IntCP’06 Nantes - France, September 2006

Combining CP and Interval Methods for solving the Direct Kinematic of a Parallel Robot under Uncertainties

C. Grandón, D. Daney and Y. Papegay

COPRIN project at INRIA - Sophia Antipolis, France.

2

Outline

• Introduction• Kinematics of a Robot

• Example of a 3-2-1 parallel manipulator

• Problem Statement

• Approaches• Formal algebraic evaluation

• Numerical methods

• Combination of both

• Preliminary results

• Conclusions and future works

3

Kinematics of a Robot

• Computing the kinematics of a robot is a central concern in robotics for control

• A kinematics model:• X: Generalized coordinates (position/orientation end effectors)

• Q: Articular coordinates (actuator control variables)

• P: Kinematics parameters (dimensional parameters)

• Two important questions• If I know the values of Q and P: Where is the robot end-

effector?

• If I want to bring the robot to a given position: What values must I give to the control variables?

0),,( PQXf

4

Kinematics of a Robot

• Serial Robot• Direct kinematics

– Closed form

• Inverse kinematics– Solve a system

• Parallel Robot• Inverse kinematics

– Closed form

• Direct kinematics– Solve a system

),( PQgX

),( PXhQ

5

Kinematics of a 3-2-1 Parallel Robot

• It is a specific version of a Gough Platform with interesting system of kinematics equations.

• X: coordinates of the three mobile attachment points

• Q: length of the legs

• P: Distance between base points, and distances between mobile points

• Bounded Uncertainties• Design [P], measurement [Q]

6

Problem Statement

• Given a set of parameters with bounded uncertainties, to compute a certified approximation of the set of solutions

• Interval Direct kinematics

• Relationships

0),,( ],[ ],[ |)hull( PXQfQqPpx

7

Approaches

• Formal algebraic evaluation• Based on the work of Manolakis 1996, Thomas et al. 2005,

and proposed in Ceccarelli et al. 1999, Ottaviano et al. 2002.

• Using Trilateration in three sub-systems in cascade

• How to handle uncertainties? -> Interval evaluation

8

Approaches

• Numerical methods and Constraint Programming• Based on Branch and Prune algorithms combining filtering,

bisection and evaluation techniques.

• Working in a 9-dimensional equation system

• How to separate solutions?

9

New Approach

• Combination of Formal and Numeric approaches• To Solve three square sub-system in cascade

• To Apply filtering techniques in each phase

• To combine with a special conditional bisection algorithm

• To use a global filtering phase at the end

• Example

10

Preliminary Results

• Applying to CaTraSys (Cassino Tracking System)• Measuring system, conceived and designed at LARM

(LAboratory of Robotics and Mechatronics) in Cassino

11

Preliminary Results

• Algorithms• Algebraic only

• Algebraic + 2B

• Algebraic + 3B

• Classic Solver

• Computed results

12

Preliminary Results

13

Conclusions

• We have presented an interval extension of the Direct Kinematics of a 3-2-1 parallel robot

• Handle uncertainties in parameters

• Obtaining certified solutions

• Combination of formal (symbolic) and numeric solvers• Sharper approximations of the solutions

• Keep information about different independent configurations

• A current application in robotic has been reported.

• Future Works?

14

Thank you for your attention