RABBIT: A Testbed for Advanced Control Theory Chevallereau, et. al.
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Transcript of RABBIT: A Testbed for Advanced Control Theory Chevallereau, et. al.
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RABBIT: A Testbed for Advanced Control Theory
Chevallereau, et. al.
Michael Mistry
2/24/04
CLMC Lab
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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
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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
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Mathematical Model
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Mathematical Model
• Flight: 7 DOF
• Single Stance: 5 DOF
• Double Stance: 3 DOF
• Single Stance Dynamics (by Lagrange):
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Impact Model
• Impact is instantaneous (and therefore double stance is instantaneous)
• Impulsive forces may result in an instantaneous change in velocities
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Dynamic Model with Impact
• Where S is the set of points where the swing leg touches the ground
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Virtual Constraints
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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.)
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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
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Virtual Leg
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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
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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.
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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
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HZD Model
• Hybrid zero dynamics of our system are:
• State is a 2 dimensional:
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Graphical Interpretation
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Graphical Interpretation
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Condition for Periodic Solution
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Energy Analysis
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State Space Orbit