Fast and Robust Legged Locomotion Sean Bailey Mechanical Engineering Design Division Advisor: Dr....
-
date post
21-Dec-2015 -
Category
Documents
-
view
215 -
download
0
Transcript of Fast and Robust Legged Locomotion Sean Bailey Mechanical Engineering Design Division Advisor: Dr....
Fast and Robust Legged Locomotion
Sean BaileyMechanical Engineering Design Division
Advisor: Dr. Mark Cutkosky
May 12, 2000
Intro Biomimesis Design Analysis Conclusions
Overview• Introduction
• Functional Biomimesis
• Robot Design
• Model Analysis
• Conclusions
Intro Biomimesis Design Analysis Conclusions
Fast, Robust Rough Terrain Traversal• Why?
– Mine clearing– Urban Reconnaissance
• Why legs?
• Basic Design Goals– 1.5 body lengths per
second– Hip-height obstacles– Simple
Intro Biomimesis Design Analysis Conclusions
Traditional Approaches to Legged Systems• Statically stable
– Tripod of support– – Slow– Rough terrain
• Dynamically stable– No support
requirements– – Fast– Smooth terrain
0 F
0 F
Intro Biomimesis Design Analysis Conclusions
Biological Example• Death-head cockroach Blaberus discoidalis
• Fast– Speeds of up to 10 body/s
• Rough terrain– Can easily traverse fractal terrain of
obstacles 3X hip height
• Stability– Static and dynamic
FunctionalBiomimesis
“Biomimetic” configuration
Extract fast rough terrain locomotion capabilities
Too complex!
Intro Biomimesis Design Analysis Conclusions
Biomimesis Options
Intro Biomimesis Design Analysis Conclusions
Biological Inspiration• Control heirarchy
– Passive component
– Active component
Intro Biomimesis Design Analysis Conclusions
MechanicalSystem
Environment
MechanicalFeedback(Preflexes)
SensoryFeedback(Reflexes)
Neural System
FeedforwardMotor Pattern
Passive DynamicSelf-Stabilization
Locomotion
Is Passive Enough?• Passive Dynamic Stabilization
– No active stabilization
– Geometry
– Mechanical system properties
Intro Biomimesis Design Analysis Conclusions
Cockroach Geometry
•Passive Compliant Hip Joint•Effective Thrusting Force
Functional Biomimesis
•Damped, Compliant Hip Flexure•Embedded Air Piston
Robot Implementation
Geometry
•Rotary Joint•Prismatic Joint
Intro Biomimesis Design Analysis Conclusions
Sprawlita• Mass - .27 kg
• Dimensions - 16x10x9 cm
• Leg length - 4.5 cm
• Max. Speed - 39cm/s 2.5 body/sec
• Hip height obstacle traversal
Intro Biomimesis Design Analysis Conclusions
Movie
• Compliant hip
• Alternating tripod
• Stable running
• Obstacle traversal
Intro Biomimesis Design Analysis Conclusions
Mechanical System Properties• Prototype: Empirically tuned properties
• Design for behavior
?Mechanical
SystemProperties
Modeling
Intro Biomimesis Design Analysis Conclusions
“Simple” Model
• Body has 3 planar degrees of freedom– x, z, theta
– mass, inertia
• 3 massless legs (per tripod)– rotating hip joint - damped torsional spring
– prismatic leg joint - damped linear spring
– 6 parameters per leg
18 parameters to tune - TOO MANY!
Full 3D model Planar model Symmetry assumption
K, B, nom
k, b, nom
Intro Biomimesis Design Analysis Conclusions
Simplest Locomotion Model
• Body has 2 planar degrees of freedom– x, z– mass
• 4 massless legs– freely rotating hip joint – prismatic leg joint - damped linear spring– 3 parameters per leg
6 parameters to tune, assuming symmetry
g
g
k, b, nom
g
Biped QuadrupedBiped
Intro Biomimesis Design Analysis Conclusions
• Time-Based Mode Transitions– Clock-driven motor pattern
– “Groucho running”1
• One “reset” mode– Two sets of legs - Two modes
– Symmetric - treat as one mode
• Mode initial conditions– Nominal leg angles
– Instant passive component compression
Modeling assumptions
1 McMahon, et al 1987
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
teTime
x0
= state trajectory
Stride Period
T T T T
g
Intro Biomimesis Design Analysis Conclusions
• Time-Based Mode Transitions– Clock-driven motor pattern
– “Groucho running”1
• One “reset” mode– Two sets of legs - Two modes
– Symmetric - treat as one mode
• Mode initial conditions– Nominal leg angles
– Instant passive component compression
Modeling assumptions
1 McMahon, et al 1987
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
teTime
x0
= state trajectory
Stride Period
g
t = 2T-
T T T T
Intro Biomimesis Design Analysis Conclusions
• Time-Based Mode Transitions– Clock-driven motor pattern
– “Groucho running”1
• One “reset” mode– Two sets of legs - Two modes
– Symmetric - treat as one mode
• Mode initial conditions– Nominal leg angles
– Instant passive component compression
Modeling assumptions
1 McMahon, et al 1987
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
teTime
x0
= state trajectory
Stride Period
t = 2T+
T T T T
g
Intro Biomimesis Design Analysis Conclusions
• Time-Based Mode Transitions– Clock-driven motor pattern
– “Groucho running”1
• One “reset” mode– Two sets of legs - Two modes
– Symmetric - treat as one mode
• Mode initial conditions– Nominal leg angles
– Instant passive component compression
Modeling assumptions
1 McMahon, et al 1987
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
teTime
x0
= state trajectory
Stride Period
t = 2T + 1/3T
T T T T
g
Intro Biomimesis Design Analysis Conclusions
• Time-Based Mode Transitions– Clock-driven motor pattern
– “Groucho running”1
• One “reset” mode– Two sets of legs - Two modes
– Symmetric - treat as one mode
• Mode initial conditions– Nominal leg angles
– Instant passive component compression
Modeling assumptions
1 McMahon, et al 1987
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
teTime
x0
= state trajectory
Stride Period
t = 2T + 2/3T
T T T T
g
Intro Biomimesis Design Analysis Conclusions
• Time-Based Mode Transitions– Clock-driven motor pattern
– “Groucho running”1
• One “reset” mode– Two sets of legs - Two modes
– Symmetric - treat as one mode
• Mode initial conditions– Nominal leg angles
– Instant passive component compression
Modeling assumptions
1 McMahon, et al 1987
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
teTime
x0
= state trajectory
Stride Period
t = 3T-
T T T T
g
Intro Biomimesis Design Analysis Conclusions
• Time-Based Mode Transitions– Clock-driven motor pattern
– “Groucho running”1
• One “reset” mode– Two sets of legs - Two modes
– Symmetric - treat as one mode
• Mode initial conditions– Nominal leg angles
– Instant passive component compression
Modeling assumptions
1 McMahon, et al 1987
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
teTime
x0
= state trajectory
Stride Period
t = 3T+
T T T T
g
Intro Biomimesis Design Analysis Conclusions
• Time-Based Mode Transitions– Clock-driven motor pattern
– “Groucho running”1
• One “reset” mode– Two sets of legs - Two modes
– Symmetric - treat as one mode
• Mode initial conditions– Nominal leg angles
– Instant passive component compression
Modeling assumptions
1 McMahon, et al 1987
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
teTime
x0
= state trajectory
Stride Period
t = 3T + 1/3T
T T T T
g
Intro Biomimesis Design Analysis Conclusions
Non-linear analysis tools• Discrete non-linear system
• Fixed points– numerically integrate to find
– exclude horizontal position information
)(1 kk xfx = state trajectory
= fixed points
)( ** xfx
xk+1 = xk = x*
Leg Set2
Leg Set1
Leg Set2
Leg Set 1
Sta
te
Time
x0
= state trajectory
Stride Period
T T T T
Intro Biomimesis Design Analysis Conclusions
Non-linear analysis tools• Floquet technique
– Analyze perturbation response
– Digital eigenvalues via linearization - examine stability
– Use selective perturbations to construct M matrix
][,...,1 Meign ll
úúúú
û
ù
êêêê
ë
é
úúúú
û
ù
êêêê
ë
é
úúúú
û
ù
êêêê
ë
é
0
0
0
???
???
???
???
4
3
2
1
4
3
2
1 d
dddddddd
dddd
úúúú
û
ù
êêêê
ë
é
0
0
0
d
kx
úúúú
û
ù
êêêê
ë
é
4
3
2
1
1
dddd
kx
pertx
perttotal xxx *
pertk
pertk
pertk Mxx
xx
xfx
*1 |)(
*x = nominal trajectory
NumericallyIntegrate
Intro Biomimesis Design Analysis Conclusions
Non-linear analysis tools• Floquet technique
][,...,1 Meign ll
)(1 kk xfx )f(xx **
)()()( **1
* pertk
pertk
pertk xfxfxxfxx
...)(
)( ** | tohx
xx
xfxf pert
k
pertk
pertk
pertk Mxx
xx
xfx
*1 |)(
pertk
pertk x
xx
xfxxx *
*1
* |)(
Perturbation Response
0.83 0.84 0.85 0.86 0.87 0.880
0.005
0.01
0.015
0.02
0.025
Perturbation Response over 3 Mode Transitions
X (meters)
Z (
me
ters
)
Nominal OrbitPerturbed Trajectory
Intro Biomimesis Design Analysis Conclusions
Intro Biomimesis Design Analysis Conclusions
• Relationships– damping vs. speed and
“robustness”
– stiffness, leg angles, leg lengths, stride period, etc
• Use for design– select mechanical properties
– select other parameters
• Insight into the mechanism of locomotion
6.5 7 7.5 8 8.5 9 9.5 101.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
Damping (N-s/m)
Recovery RateHorizontalVelocity
X_
do
t (m/s
)
1/m
ax
[eig
(M)]
Analysis trends
Intro Biomimesis Design Analysis Conclusions
Design Example RobustnessSpeed
Stiffness
Damping
Stiffness
Damping
Stiffness
Damping
Speed = 0 Speed = 13 cm/s Speed = 23.5 cm/s
Intro Biomimesis Design Analysis Conclusions
Locomotion Insight
StaticallyUnstable
Region
Initialcondition
ModeEquilibrium
Trajectory
LegExtension
Limit
Leg Pre-Compressions
• Body tends towardsequilibrium point
• Parameters andmechanical propertiesdetermine how
Intro Biomimesis Design Analysis Conclusions
• Current leg systems are either fast or can handle rough terrain• Biology suggests emphasis on good mechanical design
– enhances capability– simplifies control
• Purely clock-driven systems can be fast and robust
• Floquet technique can be used to indicate locomotion robustness• Trends can be established to improve design and provide insight
Summary and Conclusions
Intro Biomimesis Design Analysis Conclusions
Future Work• Extend findings and insights to more complex models
• Develop easily modeled 4th generation robot
• Utilize sensor feedback in high level control
• Examine other behaviors