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RABBIT: A Testbed for Advanced Control Theory Chevallereau, et. al.

RABBIT: A Testbed for Advanced Control Theory Chevallereau, et. al. Michael Mistry 2/24/04 CLMC Lab. Grizzle vs. ZMP. No trajectory tracking A disturbance will force ASIMO to “catch up” to the planned trajectory Controller creates an asymptotically stable orbit.

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RABBIT: A Testbed for Advanced Control Theory Chevallereau, et. al.

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  1. RABBIT: A Testbed for Advanced Control TheoryChevallereau, et. al. Michael Mistry 2/24/04 CLMC Lab

  2. Grizzle vs. ZMP • No trajectory tracking • A disturbance will force ASIMO to “catch up” to the planned trajectory • Controller creates an asymptotically stable orbit. • Similar to a van der Pol oscillator • Robot converges into a trajectory instead of being forced into a trajectory

  3. Grizzle vs. ZMP • RABBIT is purposefully underactuated • No ankles, no feet • ZMP does not apply • Feedback controller can be computed to be optimal with respect to any cost function • Such as minimal energy

  4. Mathematical Model

  5. Mathematical Model • Flight: 7 DOF • Single Stance: 5 DOF • Double Stance: 3 DOF • Single Stance Dynamics (by Lagrange):

  6. Impact Model • Impact is instantaneous (and therefore double stance is instantaneous) • Impulsive forces may result in an instantaneous change in velocities

  7. Dynamic Model with Impact • Where S is the set of points where the swing leg touches the ground

  8. Virtual Constraints

  9. Virtual Constraints • Cylinder walls apply constraints: • Alternatively, we can apply “virtual constraints” via control laws. Calling the output : • Then control the output to zero (using PD, etc.)

  10. Constraining the RABBIT • 4 constraints + 5 DOF = 1 DOF • Keep torso erect at a nearly vertical angle • Hip height rises and falls during step • Swing foot traces a parabolic trajectory (x,y) • Describe these constraints as functions of the angle of the virtual leg • Virtual leg is a good choice because it is monotonically increasing during a forward step

  11. Virtual Leg

  12. Constraining the RABBIT • Now express four outputs as: • Where θ(q) is a monotonically increasing scalar function of the configuration variables • i.e. virtual leg • Analogous to time • h0 represents the four quantities to be controlled • hd specifies the virtual constraints

  13. Hybrid Zero Dynamics (HZD) • Zero dynamics: the dynamics of the system compatible with the outputs being identically zero • Hybrid because swing phase is continuous but impact phase is discrete.

  14. Hybrid Zero Dynamics • Swing phase zero dynamics has one DOF: • Z is the surface of all points in the state space where outputs are zero • σZ is the angular momentum of the robot about the pivot point of the stance leg • xc is the horizontal distance between pivot point and COG

  15. HZD Model • Hybrid zero dynamics of our system are: • State is a 2 dimensional:

  16. Graphical Interpretation

  17. Graphical Interpretation

  18. Condition for Periodic Solution

  19. Energy Analysis

  20. State Space Orbit

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