Modulation of Aerodynamic Angles for Optimal Mars Descent Trajectory using Indirect Approach
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Advances in Astronautics Science and Technology https://doi.org/10.1007/s42423-020-00063-0
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ORIGINAL PAPER
Modulation of Aerodynamic Angles for Optimal Mars Descent Trajectory using Indirect Approach Megh Bhatnagar1 · R. V. Ramanan1 Received: 18 July 2019 / Revised: 2 November 2020 / Accepted: 11 November 2020 © Chinese Society of Astronautics 2020
Abstract A formulation, based on indirect approach, that uses both the aerodynamic angles: angle of attack and bank angle as control variables and maximizes the parachute deployment altitude of a Mars entry vehicle is presented. The complexity of handling the control variable ‘angle of attack’ in the indirect approach is overcome by expressing the aerodynamics coefficients as a quadratic polynomial of angle of attack. The problem is formulated as a two point boundary value problem using the Pontryagin’s principle of the optimal control theory. The solution is obtained using differential evolution technique, a heuristic optimization technique. This is an alternative formulation to the commonly used direct approach using non-linear programming. The solution procedure based on indirect approach reduces the number of unknowns drastically compared to the direct approach. The benefit of using angle of attack modulation in addition to bank angle modulation is quantified. The implication of constraints on minimum allowable altitude and maximum deceleration on the optimized trajectory is analyzed using the new formulation and the solution approach. Keywords Angle of attack · Bank angle · Mars descent trajectory · Parachute deployment altitude · Optimal control · Indirect approach
Abbreviations CD CL D H L L/D Sref g0 gr gφ h r t
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Coefficient of drag Coefficient of lift Drag force on the module, kg m/s2 Hamiltonian Lift force on module, kg m/s2 Hypersonic lift−to−drag ratio of the module Module reference area, m2 Acceleration due to gravity on earth, 9.8066 m/s2 Radial component of gravity, m/s2 Latitudinal component of gravity, m/s2 Altitude, km Radial distance, km Time, s
Velocity, m/s Flight path angle, deg Declination, deg Longitude, deg Flight azimuth, deg Gravitational constant of Mars, 42,828.28 km3 /s2 Bank angle, deg Angle of attack, deg Density, kg/m3 Rate of rotation of Mars, rad/s
Subscripts i Corresponds to initial instant of time f Corresponds to final instant of time
R. V. Ramanan [email protected] Megh Bhatnagar [email protected]
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Department of Aerospace Engineering, Indian Institute of Space Science and Technology, Thiruvananthapuram 695547, India
1 Introduction Being the close neighbor of Earth, the planet Mars has always inspired humans to explore its geography, atmosphere and the presence of any form of life. The focus has now shifted
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Advances in Astronautics Science and Technology
to lander missions from the orbiter missions. But, the atmospheric density and the gravity field of Mars pose design challenges on the landing trajectory. During landing, t
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