Metastable Legged Robot Locomotion Katie Byl Robot Locomotion

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

Overview § Background § Past projects and degree work § Ph. D Work § Stability metrics for locomotion on rough terrain: mean first-passage time (MFPT) § Metastable (long-living) dynamics § Compass-gait biped simulations § Little. Dog 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 § Bachelor’s thesis * § Dynamic Signal Analyzer (DSA) • to obtain empirical transfer function for a system • Simulink/MATLAB block for d. SPACE 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 d. SPACE boards § MATLAB code to get empirical transfer function MIT Computer Science & Artificial Intelligence Laboratory

Master’s Thesis § Activ. Lab labware for 2. 003: Modeling Dynamics and Control 1 § 1 st-order dynamics MIT Computer Science & Artificial Intelligence Laboratory

Master’s Thesis § 2 nd- and 4 th-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

Ph. D: 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

Ph. D: Legged Locomotion § Metastability § Fast mixing-time dynamics § Rapid convergence to long-living (metastable) limitcycle behavior MIT Computer Science & Artificial Intelligence Laboratory

Ph. D: Legged Locomotion § Compass gait: optimal vs one-step control MIT Computer Science & Artificial Intelligence Laboratory

Ph. D: Legged Locomotion § Little. Dog: Phase 1 (static crawl) results MIT Computer Science & Artificial Intelligence Laboratory

Ph. D: Legged Locomotion § Little. Dog 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 Kajita 03 § 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 5 x 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 Little. Dog 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