Fast and Robust Legged Locomotion

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Fast and Robust Legged Locomotion
Sean Bailey Mechanical Engineering Design
Division Advisor Dr. Mark Cutkosky
May 12, 2000
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Overview

• Introduction
• Functional Biomimesis
• Robot Design
• Model Analysis
• Conclusions

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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

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Traditional Approaches to Legged Systems

• Statically stable
• Tripod of support
• Slow
• Rough terrain
• Dynamically stable
• No support requirements
• Fast
• Smooth terrain

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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

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Biomimesis Options
Too complex!
Functional Biomimesis
Biomimetic configuration
Extract fast rough terrain locomotion capabilities
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Biological Inspiration

• Control heirarchy
• Passive component
• Active component

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Is Passive Enough?

• Passive Dynamic Stabilization
• No active stabilization
• Geometry
• Mechanical system properties

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Geometry
Cockroach Geometry
Functional Biomimesis
Robot Implementation

• Passive Compliant Hip Joint
• Effective Thrusting Force

• Damped, Compliant Hip Flexure
• Embedded Air Piston

• Rotary Joint
• Prismatic Joint

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Sprawlita

• Mass - .27 kg
• Dimensions - 16x10x9 cm
• Leg length - 4.5 cm
• Max. Speed - 39cm/s 2.5 body/sec
• Hip height obstacle traversal

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Movie

• Compliant hip
• Alternating tripod
• Stable running
• Obstacle traversal

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Mechanical System Properties

• Prototype Empirically tuned properties
• Design for behavior

?
Mechanical System Properties
Modeling
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Simple Model
K, B, ?nom
k, b, ?nom

• 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!

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Simplest Locomotion Model
k, b, ?nom
Biped
Biped
Quadruped

• 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

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Modeling assumptions

• Time-Based Mode Transitions
• Clock-driven motor pattern
• Groucho running1
• One reset mode
• Two sets of legs - Two modes
• Symmetric - treat as one mode
• Mode initial conditions
• Nominal leg angles
• Instant passive component compression

1 McMahon, et al 1987
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Modeling assumptions

• Time-Based Mode Transitions
• Clock-driven motor pattern
• Groucho running1
• One reset mode
• Two sets of legs - Two modes
• Symmetric - treat as one mode
• Mode initial conditions
• Nominal leg angles
• Instant passive component compression

t 2T-
State
x
0
Leg Set
Leg Set
Leg Set
Leg Set
2
1
2
1
Time
Stride Period
1 McMahon, et al 1987
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Modeling assumptions

• Time-Based Mode Transitions
• Clock-driven motor pattern
• Groucho running1
• One reset mode
• Two sets of legs - Two modes
• Symmetric - treat as one mode
• Mode initial conditions
• Nominal leg angles
• Instant passive component compression

1 McMahon, et al 1987
18
Modeling assumptions

• Time-Based Mode Transitions
• Clock-driven motor pattern
• Groucho running1
• One reset mode
• Two sets of legs - Two modes
• Symmetric - treat as one mode
• Mode initial conditions
• Nominal leg angles
• Instant passive component compression

1 McMahon, et al 1987
19
Modeling assumptions

• Time-Based Mode Transitions
• Clock-driven motor pattern
• Groucho running1
• One reset mode
• Two sets of legs - Two modes
• Symmetric - treat as one mode
• Mode initial conditions
• Nominal leg angles
• Instant passive component compression

1 McMahon, et al 1987
20
Modeling assumptions

• Time-Based Mode Transitions
• Clock-driven motor pattern
• Groucho running1
• One reset mode
• Two sets of legs - Two modes
• Symmetric - treat as one mode
• Mode initial conditions
• Nominal leg angles
• Instant passive component compression

1 McMahon, et al 1987
21
Modeling assumptions

• Time-Based Mode Transitions
• Clock-driven motor pattern
• Groucho running1
• One reset mode
• Two sets of legs - Two modes
• Symmetric - treat as one mode
• Mode initial conditions
• Nominal leg angles
• Instant passive component compression

1 McMahon, et al 1987
22
Modeling assumptions

• Time-Based Mode Transitions
• Clock-driven motor pattern
• Groucho running1
• One reset mode
• Two sets of legs - Two modes
• Symmetric - treat as one mode
• Mode initial conditions
• Nominal leg angles
• Instant passive component compression

1 McMahon, et al 1987
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Non-linear analysis tools

• Discrete non-linear system
• Fixed points
• numerically integrate to find
• exclude horizontal position information

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Non-linear analysis tools

• Floquet technique
• Analyze perturbation response
• Digital eigenvalues via linearization - examine
  stability
• Use selective perturbations to construct M matrix

Numerically Integrate
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Non-linear analysis tools

• Floquet technique

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Perturbation Response
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Analysis trends

• 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

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Design Example
Damping
Damping
Damping
Stiffness
Stiffness
Stiffness
Speed 0
Speed 13 cm/s
Speed 23.5 cm/s
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Locomotion Insight

• Body tends towardsequilibrium point
• Parameters andmechanical propertiesdetermine how

Mode Equilibrium
Trajectory
Statically Unstable Region
Initial condition
Leg Extension Limit
Leg Pre- Compressions
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Summary and 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

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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

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Thanks!

• Center for Design Research
• Dexterous Manipulation Lab
• Rapid Prototyping Lab
• Mark Cutkosky
• Jorge Cham, Jonathan Clark