Innovative Optimal Guidance of Spacecrafts for Soft-Landing on Atmosphere-less Celestial Bodies
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Nonlinear trajectory optimization based optimal guidance schemes are presented in this thesis for soft-landing of spacecrafts on atmosphere-less celestial bodies. First, a pseudo-spectral philosophy based multi-phase constrained fuel-optimal trajectory optimization problem for soft-landing on the closest celestial body (moon) is presented. The objective here is to find an optimal approach to successfully guide a spacecraft from the perilune of 18 km altitude of a transfer orbit to a height of 100 m over a specific landing site. The proposed approach takes into account various mission constraints in different phases from perilune to the landing site. These constraints include Phase-1 from 18 km to 7 km altitude where the lander's velocity is reduced suitably for camera imaging, Phase-2 to hold the attitude for 35 sec for vision camera processing for obtaining navigation error, and Phase-3 from the end of phase-2 to 100 m altitude over the landing site, where navigation accuracy is good (due to vision camera navigation inputs). This multi-phase constrained trajectory optimization problem has been successfully solved using the Legendre pseudo-spectral method. Owing to its computational efficiency this approach can be used as a guidance strategy using highly-efficient on-board processors that are going to be used in the not so distant future. Next, attention is focused on landing on more challenging celestial bodies such as asteroids, which are far away from earth, odd-shaped and have near-zero gravity. In parallel, attention is also focused to develop computationally efficient numerical techniques for solving nonlinear trajectory optimization problems so that those can be used as optimal guidance schemes in the upcoming processors in the very near future. Moreover, these techniques need to be fairly robust to uncertain parameters, which typically consist of uncertainties in initial conditions as well as gravitational inaccuracies of the relatively-unknown celestial body. With this motivation in mind, the Unscented Model Predictive Static Programming (U-MPSP) and a Quasi-Spectral version of it are developed in this thesis. U-MPSP is a fusion of two philosophies, namely the unscented optimal control formulation (which in turn is inspired from the unscented Kalman filter philosophy) as well as the model predictive static programming (MPSP), which is known for its computational efficiency. First, the unscented transform is utilized to construct a low-dimensional finite number of deterministic problems to cater for the infinitely-many possible values of the uncertain parameters. The philosophy of MPSP is utilized next so that the solution can be obtained in a computationally efficient manner. The resulting solution not only ensures that the mean value of the ensemble meets the terminal constraint in the sense of a hard constraint, but it also ensures that the associated co-variances are minimized. To reduce the dimension of the optimization problem even further, getting inspired by the pseudo-spectral philosophy, a Quasi-spectral version of U-MPSP (named as QS-UMPSP) is also presented next in this thesis. In QS-UMPSP design, a much lesser number of basis functions is used in the process, even though regular sampling time based large number of grid points can be selected for the associated numerical computations. As the optimization problem eventually leads to the optimal selection of coefficients of the basis functions, the overall dimension of the optimization process is significantly reduced. This is the core reason why QS-UMPSP is computationally much more efficient than U-MPSP. The significance of the U-MPSP and QS-UMPSP have been demonstrated by successfully solving the soft-landing problem on the asteroid Vesta.