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

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1 Fritz Stöckli, Prof. Dr. Kristina Shea

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

Engineering Design + Computing Laboratory

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2 Fritz Stöckli, Prof. Dr. Kristina Shea

Robotic Systems

Passive Robotic Systems

  • No actuators and control
  • No energy source necessary
  • Potential to save energy and

components

Engineering Design + Computing Laboratory

Passive dynamic walking, Mcgeer, T., 1990, International Journal of Robotics Research

Active Robotic Systems

  • Actuators and feedback control
  • High task flexibility possible
  • Responsive to environment
  • High robustness

https://www.bostondynamics.com/atlas

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3 Fritz Stöckli, Prof. Dr. Kristina Shea

Automated Topological Synthesis in Robotics

Active Dynamic Systems

  • Evolving topology and control

together

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Evolving Virtual Creatures, Karl Sims, 1994 Generative Representations for the Automated Design

  • f Modular Physical Robots, G.Hornby, H.Lipson, 2003

This Research: Passive Dynamic Systems

  • Forces, masses, … are important
  • Do not draw energy from a source
  • No feedback control

Kinematic Systems

  • Not considering causes of

motion (forces, masses, … do not matter)

Computational Design of Linkage-Based Characters, Bernhard Thomaszewski, Stelian Coros, Damien Gauge, Vittorio Megaro, Eitan Grinspun, Markus Gross

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4 Fritz Stöckli, Prof. Dr. Kristina Shea

CDS of Passive Dynamic Systems - Overview

Engineering Design + Computing Laboratory

Robotic Task Robot Topology Multi-Body System Shape Embodiment Prototype Graph Grammar Rule-Based Topology Opt 3D-Printing Simulation-Driven Parametric Opt

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5 Fritz Stöckli, Prof. Dr. Kristina Shea

Example Problem: Brachiating

Engineering Design + Computing Laboratory

Prototype Robotic Task Robot Topology Multi-Body System Graph Grammar Simulation-Driven Parametric Opt Shape Embodiment Prototype Rule-Based Topology Opt 3D-Printing

  • Brachiating: The swinging locomotion
  • f primates moving from one tree

branch to the next.

A five-link 2D brachiating ape model with life-like zero energy- cost motions, Mario Gomes, Andi L. Ruina, 2005

  • Complex, bio-inspired models of

passive dynamic brachiating exist:

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6 Fritz Stöckli, Prof. Dr. Kristina Shea

More Complex Solutions

  • Might require less space
  • Test for synthesis method

Motivation for Complex Brachiating Topologies

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Single Pendulum

  • Simplest possible solution
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7 Fritz Stöckli, Prof. Dr. Kristina Shea

Simulation-Driven Parametric Optimization

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Prototype

  • Multibody simulation
  • Arbitrary 2D systems with revolute joints
  • Closed kinematic chains possible
  • Parametric Optimization
  • Evaluation based on system trajectory

Robotic Task Robot Topology Multi-Body System Graph Grammar Simulation-Driven Parametric Opt Shape Embodiment Prototype Rule-Based Topology Opt 3D-Printing

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8 Fritz Stöckli, Prof. Dr. Kristina Shea

Multi-Body Dynamics

Engineering Design + Computing Laboratory

Equations of motion (set of ODEs) Mass matrix Motion trajectories can be calculated using numeric integreation. Formulation works for open kinematic chains only. System coordinates Forces (gravity, springs, … ) Open kinematic chain Closed kinematic chain

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9 Fritz Stöckli, Prof. Dr. Kristina Shea

Multi-Body Dynamics with Closed Kinematic Chains

Engineering Design + Computing Laboratory

Equations of motion for systems with closed kinematic chains (Differential-Algebraic System) Matrix of generalized force directions (How constraining forces act on system coordinates) Set of ODEs Vector of constraining forces Vector of Constraints (Same as in Lecture 3 “Kinematics of Mechanisms”) Set of algebraic Eqations

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10 Fritz Stöckli, Prof. Dr. Kristina Shea

Multi-Body Dynamics with Closed Kinematic Chains

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Transform into set of ODEs by taking second derivative of Initial concitions: Solve for and

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11 Fritz Stöckli, Prof. Dr. Kristina Shea

Numerical problems and Stabilization

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Numeric errors during integration can accumulate and break constraints Baumgarte Stabilization: Correct these errors during integration by replacing by (change accordingly)

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12 Fritz Stöckli, Prof. Dr. Kristina Shea

Body Coordinate Representation (2D)

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For each body 𝑗 glbal coordinates 𝑦𝑗, 𝑧𝑗, φ𝑗 mass 𝑛𝑗 and moment of inertia 𝐽𝑗 System coordinates: Forces (here gravity only):

𝑧𝑗 𝑦𝑗 φ𝑗 Center of mass

= (𝑦1, 𝑧1, φ1, … , 𝑦𝑂, 𝑧𝑂, φ𝑂)𝑈 = 𝑒𝑗𝑏𝑕(𝑛1, 𝑛1, 𝐽1, … , 𝑛𝑂, 𝑛𝑂, 𝐽𝑂) Mass matrix: = (0, − 𝑛1𝑕, 0, … ,0, −𝑛𝑂𝑕, 0,)𝑈 Vector of Constraints to model joints: Same as in Lecture 3

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13 Fritz Stöckli, Prof. Dr. Kristina Shea

Robot Topology Design Synthesis

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Prototype

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  • Robot topology represented by graph
  • Grammar rules used to automatically

generate new systems Robotic Task Robot Topology Multi-Body System Graph Grammar Simulation-Driven Parametric Opt Shape Embodiment Prototype Rule-Based Topology Opt 3D-Printing

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14 Fritz Stöckli, Prof. Dr. Kristina Shea

Origins of Transformational Grammar Rules in Linguistics

Engineering Design + Computing Laboratory

  • 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
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15 Fritz Stöckli, Prof. Dr. Kristina Shea

Graph Grammar(I)

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  • 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: Pattern to find in graph

a

Right hand side of rule: Replaces left hand side

a

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16 Fritz Stöckli, Prof. Dr. Kristina Shea

Graph Grammar(II)

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Recognize left hand side of rule in graph

a a a a a a a a

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17 Fritz Stöckli, Prof. Dr. Kristina Shea

Graph Grammar(III)

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Choose where to apply rule

a a a a a a a a

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Example: Gear Box Design

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19 Fritz Stöckli, Prof. Dr. Kristina Shea

Example: Low-pass filters

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rule #1 rule #3 rule #2 rule #1 rule #2 rule #1

rule #3 rule #1 rule #2

rule #3

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Design Rules for Passive Dynamic Systems (I)

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Graph Representation Multibody System Simulation

𝐶𝐷,1 𝐶𝑀,1 𝐶𝑆,1 𝑚1 𝑚2 𝑚3 𝑚2 𝑚3

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Design Rules for Passive Dynamic Systems (II)

Engineering Design + Computing Laboratory

𝑆𝑣𝑚𝑓 𝐵𝑒𝑒𝑀𝑆𝐶𝑝𝑒𝑧𝑈𝑝𝑀𝑆𝐶𝑝𝑒𝑧

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22 Fritz Stöckli, Prof. Dr. Kristina Shea

Symmetry for Brachiating (I)

Engineering Design + Computing Laboratory

Symmetric Graph

  • Rules generate symmetric

configurations only

  • Mirror symmetry

Symmetry

  • Is required for cyclic

brachiating

  • Similar as in walking between

left and right leg

Symmetric Multibody System

  • Rules generate symmetric

geometries only

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23 Fritz Stöckli, Prof. Dr. Kristina Shea

Symmetry for Brachiating (II)

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Symmetric Parameterization

  • Symmetry is maintained when
  • ptimization variables are varied
  • This is included in the design

rules

𝐶𝑀,1 𝐶𝑆,1 Δ𝑦𝑘2

Arbitrary Parameterization

  • Problem: Symmetry breaks

𝐶𝑀,1 𝐶𝑆,1 Δ𝑦𝑘2 Symmetry breaks

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24 Fritz Stöckli, Prof. Dr. Kristina Shea

Evaluation Criteria

Cyclic Locomotion

  • Number of successful

swings

Engineering Design + Computing Laboratory

  • Difference in states and

hand position after first and last swing

Complexity

  • Measured by the number
  • f bodies

Space Requirement

  • Lowest coordinate swept

during the whole motion

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Synthesis and Optimization

Parametric Optimization

  • For each topology generated
  • Multi-objective genetic

algorithm (pop size: 200, generations: 80) From Matlab toolbox

Engineering Design + Computing Laboratory

Topological Synthesis

  • Multi-objective burst algorithm

(burst length: 3, max iterations: 500)

  • Highly non-linear, non-convex

multi-modal optimization landscape

Cyclic locomotion: blue: good performance red: poor performance 𝑚2 𝑚3

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Intermediate Solutions after some Generations

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27 Fritz Stöckli, Prof. Dr. Kristina Shea

Results

Engineering Design + Computing Laboratory

Space requirement

  • f single pendulum

Ideal cyclic locomotion Number of bodies Evaluation Plot

  • Final populations of eight

different topologies

  • All do three successful swings
  • 3 Objectives
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Results

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Results

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More space required

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Results

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Less space required Less space required Less space required

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Shape Embodiment Design

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Prototype

  • Bodies defined by inertia properties only
  • Topology optimization needed to find

shapes of bodies Robotic Task Robot Topology Multi-Body System Graph Grammar Simulation-Driven Parametric Opt Shape Embodiment Prototype Rule-Based Topology Opt 3D-Printing

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Automated Shape Design for Multi-Body Systems

Engineering Design + Computing Laboratory

Multi-body system representation Joint Center of mass Dynamic properties: [Mass, moment of inertia, center of mass] Find shape

  • Matching dynamic properties
  • Connecting all elements
  • Avoiding collisions
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Assemble Geometric Primitives

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𝑆1 𝑆2 𝑦 𝑧

Ellipse

  • 5 variables: 𝑦, 𝑧, 𝑆1,𝑆2, 𝛽
  • Sometimes no solution

found

  • Compact

Three Circles

  • 6 variables: 𝑦, 𝑧, 𝑆1,𝑆2, 𝛽, 𝑚
  • Mass and Moment of

inertia change more independently

  • Uses more space

𝑆2 𝑦 𝑧 𝑆1 𝑚

Find shape with given dynamic properties

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Assemble Geometric Primitives

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Finite Element Design Space

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𝒏𝑈𝝇 = 𝑛0 𝒅𝑦𝑈 𝑛0 𝝇 = 𝒅𝑦𝑈𝝇 𝒏𝑈𝝇 = 𝑑𝑦0 𝑱𝑈𝝇 = 𝐽0 𝒅𝑦𝑈 𝑛0 𝝇 = 𝒅𝑧𝑈𝝇 𝒏𝑈𝝇 = 𝑑𝑧0 Feasibility/Satisfiability Problem: 0 ≤ 𝜍𝑛𝑗𝑜 ≤ 𝜍𝑓 ≤ 𝜍𝑛𝑏𝑦 𝜍𝑗 Density of element i / Material per 2D element i 𝑛0, 𝐽0, 𝑑𝑦0, 𝑑𝑧0 Desired dynamic properties 𝑩𝝇 = 𝒄

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Finite Element Design Space

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Finite Element Design Space

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Fabrication

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Prototype

  • Additive manufacturing works well for

complex shapes

  • Ball bearings for low friction

Robotic Task Robot Topology Multi-Body System Graph Grammar Simulation-Driven Parametric Opt Shape Embodiment Prototype Rule-Based Topology Opt 3D-Printing

Experiment and Optimization based Design of a Passive Walking Robot, Fabio Modica, 2016, EDAC master thesis

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Measure static/dynamic friction of bearing

Engineering Design + Computing Laboratory

Arduino Board Ball bearing Three pendulums with different dynamic properties Codewheel with 1024 counts per revolution Optical incremental encoder with two channel output

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Measurement results

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Fit Measurement to Simulation Model

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ሷ 𝜒𝐽 = −𝑛𝑕𝑚𝑡𝑗𝑜 𝜒 − 𝑒𝐺 ሶ 𝜒 Friction Torque vellocity proportional (viscous friction): Standard in robotics, good properties of ODE and control problem

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Fit Measurement to Simulation Model

Engineering Design + Computing Laboratory

ሷ 𝜒𝐽 = −𝑛𝑕𝑚𝑡𝑗𝑜 𝜒 − 𝑈𝐺 Combination of viscous friction and coulomb friction

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3D-Printed Bearing

Engineering Design + Computing Laboratory

  • Planetary gear bearing

with clearance adapted to our FDM machine

  • Very robust gait
  • Passive walker built

using FDM parts only

  • Printed in one job
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44 Fritz Stöckli, Prof. Dr. Kristina Shea

CDS of Passive Dynamic Systems - Overview

Engineering Design + Computing Laboratory

Robotic Task Robot Topology Multi-Body System Shape Embodiment Prototype Graph Grammar Rule-Based Topology Opt 3D-Printing Simulation-Driven Parametric Opt

Problem specific rules generate topologies preserving symmetry Evaluation based on system trajectories Topology optimization avoiding collision for given trajectories

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Future Work

  • Evaluation
  • Sensitivity analysis
  • Additional joint types, friction, springs, …
  • Other robotic tasks
  • Prototyping
  • Synthesis and optimization
  • Different strategies

Engineering Design + Computing Laboratory