# 4. Setting up a Problem

The Problem contains all of the information needed to solve a trajectory optimization problem. At a minimum, this is the model, objective, and initial condition. A Problem is passed to a solver, which extracts needed information, and may or may or not modify its internal representation of the problem in order to solve it (e.g. the Augmented Lagrangian solver combines the constraints and objective into a single Augmented Lagrangian objective.)

## Creating a Problem

Let's say we're trying to solve the following trajectory optimization problem:

\begin{aligned} \min_{x_{0:N},u_{0:N-1}} \quad & (x_N-x_f)^T Q_f (x_N-x_f) + \sum_{k=0}^{N-1} (x_k-x_f)^T Q (x_k - x_f) + u^T R u \\ \textrm{s.t.} \quad & x_{k+1} = f(x_k, u_k), \\ & |u_k| \leq 3 \\ & x_N = x_f \\ \end{aligned}

We'll quickly set up the dynamics, objective, and constraints. See previous sections for more details on how to do this.

using TrajectoryOptimization
using RobotDynamics
using RobotZoo: Cartpole
using StaticArrays, LinearAlgebra

# Dynamics and Constants
model = Cartpole()
n,m = RobotDynamics.dims(model)
N = 101   # number of knot points
tf = 5.0  # final time
x0 = @SVector [0, 0, 0, 0.]  # initial state
xf = @SVector [0, π, 0, 0.]  # goal state (i.e. swing up)

# Objective
Q = Diagonal(@SVector fill(1e-2,n))
R = Diagonal(@SVector fill(1e-1,m))
Qf = Diagonal(@SVector fill(100.,n))
obj = LQRObjective(Q, R, Qf, xf, N)

# Constraints
conSet = ConstraintList(n,m,N)
bnd = BoundConstraint(n,m, u_min=-3.0, u_max=3.0)
goal = GoalConstraint(xf)
add_constraint!(conSet, goal, N:N)

The following method is the easiest way to set up a trajectory optimization problem:

prob = Problem(model, obj, x0, tf, constraints=conSet, xf=xf, integration=RK4)

where the keyword arguments are, of course, optional.

This constructor has the following arguments:

• (required) model::RobotDynamics.AbstractModel - dynamics model. Can also

be a vector of RobotDynamics.DiscreteDynamics of length N-1.

• (required) obj::AbstractObjective - objective function
• (required) x0::AbstractVector - initial state
• (required) tf::AbstractFloat - final time
• (optional) constraints::ConstraintSet - constraint set. Default is no constraints.
• (optional) xf::AbstractVector - Goal state.
• (optional) N::Int - number of knot points. Default is given by length of objective.
• (optional) dt::AbstractFloat - Time step length. Can be either a scalar or a vector of length N-1. Default is calculated using tf and N.
• (optional) integration::RobotDynamics.QuadratureRule - Quadrature rule for discretizing the dynamics. Default is given by RobotDynamics.RK4.
• (optional) X0 - Initial guess for state trajectory. Can either be a matrix of size (n,N) or a vector of length N of n-dimensional vectors.
• (optional) U0 - Initial guess for control trajectory. Can either be a matrix of size (m,N) or a vector of length N-1 of n-dimensional vectors.

## Initialization

A good initialization is critical to getting good results for nonlinear optimization problems. TrajectoryOptimization.jl current supports initialization of the state and control trajectories. Initialization of dual variables (i.e. Lagrange multipliers) is not yet support but will be included in the near future. The state and control trajectories can be initialized directly in the constructor using the X0 and U0 keyword arguments described above, or using the following methods:

where, again, these can either be matrices or vectors of vectors of the appropriate size. It should be noted that these methods work on either Problems or instances of AbstractSolver.

Alternatively, the problem can be initialized with both the state and control trajectories by passing in a RobotDynamics.SampledTrajectory via initial_trajectory!.

### Getters

The following methods can be used to get information out of the problem definition

To query the state and control dimensions along the trajectory, the preferred method is to one of

RobotDynamics.state_dim(prob, k)
RobotDynamics.control_dim(prob, k)
RobotDynamics.dims(prob, k)

which return n, m, or the tuple n,m,N for the state and control dimensions at time step k, and the horizon length N.

Alternatively, you can get the state and control dimensions as vectors of length N by omitting the time step k.

### Extracting states and controls

To extract the state and control trajectories, the Problem type supports the same methods as RobotDynamics.SampledTrajectory, e.g.

states(prob)         # return a vector of state vectors
control(prob)        # return a vector of control vectors
states(prob, k)      # get a vector of the kth element of the state vector
controls(prob, i:j)  # get a vector the vectors of elements i through j of the controls vector. 
Tip

To convert a vector of vectors to a 2D array, use:

Xmat = hcat(Vector.(X)...)

Note that converting to a vector is a safe way to avoid the expensive operation of concatenating a bunch of static vectors, if the elements of X are a subtype of StaticArrays.StaticVector.

### Other Methods

The cost of the current trajectory can be evaluated using cost. The initial state and the current controls trajectory can be used to simulate the system forward (open-loop) to obtain a state trajectory via rollout!.

For MPC applications, the following methods can be useful: