Transcript BDI_talk_jun21_2007

Metastable LeggedRobot Locomotion
Katie Byl
Robot Locomotion Group
June 21, 2007
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
MIT Computer Science & Artificial Intelligence Laboratory
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
2.70
6.270
MIT Computer Science & Artificial Intelligence Laboratory
MIT BJ
6.302
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
2.70
6.270
MIT Computer Science & Artificial Intelligence Laboratory
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
MIT Computer Science & Artificial Intelligence Laboratory
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
MIT Computer Science & Artificial Intelligence Laboratory
Master’s Thesis
 ActivLab labware for 2.003:
Modeling Dynamics and Control 1
 1st-order dynamics
MIT Computer Science & Artificial Intelligence Laboratory
Master’s Thesis
 2nd- and 4th-order dynamics
Time
response
Freq.
response
MIT Computer Science & Artificial Intelligence Laboratory
Master’s Thesis
 Segway-style inverted pendulum
MIT Computer Science & Artificial Intelligence Laboratory
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
MIT Computer Science & Artificial Intelligence Laboratory
PhD: Legged Locomotion
 Metastability
 Fast mixing-time dynamics
 Rapid convergence to long-living (metastable) limitcycle behavior
MIT Computer Science & Artificial Intelligence Laboratory
PhD: Legged Locomotion
 Compass gait: optimal vs one-step control
MIT Computer Science & Artificial Intelligence Laboratory
PhD: Legged Locomotion
 LittleDog: Phase 1 (static crawl) results
MIT Computer Science & Artificial Intelligence Laboratory
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:
 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
MIT Computer Science & Artificial Intelligence Laboratory
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:
 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
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
MIT Computer Science & Artificial Intelligence Laboratory
Double-support gait creation
Pacing
MIT Computer Science & Artificial Intelligence Laboratory
Double-support gait creation
Trotting
MIT Computer Science & Artificial Intelligence Laboratory
Questions?
MIT Computer Science & Artificial Intelligence Laboratory
ZMP pacing – with smoothing
 Smoothing requested ZMP reduces overshoot
square wave
MIT Computer Science & Artificial Intelligence Laboratory
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
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
MIT Computer Science & Artificial Intelligence Laboratory
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
Compass Gait
 Limit cycle analysis
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…






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
MIT Computer Science & Artificial Intelligence Laboratory
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
MIT Computer Science & Artificial Intelligence Laboratory
Master’s Thesis
 Inverted pendulum dynamics
Bandwidth = 0.5 Hz
ζ= 0.25
(damping ratio)
MIT Computer Science & Artificial Intelligence Laboratory
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
MIT Computer Science & Artificial Intelligence Laboratory
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…
MIT Computer Science & Artificial Intelligence Laboratory
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
MIT Computer Science & Artificial Intelligence Laboratory