Kinematic Synthesis of Robotic Manipulators from Task Descriptions

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Kinematic Synthesis of Robotic Manipulators from
Task Descriptions

• June 2003
• By Tarek Sobh, Daniel Toundykov

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Envisioning Optimal Geometry
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Objectives

• Parameters considered in this work
• Coordinates of the task-points
• Spatial constraints
• Restrictions (if any) on the types of joints
• Goals
• Simplified interface
• Performance
• Modular architecture to enable additional
  optimization modules (for velocity, obstacles,
  etc.)

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Optimization Techniques

• Minimization of cost functions
• Stochastic algorithms
• Parameters space methods
• Custom algorithms developed for specific types of
  robots

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Steepest Descent Method
fi(x)0 ? S(x)?fi(x)2

• System of equations is combined into a single
  function whose zeroes correspond to the solution
  of the system
• Algorithm iteratively searches for local minima
  by investigating the gradient of the surface
  S(x).
• Points where S(x) is small provide a good
  approximation to the optimal solution.

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Manipulability Measure
wvdet(JJT)

• For performance purposes the manipulability
  measure was the one originally proposed by Tsuneo
  Yoshikawa
• Singular configurations are avoided by maximizing
  the determinant of the Jacobian matrix

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Optimization Measure
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Single Target Problem

• Cost b Manipulability-1 p Distance to
  target
• b bias to eliminate singularities
• p precision factor

• Parameters that minimize the cost yield larger
  manipulability and small positional error
• Increase of the precision factor forces the
  algorithm to reduce the positional error in order
  to compensate the overall cost growth

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Optimization for Multiple Targets

• Several single-target cost functions are combined
  into a single expression
• Single-target cost functions share the same set
  of invariant DH-Parameters however, each of
  these functions has its own copy of the joint
  variables

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Invariant DH-Parameters

• Invariant parameters depend on the types of
  joints
• When no joints are specified, the algorithm
  compares all possible configurations based on the
  average manipulability value
• Invariant DH-parameters have a dumping factor. If
  dumping is large, the dimensions of the robot
  must decrease to keep the total cost low

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Results of Optimization
Geometry that maximizes manipulability at each
target

• Shared
• DH-parameters

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Joint Vector for Target 1
?
Inverse Solution for Target 1


Joint Vector for Target N
?
Inverse Solution for Target N
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Mathematica (Wolfram Research Inc )

• Powerful mathematical and graphics tools for
  scientific computing
• Flexible programming environment
• Availability of enhancing technologies
• Nexus to Java-based applications via J/Link
  interface
• Flexible Web-integration provided by
  webMathematica software
• Potential access to distributed computing
  systems, such as gridMatematica

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CAD Module Structure
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Input Data

• The set of task points
• Configuration restrictions
• DOF value if the system should determine optimal
  types of joints by itself
• or a specific configuration, such as Cartesian,
  articulated etc.
• Precision and size-dumping factors
• Output file name

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Screenshots
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Sample I

• Design a 3-link robot for a specific parametric
  trajectory
• No configuration was given, so the software had
  to choose the types of joints
• Dimensions of the robot were severely restricted

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Sample I Trajectory
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Sample I DH-Table (PRP)
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Sample I Manipulability Ellipsoids
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Sample II

• The trajectory has been changed
• This time we require a spherical manipulator
• No significant spatial constraints have been
  provided

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Sample II Trajectory
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Sample II DH-Table (RRR)
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Sample II Manipulability Ellipsoids
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Further Research

• Work has been done to account for robot dynamics
  and velocity requirements
• Online interface to the design module
• Future research may include obstacle avoidance
  and integration with distributed computing
  architectures