# Projects

## Vision-Based Navigation system (VBN) for Autonomous Satellite Navigation in Space

Interdisciplinary Center for Security, Reliability and Trust (SnT), Université du Luxembourg

July 2021 – June 2023

Eurostars FNR-INTER Grant Number: INTER20/EUROSTARS/15254521

The project aims to create a functional VBN system prototype at TRL 7, representing a future commercial off-the-shelf product for the space industry. The prototype is intended to perform autonomous relative navigation maneuvers using custom navigation software algorithms, including AI and computer vision. The project involves developing a guidance and control algorithm that incorporates coupled orbit and attitude dynamics in the near-Earth environment. Software-in-the-loop validation of the control algorithm is done in a simulation environment such as the Mission Design Simulator (MDS), proprietary software developed by the project's collaborator, BlackSwan Space from Lithuania. Eventually, complete real-time hardware-in-the-loop tests are conducted with the integration of the control algorithm, pose estimators and robotic manipulators. The chaser and target satellites are mounted on the robotic end-effectors in the ZeroG laboratory at the University of Luxembourg. The apparent motion of the target and chaser satellites, during rendezvous, are provided as way-points to the end-effectors of the robotic manipulators.

Related publications:

Vivek Muralidharan, Carol Martinez Luna, Augustinas Zinys, Marius Klimavicius and Miguel Angel Olivares Mendez. “Autonomous control for satellite rendezvous in near-Earth orbits”, IEEE International Conference on Control, Automation and Diagnosis, Lisbon, Portugal, July 2022.

## Guidance Navigation and Control Strategy for Cislunar Rendezvous

Interdisciplinary Center for Security, Reliability and Trust (SnT), Université du Luxembourg

February 2022 – May 2023

A rendezvous technique in the cislunar space is proposed in this investigation, one that leverages coupled orbit and attitude dynamics in the Circular Restricted Three-body Problem (CR3BP). An autonomous Guidance, Navigation and Control (GNC) technique is demonstrated in which a chaser spacecraft approaches a target spacecraft in the southern 9: 2 synodic-resonant L2 Near Rectilinear Halo Orbit (NRHO), one that currently serves as the baseline for NASA's Gateway. A two-layer guidance and control approach is contemplated. First, the guidance strategy uses a nonlinear optimal controller to identify an appropriate baseline rendezvous path, both in position and orientation. As the spacecraft progresses along the pre-computed baseline path, optical sensors measure the relative pose of the chaser relative to the target. A cooperative target with ArUco markers is considered in this investigation. A Kalman filter processes these observations and offers precise state estimates. A linear controller compensates for any deviations identified from the predetermined rendezvous path. The efficacy of the GNC technique is tested by considering a complex scenario in which the rendezvous operation is conducted with a non-cooperative tumbling target. Hardware-in-the-loop laboratory experiments are conducted as proof-of-concept to validate the real-time usage of the proposed guidance and control algorithm, with observations supplemented by optical navigation techniques.

Related publications:

Vivek Muralidharan, Mohatashem R. Makhdoomi, Kuldeep R. Barad, Lina M. Amaya-Mejia, Kathleen C. Howell, Carol Martinez Luna, and Miguel Angel Olivares Mendez. “Hardware-in-the-loop Proximity Operations in Cislunar Space”, Proceedings of the International Astronautical Congress (IAC), Paris, France, September 2022.

Vivek Muralidharan, Mohatashem R. Makhdoomi, Kuldeep R. Barad, Lina M. Amaya-Mejia, Kathleen C. Howell, Carol Martinez Luna, and Miguel A. Olivares Mendez. “Rendezvous in cislunar halo orbits: Hardware-in-the-loop simulation with coupled orbit and attitude dynamics”, Under Review, Acta Astronautica, February 2023.

## Emulating On-Orbit Interactions Using Guided Robotic Motion

Interdisciplinary Center for Security, Reliability and Trust (SnT), Université du Luxembourg

July 2022 – July 2023

Ground-based facilities that emulate on-orbit interactions are key tools for developing and testing space technology. This investigation presents a ROS-based framework to emulate on-orbit operations using on-ground robotic manipulators. The framework combines Virtual Forward Dynamic Model (VFDM) for Cartesian motion control of robotic manipulators with an Orbital Dynamics Simulator (ODS) based on the Clohessy Wiltshire (CW) Model for motion in near-Earth orbits. The VFDM-based Inverse Kinematics (IK) solver is known to have better motion tracking, path accuracy, and solver convergency than traditional IK solvers. Consequently, it provides a stable Cartesian motion for manipulator-based on-orbit emulations, even at singular or near singular configurations of the robotic arms. The framework is tested on the ZeroG-Lab robotic facility to emulate two scenarios: free-floating satellite motion and free-floating interaction (collision). Results show fidelity between the simulated motion commanded by the ODS and the one executed by the robot-mounted mockups.

Related publications:

Mohatashem R. Makhdoomi, Vivek Muralidharan, Kuldeep R. Barad, Juan Sandoval, Miguel A. Olivares-Mendez, Carol Martinez Luna. “Emulating On-Orbit Interactions Using Forward Dynamics Based Cartesian Motion”, Submitted to ICRA2023, arXiv:2209.15406, September 2022.

## Guidance Strategy for Optimal Landing

Interdisciplinary Center for Security, Reliability and Trust (SnT), Université du Luxembourg

July 2022 – December 2022

An object-oriented python library is developed that delivers an optimal guidance path for landing a module on different celestial bodies such as the Earth, Moon, Mars, etc. Such libraries are potentially used for delivering training sets for machine learning algorithms that are intended for autonomous landing. Nonlinear optimizer delivers a guidance path that accounts for certain parameters not limited to the gravitational acceleration of the landing celestial body, the number of thrusters on the spacecraft landing module and their configuration, i.e., the orientation and position of these thrusters relative to the spacecraft body. Gravitational kinematic equations and attitude dynamics are evaluated to deliver a safe landing upon touchdown with the surface. Modular thrust capabilities with a variable number of thrusters and their configuration allow the user to generate a wide variety of landing cases. A sample animation shows a landing module with four thrusters in the body frame vertical direction approaching the Martian surface, starting at a perturbed position and orientation; note that the velocity at touchdown is almost zero.

## Stretching Directions in Cislunar Space: Stationkeeping and Transfer Design

Purdue University, USA

January 2018 – August 2021

The orbits of interest for potential missions are stable or nearly stable to maintain long-term presence. Near Rectilinear Halo Orbits (NRHOs) offer such stable or nearly stable orbits that are defined as part of the L1 and L2 halo orbit families in the circular restricted three-body problem. Within the Earth-Moon regime, the L1 and L2 NRHOs are proposed as long-horizon trajectories for cislunar exploration missions, including NASA's upcoming Gateway mission. These stable or nearly stable orbits do not possess well-distinguished unstable and stable manifold structures. As a consequence, existing tools for stationkeeping and transfer trajectory design that exploit such underlying manifold structures are not reliable for orbits that are linearly stable. The investigation focuses on leveraging stretching direction as an alternative for visualizing the flow of perturbations in the neighborhood of a reference trajectory. The information supplemented by the stretching directions is utilized to investigate the impact of maneuvers for two contrasting applications; the stationkeeping problem, where the goal is to maintain a spacecraft near a reference trajectory for a long period of time, and the transfer trajectory design application, where rapid departure and/or insertion is of concern.

Stationkeeping

A stationkeeping technique using x-axis control is evaluated for low perilune radius NRHOs in the L1 and L2 families, that offer candidate solutions for various cislunar missions including the Gateway mission. A systematic and straightforward approach is demonstrated using the stretching directions that describe the interaction between the flow evolving from one maneuver location to the next during coast segments and the flow evolution from the maneuver location to the target, to identify successful combinations of maneuver and target locations for stationkeeping. In general, for stationkeeping, it is an effective strategy to deliver maneuvers with minimal components in the stretching direction.

Transfer Design

For missions with potential human presence, a rapid transfer between orbits of interest is a priority. The magnitude of the state variations along the maximum stretching direction is expected to grow rapidly and, therefore, offers information to depart from the orbit. Similarly, the maximum stretching in reverse time enables arrival with a minimal maneuver magnitude. The impact of maneuvers in such sensitive directions is investigated. Further, enabling transfer design options to connect between two stable orbits. The transfer design strategy developed in this investigation is not restricted to a particular orbit but applicable to a broad range of stable and nearly stable orbits in the cislunar space, including the Distant Retrograde Orbit (DROs) and the Low Lunar Orbits (LLO) that are considered for potential missions. Examples of transfers linking a southern and a northern NRHO, a southern NRHO to a planar DRO, and a southern NRHO to a planar LLO are demonstrated.

Related publications:

Vivek Muralidharan and Kathleen C. Howell. "Stretching Directions in Cislunar Space: Applications for Departure and Transfer Design," Astrodynamics, Vol. 7, pp. 153-178, June 2023.

Vivek Muralidharan and Kathleen C. Howell. "Leveraging Stretching Directions for Stationkeeping in Earth-Moon Halo Orbits," Advances in Space Research, Vol. 69, Issue 1, pp 620-646, January 2022.

Vivek Muralidharan and Kathleen C. Howell. "Departure and Trajectory Design Applications using Stretching Directions", AAS/AIAA Astrodynamics Specialist Conference, Big Sky, Montana, USA (Virtual), August 2021.

Vivek Muralidharan and Kathleen C. Howell. "Orbit Maintenance Strategy for Earth-Moon Halo Orbits", AAS/AIAA Space Flight Mechanics Meeting, February 2021.

Vivek Muralidharan and Kathleen C. Howell. "Stationkeeping in Earth-Moon Near Rectilinear Halo Orbits", AAS/AIAA Astrodynamics Specialist Conference, South Lake Tahoe, California, USA, August 2020.

## Stationkeeping for Deep Space Missions

Mitsubishi Electric Research Laboratories (MERL), Massachusetts, USA

May 2019 – August 2019

Stationkeeping on Near-Rectilinear Halo Orbits (NRHOs) in

the Earth-Moon system

The investigation considers the control of a spacecraft along a near-rectilinear halo orbit about the Earth-Moon L2 Lagrange point for an indefinite period. For indefinite stationkeeping, it is important to minimize fuel consumption, while allowing for occasional transfer to a new orbit. The control scheme therefore consists of two components: the first component is the tracking of the nominal NRHO and the second component is an orbit correction maneuver between NRHO trajectories. The nominal NRHO is computed using a multiple-shooting technique that takes into account all forces on the spacecraft whose magnitude is larger than the dominant disturbance forces caused by navigational error. The tracking component is a linear-quadratic regulation scheme that rejects disturbances caused by orbit determination error, using a Lyapunov sublevel set that model the state covariance generated using a sequential Kalman filter. The orbit correction maneuver is computed to minimize fuel costs.

Orbital maintenance on a Quasi-Satellite Orbit (QSO) about the Martian moon, Phobos

The goal of this investigation is to perform orbital stationkeeping on a quasi-satellite orbit about the Martian moon, Phobos. The orbit is computed using a high-fidelity ephemeris model accounting for most of the significant forces within the model, such that the major sources of disturbances are due to measurement error. Two types of orbit maintenance schemes are considered. The first is based on asymptotically tracking the desired trajectory and the second is based on stabilizing to the manifold of trajectories that share the same Jacobi constant as the reference trajectory. The latter can be done because trajectories with the same Jacobi constant are in the neighborhood of one another. The results show that the trajectory-tracking scheme has lower fuel consumption when tracking must be precise and that the approach of stabilizing to a manifold has better fuel consumption at the expense of tracking.

Related publications:

Vivek Muralidharan, Avishai Weiss, and Uros V. Kalabic. “Control Strategy for Long-Term Station-Keeping on NearRectilinear Halo Orbits”, AIAA SciTech 2020 Forum, Orlando, Florida, USA, January 2020.

Vivek Muralidharan, Avishai Weiss, and Uros V. Kalabic. "Tracking neighboring quasi-satellite orbits around Phobos", World Congress of the International Federation of Automatic Control (IFAC), Berlin, Germany, July 2020.

Uros Kalabic, Vivek Muralidharan, and Avishai Weiss, “Tracking neighboring quasi-satellite orbits around Mars’s moon Phobos,” U.S. Patent US20210206517A1, published July 8, 2021.

## Orbit Maintenance for Sun-Earth/Moon Libration Point Missions

Purdue University, USA

August 2015 – December 2017

The libration point orbits in the Sun-Earth/Moon system originate in a nonlinear dynamical regime. Coupled with the unstable nature of the orbit, the impact of any perturbations is expected to increase rapidly. The feasibility of a flow-based, Cauchy-Green tensor control strategy for stationkeeping is examined. The stationkeeping process is stochastic, thus Gaussian random errors are introduced for simulation. The evolution of a velocity perturbation over time is monitored, beyond which the attainable state in the accessible region nearest to the target state is employed as feedback to compute the necessary full, three-axis corrective maneuver. The application and appropriateness of single-axis control maneuvers for orbit maintenance are also evaluated. The selection procedure for certain parameters such as tolerances and weighting values is developed to incorporate the available dynamical information, yielding a versatile and straightforward strategy. Weighting matrices within the Target Point Approach (TPA) are effective in influencing the stationkeeping costs as well as the size and direction of maneuvers. Monte Carlo simulations are run to understand the impact of maneuvers over time.

Related publications:

Vivek Muralidharan, "Orbit Maintenance Strategies for Sun-Earth/Moon Libration Point Missions: Parameter Selection for Target Point and Cauchy-Green Tensor Approaches," M.S. Thesis, Purdue University, USA, 2017.