Information about Presentation solving fk pparallelrobots mun rose 2014

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.

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

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 Memorial University RoSe, March 14, 2014

SuivantPrécédent 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

SuivantPrécédent 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

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 Memorial University RoSe, March 14, 2014

SuivantPrécédent • 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 Memorial University RoSe, March 14, 2014

SuivantPrécédent • 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

SuivantPrécédent 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 Memorial University RoSe, March 14, 2014

SuivantPrécédent 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 Memorial University RoSe, March 14, 2014

SuivantPrécédent 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 Memorial University RoSe, March 14, 2014

SuivantPrécédent • 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

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 Memorial University RoSe, March 14, 2014

SuivantPrécédent Algebraic Methods • Dyallitic Elimination -> several solutions • Resultants method -> several solutions • Groebner bases -> all exact solutions (certified) Solving the system Memorial University RoSe, March 14, 2014

SuivantPrécédent 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 Memorial University RoSe, March 14, 2014

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

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 RoSe, March 14, 2014

SuivantPrécédent 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

SuivantPrécédent 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

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

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

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

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

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 RoSe, March 14, 2014

SuivantPrécédent Results Analysis Groebner basis + Rational Univariate Representation Memorial University RoSe, March 14, 2014

SuivantPrécédent Results Analysis Groebner basis + Rational Univariate Representation Memorial University RoSe, March 14, 2014

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

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

SuivantPrécédent Assembly Modes Results Analysis Memorial University RoSe, March 14, 2014

SuivantPrécédent Assembly Modes Results Analysis Memorial University RoSe, March 14, 2014

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

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