Calculation of the Direction Cosine (GPS)

The previous sections have motivated the necessity of maintaining accurate direction cosine matrices. The following two subsections will consider two methods for maintaining the direction cosine matrix as the two reference frames experience arbitrary relative angular motion. Each technique relies on measuring the relative angular rate and integrating it (via different methods). Initial conditions for the resulting differential equations are discussed in Sections 10.3 and 11.7. Prior to integration, the angular rates should be properly compensated for biases and navigation frame rotation as discussed in Section 11.3.

The measurement of angular rates followed by integration to determine angle has the disadvantage that measurement errors will accumulate during the integration process. However, the approach has several distinct advantages. The angular rates are measurable via inertial measurements. Inertial measurements do not rely on the reception of any signal exterior to the sensor itself. Therefore, the accuracy of the measurement and integration processes will only be limited by the accuracy inherent to the instrument and of the integration process. Both of these quantities can be accurately calibrated so that the system accuracy can be reliably predicted. Also, the error accumulation through the integration of inertial quantities is a slow process that can be corrected via aiding sensors. Alternatively, level sensors and compass type instruments could be used to directly measure the Euler angles. However, level (gravity vector) sensors are sensitive to acceleration as well as attitude changes; and, yaw/heading sensors are sensitive to local magnetic fields. The resulting measurement errors are difficult to quantify accurately and reliably at the design stage.

Direction Cosine Derivatives

Eqn. (2.53) provides a differential equation which can be integrated to maintain the direction cosine matrix. Such an approach could numerically integrate each of the direction cosine elements separately; however, since the equation is linear, a closed form solution should be obtainable. Direct numeric integration of eqn. (2.53) does not enforce the orthogonality nor normality constraints on the direction cosine matrix; hence, additional normalization calculations would be required.

Lettmp20-532_thumband select T sufficiently small thattmp20-533_thumbcan be considered constanttmp20-534_thumbTo derive a closed form solution to eqn. (2.53) with initial conditiontmp20-535_thumbdefine the integrating factor


wheretmp20-541_thumbhas been used as a shorthand notation fortmp20-542_thumbto decrease the complexity of the notation. Thetmp20-543_thumbnotation is defined in eqn. (B.15).

Multiplying the integrating factor into eqn. (2.53) and simplifying, yields tmp20-547_thumb

where the last step is valid given the assumption that T is sufficiently small so thattmp20-548_thumbis constant over the period of integration. Integrating both sides, over the period of integration yields


To simplify this expression, let


wheretmp20-553_thumbPowers of the matrixtmp20-554_thumbreduce as follows:


which are verified in Exercise 2.10. Therefore, using eqn. (B.42),


Substituting eqn. (2.69) into eqn. (2.63) yields


wheretmp20-560_thumbEqn. (2.70) is properly defined theoretically, even astmp20-561_thumbbut must be implemented numerically with care.

The designer must ensure that the intervaltmp20-562_thumbis sufficiently small (i.e., the sample frequency is sufficiently fast) to satisfy the assumption above eqn. (2.62) that eachtmp20-563_thumbcan be considered constant over each period of integration.

Euler Angle Derivatives

Section 2.5 defined the Euler anglestmp20-564_thumband described their use to determine the direction cosine matrixtmp20-565_thumbThis method of determining the direction cosine matrix requires the navigation system to measure or compute the Euler angles. This section presents the method for computing the Euler angle derivativestmp20-572_thumbfrom the angular rate vectortmp20-573_thumb The navigation system could then integrate the Euler angle derivatives, to determine the Euler angles from which the direction cosine matrix can be computed.

Note that the three-tupletmp20-574_thumbis not related totmp20-575_thumbby a rotational transform because each of the components oftmp20-576_thumbis defined in a different reference frame. Instead, we have that


Figures 2.14-2.16 define the yaw, pitch, and roll rotations. In addition, the figures show thattmp20-583_thumbis the rate of rotation about the K ortmp20-584_thumbaxes, which tmp20-585_thumb are coincident; that _ is the rate of rotation about thetmp20-586_thumbaxes, which

are coincident; and, thattmp20-587_thumbis the rate of rotation about thetmp20-588_thumbaxes, which are coincident. From the facts in the previous sentence we have that


From eqn. (2.71) we have


From eqn. (2.42),


and from eqns. (2.41-2.42),


Therefore, eqn. (2.72) yields


The inverse transformation is


Neithertmp20-602_thumbnor its inverse is a rotation matrix. Eqns. (2.73) and (2.74) do not represent vector transformations between frames-of-reference. The matrixtmp20-603_thumbis singular whentmp20-604_thumb

Eqn. (2.74) is a nonlinear ordinary differential equation for the Euler angles. Numeric integration of eqn. (2.74) provides estimates of the Euler angles from which the direction cosine matrix can by computed using eqn. (2.43). This approach requires numerous trigonometric operations at the high rate of the attitude portion of the INS system.

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