Underactuated Nonlinear Systems Research Articles

Computing of LQR Technique for Nonlinear System Using Local Approximation

The main idea behind the present research is to design a state-feedback controller for an underactuated nonlinear rotary inverted pendulum module by employing the linear quadratic regulator (LQR) technique using local approximation. The LQR is an excellent method for developing a controller for nonlinear systems. It provides optimal feedback to make the closed-loop system robust and stable, rejecting external disturbances. Model-based optimal controller for a nonlinear system such as a rotatory inverted pendulum has not been designed and implemented using Newton-Euler, Lagrange method, and local approximation. Therefore, implementing LQR to an underactuated nonlinear system was vital to design a stable controller. A mathematical model has been developed for the controller design by utilizing the Newton-Euler, Lagrange method. The nonlinear model has been linearized around an equilibrium point. Linear and nonlinear models have been compared to find the range in which linear and nonlinear models’ behaviour is similar. MATLAB LQR function and system dynamics have been used to estimate the controller parameters. For the performance evaluation of the designed controller, Simulink has been used. Linear and nonlinear models have been simulated along with the designed controller. Simulations have been performed for the designed controller over the linear and nonlinear system under different conditions through varying system variables. The results show that the system is stable and robust enough to act against external disturbances. The controller maintains the rotary inverted pendulum in an upright position and rejects disruptions like falling under gravitational force or any external disturbance by adjusting the rotation of the horizontal link in both linear and nonlinear environments in a specific range. The controller has been practically designed and implemented. It is vivid from the results that the controller is robust enough to reject the disturbances in milliseconds and keeps the pendulum arm deflection angle to zero degrees.

Model-Free Controller Design for Nonlinear Underactuated Systems With Uncertainties and Disturbances by Using Extended State Observer Based Chattering-Free Sliding Mode Control

Most of the control strategies require a mathematical model or reasonable knowledge that is difficult to obtain for complex systems. Model-free control is a good alternative to avoid the difficulties and complex modeling procedures, especially if the knowledge about the system is insufficient. This paper presents a new control scheme completely independent of the system model. The proposed scheme combines sliding mode control (SMC) with intelligent proportional integral derivative (iPID) control based on a local model and extended state observer (ESO). Although the iPID control makes the proposed method model-free, it cannot guarantee that the tracking errors converge to zero asymptotically except the system is in a steady-state regime. Therefore, the SMC is added to the control scheme to ensure the convergence by minimizing the estimation errors of the observer. The proposed iPIDSMC controller is tested in the presence of different parameter variations and external disturbances on an inverted pendulum - cart (IPC), which is a highly unstable underactuated system with nonlinear coupled dynamics. The proposed controller is compared with the PID, iPID and Hierarchical Sliding Mode Control (HSMC) for a clearer evaluation. Simulation results showed that the proposed controller is extremely insensitive to parameter variations, matched and mismatched disturbances and the control signal of the proposed method is chattering-free, even though it is based on a discontinuous control action.

Combined algorithms for analytical inverse kinematics solving and control of the Q-PRR aerial manipulator*

This article presents the design and modeling of a new aerial manipulator system, called Q-PRR, composed of three joints with a fixed base in the center of mass of the multirotor considered as a whole system. This structure has a prismatic joint as a first joint which allows to keep the center of gravity of the Q-PRR as close as possible to the center of gravity of the multirotor. This will also allow to reduce the influence of arm motion on the multirotor roll thus to ensure the stability of the system on trajectory tracking with dynamic changes in the multirotor’s center of gravity. Furthermore, the configuration of the manipulator arm for the desired position of the end-effector given by the inverse kinematics model is kept without any change in the position and attitude of the multirotor. This article develops both forward and inverse kinematics models for a nonlinear underactuated system using the Denavit–Hartenberg notation. When a new algorithm is presented for the inverse kinematics based on Levenberg–Marquardt algorithm. Then, the dynamic model in the joint spaces is developed with the Lagrangian formalism. The Q-PRR is controlled using a model-free control with a comparison of two states, a free fly and disturbance forces applied to the whole system with manipulator arm movement.

A Constructive Procedure for Orbital Stabilization of a Class of Underactuated Mechanical Systems

In this note, we present a constructive procedure to solve the orbital stabilization problem of a class of nonlinear underactuated mechanical systems with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> degrees of freedom and underactuation degree one using the Immersion and Invariance technique. We define sufficient conditions to solve explicitly the partial differential equations arising in the Immersion and Invariance methodology, so that mechanical systems with gyroscopic terms can be considered. At the end, we use two practical systems as examples to illustrate the design procedure, as well as validating performance via simulations and experiments.

Robust Stabilization of Underactuated TORA System Based on Disturbance Observer and Fixed-Time Sliding Mode Control Method

A translational oscillator with a rotational actuator (TORA) is an underactuated nonlinear mechanical system with two degrees of freedom (DOF). This paper concerns the robust stabilization control problem for the system with multiple external disturbances. First, a disturbance observer is constructed based on the internal nonlinear dynamic behavior of the system. Second, a robust stabilization controller is designed by the estimated disturbances and the fixed-time sliding mode control method. The controller realizes the global robust stabilization control objective of the TORA system, and the stability of both disturbance observer and robust closed-loop control system are analyzed using the Lyapunov theorem. Finally, the effectiveness of the theoretical results are verified by numerical experiments.

A Comparative Study of Control Methods for X3D Quadrotor Feedback Trajectory Control

Unmanned aerial vehicles (UAVs), particularly quadrotor, have seen steady growth in use over the last several decades. The quadrotor is an under-actuated nonlinear system with few actuators in comparison to the degree of freedom (DOF); hence, stabilizing its attitude and positions is a significant challenge. Furthermore, the inclusion of nonlinear dynamic factors and uncertainties makes controlling its maneuverability more challenging. The purpose of this research is to design, implement, and evaluate the effectiveness of linear and nonlinear control methods for controlling an X3D quadrotor’s intended translation position and rotation angles while hovering. The dynamics of the X3D quadrotor model were implemented in Simulink. Two linear controllers, linear quadratic regulator (LQR) and proportional integral derivate (PID), and two nonlinear controllers, fuzzy controller (FC) and model reference adaptive PID Controller (MRAPC) employing the MIT rule, were devised and implemented for the response analysis. In the MATLAB Simulink Environment, the transient performance of nonlinear and linear controllers for an X3D quadrotor is examined in terms of settling time, rising time, peak time, delay time, and overshoot. Simulation results suggest that the LQR control approach is better because of its robustness and comparatively superior performance characteristics to other controllers, particularly nonlinear controllers, listed at the same operating point, as overshoot is 0.0% and other factors are minimal for the x3D quadrotor. In addition, the LQR controller is intuitive and simple to implement. In this research, all control approaches were verified to provide adequate feedback for quadrotor stability.

Adaptive fixed-time hierarchical sliding mode control for switched under-actuated systems with dead-zone constraints via event-triggered strategy

This article investigates the event-triggered-based adaptive fixed-time control problem for switched under-actuated nonlinear systems subject to dead-zone constraints. To improve the utilization of communication resources, robustness and response rate of the system under consideration, an efficient control method combining hierarchical sliding mode technique and event-triggered mechanism is proposed. On the basis of this method, a new fixed-time stability criterion is developed to ensure that the designed controller can make the considered system be fixed-time stable and particularly the settling time be independent of the initial conditions. Then, the singularity problem of the designed controller is successfully solved by employing a projection algorithm. Finally, rigorous theoretical derivations and simulation results are included to illustrate the effectiveness of the proposed method.

Flatness-Based Active Disturbance Rejection Control for a PVTOL Aircraft System with an Inverted Pendular Load

This paper presents a systematic procedure for the control scheme design for a PVTOL aircraft system with an inverted pendular load, which is a nonlinear underactuated system. The control scheme is based on the use of angular movement as an artificial control in order to propose new auxiliary control inputs. This is achieved by a linear extended state observer-based active disturbance rejection control to reject both nonmodeled dynamics and external disturbances. The flying planar inverted pendulum is then linearized around an unstable equilibrium point, and the resulting system is subdivided into two subsystems: (1) the height system, and (2) the horizontal pendulum system. For the height system, a linear extended state observer-based active disturbance rejection control is proposed in order to accomplish a take-off and landing task in the presence of external disturbances and non-linearities neglected in the linearization process. The flatness property in the horizontal-pendulum system is exploited in order to propose another active disturbance rejection control of linear nature. The flatness of the tangentially linearized model provides a unique structural property that results in an advantageous low-order cascade decomposition of the linear extended state observer design. Numerical simulations show the effectiveness of the proposed control scheme in trajectory tracking tasks in the presence of disturbances caused by crosswinds with random amplitudes.

Dynamic surface control with a nonlinear disturbance observer for multi‐degree of freedom underactuated mechanical systems

AbstractUnderactuated systems are extensively utilized in practice while attracting a huge deal of attention in theoretical studies. There are few robust control strategies for general underactuated systems because of the variety of their dynamic models. A dynamic surface control strategy with a nonlinear disturbance observer is proposed in this study, to stabilize multi‐degree of freedom underactuated systems. In such systems the number of underactuated degrees of freedom is not higher compared to the actuated ones. A disturbance observer is utilized to dispose of the uncertain disturbance and cross terms in dynamic model which may cause failure to the controller. Then, a dynamic surface control strategy is presented which is not sensitive to the diversity of dynamic models. The stability of whole system is proven by Lyapunov‐based method. The control law is successfully applied to nonlinear underactuated systems in benchmark cascade form such as two‐translational oscillator with rotational actuator, crane, wheeled inverted pendulum. The effectiveness of proposed controllers are illustrated by MATLAB simulation results. Finally, comparative studies are presented to verify the superiority of the proposed method.

Modeling and Control of a Wheeled Biped Robot.

It is difficult to realize the stable control of a wheeled biped robot (WBR), as it is an underactuated nonlinear system. To improve the balance and dynamic locomotion capabilities of a WBR, a decoupled control framework is proposed. First, the WBR is decoupled into a variable-length wheeled inverted pendulum and a five-link multi-rigid body system. Then, for the above two simplified models, a time-varying linear quadratic regulator and a model predictive controller are designed, respectively. In addition, in order to improve the accuracy of the feedback information of the robot, the Kalman filter is used to optimally estimate the system state. The control framework can enable the WBR to realize changing height, resisting external disturbances, velocity tracking and jumping. The results obtained by simulations and physical experiments verify the effectiveness of the framework.

Adaptive fuzzy PID control based on nonlinear disturbance observer for quadrotor

In this paper, the composite control of position and orientation of the quadrotor for tracking purposes are investigated. The quadrotor is an underactuated nonlinear system. So, based on the introduced virtual inputs the quadrotor is transformed into an actuated system. External disturbances and different sources of uncertainty such as input saturation are considered in the dynamic model of the quadrotor. Also, after transform implementation, some complicated terms appear which are considered as other sources of uncertainty. To guarantee the stabilization and tracking goals of the quadrotor in the perturbed model, the composite adaptive fuzzy PID-Like controller is proposed. This controller consists of three parts where the innovation of the paper appears in all terms. The main term is an adaptive fuzzy PID-like controller which able to approximate the coefficients of PID based on the decentralized fuzzy logic. Then, a H∞ compensator based on a modified Riccati-like equation is presented; which can attenuate the fuzzy approximation errors. Also, as the last term of the composite controller, the modified disturbance observer is designed that estimates the uncertainty and disturbance. The experimental result shows that the proposed method leads to resilient stabilization and tracking in comparison with conventional methods.

Neuro-adaptive fixed-time non-singular fast terminal sliding mode control design for a class of under-actuated nonlinear systems

This paper presents a fixed-time neuro-adaptive control design for a class of uncertain under-actuated nonlinear systems (UNS) using a non-singular fast terminal sliding mode control (TSMC) with a radial basis function (RBF)-based estimator to achieve the convergence and robustness against the uncertainties. The mathematical model of the considered class is reduced into an equivalent regular form. A fast TSMC is designed for the transformed form to improve the control performance and annihilate the associated singularity problem of the conventional TSMC. Lyapunov stability theory ensures the steering of the sliding manifold and system states in fixed time. RBF neural networks are adaptively estimate the nonlinear drift functions. The theoretical design, analysis, and simulations of cart-pendulum and quadcopter demonstrate the feasibility and benefits of the regular form transformation and the designed control design. Comparing the proposed synthesis with the standard literature presents the attractive nature of proposed method for such a class.

Adaptive neural finite-time hierarchical sliding mode control of uncertain under-actuated switched nonlinear systems with backlash-like hysteresis

This article investigates an adaptive finite-time tracking control for under-actuated nonlinear systems with unknown backlash-like hysteresis and arbitrary switchings. A hierarchical sliding mode (HSM) control scheme is developed considering external disturbances and hysteresis nonlinearity. In addition, unknown functions of the considered system are handled based on neural networks (NNs) approximation, and a projection algorithm is introduced to solve the singularity of the denominator. To avoid the case in which the error is repeatedly magnified, the sum of the upper bounds of the approximation error, the external disturbance and the backlash-like hysteresis model nonlinear parts are estimated.By resorting to the Lyapunov stability theory, the boundedness of all signals can be proved under arbitrary switchings. At last, two simulation examples are presented to demonstrate the validity of the proposed approach.

Design of optimized interval type-2 fuzzy logic controller based on the continuity, monotonicity, and smoothness properties for a cart-pole inverted pendulum system

In this study, an interval type-2 fuzzy logic controller design method is proposed and validated on the real-time under-actuated nonlinear inverted pendulum system for swing-up and stabilization control problems. The swing-up algorithm is designed to give a faster response and the stabilization controller is designed based on continuity, monotonicity, and smoothness theorems for worst and best cases using the Mamdani fuzzy inference system in order to show the effectiveness of the proposed method. Outstanding type reduction methods are analyzed for the stabilization controller based on the design approach to determine the best type reduction algorithm for the effective and robust control performance. Real-time experiments are conducted to investigate the capability of the proposed controller in terms of adaptation performance and robustness ability. The controller also is able to handle structured and unstructured uncertainties such as measurement noise, external payload, undesirable internal/external disturbances, and parameter uncertainties. The results show that the proposed method clearly improves the control performance of the system in six experimental tasks conducted.

Backstepping control for a class of underactuated nonlinear mechanical systems with a novel coordinate transformation in the discrete-time setting

In this technical note, a combined discrete-time controller, consisting of a partial feedback linearization controller and a backstepping controller, is proposed to manage the stabilization for a class of underactuated nonlinear mechanical systems. The traditional backstepping control method is impracticable to directly apply it to underactuated systems since they are generally not in the lower triangular form. First, a feedback controller that partially linearizes the underactuated mechanical systems is derived to facilitate the backstepping controller design. In the design, discrete-time system dynamics obtained utilizing the Euler approximation are considered. Afterward, a novel coordinate transformation is introduced for a class of underactuated systems that transform the mathematical model of the partially linearized underactuated mechanical systems into the strict-feedback form. After these steps, the conventional backstepping approach is structured in discrete-time setting. The stability of the closed-loop system dynamics with the proposed combined discrete-time controller is shown utilizing Lyapunov theory. Several computer-based simulations are executed to depict the performance and the applicability of the proposed method on the Cart–Pole system and the Inertia Wheel Pendulum. Moreover, the effectiveness of the proposed nonlinear controller is verified by numerical simulation results in comparison with a discrete-time chattering-free sliding-mode controller.

Trajectory tracking for underactuated surface vessels with time delays and unknown control directions

The problem of trajectory tracking control for underactuated surface vessels with time-delays, model uncertainty and unknown control directions has been considered in this paper. Firstly, an integration time delayed sliding mode controller is proposed, which decomposes the underactuated non-linear system into a cascading system consisting of a kinematics subsystem and a dynamics subsystem. Secondly, an adaptive backstepping control method for the heading angle system is proposed, which can drive the heading angle error into a preset interval. Then the Bouc-wen model is used to describe the hysteresis of the system, Nussbaum gain function is adopted to deal with unknown control directions, and the adaptive controller is designed by combining error transformation with Lyapunov function. Furthermore, a suitable Lyapunov-Krasovsky function is selected to analyse and prove the global stability of the closed-loop system. Finally, some numerical simulations show the effectiveness of the presented control methods.

Sliding Mode Control of Underactuated Nonlinear Systems Based on Piecewise Double Power Reaching Law

A sliding mode control (SMC) strategy using novel piecewise double power reaching law (PDPRL) for a class of fourth-order nonlinear underactuated systems (USs) is proposed to realize simultaneous control of the unactuated and the actuated part. A novel PDPRL is designed in the form of a piecewise function to reduce the chattering produced by SMC. The fixed-time convergence characteristic and the existence of maximum convergence time independent of the initial value of the sliding mode surface for the PDPRL are analyzed. Considering the uncertainties of model parameters and external disturbance in the USs, a supertwisting disturbance observer (STDO) is provided to accurately estimate the disturbance in real time, and the estimated value is compensated to the controller. The underactuated inverted pendulum system is taken to verify the effectiveness of the devised strategy, and the simulation results show that the proposed strategy can reduce chattering, suppress the disturbance, and enhance the robustness of the system.

Continuous Sliding-Mode Output-Feedback Control for Stabilization of a Class of Underactuated Systems

This article presents a robust output-feedback control scheme for the stabilization of a class of nonlinear underactuated mechanical systems in the presence of coupled disturbances. The scheme is based on the so-called coupled sliding-mode control, i.e., no transformation into the normal form is required. The proposed output approach is composed of a super-twisting controller and a high-order sliding-mode observer that robustly stabilize the origin of the system asymptotically. All of the results are proven by Lyapunov approaches. The proposed approach is validated by means of simulations and experimental results on a four-degree-of-freedom underactuated crane.

Three-Dimensional Obstacle Avoidance Strategy for Fixed-Wing UAVs Based on Quaternion Method

This work provides a generalization of the three-dimensional velocity obstacle (VO) collision avoidance strategy for nonlinear second-order underactuated systems in three-dimensional dynamic uncertain environments. A hierarchical architecture is exploited to deal with conflicting multiple subtasks, which are defined as several rotations and are parameterized by quaternions. An improved VO method considering the kinodynamic constraints of a class of fixed-wing unmanned aerial vehicles (UAV) is proposed to implement the motion planning. The position error and velocity error can be mapped onto one desired axis so that, only relying on an engine, UAVs can achieve the goal of point tracking without collision. Additionally, the performance of the closed-loop system is demonstrated through a series of simulations performed in a three-dimensional manner.

Deep Reinforcement Learning based Super Twisting Controller for Liquid Slosh Control Problem

Deep Reinforcement Learning (DRL) based parameter optimization of super twisting control (STC) for the liquid slosh control problem in a moving vehicle is proposed in this paper. The slosh control problem, including the vehicle dynamics, represents an under-actuated nonlinear dynamical system. The slosh phenomenon is modeled by a simple pendulum on a cart and STC had been designed in the literature for the system when the vehicle motion is in a straight line. In this paper, a DRL framework is designed for the first time to tune the STC parameters in order to deliver near optimal performance. The effectiveness of this proposed learning-based STC for the slosh control problem is validated in a Python simulation environment and its performance compared to that of the existing STC design without the learning.