An Inverted double pendulum is a combination of two individual pendulums which represents an example of a nonlinear and unstable dynamic system and it is also a good example of a physical system which can exhibit chaotic behavior.This document contains a first analysis of the model and the control of this system. Also presented is the installation of the electrical materials needed to control the system contain instrumenting the motor, current measurement system, motor shaft angle sensor, vision systemand MYRIO which is an embedded hardware device created by National Instruments will be used for data acquisition and control the system

Σκοπός της παρούσας διπλωματικής είναι ο έλεγχος ενός ανάστροφου εκκρεμούς φορτίου πλατφόρμας-σφαίρας (cart-ball) με χρήση ασαφούς ελέγχου ο οποίος υλοποιείται σε ηλεκτρονικό υπολογιστή στο περιβάλλον προγραμματισμού National Instruments LabVIEW 2009. Το φυσικό σύστημα αποτελείται από μία μεταλλική πλατφόρμα σταθερά προσδεδεμένη σε ένα γραμμικό άξονα της εταιρίας FESTO. Για την κίνηση του άξονα χρησιμοποιείται τριφασικός ηλεκτρικός σερβοκινητήρας, επίσης της FESTO, ο έλεγχος του οποίου γίνεται με χρήση ειδικού προγραμματιζόμενου ελεγκτή της ίδιας εταιρίας. Πάνω στην μεταλλική πλατφόρμα έχει τοποθετηθεί μία αψίδα παράλληλη στη διεύθυνση κίνησης της πλατφόρμας. Πάνω στην αψίδα έχουν δημιουργηθεί δύο οδηγοί επικαλυμμένοι με σκόνη γραφίτη σε μορφή κόλλας, η οποία τους κάνει ηλεκτρικά αγώγιμους. Ο ένας από τους οδηγούς έχει τάση 10V στα άκρα του ενώ το ένα άκρο του άλλου αγωγού είναι γειωμένο. Στόχος του αυτομάτου ελέγχου είναι να ισορροπήσει πάνω στην αψίδα μία μεταλλική σφαίρα. Η μεταλλική σφαίρα βραχυκυκλώνει τους δύο αγώγιμους οδηγούς και κατ' αυτό τον τρόπο, με την αρχή του ροοστάτη, λαμβάνεται η θέση της σφαίρας. Η μέτρηση της τιμής της θέσης της σφαίρας όπως και η θέση και η ταχύτητα της πλατφόρμας, οι οποίες λαμβάνονται από τον ελεγκτή του κινητήρα, εισάγονται στον υπολογιστή, μέσω ειδικής κάρτας Εισόδου/Εξόδου (Ε/Ε) της εταιρίας National Instruments, όπου λαμβάνονται από το LabVIEW και οδηγούνται στον ασαφή ελεγκτή που υλοποιείται στο ίδιο πρόγραμμα. Η ταχύτητα της σφαίρας λαμβάνεται επίσης υπ' όψιν στον αυτόματο έλεγχο και προκύπτει στο LabVIEW από παραγώγιση της θέσης της σφαίρας. Το ασαφές σύστημα αποτελείται από πέντε συναρτήσεις συμμετοχής για τη μεταβλητή της θέσης, τρεις συναρτήσεις συμμετοχής για τη μεταβλητή της ταχύτητας και εφτά συναρτήσεις συμμετοχής για την έξοδο του ελεγκτή που εκφράζει την επιθυμητή ταχύτητα της πλατφόρμας. Ο συμπερασμός γίνεται με χρήση 22 κανόνων. Η έξοδος του ασαφούς ελεγκτή εκφράζεται με μια τιμή από -10V έως +10V. Η έξοδος από τον ασαφή ελεγκτή οδηγείται πάλι μέσω της ειδικής κάρτας Ε/Ε στον προγραμματιζόμενο ελεγκτή του κινητήρα της FESTO ο οποίος ελέγχει κατάλληλα το ρεύμα τυλίγματος του κινητήρα ώστε να επιτευχθεί η επιθυμητή ταχύτητα. Η περίοδος του ελέγχου είναι 100 ms. Τα αποτελέσματα του ελέγχου του πιο πάνω συστήματος κρίνονται γενικά ικανοποιητικά δεδομένης της ισχυρής αστάθειας που το χαρακτηρίζει. / The subject of the thesis is the automatic control of an inverted pendulum consisting of a cart and a ball. We used fuzzy control to balance the ball on top of an arch. The controller was created in LabView 2009.

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 presents the design, construction, and control of a two-wheeled inverted pendulum (TWIP) robotic walker prototype for assisting mobility-impaired users with balance and fall prevention. A conceptual model of the robotic walker is developed and used to illustrate the purpose of this study. A linearized mathematical model of the two-wheeled system is derived using Newtonian mechanics. A control strategy consisting of a decoupled LQR controller and three state variable controllers is developed to stabilize the platform and regulate its behavior with robust disturbance rejection performance. Simulation results reveal that the LQR controller is capable of stabilizing the platform and rejecting external disturbances while the state variable controllers simultaneously regulate the system’s position with smooth and minimum jerk control. A prototype for the two-wheeled system is fabricated and assembled followed by the implementation and tuning of the control algorithms responsible for stabilizing the prototype and regulating its position with optimal performance. Several experiments are conducted, confirming the ability of the decoupled LQR controller to robustly balance the platform while the state variable controllers regulate the platform’s position with smooth and minimum jerk control.

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This report presents the construction of a two-wheeled self balancing robot with the ability to handle an uneven load. Two-wheeled self-balancing robots have proven to be a potential solution to the problem of efficient warehouse management, thanks to their ability to navigate tight spaces quickly and efficiently while balancing their load. The research questions defined include the ability to develop an algorithm to dynamically adjust the robot’s tilt angle, how an uneven load affects the robot’s stability, and whether a PID controller is sufficient in this context. The project was limited to constructing a prototype that could balance with an external load, with a budget of 1,000 Swedish kronor and a four-month timeframe. By testing and evaluating different control algorithms, the robot’s performance was presented in terms of stability and efficiency. The implementation of PID control was successful, and the robot was able to balance as a result. However, the goal of handling an uneven load was not met without the implementation of an additional algorithm to dynamically adjust the robot’s tilt angle. With these two control techniques, the robot was able to balance with and without an added load with good stability. To evaluate performance, tests were performed with the load placed centered and off-centered on the robot’s top plate. The results of the tests showed that the robot was able to dynamically adjust its tilt angle to balance with added weight without affecting stability. / Denna rapport presenterar konstruktionen av en tvåhjulig självbalanserande robot med förmågan att hantera snedfördelad last. Tvåhjuliga självbalanserande robotar har visat sig vara en potentiell lösning på problemet kring effektiv lagerhantering tack vare deras förmåga att hantera snäva utrymmen på ett snabbt och energieffektivt sätt samtidigt som lasten balanseras. Forskningsfrågorna som definierades inkluderar möjligheten att framställa en algoritm för att dynamiskt ställa in robotens lutningsvinkel, hur ojämn last påverkar robotens stabilitet, och om en PID-kontroller är tillräcklig i detta sammanhang. Projektet begränsades till att konstruera en prototyp som klarar av att balansera med extern last, med en budget på 1000 svenska kronor och en tidsram på fyra månader. Genom att testa och utvärdera olika kontrollalgoritmer presenterades robotens prestanda i termer av stabilitet och effektivitet. Roboten balanserade tack vare implementeringen av PID-reglering, men önskemålet om snedfördelad last uppfylldes inte och det krävdes ytterligare en algoritm för att dynamiskt reglera robotens lutningsvinkel. Med hjälp av dessa två reglertekniker kunde roboten balansera både med och utan adderad last med god stabilitet. Tester utfördes för att utvärdera prestandan när lasten var placerad både centrerat och ocentrerat på robotens topplatta. Resultaten visade att roboten kan dynamiskt anpassa lutningsvinkeln för att balansera med tillagd vikt utan att stabiliteten påverkas.

Reinforcement learning is conceptually based on an agent learning through interaction with its environment. This trial-and-error learning method makes the process prone to situations in which the agent is stuck in a dead-end, from which it cannot keep learning. This thesis studies a method to diminish the risk that a wheeled inverted pendulum, or WIP, falls over during training by having a Qlearning based agent estimate a PID controller before training it on the balance problem. We show that our approach is equally stable compared to a Q-learning agent without estimation training, while having the WIP falling over less than half the number of times during training. Both agents succeeds in balancing the WIP for a full hour in repeated tests. / Reinforcement learning baseras på en agent som lär sig genom att interagera med sin omgivning. Denna inlärningsmetod kan göra att agenten hamnar i situationer där den fastnar och inte kan fortsätta träningen. I denna examensuppsats utforskas en metod för att minska risken att en självkörande robot faller under inlärning. Detta görs genom att en Q-learning agent tränas till att estimera en PID kontroller innan den tränar på balanseringsproblemet. Vi visar att vår metod är likvärdigt stabil jämfört med en Q-learning agent utan estimeringsträning. Under träning faller roboten färre än hälften så många gånger när den kontrolleras av vår metod. Båda agenterna lyckas balansera roboten under en hel timme.

This bachelor thesis at the Royal Institute of Technology in Stockholm, Sweden, aims to clarify how the minimum bandwidth of a stabilised mobile inverted pendulum is affected by the position of its center of mass and the frequency of measuring the state variables. The scope is to apply basic theories in automation control and electronic systems when designing the system. The mechatronical system in the experimental trials is constructed with the main components consisting of bipolar stepper motors, stepper motor drivers (DRV8825), potentiometer and Arduino UNO. The mobile pendulum in question consists of a cart on a rail with a potentiomoter and a rod mounted to it. In the experimental trials in this paper, three different radial positions of the center of mass are analysed with regards to the bandwidth of relevant measurements. A variety of PID parameters, for both the angle and position regulation, are the essential basis on which these stability trials are performed. The results are limited to the resolution of the potentiometer outputs, the length of the rail and the stiffness of the rod. The result is presented by comparisons between data of the greatest angles which are stabilised and the corresponding maximal latency of the angle corrections. These data concludes to that the minimum required bandwidth to sustain stability decreases for an increasing length between the center of mass and the point of rotation. / Denna kandidatexamensuppsats genomfördes på Kungliga Tekniska Högskolan, i Stockholm, Sverige, och har som mål att klargöra hur den minsta bandbredden för en stabiliserad inverterad pendel påverkas av positionen av dess tyngdpunkt samt nätfrekvensen av tillståndsvariablerna. Projektet omfattar grundläggande teorier om reglerteknik och elektriska system.  Det mekatroniska systemet som används i experimenten är uppbyggt av följande nyckelkomponenter: bipolära stegmotorer, stegmotordrivare (DRV8825), potentionmeter och Arduino UNO. Den mobila pendeln består av en stång och en potentiometer som är monterad på en vagn löpandes på en räls. I de utförda experimenten i denna rapport analyseras tre radiella positioner på masscentrum med hänsyn till bandbredden och relevanta mätningar. En reglering med olika PID parametrar, för både vinkel- och positionsreglering, är den kritiska process som lade grunden för analyserna av stabiliteten. Resultaten är begränsade av upplösningen hos potentiometerns utsignal, längden av rälsen och stångens styvhet.  Resultaten presenteras genom jämförelser mellan data för de största vinklarna som stabiliseras och de motsvarande maximala fördröjningarna. Sammanfattningsvis minskar den minst nödvändiga bandbredden för att upprätthålla stabilitet för ökande längder mellan pendelns tyngdpunkt och rotationspunkten.

En damm används i huvudsak för att magasinera vatten i energiutvinningssyfte. Dammen rör sig fram och tillbaka i ett säsongsmönster mestadels beroende på skillnader i utomhustemperatur och vattentemperaturen i magasinet. Det nordiska klimatet innebär risk för isläggning i magasinet, för vilken lasten är relativt outforskad. Denna rapport syftar till ett med multipla linjära regressionsmodeller samt dynamiska regressionsmodeller avgöra vilka variabler som förklarar en specifik svensk dammkonstruktions rörelse. Dammens rörelse mäts genom att mäta dammens förflyttning kontra berggrunden med data från dammens inverterade pendlar. Av särskilt intresse är att avgöra islastens påverkan på rörelsen. Resultaten visar att multipla linjära regressions-modeller inte fullständigt lyckas modellera dammens rörelse, då de har problem med autokorrelerade residualer. Detta hanteras med hjälp av autoregressiva regressionsmodeller där de initiala förklarande variablerna inkluderas, kallat dynamisk regression. Denna rapports resultat visar att de autoregressiva parametrarna fungerar mycket väl för att förklara pendlarna, men att även tid, temperatur, det hydrostatiska trycket samt istjocklek är användbara förklarande variabler. Istjockleken visar signifikant påverkan på 5 % signifikansnivå på två av de undersökta pendlarna, vilket är ett noterbart resultat. Författarna menar att rapportens resultat indikerar att det finns anledning att fortsätta forska kring islastens påverkan på dammkonstruktioner. / A dam is a structure mainly used for storing water and generating electricity. The structure of a dam moves in a season-based pattern, mainly because of the difference in temperature between the air on outside of the dam and the water on the inside. Due to the Nordic climate, occurrences of icing on the water in the basin is fairly frequent. The effects of ice on the structural load of the dam are relatively unexplored and are the subject to this bachelor’s thesis. The goal of this project is to evaluate which predictors are significant to the movement of the dam with multiple linear regression models and dynamic regressions. The movement is measured by inverted pendulums that register the dam’s movement compared to the foundation. It is of particular interest to determine if the ice load influences the movement of the dam. The multiple regression models used to explain the dam’s movement were all discarded due to autocorrelation in the residuals. This falsifies the models, since autocorrelation means that they don’t meet the needed assumptions. To counteract the autocorrelation, dynamic models with autoregressive terms were fitted. These models showed no problem with autocorrelation. The result from the dynamic models were successful and managed to significantly explain the movement of the dam. The autoregressive terms proved to be efficient explanatory variables. The dynamic regression models also show that the time, temperature, hydrostatic pressure and ice thickness variables are also useful explanatory variables. The ice thickness shows a significant effect at the 5 % significance level on two of the investigated pendulums. The report's results indicate that there is reason to continue research on the ice load impact on dam constructions.

A robot capable of balancing itself on two wheels has been built and programmed. While balancing, the robot keeps within a limited area. The robot has a face with two eyes and a mouth, consisting of LED-matrices, which switch between six different facial expressions. The robot is programmed using Arduino boards, one of which implements PID regulators to control the motors.