An Alternative Derivation of the Quaternion Equations of Motion for Rigid-Body Rotational Dynamics

In this note, a general formulation for rigid body rotationaldynamics is developed using quaternions, also known as Eulerparameters. The use of quaternions is especially useful in multi-body dynamics when large angle rotations may be involved sincetheir use does not cause singularities to arise, as it occurs whenusing Euler angles.The equations of rotational motion in terms of quaternions ap-pear to have been ﬁrst obtained by Nikravesh et al. 1 . The au-thors in this particular paper developed an approach to model andsimulate constrained mechanical systems in which the compo-nents bodies in the mechanical system are connected by nonre-dundant holonomic constraints. Since they use unit quaternions toparametrize the angular coordinates of a rigid body, they providemany useful unit quaternion identities and show their relation tothe components of the angular velocity of the rigid body. To de-velop a suitable set of equations of motion involving quaternions,they utilize Lagrange’s equation. Of course the components of theunit quaternion must be of a unit norm and they are not all inde-pendent, and so the unit norm requirement must be imposed as aconstraint on the system. They employ the Lagrange multipliermethod to deal with this characteristic, and in so doing, they ar-rive at a mixed set of ordinary differential equations and an alge-braic equation, which comprises a differential algebraic equation DAE . The accelerations are found by inverting the DAE massmatrix, and the Lagrange multiplier is explicitly found. The gen-eralized torques in this framework are given in a four-vector for-mat, and its relation to a physically applied torque in the body-ﬁxed coordinate frame is provided by computing the virtual workof a force located at an arbitrary point of the rigid body andutilizing quaternion identities.In a later paper, Morton 2 obtains both Hamilton’s andLagrange’s equations in terms of quaternions. However, neither ofthese sets of equations is developed solely through the use ofHamiltonian or Lagrangian dynamics, as one might have ex-pected. Morton’s starting point for each of these sets ofequations—the backbone for all of his derivations—is the New-tonian equations of rotational motion that describe the rate ofchange in the angular momentum in terms of quaternions Eqs. 49 and 57 in Ref. 2 . To obtain Hamilton’s equations, Mortoncompares the rate of change in the angular momentum, expressedin terms of quaternions using Newton’s equations and what isobtained using the Hamiltonian formalism Eqs. 57 and 62bare compared in Morton’s paper 2 . He thus obtains, through thissomewhat long and involved procedure, the crucial connectionbetween the physically applied torque vector and the generalizedquaternion torques that are necessary to complete his Hamilton’sequations. In deriving Lagrange’s equations, Morton likewisestarts with the Newtonian equations of motion in terms of quater-nions Eq. 49 in Morton’s paper 2 and compares these equa-tions with what is obtained by carrying out the Lagrange deriva-tive of the kinetic energy of rotation Morton’s Eqs. 79 and 82 2 . Results obtained through comparisons between the Hamil-tonian formulation and the Newtonian formulation, regarding theconnection between the physically applied torque and the quate-rion torque, are then utilized in the Lagrange derivative to identifythe Lagrange multiplier that is needed to enforce the constraintthat the quaternion must have a unit norm. Hence, the derivationof the Lagrangian equations of rotational motion in terms ofquaternions appears to be carried out via a mixed line ofthinking—some of it Newtonian, some Hamiltonian, someLagrangian,—and the ﬁnal Hamilton and Lagrange equations areboth obtained basically through a comparison with those obtainedby the Newtonian approach.Having described in detail the two principal methods used todate in arriving at the requisite equations of motion, in this paper,we develop the Lagrange equations for rotational motion using adirect Lagrangian approach and follow a simple and straight-forward line of reasoning rooted solely within the framework ofLagrangian dynamics. The development does not require the useof Lagrange multipliers, and it yields a positive deﬁnite massmatrix involving only the quaternion components. After the un-constrained equations of motion are obtained, the so-called funda-mental equation is directly employed to get the ﬁnal equationsdescribing rotational motion. The ease, conceptual simplicity, andclarity, with which the equations are derived, are striking whencompared with the derivation of Morton 2 . No appeal to New-tonian mechanics is made, and no comparisons with results fromit are made to arrive at the requisite equations. Of special impor-tance is the simple derivation provided herein of the connectionbetween the physically applied torque and the generalized quater-nion torque. Unlike in Ref. 2 , it is obtained completely withinthe framework of Lagrangian mechanics from simple argumentsrelated to virtual work, and unlike in Ref. 1 , it is achieved in afew simple steps that require less appeal to quaternion identitiesand algebraic manipulations. The approach used in the derivationis new and relies on some simple recent results dealing with thedevelopment of the equations of motion of constrained mechani-cal systems obtained by Udwadia and Kalaba 3,4 . In addition, itprovides new insights that were unavailable before.