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Dual Quaternion Synthesis


Title: Dual Quaternion Synthesis Author: Alba Perez Last modified by: J. M. McCarthy Created Date: 9/26/2002 9:45:59 PM Document presentation format – PowerPoint PPT presentation

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Title: Dual Quaternion Synthesis

Dual Quaternion Synthesis of Constrained Robotic
Alba Perez Robotics and Automation
Laboratory University of California, Irvine
  • Introduction to Robot Design
  • What are constrained robotic systems?
  • The design problem (finite position synthesis)
  • Dual Quaternion Synthesis
  • kinematics equations
  • design equations
  • counting
  • solutions
  • Applications
  • computer aided design
  • avatar synthesis
  • crippled space robot arms

The Catholic University of America December 2,
Robotic Systems
  • Types of Joints
  • Types of robots
  • Serial Robots Parallel Robots

  • Serial Robots
  • 6 or more DOF The workspace is a portion of the
    6-dimensional space of displacements, whose
    boundary is given by the geometry of the robot
    and the limits on the joint angles.
  • Design criteria WS volume, dexterity, obstacle
  • Less than 6 DOF The workspace is a submanifold
    of the 6-dimensional space of displacements.
    Positions that seem to be within the physical
    volume of the robot may not belong to the WS.
  • Design criteria Subspace of movements,
    task-oriented design.
  • Parallel Robots
  • The WS can be seen as the intersection of the WS
    of each of the supporting serial chains. Each leg
    may impose constraints to the movement of the
    platform. Same distinction can be made between
    robots with less than 6 DOF and robots with 6 DOF
    or more.
  • Design criteria WS volume, isotropy.

Panasonic 6-dof robot
Constrained (Spatial) Robots
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 serial robots
Parallel 2-TPR robot
  • The constraints provide structural support in
    some directions, while allowing movement in the
  • 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.

Constrained Planar Robots
  • Serial chains that impose one constraint RR,
    RP, PR, PP
  • Parallel chains impose two constraints RRRR,

Two DOF systems
One DOF systems
Finite-position Synthesis
  • Kinematic Synthesis
  • Determine the mechanical constraints (i.e., links
    and joints) that provide a desired movement.
  • Finite-position Synthesis
  • Identify a set of task positions that represent
    the desired movement of the workpiece.
  • Developed for synthesis of serial open chains.
    The multiple solutions can be assembled to
    construct parallel chains.

The result is a design process for constrained
robotic systems.
Finite-position Synthesis
  • Finite-position Synthesis
  • Given (a) a robot topology, 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
  • There are different ways to formulate the set of
    design equations.

Finite Position Synthesis for Planar Robots
Approaches Graphical synthesis Analytical
constraint synthesis Complex number formulation
Two position synthesis
Three position synthesis
Finite Position Synthesis for Spatial Robots
  • 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)
  • Loop Closure Equations along the chain from a
    reference configuration to each goal
  • Robot kinematics equations, that define the set
    of positions reachable by the end-effector, are
    equated to each task position.
  • Dual Quaternion Synthesis

Dual quaternion synthesis is a combination of
Kinematic Geometry and Robot Kinematics
Equations. It is, in addition, an extension of
the complex number formulation to spatial robots.
Literature Review
Geometric features of the chain are used to
formulate the algebraic constraint equations.
(distance and angle constraints)
  • Roth, B., 1968, The design of binary cranks
    with revolute, cylindric, and prismatic
  • joints, J. Mechanisms, 3(2)61-72.
  • 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.
  • Innocenti, C., 1994, Polynomial Solution of
    the Spatial Burmester Problem.'' Mechanism
    Synthesis and Analysis, ASME DE vol. 70.
  • Nielsen, J. and Roth, B., 1995, Elimination
    Methods for Spatial Synthesis,
  • Computational Kinematics, (eds. J. P. Merlet and
    B. Ravani), Vol. 40 of Solid Mechanics and Its
    Applications, pp. 51-62, Kluwer Academic
  • 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.

RR chain
Literature Review
  • 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.
  • Tsai, L.W., and Roth, B., A Note on the Design
    of Revolute-Revolute Cranks,
  • Mechanism and Machine Theory, Vol. 8, pp. 23-31,
  • Loop closure equations along the chain from a
    reference configuration to each goal
  • Sandor, G. N., and Erdman, A. G., 1984, Advanced
    Mechanism Design Analysis
  • and Synthesis, Vol. 2. Prentice-Hall, Englewood
    Cliffs, NJ
  • .
  • Sandor, G.N., Xu, Y., and Weng, T.C., 1986,
    Synthesis of 7-R Spatial Motion Generators with
    Prescribed Crank Rotations and Elimination of
    Branching, The International Journal of Robotics
    Research, 5(2)143-156.
  • Sandor, G.N., Weng, T.C., and Xu, Y., 1988, The
    Synthesis of Spatial Motion
  • Generators with Prismatic, Revolute and Cylindric
    Pairs without Branching Defect,
  • Mechanism and Machine Theory, 23(4)269-274.

Literature Review
  • Robot kinematics equations define the set of
    positions reachable by the end-effector. Equate
    to each task position to obtain design equations
  • Park, F. C., and Bobrow, J. E., 1995,
    Geometric Optimization Algorithms for Robot
    Kinematic Design. Journal of Robotic Systems,
  • 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.
  • Lee, E., and Mavroidis, D., 2002c, Geometric
    Design of Spatial PRR Manipulators
  • Using Polynomial Elimination Techniques, Proc.
    ASME 2002 Design Eng.
  • Tech. Conf., paper no. DETC2002/MECH-34314, Sept.
    29-Oct. 2, Montreal,
  • Canada.

The Dual Quaternion form of the Kinematics
equations captures the geometry fo the screw
triangle in an efficient mathematical form.
Dual Quaternions
  • Represent elements of the group of spatial
    displacements SE(3)
  • They form the even Clifford subalgebra C(P3)
  • We can write them as four-dimensional dual
  • The dual vector S is the screw axis of the

Dual quaternion formulation
Matrix formulation
Literature Review
  • Dual algebra has been used for the kinematic
    analysis of spatial mechanisms.
  • Clifford, W., 1873, Preliminary sketch of
    bi-quaternions, Proc. London Math. Soc.,
  • Yang, A.T., and Freudenstein, F., 1964,
    Application of Dual-Number Quaternion Algebra to
    the Analysis of Spatial Mechanisms, ASME Journal
    of Applied Mechanics, June 1964, pp.300-308.
  • Sandor, G.N., 1968, Principles of a General
    Quaternion-Operator Method of Spatial Kinematic
    Synthesis, Journal of Applied Mechanics,
  • Ravani, B. and Ge Q. J.,1991, Kinematic
    Localization for World Model Calibration in
    Off-Line Robot Programming Using Clifford
    Algebra, Proc. of IEEE International Conf. on
    Robotics and Automation, Sacramento, CA, April
    1991, pp. 584-589.
  • Horsch, Th., and Nolzen, H., 1992, Local Motion
    Planning Avoiding Obstacles with Dual
    Quaternions, Proc. of the IEEE Int. Conf. on
    Robotics and Automation, Nice, France, May 1992.
  • Daniilidis, K., and Bayro-Corrochano, E.,
    1996,The dual quaternion approach to hand-eye
    calibration, Proc. of the 13th Int. Conf. on
    Pattern Recognition, 1996,1 318 -322.

Challenges of the Synthesis 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? (30 equations)
  • Using the kinematics equations, we obtain a set
    of 130 equations in 130 variables, including the
    joint angles.

Challenges of the Synthesis Problem
  • Solving the design equations
  • Set of polynomial equations have a very high
    total degree.
  • The joint variables may be eliminated to reduce
    the dimension of the problem.
  • Due to internal structure, the equations have far
    less solutions than the Bezout bound.
  • Some sample cases
  • RR chain (2 dof robot)
  • Initial total degree 210 1024.
  • Final solution six roots, with only two real
  • 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 1024 roots.

Source Hai-Jun Su
Dual Quaternion Synthesis of Constrained Robots
Dual Quaternion Synthesis of Constrained Robots
  • Create dual quaternion kinematics equations
  • Quaternion product of relative screw
    displacements from a reference position.
  • Counting
  • Compute nmax, maximum number of complete task
    positions for each topology.
  • Create design equations
  • Equate dual quaternion kinematics equations to
    nmax task dual quaternions.
  • Solve the design equations
  • Solve numerically in parameterized form (with
    joint variables)
  • Eliminate the joint variables to obtain a set of
    reduced equations
  • For those cases where it is possible, algebraic
    elimination leads to a close solution
  • Different algebraic methods (resultant, matrix
    eigenvalue, ) to create a univariate polynomial.
  • For those cases that are too big for algebraic
    elimination, numerical methods to find solutions
  • Polynomial continuation methods.
  • Newton-Raphson numerical methods.

Dual Quaternion Kinematics Equations
  • 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
  • Formulate the kinematics equations of the robot
    using dual quaternions,
  • The dual quaternion kinematics equations are
    equivalent to the product of exponentials in the
    Lie algebra of SE(3),

Dual Quaternion Design Equations
  • 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 8-dim. vector equations where
    the variables to solve for are the Plucker
    coordinates of the axes Sj in the reference
  • The equations are parameterized by the joint
    variables ?j, j1,m.

  • How many complete task positions can we define?
  • Consider a serial chain with r revolute joints
    and t prismatic joints, and n task positions.
  • Parameters
  • R joint-- 6 components of a dual vector, 6r.
  • P joint-- 3 components of a direction vector,
  • Joint variables, (rt)(n-1), measured relative
    to initial configuration.
  • Dual Quaternion design equations, 6(n-1)
  • Associated constraint equations
  • R joint-- 2 constraints (Plucker conditions),
  • P joint-- 1 constraint (unit vector), t .
  • Imposed extra constraint equations, c.

Equations 6(n-1)2rtc. Unknowns
6r3t(rt)(n-1). nmax (6 3r t -
c)/(6 - r - t) (note rt lt 6 for constrained
robotic systems)
  • How many task orientations can we define?
  • Consider a serial chain with r revolute joints,
    and nR task orientations.
  • Parameters
  • R joint-- 3 components of a dual vector, 3r.
    (no other joints contribute to orientations).
  • Joint variables, r (nR-1) (measured relative to
    initial configuration).
  • Dual Quaternion design equations, 3(nR-1)
  • Associated constraint equations
  • R joint-- 1 constraint (unit vector), r.
  • Imposed extra constraint equations, c.

Equations 3(nR-1) r c. Unknowns 3r r
(nR-1). nR ( 3 r - c ) / ( 3-r ) (note
rlt 3 for orientation-constrained robotic systems)
Count-based classification
Orientation-limited robots
Generally-constrained robots
RPRP chain
RPP chain
PRRR chain
Summary of the Results
Previous results
Suh 1969, Tsai Roth 1972.
Partial solution Tsai 1972.
Partial solution Sandor 1988, complete
Mavroidis 2000.
Partial solution Sandor 1988.
Partial solution Sandor 1988.
Applications for Kinematic Synthesis of
Constrained Robots
  • Computer-aided design of robotic systems
  • Identification of kinematic structures from video
    images (avatar synthesis)
  • Mission recovery planning for crippled robot arms.

Computer-aided Design of Robotic
Systems Extendable Synthesis and Analysis
The results of the dual quaternion synthesis need
to be provided to the mechanical designer of
spatial mechanisms in an environment that allows
him or her to define tasks, synthesize robots,
simulate the movement and rank the possible
Data I/O
Animation Bar
Object Tree
GL4Java Viewer
Info Panel
Work piece
Teach Panel
Software Synthetica 1.0, developed by Hai Jun
Su, Curtis Collins and J.M. McCarthy
Synthetica 1.0
Example RPC Robot
Task definition 5 positions
Software Synthetica 1.0, developed by Hai Jun
Su, Curtis Collins and J.M. McCarthy
Example RPC Robot
Dual quaternion synthesis 4 soutions
Identification of kinematic structures Generation
of kinematic skeletons for avatars
  • When reconstructing 3-dimensional avatars from
    images (pictures or video), a procedure is needed
    to identify the kinematic structure necessary for
    pose estimation, tracking and movement
  • Standard approach thinning algorithms based on
    finding the center of mass from a visual hull.
    Captures the three-dimensional structure but does
    not have information about the movement of the
  • Kinematic approach Synthesizing a skeleton
    composed of articulated rigid bodies joined by
    joints. It gives a compact representation of the
    movement of the avatar.

Mission Recovery Planning for Crippled Robot Arms
Many NASA missions rely on a robot arm as a
primary interface for scientific activities.
Operational failures such as an actuator failure
in the arm can eliminate the opportunity for any
scientific return.
  • Standard approach Current robotic technology
    focuses on the design and trajectory planning for
    robot arms with six or more degrees of freedom.
    Planning for and use of a crippled robot arm that
    has fewer degrees of freedom fall outside the
    capability of current thinking.
  • Kinematic synthesis approach Use kinematic
    geometry reasoning to identify the movements
    available to a crippled robot with reduced
    degrees of freedom, and plan for the appropriate
    arm configurations that allow the performance of
    critical tasks in the face of various actuator
    failure modes.

  • The dual quaternion synthesis procedure uses
    the dual quaternion kinematics equations of an
    open chain to formulate the design equations.
  • Multiple solutions can be assembled to create
    parallel robots
  • The synthesis procedure has been applied to
    general 2-5 degree of freedom serial robots.
  • Future work
  • Conditions for branching, joint limits and
    self-intersection are required for general
    parallel systems.
  • Efficient implementation in CAD software
  • User interface strategies for specifying spatial
    linkage tasks.
  • Numerical solutions that are robust relative to
    local minima.
  • Applications for the identification of
    constrained and non-constrained kinematic
  • Skeleton identification in avatars
  • Identification of movable kinematic structure in
    protein configurations.
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