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We report on the solution of a real-time scheduling problem that arises in the design of software-based operation control of aircraft. A set of tasks has to be distributed on a minimum number of machines and offsets of the tasks have to be computed. The tasks emit jobs periodically starting at their offset and then need to be executed on the machines without any delay. Also, further constraints in terms of memory usage and redundancy requirements have to be met. Approaches based on standard integer programming formulations fail to solve our real-world instances. By exploiting structural insights of the problem we obtain an IP-formulation and primal heuristics that together solve the real-world instances to optimality and outperform text-book approaches by several orders of magnitude. Our methods lead, for the first time, to an industry strength tool to optimally schedule aircraft sized problems.

Previous work by Mattikalli et al. [l] considered the stability of assemblies of frictionless contacting bodies under uniform gravity. A linear programming-based technique was described that would automatically determine a single stable orientation for an assembly (if such an orientation existed). In this paper, we include Coulomb friction at contacts between bodies and give a characterization of the entire set of stable orientations of an assembly under uniform gravity. Our characterization is based on the concept of potential stability, which describes a necessary but not sufficient condition for the stability of an assembly. Orientations that are computed as being unstable, however, are guaranteed to fall apart. Our characterization reveals that the set of stable orientations maps out a convex region on the unit-sphere of directions and corresponds to a spherical analog of a planar polygon-the region is bounded by a sequence of vertices joined by great arcs. Linear programming techniques are used to automatically find this set of vertices, yielding a description of the range of stable orientations for any assembly. For frictionless assemblies, our characterization of stable orientations is exact. For assemblies with friction, some conservative approximations associated with the use of a linearized Coulomb law are made.

A method to determine constraints on translational and rotational motion of planar and 3-D objects from their contact geometry is presented. Danslations are represented by spatial vectors and rotations by axes in space. For each of these, a geometric realization (Ma) of the space of motion parameters is created. Subspaces in Ma that represent the range of values of motion parameters that are 'disallowed' due to the contact are identified. The geometric realization makes it easier to visualize results, provides a good measure of the extent of restraints between objects, reduces computations by eliminating redundant constraints, and simplifies computation of net constraints. The proposed representation can be used effectively to automate the evaluation of motion Constraints. Motion Constraints from ContactReuleaux[S] analyzed the effect of point contact on translational and rotational motion of planar objects. Using a graphical method, he derived the field of restraint (a set of directions in which motion is pre-2178 0-8186-2720-4/92 $3.00 81992 IEEE

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