Inverted Pendulum Cart Research Articles

Quadruped Walking Algorithm Integrating Balance Control Using Cartinverted Pendulum Model and Optimization-based Whole-body Inverse Dynamics Control

Quadruped Walking Algorithm Integrating Balance Control Using Cartinverted Pendulum Model and Optimization-based Whole-body Inverse Dynamics Control

Design of two-loop FOPID-FOPI controller for inverted cart-pendulum system

The inverted cart-pendulum system (ICPS) consists in having a pendulum mounted on a sliding cart, with the pivot point fixed. This real time experiment indeed looks like a rocket and its functionality is akin to a rocket. These are the launchers and the missile guidance and control as well as construction anti-seismic measures also. The control aim in these systems is to maintain the inverted pendulum vertically stable. The system is causal but unstable and, therefore, has no minimum phase. Therefore, the right half plane pole and zero are close to each other. Therefore, the stability of the system can be considered as problematic at some points. Unfortunately, linear time- invariant (LTI) classical controllers are incapable of offering suffient loop robustness for such systems. This paper aims to project a two-loop fractional order controller (2-LFOC) design to stabilize a higher-order nonlinear inverted cart-pendulum system (ICPS). The modeling, linearization, and control of ICPS are demonstrated in this work. The control target is adjusted so that the inverted pendulum stabilizes in its upright state when the cart reaches the required point. To fulfill the control objective, two-loop FOPID-FOPI controllers are proposed, and the Levenberg Marquardt algorithm (LMA) is utilized to tune the controllers. A novel nonlinear integral of time-associated absolute-error (ITAE) based fitness formula considering the settling time and rise time is used to fit the controller parameters for 2-LFOC. A performance comparison with the PID controller in terms of different time domain parameters such as rise_time (T R ), peak_time (T P ), settling_time (T S ), maximum overshoot (OS M ), maximum undershoot (US M ) and steady-state error (E SS ) are investigated. Stability analysis using Riemann surface observation of the system compensated with the proposed controller is presented in this work. The robust behavior of the two-loop FOPID-FOPI controller is verified by the application of disturbances in the system and the Reimann surface observation.

Design and Implementation of Remote Lab PID Controller Experiment Based on IoT

The education sector has increasingly embraced distance education to ensure a safe and uninterrupted learning process, especially in wars or epidemics. This paper focuses on designing and implementing a PID (Proportional-Integral-Derivative) controller experiment in remote laboratories utilizing the Internet of Things (IoT). Also, a laboratory experiment was developed using a naturally unstable system, specifically a cart-inverted pendulum. The experiments aim to enhance students' understanding of PID controller tuning within a real-world context. This system was chosen due to its dual motion characteristics, with a linear motion for the cart and a circular motion for the pendulum. Two controllers were designed and implemented for the system to enable control and feedback. Additionally, the Blynk platform was integrated into the experiment setup, allowing for real-time visualization of the system's response, control of PID parameters, and the ability to view the video stream via platforms like Skype. For remote connectivity, NodeMCU, an IoT development board, controlled the pendulum, collected system parameters, and transmitted them to the cloud through the Internet. Moreover, the sensors and the system were mathematically modeled, and their transfer functions were extracted. That allows the students to do the PID experiment theoretically and practically and compare the results. This complete setup enables local and remote access to the experiment, ensuring students can experiment regardless of their physical location. As a result of the study, designing and implementing a remote laboratory PID controller experiment utilizing IoT technology provides students with an innovative and immersive learning experience.

LQR controller performance via particle swarm optimization and neural networks

AbstractThe inverted pendulum‐cart (IPC) control problem has long been a benchmark in the field of control systems due to its inherent instability and nonlinear dynamics. The linear quadratic regulator (LQR) control technique has been proven to be effective in stabilizing the inverted pendulum; however, the challenge lies in finding the optimal control gains that provide the best performance. This article presents a method to address the LQR control design problem for the IPC system which is compared against a particle swarm optimization based LQR and a neural network optimized LQR. The method provides a deterministic approach to finding the weighing matrices Q and R in accordance with the time domain characteristics chosen by the designer, such as settling time and maximum peak overshoot. Results from MATLAB simulations indicate that the suggested strategy has good performance.

Adaptive Elite Ant Lion Optimizer tuned optimal controller for underactuated systems

The issue of stabilization and ensuring the correct path is followed in the cart-inverted pendulum underactuated system is being addressed. In the construction of a robust Linear Quadratic Regulator (LQR) optimum controller, the optimization algorithms are best suited for tuning the LQR weighing matrices which are usually obtained by cumbersome trial and error methods. In this study, a novel Adaptive Elite Ant Lion Optimizer (AE-ALO) is suggested to tune the weighing matrices of the optimum controller. Aiming at the foible of ALO's imbalanced exploration for some intricate optimization problems, and influenced by Adaptive Particle Swarm Optimization (APSO), the amended location of antlions in ALO's elitism operator is enhanced, yielding the Adaptive Elite-ALO (AE-ALO). The proposed AE-ALO tested in 12 standard benchmark functions outperforms the existing algorithms interms of global exploration and is then used to tune the LQR weighing matrices such that the trajectory tracking error is minimized. The suggested controller's viability is demonstrated in Quanser's IP02 benchmark Cart-inverted pendulum system. The study found that using the method proposed resulted in a 10.97 % decrease in the ISE of trajectory tracking while stabilizing the pendulum in the unstable upright position, as compared to using ALO and APSO tuned LQR control schemes.

Optimizing LQR controllers: A comparative study

Linear Quadratic Regulator is one of the most common ways to control a linear system. Despite Linear Quadratic Regulator’s (LQR) strong performance and solid resilience, developing these controllers have been challenging, largely because there is no reliable way to choose the Q and R weighing matrices. In this regard a deterministic method is used for choosing them in this paper, providing the designers a precise control over performance variables. An Artificial Bee Colony (ABC) optimisation is also used to find the sub-optimal gain matrices along with an analytical approach based on neural networks. A comparative study of the three approaches is performed using MATLAB simulations. These three approaches are applied on an inverted pendulum–cart system due to its complexity and dexterity. The results show that all the three methods show comparable performances with the proposed analytical method being slightly better in terms of transient characteristics.

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.

Posture stabilizing control of quadruped robot based on cart-inverted pendulum model

Posture stabilizing control of quadruped robot based on cart-inverted pendulum model

An Extended Experimental Study on Control of Unstable and Non-Minimum Phase Plants With the Cascaded Form of a Fractional Order Compensator

With the advent and emergence of calculus with fractional order (FO), the generalized form of the integer order (IO) has created significant advancement in the field of control. The real time systems represented by the FO models are more accurate enough developing improved performance in contrast to the classical IO one. In this context, the area is still deficient of methodical design algorithms with the different available structures of the FO compensators to control the various types of any unstable and non-minimum phase (NMP) Linear Time Invariant (LTI) plants. There is dearth of detailed analytical and experimental work on the cascaded form of this FO lead compensator in the available literatures. The cascaded structure of the non-integer order compensator is therefore employed here through classical control theory to exhibit the simplified design method in frequency domain explicitly through numerical studies and MATLAB simulation results. These FO compensators with its non-integer order helps one to add the required exact magnitude and phase to the plant to be compensated at a particular frequency thus leading to the unique solution of the desired compensator parameters. This proposition has been further extended to compensate a highly non-linear unstable NMP Cart-Inverted Pendulum System. Experimental results have been appended in support of the method explained to show the applicability of this study in control system design.

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.

State estimation and optimal control of an inverted pendulum on a cart system with stochastic approximation approach

<abstract> <p>In this paper, optimal control of an inverted pendulum on a cart system is studied. Since the nonlinear structure of the system is complex, and in the presence of random disturbances, optimization and control of the motion of the system become more challenging. For handling this system, a discrete-time stochastic optimal control problem for the system is described, where the external force is considered as the control input. By defining a loss function, namely, the mean squared errors to be minimized, the stochastic approximation (SA) approach is applied to estimate the state dynamics. In addition, the Hamiltonian function is defined, and the first-order necessary conditions are derived. The gradient of the cost function is determined so that the SA approach is employed to update the control sequences. For illustration, considering the values of the related parameters in the system, the discrete-time stochastic optimal control problem is solved iteratively by using the SA algorithm. The simulation results show that the state estimation and the optimal control law design are well performed with the SA algorithm, and the motion of the inverted pendulum cart is addressed satisfactorily. In conclusion, the efficiency of the SA approach for solving the inverted pendulum on a cart system is verified.</p> </abstract>

A novel design methodology on cascaded fractional order (FO) PI-PD control and its real time implementation to Cart-Inverted Pendulum System

The tune up of fractional order (FO)-PIλDμ controllers is largely advocated in available literature which mostly requires evolutionary optimization techniques or solution of complex non-linear equations thereby leading to lack of practical solutions in terms of selecting the controller parameters. It may be quite feasible if one can consider PID control in the form of cascaded PI−PD control with which the task of attainment of the control objectives can be accomplished methodically. In extension to this, studies related to FO PIλ−PDμ controller are not too exhaustive. Generally, graphical methods are adopted to compute the controller parameters. There is still paucity of comprehensive design algorithms for these controllers to which end this work comes up with a novel and simple specification driven design methodology of the cascaded structures of FO PIλ−PDμ and [PI]λ−[PD]μ controllers following the classical control theory Keeping away from the complex and implicit design strategies they assist with an exact and unique solution to design these FO controllers in frequency domain with satisfactory dynamic performance in time domain which may not be possible with the FO-PIλDμ controllers. Illustrative examples and experimental validations aid to substantiate the reliability of the proposed method.

Swing-up control of an inverted pendulum cart system using the approach of Hedge-algebras theory

Swing-up control of an inverted pendulum cart system using the approach of Hedge-algebras theory

Analysis of a Cart-Inverted Pendulum System with Harmonic Disturbances Based on its Criterion Matrix

The control of an inverted pendulum is a classical benchmark control problem. Its dynamics resemble that of many real-world systems of interest like pendulous, missile launchers, segways, and many more. The control of this system is challenging as it is a highly unstable, highly non-linear, non-minimum phase system, and underactuated. Furthermore, the physical constraints on the track position also pose complexity in its control design. A great deal of nonlinearity is present inherently and as well as affected by the surrounding external disturbances. The paper presents an approach for analysis of a cart-inverted pendulum system with harmonic disturbances. The approach is based on the index of the criterion matrix of the system named a degeneration factor. The degeneration factor is constructed with the singular values of the criterion matrix of the system and allows us to find frequency range, where the system operates as a whole. A linear-quadratic regulator is used to control the cart-inverted pendulum system. The results are supported with an example.

Balancing inverted pendulum cart on inclines using accelerometers

Abstract The balance of inverted pendulum on inclined surfaces is the precursor to their control in unstructured environments. Researchers have devised control algorithms with feedback from contact (encoders - placed at the pendulum joint) and non-contact (gyroscopes, tilt) sensors. We present feedback control of Inverted Pendulum Cart (IPC) on variable inclines using non-contact sensors and a modified error function. The system is in the state of equilibrium when it is not accelerating and not falling over (rotational equilibrium). This is achieved when the pendulum is aligned along the gravity vector. The control feedback is obtained from non-contact sensors comprising of a pair of accelerometers placed on the inverted pendulum and one on the cart. The proposed modified error function is composed of the dynamic (non-gravity) acceleration of the pendulum and the velocity of the cart. We prove that the system is in equilibrium when the modified error is zero. We present algorithm to calculate the dynamic acceleration and angle of the pendulum, and incline angle using accelerometer readings. Here, the cart velocity and acceleration are assumed to be proportional to the motor angular velocity and acceleration. Thereafter, we perform simulation using noisy sensors to illustrate the balance of IPC on surfaces with unknown inclination angles using PID feedback controller with saturated motor torque, including valley profile that resembles a downhill, flat and uphill combination. The successful control of the system using the proposed modified error function and accelerometer feedback argues for future design of controllers for unstructured and unknown environments using all-accelerometer feedback.

Fast Terminal Sliding Control of Underactuated Robotic Systems Based on Disturbance Observer with Experimental Validation

In this study, a novel fast terminal sliding mode control technique based on the disturbance observer is recommended for the stabilization of underactuated robotic systems. The finite time disturbance observer is employed to estimate the exterior disturbances of the system and develop the finite time control law. The proposed controller can regulate the state trajectories of the underactuated systems to the origin within a finite time in the existence of external disturbances. The stability analysis of the proposed control scheme is verified via the Lyapunov stabilization theory. The designed control law is enough to drive a switching surface achieving the fast terminal sliding mode against severe model nonlinearities with large parametric uncertainties and external disturbances. Illustrative simulation results and experimental validations on a cart-inverted pendulum system are provided to display the success and efficacy of the offered method.

PSO-Based Integral Sliding Mode Controller for Optimal Swing-Up and Stabilization of the Cart-Inverted Pendulum System

An integral sliding mode controller (ISMC) which employs particle swarm optimization (PSO) algorithm to search for optimal values of the parameters of the integral sliding manifold as well as the gains of the controller is proposed in this work. We considered the swing-up and stabilization of the cart-inverted pendulum system which is assumed to be affected by uncertainties. First, we determined the swing-up and stabilization conditions of the control system by using the internal dynamics of the cart-inverted pendulum system and sliding mode dynamics. A PSO algorithm is then used to search for the optimal values of the ISMC design parameters that satisfy the stabilization condition with the aim of improving the transient performance of the control system. To mitigate the chattering phenomenon, a saturation function of the integral sliding variable was used in the discontinuous control law. Simulation results on swing-up and stabilization of the cart-inverted pendulum system revealed improvement in transient behaviour by reducing settling time (by 52.61%), overshoots (by 45.56%) and required track length for cart movement (by 68.34%).

On Internal Model Control (IMC) of MIMO Systems

Control of MIMO systems offers a challenging problem due to the cross coupling effects present between the inputs and the outputs of MIMO systems. In order to achieve an efficient control decouplers are introduced which eliminate these cross-coupling effects present in MIMO systems. With decouplers properly designed, it is possible to derive control inputs for fully-actuated and over-actuated MIMO systems to achieve reference tracking of the individual outputs. However, the reference tracking for individual outputs is almost unachievable in under-actuated MIMO systems with decoupling control methods. In dealing with this, the present work proposes an internal model control (IMC) applicable as appropriate to all classes of MIMO systems without incorporating any decoupling technique. This paper extends with two unique design methods of the IMC technique. These proposed methods enable one to attain perfect reference tracking of the individual outputs even in presence of the cross-coupling present in the MIMO systems. The conditions for the perfect reference tracking of multi outputs for under-actuated MIMO systems are derived. Different examples are considered to test the efficiency of the methods. The proposed methods are implemented on an experimental test set-up of highly non-linear and unstable cart-inverted pendulum system. The reference tracking goal and disturbance rejection capability of the control scheme are examined well in real-time. The experimental results are portrayed to establish the efficacy of the theoretical claim made.

Self-excited oscillations in an inverted cart–pendulum based on the two-relay approach

Solving the orbital stabilization of a class of underactuated systems near its open-loop unstable equilibrium point is a challenging task but useful in repetitive tasks. This paper addresses the control problem of an inverted cart–pendulum’s motion driven by a two-relay controller. This controller is tuned to set the desired amplitude and frequency using the locus of the perturbed relay system, which is a frequency-domain method. We solved the periodic motion of the pendulum occurring in its open-loop unstable equilibrium point. The experimental results showed that the proposed two-relay controller forced the pendulum into a periodic motion around the upright position.

Performance assessment of particle swarm optimisation and differential evolution algorithms on integral sliding mode control of cart-inverted pendulum

In this paper, optimal parameters of integral sliding mode controller (ISMC) that balances a cart-inverted pendulum (CIP) system in the unstable equilibrium position are determined using both PSO a...