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

3. 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

4. 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

5. Mathematical Model

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

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

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

9. Virtual Constraints

10. 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.)

11. 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

12. Virtual Leg

13. 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

14. 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.

15. 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

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

17. Graphical Interpretation

18. Graphical Interpretation

19. Condition for Periodic Solution

20. Energy Analysis

21. State Space Orbit