## Dynamic Walking 2010. Russ Tedrake. Feedback Motion Planning via Sums-of-Squares Verification. Talk

1. Walking has a lot that it can take from motion planning.
2. Discusses the limit cycle of a walker that only controlls torque at the hip
1. How do we make it robust?  It is fragile
2. Its limit cycle exists in 2D
3. How do we walk on rough terrain
4. Why is this hard?
1. Dynamic constraints
2. Underactuated (can’t follow arbitrary trajectories)
3. Limits on actuators limit torque and possible speed
5. What works well?
1. Trajectory optimization (handles constraints naturally)
2. Local orbital stabilization (local, more difficult to deal with constraints)
3. Global feasible control
4. Control dealing with regions of attraction
6. Talks about RRTs, example in inverted pendulum (dynamic constraints), and some optimizations to RRTs
7. Then moves to little dog
1. Plan in the space of “half bound” primitives, so not at the low-level
8. These randomized methods can find solutions in these large dynamics domains
9. Then attempt to locally stabilize the plans via linear feedback
10. Need to know dynamics and the form of the equations for LQR Trees to work
11. Have results for limit cycles with impact
12. The magic isn’t connecting the random point to the goal, its in computing the region around that trajectory so that information can be re-used
1. So, it doesn’t really conflict with other methods of planning, but provides a way to hold on to planning that was done previously
13. Algorithm (roughly)
1. Figure out the linearly controllable region around the goal
2. Sample a random point in state space
3. Attempt to find a trajectory that connects that point to some region that has already been solved for
4. They then “stabilize” the trajectory with time varying linear feedback
5. Compute the region around the trajectory that is also stable
14. It probabalistically covers the space (any region that can reach the goal will be covered eventually)
15. The implementation is very complicated and difficult to get right
16. Examples of it running swingup on a real robot
17. Also have it working on an actual compass gait robot
18. Have it working for getting a plane to perch (solved with another method before, but this is more stable)
19. He says he used to be against model-based approaches with walking and more was into learning, but the machinery available when you know the mechanics is so much more powerful that he prefers those methods now
1. This can be used in concert with system identification to figure out the dynamics
20. Works fine in domains with nonlinear dynamics
21. Lets you build sparse trajectory libraries
22. Says planning is the next interesting thing to talk about in that field, moving from limit cycles to optimization to planning (limit cycles don’t work in uneven terrain)
1. “Hopefully youre using RRTs in 5 years”
23. Have done up to 6 action dimension/12 state problems
24. Says the sum of squares component doesn’t scale well with high dimensionality
25. Says cost is a polynomial of a polynomial, but with high constant factors
26. Sees room for RL in deciding exactly how to move around once you fall into a funnel
27. One of the questions at the end mentions work by Sontag

# Stable Dynamic Walking over Rough Terrain.  Theory and Experiment.  Manchester, Mettin, Iida, Tedrake.  Robotics Research 2011.

1. Seems to cover mostly the same material as the above talk
2. Concern with design stabilization of non-periodic trajectories of underactuated system
1. One example is a biped walking on rough terrain
3. The goal is to do so by computing a “transverse linearization about the desired motion: a linear impulsive system which locally represents dynamics about a target trajectory” which is then stabilized using receding horizon control
4. The application domain here is a compass walker that only has hip actuation
5. Claims you can have either: stability and versatility or efficiency and naturalism, but not all four, paper tries to give all
6. For the domain they focus on, they make a provably stable controller
7. Two classes of walkers:
1. Discuss ZMP, which ensures the center of gravity is always within the foot/feet on ground.  This can be computed with standard tools, but produces conservative, inefficient, unnatural gaits
2. Passive-dynamic walkers do not have full actuation and allow gravity and other aspects of the dynamics to play a large role in locomotion
8. Dynamic walking on uneven terrain is a relatively undeveloped topic, although there are a few references, a number by the authors
9. This paper also discusses Poincare maps.  Came up before in an earlier paper, not sure which authors.
1. The problem is that these maps are only defined for periodic limit cycles, which does not exist on uneven terrain
10. The method of transverse linearization was also previously used for periodic systems, but here is extended to nonperiodic systems.
1. The advantage is the representation is continuous, as opposed to Poincare maps which occur at a few fixed locations in the limit cycle
11. Stabilizing only the transverse dynamics is useful in underactuated systems
12. Model is a nonlinear system subject to instantaneous impacts
13. Time intervals lie in periods where there is no impact, time advances a step when an impact occurs
14. Assumptions made are pretty weak
15. “The stabilization of the transverse linearization implies local exponential orbital stabilization of the original nonlinear system to the target motion.  A transverse linearization is a time-varying linear system representing of the dynamics close to the target motion.”  Stabilizing the transverse linearization implies stabilizing the original domain
16. Computing the transverse linearization can be done analytically, there is a little extra that has to be added due to the contacts
17. The MPC they are doing is LQR
18. The rollots are done to a fixed number of footsteps in the future
19. Looks like they got this to run on an actual compass-walker going down 2 (very short) stairs, although I’m not 100% sure