diff --git a/lqr.html b/lqr.html index 760cb7f1..32c7414f 100644 --- a/lqr.html +++ b/lqr.html @@ -531,7 +531,7 @@

Underactuated Robotics -

LQR with input and state constraints

+

LQR with input and state constraints

A natural extension for linear optimal control is the consideration of strict constraints on the inputs or state trajectory. Most common are linear inequality constraints, such as $\forall n, |\bu[n]| \le 1$ or @@ -548,18 +548,6 @@

Underactuated Roboticstrajectory optimization chapter.

- -

We do actually understand what the optimal policy of the - inequality-constrained LQR problem looks like, thanks to work on - "explicit MPC" Alessio09 -- the optimal policy is now - piecewise-linear (though still continuous), with each piece described by - a polytope, and the optimal cost-to-go is piecewise-quadratic on the same - polytopes. Unfortunately, the number of pieces grows exponentially with - the number of constraints and the horizon of the problem, making it - impractical to compute for all but very small problems. There are, - howeer, a number of promising approaches to approximate explicit MPC - (c.f. - Marcucci17).

@@ -910,19 +898,6 @@

Underactuated Robotics

References

    -
  1. -A. Alessio and A. Bemporad, -"A survey on explicit model predictive control", -Int. Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control , 2009. - -

  2. -
  3. -Tobia Marcucci and Robin Deits and Marco Gabiccini and Antonio Bicchi and Russ Tedrake, -"Approximate Hybrid Model Predictive Control for Multi-Contact Push Recovery in Complex Environments", -Humanoid Robots (Humanoids), 2017 IEEE-RAS 17th International Conference on , 2017. -[ link ] - -

  4. Michael Posa and Scott Kuindersma and Russ Tedrake, "Optimization and stabilization of trajectories for constrained dynamical systems", diff --git a/trajopt.html b/trajopt.html index 5aa44ad3..d6791f3d 100644 --- a/trajopt.html +++ b/trajopt.html @@ -1363,8 +1363,17 @@

    The special case of direct shooting without state constraints

    Explicit model-predictive control

    -

    Alessio09

    - +

    We do actually understand what the optimal policy of the + inequality-constrained LQR problem looks like, thanks to work on + "explicit MPC" Alessio09 -- the optimal policy is now + piecewise-linear (though still continuous), with each piece described by + a polytope, and the optimal cost-to-go is piecewise-quadratic on the same + polytopes. Unfortunately, the number of pieces grows exponentially with + the number of constraints and the horizon of the problem, making it + impractical to compute for all but very small problems. There are, + however, a number of promising approaches to approximate explicit MPC + (c.f. Marcucci17).

    +
@@ -1805,6 +1814,13 @@

The special case of direct shooting without state constraints

"A survey on explicit model predictive control", Int. Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control , 2009. +
+
  • +Tobia Marcucci and Robin Deits and Marco Gabiccini and Antonio Bicchi and Russ Tedrake, +"Approximate Hybrid Model Predictive Control for Multi-Contact Push Recovery in Complex Environments", +Humanoid Robots (Humanoids), 2017 IEEE-RAS 17th International Conference on , 2017. +[ link ] +

  • O. Junge and J. E. Marsden and S. Ober-Bloebaum,