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Metastable Locomotion for LittleDog Katie Byl Robot Locomotion Group July 11, 2007 Goal: piecewise control Repeatable open-loop motions Based on simple (approximate) physical models Good evidence for high repeatability Define a working range for each motion Improve accuracy through trials (learning) Estimate reliability for a particular motion Select short-sighted motion with best “performance” Stability metrics for locomotion on rough terrain: mean first-passage time (MFPT) Speed also a requirement MIT Computer Science & Artificial Intelligence Laboratory Is short-sighted control ok? 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 Metastable dynamics Metastability Fast mixing-time dynamics Rapid convergence to long-living (metastable) limitcycle behavior 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 Motion Control for LittleDog 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 Fastest flat-terrain walk and trot: 17 and 20+ cm/s 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 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 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 Repeatability Bounding: PITCH MIT Computer Science & Artificial Intelligence Laboratory Repeatability Bounding: COB X position MIT Computer Science & Artificial Intelligence Laboratory Sensing Bounding: Pitch via “sensor fusion” MIT Computer Science & Artificial Intelligence Laboratory Repeatability ZMP Walking: Pitch MIT Computer Science & Artificial Intelligence Laboratory Repeatability ZMP Trot-Walking: Pitch MIT Computer Science & Artificial Intelligence Laboratory Repeatability Dynamic climb: (10 consecutive trials) Pitch MIT Computer Science & Artificial Intelligence Laboratory Repeatability Dynamic climb: (10 consecutive trials) Roll MIT Computer Science & Artificial Intelligence Laboratory Repeatability Dynamic climb: (10 consecutive trials) Yaw MIT Computer Science & Artificial Intelligence Laboratory Repeatability Bounding: Force sensor (est. vert. force) MIT Computer Science & Artificial Intelligence Laboratory Repeatability Bounding: accelerometer and force sensors MIT Computer Science & Artificial Intelligence Laboratory Where to go next (post-thesis) 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