Trajectory optimization is the process of designing a trajectory that minimizes (or maximizes) some measure of performance while satisfying a set of constraints. Generally speaking, trajectory optimization is a technique for computing an open-loop solution to an optimal control problem. It is often used for systems where computing the full closed-loop solution is either impossible or impractical.
Although the idea of trajectory optimization has been around for hundreds of years (calculus of variations, brachystochrone problem), it only became practice for real-world problems with the advent of the computer. Many of the original applications of trajectory optimization were in the aerospace industry, computing rocket and missile launch trajectories. More recently, trajectory optimization has also been used in a wide variety of industrial process and robotics applications.
- 1 History
- 2 Applications
- 3 Terminology
- 4 Trajectory optimization techniques
- 5 Comparison of techniques
- 6 Software
- 7 References
Trajectory optimization first showed up in 1697, with the introduction of the Brachystochrone problem: find the shape of a wire such that a bead sliding along it will move between two points in the minimum time. The interesting thing about this problem is that it is optimizing over a curve (the shape of the wire), rather than a single number. The most famous of the solutions was computed using calculus of variations.
In the 1950s, the digital computer started to make trajectory optimization practical for solving real-world problems. The first optimal control approaches grew out of the calculus of variations, based on the research of Gilbert Ames Bliss and Bryson in America, and Pontryagin in Russia. Pontyragin's maximum principle  is of particular note. These early researchers created the foundation of what we now call indirect methods for trajectory optimization.
Much of the early work in trajectory optimization was focused on computing rocket thrust profiles, both in a vacuum and in the atmosphere. This early research discovered many basic principles that are still used today. Another successful application was the climb to altitude trajectories for the early jet aircraft. Because of the high drag associated with the transonic drag region and the low thrust of early jet aircraft, trajectory optimization was the key to maximizing climb to altitude performance. Optimal control based trajectories were responsible for some of the world records. In these situations, the pilot followed a Mach versus altitude schedule based on optimal control solutions.
One of the important early problems in trajectory optimization was that of the singular arc, where Pontryagin's maximum principle fails to yield a complete solution. An example of a problem with singular control is the optimization of the thrust of a missile flying at a constant altitude and which is launched at low speed. Here the problem is one of a bang-bang control at maximum possible thrust until the singular arc is reached. Then the solution to the singular control provides a lower variable thrust until burnout. At that point bang-bang control provides that the control or thrust go to its minimum value of zero. This solution is the foundation of the boost-sustain rocket motor profile widely used today to maximize missile performance.
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.   One interesting application shown by the U.Penn GRASP Lab is computing a trajectory that allows a quadrotor to fly through a hoop as it is thrown. 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.
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.  Like in many other applications, trajectory optimization can be used to compute a nominal trajectory, around which a stabilizing controller is built.  Trajectory optimization can be applied in detailed motion planning complex humanoid robots, such as Atlas.  Finally, trajectory optimization can be used for path-planning of robots with complicated dynamics constraints, using reduced complexity models. 
For tactical missiles, the flight profiles are determined by the thrust and lift histories. These histories can be controlled by a number of means including such techniques as using an angle of attack command history or an altitude/downrange schedule that the missile must follow. Each combination of missile design factors, desired missile performance, and system constraints results in a new set of optimal control parameters.
- Decision variables
- The set of unknowns to be found using optimization.
- Trajectory optimization problem
- A special type of optimization problem where the decision variables are functions, rather than real numbers.
- Parameter optimization
- Any optimization problem where the decision variables are real numbers.
- Nonlinear program
- A class of constrained parameter optimization where either the objective function or constraints are nonlinear.
- Indirect method
- An indirect method for solving a trajectory optimization problem proceeds in three steps: 1) Analytically construct the necessary and sufficient conditions for optimality, 2) Discretize these conditions, constructing a constrained parameter optimization problem, 3) Solve that optimization problem.
- Direct method
- A direct method for solving a trajectory optimization problem consists of two steps: 1) Discretize the trajectory optimization problem directly, converting it into a constrained parameter optimization problem, 2) Solve that optimization problem.
- The process by which a trajectory optimization problem is converted into a parameter optimization problem. This is sometimes referred to as discretization. Transcription methods generally fall into two categories: shooting methods and collocation methods.
- Shooting method
- A transcription method that is based on simulation, typically using explicit Runge--Kutta schemes.
- Collocation method (Simultaneous Method)
- A transcription method that is based on function approximation, typically using implicit Runge--Kutta schemes.
- Pseudospectral method (Global Collocation)
- A transcription method that represents the entire trajectory as a single high-order orthogonal polynomial.
- Mesh (Grid)
- After transcription, the formerly continuous trajectory is now represented by a discrete set of points, known as mesh points or grid points.
- Mesh refinement
- The process by which the discretization mesh is improved by solving a sequence of trajectory optimization problems. Mesh refinement is either performed by sub-dividing a trajectory segment or by increasing the order of the polynomial representing that segment.
- Multi-phase trajectory optimization problem
- Trajectory optimization over a system with hybrid dynamics can be achieved by posing it as a multi-phase trajectory optimization problem. This is done by composing a sequence of standard trajectory optimization problems that are connected using constraints.
Trajectory optimization techniques
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. Direct and indirect methods can be blended by an application of the covector mapping principle of Ross and Fahroo.
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). 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. 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. 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
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.
Comparison of techniques
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.
Indirect vs. direct 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. As a result, direct methods are more widely used, especially in non-critical applications. Indirect methods still have a place in specialized applications, particularly aerospace, where accuracy is critical.
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 the be solved accurately using an indirect method.
Shooting vs. collocation
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. These methods tend to be less accurate then others (due to their low-order), but are particularly robust for problems with difficult path constraints.
Orthogonal collocation methods are best for obtaining high-accuracy solutions to problems where the accuracy of the control trajectory is important. Some implementations have trouble with path constraints. These methods are particularly good when the solution is smooth.
Mesh refinement: h vs. p
It is common to solve a trajectory optimization problem iteratively, each time using a discretization with more points. A h-method for mesh refinement works by increasing the number of trajectory segments along the trajectory, while a p-method increases the order of the transcription method within each segment.
Direct collocation methods tend to exclusively use h-method type refinement, since each method is a fixed order. Shooting methods and orthogonal collocation methods can both use h-method and p-method mesh refinement, and some use a combination, known as hp-adaptive meshing. It is best to use h-method when the solution is non-smooth, while a p-method is best for smooth solutions.
Examples of trajectory optimization programs include:
- PyKEP, PyGMO (Open Source, from the European Space Agency for interplanetary trajectory optimization)
- Copernicus Trajectory Design and Optimization System 
- DIRCOL: A general-purpose trajectory optimization software based on direct collocation.
- Drake: A planning, control, and analysis toolbox for nonlinear dynamical systems.
- General Mission Analysis Tool
- GPOPS-II (General Purpose OPtimal Control Software) Solves multi-phase trajectory optimization problems. (Matlab) 
- JModelica.org (Modelica-based open source platform for dynamic optimization)
- LOTOS (Low-Thrust Orbit Transfer Trajectory Optimization Software) from Astos Solutions
- MIDACO Optimization software particularly developed for interplanetary space trajectories. (Avail. in Matlab, Octave, Python, C/C++, R and Fortran)
- OTIS (Optimal Trajectories by Implicit Simulation) 
- POST (Program to Optimize Simulated Trajectories) , 
- TrajOpt: An open-source trajectory optimization library for Matlab
- ZOOM, Conceptual Design and Analysis of Rocket Configurations and Trajectories) 
A collection of low thrust trajectory optimization tools, including members of the Low Thrust Trajectory Tool (LTTT) set, can be found here: LTTT Suite Optimization Tools.
- 300 Years of Optimal Control: From The Brachystochrone to the Maximum Principle, Hector J. Sussmann and Jan C. Willems. IEEE Control Systems, 1997.
- Bryson, Ho,Applied Optimal Control, Blaisdell Publishing Company, 1969, p 246.
- L.S. Pontyragin, The Mathematical Theory of Optimal Processes, New York, Intersciences, 1962
- Ross, I. M. A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, San Francisco, 2009.
- Daniel Mellinger and Vijay Kumar, "Minimum snap trajectory generation and control for quadrotors" International Conference on Robotics and Automation, IEEE 2011
- Markus Hehn and Raffaello D'Andrea, "Real-Time Trajectory Generation for Quadrocopters" IEEE Transactions on Robotics, 2015.
- John W. Eaton and James B. Rawlings. "Model-Predictive Control of Chemical Processes" Chemical Engineering Science, Vol 47, No 4. 1992.
- T. Chettibi, H. Lehtihet, M. Haddad, S. Hanchi, "Minimum cost trajectory planning for industrial robots" European Journal of Mechanics, 2004.
- Manoj Srinivasan and Andy Ruina. "Computer optimization of a minimal biped model discovers walking and running" Nature, 2006.
- E.R. Westervelt, J.W. Grizzle, and D.E. Koditschek. "Hybrid Zero Dynamics of PLanar Biped Walkers" IEEE Transactions on Automatic Control, 2003.
- Michael Posa, Scott Kuindersma, and Russ Tedrake. "Optimizaiton and stabilization of trajectories for constrained dynamical systems." International Conference on Robotics and Automation, IEEE 2016.
- Hongkai Dai, Andres Valenzuela, and Russ Tedrake. "Whole-body motion planning with Centroidal Dynamics and Full Kinematics" International Conference on Humanoid Robots, IEEE 2014.
- Phillips, C.A, "Energy Management for a Multiple Pulse Missile", AIAA Paper 88-0334, Jan., 1988
- John T. Betts "Practical Methods for Optimal Control and Estimation Using Nonlinear Programming" SIAM Advances in Design and Control, 2010.
- Christopher L. Darby, William W. Hager, and Anil V. Rao. "An hp-adaptive pseudospectral method for solving optimal control problems." Optimal Control Applications and Methods, 2010.
- Patterson, Michael A.; Rao, Anil V. (2014-10-01). "GPOPS-II: A MATLAB Software for Solving Multiple-Phase Optimal Control Problems Using hp-Adaptive Gaussian Quadrature Collocation Methods and Sparse Nonlinear Programming". ACM Trans. Math. Softw. 41 (1): 1:1–1:37. doi:10.1145/2558904. ISSN 0098-3500.
- I. M. Ross and M. Karpenko, "A Review of Pseudospectral Optimal Control: From Theory to Flight," Annual Reviews in Control, Vol. 36, pp. 182-197, 2012.
- Survey of Numerical Methods for Trajectory Optimization; John T. Betts Journal of Guidance, Control, and Dynamics 1998; 0731-5090 vol.21 no.2 (193-207)
- Anil V. Rao "A survey of numerical methods for optimal control" Advances in Astronautical Sciences, 2009.
- Camila C. Francolin, David A. Benson, William W. Hager, Anil V. Rao. "Costate Estimation in Optimal Control Using Integral Gaussian Quadrature Orthogonal Collocation Methods" Optimal Control Applications and Methods, 2014.
- Lloyd N. Trefethen. "Approximation Theory and Approximation Practice", SIAM 2013
- David H. Jacobson, David Q. Mayne. "Differential Dynamic Programming" Elsevier, 1970.