Underactuation Problem Research Articles

Robust adaptive optimal trajectory tracking control for underactuated AUVs with position and velocity constraints in three‐dimensional space

AbstractThe safety and optimality of underactuated autonomous underwater vehicles (AUVs) during operations are essential factors to consider. In this context, a three‐dimensional robust adaptive optimal trajectory tracking control method under position and velocity constraints, unknown dynamics, and environmental disturbances is proposed. The main features of the method are: (1) The outputs of an underactuated AUV system are redefined to handle the underactuation problem. (2) The system with position and velocity constraints is transformed into an unconstrained system by a nonlinear state‐dependent transformation. (3) A critic‐identifier architecture is constructed using adaptive dynamic programming and neural networks in a backstepping framework. Specifically, critic networks and weight update laws without requiring initial stability control are designed to solve Hamilton‐Jacobi‐Bellman equations in kinematic and dynamic subsystems, and optimal virtual and actual control laws are obtained. (4) A neural network identifier is developed to estimate unknown dynamics. Disturbances are overcome by improving the cost function and solving for optimal control of the nominal dynamic subsystem. By stability analysis, tracking errors in the AUV closed‐loop system can converge to an arbitrarily small compact set of the origin, and the other signals are uniformly ultimately bounded. Simulation comparisons demonstrate the effectiveness and superiority of the proposed method.

An enhanced coupling nonlinear control for quadrotor with suspended double‐pendulum payload

AbstractIn the field of logistics transportation, unmanned aerial vehicles are commonly utilized for cargo transport in a suspended manner, and the suspended payloads will exhibit the double‐pendulum dynamic characteristics (dPDC). However, due to the underactuation and load‐swing problems caused by the dPDC, it is challenging to design controllers for a quadrotor with a suspended double‐pendulum type payload. Existing related studies simplify the dPDC to a single‐pendulum dynamic characteristic, which results in degraded transient performance. To overcome the issue, this article presents a novel controller approach that considers the real dynamic characteristics, that is, dPDC. The proposed approach tackles the quadrotor position control and anti‐swing problem in two steps. First, a composite signal incorporating both the quadrotor position and the payload swing angles is introduced to restrain the payload swing. Second, a nonlinear controller based on the energy analysis method is proposed to overcome the coupling between rotation and translation, which addresses the underactuation problem. Theoretical analysis confirms the asymptotic stability of the system under this control strategy. Numerical and experimental results are provided to demonstrate the effectiveness of the proposed controller.

Adaptive Finite-Time Trajectory Tracking Event-Triggered Control Scheme for Underactuated Surface Vessels Subject to Input Saturation

In this paper, the trajectory tracking problem of underactuated surface vessels (USVs) that is subject to input saturation, unmodeled dynamics, and marine environmental disturbances is investigated. Firstly, USVs underactuation problem is handled by constructing a coordinate transformation. An auxiliary dynamic system is adopted to deal with input saturation. The adaptive law is presented to online update the upper bounds of the composite disturbances. The event-triggering condition is introduced to reduce drastically control execution frequency and the transmission load. By combining the auxiliary dynamic system, an adaptive technique, a first-order command filtered technique and an event-triggered condition, a novel adaptive finite time event-triggered control (AFETC) scheme is proposed, which has the following significant characteristics compared to most existing schemes: 1) the artfully constructed coordinate transformation not only avoids computational complexity but also simplifies the design; 2) the first-order command filtering vector-backstepping control method not only circumvents “explosion of complexity” in the traditional backstepping approach, but also ensures the finite-time settling of position tracking errors of USVs; 3) with the event triggered control, it reduces the control execution frequency, thereby decreasing the communication transmission load; and 4) it ensures stable tracking without requiring a priori information of the unmodelled dynamics of USVs or external disturbances. The designed robust adaptive control has proven to enable USVs to track the desired trajectory while guaranteeing the closed-loop stability in a finite time and avoiding Zeno behavior. Simulation and comparation results demonstrate the effectiveness and superiority of the proposed control strategy.

An Experimental Study on Trajectory Tracking Control of Torpedo-like AUVs Using Coupled Error Dynamics

In this paper, we propose a trajectory tracking controller with experimental verification for torpedo-like autonomous underwater vehicles (AUVs) with underactuation characteristics. The proposed controller overcomes the underactuation problem by designing the desired error dynamics in a coupled form using state variables in body-fixed and world coordinates. Unlike the back-stepping control requiring high-order derivatives of state variables, the proposed controller only requires the first derivatives of the states, which can alleviate noise magnification issues due to differentiation. We adopt time delay estimation to estimate the dynamics indirectly using control inputs and vehicle outputs, making the proposed controller relatively easy to apply without requiring the all of the vehicle dynamics. We also address some practical issues that commonly arise in experimental environments: handling measurement noises and actuation limits. To mitigate the effects of noise on the controller, a filtering technique using a moving window average is employed. Additionally, to account for the actuation limits, we design an anti-windup structure that takes into consideration the nonlinearity between the thrusting force and rotating speed of the thruster. We verify the tracking performance of the proposed controller through experimentation using an AUV. The experimental results show that the 3D motion control of the proposed controller exhibits an RMS error of 0.3216 m and demonstrate that the proposed controller achieves accurate tracking performance, making it suitable for survey missions that require tracking errors of less than one meter.

Piecewise Compensation Model Predictive Governor Combined With Conditional Disturbance Negation for Underactuated AUV Tracking Control

With respect to the accurate tracking control for autonomous underwater vehicles (AUVs), this paper proposes a novel control architecture consisting of piecewise compensation model predictive governor and conditional disturbance negation-based active disturbance rejection controller. The designed piecewise compensation mechanism is implemented to eliminate the steady tracking errors resulting from the underactuation problem of AUVs, and the model predictive governor aims to generate reference tracking velocities satisfying the constraints. Concurrently, a finite-time extended state observer is designed to improve the convergence performance of the estimation errors. In addition, the concept of conditional disturbance negation is employed to endow AUVs with the capability of taking advantage of disturbances rather than only rejecting disturbances. Finally, comprehensive analyses of the simulations and the experiments demonstrate the impressive control performance, verifying the feasibility of potential applications on real AUVs.

Virtual angle‐based adaptive control for trajectory tracking and balancing of ball‐balancing robots without velocity measurements

SummaryThis article proposes a virtual angle‐based adaptive control method for trajectory tracking and balancing of ball‐balancing robots without velocity measurements. The trajectory tracking and balancing control of ball‐balancing robots is challenging due to underactuation and uncertain nonlinearities. The hierarchical control strategy, which designs the control system using a linear combination of trajectory tracking and balancing errors, can be a solution. However, it has a local minimum problem where the convergence of tracking and balancing errors is not guaranteed even though the linear combination error is zero. Therefore, a virtual angle‐based control method is presented, which can solve the underactuation problem without the local minimum issue. In addition, a neural network‐based observer is developed to estimate the velocity information of ball‐balancing robots with uncertainties, and the input saturation problem is considered. It is proven that all tracking and balancing errors are bounded and can be made arbitrarily small in the Lyapunov sense. Finally, simulation results are provided to verify the effectiveness of the proposed control system.

Minimum-learning-parameter-based adaptive finite-time trajectory tracking event-triggered control for underactuated surface vessels with parametric uncertainties

For the problem of trajectory tracking control of underactuated surface vessels (USVs) affected by input saturation, parametric uncertainties, time-varying marine environmental disturbances, and limited transmission resources, a robust adaptive finite-time event-triggered control scheme is presented in this paper. First, a coordinate transformation is constructed to tackle the underactuation problem of the USV and enables the USV control law to be devised in vector form, which simplifies the design and makes it more suitable for computer processing. A continuous auxiliary dynamic system is constructed to handle the input saturation problem. The minimum learning parameter (MLP) technique is used to approximate the parametric uncertainties of the model. An adaptive law is devised to update online the upper bound of the combination of external disturbances and the approximation error generated by the MLP. Additionally, a finite-time control scheme is presented by introducing a robustifying term in the virtual control law, which improves the control performance; an introduced event triggering mechanism in the channel from the controller to the actuator decreases both the transmission burden and the control execution frequency without affecting the control performance. The theoretical analysis reveals that the devised robust adaptive finite-time event-triggered control scheme enables the USVs to track the desired trajectory in a finite time with a small number of control executions, while ensuring the closed-loop stability of the system and avoiding Zeno behavior. The simulations verify the effectiveness and the superiority of the presented control scheme.

Adaptive backstepping position tracking control of quadrotor unmanned aerial vehicle system

<abstract><p>In this work, an adaptive backstepping position tracking control using neural network (NN) approximation mechanism is proposed with respect to the translational system of quadrotor unmanned aerial vehicle (QUAV). Concerning the translational system of QUAV, on the one hand, it does not satisfy the matching condition and is an under-actuation dynamic system; on the other hand, it is with strong nonlinearity containing some uncertainty. To achieve the control objective, an intermediary control is introduced to handle the under-actuation problem, then the backstepping technique is combined with NN approximation strategy, which is employed to compensate the uncertainty of the system. Compared with traditional adaptive methods, the proposed adaptive NN position control of QUAV can alleviate the computation burden effectively, because it only trains a scalar adaptive parameter instead of the adaptive parameter vector or matrix. Finally, according to Lyapunov stability proof and computer simulation, it is proved that the control tasks can be accomplished.</p></abstract>

Command-Filtered Robust Adaptive NN Control With the Prescribed Performance for the 3-D Trajectory Tracking of Underactuated AUVs.

A novel robust adaptive neural network (NN) control scheme with prescribed performance is developed for the 3-D trajectory tracking of underactuated autonomous underwater vehicles (AUVs) with uncertain dynamics and unknown disturbances using new prescribed performance functions, an additional term, the radial basis function (RBF) NN, and the command-filtered backstepping approach. Different from the traditional prescribed performance functions, the new prescribed performance functions are innovatively proposed such that the time desired for the trajectory tracking errors of AUVs to reach and stay within the prescribed error tolerance band can be preset exactly and flexibly. The additional term with the Nussbaum function is designed to deal with the underactuation problem of AUVs. By means of RBF NN, the uncertain item lumped by the uncertain dynamics of AUVs and unknown disturbances is eventually transformed into a linearly parametric form with only a single unknown parameter. The developed control scheme ensures that all signals in the AUV 3-D trajectory tracking closed-loop control system are bounded. Simulation results with comparisons show the validity and the superiority of our developed control scheme.

Online recorded data-based finite-time composite neural trajectory tracking control for underactuated MSVs.

This paper presents an online recorded data-based composite neural finite-time control scheme for underactuated marine surface vessels (MSVs) subject to uncertain dynamics and time-varying external disturbances. The underactuation problem of the MSVs was solved by introducing the line-of-sight (LOS) method. The uncertain dynamics of MSVs are approximated by the composite neural networks (NNs). A modified prediction error signal is designed by virtue of online recorded data. The weight updating law of NN is driven by both tracking error and prediction error, introducing additional correction information to the weights of NN, thus improving the learning ability of the NN. Furthermore, disturbance observers can be devised to estimate the compound disturbances consisting of the approximation errors of NNs and external disturbances. Moreover, the smooth function is inserted into the design of the control scheme, and the finite-time composite neural trajectory tracking control of MSVs is achieved. The stability of the MSVs trajectory tracking closed-loop control system is guaranteed rigorously by the Lyapunov approach, and the tracking error will converge to the set of residuals around zero within a finite time. The simulation tests on an MSV verify the effectiveness of the proposed control scheme.

Distributed event-triggered adaptive output-feedback formation tracking of uncertain underactuated underwater vehicles in three-dimensional space

We present an adaptive-observer-based event-triggered design for distributed three-dimensional output-feedback formation tracking of multiple uncertain underactuated underwater vehicles (UUVs) under a directed network. It is assumed that all nonlinear functions of the vehicle dynamics are unknown and the velocity information of vehicles is immeasurable for feedback. Compared with existing distributed tracking results for UUVs, the primary contribution of this study is the development of a state-transformation-based adaptive observer using a time-varying scaling signal to achieve the output-feedback three-dimensional formation tracking in the presence of unknown nonlinearities and the underactuation constraint. Based on the state transformation, local adaptive observers using neural network approximators and their adaptive laws are designed to estimate local unmeasured velocities of UUV followers. By designing distributed nonlinear error functions and auxiliary stabilizing signals, local output-feedback tracking laws with triggering conditions are recursively constructed to ensure preselected formation performance while solving the underactuation problem of the UUV dynamics. It is shown that the formation tracking errors remain within pre-designable time-varying bounds and finally converge to a neighborhood of the origin. Simulation results are presented to show the effectiveness of the suggested output-feedback formation tracking strategy.

Composite learning tracking control for underactuated marine surface vessels with output constraints.

In this paper, a composite learning control scheme was proposed for underactuated marine surface vessels (MSVs) subject to unknown dynamics, time-varying external disturbances and output constraints. Based on the line-of-sight (LOS) approach, the underactuation problem of the MSVs was addressed. To deal with the problem of output constraint, the barrier Lyapunov function-based method was utilized to ensure that the output error will never violate the constraint. The composite neural networks (NNs) are employed to approximate unknown dynamics. The prediction errors can be obtained using the serial-parallel estimation model (SPEM). Both the prediction errors and the tracking errors were employed to construct the NN weight updating. Using approximation information, the disturbance observers were designed to estimate unknown time-varying disturbances. The stability analysis via the Lyapunov approach indicates that all signals of unmanned marine surface vessels are uniformly ultimate boundedness. The simulation results verify the effectiveness of the proposed control scheme.

Event-Triggered Composite Learning Finite-Time Trajectory Tracking Control for Underactuated MSVs Subject to Uncertainties

In this paper, a novel event-triggered composite learning finite-time control scheme is presented for underactuated marine surface vehicles (MSVs) trajectory tracking under unknown dynamics and unknown time-varying disturbances. Line-of-sight (LOS) tracking control method is employed to address the underactuation problem of MSVs. The neural networks (NNs) are untilized to approximate unknown dynamics. The serial-parallel estimation model is employed to construct the prediction error, and the prediction errors and tracking errors are fused with construct the NN weights updating. Combining the result of approximation information, the disturbances observers can be created to achieve disturbance estimation. Fractional power technology is artistically introduced to realize the finite-time trajectory tracking control of MSVs based on composite learning. The proposed control scheme ensures the simultaneous realization of high precision tracking performance and unknown information approximation. Moreover, an event-triggered mechanism is introduced to reduce the transmission load and the execution rate of actuators. It is proved that the proposed control scheme ensures all error signals of the MSVs trajectory tracking control system can converge to the neighborhood of zero within a finite time. Finally, the simulation results on an MSV verify the effectiveness and superiority of the proposed control scheme.

과소 작동 문제를 갖는 구형 이동 로봇의 동적 표면 기법 기반의 비선형 제어

본 논문은 과소 작동 문제를 갖는 구형 이동 로봇의 동적 표면 제어를 제안한다. 구형 이동 로봇의 과소 작동 문제는 제어 성능을 저조하게 하거나 하드웨어의 내구성을 손상시킬 수 있다. 이를 해결하기 위해 구형 이동 로봇 시스템의 안정화를 위한 기준각도로 가상 각도를 도입하고 동적 표면 제어 기법을 이용하여 비선형 제어기를 설계한다. 안정성 확보를 위해서 Lyapunov 안정성 이론에 의해 모든 오차는 궁극적으로 균일하게 유계가 된다. 마지막으로 모의실험을 통해 제안한 제어 시스템의 성능을 검증한다.

Neural Network Control for Trajectory Tracking and Balancing of a Ball-Balancing Robot with Uncertainty

In this paper, a neural-network-based control method to achieve trajectory tracking and balancing of a ball-balancing robot with uncertainty is presented. Because the ball-balancing robot is an underactuated system and has nonlinear couplings in the dynamic model, it is challenging to design a controller for trajectory tracking and balancing. Thus, various approaches have been proposed to solve these problems. However, there are still problems such as the complex control system and instability. Therefore, the objective of this paper was to propose a solution to these problems. To this end, we developed a virtual angle-based control scheme. Because the virtual angle was used as the reference angle to achieve trajectory tracking while keeping the balance of the ball-balancing robot, we could solve the underactuation problem using a single-loop controller. The radial basis function networks (RBFNs) were employed to compensate uncertainties, and the controller was designed using the dynamic surface control (DSC) method. From the Lyapunov stability theory, it was proven that all errors of the closed-loop control system were uniformly ultimately bounded. Therefore, the control system structure was simple and ensured stability in achieving simultaneous trajectory tracking and balancing of the ball-balancing robot with uncertainty. Finally, the simulation results are given to verify the performance of the proposed controller through comparison results. As a result, the proposed method showed a 19.2% improved tracking error rate compared to the existing method.

Observer-based fixed-time continuous nonsingular terminal sliding mode control of quadrotor aircraft under uncertainties and disturbances for robust trajectory tracking: Theory and experiment

This paper solves an accurate fixed-time attitude and position control problems of a quadrotor UAV system. The aircraft system is subject to nonlinearities, parameter uncertainties, unmodeled dynamics, and external time-varying disturbances. To deal with the under-actuation problem of the quadrotor’s dynamics, a hierarchical control structure with an inner–outer loop framework is adopted for the flight control system design. Robust nonlinear control strategies for attitude and position control are innovatively proposed based on a new continuous nonsingular terminal sliding mode control (CNTSMC) scheme. A full-order homogeneous terminal sliding surface is designed for the attitude and position states in such a way that the sliding motion is fixed-time stable independently of the system’s initial condition. Hence, this contributes to enhancing the control system robustness. A disturbance observer-based control (DOBC) approach is developed to stabilize the inner rotational subsystem (attitude-loop). This compounded control structure integrates a finite-time observer (FTO) and the CNTSMC scheme. The FTO observer is incorporated into the control framework to cope with the strong perturbations. An output-feedback control approach is adopted for the outer translational subsystem (position-loop) to ensure a velocity-free control. In this context, the CNTSMC scheme is combined with a fixed-time extended state observer (FXESO) to achieve an active disturbance rejection control (ADRC) by estimating and canceling the lumped disturbances. Therefore, within the developed control approach including the robust CNTSMC scheme, DOBC, and ADRC strategies, robust and accurate trajectory tracking control can be achieved despite uncertainties and disturbances. Stability analysis of the closed-loop system is rigorously investigated by using the Lyapunov theorem, bi-limit homogeneous theory, and the notion of input-to-state stability (ISS). Extensive experimental tests under the influence of various disturbances are conducted to corroborate the theoretical findings. To this end, an effective model-based design (MBD) framework is established to implement the developed control algorithms in real autopilot hardware. Furthermore, processor-in-the-loop (PIL) experiments are also carried out within the MBD framework. A comparative study is made involving our control algorithms and other control strategies. Overall, the obtained results show that the synthesized control system yields performance improvement regarding fixed-time tracking stability featuring fast transient, strong robustness, and high steady-state precision. Besides, the chattering effect of regular linear sliding mode control (LSMC) is significantly alleviated. Moreover, unlike conventional TSMC, the control input shows no singularity.

Fuzzy-Based Super-Twisting Sliding Mode Stabilization Control for Under-Actuated Rotary Inverted Pendulum Systems

This paper considers the stabilization problem for under-actuated rotary inverted pendulum systems (RotIPS) via a fuzzy-based continuous sliding mode control approach. Various sliding mode control (SMC) methods have been proposed for stabilizing the under-actuated RotIPS. However, there are two main drawbacks of these SMC approaches. First, the existing SMCs have a discontinuous structure; therefore, their control systems suffer from the chattering problem. Second, a complete proof of closed-loop system stability has not been provided. To address these two limitations, we propose a fuzzy-based (continuous) super-twisting stabilization algorithm (FBSTSA) for the under-actuated RotIPS. We first introduce a new sliding surface, which is designed to resolve the under-actuation problem, by combining the fully-actuated (rotary arm) and the under-actuated (pendulum) variables to define one sliding surface. Then, together with the proposed sliding surface, we develop the FBSTSA, where the corresponding control gains are adjusted based on a fuzzy logic scheme. Note that the proposed FBSTSA is continuous owing to the modified super-twisting approach, which can reduce the chattering and enhance the control performance. With the proposed FBSTSA, we show that the sliding variable can reach zero in finite time and then the closed-loop system state converges to zero asymptotically. Various simulation and experimental results are provided to demonstrate the effectiveness of the proposed FBSTSA. In particular, (i) compared with the existing SMC approaches, chattering is alleviated and better stabilization is achieved; and (ii) the robustness of the closed-loop system (with the proposed FBSTSA) is guaranteed under system uncertainties and external disturbances.

Fractional attitude-reactive control for robust quadrotor position stabilization without resolving underactuation

Quadrotors are Unmanned Aerial Systems (UAS), whose nonlinearities, uncertainties, and unavoidable aerodynamic disturbances stand for a difficult control problem. Additionally, the highly nonlinear coupling between the position and the orientation dynamics exacerbates the underactuation problem when the objective is the position tracking. In this paper, aiming at designing a robust position controller without resolving underactuation, a novel “reactive orientation control” is proposed. The orientation controller is model-free and it is build upon differintegral fractional operators to react to a continuous and bounded additive term that stands for an affine exogenous position controller. Such term indeed becomes an attitude disturbance that is purposely injected to displace the quadrotor towards the desired position of the underactuated coordinates. The closed-loop analysis shows that the norm of the transfer function between the disturbance and the position tracking error is minimized for a stable and robust position regime. Simulations and experimental results are given to discuss the feasibility of the proposed scheme.

Implementation of factitious force method for control of 5R manipulator with skid-steering platform REX

Abstract In the paper implementation of the factitious force concept for a controlling complex mobile manipulator has been presented. As the nonholonomic constraint only lack of longitudinal slippage of wheels has been chosen - in skid-steering platforms lateral slippage is necessary to change orientation of such a platform. From a control theory point of view such a system is dynamically underactuated. As a solution to a underactuation problem a method of factitious force has been proposed. This method assumes extension on the dynamics level, in the form of an additional control inputs uv, which values are equal to zero equivalently. For a mobile manipulator, consisting of platform REX and 5R robotic onboard arm, a cascaded control law has been proposed. A simulation study was conducted for a mathematical model of a considered object with real values of physical parameters, i.e. lengths, masses, inertia moments etc. obtained from the 3D model. Results obtained in simulations have shown a proper action of the control system and convergence of tracking errors, occurring in a platform and in joints of a manipulator, to zero.

Fully magnetic sliding mode control for acquiring three-axis attitude

Satellite equipped with purely magnetic attitude control system is considered. Sliding mode control is used to achieve three-axis satellite attitude. Underactuation problem is solved for transient motion. Necessary attitude is acquired by proper sliding manifold construction. Satellite motion on the manifold is executed with magnetic control system. One manifold construction approach is proposed and discussed. Numerical examples are provided.