1. Metastable Legged-Robot Locomotion
Katie Byl
Robot Locomotion Group
June 21, 2007

2. Overview Background Past projects and degree work PhD Work
Stability metrics for locomotion on rough terrain:
mean first-passage time (MFPT)
Metastable (long-living) dynamics
Compass-gait biped simulations
LittleDog Phase 1 (static) and 2 (dynamic) motions

3. Background: Past MIT Projects
2.70 (now 2007) “Intro to Design” / 6.270
Lego/LOGO instructor at Museum of Science
MIT Blackjack Team
6.302 lost-cost maglev lab kit
various UROPS and MATLAB-coding jobs
6.270
2.70
MIT BJ
6.302

4. Background: Past MIT Projects
2.70 (now 2007) “Intro to Design” / 6.270
Lego/LOGO instructor at Museum of Science
MIT Blackjack Team
6.302 lost-cost maglev lab kit
various UROPS and MATLAB-coding jobs
6.270
2.70
MIT BJ
6.302

5. Background Bachelor’s thesis * Master’s thesis * TA appointments
Dynamic Signal Analyzer (DSA)
to obtain empirical transfer function for a system
Simulink/MATLAB block for dSPACE controller
Master’s thesis *
2.003 lab creation
Inverted pendulum (segway-type)
TA appointments
2.14 (Controls); and 2.29 (MATLAB); (Modeling Dynamics and Control)
*Precision Motion Control Lab, Prof. Dave Trumper

6. Bachelor’s Thesis Dynamic Signal Analyzer (DSA)
Goal: integrated system ID for real-time controllers
Simulink/MATLAB block for dSPACE boards
MATLAB code to get empirical transfer function

7. Master’s Thesis
ActivLab labware for 2.003: Modeling Dynamics and Control 1
1st-order dynamics

8. Master’s Thesis 2nd- and 4th-order dynamics Time response Freq.

9. Master’s Thesis
Segway-style inverted pendulum

10. PhD: Legged Locomotion
Mean first-passage time (MFPT)
Goal: Exceptional performance most of the time, with rare failures (falling)
Metric: maximize distance (or time) between failures

11. PhD: Legged Locomotion
Metastability
Fast mixing-time dynamics
Rapid convergence to long-living (metastable) limit-cycle behavior

12. PhD: Legged Locomotion
Compass gait: optimal vs one-step control

13. PhD: Legged Locomotion
LittleDog: Phase 1 (static crawl) results

14. PhD: Legged Locomotion
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: 1.2 cm/sec, 4.8 cm [step ht]
Phase 2: 4.2 cm/sec, cm
Fastest recorded run, with NO COMPUTATION:
- about 3.4 cm/sec

15. PhD: Legged Locomotion
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: 1.2 cm/sec, 4.8 cm [step ht]
Phase 2: 4.2 cm/sec, cm
Fastest recorded run, with NO COMPUTATION:
- about 3.4 cm/sec

16. 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): , June 1999.

17. Double-support gait creation
3 possible leg-pairing types
Pacing left vs right
Bounding fore vs rear
Trot 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

18. Double-support gait creation
Pacing

19. Double-support gait creation
Trotting

20. Questions?

21. ZMP pacing – with smoothing
Smoothing requested ZMP reduces overshoot
square wave smoothed input

22. 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 IEEE, Sep 2003.

23. 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

24. 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 Stochastic return map
MFPT as fn of init. cond.
Metastable basin of attraction

25. 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)

26. Compass Gait
Limit cycle analysis

27. 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

28. Double-support results to date
Bounding – currently quite heuristic…
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

29. Where to go next… 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

30. Master’s Thesis Inverted pendulum dynamics Bandwidth = 0.5 Hz ζ= 0.25
(damping ratio)

31. Murphy Video Goals: Identify gait characteristics
Speculate on forces and timing
Questions relevant to LittleDog gaits
What is being optimized? (If anything?)
How important is ankle torque?
How/why do different motions segue well

32. Dog gaits video to follow…
Trotting - Efficient; most-common; rear feet follow fore feet
Gallop - Fast; support; pole-vault with front
Pacing - Asymmetric; low lateral accelerations; push-pull
Crawl - Not common; used to amble or to step carefully
Leap - used to clear obstacles; practiced often (in play)
Bound - uncommon; gallop-like except pairwise rear and front
Weave - example of learning to do a motion efficiently
video to follow…

33. Video list trot_waterprints_withpan gallop_tri_1 pacing_3
crawl_waterprints
leap_from_trot
bound_uphill_snow
dbbound_slide_snow
weave_hops
agility_frontcross