Changing Assembly Modes without Passing Parallel Singularities in Non-Cuspidal 3-R\underline{P}R Planar Parallel Robots
Ilian Bonev, Sébastien Briot, Philippe Wenger, Damien Chablat
aa r X i v : . [ c s . R O ] S e p Proceedings of the Second International Workshop on
Fundamental Issues and Future Research Directions for Parallel Mechanisms and Manipulators
September 21–22, 2008, Montpellier, FranceN. Andreff, O. Company, M. Gouttefarde, S. Krut and F. Pierrot, editors
Changing Assembly Modes without Passing Parallel Singularities in Non-Cuspidal 3-RPRPlanar Parallel Robots I LIAN
A. B
ONEV , S ´
EBASTIEN B RIOT
D´epartement de g´enie de la production automatis´ee´Ecole de technologie sup´erieure1100, rue Notre-Dame OuestMontreal, QC, Canada H3C 1K3 P HILIPPE W ENGER , D
AMIEN C HABLAT
Institut de Recherche en Communications etCybern´etique de Nantes UMR CNRS 65971, rue de la No¨e, BP 9210144312 Nantes Cedex 03 France
Abstract:
This paper demonstrates that any general 3-DOFthree-legged planar parallel robot with extensible legs canchange assembly modes without passing through parallel singu-larities (configurations where the mobile platform loses its stiff-ness). While the results are purely theoretical, this paper ques-tions the very definition of parallel singularities.
Most parallel robots have singularities that limit the motion ofthe mobile platform. The most dangerous ones are the singular-ities associated with the loss of stiffness of the mobile platform,which we call here parallel singularities . Indeed, approaching aparallel singularity also results in large actuator torques or forces.Hence, these singularities should be avoided for most applica-tions. A safe solution is to eliminate parallel singularities at thevery design stage [1,2] or to define singularity-free zones in theworkspace [3,4]. It is also possible to avoid parallel singularitieswhen planning trajectories [5,6].For a parallel robot with multiple inverse kinematics solu-tions, belonging to different working modes , a change of con-figuration of one of its legs may allow it to avoid a parallel singu-larity [7,8]. This paper addresses a recent, difficult issue that hasbeen investigated by few researchers: the possibility for a parallelrobot to move between two direct kinematic solutions, belongingto two assembly modes , without encountering a parallel singular-ity. We will focus on planar parallel robots with three extensiblelegs, referred to 3-
RPR . As shown in [9], the study of the 3- RPR planar robot may help better understand the kinematic behaviorof its more complex spatial counterpart, the octahedral hexapod.Planar parallel robots may have up to six direct kinematicsolutions (or assembly-modes). It was first pointed out that tochange its assembly-mode, a 3-
RPR planar parallel manipulator R and P stand for revolute and prismatic joints, respectively. The underlinedletter refers to the actuated joint. should cross a parallel singularity [10]. But [11] showed, usingnumerical experiments, that this statement is not true in general.In fact, an analogous phenomenon exists in serial robots, whichcan move from one inverse kinematic solution to another withoutmeeting a singularity [11]. The non-singular change of posturein serial robots was shown to be associated with the existenceof points in the workspace where three inverse kinematic solu-tions meet, called cusp points [12]. On the other hand, McA-ree and Daniel [9] pointed out that a 3- RPR planar parallel robotcan execute a non-singular change of assembly-mode if a pointwith triple direct kinematic solutions exists in its joint space. Theauthors established a condition for three direct kinematic solu-tions to coincide and showed that a non-singular assembly-modechanging trajectory in the joint space should encircle a cusp point.Wenger and Chablat [13] investigated the question of whethera change of assembly-mode must occur or not when moving be-tween two prescribed poses in the workspace. More recently,Zein et al. [14] investigated the non-singular change of assembly-mode in planar 3-
RPR parallel robots and proposed an explana-tory approach to plan non-singular assembly-mode changing tra-jectories by encircling a cusp point. Finally, the most recentresults showed that a non-singular change of assembly-mode ispossible without moving around a cusp point [15,16].In [15], an example of a 3-
RPR planar parallel robot was givenwhose workspace is divided into two portions by the singularitysurface, while having more than two assembly modes. It is there-fore obvious that the robot can change an assembly mode to atleast another one without crossing any singularity. In [16], anexample of a 3-
PRR planar parallel robot (with actuators havingparallel directions) was given and it was shown that an assemblymode can be changed by passing through a serial singularity (inwhich a leg is singular) and changing working modes.In this paper, we show that any non-architecturally singular3-
RPR planar parallel robot can change assembly-mode withoutencircling a cusp point or passing through a parallel singularity.1igure 1: Schematics of a general 3-
RPR planar parallel robot
RPR
Planar Parallel Robots
Referring to Fig. 1, we denote with A i and B i the base and plat-form revolute joints, respectively. The directed distance between A i and B i along the direction of prismatic actuator i is ρ i , whichis the active joint variable. Finally, we denote by C the center ofthe mobile platform.It is well known that the 3- RPR planar parallel robot is at aparallel singularity when the lines passing through the passiverevolute joints in each leg intersect at one point or are paral-lel. These lines represent the reciprocal screws, i.e., the reactionforces applied to the mobile platform [17].The singularity loci of this robot, defined as the set of posi-tions of point C where the robot is at singularity for a given ori-entation, were studied in detail in [18] and it was shown that theyform a conic (i.e., a hyperbola, a parabola or an ellipse), unlessthere is an architectural singularity. An architectural singularityoccurs, for example, when the mobile platform and the base formsimilar triangles, in which case there is an orientation at whichall positions correspond to singularities. A more general study ofthe singularity surface of a 3- RPR parallel robot is given in [19].In [17], it was shown that all points from this conic corre-spond to parallel singularities except for three (or two, or one)of them. These three points correspond to the poses of the plat-form in which one (or more) legs are in a serial singularity, i.e.,in which two revolute joints in a leg coincide. Such a singularitycorresponds to an uncontrollable passive motion [18]. In such aconfiguration, the reciprocal screw associated with the singularleg degenerates to two linearly independent forces. Thus, in sucha configuration, there is a parallel singularity if and only if thelines associated with the two non-singular legs pass through thecoinciding revolute joints of the singular leg. This would only bepossible for special designs in which an angle of the base triangleis equal to the corresponding angle of the platform triangle.That passing the singularity curve means passing though aparallel singularity, as shown in Fig. 2, is fairly common knowl-edge. What no one has previously pointed out is that this singu-larity curve has passages that usually correspond to serial singu-larities only. Figure 3 illustrates that virtually any general 3-
RPR planar parallel robot can cross such a singularity curve through these special passages, without even being near a parallel singu-larity (though measuring proximity to a parallel singularity is stillan open question).Since this robot can cross the singularity curves for any ori-entation without passing through a parallel singularity, it is obvi-ous that it can switch between any two assembly modes, withoutpassing through a parallel singularity.
Of course, the main purpose of this paper is purely theoretical.It is clear that, in practice, a 3-
RPR parallel robot could hardlypass a serial singularity for many reasons. The most obvious oneseems to be that a prismatic actuator that can change from posi-tive lengths to negative ones can be difficult to build. However,this problem can be easily overcome if we use
RRR legs instead,so this is not an issue. What is more difficult is to cope with the(uncontrollable) passive motion of the singular leg. If such a pas-sive motion occurs during a serial singularity, then the prescribedtrajectory of Fig. 2 can no longer be followed. Finally, it wouldbe even more difficult to drive the robot to such a configuration.Outside serial singularities, the 3-
RPR parallel robot can acceptinput errors — this would simply result in output errors. How-ever, in a serial singularity, no errors are possible — this wouldresult in jamming (the lengths of the two non-singular legs arenot independent from one another). Note that this is not the casein most other parallel robots: they can cross a serial singularitywithout any difficulty.This brings us to the essential question of what is a parallelsingularity. The mobile platform clearly does not lose stiffness inthe configuration in question. Yet, in this configuration, the num-ber of direct kinematic solutions abruptly drops to one (or two forsome special cases). Indeed, the direct kinematic problem of thisrobot comes to finding the maximum six intersection points be-tween a sextic and a circle. When an actuator has zero length, thiscircle degenerates to a point, which explains the sudden drop inthe number of solutions: there may be at most two “intersectionpoints” between a point and a sextic.This interpretation also helps understand why such a config-uration is not tolerant to input errors. In the case of non-zero leglengths, if we slightly change the lengths of the two legs corre-sponding to the sextic, the latter will slightly change but therecould still be six intersection points. If however, we do the samein the case of a zero-length leg, the point that is the degenerationof a circle, will no longer lie on the sextic.
This paper demonstrates that the problem of assembly-modechanging is still an open issue and its objectives should be bet-ter defined. Namely, the restrictions on such a change should bespecified. Can we pass a serial singularity? If the answer is posi-tive, should we be able to do this in practice or not necessarily?This paper also questions the very definition of a parallel sin-gularity, often associated with both a loss of stiffness and degen-eracy of the direct kinematics. An example of a serial singularityis given in which the direct kinematics degenerates but the mobile2latform does not lose stiffness.
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