外文翻译--机床执行机构逆转运动的研究【中英文文献译文】
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On actuator reversal motions of machine toolsTaejung Kima, Seung-kil Sonb, Sanjay E. Sarmaa,*aDepartment of Mechanical Engineering, Massachusetts Institute of Technology, 35-010, MIT, Cambridge,MA 02139, USAbEngineering Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USAReceived 30 August 2002; received in revised form 16 June 2003; accepted 17 September 2003AbstractWhen joints in a machine tool reverse the direction of their motion, non-linearities that are ignored inmachine design and control are reflected noticeably in the accuracy of machined surfaces. For example,friction characteristics of a machine tool become highly non-linear at low operating speeds, demandingsophisticated compensation. We present a theoretical treatment of the kinematics of reversals and reversalfree paths of machine tools. We visualize and compare reversal characteristics of active joints in serial,parallel, and hybrid mechanisms for various trajectories and sweeping patterns. Reversal characteristicshave implications both in the design of machine tools and in path planning.? 2003 Elsevier Ltd. All rights reserved.Keywords: Reversal; Friction; Machine tool; Kinematics1. IntroductionWhen joints in a machine tool mechanism undergo a reversal in their direction of motion,several anomalies occur, which must be taken care of to achieve high precision. This paper dealswith the kinematics of reversals in machine tool mechanisms, concentrating mostly on the theoryand the properties of reversals and reversal-free paths. These effects tend to be ignored during theinitial stages of mechanism design, and our goal is to present the theoretical basis for predictingreversals. In particular, we show that reversals are pronounced in machining even plane surfacesand straight lines with parallel machine tools and are generally difficult to avoid in multi-axis free-form machining with most types of machine tools.*Corresponding author. Tel.: +1-617-253-1925; fax: +1-617-253-7549.E-mail address: sesarma (S.E. Sarma).0094-114X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2003.09.002Mechanism and Machine Theory 39 (2004) 299322/locate/mechmtThe complications associated with reversals arise from the fact that they exacerbate non-linearities in the behavior of bearings and drive mechanisms. A classical example of a reversal-related problem is backlash in lead screws. Backlash arises when a reversal in the loading of adrive mechanism causes the point of contact in the bearing to change suddenly. In practice,backlash in precision mechanisms can be reduced by pre-loading joints to reduce play.A more fundamental non-linearity which is magnified during motion-reversals is the transitionbetween static and dynamic regimes of friction in joints. Fig. 1(a) shows the typical behavior offriction in a lubricated joint as it transitions from low-velocity to high-velocity motions. Thefriction mechanism changes from stiction effects to the Stribeck effect to more linearly viscousbehavior 2. The magnitude of the friction force increases and switches direction as the jointpasses through the zero-speed regime. Simple linear control strategies, designed for linear viscousfriction models, compensate this effect poorly, and can suffer from delays, tracking irregularitiesand loss of machining precision 3. Quadrant glitches, as shown in Fig. 1(b), are typical examplesof friction-induced errors generated during circular end-milling on a 3-axis milling machine 4.Better tracking precision can be attained in this regime by increasing the gain in the controllerevery time the anomaly occurs, or by adopting more complex control strategies. However,implementing these strategies requires more advanced controllers, and careful experimentation forparameter identification, which are both necessarily more expensive. It is best, therefore, to avoidthe occurrence of reversals during critical motions that the designer considers important.1.1. Reversal characteristicsAs described above, it would be desirable for a machine tool to be designed in such a way thatreversals do not occur in situations where they would hurt performance. In Cartesian machines,for example, reversals do not occur while the machine is executing a unidirectional motion in astraight-line trajectory. This is very important because the straight line has a special and funda-mental role in design and manufacturing as a datum and as a mating surface. Reversals do occurin Cartesian 3-axis machines, but only in places where the direction of motion is explicitly re-versed, such as in machining a circle as shown in Fig. 1(b). This constitutes good kinematicFig. 1. A typical friction characteristic in a joint and its negative effect in precision. (a) Relation between friction andvelocity. (b) Quadrant glitches (courtesy of Eugene Tung 4).300T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322behavior from the point of view of reversals: all reversals occur in an easily describable way, andreversal characteristics are position-independent and configuration-independent.Unfortunately, not all machines have these desirable properties, and reversals are especiallydifficult to predict in parallel machines. A parallel machine may suffer a reversal while prescribinga straight-line motion in its workspace. This behavior could turn out to be a severe impediment tothe use of parallel machines in machining applications. Serial Cartesian machine tools too sufferfrom reversals during free-form machining. In machining an aerodynamically critical surface, forexample, a curved trajectory may result in undesirable reversal-induced ridges. Understandingthese problems is important, and we examine a number of issues in this paper: where the reversalsoccur in a path, what the nature of reversal-free paths is, where reversals are inevitable regardlessof direction of motion, how the regions of reversal-free reachability are derived and so on. Ourtreatment is mostly theoretical, and based on the kinematics of a given machine. Much of thecontribution of this paper lies in developing the mathematical machinery necessary to describereversals and related concepts precisely and unambiguously.1.2. BackgroundThe problem of backlash has been an age-old challenge in machine design, and has received agreat deal of attention in the machine-element community. A description of ideas and techniquesin machine elements is available in 1.There has been a great deal of literature on friction models, and a comprehensive summary isavailable in 5. An elaborate model of friction presented by Canudas De Wit et al. 6 captures thedifferent effects that must be considered in reversal situations including the early work of Cou-lomb, of Dudley 7, who showed that stick-slip is not captured by Coulomb and viscous models,of Rabinowicz 9, who showed that there is a time lag between the beginning of motion and theappearance of a friction force, and of Dahl 8 who further incorporated hysteretic effects.However, even Canudas De Wit?s 6-parameter model does not capture all the subtleties of fric-tion, such as the dependence of friction on the time derivative of applied force. In addition,friction parameters have been shown to be highly variable within the workspace, and over time10. Nevertheless, this model forms the basis of a number of control methods that have beenproposed in the last few years.Researchers in the field of motion control have proposed a rich variety of compensationmethods to achieve smooth motion with high accuracy under the action of friction. Not being thefocus of this paper, we present only a rough overview here. Comprehensive summaries areavailable in 11,12. The techniques for control of systems under friction include: estimation andadaptive methods 13,14, robust compensation schemes 15, repetitive control 4, multi-loopcontrol 16, non-linear compensation (similar to sliding mode) 17, non-linear PID (whichchanges the gain depending on plant state) 18 and a broad class of pulse code modulationtechniques which are summarized in 12.There is a great deal of literature on parallel machines, and there are dozens of milling machineswhich use parallel kinematic structures. A summary is available in 19. The putative advantage ofparallel machines over their serial counterparts is that the load can be distributed more evenly totheir kinematic links and, as a result, a higher stiffness can be achieved using lighter structuralelements and less expensive actuators. However we will show that the coupled kinematics ofT. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322301parallel mechanisms can cause reversals in their joints for seemingly simple and straight tra-jectories. Furthermore, studies have shown that the errors in an actuator propagates into all thecoordinate axes 19, and this can be problematic when it is important to maintain planarity orstraightness. In addition, there is literature that shows that frequent actuator reversals can excitehigher harmonic components in the machine tool structure 20,21. This increases the necessarystructural stiffness of parallel machines and, to some extent, undermines the very basis for whichparallel mechanisms are preferred in the first place. Beyond this, however, there appears to belittle prior work in the study of reversals in serial or parallel machine tools.1.3. OutlineAn absolutely fair comparison amongst different machine tool mechanisms is not easy toestablish for reversal behavior because performance depends on the trajectories considered and onthe tasks at hand. If for example, the machine is not required to cut flat surfaces, but, say, spheresinstead, a parallel machine of a certain configuration may be more natural than a Cartesian serialmachine. Instead of making assumptions about the task that the machine is considered for, wewill define the concept of a task space which provides the capability of capturing any task surface,and we will ask how many reversals can occur in a typical path and whether it is possible toachieve the same objective with a reduced number of reversals.In Section 2, we set a mathematical framework upon which the rest of this paper is based. InSection 3, we answer fundamental questions about the behavior of reversals and develop methodsof analysis. In Section 4, we present some examples with three types of machine tools: a serial, ahybrid and a parallel machine as shown in Fig. 2. We conclude in Section 5.2. A framework for analysisIn this section, we lay down a mathematical framework upon which the rest of this paper isbased. It is assumed that what we will call a restricted inverse kinematics map is known. Thissection simply defines the term and justifies the need for this assumption.Fig. 2. The machines we consider: (a) a serial, (b) hybrid and (c) parallel machine tool.302T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322To describe the motion of a cutting tool, we attach a fiducial point on the center line of thecutting tool near the tool tip.13-tuples of the Cartesian coordinates of the fiducial point con-stitute what we refer to as the workspace of the machine tool. In general, a cutting tool, as a rigidbody, has 6 degrees of freedom: three for the translation of the fiducial point and another three forthe rotation of the cutting tool. Since we are considering milling machines, and because cuttingtools can be considered to be axi-symmetric, the sixth freedom of motion, which is the spin aboutthe centerline of a cutting tool, is typically unnecessary and redundant. Therefore, two additionalcoordinates are enough to describe the orientation of a 5-axis machine in the context of milling.All three degrees of freedom of rotation may be necessary in other applications.We refer to the number of degrees of freedom of the cutting tool restricted to our subjectiveinterest as the task space dimension of the application. The task space dimension is not necessarilyidentical to the degrees of freedom of the machine involved in the machining task. For example,we can perform 5-axis machining using 6-axis machines; the additional one degree of freedom isredundant. In 3-axis machining, only translation of a cutting tool is allowed and the task spacedimension is 3. In the path planning stage, the dimension can be further reduced by introducingtask constraints. For example in 3-axis roughing, it is common to remove material from thestock layer by layer; in this case, the task constraint is that the fiducial point should stay on aplane while a layer of material is removed, and the task space dimension of machining each layeris 2. In finishing the surface of a free-form part using 5-axis machining, the tool tip is constrainedto contact the required surface, and the task space dimension of finishing is reduced to 4. Whenplanning finishing paths, collision-free orientations of a cutting tool at each point on the designedsurface can be determined by a collision-detection pre-processor, and the remaining planning taskcan be performed on a 2-dimensional space. In this case, the task space dimension is 2.Consider a machine tool possessing N degrees of freedom and a machining strategy whose taskspace dimension is M 6N. The motion of a cutting tool restricted to the strategy can be capturedby a history of M-tuples of real numbers. We refer to the totality of the M-tuples representing therestricted postures as the task space under the strategy. The totality of the N-tuples of displace-ments of actuators in the machine is referred to as the actuation space.We assume that the kinematics of a machine tool and the constraints introduced by amachining strategy determine a smooth map f from the task space U ? RMof the strategy to theactuation space H ? RNof the machine tool,f : U ! H;u ! h fu;which we refer to as the restricted inverse kinematic map. Any redundancy must be resolved in apoint-wise manner (or holonomically) based on our assumption. We assume that the task spaceU is simply-connected and that it avoids singularities by choice. The range fU of the map is anM-dimensional subspace of the N-dimensional actuation space. The actuator displacement h canbe thought of as one of the local coordinates of the configuration space of the machine tool. Thissetting is general enough for our purpose, even though complete generality can be achieved by1Since a cutting tool has a non-zero diameter, the contact point between the tool and the surface is not necessarily onthe line of symmetry of the tool 22. However, a loose definition of the location of the fiducial point suffices for ourcurrent analysis.T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322303considering the configuration space of machine tools including both active and passive jointdisplacements. A rigorous statement of the above assumption exceeds the scope of our currenttreatment, but more details are available in 23,24. In this paper, we will show how the settingdescribed above can capture the reversal characteristics of most machines.3. Reversal analysisIn this section, we develop several concepts which are useful in judging the reversal charac-teristics of machine tools. We will ask whether a point-to-point task can be completed without anyactuator reversals or whether there is a condition that guarantees reversal-free or certainly-reversing paths. We also ask at which point in a trajectory the actuators reverse their direction ofmotion. Finally, we analyze the reversal characteristics of surface machining or sweeping. Wecompare the three types of machine tools in Section 4 using the concepts developed in this section.Most of our propositions can be understood intuitively; we omit some proofs but provide theirrationale in the main body of text.3.1. Reversal conditionsIn this section, we show various ways to interpret reversals of actuators.3.1.1. A reversal condition on a trajectory for a tracking taskA fundamental function of a machine tool is the ability to track a trajectory specified for itsend-effector. We consider a machine tool moving along a piecewise regular trajectoryu : t 2 0;T? ! ut 2 U in the task space U. If hit ? fiut has a local extremum at t t0forany i, we refer to the point ut0 as a reversal point. If there is no reversal point along a trajectory,we say that the trajectory is reversal-free.For the ith actuator to be reversed at t 2 0;T? along a regular trajectory, it is sufficient that_hit ?ddtfiut XMj1ofiouj?dujdt? 0andhit ?d2dt2fiut 6 0:This condition itself suggests a straightforward procedure for finding the reversal points along agiven regular trajectory in the task space. Plotting and counting reversal points along specifiedtrajectories is a one way to visualize or to compare the reversal characteristics of machine tools.3.1.2. Reversal-singular points and strong reversal-singular pointsAt a point u 2 U in the task space U, if there is an integer k 2 f1;2;.;Ng such thatrfk?ofkou1;ofkou2.ofkouM? 0;we refer to the point u as a reversal-singular point with respect to the kth actuator, where fkis thekth component of the inverse kinematics f. A reversal-singular-point with respect to the kth304T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322actuator is said to be strong if the Hessian matrix hkij? of the kth component fkof the inversekinematics f is definite at the point, wherehkij?o2fkouiouj:A reversal-singular-point is said to be degenerate if its corresponding Hessian matrix is singular.Degenerate reversal-singular-points are uncommon in that a slight perturbation of kinematicparameters can remove them.At a strong reversal-singular-point with respect to the kth actuator, the kth actuator dis-placement fkis either a local maximum or minimum. Therefore, the kth actuator must reverse itsdirection of motion at the strong reversal-singular point, which leads to the following proposition:Proposition 1. When the fiducial point of a machine tool passes a strong reversal-singular-point withrespect to the kth actuator, reversal of the kth actuator is unavoidable.Mere reversal-singular-points (which, for emphasis, we will sometimes refer to as non-strongreversal-singular-points), also imply a high chance of reversals because the trajectories passinga reversal-singular point must satisfy a very special condition to constitute reversal-free motions.This is especially easy to understand for a 2-dimensional task space. If a non-degenerate reversal-singular-point in a 2-dimensional task space with respect to the kth actuator is not strong, the kthcomponent fkof the inverse kinematics has a saddle point at the reversal-singular-point. Thisimplies:Proposition 2. At a non-degenerate, non-strong reversal-singular-point in a 2-dimensional taskspace, a regular trajectory must be tangential to one of the separatrices of the function fkif it isreversal-free.The proof can be easily established through the Morse Lemma. The condition stated above is arather special condition which is rarely satisfied, especially in sweeping tasks, which we discuss inSection 3.3. A plot of the reversal-singular-points is a canonical way to visualize the reversalcharacteristics of machine tools because its appearance does not depend on a particular trajectorywe may have chosen.3.1.3. Reversal free directions in the tangent spaces of the task spaceWe can specify a reversal condition in terms of tangent vectors of trajectories. We also definerelated terms for future use in this section.If a cutting tool moves in a trajectory obeying the given task constraints, the velocity_h of theactuators and the velocity _ u in the task space have the following linear relation:_h C _ u1where C is the Jacobian matrix of the restricted inverse kinematics f. The M-tuple _ u is an elementof the tangent space of the task space. The tangent space at a point u 2 U is denoted by TuU. Theunion of tangent spaces is referred to as the tangent bundle, and symbolically as TU.T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322305At every point in the task space at an instant, each actuator has two alternatives, eitherincreasing or decreasing its displacement. In other words, either_hiP0 or_hi60.2In particular, ifa trajectory is reversal-free, only one of the two inequalities must hold during its entire interval ofmotion, except for_hi 0. In other words, each actuator displacement must be monotonic along areversal-free trajectory. The permutations of these binary alternatives over all the actuators can beencoded using a binary system using the symbols i and d as the basic digits, where the symbolsrepresent the words increasing and decreasing. For example, in a 5-axis machine tool, if thefirst and the last actuator displacements are increasing and the other displacements are decreasing,we denote the case by idddi, which we refer to as an actuator permutation. The number of possibleactuator permutations is 2N, where N is the number of actuators.Conversely, an actuator permutation p imposes N inequalities in the tangent space TuU at agiven point u 2 U. Each inequality,_hiP0 or_hi60, can be thought of as a half space in thetangent space due to the linear relation shown in Eq. (1). The intersection of such half spacesforms a simply-connected cone in the tangent space. We refer to the resulting cone as the pthreversal-free cone of a given point u 2 U, and symbolically as CpuU. We say that CpuU is void ifCpuU f0g, and CpuU is degenerate if CpuU is a set of measure zero in the tangent space. If a cone isvoid, it is degenerate because f0g is a set of measure zero.An entire task space can be partitioned into two regions depending on whether the pth reversal-free cone is degenerate or not. The set of points in the task space which possess non-degeneratepth reversal-free cones is referred to as the pth monotonic region, and symbolically, as Mp. The setof points which possess non-void pth reversal-free cones is referred to as the pth extendedmonotonic region, and symbolically, as, Mp, which must be a superset of the pth monotonic regionMp. In practice, it is common that Mpand Mpare almost identical, which means that Mpn Mpisa set of measure zero.In a 2-dimensional tangent space, a reversal free cone is bounded by two rays emitted from theorigin of the tangent space; one is referred to as the right ray and the other is referred to as the leftray where the choice of right and left is, in fact, immaterial. By normalizing the right rays andthe left rays in a monotonic region into unit vectors, we can form corresponding unit vector fieldsin the monotonic region. They are referred to as the right and the left vector field, respectively;symbolically as VpRU and VpLU. The complementary permutation of an actuator permutation p isdefined as the permutation generated by reversing the signs of the corresponding inequalities anddenoted by ?p. For example, if p idddi, then ?p diiid. Fig. 3 shows examples of reversal freecones at a point.Proposition 3. We can state the following immediate consequences:(1) If a trajectory ut 2 U is reversal-free, its velocity vector _ ut 2 TutU must stay in a reversalfree cone for an actuator permutation. In other words, there should be an actuator permutationp such that _ ut 2 CputU for all t for reversal free motion.32The case hi 0 is considered to be both of the cases.3At a non-smooth point in a piecewise regular trajectory, there are two tangent vectors. In this case, we require thatboth tangent vectors satisfy the condition.306T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322(2) At a point u 2 U, CpuU and C?puU are axi-symmetric in TuU. In particular, CpuU f0g iffC?puU f0g; C?puU is degenerate iff CpuU is degenerate.(3) At a point u 2 U, pCpu TuU. In other words, the tangent space at a point is covered or tiledby the reversal-free cones of the point.(4) At a point u on the boundary of an extended monotonic region, Mp, the corresponding reversal-free cone CpuU is degenerate except for some special isolated points, which are reversal-singular-points.(5) The pth right vector field VpRU and the pth left vector field VpLU are continuous in each componentof Mpexcept at the reversal-singular-points. The vector field aVpRU 1 ? aVpLU is continuousand its value at any point u 2 Mpbelongs to the cone CpuU, where a is a continuous function on Uwhose range is the interval 0;1? ? R.(6) Mp M?p, Mp M?pand Mp? Mp.Proposition 3(1) is simply a recap of the reversal free condition in terms of reversal-free cones.If a trajectory is reversal-free, the sign of the actuator speed must be fixed during its entire intervalof motion, which implies the existence of the actuator permutation satisfying the stated condition.In other words, if the tangent vector of a trajectory is in the pth cone at an instant and if it latermoves to a different cone, say the qth cone (p 6 q), the speed of at least one of the actuatorschanges its sign at an instant during the transition. Proposition 3(2) holds because _ u 2 CpuU im-plies ?_ u 2 C?puU for any _ u 2 TuU. This is shown in Fig. 3(b). Proposition 3(3) is a recap of thesimple condition stated earlier: at every point in the task space at an instant, each actuator hastwo alternatives, either increasing or decreasing its displacement, namely either_hiP0 or_hi60and there are no other cases. Proposition 3(4) and (5) are immediate from the continuity of thegradient flow of fkfor every k in addition to its regularity except at reversal-singular-points. Recallthat fkis smooth by our assumption and that the pth cone disappears outside the pth extendedFig. 3. Example of a reversal free cone for finish-machining a parabolic surface parameterized by x u 2v, y u ? vand z u2 2v2 2uv, using a 3-axis Cartesian machine tool. In this case, the inverse kinematics map f is simplyf : U ! H, u;v ! u 2v;u ? v;u2 2v2 2uv. The corresponding Jacobian matrix is 1 2;1 ? 1;2u 2v 4v 2u?and is reduced to 1 2;1 ? 1;2 3? at u;v 1=2;1=2 2 U. Formally, Cidi1=2;1=2U f_ u; _ v j _ u 2_ vP0; _ u ? _ v60;2_ u3_ vP0g ? T1=2;1=2U. (a) A reversal free cone and its complement. (b) The union of the reversal-free cones at a pointcovers an entire tangent space at the point.T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322307monotonic region. Intuitively speaking, the reversal free cones cannot appear or disappear sud-denly. Proposition 3(6) follows the symmetry confirmed in Proposition 3(2), especially thestatement that CpuU is degenerate iffC?puU is degenerate. These are easily understood through theexample shown in Fig. 3.3.2. Reversals in point-to-point tasksGiven two end points in the task space, a point-to-point task seeks to find a trajectory thatconnects the two end points while satisfying certain conditions. We can make the followinggeneral statement in the actuation space:Proposition 4. For two given points in an actuation space, the straight segment that connects the twopoints is a reversal-free path if the segment stays in the actuation space. In particular, if the actuationspace is convex, any two points in the actuation space can be connected without any reversal.This follows obviously because any straight line in actuation space represents a monotonicchange of the actuator coordinates.However, in the task space, the question of reversal-free paths is not trivial. We now addressthis problem. For a task space, what we can assure with generality is, at most, a negative state-ment:Proposition 5. Any periodic path in the task space must have reversal points.This does not answer our original question: whether a given pair u0;u1 of points in the taskspace can be connected without any reversals by a trajectory which stays in the task space. Toanswer the question, we will construct the set of points which can be reachable without anyreversal from one of the given points, u0. The resulting set is referred to as the (reversal-free)reachability set of the point u0and symbolically as Fu0. If the other point u1belongs to the reversal-free reachability set Fu0, we can conclude that the pair of points, u0and u1, can be connected in areversal free manner. The following proposition is also useful in this regard:Proposition 6. If it is possible to connect two points with a reversal-free trajectory, there must be anextended monotonic region to which both points belong.The proof is easily constructed using the condition on the tangent vectors for reversal freemotions, Proposition 3(1).Plotting reversal-free reachability sets is also a good means of visualizing the reversal char-acteristics of machine tools. In a high dimensional task space, finding a reversal-free reachabilityset of a point is computationally-intensive. However, the problem can be dealt with effectively in a2-dimensional task space in a heuristic manner. Here, we show how we can visualize the reversal-free reachability set of a given point u02 U in a 2-dimensional task space U.Consider the set of points which are reachable from a point u02 U by a trajectory ut 2 Uwhile the velocity _ ut of the trajectory stays in the pth reversal-free cones CputU along the tra-308T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322jectory. The resulting set is referred to as a pth (reachability) leaf of the point u0and, symbolically,as Fpu0.Proposition 7. The following properties of reachability-leaves are immediate from the definition:(1) A reachability leaf is path-connected.(2) Fpu0? Mp.(3) u 2 Fpu0implies that Fpu? Fpu0and u02 F?pu.(4) Fu0 pFpu0.Proposition 7(1) follows immediately from the existence of the reversal free motion from thegiven point u0to a point in the reachability leaf. Proposition 7(2) is a recap of Proposition 6.Proposition 7(3) states that two consecutive reversal free trajectories can be combined into a singlereversal free trajectory. Proposition 7(4) states that the reversal-free reachability set of a point canbe constructed simply as the union of its leaves, which is immediate from Proposition 3(1) and (3).Consequently, the problem of finding a set of the reachability is reduced to finding thereachability-leaves of a given point and we need to show how to attain a leaf Fpu0of a point u0in a2-dimensional task space U. We now address this reduced problem. If the start point u0does notbelong to the extended monotonic region Mp, namely u062 Mp, we simply terminate the procedurebecause Fpu0? Mpand Fpu0is a path-connected set. Henceforth, we consider only the case ofu02 Mp.A procedure for finding the reachability leaf is now described. We integrate the pth right vectorfield from the start point u0. The integral line, which we denote by LR, meets the boundary of theextended monotonic region Mpat a point r.4In the same way, we define the intersection point lbetween the integral line LLof the left vector field and the extended monotonic region. This isshown in Fig. 4(a). We will construct the reachability leaf by connecting the two end points.Proposition 8. (1) The cone at r and the cone at l are degenerate if r and l are not reversal-singular-points because they are on the boundary of the extended monotonic region. (2) The directions of thecorresponding cones, CprU and CplU, are parallel to the integral lines LRand LL, respectively. (3) Thedirections must be towards the outside of the extended monotonic region.The first part is a special case of Proposition 3(4). Since the pth reversal free cones aredegenerate and non-void at r and l, the integral lines must be tangential to the degeneratedirection, which proves the second part. If the degenerate directions at r and l are toward theinside of the extended monotonic region, the left and the right integral lines cannot reach thepoints r and l which proves the third part.Now we return to our reachability leaf construction. We march along the boundary of theextended monotonic region from the point r to the left. During the march, we observe thedegenerate cone along the boundary. At the beginning, the cone is aligned towards the outside of4This step works when there are no reversal singularities in the extended monotonic region. We do not cover the rarebut special case in this paper. In such cases a brute force method can be used.T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322309the extended monotonic region. At some instant during the march, there is a chance that thedirection of the cone on the boundary is toward the inside of the extended monotonic region asshown in Fig. 4(b). If so, at that point, we march along the integral line of the right vector field asshown in Fig. 4(c). This is called a transition in the march. The transient integral line intersects theboundary of the monotonic region. If a self-intersection does not occur during the march, wecontinue to march along the boundary of the extended monotonic region. This march is continueduntil we reach the point l or until it forms a self-intersection. If a self-intersection is formed, wemark the r-to-l march as a failure and launch the same procedure from l to r using the left integrallines as transient curves. If both trials fail, we resort to a rather brute-force procedure: we inte-grate the vector field aVpRU 1 ? aVpLU from the starting point for various values of a. Integrallines of the vector field aVpRU 1 ? aVpLU are guaranteed to be reversal-free by Proposition 3(5).This heuristic marching procedure works and converges in most cases. However, the existence ofthe reversal singularities or holes hinders the universality of the marching procedure, and it can bebacked up by a brute-force procedure.We can actually prove that the above heuristic marching procedure finds correct reachabilityleaves for a special but common situation.Proposition 9. Suppose that the points r and l, which are the end points of the first two integral linesin the marching procedure, are on the most outside boundary loop of the component of the extendedmonotonic region to which the starting point u0belongs. Let B be the region bounded by the twointegral lines, LLand LR, together with the segment of the boundary of the extended monotonic regionthat connects the two points r, and l, from r to l counter-clockwise (see Fig. 4). If the region Bcontains neither holes nor reversal-singular points and if the marching procedure is successful, the setconstructed by the marching procedure is the correct reachability leaf.The proof is given in Appendix A. This proposition can be used as a check for the validity ofthe set constructed by the marching procedure. A generalization is possible for the case whenisolated reversal-singular-points are on the boundary of the region B but are not inside the region.If the r-to-l march succeeds, we check whether the left rays at the reversal-singular-points on theconstructed set is toward the outside of the region B. For the l to r march, we check the right rays.Fig. 4. Constructing a reachability leaf. (a) Integration along the right and the left vector field. (b) March along theboundary. (c) March along the right vector field.310T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322If no transition occurs, we check whether any portion of the reversal free cones at the reversal-singular-points is toward the outside of the region. The proof for this extension of Proposition 9 isreadily made once the proof of Proposition 9 is understood. Devising a marching algorithmhandling the reversal-singular-points in the middle of the region B is an open question.3.3. Reversals in surface machiningThe objective of finish machining is to sweep a 2-dimensional manifold. If we assume that theorientation of a cutting tool is pre-determined, the task space dimension is 2, for the finishingprocess. We can capture the family of sweeping paths with a vector field on the task space. In thissetting, we regard that the trajectories in the task space are chosen only from the streamlines of thegiven vector field. If every streamline of a vector field is a reversal-free trajectory, the vector field issaid to be reversal-free.A surface in the workspace of a machine tool can be specified in a parametric form, ru;v,where r is a regular map from the uv-parameter space of the surface, which we think of as the 2-dimensional task space U, to the 3-dimensional Cartesian workspace W . We require that the tooltip be constrained on the surface and that the orientation is pre-determined. Imposing theseconstraints, we can construct a map from the uv-parameter space of the surface to the actuationspace. More rigorous treatment of this model can be found in 22.A vector field on the task space, which captures a family of streamlines, can be specified with acontinuous mapping:_ u : U ! TU;u ! _ uu ? _ uu;v_ vu;u?T2 TuU:Now, given a vector field _ uu ? _ uu;v_ vu;u?Ton the task space U, the candidate for thepoints in the task space where the kth actuator reverses its direction can be determined by solvingthe following equation:_hk?ofkou? _ u ofkov? _ v ck1u;v ? _ uu;v ck2u;v ? _ vu;v 0:2This defines a curve in the task space, where cij? are known as the Jacobian matrix of the inversekinematics. We call these lines reversal lines of the vector field, which is also a visualization toolfor the reversal behavior of a machine tool. In a higher dimensional task space, the equationdefines a hyper-surface.The meaning of reversal singularities interpreted in Propositions 1 and 2 is repeated forsweeping tasks:Proposition 10. When we sweep a surface, the reversal of an actuator is not avoidable once thereappears a strong reversal-singular-point in the task space no matter what vector field is assigned tosweep the surface. If a streamline of a sweeping vector field passes a non-strong reversal singularitywith respect to the kth actuator without any reversals, the streamline must be tangential to one of theseparatrices of the function fku;v; this is a very special condition which is rarely satisfied.Therefore, once a reversal-singular-point appears in a task space, the correspondingactuator reverses its motion at the reversal-singular-point almostalways; plotting theT. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322311reversal-singular-points is a canonical way to visualize the reversal characteristics of sweepingtasks because it does not depend on the vector field we choose.We now state a positive proposition:Proposition 11. If there are no reversal-singular-points in a task space and if a monotonic region Mpcovers the task space, we can sweep the task space without any reversal with the following vectorfield:aVpRU 1 ? aVpLUwhich was mentioned in Proposition 3.This is because the vectors in the vector field stay in the pth reversal free cones as mentioned inProposition 3. An extension of this problem is to cover the task space with the minimum numberof monotonic regions, which is essentially reduced to the minimum set cover problem when wediscretize the space.3.4. Remarks on redundancyFor redundant machines, we can derive reversal-free reachability sets in a similar manner, atleast conceptually. Here, we explain the reason with an example since redundancy is a subjectiveconcept and its context is important. Consider machining a parameterized surface patch, ru;v,with the 6-axis machine shown in Fig. 2(c). We assume that the orientation of the cutting tool ispre-determined at every point on the surface as before. We regard the uv-parameter space 0;1?2ofthe surface as the task space U. When we choose a point u;v in the task space, this setting fixes 5degrees of freedom, three for translation and two for orientation of the cutting tool. In theprevious discussion, we had assumed that the extra degree of freedomthe spin rotation aboutthe tool axiswas pre-determined. Now, we let the moving platform of the machine be free tospin. Let k be the spin angle which is appropriately defined5and is chosen from a set, sayL ? R. We will refer to the space formed by the triple, u;v;k, as the extended task space andsymbolically as V ? U ? L. The inverse kinematics of the machine tool is reduced to a mapg : V ! H, u;v;k ! h from the extended task space V to the actuation space H. A smooth mapr : U ! L can be thought of as a redundancy resolver. The map, fr: U ! H, u ! gu;ru iswhat was previously called the restricted inverse kinematics.We define the reversal-free reachability set, say Gu0;v0;k? V , of a start-point u0;v0;k in theextended task space using the map g as if the extended task space were a task space in the previousdefinition. We construct Gu0;v0;kfor all the possible spin angles, k?s, and refer to their union asGu0;v0? V . The reversal-free reachability set, say Fu0;v0?L, in the task space U with a freeparameter k 2 L is simply the projection of Gu0;v0to the uv-task space U ? V in the uvk-extendedtask space V . In fact, this can be used as the definition of the reversal-free reachability set of apoint u 2 U under the redundant kinematics g:5For example, k can be an angle formed by a certain line fixed in the moving platform and a certain plane in theworkspace; the line in the moving platform should not be parallel with the tool axis.312T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322Fu?L? Projection to U ofk2LGu;k !;where V U ? L and u 2 U. The definition itself suggests a way to construct the reversal-freereachability set even though it is too expensive for high dimensional spaces. Conclusively,Proposition 12. The reversal-free reachability set, previously denoted by Fu, under a restricted in-verse kinematics frmust be contained in the reversal-free reachability set Fu?Lunder the redundantkinematics g, namely Fu?L? Fu.The proof is immediate from the definition. If a path ut t 2 0;1? in the task space U isreversal-free under the restricted inverse kinematics fr, the path ut;rut in the extended taskspace V is reversal-free under the redundant kinematics g. The end point of the path ut;rutmust be in Gu0;ru0whose projection onto U is contained in Fu0?Lby the definition.Therefore, there is a possibility to cover a wider area avoiding actuator reversal by choosing agood redundancy resolver than choosing it in an arbitrary manner. However, it should be alsomentioned that the use of a redundant machine can increase the basic susceptibility to reversal,compared to the non-redundant machine due to the increase in the number of actuators.Even in this example, we must construct the reversal-free reachability set Gu;kin a 3-dimen-sional space, which is computationally expensive and topologically complex. The study on aneffective mean for this construction deferred as our future work.4. Visualization for the three types of machine toolsThe various concepts that were developed in the previous section can be visualized to enhanceour understanding of the reversal characteristics of machine tools. In this section, we showreversal-singular-points, reversal lines, reversal points and reachability sets for the three machinetools shown in Fig. 1, which, we believe, assist high level decision on the design and the use ofmachine tools. In addition, we explain the concepts we developed in the previous section in detailwith examples.4.1. The machine tools and the surfacesFig. 5(a) shows the skeletons of the three machines which are serial, hybrid and parallel.Their actuators are numbered in the same figure. We will consider two surfaces, which arespecified as Bezier tensor product patches; one is a flat surface, and the other is a curved surface asshown in Fig. 5(b). The actual size of the surface was scaled for each machine tool so that thesurface can fit into the designed workspace. For the serial machine tool, the surface was rotatedabout the x-axis to avoid the singularity at the vertical posture. As long as the inverse kinematicsare computable, the flat surfaces were adjusted to a relatively large size so that global phenomenacan be observed. We will consider assigning both the surface normal and the vertical direction tothe orientation of the tool although Fig. 5 shows the surface normal assignment. We consider theT. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322313finish process whose task space is 2-dimensional. The inverse kinematics is given as a mapu;v 2 0;1?2! fu;v 2 RN, where N is the number of actuators, and u and v are parametersused for defining the surface. For the parallel machine, the redundancy is settled in such a waythat the line ab fixed in the moving platform as shown in Fig. 5(a) is parallel to the xz-plane.4.2. Reversal-singular-pointsReversal-singular-points can be computed by solving ofk=ou 0 and ofk=ov 0 simulta-neously for each k. We show a typical case in Fig. 6(a); in the task space shown, we showof4=ou 0 lines, of4=ov 0 lines and the level lines of the 4th component f4u;v of the inversekinematics of the hybrid machine when we machine the curved surface with the surface normalorientation. One reversal-singular-point is strong among the total of three reversal-singular-pointsas shown in Fig. 6(a). By repeating the same procedure for other actuators, we find the total of 30reversal-singular-points in the task space as shown in Fig. 6(b). Fig. 7 shows the distribution of thereversal-singular-points for the curved surface we are considering. There is a general tendency thatthe number of reversal-singular-points increases as the mechanism approaches a fully parallelmechanism and as the surface is curved. Particularly, in surface normal machining, swings inthe orientation are the major contributor to the increase in the number of reversal-singular-points.4.3. Reversal points along specified trajectoriesIn Fig. 8, we show the number of reversal points along specified trajectories. Largely circularand figure 8-like curves are traced on surfaces in the workspace. It is observed that the parallelmachine exhibits greater numbers of reversal points than the hybrid machine. More reversalpoints are found in figure 8 curves than the circular curves. For flat surfaces, the hybrid ma-chine exhibits a pattern of reversals which is quite similar to the serial machine. The comparisonbetween Figs. 7 and 8 shows that an increase in the number of reversal singularities correlatesquite well with an increase in the number of reversal points.Fig. 5. Machines and surfaces. (a) The Skeletons of the three machines with the surfaces. (b) The curved surface.314T. Kim et al. / Mechanism and Machine Theory 39 (2004) 2993224.4. Reversal lines for sweeping tasksFig. 9(a) shows streamlines of a vector field we consider in this example for a sweeping task.Reversal lines with respect to an actuator are defined by Eq. (2). In Fig. 9(b), we show reversallines with respect to the first actuator of the parallel machine tool in the uv-parameter space,Fig. 6. Finding reversal-singular-points. (a) The reversal-singular-points for the 4th actuator of the hybrid machine forsurface normal orientation assignment in uv-task space or parameter space. (b) The distribution of all the reversal-singular-points.Fig. 7. Distribution of reversal-singular-points.T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322315which are the bold lines; the circular markers are placed at the reversal-singular-points and thedashed lines are the level lines of the first component f1u;v of the inverse kinematics. It is ob-served that the reversal lines pass the reversal-singular-points and a level line is tangential to astreamline at each point along reversal lines. By solving the same equation for other actuators, wecan visualize a complete set of the reversal lines of the sweeping task as shown in Fig. 9(c).In Fig. 10, we show reversal lines for various cases; reversal lines represent the complexity in thereversal behavior of machine tools quite well. Like our previous comparisons, we observe morecomplex patterns of reversal lines for the fully-parallel machine. However, in the surface normalmachining of curved surfaces, the disadvantage is not clear; all three machines show fairlycomplex patterns of reversals in this case. As implied by Proposition 5, sweeping paths withcircular patterns are more susceptible to the actuator reversals than the largely parallel sweepingpaths.Fig. 8. The number of reversal points along specified trajectories.Fig. 9. Finding reversal lines for a vector field. (a) The streamlines of a vector field. (b) Reversal lines for an actuator inthe uv-parameter space. (c) All the reversal lines on the surface.316T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322We find few cases in Fig. 10 for flat surfaces which succeed in reversal-free sweeping. They canbe explained by Proposition 11. For example, we consider the reversal-free sweeping of theparallel machine tool, which consists of horizontal sweeping paths as shown in the bottom-rightcorner of Fig. 10. Fig. 11 shows a number of sampled points in the diiiddth monotonic region andthe corresponding reversal free cones therein, for the parallel machine. We find that the tangentvectors of any horizontal lines in the monotonic region in the indicated box belong to the reversalfree cones as shown in Fig. 11. Therefore, we can sweep the diiiddth monotonic region in theindicated box with horizontal lines without reversal. If the workspace is set as the indicated boxinside the monotonic region shown in Fig. 11, the family of horizontal lines can sweep the taskspace without any reversals.4.5. Reversal free reachability setsWe consider the surface normal machining of the curved surface using the hybrid machine toolas shown in Fig. 5(a). In Fig. 12(a), we show the reversal free cones in the diiiith monotonic regionand the corresponding reachability leaf of a point in the middle, constructed by the marchingprocedure. Repeating the same for other actuator permutations, we can construct the reachabilityFig. 10. Reversal lines for various cases.T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322317set as their union, as stated in Proposition 7, which is shown in Fig. 12(b). Each leaf was labeled inthe figure. The corresponding reachability set is shown in Fig. 12(c). Especially for machining aplanar surface using the parallel machine tool, we can construct its reversal free reachability setusing only a ruler and a compass, which is shown in Fig. 13.We make a table of the reachability sets for all the three machines in Fig. 14. It is seen that thevertical or fixed orientation produces much wider reachability set than the surface normal ori-entation. Using the hybrid machine tool, we can reach an area almost as wide as if we were usingthe serial machine tool without any reversals. On the other hand, it is observed that the parallelmachine tool can reach a much smaller area of the workspace.Fig. 11. Reversal free sweeping.Fig. 12. Finding a reachability set. (a) The reversal free cones in the diiiith extended monotonic region and the cor-responding reachability leaf. (b) The reachability set constructed as a union of reachability leaves in the parameterspace. (c) The reachability set in the task space.318T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322Fig. 14. Reversal free reachability sets.Fig. 13. The construction of the reachability set for the parallel machine tool using only a ruler and a compass.T. Kim et al. / Mechanism and Machine Theory 39 (2004) 2993223195. ConclusionsIn this paper, we investigated the reversals of the relative motion in kinematic pairs of machinetools. First, we developed the principal concepts, which were: reversal-singular-points, reversalpoints along specified trajectories, reversal free cones, reversal lines for a sweeping task andreversal free reachability sets. Then, we visualized them for various machine tools varying ori-entation patterns and the surfaces. In addition, we briefly discussed the algorithms for the visu-alization and the properties related to the developed notions.We conclude that the factors that increase the susceptibility of joint reversals are (1) the numberof joints in the mechanism which move simultaneously (2) the curvatures of the inverse kine-matics, (3) patterns of tool orientations and (4) the curvatures of the surfaces and the trajectories.Largely, the parallel machine tool showed relatively poor reversal behavior, especially for sim-plefor example, planargeometries, compared to the serial and the hybrid machine. Forcomplex situationsfor example, if all the axes move simultaneously along a curved surfaceall the three machines show high susceptibility of joint reversals, and the advantages or dis-advantages are not as clear.We are interested in several topics for future research including: (1) extension of this workconsidering passive joints, (2) the problem of finding tool orientations so that a reversal freesweeping path can be realized, if it exists, (3) the relation between reversal characteristics and therequired actuator acceleration or torque, (4) quantifying the magnitude of the adverse effectinduced by a reversal and (5) applying our results to the design of a specific machine, to planningtool paths with extra precision, and to choosing a machine tool that most fits to a particularmachining task.AcknowledgementWe acknowledge the funding of NASA and NSF, Contract # DMI9912558.Appendix AProof of the Proposition 9. We will prove the proposition only for the case when the r-to-l marchis successful. The other case can be proved in the same manner. In addition, we consider the casewhen less than two transitions occur in the march. The proof for more than one transition isreadily derived from the one-transition case.Let C be the region constructed by the marching procedure.Case A: No transition occurs. A typical case is shown in Fig. 15(a). (?) Since the march wassuccessful and no transition occurs, the degenerate cones must be toward the outside of theconstructed region C. Take an arbitrary point u 2 C. From u, we integrate the reversed leftvector field, ?VPLU, until the streamline hits the boundary of the region C. A typical reversed leftvector field is shown in Fig. 15(b). The streamline can exit the region C only through the rightinitial integral line LRbecause the vector field is tangential to LL, we have only influx along theremaining boundary segment due to (?) and there is no singular point in the region. Let u1be the320T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322exit point. Since u1is on the initial right integral line, u1is reachable from u0without reversal. Onthe other hand, u is reachable from u1along the constructed streamline without reversal.Therefore, we proved that any point in
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