Solving the Goddard problem with thrust and dynamic pressure constraints using saturation functions Knut Graichen Nicolas Petit Centre Automatique et Syst` emes, Ecole des Mines de Paris, 75272 Paris, France (e-mail: { knut.graichen , nicolas.petit }@ensmp.fr) Abstract: This paper addresses the well-known Goddard problem in the formulation of Seywald and Cliff with the objective to maximize the altitude of a vertically ascending rocket subject to dynamic pressure and thrust constraints. The Goddard problem is used to propose a new method to systematically incorporate the constraints into the system dynamics by means of saturation functions. This procedure results in an unconstrained and penalized optimal control problem which strictly satisfies the constraints. The approach requires no knowledge of the switching structure of the optimal solution and avoids the explicit consideration of singular arcs. A collocation method is used to solve the BVPs derived from the optimality conditions and demonstrates the applicability of the method to constrained optimal control problems. Keywords: Optimal control; State and input constraints; Two–point boundary value problem; Aerospace applications. 1. INTRODUCTION The classical Goddard problem presented in 1919 (God- dard, 1919) concerns maximizing the final altitude of a rocket launched in vertical direction. The problem has become a benchmark example in optimal control due to a characteristic singular arc behavior in connection with a relatively simple model structure, which makes the God- dard rocket an ideal object of study, see e.g. (Garfinkel, 1963; Munick, 1965; Tsiotras and Kelley, 1992; Seywald, 1994; Bryson, 1999; Milam, 2003). A particularly interesting formulation of the Goddard problem is given by Seywald and Cliff (Seywald and Cliff, 1992), who considered both thrust and dynamic pressure constraints. This significantly complicates the Goddard problem, since the dynamic pressure constraint represents a first–order state constraint (in the sense of (Bryson and Ho, 1969, Ch. 3.11)) in addition to the “zeroth– order” thrust constraint. Based on Pontryagin’s maximum principle, Seywald and Cliff thorougly investigated the singular arcs and the optimal switching structure of the system. The analytical effort which is required to consider the constraints and singular arcs of the Goddard problem is a well–known difficulty with constrained OCPs. In particular, the switching structure of the optimal solution must be known a–priori and leads to interior boundary conditions for the boundary value problem (BVP) derived from the optimality conditions (Bryson and Ho, 1969). In this contribution, the Goddard problem with thrust and dynamic pressure constraints (Seywald and Cliff, 1992) is used as an example to present a new method for handling constraints in optimal control by means of saturation functions. Following the ideas in (Graichen, 2006) originally developed in the context of feedforward control design, the saturation function approach takes advantage of the fact that the thrust and dynamic pressure constraints have different orders. The saturation functions are used to successively incorporate the constraints within a new system representation, which strictly satisfies the constraints. In this way, the original constrained optimal control problem (OCP) is replaced by an unconstrained one, which can be treated by the standard calculus of variations without requiring knowledge of the switching structure of the optimal solution. An additional penalty term is introduced in the cost of the derived unconstrained OCP to avoid singular arcs, which correspond to active constraints in the original constrained OCP. The penalty term has the positive side effect that the original singular arcs in the Goddard problem are circumvented. The differential–algebraic equations of the BVP stemming from the optimality conditions are numerically solved with a modified version of the collocation–based BVP solver bvp4c of Matlab. The penalty term in the BVP is thereby continuously decreased in order to approach the optimal solution of the Goddard problem. The paper is organized as follows: Section 2 summarizes the Goddard problem with the thrust and dynamic pres- sure constraints. Section 3 describes in a first step the incorporation of the thrust constraint in order to illustrate the idea of the saturation function approach. Section 4 describes the collocation method for the solution of the BVP derived from the optimality conditions and presents the numerical results for the thrust–constrained Goddard problem. Section 5 is devoted to the additional incorpo- ration of the dynamic pressure constraint by means of the saturation functions. The numerical results for the Goddard problem with thrust and dynamic pressure con- straints show the applicability of the approach as well as the accuracy of the employed collocation method. It represents a first step in the development towards a gen- eral methodology to efficiently solve constrained optimal trajectory generation problems. Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 14301 10.3182/20080706-5-KR-1001.0637