4,, ]. Here, the orientation of the end-effetor is desribed by using the ZY X Euler angles onvention (roll-pith-yaw angles φ, θ and ψ). The trajetories given to the atuators (whose angles are shown Fig. 7) as input are: q a =. sin(t), q b =. sin(t), q a =. sin(t), q b =.sin(t), q 3a =.5 sin(t), q 3b =. sin(t). We simulated the amera loated at the entre of the base frame F (with the amera plane parallel to the x Oy plane) by using a pinhole amera model beause it is simple to implement and is a good approximation of real ameras. In the Fig. 9, a amera with O as enter of projetion and the prinipal axis parallel to Z axis is shown. The distane between O and the image plane is the foal length f. The 3D point P =(X, Y, Z) is projeted on the image plane at oordinates P =(u, v). The parameters of the amera used for the simulations are:

6 end-effetor position (m) z φ θ x,y ψ end-effetor orientation (rad) norm of orientation error (rad) time (se.) Fig.. The end-effetor position and orientation vetor omputed with the new pose estimation method without any noise. time (se.) Fig.. The error norm between the omputed and real end-effetor orientations without any noise. norm of positionning error (m) end-effetor position (m) z φ θ x,y ψ end-effetor orientation (rad) time (se.) time (se.) Fig.. The error norm between the omputed and real end-effetor positions without any noise. - foal length with respet to u = 3 pixels; - foal length with respet to v = 3 pixels; - priniple point image along u = 4 pixels; - priniple point image along v = 768 pixels. Without adding any noise, we obtained the results shown in Figs., and : the norm of the error between the omputed position of the end-effetor and the real one is very low (around meters) as well as the norm of the error of the orientation (around 7 radians), therefore the new method for the pose estimation omputes the pose in a very good way. Then, we added a measurement noise that is introdued like thereafter. The extration of the Plüker oordinates of the leg line is based on the equations of the leg edges. In the simulation, they are projeted to the image plane and onverted from meter to pixel. Then, the edge line intersetions with image boundary are omputed: the oordinates of the intersetion points have to be rounded due to the pixel auray. A new equation of the edge line is then reomputed taking into aount the error introdued in the intersetion points between Fig. 3. The end-effetor position and orientation vetor omputed with the new pose estimation method with measurement noise. the edge line and the image boundary. The measurements error hoses given by a pixel auray equal to. The results obtained adding noise are shown Figs. 3, 4 and 5: it is possible to notie that the norm of the error of the positios in the order of 4 m and the norm of the error of the rotatios in the order of rad, therefore the new method for the pose estimation shows a good robustness to measurement noise. V. CONCLUSIONS In this paper a new approah for end-effetor pose estimation in the ase of a parallel robot has been proposed. This method is based on the extration of the Plüker oordinates of the leg enter line and, unlike the old method based on the leg diretion proposed in the previous papers, it is suitable to all the Parallel Kinemati Mahines families, inluding parallel robots whose legs diretions are onstant evef the endeffetor pose hanges, like MEPaM. This new method has been applied to MEPaM and some simulation results (without any noise and with measurement noise as well) of the new pose estimator have showts effiieny. REFERENCES [] J.P. Merlet. Parallel Robots. Springer, nd edition, 6.

7 norm of positionning error (m) 4 time (se.) Fig. 4. The norm of the error between the omputed end-effetor position and the real one with measurement noise. the First European Conferene on Mehanism Siene, Obergurgl, Austria, 6. [] C. Chen, T. Gayral, S. Caro, D. Chablat, G. Moroz, and S. Abeywardena. A Six Degree of Freedom Epiyli-Parallel Manipulator.. [] S. Caro, G. Moroz, T. Gayral, D. Chablat, and C. Chen. Singularity Analysis of a Six-DOF Parallel Manipulator Using Grassmann-Cayley Algebra and Grobner Bases. Symposium on Brain, Body and Mahine, Montreal, QC.,Canada,. [] L. Villani and J. De Shutter. Handbook of Robotis, hapter Ch7: Fore Control, pages Springer, 8. [3] N. Andreff, B. Espiau, and R. Horaud. Visual servoing from lines. International Journal of Robotis Researh, (8):679 7,. [4] N. Andreff, T. Dallej, and P. Martinet. Image-based visual servoing of gough-stewart parallel manipulators using legs observation. International Journal of Robotis Researh, 6(7): , 7. [5] J.M. Selig. Geometri fundamentals of robotis, 5. Springer, nd edition. norm of orientation error (rad) time (se.) Fig. 5. The norm of the error between the omputed end-effetor rotation and the real one with measurement noise. [] B. Espiau, F. Chaumette, and P. Rives. A new approah to visual servoing in robotis. IEEE Transations on Robotis and Automation, 8(3), 99. [3] R. Horaud, F. Dornaika, and B. Espiau. Visually guided objet grasping. IEEE Transations on Robotis and Automation, 4(4):55 53, 998. [4] P. Martinet, J. Gallie, and D. Khadraoui. Vision based ontrol law using 3D visual features. In Proeedings of the World Automation Congress, WAC96, Robotis and Manufaturing Systems, volume 3, pages 497 5, Montpellier, Frane, May 996. [5] E. Ozgur, N. Andreff, R. Dahmouhe, and P. Martinet. High speed parallel kinemati manipulator state estimation from legs observation. In Proeedings of the IEEE/RSJ International Conferene on Intelligent Robots and Systems (IROS 3), Tokyo Big Sight, Japan, 3. [6] N. Andreff, A. Marhadier, and P. Martinet. Vision-based ontrol of a Gough-Stewart parallel mehanism using legs observation. In Proeedings of the IEEE International Conferene on Robotis and Automation, ICRA 5, pages , Barelona, Spain, April 8-5. [7] V.E. Gough and S.G. Whitehall. Universal tyre test mahine. In Proeedings of the FISITA 9th International Tehnial Congress, pages 7 37, May 96. [8] E. Ozgur, N. Andreff, and P. Martinet. Dynami ontrol of the quattro robot by the leg edgels. In Proeedings of the IEEE International Conferene on Robotis and Automation, ICRA, Shanghai, China, May 9-3. [9] N. Andreff and P. Martinet. Vision-based kinemati modelling of some parallel manipulators for ontrol purposes. In Proeedings of EuCoMeS,

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