In this paper, we propose a time-efficient approach to generate safe, smooth and dynamically feasible trajectories for quadrotors in obstacle-cluttered environment. By using the uniform B-spline to represent trajectories, we transform the trajectory planning to a graph-search problem of B-spline control points in discretized space. Highly strict convex hull property of B-spline is derived to guarantee the dynamical feasibility of the entire trajectory. A novel non-uniform kinodynamic search strategy is adopted, and the step length is dynamically adjusted during the search process according to the Euclidean signed distance field (ESDF), making the trajectory achieve reasonable time-allocation and be away from obstacles. Non-static initial and goal states are allowed, therefore it can be used for online local replanning as well as global planning. Extensive simulation and hardware experiments show that our method achieves higher performance comparing with the state-of-the-art method.
As to the time-allocation of trajectory, which is another significant factor that affects the quality of trajectories. @cite_19 refined the control points of B-spline while total duration remains fixed. @cite_14 sampled the duration of segment until the related quadratic program is solved, but it may be not suitable to generate trajectories with many segments. Heuristic strategy may be reasonable and effective @cite_0 . For example, @cite_1 generated time-indexed path according to the density of obstacle, but the final trajectories do not strictly follow the heuristics after optimization. In this paper, we simultaneously deal with kinodynamic search and time-allocation of the trajectory, which shows noticeable benefits.
@cite_0 @cite_19 @cite_14 @cite_1
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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. 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. In this work, we propose a search-based planning method to compute dynamically feasible trajectories for a quadrotor flying in an obstacle-cluttered environment. Our approach searches for smooth, minimum-time trajectories by exploring the map using a set of short-duration motion primitives. The primitives are generated by solving an optimal control problem and induce a finite lattice discretization on the state space which can be explored using a graph-search algorithm. The proposed approach is able to generate resolution-complete (i.e., optimal in the discretized space), safe, dynamically feasibility trajectories efficiently by exploiting the explicit solution of a Linear Quadratic Minimum Time problem. It does not assume a hovering initial condition and, hence, is suitable for fast online re-planning while the robot is moving. Quadrotor navigation with online re-planning is demonstrated using the proposed approach in simulation and physical experiments and comparisons with trajectory generation based on state-of-art quadratic programming are presented. In this paper, we propose a citywide and real-time model for estimating the travel time of any path (represented as a sequence of connected road segments) in real time in a city, based on the GPS trajectories of vehicles received in current time slots and over a period of history as well as map data sources. Though this is a strategically important task in many traffic monitoring and routing systems, the problem has not been well solved yet given the following three challenges. The first is the data sparsity problem, i.e., many road segments may not be traveled by any GPS-equipped vehicles in present time slot. In most cases, we cannot find a trajectory exactly traversing a query path either. Second, for the fragment of a path with trajectories, they are multiple ways of using (or combining) the trajectories to estimate the corresponding travel time. Finding an optimal combination is a challenging problem, subject to a tradeoff between the length of a path and the number of trajectories traversing the path (i.e., support). Third, we need to instantly answer users' queries which may occur in any part of a given city. This calls for an efficient, scalable and effective solution that can enable a citywide and real-time travel time estimation. To address these challenges, we model different drivers' travel times on different road segments in different time slots with a three dimension tensor. Combined with geospatial, temporal and historical contexts learned from trajectories and map data, we fill in the tensor's missing values through a context-aware tensor decomposition approach. We then devise and prove an object function to model the aforementioned tradeoff, with which we find the most optimal concatenation of trajectories for an estimate through a dynamic programming solution. In addition, we propose using frequent trajectory patterns (mined from historical trajectories) to scale down the candidates of concatenation and a suffix-tree-based index to manage the trajectories received in the present time slot. We evaluate our method based on extensive experiments, using GPS trajectories generated by more than 32,000 taxis over a period of two months. The results demonstrate the effectiveness, efficiency and scalability of our method beyond baseline approaches.