Constrained Path Planning using Quadratic Programming

International audienceSampling-based planning algorithms have been extensively exploited to solve a wide variety of problems. In recent years, many efforts have been dedicated to extend these tools to solve problems involving constraints, such as geometric loop-closure, which lead the valid Configuration Space (CS) to collapse to a lower-dimensional manifold. One proposed solution considers an approximation of the constrained Configuration Space that is obtained by relaxing constraints up to a desired tolerance. The resulting set has then non-zero measure, allowing to exploit classical planning algorithms to search for a path connecting two given states. When constraints involve kinematic loops in the system, relaxation generally bears to undesired contact forces, which need to be compensated during execution by a proper control action. We propose a new tool that exploits relaxation to plan in presence of constraints. Local motions inside the approximated manifold are found as the result of an iterative scheme that uses Quadratic Optimization to proceed towards a new sample without falling outside the relaxed region. By properly guiding the exploration, paths are found with smaller relaxation factors and the need of a dedicated controller to compensate errors is reduced. We complete the analysis by showing the feasibility of the approach with experiments on a real platform

A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints

A computationally efficient method to solve non-convex programming problems with linear equality constraints is presented. The proposed method is based on a recursively feasible and descending sequential convex programming procedure proven to converge to a locally optimal solution. Assuming that the first convex problem in the sequence is feasible, these properties are obtained by convexifying the non-convex cost and inequality constraints with inner-convex approximations. Additionally, a computationally efficient method is introduced to obtain inner-convex approximations based on Taylor series expansions. These Taylor-based inner-convex approximations provide the overall algorithm with a quadratic rate of convergence. The proposed method is capable of solving problems of practical interest in real-time. This is illustrated with a numerical simulation of an aerial vehicle trajectory optimization problem on commercial-of-the-shelf embedded computers

Chance Constrained Optimization for Targeted Internet Advertising

We introduce a chance constrained optimization model for the fulfillment of guaranteed display Internet advertising campaigns. The proposed formulation for the allocation of display inventory takes into account the uncertainty of the supply of Internet viewers. We discuss and present theoretical and computational features of the model via Monte Carlo sampling and convex approximations. Theoretical upper and lower bounds are presented along with a numerical substantiation