We present a multi-query recovery policy for a hybrid system with goal limit cycle. The sample trajectories and the hybrid limit cycle of the dynamical system are stabilized using locally valid Time Varying LQR controller policies which probabilistically cover a bounded region of state space. The original LQR Tree algorithm builds such trees for non-linear static and non-hybrid systems like a pendulum or a cart-pole. We leverage the idea of LQR trees to plan with a continuous control set, unlike methods that rely on discretization like dynamic programming to plan for hybrid dynamical systems where it is hard to capture the exact event of discrete transition. We test the algorithm on a compass gait model by stabilizing a dynamic walking hybrid limit cycle with point foot contact from random initial conditions. We show results from the simulation where the system comes back to a stable behavior with initial position or velocity perturbation and noise.
The dynamic feasibility of solutions to motion planning problems using Rapidly Exploring Random Trees depends strongly on the choice of the distance metric used while planning. The ideal distance metric is the optimal cost of traversal between two states in the state space. However, it is computationally intensive to find the optimal cost while planning. We propose a novel approach to overcome this barrier by using a supervised learning algorithm that learns a nonlinear function which is an estimate of the optimal cost, via offline training. We use the Iterative Linear Quadratic Regulator approach for estimating an approximation to the optimal cost and learn this cost using Locally Weighted Projection Regression. We show that the learnt function approximates the original cost with a reasonable tolerance and more importantly, gives a tremendous speed up of a factor of 1000 over the actual computation time. We also use the learnt metric for solving the pendulum swing up planning problem and show that our metric performs better than the popularly used Linear Quadratic Regulator based metric.  Recent advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of stability for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses these results to efficiently combine locally-valid linear quadratic regulator (LQR) controllers into a nonlinear feedback policy which probabilistically covers the reachable area of a (bounded) state space with a region of stability, certifying that all initial conditions that are capable of reaching the goal will stabilize to the goal. We carefully investigate the algorithm on a two-dimensional model system and discuss the potential for the control of more complicated underactuated control problems like bipedal walking. Rapidly-exploring Random Trees (RRTs) are widely used to solve large planning problems where the scope prohibits the feasibility of deterministic solvers, but the efficiency of these algorithms can be severely compromised in the presence of certain kinodynamics constraints. Obstacle fields with tunnels, or tubes are notoriously difficult, as are systems with differential constraints, because the tree grows inefficiently at the boundaries. Here we present a new sampling strategy for the RRT algorithm, based on an estimated feasibility set, which affords a dramatic improvement in performance in these severely constrained systems. We demonstrate the algorithm with a detailed look at the expansion of an RRT in a swingup task, and on path planning for a nonholonomic car. We present an algorithm that probabilistically covers a bounded region of the state space of a nonlinear system with a sparse tree of feedback stabilized trajectories leading to a goal state. The generated tree serves as a lookup table control policy to get any reachable initial condition within that region to the goal. The approach combines motion planning with reasoning about the set of states around a trajectory for which the feedback policy of the trajectory is able to stabilize the system. The key idea is to use a random sample from the bounded region for both motion planning and approximation of the stabilizable sets by falsification; this keeps the number of samples and simulations needed to generate covering policies reasonably low. We simulate the nonlinear system to falsify the stabilizable sets, which allows enforcing input and state constraints. Compared to the algebraic verification using sums of squares optimization in our previous work, the simulation-based approximation of the stabilizable set is less exact, but considerably easier to implement and can be applied to a broader range of nonlinear systems. We show simulation results obtained with model systems and study the performance and robustness of the generated policies. 