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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.
IntCP06 Nantes - France, September 2006
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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

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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?

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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

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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

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Problem Statement

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

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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? -gt Interval
  evaluation

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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?

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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

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Preliminary Results

• Applying to CaTraSys (Cassino Tracking System)
• Measuring system, conceived and designed at LARM
  (LAboratory of Robotics and Mechatronics) in
  Cassino

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Preliminary Results

• Algorithms
• Algebraic only
• Algebraic 2B
• Algebraic 3B
• Classic Solver
• Computed results

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Preliminary Results

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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?

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Thank you for your attention