Trajectory replanning for quadrotors is essential to enable fully autonomous flight in unknown environments. Hierarchical motion planning frameworks, which combine path planning with path parameterization, are popular due to their time efficiency. However, the path planning cannot properly deal with non-static initial states of the quadrotor, which may result in non-smooth or even dynamically infeasible trajectories. In this paper, we present an efficient kinodynamic replanning framework by exploiting the advantageous properties of the B-spline, which facilitates dealing with the non-static state and guarantees safety and dynamical feasibility. Our framework starts with an efficient B-spline-based kinodynamic (EBK) search algorithm which finds a feasible trajectory with minimum control effort and time. To compensate for the discretization induced by the EBK search, an elastic optimization (EO) approach is proposed to refine the control point placement to the optimal location. Systematic comparisons against the state-of-the-art are conducted to validate the performance. Comprehensive onboard experiments using two different vision-based quadrotors are carried out showing the general applicability of the framework.
 We present an online method for generating collision-free trajectories for autonomous quadrotor flight through cluttered environments. We consider the real-world scenario that the quadrotor aerial robot is equipped with limited sensing and operates in initially unknown environments. During flight, an octree-based environment representation is incrementally built using onboard sensors. Utilizing efficient operations in the octree data structure, we are able to generate free-space flight corridors consisting of large overlapping 3-D grids in an online fashion. A novel optimization-based method then generates smooth trajectories that both are bounded entirely within the safe flight corridor and satisfy higher order dynamical constraints. Our method computes valid trajectories within fractions of a second on a moderately fast computer, thus permitting online re-generation of trajectories for reaction to new obstacles. We build a complete quadrotor testbed with onboard sensing, state estimation, mapping, and control, and integrate the proposed method to show online navigation through complex unknown environments. We address the controller design and the trajectory generation for a quadrotor maneuvering in three dimensions in a tightly constrained setting typical of indoor environments. In such settings, it is necessary to allow for significant excursions of the attitude from the hover state and small angle approximations cannot be justified for the roll and pitch. We develop an algorithm that enables the real-time generation of optimal trajectories through a sequence of 3-D positions and yaw angles, while ensuring safe passage through specified corridors and satisfying constraints on velocities, accelerations and inputs. A nonlinear controller ensures the faithful tracking of these trajectories. Experimental results illustrate the application of the method to fast motion (5–10 body lengths second) in three-dimensional slalom courses. There is extensive literature on using convex optimization to derive piece-wise polynomial trajectories for controlling differential flat systems with applications to three-dimensional flight for Micro Aerial Vehicles. In this work, we propose a method to formulate trajectory generation as a quadratic program (QP) using the concept of a Safe Flight Corridor (SFC). The SFC is a collection of convex overlapping polyhedra that models free space and provides a connected path from the robot to the goal position. We derive an efficient convex decomposition method that builds the SFC from a piece-wise linear skeleton obtained using a fast graph search technique. The SFC provides a set of linear inequality constraints in the QP allowing real-time motion planning. Because the range and field of view of the robot's sensors are limited, we develop a framework of Receding Horizon Planning , which plans trajectories within a finite footprint in the local map, continuously updating the trajectory through a re-planning process. The re-planning process takes between 50 to 300 ms for a large and cluttered map. We show the feasibility of our approach, its completeness and performance, with applications to high-speed flight in both simulated and physical experiments using quadrotors. We explore the challenges of planning trajectories for quadrotors through cluttered indoor environments. We extend the existing work on polynomial trajectory generation by presenting a method of jointly optimizing polynomial path segments in an unconstrained quadratic program that is numerically stable for high-order polynomials and large numbers of segments, and is easily formulated for efficient sparse computation. We also present a technique for automatically selecting the amount of time allocated to each segment, and hence the quadrotor speeds along the path, as a function of a single parameter determining aggressiveness, subject to actuator constraints. The use of polynomial trajectories, coupled with the differentially flat representation of the quadrotor, eliminates the need for computationally intensive sampling and simulation in the high dimensional state space of the vehicle during motion planning. Our approach generates high-quality trajecrtories much faster than purely sampling-based optimal kinodynamic planning methods, but sacrifices the guarantee of asymptotic convergence to the global optimum that those methods provide. We demonstrate the performance of our algorithm by efficiently generating trajectories through challenging indoor spaces and successfully traversing them at speeds up to 8 m s. A demonstration of our algorithm and flight performance is available at: http: groups.csail.mit.edu rrg quad_polynomial_trajectory_planning. In this paper, we present an improved octree-based mapping framework for autonomous navigation of mobile robots. Octree is best known for its memory efficiency for representing large-scale environments. However, existing implementations, including the state-of-the-art OctoMap [1], are computationally too expensive for online applications that require frequent map updates and inquiries. Utilizing the sparse nature of the environment, we propose a ray tracing method with early termination for efficient probabilistic map update. We also propose a divide-and-conquer volume occupancy inquiry method which serves as the core operation for generation of free-space configurations for optimization-based trajectory generation. We experimentally demonstrate that our method maintains the same storage advantage of the original OctoMap, but being computationally more efficient for map update and occupancy inquiry. Finally, by integrating the proposed map structure in a complete navigation pipeline, we show autonomous quadrotor flight through complex environments.