Equations of motion

The equations of motion for the quaternions have been discussed by several authors [7,11, 12,13] with varying degrees of completeness. Note that the explicit form for the matrix connecting the angular velocity of the object in the body-fixed frame and the time derivatives of the quaternions is not treated with a uniform notation, so care must be taken when comparing the elements of this matrix as presented by different authors. For this reason, we present the development of the equations of motion in detail.

The quaternion parameters, χ, η, ξ, and ζ for a individual body are related to the Euler angles, as described by Goldstein [14], by [7]

$$\chi = \cos(\theta /2)\cdot \cos((\psi +\phi)/2),$$

$$\eta = \sin(\theta /2)\cdot \cos((\psi -\phi)/2),$$

$$\xi = \sin(\theta /2)\cdot \sin((\psi -\phi)/2),$$

$$\zeta = \cos(\theta /2)\cdot \sin((\psi +\phi)/2).$$

The quaternions satisfy the constraint,

$$\chi ^2 + \eta ^2 + \xi ^2 + \zeta ^2 = 1$$ (1)

The connection of the quaternions with the description of the dynamics of the rigid object is through the matrix equation that connects the time derivatives of the quaternions with the principal angular velocity ω _p,

$$\left( \begin{array}{c} \dot{\xi} \\ \dot{\eta} \\ \do... ...\\ \omega _{py} \\ \omega _{pz} \\ 0 \end{array} \right)$$ (2)

The 4 x 4 matrix in this equation is orthogonal so that the transformation is singularity free.

Equations of motion for the quaternions are obtained by transforming the Euler equations for a rigid body that has the center of mass fixed and is subject to torques N, in the principal frame,

$$\dot \omega _{px} = N_x/I_x + \omega _{py} \omega _{pz}(I_y - I_z)/I_x$$

$$\dot \omega _{py} = N_y/I_y + \omega _{pz} \omega _{px}(I_z - I_x)/I_y$$ (3)

$$\dot \omega _{pz} = N_z/I_z + \omega _{px} \omega _{py}(I_x - I_z)/I_z ,$$

into a quaternion form using the following sequence of matrix operations. First, define matrices Q = (ξ, η, ζ, χ) ^T and W = ( ω_px, ω_py, ω_pz, 0 )^T so that Eq. (2) becomes

$$\dot Q_\alpha = {1 \over 2} M_{\alpha \beta} W_\beta$$ (4)

where repeated Greek indices are summed. Now

$$W_{\gamma} = 2 M^T_{\gamma \alpha}\dot Q_\alpha$$ (5)

and

$$\dot W_{\gamma}=2 M^T_{\gamma \alpha}\ddot Q_\alpha +2\dot M^T_{\gamma \alpha}\dot Q_\alpha = {\cal T}_{\gamma}$$ (6)

where $\cal T$ is obtained from the right-hand side of Eq. (3) with ${\cal T}_4=0$. This reduces to

$$\ddot Q_\beta = {1\over 2}M_{\beta \gamma} {\cal T}_\gamma -M_{\beta \gamma} \dot M^T_{\gamma \alpha} \dot Q_{\alpha},$$ (7)

which in turn simplifies to

$$\ddot Q_\beta = {1\over 2}M_{\beta \gamma} {\cal T}_\gamma -Q_\beta (\dot Q^T_\alpha \dot Q_\alpha)$$ (8)

when the conditions $Q_\alpha Q_\alpha =1$ and $Q_\alpha \dot Q_\alpha =0$ are applied.

Note that the explicit form of the matrix M depends on the order of the quaternion parameters in the matrix Q and that different authors have made different choices. The general form for the equations of motion for $Q_\alpha$ is independent of this choice but any given implementation must be internally consistent.