# Codac: constraint-programming for robotics¶

Codac (Catalog Of Domains And Contractors) is a C++/Python library providing tools for constraint programming over reals, trajectories and sets. It has many applications in state estimation or robot localization.

What is constraint programming?
In this paradigm, users concentrate on the properties of a solution to be found (e.g. the pose of a robot, the location of a landmark) by stating constraints on the variables. Then, a solver performs constraint propagation on the variables and provides a reliable set of feasible solutions corresponding to the problem. In this approach, the user concentrates on what is the problem instead of how to solve it, thus leaving the computer dealing with the how.
In the field of robotics, complex problems such as non-linear state estimation, parameter estimation, delays, SLAM or kidnapped robot problems can be solved in a very few steps by using constraint programming. Even though the Codac library is not meant to target only robotics problems, the design of its interface has been largely influenced by the needs of the above class of applications. Codac provides solutions to deal with these problems, that are usually hardly solvable by conventional methods such as particle approaches or Kalman filters.

In a nutshell, Codac is a high-level constraint programming framework providing tools to easily solve a wide range of problems.

Keywords

 constraint-programming mobile robotics interval-analysis dynamical-systems tubes localization state-estimation SLAM solver

## Getting started: 2 minutes to Codac¶

We only have to define domains for our variables and a set of contractors to implement our constraints. The core of Codac stands on a Contractor Network representing a solver.

In a few steps, a problem is solved by

1. Defining the initial domains (boxes, tubes) of our variables (vectors, trajectories)

2. Take contractors from a catalog of already existing operators, provided in the library

3. Add the contractors and domains to a Contractor Network

4. Let the Contractor Network solve the problem

5. Obtain a reliable set of feasible variables

For instance.
Let us consider the robotic problem of localization with range-only measurements. A robot is described by the state vector $$\mathbf{x}=\{x_1,x_2,\psi,\vartheta\}^\intercal$$ depicting its position, its heading and its speed. It evolves between three landmarks $$\mathbf{b}_1$$, $$\mathbf{b}_2$$, $$\mathbf{b}_3$$ and measures distances $$y_i$$ from these points.

The problem is defined by classical state equations:

$\begin{split}\left\{ \begin{array}{l} \dot{\mathbf{x}}(t)=\mathbf{f}\big(\mathbf{x}(t),\mathbf{u}(t)\big)\\ y_i=g\big(\mathbf{x}(t_i),\mathbf{b}_i\big) \end{array}\right.\end{split}$

where $$\mathbf{u}(t)$$ is the input of the system, known with some uncertainties. $$\mathbf{f}$$ and $$g$$ are non-linear functions.

First step.
Defining domains for our variables.

We have three variables evolving with time: the trajectories $$\mathbf{x}(t)$$, $$\mathbf{v}(t)=\dot{\mathbf{x}}(t)$$, $$\mathbf{u}(t)$$. We define three tubes to enclose them:

dt = 0.01                                         # timestep for tubes accuracy
tdomain = Interval(0, 3)                          # temporal limits [t_0,t_f]=[0,3]

x = TubeVector(tdomain, dt, 4)                    # 4d tube for state vectors
v = TubeVector(tdomain, dt, 4)                    # 4d tube for derivatives of the states
u = TubeVector(tdomain, dt, 2)                    # 2d tube for inputs of the system

float dt = 0.01;                                  // timestep for tubes accuracy
Interval tdomain(0, 3);                           // temporal limits [t_0,t_f]=[0,3]

TubeVector x(tdomain, dt, 4);                     // 4d tube for state vectors
TubeVector v(tdomain, dt, 4);                     // 4d tube for derivatives of the states
TubeVector u(tdomain, dt, 2);                     // 2d tube for inputs of the system


We assume that we have measurements on the headings $$\psi(t)$$ and the speeds $$\vartheta(t)$$, with some bounded uncertainties defined by intervals $$[e_\psi]=[-0.01,0.01]$$, $$[e_\vartheta]=[-0.01,0.01]$$:

x[2] = Tube(measured_psi, dt).inflate(0.01)       # measured_psi is a set of measurements
x[3] = Tube(measured_speed, dt).inflate(0.01)

x[2] = Tube(measured_psi, dt).inflate(0.01);      // measured_psi is a set of measurements
x[3] = Tube(measured_speed, dt).inflate(0.01);


Finally, we define the domains for the three range-only observations $$(t_i,y_i)$$ and the position of the landmarks. The distances $$y_i$$ are bounded by the interval $$[e_y]=[-0.1,0.1]$$.

e_y = Interval(-0.1,0.1)
y = [Interval(1.9+e_y), Interval(3.6+e_y), \      # set of range-only observations
Interval(2.8+e_y)]
b = [[8,3],[0,5],[-2,1]]                          # positions of the three 2d landmarks
t = [0.3, 1.5, 2.0]                               # times of measurements

Interval e_y(-0.1,0.1);
vector<Interval> y = {1.9+e_y, 3.6+e_y, 2.8+e_y}; // set of range-only observations
vector<Vector>   b = {{8,3}, {0,5}, {-2,1}};      // positions of the three 2d landmarks
vector<double>   t = {0.3, 1.5, 2.0};             // times of measurements

Second step.
Defining contractors to deal with the state equations.

The distance function $$g(\mathbf{x},\mathbf{b})$$ between the robot and a landmark corresponds to the CtcDist contractor provided in the library. The evolution function $$\mathbf{f}(\mathbf{x},\mathbf{u})=\big(x_4\cos(x_3),x_4\sin(x_3),u_1,u_2\big)$$ can be handled by a custom-built contractor:

ctc_f = CtcFunction(
Function("v[4]", "x[4]", "u[2]",
"(v[0]-x[3]*cos(x[2]) ; v[1]-x[3]*sin(x[2]) ; v[2]-u[0] ; v[3]-u[1])"))

CtcFunction ctc_f(
Function("v[4]", "x[4]", "u[2]",
"(v[0]-x[3]*cos(x[2]) ; v[1]-x[3]*sin(x[2]) ; v[2]-u[0] ; v[3]-u[1])"));

Third step.
Adding the contractors to a network, together with there related domains, is as easy as:
cn = ContractorNetwork()   # creating a network

for i in range (0,len(y)): # we add the observ. constraint for each range-only measurement

p = cn.create_interm_var(IntervalVector(4)) # intermed. variable (state at t_i)

# Distance constraint: relation between the state at t_i and the ith beacon position

# Eval constraint: relation between the state at t_i and all the states over [t_0,t_f]

ContractorNetwork cn;        // creating a network

for(int i = 0 ; i < 3 ; i++) // we add the observ. constraint for each range-only measurement
{
IntervalVector& p = cn.create_interm_var(IntervalVector(4)); // intermed. variable (state at t_i)

// Distance constraint: relation between the state at t_i and the ith beacon position

// Eval constraint: relation between the state at t_i and all the states over [t_0,t_f]
}

Fourth step.
Solving the problem.
cn.contract()

cn.contract();

Fifth step.
Obtain a reliable set of feasible positions: a tube, depicted in blue. The three yellow robots illustrate the three instants of observation. The white line is the unknown truth.
You just solved a non-linear state-estimation without knowledge about initial condition.
See the full example on Github: in C++ or in Python.

In the tutorial and in the examples folder of this library, you will find more advanced problems such as Simultaneous Localization And Mapping (SLAM), data association problems or delayed systems.

## User manual¶

Want to use Codac? The first thing to do is to install the library, or try it online:

Then you have two options: read the details about the features of Codac (domains, tubes, contractors, slices, and so on) or jump to the standalone tutorial about how to use Codac for mobile robotics, with telling examples.

## Tutorial for mobile robotics¶

The following tutorial is standalone and tells about how to use Codac for mobile robotic applications, with telling examples:

This software is under GNU Lesser General Public License.

For recent improvements and activities, see the Codac Github repository. You can post bug reports and feature requests on the Issues page.

## Contributors¶

 Auguste Bourgois Julien Damers Fabrice Le Bars Raphael Voges Quentin Brateau Thomas Le Mézo Cyril Bouvier Bertrand Neveu Peter Franek Gilles Trombettoni Verlein Radwan Mohamed Saad Ibn Seddik

## How to cite this project?¶

We suggest the following BibTeX template to cite Codac in scientific discourse:

@misc{codac,
author = {Rohou, Simon and Desrochers, Benoit and others},
year = {2022},
note = {http://codac.io},
title = {The {Codac} library -- {C}onstraint-programming for robotics}
}