Computational Design Synthesis of Passive Dynamic Systems Fritz - PowerPoint PPT Presentation

Computational Design Synthesis of Passive Dynamic Systems Fritz Stöckli Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory Department of Mechanical and Process Engineering ETH Zürich May 8 2019 Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 1

Robotic Systems Active Robotic Systems Passive Robotic Systems  Actuators and feedback control  No actuators and control  High task flexibility possible  No energy source necessary  Responsive to environment  Potential to save energy and  High robustness components Passive dynamic walking, Mcgeer, T., 1990, International Journal of Robotics Research https://www.bostondynamics.com/atlas Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 2

Automated Topological Synthesis in Robotics Active Dynamic Systems Kinematic Systems  Evolving topology and control  Not considering causes of motion (forces, masses, … do together not matter) Evolving Virtual Creatures, Karl Sims, 1994 Computational Design of Linkage-Based Characters, Bernhard Generative Representations for the Automated Design Thomaszewski, Stelian Coros, Damien Gauge, Vittorio Megaro, of Modular Physical Robots, G.Hornby, H.Lipson, 2003 Eitan Grinspun, Markus Gross This Research: Passive Dynamic Systems  Forces, masses, … are important  Do not draw energy from a source  No feedback control Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 3

CDS of Passive Dynamic Systems - Overview Robotic Task Graph Grammar Robot Topology Simulation-Driven Parametric Opt Multi-Body System Rule-Based Topology Opt Shape Embodiment 3D-Printing Prototype Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 4

Example Problem: Brachiating  Brachiating: The swinging locomotion Robotic Task of primates moving from one tree branch to the next. Graph Grammar Robot Topology Simulation-Driven Parametric Opt  Complex, bio-inspired models of Multi-Body System passive dynamic brachiating exist: Rule-Based Topology Opt Shape Embodiment 3D-Printing Prototype Prototype A five-link 2D brachiating ape model with life-like zero energy- cost motions, Mario Gomes, Andi L. Ruina, 2005 Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 5

Motivation for Complex Brachiating Topologies Single Pendulum More Complex Solutions  Simplest possible solution  Might require less space  Test for synthesis method Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 6

Simulation-Driven Parametric Optimization Robotic Task Graph Grammar Robot Topology Simulation-Driven Parametric Opt  Multibody simulation Multi-Body System  Arbitrary 2D systems with revolute joints  Closed kinematic chains possible Rule-Based Topology Opt  Parametric Optimization  Evaluation based on system trajectory Shape Embodiment 3D-Printing Prototype Prototype Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 7

Multi-Body Dynamics Mass matrix Equations of motion (set of ODEs) System coordinates Forces (gravity, springs, … ) Motion trajectories can be calculated using numeric integreation. Formulation works for open kinematic chains only. Closed kinematic chain Open kinematic chain Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 8

Multi-Body Dynamics with Closed Kinematic Chains Equations of motion for systems with closed kinematic chains (Differential-Algebraic System) Set of ODEs Set of algebraic Eqations Vector of Constraints (Same as in Lecture 3 “Kinematics of Mechanisms”) Vector of constraining forces Matrix of generalized force directions (How constraining forces act on system coordinates) Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 9

Multi-Body Dynamics with Closed Kinematic Chains Transform into set of ODEs by taking second derivative of Initial concitions: Solve for and Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 10

Numerical problems and Stabilization Numeric errors during integration can accumulate and break constraints Baumgarte Stabilization: Correct these errors during integration by replacing by (change accordingly) Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 11

Body Coordinate Representation (2D) φ 𝑗 𝑧 𝑗 For each body 𝑗 glbal coordinates 𝑦 𝑗 , 𝑧 𝑗 , φ 𝑗 Center of mass mass 𝑛 𝑗 and moment of inertia 𝐽 𝑗 𝑦 𝑗 = (𝑦 1 , 𝑧 1 , φ 1 , … , 𝑦 𝑂 , 𝑧 𝑂 , φ 𝑂 ) 𝑈 System coordinates: = 𝑒𝑗𝑏𝑕(𝑛 1 , 𝑛 1 , 𝐽 1 , … , 𝑛 𝑂 , 𝑛 𝑂 , 𝐽 𝑂 ) Mass matrix: = (0, − 𝑛 1 𝑕, 0 , … , 0, −𝑛 𝑂 𝑕, 0 , ) 𝑈 Forces (here gravity only): Vector of Constraints to model joints: Same as in Lecture 3 Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 12

Robot Topology Design Synthesis Robotic Task Graph Grammar  Robot topology represented by graph Robot Topology  Grammar rules used to automatically Simulation-Driven generate new systems Parametric Opt Multi-Body System Rule-Based Topology Opt Shape Embodiment 3D-Printing Prototype Prototype Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory Engineering Design + Computing Laboratory 13

Origins of Transformational Grammar Rules in Linguistics  A language is undefinable except for its grammar  proper ways to form valid statements  Generative Grammars  Noam Chomsky - 1956  Rules that collectively define a language of feasible states  A rule represents heuristic knowledge Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 14

Graph Grammar(I)  Graph rewriting system  Rules used to change graph  Application conditions: Where the rule can be applied  Application procedure: What it does to the graph  Rules represent heuristic knowledge Left hand side of rule: Right hand side of rule: Pattern to find in graph Replaces left hand side a a Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 15

a a Graph Grammar(II) Recognize left hand side of rule in graph a a a a a a Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 16

a a Graph Grammar(III) Choose where to apply rule a a a a a a Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 17

Example: Gear Box Design Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 18

Example: Low-pass filters rule #3 rule #1 rule #2 rule #1 rule #1 rule #1 rule #2 rule #3 rule #2 rule #3 Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 19

Design Rules for Passive Dynamic Systems (I) Graph Representation Simulation Multibody System 𝐶 𝐷,1 𝑚 3 𝑚 1 𝑚 3 𝐶 𝑀,1 𝐶 𝑆,1 𝑚 2 𝑚 2 Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 20

Design Rules for Passive Dynamic Systems (II) 𝑆𝑣𝑚𝑓 𝐵𝑒𝑒𝑀𝑆𝐶𝑝𝑒𝑧𝑈𝑝𝑀𝑆𝐶𝑝𝑒𝑧 Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 21

Symmetry for Brachiating (I) Symmetry  Is required for cyclic brachiating  Similar as in walking between left and right leg Symmetric Graph  Rules generate symmetric configurations only  Mirror symmetry Symmetric Multibody System  Rules generate symmetric geometries only Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 22

Symmetry for Brachiating (II) Arbitrary Parameterization 𝐶 𝑀,1 𝐶 𝑀,1 𝐶 𝑆,1  Problem: Symmetry breaks 𝐶 𝑆,1 Δ𝑦 𝑘2 Δ𝑦 𝑘2 Symmetry breaks Symmetric Parameterization  Symmetry is maintained when optimization variables are varied  This is included in the design rules Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 23

Evaluation Criteria Cyclic Locomotion  Number of successful swings  Difference in states and hand position after first and last swing Space Requirement  Lowest coordinate swept during the whole motion Complexity  Measured by the number of bodies Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 24

Synthesis and Optimization Parametric Optimization Topological Synthesis  Multi-objective burst algorithm  For each topology generated (burst length: 3, max  Multi-objective genetic iterations: 500) algorithm (pop size: 200, generations: 80) From Matlab toolbox  Highly non-linear, non-convex multi-modal optimization landscape Cyclic locomotion: 𝑚 3 blue: good performance red: poor performance 𝑚 2 Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 25

Intermediate Solutions after some Generations Fritz Stöckli, Prof. Dr. Kristina Shea Engineering Design + Computing Laboratory 26