Dimensional Synthesis of RPC Serial Robots

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Dimensional Synthesis of RPC Serial Robots
ICAR 2003 The 11th International Conference on
Advanced Robotics June 30 - July 3, 2003
University of Coimbra, Portugal
Alba Perez (maperez_at_uci.edu), J.M. McCarthy
(jmmccart_at_uci.edu) Robotics and Automation
Laboratory Department of Mechanical and Aerospace
Engineering University of California, Irvine
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Overview
Constrained robotic system A workpiece, or
end-effector, supported by one or more serial
chains such that each one imposes at least one
constraint on its movement.
Classification of constrained robotic systems
3-RPS constrained robot (category 3I, 3 degrees
of freedom)

• The constraints provide structural support in
  some directions, while allowing movement in the
  others.
• The workspace of a constrained robot has less
  that six degrees of freedom. Therefore, positions
  that lie within the physical volume of the system
  may be unreachable.

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Overview

• Kinematic Synthesis
• Determine the mechanical constraints (i.e., links
  and joints) that provide a desired movement.
• Finite-position Synthesis
• Can be interpreted as the design of constrained
  robotic systems.
• Identify a set of task positions that represent
  the desired movement of the workpiece.
• The methodology is developed for synthesis of
  serial open chains. The multiple solutions can
  be assembled to construct parallel chains.

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Overview

• Finite-position Synthesis Methodology
• Given (a) a constrained serial chain, and (b) a
  task defined in terms of a set of positions and
  orientations of a workpiece,
• Find The location of the base, the location
  of the connection to the workpiece, and the
  dimensions of each link such the the chain
  reaches each task position exactly.
• A set of design equations evaluated at each of
  the task positions is used to determine the
  mechanism.
• There exist different methodologies to create
  the set of design equations.

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Overview

• The Design Equations for Finite Position
  Synthesis can be obtained in several ways
• Geometric features of the chain are used to
  formulate the algebraic constraint equations.
  (distance and angle constraints)
• Kinematic geometry based on the screw
  representation of the composition of
  displacements. (equivalent screw triangle)
• Robot kinematics equations define the set of
  positions reachable by the end-effector. Equate
  to each task position to obtain design equations
• Solve for the base position G, the connection
  to the workpiece H, and the link dimensions
  (?j, aj) and joint parameters (?j, dj)j (i
  positions).

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Background
Geometric features of the chain are used to
formulate the algebraic constraint equations.
(distance and angle constraints)

• Chen, P., and Roth, B., 1969, Design
  Equations for the Finitely and Infinitesimally
  Separated Position Synthesis of Binary Links and
  Combined Link Chains, ASME J. Eng. Ind.
  91(1)209219.
• Suh, C.H., and RadcliÆe, C.W., 1978, Kinematics
  and Mechanisms Design. John Wiley.
• Innocenti, C., 1994, Polynomial Solution of
  the Spatial Burmester Problem.'' Mechanism
  Synthesis and Analysis, ASME DE vol. 70.
• Murray, A.P., and McCarthy, J.M., 1994, Five
  Position Synthesis of Spatial CC Dyads. Proced.
  ASME Mechanisms Conference, Minneapolis, MN,
  Sept. 1994.
• Kim, H. S., and Tsai, L. W., 2002, Kinematic
  Synthesis of Spatial 3-RPS Parallel
  Manipulators, Proc. ASME Des. Eng. Tech. Conf.
  paper no. DETC2002/MECH-34302, Sept. 29-Oct. 2,
  Montreal, Canada.

CS chain
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Background

• Kinematic geometry based on the screw
  representation of the composition of
  displacements. (equivalent screw triangle)

• Tsai, L. W., and Roth, B., 1972, Design of
  Dyads with Helical, Cylindrical, Spherical,
  Revolute and Prismatic Joints, Mechanism and
  Machine Theory, 7591-598.

• Robot kinematics equations define the set of
  positions reachable by the end-effector. Equate
  to each task position to obtain design equations

• Mavroidis, C., Lee, E., and Alam, M., 2001, A
  New Polynomial Solution to the Geometric Design
  Problem of Spatial RR Robot Manipulators Using
  the Denavit-Hartenberg Parameters, J. Mechanical
  Design, 123(1)58-67.
• Lee, E., and Mavroidis, D., 2002, Solving the
  Geometric Design Problem of Spatial 3R Robot
  Manipulators Using Polynomial Homotopy
  Continuation, ASME J. of Mechanical Design,
  124(4), pp.652-661.

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Features of this Problem

• Stating the design equations
• Methods based on geometric constraints give
  simpler equations but lack a general methodology
  to find the constraints for all kinds of chains.
• Methods based on the kinematics equations are
  general but give a more complicated set of
  equations with extra variables.

• RR chain
• 10 geometric constraints

• 5R chain
• geometric constraints?
• Using the kinematics equations, we obtain a set
  of 120 equations in 120 variables, including the
  joint angles.

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Features of the problem

• Solving the design equations
• Set of polynomial equations of high degree in
  several variables.
• The joint variables can be eliminated to reduce
  the dimension of the problem.
• Due to internal structure, they could be much
  simplified.
• Some sample cases

• RR chain (2 dof robot)
• Initial total degree 210 1024.
• Final solution six roots, with only two real
  solutions.

• RPR chain (3 dof robot)
• Initial total degree 2346 32768.
• Final solution 12 roots.

• RPS chain (5 dof robot)
• Initial total degree 262144.
• Final solution 1020 roots.

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Our Approach

• Stating the design equations
• Use dual quaternion synthesis systematic way of
  creating the design equations that allows
  elimination of the joint variables.
• Solving the design equations
• For those cases where it is possible, algebraic
  elimination leads to a close solution
• Resultant methods to create a univariate
  polynomial.
• Matrix eigenvalue methods.
• For those cases that are too big for algebraic
  elimination, numerical methods to find all
  solutions
• Polynomial continuation methods.

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Dual Quaternion Synthesis of Constrained Robots

• The robot kinematics equations of the chain are
  used to formulate design equations.
• The set of displacements of the chain are
  written as a product of coordinate
  transformations,
• Formulate the kinematics equations of the robot
  using dual quaternions,

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Dual Quaternion Synthesis of Constrained Robots

• From the dual quaternion kinematics equations,

• Create the design equations equate the
  kinematics equations to each task position
  written in dual quaternion form
• We obtain a set of vector equations where the
  variables to solve for are the Plucker
  coordinates of the axes Sj in the reference
  position.
• The equations are parameterized by the joint
  variables ?j, j1,k.

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Dual Quaternion Synthesis of Constrained Robots

• How many task positions can we define?
• Consider a serial chain with r revolute joints
  and t prismatic joints.

• Number of task positions, n.
• Dual Quaternion design equations, 6(n-1)
• Parameters
• R joint-- 6 components of a dual vector, 6j.
• P joint-- 3 components of a direction vector,
  3k.
• Associated constraint equations
• R joint-- 2 constraints (Plucker conditions),
  2j.
• P joint-- 1 constraint (unit vector), k .
• Imposed constraint equations, c.
• Joint variables, (rt)(n-1) (measured relative to
  initial configuration).

Equations 6(n-1)2rtc. Unknowns
6r3t(rt)(n-1). n (4r 2t -c)/(6-r-t)
1. (note rt lt 6 for constrained robotic
systems)
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Dual Quaternion Synthesis of Constrained Robots

• Systematic methodology to create design
  equations for any constrained robot.
• A formula for counting the maximum task
  positions we can define for each robot topology.
• Solve the parameterized design equations for
  both dimensions and inverse kinematics, or
• Create reduced algebraic equations that may be
  further reduced to find closed solutions.

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Design Example RPC Robot

• Design equations
• We can define a maximum of n5 task positions if
  we impose g.h0, w.h0.

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Design Example RPC Robot

• Parameterized design equations
• 31 equations in 31 unknowns, 16 of them are
  joint variables.
• each equation is at most of multi-degree 6.
• Reduced design equations
• Eliminate the joint variables to obtain a set of
  15 equations in 15 parameters.
• Separation of rotations and translations and
  further resultant eliminations lead to a set of
  linear equations plus a 6th degree polynomial. We
  obtain at most 6 RPC robots.

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Numerical Example RPC Robot
Task definition 5 positions
Software Synthetica 1.0, developed by Hai Jun
Su, Curtis Collins and J.M. McCarthy
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Numerical Example RPC Robot
Dual quaternion synthesis 4 soutions
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Numerical Example RPC Robot
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Conclusions

• The dual quaternion synthesis procedure uses
  the kinematics equations of an open chain to
  formulate the design equations.
• Multiple solutions can be assembled to create
  parallel robots.
• The synthesis procedure can be applied to
  general 2-5 degree of freedom serial chains.
• For simple cases, algebraic simplification is
  performed to obtain a closed solution. More
  complicated cases require polynomial continuation
  algorithms.
• The synthesis procedures are being implemented
  in the spatial design software SYNTHETICA 1.0.
  The java applet can be downloaded at
  http//synthetica.eng.uci.edu/mccarthy/Synthetica
  1.0/Synthetica.htm
• Work needed
• Strategies for specifying spatial linkage
  tasks.
• Numerical solutions that are robust relative to
  local minima.
• Conditions for branching, joint limits and
  self-intersection are required for general
  parallel systems.