There are a wide variety of applications for trajectory optimization, primarily in robotics: industry, manipulation, walking, path-planning, and aerospace. It can also be used for modeling and estimation. Trajectory optimization is often used to compute trajectories for quadrotor helicopters. These applications typically used highly specialized algorithms. Another, this time by the ETH Zurich Flying Machine Arena , involves two quadrotors tossing a pole back and forth between them, with it balanced like an inverted pendulum.
Trajectory optimization is used in manufacturing, particularly for controlling chemical processes such as in  or computing the desired path for robotic manipulators such as in . There are a variety of different applications for trajectory optimization within the field of walking robotics. For example, one paper used trajectory optimization of bipedal gaits on a simple model to show that walking is energetically favorable for moving at a low speed and running is energetically favorable for moving at a high speed.
For tactical missiles , the flight profiles are determined by the thrust and lift histories. Each combination of missile design factors, desired missile performance, and system constraints results in a new set of optimal control parameters. The techniques to any optimization problems can be divided into two categories: indirect and direct.
An indirect method works by analytically constructing the necessary and sufficient conditions for optimality, which are then solved numerically. A direct method attempts a direct numerical solution by constructing a sequence of continually improving approximations to the optimal solution.
The optimal control problem is an infinite-dimensional optimization problem, since the decision variables are functions, rather than real numbers. All solution techniques perform transcription, a process by which the trajectory optimization problem optimizing over functions is converted into a constrained parameter optimization problem optimizing over real numbers.
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Generally, this constrained parameter optimization problem is a non-linear program, although in special cases it can be reduced to a quadratic program or linear program. Single shooting is the simplest type of trajectory optimization technique.
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The basic idea is similar to how you would aim a cannon: pick a set of parameters for the trajectory, simulate the entire thing, and then check to see if you hit the target. The entire trajectory is represented as a single segment, with a single constraint, known as a defect constraint, requiring that the final state of the simulation match the desired final state of the system. Single shooting is effective for problems that are either simple or have an extremely good initialization.
Both the indirect and direct formulation tend to have difficulties otherwise. Multiple shooting is a simple extension to single shooting that renders it far more effective. Rather than representing the entire trajectory as a single simulation segment , the algorithm breaks the trajectory into many shorter segments, and a defect constraint is added between each.
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The result is large sparse non-linear program, which tends to be easier to solve than the small dense programs produced by single shooting. Direct collocation methods work by approximating the state and control trajectories using polynomial splines. These methods are sometimes referred to as direct transcription. Trapezoidal collocation is a commonly used low-order direct collocation method.
The dynamics, path objective, and control are all represented using linear splines, and the dynamics are satisfied using trapezoidal quadrature. Hermite-Simpson Collocation is a common medium-order direct collocation method.
The state is represented by a cubic-Hermite spline , and the dynamics are satisfied using Simpson quadrature. Orthogonal collocation is technically a subset of direct collocation, but the implementation details are so different that it can reasonably be considered its own set of methods. Orthogonal collocation differs from direct collocation in that it typically uses high-order splines, and each segment of the trajectory might be represented by a spline of a different order.
The name comes from the use of orthogonal polynomials in the state and control splines. Pseudospectral collocation, also known as global collocation, is a subset of orthogonal collocation in which the entire trajectory is represented by a single high-order orthogonal polynomial. As a side note: some authors use orthogonal collocation and pseudospectral collocation interchangeably.
When used to solve a trajectory optimization problem whose solution is smooth, a pseudospectral method will achieve spectral exponential convergence. Differential dynamic programming , is a bit different than the other techniques described here. In particular, it does not cleanly separate the transcription and the optimization. Instead, it does a sequence of iterative forward and backward passes along the trajectory.
Each forward pass satisfies the system dynamics, and each backward pass satisfies the optimality conditions for control. Eventually, this iteration converges to a trajectory that is both feasible and optimal. There are many techniques to choose from when solving a trajectory optimization problem. There is no best method, but some methods might do a better job on specific problems.
This section provides a rough understanding of the trade-offs between methods. When solving a trajectory optimization problem with an indirect method, you must explicitly construct the adjoint equations and their gradients. This is often difficult to do, but it gives an excellent accuracy metric for the solution. Direct methods are much easier to set up and solve, but do not have a built-in accuracy metric. Indirect methods still have a place in specialized applications, particularly aerospace, where accuracy is critical.
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One place where indirect methods have particular difficulty is on problems with path inequality constraints. These problems tend to have solutions for which the constraint is partially active. When constructing the adjoint equations for an indirect method, the user must explicitly write down when the constraint is active in the solution, which is difficult to know a priori.
One solution is to use a direct method to compute an initial guess, which is then used to construct a multi-phase problem where the constraint is prescribed. The resulting problem can then be solved accurately using an indirect method. Single shooting methods are best used for problems where the control is very simple or there is an extremely good initial guess.
For example, a satellite mission planning problem where the only control is the magnitude and direction of an initial impulse from the engines. Multiple shooting tends to be good for problems with relatively simple control, but complicated dynamics. Although path constraints can be used, they make the resulting nonlinear program relatively difficult to solve. Direct collocation methods are good for problems where the accuracy of the control and the state are similar.
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These methods tend to be less accurate than others due to their low-order , but are particularly robust for problems with difficult path constraints. These bounds represent physical or other limitations on the system. For an aircraft, for instance, the altitude must always be greater than that of the landscape, and the control available is limited by the physical capabilities of the engines and control surfaces.
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Many important classes of problems have been left out of our presentation. For example, the state variable inequality constraint given in 1. The general problem cannot be treated in complete detail using essentially elementary mathematics. However, important spe- cial cases of the general problems can be treated in complete detail using elementary mathematics.
These special cases are suciently broad to solve many interesting and important problems. Furthermore, these special cases suggest solutions to the more general problem. Therefore, complete solutions to the general problem are stated and used.
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The theoretical gap between the solution to the special cases and the solution to the general problem is discussed, and additional references are given for completeness. To introduce important concepts, mathematical style, and notation, in Chapter 2 the parameter minimization problem is formulated and conditions for local optimality are determined. By local optimality we mean that optimality can be veried about 8 Chapter 1. Introduction a small neighborhood of the optimal point.