A survey of equations of motion in terms of inertial quasi-velocities for serial manipulators

Abstract

In this work we compare equations of motion using the so-called inertial quasi-velocities. As a result of these velocities we obtain two first-order decoupled equations of motion instead of one second-order differential equation of motion. The methods presented here, solve in a way, the problem of nonlinear dynamic decoupling. The first and the second method result from diagonalized Lagrangian robot dynamics (Jain and Rodriguez, IEEE Trans Robot Autom 11: [571][572][573][574][575][576][577][578][579][580][581][582][583][584] 1995) and are known as normalized and unnormalized quasi-velocities. The third method described by Junkins and Schaub (J Astronaut Sci 45:279-295, 1997) offers eigenfactor quasi-coordinate velocities formulation for multibody dynamics. As a consequence of using transformation given by Loduha and Ravani (Trans ASME J Appl Mech 62:216-222, 1995) we obtain decoupled equations of motion in terms of modified generalized velocity components. Here we limit all these methods to serial manipulators. The novelty of this paper consists in physical interpretation of the quasi-velocities and discussion concerning equations of motion, the kinetic energy shaping, relationship between each of them and properties useful for simulation and control purposes. Also forward dynamics algorithms and their computational complexity in terms of new velocities are given. Simulation results illustrate the theoretical investigations. We conclude that all methods offer interesting possibilities for dynamic simulation and future control investigations.

Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction

“…Quasi-velocities may also be defined as the projection of the instantaneous angular and gravity center velocities on the axes of some suitable frame [47]. Usually quasivelocities are applied to decouple and simplify, in a way, the dynamics, with diagonalized Lagrangian dynamics and simplified kinetic energy form [21,38]. This is also the case, as shown below, of the quasi-velocity that is analyzed in this paper.…”

“…Many different ways of transforming the Lagrange dynamics into more suitable "canonical" forms have been proposed. Among these some are based on quantities known as quasi-velocities (or non-holonomic velocities, or generalized velocities, or generalized speeds, or pseudo-velocities, or kinematic characteristics) [32,38,46,47,49,59,61,71]. Others use generalized coordinate transformations [64].…”

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

“…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.…”

“…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).…”

“…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).…”