## 6 Installation Error Compensation for the Shearer's SINS
### Coning Error and Compensation Algorithm  
Affected by the vibration of the working environment and the angular motion of the carrier itself, when the SINS of the shearer produces sinusoidal angular vibrations with the same frequency but different phases on two mutually perpendicular input axes, there will be coning motion forming on the third axis perpendicular to the above two ones, with a constant angular velocity generated [32]. The components of the angular velocity vector in the \(b\) system are  
\[\left\{ \begin{array}{l}\omega_{x}^{b}=-2\psi\sin^{2}(\alpha/2)\\ \omega_{y}^{b}=-\psi\sin\alpha\sin(\psi t)\\ \omega_{z}^{b}=\psi\sin\alpha\cos(\psi t)\end{array} \right.. \tag{55}\]  
In Equation (55), \(\psi\) is the angular frequency of coning motion, and \(\alpha\) is the half-cone angle of coning motion. Coning motion is the most severe working environment condition, which causes an additional angular velocity--constant angular velocity--with the same property as gyroscope constant drift on the \(x_{b}\) axis of the carrier coordinate system. Because the gyroscope on the \(x_{b}\) axis of the carrier coordinate system cannot automatically separate other angular velocities with the same property as the gyroscope constant drift, the system will inevitably detect the constant angular velocity to generate a coning error which seriously affects the accuracy of the position and attitude monitoring of SINS. Therefore, the coning error must be compensated. In this paper, the equivalent rotation vector method is chosen for compensation study for the coning error. The essence of this method is the use of multiple pieces of gyroscope sampled information to fit the angular motion of the carrier in one attitude update period, so that the finite rotation of the carrier is closer to an infinite small rotation to reduce the coning error. The rotation vector is defined as  
\[\Phi=\phi\cdot\overrightarrow{n}=\left[\Phi_{x}\Phi_{y}\Phi_{z}\right]^{T}= \begin{bmatrix}\alpha\sin(\psi t)\\ \alpha\cos(\psi t)\\ 0\end{bmatrix} , \tag{56}\]  
where \(\phi=\sqrt{\Phi^{T}\Phi}\) is the magnitude of the rotation vector, and \(\overrightarrow{n}\) is the unit vector on the Euler axis. The rotation vector differential equation is an important condition for coning error compensation, and the rotation vector differential equation is  
\[\dot{\Phi}=\omega+\frac{1}{2}\Phi\times\omega+\frac{1}{\phi^{2}}\bigg{[}1- \frac{\phi\sin\phi}{2(1-\cos\phi)}\bigg{]}\Phi\times(\Phi\times\omega), \tag{57}\]  
where \(\omega=\left[\omega_{x}\omega_{y}\omega_{z}\right]^{T}\) is the angular velocity vector, and the sum of the second and third terms on the right is the generated coning error. From the above equation, it can be obtained that (i) when the motion carrier rotates on a fixed axis, the direction of the rotation vector \(\Phi\) and that of the angular velocity \(\mathbf{\omega}\) are consistent; (ii) when the last two terms on the right side of the equal sign are zero--in other words, only the angular velocity \(\mathbf{\omega}\) is used for the calculation--there will be no coning error effect, but the direction of the angular velocity is not fixed, so the rotation vector and the angular velocity cannot be guaranteed to have the same direction; (iii) if the last two terms on the right side of the equal sign are not zero, when the direction of the angular velocity and that of the rotation vector are perpendicular to each other--in other words, the motion carrier is in a coning motion--there is a serious coning error effect.