Bayesian Calibration Module

Bayesian CAD modeling

A Bayesian CAD modeler offers the possibility to describe the machines (positions, precision, kinematics and control parameters, etc.), the objects (locations and shapes), the sensors (range and accuracy) and the environment within a simple, unified and concise probabilistic description. Consequently, the main difference between a Bayesian CAD modeler and a classical one is that, when resolving calibration and inverse kinematics problems, it is possible to take into account all the uncertainties at hand.

What can we do with it?

Bayesian reasoning represents a powerful and generic tool for computing uncertainties and constraints propagation. Using Bayesian inference, we can deal with uncertainty and solve many inverse problems found in CAD and robotics:

Calibration: most of the calibration and reconstruction problems can be solved using our approach:

• Machine calibration: Finding the location of the base and the parameters of the links given sensors' data.
• Robot Positioning: Finding a location of the base of a robot to access several working locations with an optimal precision.
• Part positioning: Finding the location of a part from information read on different sensors' sources.
• Tool design: Finding the optimal tool shape to reach a given location.
• Constraint satisfaction: Finding the location of a part to satisfy a set of geometrical constraints.

Inverse kinematics: new possibilities are offered when solving inverse kinematics problems, such as:

• Best precision inversion: Finding, robot's configuration leading to the best precision of the tool point by taking into account the uncertainty on the lengths of the links, on the commanded joint values and on locations and shapes of the parts involved by this task.

How does it work?

The kernel is made of a Bayesian inference engine dedicated to geometry.  The a priori knowledge on the scene is represented as a joint distribution of all the variables necessary to describe the machines, the parts, the sensor readings and the environment. The inference engine is divided into three components:

• The symbolic module which computes the formula permitting to obtain the desired value from the joint distribution. This module uses a resolution principle based on the Bayes' formula.
• The summation module which is used to sum probabilities over a set of variables (all the unknowns). This module is based on a Monte-Carlo algorithm.
• The optimization module which is used to find the most probable values of the searched variables. This module uses a Genetic Algorithm.

Example

To explain the principle of our method, we give a simple inverse kinematics problem with a 2 dof planar arm:
Click here to view this example

What have we achieve?

We have developed the kernel and tested it on a simple Cad modeler for numerous problems:

• This Figure represents a screen copy of this CAD modeler.
  
  
• The purpose of this movie is to give a feeling of what is a probabilistic specification of a CAD scene. It presents several random draws corresponding to the specified probability distribution. As there are uncertainties on the joint's positions and link's lengths of the robot and on car's location, their relative position is only approximately known.
  
  
• These two images represent two different solutions for a kinematics problem. This problem consists to put two parts (the red and the blue cubes on the images) one against the other, using two 6 dof Rx90 arms:

The solution found when equal uncertainty levels were assumed on the two arms.

The solution found when more uncertainty on command values of the right arm is assumed. We can see that the right arm bends at maximum to minimize the influence of the uncertainty of its command values on the uncertainty of the relative position between the two parts.

• This movie describes the measurement procedure and the internal data structure used to localize an object with a range finder (laser) mounted the extremity of a robot arm.
  A set of measurements (the distance given by the laser and robot's configuration) are taken. These measurements are used to find the position of the object. For each configuration, uncertainties resulting from errors on arm's links are explicitly taken into accout.
  

What do we want to achieve?

The capability to take in account all uncertainties at hand in CAD problems would be a major breakthrough for this field. We are now convinced that it is possible. Our goal is to develop with industrial partners a Bayesian Calibration Module product to solve these problems.

What are we looking for?

The fundamental research part of this project has been achieved:

• A unified and sane theory of Bayesian inference for CAD problems has been proposed.
• A Bayesian inference engine dedicated to geometry has been developed.
• Its possibilities and utility has been demonstrated on numerous problems.

The applied research part and industrial development may start, it supposes to:

• Optimize the Bayesian inference engine.
• Integrate it to commercial CAD modelers.
• Develop first industrial applications to definitely validate the approach.

Consequently, we are looking for industrial partners interested to work with us on these three objectives.