Trajectory Tracking Problem Research Articles

Quadrotor trajectory tracking based on backstepping control and radial basis function neural networks

In this paper we introduce a novel method for solving the trajectory tracking problem for a quadrotor system based on backstepping control and radial basis function (RBF) neural networks. Backstepping controllers in quadrotor control are designed so as to guarantee Lyapunov stability of the closed-loop system based on a dynamic model of the quadrotor, derived using first principles equations. Such a model is prone to modeling errors due to various factors, including uncertainties, time varying quadrotor parameters and external disturbances acting on the vehicle, thus compromising the controller performance. To remedy this situation, we couple the controller with a radial basis function (RBF) neural network trained with the fuzzy means (FM) algorithm, which allows the proposed RBF – backstepping framework to take into account unmodeled dynamics with increased accuracy, thus improving the tracking performance. The resulting method is evaluated on a detailed quadrotor model over trajectories with distinct geometrical characteristics, and compared with a backstepping controller using a different neural network architecture and a standard backstepping control scheme. The proposed control method produces very competitive results, outperforming its rival in terms of trajectory tracking performance.

Finite-Time Model-Free Adaptive Control for Discrete-Time Nonlinear Systems

This paper addresses the fast trajectory tracking problem for discrete-time nonlinear systems, and whereby a finite-time model-free adaptive control (FMFAC) approach is innovatively devised. Here, the nonlinear systems’ dynamics are completely unknown and the widespread disturbances are considered. The nonparametric dynamic linearization technique and the adaptive extended disturbance observer are firstly deployed to achieve real-time linearization of the discrete-time nonlinear system, and an efficient finite-time performance index function is thereafter devised, and thereby contributing to a novel FMFAC control scheme. Theoretical analysis indicates the finite-time model-free tracking control within a stable region can be guaranteed, and the boundedness of the control input signal can be ensured rigorously, even though under the effect of various disturbances. The key points of the proposed FMFAC approach are model-free and possessing the fast regulation ability, and the corresponding effectiveness and superiority are validated via simulation studies.

Non-Linear Prediction Scheme for an Omnidirectional Mobile Robot with Disturbed Time Delay

This work focuses on the trajectory tracking problem of an input-delayed omnidirectional mobile robot affected by a constant disturbance in the time delay. The trajectory-tracking solution is based on a prediction strategy that represents a generalization of the well-known sub-prediction strategy developed for the linear case. It is formally proven that the prediction scheme provides the future state of the system and that a feedback law based on the future predicted state solves the trajectory tracking problem under some assumptions related to the size of the disturbance and the number of sub-predictors considered. To evaluate the prediction-based strategy, numerical simulations are carried out showing an adequate performance of the overall closed-loop system.

Motion planning and stabilization of nonholonomic systems using gradient flow approximations

Nonlinear control-affine systems described by ordinary differential equations with time-varying vector fields are considered in the paper. We propose a unified control design scheme with oscillating inputs for solving the trajectory tracking and stabilization problems under the bracket-generating condition. This methodology is based on the approximation of a gradient-like dynamics by trajectories of the designed closed-loop system. As an intermediate outcome, we characterize the asymptotic behavior of solutions of the considered class of nonlinear control systems with oscillating inputs under rather general assumptions on the generating potential function. These results are applied to examples of nonholonomic trajectory tracking and obstacle avoidance.

Trajectory Kinematics Control for a Robotis Bioloid Premium Arm

This paper addresses the problem of trajectory tracking in one arm of the Bioloid Premium humanoid robot. The kinematics modeling of the left arm was obtained using the Denavit-Hartenberg methodology. A kinematics control strategy based on state feedback and Input-Output Linearization technique was employed to achieve accurate tracking of the desired trajectories. A C language code was written to reprogram the CM-530 controller of the Bioloid Premium to allow it to interpret and execute the trajectory positions. In addition, wireless communication was established using ROS as the development platform. The experimental results demonstrate the success of the proposed approach in applying kinematic control on the robotic arm, offering an accurate and advanced alternative to the mobility options natively enabled by the ROBOTIS software.

A Robust Interval Predictive Control for Perturbed Unicycle Mobile Robots

In this paper, we design a robust control strategy to solve the trajectory-tracking problem for perturbed unicycle mobile robots. The design is composed of an Integral Sliding-Mode Control, an interval predictor-based state-feedback controller and a Model Predictive Control. The controller handles some perturbations in the kinematic model, and state and input constraints that are related to workspace constraints and saturation on the actuators, respectively. The proposed approach ensures the exponential convergence to zero of the tracking error. Some simulation results illustrate the performance of the proposed approach.

Three-dimensional trajectory tracking guidance against near-space maneuvering targets with multiple constraints under wind field

This paper presents a three-dimensional disturbance observer-based adaptive tracking guidance scheme for the trajectory tracking problem of a missile intercepting a hypersonic vehicle with the wind field, autopilot lag dynamics, and input saturation. Unlike most of the existing missile trajectory tracking guidance systems without velocity controllability and external wind field model, the wind field model is added to the dynamic model, and the velocity control channel is introduced into the three-dimensional trajectory tracking guidance problem to prevent impractical scenarios and increase tracking precision. The proposed tracking guidance methodology incorporates the sliding mode and adaptive control techniques, in which the sliding mode motion is finite time stable independently of the tracking guidance system's initial conditions. Also, the super twisting switch control law is added to cope with the chattering problem. In this context, external uncertainty disturbances and unmeasured states are lumped into lump disturbance and estimated using the backpropagation neural network extended state observer. To this end, numerical simulations demonstrate the tracking to reach zero between 3 s and 5 s against various uncertainties in engineering practical, and position tracking errors are within 0.15 m, superior to the existing tracking guidance methods.

Fixed‐time tracking control of manipulators based on a fast variable power reaching law

SummaryAs for the problem of trajectory tracking of a two‐joint series manipulator, a novel fixed‐time control scheme is proposed based on non‐singular fast terminal sliding mode (NFTSM) control and a fast variable power reaching law. The innovation of the proposed method is: (1) by employing the NFTSM surface, we solve the singularity problem existed in traditional terminal sliding mode surface. (2) The convergence rate is increased by 50% by the combination of fixed‐time control and NFTSM surface control. (3) A new fast variable power reaching law is presented, which combines the advantages of the double power reaching law and the fast power reaching law, and it effectively improves the reaching rate and reduces the amplitude by 40–60 N m. Theoretical analysis proves that the better tracking performance is obtained, and its convergence time upper limit is not affected by the initial states of the system. Finally, the effectiveness and feasibility of our controller are verified through a numerical simulation.

Output feedback stochastic MPC for tracking control of quadrotors with disturbances

AbstractIn this paper, the trajectory tracking problem of controlling a constrained quadrotor with unmeasurable system states in an environment with stochastic wind‐gust disturbance is considered. The mathematical model of the quadrotor is divided into the translational system and the rotational system, while only the measurement output of the quadrotor can be accessed. A new output‐based control method is developed for solving this problem. In the translational control system, an output feedback stochastic model predictive control (MPC) algorithm is proposed to generate the optimal control sequence with less conservativeness, by taking into account the information on the distribution of the disturbances and the uncertainty resulting from the attitude tracking error. The closed‐loop probabilistic constraints satisfaction, the recursive feasibility and the stability of the algorithm are further proved. In the rotational system, the active disturbance rejection control (ADRC) method to estimate and compensate for external disturbances is leveraged and robust control for attitude tracking is accomplished. The convergence of the disturbance estimator and the stability proof are provided. Finally, the robustness and effectiveness of the proposed control strategy are verified by an illustrative example.

Kinematic and dynamic performances of artificial swarm systems: Aggregation, collision avoidance and compact formation

A novel control method for the artificial swarm system is proposed in this paper. To reflect the biological swarm performance (aggregation), a kinematic model of the artificial swarm system is established by introducing a special potential function. Then, the dynamic model is further obtained to satisfy more accurate dynamic swarm performances — collision avoidance and compact formation. The former can be realized by diffeomorphism transformation approach and a safety zone is guaranteed accordingly. As for the compact formation, we creatively convert it into a constraint-following problem between leader and followers. Based on the dynamic model, a novel control consisting of two parts is designed, including the model-based control and the adaptive robust control. On one hand, the trajectory tracking problem of the artificial swarm system is solved. On the other hand, the uncertainties in the artificial swarm system can be compensated perfectly. The deterministic performance, uniform boundedness and uniform ultimate boundedness, is guaranteed. The effectiveness of the collaborative control is verified by the proof based on Lyapunov approach and simulations of the artificial swarm system.

Optimal guidance law design for three-dimensional trajectory tracking of missiles based on fixed-gain disturbance observer

To address the trajectory tracking problem of missiles, an optimal disturbance rejection guidance law (ODRGL) in three dimensions is proposed under the premise of uncontrollable speed. Error models of the actual and the reference trajectory are built in channels of pitch and yaw, respectively. In the yaw channel, a modeling error is introduced as the model cannot be built directly. According to the separation theorem, the controller and the observer are designed respectively. A fixed-gain disturbance observer (FGDO) is proposed to estimate the disturbance such as wind disturbances and modeling errors, and the disturbance is compensated for feed-forward. The nominal error models of pitch and yaw channels are linearized by the differential geometric linearization theory, and then linear quadratic regulator (LQR) controllers are designed for the linearized models. The simulation results show that in the presence of wind disturbances and modeling errors, the optimal guidance law proposed in this paper can stabilize the original nonlinear system and ensure energy optimization.

Trajectory tracking control of an autonomous underwater helicopter with improved prescribed performance

Aiming at the high-precision trajectory tracking problem of an autonomous underwater helicopter (AUH) under conditions of environmental disturbances, thruster loss and floating error, a prescribed performance control method based on a novel performance function is designed and can be applied in ultra-mobility tasks. The com bination of the proposed new performance function and the constraint inequality avoids large overshoot in the initial period of system and allows the expected convergence time to be set artificially. The thruster loss, expressed in thrust distribution matrix, is incorporated into general uncertainty with external disturbances. In addition, the extended state observer (ESO) is used to compensate the unknown states, providing a reference for the estimation of upper bound of general uncertainty. Simulation results show that the proposed control scheme is able to guarantee the pre-defined tracking performance limited by performance functions under general uncertainty.

Neural Adaptive Quantitative Prescribed Performance Sectionalized Event-Triggered Control for Autonomous Underwater Vehicles

In this paper, we study the quantitative design paradigm (QDP) of tracing control for a class of second-order system and further extend this to solve the trajectory tracking problem of autonomous underwater vehicles. The key merit of QDP is the capability of assigning some quantitative indices (e.g., overshoot and convergence time). To pursue performance enhancement with regard to tracking performance and bandwidth saving, a sectionalized event-triggered mechanism (SETM) incorporated with prespecified convergence time is proposed. To recognize the peculiarities of the lumped disturbances, an estimation-triggered neural network (ETNN) is designed via a property indicator, such that the disturbances that deteriorate the performance of the closed-loop system will be compensated and beneficial disturbances will be reserved otherwise, enabling less energy consumption without sacrificing the tracking performance. Theoretical analysis and simulation results are provided and analyzed, validating the performance and efficiency of the proposed solution.

Actor-critic objective penalty function method: an adaptive strategy for trajectory tracking in autonomous driving

Trajectory tracking is a key technology for controlling the autonomous vehicles effectively and stably to track the reference trajectory. How to handle the various constraints in trajectory tracking is very challenging. The recently proposed generalized exterior point method (GEP) shows high computational efficiency and closed-loop performance in solving the constrained trajectory tracking problem. However, the neural networks used in the GEP may suffer from the ill-conditioning issue during model training, which result in a slow or even non-converging training convergence process and the control output of the policy network being suboptimal or even severely constraint-violating. To effectively deal with the large-scale nonlinear state-wise constraints and avoid the ill-conditioning issue, we propose a model-based reinforcement learning (RL) method called the actor-critic objective penalty function method (ACOPFM) for trajectory tracking in autonomous driving. We adopt an integrated decision and control (IDC)-based planning and control scheme to transform the trajectory tracking problem into MPC-based nonlinear programming problems and embed the objective penalty function method into an actor-critic solution framework. The nonlinear programming problem is transformed into an unconstrained optimization problem and employed as a loss function for model updating of the policy network, and the ill-conditioning issue is avoided by alternately performing gradient descent and adaptively adjusting the penalty parameter. The convergence of ACOPFM is proved. The simulation results demonstrate that the ACOPFM converges to the optimal control strategy fast and steadily, and perform well under the multi-lane test scenario.

Data‐driven adaptive trajectory tracking control of unmanned marine vehicles under disturbances and DoS attacks

AbstractThis article researches the problem of trajectory tracking for unmanned marine vehicles (UMVs) under disturbances and denial‐of‐services (DoS) attacks in the wireless channel. An equivalent data‐driven model of UMVs with ocean disturbances is established by using partial form dynamic linearization algorithm. The disturbances and the input of UMVs have different pseudo partitioned Jacobean matrix, and the disturbances are estimated by using extended state observer, which improves the immunity of UMVs to disturbances in the environment. It is the first time that the DoS attacks are considered under the data‐driven model for UMVs, and a novel data‐driven adaptive trajectory tracking control framework is constructed. The article proposes an attack predictive compensation mechanism to mitigate their effects of DoS attacks, which follows the Bernoulli distribution. Based on it, the data‐driven adaptive trajectory tracking controller is designed such that the error of trajectory tracking is convergent under DoS attacks and external disturbances. Finally, the effectiveness of the proposed data‐driven control scheme and the predictive compensation mechanism is validated through the simulations.

Robust control for a tracked mobile robot based on a finite-time convergence zeroing neural network.

Since tracked mobile robot is a typical non-linear system, it has been a challenge to achieve the trajectory tracking of tracked mobile robots. A zeroing neural network is employed to control a tracked mobile robot to track the desired trajectory. A new fractional exponential activation function is designed in this study, and the implicit derivative dynamic model of the tracked mobile robot is presented, termed finite-time convergence zeroing neural network. The proposed model is analyzed based on the Lyapunov stability theory, and the upper bound of the convergence time is given. In addition, the robustness of the finite-time convergence zeroing neural network model is investigated under different error disturbances. Numerical experiments of tracking an eight-shaped trajectory are conducted successfully, validating the proposed model for the trajectory tracking problem of tracked mobile robots. Comparative results validate the effectiveness and superiority of the proposed model for the kinematical resolution of tracked mobile robots even in a disturbance environment.

Neural Fractional Order PID Controllers Design for 2-Link Rigid Robot Manipulator

The robotic manipulator is considered one of the complex systems that include multi-input, multi-output, non-linearity, and highly coupled. The uncertainty in the parameters and external disturbances have a negative influence on the performance of the system. Therefore, the controllers that will be designed for these systems must be able to deal with these complexities and difficulties. The Proportional, Integral, and Derivative (PID) controller is known to be simple and well robust, while the neural network has a solid ability to map complex functions. In this paper, we propose six control structures by combining the benefits of PID controller with integer and fractional order and the benefits of neural networks to produce hybrid controllers for a 2-Link Rigid Robot Manipulator (2-LRRM) handling with the problem of trajectory tracking. The Gorilla Forces Troops Optimization algorithm (GTO) was used to tune the parameters of the proposed controller schemes to minimize the Integral of Time Square Error (ITSE). In addition, the robustness of the performance of the suggested control systems is tested by altering the initial position, external disturbances and parameters and carried out using MATLAB. The best performance of the proposed controllers was the Neural Network Fractional Order Proportional Integral Derivative Controller (NNFOPID).

Podążanie za zadaną trajektorią grupy robotów kołowych z użyciem wirtualnych połączeń sprężysto-tłumiących

The article presents the solution to the problem of trajectory tracking of a self-organized group of wheeled robots in the environment without obstacles. The group of robots is tracking a trajectory realized as following a reference point by the geometric center of the group, as well as simultaneously, reaching and maintaining a given distance between neighboring robots. The proposed method is based on virtual forces from virtual spring-damper connections between robots, which allows for the trajectory tracking of the previously self-organized group while maintaining its desired shape. The presented method of control is described in detail with the description of i-th robot dynamics and was tested numerically and experimentally. The paper presents the results of numerical tests and experimental research and ends with discussion and conclusions. The paper’s results could be expanded for applications related to robotic group trajectory tracking.

Flatness-Based Nonlinear Control for Path Planning and Tracking of Sloshing Liquid Container

The movement of the liquid inside the container, known as sloshing, is usually undesired. Thus, there is the necessity to keep under control the peaks that the liquid free-surface exhibits during motion. This paper aims at providing a solution for suppressing sloshing liquid in horizontally moving cylindrical container. After introducing equivalent discrete models based on a mass-spring-damper system introduced by the literature (non-linear model), the identification and utilization of flat outputs is presented to generate rest-to-rest trajectories in sloshing liquid systems, which is ensure the equilibrium of the sloshing height at both initial and final points. Moreover, a sliding-mode controller is described to solve the trajectory tracking problem. The effectiveness of the proposed approach is demonstrated through numerical simulations comparisons with a model predictive controller (MPC). This research contributes to the advancement of control techniques for anti-sloshing technology systems, enabling enhanced stability, performance, and safety in various engineering applications.

Development of trajectory-tracking maneuvering system for automatic berthing/unberthing based on double deep Q-network and experimental validation with an actual large ferry

The development of an automatic berthing and unberthing system is essential for realizing autonomous ship navigation systems. Although numerous berthing/unberthing methods using neural networks (NNs) have been proposed in recent years, whether NNs trained via numerical simulations including modeling errors can be used for the automatic berthing/unberthing of actual ships in natural external disturbances is not clear. Hence, in this study, a maneuvering system for automatic berthing/unberthing was developed using a NN that solves the trajectory-tracking problem. The proposed method considered future targets as discrete points in learning, such as to absorb the modeling error and reduce the influence of external disturbances; thus, it achieved a more robust control compared to path planning methods. The system achieved trajectory tracking at low speeds by offline learning without teaching data to follow a target ship with the same maneuverability as its own ship and operated randomly under external disturbances. Furthermore, an online target correction logic function that locally corrects targets was developed to increase the tracking ability in external disturbances. Subsequently, a berthing and unberthing experiment was conducted using an actual large ferry to ensure the applicability of simulation-based NNs to the trajectory-tracking control of an actual ship in a real-life environment.