Optimal Thrust Profile for Planetary Soft Landing Under Stochastic Disturbances

B YAND large, the literature on optimal control deals with the minimization of a performance index that penalizes control energy, because the input appears in quadratic form as part of the running cost. Such problems are typically referred to as minimum energy problems in optimal control theory—they involve the minimization of the L-norm of an otherwise unconstrained control signal. While L minimization can be useful in addressing several optimal control problems in engineering (preventing engine overheating, avoiding high-frequency control input signals etc.), there are practical applications in which the control input is bounded (e.g., due to actuation constraints), and theL-norm is amore suitable choice to penalize. These problems are also called minimum-fuel problems, due to the nature of the running cost, which involves an integral of the absolute value of the input signal. Minimum-fuel control appears as a necessity in several settings, especially in spacecraft guidance and control [1,2], in which fuel is a limited resource. Indeed, in such applications, using the L-norm results in significantly more propellant consumption, as well as undesirable continuous thrusting. In some illustrative examples, this fuel penalty can be as high as 50% [3]. In this paper, we address a stochastic version of the so-called soft-landing problem (SLP). The objective of the SLP is to find the optimal thrust profile for a spacecraft attempting to make a soft landing on a planet, using the minimum amount of fuel. The problem was originally addressed by considering only one spatial dimension (namely, the altitude with respect to the planet), in which case its deterministic formulation offers a closed-form solution (initially obtained by Miele [4,5] during the 1960s; see also [6,7]). In more recent years, there has been renewed interest in the topic, appearing under the name powered-descent guidance (PDG), mainly due to the success of NASA’s Mars Science Laboratory program. Several results appear in the literature, treating a more complex problem involving all three spatial dimensions, more accurate modeling of the dynamics to account for planetary rotation, and several state and control constraints [8–12]. An analysis for fuel optimality is also included in [13,14]. The challenges faced in the implementation of planetary PDG controllers are the twofold: 1) the environmental uncertainty and stochastic disturbances present, and 2) the limited capabilities for onboard computation. The aim of this paper is to present an application of the framework of stochastic control using forward and backward sampling, developed in previous work by the authors [15–17], to the SLP. The problem considered within this paper is a combination of a stochastic L-optimal control problem with a first-exit type of formulation. Such a combination has not been previously considered in our work, and no results have been published containing applications of our algorithm on first-exit problems in general. The motivation for this problem formulation stems from the fact that the proposed algorithm is intended to be deployed during the very last stage of the descent (i.e., seconds before landing), when the altitude is relatively low. In this setting, it is assumed that any constraints with respect to navigating toward the landing site have been already satisfied, and the final objective is to ensure that the spacecraft touches the ground smoothly, despite environmental disturbances. In fact, to the best of our knowledge, the majority of PDG controllers involve a terminal stage (i.e., just before landing), where the trajectory is vertical (see, e.g., [18]). Furthermore, this paper illustrates the advantages of a control law stemming from our framework, as opposed to a closedloop deterministic control law that does not directly take stochasticity into consideration. Specifically, it is shown that simply employing a deterministic closed-loop controller in a model-predictive control fashion does not mitigate the risk of a crash during landing. In contrast, the proposed feedback controller, while subject to the same control structure restrictions, drastically reduces the chances of a crash, making them arbitrarily small. The proposed algorithm demonstrates superior performance, offering a much lower mean and variance for the touchdown speed. Depending on the given safety specifications, we can further reduce this mean and variance, thus gaining a more robust, safer controller, at the expense of slightly increased fuel expenditure. Finally, the nature of the algorithm allows for a complete solution of the problem a priori and off-line, thus minimizing the required onboard computing capabilities of the spacecraft.