A multi-body system is a system that has two or more independantly moving bodies under consideration. Each body has a mass, distributed about some center of mass, and an inertia around the center of mass.
For the purpose of this post, only the 2D case of a robotic arm with a linear actuator will be considered.
For representation of the positions, the points of interest can be defined in either global or local reference frames.
In the above figure, the global reference frame origin is denoted as $o_0$ and the local reference frames origins are denoted as $o_1$, $o_2$. The respecitve x and y axes are accordingly denoted as $x_0$, $x_1$, $x_2$ and $y_0$, $y_1$, $y_2$.