- Walking has a lot that it can take from motion planning.
- Discusses the limit cycle of a walker that only controlls torque at the hip
- How do we make it robust? It is fragile
- Its limit cycle exists in 2D

- How do we walk on rough terrain
- Why is this hard?
- Dynamic constraints
- Underactuated (can’t follow arbitrary trajectories)
- Limits on actuators limit torque and possible speed

- What works well?
- Trajectory optimization (handles constraints naturally)
- Local orbital stabilization (local, more difficult to deal with constraints)
- Global feasible control
- Control dealing with regions of attraction

- Talks about RRTs, example in inverted pendulum (dynamic constraints), and some optimizations to RRTs
- Then moves to little dog
- Plan in the space of “half bound” primitives, so not at the low-level

- These randomized methods can find solutions in these large dynamics domains
- Then attempt to locally stabilize the plans via linear feedback
- Need to know dynamics and the form of the equations for LQR Trees to work
- Have results for limit cycles with impact
- 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
**So, it doesn’t really conflict with other methods of planning, but provides a way to hold on to planning that was done previously**

- Algorithm (roughly)
- Figure out the linearly controllable region around the goal
- Sample a random point in state space
- Attempt to find a trajectory that connects that point to some region that has already been solved for
- They then “stabilize” the trajectory with time varying linear feedback
- Compute the region around the trajectory that is also stable

- It probabalistically covers the space (any region that can reach the goal will be covered eventually)
- The implementation is very complicated and difficult to get right
- Examples of it running swingup on a real robot
- Also have it working on an actual compass gait robot
- Have it working for getting a plane to perch (solved with another method before, but this is more stable)
- 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
- This can be used in concert with system identification to figure out the dynamics

- Works fine in domains with nonlinear dynamics
- Lets you build sparse trajectory libraries
- 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)
- “Hopefully youre using RRTs in 5 years”

- Have done up to 6 action dimension/12 state problems
- Says the sum of squares component doesn’t scale well with high dimensionality
- Says cost is a polynomial of a polynomial, but with high constant factors
**Sees room for RL in deciding exactly how to move around once you fall into a funnel**- 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.

- Seems to cover mostly the same material as the above talk
- Concern with design stabilization of non-periodic trajectories of underactuated system
- One example is a biped walking on rough terrain

- 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
- The application domain here is a compass walker that only has hip actuation
- Claims you can have either: stability and versatility or efficiency and naturalism, but not all four, paper tries to give all
- For the domain they focus on, they make a provably stable controller
- Two classes of walkers:
- 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
- Passive-dynamic walkers do not have full actuation and allow gravity and other aspects of the dynamics to play a large role in locomotion

- Dynamic walking on uneven terrain is a relatively undeveloped topic, although there are a few references, a number by the authors
- This paper also discusses Poincare maps. Came up before in an earlier paper, not sure which authors.
- The problem is that these maps are only defined for periodic limit cycles, which does not exist on uneven terrain

- The method of transverse linearization was also previously used for periodic systems, but here is extended to nonperiodic systems.
- The advantage is the representation is continuous, as opposed to Poincare maps which occur at a few fixed locations in the limit cycle

- Stabilizing only the transverse dynamics is useful in underactuated systems
- Model is a nonlinear system subject to instantaneous impacts
- Time intervals lie in periods where there is no impact, time advances a step when an impact occurs
- Assumptions made are pretty weak
- “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
- Computing the transverse linearization can be done analytically, there is a little extra that has to be added due to the contacts
- The MPC they are doing is LQR
- The rollots are done to a fixed number of footsteps in the future
- Looks like they got this to run on an actual compass-walker going down 2 (very short) stairs, although I’m not 100% sure