Triple Inverted Pendulum Research Articles

Mixed H2/H-infinity Controller Design for Triple Inverted Pendulum System

The steady increase in reliance on robotic systems in industrial applications arises the need to ensure their reliability despite the presence of external disturbances in the environment they work in and their parameters variations due to several factors like: change of temperature, wear and tear effect, and payload variation. Traditional controllers usually fail to maintain the system's stability and performance in the presence of uncertainty, while robust controllers aim to guarantee stability and desired performance across a range of possible uncertainties in the system through incorporating these uncertainties into their design process. This paper investigates the robustness of three controllers namely, H2 controller, H-infinity controller, and mixed H2/H-infinity controller, applied to the triple inverted pendulum system which translates to robustly control legged robots and robotic arms. The results show that among the three controllers, only the mixed H2/H-infinity control system is robustly stable and maintains its performance with stability margin 1.1864 and performance margin 1.1662. Index Terms— H2, H-infinity, Mixed H2/H-infinity, Robust Control, triple inverted pendulum.

Reinforcement learning to achieve real-time control of triple inverted pendulum

This work uses reinforcement learning (RL) to achieve the first-ever data-driven real-time control of an actual, not simulated, triple inverted pendulum (TIP) in a model-free way. A swing-up control task for the TIP is formulated as a Markov decision process with a dense reward function, then conducted in real time by using a model-free RL approach. To increase the sample efficiency of learning, a structure-aware virtual experience replay (VER) method is proposed; it works together with an off-policy actor-critic algorithm. The VER exploits the geometrically-symmetric property of TIPs to create virtual sample trajectories from measured ones, then uses the resulting multifold augmented dataset to effectively train actor and critic networks during the learning process. These structure-infused training data serve to obtain additional information and hence increase the convergence speed of network learning. We combine the proposed VER with a state-of-the-art actor-critic algorithm, and then validate its effectiveness through numerical simulations. Notably, the inclusion of VER amplifies computational efficiency, slashing the requisite trials, training steps, and overall duration by approximately 66.67%. Finally experiments demonstrate the real-time control capability of the proposed approach on an actual TIP system.

Reinforcement Learning for Adaptive Optimal Stationary Control of Linear Stochastic Systems

This article studies the adaptive optimal stationary control of continuous-time linear stochastic systems with both additive and multiplicative noises, using reinforcement learning techniques. Based on policy iteration, a novel off-policy reinforcement learning algorithm, named optimistic least-squares-based policy iteration, is proposed, which is able to find iteratively near-optimal policies of the adaptive optimal stationary control problem directly from input/state data without explicitly identifying any system matrices, starting from an initial admissible control policy. The solutions given by the proposed optimistic least-squares-based policy iteration are proved to converge to a small neighborhood of the optimal solution with probability one, under mild conditions. The application of the proposed algorithm to a triple inverted pendulum example validates its feasibility and effectiveness.

3단 도립진자의 구조 제안 및 LW-RCP02를 이용한 Swing-up Control의 구현

In this paper, we propose a new structure for a triple inverted pendulum with easy construction properties. We also implement a swing-up control for the proposed triple inverted pendulum using a lab-built rapid control prototyping environment known as light-weight rapid control prototyping 02 (LW-RCP02). The swing-up control of a triple inverted pendulum has a two-degree-of-freedom structure and requires a very accurate mathematical model. The proposed triple inverted pendulum adopts the same method that was developed for a double inverted pendulum by the authors’ laboratory to achieve excellent matching properties with the model. For free rotation of three pendulums, a hollow shaft revolute joint that can be used with a slip ring is newly proposed. Swing-up control of a triple inverted pendulum is implemented using Simulink and LW-RCP02. Moreover, a two-degree-of-freedom control structure combining feedforward and feedback is adopted for swing-up control of a triple inverted pendulum. We implement the parameter estimation method for the triple inverted pendulum and calculate the trajectory for feedforward control using estimated parameters and the direct transcription method. The uncertainty of feedforward control is compensated by adding a time-varying LQ feedback controller. As the sampling frequency of the implemented controller is 1KHz, real-time control is achievable using a standard PC.

A near-optimal decentralized control approach for polynomial nonlinear interconnected systems

This article tackles the decentralized near-optimal control problem for the class of nonlinear polynomial interconnected system based on a shifted Legendre polynomials direct approach. The proposed method converts the interconnected optimal control problems into a nonlinear programming one with multiple constraints. In light of the formulated NLP optimization, state and control coefficients are used to design a nonlinear decentralized state feedback controller. Overall closed-loop system stability sufficient conditions are investigated with the help of Grönwall lemma. The triple inverted pendulum case is considered for simulation. Satisfactory results are obtained in both open-loop and closed-loop schemes with comparison to collocation and state-dependent Riccati equation techniques.

Adaptive Optimal Control of Linear Periodic Systems: An Off-Policy Value Iteration Approach

This article studies the infinite-horizon adaptive optimal control of continuous-time linear periodic (CTLP) systems. A novel value iteration (VI) based off-policy adaptive dynamic programming (ADP) algorithm is proposed for a general class of CTLP systems, so that approximate optimal solutions can be obtained directly from the collected data, without the exact knowledge of system dynamics. Under mild conditions, the proofs on uniform convergence of the proposed algorithm to the optimal solutions are given for both the model-based and model-free cases. The VI-based ADP algorithm is able to find suboptimal controllers without assuming the knowledge of an initial stabilizing controller. Application to the optimal control of a triple inverted pendulum subjected to a periodically varying load demonstrates the feasibility and effectiveness of the proposed method.

On Stabilization of a Triple Inverted Pendulum via Vibration of a Support Point with an Arbitrary Frequency

The stabilization of the upper statically unstable position of a triple inverted pendulum via parametric excitation of the support has been studied. The presented results were obtained by the multiple scale method and the Floquet theory. The stability diagrams in the excitation parameters space (amplitude and frequency of the support excitation) are plotted. It is shown that stabilization is possible for low, medium, and high excitation frequencies. The influence of system parameters on stabilization zones of the upper unstable position of the pendulum is analyzed.

Enhanced electromagnetic swarm yields better optimisation

Swarm intelligence has been one of the leading techniques used by researchers worldwide for optimisation. In this paper, the fine tuning of the update equations for the swarm are done based on linkage of particle motion with a electromagnetic field and also under the influence of strategic delays. The motion of a particle in a search space is confined to free space in general, however if restricted the solution under the envelope of a magnetic field, the algorithm better converges within a electromagnetic field. Simulation studies have been performed on the triple inverted pendulum case study which showed that stability was achieved with ease when compared to classical methods of control.

On stabilization of a triple inverted pendulum via vibration of a support point with an arbitrary frequency

On stabilization of a triple inverted pendulum via vibration of a support point with an arbitrary frequency

Optimization of triple inverted pendulum control process based on motion vision

An inverted pendulum is a typical nonlinear and absolutely unstable system. In order to control the triple inverted pendulum effectively and steadily, an optimization method of inverted pendulum control process based on motion vision was proposed. The real-time motion pictures of the triple inverted pendulum in the swinging-up process were collected through the CCD camera, and the real-time motion pictures of the triple inverted pendulum were recognized, matched and optimized by using Harris algorithm. By using motion vision to control and optimize the triple inverted pendulum, the stabilization control of the triple inverted pendulum was realized.

Fractional Control of a Humanoid Robot Reduced Model with Model Disturbances

ABSTRACTThere is an open discussion between those who defend mass-distributed models for humanoid robots and those in favor of simple concentrated models. Even though each of them has its advantages and disadvantages, little research has been conducted analyzing the control performance due to the mismatch between the model and the real robot, and how the simplifications affect the controller’s output. In this article we address this problem by combining a reduced model of the humanoid robot, which has an easier mathematical formulation and implementation, with a fractional order controller, which is robust to changes in the model parameters. This controller is a generalization of the well-known proportional–integral–derivative (PID) structure obtained from the application of Fractional Calculus for control, as will be discussed in this article. This control strategy guarantees the robustness of the system, minimizing the effects from the assumption that the robot has a simple mass distribution. The humanoid robot is modeled and identified as a triple inverted pendulum and, using a gain scheduling strategy, the performances of a classical PID controller and a fractional order PID controller are compared, tuning the controller parameters with a genetic algorithm.

Simulation of a Triple Inverted Pendulum Based on Fuzzy Control

The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms. This paper studies the use of fuzzy control method to study the stability control problem of a triple inverted pendulum system. By the linear model of the system, the feedback weight matrix of the LQR optimal control and the feedback parameters of the linear optimal control are designed to determine the parameters of the fuzzy controller. The simulation results show that the proposed method can achieve the stability control of the three stage inverted pendulum, and has good dynamic performance with simple parameter selection.

Устойчивость тройного инвертированного физического маятника из статьи академика В.Н. Челомея 1983 г.

Устойчивость тройного инвертированного физического маятника из статьи академика В.Н. Челомея 1983 г.

Dynamics of quiet human stance: computer simulations of a triple inverted pendulum model

For decades, the biomechanical description of quiet human stance has been dominated by the single inverted pendulum (SIP) paradigm. However, in the past few years, the SIP model family has been falsified as an explanatory approach. Double inverted pendulum models have recently proven to be inappropriate. Human topology with three major leg joints suggests in a natural way to examine triple inverted pendulum (TIP) models as an appropriate approach. In this study, we focused on formulating a TIP model that can synthesise stable balancing attractors based on minimalistic sensor information and actuation complexity. The simulated TIP oscillation amplitudes are realistic in vertical direction. Along with the horizontal ankle, knee and hip positions, though, all simulated joint angle amplitudes still exceed the measured ones about threefold. It is likely that they could be eventually brought down to the physiological range by using more sensor information. The TIP systems’ eigenfrequency spectra come out as another major result. The eigenfrequencies spread across about . Our main result is that joint stiffnesses can be reduced even below statically required values by using an active hip torque balancing strategy. When reducing mono- and bi-articular stiffnesses further down to levels threatening dynamic stability, the spectra indicate a change from torus-like (stable) to strange (chaotic) attractors. Spectra of measured ground reaction forces appear to be strange-attractor-like. We would conclude that TIP models are a suitable starting point to examine more deeply the dynamic character of and the essential structural properties behind quiet human stance.Abbreviations and technical termsInverted pendulum body exposed to gravity and pivoting in a joint around position of unstable equilibrium (operating point)SIP single inverted pendulum: one rigid body pivoting around fixation to the ground (external joint)DIP double inverted pendulum: two bodies; external and internal joint operate around instabilityTIP triple inverted pendulum: three bodies; external and both internal joints operate around instabilityQIP quadruple inverted pendulum: four bodies, foot replaces external joint; all three internal joints operate around instabilityEigenfrequency characteristic frequency that a physical system is oscillating at when externally excited at a limited energy levelDOF degree of freedom; in mechanics: linear displacement or angle or combination thereof Mono-articular stiffness: coefficient of proportionality between mechanical displacement of a DOF and restoring force/torque component in the respective DOFBi-articular stiffness coefficient of proportionality between mechanical displacement of a DOF and restoring force/torque component in another DOFGRF ground reaction forceHAT segment including head, arms and trunkCOM centre of massCOP centre of pressure in the plane of the force platform surface

The stabilization and 3D visual simulation of the triple inverted pendulum based on CGA-PIDNN

Aiming at the triple inverted pendulum which is a strong coupling, multivariable, high-order and unsteady system, a design method of the controller based on PID neural network (PIDNN) optimized by cloud genetic algorithm (CGA) is proposed, this method is called CGA-PIDNN. CGA can be applied to learn and train the PIDNN connection weights. CGA can overcome the defect of the slow convergence rate and premature convergence for genetic algorithm (GA). PIDNN is a simple and normative network which is easy to be realized and has a good dynamic performance. The CGA-PIDNN control system of triple inverted pendulum is verified with MATLAB simulation test. The comparison results with the control effect of PIDNN control system optimized by standard GA (GA-PIDNN) are presented first. Then in LabVIEW environment, by using the combination of virtual reality technology and MATLAB, the three-dimensional (3D) animation simulation model of the triple inverted pendulum CGA-PIDNN control system is built. The simulation results indicate that CGA-PIDNN control method is effective, whose control effects are superior to those by GA-PIDNN control, it is believed that CGA-PIDNN is effective and will become a promising candidate of control methods.

Nonlinear control of triple inverted pendulum based on GA–PIDNN

The triple inverted pendulum is a nonlinear, dynamic and unsteady system. The traditional control methods of triple inverted pendulum have problems of limited control accuracy, slow responding. This kind of pendulum system is difficult to control due to the inherent instability, nonlinear behavior and difficultly in establishing a precise mathematical model. In addition, the back-propagation (BP) algorithm has the shortage of easy trapped in local minimum. The triple inverted pendulum control based on GA–PIDNN is proposed. The PID neural network (PIDNN) is a new kind of feedforward multi-layer network. Besides multi-layer forward networks traditional merit, such as approach the ability proceed together the calculation nonlinear transformation, its middle layer has the proportional (P) integral, (I) derivative, (D) dynamic characteristic. Genetic algorithm (GA) has good parallel design structure and characteristics of global optimization. The nonlinear identification model is established, and controller is designed via GA–PIDNN based on the combination of the merits GA and PIDNN in the research of triple inverted pendulum. In simulation, through the comparative study of GA–PIDNN and PIDNN optimized by BP (BP–PIDNN), simulations results show that GA is more accurate and effective.

Application of Genetic Algorithm Optimization LQR Weighting Matrices Control Inverted Pendulum

In practice, the key problem to apply LQR optimal control method is how to correctly choose the weighted matrix of performance index. At present, there is no formulaic approach for this problem. To obtain the satisfying results, people must repeat to test many times. This kind of LQR control method based on genetic algorithms, which can obtain satisfying control results at first hand, is presented for triple inverted pendulum system. The method optimizes the Q-matrix by using genetic algorithms, selects trace of the result of Riccati equation as the objective function. The control problem of triple inverted pendulum is resolved successfully. The simulation results prove that the control effect by this method is better than the other methods mentioned in the references.

The Research on Human Simulating Intelligent Control of the Inverted Pendulum Based on Feedback Linearization

In this paper, the partial feedback linearization was made for the non-linear model of the triple inverted pendulum, by means of the differential geometry. Then the simulation and analysis of tracking control and interference control were made, which combined with the human simulating intelligent control and LQR control in the system. The results show that the human simulating intelligent control can achieve better control results than the LQR control, it not only can effectively improve the stability of the system, but also the anti-interference ability is better than LQR control, it meets the control requirements of the triple inverted pendulum.

Stabilising the triple inverted pendulum by variable gain linear quadratic regulator

A variable gain linear quadratic regulator (VGLQR) for multivariable non-linear control systems is designed. This control scheme, which can be viewed as a novel gain scheduling controller, consists of linearising the non-linear system to a family of time-invariant systems about sampling points and using LQR method to obtain optimal feedback gain in every sampling interval. In addition, a fast algorithm based on Schur method for online solving the algebraic Riccati equation is discussed in detail, and a software package, which is used to solve algebraic Riccati equation at each sampling point, is developed during the present research. A triple inverted pendulum is stabilised by using the VGLQR method and the results of simulation and physical experiments are displayed. The results of experiments demonstrate that the present control scheme is of good stability and robustness.

An Asymptotically Optimal NMPC Core for Multi Degrees of Freedom Rigid Bodies

We address the applicability of nonlinear model predictive control (NMPC) to rigid bodies with multi degrees of freedom. These systems have complicated nonlinear motion equations that cause large computational time for NMPC. We propose an asymptotically optimal NMPC core based on the uniform state sampling concept. We examined the optimality and the computational cost of the proposed core using double and triple inverted pendulum models. The result showed that the proposed core calculates sub-time-optimal motions with 100 and 1,000 times faster than the competing cores with maintaining the same optimality. We applied the proposed core to a sixth inverted pendulum model. The result showed that the proposed NMPC core is able to calculate a sub-time-optimal motion of the model.