## Infeasible Start

Standard indirect methods such as iLQR cannot be initialized with a state trajectory since they are always dynamically feasible. However, for some problems an initial state trajectory is very informative and easy to generate, while supplying an initial guess for the controls is extremely difficult. For example, consider a quadrotor flying around some obstacles. Guessing a good path, or even the velocities, would be pretty easy, but supplying the control sequence to generate that path is nearly as hard as just solving the entire trajectory optimization problem.

ALTRO allows for "infeasible" starts by augmenting the discrete dynamics so that they become fully actuated, i.e. for any state we can provide a control that will acheive it. This increases the size of the control dimension by n, the number of states in the original problem, so the problem becomes more expensive to solve.

We specify that we want an infeasible start by passing the infeasible flag to the ALTRO constructor:

using Altro
prob,opts = Problems.DubinsCar(:escape)
solver = ALTROSolver(prob, opts, infeasible=true, R_inf=0.1)

where R_inf is the norm of the regularizer on the additional controls. Notice how the new control dimension is 5, since the original control and state dimensions were 2 and 3.

The initial state trajectory can be provided using one of

initial_states!(solver, X0)
initial_trajectory!(solver, Z0)

where X0::Union{SVector, Matrix, Vector{<:StaticVector}} and Z0::SampledTrajectory.

Altro.InfeasibleModelType
InfeasibleModel

An infeasible model is an augmented dynamics model that makes the system artifically fully actuated by augmenting the control vector with n additional controls. The dynamics are handled explicitly in discrete time:

$x_{k+1} = f(x_k,u_k,dt) + w_k$

where $w_k$ are the additional n-dimensional controls. In practice, these are constrained to be zero by the end of the solve.

Constructors

InfeasibleModel(model::AbstractModel)
Altro.InfeasibleConstraintType
InfeasibleConstraint

Constraints additional $infeasible'' controls to be zero. Constructors:$julia InfeasibleConstraint(model::InfeasibleModel) InfeasibleConstraint(n,m)

## Using Implicit Integrators

By leveraging the functionality of RobotDynamics.jl, Altro can easily solve problems using implicit integrators like implicit midpoint, which as a symplectic integrator has energy-conserving behaviors and also preserves any implicit norms in the dynamics (such as the norm on a quaternion representing rotations). The following example shows how to use an implicit integrator:

using RobotZoo: Cartpole
using RobotDynamics
using TrajectoryOptimization
using Altro
using LinearAlgebra
const RD = RobotDynamics

model = Cartpole()
dmodel = RD.DiscretizedDynamics{RD.ImplicitMidpoint}(model)

# Temporary "hack" to make sure it doesn't try to use the UserDefined method

tf = 2.0
N = 51
n,m = RD.dims(model)
x0 = [0,0,0,0.]
xf = [0,pi,0,0]

Q = Diagonal(fill(1.0, n))
R = Diagonal(fill(0.1, m))
Qf = Q*(N-1)
obj = LQRObjective(Q,R,Qf,xf,N)

prob = Problem(dmodel, obj, x0, tf)

solver = ALTROSolver(
prob,
)

The key here is to specify the solver option dynamics_diffmethod to be RobotDynamics.ImplicitFunctionTheorem() which takes another RobotDynamics.DiffMethod as an argument, which specified how the Jacobians of the dynamics residual should be computed. The implicit function theorem then uses the partial derivatives to compute the Jacobians with respect to the next state, which are the Jacobians requried by algorithms like iLQR.

## Disabling Octavian

By default, Altro.jl uses Octavian.jl for matrix multiplication. This typically yields very good runtime performance but can take a while to compile the first time. If you want to disable the use of Octavian, set the environment variable ALTRO_USE_OCTAVIAN = false prior to using Altro. If Altro has already precompiled, you'll need to delete the compiled cache using

rm -rf ~/.julia/compiled/v1.x/Altro/*

and then when you enter using Altro in the Julia REPL you should see it print a message that it's precompiling. You can check to see if Altro is using Octavian by checking the Altro.USE_OCTAVIAN variable in the Altro module.