The Catholic University of America

December 2, 2003

Alba Perez

Robotics and Automation Laboratory

University of California, Irvine

Dual Quaternion Synthesisof

Constrained Robotic Systems

• 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

Robotic Systems

• Types of Joints

• Types of robots

Serial Robots Parallel Robots

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• 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, 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.

Workspace

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.

• 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.

Classification of constrained serial robots

Parallel 2-TPR robot

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

Finite-position SynthesisKinematic 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 mechanism.

• There are different ways to formulate the set of design equations.

Finite Position Synthesisfor Planar Robots

Two position synthesis

Three position synthesis

Approaches:Graphical synthesisAnalytical constraint synthesisComplex number formulation

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 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.

Finite Position Synthesisfor Spatial Robots

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

RR chain

• Roth, B., 1968, “The design of binary cranks with revolute, cylindric, and prismaticjoints”, 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):209–219.

• 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 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, Montreal, Canada.

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, 7:591-598.

• 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.

S12S13S23 n1n2

n3t12/2θ12/2θ23/2θ13/2t13/2t23/2B12A12 A13

A23B23B13C13C12C23

• 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: Analysisand 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 MotionGenerators with Prismatic, Revolute and Cylindric Pairs without Branching Defect”,Mechanism and Machine Theory, 23(4):269-274.

Literature Review

R1R2{F}{M}1

{M}2

WZ

β1β2β12α1α1Rθ1θ2 α2α2Rα12

O

• 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 ManipulatorsUsing Polynomial Elimination Techniques,” Proc. ASME 2002 Design Eng.Tech. Conf., paper no. DETC2002/MECH-34314, Sept. 29-Oct. 2, Montreal,Canada.

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

Literature Review

Z0

θo α0doa0

{F} {M1}a1d1

a2d2

d3α1

α2

θ1θ2 θ3X0 Z1 Z2Z3

X4X1 X2

X3

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 vectors,

• The dual vector S is the screw axis of the transformation

• The dual numbers contain the information about the rotation and translation.

Matrix formulation:

Dual quaternion formulation: X YZ

yxz[R,d]

X YZ

S yxz

d

θ t

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., 4:381395.

•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.

1. 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.

Fixed AxisMoving Axis End EffectorG W i

PiBRR 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

Challenges of the Synthesis Problem2. 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:

Fixed AxisMoving Axis End EffectorG W i

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

BαPHG

R

HG P d

BPNR

( ) -b view along N( )a ( ) c view along G

N HRPS chain (5 dof robot): • Initial total degree: 262144.• Final solution: 1024 roots.

G W

Pθ φ

d RPR chain (3 dof robot): • Initial total degree: 23*46 = 32768.• Final solution: 12 roots.

Source: Hai-Jun Su

Dual Quaternion Synthesisof

Constrained Robots

1. Create dual quaternion kinematics equations• Quaternion product of relative screw displacements from a reference position.

2. Counting

• Compute nmax, maximum number of complete task positions for each topology.

3. Create design equations

• Equate dual quaternion kinematics equations to nmax task dual quaternions.

4. 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 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,

• The dual quaternion kinematics equations are equivalent to the product of exponentials in the Lie algebra of SE(3),

Dual Quaternion Kinematics Equations

θFaF1αF1dFAF

F d1θ1S1S2

S3A12A23a12α1θ2d2

α23a23 AMθ3d3Ma3Mα3MdMθM

Dual Quaternion Design 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 θj, j=1,…m.

• From the dual quaternion kinematics equations,

• 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, (r+t)(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)+2r+t+c. Unknowns: 6r+3t+(r+t)(n-1).

nmax = (6 + 3r + t - c)/(6 - r - t)

(note r+t < 6 for constrained robotic systems)

Counting

• 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 r< 3 for orientation-constrained robotic systems)

Counting

Count-based classification Orientation-limited robots Generally-constrained robots

Gθ UbHd

RPP chain

G(θ)

( )H d(Wφ)( )U b

RPRP chain

Wφ

Hd

Gθ Fψ

PRRR chain

Summary of the ResultsPrevious 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.

Applicationsfor

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.

Object Tree

Info Panel

Teach Panel

Work piece

GL4Java Viewer

Animation Bar

Data I/O

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.

Software: Synthetica 1.0, developed by Hai Jun Su, Curtis Collins and J.M. McCarthy

Synthetica 1.0

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2 TPR

3 RPS

Example: RPC RobotTask definition: 5 positions

Software: Synthetica 1.0, developed by Hai Jun Su, Curtis Collins and J.M. McCarthy

Example: RPC RobotDual 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 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.

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 structures:

• Skeleton identification in avatars

• Identification of movable kinematic structure in protein configurations.

Conclusions