EFFICIENT KINEMATIC SYNTHESIS OF MULTIBODY SYSTEMS. R. Sancibrian Dpt. of Structural and Mechanical Engineering University of Cantabria Spain

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1 EFFICIENT KINEMATIC SYNTHESIS OF MULTIBODY SYSTEMS R. Sancibrian Dpt. of Structural and Mechanical Engineering University of Cantabria Spain 1

2 Overview Design of complex mechanical systems Commercial software available Optimal synthesis in mechanical design Problem statement Optimization methods Types of problems encountered Deterministic approaches Hybrid algorithms Design examples 2

3 Design of complex mechanical system Automotive suspension and steering systems Well Known CAE methods Requirements Design rules (Well Known?) Construction machinery Analysis Synthesis Aircraft landing gear Parallel manipulators Design Flaps It is necessary to invent the system before to analyse it Antenna pointing mechanisms 3

4 Commercial Software available ANALYSIS Closed-source packages - Adams - LMS DADS - Working model - SIMPACK - DynaFlexPro - Neweul - SAMCEF Mecano - RecurDyn - etc. SYNTHESIS - SyMec - LINCAGES - Synthetica (Univ. of California, Irvine) - SAM 4

5 Optimal Synthesis in Mechanical Design Optimal synthesis involves the minimization of the objective function according to a set of equations given geometric and functional constraints. Objective: Path generation / function generation / rigid-body guidance Variables: dependent coordinates / design variables Constraints: assembling the mechanism / other functional constraints x d y d Desired path Generated path z 1 x P y P θ 4 δ i δ o L 5 α L 3 L 2 θ 3 θ 5 L 4 θ 1 x 0 y 0 L w T θ 2 x L θ 1 20 θ 1 -π q T 3 4 x4 y4 0 y0 L 1 20 L1 L2 L3 L4 5 Turn centre Direction 5

6 Problem statement x d y d x P y P Desired path Generated path θ 4 The objective function 1 T F qw ( ), w ( ), ( ), 2 g qw w d g qw w d L 5 α L 3 θ θ 5 3 L 4 Assembling constraints L 2 θ 2 q L θ 1 20 θ 1 -π T 3 4 x4 y4 θ 1 x 0 y 0 i i L1cos1 L2cos 20 2 L3cos3 L3cos3 L1sin1 L2sin 20 2i L3sin3 L3sin3 Φ qw, w 0 i xg x0 L1cos1 L2cos 20 2 L5cos 5 yg y0 L1sin1 L2sin 20 2i L5sin 5 d p-1 w T x 0 y0 L 1 20 L1 L2 L3 L4 5 g 4 g 3 g 2 minimize F subject to Φ q w, w 0 and gqw ( ), w 0 qw ( ), w d p d 1 d 2 d 3 Desired path d 4 g p-1 g p Generated path g 1 Constrained non-linear least-square optimization 6

7 Problem statement 7

8 Optimization methods Optimization methods Deterministic Hybrid Stochastic Direct Search Gradient-based Genetic Algorihms Evolutionary Algorithms Constrained Unconstrained - Reasonable efficiency - Accurate - Local search - Depend on the initial guess - Global search - Independent of the initial conditions - Low efficiency and accuracy 8

9 Types of problems encountered g 4 g 3 d p-1 d p d 4 d 1 d d 2 3 g p-1 g p Estimation of the synthesis error g 1 g 2 F p 1 T i i i i 1 T g d g d gd gd 2 2 i 1 Open conf. = 0 Singularities Branch defect Closed conf. 0 Non-assembly problem 9

10 Extensible-link optimization Desired path Desired path Initialization m 6, k 6 l 6, k 6 m 5, k 5 it1;i1 l 5, k 5 l 3, k 3 l 4, k 4 m 2, k 2 m 3, k 3 m 4, k 4 1 min 2 l 2, k 2 l 12, k 12 l 22, k 22 l 11, k11 m 22, k 22 m 12, k 12 m 11, k 11 ii1 itit1 l 21, k 21 m 21, k 21 Main advantages: No assembling constraints. No singularities during the optimization process. Main disadvantages: Lack of accuracy working with complex paths. Assembling problems min 1 2.? Yes End No 10

11 Exact-constraints optimization Desired path x d y d x P y P Generated path θ 4 GRG (Generalized Reduced Gradient) L 5 L 2 θ 2 w T α L 3 θ 3 θ 5 θ 20 L 1 x q T θ 1 -π 3 4 x4 y4 L 4 θ 1 x 0 y 0 0 y0 L 1 20 L1 L2 L3 L4 5 Main advantages: The assembling constraints are always fulfilled. Accuracy in the solution 1 T F qw ( ), w ( ), ( ), 2 g qw w d g qw w d ( ), T ( ), F qw w J g qw w d 0,, gq w w gq w w q w J w q w w Φ q Φ 0 q w w 1 g Φ Φ J q q w Main disadvantages: Singularities appear during the optimization. T 1 T j1 j w w J J J g d 11

12 Formulation of the goal g 4 function d p-1 d p d 1 d 2 d 4 d 3 Translation y g 4 g p-1 g 3 g p g 3 g 2 g 1 1 T F qw ( ), w ( ), ( ), 2 g qw w d g qw w d Fourier Descriptors Synthesis error TRS (Translation, Rotation and Scaling) d p-1 β d d p d 1 g p-1 g p d 2 β g d 3 Rotation g 1 g 2 d 4 x g 4 g 3 Only closed paths All kind of paths Reducing the error between the desired and generated values at the beginning of the iterations. This includes: 1) Translation 2) Rotation 3) Scaling and renumbering the precision points d p-1 d p d 1 d 2 d 3 d 4 Scaling g p-1 g p g 1 g 2 pj1 p ij 2 1 i ijp 1 i g Ag0 d 0 Ag0 d0 Fm, s s s ; j 12,,..., p i1 ipj2 2 12

13 TRS Process Translation Rotation Scaling + renumbering positions Optimization 13

14 Avoiding singularities vs. jumping singularities Avoiding singularities Jumping singularities -e> 0 Pseudoinverse The space of the design variables is insufficiently explored The space of the design variables is insufficiently explored > 0 = 0 > 0 = unexplored region 14

15 Jumping sigularities 15

16 Hybrid algorithm Questions to be answered: 1. When? 2. Who? 3. What? 4. How long? 5. How should individuals be treated after a local search? Local Search (LS) Deterministic Hybrid Evolutionary Algorithm (EA) Stochastic 16

17 Hybrid Algorithm 1 (HA1) Stochastic (EA) Initial population EA operators gen=gen+1 Offspring and fitness evaluation E Selection No Conv.? Yes Local Search Optimal Solution 17

18 Hybrid Algorithm 2 (HA2) Stochastic (EA) Initial population EA operators Deterministic (LS) Size of the population: 50 gen=gen+1 E Offspring TRS Fitness evaluation and Selection No Conv.? Yes Local Search Optimal Solution 18

19 Hybrid Algorithm 3 (HA3) Stochastic (EA) Initial population Size of the population: 50 EA operators Deterministic (LS) gen=gen+1 E Offspring TRS Fitness evaluation and Selection Local Search No Conv.? Yes Optimal Solution 19

20 Hybrid Algorithm 4 (HA4) Stochastic (EA) Initial population EA operators Deterministic (LS) TRS gen=gen+1 E Offspring Fitness evaluation and Selection No Elite? Yes No Conv.? Yes Local Search Optimal Solution 20

21 Results EA HA1 HA2 HA3 HA4

22 Optimal design of a suspension system The objective of suspension systems in road vehicles is to provide comfort and handling. Thus, kinematic parameters must be established defining the relative motion between wheel and chassis. The problem can be tackled using dimensional synthesis techniques. This work was awarded by the American Society of American Engineers (ASME) 22

23 Functional parameters Functional parameters provide the adequate behavior of the suspension system. Camber angle: is the angle between the wheel center plane and the plane vertical to the road. Toe-in angle: the angle between the longitudinal center plane in the vehicle, vertical to the road, and the line intersecting the wheel center plane with the road. Tread width alteration: the lateral displacement of the wheel contact point. w T d d d d y 23

24 Kinematic models e 20 O 2 Upper control arms O 4 l 22 Upper control arm A 2 l 23 O 3 Tie-rod O 2 A 2 A 4 O3 Tie-rod l 33 Steering knuckle l 12 α 20 l 20 l 30 A 3 A 3 Z A 0 l 10 e 1 A 4 α O 10 Y Steering knuckle 1 Y e 10 l l O X 1 A 5 A X 1 A 1 Wheel center plane Lower control arm Wheel center plane Lower control arms P Z O 5 A 0 e 1 A 6 P Double-Wishbone Multilink 24

25 Optimization 25

26 Multilink 26

27 Application to a mechanism for injection machine Injection system Evolutionary and hybrid algorithms were used in the design of die-cast injection mechanism (zamak alloy) θ 3 θ 4 L 3 L 4 L 2 θ 2 α L 5 L 1 θ 1 L 6 x 0 y 0 θ 10 θ 6 x 5 y 5 Initially rapid motion of the slider is necessary, then a slower motion, and finally a fast backward motion when the mould has been filled 27

28 Application to a mechanism for injection machine Injection mechanism Electric motor Desired y d (mm) EA solution y g (mm) Hybrid solution y s (mm) Slider position Desired EA solution Hybrid solution Input link position 28

29 Design of surgical instruments Redesign of laparoscopic instruments optimization techniques. Surgeons in laparoscopic surgery have to bear hard conditions of work. The instruments cause them injuries in hand and arms. So we redesign the surgical instruments to improve their ergonomic characteristics. 29

30 Handle design 30

31 Articulated grasper design 31

32 Prototype 32

33 Thank you for your attention 33