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### 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

1
Dual Quaternion Synthesis of Constrained Robotic
Systems
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,
2003
2
Robotic Systems
• Types of Joints
• Types of robots
• Serial Robots Parallel Robots

3
Workspace
• 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
avoidance.
• 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,
• 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
4
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
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.

5
Constrained Planar Robots
• Serial chains that impose one constraint RR,
RP, PR, PP
• Parallel chains impose two constraints RRRR,
RRRP, RRPR, RRPP, RPRP, RPPR

Two DOF systems
One DOF systems
6
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.
7
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
• A set of design equations evaluated at each of
the task positions is used to determine the
mechanism.
• There are different ways to formulate the set of
design equations.

8
Finite Position Synthesis for Planar Robots
Approaches Graphical synthesis Analytical
constraint synthesis Complex number formulation
Two position synthesis
Three position synthesis
9
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
configuration.
• Robot kinematics equations, that define the set
of positions reachable by the end-effector, are
• 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.
10
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.
91(1)209219.
• 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
Publishers.
• 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,

RR chain
11
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
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,
1973.
• Loop closure equations along the chain from a
reference configuration to each goal
configuration.
• 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.

12
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,
12(6)453-463.
• 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,

The Dual Quaternion form of the Kinematics
equations captures the geometry fo the screw
triangle in an efficient mathematical form.
13
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
vectors,
• The dual vector S is the screw axis of the
transformation

Dual quaternion formulation
Matrix formulation
14
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.,
4381395.
• 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,
35(1)40-46.
• 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.

15
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.

16
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
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 1024 roots.

Source Hai-Jun Su
17
Dual Quaternion Synthesis of Constrained Robots
18
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
• 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.

19
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
transformations,
• 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),

20
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
position.
• The equations are parameterized by the joint
variables ?j, j1,m.

21
Counting
• 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,
3t.
• 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),
2r.
• 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)
22
Counting
• How many task orientations can we define?
• Consider a serial chain with r revolute joints,
• 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)
23
Count-based classification
Orientation-limited robots
Generally-constrained robots
RPRP chain
RPP chain
PRRR chain
24
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.
25
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.

26
Computer-aided Design of Robotic
Systems Extendable Synthesis and Analysis
Software
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
solutions.
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
27
Synthetica 1.0
2 TPR
3 RPS
28
Example RPC Robot
Software Synthetica 1.0, developed by Hai Jun
Su, Curtis Collins and J.M. McCarthy
29
Example RPC Robot
Dual quaternion synthesis 4 soutions
30
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
generation.
• 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
subject
• Kinematic approach Synthesizing a skeleton
composed of articulated rigid bodies joined by
joints. It gives a compact representation of the
movement of the avatar.

31
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.

32
Conclusions
• 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