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

Metastable Legged-Robot Locomotion

Katie Byl Robot Locomotion GroupJune 21, 2007

MIT Computer Science & Artificial Intelligence Laboratory

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


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

2.706.270

MIT BJ 6.302


Background

Bachelor’s thesis * 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); 2.670 and 2.29 (MATLAB);

2.003 (Modeling Dynamics and Control)

*Precision Motion Control Lab, Prof. Dave Trumper


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


Master’s Thesis ActivLab labware for 2.003:

Modeling Dynamics and Control 1 1st-order dynamics


Master’s Thesis 2nd- and 4th-order dynamics

Timeresponse

Freq.response


Master’s Thesis Segway-style inverted pendulum


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



Metastability Fast mixing-time dynamics Rapid convergence to long-living (metastable) limit-

cycle behavior



Compass gait: optimal vs one-step control



LittleDog: Phase 1 (static crawl) results



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, 7.8 cm

Fastest recorded run, with NO COMPUTATION:

- about 3.4 cm/sec


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.


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



Pacing



Trotting


Questions?


ZMP pacing – with smoothing

Smoothing requested ZMP reduces overshoot

square wave 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.


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


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


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)


Compass Gait

Limit cycle analysis


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


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


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


Master’s Thesis Inverted pendulum dynamics

Bandwidth = 0.5 Hz

ζ= 0.25(damping ratio)


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


Dog gaits

Trotting - Efficient; most-common; rear feet follow fore feet

Gallop - Fast; 1-2-1 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…


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

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

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