Development of a Reachability Analysis Algorithm for Space Applications

In the last decades developments in space technology paved the way to more challenging missions like asteroid mining, space tourism and human expansion into the Solar System. These missions require difficult tasks such as real-time capable guidance schemes for re-entry, landing on celestial bodies and implementation of large angle maneuvers for spacecraft. There is a need for an analysis tool to increase the robustness and success of these missions. Reachability analysis contributes to this requirement by obtaining the set of all achievable states for a dynamical system starting from an initial condition with given admissible control inputs of the system. In this study, an optimal control based reachability analysis algorithm is developed for evaluating the performance of the guidance and control methods for space missions considering the desired performance index. The developed method considers a soft-landing problem for a Moon mission as the case study, and attainable area of the lander as the performance index. The method computes feasible trajectories for the lunar lander between the point where the terminal landing maneuver starts and points that constitutes the candidate landing region. The candidate landing region is discretized by equidistant points on a two dimensional plane, i.e. in downrange and crossrange coordinates, and for each grid point a distance function is defined. This distance function acts as an objective function for a related optimal control problem (OCP). Each infinite dimensional OCP is transcribed into a finite dimensional Nonlinear Programming Problem (NLP) by using Pseudo-Spectral Methods (PSM). The NLPs are solved using available tools to obtain feasible trajectories and approximated reachable sets with information about the states of the dynamical system at the grid points. The proposed method approximates reachable sets of the lander with propellant-to-reach and time-to-reach cost by solution of NLPs. A polynomial-based Apollo guidance scheme is used to compare the results for the developed method. The coefficients that define the position of the lander are obtained by solving a series of explicit equations for the given initial and final states. A model inversion based PD controller is designed to track the generated trajectory. Feasible solutions that satisfy safe landing conditions are filtered and the results are compared for the two different approaches. Finally, the uncertainties which are characterized by initial state error and system parameters are also considered. A multivariate trajectory interpolation tool is used to interpolate RS with different initial states. A Riccati equation-based controller is designed to track the previously obtained reference trajectories within presence of the uncertainties. Monte Carlo (MC) simulations are carried out to obtain safe attainable landing area of the lunar lander as probability maps. The same uncertainty set is used to verify these probability maps by propagating the uncertainties using unscented transform. The developed tool analyzes the different guidance and control methods, for the attainable landing area of the lander, under various landing scenarios, with different dynamical models and controller parameters. Numerous quality metrics are used to compare the change of characteristics of the attainable landing area and performance of the guidance and control methods, and selected design parameters

Real-time Capable Nonlinear Model Predictive Controller Design for The Upper Stage of a Launch Vehicle

In this paper, a real-time capable Nonlinear Model Predictive Controller (NMPC) is implemented for the attitude control of an upper stage launch vehicle with liquid propellant. A mass spring model is used as an analogy to simulate the disturbance generated by the sloshing propellant. For the implementation of the NMPC, an optimal control problem (OCP) is defined with finite time horizon. The objective function is minimized while satisfying constraints on the control inputs. The resulting OCP is transcribed using single shooting method to parametrize the control inputs using uniform discretization points. The continuous control inputs are obtained by linear interpolation. A dedicated discretization algorithm in FORTRAN is coupled with a solver which used quasi-Newton algorithm to generate solutions fast. Approximation of the Hessian matrix is used to reduce computational requirements. Furthermore, the algorithm can perform parallel computation of the derivatives of the objective function with respect to optimization variables. This results in a real-time capability of generating solutions in the order of milliseconds for each iteration. The algorithm is applied for attitude maneuver and disturbance rejection for the upper stage of a launch vehicle

Entwicklung einer Erreichbarkeitsanalyse Algorithmus für Weltraumanwendungen

In the last decades developments in space technology paved the way to more challenging missions like asteroid mining, space tourism and human expansion into the Solar System. These missions require difficult tasks such as real-time capable guidance schemes for re-entry, landing on celestial bodies and implementation of large angle maneuvers for spacecraft. There is a need for an analysis tool to increase the robustness and success of these missions. Reachability analysis contributes to this requirement by obtaining the set of all achievable states for a dynamical system starting from an initial condition with given admissible control inputs of the system. In this study, an optimal control based reachability analysis algorithm is developed for evaluating the performance of the guidance and control methods for space missions considering the desired performance index. The developed method considers a soft-landing problem for a Moon mission as the case study, and attainable area of the lander as the performance index. The method computes feasible trajectories for the lunar lander between the point where the terminal landing maneuver starts and points that constitutes the candidate landing region. The candidate landing region is discretized by equidistant points on a two dimensional plane, i.e. in downrange and crossrange coordinates, and for each grid point a distance function is defined. This distance function acts as an objective function for a related optimal control problem (OCP). Each infinite dimensional OCP is transcribed into a finite dimensional Nonlinear Programming Problem (NLP) by using Pseudo-Spectral Methods (PSM). The NLPs are solved using available tools to obtain feasible trajectories and approximated reachable sets with information about the states of the dynamical system at the grid points. The proposed method approximates reachable sets of the lander with propellant-to-reach and time-to-reach cost by solution of NLPs. A polynomial-based Apollo guidance scheme is used to compare the results for the developed method. The coefficients that define the position of the lander are obtained by solving a series of explicit equations for the given initial and final states. A model inversion based PD controller is designed to track the generated trajectory. Feasible solutions that satisfy safe landing conditions are filtered and the results are compared for the two different approaches. Finally, the uncertainties which are characterized by initial state error and system parameters are also considered. A multivariate trajectory interpolation tool is used to interpolate RS with different initial states. A Riccati equation-based controller is designed to track the previously obtained reference trajectories within presence of the uncertainties. Monte Carlo (MC) simulations are carried out to obtain safe attainable landing area of the lunar lander as probability maps. The same uncertainty set is used to verify these probability maps by propagating the uncertainties using unscented transform. The developed tool analyzes the different guidance and control methods, for the attainable landing area of the lander, under various landing scenarios, with different dynamical models and controller parameters. Numerous quality metrics are used to compare the change of characteristics of the attainable landing area and performance of the guidance and control methods, and selected design parameters

Safe landing area determination for a Moon lander by reachability analysis

In the last decades developments in space technology paved the way to more challenging missions like asteroid mining, space tourism and human expansion into the Solar System. These missions result in difficult tasks such as guidance schemes for re-entry, landing on celestial bodies and implementation of large angle maneuvers for spacecraft. There is a need for a safety system to increase the robustness and success of these missions. Reachability analysis meets this requirement by obtaining the set of all achievable states for a dynamical system starting from an initial condition with given admissible control inputs of the system. This paper proposes an algorithm for the approximation of nonconvex reachable sets (RS) by using optimal control. Therefore subset of the state space is discretized by equidistant points and for each grid point a distance function is defined. This distance function acts as an objective function for a related optimal control problem (OCP). Each infinite dimensional OCP is transcribed into a finite dimensional Nonlinear Programming Problem (NLP) by using Pseudospectral Methods (PSM). Finally, the NLPs are solved using available tools resulting in approximated reachable sets with information about the states of the dynamical system at these grid points. The algorithm is applied on a generic Moon landing mission. The proposed method computes approximated reachable sets and the attainable safe landing region with information about propellant consumption and time

Approximation of attainable landing area of a moon lander by reachability analysis

Developments in space technology have paved the way for more challenging missions which require advanced guidance and control algorithms for safely and autonomously landing on celestial bodies. Instant determination of hazards, automatic guidance during landing maneuvers and likelihood maximization of safe landing are of paramount importanc

Attainable Landing Area Computation of Lunar Landers with Pseudo-Spectral and Polynomial Based Guidance Methods

In this paper, we introduce two methods for computing reachable sets, namely, via Pseudo-Spectral based optimal control techniques and via a modified form of the Apollo guidance algorithm. Specifically, in the latter case, we use the Apollo guidance to generate a reference trajectory, and a Lyapunov-based PD controller to track it. We then demonstrate the effectiveness of both approaches on the soft-landing problem for a Moon mission, where we show that although both algorithms achieve similar propellant requirements, the optimal control approach generates larger reachable sets at the expense of higher computational cost

SAFE LANDING AREA DETERMINATION FOR A MOON LANDER BY REACHABILITY ANALYSIS

In the last decade developments in space technology paved the way to more challenging missions like asteroid mining, space tourism and human expansion into to Solar System. These missions result in difficult tasks such as guidance schemes for re-entry, landing on celestial bodies and implementation of large angle maneuvers for spacecrafts. There is a need for a safety system to increase robustness and success of these missions. Reachability analysis meets this requirement by obtaining the set of all achievable states for a dynamic system starting from a set of initial conditions for the admissible control inputs and possible uncertainties in the parameters a↵ecting the dynamics of the system. This paper proposes an algorithm for the approximation of nonconvex reachable sets by using optimal control. The state space is discretized by equidistant points and for each grid point a distance function is defined resulting in an optimal control problem. The continuous optimal control problem is transcribed into a Nonlinear Programming Problem (NLP) by using Pseudo Spectral methods. Finally the NLP is solved using available tools resulting in approximated reachable sets with information about the states of the dynamical system at these grid points. The algorithm is applied on a generic moon landing mission. An optical navigation system is able to detect an available landing area within the field of view of the onboard sensors. In order to ensure safe landing during the terminal landing phase, all achievable candidate landing points must be identified considering the current attitude, path constraints and control constraints. If the candidate landing area is not suitable due to terrain conditions, the control system must steer the spacecraft to another emergency landing point or implement another maneuvering scheme. The proposed method computes approximated reachable sets which provide information about this attainable safe landing region with time and propellant costs. Additionally, interpolation of 2 sets with di↵erent initial conditions is introduced. Approximated reachable sets are obtained for di↵erent initial altitudes. Trajectories belonging to 2 distinct approximated sets are then interpolated to obtain an interpolated reachable set. The Hausdor↵ distance between the interpolated reachable set and the approximated reachable set is used to assess the quality of the result