On the Hamel Coefficients and the Boltzmann–Hamel Equations for the Rigid Body

The Boltzmann–Hamel (BH) equations are central in the dynamics and control of nonholonomic systems described in terms of quasi-velocities. The rigid body is a classical example of such systems, and it is well-known that the BH-equations are the Newton–Euler (NE) equations when described in terms of rigid body twists as quasi-velocities. It is further known that the NE-equations are the Euler–Poincaré, respectively, the reduced Euler–Lagrange equations on SE(3) when using body-fixed or spatial representation of rigid body twists. The connection between these equations are the Hamel coefficients, which are immediately identified as the structure constants of SE(3). However, an explicit coordinate-free derivation has not been presented in the literature. In this paper the Hamel coefficients for the rigid body are derived in a coordinate-free way without resorting to local coordinates describing the rigid body motion. The three most relevant choices of quasi-velocities (body-fixed, spatial, and hybrid representation of rigid body twists) are considered. The corresponding BH-equations are derived explicitly for the rotating and free floating body. Further, the Hamel equations for nonholonomically constrained rigid bodies are discussed, and demonstrated for the inhomogenous ball rolling on a plane.

Dynamics modeling and performance analysis of underwater vehicles based on the Boltzmann-Hamel equations approach

The paper addresses dynamics modeling of underwater vehicles, which are typical examples of variable mass and configuration mechanical systems. Specifically, we address inertia–based propelled vehicles, whose mass changes due to change of water amount in their water tanks and their configuration changes due to movable mass enabling maneuvers. They can be additionally subjected to constraints, which are imposed due to their desired performance, tracking pre-specified motions or are control constraints implying underactuation. The paper presents a new approach to variable mass and configuration systems modeling, which is based upon Boltzmann-Hamel equations derived in quasi-coordinates. The dynamics framework based upon quasi-coordinates results in dynamic models suitable for performance analysis and model-based control applications. It serves free or constrained variable mass and configuration systems modeling. The example researched in the paper is an underwater vehicle, which is purely inertia–based propelled. The theoretical development is illustrated with simulation studies of dynamical behavior and performance of the underwater vehicle model moving a specified trajectory.

Dynamic Simulation of a Tripod Based in Boltzmann-Hamel Equations

In this paper, authors show the results of performance simulation of a 3PRS lower-mobility parallel manipulator actuated by electronic devices. The simulation includes a mechatronic model of the actuators, the dynamic response of the mechanism found using the Boltzmann-Hamel equations, and a model of the control. The dynamics of the actuators is modelled using two transfer functions found by means of a test based identification method. Authors find the explicit Dynamic equations needed for simulation using analytical mechanics. Lagrange equations are convenient and systematic in open-loop serial kinematic chains. However, for closed-loop mechanisms are cumbersome, even with the help of Lagrange multipliers. Moreover, if spatial rotations are involved, generalized coordinates are very much coupled in the expressions of Lagrange functions to be differentiated. Boltzmann-Hamel equations come to help in this regard. By using Simulink, authors could evaluate the stability of the control and the bandwidth of the manipulator.

Alternative Causality Assignment Procedures in Bond Graph for Mechanical Systems

This paper proposes to extend the set of causality assignment procedures. The proposed alternative procedures are mainly inspired by formulations developed in the mechanical domain. They enable Lagrange equations, Hamilton equations, and Boltzmann-Hamel equations to be obtained, as well as formulations with the Lagrange multipliers. In the context of system modeling a varied set of mechanical oriented equations are available in a systematic way from the bond graph representation and the proposed corresponding procedures provide an algorithmic frame for programming these mathematical formulations. The graphical features of the bond graph tool and the causality stroke concept enable formulations to be methodically obtained, formulations that can otherwise be very awkward to express. Also these procedures emphasize certain interesting properties of the bond graph tool e.g.: there is a clear distinction between the energy topology of a system and its dynamic equations; it also enables graphic structural analyses to be undertaken; and finally it can play a pedagogical role in engineering education.