Institut de Robtica i Informtica Industrial

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A branch-and-prune solver for systems of
distance constraints
J.M. Porta, L. Ros, F. Thomas, and C. Torras
Institut de Robòtica i Informàtica
Industrial Barcelona - Catalonia
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Outline

• The problem
• Classical approaches in Robotics
• Formulation with multilinear constraints
• Solving multilinear constraints
• Some solved robots and molecules

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Problem
Given a mechanism
With several loops
With any type of joints
Find
All configurations of the mechanism lying within
given ranges for its degrees of freedom
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This statement generalizes...
... the inverse kinematics of serial robots
... the direct kinematics of parallel robots
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Open problem
Central problem in Computational Kinematics
There only exist ad-hoc solutions for specific
mechanisms
No general efficient algorithm has been found
"General"
All kinds of joints
Amount of loops
Shape of the solution space
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A complete algorithm is wanted
?j
?j
?i
?i
Isolated points
Whole continua of solutions
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Outline

• The problem
• Classical approaches in Robotics
• Formulation with multilinear constraints
• Solving systems of multilinear constraints
• Some solved robots and molecules

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Classical formulationUsing a constraints graph
plataform
base
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Matrix equationsdeducible from the constraints
graph
The closure equations that correspond to a cycle
basis yield a system with trigonometric terms
whose solutions are the valid configurations of
the mechanism
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Algebraizationof the equations
Each scalar equation involves trigonometric terms
f(sin ?i, cos ?i, ..., di) 0
But the system can be algebraized using
The tangent-of-the-half angle (?i / 2)
substitution, or
The change of variables xi sin ?i yi
cos ?i xi2 yi2 1
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Polynomial system
In any case, we end up with a system of
polynomial equations that must be solved
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Classical methods
Elimination-based
Homotopy-based
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Classical methodsShortcomings
Elimination-based
Require the solution of a high degree polynomial.
Need special data structures to represent reals
with many significant digits
Homotopy-based
Large systems of complex initial value problems
must be solved
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But
Not only a good resolution technique is needed
We seek a good combination of algebraization
resolution technique
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Outline

• The problem
• Classical approaches in Robotics
• Formulation with multilinear constraints
• Solving systems of multilinear constraints
• Some solved robots and molecules

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How can we obtain a system of multilinear
equations?
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First stepTranslate the mechanism into an
equivalent bar-and-joint framework
Rigid bars
Spherical joints
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Translating rotational pairs
One-to-one correspondence of configurations
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Translating a 6R loop(Must be solved in the
inverse kinematics of a 6R robot)
6R Loop
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Advantages of the translation
The mobility of the mechanism can be deduced
using Rigidity Theory
The valid configurations of a bar-and-joint
framework satisfy a system of multilinear
equations (thus, without trigonometric terms)
The multilinearity allows applying a simple
branch-and-prune algorithm.
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Second stepUse Cayley-Menger equations
Cayley-Menger determinant of four points
0 r12 r13 r14 1
p4
r12 0 r23 r24 1
r13 r23 0 r34 1
p1
p3
r14 r24 r34 0 1
p2
1 1 1 1 0
rij squared distance from pi to pj
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Second stepUse Cayley-Menger equations
Of three points
0 r12 r13 1
r12 0 r23 1
16 A2
r13 r23 0 1
1 1 1 0
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Second stepUse Cayley-Menger equations
Conditions that the distances between n points
must satisfy so that these points are embeddable
in R3
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Outline

• The problem
• Classical approaches in Robotics
• Formulation with multilinear constraints
• Solving systems of multilinear constraints
• Some solved robots and molecules

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Algorithm prunning bisection
Prunning
Bisection
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Prunning
Thanks to the convex hull property of
multilinear functions
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Outline

• The problem
• Classical approaches in Robotics
• Formulation with multilinear constraints
• Solving systems of multilinear constraints
• Some solved robots and molecules

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Example 1Octahedral manipulator
Direct kinematics
Find all platform poses, given the lengths of
the actuated lengths
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Octahedral manipulator In a non-singular
configuration
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Stewart-Gough platform In a singular
configuration
Jacobian, almost null
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Octahedral cyclohexane
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Octahedral cyclohexane
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Octahedral cyclohexane
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Octahedral cyclohexane
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Octahedral cyclohexane
C
C
C
C
C
C
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Cyclohexane
50 miliseconds
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Whole conformational space of theCycloheptane
28 equations
25 inequations
10 minutes
7 variables
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One slice of theCyclooctane
84 equations
69 inequations
1'5 hours
11 variables
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Conformational space of the cycloctaneSeveral
slices of the 2-dim variety
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Future work
To develop algorithms able to solve problems
involving many degrees of freedom in reasonable
times To exectue the algorithms in parallel
computers or inclusters of computers