Tracking neighboring quasi-satellite orbits around Mars's moon Phobos

A method for activating at least one thruster of a propulsion system of a vehicle for trajectory-tracking control of the vehicle, the method uses a processor coupled to a memory storing executable instructions, vehicle data and historical space data, wherein the processor generates a transfer orbit for the vehicle from an initial orbit to a target orbit, the method comprising:

• computing a target orbit for the vehicle about Phobos using space data received via an input interface, based on determining a quasi-satellite orbit (QSO) with a trajectory that moves in a clockwise direction when viewed in a x-y plane in relation to Phobos, wherein determining the QSO comprises:
• generating a family of distant retrograde orbits (DROs) in a x-y plane in relation to Phobos;
• stacking each DRO of the DROs on top of each other sequentially in time, with each revolution of the vehicle around Phobos discretized into a plurality of patch points;
• implementing a multiple-shooting continuation module with an additional constraint of a fixed z-coordinate of a first patch point of the plurality of patch points, that results in a continuous, free trajectory, to obtain an initial target orbit;
• updating the multiple-shooting continuation module with the initial target orbit, such that the updated multiple-shooting continuation module includes a first target z-amplitude higher than a second target z-amplitude from the multiple-shooting continuation module;
• and
• iteratively, updating the multiple-shooting continuation module until a z-coordinate of the first patch point is equal to a desired threshold, then stop the iteration, to obtain the target orbit;
• computing a free trajectory with patch points along the free trajectory, such that each patch point along the free trajectory includes a position and a velocity;
• determining a feedback gain at each patch point of the patch points along the free trajectory, such that a state penalty function at each patch point of the free trajectory is set to match a state uncertainty function at a respective patch point of the free trajectory;
• applying the feedback gain at each patch point to map the position and the velocity at each patch point to delta v commands, in order to maintain the target orbit;
• and
• outputting the delta v commands via an output interface to activate the at least one thruster for the trajectory-tracking control of the vehicle.

A non-transitory machine-readable medium including instructions stored thereon which, when executed by processing circuitry, configure the processing circuitry to perform a method to activate at least one thruster of a propulsion system of a vehicle for trajectory-tracking control of the vehicle, the method uses a processor connected to a memory storing executable instructions, vehicle data and historical space data, wherein the processor generates a transfer orbit for the vehicle from an initial orbit to a target orbit, the method comprising:

• computing a target orbit for the vehicle about Phobos using space data received via an input interface, based on determining a quasi-satellite orbit (QSO) with a trajectory that moves in a clockwise direction when viewed in a x-y plane in relation to Phobos, wherein determining the QSO comprises:
• generating a family of distant retrograde orbits (DROs) in a x-y plane in relation to Phobos;
• stacking each DRO of the DROs on top of each other sequentially in time, with each revolution of the vehicle around Phobos discretized into a plurality of patch points;
• implementing a multiple-shooting continuation module with an additional constraint of a fixed z-coordinate of a first patch point of the plurality of patch points, that results in a continuous, free trajectory, to obtain an initial target orbit;
• updating the multiple-shooting continuation module with the initial target orbit, such that the updated multiple-shooting continuation module includes a first target z-amplitude higher than a second target z-amplitude from the multiple-shooting continuation module;
• and
• iteratively, updating the multiple-shooting continuation module until a z-coordinate of the first patch point is equal to a desired threshold, then stop the iteration, to obtain the target orbit;
• computing a free trajectory with patch points along the free trajectory, such that each patch point along the free trajectory includes a position and a velocity;
• determining a feedback gain at each patch point of the patch points along the free trajectory, such that a state penalty function at each patch point of the free trajectory is set to match a state uncertainty function at a respective patch point of the free trajectory;
• applying the feedback gain at each patch point to map the position and the velocity at each patch point to delta v commands, in order to maintain the target orbit;
• outputting the delta v commands via an output interface to activate the at least one thruster for the trajectory-tracking control of the vehicle.