Biomimetic control with a feedback coupled nonlinear oscillator Animal experiments, design tools, an

1
Biomimetic control with a feedback coupled
nonlinear oscillatorAnimal experiments, design
tools, and robot adaptation results

• Sean Bailey
• University oral examination
• June 9, 2004
• Stanford University

2
Whats it all about?

• Rhythm (cyclic tasks)
• Night and day
• Music
• Synchrony (coupling)
• Sleep wake cycle
• Dancing
• Locomotion?

3
Driving questions

• Source of rhythm and synchrony?
• Physical dynamics
• Neural dynamics
• Sensory coupling
• Functional role of coupling?
• Coordination
• Adaptation?
• General design methodology?
• Adaptive controllers for performing cyclic tasks
• Not ad hoc
• Intuitive design and analysis tools

4
Approach

• Not just biologically-inspired, but coupled
• Biology inspires engineering/robotics design
• Engineering/robotics motivates biological research

5
Specific questions

• Sensor-based behavior
• How does sensory information affect high speed
  locomotion?
• Biomimetic adaptation
• Use this new information to develop a general,
  adaptive control structureand design
  methodology?
• Implementation
• Design an adaptive controller to improve the
  running of a biomimetic hexapod?

6
Contributions

• Animal experiments
• Developed novel experimental setup for exploring
  neural function during natural behavior
• Demonstrated the use of sensory information
  during high speed running in the cockroach
• Supported model of feedback coupled nonlinear
  oscillator
• General, visual design method for cyclic dynamic
  task controllers
• Proposed explicit adaptive function for
  biomimetic control structure
• Developed several novel general design and
  analysis tools
• Adaptive controller implementation
• Designed adaptive controller for running on
  slopes for a biomimetic hexapod
• Developed a novel visualization tool for a class
  of these controllers
• Improved uphill running performance by 33

7
Outline

• Biomimetic robots and the need for adaptation
• Biological cyclic behaviors nonlinear
  oscillators
• Animal experiments
• Use of sensory feedback
• Support for a model
• Visual design method
• Develop new design and analysis tools for
  adaptation
• Biomimetic hexapod implementation and results
• Design an adaptive controller
• Discuss performance
• Conclusions and future work

8
Outline

• Biomimetic robots and the need for adaptation
• Biological cyclic behaviors nonlinear
  oscillators
• Animal experiments
• Use of sensory feedback
• Support for a model
• Visual design method
• Develop new design and analysis tools for
  adaptation
• Biomimetic hexapod implementation and results
• Design an adaptive controller
• Discuss performance
• Conclusions and future work

9
Biomimetic robotics

• Biomimetic design principles
• Geometry
• Passive visco elastic elements
• Result
• Fast, self stabilizing running
• Obstacle traversal
• Control
• Rhythmic motor patterns
• No feedback
• Performance
• Hand-tuned for flat ground
• What about the unknown?

50 bodylengths/second 3X hip height obstacles
2 bodylengths/second 1X hip height obstacles
10
The need for adaptation

• Performance degrades when conditions change
• Not just suboptimal, but unaware
• No chance of adaptation
• No connection to dynamics
• Slope running example
• Optimal frequency for flat ground
• Higher frequency for uphill
• Lower frequency for downhill

Optimal flat ground frequency
Desired adaptive behavior Frequency changes with
slope
11
Outline

• Biomimetic robots and the need for adaptation
• Biological cyclic behaviors nonlinear
  oscillators
• Animal experiments
• Use of sensory feedback
• Support for a model
• Visual design method
• Develop new design and analysis tools for
  adaptation
• Biomimetic hexapod implementation and results
• Design an adaptive controller
• Discuss performance
• Conclusions and future work

12
Studies on rhythmic behaviors in animals
Motor PatternGenerators

• Robotics perspective How are these patterns
  generated?
• Only sensory-based?
• Natural dynamics? (McGeer 1990)
• Grillner (1979)
• Animal with all sensors removed can generate
  complex motor patterns
• Cohen (1999)
• Regulated by sensory information
• Adaptive?
• van der Pol (1928)
• Studied heartbeat arrhythmias
• Significant early work modeling cyclic biological
  behaviors with nonlinear oscillators

Passive Dynamic Self-Stabilization
Feedforward Motor Pattern
Sensory Feedback
Mechanical System (muscles, limbs)
Mechanical Feedback
Environment
Touch,
Touch,
Vision
Vision
Odor
Odor
Wind
Wind
Contact
Contact
Strain
Strain
Visco
-
elastic
Response
Locomotion
13
Nonlinear oscillator properties

• Nonlinear systems
• Difficult to work with
• Superposition does not apply
• Fundamentally nonlinear
• Self sustained, stable limit cycles
• Selective frequency entrainment

Interesting behavior is a result of
nonlinearities
14
Generating motor patterns

• Nonlinear oscillator controller
• Cyclic motor signal output
• Sensory input
• Good for a controller because
• Operates open loop
• Sensory failure
• Sensory information can tune
• But only of the right frequency
• Biomimetic, not ad hoc
• Descriptive of general cyclic behaviors
• Hypothesized adaptive role

Promising as a general approach for adaptive
controllers for cyclic dynamic tasks
15
Outline

• Biomimetic robots and the need for adaptation
• Biological cyclic behaviors nonlinear
  oscillators
• Animal experiments
• Use of sensory feedback
• Support for a model
• Visual design method
• Develop new design and analysis tools for
  adaptation
• Biomimetic hexapod implementation and results
• Design an adaptive controller
• Discuss performance
• Conclusions and future work

16
Locomotion experiments overview

• Role of sensory feedback
• Motor patterns generated in isolation
• Recent work on insect locomotion
• Delays are significant
• Focused on lack of real-time feedback
• Passive mechanisms
• How are sensors used?
• At the highest speeds of locomotion
• Support nonlinear oscillator model
• Hypothesis of adaptation
• Novel
• Naturally behaving animal
• Controlled sustained perturbations
• High speed running

17
Locust studies

• Wendler (1972)
• Intact animals
• Input signal
• Sustained mechanical perturbation
• Output signal
• Electromyogram (EMG)
• Phase relationship
• Signals not independent
• Supports nonlinear oscillator model

Mechanical oscillation
EMG signal

A preferred phase
Experimental methodology
Phase
18
Neural model of experiments

• Nervous system is the controller
• Input
• Sustained cyclic mechanical perturbation (10
  seconds)
• Adds to existing feedback
• Output
• Electromyograms (EMG)

Only path between input and output is through the
nervous system
19
Why hasnt this been done?

• Running
• Apply sustained perturbations
• EMG implantation procedure
• Anatomy
• Time intensive
• Fragile
• Skill
• Data collection analysis
• Lots of programming engineering

Muscle 137
Carbonell, 1947
20
Experimental setup

• Cockroach
• Blaberus discoidalis
• Intact
• Naturally behaving (inertial treadmill)
• High speed running
• Input
• Force-displacement actuator
• Output
• EMG signal
• Real time operating system

21
Experimental setup

• Cockroach
• Blaberus discoidalis
• Intact
• Naturally behaving (inertial treadmill)
• High speed running
• Input
• Force-displacement actuator
• Output
• EMG signal
• Real time operating system

Force-displacement actuator
Cockroach Blaberus discoidalis
EMG electrodes (50mm silver wire)
b
Spherical inertial treadmill
22
Phase measure

• Input
• Fore aft perturbation
• 2 cycles per stride
• Peaks
• Output
• EMG signal
• Simple structure
• First spike of each stride
• Phase
• Normalized difference between characteristic
  features

23
Expected results

• Phase relationship
• Exists between related signals
• Control case
• No perturbation reference signal
• Expect no preferred phase
• Uniform distribution
• Perturbation case
• Expect a preferred phase
• Not uniform distribution

Control
Uniform
Perturbation
Not uniform
24
Trial data

• Each trial
• Sustained perturbation
• 10 seconds
• Many phase measures
• Visualization
• Individual phases
• Histogram
• Get mean phase
• Vector averaging
• Merge trials

Mean phase for trial
25
Results
Control
90

• Each animal
• Compare cases
• Control
• Perturbation
• Quantify results
• Rayleigh test for uniformity
• Not uniform? (plt0.05)
• Expect
• Control case uniform
• Perturbation not uniform

120
60
2
30
150
1
0
180
Rayleigh test N 9 p gt 0.05
210
330
Uniform
240
300
270
Perturbation
90
120
60
2
30
150
1
180
0
Rayleigh test N 6 p lt 0.05
210
330
300
Not uniform
240
270
26
Results

• Control case
• Expect uniform 8/9
• Perturbation case
• Expect not uniform 6/9
• Majority of animals
• 6/9 (plt0.01)
• 4/9 (plt0.001)
• Many uncontrolled factors
• Conclusions
• Sensory information influences high speed
  running
• Supports feedback coupled nonlinear oscillator
  model

27
Outline

• Biomimetic robots and the need for adaptation
• Biological cyclic behaviors nonlinear
  oscillators
• Animal experiments
• Use of sensory feedback
• Support for a model
• Visual design method
• Develop new design and analysis tools for
  adaptation
• Biomimetic hexapod implementation and results
• Design an adaptive controller
• Discuss performance
• Conclusions and future work

28
The biomimetic controller design

• Biomimetic control structure for cyclic dynamic
  systems
• Coupled dynamic system
• Open loop sensor failure operation
• Otherwise, uses sensory information
• Controller
• Input Sensory information
• Output Motor commands
• Plant (mechanical system)
• Input Motor commands
• Output Sensory information

Coupled system
Williamson, 1999
29
The Williamson method

• Method
• Considers plant and controller separately
• Predicts coupled system behavior
• Visual design by changing shapes
• Features
• Alternative to blind numerical integration
• Abstracts away from difficult nonlinear
  mathematics
• Intuitive design choices

Input
Output
Controller
Controller
(nonlinear oscillator)
(nonlinear oscillator)
Input
Output
Plant
Plant
(actuated mechanical system
(actuated mechanical system
interacting with an environment)
interacting with an environment)
Williamson, 1999
30
The Williamson method
Plant
Controller

• Frequency response
• Nonlinear Describing function analysis
• Nyquist plot
• Invert controller
• Steady state solution
• Plant 1/Controller
• Match gain, phase, and frequency

Gain
Gain
Frequency Response Plots (Bode)
Phase
Phase
Frequency
Frequency
s-plane Representation (Nyquist)
Imaginary
Imaginary
Real
Real
Imaginary
Plant 1/Controller
Invert
Real
31
The Williamson method
Plant
Controller

• Frequency response
• Nonlinear Describing function analysis
• Nyquist plot
• Invert controller
• Steady state solution
• Plant 1/Controller
• Match gain, phase, and frequency

Gain
Gain
Frequency Response Plots (Bode)
Phase
Phase
Frequency
Frequency
s-plane Representation (Nyquist)
Imaginary
Real
?
Imaginary
Plant 1/Controller
Invert
Real
32
The Williamson method
Plant
Controller

• Frequency response
• Nonlinear Describing function analysis
• Nyquist plot
• Invert controller
• Steady state solution
• Plant 1/Controller
• Match gain, phase, and frequency

A
Gain
Gain
Frequency Response Plots (Bode)
A
Phase
Phase
Frequency
Frequency
s-plane Representation (Nyquist)
Imaginary
Real
?
Imaginary
Plant 1/Controller
Invert
Real
33
The Williamson method
Plant
Controller

• Frequency response
• Nonlinear Describing function analysis
• Nyquist plot
• Invert controller
• Steady state solution
• Plant 1/Controller
• Match gain, phase, and frequency

A
Gain
Gain
Frequency Response Plots (Bode)
A
Phase
Phase
Frequency
Frequency
s-plane Representation (Nyquist)
Imaginary
A
Real
?
Imaginary
?
Plant 1/Controller
Invert
Real
34
Predicted coupled system solution

• Solution Intersection (3)
• Gain radius
• Phase angle
• Frequency infer

Controller
(intersection of gain, phase, and frequency)
w design target but not visualized
Plant
Predicted Solution
35
Visual Design by changing shapes

• Options
• Controller parameters
• Feedback - type sign
• Gains
• Controller
• Fix open-loop behavior for sensor failure
• All other parameters coupled
• Cant change
• Feedback
• Take derivitive 90o rotation shape change
• Negate 180o rotation
• Too drastic
• Gains
• Shape by scaling
• Good, but generally not enough

36
Design by changing shapes

• Options
• Controller parameters
• Feedback - type sign
• Gains
• Controller
• Fix open-loop behavior for sensor failure
• All other parameters coupled
• Cant change
• Feedback
• Take derivitive 90o rotation shape change
• Negate 180o rotation
• Too drastic
• Gains
• Shape by scaling
• Good, but generally not enough

Changing controller parameters
37
Design by changing shapes

• Options
• Controller parameters
• Feedback - type sign
• Gains
• Controller
• Fix open-loop behavior for sensor failure
• All other parameters coupled
• Cant change
• Feedback
• Take derivitive 90o rotation shape change
• Negate 180o rotation
• Too drastic
• Gains
• Shape by scaling
• Good, but generally not enough

Velocity
Position
(-) Position
90o
180o
?
?
?
Changing plant feedback
38
Design by changing shapes

• Options
• Controller parameters
• Feedback - type sign
• Gains
• Controller
• Fix open-loop behavior for sensor failure
• All other parameters coupled
• Cant change
• Feedback
• Take derivitive 90o rotation shape change
• Negate 180o rotation
• Too drastic
• Gains
• Shape by scaling
• Good, but generally not enough

Controller
Output gain
Input gain
Dynamics
Dynamics
Output gain
Input gain
Plant
Changing gains
39
Design by changing shapes

• Options
• Controller parameters
• Feedback - type sign
• Gains
• Controller
• Fix open-loop behavior for sensor failure
• All other parameters coupled
• Cant change
• Feedback
• Take derivitive 90o rotation shape change
• Negate 180o rotation
• Too drastic
• Gains
• Shape by scaling
• Good, but generally not enough

Controller
Output gain
Input gain
Dynamics
Dynamics
Output gain
Input gain
Plant
No good design options
40
Different goal, not adaptation
A

• Williamson (1999)
• Casually noted possible adaptive behavior
• Wanted constant frequency
• Behavior
• Response to changing conditions
• What is needed
• Fine tuning shaping methods
• Visual representation of changing conditions
• Predicting adaptive behaviors
• Designing for adaptation

1
0
-1
Imaginary
-2
w
w
-3
-4
z
-5
-5
-4
-3
-2
-1
0
1
2
3
Real
1.42
1.4
1.38
Behavior Response to changing conditions
1.36
My design tools for adaptation
w (radians/second)
1.34
1.32
1.3
1.28
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Damping ratio (z)
41
Analyze w contours for behavior
A

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

1
0
-1
Imaginary
-2
w
w
-3
-4
z
-5
-5
-4
-3
-2
-1
0
1
2
3
Real
42
Analyze w contours for behavior
A

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

1
0
-1
Imaginary
-2
w
w
-3
-4
z
-5
-5
-4
-3
-2
-1
0
1
2
3
Real
43
Analyze w contours for behavior

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

Imaginary
-0.6
-0.8
Real
-1
-1.2
Imaginary
-1.4
-1.6
-1.8
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
44
Analyze w contours for behavior

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

Imaginary
-0.6
-0.8
Real
-1
-1.2
Imaginary
-1.4
-1.6
-1.8
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
45
Analyze w contours for behavior

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

-1
-1.2
Imaginary
-1.4
-1.6
-1.8
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
46
Analyze w contours for behavior

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

-1
-1.2
Imaginary
-1.4
-1.6
-1.8
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
47
Analyze w contours for behavior

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

C
B
-1
-1.2
Imaginary
A
-1.4
3 regions of w contours
-1.6
-1.8
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
48
Analyze w contours for behavior

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

C
B
-1
-1.2
Imaginary
A
-1.4
3 regions of w contours
-1.6
-1.8
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
A
Relative rotation of w contours
z
49
Analyze w contours for behavior

• w contour analysis
• Lines of constant w
• Relative rotation
• Controller lines essentially radial
• Plant lines change orientation
• Behavior plant relative rotation
• Clockwise increasing
• Aligned no change
• Counterclockwise - decreasing

Imaginary
A
Relative rotation of w contours
z
Behavior
w contours predict behavior
1.4
1.38
w (radians/second)
1.36
1.34
1.32
A
B
C
1.3
1.28
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Damping ratio (z)
50
Why w contours?

• Think contour maps
• Contours represent elevation
• Frequency is design objective
• Why not explicitly represent?

C
B
-1
Imaginary
-1.2
A
-1.4
3 regions of w contours
C
-1.6
z
-1.8
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
B
A
Relative rotation of w contours
z
A
Behavior
1.4
1.38
w (radians/second)
Solution is really intersection of 2 surfaces in
3D space
1.36
1.34
1.32
A
B
C
1.3
1.28
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Damping ratio (z)
51
Three dimensional representation

• Explicitly visualize w
• Elevation
• Controller and plant represented by surfaces
• Solution
• Intersection of two surfaces

Coupled system solution
Controller
Plant
52
Three dimensional representation

• Explicitly visualize w
• Elevation
• Controller and plant represented by surfaces
• Solution
• Intersection of two surfaces

Coupled system solution
Controller
Plant
Visually intuitively represent adaptive
behavior
53
Designing coupled system frequency
w axis

• Coupled system frequency
• Elevation of intersection
• Design
• Rotate surfaces about w axis
• Change the phase
• Lower frequency
• Rotate controllercounter-clockwise
• Add phase lag
• How?
• Ideally phase delay
• Approximation intentional time delay

Coupled system solution
Controller
Plant
3D visualization makes design intuitive
54
Designing coupled system frequency
w axis

• Coupled system frequency
• Elevation of intersection
• Design
• Rotate surfaces about w axis
• Change the phase
• Lower frequency
• Rotate controllercounter-clockwise
• Add phase lag
• How?
• Ideally phase delay
• Approximation intentional time delay

Coupled system solution
Controller
Plant
3D visualization makes design intuitive
55
Intentional time delay

• Intentional time delay in the loop
• Rotates controller
• Counterclockwise (CCW)
• Approximation
• Rotates
• and smears

Controller
Output gain
Input gain
Dynamics
Dynamics
Output gain
Input gain
Plant
56
Intentional time delay

• Intentional time delay in the loop
• Rotates controller
• Counterclockwise (CCW)
• Approximation
• Rotates
• and smears

Time delay
Rotates CCW
57
Intentional time delay

• Intentional time delay in the loop
• Rotates controller
• Counterclockwise (CCW)
• Approximation
• Rotates
• and smears

Time delay
Increasing time delay
Intersection
Intersection
Intersection
Controller rotates, intersection elevation is
lowered
58
Intentional time delay

• Intentional time delay in the loop
• Rotates controller
• Counterclockwise (CCW)
• Approximation
• Rotates
• and smears

Time delay
Adaptive behavior
1.5
1.4
1.3
Increasing time delay
1.2
New design tool can fine-tune coupled system
frequency
w (radians/second)
1.1
1
0.9
0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Damping ratio (z)
59
Outline

• Biomimetic robots and the need for adaptation
• Biological cyclic behaviors nonlinear
  oscillators
• Animal experiments
• Use of sensory feedback
• Support for a model
• Visual design method
• Develop new design and analysis tools for
  adaptation
• Biomimetic hexapod implementation and results
• Design an adaptive controller
• Discuss performance
• Conclusions and future work

60
Implementing an adaptive controller

• Biomimetic hexapod
• Sprawlita
• ADAMS simulation
• Design an adaptive controller
• Desired slope behavior

61
Design

• Desired slope adaptation behavior
• Higher frequency for downhill
• Lower frequency for uphill
• Optimal flat ground frequency
• Sensor failure frequency
• Defines nonlinear oscillator parameters

Uphill
Downhill
90
85
Open-loop
80
75
Desired frequency (radians/second)
70
Optimal w flat ground
65
60
55
50
-20
-15
-10
-5
0
5
10
15
20
Slope (degrees)
Want w to decrease with slope
62
Design

• Controller parameters
• All defined by sensor failure (open loop) w
• except for time delay
• Motor pattern
• Alternating valve activation
• Always amplitude 1
• Feedback
• Rear leg extension
• Left Right (to center)
• Characteristic of all
• Remaining design options
• Gain
• Time delay

63
Naïve first step
Uphill
Downhill

• Controller
• mostly flat and radial
• Gain
• Use to get similar radii
• Little behavior effect
• Time delay
• Flat ground coupled system frequency

95
90
85
80
75
w (radians/second)
70
Wrong zero slope w
65
Desired
60
55
50
-15
-10
-5
0
5
10
15
Slope (degrees)
64
Naïve first step
Uphill
Downhill

• Controller
• mostly flat and radial
• Gain
• Use to get similar radii
• Little behavior effect
• Time delay
• Flat ground coupled system frequency

95
90
85
80
75
w (radians/second)
70
Wrong zero slope w
65
Desired
60
55
50
-15
-10
-5
0
5
10
15
Slope (degrees)
Rotate controller CCW about w axis by
intentional time delay of 0.004 seconds
Uphill
Downhill
90
85
80
75
w (radians/second)
70
Correct zero slope w
65
60
Desired
55
50
-15
-10
-5
0
5
10
15
Slope (degrees)
65
Naïve first step
Uphill
Downhill

• Controller
• mostly flat and radial
• Gain
• Use to get similar radii
• Little behavior effect
• Time delay
• Flat ground coupled system frequency

95
90
85
80
75
w (radians/second)
70
Wrong zero slope w
65
Desired
60
55
50
-15
-10
-5
0
5
10
15
Slope (degrees)
Rotate controller CCW about w axis by
intentional time delay of 0.004 seconds
Uphill
Downhill
90
85
80
Not desired behavior Why? Options?
75
w (radians/second)
Correct zero slope w
70
65
60
Poor uphill behavior
Desired
55
50
-15
-10
-5
0
5
10
15
Slope (degrees)
66
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

0.1
0
-0.1
-0.2
-0.3
15
-0.4
Imaginary
-0.5
0
-15
0
-0.6
-0.7
-0.8
Constant Phase
-0.9
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
67
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

0.1
0
Poor behavior
-0.1
-0.2
-0.3
15
-0.4
Imaginary
-0.5
0
0
Good behavior
-0.6
-0.7
-0.8
Constant Phase
-0.9
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
68
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

0.1
0
-0.1
-0.2
-0.3
15
-0.4
Imaginary
-0.5
0
-15
0
-0.6
-0.7
-0.8
Constant Phase
-0.9
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
Real
69
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

70
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

15
-0.35
-0.4
-0.45
Imaginary
-0.5
-15
Plant
-0.55
Controller
0
Coupled System
-0.6
Constant Phase
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
Real
71
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

-0.35
Plant
-0.4
Controller
-0.45
Imaginary
-0.5
-0.55
-0.6
Constant Phase
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
Real
72
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

Uphill
-0.35
Plant
-0.4
Controller
-0.45
Slope (q)
Input amplitude
Imaginary
-0.5
-0.55
-0.6
Constant Phase
Downhill
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
Real
73
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

Uphill
-0.35
Plant
-0.4
Controller
-0.45
Slope (q)
Input amplitude
Imaginary
-0.5
-0.55
-0.6
Constant Phase
Downhill
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
Real
A
Relative rotation of w contours
q
w increases with slope
w decreases with slope
No change
74
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

Uphill
-0.35
Plant
-0.4
Controller
-0.45
Slope (q)
Input amplitude
Imaginary
-0.5
-0.55
Want
-0.6
Constant Phase
Downhill
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
Real
A
Relative rotation of w contours
q
w increases with slope
w decreases with slope
No change
75
w contour analysis of adaptation behavior

• w contour analysis
• Determine source of behavior
• Procedure
• Lines of constant w
• Simplify area of interest
• w contour analysis
• Results
• Uphill, signal changes gain, not phase
• Gain change doesnt affect frequency
• Signal needed
• Phase lag increase with slope

Uphill
-0.35
Plant
-0.4
Controller
-0.45
Slope (q)
Input amplitude
Imaginary
-0.5
-0.55
Want
-0.6
Constant Phase
Downhill
-0.5
-0.45
-0.4
-0.35
-0.3
-0.25
-0.2
-0.15
Real
Design Feedback properties to get desired
behavior
A
Relative rotation of w contours
q
w increases with slope
w decreases with slope
No change
76
Feedback modification

• Rear leg extension
• Uphill changes gain, not phase
• Running analysis
• Legs extend more slowly
• Dont extend as far
• Phase constant
• New feedback
• Take advantage of slow extension
• Pulse at threshold
• Phase lag increases with slope

Threshold
Pulse feedback has desired phase-slope relation
Phase lag increases with slope
77
Pulse coupled system

• Same structure
• Controller
• Pulse input changes shape
• Plant dynamics
• Gain is constant
• Inscribed on cylinder

Controller
Sign
Input gain
Dynamics
Time delay
Pulse feedback
Valve
Dynamics
Time
Time
Output gain
Valve Signal
Plant
Different slopes only affect phase, not gain
78
Feedback gain

• Output gain of the plant
• Changes radius of the cylinder
• Changes intersection with controllerand cylinder

Controller
Sign
Input gain
Dynamics
Time delay
Dynamics
Output gain
Valve Signal
Plant
79
Feedback gain

• Output gain of the plant
• Changes radius of the cylinder
• Changes intersection with controllerand cylinder

Controller
Sign
Input gain
Dynamics
Time delay
Dynamics
Output gain
Valve Signal
Plant
80
Feedback gain

• Output gain of the plant
• Changes radius of the cylinder
• Changes intersection with controllerand cylinder

Controller
Sign
Input gain
Dynam