On the rotational equations of motion in rigid bod

Authors: Karim Sherif Karin Nachbagauer Wolfgang Steiner

Publish Date: 2015/02/28

Volume: 81, Issue: 1-2, Pages: 343-352

PDF Link

Abstract

Many models of threedimensional rigid body dynamics employ Euler parameters as rotational coordinates Since the four Euler parameters are not independent one has to consider the quaternion constraint in the equations of motion This is usually done by the Lagrange multiplier technique In the present paper various forms of the rotational equations of motion will be derived and it will be shown that they can be transformed into each other Special attention is hereby given to the value of the Lagrange multiplier and the complexity of terms representing the inertia forces Particular attention is also paid to the rotational generalized external force vector which is not unique when using Euler parameters as rotational coordinatesMultibody systems allow the modeling of entire systems of rigid and flexible bodies connected by joints and driven by forces and actuators The most common and widely used type of rotational coordinates in multibody systems that treat the orientation of either rigid or flexible bodies are Euler angles An alternative nonminimal representation of the orientation of a body is given by a socalled quaternionic parametrization In recent years quaternionic parametrizations found new attraction because they substantially simplify the mathematical formulation and they lead in contrast to parametrization groups using three variables to a singularityfree description of the rotationThe present paper deals with the different forms of the rotational equations of motion of an unconstrained rigid body in multibody systems using Euler parameters The latter parameters are unit quaternions that is a collection of four real parameters which are not independent In terms of Euler’s theorem 1 three of the four Euler parameters represent a rotation axis and the fourth is considered as single rotation about the latter axis Since the four Euler parameters are not independent one has to consider the quaternion constraint in the equations of motion This is usually done by using the Lagrange multiplier technique 2 where the derivative of the unity of the Euler parameters is related to the angular velocity A numerical time integration of the latter equations however would lead to errors in the unity of the quaternions Therefore it is essential to explicitly extend the equations by the unity of the quaternions yielding a set of DAEs instead of ODEs In this case the Lagrange multiplier has no physical meaning Alternatively one can find several approaches avoiding the Lagrange multiplier technique eg in the work by Möller and Glocker 3 in which unconstrained quaternions are used An alternative derivation of the quaternion equations of motion for a rigid body has been presented by Udwadia and Schutte 4 In 5 the equations of motion have been derived firstly in terms of quaternions using Lagrange’s formalism in order to analyze constrained mechanical systems including nonredundant holonomic constraints Moreover the latter paper shows as well how the physical torque vector can be expressed as the generalized quaternion torque vector Alternatively in 6 a different form for the generalized quaternion torque vector is presented and by analytically solving the equations of motion it has been demonstrated that the Lagrange multiplier according to the unity of the quaternion constraint is equal to twice the rotational kinetic energy see also 7Similar to the work in 6 Vadali 8 has demonstrated that Lagrangian equations can be properly reformulated such that the value of the Lagrange multiplier is zero Although the Lagrange multiplier is zero within the latter formulation it would lead to an error in the numerical time integration if the unity of the Euler parameters is no longer explicitly considered in the equations as a result of the drift phenomenon Therefore one has to renormalize the Euler parameters after each numerical time integration step on position level in order to fulfill the unity of the Euler parameters see eg 3 7 9Although one can find different forms of the generalized quaternion torque vector and as well suggestions on the amplitude of the Lagrange multiplier when using Euler parametrization the connection between the generalized quaternion torque vector and the Lagrange multiplier has not yet been presented in the open literature In the present paper particular attention is paid on various different forms of the generalized quaternion torque vector and their influence on the amplitude of the Lagrange multiplier in rotational dynamicsFurthermore in the present paper the generalized force vector is split into two parts one part takes into account the rotational dynamics of the body while the second part does not influence the rotation of the body at all This fact has been mentioned as well in the work by Udwadia and Schutte 4 in which the connection between the physically applied torque and the generalized torque associated to the Euler parameters has been derived To be more precise the derivation in 4 reveals that the generalized torque vector includes a component in the direction of the Euler parameter vector which does not contribute to the rotational dynamics of a rigid bodyThe present paper shows that all the forms of the equations of motion presented herein lead to the same rotational dynamics of the body The difference between the presented forms is given by the amplitude of the Lagrange multiplier and the mathematical complexity in the equations of motion The Euler’s equations are the starting point of consideration in order to show that the Lagrange parameter concerning the Euler parameter constraint gets zero contrary to the derivation in 8 in which the Lagrangian equations are reformulated leading to the Euler’s equations Moreover the derivation of the equations of motion for a single rigid body from d’Alembert’s variational principle in the present paper reveals various forms when using Euler parameters for the description of the rotation Of particular importance is the presented derivation of the generalized quaternion torque vector in terms of the physically applied force vector which can also be related to the approach presented in 6 10 In addition to the work in the latter mentioned publications in the present paper the different forms of the generalized quaternion torque vector are analyzed leading to new insights and their influence on the Lagrange parameter will be emphasized hereinThis section defines the kinematic variables of interest and recalls a number of fundamental kinematic relations which are needed in the present paper Fundamental derivations concerning Euler parameters and a collection of many useful quaternion identities are presented see also 6

Keywords: