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Figure 5: Ratio of the computational time between a m-th order expansion of the flow and its zeroth order counterpart. ne eg ne nee ae nn nn ne ee Re RE REE Me See ee eee The simulation results show that the DA-based higher-order filters are characterized by the same accuracy of their counterparts in real algebra. An important consideration on the computational effort can be made at this point. Let n be the dimension of the state vector to be estimated. In the frame of variational approach the number of equations that must be inte- grated together are we n‘1.’> If we assume that the integration time scales linearly with the number of equations, the ratio R, of the computational time between a m-th order computation and its zeroth order counterpart is Re=1+ an n?. As an example, a second order computation with n = 6 implies the integration of 258 equations and R, = 43. At third order the number of equations rises to 1554 and R, = 259. In Figure 5 the values of R, associated to DA expansion of the flow (represented in Eq. (38)) of the ODE system defined by equations (48)—(49) are compared with those of a variational approach. It is apparent that the DA method is orders of magni- tudes more efficient than the variational one. Furthermore, the DA method is application independent and the intricate task of finding the analytical expression of the variational equations is avoided.

Figure 5 Ratio of the computational time between a m-th order expansion of the flow and its zeroth order counterpart. ne eg ne nee ae nn nn ne ee Re RE REE Me See ee eee The simulation results show that the DA-based higher-order filters are characterized by the same accuracy of their counterparts in real algebra. An important consideration on the computational effort can be made at this point. Let n be the dimension of the state vector to be estimated. In the frame of variational approach the number of equations that must be inte- grated together are we n‘1.’> If we assume that the integration time scales linearly with the number of equations, the ratio R, of the computational time between a m-th order computation and its zeroth order counterpart is Re=1+ an n?. As an example, a second order computation with n = 6 implies the integration of 258 equations and R, = 43. At third order the number of equations rises to 1554 and R, = 259. In Figure 5 the values of R, associated to DA expansion of the flow (represented in Eq. (38)) of the ODE system defined by equations (48)—(49) are compared with those of a variational approach. It is apparent that the DA method is orders of magni- tudes more efficient than the variational one. Furthermore, the DA method is application independent and the intricate task of finding the analytical expression of the variational equations is avoided.

Related Figures (6)

Figure 1: Analogy between the floating point representation of real numbers in a computer environment (left figure) and the introduction of the algebra of Taylor polynomials in the differential algebraic framework (right figure).
Figure 1: Analogy between the floating point representation of real numbers in a computer environment (left figure) and the introduction of the algebra of Taylor polynomials in the differential algebraic framework (right figure).
Note that {7,;,¢;} € {1,...,2n}, where n is the dimension of the state space, Met is the predicted measurement, Ky; is the Kalman gain matrix, P74, is the cross-covariance matrix of the state and the measurement, and P;’, is the covariance matrix of the measurements. The oe ) Operator includes the higher-order partials of the solution flow, which map the deviations at time m to time (m+1), and h ae ) includes the higher-order partials of the measurement function. These Tither-order partials are called state transition tensors (STTs). In order to apply the HNEKF, the reference trajectory and its higher-order state transition tensors must be integrated for each time interval solving a system of differential equations up to the order m of the Taylor expansion. The differential equations up to the third-order deviation are given in Eq. (36)
Note that {7,;,¢;} € {1,...,2n}, where n is the dimension of the state space, Met is the predicted measurement, Ky; is the Kalman gain matrix, P74, is the cross-covariance matrix of the state and the measurement, and P;’, is the covariance matrix of the measurements. The oe ) Operator includes the higher-order partials of the solution flow, which map the deviations at time m to time (m+1), and h ae ) includes the higher-order partials of the measurement function. These Tither-order partials are called state transition tensors (STTs). In order to apply the HNEKF, the reference trajectory and its higher-order state transition tensors must be integrated for each time interval solving a system of differential equations up to the order m of the Taylor expansion. The differential equations up to the third-order deviation are given in Eq. (36)
Figure 2: Scheme of the DAHNAEKF-m prediction step. The exponential map related to the predetermined trajectory is computed offline while only map evaluations are performed online.
Figure 2: Scheme of the DAHNAEKF-m prediction step. The exponential map related to the predetermined trajectory is computed offline while only map evaluations are performed online.
Figure 6: Results obtained using the DAHNEKF-m, with m = 1, 2, 3, for the Earth’s artificial satellite test case.
Figure 6: Results obtained using the DAHNEKF-m, with m = 1, 2, 3, for the Earth’s artificial satellite test case.
Related topics:
Mechanical EngineeringAerospace EngineeringDifferential AlgebraOrbit Determination
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