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 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?

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

5 Kinematics of a 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

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 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 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