Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding

In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.

Constraint Embedding for Vehicle Suspension Dynamics

The goal of this research is to achieve close to real-time dynamics performance for allowing auto-pilot in-the-loop testing of unmanned ground vehicles (UGV) for urban as well as off-road scenarios. The overall vehicle dynamics performance is governed by the multibody dynamics model for the vehicle, the wheel/terrain interaction dynamics and the onboard control system. The topic of this paper is the development of computationally efficient and accurate dynamics model for ground vehicles with complex suspension dynamics. A challenge is that typical vehicle suspensions involve closed-chain loops which require expensive DAE integration techniques. In this paper, we illustrate the use the alternative constraint embedding technique to reduce the cost and improve the accuracy of the dynamics model for the vehicle.

Recursive Algorithms Using Local Constraint Embedding for Multibody System Dynamics

This paper describes a constraint embedding approach for handling of local closure constraints in multibody system dynamics. The approach uses spatial operator techniques to eliminate local-loop constraints from a system to effectively convert it into a tree-topology system. This conversion to a tree-topology allows the direct application of the host of available techniques including mass matrix factorization and inversion to be applied to the system dynamics. One application is the extension of the well-known recursive O(N) forward dynamics for solving the system dynamics of these systems. The algorithms are especially applicable to systems where the constraints are confined to small-subgraphs within the system topology. The paper provides background on the spatial operator approach, the extensions for handling embedded constraints, and concludes with examples of such constraints.