[Psychology] Mechanical Assemblies Phần 2 ppsx

38 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES FIGURE 3-5. Schematic Diagram of Matrix Transforms Applied to the Stapler. Left: The parts of the stapler have been replaced by blobs. Right: Straight-line arrows have been added to relate frames on the same part. Curved arrows have been added linking the coordinate frames of assembly features on different parts to indicate which ones are to be joined in order to assemble the parts. Double curved lines indicate the KCs that were identified in Chapter 1. FIGURE 3-6. Schematic Representation of a Transform. The transform T contains a translational part represented by vector p and a rotational part represented by matrix R. Vec- tor p is expressed in the coordinates of frame 1. Matrix R rotates frame 1 into frame 2. vectors are assumed to be column vectors, so a transposed vector is a row vector.) On a component-by-component basis, transform T is where vector p is expressed in the coordinates of the orig- inal frame and r, ; are the direction cosines of axis i in frame 1 to axis j in frame 2. Transform T can be used to calculate the coordinates of a point in the second coordinate frame in terms of the first coordinate frame. The coordinates of a point are given by 3.C. MATRIX TRANSFORMATIONS 39 Then, in general, if q is a vector in the second frame, its coordinates in the first frame are given by q'\ This says that q' is obtained by rotating q by R and then adding p. Suppose a transform T consists only of matrix R, and suppose that we want to find the coordinates of the end of a unit vector along the z axis of the rotated second frame in terms of the unrotated first frame. The calculation is This result shows that the columns of matrix R tell where the coordinate axes have rotated. That is, the first column tells where the x axis went, and so on. The ele- ments of each column are the cosines, respectively, of the x,y, and z components of the new axis expressed in the original frame. Matrix R can be generated a number of ways. One way is to rotate once about each coordinate axis. This will gen- erate one elemental rotation matrix. Matrix R can then be created by multiplying the elemental matrices into one an- other. The elemental matrices, as discussed in [Paul], are The order in which T's and R's are multiplied is impor- tant, and different sequences will create different results. For example, rotates vector u into a new orientation w by first rotating 90° about the z axis in the frame in which u is measured, then 90° about _y in the same frame. However, rotates vector u into a new orientation w' by first rotating about the y axis and then about the z axis. Equation (3-9) can also be interpreted as saying, Rotate u 90° about its original y axis, then 90° about its new z axis. Similarly, Equation (3-10) can be interpreted as saying: first rotate u 90° about its original z axis and then rotate it 90° about its new y axis. A transform that simply repositions a frame without reorienting it is A transform T that comprises a translation p x along x followed by a rotation of 90° about the new (translated) z could then be written We can also compute the inverse of a transform. In words, the inverse of T should undo what T did. If or, equivalently, if then The transform in Equation (3-16) is the inverse of the transform in Equation (3-15). Embedded in these relation- ships is the fact that, for rotation matrices, 3.C.2.b. Examples Here are some examples that illustrate the rules for using trans and rot, including the effects of doing so in different sequences. then 40 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES Equation (3-18) reminds us of the rule regarding se- quence of application of a transform. It contains the trans- forms that we will use in the examples here. <— use original axes trans(p x ,Q,Q)rvt(z,90) (3-18) use new axes We will compare this combined transform with one that contains the same matrices but does something completely different: We will calculate the effects in both cases, applying the transforms from left to right and from right to left. First, Equation (3-18) is expanded in Equation (3-20). The ac- tions are performed in both sequences in Figure 3-7. It is seen that both sequences result in the same new frame. FIGURE 3-7. Illustration of Two Ways of Interpreting Equation (3-20). Left: Performing the operations from right to left requires that the original XYZ axes be used through- out the action. Hence, we first rotate 90° about Z and then translate a distance p x along the original X axis. Right: Per- forming the operations from left to right requires that the new axes be used throughout the action. (For the first operation, new and original have the same orientation.) Hence, we first translate a distance p x along the original/new X axis and then rotate 90° about the new (translated) frame's Z axis. Second, we will perform the actions of Equation (3-19) in both sequences. This is illustrated in Figure 3-8. First, Equation (3-19) is expanded in Equation (3-21). Again, we see that the same final frame results. Of course, it is different from the frame that results from the operations in Equation (3-20). = trans(0,p x ,Q)rvt(z,90) 3.C.2.C. Composition of Transforms The main use of transforms is to permit chaining a series of them together so that we can locate a distant frame by means of several intermediate frames. This is done merely by multiplying one transform by another, as shown in Figure 3-9. The following forms are equivalent: The first thing to notice about the matrix in the fifth equation is that it follows the form of the general trans- form: a rotation matrix in the upper left, a position vector at the right, and a row of three zeroes and a one along the bottom. Thus the composition of two transforms is another transform. This means that we can continue to chain transforms in this way, obtaining another transform each time. The second thing to notice is that we can say 3.C. MATRIX TRANSFORMATIONS 41 FIGURE 3-8. Illustrating Three Ways to Interpret Equation (3-21). Left: Performing the operations (rotz, 90) trans(p x ,0,0) right to left requires using the original axes, including honoring the location of the origin when performing the rotation about (original) Z. Middle: Performing the operations frans(0, p x ,0)rot(z, 90) left to right requires using the new axes, again including honoring the location of the origin when performing the rotation about (new) Z. Right: Performing operations trans(0, p Xl O)rot(z,90) right to left requires rotating first 90° about Z and then translating a distance p x along (original) Y. These and other interpretations of Equation (3-21) give the same result. FIGURE 3-9. Illustrating the Composition of Two Trans- forms. 7~12 locates frame 2 in frame 1 coordinates. IQI locates frame 1 in frame 0 coordinates. 7"o2 locates frame 2 in frame 0 coordinates. Example rotation transform function Rz = rotz(theta) % creates rotation matrix about axis Z % input in radians ct = cos(theta) st = sin(theta) Rz = [ct -st 0 0; st ct 0 0; 0 0 1 0; 0 0 0 1] Conversion from degrees to radians Translation transform function degtorad = dtr(theta) % converts degrees to radians degtorad = theta*pi/180 function Tr = trans(;c, y, z) % creates translation matrix Note: Function Rz is an example of a rotation operation. Similar functions for rotat- ing about the other axes are easy to write using Equation (3-6) and Equation (3-8). in words what the composite transform does: It translates along PQI, then rotates by 7?oi, then translates along p\i, and finally rotates again about R\ 2 . The third thing to notice is that the composite transform 7o2 accomplishes in one leap what TQ\ followed by T\2 do one step at a time. When we write a transform, say TQ\, we are able to convert any vector expressed in frame 1 coordinates into frame 0 coordinates. We can also convert any transform ex- pressed in frame 1 coordinates so that its effect appears in frame 0 coordinates. Such a transform might be called T\I. If frame 2 is rotated in some complex way from frame 0, it may be easier to express the effect (a translation or a rotation) that we want in frame 2 coordinates and then calculate the effect in frame 0 coordinates by writing The order in which we multiply transforms is impor- tant. If T\ and TI are transforms, then This fact is used in constructing Equation (3-23), which is the basic equation of matrix transforms, as well as in the examples in Equation (3-20) and Equation (3-21). When we multiply a transform TQ\ from the right by another transform T\2, we use TQ\ as the base, effectively adding a coordinate frame T\2 to a chain of frames that begins at the left end of the chain with a base frame whose transform is /, the identity transform. Table 3-1 gives some useful MATLAB 4 functions for working with transforms. If we are careful about how we choose the subscripts of transforms, we can easily read them as a recipe for walking from frame to frame: 7} ; takes us from frame / to frame j. When we compose two transforms, as in T^ = 7}^ Tkj, we can say that subscript k is "used up" when 7}* and T^ are chained together to form 7} y . This means that frame k no longer needs to be represented explicitly because its effect has been absorbed in T ( j. T f j then carries us directly from frame / to frame j. Careful subscripting is very important in debugging complex chains of frames, especially when they are used for variation analysis. 4 MATLAB is a trademark of The Math Works, Inc. TABLE 3-1. Three Useful MATLAB Functions for Operating on Transforms 42 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES We can also express small changes in a transform using a transform. This is highly convenient because it means that we can use the same mathematics to express both the nominal location and the varied location of a frame, and hence of a part or a feature on a part. This is how we will perform variation analyses in Chapter 6. The kinds of variations that we can express this way are errors in rotation or translation, that is, errors in R or in p. These may be written as follows: FIGURE 3-10. Properties of the Error Transform. If DT is an error in 7", then the erroneous T' is expressed as r = T DT. The upper left 3x3 submatrix 8R is a differential ro- tation matrix. Its elements correspond to a small error in rotation of 80 X about *, 89 y about y, and 80 Z about z. The vector dp contains small differential translations dx, dy, and dz. We may write the differential rotation matrix as shown because, if the rotations are small enough, we may consider them to be in the form of a vector like a rotation rate vector, and the order in which they are accomplished does not matter. 5 The properties of the differential transform are illus- trated in Figure 3-10. If there is an error DT in a transform T, then the varied transform is expressed as Next, we will show how to use chains of transforms to represent assemblies of parts joined by features. 3.D. ASSEMBLY FEATURES AND FEATURE-BASED DESIGN This section takes up the topic of features in assembly. First we give some history, then we define manufacturing features and assembly features, and finally we show how to use transforms to locate features on parts and chain parts together via feature frames to create a connective assem- bly model. This will equip us to use the same mathematical framework to model assemblies linked by features having either nominal or varied locations. 5 To prove this, form rotation matrices mt(x, 89 X ), rot(y, 89 y ), and mt(z, 89 Z ), multiply them together, substitute 89 for sin 80 and 1 for cos 89, and eliminate all terms in powers of 9 above 1. Examples that use the methods in this section are given in Section 3.E.4.a. 3.C.3. Variation Transforms Here, again, the order is important. We accomplish transform T and then we apply the error DT. If the error occurs before transform T is applied, that is, if it occurs in the untransformed frame, then For completeness, we introduce the equivalent notation where Multiplying these together creates the error trans- form DT: 3.D. ASSEMBLY FEATURES AND FEATURE-BASED DESIGN 43 3.D.1. History 3.D.2. Fabrication Features Feature-based design dates at least to the early 1980s or late 1970s. It was originally an attempt to organize and simplify computer numerical control (CMC) program- ming of machine tools. Such programs were tedious to write and prone to errors. Many programs were written to cut the same basic shapes, such as drilling and cham- fering holes, carving pockets and keyways, and so on. Programming would be easier if one could use a library subroutine capable of carving, say, a general pocket of dimensions L, D, and W, simply by supplying numeri- cal values for L, D, and W, plus coordinates for the po- sition and orientation of the pocket. Pockets, keyways, and so on, soon came to be called features. Later, re- search was done to classify features and provide more comprehensive feature libraries ([Faux and Pratt], [Shah and Rogers]). Finally, it was realized that features pro- vided the opportunity to capture information beyond mere geometry. For example, a keyway feature for holding a key could be given its size based on a calculation of the likely force that the key would encounter. In this way, features rose to become carriers of design knowl- edge and intent. Below, we call pockets, lightening holes, fillets, and so on, fabrication features because they are used to define the shape of the part. Locating holes, keyways, and so on, are called assembly features be- cause they are used to define how parts join to each other. Features were later realized in terms of object-oriented programming. An object in this context is a set of data and program code (called a method) capable of express- ing an item of interest. The data for a keyway feature object might include size parameters of the keyway, while the method could draw a picture of the keyway or con- tain CNC code for cutting it. Another property of objects, called inheritance, is also useful for features. Inheritance means that objects are often subclasses of each other, and the subclasses inherit all of the properties of their su- perclasses while adding other properties that distinguish them. A hole feature is a subclass of the superclass fea- ture while a threaded hole is a subclass of the hole. Inher- itance simplifies programming the objects because only the new elements have to be added when a subclass is defined. First we will address fabrication features, then assem- bly features. Finally, we will build assembly models by connecting parts using their assembly features. Fabrication features are the regions of a part that are of importance for the purposes of creating the general shape of the part. Fabrication features present a challenge be- cause their very identity, being for the purpose of defining fabrication instructions, is different depending on the fab- rication method used. An example is shown in Figure 3-11. Here we see a series of pockets separated by walls. The identity of the feature is different depending on whether the pockets are made by removing the metal from the pocket area (say by machining) or by adding metal (say by molding or casting). In the first case, the feature is the pocket, and the rules for creating it are the rules of machin- ing. In the second case, the feature is the wall and the rules for creating it are the rules for molding. In the case of mold- ing, moreover, the walls cannot be parallel as they can be in the case of machining. Instead they must be tapered (some- times called drafting them or giving them a draft angle). The reason why this is a problem is that in many cases the designer does not know, at the time of design, what process will be used to make the part. It depends on the production quantity needed and the economics of the sit- uation. A small number of parts may be machined, but a large number is more likely to be cast or molded. De- mand may change over the life of the part. For these rea- sons, it is worth allowing the designer to use a generic feature to create the shape in the computer independent of fabrication method. Later on, someone will define the fabrication version of the feature, possibly combining or splitting the original generic features. To make this con- version efficient, it is valuable to be able to inspect a CAD design and automatically recognize the shapes that should be combined into features relevant to a specific fabrication process. This is called feature recognition. Lacking this, a FIGURE 3-11. Pocket and Wall Features. Two pockets are separated by a wall. If the pockets are made by machining, then each pocket is a feature. If the pockets are made by molding or casting, then the walls are the features, and their shape must be tapered ("drafted") so that the part will come out of the mold easily. Note that the pocket feature implicitly defines half a wall, while the wall feature implicitly defines half a pocket. 44 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES Assembly features are regions of a part that are impor- tant for assembly purposes. Assembly features are made during fabrication, so they are, or correspond to, fabrica- tion features. However, not all fabrication features become assembly features. Furthermore, assembly features carry different design intent and information in their object data and methods. On the stapler, the holes at the right-hand ends of the handle, carrier, and anvil are assembly features, as is the entire outer surface of the pin. The inner rectan- gular pocket of the carrier is an assembly feature that joins to the outer surface assembly feature of the staples. Figure 3-12 contains some simple assembly features and the corresponding information that is of interest to us for assembly purposes. Associated with each feature is a transform that holds the location and orientation of the fea- ture's coordinate frame with respect to a coordinate frame on the part. Also noted is the assembly approach and fine motion direction for a compatible mating part. When a FIGURE 3-12. Three Simple Assembly Features. Each feature is accompanied by information showing where it is and the direction from which a compatible mating part would approach. feature has one and only one approach direction, the —z axis of the local frame is used as that direction. This corresponds to a feature hierarchy for robot assembly modeling recommended in [Kim and Wu]. In Chapter 4 we will introduce a library of mating features and demon- strate how they constrain the location of mating parts. Any geometric shape that is used for assembly can be included in an assembly feature library as long as the re- quired assembly information can be represented. We will learn in Chapter 10, for example, that chamfer width and friction coefficient are important to successful assembly of two parts. The clearance between mating parts is also important, but we cannot calculate that until we know the diameter of the mating feature. This we cannot know until we have a model that permits us to say which parts mate to which other parts using which features on those parts. Some authors define assembly features as related ge- ometric shapes on mating parts, rather than as single ge- ometric shapes on individual parts as we do here. Each definition has its uses and advantages. In this book we de- fine them as individuals, reserving the flexibility to mate a feature on one part to different possible features on an- other part. Our definition also permits us to define parts individually with their features if we wish. It also permits us to consider and define variations in feature shape and location individually and later combine their effects. 3.D.4. The Disappearing Fabrication Feature person must identify the features manually. Feature recog- nition presents challenges of its own which are beyond the scope of this book. The problem is the subject of ongoing research. 3.D.3. Assembly Features It is important to realize that many fabrication features are temporary and do not appear on the finished parts. Usually these temporary features serve to hold the parts accurately while other operations, such as machining or grinding, are performed. In fact it has been said that the vast majority of part features do not survive fabrication. Examples include (a) bosses on castings that interface to clamps on machine tool beds and (b) holes or V's punched into sheet metal parts to locate them for later stamping operations. Such features are typically cut or ground off after fabrication is complete. This fact is important because of its implications for variation analysis. We learned in Chapter 2 that the KCs for subassemblies and parts are subsets of the KCs for the assembly as a whole. In this chapter we are learning that the accuracy of part locations depends on the accu- racy with which features are made and placed relative to each other on parts. The accuracy with which we want the assembly features placed on parts can be declared by 3.E. MATHEMATICAL MODELS OF ASSEMBLIES 45 imposing tolerances on the transforms that relate them to part center coordinates or to each other. But those transforms cannot always be generated in their final form. During fabrication, the part must be held and cutting operations, for example, must be performed using transforms that are relative to the features by which the parts are held. Unless an assembly feature can be used to hold a part, the transforms that generate the final feature relationships will pass to and through the fabrication fea- tures on their way from one assembly feature to another. The resulting variation in the assembly feature locations thus depends on the variation in many additional trans- form chain segments that may no longer be on the part once the fabrication features have been removed. These additional transforms are usually added by pro- cess engineers. They have some freedom to choose a set of processes and fixturing features that hopefully will achieve the final desired assembly feature accuracy. In some cases, however, their choices are restricted and achievement of the final tolerances is in doubt. Worse is the situation in which the process engineers do not understand the de- sired function of the assembly because its KCs have not been declared or made available to them. In such situa- tions, they choose a convenient set of fixturing features sufficient to make the part easily or economically from their point of view regardless of its role in the final as- sembly. In such cases it is difficult to diagnose assem- bly problems. When the diagnosis is made, correcting the problem often requires a new fabrication process, fix- tures, tools, and measurement plan, a costly consequence. This is one of many reasons why designers of assemblies must keep in close touch with fabrication experts so that they can ensure achievement of the desired final variation limits. 3.E. MATHEMATICAL MODELS OF ASSEMBLIES We are now in a position to compare a variety of loca- tion models of assemblies. These are the world coordinate model, surface-constrained models, and connective mod- els. Each uses 4x4 transforms, but only one, the con- nective model, permits us to model assemblies as chains, specifically, chains of frames. This is the kind of model we need in order to capture KCs and their delivery paths. In a world coordinate model, assemblies are placed in a world coordinate frame by expressing each part's coordi- nate frame and (x, v, z) coordinate location in the world frame. The origin of the world frame of a car or airplane, for example, is normally placed in front of the vehicle a bit beneath the ground plane. This ensures that each part and point in each part has positive coordinates. Each part may be found by estimating its world coordinates and asking for a picture on the computer screen of parts near those coordinates. Figure 3-13 shows three parts located in a world coor- dinate frame. A model like that in Figure 3-13 is often made by draw- ing each part separately and then carefully placing them in the picture until the desired surfaces touch. A variety of modeling errors could occur. In Figure 3-13b, one such error is shown, namely that part B is in the wrong position. The result is that it interpenetrates part A, an event called interference. CAD systems can detect interferences. How- ever, the same or similar interference could be caused by either part A or part B being the wrong shape even if they are in the correct location, or by part A being in the wrong location. Because this kind of model does not represent the fact that part A should assemble to part B, these kinds of errors cannot be distinguished. FIGURE 3-13. An Assembly of Three Parts in a World Coordinate Frame, (a) The parts are in their nominal loca- tions, (b) Part B is in the wrong location. It interferes with part A and no longer touches part C. 3.E.1. World Coordinate Models 46 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES In a connective assembly model, the user joins parts by connecting them at their assembly features. This can be done by applying the methods of surface constraint to sur- faces on the features. Better, yet, the frames representing the features can be constrained to each other directly. Fig- ure 3-14 shows three parts joined this way. On the left is the nominal situation while on the right a varied situa- tion, caused by an error in placing an assembly feature on part B, is shown. Note that this error can be detected even if the parts are modeled only approximately, as long as the assembly features are modeled and placed on the parts accurately. 6 By contrast, detection of errors in a world co- ordinate model like that of Figure 3-13 requires that the parts be modeled accurately, since no distinction is made when modeling them between assembly feature surfaces and other surfaces. A connective assembly model can represent parts, as- sembly features, and surfaces individually and can tell the difference between them. This makes it possible to model different kinds of variation correctly and to distinguish in the model different sources of error. Consider the situation in Figure 3-15. Part A in this figure is joined to part B by making a surface on one part coincident with a surface on the other. In Figure 3-16, part B mates to an assembly feature "f" on part A. The left sides of each figure show ap- parently identical nominal situations, while the right sides show apparently identical varied situations. However, in Figure 3-15, we cannot tell the cause of the variation be- cause it does not contain a separate and coordinated group of surfaces called an assembly feature. All sources of error 6 Other errors that could cause interferences between parts can be detected only if the parts' shapes are modeled accurately. FIGURE 3-14. Three Parts Joined by a Connective As- sembly Model, (a) The nominal situation, (b) A feature on part B is misoriented and mispositioned, causing part C to be in the wrong position. FIGURE 3-15. Two Parts Constrained by Aligning Two Surfaces. The surfaces that are aligned are indicated by the dashed line. Left: The nominal situation. Right: The situation if the constraint surface on part A is misoriented. FIGURE 3-16. Two Parts Constrained by Joining Assem- bly Features. Left: The nominal situation. Right: The situa- tion if the feature "f" on part A is misoriented. must therefore be attributed to mislocated surfaces, and all surfaces are treated identically. In fact, in some CAD systems we cannot even tell if the error is on part A or on part B. In Figure 3-16, we can represent the fact that the entire feature on part A is misoriented because we have modeled the feature explicitly. Alternatively, we can rep- resent mismanufacture of the feature leading to its having one misoriented surface. In fact, every kind of error that could occur in practice can be represented individually and unambiguously. This is a huge advantage when analyzing variations. In a surface-constrained assembly model, the user joins items by establishing relationships between different sur- faces. Two planes can be made coincident, or two cylinders can be made coaxial, for example. Such operations are of- ten used to build up parts made of elementary surfaces and simpler objects. In some CAD systems, assemblies are built up the same way. The result is that the CAD model cannot distinguish parts and their subparts from assemblies. 3.E.3. Connective Models 3.E.2. Surface-Constrained Models 3.E. MATHEMATICAL MODELS OF ASSEMBLIES 47 The connective model of assembly defines a part as hav- ing a central coordinate frame plus one or more assembly features, each feature having its own frame. A transform relates each feature's location on the part to the part's cen- tral coordinate frame. Features like those in Figure 3-12 consist of a single geometric element. They can be placed on a part by defining a transform from part center coor- dinates to the feature frame. Alternatively, the transform to the feature frame can be directed from another feature frame. When two parts join, assembly features on one part are made to coincide with assembly features on the other part. This is done by defining a feature interface trans- form that relates the frame on one part's assembly feature to the frame on the other part's assembly feature ([Gerbino and Serrano]). If the axes of these two frames are identi- cal, then the interface transform is the identity transform. If not, then typically a reorientation assembly transform must be written to account for the difference between the axes of the two feature frames. 3.E.4.a. General Mathematical Connective Assembly Model In order to know which parts mate to which other parts and to calculate where the parts are in space as a result, we ex- ploit the fact that each feature has an associated transform which tells where the feature is on the part. "Assembly" of two parts then consists of putting the features' frames together according to some procedure, and then compos- ing several transforms to express the part-to-part relation- ships. These relationships are illustrated in Figure 3-17. To find or arrive at part B from part A, one starts at the coordinate frame of part A, follows the transform to the coordinate frame of its feature F A , then goes to the trans- form of the mating feature F B on part B, then follows the transform 7> B _ S = ^B~F B f rom mat feature to part B's coordinate frame. We can express this as The first transform on the right in Equation (3-33) T A _^locates part A's feature on part A relative to the part's coordinate frame. The second transform Tp A -p B is a feature interface transform that captures the relationship FIGURE 3-17. Mating Two Parts Using Assembly Fea- tures. The mathematics of composing transforms may be used to find the location of a mating part relative to another part if we know where the assembly features are on each part. Note that in part B, transform TF B -B equals T^_ FB . between the feature frames on the two parts. The third transform T^ FB is the inverse of the transform TB-F B that locates part B's feature with respect to part B's coor- dinate frame. The inverse appears because nominally the transform TB-F B carries us from part B's origin to part B's feature FB- The inverse is what we need to carry us from the feature to the part coordinate frame. This step completes the trip from A to B. The feature interface transform can express any of several constraints between features, such as making their frame origins coincide while making the frames' z axes point toward each other, making the frames' x-y planes coincide while orienting the axes in some specific way, off- sets between feature frames, and so on. Using this tech- nique, we can model an assembly as a chain of frames. The model relates the parts in the same way as the physi- cal parts relate to each other: Their assembly features are joined. In CAD systems it is common to locate parts with re- spect to each other by constraining certain surfaces to have specific relationships with each other. For example, two planes could be made to coincide, or two cylinders could be made coaxial. This method is less general than the one described here because it fails to capture the fact that the surfaces in question belong to particular parts or features. It also prevents us from using the frame information to calculate relative part locations and variations on them. 3.E.4. Building a Connective Model of an Assembly by Placing Feature Frames on Parts and Joining Parts Using Features Next Page [...]... for Assembly," 7990 ASME Design Automation Conference, vol DE 23 -1, pp 9-16, Chicago, September 1990, also ASME Journal of Mechanical Design, vol 115, pp 723 -734, 1993 [Denavit and Hartenberg] Denavit, J., and Hartenberg, R S., "A Kinematic Notation for Lower Pair Mechanisms Based on Matrices," Journal of Applied Mechanics, vol 22 , pp 21 522 1,1955 [Faux and Pratt] Faux, I D., and Pratt, M J., Computational... Outer diameter (Chamfered) pin Inner race face Plane 1 2 Thread Threaded pin Head Plane 1 2 3 Bore (Chamfered) bore Outer diameter (Chamfered) pin Inner race face Plane 1 2 Thread Threaded pin Head Plane 1 2 3 Bore (Chamfered) bore Outer diameter (Chamfered) pin Inner race face Plane 1 2 Thread Threaded pin Head Plane 1 2 3 Bore (Chamfered) bore 1 2 outer gimbal that contains as yet no bearings No liaison... "A FeatureBased Tolerancing Model for Functional Analysis in Assemblies of Rigid Parts," Proceedings of the 2nd CIRP ICME 20 00 Seminar, Capri, Italy, June 21 -23 , 20 00 [Kim and Wu] Wu, C H., and Kim, M G., "Modeling of PartMating Strategies for Automating Assembly Operations for Robots," IEEE Transactions on Systems, Man, and Cybernetics, vol 24 , no 7, pp 1065-1074, 1994 [Landau] Landau, B., "Developing... with the pin's centerline The next example, Figure 3 -22 , shows how to position a feature and orient it differently from part center coordinate axes The third example, Figure 3 -23 , shows how to build a part and position and orient a feature on it FIGURE 3 -22 Illustrating How to Position and Orient a Feature This example extends the example in Figure 3 -21 The transform equation may be read to say: "To get... pathological variations in part C's location Constraint is discussed in Chapter 4  52 3 MATHEMATICAL AND FEATURE MODELS OF ASSEMBLIES TABLE 3 -2 A Simple Part Model PART COORDINATE_FRAME CONTAINS_FEATURE CONTAINS_FEATURE MATES_TO_PART VIA_FEATURE HOW_TO_FIND (MATES_TO_PART) NAME_A X, Y, Z FEATURE_1 & TRANSFORM (Fl) FEATURE _2 & TRANSFORM (F2) NAME_X FEATURE_1 USEJTRANSFORM ( VIA_FEATURE) TABLE 3-3 A Simple Feature... Then, following the example in Figure 3 -25 , calculate the position of the hammer with respect to the crimper 2 Find the transform T 12 for part A, a rectangular block, shown in Figure 3-35 In the figures, frame 1 is the part's origin frame, while frame 2 is the frame on the assembly feature, a round peg FIGURE 3-36 Figure for Problem 3 FIGURE 3-35 Figure for Problem 2 3 Find T34 for part B shown in Figure... other They utilize the MATLAB functions for translation and rotation that appear in Table 3-1 These examples appear in Figure 3 -21 through Figure 3 -25 The first example, Figure 3 -21 , shows how to position a feature whose axes align with part center coordinate axes FIGURE 3 -20 Sketch of Frame Relationships for General Varied Connective Assembly Model This model augments the model in Figure 3-19 by the... generated simply by sequencing liaisons 2, 3, 4, and 5 or 11, 12, 13, and 14 Take the inner gimbal, for example It must be FIGURE 3 -28 Exploded View of Seeker Head 8 Missile steering algorithms are discussed in A H Bryson and L.Y.-C Ho, Applied Optimal Control, Ginn/Blaisdell, 1969, pp.154-155 9 These figures and the table were prepared by Alexander Edsall FIGURE 3 -29 Liaison Diagram for Seeker Head 54... represented in Figure 3 -20 is given in Chapter 6 The model as presented here appears to assume that only one feature can join part pairs and that assemblies can only consist of single linear chains of parts In fact, we can represent more complex assemblies, such as those created by joining a part to two previously joined parts by recruiting features on all three parts We will present assemblies of this... to frame C, go 3 units in X and 4 units in Z (in frame A coordinates) and then rotate 90° about frame A's (relocated) Y axis." Next, Figure 3 -24 , we build a second part and place a hole on it Last, Figure 3 -25 , we assemble the parts in Figure 3 -23 and Figure 3 -24 In Chapter 6, we will revisit these examples, inserting variation transforms and determining the varied position of point F 3.E.4.C Examples: . Retaining screw J Inner bearing K Retaining screw Feature 1 2 3 4 1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 1 2 1 2 3 1 2 1 2 3 1 2 1 2 3 1 2 Feature Name Bearing bore Trunnion bore Trunnion bore Bearing . axis." Next, Figure 3 -24 , we build a second part and place a hole on it. Last, Figure 3 -25 , we assemble the parts in Figure 3 -23 and Figure 3 -24 . In Chapter 6, we will. appear in Figure 3 -21 through Figure 3 -25 . The first example, Figure 3 -21 , shows how to position a feature whose axes align with part center coordinate axes. FIGURE 3 -20 . Sketch of Frame