Automatic Synthesis of a Planar Linkage Mechanism Yoon Young Kim Seoul National University 2 Our Goal: Automatic Mechanism Synthesis??
3 Research Motivation Arrow Model at Brandeis Univ. Space Robot at MIT Binary Manipulator at Johns Hopkins Binary Manipulator at SNU More Use of Robots or artificial life-forms Motion-generating mechanisms become also important. Optimization Methodology Can Be a Critical Design Tool. 4 Issues in Mechanism Design? ( x, y, z) θ 3 θ 2 [1] Forward Kinematics Analysis Input θ Output i motion? θ 1 [2] Inverse Kinematics Analysis Output( x, yz, ) Input motion? [3] Mechanism Synthesis [Given Input & Desired Output Motion Mechanism?] The Subject of This Talk
5 Related Research Optimal Shaping and Sizing of Mechanism Design Example 1 : Design problem of hydraulic shovel mechanism Fujita et al. (2000) Design variables: linkage shape, joint positions Objective: minimize maximum forces required by cylinders 6 Related Research Example 2 : Design problem of Fingertip Haptic Display Workspace analysis of human index finger FHD (U. of Washington, 2004) Design variables: linkage lengths, angles Objective: maximum force output Constraint: cover human finger s stochastic workspace
7 What if no good initial design is available? Many trials and errors Develop a new design method not requiring baseline designs, Automatic Design Synthesis Method 8 Our Goal: Automatic Mechanism Synthesis?
9 Research Direction? 0. Synthesize a desired mechanism without an Initial Layout. 1. To begin with, we will work with 2-D link mechanisms consisting of rigid links and revolute joints only.? Rigid Link Revolute Joint 2. We will set up Mechanism Synthesis as Optimization Problem, but how? 10 Optimization Setup? 0 Prescribed Actual ( rp rp ) Minimize f = max ( t) ( t) t t t f 2 P(t) Difficulties? 1) Kinematic equations are highly nonlinear. 2) The whole motion path should be traced. 3) Gradient-based Optimizer is always preferred for efficiency.
11 Design Domain Discretization Approach? Workspace discretization Joint 1. Revolute, 2. Rigid, 3. No joint Link 12 Simply consider all possible cases? Candidate Mechanism 1 Candidate Mechanism 2...... Revolute Joint Candidate Mechanism i Rigid Joint (Rigid Connection) Candidate Mechanism N.................. By GA? not so efficient, difficult for complex mechanisms Better to use gradient-based optimizers if possible
KEY: Need a Single Unified Model Representing All Possible Kinematic Configurations by Real-valued Variables. 13 Revolute J., Rigid Joint or Disconnected How about using the modeling technique used for Topology Optimization? Vary Link Density? does not make sense in rigid-body links Rigid or No Link Our Idea? 14 Model joints by means of springs with varying stiffness Joint
15 Discretized Mechanism Configuration Our Proposition!! Rotational Spring K R Translational Spring K T Design Variables k k k k k k min Max T T T min max R R R Price: Kinematic analysis Kinematic analysis and static force equilibrium analysis (or equations of motion) 16 This spring idea really works?
17 1. Check Open Link Case At the 1 st Optimization Iteration Desired Output Motion t=0 sec t=1 sec t=2 sec? At the final Optimization Iteration Input Motion Our Model Springs with varying stiffness Design Variables t=0 sec t=1 sec t=2 sec Flying links with weak springs 18 2. Check Closed Link Case? Output Motion Known Solution Our (Mechanism) Model Animation for Optimization History Input Motion PROBLEM: The Ranges of Relative stiffness Rates of rotational springs to translational springs are hard to adjust. Then what? Springs
19 New Modeling Ideas: Better not to Use Rotational Springs, but Only Translational Springs. But How? Replace Links with Rigid Blocks K K ij1 ij 2 Links are Now Modeled by Rigid Blocks, and No Rotational Springs are used. 20 This idea Can Be Useful? i.e., all possible link combinations (Revolute Joint, or Rigid Joint, or disconnected Joint ) by adjusting stiffness values? Varying Translational Stiffness K ε = = 1 min max k kij k
21 1) Revolute Joint (at Location 1) F F 1 2 1 2 kij = k max kij = k min (Overlapping does not interfere with actual motion.) 22 Interpretation: Rigid Block with Springs Link with Revolute Joints kij = k max kij = k min 1 2
23 2) Rigid Joint Interpretation kij = k max 24 3) No Connection (or Disconnected Joint) Interpretation kij = k min
Now Back to Optimization Formulation for Mechanism Synthesis 25 P(t) K K ij1 ij 2 0 Prescribed Actual ( rp rp ) Minimize f = max ( t) ( t) k N t t t min max with i, = 1,2,, f k k k i N Subject to the field equations 2 26 Equation of Motion or Force Equilibrium Equation MY +Φ λ = Q T Z (1) Since velocities and accelerations are zero for static equilibrium, the equation yields equilibrium equations as T Φ Zλ Q = 0 (1') Φ = 0 (2) Incremental equations for (1 ) and (2) can be written as T ( Φ λ) Z Q Z Φ Z Z Φ 0 T Z ΔZ R = Δλ Φ where, Φ R = T Zλ Q
27 Mechanism Synthesis : Case Study Open linkage Output Motion? o Motion 2:-20 Input Motion o Motion 1:30 Time Time Prescribed Actual ( r t r t ) max ( ) ( ) Objective: Minimize P P t0 t tf Number of Design Variables: 19(open), 8(closed) Closed linkage Output Motion Input Motion?o Motion :30 Time 2 28 Need for 2 nd -Generation Rigid Block Model We may need anchored revolute joints inside the configuration region to expand the design space. Then, what should we do? Desired output motion? Input motion
29 Let s Introduce Anchoring Springs m Connecting Springs Anchoring Springs 30 Check effectiveness of Anchoring Springs k anchoring m k connecting kpiling Case 1 kconnecting = 1, kpiling = 4 Case 2 10 k = 1, k = 10 k 4 connecting piling connecting
31 Now, Application Problem Definition Discretized Design Model Input Motion? o Motion :30 Time P(t) t0 t tf Output Motion (click!) Prescribed Actual ( rp t rp t ) Minimize max ( ) ( ) 2 Connecting Springs: Anchoring Springs: 32 Convergence History Objective Function Convergence History Design variables Maximum of Distance Stiffness Synthesized Mechanism
33 Alternative anchoring spring model: model 2 Model 2 Model 1 Connecting Springs anchoring Springs m 34 Linear motion mechanism Problem Definition Discretized Design Model Input Motion t0 t tf? o Motion :30 Time Output Motion (click) Target Actual ( rp t rp t ) Minimize max ( ) ( ) Number of Design Variables: 59 2 Our Modeling
35 Convergence History Objective Function Design variables Synthesized Mechanism 36 Door-opening mechanism design test Problem Definition Input Motion o Motion :30 Time A B * B *? Output Motion (click) A Discretized Design Model Connecting Springs: y y x x A * * A B B Minimize max r *( t) + r *( t) t0 t t f AA BB x x y y * * A A B B anchoring Springs:
37 Convergence History Objective Function Design variables Synthesized Mechanism 38 Conclusions Automatic Synthesis of Rigid-link and Revolute Joint Mechanism Was Investigated. A Synthesis Model Consisting of Rigid Blocks and Springs Is Proposed. Some Successful Test Results Were Obtained. K K ij1 ij 2 Remark: still a long way to go, it is just a beginning.
39 Based on : Y.Y.Kim et al, Automatic Synthesis of a Planar Linkage Mechanism with Revolute Joints by Using Spring- Connected Rigid Block Models, ASME J. Mech. Design, Accepted (2006) 40 Mechanism synthesis using ADAMS Step 1: Discretize by Block links and dummy links Point with desired motion Green links: dummy links Dummy motion
41 Connecting Rigid-body block by springs? Use Bushing Elements K K ij1 ij 2 42 Bushing Elements Connecting Blocks by bushing element K K ij1 ij 2 Point with desired motion Block motion
Step 2: Bushing stiffness varies as a function of a design variable 43 Create stiffness variables K K ij1 ij 2 Create auxiliary design variables Make bushing stiffness vary as a ftn. of a design variable 44 Mechanism synthesis using ADAMS Step 3: Define objective function Objective function name Marker number of output point of Block link Marker number of output point of dummy link * DM: Distance between output point of block link and dummy link
45 Mechanism synthesis using ADAMS Step 4: Set and solve optimization problem