Self-stabilizing platform: ZTäBiLAjZöR

(1)

INOM

EXAMENSARBETE TEKNIK, GRUNDNIVÅ, 15 HP

STOCKHOLM SVERIGE 2016,

Self-stabilizing platform

ZTäBiLAjZöR

ANDERS KARLSSON

JONATHAN CRESSELL

(2)

(3)

Self-stabilizing platform

ZTäBiLAjZöR

ANDERS KARLSSON JONATHAN CRESSELL

Bacherlor’s Thesis in Mechatronics Supervisor: Martin Edin Grimheden Examiner: Martin Edin Grimheden Approved: 2016-06-07

(4)

(5)

Abstract

The use of accelerometers and gyroscopes has greatly increased in recent years, diﬀerent applications of these are seen in everything from smartphones to mo- torcycles. The stabilizing mechanisms used for cameras may however be more well-known. Irrespective of purpose, motorized stabilizing mechanisms commonly utilize an inertial measurement unit to determine real-time position and movement. This position is then analysed and an action is chosen. To improve performance, control science is often applied.

The purpose of this report is to analyze how a common PID controller will react to variations in load. This report will describe the construction and development of a self-stabilizing platform, controlled by two PID controllers, based on the Arduino platform. Theory behind the included components is presented and the methods of modeling, construction and testing is discussed.

An IMU consisting of three accelerometers and three gyroscopes is used to measure the tilt of the platform, a Kalman filter is then used to suppress the signal noise to reasonable levels. This was successfully achieved using Open- source code from the Arduino community.

Two DC-motors adjust the angle of the platform, one for each axis, with a PID controller each. The system was modelled for each motor to determine the PID controller parameters that would satisfy the requirements set regard- ing speed and precision. A fast system was prioritized to assure small angles and reduce the torque on the motor shafts. This resulted in a system with roughly 0.1 seconds of rise time, 1.3 seconds settling time and 18% overshoot at a 10° step.

By conducting step response tests using the demonstrator, the research question could be answered. It was found that a PID controller tuned for a specific system will have a performance drop but not become unstable when load is varied, up to the level of load that causes too large amounts of torque for the motors to counter.

(6)

(7)

Sammanfattning

Själv-stabiliserande plattform

Användningen av accelerometrar och gyroskop har ökat de senaste åren, olika tillämpningar finns i allt från smarta telefoner till motorcyklar. Däremot är till exempel stabiliseringsmekanismen använd till kameror bättre känd. Oav- sett syfte är det vanligt att motoriserade stabiliseringsmekanismer använder en tröghetsmätningsenhet, bättre känd som en IMU, för att avgöra position och rörelse i realtid. Denna position är sedan analyserad och en handling väljs. För att förbättra prestanda appliceras ofta reglerteknik på systemet.

Syftet med denna rapport är att undersöka hur en normal PID-regulator kom- mer att reagera vid variationer i belastning. Rapporten beskriver konstruktio- nen och utvecklingen av en själv-stabiliserande plattform, styrd av två PID- regulatorer, baserad på Arduino-plattformen. Bakomliggande teori kring kom- ponenterna presenteras och metoder för modellering, konstruktion och testning diskuteras.

En IMU bestående av tre accelerometrar och tre gyroskop används för att mäta plattformens lutning, ett Kalman-filter dämpar sedan bruset i signalen till godtagbara nivåer. Detta åstadkoms med hjälp av kod från Open-source källor från Arduino-communityt.

Två DC-motorer justerar vinkeln på plattformen, en för var axel, med en PID- regulator var. Systemet modellerades för vardera av motorerna, för att bestäm- ma parametrar för PID-regulatorerna som kunde uppfylla de satta kraven på hastighet och precision. Vikt lades på att systemet skulle vara snabbt för att garantera små vinklar och minimalt vridmoment på motorernas utgående axlar.

Detta resulterade i ett system med c:a 0.1 sekunders stigtid, insvängningstid på 1.3 sekunder och ett översläng på hela 18% vid ett steg på 10°.

Forskningsfrågan kunde besvaras med tester genomförda på demonstratören.

Det gavs att en PID-regulator anpassad för ett visst system endast får sämre prestanda utan att bli instabil vid förändringar i belastning, upp till den vikt som hindrar motorerna från att rotera på grund av för stort vridmoment.

(8)

(9)

Preface

We would like to present our thanks to our supervisor Martin Edin Grimheden for assisting us throughout this project. We would also like to thank Jan Stamer for helping us in the workshop, Staﬀan Qvarnström for helping with the demonstrator, Tomas Östberg for lending us tools and materials and Lei Feng for helping us with the control science.

Anders Karlsson Jonathan Cressell Stockholm, May, 2016

(10)

(11)

Nomenclature

Symbols

Symbols Description

b Dynamic friction constant [Nm·s]

Bt Control matrix

e Static error

e_{EM F} Back EMF

Ft State transition matrix

G Transfer function

g Gravitation constant [m/s²]

i Current [A]

J Total moment of inertia per motor [kg·m²] J₁ Motor 1 moment of inertia [kg·m²]

J2 Motor 2 moment of inertia [kg·m²] J_r Rotor moment of inertia [kg·m²] K_e Back EMF constant [mV/rpm]

KD Derivative part parameter K_I Integrative part parameter K_P Proportional part parameter

Kt Torque constant [mN/A]

L Inductance [H]

l Distance from rotational axis to center of mass [m]

m Mass [kg]

Qt Covariance matrix

R Resistance [Ω]

T Motor torque [Nm]

t Time [s]

u Input signal

ut Control vector

V Voltage [V]

V_N Nominal voltage [V]

w_t Vector containing noisy terms at time t

(14)

x_t₋₁ State matrix at time t− 1

θ Angle [°]

θ˙ Angular velocity [rad/s]

θ¨ Angular accerelation [rad/s²]

Abbreviations

Abbreviation Description

CAD Computer Aided Design

EMF Electro-Motive Force

I²C Inter-Integrated Circuit IMU Inertial Measurement Unit

MDC Motion-Disturbance Compensating PID ProportionalIntegralDerivative

PLA Polylactide

PWM Pulse Width Modulation

Rpm Revolutions per minute

xii

(15)

Chapter 1

Introduction

This chapter explains the background to this project, along with its purpose and limitations.

1.1 Background

The need for stabilizing mechanisms is wide-spread. Various techniques are applied in search of the ideal solution. A common application is camera stabilization, less common are perhaps self-leveling anti-motion sickness seats or surgery platforms [Grober, D.E., 2009]. These systems share the importance of maintaining a constant position or direction relative to a point of reference, regardless of disturbance or movement in space. A method for measuring this position is using an Inertial Measurement Unit (IMU) consisting of gyroscopes and accelerometers, commonly used in self-stabilizing platforms.

The Motion-Disturbance Compensating (MDC), or self-stabilizing, platform aims to correct angular deviations using motors. The signal from the measuring unit re- quires filtering and in order to obtain any reasonable level of system rapidity control science must be applied. The motors are seen as two individual systems, one for the x-axis and one for the y-axis. A ProportionalIntegralDerivative (PID) controller is well suited for systems with one signal in, voltage, and one signal out, angle, such as the one in question.

1.2 Purpose

This report aims to investigate the robustness of a PID controlled self-stabilizing platform, in order to answer the research question:

“How will a PID controlled system react to variations in load?”

(16)

CHAPTER 1. INTRODUCTION

The result of this could help widening the range of methods used to create stabilizing mechanisms. In doing so, solutions to problems such as the previously mentioned surgery platform could be found.

1.3 Scope

The demonstrator will be limited to two degrees of freedom, rotation around the x-axis, roll, and y-axis, pitch. Rotation around the z-axis, yaw, and linear transla- tion movement will be disregarded as it was found to be irrelevant for a stabilizing platform. Open-Source code will be used for the sensor signal filter and the Inter- intergrated circuit (I²C) communication bus and will as such be explained only briefly. Further, the demonstrator will be constructed as a small scale test rig for small loads. This project is a bachelor’s thesis.

1.4 Method

A mathematical model of the system was created in order to theoretically design a control system that could later be applied to the demonstrator, which was constructed in order to test the theory. Prior to construction a realistic 3D-model was drawn up using Solid Edge in order to determine values such as the distance to the centre of mass and create models for 3D-printing. Necessary PID controller parameters were evaluated using MATLAB Simulink.

2

(17)

1.5. SIMILAR PROJECTS

1.5 Similar projects

Self-stabilizing platforms are nowadays a relatively popular home-project and many diﬀerent types were found during research. Most of these projects use servo motors, see figure 1.1, in other words, most are not feedback systems. The downside of servo motors in this application is that they cannot be regulated outside their built in regulator, which in most cases makes the construction slow [Rivello, 2011]. The great advantage is its simplicity, the angle of the measurement unit can be fed directly to the servo motor [Rony Chakraborty, 2014].

Figure 1.1: Typical construction, Source:[Rivello, 2011]

A less common approach to this problem is using DC-motors and implementing a separate controller to the system [S. Narkar, S. Bhalekar, T. Nawge, K. Parab, P.

Pati, 2013]. An advantage of this is that they can be regulated with an independent controller and it can, for example, be tuned for an aggressive fast response or a defensive slower response. However, this will greatly increase the complexity of the system and as such require a higher level of understanding of the problem.

(18)

(19)

Chapter 2

Theory

The theory chapter provides basic knowledge necessary to understand the problem and the following solution.

2.1 Stabilizing platforms

A self-stabilizing platform usually takes in account at least two angles; roll and pitch. The purpose of such a platform is for it to maintain close-to zero degrees of angle towards the direction of gravity, in other words, to stay horizontal. In order to achieve this each angle has to be measured. This can be done in a number of ways.

A rotary encoder can be used to measure the rotation of the motor shaft and have the motor adjust accordingly [M. Ruderman, J. Krettek, F. Homann, T. Bertram, 2008]. Another option is to use a device capable of measuring the platforms current angle, such as a gyroscope and accelerometer combination. This also enables the system to cope with the mounting (base) platform being tilted, unlike the encoder solution which can only adjust when the motor is rotated as pictured in figure 2.1.

Figure 2.1: a) Platform not adjusting when mount is tilted b) Platform adjusting when mount is tilted, Drawn in:[Microsoft Powerpoint 2016]

(20)

CHAPTER 2. THEORY

2.2 Inertial measurement unit

An IMU of the scale used for this project often consists of a combination of three accelerometers and three gyroscopes placed orthogonally to each other, representing a coordinate system. An accelerometer measures G-forces, or inertial acceleration.

A gyroscope measures its rotational position in relation to an arbitrarily chosen coordinate system. The reason for using both a gyroscope and accelerometer becomes clear when considering their weaknesses. A gyroscope will have an accumulating error and can as such often not be used alone while an accelerometer has a problem with the gravity component. These two are fortunately easily combined and will compensate each others weaknesses very well [Gabor Paller, 2011].

2.3 Pulse Width Modulation

Pulse Width Modulation (PWM) allows for digital signals to appear to be analogue.

A digital signal is either a zero or a one, while an analogue signal has a larger spectrum. Rather than having one output channel for each voltage, a PWM signal is used [Johansson, 2013a]. The PWM signal is a pulse of ones and zeros allowing the voltage to be set to a percentage of the maximum output voltage of the used channel. A PWM signal is usually between 0 and 255, where 0 is 0% of the available output voltage and 255 is 100%. Over a pulse, a signal of 127 will result in the signal sent being one for half of the period, and zero for the rest. This will be interpreted by the motor as a voltage of half the maximum voltage. Presented in the figure 2.2 is a system with output voltage of 5V, where for example the 25% duty cycle results in an output signal of 1.25V. PWM is thus normally used to control the rotation speed of DC motors, which also is the case in this project.

Figure 2.2: PWM signal in an Arduino, Source:[Arduino.cc, 2016c]

6

(21)

2.4. PID CONTROLLER

Further, the frequency of the pulse can be adjusted depending on the requirements of the system. Typical frequency is 490Hz [Arduino.cc, 2016a], which is audible to the human ear. By increasing the PWM frequency, the noise can be removed.

2.4 PID controller

The PID control system consists of a proportional (P) part, integral (I) part and derivative (D) part, and is a very common feedback control system [Lee Payne, 2014]. It is commonly applied to systems that can be mathematically modelled and have a one to one input-output relation.

The proportional part is simple; the error signal is proportional to the control signal. If the error is big, so is the control signal. The problem with this is that as the error gets smaller so does the control signal and the error will therefore never be fully eliminated (theoretically), this is called a steady state error. In order to reduce this error, the proportional constant can be increased, though this will reduce the stability of the system.

To properly eliminate the steady state error an integral part is required. The integral part will historically register the error between the reference and output signals and increase the input signal accordingly. This can however also result in decreased stability, why a derivative part is introduced [Paul Avery, 2009].

The derivative part can increase stability by predicting the movement of the system, the prediction is however then clearly sensitive to disturbances and measurement noise. As such, signal filtering becomes more important.

Conclusively, the PID will be the sum of the above presented three parts as u(t) = K_pe(t) + K_I

∫ _t

0

e(τ ) dτ + K_D (d

dte(t) )

(2.1) where K_P, K_I and K_D are the tuning parameters for proportional, integral and derivative control respectively, and e(t) is the error between reference signal and output signal.

(22)

CHAPTER 2. THEORY

2.5 DC Motor

Direct Current motor, commonly known as DC-motor, is as the name suggests an electrical motor powered by DC. The most common DC-motors have an odd number of poles, where the ingoing current direction determines the outgoing rotational direction.

Figure 2.3: Brushed DC-motor, Source:[Zgcmotor.com, 2015]

Small models often consist of a permanent magnet as stator and poles connected to the rotor made up of coils that create a magnetic field [Giorgos Lazaridis, 2010].

These electromagnetic fields change direction throughout one cycle. As seen in figure 2.3 the brushes are in contact with the commutator, which leads the current to diﬀerent coils depending of the rotors orientation.

Figure 2.4: DC motor circuit, Source:[ctms.engin.umich.edu, 2016]

8

(23)

2.6. KALMAN FILTER

The electrical circuit of a DC motor can be modelled as depicted in figure 2.4.

Applying Kirchoﬀ’s law of voltage [Johansson, 2013b] results in V − eEM F = L

(di dt

)

+ Ri (2.2)

where V is the voltage, e_{EM F} is the back EMF, L is the inductance, i is the current and R is the circuit resistance. Furthermore, the back EMF can be written as

e_{EM F} = K_eθ˙ (2.3)

with Ke being the motor voltage constant and ˙θ the angular velocity. From these equations the relation between voltage and angular velocity can be derived. Since the angular velocity is proportional to the ingoing voltage, a DC motor is easily controlled using Pulse-Width Modulation (PWM) and a H-bridge, and as such well suited for the task at hand.

2.6 Kalman filter

The angle readings from an IMU carry much noise, which results in them requiring filtering before a reasonably true angle can be measured. Without filtering, the variations in the measured angle may result in deceiving a controller in to reacting to untrue movement. To remove these noisy readings a Kalman-filter, first devel- oped by NASA in collaboration with Dr. R. Kalman for the Apollo program [L.A.

McGee, S.F. Schmidt, 1985], is a good choice. The Kalman-filters main advantages are the quality of its estimation and relatively low complexity [Zaknich, 2005]. A disadvantage is that it only works for gaussian linear systems, for non-gaussian and non-linear systems a particle filter would be preferable [M.S. Arulampalam, S.

Maskell, N. Gordon, T. Clapp, 2002].

For most systems it is possible to calculate an expected outcome of a specific action, in other words a prediction for the sample space of the next state. When the prediction is made and the next state is initiated, noisy readings from the sensor gives another sample space. These two combined give a good approximation of the systems real state [Ramsey Faragher, 2012]. The algorithm is recursive, which narrows the sample space continuously. This is the basic thought of the method called Kalman-filtering and is visualized in figure 2.5. The fact that the Kalman filter does not require any history other than its previous state makes it well suited for continuously changing systems such as a self-stabilizing platform [Tim Babb, 2015].

(24)

CHAPTER 2. THEORY

Figure 2.5: The Kalman cycle, Source:[Codeproject.com, 2016]

This method would not be very useful if it were not for the key properties of the Gaussian distribution. The product of two or more distributions returns another Gaussian distribution with a new mean value and variance [Tim Babb, 2015]. As mentioned above the required information has already been obtained, in the form of predictions made from the previous state and measurements from the current state. Both are Gaussian distributed, and together the current states sample space narrows.

10

(25)

Chapter 3

Demonstrator

In this chapter the design of the prototype is described with further details on each component.

3.1 Problem formulation

A number of aspects had to be considered in the construction of the demonstrator. Firstly, in order to achieve a stable system, excessive moment of inertia on the motor shaft had to be reduced. This ruled out a number of otherwise interesting construction ideas that could have been realized by using stronger and faster systems, that in turn would require more resources. Secondly, the platform needs to be able to move fairly freely in both axes. Lastly, the IMU had to be mounted on the top plate in order to provide just measurements of the platforms position and movement.

3.2 Software

The software consists of the IMU signal analysis and a Kalman filter, followed by PID controllers for each motor. The raw IMU data is pulled through the Kalman filter in order to obtain a more stable signal, each filtered angle value is then compared to the reference value, zero, followed by a PID controller. The direction of the motor depends on the output signal from the PID controller. After this comparison is made, the output signal is sent along with the direction information to the H- bridge, which then results in the motors behaving accordingly. This is graphically presented in figure 3.1.

(26)

CHAPTER 3. DEMONSTRATOR

Figure 3.1: Programming flowchart, Drawn in:[draw.io]

3.3 Electronics

The components presented in this section are discussed individually.

3.3.1 DC Motor

The motors used in this project are Faulhaber 2842S006C with an output power of 5,33 W each. They are equipped with gears of a ratio 14:1, significantly increasing the torque at the cost of rotation speed. Considering the rotation speed pre-gearing is 5100rpm, the geared motor will be fast enough for the provided purpose. Table 3.1 contains further key information from the motor data sheet [MicroMo Electronics Inc., 2016].

12

(27)

3.3. ELECTRONICS

Table 3.1: Faulhaber 2842S006C specifications

Denotation Value Source

Nominal voltage VN 6 [V ] Datasheet

Terminal resistance R 1.6 [Ω] Datasheet

Induction L 145 [µH] Datasheet

Back EMF constant K_e 1.150 [mV /rpm] Datasheet

Torque constant K_t 10.9 [mN m/A] Datasheet

Rotor inertia J_r 9.7026· 10⁻⁷ [kg· m²] Datasheet Inertia motor 1 J₁ 2.538· 10⁻³ [k· gm²] Solid Edge Inertia motor 2 J₂ 2.797· 10⁻³ [kg· m²] Solid Edge

Dynamic friction constant b 9.8382· 10⁻⁴ [N m· s] Calculated (appendix A.1)

3.3.2 Micro-controller - Arduino NANO

The Arduino NANO is a smaller version of Arduino UNO, having largely the same specifications. The NANO was chosen over other Arduino designs for its size, see figure 3.2.

Figure 3.2: The Arduino NANO, Source:[etechpk.net, 2016]

The Arduino is an open source based platform designed for simplicity. The NANO is provided with a ATmega328 micro-controller, which can be called its brain. Fur- thermore, the NANO is equipped with eight analog pins, 14 digital pins, of which six are PWM enabled, and a mini-USB port for communicating and programming. All Arduino models are programmed with included software, which is based on C/C++

(28)

The PWM ports on the Arduino are set up in pairs and have diﬀerent frequen- cies at which the signal is sent [Arduino.cc, 2016a]. Too low a frequency will impair the performance of the system, and the fluctuating voltage will even be audible to the human ear. The default setting of the NANO was therefore modified in order to increase the frequency of used ports [Arduino.cc, 2016d].

3.3.3 H-bridge - L298N

An H-bridge, sometimes called motor driver, allows pole switching, which practically means changing the motors rotational direction. Its main purpose is however to allow for an external power supply to the motors. The L298N, displayed in figure 3.3, was chosen because of its capability to control two motors simultaneously, provide each motor with a maximum current of 2A and also supply 5V output to the Arduino. The motor voltage supply can handle voltages from 5 to 35 V [John Boxall, 2014].

Figure 3.3: The L298N, Source:[artofcircuits.com, 2016]

3.3.4 IMU - MPU6050

The MPU6050 is a combination of a three-axis gyroscope and a three-axis accelerometer [Sparkfun.com, 2016] and is probably the most important part of the stabilizing mechanism. It provides information of its angular position and acceleration. With its small size shown in figure 3.4 and vast Open-source library it was considered ideal for this project.

14

(29)

3.4. HARDWARE

Figure 3.4: The Sparkfun MPU6050, Source:[sparkfun.com, 2016]

3.4 Hardware

The full construction displayed in figure 3.5 consists of five main parts, the handle, motor mounts, shaft brackets, platform mount and platform. Most of these parts were 3D-printed using an Ultimaker 2 from CAD models created in Solid Edge ST7.

Figure 3.5: Main construction

(30)

The handle was ergonomically designed for optimum grip and ease of use, made out of PLA-plastic using a 3D printer. The construction was made portable by mounting the Arduino NANO and H-bridge on the handle. The four almost equal motor mounts fit motor 2 to the handle and motor 1 to motor 2. These parts were 3D-printed with an Ultimaker 2 in PLA-plastic.

To transfer the motor torque from the shaft to the motor attachments nave-like brackets were constructed. By a radially located screw these were locked on to the flat side of shaft. The axle brackets were mainly made in a turning machine.

The long arm attached to the platform was designed for simplicity and maximum angular displacement. A rectangular slot by the platform mounting point created a fitting for the IMU. The platform was made of Plexiglas and was shaped in a laser cutter. For reference, the side length was 200 mm and its thickness was 5 mm.

16

(31)

3.5. FINAL DESIGN

3.5 Final design

The demonstrator is a handheld construction and cannot stand on its own. As seen in figure 3.6 a docking block has been constructed to simplify testing and demon- stration. The block was 3D-printed using very little material, the walls were only 0.8mm thick and the inside filled to 5%.

Figure 3.6: Demonstrator

(32)

(33)

Chapter 4

Calculations

This chapter illustrates the calculations, simplifications and mechanics of the con- structions transfer function.

4.1 Transfer function

In order to determine suitable parameters for the PID controller using MATLAB PID tuner, a transfer function was required. This transfer function could then be used to simulate various tests in the same program. Briefly described, the transfer function translates the input signal to an output signal. In the current system, the input signal is the PWM signal, or voltage, and the output signal is the angle read from the IMU.

In order to construct the transfer function, a number of simplifications were made.

Firstly, the static friction of DC motors is non-linear and therefore needed to be disregarded in order for a linear transfer function to be determinable. Secondly, the angular fluctuations of the platform are considered small enough for the sinusoidal value of θ to be almost equal to θ. Lastly, the transfer function is calculated without any load on the platform.

Furthermore, the system was divided in to two subsystems, one per motor, as the mechanics of these diﬀer.

(34)

CHAPTER 4. CALCULATIONS

4.1.1 Motor 1 mechanics

Motor 1 is mounted on the shaft of motor 2. On the shaft of Motor 1, the platform is attached.

Figure 4.1: Motor 1 torque balance

Presented in figure 4.1 are the moment of inertia from the platform and its mounting J₁, motor torque T , rotation angle θ, angular velocity ˙θ and dynamic friction b.

4.1.2 Motor 2 mechanics

Motor 2 is mounted on the handle and has motor 1 attached to its shaft as shown in figure 4.2.

Figure 4.2: Motor 2 torque balance

The equation describing Motor 2 diﬀers from Motor 1 only in the moment of inertia J2.

20

(35)

4.1. TRANSFER FUNCTION

4.1.3 Resulting calculations

If the magnetic field is assumed to be constant, the motor torque can be calculated by

T = K_ti (4.1)

This results in the motor torque being proportional to the current i by the torque constant Kt [Johansson, 2013c].

Figure 4.3: Torque by gravity on platform

As shown in figure 4.2 and 4.3 the torque equation around the shaft will be calculated by

(36)

CHAPTER 4. CALCULATIONS

where ¨θ is the angular acceleration, m is the mass of the construction moved by the motor in question, g is the gravitational acceleration constant, l is the vertical distance to the centre of mass and sin(θ)≈ θ as the angle is considered small.

The moment of inertia J₁ and J₂ caused by the construction, obtained from the 3D-model, can be added to the moment of inertia from the DC-motors rotor Jr, which can be found in table 3.1. Thus, the moment of inertia is only denoted as J in this section.

Combining equations (4.1) and (4.2) results in

Kti = J ¨θ + b ˙θ + mglθ (4.3) which can then be Laplace transformed along with a combination of equation (2.2) and (2.3) resulting in

KtI(s) = bsθ(s) + J s²θ(s) + mglθ(s) (4.4) V (s) = LsI(s) + RI(s) + K_esθ(s). (4.5) By combining equation (4.4) and (4.5), eliminating I(s) the following equation is given

V (s) =

[(Ls + R)(bs + J s²+ mgl)

K_t + Kes

]

θ(s) (4.6)

with which the transfer function can be identified. With the input signal being the voltage V and the output signal being the angle θ the transfer function is

G(s) = θ(s)

V (s) = K_t

(Ls + R)(bs + J s²+ mgl) + KeKts. (4.7) Each motor will have its own transfer function, due to the diﬀerence in moment of inertia as mentioned previously.

22

(37)

4.2. SYSTEM REQUIREMENTS

4.2 System requirements

In order for the system to be able to cope with the torque caused by the platform, the center of mass must be in line with the rotational axes. To achieve this despite disturbances, a fast system is required. Translated to control science, the rise time must be low. This causes the overshoot to increase, which in turn must be limited.

Lastly, the steady state error must be small for the platform to be considered level.

As a result, requirements were set up as presented in table 4.1 for a step of 10°.

Table 4.1: System requirements

Requirement Value Risetime < 1 s Overshoot < 10 % Settling time < 3 s Steady state error < 2°

(38)

(39)

Chapter 5

Test and results

In this chapter the methodology surrounding the tests is explained, followed by a presentation of the results.

5.1 Tests

Tests are required to answer the research questions, as well as to determine the performance of and fine-tune the PID parameters. Further, a test was performed to identify the signal noise post filtering.

5.1.1 Signal filtering

The signal noise reduction achieved by Kalman filtering had to be determined. The IMU was placed in a locked position and the signal was read over a period of time.

The raw angular data was compared to the filtered angle data as well as the mean of the raw data for reference.

5.1.2 Step response

When testing a system of this character step response tests are commonly used.

A step response test is conducted by allowing the system to enter a steady state, upon which a change in the reference signal is made. In the case of a stabilizing platform, the test should monitor performance when a disturbance occurs. This can be modelled by varying the reference signal by a set amount and recording the time and angle deviation. All tests were repeated multiple times to assure reliable results, and a test that closely corresponded with the mean values of all tests was selected for each result.

A number of tests were conducted with the PID parameters determined and applied to the model. The angle was set to 10° in one or both axes after which is was reset to 0°, to monitor the step response. Both the setting and removal of the reference

(40)

CHAPTER 5. TEST AND RESULTS

In order to investigate the reliability of the PID controller when the characteris- tics of the system changes, a weight of 200g was placed on the platform and the tests were repeated without adjusting the controller parameters. The weight was then increased in order to investigate the limits of the system.

5.2 Results

With the help of the MATLAB Simulink PID Tuner, a set of suitable parameters could be determined. These were further fine-tuned on the demonstrator to achieve performance that satisfy the system requirements presented in section 4.2. The final values were as presented in table 5.1. It was also found that weights above 290g will prevent the system from eliminating the steady state error by causing too large amounts of torque.

Table 5.1: Chosen system parameters

Parameter Controller motor 1 Controller motor 2

K_P 18.94 15.94

K_I 0.54 0.58

K_D 25.48 25.65

26

(41)

5.2. RESULTS

A number of identical tests provided a mean value for the raw angle. The filtered angle and raw data from one test was then compared to this as presented in figure 5.1.

0 2000 4000 6000 8000 10000

Time (ms) -3.4

-3.2 -3 -2.8 -2.6 -2.4 -2.2

Angle (deg)

Kalman Raw Mean raw

Figure 5.1: Signal readings steady IMU

The data was sampled at a rate of 5ms. The raw data peaked at± 0.57°, while the Kalman filtered value only had a deviation of ± 0.12°.

(42)

5.2.2 Step response

The results from the step response tests were as shown below.

0 1000 2000 3000 4000

Time (ms) -0.5

0 0.5 1 1.5

Amplitude

Motor 2 clockwise Motor 2 counterclockwise

(a) X

0 1000 2000 3000 4000

Time (ms) -0.5

0 0.5 1 1.5

Amplitude

(b) Y

0 1000 2000 3000 4000

Time (ms) -0.5

0 0.5 1 1.5

Amplitude

Motor 1 clockwise Motor 1 counterclockwise Motor 2 clockwise Motor 2 counterclockwise

(c) X and Y

Figure 5.2: Step response without load

Figure 5.2 shows results from the tests without additional load on the platform.

Rise time ranges from 60 to 103ms, settling time was around 1300ms and the overshoot was roughly 16% when tilted in one axis, 30% when in two. The steady state error was between 0.07 and 0.34°.

28

(43)

5.2. RESULTS

After placing the weight on the platform, the performance was as shown in figure 5.3.

0 1000 2000 3000 4000

Time (ms) -0.5

0 0.5 1 1.5

Amplitude

(a) X

0 1000 2000 3000 4000

Time (ms) -0.5

0 0.5 1 1.5

Amplitude

(b) Y

0 1000 2000 3000 4000

Time (ms) -0.5

0 0.5 1 1.5

Amplitude

Motor 1 clockwise Motor 1 counterclockwise Motor 2 clockwise Motor 2 counterclockwise

(c) X and Y

Figure 5.3: Step response with 200g load

While the rise time remained around 100ms, overshoot increased to around 40% for all scenarios, and settling time came close to 3 seconds with a steady state error of 0.2°to 0.8°.

(44)

An overview of the key values from both simulations and demonstrator tests are shown in table 5.2.

Table 5.2: System performance

Motor 1 Motor 2 Requirement Risetime 0.103 s 0.085 s < 1 s

Overshoot 18.7% 15% < 10 %

Settling time 1.345 s 1.104 s < 3 s Steady state error 0.07° 0.22° < 2°

30

(45)

Chapter 6

Discussion and conclusions

This chapter consists of discussions concerning the results obtained in the various parts of the project, followed by conclusions.

6.1 Discussion

The discussion is held in regard to achieving a functional system and obtaining a just answer to the research question.

6.1.1 Control science

Since the controller was designed for a system without load, performance was expected to drop. This could be tuned further in order to achieve a less sensitive system, though that would much likely be at the cost of the original systems performance.

Higher performance could be obtained by discretizing the controllers and thus matching the measurement frequency with the PID output frequency, rather than using a delay which freezes the program for a short period. This would however require a significantly higher system complexity.

The overshoot was higher than the specified requirements all scenarios, while the speed requirements were met with much margin. Thus, by lowering the speed a more stable system and preferable performance could be achieved.

6.1.2 Tests

The tests were satisfactory in the aspect of analyzing the system, however since the test data was retrieved from the IMU, reliability can be questioned. The IMU readings tend to spike upon sudden movements, resulting in untrue readings. The test results with load were also not ideal due to the performance without load not

(46)

CHAPTER 6. DISCUSSION AND CONCLUSIONS

As shown in section 5.2.1 the Kalman filter reduces signal noise by a factor of 5.

This is a significant diﬀerence and greatly improves the performance of the controller. Had the noise been greater, a low-pass filter may have been required for the derivative part of the controller to stabilize. As the Kalman filter is anticipative, untrue readings may appear momentarily upon sudden accelerations, causing trou- ble when testing. This too was preferred to the higher complexity or computational strength required by other options.

6.1.4 Error sources

When determining the transfer function, a number of simplifications were made.

Firstly, the static friction of the DC motors was disregarded from, something that would have a noticeable impact on performance at small angles. This could be ac- counted for by applying a non-linear model, which would also allow for the torque equation to be corrected, where the sinusoidal function was simplified.

Further error sources worth mentioning are the approximations made when determining the dynamic friction constant and the moment of inertia for both motors.

The data sheet for the motors also specifies that the coil resistance can vary by

±12%. There is also a small, barely noticeable, step in the gear which will have impact on performance when changing directions.

As mentioned in section 6.1.3, the readings from the IMU may not always rep- resent the actual angle, especially when there is large and sudden overshoot. As it was not certain that the values were exact, slightly rounded values were presented in the results.

Having assumed small angles also resulted in the mathematical model not being very accurate at larger errors, and the system would therefore not have the expected performance.

Lastly, the motors have diﬀerent current to angular velocity ratios depending on the rotational direction. This resulted in there being larger overshoot when starting in the direction where the motors were stronger, and a slower system when in the opposite direction.

32

(47)

6.2. CONCLUSIONS

6.2 Conclusions

The purpose of this report was to answer a research question.

How will a PID-controlled system react to variations in load?

The system remained stable despite the load being increased, as seen in chapter 5.2, though performance decreased. Overshoot increased greatly, explained by the increased moment of inertia and torque requirement. The impact on performance is greater when the system moves in both directions compared to when limited to one direction. By optimizing the controller parameters, a less sensitive system can be achieved.

(48)

(49)

Chapter 7

Recommendations and future work

In this section a few recommendations for the project are presented, along with ideas for future work.

7.1 Recommendations

Since the transfer function determined in this project was limited to be linear, fairly large simplifications were made. These are relevant to the performance of the system, as such it would be recommended to create a non-linear model in MATLAB Simulink. This would improve controller tuning abilities greatly.

A higher quality IMU could improve readings, and a motor without gears would remove the step. For anyone interested in the subject the Open-Source community related to Arduino can provide examples of code for signal filtering, IMU readings, PID control.

7.2 Future work

The assertion in section 4.1.3 that the angle is small causes instability when it is not, this can be resolved by implementing defensive and aggressive parameters that will switch depending on the platforms current angle, with defensive parameters at larger angles.

Performance can also be improved by matching the PWM output frequency with the rest of the system, so that the controller calculates new signals at the same pace as the PWM sends new signals. This would increase accuracy.

Naturally, a lot of time can be spent improving the controller parameters for both controllers by trial and error.

(50)

(51)

Bibliography

[Arduino.cc, 2016a] Arduino.cc (2016a). analogWrite(). Available from: https:

//www.arduino.cc/en/Reference/AnalogWrite [cited 2016-05-07].

[Arduino.cc, 2016b] Arduino.cc (2016b). ARDUINO NANO. Available from: http:

//www.arduino.cc/en/Main/ArduinoBoardNano [cited 2016-05-06].

[Arduino.cc, 2016c] Arduino.cc (2016c). Picture: PWM. Available from: https:

//www.arduino.cc/en/uploads/Tutorial/pwm.gif [cited 2016-06-03].

[Arduino.cc, 2016d] Arduino.cc (2016d). Secrets of Arduino PWM. Available from:

https://www.arduino.cc/en/Tutorial/SecretsOfArduinoPWM [cited 2016-05- 07].

[artofcircuits.com, 2016] artofcircuits.com (2016). Picture: L298N. Available from:

http://artofcircuits.com/wp-content/uploads/2013/10/L298N-module.

jpg [cited 2016-05-05].

[Codeproject.com, 2016] Codeproject.com (2016). Picture: Kalman-cycle. Avail- able from: http://www.codeproject.com/KB/recipes/865935/cycle.png [cited 2016-05-05].

[ctms.engin.umich.edu, 2016] ctms.engin.umich.edu (2016). Picture: DC- Circuit. Available from: http://ctms.engin.umich.edu/CTMS/Content/

MotorPosition/Simulink/Modeling/figures/motor.png [cited 2016].

[etechpk.net, 2016] etechpk.net (2016). Picture: ARDUINO NANO. Available from: http://www.etechpk.net/wp-content/uploads/2016/02/ARDUINO_

NANO_03.png [cited 2016-05-05].

[Gabor Paller, 2011] Gabor Paller (2011). Better motion control using accelerometer/gyroscope sensor fusion. Available from: http://www.slideshare.net/paller/

better-motion-control-using-accelerometergyroscope-sensor-fusion [cited 2016-05-05].

[Giorgos Lazaridis, 2010] Giorgos Lazaridis (2010). How DC Motors are made and how they work. Available from: http://pcbheaven.com/wikipages/How_DC_

(52)

BIBLIOGRAPHY

[Grober, D.E., 2009] Grober, D.E. (2009). Autonomous, self leveling, self correcting anti-motion sickness chair, bed and table. US Patent 7,490,572. Available from:

https://www.google.com/patents/US7490572 [cited 2016-05-07].

[Johansson, 2013a] Johansson, H. (2013a). Elektroteknik. In Johansson, H., editor, Elektroteknik, page 7:45, Stockholm, SWE. Universitets service AB.

[Johansson, 2013b] Johansson, H. (2013b). Elektroteknik. In Johansson, H., editor, Elektroteknik, page 1:16, Stockholm, SWE. Universitets service AB.

[Johansson, 2013c] Johansson, H. (2013c). Elektroteknik. In Johansson, H., editor, Elektroteknik, pages 7:12–7:13, Stockholm, SWE. Universitets service AB.

[John Boxall, 2014] John Boxall (2014). Tutorial - L298N Dual Motor Controller Module 2A and Arduino. Available from: http://tronixlabs.com.au/news/

tutorial-l298n-dual-motor-controller-module-2a-and-arduino/ [cited 2016-05-02].

[L.A. McGee, S.F. Schmidt, 1985] L.A. McGee, S.F. Schmidt (1985). Discov- ery of the Kalman Filter as a Practical Tool for Aerospace and Industyr.

Available from: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/

19860003843.pdf [cited 2016-05-03].

[Lee Payne, 2014] Lee Payne (2014). The Modern Industrial Workhorse: PID Con- trollers. Available from: http://www.techbriefs.com/component/content/

article/ntb/features/feature-articles/20013 [cited 2016-05-06].

[M. Ruderman, J. Krettek, F. Homann, T. Bertram, 2008] M. Ruderman, J. Kret- tek, F. Homann, T. Bertram (2008). Optimal State Space Control of DC Motor. Available from: https://www.researchgate.net/publication/

216232528_Optimal_State_Space_Control_of_DC_Motor [cited 2016-05-07].

[MicroMo Electronics Inc., 2016] MicroMo Electronics Inc. (2016). Se- ries 2842 ... C. Available from: http://www.me.mtu.edu/~wjendres/

ProductRealization1Course/Motor_Specs.pdf [cited 2016-05-04].

[M.S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, 2002] M.S. Arulampalam, S. Maskell, N. Gordon, T. Clapp (2002). Available from: http://www.cse.

psu.edu/~rtc12/CSE598G/papers/ParticleFilterTutorial.pdf [cited 2016- 05-02].

[Paul Avery, 2009] Paul Avery (2009). Introduction to PID control. Avail- able from: http://machinedesign.com/sensors/introduction-pid-control [cited 2016-05-06].

[Ramsey Faragher, 2012] Ramsey Faragher (2012). Understanding the Ba- sis of the Kalman Filter Via a Simple and Intuitive Derivation . Available from: http://www.cl.cam.ac.uk/~rmf25/papers/Understanding%

20the%20Basis%20of%20the%20Kalman%20Filter.pdf [cited 2016-04-22].

38

(53)

[Rivello, 2011] Rivello (2011). Self balancing waiter tray. (or almost it) . Avail- able from: http://forum.arduino.cc/index.php/topic,68755.0.html [cited 2016-05-06].

[Rony Chakraborty, 2014] Rony Chakraborty (2014). MPU6050 DMP with Pro- tonBasic. Available from: http://www.ekushebangla.com/DMP_MPU6050.html [cited 2016-05-06].

[S. Narkar, S. Bhalekar, T. Nawge, K. Parab, P. Pati, 2013] S. Narkar, S.

Bhalekar, T. Nawge, K. Parab, P. Pati (2013). Self Balancing Platform.

Available from: http://www.ijctee.org/NSPIRE2013/IJCTEE_0313_Special_

Issue_24.pdf [cited 2016-05-06].

[sparkfun.com, 2016] sparkfun.com (2016). Picture: MPU6050. Avail- able from: https://cdn.sparkfun.com//assets/parts/6/3/5/5/11028-04.

jpg [cited 2016-05-05].

[Sparkfun.com, 2016] Sparkfun.com (2016). SparkFun Triple Axis Accelerometer and Gyro Breakout - MPU-6050. Available from: https://www.sparkfun.com/

products/11028 [cited 2016-05-06].

[Tim Babb, 2015] Tim Babb (2015). How a kalman filter works, in pictures. Available from: http://www.bzarg.com/p/

how-a-kalman-filter-works-in-pictures/ [cited 2016-04-01].

[Zaknich, 2005] Zaknich, A. (2005). Principles of Adaptive Filters and Self-learning Systems. Springer-Verlag, Austrailia, 1nd edition.

[Zgcmotor.com, 2015] Zgcmotor.com (2015). Picture: DC-motor. Available from: http://www.zgcmotor.com/upload/Image/20150521/20150521150754_

33455.jpg [cited 2016-05-05].

(54)

Self-stabilizing platform: ZTäBiLAjZöR

Self-stabilizing platform

ZTäBiLAjZöR

ANDERS KARLSSON

JONATHAN CRESSELL

Self-stabilizing platform

Abstract

Sammanfattning

Preface

Contents

Nomenclature

Chapter 1

Introduction

Chapter 2

Theory

Chapter 3

Demonstrator

Chapter 4

Calculations

Chapter 5

Test and results

Chapter 6

Discussion and conclusions

Chapter 7

Recommendations and future work

Bibliography