Dynamic transformations, linearization and chains of mechanical systems

We consider dynamic transformations of systems which are defined as transformations depending on the variables and their time derivatives up to finite order. Two systems are called dynamically equivalent if they have the same sets of trajectories, up to dynamic transformations. A system is called free if it is dynamically equivalent to a system which consists of a finite number of variables and no constraints whatsoever (neither differential nor algebraic). This notion is close to the notion of “differentially flat” system and means that the set of trajectories of the system is parametrized by a finite number of arbitrary functions of time. We give certain general criteria for chains of subsystems to form a free system. In particular, we show how to interconnect two (or finite number) of free systems so that they form a free system. We analyse chains of mechanical systems with nonholonomic “joints” (chain of particles, chain of trailers with rigid and elastic joints). We derive their equations of motion with controlling forces using D´Alembert principle. We show that these equations of motion form free systems which enlarges the class of systems already known to be “differentially flat” (as kinematic equations of a car with trailers). We indicate possible control applications of these properties