Closed Multibody Systems

In closed multibody systems, the joint coordinates depend on each other due to the closure conditions of the kinematic loops and are therefore no longer minimal coordinates as in open multibody systems. The closure conditions are formulated by cutting the loops at the so-called secondary joints leading to implicit constraints in the primary joint coordinates of the spanning tree. With the help of explicit cloure conditions, the primary joint coordinates can be expressed by suitably defined minimal coordinates. To do this, a nonlinear system of equations must be solved at the position level, which is generally only possible numerically. These relationships are first developed for a single multibody loop and then for systems with multiple loops. With the help of the implicit closure conditions, the equations of motion are formulated as differential-algebraic equations in the primary joint coordinates.With the help of the explicit cloure conditions, the equations of motion are set up as ordinary differential equations in the minimal coordinates.