Browsing by Author "Miele, Angelo"
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Item An efficient computational scheme for solving nonlinear, two-point boundary-value problems via the method of adjoint variables(1981) Coker, Estelle M.; Miele, Angelo; Bayazitoglu, Yildiz; Wierum, Frederic A.A method for solving nonlinear differential equations of the form x - (x,t) =, £ t <_ 1, subject to boundary conditions of the form w(x()) = , ÿ(x(l)) = / is developed. It is assumed that t is a scalar, x and are n-vectors, u is a p-vector, and ^ is a q-vector, with p + q ** n. The method is based on the consideration of the performance index P, the cumulative error in the differential equations and the boundary conditions. The differential equations and the boundary conditions are linearized about a nominal function x(t); the linearized system is embedded into a more general system by means of a scaling factor a, <_ a <_ 1, applied to each forcing term. The variations per unit stepsize A(t) = Ax(t)/a are governed by a system of n linear differential equations, subject to p separated initial conditions and q separated final conditions. Then, the system is solved employing the method of adjoint variables. The scaling factor a is determined by a bisection process, starting from a = 1, so as to ensure the decrease of the performance index P. Convergence to the desired solution is achieved when the inequality P £ e is met, where e is a small, preselected number. Two updating schemes are considered, called Scheme (a) and Scheme (b) for easy identification. In Scheme (a), the initial point x() is updated according tox(O) =x() +aA(), and the new nominal function x(t) is obtained by forward integration of the nonlinear differential equations. In Scheme (b), the function xCt) is updated according to x(t) = x(t) + aA(t). Four numerical examples are solved using the ITEL AS/6 computer of Rice University. The computational scheme developed here for the method of adjoint variables is particularly efficient, in that it minimizes the algorithmic work per iteration, namely, the number of integrations to be performed in order to solve the linear, two-point boundary-value problem. In the method developed by Roberts and Shipman (Ref. 4), the number of integrations is n, where n is the number of state variables. In this thesis, we show that the number of integrations can be reduced to q, where q < n is the number of final conditions.Item An updating rule for the penalty constant used in the penalty function method for mathematical programming problems(1972) Coggins, Gary Max; Miele, AngeloThis thesis considers the problem of minimizing a function f(x) subject to a constraint cp(x) = 0, where f is a scalar quantity, x an n-vector, and cp a q-vector, with q being less than n. The penalty function method is investigated; that is, the vector x is viewed as unconstrained, and the function f(x) is replaced by the penalty function U(x, k) = f(x) + kP(x). Here, the penalty constant k is a positive, scalar quantity, and P(x) = cpT(x)cp(x) is the norm of the constraint error. Crucial to the penalty function method is the prediction of the rate pi = k*/k at which the penalty constant should be increased when shifting from one cycle of the algorithm to the next. Here, k denotes the penalty constant of the present cycle, and k* denotes the penalty constant of the next cycle. In this dissertation, two updating rules are compared: (a) the penalty constant is increased at a constant rate pi = const, and (b) the penalty constant is increased at a variable rate pi = 10j(P/P^), where P is the constraint error at the beginning of a cycle and P is the desired constraint error at convergence. In order to evaluate these updating rules, six numerical examples are investigated. The first example deals with a quadratic function subject to linear constraints; the remaining examples deal with nonquadratic functions subject to nonlinear constraints. Each example is solved with the conjugate gradient algorithm and the modified quasilinearization algorithm for several starting values of the penalty constant. From the numerical experiments, the following conclusions, concerning the number of iterations for convergence, arise: (i) the variable-rate updating rule is superior to the constant-rate updating rule in both the conjugate gradient algorithm and the modified quasilinearization algorithm; and (ii) the above superiority is more pronounced in the conjugate-gradient algorithm than in the modified quasilinearization algorithm.Item Ascent performance feasibility for next-generation spacecraft(1998) Mancuso, Salvatore Massimo; Miele, Angelo; Spanos, Pol D.This thesis deals with the optimization of the ascent trajectories for single-stage suborbital (SSSO), single-stage-to-orbit (SSTO), and two-stage-to-orbit (TSTO) rocket-powered spacecraft. The maximum payload weight problem has been solved using the sequential gradient-restoration algorithm. For the TSTO case, some modifications to the original version of the algorithm have been necessary in order to deal with discontinuities due to staging and the fact that the functional being minimized depends on interface conditions. The optimization problem is studied for different values of the initial thrust-to-weight ratio in the range 1.3 to 1.6, engine specific impulse in the range 400 to 500 sec, and spacecraft structural factor in the range 0.08 to 0.12. For the TSTO configuration, two subproblems are studied: uniform structural factor between stages and nonuniform structural factor between stages. Due to the regular behavior of the results obtained, engineering approximations have been developed which connect the maximum payload weight to the engine specific impulse and spacecraft structural factor; in turn, this leads to useful design considerations. Also, performance sensitivity to the scale of the aerodynamic drag is studied, and it is shown that its effect on payload weight is relatively small, even for drag changes approaching $\pm 50\%.$ The main conclusions are that: the design of a SSSO configuration appears to be feasible; the design of a SSTO configuration might be comfortably feasible, marginally feasible, or unfeasible, depending on the parameter values assumed; the design of a TSTO configuration is not only feasible, but its payload appears to be considerably larger than that of a SSTO configuration. Improvements in engine specific impulse and spacecraft structural factor are desirable and crucial for SSTO feasibility; indeed, it appears that aerodynamic improvements do not yield significant improvements in payload weight.Item Blunt-nosed missile shapes of minimum ballistic factor(1968) Heideman, J. C. (John C.); Miele, AngeloThe class of slender, axisymmetric missiles of given length, base radius, and specific weight is considered under the assumptions that the pressure distribution is modified Newtonian and the surface-averaged skin-friction coefficient is constant. The problem of minimizing the ballistic factor is formulated and solved via the calculus of variations for the case where the nose radius of the missile has a specified value, not necessarily zero.Item Combined gradient-restoration algorithm for optimal control problems(1983) Basapur, Venkatesh K.; Miele, Angelo; Douglas, Andrew S.; Merwin, John E.The problem of minimizing a functional I subject to differential constraints, nondifferential constraints, and general boundary conditions is considered in this thesis. It consists of finding the state x(t), the control u(t), and the parameter-rr so that the functional I is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. A combined gradient-restoration algorithm is developed. This is an iterative algorithm characterized by variations Ax(t), Au(t), Air leading toward satisfaction of the optimality conditions, while simultaneously leading toward constraint satisfaction. The variations Ax(t), Au(t), Air are generated by requiring the first variations of the augmented functional J and the constraint error P to be negative. The procedure leads to a linear, two-point boundary-value problem, which is solved via the method of particular solutions. The descent properties of the algorithm are studied, and schemes to determine the optimum stepsize are discussed. In order to improve the convergence characteristics, the inclusion of a restoration phase is studied. In this connection, three versions of the algorithm are studied: the combined gradient-restoration algorithm (CGRA); the combined gradient-restoration algorithm with alternate restoration (CGRA-AR); and the combined gradient-restoration algorithm with complete restoration (CGRA-CR). A comparison of these versions with the sequential gradient-restoration algorithm (SGRA) is also made. Three numerical examples are presented to illustrate the different approaches. Key Words. Numerical methods, optimal control, gradient methods, combined gradient-restoration algorithm, differential constraints, nondifferential constraints, general boundary conditions.Item Comparison of gradient-restoration algorithms for optimal control problems with nondifferential constraints and general boundary conditions(1994) Ko, Shuh-Hung; Miele, AngeloThe problem considered here involves a functional I subject to differential constraints, nondifferential constraints, and general boundary conditions. It consists of finding the state x(t), control u(t), and parameter $\pi$ so that the functional I is minimized, while the differential constraints, nondifferential constraints, and boundary conditions are satisfied to a predetermined accuracy. Here, I is a scalar, x an n-vector, u an m-vector, and $\pi$ a p-vector. Four types of gradient-restoration algorithms are considered, and their relative efficiency in terms of the number of iterations for convergence and CPU time is evaluated. The algorithms considered are as follows: sequential gradient-restoration algorithm, complete restoration (SGRA-CR); sequential gradient-restoration algorithm, incomplete restoration (SGRA-IR); combined gradient-restoration algorithm, no restoration (CGRA-NR); and combined gradient-restoration algorithm, incomplete restoration (CGRA-IR).Item Determination of optimal parameters for aircraft take-off guidance strategies in the presence of windshear(1988) Kamaric, Emir; Miele, AngeloLow-altitude windshear is a threat to the safety of aircraft in take-off and landing: over the past 20 years, some 30 aircraft accidents have been attributed to windshear. The Aero-Astronautics Group of Rice University under the direction of Dr. Angelo Miele has determined the optimal capability of an aircraft in take-off under windshear conditions and has developed near-optimal guidance strategies. In this thesis, using the above information, the values of the parameters for different guidance strategies are determined in order to provide the best approximation to the optimal capabilities. The calculation is treated as a mathematical programming problem, solved by the Nelder-Mead simplex algorithm. Then, the optimal parameters data are applied to guidance strategies. The resulting guidance trajectories are presented in graphical form along with the optimal trajectories for comparison.Item Fuel optimal Mars transfer trajectories(2005) Benzin, Kathryn C.; Miele, AngeloThis thesis research examines the fuel optimal trajectories for a spacecraft going to Mars and returning to Earth. Challenges encountered include the defining of equations of motion and optimality conditions, formulation of constraints, and overcoming difficulties deriving from large distances and times involved and the accuracy required. In addition to calculating an optimal trajectory, two different arrival orbits at Mars are compared: a clockwise and counterclockwise Martian orbit. The optimal trajectories are computed using a mathematical optimization algorithm SNOPT, developed at Stanford University and UC San Diego. A solution method is recommended where an initial guess is generated analytically via a patched conic approach. The problem is solved in two steps: first with relaxed constraints, then using that result to find the optimal solution. This approach is proven by separately solving two different trajectories: an Earth to Mars trajectory and a Mars to Earth trajectory. The results demonstrate that the two arrival conditions are very similar in most aspects, including planetary phase angles and total trajectory time. The trajectory using the counterclockwise Mars orbit has a slight advantage in propellant usage (DeltaV), but the difference is less than 1%. These transfers also have symmetries between the outbound and return portions of the trajectories. Both trajectories should be available for consideration for a mission to Mars depending on other mission requirements.Item Hypersonic power-law bodies of maximum lift-to-drag ratio(1966) Huang, Ho-Yi; Miele, AngeloThe problem of maximizing the lift-to-drag ratio of a slender, flat-top hypersonic body is investigated under the assumptions that the pressure distribution is Newtonian and the skin-friction coefficient is constant. Direct methods are employed, and the analysis is confined to the class of bodies whose transversal contour is semicircular and whose longitudinal contour is a power law. First, unconstrained configurations are considered, and the combination of power law exponent and the thickness ratio maximizing the lift-to-drag ratio is determined. It is found that the maximum lift-to-drag ratio LID = 0.360 Cf1/3 and corresponds to a conical configuration of thickness ratio t/l = 1.18 Cf1/3, where Cf is the skin-friction coefficient. Next, constrained configurations are considered, that is, conditions are imposed on the length, the thickness, the volume, the wetted area, and the center of pressure. For each combination of constraints, an appropriate similarity parameter is introduced, and the optimum power law exponent, thickness ratio, and the lift-to-drag ratio are determined as functions of the similarity parameter.Item Hypersonic, two-dimensional power-law wings of maximum lift-to-drag ratio(1966) Wilson, William LeRoy; Miele, AngeloThe problem of maximizing the lift-to-drag ratio of a slender, flat-top hypersonic wing is investigated under the assumptions that the pressure distribution is Newtonian and the skin-friction coefficient is constant. Direct methods are employed, and the analysis is confined to the class of two-dimensional wings whose chordwise contour is a power law. First, unconstrained configurations are considered, and the combination of power law exponent and thickness ratio maximizing the lift-to-drag ratio is determined. It is found that the maximum lift-to-drag ratio is L/D = 0.529 Cf1/3 and corresponds to a wedge of thickness ratio t/l = 1.26Cf1/3, where Cf is the skin-friction coefficient. Next, constrained configurations are considered, that is, conditions are imposed on the length, the thickness, the enclosed area, and the center of pressure. For each combination of constraints, an appropriate similarity parameter is introduced, and the optimum power law exponent, thickness ratio, and the lift-to-drag ratio are determined as functions of the similarity parameter.Item Investigation of a gamma guidance scheme for flight in windshear(1992) Aizawa, Takeshi; Miele, AngeloThis thesis refers to windshear recovery systems, designed to increase the survival capability of an aircraft. It is known that optimal trajectories in take-off are difficult to implement because of the lack of global information on the wind flow field and the lack of enough time and computing capability onboard. Consequently, one is forced to employ local/prior information on the windshear and the downdraft. An investigation of the gamma guidance scheme is presented with reference to the take-off problem. Attention is focused on the feedback control form of the gamma guidance law. This law is modified so that not only the aircraft can react to unfavorable shears, but can take advantage of favorable shears. Indeed, every unfavorable shear (central core of the downburst) is both preceded and followed by favorable shears. Therefore, a systematic investigation of the effect of parameters on survival capability is presented for two windshear models. The results indicate that, through a proper choice of the parameters, the windshear efficiency of the gamma guidance law is within 5% of that of an optimal trajectory. (Abstract shortened with permission of author.)Item Method of independent multipliers for minimizing unconstrained functions(1969) Cantrell, Joel Wood; Miele, AngeloA new accelerated gradient method for finding the minimum of a function f(x) whose variables are unconstrained is presented. The new algorithm can be stated where ax is the change in the position vector x, g(x) is the gradient of the function f(x), and a and fi are scalars chosen at each step so as to yield the greatest decrease in the function. The symbol Occ denotes the change in the position vector for the iteration preceding that under consideration. It is shown that, for a quadratic function, the present algorithm reduces to the Fletcher-Reeves algorithm; thus, quadratic convergence is assured. However, for a nonquadratic function, initial convergence of the present method is much faster than that of the Fletcher- Reeves method because of the extra degree of freedom available. For a test problem, the number of iterations was about 40-50% that of the Fletcher-Reeves method and the computing time about 60-75% that of the Fletcher-Reeves method, using comparable search techniques.Item Modifications and alternatives to the cubic interpolation process for one-dimensional search(1978) Gonzalez Castro, Salvador; Miele, AngeloIn this thesis, the numerical solution of the problem of minimizing a unimodal function f(a) is considered, where a is a scalar. Two modifications of the cubic interpolation process are presented, so as to improve the robustness of the method and force the process to converge in a reasonable number of iterations, even in pathological cases. Modification Ml includes the ncnoptional bisection of the interval of interpolation at each iteration of the process. Modification M2 includes the optional bisection of the interval of interpolation: this depends on whether the slopes f(a) and f(aQ) at the terminal points agand a of two consecutive iterations have the same sign or opposite sign. An alternative to the cubic interpolation process is also presented. This is a Lagrange interpolation scheme in which the quadratic approximation to the derivative of the function is considered. The coefficients of the quadratic are determined from the values of the slope at three points: aj, a2f. and CX3 = (aj + a2l/2, where a! and a2 are the endpoints of the interval of interpolation. The proposed alternative is investigated in two versions, Version Al and Version A2. They differ in the way in which the next interval of interpolation is chosen; for Version Al, the choice depends on the sign of the slope fa(a); for Version A2, the choice depends on the signs of the slopes fa(cto) and ^(13). Twenty-nine numerical examples are presented. The numerical results show that both modifications of the cubic interpolation process improve the robustness of the process. They also show the promising characteristics of Version A2 of the proposed alternative. Therefore, the one-dimensional search schemes described here have potential interest for those minimization algorithms which depend critically of the precise selection of the stepsize, namely, the conjugate gradient method and the variable metric method.Item Multiple-subarc approach for solving minimax problems of optimal control(1981) Venkataraman, Panchapakesan; Miele, Angelo; Wierum, Frederic A.; Bayazitoglu, YildizNumerical solutions of minimax problems of optimal control are obtained through a multiple-subarc approach, used as a sequel to a single-subarc approach. The problems are solved by means of the sequential gradient-restoration algorithm. Firsts transformation technique is employed in order to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations. The transformation requires the proper augmentation of the state vector x(t),the control vector u(t),and the parameter vector ir. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the vector parameter is being optimized. The transformation technique is then employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer. The algorithm developed in the thesis belongs to the class of sequential gradient-restoration algorithms. The sequential gradient-restoration algorithm is made up of a sequence of two-phase cycles,each cycle consisting of a gradient phase and a restoration phase. The principal property of this algorithm is that it produces a sequence of feasible suboptimal solutions. Each feasible solution is characterized by a lower value of the minimax performance index than any previous feasible solution. To facilitate numerical implementation, the intervals of integration are normalized to unit length. Several numerical examples are presented to illustrate the present approach. For comparison purposes, the analytical solutions, the single-subarc solutions, and the multiple-subarc solutions are presented. Key Words. MLnimax problems, ndnimax optimal control, numerical methods, continuous approach, single-subarc approach, multiple-subarc approach, transformation techniques, sequential gradient-restoration algorithms.Item Near-optimum guidance schemes for abort landing flight in a windshear(1988) Wang, Jyhshing Jack; Miele, AngeloIn this thesis, near-optimum guidance schemes for abort landing trajectories in a windshear are investigated. These near-optimum guidance schemes are based on the properties of the optimum trajectories. The survival capability in a severe windshear is studied. The optimal trajectories are used as ideal benchmarks against which the goodness of any guidance scheme can be measured.Item Numerical experiments on quadratically convergent algorithms for function minimization(1970) Levy, Alejandro Velasco; Miele, AngeloThe nine quadratically convergent algorithms for function minimization appearing in Ref. 1 are tested through numerical examples. Both a quadratic function and a nonquadratic function are investigated. For the quadratic function, the results show that, if high-precision arithmetic together with high accuracy in the one-dimensional search is employed, all the algorithms behave identically: they all produce the same sequence of points and they all lead to the minimal point in the same number of iterations (this number is equal at most to the number of variables). For the nonquadratic function, the results show that some of the algorithms behave identically and, therefore, any one of them can be considered to be representative of the entire class. The effect of different restarting conditions on the convergence characteristics of the algorithms is studied. Proper restarting conditions for faster convergence are given.Item Numerical experiments on transformation techniques for minimax problems of optimal control(1982) Kuo, Yan Min; Miele, Angelo; Bowen, Ray M.; Thrall, Robert M.A transformation technique is employed in order to convert mini max problems of optimal control into the Mayer-Bolza problem of the calculus of variations. The transformation requires the proper augmentation of the state vector x(t), the control vector u(t), and the parameter vector TT. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the vector parameter in being optimized. The transformation technique is then employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer. The algorithm considered in this thesis belongs to the class of sequential gradient-restoration algorithms. The sequential gradient restoration algorithm is made up of a sequence of two-phase cycles, each cycle consisting of a gradient phase and a restoration phase. The principal property of this algorithm is that it produces a sequence of feasible suboptimal solutions. Each feasible solution is characterized by a lower value of the minimax performance index than any previous feasible solution. To facilitate numerical implementation, the interval of integration is normalized to unit length. Four test problems characterized by known analytical solutions are solved numerically. It is found that the combination of transformation technique and sequential gradient-restoration algorithm yields numerical solutions which are quite close to the known analytical solutions. In particular, the converged values of the minimax performance index agree well with the known analytical values.Item On properties of the Hohmann transfer(2004) Mathwig, Jarret; Miele, AngeloIn this work, we present a complete study of the Hohmann transfer maneuver between two circular coplanar orbits. After revisiting its known properties, we present a number of supplementary properties which are essential to the qualitative understanding of the maneuver. Specifically, along a Hohmann transfer trajectory, there exists a point where the path inclination is maximum: this point occurs at midradius and is such that the spacecraft velocity equals the local circular velocity. This implies that, in a Hohmann transfer, the spacecraft velocity is equal to the local circular velocity three times: before departure, at midradius, and after arrival. In turn, this allows the subdivision of the Hohmann transfer trajectory into a region where the velocity is subcircular and a region where the velocity is supercircular, with the transition from one region to another occurring at midradius. Also, we present a simple analytical proof of the optimality of the Hohmann transfer and complement it with a numerical study via the sequential gradient-restoration algorithm. Finally, as an application, we present a numerical study of the transfer of a spacecraft from the Earth orbit around the Sun to another planetary orbit around the Sun for both the case of an ascending transfer (orbits of Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto) and the case of a descending transfer (orbits of Mercury and Venus).Item Optimal combat maneuvers of a next-generation jet fighter aircraft(1998) Dabney, James Bruster; Miele, AngeloThis thesis deals with the optimization of four classes of combat maneuvers for a next-generation jet fighter aircraft: climb maneuvers, fly-to-point maneuvers, pop-up attack maneuvers, and dive recovery maneuvers. For the first three classes of maneuvers, the optimization criterion is the minimization of the flight time, resulting in a Mayer-Bolza problem of optimal control; for the fourth class, the optimization criterion is the minimization of the maximum altitude loss during dive recovery, resulting in a Chebyshev problem of optimal control. Each class of problems is solved using the sequential gradient-restoration algorithm for optimal control. Among the four classes of combat maneuvers studied, only dive recovery benefits from the ability of a next-generation fighter aircraft to maneuver at extremely high angles of attack. For the other three classes, relatively low angles of attack are required. The optimal climb trajectories are characterized by three distinct segments: a central segment often flown with a load factor of nearly 1 and two terminal segments (dive or zoom) to and from the central segment. The central and final segments are nearly independent of the initial conditions, instead being dominated by the final conditions. The optimal fly-to-point trajectories consist of three segments: turning, characterized by relatively high load factor; level acceleration at maximum thrust; and finally, resumption of steady-state cruising. The effects of the heading change magnitude and the load factor limit are discussed. The optimal pop-up trajectories consist of three segments flown at maximum power: pitch-up, zoom, and pitch-down. The effects of using the afterburner, heading change magnitude, and dive angle magnitude are discussed. The optimal dive recovery trajectories consist of one to three segments, depending on initial speed and flight path angle. All the optimal trajectories conclude with a pitch-up at the maximum available load factor. For very low initial speed, the pitch-up is preceded by a brief supermaneuver segment. For very low initial speed coupled with very high initial flight path angle, the supermaneuver segment is preceded by a dive initiation segment. The optimal trajectories reported here serve two purposes. First, they can benefit aircraft designers by highlighting those flight characteristics that are most beneficial in combat. Second, they can benefit aircraft pilots as the basis for guidance trajectories that approximate the optimal trajectories.Item Optimal control of ship maneuvers for course change, sidestep, and collision avoidance(1999) Chao, Chi-Shun; Miele, AngeloWe consider a ship subject to kinematic, dynamic, and moment equations and steered via the rudder under the assumptions that the rudder angle and rudder angle time rate are subject to upper and lower bounds. We formulate and solve four Mayer problems of optimal control, the optimization criterion being the minimum time. Also, we formulate and solve four Chebyshev problems of optimal control, the optimization criterion being the maximization with respect to the control history of the minimum value with respect to time of the distance between two identical ships, one maneuvering and one moving in a predetermined way. Problems P1 and P2 deal with course change maneuvers. In Problem P1, a ship initially in quasi-steady state must reach the final point with a given yaw angle and zero yaw angle time rate. Problem P2 differs from Problem P1 in that quasi-steady state is required at the final point. Problems P3 and P4 deal with sidestep maneuvers. In Problem P3, a ship initially in quasi-steady state must reach the final point with a given lateral distance, zero yaw angle, and zero yaw angle time rate. Problem P4 differs from Problem P3 in that quasi-steady state is required at the final point. Problems P5 and P6 deal with collision avoidance maneuvers without cooperation, while Problems P7 and P8 deal with collision avoidance maneuvers with cooperation. In Problems P5 and P7, the maneuvering ship must reach the final point with a given lateral distance, zero yaw angle, and zero yaw angle time rate. Problems P6 and P8 differ from Problems P5 and P7 in that quasi-steady state is required at the final point. The above Mayer problems and Chebyshev problems, transformed into Lagrange problems via suitable transformations, are solved via the sequential gradient-restoration algorithm in conjunction with a new singularity avoiding transformation which accounts for the bounds on rudder angle and rudder angle time rate. The optimal control histories involve multiple subarcs along which either the rudder angle is kept at one of the extreme positions or the rudder angle time rate is held at one of the two extreme values. If quasi-steady state is not required at the final point, the total number of subarcs ranges from 4 to 8, depending on the particular problem. If quasi-steady state is required at the final point, the total number of subarcs ranges from 6 to 10, depending on the particular problem: the higher number of subarcs, is due to the additional requirements that the lateral velocity and rudder angle vanish at the final point.