Single Inverted Pendulum Research Articles

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.

Coordination tending towards an anti-phase relationship determines greater sway reduction during entrainment with a simulated partner

The increased risk of falls in the older aged population demands the development of assistive robotic devices capable of effective balance support. For the development and increased user acceptance of such devices, which provide balance support in a human-like way, it is important to understand the simultaneous occurrence of entrainment and sway reduction in human-human interaction. However, sway reduction has not been observed yet during a human touching an external, continuously moving reference, which rather increased human body sway. Therefore, we investigated in 15 healthy young adults (27.20±3.55 years, 6 females) how different simulated sway-responsive interaction partners with different coupling modes affect sway entrainment, sway reduction and relative interpersonal coordination, as well as how these human behaviours differ depending on the individual body schema accuracy. For this, participants were lightly touching a haptic device that either played back an average pre-recorded sway trajectory (“Playback”) or moved based on the sway trajectory simulated by a single-inverted pendulum model with either a positive (Attractor) or negative (Repulsor) coupling to participant's body sway. We found that body sway reduced not only during the Repulsor-interaction, but also during the Playback-interaction. These interactions also showed a relative interpersonal coordination tending more towards an anti-phase relationship, especially the Repulsor. Moreover, the Repulsor led to the strongest sway entrainment. Finally, a better body schema contributed to a reduced body sway in both the “reliable” Repulsor and the “less reliable” Attractor mode. Consequently, a relative interpersonal coordination tending more towards an anti-phase relationship and an accurate body schema are important to facilitate sway reduction.

Design of Approximate Explicit Model Predictive Controller Using Parametric Optimization

AbstractThis paper introduces a new technique, called state-parameterized nonlinear programming control (sp-NLPC), for designing a feedback controller that can stabilize intrinsically unstable nonlinear dynamical systems using parametric optimization. Stability-preserving constraints are included in the optimization problem solved offline by the predictive parameterized Pareto genetic algorithm (P3GA), a constrained nonlinear parametric optimization algorithm. The optimal control policy is approximated from P3GA output using radial basis function (RBF) metamodeling. The sp-NLPC technique requires fewer assumptions and is more data-efficient than alternative methods. Two nonlinear systems (single and double inverted pendulums on a cart) are used as benchmarking problems. Performance and computational efficiency are compared to several competing control design techniques. Results show that sp-NLPC outperforms and is more efficient than competing methods. A qualitative investigation on phase plane analysis for the controlled systems is presented to ensure stability. The approximating state-dependent solution for the control input lends itself to applications of control design for control co-design (CCD). Such extensions are discussed as part of future work.

Lower limb joint-specific contributions to standing postural sway in persons with unilateral lower limb loss.

Individuals with lower limb loss are at an increased risk for falls, likely due to impaired balance control. Standing balance is typically explained by double- or single-inverted pendulum models of the hip and/or ankle, neglecting the knee joint. However, recent work suggests knee joint motion contributes toward stabilizing center-of-mass kinematics during standing balance. To what extent do hip, knee, and ankle joint motions contribute to postural sway in standing among individuals with lower limb loss? Forty-two individuals (25 m/17f) with unilateral lower limb loss (30 transtibial, 12 transfemoral) stood quietly with eyes open and eyes closed, for 30 s each, while wearing accelerometers on the pelvis, thigh, shank, and foot. Triaxial inertial measurement units were transformed to inertial anterior-posterior components and sway parameters were computed: ellipse area, root-mean-square, and jerk. A state-space model with a Kalman filter calculated hip, knee, and ankle joint flexion-extension angles and ranges of motion. Multiple linear regression predicted postural sway parameters from intact limb joint ranges of motion, with BMI as a covariate (p < 0.05). With eyes open, intact limb hip flexion predicted larger sway ellipse area, whereas hip flexion and knee extension predicted larger sway root-mean-square, and hip flexion, knee extension, and ankle plantarflexion predicted larger sway jerk. With eyes closed, intact limb hip flexion remained the predictor of sway ellipse area; no other joint motions influenced sway parameters in this condition. Hip, knee, and ankle motions influence postural sway during standing balance among individuals with lower limb loss. Specifically, increasing intact-side hip flexion, knee extension, and ankle plantarflexion motion increased postural sway. With vision removed, a re-weighting of lower limb joint sensory mechanisms may control postural sway, such that increasing sway may be regulated by proximal coordination strategies and vestibular responses, with implications for fall risk.

Visual Modulation of Human Responses to Support Surface Translation.

Vision is known to improve human postural responses to external perturbations. This study investigates the role of vision for the responses to continuous pseudorandom support surface translations in the body sagittal plane in three visual conditions: with the eyes closed (EC), in stroboscopic illumination (EO/SI; only visual position information) and with eyes open in continuous illumination (EO/CI; position and velocity information) with the room as static visual scene (or the interior of a moving cabin, in some of the trials). In the frequency spectrum of the translation stimulus we distinguished on the basis of the response patterns between a low-frequency, mid-frequency, and high-frequency range (LFR: 0.0165-0.14 Hz; MFR: 0.15–0.57 Hz; HFR: 0.58–2.46 Hz). With EC, subjects’ mean sway response gain was very low in the LFR. On average it increased with EO/SI (although not to a significant degree p = 0.078) and more so with EO/CI (p < 10−6). In contrast, the average gain in the MFR decreased from EC to EO/SI (although not to a significant degree, p = 0.548) and further to EO/CI (p = 0.0002). In the HFR, all three visual conditions produced, similarly, high gain levels. A single inverted pendulum (SIP) model controlling center of mass (COM) balancing about the ankle joints formally described the EC response as being strongly shaped by a resonance phenomenon arising primarily from the control’s proprioceptive feedback loop. The effect of adding visual information in these simulations lies in a reduction of the resonance, similar as in the experiments. Extending the model to a double inverted pendulum (DIP) suggested in addition a biomechanical damping effective from trunk sway in the hip joints on the resonance.

Multi-Objective Optimal Feedback Controls for Under-Actuated Dynamical System

This paper presents a study of optimal control design for a single-inverted pendulum (SIP) system with the multi-objective particle swarm optimization (MOPSO) algorithm. The proportional derivative (PD) control algorithm is utilized to control the system. Since the SIP system is nonlinear and the output (the pendulum angle) cannot be directly controlled (it is under-actuated), the PD control gains are not tuned with classical approaches. In this work, the MOPSO method is used to obtain the best PD gains. The use of multi-objective optimization algorithm allows the control design of the system without the need of linearization, which is not provided by using classical methods. The multi-objective optimal control design of the nonlinear system involves four design parameters (PD gains) and six objective functions (time-domain performance indices). The Hausdorff distances of consecutive Pareto sets, obtained in the MOPSO iterations, are computed to check the convergence of the MOPSO algorithm. The MOPSO algorithm finds the Pareto set and the Pareto front efficiently. Numerical simulations and experiments of the rotary inverted pendulum system are done to verify this design technique. Numerical and experimental results show that the multi-objective optimal controls offer a wide range of choices including the ones that have comparable performances to the linear quadratic regulator (LQR) control.

Calculation of the critical delay for the double inverted pendulum

Single and double inverted pendulum systems subjected to delayed state feedback are analyzed in terms of stabilizability. The maximum (critical) delay that allows a stable closed-loop system is determined via the multiplicity-induced-dominancy property of the characteristic roots, that is the dominant (rightmost) roots are associated with higher multiplicity under certain conditions of the system parameters. Other methods such as tracking the changes of the D-curves with increasing delay and the Walton–Marshall method are also demonstrated for the example of the single pendulum. For the double inverted pendulum subjected to full state feedback, the number of control gains is four, and application of numerical methods requires therefore high computational effort (i.e. optimization in a four-dimensional space). It is shown that, with the multiplicity-induced-dominancy–based approach, the critical delay and the associated control gains can be determined directly using the characteristic equation and its derivatives.

Integrated SolidWorks and Simscape platform for the design and control of an inverted pendulum system

Abstract A single inverted pendulum on a cart (SIPC) is designed and modeled physically using SolidWorks. The model is then exported to the Simulink environment to form a Simscape model for simulation and test purposes. This type of modeling uses a physical grid tactic to model mechanical structures. It requires connection of the physical elements with physical signal converter to define the implicit system dynamics to be modeled. The integration between the SolidWorks and Simscape eliminates the need of deriving the mathematical model and provides a platform for the rapid controller design for the system. State feedback control scheme is proposed, designed, and tuned aiming to maintain the pendulum in the upright place while tracking the desired cart position. Several simulation cases are studied to prove the controller abilities. In order to examine the controller robustness, disturbance rejection and noise attenuation capabilities are also discovered.

Real-time implementation and analysis of a modified energy based controller for the swing-up of an inverted pendulum on a cart

In this paper we derive a modified energy based swing-up controller using Lyapunov functions. During the derivation, all effort has been made to use a more complex dynamical model for the single inverted pendulum (SIP) system than the simplified model that is most commonly used. We consider the electrodynamics of the DC motor that drives the cart, and incorporate viscous damping friction as seen at the motor pinion. Furthermore, we use a new method to account for the limitation of having a cart-pendulum system with a finite track length. Two modifications to the controller are also discussed to make the method more appropriate for real-time implementation. One of the modifications improves robustness using a modified Lyapunov function for the derivation, while the other one incorporates viscous damping as seen at the pendulum axis. We present both simulation and real-time experimental results implemented in MATLAB Simulink.

Dynamic Posture Stabilization of a Biped Robot SUBO-1 on Slope-Changing Grounds

This paper describes a model-based dynamic posture stabilization of a biped robot on slope changing grounds. The ground slope generally is changing when a biped robot is walking outdoors on uneven grounds. Hence, if biped robots are used outside, their robustness against slope-changing grounds is necessary for stable walking. In this paper, a Zero Moment Point (ZMP) controller on level ground was designed as a basic posture stabilizer, and a disturbance observer (DOB) was additionally designed to reject a disturbance due to the ground slope. A simple single inverted pendulum model with a flexible joint is used for the ZMP controller and DOB designs. Finally, the performance of the proposed ZMP-DOB controller was verified through MATLAB simulations and walking in place experiments using a biped robot, SUBO-1 on a slope-changing treadmill.

입·출력 제약을 가진 회전형 도립진자의 견실 스윙업 제어

This paper proposes a robust swing-up strategy that considers input/output constraints for a single rotary inverted pendulum. For the swing-up of a pendulum, we adopt a 2-DOF control structure that consists of a nonlinear feedforward controller and an optimal feedback controller. In order to maximize the performance of the actuator, we formulate the problem of generating feedforward trajectories as a nonlinear optimal control problem subject to constraints. The robustness cannot be guaranteed by only the feedforward control because of the backlash generated in the actuator. Therefore we implemented an optimal feedback controller using an extended Kalman filter as the observer for the controller’s robustness. Through an experiment, we show that the proposed method has robustness against disturbance and swing a pendulum upward effectively satisfying input/output constraints.

Robust and novel two degree of freedom fractional controller based on two-loop topology for inverted pendulum

A rotary single inverted pendulum (RSIP) typically represents a space booster rocket, Segway and similar systems with unstable equilibrium. This paper proposes a novel two degree of freedom (2-DOF) fractional control strategy based on 2-loop topology for RSIP system which can be extended to control the systems with unstable equilibrium. It comprises feedback and feed-forward paths. Primary controller relates the perturbation attenuation while the secondary controller is accountable for set point tracking. To tune the parameters of proposed fractional controller a simple graphical tuning method based on frequency response is used. The study will serve the outstanding experimental results for both, stabilization and trajectory tracking tasks. The study will also serve to present a comparison of the performance of the proposed controller with the 1-DOF FOPID controller and sliding mode controller (SMC) for the RSIP system. Further to confirm the usability of the proposed controller and to avoid the random perturbations sensitivity, robustness, and stability analysis through fractional root-locus and Bode-plot is investigated.

Stabilization of unstable equilibrium point of rotary inverted pendulum using fractional controller

Many studies have been conducted by researchers globally in the recent past on the control schemes to enhance the performance of rotary single inverted pendulum (RSIP). The present paper seeks to investigate into a method to stabilize a pendulum link in upright position with a focus to minimize the deviation in rotational arm. Based on the fractional calculus, the proposed control scheme consists of two loop fractional controller in order to intensify the potentiality of RSIP. To tune the parameters of proposed fractional controller a simple graphical tuning method based on frequency response is used. The same parameters will also be tuned using dynamic particle swarm optimization (dPSO). The study will also serve to present a comparison of the performance of the proposed controller with the dynamic particle swarm optimization (dPSO) based fractional controller. Back calculation anti-windup technique will also be incorporated to compensate the saturation non-linearity. Further to confirm the usability of the proposed controller and to avoid the random perturbations sensitivity and robustness is investigated.

Fuzzy sliding-mode control with low pass filter to reduce chattering effect: an experimental validation on Quanser SRIP

Stabilization of a single rotary inverted pendulum (SRIP) system is a benchmark nonlinear control problem. Inherent instability of SRIP in upright position and the presence of modelling uncertainty pose a challenge for control engineers. Sliding-mode control (SMC) is the most widely accepted robust control technique. Various versions of SMCs to stabilize SRIP have been reported in literature. Advantages of SMC come at the cost of chattering of input, which inhibits its usage in many practical applications. Several works have been done to reduce the chattering in SMC. Application of fuzzy logic in SMC to reduce chattering has been analysed extensively in the literature. Even though implementation of fuzzy SMC (FSMC) in theory/simulations completely eliminates chattering, practical implementation of FSMC does not reduce chattering to zero. In this paper, a filtered FSMC is proposed, which filters the high frequency component in the system input. Experimental results of SMC, FSMC and a fuzzy sliding-mode controller with filter (FSMCF) are given in this paper. Results indicate that filtering the output of FSMCF reduces the chattering effect in the system without affecting the closed loop dynamics.

Performance Analysis and Experimental Validation of 2-DOF Fractional-Order Controller for Underactuated Rotary Inverted Pendulum

Several numbers of controllers are developed and implemented to enhance the performance of rotary single inverted pendulum (RSIP). This paper addresses a new two-degree-of-freedom (2-DOF) fractional control strategy for RSIP, which is a composition of feedback and feed-forward paths. Primary controller relates the perturbation attenuation, while the secondary controller is accountable for set point tracking. The pole placement technique is used for the design of 2-DOF proportional integral derivative (2-DOF PID) controller. In order to intensify the potentiality of 2-DOF PID controller, it is supplemented with fractional calculus. The tuning of fractional parameters is done by frequency domain analysis using the Nyquist plot. The proposed 2-DOF fractional-order PID controller is materialized on RSIP system which out turns the outstanding experimental results for both stabilization and trajectory tracking tasks. The system is investigated for stability, sensitivity and robustness, which confirms the ability of the proposed controller to reject the external random perturbations.

Cyber-Physical System Implementation into the Distributed Control System

Abstract: The purpose of this paper is to present the Single inverted pendulum laboratory model with LSM as a cyber-physical system (CPS). The implementation of the model application into the Distributed Control System infrastructure at the DCAI, FEEI, TU is first described. Pre-prepared program modules from the custom Simulink library Inverted Pendula Modeling and Control, developed by the authors of this paper, are used in subsequent modeling and control design. The contribution to the simulation model level of the DCS infrastructure is described in detail - the nonlinear mathematical model obtained from the library is modified to include the velocity loop, and the necessary parameters are obtained via experimental identification. Finally, a control strategy based on a hybrid control structure is designed and swing-up/stabilizing control algorithms are verified on the laboratory model.

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

Sliding mode with neuro-fuzzy network controller for inverted pendulum

This paper presents a sliding mode with fuzzy–neural network controller for nonlinear systems. The proposed controller combines the advantages of second order sliding mode control and fuzzy neural. The reaching conditions, stability and robustness of the system with the proposed controller are guaranteed. The sliding mode control (SMC) system which is insensitive to uncertainties, including parameter variations and external disturbances in the whole control process. The fuzzy–neural network mainly self–tuning fuzzy inference system (STFIS) is used to approximate the unknown system functions and switch item. Finally, the sliding–mode with fuzzy–neural network control is used to control single inverted pendulum and confirms the validity of the proposed approach. Results of simulations containing tests of robustness are presented.

Control System Design of Rotary Inverted Pendulum

In this paper we study the control system of single stage rotary inverted pendulum, and put forwards the controller design based on the core of STM32. In control strategy we use the classical control theory-PID control algorithm, which realizes the closed-loop control of rotating arm and swing rod for the single stage rotary inverted pendulum. The final test results show that the control strategy is effective.

Modeling and PID Control of Single-Stage Inverted Pendulum System

The dynamic model is obtained based on researching the structure of single inverted pendulum system in this paper. Mathematical model of inverted pendulum that is close to the working point is deduced by linearization. A PID control algorithm is put forward by analyzing the factor of influencing inverted pendulum stability. The effectiveness of proposed algorithm is verified by simulation. This algorithm has the features of high control precision and good stability.