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Metastable Locomotion
for LittleDog
Katie Byl
Robot Locomotion Group
July 11, 2007
Goal: piecewise control
 Repeatable open-loop motions
 Based on simple (approximate) physical models
 Good evidence for high repeatability
 Define a working range for each motion
 Improve accuracy through trials (learning)
 Estimate reliability for a particular motion
 Select short-sighted motion with best “performance”
 Stability metrics for locomotion on rough terrain:
mean first-passage time (MFPT)
 Speed also a requirement
MIT Computer Science & Artificial Intelligence Laboratory
Is short-sighted control ok?
 Mean first-passage time (MFPT)
 Goal: Exceptional performance most of the time,
with rare failures (falling)
 Metric: maximize distance (or time) between failures
MIT Computer Science & Artificial Intelligence Laboratory
Metastable dynamics
 Metastability
 Fast mixing-time dynamics
 Rapid convergence to long-living (metastable) limitcycle behavior
MIT Computer Science & Artificial Intelligence Laboratory
Markov Process
 The transition matrix for a stochastic system prescribes state-to-state
transition probabilities
 For metastable systems, the first (largest) eigenvalue of its transpose is
1, corresponding to the absorbing FAILURE state
 The second largest eigenvalue is the inverse MFPT, and the
corresponding vector gives the metastable distribution
F
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MFPT and Metastability
 Fast mixing-time dynamics
 Rapidly either fails (falls) or converges to long-living (metastable)
limit-cycle behavior
add Gaussian
noise; sigma=.2
Deterministic return map
MFPT as fn of init. cond.
Metastable basin of attraction
MIT Computer Science & Artificial Intelligence Laboratory
Stochastic return map
MFPT and Metastability
 Example for a DETERMINISTIC system with high sensitivity to initial
conditions (as shown by steep slope of the return map)
 Green shows where the “metastable basin” is developing
 MFPT and density of metastable basin give us better intuition for the
system dynamics (where the exact initial state is not known)
MIT Computer Science & Artificial Intelligence Laboratory
Motion Control for LittleDog
 LittleDog Phase 2: dynamic, ZMP-based gaits
 All 6 teams passed Phase 1 metrics (below)
 3 teams (at most) can pass Phase 2
 Phase 1:
 Phase 2:
1.2 cm/sec,
4.8 cm [step ht]
4.2 cm/sec,
7.8 cm
Fastest recorded run, with NO COMPUTATION:
- about 3.4 cm/sec
Fastest flat-terrain walk and trot: 17 and 20+ cm/s
MIT Computer Science & Artificial Intelligence Laboratory
Sequencing motions: Funnels
 R. R. Burridge, A. A. Rizzi, and D. E. Koditschek. Sequential composition
of dynamically dexterous robot behaviors. International Journal of
Robotics Research, 18(6):534-555, June 1999.
MIT Computer Science & Artificial Intelligence Laboratory
Double-support gait creation
 3 possible leg-pairing types
 Pacing
 Bounding
 Trot
left vs right
fore vs rear
diagonal pairings
 ZMP method: Aim for COP near “knife-edge”
 Not simply planning leg-contacts…
 Plan [model] COB accelerations and ground forces directly
Pacing
Trotting
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Double-support gait creation
Pacing
MIT Computer Science & Artificial Intelligence Laboratory
ZMP pacing – with smoothing
 Smoothing requested ZMP reduces overshoot
square wave
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smoothed input
Phase 2: dynamic gaits
 Control of ZMP using method in Kajita03
 S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiware, K. Harada, K. Yokoi, and H. Hirukawa. Biped
walking pattern generation by using preview control of zero-moment point. In ICRA IEEE
International Conference on Robotics and Automation, pages 1620-1626. IEEE, Sep 2003.
MIT Computer Science & Artificial Intelligence Laboratory
Motivation – Phase 2
 Opportunity for science in legged robots
 Dynamic gaits [Phase 2]
• Speed
• Agility
 Precision motion planning (vs CPG)
• Optimal to respond to variations in terrain
 Wheeled locomotion analogy:
 Tricycle = static stability [Phase 1]
 Bicycle = dynamic and fast
 Unicycle = dynamic and agile
MIT Computer Science & Artificial Intelligence Laboratory
Double-support results to date
 Bounding – currently quite heuristic…
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Plan a “step” in COP, to REAR legs for Δt
At start of Δt, tilt body up
Push down-and-back with rear legs
Simultaneously extend fore legs
Recover a zero-pitch 4-legged stance
Plan a “step” in COP, to FORE legs
 Intended “lift” of rear legs - actually dragged
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Repeatability
 Bounding: PITCH
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Repeatability
 Bounding: COB X position
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Sensing
 Bounding: Pitch via “sensor fusion”
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Repeatability
 ZMP Walking: Pitch
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Repeatability
 ZMP Trot-Walking: Pitch
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Repeatability
 Dynamic climb: (10 consecutive trials) Pitch
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Repeatability
 Dynamic climb: (10 consecutive trials) Roll
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Repeatability
 Dynamic climb: (10 consecutive trials) Yaw
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Repeatability
 Bounding: Force sensor (est. vert. force)
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Repeatability
 Bounding: accelerometer and force sensors
MIT Computer Science & Artificial Intelligence Laboratory
Where to go next (post-thesis)
 Optimization of double-support
 Gradient methods, in general
 Actor-critic, in particular
 Attempt “unipedal” support?
 Is there a practical use in Phase 2?
 Is this interesting science?
 Potential for significant airborne phase
 Plan now for 5x more compliant BDI legs
MIT Computer Science & Artificial Intelligence Laboratory