Whitman and Atkeson

 Present a decoupled controller for a simulated three-dimensional biped.  Dynamics broke down into multiple subsystems that are controlled seperately.  Policies are brought back into simplified states and control action back onto the full system. 2Cognitive Robotics 2010

 Coordination of multiple policies:  Time till touch down  Dynamic programming simultaneously and globally optimize:  Foot placement  Step timing  Body motion 3Cognitive Robotics 2010

 Paradigm: high degree of freedom (DoF) systems generate a nominal trajectory stabilized with a feedback controller.  Only functions in a small tube or within the state space.  Produce policies that are valid for a large region of the state space (Brice et. al. 2006)  Library of multiple trajectories. 4Cognitive Robotics 2010

 Dynamically stable walking trajectories based on the zero moment point (ZMP).  A desired ZMP trajectory was chosen before the specified footstep locations and timing. Then the CoM trajectory is calculated based on the desired ZMP trajectory. Kajita et. al. (2006) 5Cognitive Robotics 2010

 State space  (1) Pick a new action, best or random.  (2) Update the value function.  Compass gait walker  Point mass on two rigid legs. 6Cognitive Robotics 2010

 Five rigid legs  W: 78 kg (50 torso, 14 legs)  Length; Legs: 0,81 m  CoM: 1,00 m above ground  12 DoF  6 torso; 2 x 2 for hip and 1 for each knee.  3 Pitch joints for ankle.  u = coefficient of friction = 1,0 7Cognitive Robotics 2010

 High dimensional state/action space  Lower dimensional:  Each joint in sagittal/coronal plane. ▪ Dynamics decoupled and control separately. ▪ Left and right coupling at the same time. ▪ Double support is ignored (1% - 2% step) because compass gait. 8Cognitive Robotics 2010

 7 DoF  3 on the torso  4 on the hip and knees.  5 dimensional  Torque at the pitch hip, knees and ankle.  Simplified  Full system 9Cognitive Robotics 2010

 Simplify system  Origin at ankle: -2 DoF.  Ignoring the swing leg: -2 DoF.  Stance knee straight: -1 DoF  Torso at an constant angle: -1 DoF Total DoF will be 1: a two link inverted pendulum with the upper link at a fixed angle. 10Cognitive Robotics 2010

L = 0,81 m L 2 = 0,40 m L 1x = 0,4sin( ϕ ) L 1y = 0,4cos( ϕ ) ϕ = 0,1 rad M = 50 kg m = 14 kg I = ml 2 /3 τ = torque 11Cognitive Robotics 2010

 V actual velocity  V des desired velocity (1,0 m/s)  Fx = ground reaction force.  To full state; 3d vector from stance foot to stance hip in sagittal plane.  Proportional-derivative (PD) used.  K p = 1500 Nm; K d = 150Nm-s; K p = 1000 Nm; K d = 150Nm-s  Leg straight and torso at certain angle. 12Cognitive Robotics 2010

 Shape are similar.  Full model at higher frequency.  Difference in speed and touch in variation touch down model.  Torso bobs forward  Torque at hip applied. 13Cognitive Robotics 2010

 Swing leg controlled by stance leg.  Dynamics and controller are known.  Time and angle at touch down.  Error a few msec. 14Cognitive Robotics 2010

 Bending knee (5cm) above ground: inverse kinematics.  Spline at current angle and velocity.  Match velocity swing and stance leg. 15Cognitive Robotics 2010

 5 DoF  4 Dimensional action space hip and ankle.  Simplified dynamics nearly the same  A third state is added; estimated time until touchdown.  Angle of touchdown variable.  Desired velocity of zero  Added y 2 : legs close to vertical. 16Cognitive Robotics 2010

 Periods match because of period design parameter.  Impact touchdown counters with rolling torso to vertical. 17Cognitive Robotics 2010

 Ankle twist joints are used.  Servoing K p = 500 and K d = 30 Nm-s the joints to zero.  Coupling large because shin axis is in line with coronal plane torques. 18Cognitive Robotics 2010

 Result of pertubation depends on the timing, location and direction of the pertubation.  Unperturbed step: 56 sec.  Front to right not sensible, back to left.  Mid more stable then at the beginning or end. 19Cognitive Robotics 2010

 Friction affect slipping in forward direction.  Changing the height of the perturbation has a significant effect on perturbing torque around the stance foot: tipping  Changing height +20 has shows the location of the perturbation (right). 20Cognitive Robotics 2010

 Two ways:  Torso lean angle  Desired velocity.  Desired velocity 1,0 m/s and lean angle 0,1 rad..  Simplified system loses energy from touchdown. 21Cognitive Robotics 2010

 Change sagital policy forward speed  Estimates of touchdown ankle more accurate.  Changing policies and lean angles in tandem.  First policy 1,0 m/s and second 0,25 m/s.  Little energy is los on short steps at slow speeds. 22Cognitive Robotics 2010

 The couplings between the subsystems functioning in a full system properly.  Study simulation researchers believe it is well suited in real hardware, because simplified.  The control architecture is modular.  Also produce other types of walking. 23Cognitive Robotics 2010

 Future work:  Correspond full dynamics to simplified dynamics  Insert torso dynamics for more accurate touch down model.  Reverse scenario: coronal policy determine the touch down.  Provide a good mechanism which policy was most important to. 24Cognitive Robotics 2010

 Able to generate policies that are valid for a large region of the high-dimensional state of the full system.  Allowing to react on large perturbations. 25Cognitive Robotics 2010

Advantages  Simplified systems are closely related to the full systems.  Simultaneously adjustment. Disadvantages:  Only simulation is used, perturbations in real world?  No double support phase is used in their simulation.  No torso dynamics in simplified system.  Discussion… 26Cognitive Robotics 2010

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