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Inverse kinematics is a fundamental technique for motion and positioning control in robotics, typically applied to end-effectors. In this paper, we extend the concept of inverse kinematics to guiding vector fields for path following in autonomous mobile robots. The desired path is defined by its implicit equation, i.e., by a collection of points belonging to one or more zero-level sets. These level sets serve as a reference to construct an error signal that drives the guiding vector field toward the desired path, enabling the robot to converge and travel along the path by following such a vector field. We start with the formal exposition on how inverse kinematics can be applied to guiding vector fields for single-integrator robots in an m-dimensional Euclidean space. Then, we leverage inverse kinematics to ensure that the level-set error signal behaves as a linear system, facilitating control over the robot's transient motion toward the desired path and allowing for the injection of feed-forward signals to induce precise motion behavior along the path. We then propose solutions to the theoretical and practical challenges of applying this technique to unicycles with constant speeds to follow 2D paths with precise transient control. We finish by validating the predicted theoretical results through real flights with fixed-wing drones., Comment: Yu Zhou and Jes\'us Bautista contributed equally to this work. In the proceedings of the IEEE International Conference on Robotics and Automation (ICRA) 2025

Human-multi-robot shared control leverages human decision-making and robotic autonomy to enhance human-robot collaboration. While widely studied, existing systems often adopt a leader-follower model, limiting robot autonomy to some extent. Besides, a human is required to directly participate in the motion control of robots through teleoperation, which significantly burdens the operator. To alleviate these two issues, we propose a layered shared control computing framework using human-influenced guiding vector fields (HI-GVF) for human-robot collaboration. HI-GVF guides the multi-robot system along a desired path specified by the human. Then, an intention field is designed to merge the human and robot intentions, accelerating the propagation of the human intention within the multi-robot system. Moreover, we give the stability analysis of the proposed model and use collision avoidance based on safety barrier certificates to fine-tune the velocity. Eventually, considering the firefighting task as an example scenario, we conduct simulations and experiments using multiple human-robot interfaces (brain-computer interface, myoelectric wristband, eye-tracking), and the results demonstrate that our proposed approach boosts the effectiveness and performance of the task.

We propose a self-contained, resilient and fully distributed solution for locating the maximum of an unknown 3D scalar field using a swarm of robots that travel at constant speeds. Unlike conventional reactive methods relying on gradient information, our methodology enables the swarm to determine an ascending direction so that it approaches the source with arbitrary precision. Our source-seeking solution consists of three algorithms. The first two algorithms run sequentially and distributively at a high frequency providing barycentric coordinates and the ascending direction respectively to the individual robots. The third algorithm is the individual control law for a robot to track the estimated ascending direction. We show that the two algorithms with higher frequency have an exponential convergence to their eventual values since they are based on the standard consensus protocol for first-order dynamical systems; their high frequency depends on how fast the robots travel through the scalar field. The robots are not constrained to any particular geometric formation, and we study both discrete and continuous distributions of robots within swarm shapes. The shape analysis reveals the resiliency of our approach as expected in robot swarms, i.e., by amassing robots we ensure the source-seeking functionality in the event of missing or misplaced individuals or even if the robot network splits into two or more disconnected subnetworks. In addition, we also enhance the robustness of the algorithm by presenting conditions for \emph{optimal} swarm shapes, in the sense that the ascending directions can be closely parallel to the field's gradient. We exploit such an analysis so that the swarm can adapt to unknown environments by morphing its shape and maneuvering while still following an ascending direction., Comment: 15 pages, submitted version to T-RO. This version does not contain the field experiments. arXiv admin note: text overlap with arXiv:2309.02937

In recent years, there has been a significant amount of research on algorithms and control methods for distributed collaborative robots. However, the emergence of collective behavior in a swarm is still difficult to predict and control. Nevertheless, human interaction with the swarm helps render the swarm more predictable and controllable, as human operators can utilize intuition or knowledge that is not always available to the swarm. Therefore, this paper designs the Dynamic Visualization Research Platform for Multimodal Human-Swarm Interaction (DVRP-MHSI), which is an innovative open system that can perform real-time dynamic visualization and is specifically designed to accommodate a multitude of interaction modalities (such as brain-computer, eye-tracking, electromyographic, and touch-based interfaces), thereby expediting progress in human-swarm interaction research. Specifically, the platform consists of custom-made low-cost omnidirectional wheeled mobile robots, multitouch screens and two workstations. In particular, the mutitouch screens can recognize human gestures and the shapes of objects placed on them, and they can also dynamically render diverse scenes. One of the workstations processes communication information within robots and the other one implements human-robot interaction methods. The development of DVRP-MHSI frees researchers from hardware or software details and allows them to focus on versatile swarm algorithms and human-swarm interaction methods without being limited to fixed scenarios, tasks, and interfaces. The effectiveness and potential of the platform for human-swarm interaction studies are validated by several demonstrative experiments.

In obstacle-dense scenarios, providing safe guidance for mobile robots is critical to improve the safe maneuvering capability. However, the guidance provided by standard guiding vector fields (GVFs) may limit the motion capability due to the improper curvature of the integral curve when traversing obstacles. On the other hand, robotic system dynamics are often time-varying, uncertain, and even unknown during the motion planning process. Therefore, many existing kinodynamic motion planning methods could not achieve satisfactory reliability in guaranteeing safety. To address these challenges, we propose a two-level Vector Field-guided Learning Predictive Control (VF-LPC) approach that improves safe maneuverability. The first level, the guiding level, generates safe desired trajectories using the designed kinodynamic GVF, enabling safe motion in obstacle-dense environments. The second level, the Integrated Motion Planning and Control (IMPC) level, first uses a deep Koopman operator to learn a nominal dynamics model offline and then updates the model uncertainties online using sparse Gaussian processes (GPs). The learned dynamics and a game-based safe barrier function are then incorporated into the LPC framework to generate near-optimal planning solutions. Extensive simulations and real-world experiments were conducted on quadrotor unmanned aerial vehicles and unmanned ground vehicles, demonstrating that VF-LPC enables robots to maneuver safely.

We design a distributed coordinated guiding vector field (CGVF) for a group of robots to achieve ordering-flexible motion coordination while maneuvering on a desired two-dimensional (2D) surface. The CGVF is characterized by three terms, i.e., a convergence term to drive the robots to converge to the desired surface, a propagation term to provide a traversing direction for maneuvering on the desired surface, and a coordinated term to achieve the surface motion coordination with an arbitrary ordering of the robotic group. By setting the surface parameters as additional virtual coordinates, the proposed approach eliminates the potential singularity of the CGVF and enables both the global convergence to the desired surface and the maneuvering on the surface from all possible initial conditions. The ordering-flexible surface motion coordination is realized by each robot to share with its neighbors only two virtual coordinates, i.e. that of a given target and that of its own, which reduces the communication and computation cost in multi-robot surface navigation. Finally, the effectiveness of the CGVF is substantiated by extensive numerical simulations., Comment: Published on IEEE Transactions on Automatic Control, 2024

In this paper, we propose a distributed guiding-vector-field (DGVF) algorithm for a team of robots to form a spontaneous-ordering platoon moving along a predefined desired path in the n-dimensional Euclidean space. Particularly, by adding a path parameter as an additional virtual coordinate to each robot, the DGVF algorithm can eliminate the singular points where the vector fields vanish, and govern robots to approach a closed and even self-intersecting desired path. Then, the interactions among neighboring robots and a virtual target robot through their virtual coordinates enable the realization of the desired platoon; in particular, relative parametric displacements can be achieved with arbitrary ordering sequences. Rigorous analysis is provided to guarantee the global convergence to the spontaneous-ordering platoon on the common desired path from any initial positions. 2D experiments using three HUSTER-0.3 unmanned surface vessels (USVs) are conducted to validate the practical effectiveness of the proposed DGVF algorithm, and 3D numerical simulations are presented to demonstrate its effectiveness and robustness when tackling higher-dimensional multi-robot path-navigation missions and some robots breakdown.

We present a solution for locating the source, or maximum, of an unknown scalar field using a swarm of mobile robots. Unlike relying on the traditional gradient information, the swarm determines an ascending direction to approach the source with arbitrary precision. The ascending direction is calculated from measurements of the field strength at the robot locations and their relative positions concerning the centroid. Rather than focusing on individual robots, we focus the analysis on the density of robots per unit area to guarantee a more resilient swarm, i.e., the functionality remains even if individuals go missing or are misplaced during the mission. We reinforce the robustness of the algorithm by providing sufficient conditions for the swarm shape so that the ascending direction is almost parallel to the gradient. The swarm can respond to an unexpected environment by morphing its shape and exploiting the existence of multiple ascending directions. Finally, we validate our approach numerically with hundreds of robots. The fact that a large number of robots always calculate an ascending direction compensates for the loss of individuals and mitigates issues arising from the actuator and sensor noises., Comment: 7 pages, CDC 2024, accepted version

Motion planning has been an important research topic in achieving safe and flexible maneuvers for intelligent vehicles. However, it remains challenging to realize efficient and optimal planning in the presence of uncertain model dynamics. In this paper, a sparse kernel-based reinforcement learning (RL) algorithm with Gaussian Process (GP) Regression (called GP-SKRL) is proposed to achieve online adaption and near-optimal motion planning performance. In this algorithm, we design an efficient sparse GP regression method to learn the uncertain dynamics. Based on the updated model, a sparse kernel-based policy iteration algorithm with an exponential barrier function is designed to learn the near-optimal planning policies with the capability to avoid dynamic obstacles. Thereby, batch-mode GP-SKRL with online adaption capability can estimate the changing system dynamics. The converged RL policies are then deployed on vehicles efficiently under a safety-aware module. As a result, the produced driving actions are safe and less conservative, and the planning performance has been noticeably improved. Extensive simulation results show that GP-SKRL outperforms several advanced motion planning methods in terms of average cumulative cost, trajectory length, and task completion time. In particular, experiments on a Hongqi E-HS3 vehicle demonstrate that superior GP-SKRL provides a practical planning solution., Comment: 10 pages, 12 figures, submitted to RAL

This paper focuses on the motion planning for mobile robots in 3D, which are modelled by 6-DOF rigid body systems with nonholonomic kinematics constraints. We not only specify the target position, but also bring in the requirement of the heading direction at the terminal time, which gives rise to a new and more challenging 3D motion planning problem. The proposed planning algorithm involves a novel velocity vector field (VF) over the workspace, and by following the VF, the robot can be navigated to the destination with the specified heading direction. In order to circumvent potential collisions with obstacles and other robots, a composite VF is designed by composing the navigation VF and an additional VF tangential to the boundary of the dangerous area. Moreover, we propose a priority-based algorithm to deal with the motion coupling issue among multiple robots. Finally, numerical simulations are conducted to verify the theoretical results.

In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the vector field vanishes creates a topological obstruction to global convergence to the desired path and some associated topological analysis has been conducted in our previous work. In this paper, we strengthen the result in our previous work by showing that the domain of attraction of the desired path, which is a compact asymptotically stable one-dimensional embedded submanifold of an $n$-dimensional ambient manifold $\mathcal{M}$, is homeomorphic to $\mathbb{R}^{n-1} \times \mathbb{S}^1$, and not just homotopy equivalent to $\mathbb{S}^1$. This result is extended for a $k$-dimensional compact manifold for $k \ge 2$.

Microscopic strategy update rules play an important role in the evolutionary dynamics of cooperation among interacting agents on complex networks. Many previous related works only consider one \emph{fixed} rule, while in the real world, individuals may switch, sometimes periodically, between rules. It is of particular theoretical interest to investigate under what conditions the periodic switching of strategy update rules facilitates the emergence of cooperation. To answer this question, we study the evolutionary prisoner's dilemma game on regular networks where agents can periodically switch their strategy update rules. We accordingly develop a theoretical framework of this periodically switched system, where the replicator equation corresponding to each specific microscopic update rule is used for describing the subsystem, and all the subsystems are activated in sequence. By utilizing switched system theory, we identify the theoretical condition for the emergence of cooperative behavior. Under this condition, we have proved that the periodically switched system with different switching rules can converge to the full cooperation state. Finally, we consider an example where two strategy update rules, that is, the imitation and pairwise-comparison updating, are periodically switched, and find that our numerical calculations validate our theoretical results.

We propose coordinating guiding vector fields to achieve two tasks simultaneously with a team of robots: first, the guidance and navigation of multiple robots to possibly different paths or surfaces typically embedded in 2D or 3D; second, their motion coordination while tracking their prescribed paths or surfaces. The motion coordination is defined by desired parametric displacements between robots on the path or surface. Such a desired displacement is achieved by controlling the virtual coordinates, which correspond to the path or surface's parameters, between guiding vector fields. Rigorous mathematical guarantees underpinned by dynamical systems theory and Lyapunov theory are provided for the effective distributed motion coordination and navigation of robots on paths or surfaces from all initial positions. As an example for practical robotic applications, we derive a control algorithm from the proposed coordinating guiding vector fields for a Dubins-car-like model with actuation saturation. Our proposed algorithm is distributed and scalable to an arbitrary number of robots. Furthermore, extensive illustrative simulations and fixed-wing aircraft outdoor experiments validate the effectiveness and robustness of our algorithm., Comment: Evolved paper from arXiv:2103.12372. Accepted to IEEE Transactions on Robotics. Supplementary video: https://www.bilibili.com/video/BV16e4y147xp/

Accurately following a geometric desired path in a two-dimensional space is a fundamental task for many engineering systems, in particular mobile robots. When the desired path is occluded by obstacles, it is necessary and crucial to temporarily deviate from the path for obstacle/collision avoidance. In this paper, we develop a composite guiding vector field via the use of smooth bump functions, and provide theoretical guarantees that the integral curves of the vector field can follow an arbitrary sufficiently smooth desired path and avoid collision with obstacles of arbitrary shapes. These two behaviors are reactive since path (re)-planning and global map construction are not involved. To deal with the common deadlock problem, we introduce a switching vector field, and the Zeno behavior is excluded. Simulations are conducted to support the theoretical results.

Recently, an evolutionary game dynamics model taking into account the environmental feedback has been proposed to describe the co-evolution of strategic actions of a population of individuals and the state of the surrounding environment; correspondingly a range of interesting dynamic behaviors have been reported. In this paper, we provide new theoretical insight into such behaviors and discuss control options. Instead of the standard replicator dynamics, we use a more realistic and comprehensive model of replicator-mutator dynamics, to describe the strategic evolution of the population. After integrating the environment feedback, we study the effect of mutations on the resulting closed-loop system dynamics. We prove the conditions for two types of bifurcations, Hopf bifurcation and Heteroclinic bifurcation, both of which result in stable limit cycles. These limit cycles have not been identified in existing works, and we further prove that such limit cycles are in fact persistent in a large parameter space and are almost globally stable. In the end, an intuitive control policy based on incentives is applied, and the effectiveness of this control policy is examined by analysis and simulations., Comment: 19 pages, 6 figures

A path-following control algorithm enables a system's trajectories under its guidance to converge to and evolve along a given geometric desired path. There exist various such algorithms, but many of them can only guarantee local convergence to the desired path in its neighborhood. In contrast, the control algorithms using a well-designed guiding vector field can ensure almost global convergence of trajectories to the desired path; here, "almost" means that in some cases, a measure-zero set of trajectories converge to the singular set where the vector field becomes zero (with all other trajectories converging to the desired path). In this paper, we first generalize the guiding vector field from the Euclidean space to a general smooth Riemannian manifold. This generalization can deal with path-following in some abstract configuration space (such as robot arm joint space). Then we show several theoretical results from a topological viewpoint. Specifically, we are motivated by the observation that singular points of the guiding vector field exist in many examples where the desired path is homeomorphic to the unit circle, but it is unknown whether the existence of singular points always holds in general (i.e., is inherent in the topology of the desired path). In the $n$-dimensional Euclidean space, we provide an affirmative answer, and conclude that it is not possible to guarantee global convergence to desired paths that are homeomorphic to the unit circle. Furthermore, we show that there always exist \emph{non-path-converging trajectories} (i.e., trajectories that do not converge to the desired path) starting from the boundary of a ball containing the desired path in an $n$-dimensional Euclidean space where $n \ge 3$. Examples are provided to illustrate the theoretical results., Comment: 16 pages, 8 figures, TAC

In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold of a finite-dimensional smooth manifold of an autonomous system is homeomorphic to its tubular neighborhood. The compactness of the attractor is crucial, without which this result is false; two counterexamples are provided to demonstrate this.

This paper proposes a novel distributed technique to induce collective motions in affine formation control. Instead of the traditional leader-follower strategy, we propose modifying the original weights that build the Laplacian matrix so that a designed steady-state motion of the desired shape emerges from the agents' local interactions. The proposed technique allows a rich collection of collective motions such as rotations around the centroid, translations, scalings, and shearings of a reference shape. These motions can be applied in useful collective behaviors such as \emph{shaped} consensus (the rendezvous with a particular shape), escorting one of the team agents, or area coverage. We prove the global stability and effectiveness of our proposed technique rigorously, and we provide some illustrative numerical simulations., Comment: 6 pages, submitted to CDC 2021

It is essential in many applications to impose a scalable coordinated motion control on a large group of mobile robots, which is efficient in tasks requiring repetitive execution, such as environmental monitoring. In this paper, we design a guiding vector field to guide multiple robots to follow possibly different desired paths while coordinating their motions. The vector field uses a path parameter as a virtual coordinate that is communicated among neighboring robots. Then, the virtual coordinate is utilized to control the relative parametric displacement between robots along the paths. This enables us to design a saturated control algorithm for a Dubins-car-like model. The algorithm is distributed, scalable, and applicable for any smooth paths in an $n$-dimensional configuration space, and global convergence is guaranteed. Simulations with up to fifty robots and outdoor experiments with fixed-wing aircraft validate the theoretical results., Comment: Accepted to 2021 IEEE International Conference on Robotics and Automation (ICRA)

Most of the existing path-following navigation algorithms cannot guarantee global convergence to desired paths or enable following self-intersected desired paths due to the existence of singular points where navigation algorithms return unreliable or even no solutions. One typical example arises in vector-field guided path-following (VF-PF) navigation algorithms. These algorithms are based on a vector field, and the singular points are exactly where the vector field diminishes. In this paper, we show that it is mathematically impossible for conventional VF-PF algorithms to achieve global convergence to desired paths that are self-intersected or even just simple closed (precisely, homeomorphic to the unit circle). Motivated by this new impossibility result, we propose a novel method to transform self-intersected or simple closed desired paths to non-self-intersected and unbounded (precisely, homeomorphic to the real line) counterparts in a higher-dimensional space. Corresponding to this new desired path, we construct a singularity-free guiding vector field on a higher-dimensional space. The integral curves of this new guiding vector field is thus exploited to enable global convergence to the higher-dimensional desired path, and therefore the projection of the integral curves on a lower-dimensional subspace converge to the physical (lower-dimensional) desired path. Rigorous theoretical analysis is carried out for the theoretical results using dynamical systems theory. In addition, we show both by theoretical analysis and numerical simulations that our proposed method is an extension combining conventional VF-PF algorithms and trajectory tracking algorithms. Finally, to show the practical value of our proposed approach for complex engineering systems, we conduct outdoor experiments with a fixed-wing airplane in windy environment to follow both 2D and 3D desired paths., Comment: Accepted for publication in IEEE Trransactions on Robotics (T-RO)

Vector field guided path following (VF-PF) algorithms are fundamental in robot navigation tasks, but may not deliver the desirable performance when robots encounter singular points where the vector field becomes zero. The existence of singular points prevents the global convergence of the vector field's integral curves to the desired path. Moreover, VF-PF algorithms, as well as most of the existing path following algorithms, fail to enable following a self-intersected desired path. In this paper, we show that such failures are fundamentally related to the mathematical topology of the path, and that by "stretching" the desired path along a virtual dimension, one can remove the topological obstruction. Consequently, this paper proposes a new guiding vector field defined in a higher-dimensional space, in which self-intersected desired paths become free of self-intersections; more importantly, the new guiding vector field does not have any singular points, enabling the integral curves to converge globally to the "stretched" path. We further introduce the extended dynamics to retain this appealing global convergence property for the desired path in the original lower-dimensional space. Both simulations and experiments are conducted to verify the theory., Comment: Accepted by 2020 IEEE 59th Conference on Decision and Control (CDC). This is the full version

Using a designed vector field to control a mobile robot to follow a given desired path is intuitive and practical, and to build a rigorous theory to guide its implementation is essential. In this paper, we study the properties of a general 3D vector field for robotic path following. We propose and investigate assumptions that turn out to be crucial for this method, but have been rarely explicitly stated in related works. We derive conditions under which the local path-following error vanishes exponentially in a sufficiently small neighborhood of the desired path, which is key to show the local input-to-state stability (local ISS) property of the path-following error dynamics. The local ISS property then justifies the control algorithm design for a fixed-wing aircraft model. Our approach is effective for any sufficiently smooth desired path in 3D, bounded or unbounded; note that the case for unbounded desired paths has not been sufficiently discussed in the literature. Simulations are conducted to verify the theoretical results.

Circumnavigation control is useful in real-world applications such as entrapping a hostile target. In this paper, we consider a heterogeneous multi-robot system where robots have different physical properties, such as maximum movement speeds. Instead of equal-spacings, dynamic spacings according to robots' properties, which are termed utilities in this paper, will be more desirable in a scenario such as target entrapment. A distributed circumnavigation control algorithm based on utilities is proposed for any number of mobile robots from random 3D positions to circumnavigate a target. The dynamic spacings are subject to the variation of robots' utilities. The robots can only obtain the angular positions and utilities of their two neighbouring robots, so the control law is distributed. Theoretical analysis and experimental results are provided to prove the stability and effectiveness of the proposed control algorithm., Comment: Conditional accepted by 2018 RoboCup Symposium