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