Fuel-optimal three-dimensional trajectories from Earth to Mars for spacecraft powered by a low-thrust rocket with variable speci c impulse capability are presented. The problem formulation treats the spacecraft mass as a state variable, thus coupling the spacecraft design to the trajectory optimization. Gravitational effects of the sun, Earth, and Mars are included throughout an entire trajectory. To avoid numerical sensitivity, the trajectory is divided into segments, each de ned with respect to a different central body. These segments are patched at intermediate time points, with proper matching conditions on the states and costates. The optimization problem is solved using an indirect multiple shooting method. Details of trajectories for the outbound legs of crewed missions to Mars, with trip times of 145 and 168 days, are shown. Effects due to variations in the trip time, departure and arrival orbit inclinations, initial fuel mass, and power level are investigated.
The determination of fuel-optimal, planar, Earth-Mars trajectories of spacecraft using low-thrust, variable speci c impulse I sp propulsion is discussed. The characteristics of a plasma thruster currently being developed for crewed/cargo missions to Mars are used. This device can generate variable I sp within the range of 1000-35,000 s, at constant power. The state equations are written in rotating, polar coordinates, and the trajectory is divided into two phases, patched together at an intermediate point between the Earth and Mars. The gravitationaleffects of the sun, Earth, and Mars are included in the two phases. The formulation of the problem treats the spacecraft mass as a state variable, thus, coupling the spacecraft design to the trajectory design. The optimal control problem is solved using an indirect, multiple shooting method. Results for a 144-daycrewed mission to Mars are presented. The variation of the I sp during spacecraft's escape from the Earth's gravitational eld shows an interesting periodic behavior with respect to time. The results obtained are also compared with those obtained by assuming a three-phase trajectory, with the Earth, sun, and Mars, in uencing the spacecraft, one per phase, in sequence.
The Altair Lunar Lander is the linchpin in the Constellation Program (CxP) for human return to the Moon. Altair is delivered to low Earth orbit (LEO) by the Ares V heavy lift launch vehicle, and after subsequent docking with Orion in LEO, the Altair/Orion stack is delivered through translunar injection (TLI). The Altair/Orion stack separating from the Earth departure stage (EDS) shortly after TLI and continues the flight to the Moon as a single stack.Altair performs the lunar orbit insertion (LOI) maneuver, targeting a 100-km circular orbit. This orbit will be a polar orbit for missions landing near the lunar South Pole. After spending nearly 24 hours in low lunar orbit (LLO), the lander undocks from Orion and performs a series of small maneuvers to set up for descending to the lunar surface. This descent begins with a small deorbit insertion (DOI) maneuver, putting the lander on an orbit that has a perilune of 15.24 km (50,000 ft), the altitude where the actual powered descent initiation (PDI) commences.At liftoff from Earth, Altair has a mass of 45 metric tons (mt). However after LOI (without Orion attached), the lander mass is slightly less than 33 mt at PDI. The lander currently has a single descent module main engine, with TBD lb f thrust (TBD N), providing a thrust-to-weight ratio of approximately TBD Earth g's at PDI.LDAC-3 (Lander design and analysis cycle #3) is the most recently closed design sizing and mass properties iteration. Upgrades for loss of crew (LDAC-2) and loss of mission (LDAC-3) have been incorporated into the lander baseline design (and its Master Equipment List). Also, recently, Altair has been working requirements analyses (LRAC-1). All nominal data here are from the LDAC-3 analysis cycle. All dispersions results here are from LRAC-1 analyses. Descent PhaseThere are three subphases comprising the descent phase of the Altair mission: the braking burn (BB), the approach, and terminal (note a short pitch-up maneuver will be executed near the beginning of approach). The descent subphases are depicted shown in Figure 1.Implicit guidance algorithms will be used to design the reference trajectories in all these descent subphases. In this approach, we can define, in advance of the mission, a reference trajectory as a vector polynomial function of time that evolves backward from the target state. But the reference trajectory cannot be expected to intersect the initial state of the vehicle due https://ntrs.nasa.gov/search.jsp?R=20100035768 2018-05-13T06:16:06+00:00Z to navigation and control dispersions. Implicit guidance will generate acceleration commands that consist of that computed using the reference trajectory plus two feedback terms. The first "feedback" acceleration correction is proportional to the difference between the actual and reference vehicle's positions. The second term is proportional to the difference between the actual and reference vehicle's velocities. Implicit guidance algorithm will "drive" the vehicle to achieve the target state in the presences of controller errors, na...
A genetic algorithm is used cooperatively with the Davidon-Fletcher-Powell penalty function method and the calculus of variations to optimize low-thrust, Mars-to-Earth trajectories for the Mars Sample Return Mission. The return trajectory is chosen thrust-coast-thrust a priori, has a xed time of ight, and is subject to initial and nal position and velocity equality constraints. The global search properties of the genetic algorithm combine with the local search capabilities of the calculus of variations to produce solutions that are superior to those generated with the calculus of variations alone, and these solutions are obtained more quickly and require less user interaction than previously possible. The genetic algorithm is not hampered by ill-behaved gradients and is relatively insensitive to problems with a small radius of convergence, allowing it to optimize trajectories for which solutions had not yet been obtained. The use of the calculus of variations within the genetic algorithm optimization routine increased the precision of the nal solutions to levels uncommon for a genetic algorithm. NomenclatureC p = position constraint weighting vector in heliocentric Cartesian coordinates, m ¡1 C v = velocity constraint weighting vector in heliocentric Cartesian coordinates, s/m F i = absolute tness of i th genome, dimensionless f i = normalized tness of i th genome, dimensionless G p = difference between actual and desired nal position vectors, m G v = difference between actual and desired nal velocity vectors, m/s H = Hamiltonian matrix, dimensionless m initial = initial mass of the spacecraft at the Martian sphere of in uence, kg m propellant = mass of propellant used during heliocentric transfer, kg n = number of genomes in the current generation, dimensionless P = penalty function value, dimensionless
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