Automatic Reasoning for Design Under Geometric Constraints M. Shpitalni, H. Lipson Laboratory for Computer Graphics and CAD Faculty of Mechanical Engineering Technion, Haifa, Israel 32000 e-mail: shefi@tx.technion.ac.il Abstract Parametric design is very stable but requires a predefined dimensioning and ordering scheme, thus limiting flexibility and precluding sketch input. Variational geometry design, while general and flexible, necessitates intensive use of numerical solvers to solve many simultaneous nonlinear equations. Frequently the solvers cannot solve these equations. A new system, based on an original theory for automatic constraint analysis, has been developed for solving sets of two-dimensional geometric constraints in product design. The proposed system offers the flexibility of variational based design along with the stability of parametric design. The solution strategy is based upon breaking down the problem into a sequence of construction steps. When no sequential construction is found, auxiliary geometrical constructions are automatically generated based on rules for relocating constraints. Thus, an apparently simultaneous constraint set is converted into a set that can be constructed sequentially by decomposing strongly connected components of the original constraint graph. This new approach has been implemented in a system for designing sheet metal parts. Keywords : CAD, Conceptual Design, Geometry Constraint 1. Introduction Two design paradigms are used and explored for design and manufacturing applications: feature-based design and constraint-based design [3]. In constraint-based design, shapes are specified by means of geometrical constraints that relate shape features to shape parameters. The constraints are typically specified based on predefined topological arrangements of features (sometimes entered by means of a sketch) to provide a context for the problem. Studies of constraint solution range from pure geometry, e.g., [1], to kinematics [6], engineering constraints [5], and theorem proving [2]. When solving geometrical constraint problems, a solver must provide an instance of the given topology that exactly satisfies the given constraints. Two major approaches can be identified: (1) parametric geometry, in which constraints are given so that the desired shape can be constructed sequentially according to a predefined scheme and order, and (2) variational geometry, where constraints are given by an arbitrary scheme in no particular order. The solver must then derive a solution strategy automatically in order to construct the desired shape. Since the parametric approach uses a predefined scheme of dimensions and a predefined evaluation order, the solution process is more stable and controllable, and is therefore more common in commercial CAD systems. However, the need to adhere to a prespecified dimensioning scheme and order limits the freedom of the designer to modify the definition of the shape. Moreover, the desire to allow more flexible input methods for sketching [8] and conceptual design necessitates that dimensioning of the design be independent of a specific dimensioning scheme and order. However, lack of a solution strategy makes variational geometry problems more difficult to solve because the solver must be capable of the following: deriving a solution sequence; handling large sets of simultaneous non-linear equations; managing multiple solutions; and determining user intent as the most plausible solution. Variational geometry problems are usually solved by using either numerical solvers or sequential construction solvers based on applying precoded steps, which may resort to numerical methods in special cases. In essence, numeric solvers are more general but construction solvers are more stable. This paper describes a novel solution technique which allows solving more complex problems using construction solvers without resorting to numerical methods. The new solver is based on automatic generation of auxiliary geometrical constructions and constraints. While the additional constraints do not actually add any new mathematical information, they do simplify the automatic search for a solution strategy as well as significantly enhance the analytical capabilities of the sequential solver. The new solver has been implemented successfully in a CAD system [10] for design of sheet metal parts. This paper first describes existing techniques for solving geometrical constraint problems and places the proposed method in context. Then, the new method is fully described and demonstrated through examples and implementation. 2. Geometrical constraint solvers Variational geometrical constraints are usually specified based on an initial sketch that reveals the topological relationships among the geometrical entities. Constraint problems are traditionally solved using a numeric approach, whereby geometrical constraints are translated into general algebraic expressions solved using iterative methods, such as Newton-Raphson or homotopy [7]. However, despite their generality, numerical solvers often prove to be inadequate for large sets of non-linear equations and rely to a great extent on the quality of the initial sketch. Numeric solvers are also unable to explore the solution space and provide almost no information upon failure. The alternative of solving these equations using symbolic algebra does not appear to be practical [9]. Consequently, a different approach has been adopted, based on constructive constraint solvers used for satisfying constraints using a constructive sequence of steps. This process resembles