Η παρούσα διπλωματική αφορά στον έλεγχο του ανεστραμμένου εκκρεμούς. Γίνεται μια σύντομη σύνοψη της σχετικής θεωρίας και ακολούθως ασχολείται αρχικά με τη σταθεροποίηση του γραμμικοποιημένου, γύρω από την ασταθή κατακόρυφη θέση ισορροπίας, συστήματος και στη συνέχεια με τη μελέτη δύο διαφορετικής λογικής ελέγχων για την πλήρη λύση του προβλήματος ελέγχου του εκκρεμούς, δηλαδή αφ’ ενός την ανύψωση και αφ’ ετέρου τη σταθεροποίηση του. Γίνεται και μια προσπάθεια εκτίμησης της περιοχής ελκτικότητας του συστήματος για κάποιους ελέγχους / In the present thesis the control of an inverted pendulum is presented. A brief summary of the relevant theory is presented. In the following the stabilization of the linearized system around the unstable upright position is developed. Furthermore, the study of two different control approaches for the complete solution of the pendulum; the swing up phase and stabilization phase. Lastly an effort for the rough estimation of the region of attraction is attempted

Bakgrund: Transfemoralt amputerade har nedsatt balans och ökad fallrisk, men väldigt lite forskning är gjord om detta. Inverted pendulum model (IPM) är en balansmodell för icke-amputerade som bygger på ett känt samband mellan Center of Pressure (CoP) och Center of Mass (CoM). Syftet med denna studie är att kinetiskt validera den inverterade pendelmodellen för transfemoralt amputerade. Metod: I studien deltog amputerade (n=5) och en matchande kontrollgrupp (n=5). Man samlade in data genom att deltagarna fick stå stilla på två stycken kraftplattor under tre styckern villkor; öppna ögon, stängda ögon och weight-bearing feedback. Man undersökte sedan korrelationen mellan avståndet från CoM till CoP och CoMacc i både anterioposterior (A/P) riktning samt medio-lateral (M/L) riktning och analyserade datan med trevägsvariansanalys (ANOVA). Resultat: Resultatet visade att det fanns en signifikant interaktionseffekt mellan villkor och position i M/L-riktning. I A/P-riktning fanns det en signifikant interaktionseffekt mellan grupp och position samt villkor och position. Slutsats: Resultatet innebär att IPM är kinetiskt validerat i A/P-riktning sett till hela kroppen men inte på den amputerade sidan. Hur det intakta benet förhåller sig till IPM för amputerade är tvivelaktigt. / Background: It is known that transfemoral prosthesis users lack normal balance control and are more likely to fall. Research on this topic is insufficient. The Inverted Pendulum Model (IPM) is a commonly used biomechanical model for assessment of balance and postural control for healthy individuals based on an assumption that Center of Pressure (CoP) and Center of Mass (CoM) are inter-dependent. The aim of the study is to validate IPM kinetically for transfemoral prosthesis users. Method: Amputees (n=5) and a control group (n=5) participated. During data collection, participants stood on two force plates with eyes open, eyes closed and with weight-bearing feedback. Correlation of the distance CoP-CoM and CoMacc were calculated for anteroposterior and mediolateral directions and evaluated with three-way ANOVA. Result: Results showed significant interaction effects between condition and position plus group and position in anteroposterior direction and condition and group in mediolateral direction. Conclusion: Results indicate kinetic validity of IPM for transfemoral amputees when looking at the whole body but not at the amputated side in the A/P direction. Kinetic validity of IPM for the intact leg is questionable.

Neste trabalho, uma análise quantitativa e qualitativa para a estabilização dinâmica de um pêndulo invertido com uma força externa senoidal aplicada no ponto de suspensão é feita. Inicialmente, a perturbação externa é composta de um único cosseno, então uma generalização é feita, usando uma soma de N cossenos com diferentes amplitudes e frequências. Aproximações são testadas e o tempo durante o qual o pêndulo invertido permanece estável é explorado quando N é grande, a fim de recuperar o padrão do caso onde N = 1. O caso específico de oscilações periódicas e quase periódicas, quando N = 2, é analisado e diagramas de estabilidade considerando diferentes frequências e amplitudes são estudados. Depois, um ruído Gaussiano additivo é adicionado ao sistema para que a degradação dos diagramas de estabilidade gerados por variâncias diferentes possam ser estudados. Todos os pontos deste trabalho são corroborados por simulações, as quais integram numericamente as equações de movimento do sistema através do método de Runge-Kutta de quarta ordem. Os algoritmos e detalhes extras dos métodos de integração usados são explorados numa publicação deste trabalho, a qual está apresentada, nesta dissertação, como um apêndice. / In this work a quantitative and qualitative analysis of the dynamical stabilization of an inverted pendulum with a sinusoidal external perturbation applied at the suspension point is made. Initially, the external perturbation is composed of a single cosine, then a generalization is made using a sum of N cosines with different amplitudes and frequencies. Approximations are tested, and the time for which the inverted pendulum remains stable is explored when N is large, in order to recover the pattern of the case when N = 1. The specific case of periodic and almost periodic oscillations, when N = 2, is analysed and stability diagrams considering different frequencies and amplitudes are studied. Later, an additive Gaussian noise is added to the system so the degradation of the stability diagrams generated by different variances can be studied. All points of this work are corroborated by simulations, which numerically integrate the system’s equation of motion through a fourth order Runge-Kutta method. Algorithms and extra details on the integration methods used are explored in a publication of this work, which is presented in this thesis as an appendix.

Neste trabalho, uma análise quantitativa e qualitativa para a estabilização dinâmica de um pêndulo invertido com uma força externa senoidal aplicada no ponto de suspensão é feita. Inicialmente, a perturbação externa é composta de um único cosseno, então uma generalização é feita, usando uma soma de N cossenos com diferentes amplitudes e frequências. Aproximações são testadas e o tempo durante o qual o pêndulo invertido permanece estável é explorado quando N é grande, a fim de recuperar o padrão do caso onde N = 1. O caso específico de oscilações periódicas e quase periódicas, quando N = 2, é analisado e diagramas de estabilidade considerando diferentes frequências e amplitudes são estudados. Depois, um ruído Gaussiano additivo é adicionado ao sistema para que a degradação dos diagramas de estabilidade gerados por variâncias diferentes possam ser estudados. Todos os pontos deste trabalho são corroborados por simulações, as quais integram numericamente as equações de movimento do sistema através do método de Runge-Kutta de quarta ordem. Os algoritmos e detalhes extras dos métodos de integração usados são explorados numa publicação deste trabalho, a qual está apresentada, nesta dissertação, como um apêndice. / In this work a quantitative and qualitative analysis of the dynamical stabilization of an inverted pendulum with a sinusoidal external perturbation applied at the suspension point is made. Initially, the external perturbation is composed of a single cosine, then a generalization is made using a sum of N cosines with different amplitudes and frequencies. Approximations are tested, and the time for which the inverted pendulum remains stable is explored when N is large, in order to recover the pattern of the case when N = 1. The specific case of periodic and almost periodic oscillations, when N = 2, is analysed and stability diagrams considering different frequencies and amplitudes are studied. Later, an additive Gaussian noise is added to the system so the degradation of the stability diagrams generated by different variances can be studied. All points of this work are corroborated by simulations, which numerically integrate the system’s equation of motion through a fourth order Runge-Kutta method. Algorithms and extra details on the integration methods used are explored in a publication of this work, which is presented in this thesis as an appendix.

The diploma thesis deals with control of rotary inverted pendulum - Furuta pendulum. Solution for power electronics, sensors and coupling with PC is described, identification of parameters and nonlinear simulation model in Matlab/Simulink and SimMechanics toolbox is presented. Second type Lagrange equation is used for determination of equations of motion. Controll system based on state-space model of mechanism and LQR algorithm for design of state-space controller is used and switching between swing-up cotroller a stabilizing state-space control is achieved according to actual angular position of pendulum's angle. Input integrator eliminating steady state error was used with success.

The focus of this document is on the design, modeling, and control of a self-balancing two wheel robot, hereafter referred to as the balance bot, driven by independent brushless DC (BLDC) motors. The balance bot frame is composed of stacked layers allowing a lightweight, modular, and rigid mechanical design. The robot is actuated by a pair of brushless DC motors equipped with Hall effect sensors and encoders allowing determination of the angle and angular velocity of each wheel. Absolute orientation measurement is accomplished using a full 9-axis IMU consisting of a 3-axis gyroscope, a 3-axis accelerometer, and a 3-axis magnetometer. The control algorithm is designed to minimize deviations from a set point specified by an external radio remote control, which allows the remote operator to steer and drive the bot wirelessly while it remains balanced. Multiple dynamic models are proposed in this analysis, and the selected model is used to develop a linear-quadratic regulator based state-feedback controller to perform reference tracking. Controller tracking performance is improved by incorporating a prefilter stage between the setpoint command from the remote control and the state-feedback controller. Modeling of the actuator dynamics is considered brie y and is discussed in relation to the control algorithm used to balance the robot. Electrical and software design implementations are also presented with a focus on effective implementation of the proposed control algorithms. Simulated and physical testing results show that the proposed balance bot and controller design are not only feasible but effective as a means of achieving robust performance under dynamic tracking profiles provided by the remote control.

In this paper the two reinforcement learning algorithmsQ-learning and deep Q-learning (DQN) are used tobalance an inverted pendulum. In order to compare the two, bothalgorithms are optimized to some extent, by evaluating differentvalues for some parameters of the algorithms. Since the differencebetween Q-learning and DQN is a deep neural network (DNN),some benefits of a DNN are then discussed.The conclusion is that this particular problem is simple enoughfor the Q-learning algorithm to work well and is preferable,even though the DQN algorithm solves the problem in fewerepisodes. This is due to the stability of the Q-learning algorithmand because more time is required to find a suitable DNN andevaluate appropriate parameters for the DQN algorithm, than tofind the proper parameters for the Q-learning algorithm. / I denna rapport används två algoritmer inom förstärkningsinlärning och djup Q-inlärning (DQN), för att balancera en omvänd pendel. För att jämföra dem så optimeras algoritmerna i viss utsträckning genom att testa olika värden för vissa av deras parametrar. Eftersom att skillnaden mellan Q-inlärning och DQN är ett djupt neuralt nätverk (DNN) så diskuterades fördelen med ett DNN. Slutstatsen är att för ett så pass enkelt problem så fungerar Q-inlärningsalgoritmen bra och är att föredra, trots att DQNalgoritmen löser problemet på färre episoder. Detta är pågrund av Q-inlärningsalgoritmens stabilitet och att mer tid krävs för att hitta ett passande DNN och hitta lämpliga parametrar för DQN-algoritmen än vad det krävs för att hitta bra parametrar för Q-inlärningsalgoritmen. / Kandidatexjobb i elektroteknik 2021, KTH, Stockholm

O estudo da estabilidade e desempenho em sistemas de controle é um tópico relevante na teoria de sistemas. Quando são assumidas incertezas no modelo da planta, existe uma maior dificuldade para garantir um nível de desempenho adequado do sistema dinâmico e a estabilidade pode ser comprometida. Neste trabalho são utilizados um regulador linear quadrático robusto e um filtro de Kalman robusto combinados em uma única formulação para tratar de sistemas dinâmicos incertos em tempo real. O caso de estudo selecionado é o pêndulo invertido. Seus principais desafios de controle encontrados na literatura: estabilização, seguimento e levantamento-captura, serão considerados. Os algoritmos utilizados são motivados pelo fato de que problemas estocásticos podem ser resolvidos por meio de argumentos determinísticos, baseados nos conceitos de função penalidade e mínimos quadrados regularizados. Desta forma, é possível a obtenção do melhor desempenho em contrapartida à máxima influência de incerteza admissível. A análise de desempenho do controlador robusto é realizada por meio de ensaios práticos incluindo incertezas na planta, ruído nos sensores e distúrbios no sinal de controle do pêndulo. / The study of stability and performance in control systems is a relevant topic in systems theory. When uncertainties are considered in the model of the plant, there is a greater difficulty in ensuring an appropriate performance level of the dynamic system, plus, the stability could be compromised as well. In this dissertation a robust linear quadratic regulator and a robust Kalman filter are used in a unified manner to deal with uncertain dynamic systems in real time. The selected case study is the inverted pendulum. Its main control challenges found in the literature will be considered: stabilization, tracking and catching swing-up. The used algorithms are motivated by the fact that stochastic problems can be solved through deterministic arguments based on the concepts of penalty function and regularized least-squares. Thus, it is possible to obtain an optimal performance for the maximum acceptable uncertainty. The performance analysis of the robust control is carried out by practical experiments including uncertainties in the plant, noise in the sensors and disturbance in the pendulum control signal.

This thesis introduces a new measure of balance for bipedal robotics called the foot placement estimator (FPE). To develop this measure, stability first is defined for a simple biped. A proof of the stability of a simple biped in a controls sense is shown to exist using classical methods for nonlinear systems. With the addition of a contact model, an analytical solution is provided to define the bounds of the region of stability. This provides the basis for the FPE which estimates where the biped must step in order to be stable. By using the FPE in combination with a state machine, complete gait cycles are created without any precalculated trajectories. This includes gait initiation and termination. The bipedal model is then advanced to include more realistic mechanical and environmental models and the FPE approach is verified in a dynamic simulation. From these results, a 5-link, point-foot robot is designed and constructed to provide the final validation that the FPE can be used to provide closed-loop gait control. In addition, this approach is shown to demonstrate significant robustness to external disturbances. Finally, the FPE is shown in experimental results to be an unprecedented estimate of where humans place their feet for walking and jumping, and for stepping in response to an external disturbance.

This thesis introduces a new measure of balance for bipedal robotics called the foot placement estimator (FPE). To develop this measure, stability first is defined for a simple biped. A proof of the stability of a simple biped in a controls sense is shown to exist using classical methods for nonlinear systems. With the addition of a contact model, an analytical solution is provided to define the bounds of the region of stability. This provides the basis for the FPE which estimates where the biped must step in order to be stable. By using the FPE in combination with a state machine, complete gait cycles are created without any precalculated trajectories. This includes gait initiation and termination. The bipedal model is then advanced to include more realistic mechanical and environmental models and the FPE approach is verified in a dynamic simulation. From these results, a 5-link, point-foot robot is designed and constructed to provide the final validation that the FPE can be used to provide closed-loop gait control. In addition, this approach is shown to demonstrate significant robustness to external disturbances. Finally, the FPE is shown in experimental results to be an unprecedented estimate of where humans place their feet for walking and jumping, and for stepping in response to an external disturbance.