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In this paper, we utilize the variational structure to study the existence and asymptotic profiles of ground states in multi-population ergodic Mean-field Games systems subject to some local couplings with mass critical exponents. Of concern the attractive and repulsive interactions, we impose some mild conditions on trapping potentials and firstly classify the existence of ground states in terms of intra-population and interaction coefficients. Next, as the intra-population and inter-population coefficients approach some critical values, we show the ground states blow up at one of global minima of potential functions and the corresponding profiles are captured by ground states to potential-free Mean-field Games systems for single population up to translations and rescalings. Moreover, under certain types of potential functions, we establish the refined blow-up profiles of corresponding ground states. In particular, we show that the ground states concentrate at the flattest global minima of potentials.

Mean-field Games systems (MFGs) serve as paradigms to describe the games among a huge number of players. In this paper, we consider the ergodic Mean-field Games systems in the bounded domain with Neumann boundary conditions and the decreasing Choquard coupling. Our results provide sufficient conditions for the existence of solutions to MFGs with Choquard-type coupling. More specifically, in the mass-subcritical and critical regimes, the solutions are characterized as global minimizers of the associated energy functional. In the case of mass supercritical exponents, up to the Sobolev critical threshold, the solutions correspond to local minimizers. The proof is based on variational methods, in which the regularization approximation, convex duality argument, elliptic regularity and Hardy-Littlewood-Sobolev inequality are comprehensively employed.

In this paper, we investigate the concentration behaviors of ground states to stationary Mean-field Games systems (MFGs) with the nonlocal coupling in $\mathbb R^n$, $n\geq 2.$ With the mass critical exponent imposed on Riesz potentials, we first discuss the existence of ground states to potential-free MFGs, which corresponds to the establishment of Gagliardo-Nirenberg type's inequality. Next, with the aid of the optimal inequality, we classify the existence of ground states to stationary MFGs with Hartree-type coupling in terms of the $L^1$-norm of population density defined by $M$. In addition, under certain types of coercive potentials, the asymptotics of ground states to ergodic MFGs with the nonlocal coupling are captured. Moreover, if the local polynomial expansions are imposed on potentials, we study the refined asymptotic behaviors of ground states and show that they concentrate on the flattest minima of potentials.

Aerial swarm systems possess immense potential in various aspects, such as cooperative exploration, target tracking, search and rescue. Efficient, accurate self and mutual state estimation are the critical preconditions for completing these swarm tasks, which remain challenging research topics. This paper proposes Swarm-LIO2: a fully decentralized, plug-and-play, computationally efficient, and bandwidth-efficient LiDAR-inertial odometry for aerial swarm systems. Swarm-LIO2 uses a decentralized, plug-and-play network as the communication infrastructure. Only bandwidth-efficient and low-dimensional information is exchanged, including identity, ego-state, mutual observation measurements, and global extrinsic transformations. To support the plug-and-play of new teammate participants, Swarm-LIO2 detects potential teammate UAVs and initializes the temporal offset and global extrinsic transformation all automatically. To enhance the initialization efficiency, novel reflectivity-based UAV detection, trajectory matching, and factor graph optimization methods are proposed. For state estimation, Swarm-LIO2 fuses LiDAR, IMU, and mutual observation measurements within an efficient ESIKF framework, with careful compensation of temporal delay and modeling of measurements to enhance the accuracy and consistency., Comment: 23 Pages

This paper is devoted to the study of Mean-field Games (MFG) systems in the mass critical exponent case. We firstly establish the optimal Gagliardo-Nirenberg type inequality associated with the potential-free MFG system. Then, under some mild assumptions on the potential function, we show that there exists a critical mass $M^*$ such that the MFG system admits a least energy solution if and only if the total mass of population density $M$ satisfies $M

In this paper, we propose a novel swashplateless-elevon actuation (SEA) for dual-rotor tail-sitter vertical takeoff and landing (VTOL) unmanned aerial vehicles (UAVs). In contrast to the conventional elevon actuation (CEA) which controls both pitch and yaw using elevons, the SEA adopts swashplateless mechanisms to generate an extra moment through motor speed modulation to control pitch and uses elevons solely for controlling yaw, without requiring additional actuators. This decoupled control strategy mitigates the saturation of elevons' deflection needed for large pitch and yaw control actions, thus improving the UAV's control performance on trajectory tracking and disturbance rejection performance in the presence of large external disturbances. Furthermore, the SEA overcomes the actuation degradation issues experienced by the CEA when the UAV is in close proximity to the ground, leading to a smoother and more stable take-off process. We validate and compare the performances of the SEA and the CEA in various real-world flight conditions, including take-off, trajectory tracking, and hover flight and position steps under external disturbance. Experimental results demonstrate that the SEA has better performances than the CEA. Moreover, we verify the SEA's feasibility in the attitude transition process and fixed-wing-mode flight of the VTOL UAV. The results indicate that the SEA can accurately control pitch in the presence of high-speed incoming airflow and maintain a stable attitude during fixed-wing mode flight. Video of all experiments can be found in youtube.com/watch?v=Sx9Rk4Zf7sQ, Comment: 8 pages, 13 figures

This paper introduces a novel targetless method for joint intrinsic and extrinsic calibration of LiDAR-camera systems using plane-constrained bundle adjustment (BA). Our method leverages LiDAR point cloud measurements from planes in the scene, alongside visual points derived from those planes. The core novelty of our method lies in the integration of visual BA with the registration between visual points and LiDAR point cloud planes, which is formulated as a unified optimization problem. This formulation achieves concurrent intrinsic and extrinsic calibration, while also imparting depth constraints to the visual points to enhance the accuracy of intrinsic calibration. Experiments are conducted on both public data sequences and self-collected dataset. The results showcase that our approach not only surpasses other state-of-the-art (SOTA) methods but also maintains remarkable calibration accuracy even within challenging environments. For the benefits of the robotics community, we have open sourced our codes.

We analyze the existence, linear stability, and slow dynamics of localized 1D spike patterns for a Keller--Segel model of chemotaxis that includes the effect of logistic growth of the cellular population. Our analysis of localized patterns for this two-component reaction-diffusion (RD) model is based, not on the usual limit of a large chemotactic drift coefficient, but instead on the singular limit of an asymptotically small diffusivity $d_2=\epsilon^2\ll 1$ of the chemoattractant concentration field. In the limit $d_2\ll 1$, steady-state and quasi-equilibrium 1D multi-spike patterns are constructed asymptotically. To determine the linear stability of steady-state $N$-spike patterns we analyze the spectral properties associated with both the ''large'' ${\mathcal O}(1)$ and the ''small'' $o(1)$ eigenvalues associated with the linearization of the Keller--Segel model. By analyzing a nonlocal eigenvalue problem characterizing the large eigenvalues, it is shown that $N$-spike equilibria can be destabilized by a zero-eigenvalue crossing leading to a competition instability if the cellular diffusion rate $d_1$ exceeds a threshold, or from a Hopf bifurcation if a relaxation time constant $\tau$ is too large. In addition, a matrix eigenvalue problem that governs the stability properties of an $N$-spike steady-state with respect to the small eigenvalues is derived. From an analysis of this matrix problem, an explicit range of $d_1$ where the $N$-spike steady-state is stable to the small eigenvalues is identified. Finally, for quasi-equilibrium spike patterns that are stable on an ${\mathcal O}(1)$ time-scale, we derive a differential algebraic system (DAE) governing the slow dynamics of a collection of localized spikes. Unexpectedly, our analysis is rather closely related to the analysis of spike patterns for the Gierer--Meinhardt RD system., Comment: 55 pages

Occupancy mapping is a fundamental component of robotic systems to reason about the unknown and known regions of the environment. This article presents an efficient occupancy mapping framework for high-resolution LiDAR sensors, termed D-Map. The framework introduces three main novelties to address the computational efficiency challenges of occupancy mapping. Firstly, we use a depth image to determine the occupancy state of regions instead of the traditional ray-casting method. Secondly, we introduce an efficient on-tree update strategy on a tree-based map structure. These two techniques avoid redundant visits to small cells, significantly reducing the number of cells to be updated. Thirdly, we remove known cells from the map at each update by leveraging the low false alarm rate of LiDAR sensors. This approach not only enhances our framework's update efficiency by reducing map size but also endows it with an interesting decremental property, which we have named D-Map. To support our design, we provide theoretical analyses of the accuracy of the depth image projection and time complexity of occupancy updates. Furthermore, we conduct extensive benchmark experiments on various LiDAR sensors in both public and private datasets. Our framework demonstrates superior efficiency in comparison with other state-of-the-art methods while maintaining comparable mapping accuracy and high memory efficiency. We demonstrate two real-world applications of D-Map for real-time occupancy mapping on a handle device and an aerial platform carrying a high-resolution LiDAR. In addition, we open-source the implementation of D-Map on GitHub to benefit society: github.com/hku-mars/D-Map., Comment: Supplementary material included. Accepted for publication in IEEE Transactions on Robotics (T-RO)

To describe the cellular self-aggregation phenomenon, some strongly coupled PDEs named as Keller-Segel (KS) and Patlak-Keller-Segel (PKS) systems were proposed in 1970s. Since KS and PKS systems possess relatively simple structures but admit rich dynamics, plenty of scholars have studied them and obtained many significant results. However, the cells in general direct their movement in liquid. As a consequence, it seems more realistic to consider the influence of ambient fluid flow on the chemotactic mechanism. Motivated by this, He et al. (SIAM J. Math. Anal., Vol. 53, No. 3, 2021) proposed a coupled Patlak-Keller-Segel-Navier-Stokes system that features the effect of the friction induced by the cells on the ambient fluid flow. In their pioneer work, the global existence of solutions of such system in 2D was established when the initial mass is strictly less than a threshold, which is referred to as the subcritical case. The last two authors and Zhou (Indiana Univ. Math. J., Vol. 72, No. 1, 2023) extended their result to the critical case. To our best knowledge, this system has only been studied in either the whole space or periodic domains. In this paper, we consider the chemotaxis-fluid system in two-dimensional bounded domains, in which the boundary conditions are Neumann conditions for the cell density and the chemical concentration, and the Navier slip boundary condition with zero friction for the fluid velocity. We prove that the solution of the system exists globally in time with the subcritical mass. Concerning the critical mass case, we construct the boundary spot equilibrium rigorously via the inner-outer gluing method. In particular, we develop the global $W^{2,p}$ theory for the 2D stationary Stokes system subject to Navier boundary conditions and further establish semigroup estimates of the nonstationary counterpart by analyzing the Stokes eigenvalue problem., Comment: 58 pages

Autonomous exploration is a crucial aspect of robotics that has numerous applications. Most of the existing methods greedily choose goals that maximize immediate reward. This strategy is computationally efficient but insufficient for overall exploration efficiency. In recent years, some state-of-the-art methods are proposed, which generate a global coverage path and significantly improve overall exploration efficiency. However, global optimization produces high computational overhead, leading to low-frequency planner updates and inconsistent planning motion. In this work, we propose a novel method to support fast UAV exploration in large-scale and cluttered 3-D environments. We introduce a computationally low-cost viewpoints generation method using novel occlusion-free spheres. Additionally, we combine greedy strategy with global optimization, which considers both computational and exploration efficiency. We benchmark our method against state-of-the-art methods to showcase its superiority in terms of exploration efficiency and computational time. We conduct various real-world experiments to demonstrate the excellent performance of our method in large-scale and cluttered environments.

We address the theoretical and practical problems related to the trajectory generation and tracking control of tail-sitter UAVs. Theoretically, we focus on the differential flatness property with full exploitation of actual UAV aerodynamic models, which lays a foundation for generating dynamically feasible trajectory and achieving high-performance tracking control. We have found that a tail-sitter is differentially flat with accurate aerodynamic models within the entire flight envelope, by specifying coordinate flight condition and choosing the vehicle position as the flat output. This fundamental property allows us to fully exploit the high-fidelity aerodynamic models in the trajectory planning and tracking control to achieve accurate tail-sitter flights. Particularly, an optimization-based trajectory planner for tail-sitters is proposed to design high-quality, smooth trajectories with consideration of kinodynamic constraints, singularity-free constraints and actuator saturation. The planned trajectory of flat output is transformed to state trajectory in real-time with consideration of wind in environments. To track the state trajectory, a global, singularity-free, and minimally-parameterized on-manifold MPC is developed, which fully leverages the accurate aerodynamic model to achieve high-accuracy trajectory tracking within the whole flight envelope. The effectiveness of the proposed framework is demonstrated through extensive real-world experiments in both indoor and outdoor field tests, including agile SE(3) flight through consecutive narrow windows requiring specific attitude and with speed up to 10m/s, typical tail-sitter maneuvers (transition, level flight and loiter) with speed up to 20m/s, and extremely aggressive aerobatic maneuvers (Wingover, Loop, Vertical Eight and Cuban Eight) with acceleration up to 2.5g.

The emergence of low-cost, small form factor and light-weight solid-state LiDAR sensors have brought new opportunities for autonomous unmanned aerial vehicles (UAVs) by advancing navigation safety and computation efficiency. Yet the successful developments of LiDAR-based UAVs must rely on extensive simulations. Existing simulators can hardly perform simulations of real-world environments due to the requirements of dense mesh maps that are difficult to obtain. In this paper, we develop a point-realistic simulator of real-world scenes for LiDAR-based UAVs. The key idea is the underlying point rendering method, where we construct a depth image directly from the point cloud map and interpolate it to obtain realistic LiDAR point measurements. Our developed simulator is able to run on a light-weight computing platform and supports the simulation of LiDARs with different resolution and scanning patterns, dynamic obstacles, and multi-UAV systems. Developed in the ROS framework, the simulator can easily communicate with other key modules of an autonomous robot, such as perception, state estimation, planning, and control. Finally, the simulator provides 10 high-resolution point cloud maps of various real-world environments, including forests of different densities, historic building, office, parking garage, and various complex indoor environments. These realistic maps provide diverse testing scenarios for an autonomous UAV. Evaluation results show that the developed simulator achieves superior performance in terms of time and memory consumption against Gazebo and that the simulated UAV flights highly match the actual one in real-world environments. We believe such a point-realistic and light-weight simulator is crucial to bridge the gap between UAV simulation and experiments and will significantly facilitate the research of LiDAR-based autonomous UAVs in the future., Comment: 8 pages, 13 figures

Reconstructing an accurate and consistent large-scale LiDAR point cloud map is crucial for robotics applications. The existing solution, pose graph optimization, though it is time-efficient, does not directly optimize the mapping consistency. LiDAR bundle adjustment (BA) has been recently proposed to resolve this issue; however, it is too time-consuming on large-scale maps. To mitigate this problem, this paper presents a globally consistent and efficient mapping method suitable for large-scale maps. Our proposed work consists of a bottom-up hierarchical BA and a top-down pose graph optimization, which combines the advantages of both methods. With the hierarchical design, we solve multiple BA problems with a much smaller Hessian matrix size than the original BA; with the pose graph optimization, we smoothly and efficiently update the LiDAR poses. The effectiveness and robustness of our proposed approach have been validated on multiple spatially and timely large-scale public spinning LiDAR datasets, i.e., KITTI, MulRan and Newer College, and self-collected solid-state LiDAR datasets under structured and unstructured scenes. With proper setups, we demonstrate our work could generate a globally consistent map with around 12% of the sequence time., Comment: 8 pages, 11 figures

Accurate self and relative state estimation are the critical preconditions for completing swarm tasks, e.g., collaborative autonomous exploration, target tracking, search and rescue. This paper proposes Swarm-LIO: a fully decentralized state estimation method for aerial swarm systems, in which each drone performs precise ego-state estimation, exchanges ego-state and mutual observation information by wireless communication, and estimates relative state with respect to (w.r.t.) the rest of UAVs, all in real-time and only based on LiDAR-inertial measurements. A novel 3D LiDAR-based drone detection, identification and tracking method is proposed to obtain observations of teammate drones. The mutual observation measurements are then tightly-coupled with IMU and LiDAR measurements to perform real-time and accurate estimation of ego-state and relative state jointly. Extensive real-world experiments show the broad adaptability to complicated scenarios, including GPS-denied scenes, degenerate scenes for camera (dark night) or LiDAR (facing a single wall). Compared with ground-truth provided by motion capture system, the result shows the centimeter-level localization accuracy which outperforms other state-of-the-art LiDAR-inertial odometry for single UAV system.

The strong Allee effect plays an important role on the evolution of population in ecological systems. One important concept is the Allee threshold that determines the persistence or extinction of the population in a long time. In general, a small initial population size is harmful to the survival of a species since when the initial data is below the Allee threshold the population tends to extinction, rather than persistence. Another interesting feature of population evolution is that a species whose movement strategy follows a conditional dispersal strategy is more likely to persist. In other words, the biased movement can be a benefit for the persistence of the population. The coexistence of the above two conflicting mechanisms makes the dynamics rather intricate. However, some numerical results obtained by Cosner et. al. (SIAM J. Appl. Math., Vol. 81, No. 2, 2021) show that the directed movement can invalidate the strong Allee effect and help the population survive. To study this intriguing phenomenon, we consider the pattern formation and local dynamics for a class of single species population models of that is subject to the strong Allee effect. We first rigorously show the existence of multiple localized solutions when the directed movement is strong enough. Next, the spectrum analysis of the associated linear eigenvalue problem is established and used to investigate the stability properties of these interior spikes. This analysis proves that there exists not only unstable but also linear stable steady states. Then, we extend results of the single equation to coupled systems, and also construct several non-constant steady states and analyze their stability. Finally, numerical simulations are performed to illustrate the theoretical results., Comment: 47 pages, 10 figures; revised according to reports and published

In practical applications, autonomous quadrotors are still facing significant challenges, such as the detection and avoidance of very small and even dynamic obstacles (e.g., tree branches, power lines). In this paper, we propose a compact, integrated, and fully autonomous quadrotor system, which can fly safely in cluttered environments while avoiding dynamic small obstacles. Our quadrotor platform is equipped with a forward-looking three-dimensional (3D) light detection and ranging (lidar) sensor to perceive the environment and an onboard embedded computer to perform all the estimation, mapping, and planning tasks. Specifically, the computer estimates the current pose of the UAV, maintains a local map (time-accumulated point clouds KD-Trees), and computes a safe trajectory using kinodynamic A* search to the goal point. The whole perception and planning system can run onboard at 50Hz with careful optimization. Various indoor and outdoor experiments show that the system can avoid dynamic small obstacles (down to 20mm diameter bar) while flying at 2m/s in cluttered environments. Our codes and hardware design are open-sourced on Github., Comment: 8 pages, 12 figures

This paper investigates the Keller-Segel model with quadratic cellular diffusion over a disk in $\mathbb R^2$ with a focus on the formation of its nontrivial patterns. We obtain explicit formulas of radially symmetric stationary solutions and such configurations give rise to the ring patterns and concentric airy patterns. These explicit formulas empower us to study the global bifurcation and asymptotic behaviors of these solutions, within which the cell population density has $\delta$-type spiky structures when the chemotaxis rate is large. The explicit formulas are also used to study the uniqueness and quantitative properties of nontrivial stationary radial patterns ruled by several threshold phenomena determined by the chemotaxis rate. We find that all nonconstant radial stationary solutions must have the cellular density compactly supported unless for a discrete sequence of bifurcation values at which there exist strictly positive small-amplitude solutions. The hierarchy of free energy shows that in the radial class the inner ring solution has the least energy while the constant solution has the largest energy, and all these theoretical results are illustrated through bifurcation diagrams. A natural extension of our results to $\mathbb R^2$ yields the existence, uniqueness and closed-form solution of the problem in this whole space. Our results are complemented by numerical simulations that demonstrate the existence of non-radial stationary solutions in the disk.