n Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part one: Single Rigid Bodies

Abstract

This paper presents the explicit dynamic equations of a mechanical system. The equations are presented so that they can easily be implemented in a simulation software or controller environment and are also well suited for system and controller analysis. The dynamics of a general mechanical system consisting of one or more rigid bodies can be derived from the Lagrangian. We can then use several well known properties of Lie groups to guarantee that these equations are well dened. This will, however, often lead to rather abstract formulation of the dynamic equations that cannot be implemented in a simulation software directly. In this paper we close this gap and show what the explicit dynamic equations look like.These equations can then be implemented directly in a simulation software and no background knowledge on Lie theory and dierential geometry on the practitioner's side is required. This is the rst of two papers on this topic. In this paper we derive the dynamics for single rigid bodies, while in the second part we study multibody systems. In addition to making the equations more accessible to practitioners, a motivation behind the papers is to correct a few errors commonly found in literature.For the rst time, we show the detailed derivations and how to arrive at the correct set of equations. We also show through some simple examples that these correspond with the classical formulations found from Lagrange's equations. The dynamics is derived from the BoltzmannHamel equations of motion in terms of local position and velocity variables and the mapping to the corresponding quasi-velocities. Finally we present a new theorem which states that the BoltzmannHamel formulation of the dynamics is valid for all transformations with a Lie group topology. This has previously only been indicated through examples, but here we also present the formal proof.

n Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part Two: Multibody Systems

“…We note that the inertia matrix depends on the conguration x of the multibody system, which is dierent from the single rigid body case presented in From (2012).…”

“…The mapping between the time derivative of the position variableẋ and the velocity variable v is given by S(x) as (From, 2012) …”

“…We refer to Duindam and Stramigioli (2008) and From (2012) for more detail on local velocity and position variables.…”

“…However, most formulations focus on single rigid bodies, and the extension to multibody systems is often dealt with using an ab- This the the second paper on this topic. The rst paper From (2012) derived the dynamic equations for single rigid bodies using the Boltzman-Hamel equations. In this paper this work is extended to multibody systems.…”

“…The formulation is based on From (2012) where the single rigid body dynamics was derived and the explicit equations for several dierent conguration spaces were shown. In this paper we will show that when the dynamics is derived in terms of quasivelocities in this way we can simply pick the appropriate transformation from the dynamics of single rigid bodies that we found in From (2012) and stack these into one big block-diagonal matrix to obtain the multibody dynamics. The dynamic coupling will then arise naturally from the derivation.…”