【摘要】 本文提出了复合四元数的概念,并用它列写出欧拉动力学方程,得出了用复合四元数求解欧拉动力学方程的一般方法。并在一类广泛应用的陀螺系统问题中,利用拉格朗日乘子法,化非线性方程为等价线性力程,从而使问题大为简化。本文所提出的方法是和方法的推广。更多还原

【Abstract】 Quaternions have been used successfully for attitude computations and kinematics of rigid body, and have been shown to be more advantageous than the method of Euler’s angles and direction cosines.But how to use quaternions in dynamics of rigid body, it is only started recent years. In this paper, the composite quaternion is introduced and used for formulating Euler’s dynamical equations, then a general method for solving Euler’s dynamical equations is obtained. Especially, by means of the method.of Lagrange multipliers, a set of nonlinear equations for a kind of gyro-system in wide use is converted into another set of linear equations. Thus the question can be greatly simplified both in expression and solution.This method is a generalization of the method ofIn this paper, we use the product A* and A, to make up a composite quaternion as its components and direvatives areUsisg these relationships, the Euler’s dynamical equations can be rewritten in the form of quaternionsEqs.( 3 )are bilinear form in terms of quaternion components and their derivatives. In this paper, we use composite quaternions to study a kind of widely applied gyro-system which uses the dynamical symmetric gyroscope as basic gyroscope and possesses spinning integration. The Euler’s dynamical equations of this system have been uncoupled among composite quaternions Ui, and can be written asWhere{S} = Cs0 s1 s2 s3]T, siss correspond to the generalized forces, which are linear functions of Solving these equations and the constraint equation The corresponding generalized constraint forces can be formed as aλi(i = 0, 1, 2, 3) by choosing multiplier a properly. Then, λi can be regarded as independent variables. In the end, we obtain a set of new equationsEq3. ( 6 ) are linear in λi, and, generally, they are linear equations with waria-ble coefficients. If conform to the constraint equation, Eqs. (6) have the same solutions with nonlinear Eqs. (4). These new equations have been .resolved from the bilinear equations given above,and be changed into equivalent linear equations The presented method may be extended to precession motion of high spinning gyroscope For this purpose,,we rewrite the precession equations in quaternion formsIn gsneral, the precession equations do not possess spinning integration. For this reason, we must get two relationships between the quaternion components. Toorientate the axis of the gyroscope, we chose the Laysal’ s angles α, β. Betweensλivs, in addition to constraint equation , there is a relationship λ0λ3- Xλ1λ2=0 yet. Rewriting the latter in differential constraint form,and multipling it by the second Lagrange multiplier b, we obtainAccording to this equation and Eq. ( 5 ),we can get the corresponding generalized constraint forcesThese equations can be used conveniently to solve the precession motion of system of inertial navigation, such as gyropendulum, gyrocompass, gyrohorizoncom.pass. etc.To verify the correctness of the method in this paper, we give a gravity symmetric gyroscope as example. Its equivalent linear equations in terms of quaternions are written, and iis three first integrals are obtained. These results are quite in accord with the results of usual algorithm.更多还原