Practical Numerical Trajectory Optimization via Indirect Methods

Numerical trajectory optimization is helpful not only for mission planning but also design

space exploration and quantifying vehicle performance. Direct methods for solving the opti-

mal control problems, which first discretize the problem before applying necessary conditions

of optimality, dominate the field of trajectory optimization because they are easier for the

user to set up and are less reliant on a forming a good initial guess. On the other hand,

many consider indirect methods, which apply the necessary conditions of optimality prior to

discretization, too difficult to use for practical applications. Indirect methods though provide

very high quality solutions, easily accessible sensitivity information, and faster convergence

given a sufficiently good guess. Those strengths make indirect methods especially well-suited

for generating large data sets for system analysis and worth revisiting.

Recent advancements in the application of indirect methods have already mitigated many

of the often cited issues. Automatic derivation of the necessary conditions with computer

algebra systems have eliminated the manual step which was time-intensive and error-prone.

Furthermore, regularization techniques have reduced problems which traditionally needed

many phases and complex staging, like those with inequality path constraints, to a signifi-

cantly easier to handle single arc. Finally, continuation methods can circumvent the small

radius of convergence of indirect methods by gradually changing the problem and use previ-

ously found solutions for guesses.

The new optimal control problem solver Giuseppe incorporates and builds upon these

advancements to make indirect methods more accessible and easily used. It seeks to enable

greater research and creative approaches to problem solving by being more flexible and

extensible than previous solvers. The solver accomplishes this by implementing a modular

design with well-defined internal interfaces. Moreover, it allows the user easy access to and

manipulation of component objects and functions to be use in the way best suited to solve

a problem.

A new technique simplifies and automates what was the predominate roadblock to using

continuation, the generation of an initial guess for the seed solution. Reliable generation of

a guess sufficient for convergence still usually required advanced knowledge optimal contrtheory or sometimes incorporation of an entirely separate optimization method. With the

new method, a user only needs to supply initial states, a control profile, and a time-span

over which to integrate. The guess generator then produces a guess for the “primal” problem

through propagation of the initial value problem. It then estimates the “dual” (adjoint)

variables by the Gauss-Newton method for solving the nonlinear least-squares problem. The

decoupled approach prevents poorly guessed dual variables from altering the relatively easily

guess primal variables. As a result, this method is simpler to use, faster to iterate, and much

more reliable than previous guess generation techniques.

Leveraging the continuation process also allows for greater insight into the solution space

as there is only a small marginal cost to producing an additional nearby solutions. As a

result, a user can quickly generate large families of solutions by sweeping parameters and

modifying constraints. These families provide much greater insight in the general problem

and underlying system than is obtainable with singular point solutions. One can extend

these analyses to high-dimensional spaces through construction of compound continuation

strategies expressible by directed trees.

Lastly, a study into common convergence explicates their causes and recommends mitiga-

tion strategies. In this area, the continuation process also serves an important role. Adaptive

step-size routines usually suffice to handle common sensitivity issues and scaling constraints

is simpler and out-performs scaling parameters directly. Issues arise when a cost functional

becomes insensitive to the control, which a small control cost mitigates. The best perfor-

mance of the solver requires proper sizing of the smoothing parameters used in regularization

methods. An asymptotic increase in the magnitude of adjoint variables indicate approaching

a feasibility boundary of the solution space.

These techniques for indirect methods greatly facilitate their use and enable the gen-

eration of large libraries of high-quality optimal trajectories for complex problems. In the

future, these libraries can give a detailed account of vehicle performance throughout its flight

envelope, feed higher-level system analyses, or inform real-time control applications.