Underactuated Systems Research Articles

Exact Linearization of Minimally Underactuated configuration Flat Lagrangian Control Systems by Quasi-Static Feedback of Classical States

We study the exact linearization of configuration flat Lagrangian control systems with p degrees of freedom and p - 1 inputs by quasi-static feedback of classical states. First, we present a detailed analysis of the structure of the parameterization of the system variables by the flat output. Based on that, we systematically construct a linearizing quasi-static feedback law of the classical state such that the closed-loop system shows the behavior of decoupled integrator chains. Our approach shows that the construction of a generalized Brunovsky state can be completely circumvented. Furthermore, we present a method for determining the lengths of the integrator chains achieved by quasi-static feedback laws that allow for rest-to-rest transitions.

On the Exact Linearization of Minimally Underactuated Configuration Flat Lagrangian Systems in Generalized State Representation

In this paper, we examine the exact linearization of Configuration flat Lagrangian control systems in generalized state representation with p degrees of freedom and p- 1 control inputs by quasi-static feedback of its generalized state. We formally introduce generalized Lagrangian control systems, which are obtained when Configuration variables are considered as inputs instead of forces. This work presents all possible lengths of integrator chains achieved by an exact linearization with a quasi-static feedback law of the generalized state that allows for rest-to-rest transitions. We show that such feedback laws can be systematically derived without using Brunovsky states.

Integral IDA-PBC of Underactuated Mechanical Systems with Actuator Dynamics and Uncertain Coupling

This work investigates the passivity-based control of a class of underactuated mechanical systems with first-order actuator dynamics and constant but uncertain coupling. The main contribution is a new dynamic extension of the interconnection-and-damping-assignment passivity-based control methodology that compensates for the effect of the model uncertainty. The effectiveness of the controller is demonstrated with numerical simulations on a ball-on-beam system.

Investigating Integral Reinforcement Learning to Achieve Asymptotic Stability in Underactuated Mechanical Systems

Investigating Integral Reinforcement Learning to Achieve Asymptotic Stability in Underactuated Mechanical Systems

Integral IDA-PBC for Underactuated Mechanical Systems Subject to Matched and Unmatched Disturbances

Integral IDA-PBC for Underactuated Mechanical Systems Subject to Matched and Unmatched Disturbances

Integral Controlled Lagrangians for Underactuated Mechanical Systems Subject to Position-Dependent Matched Disturbances

Integral Controlled Lagrangians for Underactuated Mechanical Systems Subject to Position-Dependent Matched Disturbances

A Comparative Analysis of Metaheuristic Algorithms Tuned Supper Twisting Sliding Mode Control of a Self-balancing Segway

A self-balancing two-wheel Segway is a multivariable, nonlinear, coupled and unstable personal transport that is among the benchmarks of under-actuated systems to test different control schemes. The proper operation of the system needs stable and robust controllers for the balancing and direction of the Segway. Stable and robust performance of such systems can be achieved using adaptive, optimal and non-linear control schemes. A conventional sliding mode controller is a nonlinear controller, considered a robust controller except for the chattering problem that can be solved using higher order sliding mode controllers such as a supper twisted sliding mode controller (STSMC). In this research, STSMC is proposed for balancing and PID controller for direction control of a self-balancing Segway. To solve the impacts of design technique, skill, and experience of the designer on the control schemes, controllers are tuned using metaheuristic optimization algorithms, namely Grey wolf algorithm (GWO), Genetic Algorithm (GA), and Particle Swarm Optimization (PSO) technique. The overall system has been implemented in MATLAB/Simulink, and performance evaluations have been executed for the time domain specifications of settling and rise times as comparison criteria for the Segway operating conditions of variable driver mass and tilt angles. The results show relatively quick responses - for initial pitch angel of 0.3 rad, and initial yaw angle of 0.1 rad – of GWO-STSMC and GWO-PID controllers. For variable driver mass and constant initial tilt angles, the pitch response is found to be less sensitive to the increase in mass of the driver for GWO-STSMC, while the yaw angle response shows a slight sensitivity decrease for GWO-PID controller. For variable initial tilt angles and constant mass of the drive, both GWO-STSMC and GWO-PID controllers show the capability of tracking the reference pitch and yaw angles.

Dynamic-Event-based Hierarchical Fuzzy Prescribed Performance Control for Underactuated Systems with Uncertain Dead Zone

Dynamic-Event-based Hierarchical Fuzzy Prescribed Performance Control for Underactuated Systems with Uncertain Dead Zone

Composite Fuzzy-Based Control Strategy for Uncertain Underactuated System

Composite Fuzzy-Based Control Strategy for Uncertain Underactuated System

Integral passivity‐based control of underactuated mechanical systems with state‐dependent matched disturbances

AbstractThis work investigates the energy shaping control of a class of underactuated mechanical systems subject to state‐dependent linearly parameterized matched disturbances. To this end, a new integral interconnection and damping‐assignment passivity‐based design is proposed. The controller design is developed for systems subject to either momenta‐dependent disturbances or position‐dependent disturbances or to both simultaneously. The effectiveness of the proposed approach is demonstrated with numerical simulations on three examples: an Acrobot system with nonlinear friction; a ball‐on‐beam system with constant and position‐dependent disturbances; a disk‐on‐disk system with a complete set of state‐dependent disturbances. In all examples, the proposed controller shows better performance compared to previous implementations.

Adaptive optimal control of under-actuated robotic systems using a self-regulating nonlinear weight-adjustment scheme: Formulation and experimental verification.

This paper formulates an innovative model-free self-organizing weight adaptation that strengthens the robustness of a Linear Quadratic Regulator (LQR) for inverted pendulum-like mechatronic systems against perturbations and parametric uncertainties. The proposed control procedure is devised by using an online adaptation law to dynamically adjust the state weighting factors of LQR's quadratic performance index via pre-calibrated state-error-dependent hyperbolic secant functions (HSFs). The updated state-weighting factors re-compute the optimal control problem to modify the state-compensator gains online. The novelty of the proposed article lies in adaptively adjusting the variation rates of the said HSFs via an auxiliary model-free online self-regulation law that uses dissipative and anti-dissipative terms to flexibly re-calibrate the nonlinear function's waveforms as the state errors vary. This augmentation increases the controller's design flexibility and enhances the system's disturbance rejection capacity while economizing control energy expenditure under every operating condition. The proposed self-organizing LQR is analyzed via customized hardware-in-loop (HIL) experiments conducted on the Quanser's single-link rotational inverted pendulum. As compared to the fixed-gain LQR, the proposed SR-EM-STC delivers an improvement of 52.2%, 16.4%, 55.2%, and 42.7% in the pendulum's position regulation behavior, control energy expenditure, transient recovery duration, and peak overshoot, respectively. The experimental outcomes validate the superior robustness of the proposed scheme against exogenous disturbances.

Gain-Scheduled Model Predictive Control for Cart–Inverted-Pendulum with Friction and Disturbances

The cart–inverted pendulum system (CIPS) is a typical example of underactuated mechanical systems. For the CIPS with friction and disturbances, a gain-scheduled model predictive control method is proposed to achieve the upright stabilization objective of the single inverted pendulum (SIP) while controlling the cart to reach a desired new position. To this end, first, a dynamic equation of the CIPS with friction and disturbances is formulated based on the Newton–Euler equation. On the basis of the dynamic equation of the CIPS, its motion characteristics and control process are analyzed. Next, the given dynamic equation of the CIPS is linearized to obtain a series of linearized models at seven different pendulum angles. Then, seven model predictive controllers (MPCs) are designed based on the above-linearized models, respectively. Introducing the idea of the gain-schedule, a gain-scheduled MPC (GSMPC) is designed to switch one of these seven MPCs to realize the control objective of the CIPS, according to the actual pendulum angle of the SIP during the control process. Finally, multi-group simulations that consider the friction and disturbances of the CIPS are implemented to demonstrate the effectiveness of the proposed gain-scheduled model predictive control method.

Fully Distributed Event-Triggered Formation Control for Multiple Quadrotors

This paper studies fully distributed formation control problems for multi-quadrotor systems with external disturbances via an event-triggered approach. For leader-follower multi-quadrotor systems, a fully distributed event-triggered formation control (DETFC) framework is proposed by utilizing an adaptive sliding-mode control approach, which does not rely on any global information of the network topology. The finite-time reachability of the sliding-mode surface can be guaranteed for the states of the nonlinear, coupled and underactuated system subject to external disturbances. Meanwhile, the zeno behavior of the proposed event-triggered protocol is proved to be excluded. A novel dynamic sliding-mode surface is designed to guarantee the formation performance as the quadrotor state trajectories move on the constructed sliding manifold. Via Lyapunov stability theory, sufficient conditions to ensure the formation results are derived for multi-quadrotor systems. Simulations and experiments are conducted to validate the effectiveness of the proposed control scheme.

Attitude optimization by extremum seeking for rigid bodies actuated by internal rotors only

SummaryIn this paper, we investigate the stabilizing effect of an extremum seeking control law on a class of underactuated Lagrangian control systems with symmetry. Our study is motivated by the problem of attitude optimization for satellites with reaction wheels. The goal is to minimize the value of an analytically unknown configuration‐dependent objective function without being reliant on measurements of the current configuration and velocity. This work extends our previous work on extremum seeking control for fully actuated mechanical systems on Lie groups in the absence of dissipation. We show that the earlier method for fully actuated systems can also be successfully applied to a certain class of underactuated systems, which are controlled by internal momentum exchange devices (reaction wheels) so that the total momentum is conserved. Conservation of momentum allows us to reduce the control system to a system on a smaller state manifold. The reduced control system is not of Lagrangian form but fully actuated. For the reduced control system we prove that the extremum seeking method leads to practical uniform asymptotic stability. We illustrate our findings by the example of a rigid body with internal rotors.

Stable cargo transportation for underactuated partial space elevator with model uncertainties and disturbances

An adaptive neural network energy-based control approach is proposed in this paper to address the issue of stabilizing a partial space elevator during cargo transportation. The method handles the challenges presented by external disturbances and model uncertainties. The dynamic model of the partial space elevator, with a typical multi-input and multi-output underactuated system, is presented. A control scheme using energy-based techniques is proposed to guarantee the stable configuration of the system, and auxiliary kinematic vector variables are designed to incorporate the uncontrollable variables to constitute controllable state subsets. Under these frameworks, adaptive control schemes are formulated using multi-layer neural networks to mitigate model uncertainty. Robust compensators are designed to effectively counteract the unknown boundaries of external disturbances and approximation errors. The stability of the closed control system is rigorously demonstrated through the application of the Lyapunov function and LaSalle's invariance principle. Numerical simulations demonstrate that the proposed control method is highly effective in suppressing oscillations during the transfer period of a partial space elevator.

Neuro-Fuzzy-Based Adaptive Control for Autonomous Drone Flight

Adaptive control is the capability of a control system to modify its operation and achieve the best possible operation mode. A quadcopter is a nonlinear, unstable and under-actuated dynamic system, thus providing a challenge to control engineers in controlling and stabilising it during flight. This paper proposes the design, development, and application of an intelligent adaptive hybrid controller to control and stabilise the drone. The training data for adaptive neuro-fuzzy inference systems (ANFIS) are generated by the Linear Quadratic Regulator (LQR) under white-noise disturbance. The trained ANFIS is subsequently used to estimate the parameters of the control distribution matrix for the actual fault condition and the reconfiguration is carried out by computing new feedback gain using the pseudo-inverse technique. For the simple adaptive controller, LQR is also used to generate the desired trajectories of the reference model. In both experiments, the extended Kalman filter is implemented due to its non-linearity benefit. We demonstrate the performance of the proposed approach as a representative case study. The preliminary numerical simulation results further indicate that the proposed method is promising compared to conventional control techniques to control and stabilise a quadcopter drone.

Self-balancing robot: modeling and comparative analysis between PID and linear quadratic regulator

<p>A two-wheeled self-balancing robot (TWSBR) is an underactuated system that is inherently nonlinear and unstable. While many control methods have been introduced to enhance the performance, there is no unique solution when it comes to hardware implementation as the robot’s stability is highly dependent on accuracy of sensors and robustness of the electronic control systems. In this study, a TWSBR that is controlled by an embedded NI myRIO-1900 board with LabVIEW-based control scheme is developed. We compare the performance between proportional-integral-derivative (PID) and linear quadratic regulator (LQR) schemes which are designed based on the TWSBR’s model that is constructed from Newtonian principles. A hybrid PID-LQR scheme is then proposed to compensate for the individual components’ limitations. Experimental results demonstrate the PID is more effective at regulating the tilt angle of the robot in the presence of external disturbances, but it necessitates a higher velocity to sustain its equilibrium. The LQR on the other hand outperforms PID in terms of maximum initial tilt angle. By combining both schemes, significant improvements can be observed, such as an increase in maximum initial tilt angle and a reduction in settling time.</p>

Tuning of Passivity-Based Controllers for Mechanical Systems

This article describes several approaches for tuning the parameters of a class of passivity-based controllers for standard nonlinear mechanical systems. In particular, we are interested in tuning controllers that preserve the mechanical system structure in the closed loop. To this end, first, we provide tuning rules for stabilization, i.e., the rate of convergence (exponential stability) and stability margin (input-to-state stability). Then, we provide guidelines to remove the overshoot. In addition, we propose a methodology to tune the gyroscopic-related parameters. We also provide remarks on the damping phenomenon to facilitate the practical implementation of our approaches. We conclude this article with experimental results obtained from applying our tuning rules to a fully actuated and an underactuated mechanical system.

Detumbling of Underactuated Tethered Satellite System Based on Hierarchical Sliding Mode Control

Millions of space debris pose a significant risk to operational spacecraft, making active debris removal (ADR) missions an urgent priority. Net capturing is a promising method to remove space debris objects due to its multiple advantages. However, once a target has been captured, a combined tethered system comprising the chaser, the target, and the tethers is formed, which requires effective stabilization. In this paper, stabilization of the entire postcapture system is realized by solely controlling the active chaser satellite. Such a control strategy can attenuate system hardware facilities compared with those by applying additional tether control such as using a winding mechanism, which were mostly discussed in the literature. Further, for the first time, considering the uncertainties during the capture process, including inaccurate target parameters, initial attitude error of the target, and external disturbances, a hierarchical sliding mode controller (HSMC) with two sublayers is developed to detumble this underactuated system. Additionally, a hybrid position controller is proposed to eliminate the undesired swing motion within the system. Numerical simulations demonstrate that HSMC method delivers better performance over the proportional-derivative controller in efficiency and robustness for postcapture stabilization of the combined tethered system. Overall, our proposed control strategy offers a promising solution for future ADR missions, contributing to a safer and more sustainable space environment.

Damping of Oscillations of a Rotary Pendulum System

This paper describes an innovative design based on the spectral approach of a novel shaper that eliminates frequency components that induce unwanted residual oscillations in various flexible mechanical systems, such as tower cranes or chain carousels, which are vital to many manufacturing and material-handling processes. However, their physical structure leads to flexible effects that limit their usefulness. Apart from the circular motion problem, control is provided by a single actuator, which makes it a so-called underactuated system. The input signal needs to be modified so that the spectral components from several interconnected degrees of freedom are considered together during shaper design, which increases the complexity of this task since one of its components induces nonlinear behavior. This means that traditional shaping techniques, based on linear theory, fail to provide good performance over the whole input range. The underdamped dynamics of the model and the effect of nonlinearities on the spectrum of the final signal are examined; the proposed method for application as a command shaping control technique is applied; and its effectiveness is analyzed by simulation and real-time implementation. The theoretical results verified on an experimental crane system confirm the expected oscillation phenomenon and show that the designed nonlinear shaper can reduce the payload swing significantly.