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Abstract

Conceptual design is the initial stage throughout the product life cycle, whose main purposes include function creation, function decomposition, and function and subfunction designs. At this stage, the information about product function and structure has the characteristics of imprecision, incompleteness, being qualitative, and so forth, which will affect the validity of conceptual design. In this paper, the signed directed graph is used to reveal the inherent causal relationship and interactions among the variables and find qualitative interactions between design variables and design purpose with the help of causal sequence analysis and constraint propagation. In the case of incomplete information, qualitative reasoning, which has the function of qualitative behavior prediction, can improve conceptual design level aided by the computer. To some extent, qualitative reasoning plays a supplementary role in evaluating scheme and predicting function. At last, with the problem of planar four-bar mechanism design, a qualitative reasoning flowchart based on the Signed Directed Graph is introduced, and an analysis is made of how to adjust design parameters to make the trajectory of a moving point reach to the predetermined position so as to meet the design requirements and achieve the effect that aided designers expect in conceptual design.

1. Introduction

Rapidly growing complexities of the product in function and structure are enabling engineers to adopt multiple design means and combine them together to shorten the time to market. Exploring advanced conceptual design theories and methodologies and their applications is used to improve the product functions and performances.

According to the requirements of the various stages of the product life cycle [1], conceptual design involves such work as product creation, functional decomposition, and function and subfunction designs, realizing carrier program and idea of systematic design that meet the function and working principle [2]. Conceptual design is the initial stage of the design process, which aims at obtaining the basic form or shape information and related description of the product. In conceptual design, detailed design information is not likely to be obtained; instead, only a rough description of the system is needed [3].

The key technologies of Computer-aided conceptual design are the product information modeling and reasoning techniques, in which design object abstract description and expression are the important part. Currently used in the conceptual design stage reasoning techniques are knowledge-based reasoning, artificial neural networks, case-based reasoning, and qualitative reasoning. Seely Brown and John de Kleer propose the theory based on the concept of “stream” theory, and B. J. Kuipers uses constraint-based qualitative simulation theory described by qualitative differential equations [4]. Cao [5] researches the function, behavior, and structure framework which can be further achieved by product conceptual design process modeling. Von-Wun and Wang [6] designed a compression spring design system with qualitative and quantitative techniques.

The information obtained during the conceptual design is usually incomplete, imprecise, and fuzzy. It is very difficult and sometimes even impossible to set up an accurate quantitative model using conventional numerical analysis tools. We need not establish a precise mathematical model of the problem at many times; instead, we just need rely on the understanding of the principles behind the research questions and qualitative analysis to get the results we want. The foundation of such a principle understanding of the problem is the application of qualitative knowledge, which is very useful in practical problem solving. In this process, we are often interested in qualitative knowledge about the nature of the system, while numerous, redundant, precise quantitative knowledge is usually not necessary. And the conceptual design is a design-analysis-redesign spiral process; a lot of repetitive designs are inevitable. Take advantage of the knowledge of the initial design phase to build qualitative modeling, qualitatively giving the system's behavior prediction, and approximate direction of the design can effectively reduce repetitive design process, improve efficiency, and reduce design costs [7].

In many cases, such as in the conceptual design of the transmission program, what need to do are to determine the transmission type, to select the drive level, and to make a preliminary analysis of the eventual impact of the transmission scheme on the product [8]. In this analysis process, it is not necessary or possible to get the design details. At the conceptual design stage, a large amount of experience knowledge of designers is to be used, and it is not feasible to adopt numerical analysis tools like the finite element to provide effective help for designers. Qualitative reasoning is one of the reasoning methods in artificial intelligence, which uses qualitative information on the system's structure, behavior and function description. It studies the relationship among them and causality, draw qualitative interpretation, in order to mimic human commonsense qualitative reasoning [9].

The knowledge of qualitative reasoning mainly comes from the objective laws described in terms of mathematical formulas. Through the establishment of qualitative constraint equations, the system behavior change caused by some local changes can be qualitatively predicted. For the parts that do not meet the design requirements, qualitative reasoning can give directional suggestions for improvement.

Conceptual design involves the application of knowledge of various types [10, 11], including empirical knowledge, common sense, and structural knowledge, which generally requires the use of multilayer knowledge representation patterns of metaknowledge, qualitative knowledge, and mathematical models and methods [12]. Therefore, in conceptual design, different corresponding representations need to be adopted according to the characteristics of different types of knowledge, with comprehensive considerations in terms of representation ability, reasoning efficiency, maintainability, and so on.

Combined with the design process, the types of knowledge and structures the conceptual design stage involves are shown in Figure 1. The paper uses the qualitative reasoning method based on the Signed Directed Graph to organize the reasoning process from requirements analysis to scheme solving.

Figure 1:

The knowledge model of conceptual design.

2. Signed Directed Graph

Signed Directed Graph is a kind of graph which is constituted by the directed connections among the nodes, referred to as SDG [12, 13], wherein the node can represent a variable, operating member or an event. The SDG model considers the links among the design variables and interactions. The state variables in the object being studied can be represented as nodes, connected with the edge between the nodes associated. The influence among the nodes can be expressed by the qualitative description method, indicating whether each node has incremental or decremental influence on other related nodes, and indicating what impact the state of a node has on the state of other related nodes. SDG can reveal the inherent causal relationship and interactions among the variables of the system, and thus it is suitable for the analysis of problems with root causes and multiple causal relationships. As a deep knowledge model, SDG has the ability to accommodate large-scale potential information.

Definition 1. The SDG model consists of nodes, directed branches, and branch sign among nodes, which is actually a collection of the three things.

The mathematic description of the SDG mode is what follows:

γ = (G, ψ),

(1)

where G stands for directed graph and ψ stands for sign set.

The directed graph G consists of the following four items:

G = (V, E, \partial^{+}, \partial^{-}),

(2)

where node set V = {v_i}, branch set E = {e_k}, adjacent associated character ∂ ⁺: E → V stands for the start node of branch, and ∂ ⁻: E → V stands for the end node of branch.

Sign set ψ can be written as

ψ : E \in {+, -} .

(3)

Sign of branch can be written as

ψ (e_{k}) = ψ (v_{i}, v_{j}) .

(4)

Definition 2. The signed branch indicates the positive and negative influence among nodes and represents the constraint propagation path. If the initial node of the branch increases or decreases, resulting in the synchronic increase or decrease of the end node, then the branch effect is called incremental influence or positive influence, abbreviated as “+,” usually indicated with a solid line with arrows. If the initial node and the end node change in the opposite way, then the branch effect is called decremental influence or negative influence, abbreviated as “−,” indicated with a dotted line with the arrow.

Definition 3. Compatible path: in γ = (G, ψ), concerning the instantaneous sample ϕ, if ϕ(∂ ⁺e^k)ϕ(e^k)ϕ(∂ ⁻e^k) = +, the branch e^k is called a compatible branch, and the end-to-end compatible branches form a compatible path. It can also be further expanded as a branch combination that meets ϕ(∂ ⁺e_i)ϕ(e_i) … ϕ(e_j)ϕ(∂ ⁻e_j) = +.

3. Qualitative Reasoning Model of SDG

3.1. SDG Model

The key to the SDG reasoning is to construct the corresponding SDG model, namely, the process of knowledge acquisition and knowledge representation in the normal sense. Through the method based on experiential knowledge or mathematical model to get the constraint relations among variables [14], to determine the qualitative influence among variables is actually to determine the state of SDG branch, and then the influence relationship among variables can be described clearly.

To convert the SDG model directly into the corresponding reasoning model involves the following steps.

According to the SDG model structure, search for all compatible paths from the node with its state known to the parameter node concerned.

Except the node with its state known in compatible paths, add rules to each remaining node n_i according to the following forms.

Ifk₁, k₂ … k_n are forward nodes of n_i, and relationship among them are “and”. Then only all these forward nodes meet the requirement at the same time, can we infer the state of node n_i:

[(node, k₁, status) and (arc, k₁, n_i, influence)] and [(node, k₂, status) and (arc, k₂, n_i, influence)]… and [(node, k_n, status) and (arc, k_n, n_i, influence)].

If k₁, k₂ … k_n are forward nodes of n_i, and relationship among them are “or”. As long as one node among these forward nodes meets the requirement at the same time, can we infer the state of node n_i:

[(node, k₁, status) and (arc, k₁, n_i, influence)] or [(node, k₂, status) and (arc, k₂, n_i, influence)] … or [(node, k_n, status) and (arc, k_n, n_i, influence)].

Qualitative reasoning includes two factors: causal sequence analysis and constraint propagation [15]. The most important part of the knowledge in the analysis of the system is causality. In general cases, this causal relationship is expressed as a function relation, but the function does not show the direction of causality [16]. Therefore, we need to use the causal sequence theory to get the causal diagram among variables after obtaining the qualitative equation [17]. This paper studies two main conditions among qualitative reasoning.

When some variables in the system change, we need to find out how the rest variables are influenced, that is, making the analysis from reason to result.

When some variables in the system change, we need to find out what factors cause these changes, that is, making the analysis from result to reason.

In the context of qualitative reasoning, the mechanism of both forward and backward ones is explored, which are two fundamentally different approaches to reasoning [18]. With forward reasoning, propositions are combined with rules to deduce new propositions. Forward reasoning is of special interest in situations where no specific goals are obtainable, and where most rules and the antecedent portion to be considered are well known. As opposed to forward reasoning, backward reasoning works in a consequence-driven way.

In the first case, we use forward reasoning. Based on the established SDG model, we search backward all the pathways which use it as the predecessor, make a qualitative reasoning of these pathways, and find out the status of the affected nodes, a process of constraint propagation.

In the second case, we use backward reasoning, starting from the changed nodes by the depth-first search method to find out all possible cause nodes and identify one or more paths from the target node to cause nodes. When the causality is transmitted along these valid paths, the tendency of the variables and the cause of the corresponding state can be explained.

3.2. Knowledge Representation of SDG

Knowledge-based reasoning systems need to express the information contained in the SDG model, and in this paper knowledge representation is achieved through the frame representation and production rules. The main structure of qualitative analysis is the description of the SDG model. The corresponding knowledge representations are listed as follows:

(deftemplate arc

(slot predecessor (default null))

(slot successor (default null))

(slot relation (default null)))

(deftemplate node

(slot name (default null))

(slot father (default null))

(slot status (default null))

(slot var1 (default null))

(slot var2 (default null))

(multislot predecessor (default (create$)))

(slot mark (default null)))

(deftemplate causality

(multislot mode (default (create$)))

(multislot relation (default (create$)))

(multislot influence (default (create$)))

(multislot stack (default (create$)))).

3.3. Framwwork of SDG Model

The paper uses frame representation and production rule method for knowledge representation and knowledge-based reasoning system to complete the sequence of causal reasoning and explanation. Figure 2 shows the main flowchart of SDG qualitative reasoning model.

Figure 2:

The qualitative reasoning flowchart based on SDG.

In the process of qualitative reasoning, the main algorithms are constraint propagation algorithm and backward reasoning calculus algorithms, which show in Figures 3 and 4, respectively.

Figure 3:

The flowchart of constraint propagation.

Figure 4:

The flowchart of forward verification to backward reasoning.

4. Qualitative Reasoning Model of Planar Four-Bar Mechanism

4.1. Constraint Relationship among Components

In the conceptual design of the planar four-bar mechanism, generally six design variables are concerned: active rod, connecting rod, followed rod, basement rod, fixed rod, and the angle between the fixed rod and the connecting rod, which are expressed by a, e, g, c, d, f, and Φ, respectively. For the sake of simplicity, it is agreed that the x-coordinate of the system coincides with the basement rod, with the origin A on left of d, as shown in Figure 5.

Figure 5:

Four-bar mechanism.

The paper uses the position method to analyze the four-bar mechanism. As shown in Figure 5, given the rotation angle θ₁ of the active rod a, components e, g, c, d, f, and Φ, and the coordinates X_A, Y_A, X_D, and Y_D of the fixed hinge points A and D, other parameters can be obtained through the displacement constraint analysis, and the assembly relations in the four-bar mechanism are the following

θ_{2} = θ_{3} - θ_{4},

(1)

X_{C} = X_{B} + (e + g) \cos θ_{2},

(2)

Y_{C} = Y_{B} + (e + g) \sin θ_{2},

(3)

α = \arctan [\frac{Y_{C} - Y_{D}}{X_{C} - X_{D}}] = \arctan [\frac{Y_{C}}{X_{C} - d}] .

(4)

The link curve is the trajectory of any point P on the link planar. For any given θ₁, the displacement coordinate of point P is as follows:

X = X_{B} + e \cos θ_{2} + f \cos θ,

(5)

Y = Y_{B} + e \sin θ_{2} + f \sin θ,

(6)

θ = ϕ + θ_{2} .

(7)

4.2. SDG Model of the Four-Bar Mechanism

The relationship among the variables of four bar mechanism is more complicated. The paper considers the qualitative influence relationship among them. Equation (8) represents the more accurate constraint among other variables with angle constraint premise:

\begin{matrix} if θ_{1} \in (0, \frac{π}{2}) then X_{B} - d < 0, \\ if θ_{1} \in (0, π) then Y_{B} > 0, \\ if θ_{3} \in (0, π) then [{(e + g)}^{2} + b^{2} - c^{2}] > 0, \\ if θ_{3} \in (0, π) then [b^{2} + c^{2} - {(e + g)}^{2}] > 0, \\ if θ_{2} \in (0, \frac{π}{2}) then X_{C} - d > 0, \\ if θ_{2} \in (0, \frac{π}{2}) then Y_{C} > 0 . \end{matrix}

(8)

The above formula can reveal the causal dependency among variables in the structure, which is the basis of constraint propagation and qualitative reasoning. Therefore, the qualitative relationship among variables can be obtained from the above constraints in the four-bar mechanism. Based on the influence relations among variables, related variables can be connected, and then the SDG mode of four-bar mechanism is obtained, as shown in Figure 6.

Figure 6:

SDG model of four-bar mechanism.

4.3. Qualitative Analysis of the Four-Bar Mechanism's Trajectory

In the design of the four-bar mechanism, we often encounter the problem that the point's trajectory in the initial design is not in a predetermined position so that it may not meet our requirements. Then we need to find the reasons that affect it, make adjustments accordingly sometimes, or need to find out what influence will happen to the remaining variables when certain variables change sometimes.

In the design of the four-bar mechanism, the initial design result is obtained by the connecting rod curve trajectory synthesis. The trajectory of point P on fixed rod is shown by the solid line in Figure 7.

Figure 7:

Initial design of four-bar mechanism by curve trajectory synthesis.

As seen from Figure 7, base point P has some gap with actual requirements on the positions of P1, P2, P3, and P4. The values of θ1 on P1, P2, P3, and P4 are, respectively, 30°, 130°, 230°, 300°. The analysis starts from p1 position, which corresponds to θ1 of 30°, and other parameters also meet the constraints of the SDG model. The position of P involves two main variables, namely, the abscissa X and ordinate Y. Then abscissa X will be discussed as an example.

It is known from Figure 7 that a, e, and f are directly influencing factors, considering other indirect factors, and then the main factors of P1 in a horizontal deviation may be collection V = {a, e, d, c, f, g, θ₁, θ, θ₄}, and thus they are likely to become factors of disturbance and may be possible cause nodes. The four-bar mechanism, as the adjustment object, must meet the following requirements.

The states of target variables through different paths must be consistent.

The disturbance of the variables in different locations can make the mechanism shift toward a more rational direction, which is reasonable consistency of change.

Factor a in influencing factor set V is selected for the analysis. Since we want to know how to adjust factor a to meet our requirements, we need to start from the constructed four-bar mechanism SDG model and find all compatible paths between this node and abscissa X of the concerned node. The compatible path that can be found is shown in Figure 8.

Figure 8:

Compatible path between disturbance factor and variable x.

Then, reasoning needs to be done along the above compatible path. If the state of the target node variable derived from reasoning is consistent, then appropriate adjustment to disturbance variables can be received; otherwise, it is unable to make a judgment based on qualitative information. Some rules summarized from compatible paths are as follows.

Rule 1.

If (node, a, +1) and (arc, a, X_B, positive) then (node, X_B, +1).

Rule 2.

If (node, a, +1) and (arc, a, Y_B, positive) then (node, Y_B, +1).

Rule 3.

If [[(node, X_B, +1) and (node, Y_B, +1)] or [(node, X_B, +1) and (node, Y_B, 0)] or [(node, X_B, 0) and (node, Y_B, 1)]] and (arc, X_B, b, positive) and (arc, Y_B, b, positive) then (node, b, +1).

Rule 4.

If (node, X_B, +1) and (arc, X_B, θ₄, negative) and (node, Y_B, +1) and (arc, Y_B, θ₄, positive) then (node, θ₄, uncertain).

Since uncertain situations appear in the fourth rule reasoning, it is impossible to tell how to do the adjustment of factor a to meet our needs according to our qualitative knowledge. Then we need to re-select a disturbance factor from the possible cause set. Factor g is be selected as a new disturbance variable, using the same method to find the compatible path from node g to abscissa X of node P, as shown in Figure 9. Some rules summarized are as follows.

Figure 9:

Compatible path between disturbance node g and variable x.

Rule 5.

If (node, g, +1) and (arc, g, θ₃, negative) then (node, θ₃, −1).

Rule 6.

If (node, θ₃, −1) and (arc, θ₃, θ₂, positive) then (node, θ₂, −1).

Rule 7.

If (node, θ₂, −1) and (arc, θ₂, θ, positive) then (node, θ, −1).

Rule 8.

If [[(node, θ₂, −1) and (node, θ, −1)] or [(node, θ₂, −1) and (node, θ, 0)] or [(node, θ₂, 0) and (node, θ, −1)]] and (arc, θ₂, X, negative) and (arc, θ, X, negative) then (node, X, +1).

The situation of the node's uncertain state does not appear in the above rules, so we can change the state of the abscissa X by adjusting node g. Also, it can be known from the above rules that abscissa X increases with the increase in node g and decreases with the decrease in node g.

Similar to the analysis process of abscissa X, we can use the same method to deal with Y of node p. It is known from the analysis that ordinate Y decreases with the increase in node g and increases with the decreases in node g. In view of the initial design situation, the X value at the location of the point P1 is larger than the actual position, and Y is smaller than the actual position. P1 is at the bottom right location of the desired position. In accordance with the requirements of the designer, X needs to become smaller, and Y needs to become bigger. Therefore, we can meet the requirement by reducing factor g, namely, the distance between the fixed rod and hinge B.

After the adjustment of factor g, though it may be satisfied that the position of the connecting rod at the point P1 is closer to a predetermined position, the requirements of the connecting rod at other predetermined positions may be affected, and there exist the following effects.

The connecting rod at other predetermined positions has not changed.

The connecting rod at other predetermined positions has changed, but it is closer to the desired position.

The connecting rod at other predetermined positions has changed, but on the whole the number of points does not meet the predetermined positions that are greatly reduced.

In order to judge the adjustment effect of the disturbance variables, it is necessary to verify the influence on the rest predetermined positions. For other possible influencing factors, a similar analysis process can be applied to obtain the state of the rest predetermined positions and the number of unsatisfied positions. It is the aim to have a minimal number of positions that do not meet the design accuracy and find the adjustment program that can best meet the user's needs. When there are two or more adjustment programs that can meet this requirement, we need to select the one in which the position that can meet the design accuracy is close to the predetermined position as much as possible. Through a series of procedures, it is known that adjusting factor g makes positions P2, P3, and P4 move toward the expected direction, and there is disturbance to the rest desired predetermined locations, but changes are within the allowable range of accuracy, so adjusting g, namely, the distance between the fixed rod and the hinge C, can meet our requirements.

The same approach can also be adopted to analyze the case of the disturbance of two or more components, which, due to the space limitation, will not be discussed.

5. Conclusion

The product information in the conceptual design stage is qualitative, imprecise, uncertain, incomplete, conceptual design itself belongs to the category of experience design. For those problems that are difficult to access knowledge and not easy to have complete description, the method based on knowledge can also realize the corresponding reasoning, but it is difficult to maintain as knowledge increases knowledge. On the other hand, those problems are relatively easy to deal with in qualitative reasoning. Since the knowledge of qualitative reasoning comes from the objective laws described by mathematical formulas, all the qualitative descriptions of the objective laws can be derived by qualitative reasoning, and thus knowledge is comprehensive and easy to acquire.

Through the directed graph, the SDG qualitative analysis expresses the inner causal relationship and interacting factors among design variables involved in the design system, SDG qualitative analysis system is suitable for the analysis of the root causes and multiple causality problems. Accurate mathematical models are not needed in modeling instead, just experience and knowledge combined with some of the objective laws are needed to lead to corresponding qualitative models, which can be converted into corresponding rules for qualitative reasoning, and then aid the designer. Based on SDG, a qualitative reasoning system framework has been established, which, after further improvement, can be used conveniently for the establishment of other qualitative reasoning systems of objective laws and can serve as a powerful qualitative analysis tool for designer.

Qualitative reasoning, of course, involves many other aspects [19] that need to be further discussed, conceptual design involves experience knowledge, qualitative knowledge, and quantitative knowledge, which needs multilevel design knowledge from quantitative knowledge to abstract knowledge. SDG qualitative reasoning provides only one of the levels. How to well represent and use such knowledge is the key to constructing the conceptual design support system, and it remains to be solved in the future.

Footnotes

Acknowledgments

The authors are grateful for detailed and insightful comments from the anonymous reviewers which have been invaluable in revising the initial paper. This research was supported by the Fundamental Research Funds for the Central Universities (72124683).

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