英语翻译--原文.pdf

On actuator reversal motions of machine tools Taejung Kim a, Seung-kil Sonb, Sanjay E. Sarma a,* a Department of Mechanical Engineering, Massachusetts Institute of Technology, 35-010, MIT, Cambridge, MA 02139, USA b Engineering Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Received 30 August 2002; received in revised form 16 June 2003; accepted 17 September 2003 Abstract When joints in a machine tool reverse the direction of their motion, non-linearities that are ignored in machine design and control are refl ected noticeably in the accuracy of machined surfaces. For example, friction characteristics of a machine tool become highly non-linear at low operating speeds, demanding sophisticated compensation. We present a theoretical treatment of the kinematics of reversals and reversal free 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 characteristics have implications both in the design of machine tools and in path planning. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Reversal; Friction; Machine tool; Kinematics 1. Introduction When 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 deals with the kinematics of reversals in machine tool mechanisms, concentrating mostly on the theory and the properties of reversals and reversal-free paths. These eff ects tend to be ignored during the initial stages of mechanism design, and our goal is to present the theoretical basis for predicting reversals. In particular, we show that reversals are pronounced in machining even plane surfaces and straight lines with parallel machine tools and are generally diffi cult 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: (S.E. Sarma). 0094-114X/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2003.09.002 Mechanism and Machine Theory 39 (2004) 299322 /locate/mechmt The 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 a drive 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 magnifi ed during motion-reversals is the transition between static and dynamic regimes of friction in joints. Fig. 1(a) shows the typical behavior of friction in a lubricated joint as it transitions from low-velocity to high-velocity motions. The friction mechanism changes from stiction eff ects to the Stribeck eff ect to more linearly viscous behavior 2. The magnitude of the friction force increases and switches direction as the joint passes through the zero-speed regime. Simple linear control strategies, designed for linear viscous friction models, compensate this eff ect poorly, and can suff er from delays, tracking irregularities and loss of machining precision 3. Quadrant glitches, as shown in Fig. 1(b), are typical examples of 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 controller every time the anomaly occurs, or by adopting more complex control strategies. However, implementing these strategies requires more advanced controllers, and careful experimentation for parameter identifi cation, which are both necessarily more expensive. It is best, therefore, to avoid the occurrence of reversals during critical motions that the designer considers important. 1.1. Reversal characteristics As described above, it would be desirable for a machine tool to be designed in such a way that reversals 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 a straight-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 occur in 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 kinematic Fig. 1. A typical friction characteristic in a joint and its negative eff ect in precision. (a) Relation between friction and velocity. (b) Quadrant glitches (courtesy of Eugene Tung 4). 300T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322 behavior from the point of view of reversals: all reversals occur in an easily describable way, and reversal characteristics are position-independent and confi guration-independent. Unfortunately, not all machines have these desirable properties, and reversals are especially diffi cult to predict in parallel machines. A parallel machine may suff er a reversal while prescribing a straight-line motion in its workspace. This behavior could turn out to be a severe impediment to the use of parallel machines in machining applications. Serial Cartesian machine tools too suff er from reversals during free-form machining. In machining an aerodynamically critical surface, for example, a curved trajectory may result in undesirable reversal-induced ridges. Understanding these problems is important, and we examine a number of issues in this paper: where the reversals occur in a path, what the nature of reversal-free paths is, where reversals are inevitable regardless of direction of motion, how the regions of reversal-free reachability are derived and so on. Our treatment is mostly theoretical, and based on the kinematics of a given machine. Much of the contribution of this paper lies in developing the mathematical machinery necessary to describe reversals and related concepts precisely and unambiguously. 1.2. Background The problem of backlash has been an age-old challenge in machine design, and has received a great deal of attention in the machine-element community. A description of ideas and techniques in machine elements is available in 1. There has been a great deal of literature on friction models, and a comprehensive summary is available in 5. An elaborate model of friction presented by Canudas De Wit et al. 6 captures the diff erent eff ects 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 the appearance of a friction force, and of Dahl 8 who further incorporated hysteretic eff ects. 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 time 10. Nevertheless, this model forms the basis of a number of control methods that have been proposed in the last few years. Researchers in the fi eld of motion control have proposed a rich variety of compensation methods to achieve smooth motion with high accuracy under the action of friction. Not being the focus of this paper, we present only a rough overview here. Comprehensive summaries are available in 11,12. The techniques for control of systems under friction include: estimation and adaptive methods 13,14, robust compensation schemes 15, repetitive control 4, multi-loop control 16, non-linear compensation (similar to sliding mode) 17, non-linear PID (which changes the gain depending on plant state) 18 and a broad class of pulse code modulation techniques which are summarized in 12. There is a great deal of literature on parallel machines, and there are dozens of milling machines which use parallel kinematic structures. A summary is available in 19. The putative advantage of parallel machines over their serial counterparts is that the load can be distributed more evenly to their kinematic links and, as a result, a higher stiff ness can be achieved using lighter structural elements and less expensive actuators. However we will show that the coupled kinematics of T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322301 parallel 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 the coordinate axes 19, and this can be problematic when it is important to maintain planarity or straightness. In addition, there is literature that shows that frequent actuator reversals can excite higher harmonic components in the machine tool structure 20,21. This increases the necessary structural stiff ness of parallel machines and, to some extent, undermines the very basis for which parallel mechanisms are preferred in the fi rst place. Beyond this, however, there appears to be little prior work in the study of reversals in serial or parallel machine tools. 1.3. Outline An absolutely fair comparison amongst diff erent machine tool mechanisms is not easy to establish for reversal behavior because performance depends on the trajectories considered and on the tasks at hand. If for example, the machine is not required to cut fl at surfaces, but, say, spheres instead, a parallel machine of a certain confi guration may be more natural than a Cartesian serial machine. Instead of making assumptions about the task that the machine is considered for, we will defi ne 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 to achieve 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. In Section 3, we answer fundamental questions about the behavior of reversals and develop methods of analysis. In Section 4, we present some examples with three types of machine tools: a serial, a hybrid and a parallel machine as shown in Fig. 2. We conclude in Section 5. 2. A framework for analysis In this section, we lay down a mathematical framework upon which the rest of this paper is based. It is assumed that what we will call a restricted inverse kinematics map is known. This section simply defi nes the term and justifi es 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) 299322 To describe the motion of a cutting tool, we attach a fi ducial point on the center line of the cutting tool near the tool tip. 1 3-tuples of the Cartesian coordinates of the fi ducial point con- stitute what we refer to as the workspace of the machine tool. In general, a cutting tool, as a rigid body, has 6 degrees of freedom: three for the translation of the fi ducial point and another three for the rotation of the cutting tool. Since we are considering milling machines, and because cutting tools can be considered to be axi-symmetric, the sixth freedom of motion, which is the spin about the centerline of a cutting tool, is typically unnecessary and redundant. Therefore, two additional coordinates 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 subjective interest as the task space dimension of the application. The task space dimension is not necessarily identical 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 is redundant. In 3-axis machining, only translation of a cutting tool is allowed and the task space dimension is 3. In the path planning stage, the dimension can be further reduced by introducing task constraints. For example in 3-axis roughing, it is common to remove material from the stock layer by layer; in this case, the task constraint is that the fi ducial point should stay on a plane while a layer of material is removed, and the task space dimension of machining each layer is 2. In fi nishing the surface of a free-form part using 5-axis machining, the tool tip is constrained to contact the required surface, and the task space dimension of fi nishing is reduced to 4. When planning fi nishing paths, collision-free orientations of a cutting tool at each point on the designed surface can be determined by a collision-detection pre-processor, and the remaining planning task can 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 task space dimension is M 6N. The motion of a cutting tool restricted to the strategy can be captured by a history of M-tuples of real numbers. We refer to the totality of the M-tuples representing the restricted 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 a machining strategy determine a smooth map f from the task space U ? RMof the strategy to the actuation 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 a point-wise manner (or holonomically) based on our assumption. We assume that the task space U is simply-connected and that it avoids singularities by choice. The range fU of the map is an M-dimensional subspace of the N-dimensional actuation space. The actuator displacement h can be thought of as one of the local coordinates of the confi guration space of the machine tool. This setting is general enough for our purpose, even though complete generality can be achieved by 1 Since a cutting tool has a non-zero diameter, the contact point between the tool and the surface is not necessarily on the line of symmetry of the tool 22. However, a loose defi nition of the location of the fi ducial point suffi ces for our current analysis. T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322303 considering the confi guration space of machine tools including both active and passive joint displacements. A rigorous statement of the above assumption exceeds the scope of our current treatment, but more details are available in 23,24. In this paper, we will show how the setting described above can capture the reversal characteristics of most machines. 3. Reversal analysis In 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 any actuator 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 of motion. Finally, we analyze the reversal characteristics of surface machining or sweeping. We compare 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 their rationale in the main body of text. 3.1. Reversal conditions In this section, we show various ways to interpret reversals of actuators. 3.1.1. A reversal condition on a trajectory for a tracking task A fundamental function of a machine tool is the ability to track a trajectory specifi ed for its end-eff ector. We consider a machine tool moving along a piecewise regular trajectory u : t 2 0;T? ! ut 2 U in the task space U. If hit ? fiut has a local extremum at t t0for any 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 suffi cient that _ hit ? d dt fiut X M j1 ofi ouj ? duj dt ? 0and hit ? d2 dt2 fiut 6 0: This condition itself suggests a straightforward procedure for fi nding the reversal points along a given regular trajectory in the task space. Plotting and counting reversal points along specifi ed trajectories 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 points At a point u 2 U in the task space U, if there is an integer k 2 f1;2;.;Ng such that rfk? ofk ou1 ; ofk ou2 . ofk ouM ? 0; we refer to the point u as a reversal-singular point with respect to the kth actuator, where fkis the kth component of the inverse kinematics f. A reversal-singular-point with respect to the kth 304T. Kim et al. / Mechanism and Machine Theory 39 (2004) 299322 actuator is said to be strong if the Hessian matrix hk ij? of the kth component fk of the inverse kinematics f is defi nite at the point, where hk ij ? o2fk ouiouj : 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