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

This paper discusses the combination of formal algebraic evaluation and numerical methods for solving the direct kinematics of a parallel robot under uncertainties. It presents preliminary results and potential future works.

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

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  1. 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. IntCP’06 Nantes - France, September 2006

  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?

  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 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

  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

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