Modelling and Analysis of a Screw Joint Test Rig

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Master of Science Thesis in Electrical Engineering

Department of Electrical Engineering, Linköping University, 2017

Modelling and analysis of a

screw joint test rig

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Master of Science Thesis in Electrical Engineering Modelling and analysis of a screw joint test rig

Beatrice Fallsberg LiTH-ISY-EX--17/5056--SE Supervisor: Erik Hedberg

isy, Linköpings universitet

Johan Nåsell

Atlas Copco

Examiner: Johan Löfberg

isy_{, Linköpings universitet}

Division of Automatic Control Department of Electrical Engineering

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Abstract

Today, tightening tools are widely used in the industry and on the market you will find several products that have been assembled with the help of tightening tools. For example, tightening tools are used in the automotive industry and when assembling computer hardware.

It is important that the tightening tools are robust and of high quality in or-der to fulfil the demanded requirements. High assembly speed has entailed an extensive use of tightening tools. To ensure that the tightening tools have the desired properties, tightening tools are tested continuously using so-called test systems. This puts high demands on the test systems since it is of importance that the tightening tools can be tested in a simple, fast and reliable way as well as repeatedly. Therefore, Atlas Copco would like to investigate whether a test system constructed with an electrical motor is a good choice.

The idea of this thesis is to investigate possibilities and limitations in a test system consisting of an electrical motor that emulates the behaviour of a screw joint. To be able to investigate hardware limitations a test rig is constructed and then modelled in MATLAB. Further, simulations have been carried out in order to analyse the possibilities and limitations of such a test rig.

The conclusion is that the implemented LQ controller seems to be able to control the braking motor sufficiently like a screw joint.

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Acknowledgments

First of all, I would like to thank Atlas Copco for giving me the opportunity to do this master thesis. I would like to show my gratitude to all who have supported me with knowledge and equipment during this master thesis. Especially thanks to my supervisor Johan Nåsell and the manager of the tightening technique R&D group Maria Södergren.

I also want to express my gratitude to my supervisor Erik Hedberg and my examiner Johan Löfberg at Linköpings University, for always being available for discussions.

Finally, I would like to thank my partner Goran Katinic for inspiring me and for always believing in me.

Linköping, June 2017 Beatrice Fallsberg

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Notation

Acronyms

Acronym Description

dac _{Digital to Analog Converter}

dc Direct Current

lq Linear Quadratic

Symbols

Symbol Description

w Angular velocity for the braking motor wt,i Angular velocity from the tightening tool αt,i Angular acceleration of the tightening tool

i Current to the motor

Q1 Design parameters for the controller

Q2 Design parameters for the controller

∆θ Difference in angle between the braking motor and the driving motor

β Filter parameter

Tideal Ideal torque from reference generator

v2 Measurement noise

km Motor constant

h(w) Motor friction

Jm Motor inertia

nr Number of revolutions during rundown d Prevailing torque/rundown torque erelative Relative error for the estimated models

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x Notation

Symbols

Symbol Description

R1 Resistor to the operational amplifier

Jrot Rotational inertia

G Shear modulus

ks Stiffness constant

ks,total Stiffness constant for the axis between the braking mo-tor and the tightening tool

v1 System disturbance

Vof f set The offset voltage for the operational amplifier r The radius for a cylinder

∆θr The reference signal for the controller

t2,T The time when the torque reaches the final target

torque

t1,T The time when the torque reaches the first target

torque

tsnug The time when the torque reaches the rundown torque T (θ) The friction model of the torque

T Torque at motor shaft

km,f Viscous friction constant

V1 Voltage in to the operational amplifier

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1

Introduction

1.1 Background

Atlas Copco Industrial Technique AB sells handheld tightening tools. It is impor-tant that the tightening tools have high performance and that the performance can easily be verified regularly to ensure that the screw joint possess the desired performance. Many industries have a high speed in the assembly which means that it is important that the performance of the tightening tools can be tested in a simple, fast and reliable way as well as repeatedly. The tightening tools are tested with a so-called test joint which verifies that the tightening tools have a certain performance.

Today there is an increasing need for more advanced test systems that can apply a well-defined load. Atlas Copco wants to improve their test methods in order to meet higher requirements regarding the endurance and precision of the tightening tools. A more advanced test system might be achieved if an electrical motor is used as a test joint. The test joint together with the tightening tool will be a system consisting of two motors working against each other, which makes it important that the system is stable as well as that the test joint emulates a screw joint.

1.2 Problem formulation

The purpose of this thesis is to investigate the possibilities and the limitations regarding the hardware limitations and the control structure for a test rig that emulates a screw joint. To analyse the possibilities and limitations, a small minia-ture test rig is constructed consisting of a braking and a driving motor. The test rig will be modelled as well as used to identify hardware restrictions. When the test rig is modelled a control strategy for the braking motor needs to be derived

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2 1 Introduction

for the braking motor to emulate a screw joint.

If the results show good potential, then Atlas Copco has a basis for future work.

1.3 Method

To achieve the objectives of this thesis a miniature test rig is constructed. The lim-itations and the dynamics of the test rig will be modelled and together with the control structure for the braking motor it will result in a simulation environment where the analysis of the screw joint test rig can be carried out.

1.4 Limitations

This thesis stretches over a limited period of time and therefore the driving motor is modelled as a filter and a predefined angular velocity. It would be interesting to model the driving motor as a control system to evaluate the performance of the test rig when the braking and the driving motor are working against each other. However, this will not be done in this thesis, due to the limited time span.

1.5 Outline

This thesis is organized in several chapters. Chapter 1 presents the background and the problem formulation as well as related work. Chapter 2 is a description of the system analysed in this thesis. Chapter 3 presents the theoretical background needed for the modelling of the test rig as well as the modelling of the screw joint torque. Chapter 4 describes the theory needed to develop the control structure for the test rig. In chapter 5 results from the simulations and the analysis of the test rig are presented. Chapter 6 contains the conclusions drawn from the results presented in chapter 5 and future work in the area covered by this thesis.

1.6 Related work

Electrical motors are common in many industrial applications, for example in assembly systems described in Lagerholm and Molinder (2014) and in an electro-hydraulic system used in André (2014). Electrical motors are also used in test systems as in Žalman and Macko (2005), where an electromotor is used for em-ulation of mechanical loads. In Žalman and Macko (2005) a load emulator is designed and implemented with the goal that users of the emulator easily should be able to verify the design of the control algorithm by choosing desired type of load and parameters of the emulator. In a study by Kyslan andĎurovský (2012) a dynamometer is emulating the behaviour of mechanical loads to test new con-trol algorithms of machines. Kyslan andĎurovský (2012) develop an advanced torque controller for the dynamometer where the controller contains the model

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1.6 Related work 3

of the emulated load. In Wang et al. (2016) an electrical load simulator is devel-oped with improved dynamic performance and stability.

Atlas Copco has had a previous Master’s thesis, Griph (2016), that investi-gated and evaluated if it is possible to control an electrical motor emulating a screw joint. In Griph (2016) several controllers were developed and it was cluded that the LQ controller had best performance compared to the other con-trollers developed in that thesis. Therefore, an LQ controller will be developed for the test rig in this thesis.

In this thesis two electrical direct current motors will be used. A simple model of a DC motor is described in Glad and Ljung (2004). The drivers for the DC motors used in this thesis are modelled as black box models also described in Glad and Ljung (2004). For the modelling of the screw joint, knowledge about a screw joint is needed. In Swedish Fasteners Network (2017) information about a screw joint is presented as well as in the Master’s thesis Griph (2016).

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2

System description

The system consists of two DC motors, one driving DC motor and one braking DC motor. For each of these DC motors there is a driver that can be configured as either a current controller or a speed controller. The DC motors used are of type Maxon DC motor - RE40 - 218008 and the drivers are of type ESCON 10/70. An Arduino Due is used as a signal generator in order to model the test rig and to identify limitations of the hardware and the test rig. A support package for Arduino and Simulink is used and therefore the control structure is implemented in Simulink. Figure 2.1 illustrates an overview some of the components that the test rig is built up with. Figure 2.2 illustrates a setup of the test rig.

Figure 2.1:Block diagram illustrating the test rig.

The driving motor is supposed to represent the tightening tool and therefore the driver for the driving motor is configured to be a speed controller because there is a speed controller in the tightening tool that the test rig is built for. The braking motor will emulate a screw joint and as studied in Griph (2016) a LQ controller was able to control a motor sufficiently like a screw joint. Therefore a LQ controller will be implemented to control the braking motor. The driver for the braking motor will be configured as a current controller and the motors will be connected with a torque sensor to be able to measure the torque of the shaft

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6 2 System description

between the motors.

Figure 2.2:The picture illustrates the setup of the test rig.

2.1 Tightening tool

There are several different tightening tools that are suitable for different appli-cations. Figure 2.3 illustrates a battery driven tightening tool from Atlas Copco. Often a tightening tool can be configured to different tightening algorithms. In this thesis a two-step tightening algorithm is implemented for the tightening tool. The two-step tightening algorithm is illustrated in Figure 2.4. The tightening algorithm contains two parts, one rundown part and one tightening part. These parts together are called the tightening process. In the rundown part the screw joint is placed in its position. Once the screw joint is placed, the tightening part begins. This transition between the rundown and the tightening part is called snug.

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2.1 Tightening tool 7

Figure 2.3:The picture illustrates a battery driven tightening tool from Atlas Copco. Image: Atlas Copco.

The two-step tightening algorithm is a torque feedback, which means that the times tsnug, t1,T and t2,T, shown in Figure 2.4, are not known in the beginning of

the tightening process. The times corresponds to when the torque has reached different values. During the rundown part the torque is constant and when the tightening part begins the torque is linearly increasing, see Figure 2.5. The time tsnug and the angle θsnug are defined as when the torque becomes non-constant. t1,T and θ1,T correspond to when the torque reaches the first target torque. t2,T

and θ2,T corresponds to when the torque reaches the final target torque.

Once the torque reaches a predefined threshold, θsnug is attained, see Fig-ure 2.5. When tsnug is reached, the angular velocity will start ramping down to first speed, see Figure 2.4. During the ramp down to first speed and during the time it stays constant at first speed the torque will start increasing as a linear function. When the first torque target is reached, the angular velocity will start ramping down to zero angular velocity. After a while the angular velocity will start ramping up to second speed and once the second speed is reached it will be held constant until the final torque target is reached. When the final torque target is reached, the angular velocity will ramp down as fast as possibly to zero angular velocity. During the last ramp down the torque will go over the final torque target, hence with this algorithm there will be an overshoot in the torque.

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Figure 2.4:Illustration of the angular velocity profile for the tightening tool. When the tightening tool is ramping up or holding a constant angular velocity the controller for the tightening tool is a speed controller, whereas when the tight-ening tool is ramping down the controller is switching to a current controller to be able to ramp down as fast as possible. The switch to a current controller during ramp downs is not modelled in this thesis.

Figure 2.5:The plot illustrates the screw joint torque.

2.2 Screw joint simulator

The screw joint simulator consists of the braking motor, its driver and its con-troller. The aim for the controller in the screw joint simulator is to control the

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2.3 Hardware 9

braking motor so it emulates a screw joint. If one considers the braking motor and its control system as a system of its own, the tightening tool’s angular veloc-ity acts as a system disturbance. Therefore, the angular velocveloc-ity from the tighten-ing tool is modelled as system disturbance in the LQ controller. This enables the LQ controller to control the braking motor regardless of the influence from the tightening tool.

The control system for the braking motor consists of a LQ controller, a ref-erence generator and an observer. The observer’s task is to estimate the system states used in the LQ controller. The task for the reference generator is to gener-ate the reference signal for the braking motor to follow.

The objective for the LQ controller is to control the braking motor so the torque between the braking motor and the driving motor emulates the torque of a screw joint. Therefore, the reference generator contains a function of an ideal screw joint torque. This means that the reference generator provides a reference signal for the braking motor in order for it to mimic the ideal screw joint torque.

2.3 Hardware

An Arduino Due is used as a signal generator and a DEWE 43 is used for collec-tion of data for modelling as well as monitoring internal signals. The reference signal to the drivers is a 12-bit differential voltage between -10V and 10V and the Arduino Due has two analog 12-bit outputs between 0.5V and 2.75V. In or-der to use a larger voltage range an operational amplifier will be designed and implemented as an external electrical interface for the Arduino Due. More infor-mation about each subsystem is provided below.

2.3.1 DC - motor

The motors used in this thesis are Maxon DC motors with model number RE40 -218008. It is a brushed motor that can apply a continuous torque of 0.18Nm. In this thesis the control signal will be the current to the motor. The current to the motor is limited to 2.6A in continuous drive while the stall current is 28A. The terminal constant is 41.5 seconds and since the tightening process is so short in comparison with the terminal time constant, the target torque of the tightening algorithm will be 0.6Nm. Figure 2.6 shows the motors used in this test rig.

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Figure 2.6:The picture illustrates the two motors and the axis between them. The motors are connected via the torque sensor.

2.3.2 Escon driver

The drivers that are used can be configured to have a set value controlled by a PWM signal or an analog signal. For the study in this thesis the driver for both the braking and the driving motor will be configured to have analog set values. The analog input for the drivers is a 12-bit signal with voltage range between −_{10V and 10 V. While analysing the driver it emerged that the analog set value} has a resolution of 5.6mV, one can then count backwards and conclude that its resolution of 12-bits corresponds to a voltage range of 25V. Because the designed operational amplifier only provides a voltage range of 0.37V and 9.05V the analog set value is limited to that voltage range and therefore the 12-bit resolution is not obtained. Figure 2.7 shows the driver used in the test rig.

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2.3 Hardware 11

Figure 2.7:The picture illustrates the driver for the DC motor.

2.3.3 Arduino board

The Arduino Due has two 12-bit analog outputs with a digital-to-analog con-verter (DAC) built in. The DAC outputs can deliver voltage between 0.55V and 2.75V. With Simulink hardware support for Arduino a program in Simulink is constructed where the resolution is set to 12-bit. In the Simulink hardware sup-port for Arduino there is an output block for the DAC output which takes val-ues between 0 and 4095, where 0 corresponds to voltage 0.55V and 4095 corre-sponds to voltage 2.75V on the DAC output. This gives the resolution in voltage (2.75 − 0.55)/4096 = 0.54mV from the DAC output. There is also an enable pin in the Simulink program in order to enable the driver. Figure 2.8 shows the program used to generate the signals.

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To find out the maximum rate at which this program can be run on the proces-sor in the Arduino Due, an overrun signal is monitored while a sinusoidal signal or a white noise signal is generated. Starting at a large step length and decreasing the step length, it was concluded that the maximal sampling rate for the Simulink program should be 10kHz.

2.3.4 Electrical interface - operational amplifier

The Escon driver can be configured to be controlled by a 12-bit analog set value between −10 and 10 V differential. An operational amplifier has been designed and implemented. The input to the operational amplifier is the output from the DAC on the Arduino Due which has the voltage interval 0.55V and 2.75V. The resulting output from the operational amplifier has a voltage range of 0.37V and 9.05V. The resolution from the operational amplifier is thereby (9.05−0.35)/4096 = 2.12mV. Figure 2.9 shows the Arduino Due and the constructed external electri-cal interface.

Figure 2.9: The picture illustrates the Arduino Due with the external elec-trical interface.

Operational amplifier

Operational amplifiers are used in analog electronics to amplify signals and to adjust their output range. Figure 2.10 shows the symbol for an operational am-plifier together with its inputs and outputs.

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2.3 Hardware 13

−

+

Figure 2.10:Operational amplifier

In Figure 2.10 there are two inputs, one inverted input (-) and one non-inverted input (+) as well as one output. There are also two inputs for the power supply but they are not shown in the figure. To obtain the desired gain and offset several resistances are used. Figure 2.11 illustrates an electrical schematic that gives the static gain and offset.

Vout= − R2 R1V1 + R4 R3+ R4 (1 +R2 R1 )Vof f set (2.1) − + V1 R1 R4 R3 Vof f set R2 C

Figure 2.11:Operational amplifier with resistors.

The amplifier that is constructed in this thesis has the resistor values shown in table 2.1 that results in the output voltage described by the Equation (2.2), C is chosen to 0.068F and Vof f set= 3.3V.

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Table 2.1: Table with the resistor values that are used in the operational amplifier. Resistance value [kΩ] R1 12 R2 47 R3 100 R4 220

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3

Modelling

There are several methods for modelling the dynamics of a system. In this thesis two relatively common ways of modelling dynamic systems are used, black box modelling and physical modelling.

3.1 Modelling of the drivers

The drivers are modelled as black box models. The following sections presents the theory of black box modelling, followed by the procedure that was applied when modelling the drivers. Both drivers run at a frequency of 2.3kHz.

3.1.1 Black box model

In this thesis a black box refers to a system where only the system’s input and out-put are observed, the dynamics in between are illustrated with a black box, see 3.1. The dynamics between the input and the output of a system can be described both as a linear or nonlinear discrete or continuous differential equation. Most systems contain a combination of linear and nonlinear dynamics but it is usu-ally enough to approximate the dynamics of the system with a linear differential equation.

Black box

u y

Figure 3.1:Block diagram of a black box model.

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16 3 Modelling

y(t + nt) + a1y(t − (n − 1)T ) + . . . +any(t) =

b0u(t + nT )+b1u(t + (n − 1)T ) + . . . + bnu(t)

(3.1) y(t) is the output, u(t) is the input, a1, ..., anand b0, ..., bkare constant parame-ters and T is the sampling interval.

The dynamics illustrated with the black box can for example be estimated and presented as a transfer function, see (3.2) where y(t) = G(q)u(t).

G(q) = b0q n_{+ b} 1qn−1+ . . . + bn qn_{+ a} 1qn−1+ . . . + an (3.2) One type of black box model is the output error model. The output error model does not describe the disturbance characteristics separately from the dy-namics of the system, see Figure 3.2. The estimation of the parameters a1, ..., an and b0, ..., bkwill be done in the system identification toolbox in MATLAB. When estimating a black box model in the system identification toolbox the transfer function is described as follows:

y(t) = B

Fu(t − nk) + e(t) (3.3)

The design parameters are the orders [nb, nf , nk], where nb+ 1 is the order of the B polynomial, nf is the order of the F polynomial and nk is the time delay expressed in number of samples. The input and output data and the sampling rate that they were collected with are required for the estimation. In the follow-ing sections information about the data collection, needed for the modellfollow-ing, is presented.

B F

e

u y

Figure 3.2:Block diagram illustrating an output error model.

3.1.2 Data collection

When modelling a system it is important to choose a suitable input signal to the system, so that all dynamics of the system can be observed in the output signal. Therefore, it is important to know the bandwidth of the system in order for the input signal to contain higher frequencies than the bandwidth. It is also necessary to use a suitable sampling rate for the Simulink program so that the processor in the Arduino Due has time to run all its instructions. For the system in this thesis a suitable sampling rate for the Arduino Due is 10kHz, see Section 2.3.3.

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3.1 Modelling of the drivers 17

3.1.3 Determining the bandwidth

The drivers for the braking and the driving motors will be modelled as black box models. Since the driver of the braking motor is configured as a current controller while the driver of the driving motor is configured as a speed controller, they will have different bandwidths. The definition of the bandwidth can be found in Glad and Ljung (2006) and is when the amplitude of the signal has decreased by a factor of 1/

√

2. The bandwidth was determined by using a sine signal, starting at a low frequency and then increasing the frequency until the amplitude had decreased by a factor of 1/

√

2. This procedure resulted in a bandwidth of 400Hz for the current controller and 8.5Hz for the speed controller.

3.1.4 Input signal for modelling

When collecting data for the modelling of the drivers it is important to be aware of the resolution of the signals. The resolution of the analog signal from the operational amplifier is 2.12mV and the observed resolution to the analog input port at the driver is 5.6mV. This means that the signal needs 0.0056/0.00212 = 2.64 steps out from the operational amplifier to obtain one step at the input port at the driver. This means that the input signal for the modelling must be larger than 5.6mV. Each digital step corresponds to 2.12mV out from the operational amplifier which results in a decrease of digital steps that the input port of the driver can react to. Consequently, it is possible to take 776 steps clockwise or counter-clockwise. When collecting data for the modelling the drivers a band limited white Gaussian noise is used. Due to the bandwidth of the controllers, the bandwidth of the white Gaussian noise is set to 1.4kHz and 100Hz for the current and the speed controller respectively.

3.1.5 Model validation

Models are subject to different performance requirements depending on what they will be used for. In this thesis, the purpose of the models of the drivers is to obtain knowledge of how their dynamics affect the performance of the test rig. It is enough that the models describe the dynamics of the drivers adequately, hence the relative error in the simulated output signal will be studied, see Equation (3.4).

erelative(t) =

y(t) − ˆy(t)

y(t) (3.4)

Where erelative(t) is the relative error, y(t) is the output signal and ˆy(t) is the esti-mated output signal. The time discrete model can be transformed to a continuous model and then its bode diagram can be illustrated. Cross validation will be used which means that the models will not be validated with the same data that is used for the estimations.

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18 3 Modelling

3.1.6 The estimated models

Under the respective sections the estimated models and their validation are pre-sented.

Speed controller

In Equation (3.5) the estimated model of the speed controller is presented. It is estimated with input and output data collected at sampling rate 50kHz and model orders [1, 2, 6]. G(q) = 3.267 ∗ 10 −₆ q−6 1 − 1.996q−₁ + 0.996q−₂ (3.5)

Figure 3.3 shows the simulated output plotted together with the output from the real system and the relative error. Figure 3.4 is the same plot as 3.3 but zoomed in, the relative error is below 10%.

0 2 4 6 8 10 12 Sample 105 -2000 -1500 -1000 -500 0

angular velocity, [rad/s]

Speed controller, oe126

yhat y 0 2 4 6 8 10 12 Sample 105 -10 -5 0 5 10 % Relative error

(y - yhat)/y, relative error

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3.1 Modelling of the drivers 19 1.5 2 2.5 3 3.5 4 Sample 105 -2000 -1500 -1000 -500 0

Speed controller, oe126

yhat y 1.5 2 2.5 3 3.5 4 Sample 105 -10 -5 0 5 10 % Relative error

Figure 3.4:Simulation output and the relative error, zoomed in.

-150 -100 -50 0 50 Magnitude (dB) 10-1 100 101 102 103 104 105 -1080 -720 -360 0 Phase (deg) Bode Diagram Frequency (Hz) System: G_speed I/O: u1 to y1 Frequency (Hz): 8.58 Magnitude (dB): -3.4

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20 3 Modelling

In Figure 3.5 the bode diagram is illustrated, the bandwidth of the model is 8.55Hz which corresponds to the bandwidth of the system that is 8.5Hz. Figure 3.6 shows the zeros and poles of the speed controller. All the zeros and poles are inside or at the region of the unit circle.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Pole-Zero Map Real Axis Imaginary Axis

Figure 3.6:Zeros and poles of the speed controller, zeros are illustrated as a circle and poles are illustrated as a cross.

Current controller

The estimated model of the current controller is described in Equation (3.6). It is estimated with input and output data collected at a sampling rate of 50kHz and model orders [1, 2, 2].

G(q) = 0.04246q

−₂

1 − 1.109q−₁

+ 0.1534q−₂ (3.6)

In Figure 3.7 the simulated output is plotted together with the output from the real system and the relative error. Figure 3.8 is the same Figure as 3.7 but for a shorter time span, it shows that the relative error is below 15%.

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3.1 Modelling of the drivers 21 0 0.5 1 1.5 2 Sample 106 0 0.2 0.4 0.6 0.8 current, [A]

Current controller, oe122

yhat y 0 0.5 1 1.5 2 Sample 106 -20 -10 0 10 20 % Relative error

Figure 3.7: Simulation output and the relative error for the current con-troller.

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22 3 Modelling 1.5 1.5005 1.501 1.5015 1.502 1.5025 1.503 1.5035 1.504 1.5045 1.505 Sample 106 0.4 0.5 0.6 current, [A]

Current controller, oe122

yhat y 1.5 1.5005 1.501 1.5015 1.502 1.5025 1.503 1.5035 1.504 1.5045 1.505 Sample 106 -10 -5 0 5 10 % Relative error

Figure 3.8: Simulation output and the relative error zoomed in for the cur-rent controller.

The bode diagram is illustrated in Figure 3.9, and the bandwidth of the model, which is 400Hz, corresponds to the bandwidth of the system. Figure 3.10 shows the zeros and the poles of the current controller, the model is stable since all the poles are inside the unit circle.

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3.1 Modelling of the drivers 23 -40 -30 -20 -10 0 Magnitude (dB) 101 102 103 104 105 -450 -360 -270 -180 -90 0 Phase (deg) Bode Diagram Frequency (Hz) System: G I/O: u1 to y1 Frequency (Hz): 416 Magnitude (dB): -3.25

Figure 3.9:Bode diagram for the current controller.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Pole-Zero Map Real Axis Imaginary Axis

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24 3 Modelling

In this section, the models of drivers are presented as discreet transfer func-tion. In the simulations carried out in Section 5, the models of the drivers are transformed into continuous transfer functions.

3.2 Physical model

Physical models consist of mathematical equations derived from laws of physics that describe dynamics of the system. Of course, not all the dynamics in a sys-tem are described by the equations but the most important dynamics should be described by the equations, for the model to have sufficient performance.

3.2.1 Stiffness

The axis between the driving and braking motor is modelled as a torsional spring, see Figure 3.11. Equation (3.7) describes the relation between the torque on the shaft T and the difference in angle ∆θ.

Figure 3.11:Illustrating the spring that is modelling the stiffness in the axis between the screw joint test rig and the tightening tool.

T = ks∆θ (3.7)

where ksis the stiffness constant. The stiffness constant of a cylinder with radius r and length l is calculated according to the equations:

ks= JrotG l Jrot = π

r4 2

where Jrot is the rotational inertia and G is the shear modulus. The shaft be-tween the motors is assembled by different cylinders with varying radiuses and lengths. First the stiffness constant is calculated in each individual cylinder and afterwards the resulting stiffness constant is calculated according to equation:

ks,total= 1 1 ks,1+ 1 ks,2 + 1 ks,3 + 1 ks,4

where ks,1, ks,2, ks,3and ks,4are the stiffness constants for the different cylinders. See appendix B.3 for the radius and length that is used. The resulting stiffness constant is calculated to 110 Nm/rad

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3.3 Modelling of the tightening tool 25

3.2.2 Modelling of the braking motor

Figure 3.12 illustrates an electric motor where the transmission from electrical motion to rotational motion is shown. The driver for the braking motor is a cur-rent controller and therefore controlling the curcur-rent to the motor. This means that the current does not need to be a state in the motor model. It is only the angular velocity that is needed as a state in the physical model of the DC motor.

v(t) R I L e(t) ω, θ, T Jm Figure 3.12:DC - motor

Equation (3.8) describes the dynamics between the current, i, the friction, h(w), and the torque, T .

Jm dw

dt = kmi − h(w) − T (3.8)

T and h(w) can be expressed as:

h(w) = km,fw (3.9)

T = ks∆θ (3.10)

Where Jm is the motor inertia, kmis the torque constant, km,f is the viscous fric-tion constant and ksis the spring stiffness. Equation (3.8) can be rewritten as:

Jm dw

dt = kmi − km,fw − ks∆θ (3.11)

See appendix A for the values of the parameters.

3.3 Modelling of the tightening tool

The driving motor represents the tightening tool and is modelled as a speed con-troller, see Section 3.1.6. The input to the speed controller is the angular velocity reference described in Section 2.1. The speed controller in the driver is slower than the speed controller in a real tightening tool. As stated in Section 2.1 the controller during the tightening tool is switching from a speed controller to a cur-rent controller during the ramp downs, in order to ramp down as fast as possible.

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26 3 Modelling

This dynamic with switching controllers will not be modelled and simulated in this thesis.

In a real tightening tool the reference angular velocity is based on torque mea-surements. In this study the angular velocity is predefined in the simulations. When simulating with the model of the speed controller the predefined angular velocity is reconstructed with an extended constant second speed. While simulat-ing with the reconstructed reference the torque is studied and once it reaches the final torque the simulation is stopped. The time elapsed to reach final torque is saved and with that information a new angular velocity reference is constructed with the final ramp down. With this new angular velocity reference the final torque is reached. In a real tightening tool θsnug is defined with a threshold, see Section 2.1. In this thesis θsnug is modelled as a predefined number of revolu-tions, nr, as follows:

θsnug = 2πnr (3.12)

Depending on if the screw joint is hard or soft θsnug is between 30 to 720 degrees, which means that nris between 0.08 to 2. But since the speed controller that models the driver is slower than a regular tightening tool nr is modelled larger in order for the rundown velocity to be constant before θsnugis reached.

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4

Control structure

A Linear Quadratic controller will be derived for the braking motor. LQ control is a control strategy that uses state feedback which requires the states of the sys-tem to be measured as an output signal or estimated with an observer. The first section of this chapter describes the theory needed for the control structure and in the second section the controller for the braking motor is presented.

4.1 Linear Quadratic control

Linear Quadratic control (LQ) is a linear optimal control that minimizes a quadratic cost function (4.4). It is a state feedback controller, therefore it requires all the states to be observable, either measured or estimated. More information about the LQ controller can be found in Glad and Ljung (2003). The LQ control needs the system to be presented in state space form, a general state space form is the following:

˙x = Ax + Bu + N v1 (4.1a)

z = Mx (4.1b)

y = Cx + Du + v2 (4.1c)

where A, B, N , M, C and D are matrices. x is a vector consisting of the states of the system, u is the control signal, v1is system disturbance, v2is measurement

noise, z is the control variable and y is the output signal. The optimal control signal is given by:

u = −Lx + Lrr (4.2)

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28 4 Control structure

where L is a static feedback vector or matrix given by:

L = Q₂−1BTS (4.3)

where B is given by (4.1) and S is the unique positive semidefinite symmetric solution to the algebraic riccati matrix equation:

ATS + SA + MTQ1M − SBQ2 −₁

BTS = 0

where Q1and Q2are used as design parameters, they penalize the states and

the control signal respectively. A larger value for Q1makes it more expensive to

have a large error, while a larger value for Q2makes it more expensive to have a

large control signal. min(kek2_Q 1+ kuk 2 Q2) = min Z (eT(t)Q1e(t) + uT(t)Q2u(t))dt (4.4) Lris given by: Lr = (M(BL − A) −₁ )B−1

where A, B and M is given by (4.1) and L is given by (4.3). Lr is constructed so that the closed loop gets a static gain of one. r in (4.2) is the reference signal.

4.1.1 Observer

An observer is needed to estimate the states if not all the states are measured. There are several different observers and the observer that will be used is de-scribed in Glad and Ljung (2003) as:

˙ˆx = A ˆx + Bu + K(y − ˆy) (4.5)

where ˆy = C ˆx + Du, K is called the Kalman gain. y − ˆy describes the goodness of the estimation. This can be seen if x − ˆx = 0 ⇔ x = ˆx then ˆy = C ˆx + Du = Cx + Du and y = Cx + Du which is the same as y = ˆy. This means that the choice of K is important because K affects the observer sensitivity to measurement noise and its ability to handle the effects of system disturbances. v1 and v2 are the

system disturbances and the measurement noise respectively, their intensities are defined as:

R1 = E(v1v1T)

R2 = E(v2v2T)

R12 = E(v1v2T)

(4.6)

The Kalman gain, K, is then given by:

K = (P CT + N R12)R −₁

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4.2 Controller for the braking motor 29

where C is given by (4.1), R12 and R2 are given by (4.6) and P is the unique

positive semidefinite symmetric solution to the algebraic riccati matrix equation: AP + P AT −_{(P C}T _{+ N R}₁₂_)R−1

2 (P CT + N R12)T + N R1NT = 0 (4.8)

A, N and C are given by (4.1), R1, R2and R12are given by (4.6).

If the states are estimated the optimal state feedback is:

u = −L ˆx + Lrr

4.1.2 Controllability

It is important that the states of a system are controllable, otherwise the states cannot be controlled. There are several ways to investigate whether the states of a system are controllable or not. The definition of controllability can be found in Glad and Ljung (2003). A system in state space representation is said to be controllable if there exists a matrix L so that A − BL is stable i.e. all its eigenvalues are in the stability region.

4.1.3 Detectability

Sometimes it is important to observe the states of a system and in such a case the definition of detectability can be used to verify if the states are detectable. When using an observer for estimation of the states it is only possible to estimate the states if they are detectable. The definition of detectability is found in Glad and Ljung (2003). A system in state space representation is said to be detectable if there exists a matrix K so that A − K C is stable i.e. all its eigenvalues are in the stability region.

4.2 Controller for the braking motor

To be able to simulate the braking motor as a screw joint a LQ controller will be used as well as a reference generator. The reference generator calculates the reference signal, ∆θr, so that the torque at the braking motor emulates the torque of a screw joint. In Figure 4.1 the LQ controller and the reference generator are illustrated in a block diagram.

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30 4 Control structure Lr Driver u Motor Observer L Ref.gen ∆θr ur y ˆ x ˆ x −

Figure 4.1:Block diagram of the LQ controller and the reference generator. The control signal u for the motor is the current, ur is the reference current to the driver and the output signals from the system are the torque at the motor shaft T and the angular velocity of the braking motor w. The equations that mod-els the braking motor and the shaft between the braking motor and the tightening tool are (3.11) and (3.10):

Jm dw

dt = kmi − km,fw − ks∆θ T = ks∆θ

The system disturbance v1 is modelled as white noise, this is a requirement

from the definition of the LQ controller. When simulating, the system distur-bance v1 is the angular velocity of the tightening tool which consists of ramps

and constant angular velocity. To make sure that the controller can handle the system disturbance one can model the influence from the tightening tool as fil-tered white noise. The filfil-tered system disturbance is then a state in the state space model for the controller.

In order for the LQ controller to handle system disturbance consisting of both ramps and constant angular velocity the system disturbance must be filtered with a filter of order two. This can be seen if one studies the final value theorem with the transfer function from the system disturbance to the output signals. If one then let the system disturbance be a ramp one can see that the system distur-bance needs to be filtered with a filter of order two to compensate for the ramp in the system disturbance. The following equation models the influence from the tightening tool:

wt,i = β2

(p + β)2v1 (4.9)

where wt,i is the angular velocity from the tightening tool, β is a design pa-rameter for the filter. β needs to be larger than zero for the model of the influence from the tightening tool to be stable.

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4.2 Controller for the braking motor 31 The states of the system are the angular velocity of the motor w, the difference in the angle between the tightening tool and the braking motor ∆θ, the angular velocity of the tightening tool wt,i and the acceleration in angular velocity αt,i. Equations (3.11), (3.10) and (4.9) are rewritten in state space representation as follows:                    w ∆θ wt,i αt,i                    =                    x1 x2 x3 x4                    (4.10)                    ˙x1 ˙x2 ˙x3 ˙x4                    =                       −km,f Jm −ks Jm 0 0 1 0 −₁ ₀ 0 0 0 1 km,f Jm 0 −β 2 ₋_2β                       x +                     km Jm 0 0 0                     u +                    0 0 0 β2                    v1 (4.11) y =        0 ks 0 0 1 0 0 0        x (4.12)

In the state space representation of the system, the last two states are not controllable, but all states are detectable. It is reasonable that the last two states are not controllable because they are states of the system disturbance.

4.2.1 Reference generator

The reference generator generates the reference signal ∆θr to the braking motor to make it emulate a screw joint. The relation between the stiffness constant, the difference in angle and the torque between the braking motor and the tightening tool is:

T = ks∆θ (4.13)

By controlling the difference in angle ∆θ, the aim is to make T to look like the torque of a screw joint. Therefore, it is necessary to know how the torque of a screw joint looks, which is described in the next section.

Screw joint torque

The screw joint torque is the torque emanating to the tightening tool. In a tight-ening, the tightening tool needs to apply a torque in order for the screw to be tightened as desired. The tightening tool needs to apply a torque that compen-sates for both the screw inertia as well as for the friction that arises when the screw is tightened.

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32 4 Control structure

The motor inertia in the braking motor is larger than the inertia in a screw joint. This could potentially be a problem, but during simulations it has been found that the inertia of the braking motor has little effects and can be neglected.

The controller should be able to control the braking motor so the torque at the shaft between the braking motor and the tightening tool emulates the torque from a screw joint. Therefore, an ideal appearance of the screw joint’s torque is used in the reference generator which is modelled as:

Tideal= Jsdw

dt + T (θ) (4.14)

Jsdw_dt is the screw joint inertia and T (θ) describes the friction. The friction in a tightening is complex and the model of the friction used in this thesis is an approximation of the friction in a tightening. The approximation of the friction plotted against angle, is that the friction starts ramping up to its constant value before the snug and after that the friction is linearly increasing. Hence, it will be modelled as: T (θ) =            k1θ, for 0 ≤ θ < θramp

d, for θramp≤θ < θsnug kθ + m, for θ ≥ θsnug

(4.15)

θrampis the angle when the model of the friction starts being constant during the rundown part. θsnug is the angle, where the rundown part ends and the tightening of the joint starts. d is the constant torque from the friction during the rundown part of a tightening process. m can be calculated as m = d − kθsnug. Figure 2.5 illustrates the function modelling the ideal torque.

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4.2 Controller for the braking motor 33

In this thesis θsnug is modelled as a predefined value, see Section 3.3. The relation between the angle of the tightening tool θtand the angle of the braking motor θ is:

θ = T

ks,total

+ θt (4.16)

∆θr is the reference signal and is defined as ∆θ = θ − θt. From Equation (4.16) the reference signal can be described as:

∆θr = T ks,total

(4.17) The aim is to control the braking motor so that T follows Tidealand therefore the reference signal ∆θr is generated as:

∆θr = Tideal ks,total

(4.18) where Tidealis defined in Equation (4.14) and ks,total is calculated in Section 3.2.1.

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5

Results and discussion

In this chapter, the results from this thesis are presented and discussed. In Sec-tion 5.2 the results from the model of the tightening tool are presented and dis-cussed. In order to analyse possibilities and limits of the test rig hardware limita-tions, such as the frequency and the resolution, are modified during simulations to see what impact they have on the performance of the test rig, see Section 5.3. There are always model errors present in a model and to see how sensitive the con-troller is to such errors a model error in the stiffness constant was implemented, see the result in chapter 5.4. It is also interesting to see how the prevailing torque impact the performance of the test rig, see Section 5.5. In Section 5.6 the model of the tightening tool is redesigned to investigate how the tightening tool affects the performance of the test rig. Finally, Section 5.7 presents the results from the analysis regarding the importance of optimizing the controller with the angular velocity from the tightening tool.

5.1 Notes on simulations and analysis

During all simulations carried out in this thesis several internal and external sig-nals have been studied. In all simulations, the restriction in the current to motor is satisfied and therefore the control signal is not illustrated in each section. The aim of this thesis is to investigate the possibilities and limitations when construct-ing a test rig to make sure that the tightenconstruct-ing tool fulfils the desired performance. The aim for the LQ controller is to control the braking motor so the torque be-tween the tightening tool and the braking motor emulates the torque of a screw joint. This is the same torque as the torque at the shaft of the braking motor. Therefore, the most interesting signals to study, given that the other signals seem reasonable, are the torque at the shaft of the braking motor as well as the ideal torque from the reference generator. The simulations carried out in this chapter

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36 5 Results and discussion

are simulated with no restrictions in resolution for the driving motor, since the resolution of the tightening tool the test rig is constructed for is better than the hardware used in this thesis.

5.1.1 Procedure for analysis

It is important to have knowledge of how different hardware components affect the performance of the test rig. It is also of importance to know the performance of the test rig when several hardware components are modelled, in order to dis-cuss the possibilities of a full-size test rig but also to make necessary modifica-tions to achieve the desired performance.

The control structure is based on the fact that the torque between the motors should follow the ideal torque modelled in the reference generator. If the torque between the motors does not follow the torque modelled in the reference genera-tor, it is important to investigate that deviation. This could potentially arise due to the LQ controller not being good enough for such a purpose or not being able to handle the impact from the tightening tool, any other subsystem or hardware limitations.

It is important to be aware of the approximations made in this thesis, in order to discuss or execute tests that expose the LQ controller to situations more similar to reality. Hence, to get an idea of the performance of the full-size test rig.

5.2 The tightening tool

In the test rig, the driver for the driving motor is configured as a speed controller that is simulating the dynamic of a tightening tool. During the modelling of the speed controller it was found that the speed controller has a rise time of 44ms, see Figure 5.1. If one compares the speed controller with a tightening tool it becomes apparent that the speed controller is very slow since the rise time of a tightening tool is about 8ms. Since the idea is to use this test rig for a tightening tool, it is important to investigate how the performance of the test rig is affected by the angular velocity from the tightening tool. Therefore, the angular velocity reference of the tightening tool is filtered using a filter with a faster response time compared to the speed controller for some of the simulations.

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5.2 The tightening tool 37 0 0.05 0.1 0.15 0.2 0.25 0.3 time, [s] 0 0.5 1

Step response - speed controller

unit step step response 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 time, [s] 0 0.5 1

unit step step response

Figure 5.1: Step response for the speed controller, the speed controller is slow and has a static gain over one.

In Figure 5.2 the angular velocity reference of the tightening tool is plotted together with the output from the faster filter and the speed controller. In the Figure, it is obvious that the speed controller is quite slow compared to the faster filter. As mentioned before, the speed controller for the model of the tightening tool is substituted with a filter using a faster response time for some simulations to investigate how the tightening tool affects the performance of the test rig. This is because a filter with a faster response time is more similar to a tightening tool. The reason why it is not an appropriate choice to use the angular velocity ref-erence as model for the tightening tool, is because in a real tightening tool the edges in the angular velocity reference are smoothed out. Therefore, a filter with a faster response time is more similar to a real tightening tool than the angular velocity reference. A low pass filter with faster response time is constructed in MATLAB and used in simulations. Note that the restriction in frequency remains unchanged. The resulting angular velocity out from the redesigned filter of the tightening tool is presented in Figure 5.2. In Section 5.7 filters with even faster response time for the model of the tightening tool are investigated to see how this affects the performance of the test rig.

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38 5 Results and discussion 0 0.5 1 1.5 2 2.5 3 time, [s] 0 20 40 60 80

Angular velocity

angular velocity reference filtered with faster filter filtered with speed controller

1.18 1.2 1.22 1.24 1.26 1.28 1.3 time, [s] 0 10 20 30

Figure 5.2:The angular velocity reference for the tightening tool, the angu-lar velocity out from the faster filter and the anguangu-lar velocity out from the speed controller. Clearly the angular velocity out from the faster filter bet-ter follows the angular velocity reference compared to the output from the speed controller. Hence, the faster filter is a better model of a tightening tool than the speed controller.

5.3 Impact from hardware

In this thesis, the test rig described in Section 2 is modelled. One aim with this thesis is to investigate how different hardware limitations in the test rig affect the performance of the test rig. Therefore, simulations have been carried out where the impact from restrictions in frequency and resolution are analysed. Further, the impact from the driver for the braking motor that is a subsystem in the test rig is analysed. It is important to know how the hardware limitations affect the test rig, so one can make necessary modifications to achieve the desired performance of a future test rig.

The simulations that have been done in this section are described in appendix B. The performance of the test rig is not affected in a significant way whether the frequency of the tightening tool is 2.3kHz or 4kHz, see Figure 5.3 and lines T 4 and T 5. The reason why the frequency 4kHz is simulated is because it is a typical frequency for a tightening tool this test rig is developed for. If one compares lines T 5 and T 6 one can see that the torque T 4 oscillates marginally less when the frequency of the current controller is 10kHz compared to 2.3kHz.

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5.3 Impact from hardware 39 0 10 20 30 40 50 60 70 80 angle, [rad] 0 0.2 0.4 0.6 torque, [Nm]

Torque against angle T1 T2 T3 T4 T5 60 62 64 66 68 70 72 74 76 angle, [rad] 0.2 0.4 0.6 torque, [Nm] T1 T2 T3 T4 T5

Figure 5.3:The torque plotted against angle for test 1 to test 5. It is clear that T3, T4 and T5 have different slopes during the tightening part compared to T1, which is the simulation without any limitations and the driver model.

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40 5 Results and discussion 0 10 20 30 40 50 60 70 80 angle, [rad] 0 0.2 0.4 0.6 torque, [Nm]

Torque against angle T1 T6 T7 T8 T9 T10 60 65 70 75 angle, [rad] 0.2 0.4 0.6 torque, [Nm] T1 T6 T7 T8 T9 T10

Figure 5.4: The torque plotted against angle for tests 1, 6, 7, 8, 9 and 10. One can see that test 6 and 9 have different slopes during the tightening part. It was found that the static gain of the current controller is causing the difference in slopes during the tightening part. The conclusion is therefore that it is important that the driver for the braking motor has static gain one.

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5.3 Impact from hardware 41 0 10 20 30 40 50 60 70 80 angle, [rad] 0 0.02 0.04 torque, [Nm] Difference in torque T1 T2 T3 T4 T5 0 10 20 30 40 50 60 70 80 angle, [rad] 0 0.02 0.04 0.06 torque, [Nm] T1 T6 T7 T8 T9 T10

Figure 5.5: The difference in torque against angle for test 1 to 10. In the upper figure, it is only in test 1 that the difference in torque is not increasing during the tightening part. In test 2 to 5 in the upper figure, one can see that the difference in torque increases during the tightening part. In the lower figure, it is only in test 6 and 9 that the difference in torque increases during the tightening. Test 1, 7, 8 and 10 have in common that the driver for the braking motor is modelled with a static gain of one or removed. Therefore, the conclusion is that if the driver for the braking motor does not have a static gain of one, the performance of the test rig is reduced.

One interesting thing is that if one compares line T 1 with T 2, T 3, T 4, T 5 and T 6 in Figures 5.3 and 5.4 is it clear that the ramps in the torque have different slopes. This can also be seen in Figure 5.5, which shows that for T 2, T 3, T 4, T 5, T 6 and T 9 the torque at the braking motor does not have the same slope as the ideal torque from the reference generator.

To see if the current controller’s static gain of 0.9503, see Figure 5.6, causes this behaviour a simulation without the current controller, its frequency and res-olution were made, see line T 8 in Figure 5.4. T 8 has clearly a slope more similar the ideal torque. To rule out that it is not the frequency in the driver for the brak-ing motor that causes this, a simulation with only limitations in the frequency of the current to the braking motor was done, see line T 9. Line T 9 has the same slope as T 8, which indicates that it is not the frequency of the braking motor’s driver that causing this behaviour.

The current controller is a quite fast filter but to make sure that it is nothing else in the filter that is causing this behaviour, one simulation was made where

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the filter was replaced with a static gain of the same value as the static gain of the filter, see line T 10. T 10 has the same behaviour as T 7 and therefore it is most likely the static gain of the filter that is causing this behaviour. The solution to this problem was to place a static gain of 1/0.093 before the current controller and the resulting torque from that simulation is T 11 which clearly follows the ideal torque better than line T 7.

To see how the resolution of the driver of the braking motor and the resolution of the Arduino Due affects the torque at the motor shaft, a simulation was made without any restriction in resolution of those components, see line T 12 in Figure 5.7. Figure 5.7 shows that limitations in resolution in the driver for the braking motor and the Arduino Due affect the torque between the motors to such a small extent that the conclusion drawn is that the limitations in resolution of those components do not individually affect the performance of the test rig.

In Figure 5.8 the control signals from simulations test 1 to test 10 are shown. During the rest of the simulations in this thesis the static gain of the LQ controller is compensated to be one. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 time, [s] 0 0.5 1 current, [A]

Step response - current controller

unit step step response 0.049 0.0495 0.05 0.0505 0.051 0.0515 0.052 0.0525 0.053 0.0535 0.054 time, [s] 0 0.5 1 current, [A] unit step step response

Figure 5.6:Step response for the current controller, the step response shows that the current controller has a short response time and a static gain less than one.

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5.3 Impact from hardware 43 -10 0 10 20 30 40 50 60 70 80 angle, [rad] 0 0.2 0.4 0.6 0.8 torque, [Nm]

Torque against angle T1 T11 T12 59 60 61 62 63 64 angle, [rad] 0.06 0.08 0.1 torque, [Nm] T1 T11 T12

Figure 5.7:The torque plotted against angle for tests 1, 11 and 12. Test 11 is with restrictions in resolution of the driver for the braking motor and the Arduino Due. Test 12 is without any restrictions in resolution. The conclu-sion drawn is that the limitations in resolution in the driver for the braking motor and the Arduino Due do not individually affect the performance of the test rig.

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44 5 Results and discussion 0 0.5 1 1.5 2 2.5 3 3.5 time, [s] 0 1 2 3 4 5 6 7 8 9 current, [A] Control signal T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

Figure 5.8:The control signal during test 1 to 10. It can be seen that the con-trol signal in these tests is within ±28A and that the tightening process takes about 3s. The conclusion is therefore that the control signal stays within its permitted interval.

5.4 Reference generator

In the analysis of the reference generator, the tightening tool is modelled as de-scribed in appendix B.2. In the reference generator during rundown the ideal torque, plotted against angle, consists first of a ramp and then a constant value, see Figure 4.2. In reality, the screw joint torque, plotted against angle, during run-down becomes constant instantaneously when the tightening tool overcomes the friction. The model of the ideal torque which is plotted against angle in the ref-erence generator is an approximation because the LQ controller did not perform well when the ideal torque during rundown contained an instantaneous step. If the ramp is steeper the model of the friction becomes closer to the reality. There-fore, it is interesting to see how the LQ controller for the braking motor handles steeper ramps in the beginning of the rundown part.

To evaluate how well the LQ controller handles steeper ramps in the begin-ning of the rundown part, the angle θrampis changed to 1 and 0.1 degree com-pared to previous simulations where θrampwas set to 5 degrees, see Figure 5.9.

Another approximation is made in the reference generator. In reality the screw joint torque drops down to zero when the angular velocity is zero between first target torque and final target torque, see Figure 5.10. This is not modelled

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5.4 Reference generator 45

in this thesis, instead the ideal torque in the reference generator is modelled as in Figure 5.11. If it was modelled as in Figure 5.10, the ideal torque of the reference generator would have a steep ramp up to the first target torque when the angular velocity reference begins to ramp up to the second angular velocity.

To evaluate how well the LQ controller handles a steep ramp up to first tar-get torque, the constant torque during rundown is changed to 50% of the final target torque, see the lower part in Figure 5.9. This is because the final target torque is 0.6Nm and the first target torque is 0.3Nm, if the constant torque dur-ing rundown is set to 50% of final target torque the ramp in the beginndur-ing of the rundown part is then similar to the ramp in torque when the angular velocity reference begins to ramp up to second angular velocity.

When the prevailing torque is 10% of the final target one can see in Figure 5.12 that the performance in motor torque is almost the same regardless the val-ues of θramp. The LQ controller has problem to handle the situation when the prevailing torque has a steep ramp to its constant value of 50% of the final target, see Figure 5.13. The conclusion is therefore, that the LQ controller might not be able to handle the ramp in torque, from zero to first target torque when the angular velocity starts ramping up to second angular velocity.

0 2 4 6 8 10 angle, [degree] 0 0.02 0.04 0.06 torque, [Nm]

Ideal torque, prevailing torque 10% of final T

5 degrees 1 degrees 0.1 degree 0 2 4 6 8 10 angle, [degree] 0 0.1 0.2 0.3 torque, [Nm]

Ideal torque, prevailing torque 50% of final T

5 degrees 1 degrees 0.1 degree

Figure 5.9:In the upper figure the ideal torque from the reference generator is illustrated. In this simulation, the prevailing torque is 10% of the final target torque. In the lower figure the ideal torque from a simulation when the prevailing torque is 50% of the final target torque is shown.

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Figure 5.10: The angular velocity and an approximation of the screw joint torque plotted against time.

Figure 5.11:The angular velocity and the screw joint torque plotted against time.

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5.4 Reference generator 47 0 50 100 150 200 250 300 350 angle, [degree] 0 0.02 0.04 0.06 0.08 torque, [Nm]

Torque against angle

5 degrees 1 degrees 0.1 degrees 0 1 2 3 4 5 6 7 8 angle, [degree] 0 0.02 0.04 0.06 torque, [Nm] 5 degrees 1 degrees 0.1 degrees

Figure 5.12: The torque at the motor shaft when the prevailing torque is 10% of the final target torque and the angle θrampis 5, 1 and 0.1 degrees. The figure shows that the performance in the motor torque is almost the same regardless of whether the angle θrampis 5, 1 and 0.1 degree when the prevailing torque is 10% of the final target torque.

Modelling and Analysis of a Screw Joint Test Rig

Master of Science Thesis in Electrical Engineering

Department of Electrical Engineering, Linköping University, 2017