Modeling and Control of Mechanical Systems in Terms of Quasi-Velocities

Multi-body systems’ (MBS) dynamics are often described by the second-order nonlinear equations parameterized by a configuration-dependent inertia matrix and the nonlinear vector containing the Coriolis and centrifugal terms. Since these equations are the cornerstone for simulation and control of robotic manipulators, many researchers have attempted to develop efficient modelling techniques to derive the equations of motion of multi-body systems in novel forms. A unifying idea for most modeling techniques is to describe the equations of motion in terms of general coordinates and their time–derivatives. In classical mechanics of constrained systems, a generalized velocity is taken to be an element of tangential space of configuration manifold, and a generalized force is taken to be the cotangent space. However, neither does space possess a natural metric as the generalized coordinates or the constrains may have a combination of rotational and translational components. As a result, the corresponding dynamic formulation in not invariant and a solution depends on measure units or a weighting matrix selected Aghili (2005); Angeles (2003); Lipkin and Duffy (1988); Luca and Manes (1994); Manes (1992). There also exist other techniques to describe the equations of motion in terms of quasi–velocities, i.e., a vector whose Euclidean norm is proportional to the square root of the system’s kinetic energy, which can lead to simplification of these equations Aghili (2008; 2007); Bedrossian (1992); Gu (2000); Gu and Loh (1987); Herman (2005); Herman and Kozlowski (2006); Jain and Rodriguez (1995); Junkins and Schaub (1997); Kodischeck (1985); Kozlowski (1998); Loduha and Ravani (1995); Papastavridis (1998); Rodriguez and Kertutz-Delgado (1992); Sinclair et al. (2006); Spong (1992). A recent survey on some of these techniques can be found in Herman and Kozlowski (2006). In short, the square–root factorization of mass matrix is used as a transformation to obtain the quasi–velocities, which are a linear combination of the velocity and the generalized coordinates Herman and Kozlowski (2006); Papastavridis (1998). It was shown by Kodistchek Kodischeck (1985) that if the square–root factorization of the inertia matrix is integrable, then the robot dynamics can be significantly simplified. In such a case, transforming the generalized coordinates to quasi–coordinates by making use of the integrable factorization modifies the robot dynamics to a system of double integrator. Then, the cumbersome derivation of the Coriolis and centrifugal terms is not required. It was later realized by Gu et al. Gu and Loh (1987) that such a transformation is a canonical transformation because it satisfies Hamilton’s equations. Rather than deriving the mass matrix of MBS first and then obtaining its factorization, Rodriguez et al. Rodriguez and Kertutz-Delgado (1992)