Constraint-bounded design search

The design process requires continual checking of the consistency of design choices against given sets of goals that have been fulfilled. Such a check is generally performed by comparing abstract representations of design goals with these of the sought real building objects (RBO) resulting from complex intellectual activities closely related to the designer's culture and to the environment in which he operates. In this chapter we define a possible formalization of such representations concerning the goals and the RBO that are usually considered in the architectural design process by our culture in our environment. The representation of design goals is performed by expressing their objective aspects (requirements) and by defining their allowable values (performance specifications). The resulting system of requirements defines the set of allowable solutions and infers an abstract representation of the sought building objects (BO) that consists of the set of characteristics (attributes and relations) which are considered relevant to represent the particular kind of RBO with respect to the consistency check with design goals. The values related to such characteristics define the performances of the RBO while their set establishes its behaviour. Generally speaking, there is no single real object corresponding to an abstract representation but the whole class of the RBO that are equivalent with respect to the values assumed by the considered characteristics. The more we increase the number of these, as well as their specifications, the smaller the class becomes until it coincides with a single real object - given that the assessed specifications be fully consistent. On the other hand, the corresponding representation evolves to the total prefiguration of the RBO. It is not therefore possible to completely define a BO representation in advance since this is inferred by the considered goals and is itself a result of the design process. What can only be established in advance is that any set of characteristics assumed to represent any RBO consists of hierarchic, topological, geometrical and functional relations among the parts of the object at any level of aggregation (from components to space units, to building units, to the whole building) that we define representation structure (RS). Consequently the RS may be thought as the elementary structures that, by superposition and interaction, set up the abstract representation that best fit with design goals

Planning rigid body motions using elastic curves

This paper tackles the problem of computing smooth, optimal trajectories on the Euclidean group of motions SE(3). The problem is formulated as an optimal control problem where the cost function to be minimized is equal to the integral of the classical curvature squared. This problem is analogous to the elastic problem from differential geometry and thus the resulting rigid body motions will trace elastic curves. An application of the Maximum Principle to this optimal control problem shifts the emphasis to the language of symplectic geometry and to the associated Hamiltonian formalism. This results in a system of first order differential equations that yield coordinate free necessary conditions for optimality for these curves. From these necessary conditions we identify an integrable case and these particular set of curves are solved analytically. These analytic solutions provide interpolating curves between an initial given position and orientation and a desired position and orientation that would be useful in motion planning for systems such as robotic manipulators and autonomous-oriented vehicles