I Introduction
Unmanned aerial vehicles, especially quadrotors have gained widespread popularity in various applications, such as inspection, precision agriculture, and search and rescue. Among the tasks, autonomous exploration, in which the vehicle explores and maps the unknown environment to gather information, is a fundamental component to enable effective operation. Despite the various exploration planning methods presented in recent years, most of them are demonstrated to have deficient efficiency, prohibiting them from realworld usage. First of all, existing planners plan exploring motions in greedy manners, such as maximizing the immediate information gain, or navigating to the closest unknown region. The greedy strategies ignore global optimality and therefore result in low overall efficiency. Besides, most methods generate rather conservative motions in order to guarantee information gain and safety simultaneously in previously unknown environments. Lowspeed exploration, however, disallows quadrotors to fully exploit their dynamic capability to fulfill the missions. Last but not least, many methods suffer from high computational overhead and can not response quickly and frequently to environmental changes. However, to enable faster exploration, it is desirable to replan new motion immediately whenever new information of the environment is available.
Motivated by the above facts, this paper proposes FUEL, a hierarchical framework that can support Fast UAV ExpLoration in complex environments. We introduce a frontier information structure (FIS), which contains essential information in the entire space required by exploration planning. The structure can be updated efficiently and incrementally when new information is collected, so it is capable of supporting high frequency planning. Based on the FIS, a hierarchical planner generates exploring motion in three coarsetofine steps. It starts by finding a global exploration tour that is optimal in the context of accumulated environment information. Then, viewpoints on a local segment of the tour are refined, further improving the exploration rate. Finally, a safe, dynamically feasible and minimumtime trajectory is generated. The planner produces not only efficient global coverage path, but also safe and agile local manuevers. Moreover, the planner is triggered whenever new regions are explored, so that the quadrotor always responses promptly to any environmental changes, leading to consistently fast exploration. We compare our method with classic and stateoftheart methods in simulation. Results show that in all cases our method achieves complete exploration in much shorter time (38 times faster). What’s more, we conduct fully onboard realworld exploration in various challenging environments. Both the simulation and realworld tests demonstrate unprecedented performance of our method compared to stateoftheart ones. To benefit the community, we will make the source code public. The contributions of this paper are summarized as follows: 1) An incrementally updated FIS, which captures essential information of the entire explored space and facilitates exploration planning in high frequency. 2) A hierarchical planning method, which generates efficient global coverage paths, and safe and agile local maneuvers for highspeed exploration. 3) Extensive simulation and realword tests that validate the proposed method. The source code of our system will be made public.
Ii Related Work
Iia Exploration Path Planning
Robotic exploration, which deals with the problem of mapping unknown environments quickly and completely, has been studied for years. Among the various proposed methods, the frontierbased approaches are one type of classic approaches. The methods are first introduced in [33] and evaluated more comprehensively afterwards in [15]. To detect frontiers in 3D space, a stochastic differential equationbased method is proposed in [26]. In the original method[33], the closest frontier is selected as the next target. [5] presented a different scheme of frontier selection to enable higher flight speed, which improves the exploration rate. In [8], a differentiable measure of information gain based on frontier is introduced, allowing path to be optimized with gradient information. Samplingbased exploration, as another type of major approaches, generate viewpoints randomly to explore the space. These methods are closely related to the concept of next best view (NBV) [6], which computes covering views repeatedly to obtain a complete model of a scene. [1] first used NBV in 3D exploration, in which it expands RRTs with accessible space and executes the edge with highest information gain in a receding horizon fashion. The method was extended to consider uncertainty of localization[20], visual importance of different objects [7] and inspection tasks [2] later. To avoid discarding the expanded trees directly, roadmaps are constructed in [31, 30] to reuse previous knowledge. [24] maintains and refines a single tree continuously using a rewiring scheme inspired by RRT*. To achieve faster flight, [9] samples safe and dynamically feasible motion primitives directly and execute the most informative one. There are also methods combining the advantages of frontierbased and samplingbased approaches. [3, 25] plan global path toward frontiers and sample path locally. [3] also presented a gradientbased method to optimize the local path. [18] samples viewpoints around frontiers and finds the global shortest tour passing through them. [27] generates inspection paths that cover the frontier completely using a samplingbased algorithm. Most of existing methods make decision greedily and does not account for the dynamics of the quadrotor, which leads to inefficient global tours and conservative maneuvers. In contrast, we plan tours that efficiently cover the entire environment and generate dynamically feasible minimumtime trajectories to enable agile flight.
IiB Quadrotor Trajectory Planning
Trajectory planning for quadrotor has been widely studied recently. There are two major categories of approaches, the hardconstrained and softconstrained ones. Hardconstrained methods are pioneered by minimumsnap trajectory[17], which generates polynomial trajectories of the flat outputs of quadrotors through quadratic programming(QP). After it, closedform solution was presented [23]. Based on [17], [4, 12, 10] extract convex shapes safe regions and solve QP within them to generate smooth and safe trajectories. To obtain a more reasonable time allocation, fast marching[12], kinodynamic search[10] and mixed integerbased methods [28] are proposed. [12] also introduced an efficient Bézier curvebased method to guarantee feasibility. Softconstrained methods typically formulate trajectory generation as nonlinear optimization trading off several objectives. Recently [19, 11, 29, 35, 34, 36] applied them to the local replanning problem, in which the methods demonstrate attractive performance. The methods were revived by [22] and extended to continuoustime quadrotor trajectories[19] later. To relieve the issue of local minima, [11] initializes the optimization with collisionfree paths. [29] introduces uniform Bsplines and utilizes the continuity and locality properties of them for replanning. More recently, [35] further exploited Bsplines and demonstrated fast flight in field tests. [35] is further improved with topological guiding paths and perceptionawareness in [34, 36]. In this paper, we base our trajectory planning on [35] but extend it to optimize all parameters of Bsplines. In this way, the total trajectory time can be minimized so that the unknown space is explored with higher navigation speed.
Iii System Overview
The proposed framework operates upon a voxel grid map. As illustrated in Fig.2, it is composed of an incremental update of the FIS (Sect.IV) and a hierarchical exploration planning approach (Sect.V). Whenever the map is updated using sensor measurements, it is examined whether any frontier clusters are influenced. If so, FISs of influenced clusters are removed while new frontiers along with their FISs are extracted (Sect.IV). After that, the exploration planning is triggered, which finds global exploration tour, refines local viewpoints, and generates trajectory to a selected viewpoint successively (Sect.V). The exploration is considered finished if no frontier exists.
Iv Incremental Frontier Information Structure
As presented in classic frontierbased exploration [33], frontiers are defined as knownfree voxels right adjacent to unknown voxels, which are grouped into clusters to guide the navigation. Traditionally, the extracted information is too coarse to do finegrained decision making. Besides, frontiers are retrieved by processing the entire map, which is not scalable for large scenes and high planning frequencies. In this work, we extract richer information from frontiers to enable more elaborate planning, and develop an incremental approach to detect frontiers within locally updated map.
Iva Frontier Information Structure
A frontier information structure is computed when a new frontier cluster is created. It stores all cells belonging to the cluster and the average position of . The axisaligned bounding box (AABB) of the cluster is also computed, in order to accelerate the detection of frontier changes (Sect.IVB). To serve the exploration planning (Sect.V), candidate viewpoints are generated around the cluster. Besides, a doubly linked list containing connection costs between and all other clusters is computed. Data stored by a FIS is listed in Tab.I.
IvB Incremental Frontier Detection and Clustering
As depicted in Fig.3, every time the map is updated by sensor measurements, the AABB of the updated region is also recorded, within which outdated frontier clusters are removed and new ones are searched. It starts with going through all clusters and returning only those whose AABBs () intersect with
. Then, precise checks are conducted for the returned clusters, among which the ones containing cells that are no longer frontier are removed. These two processes are inspired by the broad/narrow phase collision detection algorithms, which eliminate most unaffected clusters in a fast way and significantly reduce the number of expensive precise checks. After the removal, new frontiers are searched and clustered into groups by the region growing algorithm, similar to classic frontierbased method. Among the groups, the ones with small number of cells typically resulting from noisy sensor observations are ignored. The remaining groups, however, may contain largesize clusters which are not conducive to distinguishing distinctive unknown regions and making elaborate decisions. Therefore, we perform Principal Component Analysis (PCA) for each cluster and split it into two uniform ones along the first principal axis, if the largest eigenvalue exceeds a threshold. The split is conducted recursively so that all large clusters are divided into small ones.
Data  Explanation 

Frontier cells that belong to the cluster  
Average position of  
Axisaligned bounding box of  
Viewpoints covering the cluster  
Doubly linked list of connection costs to all other clusters 
IvC Viewpoint Generation and Cost Update
Intuitively, a frontier cluster implies a potential destination to explore the space. However, unlike previous approaches which simply navigating to the center of a cluster, we desire more elaborate decision making. To this end, when a cluster is created, we generate a rich set of viewpoints covering it, where . are found by uniformly sampling points in the cylindrical coordinate system whose origin locates at the cluster’s center, as displayed in Fig.4. For each of the sampled points lying within the free space, the yaw angle is determined as the one maximizing sensor coverage to the cluster, by using an yaw optimization method similar to[31]. The coverage is evaluated as the number of frontier cells that comply with the sensor model and are not occluded by occupied voxels. Then, viewpoints with coverage higher than a threshold are reserved and sorted in descending order of coverage. We reserve at most viewpoints in () to make the local viewpoint refinement (Sect.V) tractable. To perform global planning of exploration tour (Sect.V), a connection cost between each pair of clusters is required. Let denotes a time lower bound when moving between two viewpoints and , it is computed by:
(1)  
where denote the shortest path between and , and are the limits of velocity and angular rate of yaw. For each pair
, we select the viewpoints with highest coverage and estimate the cost as
, in which is searched using the A* algorithm. Note that computing connection costs between all pairs of clusters from scratch requires A* searching, which is considerably expensive. Fortunately, the costs can also be computed in an incremental manner. When any outdated clusters are removed (Sect.IVB), associated cost items in of all remaining FISs are erased. After that, connection costs from each new cluster to all other clusters are computed and inserted into . Suppose there are new clusters in each frontier detection, the above update scheme takes time. Practically, is small and can be regarded as a constant factor, resulting in a linear time update of connection costs.V Hierarchical Exploration Planning
Instead of adopting greedy exploration strategies or generating conservative maneuvers, we produce global path to cover the frontiers efficiently and plan safe and agile motions for faster flight. Our planner takes inspiration from the recent hierarchical quadrotor planning paradigm [4, 12, 35], and make decisions in three phases, as showed in Fig.5.
Va Global Exploration Tour Planning
Our exploration planning begins with finding a global tour to cover existent frontier clusters efficiently. Inspired by[18], we formulate it as a variant of the Traveling Salesman Problem (TSP), which computes an openloop tour starting from the current viewpoint and passing viewpoints at all clusters. We reduce this variant to an standard Asymmetric TSP (ATSP) that can be solved quickly by available algorithms, by properly designing the engaged cost matrix . Assume there are clusters totally, corresponds to a dimensions square matrix. The major part is the block composed of the connection cost between each pair of frontier clusters, which is computed as:
(2)  
As mentioned in Sect.IVC, this information is maintained when frontiers are detected. Thus, the block can be filled without extra overhead. The first row and column of are associated with the current viewpoint and clusters. Starting from , the cost to the th cluster is evaluated by:
(3)  
here a motion consistency cost is introduced, which is generally computed as:
(4) 
where is the current velocity. In some cases, multiple tours have comparable time lower bound, so backandforth maneuvers may be generated in successive planning steps and slow down the progress. We eliminate this inconsistency with , which penalizes large changes in flight direction. Our problem is different from standard TSP whose solution is a closedloop tour. However, we can reduce it to an ATSP by assigning zero connection costs from other clusters to the current viewpoint:
(5) 
In this way, going back to the current viewpoint in any closedloop tours contributes no extra cost, so each closedloop tour always contains an openloop one with identical cost. As a result, we can obtain the optimal openloop tour by finding the optimal closedloop one and retrieving its equalcost openloop tour.
VB Local Viewpoint Refinement
The global tour planning finds a promising order to visit all clusters. Nonetheless, it involves only single viewpoint of each cluster, which are not necessarily the best combination among all viewpoints. To this end, a richer set of viewpoints on a truncated segment of the global tour are considered to further improve the exploration rate, by using a graph search approach, as depicted in Fig.6. We found consecutive clusters whose viewpoints on the global tour are closer than to the current position. To simplify the notation, suppose that are the considered clusters. We create graph nodes for their viewpoints and the current viewpoint . Then each node is connected to other nodes associated with the next cluster with a directed edge, which compose a directed acyclic graph capturing possible variation of the truncated tour. We utilize the Dijkstra algorithm to search for the optimal local tour that minimizes the cost:
(6)  
which also consists of time lower bounds and motion consistency.
VC Minimumtime Bspline Trajectory
Given the discrete viewpoints, continuous trajectories are required for smooth navigation. Our quadrotor trajectory planning is based on a stateoftheart method[35], which generates smooth, safe and dynamically feasible Bspline trajectories. We go one step further to optimize all parameters of Bsplines, so that the total trajectory time is minimized to enables the quadrotor to fully utilize its dynamic capability. As the quadrotor dynamics are differentially flat[17], we plan trajectories for the flat outputs . Let where be the control points of a degree uniform Bspline, and be the knot span. We find the Bspline that tradesoff smoothness and total trajectory time, and satisfies safety, dynamic feasibility and boundary state constraints. It can be formulated as an following optimization problem:
(7) 
Similar to [35], is the elastic band smoothness cost:
(8) 
in which is the penalty matrix:
(9) 
is the total trajectory time depending on and the number of Bspline segments:
(10) 
, and are the penalties to ensure safety and dynamic feasibility. Given the following function:
(11) 
is evaluated as:
(12) 
where is the distance of point to the nearest obstacle, which can be obtained from the Euclidean signed distance field (ESDF) maintained by the mapping module. and penalize infeasible velocity and acceleration:
(13) 
(14) 
in which the control points of derivatives are utilized:
(15)  
(16) 
In Equ.12, 13 and 14, the convex hull property of Bspline is utilized to ensure the feasibility efficiently. For brevity we refer the reader to [35] for more details. Lastly, we set the 0th to 2th order derivatives at the start to the instantaneous state for smooth motion. The 0th order derivative at the end is also determined by the viewpoint to be visited. In implementation we use cubic Bsplines, so the associated cost is:
(17)  
Vi Results
Via Implementation Details
. In global tour planning, the ATSP is solved using a LinKernighanHelsgaun heuristic solver
[14]. In local viewpoint refinement, we reserve at most viewpoints in each FIS and truncate the global tour within radius . For trajectory optimization, we use , , , , and and solve the problem with a general nonlinear optimization solver NLopt^{2}^{2}2https://nlopt.readthedocs.io/en/latest/. Cubic Bspline () is used as the trajectory representation. In all field experiments, we localize the quadrotor by a visualinertial state estimator[21]. A volumetric mapping framework[13] is utilized to map the environments. Similar to [32], it builds an occupancy grid map and also maintains an ESDF to facilitate the trajectory planning. We use a geometric controller[16] for tracking control of the trajectory. We equipped our customized quadrotor platform with an Intel RealSense Depth Camera D435i. All the above modules run on an Intel Core i78550U CPU.ViB Benchmark and Analysis
Scene  Method  Exploration time (s)  Flight distance (m)  

Avg  Std  Max  Min  Avg  Std  Max  Min  
Bridge  Classic[33]  575  53  643  511  250  42  285  190  
Rapid[5]  288  15  305  264  286  13  303  269  
NBVP[1]  857  117  1018  740  322  47  377  261  
Proposed  104  1.5  105  102  165  3.8  170  161  

Classic[33]  814  104  961  721  419  63  509  373  
Rapid[5]  669  68  766  613  469  32  514  440  
NBVP[1]  1037  152  1253  925  1539  262  1898  1279  
Proposed  168  16  192  156  280  20  310  264 
We test our proposed framework in simulation. We benchmark it in a bridge scenario and a large maze scenario. Three methods are compared: the NBVP [1], the classic frontier method[33] and the rapid frontier method[5]. Note that no open source code is available for [5], so we use our implementation. In all tests the dynamic limits are set as m/s and rad/s for all methods. The FOVs of the sensors are set as with a maximum range of m. In both scenarios each method is run for 3 times with the same initial configuration.
ViB1 Bridge Scenario
Firstly, we compare the four methods in a space containing a bridge, as showed in Fig.7. We show the fastest exploration progress of each method among all tests in Fig.8 and list the statistics in Tab.II
, which indicates that we achieve much shorter exploration time and smaller time variance. As can be seen, the overall exploration path of our method is significantly shorter, primarily because we plan tour globally. The executed path is also smoother, since we refine motion locally and generate smooth trajectory. Finally, we are able to navigate at a higher flight speed, owing to the minimumtime trajectory planning.
ViB2 Large Maze Scenario
We also compare the methods quantitatively in a large maze environment shown in Fig.9. The explored space is large. The resulting statistics and exploration progress are also presented in Tab.II and Fig.8. In this scenario, all the benchmarked methods take long time to reach full coverage, due to the complexity of the scene. In contrast, our method complete the exploration 4+ times faster in average. Path executed by the four methods after completion are displayed in Fig.9.
ViC Field Exploration Tests
To further validate the proposed method, we conduct extensive field experiments in both indoor and outdoor environments. In all tests we set the dynamics limits as m/s, m/s and rad/s. Note that we do not use any external device for localization and only rely on the onboard state estimator. First, we present fast exploration tests in two indoor scenes. The first scene is shown in Fig.1, within which we deploy dozens of obstacles and the quadrotor should perform 3D maneuvers to map the unknown space and avoid obstacles simultaneously. We bound the space to be explored with a box. One sample map and the flight trajectory is presented in Fig.1. The second indoor scene is a larger environment including two rooms, where one room is similar to scene 1 and the other is a part of an office containing tables and chairs. The space is bounded by a box. The quadrotor starts by exploring the large room, after which it proceeds to the small one. The second scene, the online generated map and trajectory are shown in Fig.10. Note that in the two scenes the quadrotor starts out at a spot with low visibility, so it only maps a small region of the environment at the beginning. Finally, to validate our method in natural environments, we conduct exploration tests in a forest. The size of the area to explore is . The experiment environment and the associated results are displayed Fig.11. The above experiments demonstrate the capability of our method in complex realworld scenarios. They also show the merits of our autonomous quadrotor system, among which the state estimation[21] and mapping modules[13] are also crucial to fullfil the realworld tasks. Video demonstration of all experiments is available (Fig.1), we refer the readers to it for more details.
Vii Conclusions
In this paper, we propose a hierarchical framework for rapid autonomous quadrotor exploration. An incrementally maintained FIS is introduced to provide the exploration planning with essential information. Based on FIS, a hierarchical planner plans exploration motions in three sequential steps, which finds efficient global tours, selects a local set of optimal viewpoints, and generates minimumtime local trajectories. The method makes decisions in high frequency to response quickly to environment changes. Both benchmark and realworld tests show the competence of our method. As most methods do, currently our method assumes that state estimation is good enough and the side effects of pose drift can be ignored. However, error in state estimation exists generally and has great influence on the generated map especially in largescale environments. In the future we plan to consider the state estimation uncertainty in our method and conduct autonomous exploration in larger scenes.
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