Utilization of the Moore-Penrose inverse in the modeling of overconstrained mechanisms with frictionless and frictional joints

Marek Wojtyra , Marcin Pękal , Janusz Frączek

Abstract

The indeterminate equations that describe overconstrained mechanisms are often solved using the Moore-Penrose inverse. Some limitations of this approach are investigated here. Firstly, frictionless systems are considered. The problem of solvability of accelerations and joint reactions is studied—the non-uniqueness of reactions and uniqueness of accelerations is discussed. Next, the dependence of the results on the selection of the physical units is examined. It is checked which elements of the solution are physically non-equivalent after changing the units; relationships between different solutions are derived. Secondly, frictional systems are considered. Joint friction dependent and independent on normal load is studied. Fixed point iterations and Newton's method are applied to solve nonlinear equations of motion. The Moore-Penrose inverse is employed to conduct calculations. The null space solution components are considered, and necessary amendments in the iterative processes termination criteria are discussed. The origins of non-uniqueness of accelerations and Lagrange multipliers are analyzed. The unit-sensitivity of frictional system models is addressed. Finally, an illustrative example is given, and conclusions are drawn—limitations and possible improvements of the Moore-Penrose inverse approach are discussed.