Geometric Methods for Analysis and Control of Biomimetic Robotic Locomotion

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Geometric Methods for Analysis and Control of
Biomimetic Robotic Locomotion
Short Course, University of Verona August
25-29 Organized by Prof. Paolo Fiorini
Joel W. Burdick Mechanical Engineering,
BioEngineering California Institute of Technology
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Goal of Introductory Lecture

• Biomimetic Locomotion what is it, why study it?
• Examples (videos)
• Key Issues
• Key Ideas (without much math)
• Motivate organization of remaining lectures

Have some fun!
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Movement is a key requirement for many autonomous
systems

• Traditional Design Approach
• Thrust device(s)
• Steering device(s)
• Limitations
• Inappropriate for some uses/terrains
• Often not maneuverable
• at low speeds,
• in tight spaces
• complex environments
• Inappropriate design

Global Hawk
Bomb/Swat Robot
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Look to Nature!
Biomimetic Locomotion robotic movement that
mimics natural patterns of movement.

• Advantages
• Move by shape deformation
• Nature gives clues
• Diversity of regimes

• Disadvantages
• Unknown models/principles
• Complex dynamics/control
• Diversity of models

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Hyper-Redundant Robotic Systems(Chrikjian and
Burdick, 1988-1994)

• snake-like robots
• niche applications inspection, medicine,
• algorithms for coordinating large DOF

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Robotic Gastroscope(System overview)

• Analogous to segmented worm
• Grippers provide traction
• Extensors cause displacement
• Many possible gaits

Generation 5 prototype
Testing robot in live pig intestine
Gait
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Tissue Elasticity
Quasi-static analysis
In vivo porcine intestine
Derived Constitutive relations
Deformation and stress from gripper inflation
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Results
Blood Flow Rate

Analytical Model
Experimental Verification
Blood volume flow rate (in3/sec) vs. Luminal
intestinal pressure (psi)
0
0.67
0
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Limitations in Biomimetic Locomotion Research

• Current approaches
• morphology restricted.
• unpredictive (not rigorous)
• robustness?
• scaling?
• Cost effective and robust biomimetic locomotion
  will require a more universal approach

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Non-Traditional Biomimetic Locomotors
Limitations are much more significant for
non-traditional biomimetic systems
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Home/Personal Robots

• Remote presence in your home
• eldercare
• simple security
• personal messages
• plaything, playmate

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Control Issues
q2
Controllability does there exist an control
action that takes vehicle from q1 to q2 ?
Trajectory Planning (open loop control) compute
control inputs, u, taking vehicle from q1 to q2
q1
Feedback control find u(q) to stabilize robot to
a point or trajectory in presence of disturbances

• Need a model!
• Compatible with geometric nonlinear control
  theory
• Doesnt need to capture all effects accurately

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The Snakeboard(a poster-boy for biomimetic
locomotion!)

• Controllable
• move in any direction?
• Trajectory generation
• how to wiggle from A to B?
• Control
• get back on track

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Carangiform/Thunniform swimming(R. Mason, K.
Morgansen, S. Kelly, R. Murray)
Some of the most impressive aquatic swimmers
Involves complex vorticity generation control
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Schematic of Experiment(R. Mason, K. Morgansen)
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Fish Fin Motor Project
Problem Personal water-craft manufacturers are
facing an environmental regulation
crisis. Solution? move away from motor/prop
paradigm.
Long term agenda Sensor-guided, fin-assisted
control for personal watercraft
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Minimalist Legged Locomotion(Goodwine, Kajita,
Vela)

• Characteristics
• 2 DOF planar legs
• capable of general motion

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The Ideal State
Properties of a comprehensive engineering
framework

• not constrained to a given morphology
• doesnt depend on DOF
• handles diverse environments
• can be implemented in software tools
• rigorous enough to predict and enable robust
  performance

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Investigative Life Cycle (hard problems)
Typical birth and maturation cycle
End result new set of tools/products
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How to we get there?

• Systematic Mathematical Study
• Look at similarities, not differences
• Take a control theoretic perspective
• develop canonical forms of equations of motion
• develop a control theory for these equation
• derive trajectory generation from theory
• derive feedback algorithms from theory
• Not done in isolation (larger context for theory)

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Parallel Developments

• Biomechanical Modeling ½ Theoretical Mechanics
• Geometrical picture of non-holonomic systems is
  recent advance.
• Geometrical Mechanics picture of fluids still
  emerging.
• Biomimetic Locomotion ½ Nonlinear Control
• Nonlinear feedback control of under-actuated
  nonlinear systems is in its infancy.
• Feedback control of discontinuous systems?

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Where are we?

• Canonical dynamical equations
• good progress toward unifying many locomotion
  classes. Fluid mechanical dynamic legged
  systems still require significant work.
• Controllability theory
• Many cases worked out
• Uniform trajectory generation algorithms
• Probably, a paradigm for kinematic systems has
  emerged
• Uniform feedback control techniques
• Probably. Averaging paradigms seem promising
• Some systems (i.e. fluid systems) will require
  substantially new results.

Bloch, Bullo, Burdick, Goodwine, Kelly, Koiller,
Krishnaprasad, Leonard, Marsden, Mason,
Morgansen, Murray, Ostrowski, Radford, Tsakiris,
Vela, .
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A Brief Incomplete History

• Early efforts (modelling mechanics)
• Wilczek Shapere (89), Koiller (92)
  micro-swimmers
• Krishnaprasad Tsakris (94), Kelly Murray
  (95) robotics
• Lewis, Ostrowski, Burdick, Murray (94) the
  snakeboard
• Bloch, Krishnaprasad, Marsden, Murray (96)
• nonholonomic systems with symmetry
• Expansion of basic paradigm, analysis of control
  issues, .
• Bloch, Bullo, Burdick, Goodwine, Kelly,
  Krishnaprasad, Leonard, Marsden, Mason, Murray,
  Ostrowski, Radford, Tsakiris, Zefran, Zenkov.
• Kelly Murray (95), Ostrowski Burdick (97)
  controllability
• Zenkov, Bloch, Marsden (97) stability
• Goodwine Burdick (97-01) legged locomotion
• Kelly Murray (96-98) simplified fluid
  locomotion

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Principles of Biomimetic Locomotion
Generated by coupling of periodic body
deformations to external constraints. Reaction
forces generate net movement
Non-holonomic constraints
Creeping undulatory
sidewinding
Piecewise holonomic constraints
Viscous constraints
Fluid Mechanical constraints
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(conventional) Lagrangian Mechanics
System Lagrangian (kinetic potential energy)
velocity constraints
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Example Hilare robot
position
shape
Velocity Constraints
Decoupling!
Kinematic constraints determine motion!
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Example the Snakeboard
Simplified Snakeboard Geometry
Velocity Constraints
The kinematic constraints alone do not determine
motion
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Basic Geometric Principles
It is the interaction between shape change and
constraints that generates motion
shape
position
r2
Shape Space
Lie Group
The Configuration Space of all biomimetic
locomotors is a (trivial) principal fiber bundle
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Basic Geometric Principles
position
r2
shape
Q when body wiggles, how far does it
displace? A relation between shape change and
robot displacement is described by the geometric
phase of the constraint connection one-form
Hilare
Snakeboard
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Symmetries
r2
Left Action of G on Q (e.g., frame change)
Lagrangian (and constraints) invariant w.r.t.
action
Symmetries conservation laws
(Noether) (not necessarily in presence of
constraints).
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Symmetries Non-holonomic Constraints (Bloch,
Krishnaprasad, Marsden, Murray 96)
Hilare
Snakeboard
Needed for control analysis
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Extension to Other Biomimetic Propulsors

• Highly Viscous Constraints (Slug-like)
  Kelly Murray 1995
• Kinematics Snakes
  Ostrowski Burdick 1996
• Legged Systems (quasi-static)
  Goodwine Burdick 97-00
• Fluid Systems
• C-space QSE(3) x M x (fluid state)
• incompressible fluid

Lie Group

• SE(m) symmetry
• symmetries of
• fluid/body constraints
• flow at domain boundaries

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Propulsion in Idealized Fluids(a robot amoeba?)

• Microscopic (real) amoeba low Reynolds no.
  (creeping) flow
• Macroscopic (robot) amoeba well modelled by
  potential flow.

Fluid potential for surface with dk deformation
modes
Subject to boundary conditions
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Due to SE(m)-invariance
connection
where
Example
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Trajectory Generation
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Control/Controllability
Nonlinear affine control systems have the form
Control vector fields
drift
Let
The system is accessible if
(driftless STLC Accessible)
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Locomotive Gaits
A locomotive gait is a cyclic pattern of shape
change resulting in net vehicle movement.
drive
rotate
Slide (parallel park)
Closely related to geometric phase and
controllability
Kinematic case Kelly Murray 95 General Case
Ostrowski Burdick 96
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Accessible Motions
u1, u2
y
x
Control affine model
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Time Periodicity/Averaging
Time periodicity, like a symmetry, can be reduced
Original Function F(x,t)F(x,tT)
Averaged Function F(x,t)s0T F(x,t)

• Biomimetic Locomotors use gaits (periodic
  controls)
• Net behavior determined by averaged system
• Design controls for averaged system, but still
  can prove properties!

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Open Loop Behavior
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Perturbed Behavior
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Closed Loop Behavior
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Feedback control results
Open Loop
Feedback
Simulation
Experiment
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Snakeboard Feedback Control
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Legged Locomotion (and grasping)

• Want
• general methods that span morphologies
• compatible with geometric framework
• handles underactuated systems
• Approach
• stratified configuration spaces
• stratified control theory

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Stratified Configuration Spaces
Definition A set X is regularly stratified if
it can be decomposed into a finite union of
disjoint smooth manifolds, called strata
(satisfying the Whitney condition).
S0 denotes the c-space Highest codim strata is
bottom strata
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Controllability
Equations of Motion on SI are smooth
Discontinuities occur across strata
Gait ordered sequence of strata

• Gait Distribution Let
• If then
• Else, if then

Proposition If dim(Dn) dim(TxSB), the system
is small time locally gait controllable at x.
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Minimalist Hexapod Example (Tripod Gait)
Straight trajectory
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Hexapod (continued)
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Application to Manipulation
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The Kinematic Biped
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Summary

• Common Features
• Principal Bundle c-space
• Symmetries
• Constraints
• Periodic motions
• A Biomimetic control theory should synthesize
  appropriate mathematical tool, with the goal of
  simplicity, intuition, but also provable
  robustness.

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Tentative Outline

• Monday
• AM (1) Motivation and Overview
• PM (1) Review of Manifolds, Lie Groups, and Lie
  Algebras
• (2) Review of Lagrangian Mechanics
• Tuesday
• AM (1, 2) Reduction and Reduced Mechanical
  Systems (no constraints)
• PM (1) Reduction with constraints (2)
  Controllability
• Wednesday
• AM (1) Stratified Systems (2) Review of
  Averaging
• Thursday
• AM (1,2) Averaging Based Control
• AM (1) Fluid systems (2) New directions and
  open issues
• Friday
• Special Lecture on Neural Prosthetics

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Feedback Control and the Lateral Line Sensor

• Feedback control requires an estimator for
  fluid state
• Fish use a lateral line sensor
• Build artificial lateral line?
• JPL/CIT (F. Noca and C. Assad)
• CIT (Y.C. Tai)
• Work with S. Coombs (Parma Inst.) to make new
  lateral line measurements based on theoretical
  predictions.

Neuromast
Florida Gar Lateral Line
Xenopus 25 micron/sec !