Analytical landing trajectories for embedded autonomy

This paper considers an optimal guidance law for the initial braking phase of a soft landing mission on a celestial body without atmosphere in which boundary conditions on height and velocity are specifed. The optimal lander attitude for the minimum fuel landing problem is found. An analytic optimal trajectory is achieved by expanding the thrust acceleration, gravitational acceleration and the cosine of the vertical attitude angle to a high-order polynomial. Coefficients of these polynomials are obtained from the boundary conditions. A fixed gain control law and a direct adaptive control law are then developed to track the analytical reference trajectory. Finally, a mission scenario is presented to illustrate the accuracy of the analytical trajectory and validity of the control laws developed. The use of direct adaptive control for embedded autonomy will be directly contrasted against a traditional fixed gain controller, using a Lunar landing scenario. The advantage of the direct adaptive control approach is that it does not require system monitoring to detect thruster failure and can adjust its gain automatically. As such, direct adaptive control combined with the developed analytical solution enables autonomy to be embedded within the lander guidance and control system. In addition, it is shown that direct adaptive control increases the probability of lander survival through faster transient response and stability than a traditional fixed gain controller with system level failure detection and recovery

Adaptive backstepping control for optimal descent with embedded autonomy

Using Lyapunov stability theory, an adaptive backstepping controller is presented in this paper for optimal descent tracking. Unlike the traditional approach, the proposed control law can cope with input saturation and failure which enables the embedded autonomy of lander system. In addition, this control law can also restrain the unknown bounded terms (i.e., disturbance). To show the controller’s performance in the presence of input saturation, input failure and bounded external disturbance, simulation was carried out under a lunar landing scenario

Kalman Filtering for a Quadratic Form State Equality Constraint

International audienceThe main of focus of this note is to extend the NCKF to a situation for a general quadratic form state constraint using the standard Lagrangian multiplier technique and to obtain a mathematically robust method for finding solutions of Lagrangian multiplier, using an eigenvalue decomposition of a companion matrix of a polynomial function in place of a less robust Newton-Raphson iteration. An analytical criterion derived from the second-order optimal sufficient condition is used to select the Lagrangian multiplier corresponding to the minimum point of the performance index. The proposed filter is mathematically equivalent to the norm-constrained filter when the quadratic-form matrix is positive-semidefinit

Analytical Solutions of Generalized Triples Algorithm for Flush Air-Data Sensing Systems

International audienceInthiswork, thetriplesalgorithmforFADSsystemsisconsidered. Theprimarygoalistoextendthe triples algorithm to the general case, and to obtain closed-form solutions of angles of attackand sideslip simultaneously by using two different triples. The triple formulation is transformed toquadratichomogenousequationsofnon-dimensionalvelocitycomponents. ByapplyingBuchberger’salgorithm, a powerful tool in algebra geometry, an element of a Groebner basis for the homogenousequations is derived, in the form of a univariate polynomial equation. The degree of the univariateequation is less than or equal to four; thus, closed-form solutions can be obtained. The closed-formsolutions for the univariate polynomial equation are then substituted into the quadratic homogenouspolynomial equations to obtain the solution of non-dimensional velocity, which provides analyticalexpressions of angles of attack and sideslip