Object-Oriented Representation of Electro-Mechanical Assemblies Using UML

ISO 10303-Part 44 [10] provides some limited assembly design representations that capture assembly structure and kinematic joint information during the design process. The assembly model establishes a neutral representation of assemblies of products, which are composed of sets of components. In this model, complete products are called “assemblies,” and the components of the lowest levels in the assemblies are called “parts.” The model focuses on the hierarchy of the product, and on the position and orientation between parts. The application fields of interest to the assembly model are the kinematic analysis of assemblies, the animation of assemblies for digital-mockup technologies, assembly/disassembly process planning from the viewpoint of CAD and CAM systems, and tolerance analysis and synthesis. The product (assembly) data representation is divided into three major schemas:

The Product structure schema defines a product in terms of a nested decomposition into its constituents. It also defines mechanisms for expressing the composition relationship. Product structures are modeled by directed acyclic graphs (DAG). Nodes represent product definitions (PD) and arcs represent relationships (PDRs). Refer to

Figure 1. Product definition usage (PDU) refers to part/whole or composed-of relationships. It is a subtype of PDR and defines two subtypes:

The entity ACU defines the following subtypes:

This product structure schema supports at least two forms of product structure:

The BOM data structure is used to represent the assembly aspects of a product structure. It includes only one instance of a single product definition for each kind of product that participates in the assembly structure. A part list structure is a specific form of a bill-of-material that can be represented by a tree (guided by the BOM's DAG structure). A bill-of-material structure may require a more general DAG. The parts list data structure individualizes the relationship between lower-level parts of the product structure with higher-level assemblies in which they are contained.

Product concept schema describes a product as a set of specifications derived from customer needs analysis. It is limited to specification description only; it does not represent the product evolution process from the conceptual to the detailed design phase.

Configuration management schema manages the structure of the configuration of assemblies and components.

In summary, a primary feature of the entities defined in this part of ISO 10303 is that it provides the data modeler with various types of product structure data structures (e.g., BOM, parts list, etc.,) using the primitive entities. This facilitates meaningful information exchange between users without resorting to costly data model mapping tasks. However, ISO 10303 does not adequately address the following:

The ISO working group (TC 184/SC4/WG12) (JNC proposal [2]) has proposed¹ several enhancements to STEP's assembly representation. In the WG12 proposal, the detailed geometric information not only for hierarchical relationships but for peer to peer relationships among component parts via an assembly feature is introduced. Geometric constraints among component parts at the detailed geometric element level are also enabled. The WG12 proposal introduces more information on component association and includes detailed information about appropriate assembly features involved in component association.²

Following ISO definitions, an assembly has been defined as a set of components and is represented in a similar hierarchical fashion. The assembly model includes:

feature associations and the assembly features describe the details of the component's associations.

The basic structure of the working group's proposed assembly model is shown in Figure 2. In the figure, boxes describe the components (assemblies, sub-assemblies, and parts), open circles the hierarchical relationships among the component definitions, and filled circles the associations between components. Note in the figure that the component association between sub-assembly3 and sub-assembly4 could be derived from the specified component association between part4 and part1, and hence may be redundant.

It should be noted that the JNC proposal does not cover configuration management of the assemblies and components. Although the proposal outlines the possible applications of the proposed assembly representation in four fields - kinematic analysis of assemblies, animation of assemblies, assembly/disassembly process planning, and tolerance analysis and synthesis - actual application methodologies are not identified or reported.

NIST has been actively involved in identifying and developing representational methodologies for the next generation of assembly related standards. The Open

Assembly Design Environment (OpenADE) [7] project seeks ways to assist designers with assembly considerations throughout the different phases of the complete product realization process, from conception to assembly analysis and final process plan development. Readers are encouraged to refer to reference [11] for a brief summary of several ongoing research activities at NIST regarding assembly related activities. In the following sections, we discuss several projects relevant to the proposed OAM.

This section introduces the OAM assembly model schema, represented as a UML class diagram. The assembly model's class diagram is made up of multiple sub-diagrams, each of which is represented by a Package. The concept of a UML Package is that of a collection of multiple classes [8] that can be used as a namespace. The following section defines the main schema of the assembly model which describes the relationships between Assembly, Artifact, AssemblyAssociation, Part, and OAMFeature. In the succeeding sections we describe the class diagrams of AssemblyAssociation, ArtifactAssociation, OAMFeature, AssemblyFeature, Tolerance, and AssemblyFeatureAssociationRepresentation. The lower level class hierarchies of these classes are also explained. The instance diagrams are provided in Section 4. These diagrams will further help illustrate the concepts presented here.

The main assembly model schema is shown in Figure 4. The model incorporates information about assembly relationship and component composition (part-of). The assembly relationship is represented by a class named AssemblyAssociation. This class is described in detail in the package AssemblyAssociation, shown at the bottom of the diagram and described in Section 3.3

The component composition of the assembly is modeled using the part-of relationship. An Assembly is decomposed into sub-assemblies and parts. A Part is the lowest level component which cannot be further decomposed. The diagram shows that an assembly or a subassembly is made up of at least two parts.

Each assembly component, whether subassembly or part, is made up of one or more features, represented in the model by the OAMFeature class, where OAM refers to Open Assembly Model. The details of the OAMFeature definition are described in the package OAMFeature. The Assembly and Part classes are sub-classes of the NIST-CPM Artifact class, inheriting its definition. OAMFeature is a sub-class of the NIST-CPM Feature class.

The package AssemblyAssociation, shown in Figure 5, represents the component assembly relationship of an assembly. It is the aggregation of one or more Artifact Associations. The class ArtifactAssociation is defined in the package ArtifactAssociation.

Figure 6 shows the package ArtifactAssociation which describes the assembly relationships between artifacts. An ArtifactAssociation class represents the assembly relationship between one or more artifacts. For most cases, the relationship involves two or more artifacts. In some cases, however, it may involve only one artifact to represent a special situation. Such a case may occur when an artifact is to be fixed in space for anchoring the entire assembly with respect to the ground. It can also occur when kinematic information between an artifact at an input point and the ground is to be captured. Such cases can be regarded as relationships between the ground and an artifact. Hence, we allow the artifact association with one artifact associated in these special cases. In Figure 6, this situation is denoted by the multiplicity $1..*$ at the Artifact end of the association with ArtifactAssociation. The detailed information of ArtifactAssociation is defined using assembly features, assembly feature associations, and assembly feature association representations, which will be described later in Section 3.6.

The ArtifactAssociation is further specialized into the following classes: PositionOrientation, RelativeMotion and Connection.

PositionOrientation represents the relative positions and orientation between two or more artifacts which are not physically connected to each other. This entity is used to describe constraints on the relative position and orientation of two or more artifacts.

RelativeMotion represents the relative motions between two or more artifacts which are not physically connected to each other directly. This entity is used to describe the constraints on the relative motions between two or more artifacts. For instance, the relative motion of a robot hand against the base of a robot can be represented by this class. In this case, the robot hand is not connected directly to the base.

Connection represents the connection between artifacts which are physically connected with each other. As Figure 6 shows, the Connection is further specialized as FixedConnection, MovableConnection or IntermittentConnection.

FixedConnection represents a connection in which the participating artifacts are physically connected and fixed. It is used to describe the type and/or properties of fixed joints. For example, attachment (fastening) with screw, pin, or rivet, bonding such as welding, adhering, brazing, or soldering, and fitting (clearance fit, tight or interference fit) are examples of fixed connections.

MovableConnection represents the connection in which the participating artifacts are physically connected and movable with respect to one another. It is used to describe the type and/or properties of kinematic joints. Typical examples of movable connections are kinematic pairs such as shaft-bearing joints or gear joints.

IntermittentConnection represents the connection in which the participating artifacts are physically connected only intermittently. It is used to describe the type and/or properties of intermittently-connected components. For example, limit switches, contacts using bimetal, or pressure valves with springs are examples of intermittent connection.

The class OAMFeature is a sub-class of the Feature class defined in NIST-CPM. It inherits the function and behavior information from Feature. OAMFeature has tolerance information which is represented by the class Tolerance. OAMFeature has two sub-classes: AssemblyFeature and CompositeFeature.

CompositeFeature represents a composite feature that can be decomposed into multiple simple features.

AssemblyFeature specifies the relationships between a pair of assembled components (e.g., geometrical entities such as faces, edges or vertices of any parts of entities). It is a portion of a constituent and is used for defining the connectivity relationship between constituents of an assembly model.

Figure 8 shows the relationship between classes that are related to assembly features. The class AssemblyFeatureAssociation refers to the assembly relationship between one or more assembly features. This assembly relationship is represented by the class AssemblyFeatureAssociationRepresentation. The diagram also shows that the ArtifactAssociation is the aggregation of AssemblyFeatureAssociation. Detailed description of these classes are given below.

AssemblyFeature, a sub-class of OAMFeature, is defined to represent assembly features. Assembly features are a collection of geometric entities of artifacts. They may be partial shape elements of any artifact. For example, consider a shaft-bearing connection. A bearing's hole and a shaft's cylinder can be viewed as the assembly features that describe the physical connection between the bearing and the shaft. We can also think of geometric elements such as planes, screws and nuts, spheres, cones, and toruses as assembly features.

AssemblyFeatureAssociation represents the association between mating assembly features through which relevant artifacts are associated. Since associated artifacts can have multiple feature-level associations when assembled, one artifact association may have several assembly features associations at the same time. That is, an artifact association is the aggregation of assembly feature associations as shown in Figure 8. This relationship is also identified by the multiplicity symbol $1..*$ (see Figure 8). Any assembly feature association relates in general to two or more assembly features. However, as in the special case mentioned earlier (see Section 3.4) where an artifact association involves only one artifact, it may involve only one assembly feature when the relevant artifact association has only one artifact. Hence, the AssemblyFeature end of the association with AssemblyFeatureAssociation in Figure 8 is denoted by the multiplicity symbol $1..*$, not by the multiplicity symbol $2..*$.

The specific contents of any assembly feature association is described by the AssemblyFeatureAssociationRepresentation (see Figure 9). The assembly feature association representation is an aggregation of parametric assembly constraints, a kinematic pair, and/or a relative motion between assembly features as shown in Figure 9.

ParametricAssemblyConstraint represents the parametric assembly constraints of assembly feature associations. It is used to specify explicit geometric constraints between artifacts of an assembled product. The parametric assembly constraints are intended to control the position and orientation of artifacts in an assembly. Parametric assembly constraints are defined in ISO 10303 Part 108 (under ballot) and a brief description is given in the next section.

KinematicPair defines the kinematic constraints between two adjacent artifacts (links) at a joint. The kinematic structure schema in ISO 10303 Part 105 defines the kinematic structure of a mechanical product in terms of links, pairs, and joints. The kinematic pair represents the geometric aspects of the kinematic constraints of motion between two assembled components. The description of the kinematic structure is given later in Section 3.9.

KinematicPath represents the relative motion between artifacts. The kinematic motion schema in ISO 10303 Part 105 defines kinematic motion. It is also used to represent the relative motion between artifacts. Details of this are provided in Section 3.10.

In Section 3.8, detailed descriptions of the parametric assembly constraint, kinematic pair, and kinematic path are provided. These three packages defined in the ISO standards are of considerable complexity, and thus the descriptions below are not exhaustive. Only the necessary portions are modeled using UML to explain how they can be utilized to represent electro-mechanical assemblies in conjunction with the proposed UML model. Note that we have introduced some new classes and used different class names from the original ones defined in ISO standards for increased clarity.

The classes in Figure 10 represent parametric assembly constraints. The parent class of Parametricassemblyconstraint defines the assembly constraints (position and orientation) between two parts in an assembly. From this class we specialize specific types of assembly constraints, as shown in Figure 10. The constrained geometric elements can be a point, a line, a circle, a plane, a cylindrical surface, a conical surface or a spherical surface of the constrained parts. Each type of assembly constraints applies to specific types of geometric elements. One of the two elements may be the reference element when applying the constraint.

Parallel specifies that two geometric elements are mutually parallel. ParallelWithDimension constrains two parallel elements with a distance. SurfaceDistanceWithDimension specifies the distance between two surfaces.

The angle between two elements is constrained by AngleWithDimension. Perpendicular constrains two elements to be mutually perpendicular. Incidence represents that one of two elements is included in the other when it is considered as a point set. Coaxial means that the axes of the two elements are coincident. Tangent specifies that two elements are tangent to each other. Finally, FixedComponent is provided to fix the position and orientation of one part thus anchoring its own assembly in space. This constraint can be viewed as an exceptional case as it involves only one part; implicitly, the other part is the ground (i.e., the world coordinate system).

KinematicPair defines the kinematic constraints between two adjacent artifacts (links). Figure 11 shows the relationship between classes that are used to represent the kinematic constraints of kinematic pairs. Specific types of kinematic pairs are specialized from KinematicPair, PairValue, and PairRange.

KinematicPair defines the kinematic constraints between two adjacent links (artifacts) at a joint. PairRange specifies the allowable configuration range of the two links in the form of upper and lower bounds. PairValue specifies the current configuration (value) of the two links between the two bounds. PairFrame represents a coordinate system attached to a link. A kinematic pair needs two coordinate systems to describe its kinematic behavior. The two coordinate systems are attached to the two relevant links, respectively. Thus, the multiplicity at the PairFrame end is shown as 2 in Figure 11.

As mentioned above, specific classes of kinematic pairs are derived from the classes shown in Figure 11. Figure 12 shows the derived types of kinematic pairs. The kinematic pairs cover the following types: revolute, prismatic, screw, cylindrical, spherical, universal, planar, gear, rack and pinion, unconstrained, fully constrained, point on surface, sliding surface, rolling surface, point on planar curve, sliding curve, rolling curve.

Each kinematic pair has two specific classes derived from PairValue and PairRange respectively, to represent the current configuration (value) and the allowable range of the pair variable. For example, the revolute pair class diagram in Figure 13 shows the derived classes RevolutePair, RevolutePairValue, RevolutePairRange, and their associations.

The classes RevolutePairValue and RevolutePairRange are specialized, respectively, from PairValue and PairRange. Other types of kinematic pairs also have specific type of derived classes defined for their own purpose.

KinematicPath, shown in Figure 14, provides the description of kinematic motion. It is the aggregation of path elements along which the motion is to take place. A path element can specify different types of paths, as explained below. Note that since the kinematic path is composed of a set of path elements, it can describe a composite path as well as a simple path.

PathElement is a path segment with two path nodes, which represent the "from" node and "to" node, respectively. PathNode is used to define the start and end locations of a path. At each path node, the position and rotation of a frame along the "path" need to be defined. This linear transformation is defined by Transformation, which can be expressed by a $4\; \backslash times\; 4$ homogeneous transformation matrix.

The path element is further classified into different types of motion according to the intermediate motion characteristics between the "from" and "to" nodes, as shown in Figure 14. PointToPointPath allows free motion between the two nodes; it does not have a prescribed interpolation and leaves the determination of the intermediate motion to specific applications. LinearPath incorporates piecewise linear interpolation between the beginning and end nodes of the path element. CircularPath describes a circular arc path which passes through the beginning and end points of the path element, and also a third point (via point) supplied separately. CurveBasedPath describes the path using a predefined trajectory curve.

Tolerancing is a critical issue in the design of electro-mechanical assemblies. Tolerancing includes both tolerance analysis and tolerance synthesis. In the context of electro-mechanical assembly design, tolerance analysis refers to evaluating the effect of variations of individual part or subassembly dimensions on designated dimensions or functions of the resulting assembly. Tolerance synthesis refers to allocation of tolerances to individual parts or sub-assemblies based on tolerance or functional requirements on the assembly. Tolerance design is the process of deriving a description of geometric tolerance specifications for a product from a given set of desired properties of the product. Existing approaches to tolerance analysis and synthesis entail detailed knowledge of the geometry of the assemblies and are mostly applicable only during advanced stages of design, leading to a less than optimal design. During the design of an assembly, both the assembly structure and the associated tolerance information evolve continuously; significant gains can thus be achieved by effectively using this information to influence the design of that assembly. Any proactive approach to assembly or tolerance analysis in the early design stages will involve making decisions with incomplete information models. In order to carry out early tolerance synthesis and analysis in the conceptual product design stage, we include function, tolerance, and behavior information in the assembly model; this will allow analysis and synthesis of tolerances even with the incomplete data set. In order to achieve this we define a class structure for tolerance specification and we describe this in Figure 15 .

DimensionalTolerance typically controls the variability of linear dimensions that describe location, size and angle, also known as tolerancing of perfect form. This class is included in the model to accommodate the ISO tolerance standard.

GeometricTolerance is the general term applied to the category of tolerances used to control shape, position, and runout. It enables tolerances to be placed on attributes of features, where a feature is one or more ‘pieces of part surface’; feature attributes include size (for certain features), position (certain features), form (flatness, cylindricity, etc.), and relationship (e.g. perpendicular-to).

FormTolerance is applicable to single features or elements of single features.

ProfileTolerance is applicable to profiles. A profile is the outline of an object in a given plane. Profiles are formed by projecting a three-dimensional figure onto a plane or by taking a cross section through the figure.

RunoutTolerance is a composite tolerance used to control the functional relationship of one or more features of a part to a datum axis.

OrientationTolerance is applicable to feature orientations. Angularity, parallelism, perpendicularity, and in some cases, profile are orientation tolerances applicable to two or more features. These tolerances control the orientation of a feature with respect to another feature.

LocationTolerance deals with the position, concentricity, and symmetry used to control the following relationships:

A planetary gear system is used to illustrate our model. The solid model of the planetary gear system is shown in Figure 22.

The planetary gear system consists of many components. Figure 23 shows the exploded view of the above solid model.

The list of all the components of the planetary gear system shown in Figure 23 is given in Table 1.

Before proceeding further with our case-study example, we first need to define an assembly hierarchy for the planetary gear system. The planetary gear system is assumed to be composed of two parts, namely, the input-housing and the sungear, and three sub-assemblies, as shown in Figure 24. The three sub-assemblies are: (1) the output end assembly that contains the two bearings, the washer, and the output housing; (2) the ring gear assembly that consists of the ring gear and ring-gear pin; and (3) the planet gear holder assembly that consists of the three planet gears and the planet carrier assembly, which further decomposes into the output shaft and the three planet-gear pins.

. Notice that the hierarchy in Figure 24 is introduced to verify and demonstrate the proposed UML assembly model. The sequence of part assembly descriptions does not imply the actual assembly sequence. The assembly sequencing task is outside the scope of this report.

The hierarchical relationships between the components of the planetary gear system can be represented as an instance diagram as shown in Figure 25. The elements are underlined to show that they are instances. The names take the form of "instance name:class name". The root node is the entire assembly, the interior nodes are sub-assemblies, and the leaf nodes are component parts.

Information besides the hierarchical relationship between artifacts is provided by an instance of AssemblyAssociation, which is an aggregation of instances of ArtifactAssociation. The artifact associations hold relational information such as mating conditions, kinematic pairs, and associations between assembly features. For the gear head assembly shown in Figure 25, a graph of artifact associations is shown in Figure 26. The dotted lines indicate the artifact associations and the solid lines portray the hierarchical assembly relationships. The details of the artifact associations shown in Figure 26 are discussed in the following sections. First we explain the assembly relationship for each subassembly, and then we describe the entire assembly.

Figure 46 shows the kinematic diagram of the planetary gear system, where only the parts (kinematic links) involved in motion (or force) transmission are illustrated. The input motion is transmitted into the sungear through the input shaft that is part of the sungear. Three planet gears are meshed with the sungear, and with the fixed ring gear. The planet carrier assembly (i.e. output shaft + planet-gear pins) holds the three planet gears and rotates with the output shaft that is part of the planet carrier. The output shaft is connected with a bearing. For the input shaft, we assume that it is sustained by an unknown support (or ground).

Table 11 shows the kinematic pairs and the associated parts that are identified from the planetary gear system. For convenience, numbers are used to distinguish the three planet gears and the kinematic pairs of the same type. As shown in Table 11, two types of kinematic pairs (GearPair and RevolutePair) are used in the planetary gear system.

Before proceeding further with the foregoing example, we should first assign necessary frames to each link (part) of the gear system. In general, two coordinate systems, or frames, are needed to describe the kinematic behavior of any kinematic pair, each attached to a link of the pair. Considering that a binary link is associated with two kinematic pairs (one with the preceding link and the other with the following link), there are two frames associated with a link. If any link has more kinematic pairs than two, as many frames are necessary as the number of involved kinematic pairs.

The assignment of frames is actually arbitrary and depends on individual applications. In this example, assume that the necessary frames are assigned as shown in Figure 47. Figure 47(a) and (b) are depicted from positive $z\_1$ and $z\_4$ axes, respectively. Each view shows each of the two kinematic loops of the mechanism, and includes the links involved in each loop. Since the mechanisms in the planetary gear system are planar mechanisms, and the coordinate systems need not be attached directly to each link, the coordinate systems can be thought of as being on any one plane in this example.

Table 12 shows the frames and the pair variables of each kinematic pair of the planetary gear system. The frame $\backslash \{x\_i,\; y\_i,\; z\_i\backslash \}$ is attached to the preceding link of the pair, and $\backslash \{u\_j,\; v\_j,\; w\_j\backslash \}$ to the following link. A pair variable is a variable parameter of the SU-parameters (see

Appendix A and Reference [18]) defining a kinematic pair. Except the pair variable, other parameters have fixed values. For a revolute pair, the pair variable is defined by the angle between $x\_i$ and $u\_i$. In the case of a gear pair, the pair variable is given by the angle from $x\_i$ to a common perpendicular between the two frames involved. In Table 12, only one planet gear (Planet Gear 1) and its associated pairs are assigned the necessary coordinate systems for brevity as the three planetary gear are configured symmetrically in space and they play a similar role in the mechanism. The frames associated with the other planet gears are therefore not presented specifically and just denoted as ‘...’ in Table 12. In fact, they are not necessary for the analysis of the gear system due to the symmetry.

The total number of associated frames on a link depends on the type and usage of the link, as shown in Table 13. The frame $\backslash \{u\_i\; v\_i\; w\_i\backslash \}$ is related with the preceding pair of a link (in conjunction with the previous link), and $\backslash \{x\_j\; y\_j\; z\_j\backslash \}$ with the following pair (in conjunction with the following link). The unknown support and bearing in Table 13 contains only one frame $\backslash \{x\_i\; y\_i\; z\_i\backslash \}$. These parts are assumed grounded (fixed) links. Hence, they have no associated links preceding them, and thus frame $\backslash \{u\_i\; v\_i\; w\_i\backslash \}$ is not necessary for them. The sun gear, ring gear, and planet carrier will in fact have more frames since they are connected to three planet gears. That is, they are quaternary links and will have four frames in all. However, their configuration in this example is symmetric and thus the frames of the other pairs are left out for brevity. In the case of the ring gear, there are three frames listed. The frame pair $\backslash \{u\_3\; v\_3\; w\_3\backslash \}$ and $\backslash \{x\_1\; y\_1\; z\_1\backslash \}$ is used in the first loop (Figure 47(a)), and frame pair $\backslash \{u\_3\; v\_3\; w\_3\backslash \}$ and $\backslash \{x\_4\; y\_4\; z\_4\backslash \}$ in the second loop (Figure 47(b)). The frames $\backslash \{x\_1y\_1z\_1\backslash \}$ and $\backslash \{x\_4y\_4z\_4\backslash \}$ represent the same ground frames to which the ring gear is fixed. The reason that two different frames are introduced is for convenience in assigning the frames. We can use the same frame for the ground, if necessary. The planet gears are ternary links. They are respectively connected to sungear, ring gear, and planet carrier, respectively. Thus, there exist three frames for a planet gear, as shown in Table 13.

In the previous subsection 4.3.5, the kinematic pairs were illustrated in the instance diagrams of the planetary gear assembly. To demonstrate the kinematic relationships in more detail, another instance diagram is depicted in Figure 48. When the instances are of the same class, only the first occurrence of them is denoted with the class name. This diagram includes all the parts or assemblies, and other information that are related to the kinematics of the planetary gear system. Notice in Figure 48 that the three planet gear pins and the output shaft are attached by fixed connections (see Section 4.3.3). Thus they are considered to be one rigid body (we name it planet carrier) for the purpose of kinematic analysis in this section.

The first row in Figure 48 shows the individual parts or assemblies (instances of Part or Assembly) that take part in the kinematic relationship. Each part (or assembly) has its own assembly features that are defined to represent the shape aspects of the kinematic pairs. A cylinder/hole-type pair can be defined as assembly features for a revolute pair, and a teeth/teeth-type pair for a gear pair. For example, there are four assembly features for the sungear (ss: cylinder type; $t1$, $t2$, and $t3$: teeth type). The cylinder type is used to represent the revolute pair with the unknown support, and the three teeth types are respectively used to represent the gear pairs with the three planet gears around the sungear.

The assembly feature association captures the topological relationship of kinematic pairs

(AFAs in Figure 48). First, it associates two (or more when necessary) assembly features that take part in an assembly relationship (specifically, kinematic relationship). By consulting the ownership of an assembly feature, it can finally specify the links (parts) that take part in the relationship. For example, the second instance of assembly feature association in Figure 48 relates assembly features $t1$ and $t7$ to represent the gear-pair relationship between the sun gear and the mating planet gear (planetGear1). Second, it refers to an assembly feature association representation (AFAR in Figure 48), which describes the details of the relationship. In this kinematic example, the relationship to be represented is a kinematic pair (GearPair or RevolutePair). For instance, the second instance of assembly feature association in Figure 48 is associated with assembly feature association representation, by which the kinematic pair gp1:GearPair is associated (see Figure 48).

The kinematic pairs (instances of GearPair and RevolutePair in Figure 48) contain their specific kinematic information (constraints of the pair) to describe their own behavior. For example, the instance diagram about rp1:RevolutePair (unknown support – sun gear pair) in Figure 48 is illustrated in detail in Figure 49. An instance of RevolutePairRange specifies the lower and upper bounds of the rp1:RevolutePair. The current value of the pair variable $\backslash theta\_1$ is specified by an instance of RevolutePairValue. The two frames required for this revolute pair are provided by two instances of PairFrame. Typically, the pair variable $\backslash theta\_1$ is measured by the angle from $x\_1$ to $u\_1$, as shown in Figure 47(a).

As another example, the instance diagram for gp1:GearPair is shown in Figure 50. The gp1:GearPair contains some key geometric information of the gears such as radii, gear ratio, bevel and helical angles (number 1 denotes the first link, and number 2 the second). Such information is necessary when deriving the kinematic equation of the gear pair. An instance of GearPairRange sets limits on the range of gears' rotation angle. The current rotation angle of the first gear is specified by $\backslash theta\_2$ of an instance of GearPairValue; the rotation angle of the second gear can be derived from the rotation angle of the first gear by multiplying it with the gear ratio.

Instance diagrams for the other kinematic pairs can also be drawn in a similar manner. As mentioned earlier, gp2 and gp3 will have the same structure as gp1 since they are symmetrically configured with gp1. The gear-pair instances gp4, gp5, and gp6 will also have similar instance diagram to that of gp1 (see Figure 50), except that they are internal gearing: their gear ratios are negative values. For other revolute pairs rp2, rp3, rp4, and rp5, instance diagram similar to that of rp1 (see Figure 49) can be depicted.

Tolerance chains are important in a product assembly. The chains are used in the analysis for assembleability and functions such as clearance, tightness, smooth motion, and flow rate. This subsection describes the modeling of tolerance chains. Two specific tolerance chains are found in the planetary gear system assembly. The first one is in the axial direction. Three tolerances are in this chain. They are all defined on the part of the sungear and the input shaft. This chain of the tolerances determines the magnitude of clearance or interference between the Surface B of the sungear and the Surface A on the planetary gear holder, as shown in Figure 51. (Note that Surface B is the end surface of the sungear). Figure 52 shows the chain of the three dimensions and the associated tolerances: $12.95\; \backslash text\{\; mm\}\; \backslash pm\; 0.01\; \backslash text\{\; mm\}$ on the sungear, $13.59\; \backslash text\{\; mm\}\; \backslash pm\; 0.03\; \backslash text\{\; mm\}$ on the shank, and $10.8\; \backslash text\{\; mm\}\; \backslash pm\; 0.1\; \backslash text\{\; mm\}$ on the connector. The total allowable variation of the sungear's length is, therefore, $0.14\; \backslash text\{\; mm\}$. These three dimensional tolerances can be represented using objects derived from the DimensionalTolerance class described in Section 3.11. Note that not all the dimensions of this part are shown in Figure 52. Only those dimensions related to the tolerance chain are shown.

Figure 52 shows an engineering drawing of the sungear. The tolerance instance diagram (Figure 53) shows how the tolerances are represented using UML as discussed in Section 3.11. Figure 53 shows an integrated view of the part, its tolerances, and the assembly features.

In addition to dimensional tolerances, geometric tolerances are also used to control three features of this part. The end surface (Surface B) has two tolerances: a perpendicularity tolerance and a flatness tolerance. Both are for the control of geometric variation of Surface B. The perpendicularity tolerance is used to control the wobbling motion of the surface while the gear is rotating around the datum axis (Datum A1). The flatness tolerance is used to control the proper clearance between Surface B and Surface A on the planetary gear holder. A cylindricity tolerance is applied to the surface of the shank. It is used to control the form variation of the surface. These three tolerances are form tolerances. A fourth tolerance is the coaxiality tolerance. The coaxiality of the connector is controlled by a position tolerance. The detail of coaxiality tolerancing can be found in the dimensioning and tolerancing standard [19]. The three form tolerances are instances of PerpendicularityTolerance, FlatnessTolerance, and CylindricityTolerance classes respectively, described in Section 3.11.1. The position tolerance is an instance of the PositionTolerance class described in Section 3.11.3.

The second tolerance chain is in the radial direction of the planetary gear system. This chain includes three parts: the sun gear, the planetary gears, and the ring gear. This chain starts from the axis of the sun gear. The pitch diameter of the gear is $22.23\; \backslash text\{\; mm\}\; \backslash pm\; 0.03\; \backslash text\{\; mm\}$, as shown in Figure 52.

The second set next to the sun gear in the tolerance chain is the set of three planetary gears. Figure 56 shows the engineering drawing of a planetary gear. The tolerance instance diagram (Figure 55) shows how the tolerances are represented using UML. Figure 57 shows an integrated view of the part, its tolerances, and the assembly features. Table 14 tabulates the features, tolerances, and control frames with the artifact association.

The pitch diameter of the planetary gear is $10.16\; \backslash text\{\; mm\}\; \backslash pm\; 0.01\; \backslash text\{\; mm\}$. The planetary gears have the same tolerance and are assembled with the sungear in the radial direction. There is a form tolerance applied to the planetary gear. A cylindricity tolerance of $0.02\; \backslash text\{\; mm\}$ controls the inner cylindrical surface of the gear. It is used to ensure a tight fit of the gear and the pin, as shown in Figure 51.

(The features postfixed with * are the assembly features that have been used to represent the assembly relationships in previous sections. Their associated artifacts are also shown in the table.)

(Planet gears 2, 3 also have similar tolerances to those of planet gear 1.)

The third part in the chain is the ring gear, which is assembled with the planetary gears. The pitch diameter of the ring gear is 42.62 mm $\backslash pm\; 0.01$ mm. Figure 58 shows the engineering design of the ring gear. The tolerance instance diagram (Figure 59) shows how the tolerances are represented using the UML model discussed in Section 3.11.

Figure 60 shows an integrated view of the part, its tolerances, and the assembly features. Table 16 tabulates the features, tolerances, and control frames with the artifact association.

With the dimensional tolerance on the pitch circle of a planetary gear shown in Figure 56 and the dimensional tolerance on the pitch diameter of the sun gear, tolerance analysis can be performed in this chain. Examples are clearance or interference of this set of gears. In addition to dimensional tolerances, there is a geometric tolerance on the part. A parallelism tolerance is applied to a surface of the shape of a ring where the ring gear and the output housing meet. Within a tolerance of 0.05 mm, the surface has to be parallel to Datum A3, which is established from the bottom surface of the ring gear. The bottom surface meets with the input housing. This tolerance is to ensure that in the input housing, the ring gear and the output housing are properly aligned along the rotating axes of the gears.

The chain starts from the axis of the planetary gear system, the same as the axis of the sun gear. The radius of the sun gear is 11.125 mm and the radial tolerance is, therefore, $\backslash pm\; 0.015$ mm. The diameter of a planetary gear is 10.16 mm, and its dimensional tolerance is $\backslash pm\; 0.01$ mm. In the case that both gears are first assembled together, the combined nominal dimension is 21.285 mm (11.125 mm + 10.16 mm), and the combined tolerance is $\backslash pm\; 0.025$ mm (0.015 mm + 0.01 mm). The nominal radius of the ring gear is 21.31 mm, and the radial tolerance is $\backslash pm\; 0.01$ mm.

When all three types of gears are assembled, they can be in an extreme tight situation, an extreme loose situation, or any situation in between. In the extreme tight situation, the ring gear radius will be 21.30 mm (21.31 mm – 0.01 mm), and the stack-up is 21.31 mm (21.285 mm + 0.025 mm). In this case, the interference is 0.01 mm. However, in the extreme loose situation, the ring gear radius will be 21.32 mm (21.31 mm + 0.01 mm), and the stack-up will be 21.26 mm (21.285 mm – 0.025 mm). There will be a 0.055 mm clearance among these three kinds of gears.

This section describes geometric tolerances in the design. The geometric tolerances are on the three parts: the planetary gear holder, the output housing, and the input housing. Figure 62, Figure 65, and Figure 68 show the drawings of these three parts. Note that not all the dimensions are shown in the drawings. The shown dimensions are primarily applied to assembly and tolerances. Others are shown for showing the proportion of a feature relative to the entire part. Figure 62 shows the engineering drawing of the planetary gear holder. The tolerance instance diagram (Figure 63) shows how the tolerances are represented using the UML model discussed in Section 3.11. Figure 64 shows an integrated view of the part, its tolerances, and the assembly features. Table 17 tabulates the features, tolerances, and control frames with the artifact association.

All the dimensions in this design are with a default tolerance of 0.05 mm, represented by objects derived from the dimensional tolerance class. Datum axis A is the rotating axis and can be presented using an object derived from the Datum class defined in Section 3.11. In addition to the datum, three geometric tolerances exist on the drawing. Two geometric tolerances are applied to the end surface adjacent to planetary gears. Since the surface provides support to hold gears, both the instances of PerpendicularityTolerance and FlatnessTolerance are applied. The perpendicular tolerance makes a reference to Datum A. The third tolerance is an instance of TotalRunoutTolerance. It is applied to the output shaft with the tolerance value of 0.1 mm. The runout is with respect to Datum A, the shaft/gear axis, to minimize wobbling motion while the shaft is rotating. A surface profile tolerance is applied to the curved surface of the key way. It is used to ensure the key and the shaft that are tightly fit together.

Figure 65 shows the engineering drawing of the input housing. Figure 66 is the tolerance instance diagram, which shows how the tolerances are represented using the UML model discussed in Section 3.11. Figure 67 shows an integrated view of the part, its tolerances, and the assembly features. Table 18 tabulates the features, tolerances, and control frames with the artifact association.

The dimension of the 25.53 mm hole has a statistical tolerance. The statistical tolerance can be represented by an object instantiated from the StatisticalControl class in Section 3.11. The mean and standard deviation can be captured using the class described in Section 3.11.6. The variation of the diameter of the counter-bored hole is also statistically controlled, but the depth of the hole is not because it is not a critical dimension. Datum A to be established from the bottom surface is shown in the front view. Datums B and C are shown in the bottom view of the input housing. One position tolerance is applied to a pattern of four holes, each with a counter-bored hole. The position is measured in a reference to a coordinate system established by Datums A, B, and C. The tolerance is in the condition of the regardless of feature size. Another position tolerance is similarly applied to another pattern of four 5.39 mm holes. The positions of these holes are critical in assembly because the input housing and the ring gear have to be properly aligned and, then, assembled

Figure 68 shows the engineering drawing of the output housing. Figure 69 has the tolerance instance diagram, which shows how the tolerances are represented using the UML model discussed in Section 3.11. Figure 70 shows an integrated view of the part, its tolerances, and the assembly features. Table 19 tabulates the features, tolerances, and control frames with the artifact association.

In the cross-sectional view in Figure 68, the hole with a diameter of the 28.58 mm has a statistical control on the dimension and another statistical control in the position tolerance. Because this hole will contain bearings, both dimension and position have to be toleranced to ensure a tight fit. The depth of the hole is not controlled since it is not critical to the fit between this part and bearings. Datum A to be established from the bottom surface is shown in the front view. Datums B and C are shown in the bottom view of the input housing. Also, the direction and location of the cross-section is shown in this view. One position tolerance is applied to a pattern of four holes, each with the diameter of 5.38 mm. The position is measured in a reference to a coordinate system established by Datums A, B, and C. The tolerance is at the maximum material condition of the hole. Another position tolerance is applied to another pattern of four 7.19 mm holes. The tolerance is also at the maximum condition of the hole. This tolerance is statistically controlled because the positions of these holes are critical in assembly. The output housing and the ring gear have to be properly aligned and assembled. For alignment, there are two pin holes to assist the alignment. Their position is also similarly toleranced with an instance of StatisticalControl.

Both dimensional and geometric tolerances are described in these two subsections. The tolerances can be captured by objects instantiated from the appropriate tolerance classes, specified in Section 3.11.