Solving the forward kinematics of parallel manipulators is a hard problem since it involves finding the right implicit kinematics modeling to solve the problem, either displacement based equations or position based equations. Then, a review of the various solving techniques can lead to one, some or all the results. This presentation will also show limitations of each methods.
Presentation solving fk pparallelrobots mun rose 2014
1. SuivantPrécédent
RoSe, March 14, 2014
Solving the forward kinematics of
parallel robots, a review of
available methods
Memorial University of Newfounland
Solving the forward kinematics of
parallel robots, a review of
available methods
Memorial University of Newfounland
Memorial University
2. SuivantPrécédent
Introduction
Kinematics formulation
Forward Kinematics Problem
Solving the system
Results and Analysis
Summary
Introduction
Kinematics formulation
Forward Kinematics Problem
Solving the system
Results and Analysis
Summary
Outline
Memorial University
RoSe, March 14, 2014
3. SuivantPrécédent
The truly parallel manipulator
- Gough platform (Stewart platform)
- one fixed base
- one mobile plateform
- 6 kinematics chains
Each kinematics chain
- with one prismatic actuator
- through universal or ball joints
The truly parallel manipulator
- Gough platform (Stewart platform)
- one fixed base
- one mobile plateform
- 6 kinematics chains
Each kinematics chain
- with one prismatic actuator
- through universal or ball joints
Introduction
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Pros
- More rigid
- Less massive
- Larger accelerations
- Larger payloads
Cons
- Limited workspace
- Non-linear modeling
- Difficult control
Pros
- More rigid
- Less massive
- Larger accelerations
- Larger payloads
Cons
- Limited workspace
- Non-linear modeling
- Difficult control
Introduction
Memorial University
RoSe, March 14, 2014
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Inverse kinematics problem
DEF : Given the generalized coordinates of the manipulator end-
effector X, find the joint positions L.
Explicit solution.
Real solution # = 2
Inverse kinematics problem
DEF : Given the generalized coordinates of the manipulator end-
effector X, find the joint positions L.
Explicit solution.
Real solution # = 2
Introduction
Memorial University
RoSe, March 14, 2014
6. SuivantPrécédent
Forward kinematics problem
DEF : Given the joint positions L, find the generalized coordinates
X of the manipulator end-effector.
a difficult problem (Roth)
Proven:
40 complex solutions (Lazard)
Real solution # ≤ complex #
Forward kinematics problem
DEF : Given the joint positions L, find the generalized coordinates
X of the manipulator end-effector.
a difficult problem (Roth)
Proven:
40 complex solutions (Lazard)
Real solution # ≤ complex #
Introduction
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• The kinematics model variables
• L – joint variables
• X - position and orientation
– End-effector generalized coordinates
• Principle : any rigid boby can be
positioned by 3 distinct points.
• The 3 platform distinct points:
– 3 joint centers B1, B2, B3.
• The 9 variables are set as :
Kinematics
Formulation
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RoSe, March 14, 2014
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• The kinematics model variables
• L – joint variables
• X - position and orientation
– End-effector generalized coordinates
Displacement based equation system
Principle : vectorial formulation distance
constraints and norm square
Kinematics
Formulation
Memorial University
RoSe, March 14, 2014
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Position based equations
• Nine variables :
– the first 3 mobile platform joints
– {x1,y1,z1,x2,y2,z2,x3,y3,z3}
• From the IKP
• The 3 first legs: norm between Ai
and Bi
• The 3 other legs:
– CB4, CB5 and CB6 are written in
terms of variables
– Norm between Ai and Bi
Kinematics
Formulation
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RoSe, March 14, 2014
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Displacement based equations
Models
• Trigo with translation and angles
(Dieudonne)
• Translation and three trigonometric
identity (Merlet)
• Translation and the tangent angle
variable change (Griffis & Duffy)
• Translation and the rotation matrix
• Translation and rotation Groebner bases
• Translation and quaternion
• Translation and dual quaternion
•
Forwards
Kinematics
Problem
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Position based equations
Models
• three point model with platform
dimensional constraints
• three point model with platform
constraints with pointing axis
• the three point model with constraints
and function recombination
• the six point model
Forwards
Kinematics
Problem
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• Transformation into an
optimization problem
– For optimization techniques
– One objective function
– Derived from the IKP
– Let lgi be the leg length of
kinematics chain i (input of
the problem).
– augmented by one constraint
– set : the platform fixed
distances between the three
selected joint points : B1;B2
and B3 distinct points
where
where
Middle East Technical University
Forwards
Kinematics
Problem
Memorial University
RoSe, March 14, 2014
13. SuivantPrécédent
Numeric Methods
• Secant Method -> one solution
• Newton method -> one solution
• Continuation method with
homothopy -> several solutions
• Dyallitic Elimination -> several
solutions
• Interval analysis -> all solutions or
no answer (certified)
• Geometric Iterative Method -> one
solution
Solving the
system
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Algebraic Methods
• Dyallitic Elimination -> several
solutions
• Resultants method -> several
solutions
• Groebner bases -> all exact
solutions (certified)
Solving the
system
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Optimization Techniques
• Genetic Algorithm -> several
solutions
• Simulated Annealing -> some
solutions
• Hybrid Genetic Algorithm and
Simulated Annealing -> all
solutions
• G3-PCX -> all solutions
Solving the
system
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RoSe, March 14, 2014
16. SuivantPrécédent
Newton's Method
- We have one solution
- Dieudonne in 1972
Observations
- Quadratic convergence
- Small computation times
- May not converge
- Jacobian inversion
- Numeric instabilities
Example
- Very fast method for control
- on singularity free SSM: 5% failures
- Needs convergence test as the Kantorovich theorem
Newton's Method
- We have one solution
- Dieudonne in 1972
Observations
- Quadratic convergence
- Small computation times
- May not converge
- Jacobian inversion
- Numeric instabilities
Example
- Very fast method for control
- on singularity free SSM: 5% failures
- Needs convergence test as the Kantorovich theorem
Solving
Methods
Memorial University
RoSe, March 14, 2014
17. SuivantPrécédent
Interval Analysis
- All solutions
- Merlet in 2005
Observations
- Quadratic convergence
- Long computation times
- May not converge
- Jacobian inversion
- Accounts for imprecision
Example
- Needs Newton's method
- On singularity free SSM: 5% failures
- Needs enclosure test as with the Kantorovich theorem
Interval Analysis
- All solutions
- Merlet in 2005
Observations
- Quadratic convergence
- Long computation times
- May not converge
- Jacobian inversion
- Accounts for imprecision
Example
- Needs Newton's method
- On singularity free SSM: 5% failures
- Needs enclosure test as with the Kantorovich theorem
Solving
Methods
Memorial University
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Continuation method with homothopy
- Raghavan in 1993
- We have solutions for a simple equation system F(X) = 0
- We wish solutions for similar G(X) = 0
- Continuation: H(X, λ) = G(X)+ λ (F(X)−G(X))
- λ {0,…, 1}∈
Observations
- May miss solutions
- May add solutions
- Crossing solutions
- Needs iterative method
Example
- Problem going from the SSM to the 6-6
Continuation method with homothopy
- Raghavan in 1993
- We have solutions for a simple equation system F(X) = 0
- We wish solutions for similar G(X) = 0
- Continuation: H(X, λ) = G(X)+ λ (F(X)−G(X))
- λ {0,…, 1}∈
Observations
- May miss solutions
- May add solutions
- Crossing solutions
- Needs iterative method
Example
- Problem going from the SSM to the 6-6
Solving
Methods
Memorial University
RoSe, March 14, 2014
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Dyallitic Elimination - Numeric
- Isolation to a univariate equation
- Husty in 1994
Observations
- Perhaps all solutions
- Complex solutions may become real solutions
- Spurious solutions are added
Example
- Simpler parallel robots: OK
- Problem: 40 solutions for the SSM
Dyallitic Elimination - Numeric
- Isolation to a univariate equation
- Husty in 1994
Observations
- Perhaps all solutions
- Complex solutions may become real solutions
- Spurious solutions are added
Example
- Simpler parallel robots: OK
- Problem: 40 solutions for the SSM
Solving
Methods
Memorial University
RoSe, March 14, 2014
20. SuivantPrécédent
Resultants - Algebraic
- Isolation to a univariate equation
- Husty in 1994
Observations
- Perhaps all solutions
- Spurious solutions are added
- Requires elimination step with IKP
Example
- Simpler parallel robots: OK
- Problem: 40 solutions for the SSM
Resultants - Algebraic
- Isolation to a univariate equation
- Husty in 1994
Observations
- Perhaps all solutions
- Spurious solutions are added
- Requires elimination step with IKP
Example
- Simpler parallel robots: OK
- Problem: 40 solutions for the SSM
Solving
Methods
Memorial University
RoSe, March 14, 2014
21. SuivantPrécédent
Resultants - Algebraic
- Solving for Res(f,g,x1) = 0 equivalent to det(M) = 0
- In certain instances, the head terms of the polynomials cancel
→ the cancellation of the determinant
→ it adds one extraneous root.
- The resultant method is equivalent to the dyalletic method
Resultants - Algebraic
- Solving for Res(f,g,x1) = 0 equivalent to det(M) = 0
- In certain instances, the head terms of the polynomials cancel
→ the cancellation of the determinant
→ it adds one extraneous root.
- The resultant method is equivalent to the dyalletic method
Solving
Methods
Memorial University
RoSe, March 14, 2014
22. SuivantPrécédent
Groebner Bases - Algebraic
- calculation of Groebner basis: canonical form of ideal
- conversion to a Rational Univeariate Representation
- Lazard, Faugere and Rouillier in 1996 – 2000 period
Observations
- All exact solutions
- Rational or integer coefficients
- Requires solving the Univariate equation
Example
- 36 solutions for the SSM
- 6-6 computation times: 1 min in Maple
Groebner Bases - Algebraic
- calculation of Groebner basis: canonical form of ideal
- conversion to a Rational Univeariate Representation
- Lazard, Faugere and Rouillier in 1996 – 2000 period
Observations
- All exact solutions
- Rational or integer coefficients
- Requires solving the Univariate equation
Example
- 36 solutions for the SSM
- 6-6 computation times: 1 min in Maple
Solving
Methods
Memorial University
RoSe, March 14, 2014
23. SuivantPrécédent
Genetic Algorithms
- We have one solution
- Boudreau in 1996
Observations
- Heuristic computation times
- May not converge
- Modeling issue
- Starting solution dependant
Example
- May find many solutions
through repeated trials
- Smaller robots
Genetic Algorithms
- We have one solution
- Boudreau in 1996
Observations
- Heuristic computation times
- May not converge
- Modeling issue
- Starting solution dependant
Example
- May find many solutions
through repeated trials
- Smaller robots
Solving
Methods
Memorial University
RoSe, March 14, 2014
24. SuivantPrécédent
Joint variables L := [1250;
1250; 1250; 1250; 1250;
1250]
Case with 16 real results
confirmed by algebraic
method
FKP ROOT CERTIFIED RESULTSCONFIGURATION TABLE
Middle East Technical University
IBM compatible PC with 1.74 GHz dual core processors with Linux
Results
Analysis
Memorial University
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27. SuivantPrécédent
• Success rates: SA is 52 %, others 100 %
• Solving: G3-PCX obtained all 16 solutions
• G3-PCX outperformed the others on all accounts
• Population size of 200 : better response times
Middle East Technical University
Optimization Techniques
Memorial University
Results
Analysis RoSe, March 14, 2014
28. SuivantPrécédent
• Success rates: SA is 52 %, others 100 %
• Solving: G3-PCX obtained all 16 solutions
• G3-PCX outperformed the others on all accounts
• Population size of 200 : better response times
Middle East Technical University
Optimization Techniques
Memorial University
Results
Analysis RoSe, March 14, 2014
31. SuivantPrécédent
Solving methods
• Newton’s method
– With Kantorovich
– Very fast calculations
• Interval Analysis
– Certified solutions
– But long computation times
• Algebraic methods (Groebner)
– All exact solutions
– But long computation times
– For checking purposes
G3-PCX Genetic Algorithm
– All solutions
– Not very precise
Memorial University
Summary RoSe, March 14, 2014
