## Qualitative Hybrid Control of Dynamic Bipedal Walking. Ramamoorthy, Kuipers. RSS 2006.

1. Approach is concerned with dynamic walking on uneven terrain
2. A common approach is to formulate the walking problem as a linear system which can be solved with a number of methods.  This is problematic because it reduces energy efficiency, overly constrains the type of gaits that can be found, and requires constant compensation for what the body naturally does (that introduces non linearity) which introduces other forms of complexity
3. Dynamic walkers utilize, instead of overcome the inherent nonlinearities involved in legged locomotion
4. The idea is to only introduce small corrections into the trajectories that will naturally occur in the system
5. Discuss motivation based on studies of how animals walk, and how humans learn to walk.  Idea is that planning is done in some simpler subspace and then mapped back to the original problem.  This paper is motivated by trying to find simple methods of walking on uneven terrain that can then be mapped back to more complex versions of the problem.
6. Talk about Poincare maps and limit cycles for walking on smooth terrain, but these approaches don’t transfer well to walking on uneven terrain.
7. Method here proposes to use trajectory segments that can be searched over in order to walk on complex terrain
8. They present their mathematical representation of a pendulum walker – three point masses, each at the hip and feet
9. In the swing phase, one foot is planted and there are only two moving masses, which is actually equivalent to the acrobot (only the hip is actuated).  The distinction is that it is assumed the legs can retract or extend instantaneously to clear obstacles.
10. Here, impulses are only provided during double support (when both feet contact the ground), so the problem is very highly constrained.
11. It is modeled as a discrete system, with systems being when one or two feet are planted on the ground
12. Talk about the fact that a driven pendulum is chaotic, so getting everything to synchronize correctly (between the two separate systems of swinging and standing, and dealing with complications due to obstacles) may be difficult
1. The transition to chaos is gradual, “As the driving amplitude is smoothly varied from a small value, only a few trajectories near critical points and the separatrix are affected.”
13. They want to try and avoid situations that could become chaotic, do so by maintaining constraints that keep the system non-chaotic
14. Again, forces are only applied during the double support phase.  After that the leading leg is retracted (and presumably later expanded)
15. The method requires solution of a sequential quadratic program, and the method is admittedly not appropriate for on-line use, and then using some method of storing the policy
16. The result settles into a stable limit cycle when walking on flat terrain
17. Have some experiments walking on terrain that is flat, inclining, and of random heights