# 1. Setting Up a Dynamics Model

TrajectoryOptimization relies on the interface defined by RobotDynamics.jl to define the forced dynamics required to solve the problem. Please refer to the documentation for more details on setting up and defining models. We present a simple example here.

Assume we want to find an optimal trajectory for the canonical cartpole system. We can either import an existing model defined in RobotZoo.jl or use RobotDynamics.jl to define our own. Defining our own model is pretty straight-forward:

```
using RobotDynamics
using ForwardDiff
using FiniteDiff
using StaticArrays
RobotDynamics.@autodiff struct Cartpole{T} <: RobotDynamics.ContinuousDynamics
mc::T
mp::T
l::T
g::T
end
Cartpole() = Cartpole(1.0, 0.2, 0.5, 9.81)
RobotDynamics.state_dim(::Cartpole) = 4
RobotDynamics.control_dim(::Cartpole) = 1
function RobotDynamics.dynamics(model::Cartpole, x, u)
mc = model.mc # mass of the cart in kg (10)
mp = model.mp # mass of the pole (point mass at the end) in kg
l = model.l # length of the pole in m
g = model.g # gravity m/s^2
q = x[ SA[1,2] ] # SA[...] creates a StaticArray.
qd = x[ SA[3,4] ]
s = sin(q[2])
c = cos(q[2])
H = @SMatrix [mc+mp mp*l*c; mp*l*c mp*l^2]
C = @SMatrix [0 -mp*qd[2]*l*s; 0 0]
G = @SVector [0, mp*g*l*s]
B = @SVector [1, 0]
qdd = -H\(C*qd + G - B*u[1])
return [qd; qdd]
end
function RobotDynamics.dynamics!(model::Cartpole, xdot, x, u)
xstatic = SA[x[1], x[2], x[3], x[4]]
ustatic = SA[u[1]]
xdot .= RobotDynamics.dynamics(model, xstatic, ustatic)
return nothing
end
```

with our dynamics model defined, we are ready to start setting up the optimization problem.

For best performance, use StaticArrays.jl, which offers loop-unrolling and allocation-free methods for small to medium-sized matrices and vectors. For systems with large state vectors, prefer to use the in-place methods (the ones that end in `!`

).