Optimal Landing Strategy on a Uniformly Rotating Homogeneous Rectangular Parallelepiped
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Optimal Landing Strategy on a Uniformly Rotating Homogeneous Rectangular Parallelepiped Deep Parikh1,2
· Ashish Tewari1
Accepted: 3 November 2020 © American Astronautical Society 2020
Abstract Low-thrust, optimal strategies are investigated for making a smooth landing on a uniformly rotating, homogeneous rectangular parallelepiped while avoiding the sharp corners during the approach. The individual effects of principal spherical harmonic coefficients on the stability against impact are determined numerically. An iterative predictor-corrector algorithm is utilized to find a direct and retrograde family of equatorial orbits. Stability analysis of equatorial orbits confirms the fact that retrograde orbits are less prone to disturbances than direct orbits. For an optimal landing, each approach trajectory begins from a stable equatorial orbit, and terminates at a prescribed landing point. The optimality conditions are given by Euler-Lagrange equations, and the associated two-point boundary value problem is solved by a collocation method with additional path constraints, and its results are compared with those of a direct nonlinear programming search technique. It is observed that a smaller energy expenditure is required for a landing made further away from the initial location such that sufficient time is allowed for the spacecraft to remain in an unforced orbital trajectory for a majority of the trajectory. A sample inclined orbit is also studied for a possible non-planar optimal approach in the body-fixed frame, and further investigated for various landing locations. Keywords Astrodynamics · Guidance · Control and dynamics · Spacecraft guidance and control · Trajectory optimization · Orbital stability Nomenclature A a, b, c Cn,m , Sn,m e
system dynamics matrix semi-dimensions along x,y and z axes, respectively, m spherical harmonic coefficients of degree n and order m orbital eccentricity
Deep Parikh
[email protected] 1
Indian Institute of Technology Kanpur, Kanpur 208 016, Uttar Pradesh, India
2
MathWorks India Private Limited, Bangalore 560 103, Karnataka, India
The Journal of the Astronautical Sciences
e1−4 f G i Ji M Pn,m R0 r rp T U Ux , Uy , Uz Uxx , Uxy , Uxz λ λ1−4 μ Φrr φ ωr ωp Subscripts t (x,y,z)
eigenvalues of the state transition matrix true anomaly, rad gravitational constant, m3 /(kgs2 ) orbital inclination, deg Jacobi integral, m2 /s2 mass of rectangular parallelepiped, kg associated Legendre functions of degree n and order m characteristic radius, m radius of an elemental mass, m periapsis radius, m period of rotation, h gravitational potential, m2 /s2 gravitational attraction along x,y and z axis respectively, m/s2 gradient of x component of gravitational attraction along x,y and z axis respectively, 1/s2 [maximum of {a, b, c}] × 10−7 longitude of an elemental mass, deg costate variables standard gravitational parameter, m3 /s2 reduced order state transition matrix latitude of an elemental mass, deg right-ascension of ascending node, deg rotational rate, rad/s argument of peria
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