Engineering Notes Mars Entry Mission Bank Profile Optimization Geethu Lisba Jacob, ∗ Geethu Neeler, ∗ and R. V. Ramanan † Indian Institute of Space Science and Technology, Kerala 695547, India DOI: 10.2514/1.G000089 Nomenclature C D = coefficient of drag C L = coefficient of lift D = drag force on the module, kg km∕s 2 g r = radial component of gravity acceleration, m∕s 2 g ϕ = latitudinal component of gravity acceleration, m∕s 2 g 0 = acceleration due to gravity on the Earth, 9.8066 m∕s 2 H = Hamiltonian h = altitude, km J 2 = zonal gravity coefficient of Mars, 0.001964 L = lift force on the module, kg km∕s 2 L∕D = vehicle hypersonic lift-to-drag ratio M = mass, kg R = equatorial radius of Mars, 3397 km r = radial distance, km S ref = module reference area, m 2 t = time, s v = velocity, km∕s β = ballistic coefficient m∕C D S ref , kg∕m 2 γ = flight-path angle, deg δ = latitude, deg λ = longitude, deg μ = gravitational constant of Mars, 42; 828.28 km 3 ∕s 2 ρ = density, kg∕m 3 σ = bank angle, deg ψ = flight azimuth, deg ω = rate of rotation of Mars, rad∕s Subscripts i = corresponds to initial instant of time f = corresponds to final instant of time I. Introduction T HE activities related to Mars exploration have picked up momentum in the recent past. The quest for human settlement, ever dwindling resources on the Earth, and the curiosity to unravel the mysteries surrounding the universe attract the attention of mankind to Mars, our neighbor in the solar system. Landing on Mars is particularly challenging due to the aerodynamically unfriendly combination of the atmospheric density and the gravity of Mars. The Martian atmosphere is dense enough to cause significant aerodynamic heating, introducing the requirement of extensive thermal protection systems. At the same time, the atmospheric density is insufficient to produce sufficient aerodynamic resistance to decelerate the vehicle to safe velocities for touchdown. Therefore, there is a need for decelerators such as parachutes and thrusters for soft landing on Mars [1–3]. The parachute deployment altitude must be as high as possible to 1) have sufficient time for deceleration process and 2) to provide flexibility deal with the unknown landing site and the related topography in the cases of off-nominal performances. The concept of rotating the direction of the lift vector has been used to optimally achieve many mission objectives, such as 1) maximizing the parachute deployment altitude, 2) target site landing, and 3) skip entry trajectory planning and guidance, etc. The problem of achieving the target landing site is formulated using optimal control theory in [4] with bank angle as the control parameter. However, to get a closed-form control law, the flight-path angle rate is set to zero in that study. In a skip entry problem, the magnitude of the bank angle is used in the skip phase to satisfy the downrange requirement to the landing site [5]. The problem is formulated as a nonlinear univariate root-finding problem. However, there are only a limited number of studies addressing the problem of parachute deployment altitude maximization for Mars entry trajectories [6–8]. The Mars science laboratory entry module used a three-segment bank profile to meet the parachute deployment constraints. All the other missions were either ballistic or used a full lift-up profile [2]. Shuang and Yuming [6] use truncated dynamics similar to the one used in [4] and the optimal control theory to derive the closed-form bank angle control law. The authors of [6] use the direct collocation method to transform the optimal control problem into a nonlinear programming (NLP) problem, and to demonstrate the methodology maximizing the parachute deployment, altitude is used as the objective. Grant and Mendeck [7] demonstrated the use of particle swarm methodology for generating entry mission design options that satisfy the following objectives: 1) maximize the parachute deployment altitude and 2) minimize the range error ellipse length in an automated manner. Although there is no explicit statement on the method of solution, it is presumed that they used a direct method of NLP approach to determine the optimal bank profile. Cerimele and Lafluer [8] carried out extensive parametric study for various combinations of ballistic coefficient, lift-to-drag ratio, entry velocity, and terminal Mach number, etc. for parachute deployment altitude maximization. The bank profile design that maximizes the altitude is obtained using the nonlinear programming approach in the presence of the complete dynamics. The authors assume a 10-point bank angle profile at 10 discrete solution points and refine using particle swarm optimization. The bank angle profile is expressed as a function of the relative velocity. The bank angle variation with respect to time is obtained as a byproduct. Constant ballistic coefficient and L∕D ratio during descent are assumed, which are valid through the hypersonic regime. One of the conclusions of this paper [8] is that there are multiple bank profiles that nearly achieve the same maximum altitude. However, the reason for the existence of such multiple bank profiles is not discussed. It is well known that the accuracy of the solution in nonlinear programming depends on the number of discrete solution points, and any increase in the number of discrete points makes the computational time very large. As an alternative to the NLP approach, herein, the problem of maximizing the parachute deployment altitude is formulated using the optimal control theory in the presence of complete dynamics. This indirect method of solution helps express the control law as a function of the costate variables that vary with time. A solution procedure that Received 13 June 2013; revision received 22 October 2013; accepted for publication 27 October 2013; published online 9 April 2014. Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-3884/14 and $10.00 in correspondence with the CCC. *Undergraduate Student, Department of Aerospace Engineering; currently Scientist, Vikram Sarabhai Space Center, Thiruvananthapuram. † Adjunct Professor, Department of Aerospace Engineering, Thiruvanan- thapuram; rvramanan@iist.ac.in. 1305 JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 37, No. 4, July–August 2014 Downloaded by INDIAN INSTITUTE OF SPACE on June 25, 2014 | http://arc.aiaa.org | DOI: 10.2514/1.G000089