Functional Part Families and Design Change for Mechanical Assemblies

We consider two questions related to functional part families: a) how to characterize function in a computational framework, and b) how does the structure-to-function model generalize when the design changes, e.g. by changing the set of design variables? For the first, we observe that function is defined on the space of behaviours of the part, whereas structure is defined in the space of design parameters. For mechanical assemblies, as the design parameters change, their effect on the motion parameters can be complex, and cannot be automated in full generality. Thus, the mapping from structure-to-function involves considerable designer knowledge. For computational purposes, we quantify this function by defining part-family-specific Configuration Space (C-space) constructions, and also a metric that operates on these C-spaces to define each function. When the design is changed, either by changing the design space (structure), or by the user expectation (function), can existing design knowledge from the earlier part family migrate to the new product family? We make a start towards exploring how this knowledge can be modified when the part family is evolved, for example by introducing additional design variables, or by changing functional roles. Using examples from several lock designs, we present a small prototype for this process of modeling function and design change, implemented on a commercial CAD engine. ∗Address all correspondence to this author. 1 Function in Design Process Functional considerations permeate every stage of design, from the earliest conceptual thinking to detailed design optimizations. Yet the concept of function remains unclear in the design community [1]. Certainly, function is related to the design object’s behaviour, loosely defined as the totality of the object’s interactions with itself and its environment. At other times, it is related to the user’s expectations of the part, which highlight certain aspects among the behaviours. In this work we interpret function in terms of user expectations and consider the behaviours as a discrete set B, then a subset F that meets the design intent may constitute the “function”. For computational work on function, we quantify this function in terms of a performance metric that reflects the degree to which the function meets the user’s expectations. We consider the class of mechanism assemblies, in which the relative motion of different parts is captured through a configuration space (C-space) [2], which is used as an index to retrieve the intended behaviours of the existing mechanisms [3]. The Function-Behavior-Structure (FBS) [4] framework relates the function of a design object to its behaviors and its structural descriptions. However, functions are not independent of the user. Let us consider the example of a padlock. Initially, its function was seen as that of providing an open-able ring (topological torus). However, once it is instantiated, other aspects of its behaviour such as its weight, or how noisy it is, may become part of the user expectations. Thus, function may be determined by a) mapping 1 Copyright c © 2008 by ASME the structure/geometry into a comprehensive set of behaviours, and b) defining performance metrics on some or all of these behaviours. Both these problems are intractable for general designs, where determining all behaviours is problem-specific and does not generalize at all; similarly defining performance metrics is also usage specific. One of the important questions is that the nature of the design variable space itself changes as function proceeds. For example, some inter-relations between the variables may prove to be untenable based on functional considerations, so that singularities emerge in the design space. Also, as the functional aspects become clear, the relative importance of different design variables change, and the nature of approximations used in arriving at a mathematical model can be changed. These would also affect the part family structure-to-function mapping computations. Figure 1. Design Process: A mapping from design space to structure space and searching the design space based on performance evaluation of the intended behaviours. In this work we consider these problems of identifying function and relating this to design change in the context of functional part families. Initially, a functional part family is defined by a shared structure and a shared function, defined in terms of a set of performance metrics. However, later, either the structure or the function may change. Here we consider one kind of design change where the set of parameters related to a design space is expanded from the existing design space to the new design space by introducing new design parameter into the existing design space and show how the performance metrics would change in functional part families and also we examine how the design constraints vary based on functional constraints. Initially we identify a set of design parameters in a design space (D) and each design vector v in the design space can be mapped to an unique structure in structure space (S). The main aspect here involves identifying the set of intended behaviours (Bi) in the whole set of behaviours (B), and then defining performance metrics on these intended behaviors in performance space (P). These intended behaviors can also be termed as the performative behaviours(Bp) since the performance metrics can be defined only on this subset. These performative behaviours are similar to what Gero [5] calls expected behaviours. Thus to re-phrase the F-B-S model of John Gero [5], we may say that our model involves D-S-B-P, which involves mapping from D to S, S to B, and then evaluation from B to P. The results of evaluation are then used to search in design space and come up with a set of improved S. This process is shown in Fig. 1. In this paper we characterize function based on this frame work for functional part families. 1.1 Function in Part Families Part family is a set of parts that serve a related set of market applications they are functionally similar, and share a common technology base, and lead to better processes for life-cycle design [6]. Functional commonalities across product families have been considered by [7, 8], but even here, not much progress has been made in mapping the structural similarities onto function. Here, we distinguish between two types of part similarities: (i) Functional Part Family, which shares an embodiment and (ii) Geometric Part Family, which shares the same geometric structure, differing only in dimensional parameters. Lock A Lock B Lock C Lock D Lock E Lock F Figure 2. Lock Family: Functional part families are evaluated based on the same set of qualities. Though the locks shown are varying in geometry but their shared functions can be locking, strength of the lock, ease of locking etc. 2 Copyright c © 2008 by ASME Definition 1. Functional Part Family (FPF) is a set of members, in which each member share the same qualitative nature and semantics of functions but the specific performance metric may be different since the design vectors can be different. Definition 2. Geometric Part Family (GPF) is a set of members, in which each member will share the same set of performance measures P and consequently they also have the same design vector space Ω. A FPF is that set of related parts where knowledge of function can be transferred in some meaningful way. Since this cannot be defined clearly, we adopt the notion that an FPF is a set of designs that are evaluated based on the same set of qualities; i.e. the actual metrics may be different (since the design spaces or embodiments differ) but the semantics of what is being measured (e.g. strength, ease of locking) remain the same. Hence the primary function of the six locks shown in Fig. 2 is “locking”. Within a GPF, different instances arise as a result of variations in a small number of design variables. We show that given any set of design variables one can generate the C-space, and also that the performance metrics can be evaluated on this resulting C-space. Thus we can explore different designs that arise within the constraints defined by the designer. 1.2 Configuration Space Configuration Space (C-space) is the space of independent variables describing relative motions of sub-parts [9]. The Cspace of a kinematic pairs can be defined interms of the shapes and degrees of freedom of its parts. In Fig. 3(c), the hatched region correspond to invalid object positions; the free region correspond to valid object positions; and the boundary separating these two regions is the contact space. Contact space is the trajectory where the two object touch with geometric features vertex,edge and forms feature contacts vertex-vertex, vertex-edge, edge-edge type. Definition 3. Configuration Space C of a body w.r.t another is the space of all configurations −→u ∈ C the bodies can have w.r.t one another. The Obstacle space of body B w.r.t A, OSA(B) is defined as the set of motions that cause a collision between A and B. OSA(B) = {−→u |∃(x∈ A)T A B ( −→u )x∈ B}. δOSA(B) is the boundary of this obstacle space. Computing the (C-space) for general motions remains an intractable problem [10]. Further, given a C-space, obtaining successful abstractions on it i.e. segmenting the free-space into behaviourally significant regions -e.g., using topologically different contact types [11], remains a considerable challenge. In the situation involving part families, we assume that the C-space model has been worked out for some existing members in the (a) (b) (c) Figure 3. Configuration Space a) Mechanisms in the lock can be decomposed into a key-barrel fragment and a latch-bolt fragment b) C-space for key-barrel design fragment rotation θ of key results in X translation movement in latch. due to the hysteresis loses the X varies as shown.c) C-Space for latch-bolt design fragment relates horizontal motion of latch (X ) with vertical motion of u-bolt (Y ). family, so that these can now be extended to the new part being designed or the existing part being analyzed. Also, the qualitatively important aspects of the C-space are also computable based on similar examples. Thus the problem is considerably simpler for part families. Consider six different padlock designs Fig.2, in which A, B and C exhibit Bolt-Latch design fragment while Fig.3a, relates X to the bolt movement Y . In particular, we make the following claims: • For mechanical designs involving movable parts, the functionality can be captured through certain operations defined on the C-space. • It is possible to construct such Configuration Spaces for members of a part family taking into account the variation in dimensions and form (section 2.1). • Measures of performance, defined in terms of certain behaviours, can be related to metrics defined on the configuration space or other functions of the design variables (section 2.2). By relating the structure, definable in terms of design variables, to a set of behaviours which are evaluated using the performance metric, one has created the basis for optimizing these variables based on function. 1.3 Performance Measures Definition 4. Performance Measure P is a set of real valued functions from a set of behaviours to R. A set of performance measures πi are defined for the set of behaviours intended for the design. 2 Example : Padlock For the padlock the motion of the U-bolt (X) w.r.t the motion of Latch (Y ) defines the C-space shown in Fig.3c. The hatched region corresponds to collision configurations; the white region corresponds to “free space”. 3 Copyright c © 2008 by ASME