The Embedded Projection Method: A general index reduction procedure for constrained system dynamics

The ‘Embedded Projection Method’ (EPM) is an attempt to eliminate most of the difficulties related to the solution of high-index differential-algebraic equations (DAEs), as those typical of the dynamics of constrained systems. The intimate coupling between the algebraic and the differential parts lies at the origin of these difficulties, therefore we uncouple these components, introduce an unconstrained, modified state variable, and complement the resulting ordinary differential equation (ODE) by additional algebraic variables. As a result, a general, consistent index reduction from arbitrarily high values to one is obtained, allowing the use of any suitable ODE integration algorithm, without the need for a specialized DAE solver. The EPM solution is intrinsically endowed with greater accuracy and stability than conventional counterparts, thus balancing the higher complexity of the procedure. We illustrate the EPM including some applications of holonomic and non-holonomic mechanical systems.