Underactuated Mechanical Systems Research Articles

An improved Udwadia–Kalaba approach for controller design in underactuated mechanical systems

An improved Udwadia–Kalaba approach for controller design in underactuated mechanical systems

Identification of linear flat outputs using neural networks—Examples of two-degree-of-freedom underactuated mechanical systems

This paper proposes a neural networks-based approach of finding flat output of linearized underactuated mechanical systems (UMS). Given that differential flatness and controllability are equivalent for linear systems, the problem is equivalent to finding the Brunovsky canonical form of linearized UMSs. We use a two degree-of-freedom (2DOF) system to illustrate the theoretical development. The proposed method identifies the local flat output of nonlinear mechanical systems from the measurements only, without a detailed mathematical model. The identification allows us to combine the well-known active disturbance rejection control (ADRC) and differential flatness control. A time-domain direct identification (TDDI) algorithm and its variant based on the algebraic method are proposed for flat output identification (FOID). The neural networks for implementing the TDDI algorithm called FOID-NN are created to evaluate flat output candidates and to use the flat output to reconstruct the system states in terms of a linear mapping. The neural networks are trained with the loss functions defined by reconstruction errors. Two special layers, namely tracking differentiator (TD) and algebraic layer, are inserted in the FOID-NN to handle noisy signal differentiations. Simulations of a cart–pole system and a stable 2DOF nonlinear UMS are carried out to show the range of applications of the FOID-NN. The experimental data of an underactuated rotary crane are used to validate the identified flat output.

Passivity-based control of underactuated mechanical systems with Coulomb friction: Application to earthquake prevention

Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. Characteristic examples are nonautonomous and discontinuous systems due to the presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists of designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). Finally, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid’s diffusion.

Enhanced free reaching phase SMC for UMS

In this paper, a new design method for sliding mode control (SMC) is presented for nonlinear underactuated mechanical systems. The aim is to eliminate the reaching phase in SMC and avoid the chattering phenomenon while achieving fast and robust tracking for a class of underactuated mechanical systems (UMS). An alternative method is presented to express the sliding domain equations by incorporating tracking errors. The fundamental concept behind the suggested control scheme is to adjust the tracking errors, enabling the system response to commence on the sliding surface regardless of the initial conditions. This modification guarantees the elimination of the reaching phase, avoiding the chattering phenomenon, and ensuring that the tracking error converges to zero. The stability analysis of the proposed approach is conducted using the Lyapunov method. Through numerical simulations, the validity of the approach is confirmed by implementing the control scheme on a crane system. Subsequently, the performance of the proposed approach is compared with other control methods, emphasizing its effectiveness in handling uncertainties. This comparative analysis aims to emphasize the advantages and efficiency of the proposed control strategy over alternative methods, in particular, to address uncertainties and achieve the desired control objectives.

Overview of Bridge Crane Control

This paper summarizes the research hotspot of underactuated mechanical system in nonlinear control field, and emphasizes its wide application in practical engineering, including space vehicle, inverted pendulum and navigation system. The Chinese government has supported the construction crane industry through a series of policies and regulations, providing guidance for its safe production and technical specifications. In the context of social development, the application demand of bridge gantry cranes in narrow environments is increasing, and the future development trend will focus on the direction of intelligence, science and technology and humanization. In recent decades, the research results on the positioning and pendulum of bridge crane at home and abroad are divided into open loop and closed loop control methods, and the control methods are described in detail in this paper.

Adaptive robust control for fuzzy underactuated mechanical systems: A Stackelberg game-theoretic optimization approach

This article proposes a Stackelberg game-theoretic optimization approach for adaptive robust controller design of fuzzy underactuated mechanical systems (UMSs). As a commonly encountered challenge, several factors render it difficult to further improve the control performance of these UMSs, such as the coupling effects and nonlinearities resulting from the underactuated structure. Meanwhile, various sources of unexpected time-varying uncertainties can further add to the difficulty in the controller design procedure pipeline. To overcome the above-mentioned issues and further enhance the system performance, a dual-parameter optimization framework is presented here for adaptive robust controller design, leveraging fuzzy set theory and Stackelberg game. In this development, the time-varying uncertainty is characterized via suitable membership functions; and thus the appropriate fuzzy UMS is established. Then, an adaptive robust controller with a leakage-type adaptation law is presented to tackle the uncertainty appropriately. To further improve the system performance considering the trade-off among the transient performance; the steady-state performance; and the control cost; a Stackelberg game-based dual-parameter optimization method is proposed to determine the Stackelberg strategy. It is rigorously proved in the developments here that the proposed controller design and parameters optimization method guarantees the desired deterministic system performance, i.e., uniform boundedness (UB) and uniform ultimate boundedness (UUB). Moreover, a series of numerical simulations based on a flywheel inverted pendulum (FIP) are performed, which illustrate the effectiveness and superiority of the proposed method.

Adaptive total energy shaping of a class of manipulators

AbstractPoint‐to‐point control of underactuated mechanical systems is a challenging problem, especially if some of the system's parameters are uncertain. A popular method to stabilize underactuated manipulators is total energy shaping, in which a new Hamiltonian function is assigned to the closed‐loop system while a set of partial differential equations (PDEs) should be solved. In this study, an adaptive total energy shaping controller based on the approach called interconnection and damping assignment passivity‐based control (IDA‐PBC) for some underactuated robots is designed. The proposed method can overcome uncertainties in the dynamical parameters and also the parameters of the input mapping matrix under the satisfaction of the PDEs. The stability of the desired pose is ensured by the Lyapunov approach via suitable design of adaptation laws without the requirement for the persistence of excitation condition or other similar conditions. The results are applied to a 3‐DOF underactuated manipulator and verified by simulations.

Stabilization and tracking control of underactuated ball and beam system using metaheuristic optimization based TID-F and PIDD2-PI control schemes.

In this paper, we propose two different control strategies for the position control of the ball of the ball and beam system (BBS). The first control strategy uses the proportional integral derivative-second derivative with a proportional integrator PIDD2-PI. The second control strategy uses the tilt integral derivative with filter (TID-F). The designed controllers employ two distinct metaheuristic computation techniques: grey wolf optimization (GWO) and whale optimization algorithm (WOA) for the parameter tuning. We evaluated the dynamic and steady-state performance of the proposed control strategies using four performance indices. In addition, to analyze the robustness of proposed control strategies, a comprehensive comparison has been performed with a variety of controllers, including tilt integral-derivative (TID), fractional order proportional integral derivative (FOPID), integral-proportional derivative (I-PD), proportional integral-derivative (PI-D), and proportional integral proportional derivative (PI-PD). By comparing different test cases, including the variation in the parameters of the BBS with disturbance, we examine step response, set point tracking, disturbance rejection analysis, and robustness of proposed control strategies. The comprehensive comparison of results shows that WOA-PIDD2-PI-ISE and GWO-TID-F- ISE perform superior. Moreover, the proposed control strategies yield oscillation-free, stable, and quick response, which confirms the robustness of the proposed control strategies to the disturbance, parameter variation of BBS, and tracking performance. The practical implementation of the proposed controllers can be in the field of under actuated mechanical systems (UMS), robotics and industrial automation. The proposed control strategies are successfully tested in MATLAB simulation.

Study of Q-learning and deep Q-network learning control for a rotary inverted pendulum system

The rotary inverted pendulum system (RIPS) is an underactuated mechanical system with highly nonlinear dynamics and it is difficult to control a RIPS using the classic control models. In the last few years, reinforcement learning (RL) has become a popular nonlinear control method. RL has a powerful potential to control systems with high non-linearity and complex dynamics, such as RIPS. Nevertheless, RL control for RIPS has not been well studied and there is limited research on the development and evaluation of this control method. In this paper, RL control algorithms are developed for the swing-up and stabilization control of a single-link rotary inverted pendulum (SLRIP) and compared with classic control methods such as PID and LQR. A physical model of the SLRIP system is created using the MATLAB/Simscape Toolbox, the model is used as a dynamic simulation in MATLAB/Simulink to train the RL agents. An agent trainer system with Q-learning (QL) and deep Q-network learning (DQNL) is proposed for the data training. Furthermore, agent actions are actuating the horizontal arm of the system and states are the angles and velocities of the pendulum and the horizontal arm. The reward is computed according to the angles of the pendulum and horizontal arm. The reward is zero when the pendulum attends the upright position. The RL algorithms are used without a deep understanding of the classical controllers and are used to implement the agent. Finally, the outcome indicates the effectiveness of the QL and DQNL algorithms compared to the conventional PID and LQR controllers.

Online Trajectory Planning Control for a Class of Underactuated Mechanical Systems

Efficient control of underactuated systems has always been a great challenge, since one has to stabilize both actuated and unactuated states with fewer control inputs. During the past decades, great efforts have been devoted to improving the transient performance and robustness of the system, which are very crucial in plenty of application scenarios. To motivate corresponding research, an online trajectory planning method is proposed for a class of underactuated mechanical systems in this paper. Specifically, reference trajectories are first generated for actuated states. Based on that, appropriate online adjustments are made to these reference trajectories, so as to realize efficient stabilization of the unactuated states. Different from traditional off-line planning methods, the proposed one utilizes real-time feedback to counteract various unfavorable conditions, which endows it stronger robustness and better transient performance. Rigorous Lyapunov-based analysis is performed to show that the closed-loop system is asymptotically stable around the desired equilibrium point. Finally, the proposed method is applied to two underactuated mechanical systems, with simulation/experiment results provided to demonstrate its superior performance.

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

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

On Contractive Port-Hamiltonian Systems With State-Modulated Interconnection and Damping Matrices

In this paper, for the input-state-output class of port-Hamiltonian systems the contraction property is characterized. Recently, some results on contraction of port-Hamiltonian systems with constant interconnection and damping matrices have been published. This paper, extends these results for state-modulated interconnection and damping matrices. In this regard, the powerful method of IDA-PBC is extended for tracking designs. Controller design for under-actuated mechanical systems with non-constant mass matrices usually leads to non-constant (state-modulated) interconnection and damping matrices for the closed-loop system. Therefore, the result of this paper can be used for tracking controller design for under-actuated mechanical systems. To show the potency of proposed results, a tracking controller for cart-pole system is designed and simulated.

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

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.

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.

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.

An Adaptive Controller Based on Interconnection and Damping Assignment Passivity-Based Control for Underactuated Mechanical Systems: Application to the Ball and Beam System

In this paper, an adaptive technology and the interconnection and damping assignment passivity-based control method are combined to solve the stabilization problem for underactuated mechanical systems with uncertainties (including matched and unmatched). These uncertainties include unknown friction coefficients and unknown terms in kinetic energy and potential energy. A novel adaptive interconnection and damping assignment passivity-based control scheme is proposed and an adaptive stabilization controller is designed to make the closed-loop system locally stable. Verification is conducted on the ball and beam system. The locally asymptotic stability is demonstrated using the LaSalle’s invariance principle and approximate linearization. The effectiveness of the proposed control law is verified through numerical simulations.

Predefined-Time Adaptive Fault-Tolerant Control for Switched Odd-Rational-Power Multi-Agent Systems

This paper studies the distributed adaptive neural network (NN) predefined-time (PT) fault-tolerant consensus tracking control for unknown non-strict feedback nonlinear multi-agent systems (MASs) with time-varying full state constraints (TVFSCs), arbitrary switching behavior functions (possibly asynchronous), actuator faults and high powers. Compared with existing results, the convergence time estimation bound in this paper is less conservative due to the upper bound of convergence time presents as a tuning parameter. This paper avoids the control gain of each virtual controller to appear in the next virtual controller by using separable function technique, so it can reduce the control action complexity. Besides, this paper does not use any filters or piecewise continuous function in the controller design process, which can effectively avoid the potential singularity problem. An adaptive NN fault-tolerant consensus tracking controller is designed via backstepping technique and tangent barrier Lyapunov functions (BLF-Tans), which can ensure that all the closed-loop signals are bounded within a PT and the TVFSC bounds are not violated. Finally, numerical simulation and practical example further verify the effectiveness of the presented control algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Many practical (underactuated, weakly coupled and unstable mechanical systems) control systems can be modeled as uncertain high-order MAS with high powers. Due to the influence of environment and other factors, the values of some variables in the practical systems are limited and cannot be arbitrarily selected. In addition, the actual systems may fail due to human factors and in most of the relevant literatures, it is often assumed that the powers are equal to 1 and only achieve the controlled systems asymptotic stability as time approaches infinity. Therefore, in this paper, an adaptive PT fault-tolerant control scheme is developed for MASs with high powers, time-varying full state constraints and actuator faults. The effectiveness of the control scheme is illustrated based on simulation study.

Switched Energy and Neural Network-Based Approach for Swing-Up and Tracking Control of Double Inverted Pendulum

The double inverted pendulum (DIP) system is a benchmark underactuated mechanical system. The control problem of the DIP system is challenging since it not only has high nonlinearity, but also has underactuation degree equal to two. In this article, a switched energy-based swing-up and neural network (NN) controller is proposed to solve the swing-up and tracking control problem of the DIP system when the desired position is time varying. The implementation of the proposed controller can be divided into three steps. In step one, the energy-based control method is adopted to drive the first pendulum from the downward position to the neighborhood of the upright position. In step two, the sliding mode control method is adopted to stabilize the first pendulum. Meanwhile, a method combining energy-based control and “equivalent cart” is adopted to swing up the second pendulum. In step three, on the basis of approximate nonlinear output regulation theory, the NN controller is used for achieving satisfactory position tracking performance. Finally, the effectiveness of our design is verified by experimental results.