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Aiming at increasing the convergence rate and the accuracy simultaneously, an hp-adaptive Radau pseudospectral method is presented to generate a re-entry launch vehicle's optimal re-entry trajectory. The method determines the number of mesh intervals, the width of the each mesh interval, and the degree of the polynomial in each mesh interval iteratively until a user-specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by dividing a segment into more mesh intervals, while in regions of relatively low curvature, convergence is achieved by increasing the degree of the approximating polynomial within a mesh interval. Simulation results show that the optimized trajectory obtained by the method satisfies the path constraints and the boundary constraints successfully. Moreover, the hp-adaptive Radau pseudospectral method is shown to be more efficient than either a global Radau pseudospectral method or a fixed-low-order Radau pseudospectral method. The results indicate that the hp-adaptive Radau pseudospectral method can be applied for real-time trajectory generation due to its high efficiency and high precision.

In this paper, the adaptive simplified spherical simplex unscented Kalman filter was proposed to calculate angular velocity in gyro-free strapdown inertial navigation system. Firstly, a general angular velocity calculation modeling method with time-varying process noise was proposed, which was not limited to a certain kind of accelerometer configuration. Then aiming at the issues of large amount of calculation of unscented Kalman filter and the time variation of the process noise, and based on the characteristics of additive noise and linear state equation, the adaptive simplified spherical simplex unscented Kalman filter was proposed to estimate the angular velocity. The sampling points were decreased in this method through adopting the spherical simplex sampling strategy and not augmenting the state, thus improving the calculation efficiency. Meanwhile, Sage–Husa suboptimal maximum a posteriori noise estimator was brought in to estimate the process noise in real time in order to settle the problem of filter divergence induced by the time variation. Lastly, the proposed algorithm was simulated and also contrasted with the integration method, the evolution method and the conventional adaptive UKF algorithm. The simulation results indicated that the adaptive simplified spherical simplex unscented Kalman filter algorithm has higher precision than the integration method and evolution method and has higher efficiency than the AUKF, which could effectively improve the calculation precision and meanwhile guarantee the calculation efficiency.

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