Gregory Lantoine

Using Multicomplex Variables for Automatic Computation of High-Order Derivatives

The computations of the high-order partial derivatives in a given problem are often cumbersome or not accurate. To combat such shortcomings, a new method for calculating exact high-order sensitivities using multicomplex numbers is presented. Inspired by the recent complex step method that is only valid for firstorder sensitivities, the new multicomplex approach is valid to arbitrary order. The mathematical theory behind this approach is revealed, and an efficient procedure for the automatic implementation of the method is described. Several applications are presented to validate and demonstrate the accuracy and efficiency of the algorithm. The results are compared to conventional approaches such as finite differencing, the complex step method, and two separate automatic differentiation tools. The multicomplex method performs favorably in the preliminary comparisons and is therefore expected to be useful for a variety of algorithms that exploit higher order derivatives.

A Hybrid Differential Dynamic Programming Algorithm for Robust Low-Thrust Optimization

Low-thrust propulsion is becoming increasingly considered for future space missions, but optimization of the resulting trajectories is very challenging. To solve such complex problems, differential dynamic programming is a proven technique based on Bellman’s Principle of Optimality and successive minimization of quadratic approximations. In this paper, we build upon previous and existing optimization strategies to present an alternative hybrid variant of differential dynamic programming for robust low-thrust optimization. It uses first- and second-order state transition matrices to take advantage of an efficient discretization scheme and obtain the partial derivatives needed to perform the minimization. Unlike the traditional formulation, the state transition approach provides valuable constraint sensitivities and furthermore is naturally amenable to parallel computation. The method includes also a smoothing strategy to improve robustness of convergence when starting far from the optimum, as well as the capability to handle efficiently both soft and hard constraints. Procedures to drastically reduce the computation cost are mentioned. Preliminary numerical results are presented and compared to existing algorithms to illustrate the performance and the accuracy of our approach.

A Hybrid Differential Dynamic Programming Algorithm for Constrained Optimal Control Problems. Part 2: Application

In the first part of this paper series, a new solver, called HDDP, was presented for solving constrained, nonlinear optimal control problems. In the present paper, the algorithm is extended to include practical safeguards to enhance robustness, and four illustrative examples are used to evaluate the main algorithm and some variants. The experiments involve both academic and applied problems to show that HDDP is capable of solving a wide class of constrained, nonlinear optimization problems. First, the algorithm is verified to converge in a single iteration on a simple multi-phase quadratic problem with trivial dynamics. Successively, more complicated constrained optimal control problems are then solved demonstrating robust solutions to problems with as many as 7 states, 25 phases, 258 stages, 458 constraints, and 924 total control variables. The competitiveness of HDDP, with respect to general-purpose, state-of-the-art NLP solvers, is also demonstrated.

Near Ballistic Halo-to-Halo Transfers between Planetary Moons

Intermoon transfers are important components of planetary tour missions. However, these transfers are challenging to design due in part to the chaotic environment created by the multi-body dynamics. The specific objective of this work is to develop a systematic methodology to find fuel optimal, near ballistic Halo-to-Halo trajectories between planetary moons, and we achieve this goal by combining dynamical systems theory with a variety of nonlinear programming techniques. The spacecraft is constrained to start at a Halo orbit of a moon and end at another Halo orbit of a second moon. Our approach overcomes the obstacles of the chaotic dynamics by combining multiple “resonant-hopping” gravity assists with manifolds that control the low-energy transport near the Halo orbits of the moons. To help construct good initial guesses, contours of semimajor axes that can be reached by falling off a Halo orbit are presented. An empirical relationship is then derived to find quickly the boundary conditions on the Halo orbits that lead to ballistic capture and escape trajectories, and connect to desired resonances. The overall optimization procedure is broken into four parts of increasing fidelity: creation of the initial guess from unstable resonant orbits and manifolds, decomposition and optimization of the trajectory into two independent ideal three-body portions, end-to-end refinement in a patched three-body model, and transition to an ephemeris model using a continuation method. Each step is based on a robust multiple shooting approach to reduce the sensitivities associated with the close approach trajectories. Numerical results of an intermoon transfer in the Jovian system are presented. In an ephemeris model, using only 55 m/s and 205 days, a spacecraft can transfer between a Halo orbit of Ganymede and a Halo orbit of Europa.

Quadratically Constrained Linear-Quadratic Regulator Approach for Finite-Thrust Orbital Rendezvous

This paper focuses on the design of a quadratically constrained linear-quadratic regulator for finite-thrust orbital rendezvous. The original linear-quadratic optimal control problem is subject to maximum thrust magnitude and quadratic collision avoidance constraints. Thrust arcs are approximated by impulsive velocity increments and the Yamanaka–Ankersen transition matrix propagates the state vector. An explicit closed-loop solution is obtained by performing high-order series expansions of the Hamilton–Jacobi–Bellman equation on subregions of the state space associated with specific sets of active constraints. The algorithm is computationally efficient because the Lagrange multipliers are expressed as polynomial functions of the states and can be computed offline. A rendezvous in an elliptical orbit is considered to demonstrate the application of this method.

Jovian Orbit Capture and Eccentricity Reduction Using Electrodynamic Tether Propulsion

The use of electrodynamic tethers for propulsion and power generation is attractive for missions to the outer planets, which are traditionally handicapped by large propellant requirements, large times of flight, and a scarcity of power available. In this work, the orbital dynamics of a spacecraft using electrodynamic tether propulsion during the mission phases of capture, apojove pump down and perijove pump up, in the Jovian system are investigated. The main result is the mapped design space involving mission duration, tether length, and minimum perijove radius. Phase-free flyby sequences are also included, which provide performance upper bounds for a given mission architecture. It is found to be advantageous to use inbound-only flybys of the Galilean moons during capture. Flybys during the apojove pump down phase are only useful in conjunction with a perijove-raising mechanism at apoapse such as solar perturbations or a small propulsive maneuver. The electrodynamic tether system is also shown to be capab...

Periodic Orbits and Equilibria Near Jovian Moons Using an Electrodynamic Tether

Various researchers have proposed the use of electrodynamic tethers for power generation and capture from interplanetary transfers. In this paper, the effect of tether forces on periodic orbits in the Jupiter–Io system is investigated. A series of simplifications to the Lorentz force-perturbed circular-restricted three-body problem allows the development of a conservative formulation that admits a Jacobi integral. Although the conservative approximation introduces a modest magnitude error in the regions of interest, the correct perturbation direction is preserved. The presence of the Jacobi integral is amenable to the search for equilibria, periodic orbits, and the use of dynamic tools such as zero-velocity curves. The perturbed equations of motion lead to modified equilibrium positions, which are found at both Io and Metis. New families of modified Lyapunov orbits are generated at Io as functions of tether size and Jacobi integral from preexisting families as well as the new modified equilibrium points. ...

A Unied Framework for Robust Optimization of Interplanetary Trajectories

A framework intended for robust optimization of interplanetary trajectories is described. The trajectory is divided into multiple phases, and each phase is discretized into segments. The resulting discrete problem can be solved by several state-of-the-art solvers directly integrated in the framework. In addition, a wide variety of constraints and dynamical functions are available, which makes the problem modeling highly exible. An interactive visualization capability is also included to monitor the optimization process in real time. This framework is implemented in a general-purpose trajectory optimization tool prototype, and several numerical examples from real-world mission design activities are presented.

Psyche: Journey to a metal world

In September 2015, NASA selected five mission concepts from a field of 27 to proceed to the next stage (step 2) of the latest Discovery mission competition. Each team submitted a Mission Concept Study to NASA in August 2016, and in January of 2017 NASA selected Psyche and a second mission for flight. This paper describes Psyche, a unique investigation of a metal world, which is the only one of the original five mission concepts studied in detail to propose the use of Electric Propulsion (EP) to accomplish its mission objectives. Psyche will harness commercially developed EP and space power systems with strong system-level heritage to accomplish a deep space NASA science mission at comparatively low technical-risk and cost-risk. This paper describes the Psyche mission concept and the unique Solar Electric Propulsion (SEP) architecture that allows the use of SSLs commercial SPT-140 Hall thruster propulsion system at solar distances of up to 3.3 AU with only minimal modifications. Building on previous work analyzing SEP systems for Discovery-class missions, this paper describes the heritage, design, and testing which have been conducted on the power and propulsion systems to develop the Psyche mission, addresses the differences between GEO and deep-space environments, and describes actions taken to ensure that GEO heritage systems can be operated reliably in deep-space.

VAMOS: a SmallSat mission concept for remote sensing of Venusian seismic activity from orbit

The apparent youthfulness of Venus’ surface features, given a lack of plate tectonics, is very intriguing; however, longduration seismic observations are essentially impossible given the inhospitable surface of Venus. The Venus Airglow Measurements and Orbiter for Seismicity (VAMOS) mission concept uses the fact that the dense Venusian atmosphere conducts seismic vibrations from the surface to the airglow layer of the ionosphere, as observed on Earth. Similarly, atmospheric gravity waves have been observed by the European Venus Express’s Visible and Infrared Thermal Imaging Spectrometer (VIRTIS) instrument. Such observations would enable VAMOS to determine the crustal structure and ionospheric variability of Venus without approaching the surface or atmosphere. Equipped with an instrument of modest size and mass, the baseline VAMOS spacecraft is designed to fit within an ESPA Grande form factor and travel to Venus predominantly under its own power. Trade studies have been conducted to determine mission architecture robustness to launch and rideshare opportunities. The VAMOS mission concept was studied at JPL as part of the NASA Planetary Science Deep Space SmallSat Studies (PSDS3) program, which has not only produced a viable and exciting mission concept for a Venus SmallSat, but has also examined many issues facing the development of SmallSats for planetary exploration, such as SmallSat solar electric propulsion, autonomy, telecommunications, and resource management that can be applied to various inner solar system mission architectures.