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.– Similar to a van der Pol oscillator– Robot converges into a trajectory instead of

being forced into a trajectory

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

Mathematical Model

Mathematical Model

• Flight: 7 DOF

• Single Stance: 5 DOF

• Double Stance: 3 DOF

• Single Stance Dynamics (by Lagrange):

Impact Model

• Impact is instantaneous (and therefore double stance is instantaneous)

• Impulsive forces may result in an instantaneous change in velocities

Dynamic Model with Impact

• Where S is the set of points where the swing leg touches the ground

Virtual Constraints

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.)

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

Virtual Leg

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

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.

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

HZD Model

• Hybrid zero dynamics of our system are:

• State is a 2 dimensional:

Graphical Interpretation

Graphical Interpretation

Condition for Periodic Solution

Energy Analysis

State Space Orbit