Trajectory optimization problems

Table of Contents

• Trajectory optimization
  • Transcription
• Stage models
• Example

Trajectory optimization

The objective of this library is to model and solve optimal control problems (OCPs) of the form

\begin{align} \min_{x,u}~& \int_0^T \ell(x, u)\, dt + \ell_\mathrm{f}(x(T)) \\ \subjectto & \dot x (t) = f(x(t), u(t)) \\ & g(x(t), u(t)) = 0 \\ & h(x(t), u(t)) \leq 0 \end{align}

Transcription

A transcription translates the continuous-time OCP to a discrete-time, finite-dimensional nonlinear program. Aligator allows us to consider transcriptions with implicit discrete dynamics: \begin{aligned} \min_{\bmx,\bmu}~& J(\bmx, \bmu) = \sum_{i=0}^{N-1} \ell_i(x_i, u_i) + \ell_N(x_N) \\ \subjectto & f(x_i, u_i, x_{i+1}) = 0 \\ & g(x_i, u_i) = 0 \\ & h(x_i, u_i) \leq 0 \end{aligned}

In aligator, trajectory optimization problems are described using the class TrajOptProblemTpl. Each TrajOptProblemTpl is described by a succession of stages (StageModelTpl) which encompass the set of constraints and the cost function (class CostAbstractTpl) for this stage.

Additionally, a TrajOptProblemTpl must provide an initial condition \( x_0 = \bar{x} \), a terminal cost $$ \ell_{\mathrm{f}}(x_N) $$ on the terminal state \(x_N \); optionally, a terminal constraint \(g(x_N) = 0, h(x_N) \leq 0 \) on this state may be added.

Stage models

A stage model (StageModelTpl) describes a node in the discrete-time optimal control problem: it consists in a running cost function, and a vector of constraints (StageConstraintTpl), the first of which must describe system dynamics (through a DynamicsModelTpl).

Example

Define and solve an LQR (Python API):

#!/usr/bin/env python
# -*- coding: utf-8 -*-
#
# SPDX-License-Identifier: BSD-2-Clause
# Copyright 2023 Inria
 
"""Formulating and solving a linear quadratic regulator with Aligator."""
 
import matplotlib.pyplot as plt
import numpy as np
import tap
 
import aligator
from aligator import constraints, dynamics, manifolds
from aligator.utils.plotting import plot_convergence
 
 
class Args(tap.Tap):
    term_cstr: bool = False
    bounds: bool = False
 
 
args = Args().parse_args()
np.random.seed(42)
 
nx = 3  # dimension of the state manifold
nu = 3  # dimension of the input
space = manifolds.VectorSpace(nx)
x0 = space.neutral() + (0.2, 0.3, -0.1)
 
# Linear discrete dynamics: x[t+1] = A x[t] + B u[t] + c
A = np.eye(nx)
A[0, 1] = -0.2
A[1, 0] = 0.2
B = np.eye(nx)[:, :nu]
B[2, :] = 0.4
c = np.zeros(nx)
c[:] = (0.0, 0.0, 0.1)
 
# Quadratic cost: ½ x^T Q x + ½ u^T R u + x^T N u
Q = 1e-2 * np.eye(nx)
R = 1e-2 * np.eye(nu)
N = 1e-5 * np.eye(nx, nu)
 
Qf = np.eye(nx)
if args.term_cstr:  # <-- TODO: should it be `not term_cstr`?
    Qf[:, :] = 0.0
 
 
# These matrices define the costs and constraints that apply at each stage
# (a.k.a. node) of our trajectory optimization problem
rcost0 = aligator.QuadraticCost(Q, R, N)
term_cost = aligator.QuadraticCost(Qf, R)
dynmodel = dynamics.LinearDiscreteDynamics(A, B, c)
stage = aligator.StageModel(rcost0, dynmodel)
if args.bounds:
    u_min = -0.18 * np.ones(nu)
    u_max = +0.18 * np.ones(nu)
    ctrl_fn = aligator.ControlErrorResidual(nx, np.zeros(nu))
    stage.addConstraint(ctrl_fn, constraints.BoxConstraint(u_min, u_max))
 
 
# Build our problem by appending stages and the optional terminal constraint
nsteps = 20
problem = aligator.TrajOptProblem(x0, nu, space, term_cost)
 
for i in range(nsteps):
    problem.addStage(stage)
 
xtar2 = 0.1 * np.ones(nx)
if args.term_cstr:
    term_fun = aligator.StateErrorResidual(space, nu, xtar2)
    problem.addTerminalConstraint(term_fun, constraints.EqualityConstraintSet())
 
# Instantiate a solver separately
mu_init = 1e-1 if args.bounds else 1e-4
verbose = aligator.VerboseLevel.VERBOSE
tol = 1e-8
solver = aligator.SolverProxDDP(tol, mu_init, verbose=verbose)
 
his_cb = aligator.HistoryCallback()
solver.registerCallback("his", his_cb)
print("Registered callbacks:", solver.getCallbackNames().tolist())
solver.max_iters = 20
solver.rollout_type = aligator.ROLLOUT_LINEAR
 
u0 = np.zeros(nu)
us_i = [u0] * nsteps
xs_i = aligator.rollout(dynmodel, x0, us_i)
prob_data = aligator.TrajOptData(problem)
problem.evaluate(xs_i, us_i, prob_data)
 
solver.setup(problem)
solver.run(problem, xs_i, us_i)
res = solver.results
ws = solver.workspace
print(res)
 
plt.subplot(121)
fig1: plt.Figure = plt.gcf()
 
lstyle = {"lw": 0.9, "marker": ".", "markersize": 5}
trange = np.arange(nsteps + 1)
plt.plot(res.xs, ls="-", **lstyle)
 
if args.term_cstr:
    plt.hlines(
        xtar2,
        *trange[[0, -1]],
        ls="-",
        lw=1.0,
        colors="k",
        alpha=0.4,
        label=r"$x_\mathrm{tar}$",
    )
plt.hlines(
    0.0,
    *trange[[0, -1]],
    ls=":",
    lw=0.6,
    colors="k",
    alpha=0.4,
    label=r"$x=0$",
)
plt.title("State trajectory $x(t)$")
plt.xlabel("Time $i$")
plt.legend(frameon=False)
 
plt.subplot(122)
plt.plot(res.us, **lstyle)
if args.bounds:
    plt.hlines(
        np.concatenate([u_min, u_max]),
        *trange[[0, -1]],
        ls="-",
        colors="tab:red",
        lw=1.8,
        alpha=0.4,
        label=r"$\bar{u}$",
    )
plt.xlabel("Time $i$")
plt.title("Controls $u(t)$")
plt.legend(frameon=False, loc="lower right")
plt.tight_layout()
 
 
fig2: plt.Figure = plt.figure()
ax: plt.Axes = fig2.add_subplot()
niter = res.num_iters
ax.hlines(
    tol,
    0,
    niter,
    colors="k",
    linestyles="-",
    linewidth=2.0,
)
plot_convergence(his_cb, ax, res)
ax.set_title("Convergence (constrained LQR)")
ax.legend(
    [
        "Tolerance $\\epsilon_\\mathrm{tol}$",
        "Primal error $p$",
        "Dual error $d$",
    ]
)
fig2.tight_layout()
 
plt.show()