Multibody kinematics. A topological formulation based on structural-group coordinates

Kinematic analysis plays a fundamental role in multibody dynamics. Moreover it is frequently used in synthesis problems, as a first stage in the design of mechanical systems and, in other cases, the interest in the multibody system is purely kinematic (position analysis, range of movement, etc.). To perform a kinematic analysis, the configuration of the multibody system has to be described using a specific set of coordinates q: topological formulations use relative coordinates and global formulations use Cartesian or natural coordinates. When closed kinematic chains are present, the number of coordinates that describe the system is larger than its mobility, so it becomes necessary to define a vector Φ(q,t) of constraint equations that relate the q coordinates to each other (1). Once this system of equations is obtained, the kinematic analysis can be performed. To do so, Eq.(1) is differentiated with respect to time and the system of constraint equations at velocity level is obtained (2). In (2), Φq is the Jacobian matrix of the constraint vector. The constraint equations at acceleration level can be obtained by differentiating (2) with respect to time (3). In (2) and (3), the sub index t means a partial derivative with respect to time, ?̇? and ?̈? represents the vectors of dependent velocities and accelerations. Φ(q, t) = 0 (1)