Closed-Form Solution of Seventh-Order Navigation System for Unaided Flyout

Introduction S EVERAL authors have generated time-dependent solutions to various sets of navigationerror state differential equations governing the unaided  yout of a guidedweapon.1 i 5 Developed herein is a closed-formsolution to the seventh-ordersystem of horizontalplane navigation error state equations governing the  yout of a guided weapon before its  rst navigational update. During the unaided navigation,which is usually preceded by a transfer alignment process, the vertical channel is assumed to be stabilized by barometric or radar altimetermeasurements.This solution representsan enhancement to the previously derived results i 5 in that the appropriatenumberof error states are analyzedfor the navigationproblem under consideration,a moving guidedweapon is assumed, the exact eigenvalues are extracted from the system F matrix without invoking any approximations,and the appropriatedriving terms (the three gyro biases) are treated in the correct manner. The initial step in the solution process involves extracting the eigenfrequencies of the system dynamics matrix F, a process that yields frequencies near the Schuler frequency, a frequency about 15 times smaller than the Schuler frequency (the 24-h variation), and the zero frequency (pole at the origin). The seven system eigenvectors are then derived by solving the eigenvector equation in the standard fashion.6 Next, the problemis restricted to a weapon  yout in an easterly (or westerly) direction.This restriction is necessary to permit a closed-form analytic solution when later the initial conditions are incorporatedinto the problem.Because yout in an easterly direction results in the largest crosstrack error, this restriction encompassestheworst-casescenario.The next step involvesdetermining the particular solution to the system dynamics equation (i.e., the solution that results from the three gyro bias inputs), by employing the method of undetermined coef cients. The total solution, at this point, is a sum of the homogeneoussolution,which is a linear combinationof the seven linearly independentsolutionsassociatedwith theeigenvalues,and theparticularsolution.The  nal step in the solution process is the incorporationof initial conditions,which permits determination of the seven arbitrary coef cients present in the linearly independentsolutionsconstitutingthe homogeneoussolution. In terms of providing a practical usage of the equations derived herein, it is demonstrated that the exact solution for the crosstrack error can be utilized to construct an alignment quality indicator (AQI) that can be invoked during the transfer alignment process for the guided weapon. In particular, an AQI should have the ability to predict, at any time during the alignment process, the uncertainty in the crosstrack error that would result if the weapon were to be launched at that time and then  y unaidedat a constantvelocity and heading for some prescribedperiod of time. The period of time that the weapon  ies unaided is the time to the  rst navigational update [terrain scene or global positioning system (GPS)]. In calculating the expectedcrosstrackerror at the  rst navigationalupdatetime, the AQI must utilize the current quality of alignment as providedby the appropriatevariances found in the weaponKalman  lter covariance matrix.