Geometric optimal control and applications to aerospace
 Jiamin Zhu^{1},
 Emmanuel Trélat^{1}Email authorView ORCID ID profile and
 Max Cerf^{2}
https://doi.org/10.1186/s4073601700334
© The Author(s) 2017
Received: 23 January 2017
Accepted: 4 July 2017
Published: 20 July 2017
Abstract
This article deals with applications of optimal control to aerospace problems with a focus on modern geometric optimal control tools and numerical continuation techniques. Geometric optimal control is a theory combining optimal control with various concepts of differential geometry. The ultimate objective is to derive optimal synthesis results for general classes of control systems. Continuation or homotopy methods consist in solving a series of parameterized problems, starting from a simple one to end up by continuous deformation with the initial problem. They help overcoming the difficult initialization issues of the shooting method. The combination of geometric control and homotopy methods improves the traditional techniques of optimal control theory.
A nonacademic example of optimal attitudetrajectory control of (classical and airborne) launch vehicles, treated in details, illustrates how geometric optimal control can be used to analyze finely the structure of the extremals. This theoretical analysis helps building an efficient numerical solution procedure combining shooting methods and numerical continuation. Chattering is also analyzed and it is shown how to deal with this issue in practice.
Keywords
Introduction
Generally, a space vehicle is modeled as a solid body. The motion combines the translation of the center of gravity (COG) defining the trajectory and the body rotation around its center of gravity defining the attitude. A usual simplification consists in assuming that the translation and the rotation motions are independent, because the attitude time scale is often much shorter than the trajectory time scale so that the attitude control can be considered as nearly perfect, i.e., instantaneous or with a short response time. With this assumption the trajectory problem (also called the guidance problem) and the attitude problem (also called the control problem) can be addressed separately. This uncoupling of the guidance and the control problem is valid either when the torque commands have a negligible effect on the CoG motion or when the control time scale is much shorter than the guidance time scale. Most space vehicles fall into one of these two categories. The main exceptions are atmospheric maneuvering vehicles such as cruise or antiballistic missiles and airborne launchers. Such vehicles have to perform large reorientation maneuvers requiring significant durations. These maneuvers have a sensible influence of the CoG motion and they must be accounted for a realistic trajectory optimization. In these cases, the rotation and the translation motions are coupled, the command are then the nozzle or the flap deflections depending on the vehicle control devices. For a propelled launcher, the motion is controlled by the thrust force which is nearly aligned with the roll axis. We call such exception problems the attitudetrajectory or coupled problems. We refer readers interested by aerospace missions to Section 2 for a general introduction on the applications to aerospace missions, of which the objective is to give a global view on how space missions are translated into optimal control problems.
The purpose of this article is to show how to address optimal control problems in aerospace using modern techniques of geometric optimal control and how to build solution algorithms based on continuation techniques. In particular, we make a brief survey on the chattering phenomenon (also called Fuller phenomenon), and explain how to dealt with the chattering phenomenon with numerical continuations. The chattering phenomenon, which appears systematically in aerospace applications (in trajectory optimization [91] and in attitudetrajectory optimization problems [92, 93]), is the situation where the optimal control switches an infinite number of times over a compact time interval.
The geometric optimal control (stated in the early 1980s and having widely demonstrated its advantages over the classical theory of the 1960s) and the continuation techniques (which are not new, but have been somewhat neglected until recently in optimal control) are powerful approaches for aerospace applications. In this article, the main techniques of optimal control theory, including the Pontryagin Maximum Principle, the firstorder and higher order optimality conditions, the associated numerical methods, and the numerical continuation principles will be recalled. Most mathematical notions presented here are known by many readers, and can be skipped at the first reading.
After recalling some applications of the geometric control techniques and the continuation in trajectory optimization problems, we present detailed analyses of a nonacademic attitudetrajectory problem that we have studied during these years. This example deals with a minimum time maneuver of a coupled attitudetrajectory dynamic system. Due to the system high nonlinearity and the existence of a chattering phenomenon (see Sections 3.4 and Section 7 for details), the standard techniques of optimal control do not provide adequate solutions to this problem. Through this example, we will show step by step how to build efficient numerical procedures with the help of theoretical results obtained by applying geometric optimal control techniques. More precisely, we will explain how the geometric control techniques are used to analyze the extremals of the problem and to prove the existence of chattering phenomenon, and how the numerical continuation methods are used to overcome the chattering and to design the numerical resolution method.
Structure of the paper. In Section 2, several optimal control problems stemming from various aerospace missions are systematically introduced as motivation. In Section 3, we provide a brief survey of geometric optimal control, including the use of Lie and Poisson brackets with first and higher order optimality conditions. In Section 4, we recall classical numerical methods for optimal control problems, namely indirect and direct methods. In Section 5, we recall the concept of continuation methods, which help overcoming the initialization issue for indirect methods. In Section 6, we shortly give some applications of geometric optimal control and of continuation for space trajectory optimization problems. In Section 7, we detail a full nonacademic example in aerospace (an attitudetrajectory problem), in order to illustrate how to solve optimal control problems with the help of geometric optimal control theory and the continuation methods.
Applications to aerospace problems
Transport in space gives rise to a large range of problems that can be addressed by optimal control and mathematical programming techniques. Three kinds of problems can be distinguished depending on the departure and the arrival point: ascent from the Earth ground to an orbit, reentry from an orbit to the Earth ground (or to another body of the solar system), transfer from an orbit to another one. A space mission is generally composed of successive ascent, transfer and reentry phases, whose features are presented in the following paragraphs.
For farther solar system travels successive flybys around selected planets allow “free” velocity gains. The resulting combinatorial problem with optional intermediate deep space maneuvers is challenging.
The simulation task consists in integrating the dynamics differential equations derived from mechanics laws. The vehicle is generally modeled as a solid body. The motion combines the translation of the center of gravity defining the trajectory and the body rotation around its center of gravity defining the attitude. The main forces and torques originate from the gravity field (always present), from the propulsion system (when switched on) and possibly from the aerodynamics shape when the vehicle evolves in an atmosphere. In many cases a gravity model including the first zonal term due to the Earth flattening is sufficiently accurate at the mission analysis stage. The aerodynamics is generally modeled by the drag and lift components tabulated versus the Mach number and the angle of attack. The atmosphere parameters (density, pressure, temperature) can be represented by an exponential model or tabulated with respect to the altitude. A higher accuracy may be required on some specific occasions, for example to forecast the possible fallout of dangerous space debris, to assess correctly low thrust orbital transfers or complex interplanetary space missions. In such cases the dynamical model must be enhanced to account for effects of smaller magnitudes. These enhancements include higher order terms of the gravitational field, accurate atmosphere models depending on the season and the geographic position, extended aerodynamic databases, third body attraction, etc, and also other effects such as the solar wind pressure or the magnetic induced forces.
Complex dynamical models yield more representative results at the expense of larger computation times. In view of trajectory optimization purposes the simulation models have to make compromises between accuracy and speed. A usual simplification consists in assuming that the translation and the rotation motions are independent. With this assumption the trajectory problem (also called the guidance problem) and the attitude problem (also called the control problem) can be addressed separately. This uncoupling of the guidance and the control problem is valid either when the torque commands have a negligible effect on the CoG motion or when the control time scale is much shorter than the guidance time scale. Most space vehicles fall into one of these two categories. The main exceptions are atmospheric maneuvering vehicles such as cruise or antiballistic missiles and airborne launchers.
Such vehicles have to perform large reorientation maneuvers requiring significant durations. These maneuvers have a sensible influence of the CoG motion and they must be accounted for a realistic trajectory optimization.
Another way to speed up the simulation consists in splitting the trajectory into successive sequences using different dynamical models and propagation methods. Ascent or reentry trajectories are thus split into propelled, coast and gliding legs, while interplanetary missions are modeled by patched conics. Each leg is computed with its specific coordinate system and numerical integrator. Usual state vector choices are Cartesian coordinates for ascent trajectories, orbital parameters for orbital transfers, spherical coordinate for reentry trajectories. The reference frame is usually Galilean for most applications excepted for the reentry assessment. In this case an Earth rotating frame is more suited to formulate the landing constraints. The propagation of the dynamics equations may be achieved either by semianalytical or numerical integrators. Semianalytical integrators require significant mathematical efforts prior to the implementation and they are specialized to a given modelling. For example averaging techniques are particularly useful for long timescale problems, such as low thrust transfers or space debris evolution, in order to provide high speed simulations with good differentiability features. On the other hand numerical integrators can be applied very directly to any dynamical problem. An adequate compromise has then to be found between the timestep as large as possible and the error tolerance depending on the desired accuracy.
The dynamics models consider first nominal features of the vehicle and of its environment in order to build a reference mission profile. Since the real flight conditions are never perfectly known, the analysis must also be extended with model uncertainties, first to assess sufficient margins when designing a future vehicle, then to ensure the required success probability and the flight safety when preparing an operational flight. The desired robustness may be obtained by additional propellant reserves for a launcher, or by reachable landing areas for a reentry glider.
The optimization task consists in finding the vehicle commands and optionally some design parameters in order to fulfill the mission constraints at the best cost. In most cases, the optimization deals only with the path followed by one vehicle. In more complicated cases, the optimization must account for moving targets or other vehicles that may be jettisoned parts of the main vehicle. Examples or such missions are debris removal, orbital rendezvous, interplanetary travel or reusable launchers with recovery of the stages after their separation.
For preliminary design studies, the vehicle configuration is not defined. The optimization has to deal simultaneously with the vehicle design and the trajectory control. Depending on the problem formulation the optimization variables may thus be functions, reals or integers.
In almost all cases an optimal control problem must be solved to find the vehicle command law along the trajectory. The command aims at changing the magnitude and the direction of the forces applied, namely the thrust and the aerodynamic force. The attitude time scale is often much shorter than the trajectory time scale so that the attitude control can be considered as nearly perfect, i.e., instantaneous or with a short response time. The rotation dynamics is thus not simulated and the command is directly the vehicle attitude. If the rotation and the translation motions are coupled, the 6 degrees of freedom must be simulated. The command are then the nozzle or the flap deflections depending on the vehicle control devices. The choice of the attitude angles depends on the mission dynamics. For a propelled launcher, the motion is controlled by the thrust force which is nearly aligned with the roll axis. This axis is orientated by inertial pitch and yaw angles. For a gliding reentry vehicle, the motion is controlled by the drag and lift forces. The angle of attack modulates the force magnitude while the bank angle only acts on the lift direction. For orbital maneuvering vehicles, the dynamics is generally formulated using the orbital parameters evolution, e.g., by Gauss equations, so that attitude angles in the local orbital frame are best suited.
If the trajectory comprises multiple branches or successive flight sequences with dynamics changes and interior point constraints, discontinuities may occur in the optimal command law. This occurs typically at stage separations and engine ignitions or shutdowns. The commutation dates between the flight sequences themselves may be part of the optimized variables, as well as other finite dimension parameters, leading to a hybrid optimal control problem. A further complexity occurs with path constraints relating either to the vehicle design (e.g., dynamic pressure or thermal flux levels), or to the operations (e.g., tracking, safety, lightening). These constraints may be active along some parts of the trajectory, and the junction between constrained and unconstrained arcs may raise theoretical and numerical issues.
The numerical procedures for optimal control problems are usually classified between direct and indirect methods. Direct methods discretize the optimal control problem in order to rewrite it as a nonlinear large scale optimization problem. The process is straightforward and it can be applied in a systematic manner to any optimal control problem. New variables or constraints may be added easily. But achieving an accurate solution requires a careful discretization and the convergence may be difficult due to the large number of variables. On the other hand indirect methods are based on the Pontryagin Maximum Principle which gives a set of necessary conditions for a local minimum. The problem is reduced to a nonlinear system that is generally solved by a shooting method using a Newtonlike algorithm. The convergence is fast and accurate, but the method requires both an adequate starting point and a high integration accuracy. The sensitivity to the initial guess can be lowered by multiple shooting which breaks the trajectory into several legs linked by interface constraints, at the expense of a larger nonlinear system. The indirect method requires also prior theoretical work for problems with singular solutions or with state constraints. Handling these constraints by penalty method can avoid numerical issues, but yields less optimal solutions.
In some cases the mission analysis may address discrete variables. Examples of such problems are the removal of space debris by a cleaner vehicle or interplanetary travels with multiple flybys. For a debris cleaning mission the successive targets are moving independently of the vehicle, and the propellant required to go from one target to another depends on the rendezvous dates. The optimization aims at selecting the targets and the visiting order in order to minimize the required propellant. The path between two given targets is obtained by solving a timedependent optimal control problem. The overall problem is thus a combinatorial variant of the wellknown Traveling Salesman Problem, with successive embedded optimal control problems.
For an interplanetary mission successive flybys around planets are necessary to increase progressively the velocity in the solar system and reach far destinations. Additional propelled maneuvers are necessary either at the flyby or in the deep space in order to achieve the desired path. An impulsive velocity modelling is considered for these maneuvers in a first stage. If a low thrust engine is used, the maneuver assessment must be refined by solving an embedded optimal control problem. The optimization problem mixes discrete variables (selected planets, number of revolutions between two successive flybys, number of propelled maneuvers) and continuous variables (flybys dates, maneuver dates, magnitudes and orientations).
Multidisciplinary optimization deals with such problems involving both the vehicle design and the mission scenario. The overall problem is too complex to be address directly, and a specific optimization procedure must be devised for each new case. A bilevel approach consists in separating the design and the trajectory optimization. The design problem is generally non differentiable or may present many local minima. It can be addressed in some cases by mixed optimization methods like branch and bound, or more generally by metaheuristics like simulated annealing, genetic algorithms, particle swarm, etc. None is intrinsically better than another and a specific analysis is needed to formulate the optimization problem in a way suited to the selected method. These algorithms are based partly on a random exploration of the variable space. In order to be successful the exploration strategy has to be customized to the problem specificities. Thousands or millions of trials may be necessary to yield a candidate configuration, based on very simplified performance assessment (e.g., analytical solutions, impulsive velocities, response surface models etc.). The trajectory problem is then solved for this candidate solution in order to assess the real performance, and if necessary iterate on the configuration optimization with a corrected the performance model. Metaheuristics may also be combined with multiobjective optimization approaches since several criteria have to be balanced at the design stage of a new space vehicle. The goal is to build a family of launchers using a common architecture of propelled stages with variants depending the targeted orbit and payload. By this way the development and manufacturing costs are minimized while the launcher configuration and the launch cost can be customized for each flight.
Geometric optimal control
Geometric optimal control (see, e.g., [1, 74, 85]) combines classical optimal control and geometric methods in system theory, with the goal of achieving optimal synthesis results. More precisely, by combining the knowledge inferred from the Pontryagin Maximum Principle (PMP) with geometric considerations, such as the use of Lie brackets and Lie algebras, of differential geometry on manifolds, and of symplectic geometry and Hamiltonian systems, the aim is to describe in a precise way the structure of optimal trajectories. We refer the reader to [74, 85] for a list of references on geometric tools used in geometric optimal control. The foundations of geometric control can be dated back to the Chow’s theorem and to [24, 25], where Brunovsky found that it was possible to derive regular synthesis results by using geometric considerations for a large class of control systems. Apart from the main goal of achieving a complete optimal synthesis, geometric control aims also at deriving higherorder optimality conditions in order to better characterize the set of candidate optimal trajectories.
In this section, we formulate the optimal control problem on differentiable manifolds and recall some tools and results from geometric optimal control. More precisely, the Lie derivative is used to define the order of the state constraints, the Lie and Poisson brackets are used to analyze the singular extremals and to derive higher order optimality conditions, and the optimality conditions (order one, two and higher) are used to analyze the chattering extremals (see Section 3.4 for the chattering phenomenon). These results will be applied in Section 7 on a coupled attitude and trajectory optimization problem.
3.1 Optimal control problem
where the mappings f:M×N→TM, \(f^{0}: M \times N \rightarrow \mathbb {R}\), and \(g: \mathbb {R} \times M \rightarrow \mathbb {R}\) are smooth, and where the controls are bounded and measurable functions defined on [0,t _{ f }(u)] of \(\mathbb {R}^{+}\), taking values in U. The final time t _{ f } may be fixed or not. We denote \(\mathcal {U}\) the set of admissible controls such that the corresponding trajectories steer the system from an initial point of M _{0} to a final point in M _{1}.
For each x(0)∈M _{0} and \(u \in \mathcal {U}\), we can integrate the system (1) from t=0 to t=t _{ f }, and assess the cost C(t _{ f },u) corresponding to x(t)=x(t;x _{0},u(t)) and u(t) for t=[0,t _{ f }]. Solving the problem (\(\mathcal {P}_{0}\)) consists in finding a pair (x(t),u(t))=(x(t;x _{0},u(t)),u(t)) minimizing the cost. For convenience, we define the endpoint mapping to describe the final point of the trajectory solution of the control system (1).
Definition 1
If the optimal control problem has a solution, we say that the corresponding control and trajectory are minimizing or optimal. We refer to [32, 84] for existence results in optimal control.
Next, we introduce briefly the concept of Lie derivative, and of Lie and Poisson brackets (used in Section 3.3.3 for higher order optimality conditions). These concepts will be applied in Section 7 to analyze the pullup maneuver problem.
3.2 Lie derivative, lie bracket, and poisson bracket
In general, the order of the state constraints in optimal control problems is defined through Lie derivatives as we will show on the example in Section 7.1.5.
Definition 2

[·,·] is a bilinear operator;

[X,Y]=−[Y,X];

[X+Y,Z]=[X,Z]+[Y,Z];

[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0 (Jacobi identity);

[α X,β Y]=α β[X,Y]+α(L _{ X } β)Y−β(L _{ Y } α)X.
These properties show that the vector fields (as differential operators) form a Lie algebra. A Lie algebra over \(\mathbb {R}\) is a real vector space \(\mathcal {G}\) together with a bilinear operator \([\cdot,\cdot ]: \mathcal {G} \times \mathcal {G} \to \mathcal {G}\) such that for all \(X,Y,Z \in \mathcal {G}\) we have [X,Y]=−[Y,X] and [X+Y,Z]=[X,Z]+[Y,Z].
We will see in Section 3.3 (and also in Section 7) that the Lie brackets and the Poisson brackets are very useful for deriving higher order optimality conditions in simpler form and for calculating the singular controls.
3.3 Optimality conditions
This section gives an overview of necessary optimality conditions.
For the firstorder optimality conditions, we recall the Lagrange multipliers method for the optimal control problem without control constraints. Such constraints can be accounted in the Lagrangian with additional Lagrange multipliers [23]. This method leads to weaker results than the Pontryagin Maximum Principle which considers needlelike variations accounting directly for the control constraints.
In some cases, the firstorder conditions do not provide adequate information of the optimal control, and the higher order optimality conditions are needed. Therefore we recall the second and higher order necessary optimality conditions that must be met by any trajectory associated to an optimal control u. These conditions are especially useful to analyze the singular solutions because the firstorder optimality conditions do not provide any information in such cases.
3.3.1 Firstorder optimality conditions
3.3.1.0 Lagrange multipliers rule
If we define as usual the intrinsic secondorder derivative \(Q_{t_{f}}\) of the Lagrangian as the Hessian \(\frac {\partial ^{2} L_{t_{f}}}{\partial ^{2} u}(u,\psi,\psi ^{0})\) restricted to the subspace \(\ker \frac {\partial L_{t_{f}}}{\partial u}\), a secondorder necessary condition for optimality is the nonpositivity of \(Q_{t_{f}}\) (with ψ ^{0}≤0), and a secondorder sufficient condition for local optimality is the negative definiteness of \(Q_{t_{f}}\).
These results are weaker to those obtained with the PMP. The Lagrange multiplier (ψ,ψ ^{0}) is in fact related to the adjoint vector introduced in the PMP. More precisely, the Lagrange multiplier is unique up to a multiplicative scalar if and only if the trajectory x(·) admits a unique extremal lift up to a multiplicative scalar, and the adjoint vector (p(·),p ^{0}) can be constructed such that (ψ,ψ ^{0})=(p(t _{ f }),p ^{0}) up to some multiplicative scalar. This relation can be observed from the proof of the PMP. The Lagrange multiplier ψ ^{0}=p ^{0} is associated with the instantaneous cost. The case with p ^{0} null is said abnormal, which means that there are no neighboring trajectories having the same terminal point (see, e.g., [2, 85]).
3.3.1.0 Pontryagin maximum principle
The Pontryagin Maximum Principle (PMP, see [70]) for the problem (\(\mathcal {P}_{0}\)) with control constraints and without state constraints is recalled in the following statement.
Theorem 1
where T _{ x } M _{0} (resp., T _{ x } M _{1}) denote the tangent space to M _{0} (resp., M _{1}) at the point x.
The quadruple (x(·),p(·),p ^{0},u(·)) is called the extremal lift of x(·). An extremal is said to be normal (resp., abnormal) if p ^{0}<0 (resp., p ^{0}=0). According to the convention chosen in the PMP, we consider p ^{0}≤0. If we adopt the opposite convention p ^{0}≥0, then we have to replace the maximization condition (7) with a minimization condition. When there are no control constraints, abnormal extremals project exactly onto singular trajectories.
The proof of the PMP is based on needlelike variations and uses a conic implicit function theorem (see, e.g., [1, 52, 78]). Since these needlelike variants are of order one, the optimality conditions given by the PMP are necessary conditions of the firstorder. For singular controls, higher order control variations are needed to obtain optimality conditions. A singular control is defined precisely as follows.
Definition 3
Assume that M _{0}={x _{0}}. A control u defined on [0,t _{ f }] is said to be singular if and only if the Fréchet differential \(\frac {\partial E}{\partial u}(x_{0},t_{f},u)\) is not of full rank. The trajectory x(·) associated with a singular control u is called singular trajectory.
In practice the condition \(\frac {\partial ^{2} H}{\partial u^{2}}(x(\cdot),p(\cdot),p^{0},u(\cdot))=0\) (the Hessian of the Hamiltonian is degenerate) is used to characterize singular controls. An extremal (x(·),p(·),p ^{0},u(·)) is said totally singular if this condition is satisfied. The is especially the case when the control is affine (see Section 3.3.3).
The PMP claims that if a trajectory is optimal, then it should be found among projections of extremals joining the initial set to the final target. Nevertheless the projection of a given extremal is not necessarily optimal. This motivates the next section on secondorder optimality conditions.
3.3.2 Secondorder optimal conditions
The literature on first and/or secondorder sufficient conditions with continuous control is rich (see, e.g., [42, 61, 62, 65, 89]), which is less the case for discontinuous controls (see, e.g., [68]). We recall hereafter the Legendre type conditions with Poisson brackets to show that geometric optimal control allows a simple expression of the secondorder necessary and sufficient conditions (see Theorem 2).
3.3.2.0 Legendre type conditions

If a trajectory x(·), associated to a control u, is optimal on [0,t _{ f }]in L ^{ ∞ } topology, then the Legendre condition holds along every extremal lift (x(·),p(·),p ^{0},u(·))of x(·), that is$${}\frac{\partial^{2} H}{\partial u^{2}} (x(\cdot),p(\cdot),p^{0},u(\cdot)).(v,v) \leq 0, \quad \forall v\in \mathbb{R}^{m}. $$

If the strong Legendre condition holds along the extremal (x(·),p(·),p ^{0},u(·)), that is, there exists ε _{0}>0such thatthen there exists ε _{1}>0such that x(·)is locally optimal in L ^{ ∞ } topology on [0,ε _{1}]. If the extremal is moreover normal, i.e., p ^{0}≠0, then x(·)is locally optimal in C ^{0} topology on [0,ε _{1}].$${}\frac{\partial^{2} H}{\partial u^{2}} (x(\cdot),p(\cdot),p^{0},u(\cdot)).(v,v) \leq  \epsilon_{0} \v\^{2}, \quad \forall v\in \mathbb{R}^{m}, $$
The C ^{0} local optimality and L ^{ ∞ } local optimality are respectively called strong local optimality and weak local optimality^{2}. The Legendre condition is a necessary optimality condition, whereas the strong Legendre condition is a sufficient optimality condition. We say that we are in the regular case whenever the strong Legendre condition holds along the extremal. Under the strong Legendre condition, a standard implicit function argument allows expressing, at least locally, the control u as a function of x and p.
In the totally singular case, the strong Legendre condition is not satisfied and we have the following generalized condition [1, 51].
Theorem 2

If a trajectory x(·), associated to a piecewise smooth control u, and having a totally singular extremal lift (x(·),p(·),p ^{0},u(·)), is optimal on [0,t _{ f }] in L ^{ ∞ } topology, then the Goh condition holds along the extremal, that iswhere {·,·} denotes the Poisson bracket on T ^{∗} M. Moreover, the generalized Legendre condition holds along every extremal lift (x(·),p(·),p ^{0},u(·)) of x(·), that is$$\left\{\frac{\partial H}{\partial u_{i}},\frac{\partial H}{\partial u_{j}} \right\} = 0, $$$${}\left\{ \!\left\{ H, \frac{\partial H}{\partial u}.v \right\},\frac{\partial H}{\partial u}.v \right\} + \left\{ \frac{\partial^{2} H}{\partial u^{2}}.(\dot{u},v),\frac{\partial H}{\partial u}.v \right\} \leq 0, \ \forall v\in \mathbb{R}^{m}. $$

If the Goh condition holds along the extremal lift (x(·),p(·),p ^{0},u(·)), if the strong Legendre condition holds along the extremal (x(·),p(·),p ^{0},u(·)), that is, there exists ε _{0}>0 such thatand if moreover the mapping \(\frac {\partial f}{\partial u}(x_{0},u(0)):\mathbb {R}^{m} \mapsto T_{x_{0}}M\) is onetoone, then there exists ε _{1}>0 such that x(·) is locally optimal in L ^{ ∞ } topology on [0,ε _{1}].$$\begin{aligned} &\left\{ \left\{ H, \frac{\partial H}{\partial u}.v \right\},\frac{\partial H}{\partial u}.v \right\}\\ &\qquad+ \left\{ \frac{\partial^{2} H}{\partial u^{2}}.(\dot{u},v),\frac{\partial H}{\partial u}.v \right\} \leq  \epsilon_{0} \v\^{2}, \ \forall v\in \mathbb{R}^{m}, \end{aligned} $$
As we have seen, the Legendre (or generalized Legendre) condition is a necessary condition, while the strong (or strong generalized Legendre) condition is a sufficient condition. However, these sufficient conditions are not easy to verify in practice. This leads to the next section where we explain how to use the socalled conjugate point along the extremal to determine the time when the extremal is no longer optimal.
3.3.2.0 Conjugate points
We consider here the simplified problem (\(\mathcal {P}_{0}\)) with \(M=\mathbb {R}^{n}\), M _{0}={x _{0}}, M _{1}={x _{1}}, and \(U=\mathbb {R}^{m}\). Under the strict Legendre assumption assuming that the Hessian \( \frac {\partial ^{2} H}{\partial u^{2}} (x,p,p^{0},u)\) is negative definite, the quadratic form \(Q_{t_{f}}\) is negative definite if t _{ f }>0 is small enough.
Definition 4
The first conjugate time is defined by the infimum of times t>0 such that Q _{ t } has a nontrivial kernel. We denote the first conjugate time along x(·) by t _{ c }.
The time t _{ c } is a conjugate time along x(·)if and only if the mapping \(\exp _{x_{0}}(t_{c},\cdot)\) is not an immersion at p _{0}, i.e., the differential of the mapping \(\exp _{x_{0}}(t_{c},\cdot)\) is not injective.
Essentially this result states that computing a first conjugate time t _{ c } reduces to finding the zero of some determinant along the extremal. In the smooth case (the control can be expressed as a smooth function of x and p), the survey article [15] provides also some algorithms to compute first conjugate times. In case of bangbang control, a conjugate time theory has been developed (see [79] for a brief survey of the approaches), but the computation of conjugate times remains difficult in practice (see, e.g., [60]).
When the singular controls are of order one (see Definition 5), the secondorder optimality condition is sufficient for the analysis. For higher order singular controls, higher order optimality conditions are needed which are recalled in the next section.
3.3.3 Order of singular controls and higher order optimality conditions
In this section we recall briefly the order of singular controls and the higher order optimality conditions. They will be used in Section 7.1 to analyze the example, which exhibits a singular control of order two. It is worth noting that when the singular control is of order 1 (also called minimal order in [16, 34]), these higher order optimality conditions are not required.
where f, g _{1} and g _{2} are smooth vector fields on M. We assume that M _{1} is accessible from x _{0}, and that there exists a constant \(B_{t_{f}}\) such that for every admissible control u, the corresponding trajectory x _{ u }(t) satisfies \(\phantom {\dot {i}\!}\ x_{u}(t) \ \leq B_{t_{f}}\) for all t∈[0,t _{ f }]. Then, according to classical results (see, e.g., [32, 84]), there exists at least one optimal solution (x(·),u(·)), defined on [0,t _{ f }].
We call Φ (as well as its components) the switching function. We say that an arc (restriction of an extremal to a subinterval I) is regular if ∥Φ(t)∥≠0 along I. Otherwise, the arc is said to be singular.
Following [45], we give here below a precise definition of the order of a singular control. The use of Poisson (and Lie) brackets simplifies the formulation of the higher order optimality conditions. This is one of the reasons making geometric optimal control theory a valuable tool in practice.
Definition 5
 1.the first (2q−1)th time derivatives of h _{ i }, i=1,2, do not depend on u and$$\frac{d^{k}}{dt^{k}}(h_{i}) = 0,\quad k=0,1,\cdots,2q1, $$
 2.the 2qth time derivative of h _{ i }, i=1,2, depends on u linearly andalong I.$$\frac{\partial}{\partial u_{i}} \frac{d^{2q}}{dt^{2q}}(h_{i}) \neq 0, \quad \det \left(\frac{\partial}{\partial u} \frac{d^{2q}}{dt^{2q}}\Phi \right) \neq 0,\quad i=1,2, $$
The condition of a nonzero determinant guarantees that the optimal control can be computed from the 2qth time derivative of the switching function. Note that this definition requires that the two components of the control have the same order.
We next recall the Goh and generalized LegendreClebsch conditions (see [51, 56, 58]). It is worth noting that in [58], the following higherorder necessary conditions hold even when the components of the control u have different orders.
Lemma 1
Corollary 1
In the next section, we recall the chattering phenomenon that may happen in the optimal control problem, manely when there exit singular controls of higher order in the problem. This phenomenon is actually not rare as illustrated in [90] by many examples (in astronautics, robotics, economics, and etc.). These examples are mostly single input systems. The existence of chattering phenomenon for biinput control affine systems is also proved in [93].
3.4 Chattering phenomenon
Numerical methods in optimal control
The numerical procedures for optimal control problems are usually classified between direct and indirect methods. Direct methods discretize the optimal control problem in order to rewrite it as a nonlinear large scale optimization problem. The process is straightforward and it can be applied in a systematic manner to any optimal control problem. New variables or constraints may be added easily. But achieving an accurate solution requires a careful discretization and the convergence may be difficult due to the large number of variables. On the other hand indirect methods are based on the Pontryagin Maximum Principle which gives a set of necessary conditions for a local minimum. The problem is reduced to a nonlinear system that is generally solved by a shooting method using a Newtonlike algorithm. The convergence is fast and accurate, but the method requires both an adequate starting point and a high integration accuracy. The sensitivity to the initial guess can be lowered by multiple shooting which breaks the trajectory into several legs linked by interface constraints, at the expense of a larger nonlinear system. The indirect method requires also prior theoretical work for problems with singular solutions or with state constraints. Handling these constraints by penalty method can avoid numerical issues, but yields less optimal solutions. The principles of both indirect and direct methods are recalled hereafter.
4.1 Indirect methods
In indirect approaches, the Pontryagin Maximum Principle (firstorder necessary condition for optimality) is applied to the optimal control problem in order to express the control as a function of the state and the adjoint. This reduces the problem to a nonlinear system of n equations with n unknowns generally solved by Newtonlike methods. Indirect methods are also called shooting methods. The principle of the simple shooting method and of the multiple shooting method are recalled. The problem considered in this section is (\(\mathcal {P}_{0}\)).
This problem can be solved by Newtonlike methods or other iterative methods.
The multiple shooting method improves the numerical stability at the expense of a larger nonlinear system. An adequate node number must be chosen making a compromise between the system dimension and the convergence domain.
4.2 Direct methods
From a more general point of view, a finite dimensional representation of the control and of the state has to be chosen such that the differential equation, the cost, and all constraints can be expressed in a discrete way.
The numerical resolution of a nonlinear programming problem is standard, by gradient methods, penalization, quasiNewton, dual methods, etc. (see, e.g., [9, 50, 57, 81]). There exist many efficient optimization packages such as IPOPT (see [86]), MUSCODII (see [39]), or the Minpack project (see [66]) for many optimization routines.
Alternative variants of direct methods are the collocation methods, the spectral or pseudospectral methods, the probabilistic approaches, etc.
4.3 Comparison between methods
Pros and cons for direct and indirect methods
Direct methods  Indirect methods  

A priori knowledge of the solution structure  Not required  Required 
Sensible to the initial condition  Not sensible  Very sensible 
Handle the state constraints  Easy  Difficult 
Convergence speed and accuracy  Relatively slow and inaccurate  Fast and accurate 
Computational aspect  Memory demanding  Parallelizable 
In practice no approach is intrinsically better than the other. The numerical method should be chosen depending on the problem features and on the known properties of the solution structure. These properties are derived by a theoretical analysis using the geometric optimal control theory. When a high accuracy is desired, as is generally the case for aerospace problems, indirect methods should be considered although they require more theoretical insight and may raise numerical difficulties.
Whatever the method chosen, there are many ways to adapt it to a specific problem (see [85]). Even with direct methods, a major issue lies in the initialization procedure. In recent years, the numerical continuation has become a powerful tool to overcome this difficulty. The next section recalls some basic mathematical concepts of the continuation approaches, with a focus on the numerical implementations of these methods.
Continuation methods
5.1 Existence results and discrete continuation
The basic idea of continuation (also called homotopy) methods is to solve a difficult problem step by step starting from a simpler problem by parameter deformation. The theory and practice of the continuation methods are wellspread (see, e.g., [3, 71, 87]). Combined with the shooting problem derived from the PMP, a continuation method consists in deforming the problem into a simpler one (that can be easily solved) and then solving a series of shooting problems step by step to come back to the original problem.
One difficulty of homotopy methods lies in the choice of a sufficiently regular deformation that allows the convergence of the homotopy method. The starting problem should be easy to solve, and the path between this starting problem and the original problem should be easy to model. Another difficulty is to numerically follow the path between the starting problem and the original problem. This path is parametrized by a parameter denoted λ. When the homotopic parameter λ is a real number and when the path is linear^{3} in λ, the homotopy method is rather called a continuation method.
The choice of the homotopic parameter may require considerable physical insight into the problem. This parameter may be defined either artificially according to some intuition, or naturally by choosing physical parameters of the system, or by a combination of both.
A zero path is a curve c(s)∈G ^{−1}(0) where s represents the arc length. We would like to trace a zero path starting from a point Z _{0} such that G(Z _{0},0)=0 and ending at a point Z _{ f } such that G(Z _{ f },1)=0.
The first question to address is the existence of zero paths, since the feasibility of the continuation method lies on this assumption. The second question to address is how to numerically track such zero paths when they exist.
Existence of zero paths
The local existence of the zero paths is answered by the implicit function theorem. Some regularity assumptions are needed, as in the following statement (which is the contents of [46, Theorem 2.1]).
Theorem 3

Given any (Z,λ)∈{(Z,λ)∈Ω×[0,1] ∣ G(Z,λ)=0}, the Jacobian matrixis of maximum rank N;$$G^{\prime} = \left(\frac{\partial G}{\partial Z_{1}},\cdots,\frac{\partial G}{\partial Z_{N}},\frac{\partial G}{\partial \lambda} \right), $$

Given any Z∈{Z∈Ω ∣ G(Z,0)=0}∪{Z∈Ω ∣ G(Z,1)=0}, the Jacobian matrixis of maximum rank N;$$G^{\prime} = \left(\frac{\partial G}{\partial Z_{1}},\cdots,\frac{\partial G}{\partial Z_{N}}\right) $$
Then {(Z,λ)∈Ω×[0,1] ∣ G(Z,λ)=0} consists of the paths that is either a loop in \(\bar {\Omega } \times [0,1]\) or starts from a point of \(\partial \bar {\Omega } \times [0,1]\) and ends at another point of \(\partial \bar {\Omega } \times [0,1]\), where \(\partial \bar {\Omega }\) denotes the boundary of \(\bar {\Omega }\).
Now we provide basic arguments showing the feasibility of the continuation method (see Section 4.1 of [85] for more details).
where E is the endpoint mapping defined in Definition 1.

there are no minimizing abnormal extremals;

there are no minimizing singular controls: by Definition 3, the control u is not singular means that the mapping \({dE}_{x_{0},t_{f},\lambda }(u)\) is surjective;

there are no conjugate points (by Definition 4 the quadratic form \(Q_{t_{f}}\) is not degenerate). The absence of conjugate point can be numerically tested (see, e.g., [15]).
We will see that these assumptions are essential for the local feasibility of the continuation methods.
where \(Q_{t_{f},\lambda }\) is the Hessian \(\frac {\partial ^{2} L_{t_{f},\lambda }}{\partial ^{2} u}(u,\psi,\psi ^{0})\) restricted to \(\ker \frac {\partial L_{t_{f},\lambda }}{\partial u}\), and \({dE}_{x_{0},t_{f},\lambda }(u)^{\ast }\) is the transpose of \({dE}_{x_{0},t_{f},\lambda }(u)\).
We observe that the matrix (12) is invertible if and only if the linear mapping \({dE}_{x_{0},t_{f},\lambda }(u)\) is surjective and the quadratic form \(Q_{t_{f},\lambda }\) is nondegenerate. These properties correspond to the absence of any minimizing singular control and conjugate points, which are the assumptions done for the local feasibility of the continuation procedure.
On the one hand, according to the PMP, the optimal control u satisfies the extremal Eqs. 6, and thus u _{ λ }=u _{ λ }(t,p _{0,λ }) is a function of the initial adjoint p _{0,λ }. On the other hand, the Lagrange multipliers are related to the adjoint vector by p(t _{ f })=ψ, and thus ψ _{ λ }=ψ _{ λ }(p _{0,λ }) is also a function of p _{0,λ }. Therefore, the shooting function defined by S(p _{0},λ)=G(u(p _{0}),ψ(p _{0}),λ) has an invertible Jacobian if the matrix (12) is invertible. We conclude then that the assumptions (1)(3) mentioned above are sufficient to ensure the local feasibility.
Despite of local feasibility, the zero path may not be globally defined for any λ∈[0,1]. The path could cross some singularity or diverge to infinity before reaching λ=1.
The first possibility can be eliminated by assuming (2) and (3) over all the domain Ω and for every λ∈[0,1]. The second possibility is eliminated if the paths remain bounded or equivalently by the properness of the exponential mapping (i.e., the initial adjoint vectors p _{0,λ } that are computed along the continuation procedure remain bounded uniformly with respect to λ). According to [21, 82], if the exponential mapping is not proper, then there exists an abnormal minimizer. By contraposition, if one assumes the absence of minimizing abnormal extremals, then the required boundedness follows.
For the simplified problem (11), where the controls are unconstrained and the singular trajectories are the projections of abnormal extremals, if there are no minimizing singular trajectory nor conjugate points over Ω, then the continuation procedure (13) is globally feasible on [0,1].
In more general homotopy strategies, the homotopic parameter λ is not necessarily increasing monotonically from 0 to 1. There may be turning points (see, e.g., [87]) and it is preferable to parametrize the zero path by the arc length s. Let c(s)=(Z(s),λ(s)) be the zero path such that G(c(s))=0. Then, a turning point of order one is the point where \(\lambda ^{\prime } (\bar {s})= 0\), \(\lambda ^{\prime \prime } (\bar {s}) \neq 0\). In [27], the authors indicate that if \(\lambda =\lambda (\bar {s})\) is a turning point of order one, then the corresponding final time t _{ f } is a conjugate time, and the corresponding point \(E_{x_{0},t_{f},\lambda }(u(x_{0},p_{0},t_{f},\lambda))\) is the corresponding conjugate point ^{4}. By assuming the absence of conjugate points over Ω for all λ∈[0,1], the possibility of turning points is discarded.
Unfortunately, assuming the absence of singularities is in general too strong, and weaker assumptions do not allow concluding to the feasibility of the continuation method. In the literature, there are essentially two approaches to tackle this difficulty. The first one is of local type. One detects the singularities or bifurcations along the zero path (see, e.g., [3]). The second one is of global type, concerning the socalled globally convergent probabilityone homotopy method. We refer the readers to [35, 87] for more details concerning this method.
Numerical tracking the zero paths There exists many numerical algorithms to track a zero path. Among these algorithms, the simplest one is the so called discrete continuation or embedding algorithm. The continuation parameter denoted λ, is discretized by \(\phantom {\dot {i}\!}0= \lambda ^{0} < \lambda ^{1} < \cdots < \lambda ^{n_{l}} = 1\) and the sequence of problems G(Z,λ ^{ i })=0, i=1,⋯,n _{ l } is solved to end up with a zero point of F(Z). If the increment △λ=λ ^{ i+1}−λ ^{ i } is small enough, then the solution Z ^{ i } associated to λ ^{ i } such that G(Z ^{ i },λ ^{ i })=0 is generally close to the solution of G(Z,λ ^{ i+1})=0. The discrete continuation algorithm is detailed in Algorithm 1.
In some cases the parameter λ may be ill suited to parameterize the zero path, and thus causes a slow progress or even a failure of the discrete continuation. Two enhancements (predictorcorrector methods and piecewiseLinear methods) have been proposed in the literature.
5.2 Predictorcorrector (PC) continuation
A natural parameter for the zero curve (Z,λ) is the arclength denoted s.
where \(J_{G} = \frac {\partial G(Z(s),\lambda (s))}{\partial (Z,\lambda)}\) is the Jacobian, and \(t(J_{G})=\frac {dc(s)}{ds}\) is the tangent vector of the zero path c(s).
The PC continuation is described by Algorithm 2.
When the optimal control problem is regular (in the sense of the Legendre condition are defined) and the homotopic parameter is a scalar, one can use the so called differential continuation or differential pathfollowing. This method consists in integrating accurately t(J _{ G }) satisfying (14) (see details in [26]). The correction step is replaced by the mere integration of an ordinary differential equation with the help of automatic differentiation (see, e.g., [5, 29]).
5.3 Piecewiselinear (PL) continuation
The main advantage of the PL method is that it only needs the zero paths to be continuous (smoothness assumption of G is not necessary). For a detailed description of the PL methods, we refer the readers to [3, 4, 47].
Here we present the basic idea of the PL methods, which are also referred to as a simplicial methods. A PL continuation consists of following exactly a piecewiselinear curve \(c_{\mathcal {T}}(s)\) that approximates the zero path c(s)∈G ^{−1}(0).
The approximation curve \(c_{\mathcal {T}}(s)\) is a polygonal path relative to an underlying triangulation \(\mathcal {T}\) of \(\mathbb {R}^{N+1}\), which is a subdivision of \(\mathbb {R}^{N+1}\) into (N+1)simplices. ^{5}

\(G_{\mathcal {T}}(v) = G(v)\) for all vertices of \(\mathcal {T}\);

for any N+1simplex \(\sigma = [v_{1},v_{2},\cdots,v_{N+2}] \in \mathcal {T}\), the restriction \(G_{\mathcal {T}} _{\sigma }\) of \(G_{\mathcal {T}}\) to σ is an affine map.
In aerospace applications, where the continuation procedure is in general differentiable, the PL methods are usually not as efficient as the PC methods or the differential continuation that we present in next sections. Nevertheless when singularities exist in the zero path, the PL method is probably the most efficient one.
In the next section, we recall briefly some successful applications of the geometric optimal control techniques and the numerical continuation to trajectory problems, including orbital transfer problems and atmospheric reentry problems. Note that attitude problems, namely the controllability problems, have also been treated by geometric control theory, see e.g. [19, 38]. We refer the readers to the book [20] and its reference for more details.
Applications to trajectory problems
In this section, we recall applications of the geometric optimal control theory and numerical continuation methods in trajectory problems, namely in orbital transfer problems and in atmospheric reentry problems. The aim of this section is to show that the continuation and the geometric optimal control theory have been applied successfully to trajectory problems. Indeed, they are powerful and efficient tools to combine with traditional optimal control theory.
6.1 Orbital transfer problems
where r(·) is the position of the spacecraft, μ is the gravitational constant of the planet, T(·)≤T _{ max } is the bounded thrust, and m(·) is the mass with β a constant depending on the specific impulse of the engine.
Controllability properties ensuring the feasibility of the problem have been studied in [15, 20], based on the analysis of Lie algebra generated by the vector fields of the system.
The minimum time low thrust transfer is addressed for example in [28]. It is observed that the domain of convergence of the Newtontype method in the shooting problem becomes smaller when the maximal thrust decreases. Therefore, a natural continuation process consists in starting with larger values of the maximal thrust and then decreasing step by step the maximal thrust. In [28], the authors started with the maximal thrust T _{ max }=60 N and achieved the continuation up to T _{ max }=0.14 N.
The minimum fuel consumption orbit transfer problem has also been widely studied. With the cost functional \(\int _{0}^{t_{f}} \ T(t)\ dt\), the problem is more difficult than minimizing the time, since the optimal control is discontinuous. In [48, 49], the authors propose a continuation on the cost functional, starting from the minimumenergy problem. The cost functional is defined by \(\int _{0}^{t_{f}} \left ((1\lambda)\ T(t)\^{2} + \lambda \ T(t)\ \right) dt\), where λ∈[0,1] is the homotopy parameter. When λ=0 (minimumenergy), the control derived from the PMP is continuous and the shooting problem is thus easier to solve. The authors prove the existence of a zero path from λ=0 and to λ=1. This continuation approach is later applied in [33] for studying the L ^{1}minimization of trajectory optimization problem. Such minimumfuel lowthrust transfers are very important for deep space explorations, since all the propellant must be carried on board by the satellite. Similar continuation procedures have also been applied to the wellknown Goddard’s problem, and to its threedimensional variants ([12, 14]). The possible singular arcs (along which the norm of the thrust is neither zero nor maximal) have thus been analyzed and numerically computed.
Continuation procedures are also valuable for highthrust orbital transfer problems. In [31], the authors proposed a continuation procedure starting from a flat Earth model with constant gravity. The variable gravity and the Earth curvature are introduced step by step by homotopy parameters. The theoretically analysis of the flat Earth model shows that the solution structure consists in a single boost followed by a coast arc. This helps solving the starting problem in a direct way, before coming back by continuation to the real round Earth problem. The round Earth solution exhibits a different solution structure (boost – coast – boost) which appears progressively along the continuation process.
6.2 Atmospheric reentry problem
An atmospheric reentry typically begins at an altitude of 120k m and ends with a landing phase. The final landing phase until the touchdown is generally studied apart and it is highly dependent on the mission specifications (ground or sea landing, manned or unmanned flight, etc). The socalled atmospheric leg aims at reducing the vehicle energy before the final landing phase. No fuel is used and the braking has to be fully achieved by aerodynamics while satisfying the state constraints, in particular on the thermal flux. The final conditions specify a target position at a low altitude, typically less than 20 k m.
The vehicle is considered as a glider submitted to the gravity and the aerodynamic forces, the control u being the bank angle and possibly the angle of attack. The optimal control problem consists thus in steering the space shuttle from given entry conditions to targeted final conditions while minimizing the total heat and satisfying state constraints on the thermal flux, the normal acceleration, and the dynamic pressure. We refer the readers to [22, 83] for a formulation of this problem. The control u acts on the lift force orientation, changing simultaneously the descent rate and the heading angle.
A practical guidance strategy consists in following the constraint boundaries, successively : thermal flux, normal acceleration, and dynamic pressure. This strategy does not care about the cost functional and it is therefore not optimal. Applying the Pontryagin Maximum Principle with state constraints is not promising due to a narrow domain of convergence of the shooting method. Finding a correct guess for the initial adjoint vector proves quite difficult. Therefore direct methods are generally preferred for this atmospheric reentry problem (see, e.g., [6, 7, 69]).
Here we recall two alternative approaches to address the problem by indirect methods.
The first approach is to analyze the control system using geometric control theory. For example, in [17, 18, 83], a careful analysis of the control system provides a precise description of the optimal trajectory. The resulting problem reduction makes it tractable by the shooting method. More precisely, the control system is rewritten as a singleinput controlaffine system in dimension three under some reasonable assumptions. Local optimal syntheses are derived from extending existing results in geometric optimal control theory. Based on perturbation arguments, this local nature of the optimal trajectory is then used to provide an approximation of the optimal trajectory for the full problem in dimension six, and finally simple approximation methods are developed to solve the problem.
A second approach is to use the continuation method. For example, in [55], the problem is solved by a shooting method, and a continuation is applied on the maximal value of the thermal flux. It is shown in [11, 54] that under some appropriate assumptions, the change in the structure of the trajectory is regular, i.e., when a constraint becomes active along the continuation, only one boundary arc appears. Nevertheless it is possible that an infinite number of boundary arcs appear (see, e.g., [72]). This phenomenon is possible when the constraint is of order three at least. By using a properly modified continuation procedure, the reentry problem was solved in [55] and the results of [18] were retrieved.
Now we have shown by examples in trajectory problems that the geometric optimal control theory can be used to analyze the problem and the numerical continuation can be used to design efficient numerical resolution methods. In the next section, we will show step by step by a nonacademic attitudetrajectory problem how the analysis and the design of numerical continuation procedure are done.
Application to attitudetrajectory optimal control
In this section, the nonacademic attitudetrajectory optimal control problem for a launch vehicle (classical and airborne) is analyzed in detail. Through this example, we illustrate how to analyze the (singular and regular) extremals of the problem with Lie and Poisson brackets, and how to elaborate numerical continuation procedures adapted to the solution structure. Indeed the theoretical analysis reveals the existence of a chattering phenomenon. Being aware of this feature is essential to devise an efficient numerical solution method.
7.1 Geometric analysis and numerical continuations for optimal attitude and trajectory control problem (\(\mathcal {P}_{S}\))
The problem is formulated in terms of dynamics, control, constraints and cost. The Pontryagin Maximum Principle and the geometric optimal control are then applied to analyze the extremals, revealing the existence of the chattering phenomenon.
7.1.1 Formulation of (\(\mathcal {P}_{S}\)) and difficulties
7.1.1.0 Minimum time attitudetrajectory control problem (\(\mathcal {P}_{S}\))
In this section, we formulate an attitudetrajectory minimum time control problem, denoted by (\(\mathcal {P}_{S}\)).
The trajectory of a launch vehicle is controlled by the thrust which can only have limited deflection angles with the vehicle longitudinal axis. Controlling the thrust direction requires controlling the vehicle attitude. When the attitude dynamics is slow, or when the orientation maneuver is large, this induces a coupling between the attitude motion and the trajectory, as explained in Section 6.
When this coupling is not negligible the dynamics and the state must account simultaneously for the trajectory variables (considering the launch vehicle as a mass point) and the attitude variables (e.g., the Euler angles or the quaternion associated to the body frame).
The objective is then to determine the deflection angle law driving the vehicle from given initial conditions to the desired final attitude and velocity, taking into account the attitudetrajectory coupling.
where (v _{ x }, v _{ y }, v _{ z }) represents the velocity, (g _{ x }, g _{ y }, g _{ z }) represents the gravity acceleration, θ (pitch), ψ (yaw), ϕ (roll) are the Euler angles, a is the ratio of the thrust force to the mass, and b is the ratio of the thrust torque to the transverse inertia of the launcher (a and b are assumed constant). \(u=(u_{1},u_{2}) \in \mathbb {R}^{2}\) is the control input of the system satisfying \(u=u_{1}^{2}+u_{2}^{2} \leq 1\). See more details of the model and the problem formulation in [92].
where θ _{ f }, ψ _{ f }, ϕ _{ f }, ω _{ xf } and ω _{ yf } are desired final values of the state variables.
The first two conditions in (19) define a final velocity direction parallel to the longitudinal axis of the launcher, or in other terms a zero angle of attack.
The initial and final conditions are also called terminal conditions.
7.1.1.0 Difficulties
The problem (\(\mathcal {P}_{S}\)) is difficult to solve directly due to the coupling of the attitude and the trajectory. The system is of dimension 8 and its dynamics contains both slow (trajectory) and fast (attitude) components. In fact, the attitude movement is much faster than the trajectory movement. This observation is being particularly important in order to design an appropriate solution method. The idea is to define a simplified starting problem and then to apply continuation techniques. However the essential difficulty of this problem is the chattering phenomenon making the control switch an infinite number of times over a compact time interval. Such a phenomenon typically occurs when trying to connect bang arcs with higherorder singular arcs (see, e.g., [44, 63, 90, 91], or Section 3.4).
In a preliminary step, we limited ourselves to the planar problem, which is a singleinput control affine system. This planar problem is close to real flight conditions of a launcher ascent phase. We have used the results of M.I. Zelikin and V.F. Borisov [90, 91] to understand the chattering phenomenon and to prove the local optimality of the chattering extremals. We refer the readers to [93] for details.
In a second step using the Pontryagin Maximum Principle and the geometric optimal control theory (see [1, 74, 85]), we have established an existence result of the chattering phenomenon for a class of biinput control affine systems and we have applied the result to the problem (\(\mathcal {P}_{S}\)). More precisely, based on Goh and generalized LegendreClebsch conditions, we have proved that there exist optimal chattering arcs when connecting the regular arcs with a singular arc of order two.
7.1.2 Geometric analysis for (\(\mathcal {P}_{S}\))
7.1.2.0 Singular arcs and necessary conditions for optimality
Note that the abnormal extremals correspond to p ^{0}=0 in the PMP. We suspect the existence of optimal abnormal extremals in (\(\mathcal {P}_{S}\)) for certain (nongeneric) terminal conditions. In the planar version of (\(\mathcal {P}_{S}\)) studied in [93], it is proved that there is no abnormal minimizer (p ^{0}≠0) if the optimal control switches at least two times. We expect that the same property is still true here. We are able to prove that the singular extremals of (\(\mathcal {P}_{S}\)) are normal, however, we are not able to establish a clear relationship between the number of switchings and the existence of abnormal minimizers as in [93]. Thus, in our numerical simulations later, we will assume that there is at least one normal extremal for problem (\(\mathcal {P}_{S}\)) and compute it.
where \(T_{x(t_{f})}M_{1}\) is the tangent space to M _{1} at the point x(t _{ f }). The final time t _{ f } being free and the system being autonomous, we have also h _{0}(x(t),p(t))+∥Φ(t)∥+p ^{0}=0, ∀t∈[0,t _{ f }].
We say that an arc (restriction of an extremal to a subinterval I) is regular if ∥Φ(t)∥≠0 along I. Otherwise, the arc is said to be singular. An arc that is a concatenation of an infinite number of regular arcs is said to be chattering. The chattering arc is associated with a chattering control that switches an infinite number of times, over a compact time interval. A junction between a regular arc and a singular arc is said to be a singular junction.
We next compute the singular control, since it is important to understand and explain the occurrence of chattering. The usual method for to computing singular controls is to differentiate repeatedly the switching function until the control explicitly appears. Note that here we need to use the notion of Lie bracket and Poisson bracket (see Section 3.2).
along I.
We will see later that the solutions of the problem of order zero (defined in the following Section) lie on this singular surface S.
Finally, the possibility of chattering in problem (\(\mathcal {P}_{S}\)) is demonstrated in [92]. A chattering arc appears when trying to connect a regular arc with an optimal singular arc. More precisely, let u be an optimal control, solution of (\(\mathcal {P}_{S}\)), and assume that u is singular on the subinterval (t _{1},t _{2})⊂[0,t _{ f }] and is regular elsewhere. If t _{1}>0 (resp., if t _{2}<t _{ f }) then, for every ε>0, the control u switches an infinite number of times over the time interval [t _{1}−ε,t _{1}] (resp., on [t _{2},t _{2}+ε]). The condition (22) was required in the proof.
The knowledge of chattering occurrence is essential for solving the problem (\(\mathcal {P}_{S}\)) in practice. Chattering raises indeed numerical issues that may prevent any convergence, especially when using an indirect approach (shooting). The occurrence of the chattering phenomenon in (\(\mathcal {P}_{S}\)) explains the failure of the indirect methods for certain terminal conditions (see also the recent paper [30]).
7.1.3 Indirect method and numerical continuation procedure for (\(\mathcal {P}_{S}\))
The principle of the continuation procedure is to start from the known solution of a simpler problem (called hereafter the problem of order zero) in order to initialize an indirect method for the more complicated problem (\(\mathcal {P}_{S}\)). This simple lowdimensional problem will then be embedded in higher dimension, and appropriate continuations will be applied to come back to the initial problem.
The problem of order zero defined below considers only the trajectory dynamics which is much slower than the attitude dynamics. Assuming an instantaneous attitude motion simplifies greatly the problem and provides an analytical solution. It is worth noting that the solution of the problem of order zero is contained in the singular surface S filled by the singular solutions for (\(\mathcal {P}_{S}\)), defined by (23).
7.1.3.0 Auxiliary problems
We define the problem of order zero, denoted by (\(\mathcal {P}_{0}\)) as the “subproblem” of problem (\(\mathcal {P}_{S}\)) reduced to the trajectory dynamics. The control for this problem is directly the vehicle attitude, and the attitude dynamics is not simulated.
The Euler angles θ ^{∗}∈(−π,π) and ψ ^{∗}∈(−π/2,π/2) are retrieved from the components of the vector \(\vec {e}^{\ast }\) since \( \vec {e}^{\ast } = \left (\sin \theta ^{\ast } \cos \psi ^{\ast }, \sin \psi ^{\ast }, \cos \theta ^{\ast } \sin \psi ^{\ast }\right)^{\top } \).
A natural continuation strategy consists in changing continuously these terminal conditions (24)(26) to come back to the terminal conditions (20) of (\(\mathcal {P}_{S}\)).
Unfortunately the chattering phenomenon may prevent the convergence of the shooting method. When the terminal conditions are in the neighborhood of the singular surface S, the optimal extremals are likely to contain a singular arc and thus chattering arcs causing the failure of the shooting method. In order to overcome the numerical issues we define a regularized problem with a modified cost functional.
for the biinput controlaffine system (17), under the control constraints −1≤u _{ i }≤1, i=1,2, and with the terminal conditions (20). The constant K>0 is arbitrary. We have replaced the constraint \(u_{1}^{2}+u_{2}^{2}\leq 1\) (i.e., u takes its values in the unit Euclidean disk) with the constraint that u takes its values in the unit Euclidean square. Note that we use the Euclidean square (and not the disk) because we observed that our numerical simulations worked better in this case. This regularized optimal control problem with the cost (27) has continuous extremal controls and it is therefore well suited to a continuation procedure.
7.1.3.0 Numerical continuation procedure

First, we embed the solution of (\(\mathcal {P}_{0}\)) into (\(\mathcal {P}_{R}\)). For convenience, we still denote (\(\mathcal {P}_{0}\)) the problem (\(\mathcal {P}_{0}\)) formulated in higher dimension.

Then, we pass from (\(\mathcal {P}_{0}\)) to (\(\mathcal {P}_{S}\)) by means of a numerical continuation procedure, involving three continuation parameters. The first two parameters λ _{1} and λ _{2} are used to pass continuously from the optimal solution of (\(\mathcal {P}_{0}\)) to the optimal solution of the regularized problem (\(\mathcal {P}_{R}\)) with prescribed terminal attitude conditions, for some fixed K>0. The third parameter λ _{3} is then used to pass to the optimal solution of (\(\mathcal {P}_{S}\)) (see Fig. 13).
Note again that there is no concern using \(S_{\lambda _{1}}\) as shooting function for (\(\mathcal {P}_{R}\)). This would not be the case for (\(\mathcal {P}_{S}\)) : if \(S_{\lambda _{1}}=0\), then together with ω _{ x }(t _{ f })=0 and ω _{ y }(t _{ f })=0, the final point (x(t _{ f }),p(t _{ f })) of the extremal would lie on the singular surface S defined by (23) and this would cause the failure of the shooting method. On the opposite, for problem (\(\mathcal {P}_{R}\)), even when x(t _{ f })∈S, the shooting problem is smooth and it can still be solved.
The solution of (\(\mathcal {P}_{0}\)) is a solution of (\(\mathcal {P}_{R}\)) for λ _{1}=0, corresponding to the terminal conditions (24)(25) (the other states at t _{ f } being free). By continuation, we vary λ _{1} from 0 to 1, yielding the solution of (\(\mathcal {P}_{R}\)), for λ _{1}=1. The final state of the corresponding extremal gives some unconstrained Euler angles denoted by θ _{ e }=θ(t _{ f }), ψ _{ e }=ψ(t _{ f }), ϕ _{ e }=ϕ(t _{ f }), ω _{ xe }=ω _{ x }(t _{ f }) and ω _{ ye }=ω _{ y }(t _{ f }).
The shooting function \(S_{\lambda _{3}}\) is defined similarly to \(S_{\lambda _{2}}\), replacing H _{ K }(t _{ f }) with H _{ K }(t _{ f },λ _{3}). The solution of (\(\mathcal {P}_{S}\)) is then obtained by varying λ _{3} continuously from 0 to 1.
This last continuation procedure fails in case of chattering, and thus it cannot be successful for any arbitrary terminal conditions. In particular, if chattering occurs then the λ _{3}continuation is expected to fail for some value \(\lambda _{3} = \lambda _{3}^{\ast }<1\). In such a case this value of λ _{3} corresponds to a suboptimal solution of (\(\mathcal {P}_{S}\)), which is practically valuable since it satisfies the terminal conditions with a reduced final time (also not minimal), with a continuous control. The numerical experiments show that this continuation procedure is very efficient. In most cases, optimal solutions with prescribed terminal conditions can be obtained within a few seconds (without parallel calculations).
7.1.4 Direct method
In this section we envision a direct approach for solving (\(\mathcal {P}_{S}\)), with a piecewise constant control over a given time discretization. The solutions obtained with such a method are suboptimal, especially when the control is chattering (the number of switches being limited by the time step).

Step 1: we solve (\(\mathcal {P}_{S}\)) with the initial condition x(0)=x _{0} and the final conditionsThese final conditions are those of the planar version of (\(\mathcal {P}_{S}\)) (see [93] for details). This problem is easily solved by a direct method without any initialization care (a constant initial guess for the discretized variables suffices to ensure convergence).$${}\omega_{y}(t_{f})=0,\quad \theta (t_{f}) = \theta_{f},\quad v_{z} (t_{f}) \sin \theta_{f}  v_{x} (t_{f}) \cos \theta_{f} =0. $$

Then, in Steps 2, 3, 4 and 5, we add successively (and step by step) the final conditions$$v_{z} (t_{f}) \sin \psi_{f} + v_{y} (t_{f}) \cos \theta_{f} \cos \psi_{f} =0, $$and for each new step we use the solution of the previous one as an initial guess.$$\psi (t_{f})= \psi_{f},\quad \phi (t_{f})=\phi_{f},\quad \omega_{x}(t_{f}) = \omega_{xf}, $$
At the end of this process, we have obtained the solution of (\(\mathcal {P}_{S}\)).
7.1.5 Comparison of the indirect and direct approaches
So far, in order to compute numerically the solutions of (\(\mathcal {P}_{S}\)), we have implemented two approaches. The indirect approach, combining shooting and numerical continuation, is timeefficient when the solution does not contain any singular arcs.
Depending on the terminal conditions, the optimal solution of (\(\mathcal {P}_{S}\)) may involve a singular arc of order two, and the connection with regular arcs generates chattering. The occurrence of chattering causes the failure of the indirect approach. For such cases, we have proposed two alternatives. The first alternative is based on an indirect approach involving three continuations. The last continuation starting from a regularized problem with smooth controls aims at coming back to the original problem that may be chattering. When chattering appears the continuation fails, but the last successful step provides a valuable smooth solution meeting the terminal conditions.
Note that with both approaches, no a priori knowledge of the solution structure is required (in particular, the number of switches is unknown).
Note also that since our aim is to show how to apply the geometric optimal control techniques and the numerical continuation methods, we do not make more detailed comparisons. We refer the interested readers to section 6.2 of [92] for a more detailed comparison. In fact, in aerospace applications, indirect methods are preferred because they provide, in general, more precise optimal trajectories. This is especially important in deepspace trajectory planning missions.
As a conclusion about this example (\(\mathcal {P}_{S}\)), we can emphasize that the theoretical analysis has revealed the existence of singular solutions with possible chattering. This led us to introduce a regularized problem in order to overcome this essential difficulty. On the other hand a continuation procedure is devised considering the dynamics slowfast rates. This procedure is initiated by the problem of order zero reduced to the trajectory dynamics.
In the next section, we extend this approach to a more complicated problem (optimal pullup maneuvers of airborne launch vehicles), in order to further illustrate the potential of continuation methods in aerospace applications.
7.2 Extension to optimal pullup maneuver problem (\(\mathcal {P}_{A}\))
The pullup maneuver consists in an attitude maneuver such that the flight path angle increases up to its targeted value, while satisfying the state constraints on the load factor and the dynamic pressure. In this section, we address the minimum timeenergy pullup maneuver problem for airborne launch vehicles with a focus on the numerical solution method.
where (r _{ x }, r _{ y }, r _{ z }) is the position, m is the mass, (L _{ x }, L _{ y }, L _{ z }) is the lift force, and (D _{ x }, D _{ y }, D _{ z }) is the drag force.
Along a boundary arc, we must have \(h_{i} = \langle p,\hat {g}_{i}(x) \rangle = 0\), i=1,2. Assuming that only the first constraint (which is of order 2) is active along this boundary arc, and differentiating twice the switching functions h _{ i }, i=1,2, we have \(d^{2} h_{i} = \langle p, \text {ad}^{2}\hat {f}.\hat {g}_{i} (x) \rangle dt^{2}  d\eta _{1} \cdot (\text {ad}\hat {f}.\hat {g}_{i}).c_{1} dt\). Moreover, at an entry point occurring at t=τ, we have \({dh}_{i}(\tau ^{+})={dh}_{i}(\tau ^{}) d\eta _{1} \cdot (\text {ad}\hat {f}.\hat {g}_{i}).c_{1} =0\), which yields d η _{1}. A similar result is obtained at an exit point.
The main drawback of this formulation is that the adjoint vector p is no longer absolutely continuous. A jump d η may occur at the entry or at the exit point of a boundary arc, which complexifies significantly the numerical solution.
An alternative approach to address the dynamic pressure state constraint, used in [37, 41], is to design a feedback law that reduces the commanded throttle based on an error signal. According to [41], this approach works well when the trajectory does not violate too much the maximal dynamic pressure constraint, but it may cause instability if the constraint is violated significantly. In any case the derived solutions are suboptimal.
Another alternative is the penalty function method (also called soft constraint method). The soft constraint consists in introducing a penalty function to discard solutions entering the constrained region [40, 64, 84]. For the problem (\(\mathcal {P}_{A}\)), this soft constraint method is well suited in view of a continuation procedure starting from an unconstrained solution. This initial solution generally violates significantly the state constraint. The continuation procedure aims at reducing progressively the infeasibility.
The optimal control given by (36) is regular unless K=0 and ∥h(t)∥=0, in which case it becomes singular. As before the term \(K \int _{0}^{t_{f}} \u(t)\^{2} dt\) in the cost functional (34) is used to avoid chattering [44, 63, 72, 90, 91], and the exact minimum time solution can be approached by decreasing step by step the value of K≥0 until the shooting method possibly fails due to chattering.
 (a)
the position of the launcher is added to the state vector;
 (b)
the gravity acceleration \(\vec {g}\) depends on the position and the aerodynamic forces (lift force \(\vec {L}\) and the drag force \(\vec {D}\)) are considered;
 (c)
the cost functional is penalized by the state constraints violation;
We use this solution as the initialization of the continuation procedure for solving (\(\mathcal {P}_{A}\)).
The parameter λ _{4} acts only on the dynamics. Applying the PMP, λ _{4} appears explicitly in the adjoint equations, but not in the shooting function.
Finally, regarding the point (c), the penalty parameter K _{ p } in the cost functional (27) has to be large enough in order to produce a feasible solution. Unfortunately, too large values of K _{ p } may generate ill conditioning and raise numerical difficulties. In order to obtain an adequate value for K _{ p }, a simple strategy [43, 80] consists in starting with a quite small value of K _{ p }=K _{ p0} and solving a series of problems with increasing K _{ p }. The process is stopped as soon as ∥c(x(t))∥<ε _{ c }, for every t∈[0,t _{ f }], for some given tolerance ε _{ c }>0.
For convenience, we define the exoatmospheric pullup maneuver problem (\(\mathcal {P}_{A}^{exo}\)) as (\(\mathcal {P}_{A}\)) without state constraints and without aerodynamic forces and the unconstrained pullup maneuver problem (\(\mathcal {P}_{A}^{unc}\)) as (\(\mathcal {P}_{A}\)) without state constraints.

First, we embed the solution of (\(\mathcal {P}_{0} \)), into the larger dimension problem (\(\mathcal {P}_{A}\)). This problem is denoted (\(\mathcal {P}_{0}^{H}\)).

Then, we pass from (\(\mathcal {P}_{0}^{H}\)), to (\(\mathcal {P}_{A}\)) by using a numerical continuation procedure, involving four continuation parameters: two parameters λ _{1} and λ _{2} introduce the terminal conditions (32)(33) into (\(\mathcal {P}_{A}^{exo}\)); λ _{4} introduces the variable gravity acceleration and the aerodynamic forces in (\(\mathcal {P}_{A}^{unc}\)); λ _{5} introduces the soft constraints in (\(\mathcal {P}_{A}\)).
The attitude angles θ _{ e }, ψ _{ e }, ϕ _{ e }, ω _{ xe }, and ω _{ ye } are those obtained at the end of the first continuation on λ _{1}. θ ^{∗}, ψ ^{∗} are the explicit solutions of (\(\mathcal {P}_{0}^{H}\)).
These successive continuations are implemented using the PC continuation combined with the multiple shooting method. Some additional enhancements regarding the inertial frame choice and the Euler angle singularities help improving the overall robustness of the solution process.
When the step length h _{ s } is small enough, this approximation yields a predicted point (15) very close to the true zero.
Change of frame Changing the inertial reference frame can improve the problem conditioning and enhance the numerical solution process. The new frame \(S_{R}^{\prime }\) is defined from the initial frame S _{ R } by two successive rotations of angles (β _{1},β _{2}). The problem (\(\mathcal {P}_{A}\)) becomes numerically easier to solve when the new reference frame \(S_{R}^{\prime }\) is adapted to the terminal conditions. However we do not know a priori which reference frame is the best suited. We propose to choose a reference frame associated to (β _{1},β _{2}) such that \(\psi ^{\prime }_{f}=\psi ^{\prime }_{0}\) and \( \psi ^{\prime }_{f} + \psi ^{\prime }_{0} \) being minimal (the subscribe ′ here means the new variable in \(S_{R}^{\prime }\)). This choice centers the terminal values on the yaw angle on zero. Thus we can hope that the solution remains far from the Euler angle singularities occurring when ψ→π/2+k π.
This frame rotation defines a nonlinear state transformation, which acts as a preconditionner. We observe from numerical experiments that it actually enhances the robustness of the algorithm. The reader is referred to [93] for more details of the change of frame.
These Eqs. (38) are used close to the singularities.
Algorithm We describe the whole numerical strategy of solving (\(\mathcal {P}_{A}\)) in the following algorithm.
7.3 Numerical results of solving (\(\mathcal {P}_{A}\))
The Algorithm 3 is first applied to a pullup maneuver of an airborne launch vehicle just after its release from the carrier. We present some statistical results showing robustness of our algorithm. A second example considers the threedimensional reorientation maneuver of a launch vehicle upper stage after a stage separation.
7.3.1 PullUp maneuvers of an airborne launch vehicle (AVL)
We observe on Fig. 20 a boundary arc on the load factor constraint near the maximal level \(\bar {n}_{max}=2.2 g\). This corresponds on Fig. 19 to the switching function \(h(t)=b(p_{\omega _{y}},p_{\omega _{x}})\) being close to zero. Comparing Figs. 18 and 19, we see that the control follows the form of the switching function. On the other hand, the state constraint of the dynamic pressure is never active.
We observe also on Fig. 19 a steeper variation of p _{ θ }(t) at t=5.86 s. The penalty function P(x) starts being positive at this date and adds terms in the adjoint differential equation.
Running this example requires 24.6 s to compute the optimal solution, with CPU: Intel(R) Core(TM) i52500 CPU 3.30GHz; Memory: 3.8 Gio; Compiler: gcc version 4.8.4 (Ubuntu 14.04 LTS). The number of nodes for the multiple shooting has been set to 3 from experiments. Passing to four node increases the computing time to 31.2 s without obvious robustness benefit.
We next present some statistical results obtained with the same computer settings.
7.3.1.0 Statistical results
Parameter ranges
v _{0}  θ _{ v0}  ψ _{ v0}  θ _{0}  ψ _{0} 

fixed 0.8 Mach  [−10,0]°  fixed 0°  [−10,10]°  fixed 0° 
θ _{ f }  ψ _{ f }  ω _{ x0}  ω _{ y0}  θ _{0}−θ _{ v0} 
[20,80]°  [−10,10]°  [−2,2]°/s  [−2,2]°/s  [0,10]° 
For each variable, we choose a discretization step and we solve all possible combinations resulting from this discretization (factorial experiment). The total number of cases is 1701. All cases are run with the penalty parameter varying from K _{ p0}=0.1 to K _{ p1}=100 during the third continuation. For each continuation stage the number of simulations is limited to 200.
The 1701 cases are run for different settings of the number of nodes (N=0 or N=2) and of the regularization parameter (K=800 or K=1000).
Statistical results (N=2 and K=8×10^{2})
Planar  Nonplanar  

Number of cases  567  1134 
Rate of success (%)  89.07  80.04 
Number of failure cases  
 In λ _{1}continuation  0  14 
 In λ _{2}continuation  21  172 
 In λ _{4}continuation  41  26 
 In λ _{5} + K _{ p }continuation  0  10 
Average execution time (s)  
 Total  26.94  44.05 
 In λ _{1}continuation  0.49  0.48 
 In λ _{2}continuation  2.07  2.37 
 In λ _{4}continuation  2.54  2.99 
 In λ _{5} + K _{ p }continuation  23.16  37.35 
Statistical results (N=2 and K=1×10^{3})
Planar  Nonplanar  

Number of cases  567  1134 
Rate of success (%)  85.89  86.94 
Number of failure cases  
 In λ _{1}continuation  0  4 
 In λ _{2}continuation  36  120 
 In λ _{4}continuation  44  16 
 In λ _{5} + K _{ p }continuation  0  8 
Average execution time (s)  
 Total  26.55  47.96 
 In λ _{1}continuation  0.49  0.51 
 In λ _{2}continuation  2.12  2.40 
 In λ _{4}continuation  2.71  2.74 
 In λ _{5} + K _{ p }continuation  22.60  42.28 
Statistical results (N=0 and K=8×10^{2})
Planar  Nonplanar  

Number of cases  567  1134 
Rate of success (%)  83.95  74.96 
Number of failure cases  
 In λ _{1}continuation  4  10 
 In λ _{2}continuation  29  210 
 In λ _{4}continuation  21  24 
 In λ _{5} + K _{ p }continuation  37  40 
Average execution time (s)  
 Total  28.93  33.36 
 In λ _{1}continuation  0.47  0.57 
 In λ _{2}continuation  1.17  1.71 
 In λ _{4}continuation  10.80  10.51 
 In λ _{5} + K _{ p }continuation  18.17  21.56 
Tables 34 show the results with a multiple shooting using 2 nodes, with different values of the regularization parameter K. The algorithm appears fairly robust with respect to the terminal conditions. The choice of the regularization parameter K affects the resolution results: (i) the rate of success increases (resp. decreases) in the nonplanar case (resp. planar case) when K increases from K=800 to K=1000; (ii) in term of the execution time, we see that in both cases, it is faster to get a result in planar case than in nonplanar case, and most time is devoted to deal with the state constraints during the last continuation.
We see that the value of K should neither be too large nor too small. From Tables 3, 4 and 5, we observe also that the λ _{2}continuation causes most failures in the nonplanar case. The success rate could be possibly improved by adapting the K value.
Tables 3 and5 compare the multiple and the single shooting method (N=0). The multiple shooting method (N=2) clearly improves the robustness of the algorithm, without significant increase of the execution time.
7.3.2 Reorientation maneuver of a launch vehicle
Along multiburn ascent trajectories, the control (Euler angles) exhibit jumps at the stage separations (see for example [59, Figure 3]). In this case, a reorientation maneuver is necessary to follow the optimal thrust direction. For this reason, we apply the above algorithm as well to the maneuver problem of the upper stages of the launch vehicles.
Opposite to the airborne launch vehicle’s pullup maneuvers, these reorientation maneuvers are in general threedimensional and of lower magnitude. They occur at high altitudes (typically higher than 50 km since a sufficiently low dynamic pressure is required to ensure the separation safety) and high velocity (since the first stage has already delivered a large velocity increment).
The maneuver occurs in vacuum so that no state constraints apply. Finding the minimum time maneuver corresponds to solving the problem (\(\mathcal {P}_{S}\)).
The maneuver duration t _{ f } is about 175 s due to the large direction change required on the velocity. During a real flight the velocity direction change is much smaller and the maneuver takes at most a few seconds. Our purpose when presenting this “unrealistic” case is rather to show that the proposed algorithm is robust in a large range of system configurations and terminal conditions.
Conclusion
The aim of this article was to show how to apply techniques of geometric optimal control and numerical continuation to aerospace problems. Some classical techniques of optimal control have been recalled, including Pontryagin Maximum Principle, first and secondorder optimality conditions, and conjugate time theory. Techniques of geometric optimal control have then been recalled, such as higherorder optimality conditions and singular controls.
A quite difficult problem has illustrated in detail how to design an efficient solution method with the help of geometric optimal control tools and continuation methods. Some applications in space trajectory optimization have also been recalled.
Though geometric optimal control and numerical continuation provide a nice way to design efficient approaches for many aerospace applications, the answer to “how to select a reasonably simple problem for the continuation procedure” for general optimal control problems remains open. A deep understanding of the system dynamics is necessary to devise a simple problem that is “physically” sufficiently close to the original problem, while being numerically suited to initiate a continuation procedure.
In practice, many problems remain difficult due to the complexity of reallife models. In general, a compromise should be found between the complexity of the model under consideration and the choice of an adapted numerical method.
As illustrated by the example of airborne launch vehicles, many state and/or control constraints should also be considered in a reallife problem, and such constraints makes the problem much more difficult. For the airborne launch problem a penalization method combined with the previous geometric analysis proves satisfying. But this approach has to be customized to the specific problem under consideration. A challenging task is then to combine an adapted numerical approach with a thorough geometric analysis in order to get more information on the optimal synthesis. We refer the readers to [85] for a summary of open challenges in aerospace applications.
Endnotes
^{1} Given any x∈M, \(T^{\ast }_{x} M\) is the cotangent space to M at x.
^{2} If the final time t _{ f } is fixed, then \(\bar {x}(\cdot)\) is said to be locally optimal in L ^{ ∞ } topology (resp. in C ^{0} topology), if it is optimal in a neighborhood of u in L ^{ ∞ } topology (resp. in a neighborhood of \(\bar {x}(\cdot)\) C ^{0} topology).
If the final time t _{ f } is not fixed, then a trajectory \(\bar {x}(\cdot)\) is said to be locally optimal in L ^{ ∞ } topology if, for every neighborhood V of u in L ^{ ∞ }([0,t _{ f }+ε],U), for every real number η so that η≤ε, for every control v∈V satisfying E(x _{0},t _{ f }+η,v)=E(x _{0},t _{ f },u) there holds C(t _{ f }+η,v)≥C(t _{ f },u). Moreover, a trajectory \(\bar {x}(\cdot)\) is said to be locally optimal in C ^{0} topology if, for every neighborhood W of \(\bar {x}(\cdot)\) in M, for every real number η so that η≤ε, for every trajectory x(·), associated to a control v∈V on [0,t _{ f }+η], contained in W, and satisfying \(x(0) = \bar {x}(0) = x_{0}\), \(x(t_{f}+\eta) = \bar {x}(t_{f})\), there holds C(t _{ f }+η,v)≥C(t _{ f },u).
^{3} meaning that in some coordinates, for λ∈[0,1], the path consists in a convex combination of the simpler problem and of the original problem
^{4} There, the endpoint mapping has been implemented with the exponential mapping \(E_{x_{0},t_{f},\lambda }(u)=\exp _{x_{0},\lambda }(t_{f},p_{0})\) with initial condition (x(0),p(0))=(x _{0},p _{0}).
^{5} Let \(v_{1}, \cdots, v_{j+1} \in \mathbb {R}^{N+1}\), j≤N+1, be affinely independent points, i.e., v _{ k }−v _{1}, k=2,⋯,j+1 are linearly independent. A jsimplex in \(\mathbb {R}^{N+1}\) is defined by the convex hull of the set v _{1},⋯,v _{ j+1}. The convex hull of any subset w _{1},⋯,w _{ r+1}⊂v _{1},⋯,v _{ j+1} is an rface.
Declarations
Acknowledgment
The second author acknowledges the support by FA95501410214 of the EOARDAFOSR.
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