Metastable Legged-Robot 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

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

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

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); 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 Time response Freq.

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

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

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

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

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, 7.8 cm Fastest recorded run, with NO COMPUTATION: - about 3.4 cm/sec

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, 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

Double-support gait creation Pacing

Double-support gait creation 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 video to follow… 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