Simulation of dynamic and static behavior of an electrically powered lift gate

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Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Simulation of dynamic and static behavior of an

electrically powered lift gate

Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping

av

Frida Boberg

LiTH-ISY-EX--08/4008--SE

Linköping 2008

Department of Electrical Engineering Linköpings tekniska högskola

Linköpings universitet Linköpings universitet

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Simulation of dynamic and static behavior of an

electrically powered lift gate

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Frida Boberg

LiTH-ISY-EX--08/4008--SE

Handledare: Daniel Axehill

isy, Linköpings universitet

Carsten Haubenschild

Conti Temic microelectronic GmbH, Germany

Ralf Wagner

Conti Temic microelectronic GmbH, Germany

Examinator: Jacob Roll

isy, Linköpings universitet

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Avdelning, Institution

Division, Department

Division of Automatic Control Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2008-02-22 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version

http://www.control.isy.liu.se http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11627 ISBN — ISRN LiTH-ISY-EX--08/4008--SE

Serietitel och serienummer

Title of series, numbering

ISSN

—

Titel

Title

Simulering av dynamiskt och statiskt beteende hos en elektriskt driven baklucka Simulation of dynamic and static behavior of an electrically powered lift gate

Författare

Author

Frida Boberg

Sammanfattning

Abstract

Continental Automotive Systems is a German company that develops control sys-tems for different applications in cars. A control system for electrically powered lift gates that are opened or closed on the driver’s command is one of the products developed. The drive system itself is composed of a spindle that is driven by a DC-motor over a gear and a spring. When developing the control system it is con-venient to use a simulation model instead of having to implement it on the system

every time. The simulation analytically describes how the system is behaving.

In this thesis a simulation model of a drive system and a lift gate is developed and evaluated. The model parameters are estimated using System Identification Toolbox in Matlab.

Nyckelord

Keywords simulation, modeling, DC-motor, nonlinear gray box, parameter estimation,

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Abstract

Continental Automotive Systems is a German company that develops control sys-tems for different applications in cars. A control system for electrically powered lift gates that are opened or closed on the driver’s command is one of the products developed. The drive system itself is composed of a spindle that is driven by a DC-motor over a gear and a spring. When developing the control system it is con-venient to use a simulation model instead of having to implement it on the system every time. The simulation analytically describes how the system is behaving. In this thesis a simulation model of a drive system and a lift gate is developed and evaluated. The model parameters are estimated using System Identification Toolbox in Matlab.

Sammanfattning

Continental Automotive Systems är ett tyskt företag som utvecklar styrsystem för olika tillämpningar i bilar. Bland annat utvecklas ett styrsystem till eldrivna bakluckor som öppnas och stängs av föraren per knapptryck. Själva drivanordnin-gen består av en skruv som drivs av en likströmsmotor över en utväxling och en fjäder. Då man vill utveckla styrsystemet utan att behöva implementera det på systemet varje gång är en simuleringsmodell av drivanordningen och luckan ett bra hjälpmedel. Denna simuleringsmodell kan då analytiskt beräkna hur systemet uppför sig. I detta examensarbete har en simuleringsmodell av en drivanordning med tillhörande lucka utvecklats och utvärderats. Modellparametrarna estimer-ades med hjälp av System Identification Toolbox i Matlab.

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Acknowledgments

I would first of all like to thank everyone at Conti Temic microelectronic in Mark-dorf for giving me support in writing this master thesis. Especially thanks to Carsten Haubenschild, who showed a lot of patience and always was prepared to help me and discuss with me on a daily basis. Thanks also to the colleagues at Suspa GmbH, who supplied detailed information about the drive system. At Linköping University, I want to thank Daniel Axehill and Jacob Roll for their sup-port and encouragement especially in the end phase of the work. And a special thanks to Johan Ross for helping me with proof-reading and for supporting me during the whole course of my academic studies.

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Notational Conventions

List of basic variables and symbols used in this document

nM motor speed iM current uM voltage

ML torque load on motor ML0 constant load factor ηM motor efficiency factor c1, c2 motor constants

Φ magnetic flow

J motor mass moment of inertia R resistance

L inductance i gear ratio b backlash

JG gear mass moment of inertia ηG gear efficiency

nG gear speed

ωG gear angular velocity MSL torque load on gear MGL0 gear friction torque

MG0 gear constant maximum torque

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4 Notational Conventions

srev spindle raise per revolution MS0spindle constant torque factor FS force load from lift gate

vS spindle velocity k spring constant FF spring force FF 0 tension force lS spindle raise

lmaxmaximum spindle raise φ lift gate angular displacement MW torque from gravity force JL lift gate mass moment of inertia MS torque load on spindle

MF spring torque M2torque from sealing

rAP(φ) moment arm between hinge and drive application point on lift gate rGP(φ) moment arm between hinge and gravity point

α angle to horizontal β angle to gravity point γ road slope

Abbreviations

CAS Continental Automotive Systems P W M Pulse-width modulation SIT B System identification toolbox ECU Electrical control unit

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Chapter 1

Introduction

1.1 Purpose

The purpose of this thesis work is to model and simulate the dynamic behavior of an electrically powered lift gate, e.g., the lift gate itself and its drive system. The resulting model is going to be used for simulation during development of the the lift gate control system software. The simulation model has to be based on physical equations and be adjustable to different lift gate dimensions and differ-ent actuators. After adaptation to ambidiffer-ent conditions the model is verified and validated. The model should simulate both static and dynamic behavior of the power lift gate system. Thereby, development of the control unit software can be tested and evaluated without having to apply it on the real system every time or even when no real system is available. The practical advantage with simulation is mainly that tests can be done without endangering the hardware.

The thesis is carried out at Conti Temic microelectronic GmbH, a part of Continental Automotive Systems Division (CAS), located in Markdorf, Germany.

1.2 Background

CAS has put together a concept for a lift gate power closing system with an anti-trap function. The anti-anti-trap function is required when automatic closing of doors or windows is present [4]. The motor current is evaluated by means of an algorithm to identify an obstacle in the way of movement. The control unit accounts for that the closing lift gate reverses automatically when an obstacle is identified. The motor current is therefore particularly important to emulate satisfactorily.

To be able to develop the software in the control system and facilitate updates, a knowledge of how the system behaves under different conditions or with various dimensions is useful. To retrieve this information simulations can be performed. One way to do this is by modeling with Matlab and Simulink, which are powerful modeling and simulation tools. The system dynamics are described by mathe-matical equations and the models evaluated through simulation. The models are

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6 Introduction

created as block diagrams that depict the time-dependent relationships among input, states and output.

1.3 Limitations

The model developed in this work has been influenced by one specific actuator configuration but is supposed to be adaptable to any actuator and lift gate system. The equations are based on Newtonian mechanics which means that all parts are assumed to be rigid bodies. Dynamical inconsistencies due to the difference between the two drives on one lift gate, such as distortion, are not considered in this work.

The model itself does not simulate the battery which generates the voltage source. Instead an ideal step source is used. The actual motor speed encoder that is comprised in the drive system is not modeled but the motor speed is modeled as an output signal from the motor model. The DC-Motor model does not have any compensation for internal loss which means that an efficiency factor must be considered to make it more realistic.

1.4 Substantial goals

The requirements on the model developed in this thesis work are e.g., to model the system in a way that makes it possible to easily change the physical parameters to apply the model on other drive and lift gate dimensions. The static behavior as well as the dynamical behavior are to be modeled and simulated. Following conditions are to be fulfilled:

• The model must be able to simulate static behavior • The model must be able to simulate dynamic behavior • Voltage and road slope conditions must be covered

• The model must be adjustable to various spindle drive configurations • The parameters must be based on physical quantities

• Obstacles and mechanism stiffness must be considered

• The model must support motor speed control and anti-trap simulation • The model must be able to process measurement data

1.5 Method

Matlab is used to generate the system parameters by a script and the model is built in Simulink. The mathematical relations describing the system are imple-mented in Simulink as graphical blocks, one for each subsystem. See Figure 1.1.

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1.6 Structure of the thesis 7

The corresponding model is ideal and a number of extensions were made to get simulation results which fit well with measured data. For example, the clearance between the gear teeth is represented by a backlash block and hence the spindle movement starts with a short time delay in relation to the motor rotation. The system parameters and dimensions are loaded from a Matlab script and can eas-ily be replaced by another script to represent any other linear drive and lift gate system. The total system can be triggered by a voltage step signal or measured PWM-signal. Verification is done by comparison between measurement data and simulation results. The model parameters are chosen to make the simulation model fit its purposes, by the help of system identification.

1.6 Structure of the thesis

The thesis begins with Chapter 2 explaining the drive in the actuator system to be modeled. Also the theory used for modeling is addressed and system identification theory described. In Chapter 3 the whole drive and lift gate system is divided into subsystems and the equations are set up. This chapter also deals with the Simulink implementation of the submodels. In Chapter 4, the chosen sensors for measurements are explained and then simulation is performed with verification. Also parameter estimation is performed with validation. Finally the results are discussed in Chapter 5.

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Chapter 2

Background theory

2.1 Spindle drive actuator

The objective is to model a linear automatic actuator system that is composed of a spindle drive and a car lift gate. What is here called spindle drive or linear drive consists of four main components: a DC-motor, a gearing, a spindle and a spring. The drive is manufactured by Suspa Holding GmbH, Germany. Two spindle drives (one on each side) and its control unit are mounted in a Chrysler Pacifica, see Figure 2.1, and on a test assembly as well. The actuator is attached at one end on the car body and at the other on the lift gate.

Figure 2.1. Picture of drive mounted on the Chrysler Pacifica [2]

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10 Background theory

For a cut through picture of the actuator see Figure 2.2. In the test car the electrical control unit (ECU) is situated in the boot. The drive system is controlled by the ECU software to open or close or to reverse if an obstacle is situated in its way of movement. For more information on the anti-trap function, see Section 1.2.

Figure 2.2. Cross section view of the spindle drive [3]

In the following section the function of the actuator is described.

2.1.1 Description of the actuator

The DC-Motor is a 22 W att motor with graphite brushes, originated from the manufacturer Maxon. The motor is controlled by the lift gate control unit and its voltage supplied by the car battery. The task of the motor is to supply rotation to the spindle over a gearing. The gearing has a prestressed ball bearing and ratio 29:1, i.e., the motor speed is translated to a slower gear speed. This is also the speed at which the spindle, called Speedy by the manufacturer Eichenberger Gewinde, is rotating. The spindle thereby pushes the lift gate open with a pitch of 10 mm/rev. The spring is a compressed coil spring which always exerts a lifting force on the lift gate. By closing, this lifting force must be overcome by the pulling force of the spindle, i.e., the motor must turn in the other direction. When the actuator is supplied a positive voltage it opens the lift gate and by negative voltage it closes the lift gate.

To keep the lift gate in opened position, the weight force must be balanced out by the spindle drive force which pushes the lift gate open. By closing, the lift gate weight helps closing and the spindle drive motor then works in the opposite direction to prevent the lift gate to fall back by its own.

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2.2 Equations from physics 11

In this work, the point of view is that the motor sees the gearing as a load. The gearing in its turn sees the spindle as a load, and finally the spindle sees the load from the lift gate. Another way to see it would have been to sum up all loads into one and let it act on the motor as a single load.

2.2 Equations from physics

This section explains the mathematical equations used throughout the thesis for physical modeling. For more details on mechanics see [8].

2.2.1 Kirchoff’s voltage law

The Kirchoff’s voltage law, also known as Kirchoff’s second law is based on the conservation of energy and it says that the sum of electrical potential in a closed circuit is zero. This law is applied in Section 3.1.1 when the expression for the elec-tromechanical DC-motor model is set up. The voltage potential over a resistance is

uR(t) = Ri(t) (2.1)

where u is the voltage, R the resistance and i the current flowing through the resistance. For an inductance in a circuit the voltage over it is expressed as

uL(t) = L di(t)

dt (2.2)

where L is the inductance. The induced voltage due to the rotation of the motor is calculated from the magnetic flux, Φ, a motor constant, c2, and the rotational

velocity of the motor, n, according to

uind(t) = 2πn(t)c2Φ (2.3)

The torque generated from the motor current is also expressed with the magnetic flux and a motor constant according to

Mi(t) = c1Φi(t) (2.4)

The equations for the electromechanical motor model are all found in [5] and can also be found in introductory books on electrical engineering.

2.2.2 Newton’s second law

The movement of a rigid body can be divided into translation and rotation around an axis. In this work, the motor and the lift gate both rotate, and translation does not need to be considered. For an object rotating about a fixed symmetry axis, the angular momentum, LM, is expressed

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where J is the moment of inertia and ω is the angular velocity. The time derivative of the angular momentum is the sum of torques in the direction of the rotational axis, i.e.

dLM

dt =

X

M (2.6)

This leads to the known expression for rotation around an axis due to the torque around it.

Jdω

dt =

X

M (2.7)

where dω_dt is the angular acceleration.

2.2.3 Steiner’s theorem

The mass moment of inertia of a rigid body around an axis parallel to its center of mass axis can be calculated according to Steiner’s theorem, also known as the Parallel axis theorem. For a rectangular plate, see Figure 2.3, the moment of inertia around its center off mass is

JCM = 1 12m(L 2 L+ T 2 ) (2.8)

where LL is the length of the prism and T its thickness. m is the mass of the

body. The moment of inertia according to Steiner’s theorem is described by the following expression

Jp= JCM+ mr2 (2.9)

where r is the perpendicular distance between the body’s center of mass and an arbitrary parallel axis. This relationship is later used in Section 3.2.1 to estimate the mass moment of inertia of the lift gate.

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2.3 System identification 13

2.3 System identification

System identification is the derivation of dynamical models from measured data,

i.e., finding the system parameters. When doing so, one can choose between white,

gray or black box models. The so called white box model is strictly described by physical equations and therefore complex in its structure and time demanding to derive. In contrast, the black box model uses no a priori information, the system is estimated from input and output signals only. In this work the middle way, i.e., gray box model is used to estimate the system parameters. The gray box has some knowledge of what happens inside the system. One or more parameters are still unknown and these are found by parameter estimation [6].

The model parameters can be estimated by the use of statistical methods. One way to do this is by adjusting the model to observation data by means of prediction error optimization. The prediction error is the deviation between measurement and prediction of the model. To find an optimal parameter estimate, a state space model must first be set up. Generally a state space differential equation is expressed as [7]

˙

x(t) = f (x(t), u(t), p) (2.10)

y(t) = h(x(t), u(t), p) + e(t) (2.11)

where x(t) corresponds to the state vector, u(t) is the input signal and p the parameter vector. y(t) is the observation, i.e., the measurement expressed in state variables, input and parameters. The term e(t) is the noise which is also contained in the measurement. The model prediction of the measurement can be calculated from known measurement data and is expressed as

ˆ

y(t|p) = h(ˆx(t), u(t), p) (2.12)

where ˆx(t) is the estimate of the state vector.

As mentioned above the prediction error is the deviation between the measure-ment and the prediction and is hence expressed as

(t, p) = y(t) − ˆy(t|p) (2.13)

The ability of the model to predict the reality can be evaluated by a scalar function

l((t, p)). To estimate the parameters, the cost function

VN(p) = 1 N N X t=1 l((t, p)) (2.14)

is minimized with respect to the parameter vector p. The function l((t, p)) should be scalar and positive as well. Since the system here has multiple outputs, the function (t, p) is a vector and a squared quadratic norm is chosen as l((t, p)), e.g.

l() = N

X

i=1

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In this work, System Identification Toolbox (SITB) in Matlab is used to generate the parameter estimates. A state-space model is set up and implemented in Matlab as a nonlinear gray box model, see Section 4.5.1. This model is used for parameter estimation with the pem-function in Matlab’s SITB, see Section 4.6.

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Chapter 3

Modeling

3.1 Subsystems

To get a better view of the drive system to be modeled, the whole system is divided into subsystems which can be modeled, tested and verified separately. The whole system consists of the chosen subsystems DC-motor, gear, spindle, spring and finally the lift gate, see Figure 3.1. Another subsystem which also could be considered would be a damper, because the air in the drive has a damping effect on the whole system. In this work it was chosen not to consider any damping effects in agreement with the spindle drive manufacturer Suspa.

Figure 3.1. Division of the linear drive system into subsystems

3.1.1 DC-Motor

The DC-Motor is based on the electromechanical model shown in Figure 3.2. The motor takes a voltage and a torque load ML as input and generates motor speed nM and current iM as output. The current is an interesting output signal, because

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16 Modeling

it is used by the software in the anti-trap function. In this work an existing Simulink model is used and it only needs some parameter adjustments to serve as an appropriate model. Parameters as mass moment of inertia, resistance and inductance are given from data sheets. The inductance L has a small influence when the current rises or falls. The mechanical load, ML, is fed back from the

gear model. Higher load leads to lower motor speed and higher current. Losses

c1⋅ Φ c2⋅ Φ −Uq Ia MB nM −ML 1 2 p⋅ πJ U0 1/R MM pL/R 1+

Figure 3.2. The electromechanical motor model [9]

due to friction are considered in the Simulink model through the motor efficiency,

ηM. The simulation model does not consider the ripple current, i.e., a small level

of variation arising from commutation, i.e., when current is transmitted from the brush to the corresponding coil.

Output: Motor speed nM, motor acceleration ˙nM and current iM

Input: Input voltage uM and load torque ML from the gear

Model: The first equation is derived according to Section 2.2.1.

Ld

dt(iM) + RiM+ c2ΦnM − uM = 0 (3.1)

The second equation describes the torque relation in the motor, see also Section 2.2.2.

2πJ d

dt(nM) = c1ΦiM− ML− ML0 (3.2)

where J is the motor mass moment of inertia and the torque ML is the

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3.1 Subsystems 17

and magnetic flux Φ in the motor. ML0 is a constant load factor which is

dependent on the rotational direction of the motor.

ML0= (1 − ηM)M0sgn(nM) (3.3)

Here the sign function sgn is the sign of the motor speed nM. The sign

function is defined as sgn(x) =    −1 x < 0 0 x = 0 1 x > 0 (3.4)

Parameters: Motor constants ck, magnetic flow Φ, resistance R, inductance L,

efficiency ηM, load constant M0

The DC-Motor takes the voltage trigger source and torque load from the gear as inputs and generates motor speed and current as outputs.

3.1.2 Gear

The gear changes the motor speed with a factor 1_i. The torque is increased with a factor i and the gear mass contributes with a moment of inertia JG.

Nonlin-ear effects like backlash, i.e., the error in motion that occurs when gNonlin-ears change directions, are considered in the Simulink model. Friction is considered through gear efficiency ηG. In the gear model, the backlash is represented by a block in the

rotational displacement. The rotational displacement is then differentiated to get the gear speed, i.e., the output gear angular velocity. The backlash parameter, b, is defined as a fraction of one revolution, i.e.,

b = ∆θ

2π (3.5)

where ∆θ is the angular deviation.

Output: Gear angular velocity ωG, gear angular acceleration ˙ωG and torque

load on DC-motor ML

Input: Motor speed nM, motor acceleration ˙nM and spindle torque load MSL

Model: The angular velocity is ideally described through (at startup and direc-tion change there is a delay due to backlash)

nG=

nM

i (3.6)

and according to Newton’s second law of motion [10], see also Section 2.2.2

JGω˙G+ MSL+ MGL0 = iML (3.7)

where MSLis the torque which is exerted to the spindle. The friction torque, MGL0, is expressed with the efficiency ηG and is also dependent of the

di-rection according to

MGL0= (1 − ηG)MG0sgn(nG) (3.8)

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18 Modeling

Parameters: Efficiency ηG, gear ratio i, maximum torque MG0, backlash b

3.1.3 Spindle

The gear angular velocity generated from the gear model ωG is transformed to a

linear velocity vSin the spindle model. The spindle raises a length unit per

revolu-tion, srev. Nonlinear effects, like friction between the spindle and the spindle nut,

are considered via an efficiency factor in the Simulink model, see the torque trans-formation in Equation 3.10. The transmission from rotational to linear movement is ideal in the spindle model.

Output: Spindle velocity vS, spindle acceleration aS and torque load MSL on

gear

Input: Angular velocity ωG, angular acceleration ˙ωG, force load from lift gate

FS

Model: The spindle velocity vS is described through [1]

vS =

ωGsrev

2π (3.9)

The torque load that the spindle exerts on the gear, MSLis described by the

force load from the lift gate FS

MSL= (1 − ηS)MS0+ FS

srev

2π (3.10)

where ηS is the spindle efficiency and MS0is

MS0= FS0

srev

2π sgn(ωG) (3.11)

where FS0 is a constant force factor.

Parameters: Spindle raise srev, force factor FS0and efficiency ηS

3.1.4 Spring

The spring is situated parallel to the spindle inside the drive system. It has a spring constant, k, and generates a force FF on the lift gate. The spring always

exerts a lifting force on the lift gate, and the force is higher when the lift gate is closed, due to the tension force from the compressed spring. In the spring model the displacement is calculated by integrating the spindle velocity. The spring force is then dependent on the displacement.

Output: Spring force FF and displacement lS

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3.1 Subsystems 19

Model: The spring force FF is described through

FF = FF 0+ k(lmax− lS) − F (3.12)

where FF 0 is the tension force of the spring, lmaxis the maximum raising of

the spindle, i.e., when the lift gate is completely opened. lS is the variable

raising at certain time instants. The raising of the spindle is calculated by integrating the spindle velocity according to

lS =

Z

vSdt (3.13)

F is an extra force which was added to represent what happens when the

end position of the lift gate is reached, i.e., the lift gate is stopped by a force corresponding to the physical restriction that it can not be moved farther. It is implemented as a spring force in the opposite direction to the tension force, with spring constant k3.

F =

k3lS lS ≥ lmax

0 lS < lmax

(3.14)

Parameters: Spring constant k, tension force FF 0, maximum spindle

displace-ment lmaxand end force spring constant k3

3.1.5 Lift gate

The lift gate model takes the spindle displacement, lS, acceleration, aS, and

gen-erates the lift gate angular displacement, φ, and load force on the spindle, FS. The

lift gate is opened or closed by the torque generated from the forces applied on it. These forces are the gravity force from the lift gate mass, FW, and the spring

force, FF. The angular acceleration of the lift gate, ¨φ, and the mass moment of

inertia, JL, are taken into consideration when expressing the torque equations in

the dynamic case. For a derivation of JL see Section 3.2.1.

Output: Angular position φ and force load FS on to spindle

Input: Spring force FF, spindle length lS and spindle acceleration aS

Model: The torques around the lift gate sum up to an angular acceleration of the mass according to Newton’s second law of motion [10]. Also see Section 2.2.2.

JLφ = M¨ S(φ) + MF(φ) − MW(φ) + M2(φ) (3.15)

where MS is the torque exerted on the spindle. MF and MW are torque

components that the forces from spring and gravity exert. M2 is defined as

the torque exerted by the sealing on the car body initially while opening and it decays as a spring force. Torque is defined as moment arm times force and here the moment arm is dependent on the angular displacement of the lift

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20 Modeling

gate. The force load exerted on the spindle is expressed as exerted torque divided by the moment arm rAP(φ)

FS =

MS(φ)

rAP(φ)

(3.16) For a derivation of the moment arm rAP(φ) see Section 3.2.5.

The torque M2 from the sealing is defined as

M2=

(

rF 2(F20− k2lS) lS ≤Fk202

0 lS >Fk202

(3.17) where rF 2 is a constant moment arm defined as one point where the initial

force F20 from the sealing is applied. k2 is a spring constant. With a large

value for the spring constant it is obvious that this torque is only active for a short displacement of the lift gate.

Parameters: The mass moment of inertia of the lift gate JL, lift gate mass

m, initial sealing force F20, moment arm for sealing force rF 2 and spring

constant k2. More parameters that influence the kinematics of the lift gate

are explained in next section.

3.2 Derivations

In the following section, expressions for the torque relations are derived, based on planar kinetics of rigid bodies. First a couple of points on the actuator lift gate system are declared. Also see Figure 3.3 where the points are defined.

H is the hinge point, around which the lift gate rotates CP is the car body point, at which the actuator is fastened AP is the lift gate point, at which the actuator is fastened GP is the gravity point for the lift gate weight

3.2.1 Lift gate mass moment of inertia

The mass moment of inertia of the lift gate could either be approximated from measurement data or expressed from mechanical tables on inertias for rigid bodies, as is done here.

The shape of the lift gate is approximately like a prism. The mass moment of inertia, JL, is calculated according to Steiner’s Theorem in Section 2.2.3

JL=

m(L2_L+ T2)

12 + mr

2 _(3.18)

where m is the mass, LL the length of the prism and T its thickness. r is the

perpendicular distance from the mass point to the hinge around which the lift gate rotates. The values for these parameters are gathered from dimensions of the lift gate test assembly.

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3.2 Derivations 21

Figure 3.3. Geometry of the lift gate

3.2.2 Lift gate angular displacement

The angular displacement of the lift gate is calculated from the simulated spindle displacement according to law of cosine. This derivation is implemented in the kinematics block of the lift gate model. The points of the car body that are used for calculations are the hinge H, the car body point CP and the actuator lift gate point AP . We can express the distance between H and CP as s and the distance between H and AP as rAP 0. Both distances are constant. The distance between

CP and AP varies with the spindle length, lS, and this variable is defined as

l = l0+ lS. The law of cosine is then expressed as

2srAP 0cos(φ + θ) = s2+ rAP 02 − l

2 _(3.19)

where θ is the constant angle between the vectors CP-H and AP-H when the lift gate is in its closed position, i.e., where φ = 0 and lS = 0. The expression for φ

used in the kinematics block is then

φ = −θ + arccos(s

2_{+ r}2

AP 0− (l0+ lS)2

2srAP 0

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22 Modeling

with the spindle displacement lS as the only dependent variable.

3.2.3 Torque angular dependence

The torque around the hinge depends not only on the size of the forces acting on the lift gate but also on the actual angular displacement of the lift gate at every moment, i.e., the angle φ. Here the dependencies for the torques are expressed. First the torque from the gravity force is explained in Section 3.2.4 and then the torque from the spring force is explained in Section 3.2.5.

In this text, an orthogonal reference frame with the earth horizontal line as ˆx

and the negative gravity line as ˆy is the chosen coordinate system. For

simplifi-cation the angle α is introduced, which is defined as the angle between the vector

AP(φ0) − H and the horizontal, ˆx, see Figure 3.3. Note that in the figure no tilt

is present. α is expressed in a constant term, α0, and in the road slope angle, γ,

as

α = α0− γ (3.21)

3.2.4 Torque from gravity force

The torque from the gravity force is expressed with the moment arm vector,

rGP. This vector is the difference GP-H between the point of gravity, GP =

(GPx, GPy), and the hinge point, H = (Hx, Hy).

rGP(φ) = rGP 0cos(φ + β − α)ˆx + rGP 0sin(φ + β − α)ˆy (3.22)

where β is the angle between the lift gate vector rGP and the point of gravity

GP , see Figure 3.3. rGP 0 is the length of the lift gate vector. This distance is

independent of the angle φ.

rGP 0=

q

(GPx− Hx)2+ (GPy− Hy)2 (3.23)

Now when we have the moment arm, the torque can be calculated as a vector cross product

Mw(φ) = rGP(φ) × Fw (3.24)

where the gravity force, Fw, is constant (when no other disturbances like wind or

snow are present) and always directed in the negative ˆy-direction

Fw= −Fwyˆ (3.25)

The torque is considered to be directed out of the ˆx, ˆy-plane and thereby, we can

leave out the vector in the expression. The direction of the gravity torque, Mw,

is clockwise. Because the gravity force is directed in the (negative) ˆy-direction,

only the ˆx-part of rGP(φ) influence the magnitude of the torque. As a result the

torque can shorter be calculated as

Mw(φ) = −rGP 0Fwcos(φ + β − α) (3.26)

In the lift gate model, the factor rGP 0cos(φ + β − α) is derived in the trigonometry

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3.2 Derivations 23

3.2.5 Torque from spring force

In the same way as the moment arm vector is derived in Section 3.2.4 above, it can be found for the torque that the spring force excerts as well. Here we call the moment arm vector, rAP(φ), and its expressed as the difference between

the point of application vector, AP = (APx, APy), and the hinge point vector,

H = (Hx, Hy).

rAP(φ) = AP(φ) − H (3.27)

rAP(φ) = rAP 0cos(φ − α)ˆx + rAP 0sin(φ − α)ˆy (3.28) rAP 0 is the length of the distance between the hinge, H, and point of application

on the lift gate, AP .

rAP 0=

q

(APx− Hx)2+ (APy− Hy)2 (3.29)

The magnitude of the lifting spring force on the lift gate, FF, is simulated in the

Simulink model. The direction is expressed by the vectors pointing at the points of application as

ˆ

d = 1

|rAP(φ) − (CP − H)|

(rAP(φ) − (CP − H)) (3.30)

Then the spring force is expressed as follows

Fd= (FF)ˆd (3.31)

and the torque is calculated with the cross product

MF(φ) = rAP(φ) × Fd (3.32)

Since ˆd has a term which is parallel to rAP(φ) it can be left out in the resulting

torque vector. Only the part which is parallel to H − CP is considered. The resulting torque now is calculated as

MF(φ) = rAP(φ) ×

FF

|rAP(φ) − (CP − H)|

(H − CP) (3.33)

The resulting torque from the spring forces is now given by

MF(φ) = FFrAP 0[(Hy− CPy) cos(φ − α) − (Hx− CPx) sin(φ − α)] p(Hx+ rAP 0cos(φ − α) − CPx)2+ (Hy+ rAP 0sin(φ − α) − CPy)2 (3.34) Let us define rAP(φ) = rAP 0[(Hy− CPy) cos(φ − α) − (Hx− CPx) sin(φ − α)] p(Hx+ rAP 0cos(φ − α) − CPx)2+ (Hy+ rAP 0sin(φ − α) − CPy)2 (3.35) In the lift gate model FF is multiplied by rAP(φ) in the trigonometry block, i.e.,

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24 Modeling

3.2.6 Spindle raise angular dependence

The spindle raise l is expressed by trigonometric functions of the angular displace-ment of the lift gate, φ. The spindle raises in its linear direction and at the same time rotates with the lift gate movement.

l(φ) = |AP(φ) − CP| (3.37)

The the vector to the point of application on the lift gate, AP(φ), is expressed by the vector product

AP(φ) = rAP(φ) + H (3.38)

As a result we get the expression for the raise

l(φ) =

q

(rAP 0cos(φ − α) − CPx+ Hx)2+ (rAP 0sin(φ − α) − CPy+ Hy)2

(3.39) Here we identify the denominator in the expression (3.35) as l(φ). Then we can express l(φ) as

l(φ) = rAP 0[(Hy− CPy) cos(φ − α) − (Hx− CPx) sin(φ − α)]

rAP(φ)

(3.40) In the Simulink model the angular displacement is generated from the spindle raise through a function block according to the law of cosine.

3.2.7 Lift gate angular acceleration

The spindle acceleration can be expressed by a vector aSwhich is directed parallel

to ˆd.

aS = aSdˆ (3.41)

If the scalar product is taken with the unit vector in the direction of the angular displacement, ˆΦ, we get the acceleration in that direction

aS· ˆΦ = rAP 0φ¨ (3.42)

where the unit vector ˆΦ is expressed in rectangular coordinates. See also Figure 3.4.

ˆ

Φ = − sin(φ − α)ˆx + cos(φ − α)ˆy (3.43) From Equation (3.42) we now get the angular acceleration of the lift gate by dividing by rAP 0. This results in the expression

¨

φ = aS

rAP 0lS(φ)

((CPx− Hx) sin(φ − α) − (CPy− Hy) cos(φ − α)) (3.44)

In the Simulink model the angular acceleration is generated from the spindle ac-celeration and the angular displacement through a function block.

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3.3 Extensions 25

Figure 3.4. Vectors describing the acceleration of the lift gate

3.3 Extensions

By initial opening of the lift gate, not only the spindle and spring apply a lifting force on the lift gate. Also the force from the rubber strip that supports the lift gate when closed has an effect on the lifting force. This is implemented in the lift gate model by an extra spring force that is dependent on the spindle stroke.

When the lift gate reaches its upper end point, there is a force stopping it from moving further. This can also be described as a spring force, which is done in the spring model. This spring force has a very high spring constant and it only applies for a short time, until the lift gate reverses a short distance. In the Simulink implementation the force is switched on according to the spindle stroke. When the spindle reaches its fully extended position an extra force in its opposite direction is applied and the lift gate stops close to that position after reversing.

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Chapter 4

Model evaluation and

parameter estimation

4.1 Measurements

To be able to identify the system parameters, perform verification and validate the models, observations of the systems behavior are needed. By performing mea-surements on the system, observation data is collected. On the test equipment there is a data monitor which collects sensor signals from sensors measuring volt-age, current, motor position and motor temperature. These existing sensors are supplemented by some other sensors for the purpose of this work.

A first look at the models indicates that the input, applied voltage on the drive, and output, angular displacement of the lift gate, are the most obvious signals to measure. The motor current is important as well because it is used by the software in the anti-trap function. Since the surrounding temperature is important to the adaptability of the model to the real system it also has to be measured. The functionality of the contained mechanical and electrical devices is differently influenced by extreme temperatures.

The road slope influences the torque applied on the lift gate and is therefore an important parameter. The road slope is measured by an accelerometer placed inside the car. The forces on the lift gate are measured through an accelerometer, which measures the position where the lift gate is situated.

Measurements are performed both in stationary states as well as in dynamic states, when the lift gate is moving. External forces on the lift gate, e.g., the influence from an obstacle can be recorded during measurement and reproduced in the simulation model as extra forces.

When it is impossible to arrange the desired environment conditions on the actual car lift gate, for example extreme temperatures, the test assembly must be placed in a temperature chamber.

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28 Model evaluation and parameter estimation

4.2 Sensors

There are different kinds of sensors available for these types of measurements. The choice between analog or digital sensors had to be done and was decided in favor of the analog ones. Temperature is measured with an analog output temperature sensor. The road slope is found with an accelerometer which measures gravity. The angular displacement of the lift gate is measured by a angular position poten-tiometer sensor. The acceleration of the lift gate is measured by an accelerometer mounted on the lift gate. All sensors except for the potentiometer are collected in a sensor box and mounted on the lift gate, thus moving with the lift gate motion. To be able to measure the road slope, an accelerometer must be placed inside the car trunk. The angle potentiometer is positioned on one of the lift gate hinges see Figure 4.1.

Figure 4.1. Placement of the sensor box and potentiometer

4.2.1 Ambient temperature

Extreme temperature is a critical factor for the functionality of the system com-ponents and hence the reliability of the model. For example, the efficiency of the DC-motor is lower for low temperatures because it leads to slower movement of the gear [5]. The temperature sensor used here is an analog output temperature sensor with range from −55◦C to +130◦ C.

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4.2 Sensors 29

4.2.2 Road slope

The road slope influences the direction of the lift gate gravity force, relative to the other forces acting on the lift gate. Therefore the road slope is important to the total torque around the hinge. The road slope is found with an accelerometer sensor placed on the car body, e.g., in the trunk.

4.2.3 Angular displacement of the lift gate

The angular position of the lift gate is in fact the controlled output and is therefore an important signal to measure. The position is measured with a angular position potentiometer sensor which is placed on the lift gate hinge axis. The resistance of the potentiometer is controlled by the lift gate movement. Also the accelerometer placed on the lift gate can be used to retrieve the position of the lift gate.

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4.2.4 Sensor box

The sensor circuit board is laid out according to the connection diagram in Figure 4.2. The connections were made by the recommendations in the sensor data sheets.

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4.3 Simulation 31

The printed circuit board is then embedded in a box which is mounted on the lift gate test assembly. See Figure 4.3 for a picture of the sensor box.

Figure 4.3. Sensor box

4.3 Simulation

The Simulink model can be run by either an ideal source voltage, usually a step signal, or with measured voltage from the real system. The input voltage of the real system is a pulse width modulated (PWM) signal. PWM signals are used to control the amount of power sent to a load. To use microcontrollers that can output PWM signals is a common way to control electrical motors. The input signal used for simulation in this work is a measured voltage signal which is proportional to the control PWM signal.

Model verification is possible by taking parts of the model out of its concept and expose them to various tests which have certain expected outcomes. For example, the lift gate model was tested by removing all forces acting on it and letting it drop from an angle. It then oscillated around an equilibrium. To verify the whole model it is suitable to compare, e.g., the modeled current or modeled lift gate displacement with measured signals, by the same input voltage. The parameters are loaded from a Matlab script and can easily be individually changed in the Matlab prompt for further tests for adjustments.

4.4 Verification

When using the measured voltage as input in the simulation model, a certain similarity is expected between the simulation and real system observation for a

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sufficient good model. The verification was carried out in this way for different scenarios, like opening, closing and with external disturbance force.

The measurements were carried out on a test assembly, which was built accord-ing to the dimensions of a car where the drive system was applied. For a picture of the test assembly see Figure 4.4.

Figure 4.4. Picture of the test assembly

4.4.1 Comparison between model and system

In this section the output of the model is compared with measured signals. Mea-surements from an opening procedure are used for this purpose. We take a look at measurements of some signals and compare with result from the model with the same input voltage. Figure 4.5 shows the PWM proportional input voltage together with simulated and measured current in the same plot.

By changing the mass moment of inertia parameter J for the motor, the current simulation curve is adjusted toward the measured one. The value given in the data sheet for the motor was changed a factor 30 which indicates that the given value must have been out of line. A small value for the inertia parameter leads to a faster system, i.e., the motor starts up faster and hence the current signal reacts too fast. By increasing the value and comparing the simulation result with measurements a better value is found. The resistance R in the motor model influences the modeled stall current, i.e., the statical value of the current. For low values of R the current is increased and for higher decreased.

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4.5 Parameter estimation 33

Figure 4.5. Current and supply voltage when lift gate is opened

4.5 Parameter estimation

To make the simulation results resemble the real physical system, the model pa-rameters must be set accordingly. Most papa-rameters in this thesis are given by data sheets and other documentation. In Section 4.5.2 the parameters that can be esti-mated are listed. The unknown parameters can be estiesti-mated from measurements, either by hand or by statistical methods. For example, the lift gate mass moment of inertia can be calculated approximately by its known physical measurements, its mass and theoretical expressions for mass moment of inertia, see Section 3.2.1. Some parameters change with time. If they are to be modeled as constants or not, depends on the time frame considered. For example, friction parameters like efficiency are not constant in a long time perspective. In this work only a short time perspective is considered.

In this section a state space model is set up and identified according to the parameter estimation theory in Section 2.3. The derived state space model is a simplification of the original Simulink model derived earlier. The extensions from the former section are not considered, i.e., the initial force from the strip and the force when the lift gate reaches its end position are left out. Also the torque load rotational direction dependence is left out. The state space model describes the relations between current iM, motor speed nM and motor position posM.

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4.5.1 State space model

The states, x(t), in the model are chosen as

x1 : Motor current iM x2 : Motor speed nM x3 : Motor position posM

We begin by eliminating the variables in the differential equations from the sys-tem description in Section 3.1. The variables are substituted by the corresponding states. The first and third state equations are straightforward to derive, but the second one needs careful rewriting since the term ML is also dependent on the

motor acceleration. ˙ x1= −R Lx1− c2Φ L x2+ 1 Lu (4.1) ˙ x2= 1 2πJ(c1Φx1− ML− (1 − ηM)ML0) (4.2) ˙ x3= x2 (4.3)

Equation (4.3) simply means that the derivative of the motor position is the motor speed. In the second state, Equation (4.2) the term ML is dependent on states x2 and x3, and also on the motor acceleration, ˙x2 which is derived next. From

Equation (3.7) the load from the gear on the motor can be expressed as

ML=

1

i(

JG2π

i x˙2+ MSL+ (1 − ηG)MGL0) (4.4)

here MGL0 is a constant and MSL is the torque load from the spindle. From

Equation (3.10) we get

MSL= (1 − ηS)MS0+ FS

srev

2π (4.5)

MS0 is constant and FS is the force load from the lift gate on one spindle. This

term depends on the lift gate torque according to Equation (3.15)

FS = 1 2 JLφ − M¨ F+ Mw− M2 rAP(x3) (4.6) and the lift gate angular acceleration is expressed as

¨ φ = 1 i srev rAP(x3) ˙ x2 (4.7)

The total torque from the two spring forces is

MF = 2rAP(x3)FF (4.8)

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4.5 Parameter estimation 35

FF = FF 0+ klmax− klS(x3) − F (4.9)

and F is considered zero here (no consideration is taken to the force F which only acts when the lift gate reaches its opened position). The torque from the lift gate weight force, Fw is

Mw= rGP(x3)Fw (4.10)

M2was defined in Section 3.1.5 as the torque acting around the hinge for a short

time when opening or closing. It is therefore considered to be zero in the state space model. By inserting Equations (4.4) - (4.10), Equation (4.2) can be written as ˙ x2= 1 Jtot(x3) (c1Φx1+ srev 2πiFF(x3) − srev 4πi rGP(x3) rAP(x3) mg − Mtot) (4.11)

Here all the constant torque terms are simplified to one, i.e., Mtot. Mtot= (1 − ηM)M0+

1

i(1 − ηG)MG0+

1

i(1 − ηS)MS0 (4.12)

The total moment of inertia, Jtot, can in detail be expressed as Jtot(x3) = 2πJ + JG i2 + JL 4π _s rev irAP(x3) 2 (4.13)

4.5.2 System parameters

Here follows a list of parameters that can be estimated in the nonlinear gray box model.

DC-Motor

• resistance R [Ohm] • inductance L [H]

• rotor mass moment of inertia J [kgm2_]

• torque constants c1Φ [N m/A]

• speed constant c2Φ [V /rps]

• efficiency ηM [−]

Gear

• gear ratio i [−]

• gear mass moment of inertia JG [kgm2]

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Spindle

• spindle length raise srev[m/rev]

• efficiency ηS [−] Spring • spring constant k [N/m] • tension force F0[N ] Lift Gate • mass [m]

• mass moment of inertia JL [kgm2]

In this work only some of the parameters are estimated. The experiments could be extended to estimate all of them, if more time were at hand. The choice was made to estimate DC-motor parameters only. The reason is that those parameters are tightly related to the motor current, which is important to simulate as well as possible. Two estimation runs will have different parameters choices and the last run will use an even more simplified model.

4.5.3 Given parameters

Parameters given from data sheets are all to be considered as guidelines and there is a possibility that they may be adjusted according to the estimation results. The motor parameters are all given from the producers’ data sheets. The same goes for the gear and spindle parameters. The lift gate mass is given as the weight placed on the lift gate test assembly and the mass moment of inertia is calculated according to the test assembly dimensions as was done in Equation 3.18.

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4.6 Validation 37

4.6 Validation

When validating an estimated model it is common to retrace the imperfections to two kinds of errors, the bias and the variance errors. The variance error arises due to the noise that influences the system. If the noise is not reproduced, another experiment with the same input might not give the same output. Longer sequences of measurements lead to smaller variance errors. The bias error depends on defects in the model structure. These variations appear if the model is adjusted to data collected under different conditions. This means that the model is not capable of describing the system, even if it is adjusted to noise-free measurements. Good model quality can then be expressed as a model with small variance and bias error [7].

4.6.1 Estimation of some motor constants

In this section an estimation of the motor parameters is performed, using estima-tion data from an opening procedure. In next secestima-tion, another choice of parameters are estimated from the same measurement data. The data was first cut to remove the initial influence of the rubber strip force, which was not considered in the gray box model. The data was also decimated since the sample rate of the measure-ment was unnecessarily high. The decimation also speeds up the time it takes for Matlab to generate a result. The estimated parameters are listed in Table 4.1 with their estimated values, estimated standard deviation and initial values. The initial values are the values taken from the DC-motor data sheets. The most

Parameter estm value standard dev init (table) value

Resistance [Ohm] 2.12797 0.0217914 0.708

Inductance [H] 4.20318e-5 61.14862e-5 4.37e-5

Motor inertia [kgm2] 1.47e-5 2.81633e-7 1.28e-5

Motor load [Nm] 0.001 0.000434814 0.00124

Table 4.1. Estimation result with standard deviation and table values

noticeable properties in this table is first the resistance, whose value is three times as large as the initial table value. A possible reason for this large difference could be the fact that motor parameter values can vary even for motors from the same production line. It must also be reminded, that the model does not consider the power electronics and resistances that exist in the real system, e.g., in the wires between the voltage source and the motor. The second characteristic of impor-tance is the inducimpor-tance, whose estimated standard deviation is more than ten times larger than its own value. This estimation of the inductance is hence not trustworthy. A reason for this could be that the inductance is of less importance for the used measurement data. In other words, maybe a simpler model might work for simulation under similar circumstances. We will return to this subject later when the third estimation is performed.

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signals is shown in Figure 4.6. The comparison is made between measurement data and a pure simulation of the model without any prediction. The modeled

                        

Figure 4.6. Measurement compared to pure simulation without prediction

current is too high both in the early and later part of the run. The modeled motor speed has its stationary value too high compared to measurement data. The modeled position has some deviation from measurement data.

In the next plot, Figure 4.7, a simulation of the estimated model is compared with the measurements it was generated from. Here the match is much better for all three outputs. The conclusion is that the parameters are well estimated and the parameter estimation leads to a better model. The modeled current has a rather good level for the start-up but still its stationary level is too high. Generally the start-up behavior of an electromechanical systems is hard to model because of the unpredictability of the stiction forces, i.e., the static friction forces of the system. According to the estimation error plot in Figure 4.8, the prediction error is not white noise as desired for a good prediction model of the real measured signals. The fluctuation of the motor speed signal can be explained by the imprecise and time discrete Hall sensor signal. In the signal processing of the measurement signal a rounded integer value is generated. If the noise covariance matrix has small values it is an indication of that the model captures the estimation data in a good way [6]. One must also take into consideration the relation between the element values in the noise covariance matrix and the magnitude of the signals. At a first glance it looks like the motor speed is less well modeled, speed corresponds to the second

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4.6 Validation 39                           

Figure 4.7. Measurement compared to estimated model in Section 4.6.1

variable. But in relation to the absolute value of the speed its variance is not that high. The current has the highest relative error which can also be seen in the estimation plot, Figure 4.7.

>> nlgr_est.NoiseVar =

0.0491 0.2708 -0.0694 0.2708 4.5906 -0.3842 -0.0694 -0.3842 0.4010

The covariance matrix describes how good the parameter estimation is. Small values in the diagonal tell us that this particular parameter is important to the system dynamics in the chosen model structure [6]. For the estimation done here the covariance matrix is

>> nlgr_est.CovarianceMatrix = 1.0e-003 * 0.4749 0.0012 -0.0000 -0.0084 0.0012 0.0004 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0084 -0.0000 0.0000 0.0002

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40 Model evaluation and parameter estimation             

Figure 4.8. Prediction errors for the first estimation run in Section 4.6.1. Ideally the

prediction errors should be white noise. The estimation is carried out to minimize the errors and these plots show how the predicted errors are distributed around zero.

As was done above, evaluation of the covariance matrix must also be done with consideration taken to the absolute values of the parameters. When this is done the result tells that the inductance has the largest variance, i.e., the highest un-certainty. This gives us a reason to examine whether the inductance is redundant in the simulation model, which is done in Section 4.6.3.

4.6.2 Estimation of remaining motor constants

In this section we estimate the remaining motor constants that were left out in the first estimation run. We keep the resistance and inductance as estimation parameters for a comparison. These parameters are kept because of inconsistences and large estimated standard deviation. The values for inertia and load are taken from the data sheets. In Chapter 4.5.2 all DC-Motor parameters are listed and we see that the motor constants, c1Φ, c2Φ and the efficiency, ηM, were not considered

in the above estimation. The same data set was used for this purpose as described in Section 4.6.1. The estimated parameter values are listed in Table 4.2. The simulation result with these estimated parameters is shown in the next plot, Figure 4.9.

Both estimated models are good at describing the motor speed and even better at describing the position. But the current is in both estimations rather badly

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4.6 Validation 41 Parameter estm value standard dev init (table) value

Resistance [Ohm] 2.11041 0.0387357 0.708

Inductance [H] 2.47601e-5 38.388e-5 4.37e-5

Motor constant 1 [Nm/A] 0.0118198 0.00029829 0.0122 Motor constant 2 [V/rps] 0.0776828 0.000627679 0.0769

Efficiency [-] 0.960724 0.0283763 0.82

Table 4.2. Estimation result from second run with standard deviation and table values

                             

Figure 4.9. Measurement compared to estimated model, second run in Section 4.6.2.

The position has the best fit and the current has the lowest fit. The dynamics at startup and the statical level are less satisfactory depicted.

modeled. Especially the dynamic part at startup is not depicted properly. Also for the second model the inductance has the highest relative variance. Next, we are therefore examining the estimation results for a model with no inductance considered.

4.6.3 Estimation without inductance

The nonlinear gray box model needs some rearranging when the inductance is taken out of consideration. We then get a second order system, because there is a static relation between the current, i.e., the former state x1 and the speed, i.e.,

Simulation of dynamic and static behavior of an electrically powered lift gate

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Simulation of dynamic and static behavior of an

electrically powered lift gate

Simulation of dynamic and static behavior of an

electrically powered lift gate

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Abstract

Sammanfattning

Acknowledgments

Contents

List of Figures

Notational Conventions

Chapter 1

Introduction

1.1

Purpose

1.2

Background

1.3

Limitations

1.4

Substantial goals

1.5

Method

1.6

Structure of the thesis

Chapter 2

Background theory

2.1

Spindle drive actuator

2.1.1

Description of the actuator

2.2

Equations from physics

2.2.1

Kirchoff’s voltage law

2.2.2

Newton’s second law

2.2.3

Steiner’s theorem

2.3

System identification

Chapter 3

Modeling

3.1

Subsystems

3.1.1

DC-Motor

3.1.2

Gear

3.1.3

Spindle

3.1.4

Spring

3.1.5

Lift gate

3.2

Derivations

3.2.1

Lift gate mass moment of inertia

3.2.2

Lift gate angular displacement

3.2.3

Torque angular dependence

3.2.4

Torque from gravity force

3.2.5

Torque from spring force

3.2.6

Spindle raise angular dependence

3.2.7

Lift gate angular acceleration

3.3

Extensions

Chapter 4