IN
DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS
,
STOCKHOLM SWEDEN 2019
Get a Grip
Dynamic force adjustment in robotic gripper
ELLEN ANDERSON
MARTIN GRANLÖF
Get a Grip
Dynamic force adjustment in robotic gripper
ELLEN ANDERSON elland@kth.se MARTIN GRANL ¨OF mgranlof@kth.se
Bachelor’s Thesis at ITM Supervisor: Nihad Subasic
Examiner: Nihad Subasic
Abstract
Autonomous mobile robots are on the rise and are to be expected on the market in about 5-10 years. Several chal-lenges need to be solved for this to happen, and the most crucial ones are to develop versatile and safe robots. The Get a Grip robot is a dynamic force adjustment grip-per using inputs from two different sensory systems. The construction of the robot consists of two parallel gripper plates moved by a rack and pinion gear attached to a di-rect current (DC) motor. Embedded into one of the plates is a Force Sensitive Resistor (FSR) for input of the grip-per’s exerted force. Mounted to the other plate is a self constructed Slip sensor used for measuring the occurrence of slip and slip rate. A surrounding crane for mounting of the gripper and lifting was also constructed.
The idea of this bachelor’s thesis project is to enable lifting of objects with unknown weight without the gripper exert-ing more force than necessary. This is somethexert-ing that will be useful in both industrial applications and in household robots in the future.
In order to realize the concept two different methods for cal-culating the gripper’s applied force were tested, one using motor current and the other using a FSR sensor. Through testing it was concluded that the FSR sensor was the method giving better accuracy and consistency.
Proportional–Integral–Derivative (PID) controllers were then tested for both setting force references for the gripper using the Slip sensor as input, and controlling the exerted force in the gripper using the FSR as input. The results led to two PID controllers thought to be sufficient as starting points for further testing of the complete system.
Keywords
Mechatronics, Robot Gripper, Force Control, Slip sensor, PID controller
Referat
Dynamisk kraftsanpassning i robotklo
Mobila autonoma robotar f¨orv¨antas vara p˚a marknaden in-om de n¨armaste 5-10 ˚aren. F¨or att det h¨ar ska ske ¨ar det m˚anga utmaningar som beh¨over l¨osas och de mest kritiska ¨ar att utveckla m˚angsidga och s¨akra robotar.Get a Grip-roboten ¨ar en dynamisk kraftanpassande
ro-botklo som tar insignaler fr˚an tv˚a olika sensorsystem. Kon-struktionen best˚ar av tv˚aparallella plattor som f¨orflyttas av kuggst¨anger och kugghjul drivna av en DC motor. Inbyggt i en av kloplattorna finns en tryckk¨anslig kraftsensor (FSR) monterad f¨or att registrera kraften som klon genererar. P˚a den andra kloplattan sitter en egenkonstruerad glidsensor som registrerar om glidning sker och sj¨alva glidhastighet. En kran f¨or att montera klon och lyfta den konstruerades ¨aven.
Id´en bakom detta kandidatexamens projektet ¨ar att klon ska kunna lyfta ett objekt med ok¨and vikt utan att anv¨anda mer kraft ¨an n¨odv¨andigt. Det ¨ar n˚agot som kommer va-ra anv¨andbart b˚ade vid industriella till¨ampningar och hos hush˚allsrobotar i framtiden.
F¨or att realisera konceptet testades tv˚a olika metoder f¨or att estimera kraften klon genererar, den f¨orsta genom mo-torstr¨ommen och den andra genom en FSR sensor. Tester genomf¨ordes f¨or b˚ada metoderna och slutsatsen blev att FSR sensorn gav b¨ast noggranhet och var mest konsekvent. PID-regulatorn, f¨or best¨amning av kraftreferens, med in-signal fr˚an glidsensorn och PID-regulatorn, f¨or genererad klokraft, med insignal fr˚an FSR:n testades separat. Resul-tatet blev tv˚a PID-regulatorer som ans˚ags tillr¨ackliga f¨or forts¨atta tester med b˚ada regulatorerna tillsammans. Nyckelord
Mechatronik, Robotklo, Kraftreglering, Glidsensor, PID-regulator
Acknowledgements
During these five months of our bachelor degree we have received loads of help from others whom we would like to thank.
First we would like to thank our supervisor Nihad Subasic for his feedback through the project and Staffan Qvarnstr¨om and Thomas ¨Ostberg for providing us with the components for the construction and with guidance when needed. We would like to thank the course assistants, Seshagopalan Thorapalli Muralidharan and Sresht Iyer, for their willingness to help with all sorts of problems, even outside the scheduled time.
Secondly, we would like to thank the Prototype Center for providing us with ma-terials and help with the laser cutters and the 3D-printers when constructing the demonstrator. We also like to thank Ulf Gustavsson for helping us chamfer our motor shafts.
Finally we would like to thank our fellow course mates for all the feedback and help during these months, both at the seminars and in the lab. We especially want to thank Group 2 and Group 29 for their help regarding a component in the con-struction that we had a trouble with constructing.
Martin Granl¨of & Ellen Anderson Royal Institute of Technology, Stockholm, May 2019
Contents
1 Introduction 1 1.1 Background . . . 1 1.2 Purpose . . . 2 1.3 Scope . . . 2 1.4 Method . . . 3 2 Theory 5 2.1 Microcontroller . . . 5 2.2 Robot Grippers . . . 62.3 Force Modulation Methods . . . 7
2.3.1 Motor Current Regulation . . . 7
2.3.2 FSR sensor . . . 8 2.4 Slip sensors . . . 9 2.4.1 Mechanical roller . . . 9 2.5 Control Theory . . . 10 2.5.1 PID Controller . . . 10 2.5.2 Cascade Controller . . . 11 2.5.3 Filtering . . . 11 3 Demonstrator 13 3.1 Hardware and electronics . . . 13
3.1.1 Crane . . . 13 3.1.2 Robot Gripper . . . 14 3.1.3 Slip sensor . . . 14 3.1.4 Circuit . . . 15 3.1.5 Motor driver . . . 16 3.1.6 FSR . . . 16 3.2 Software . . . 16 3.2.1 Control System . . . 17 3.2.2 Crane . . . 18 3.3 Testing . . . 18 3.3.1 Current Regulation . . . 18
3.3.3 Force Regulation . . . 19
3.3.4 System Performance . . . 19
4 Result 21 4.1 Current Regulation . . . 21
4.2 Force Reference Regulation . . . 23
4.3 Force Regulation . . . 24
4.4 System Performance . . . 25
5 Discussion and conclusions 27 5.1 Discussion . . . 27
5.1.1 Current Regulation . . . 27
5.1.2 Force Reference Regulation . . . 28
5.1.3 Force Regulation . . . 28
5.1.4 System Performance . . . 29
5.2 Conclusion . . . 29
6 Recommendations and Future work 31 Bibliography 33 A Test Results 37 A.1 FSR Calibration . . . 37
A.2 Force Reference Regulation . . . 39
A.3 Force Regulation . . . 42
A.4 System Performance . . . 45
B Code 47 B.1 Arduino Code . . . 47
List of Figures
2.1 Top down view of the chip-set Arduino Uno rev 3. [5] . . . 5
2.2 Properties of different drive systems in impactive grippers [7] . . . 6
2.3 Right: Spindle drive. Left: Rack and pinion drive [7] . . . 7
2.4 Torque-Current curve for a DC motor [9] . . . 7
2.5 Present resistance due to applied force. Curve in logarithmic scale [11]. 8 2.6 A FSR Voltage Divider [11] . . . 9
2.7 Cross sectional view of mechanical roller used in a sensory system for measuring slip [12] . . . 9
2.8 Figure of two systems in series with each other [17]. Made in Power-point[19] . . . 11
2.9 Figure of a cascade regulated system with the inner feedback loop inside the dotted box [17]. Made in Powerpoint[19] . . . 11
3.1 Image of the crane and gripper platform. Made in Solid Edge ST10, and rendered in KeyShot [23] . . . 13
3.2 Image of the gripper with the two plates attached through the rails in the gripper platform to the racks on the opposite side. On one of the plates is the Slip sensor mounted and on the other the FSR sensor. Made in Solid Edge ST10 and rendered in KeyShot[23] . . . 14
3.3 Picture of the Slip sensor with its mounting structure attached. Made in Solid Edge ST10 and rendered in KeyShot[23] . . . 15
3.4 Figure of the complete circuitry of the project. Created with Fritzing[21] 15 3.5 Picture of the Adafruit TB6612 motor driver[22] . . . 16
3.6 Flowchart of the algorithm for the Get a Grip project. Made in draw.io[24] 17 4.1 Graph showing exerted force as a function of motor current. The blue circles are the measured values from test 1 and the red line is the linear adaptation to these points. The equation of the linear adaptation is shown above the graph. Made in Matrix Laboratory (Matlab)[26] . . . . 21
4.2 Graph showing exerted force as a function of motor current. The blue circles are the measured values from test 2 and the red line is the linear adaptation to these points. The equation of the linear adaptation is shown above the graph. Made in Matlab[26] . . . 22
4.3 Graph showing exerted force as a function of motor current. The blue circles are the measured values from test 3 and the red line is the linear adaptation to these points. The equation of the linear adaptation is shown above the graph. Made in Matlab[26] . . . 22 4.4 Graph comparing the force reference over time with and without
pre-filtering. The blue line is the force reference when using a moving av-erage low pass filter and the orange line is without the filter. Made in
Excel[25] . . . 23
4.5 Graph of the force reference plotted over time. Made in Excel[25] . . . . 23 4.6 Graph of the exerted force plotted over time. Made in Excel[25] . . . 24 4.7 Graph comparing exerted force between different controllers plotted over
time. The blue line uses a proportional control coefficient of 60 and the orange 70. Made in Excel[25] . . . 24 4.8 Graph of the exerted force plotted over time. Made in Excel[25] . . . 24 4.9 Graph showing how the exerted force tracks the force reference during a
lift sequence with an object weighing 250 grams. Made in Excel[25] . . . 25 4.10 Graph showing how the exerted force tracks the force reference during a
lift sequence with an object weighing 300 grams. Made in Excel[25] . . . 25 A.1 Graph of the resistance in the FSR plotted over the applied force. The
blue circles are the values in resistance taken over three separate tests and the red line is the curve adaptation. Tests performed 2019-04-04. Made in Matlab[26] . . . 37 A.2 Graph of the resistance in the FSR plotted over the applied force. The
blue circles are the average values in resistance taken over three separate tests and the red line is the curve adaptation. Tests performed 2019-04-22. Made in Matlab[26] . . . 38 A.3 Graph of the resistance in the FSR plotted over the applied force. The
blue circles are the average values in resistance taken over three separate tests and the red line is the curve adaptation. Tests performed 2019-04-29. Made in Matlab[26] . . . 38 A.4 Graph of the force reference over time using only a proportional regulator
and no prefiltering. Made in Excel[25] . . . 39 A.5 Graph of the force reference over time using only a integral regulator
and no prefiltering. Made in Excel[25] . . . 39 A.6 Graph of the force reference over time using only a derivative regulator
and no prefiltering. Made in Excel[25] . . . 39 A.7 Graph of the force reference over time using only a proportional regulator
and a EMWA with α = 0, 2. Made in Excel[25] . . . 40 A.8 Graph of the force reference over time using only a integral regulator
and a EMWA with α = 0, 2. Made in Excel[25] . . . 40 A.9 Graph of the force reference over time using only a derivative regulator
A.10 Graph of the force reference over time using a proportional integral reg-ulator and a EMWA with α = 0, 2. Made in Excel[25] . . . 41 A.11 Graph of the force reference over time using a integral derivative
regu-lator and a EMWA with α = 0, 2. Made in Excel[25] . . . 41 A.12 Graph of the force reference over time using a proportional integral
derivative regulator and a EMWA with α = 0, 2. Made in Excel[25] . . . 41 A.13 Graph of the exerted force over time using a proportional regulator.
Made in Excel[25] . . . 42 A.14 Graph of the exerted force over time using a proportional integral
regu-lator. Made in Excel[25] . . . 42 A.15 Graph of the exerted force over time using a proportional integral
regu-lator. Made in Excel[25] . . . 42 A.16 Graph of the exerted force over time using a proportional integral
deriva-tive regulator. Made in Excel[25] . . . 43 A.17 Graph of the exerted force over time using a proportional integral
deriva-tive regulator. Made in Excel[25] . . . 43 A.18 Graph of the exerted force over time using a proportional integral
regu-lator. Made in Excel[25] . . . 43 A.19 Graph of the exerted force over time using a proportional integral
regu-lator. Made in Excel[25] . . . 44 A.20 Graph of the exerted force over time using a proportional integral
regu-lator. Made in Excel[25] . . . 44 A.21 Graph of the exerted force over time using a proportional integral
deriva-tive regulator. Made in Excel[25] . . . 44 A.22 Graph of the exerted force over time using a proportional integral
regu-lator. Made in Excel[25] . . . 45 A.23 Graph of the exerted force over time using a proportional integral
deriva-tive regulator. Made in Excel[25] . . . 45 A.24 Graph showing how the exerted force tracks the force reference during a
lift sequence with an object weighing 100 grams. Made in Excel[25] . . . 45 A.25 Graph showing how the exerted force tracks the force reference during a
lift sequence with an object weighing 150 grams. Made in Excel[25] . . . 46 A.26 Graph showing how the exerted force tracks the force reference during a
List of Abbreviations
CPU Central Processing Unit DC Direct CurrentEMA Exponentially Moving Average
EWMA Exponentially Weighted Moving Average FSR Force Sensitive Resistor
I/O Input/Output
I2C Inter-Integrated Circuit LCD Liquid Crystal Display Matlab Matrix Laboratory
PID Proportional–Integral–Derivative PWM Pulse Width Modulation
USB Universal Serial Bus
Nomenclature
α Weighted Factor xi Current Average
xi−1 Previous Average
dg Diameter of Gear to Convert Rotational Movement to Linear
dr Outer Diameter of Roller e(t) Static Error I Motor Current KD Derivative Coefficient KI Integral Coefficient Kp Proportional Coefficient
n Amount of Data Points nr Available Rotation
r(t) Reference Signal
T Motor Torque
u(t) Control Signal xi Current Value
xi−n First Measured Value
y(t) Output Signal
z Aforementioned Intermediate Signal
zref Aforementioned Intermediate Reference Signal
F Gripper Force
Chapter 1
Introduction
1.1
Background
According to the Gartner Hype Curve for Emerging Technology from 2018 au-tonomous mobile robots are on the rise and are to be expected on the market in about 5-10 years [1]. Autonomous robots are freely moving and adaptable robots that are able to execute tasks in complex environments, without human supervi-sion. The main challenges are to make the robots able to operate even if crucial parameters are unknown [2]. For example, a task asked of a robot in future homes could be to pick up an egg and later a package of milk without cracking one and dropping the other. The force control of the robot grippers will in these cases be of great importance.
Most of the robots used today are found in the industry and are specialized in a certain area, with very specific and predetermined tasks. The grip force in the gripper depends on the known weight of the object that is being manipulated. An easy way to do this is by having tactile sensors built in to the grippers to evaluate the grip force applied by the grippers on the workpiece. However, in industrial envi-ronments it might in some cases be inappropriate to rely on fragile, external sensors and the grip force can instead be controlled by changing the current through the motor in the gripper [3].
Regardless of the force modulation method, input from an external party, e.g., a human who knows the weight of the workpiece, will be required to set the force needed to grasp the object. This works well in the industry but for future au-tonomous robots they have to manage varying tasks without relying on an external party.
CHAPTER 1. INTRODUCTION
1.2
Purpose
In this bachelor’s thesis project a robot gripper, that takes the weight of an object in to account when lifting it, is built. The purpose of the gripper is to only exert as much force necessary to create the required friction to safely lift, without the weight of the object being set beforehand.
To control the grip force, different force modulation methods can be used. In one of these methods, the proportionality between the current through the motor and its torque is used to regulate the grip force. It will be examined how consistent and accurate this relationship is and if it is a suitable method for this project.
In order for the gripper to lift an object with unknown weight the grip force is regulated by using input from a slip sensor which registers if the object is slipping, and then adjusts the grip force accordingly. It will be analyzed which PID controller gives the optimal performance for this application.
To summarize; The main purpose of this project is to answer the following research questions:
• How consistent and accurate is the relationship between the torque and the motor current in the gripper?
• Which PID controller gives optimal performance of the system in regards to overshoot, rise time and reaction time?
1.3
Scope
With the given time and budget the main focus of this project is on force modu-lation methods and detection of slip in a robotic gripper. Tactile robotic grippers already exist and the purpose of this project is not to compete with these in regards to performance but with simplicity and cost.
A robotic crane will be built to enable a consistent platform for testing the gripper. The crane itself will not be equipped with a sensor for object identification and therefore the object will be placed at a preset location and a hard coded signal will be used to initiate gripping phase.
Constraints on the workpiece will be rigid, uniform shape and texture. This leads to the main unknown parameter being the weight of the object and the coefficient of friction. Since the system uses the input signal from the slip sensor, the unknown coefficient of friction will not be significant in the regulation of the grip force, which makes the construction more robust. The weight interval of the objects tested will
1.4. METHOD
be kept within the range 100 grams to 500 grams since it is considered a suitable interval for testing due to the components available.
1.4
Method
The project can be separated into two steps, corresponding to each of the research questions.
The first step is to construct the gripper which will be of a simple design with two parallel plates moving towards each other. The force modulation method by regulating the current through the motor will be tested using a force sensor embed-ded into one of the claws of the gripper. The grip force will be registered through the force sensor for different motor voltages and the test results will be compared to investigate the accuracy and consistency of the method.
In the second step the weight of the object will be unknown and the gripper will use the ”trial and error”-concept. By receiving input from the slip sensor the force is increased if slippage is detected until no slip occurs. The task of determining slip will be performed by a sensor using a mechanical roller built into one of the claws in the gripper. During the prehension phase the roller will be in direct contact with the lifted object. If the roller starts turning when the object is manipulated upwards the sensor will tell the system that slippage is occurring.
Chapter 2
Theory
2.1
Microcontroller
There are currently many different microcontrollers on the market but in general they all consist of three main parts; a Central Processing Unit (CPU), Memory and Input/Output (I/O) ports. All of these three parts are connected through buses for internal system communication. The I/O ports enable communication between the microcontroller and the external components by collecting and transmitting signals. The CPU is the brain of the circuit and is where the signals are interpreted and processed. The memory is where the instructions given by the operator is stored, e.g. what signals to transmit when a certain combination of signals have been col-lected.
The performance of the microcontroller is also manly decided by these three com-ponents. The CPU sets the limit for the speed of the system, the memory limits the size of the program that can be run on the chip-set and the I/O ports limit the number of unique input and output signals that can be managed at the same time[4]. The microcontroller that will be used in this project is an Arduino Uno rev 3 and is shown in Figure 2.1. The Uno rev 3 is based around the ATmega328 micropro-cessor and has a total of 14 digital I/O ports where 6 of these are digital PWM ports. The card also has 6 analog ports and 2 power ports, one 3,3V and one 5V, and corresponding ground pins. The controller runs at an internal clockspeed of 16MHz and has 35kB of Flash Memory[5].
CHAPTER 2. THEORY
2.2
Robot Grippers
The gripper is the link between the robot arm and the object that is to be manip-ulated. As robots can be asked to perform different tasks it is required that the gripper is versatile since switching to a new gripper for each of these operations is not cost effective[6].
There are different methods for prehension and holding an object which suits some objects better than others, e.g. using needles or vacuum suction. One of the most commonly used methods is the impactive method where the gripper consists of claws or jaws and the adhesion is obtained merely by the frictional forces between the gripper and the surface of the object.
Impactive grippers can be powered by various drive systems, which gives the grip-per diverse progrip-perties, see Figure 2.2 where full circle means advantageous and empty circle means unfavourable [7]. Electric motor powered grippers have several advantages [8]:
Figure 2.2. Properties of different drive systems in impactive grippers [7]
• Position control: The range that is needed to pick up objects can easily be controlled
• Detect grip: Through motor encoders it can be determined when the work-piece is grasped.
• Force control: The current in the electric motor is directly proportional to the torque the motor applies, which gives the gripper the ability to control the force that impacts the workpiece.
• Efficiency on power and maintenance: Many companies shift from pneumatic drives to electric due to the reduced operating cost and energy savings it
2.3. FORCE MODULATION METHODS
provides. Electric motor grippers tend to be cleaner since they do not need any fluids to work, and have no general operation problems like leaks and losses that might occur when using compressed air.
The lack of fluids in the construction also simplifies the design of the gripper. Most of the force converters in electromechanical powered grippers are based on spindle or rack and pinion gears, shown in Figure 2.3, where the prime mover usually is any form of electrically commutated DC servo motor [7]. Because of all these factors electric grippers are getting more common in the industry and are thought to become even more frequently occurring in the future.
Figure 2.3. Right: Spindle drive. Left: Rack and pinion drive [7]
2.3
Force Modulation Methods
2.3.1 Motor Current Regulation
When using a DC motor the proportionality between the current and torque can be used to change the force a gripper is exerting on an object. This relationship is given by equation 2.1
T = I · kT (2.1)
where T is the output torque of the DC motor, I is the current through the motor and kT is the torque constant which is specific to the motor. This equation can be
represented by a linear curve where the torque constant is the slope of this curve, shown in Figure 2.4
CHAPTER 2. THEORY With the output torque the force applied by the gripper can be calculated using the diameter, dg, of the gear which converts the rotational movement into a linear
motion, according to equation 2.2 [9]
T = F · dg (2.2)
To find the relationship between the current and the exerted force, tests can be done using a force senor in the gripper to measure this force at different current inputs. By adding the results into a graph and using a curve fitting method an expression for the relationship between the current and the force can be found. This expression can later be used to choose the specific current which will make the gripper apply the desired grip force.
However, this force modulation method has in practise some problems when the gripper is used during a long period of time. Due to the thermal changes when the motor temperature increases, parameters in the motor may change and the eval-uated expression might not be applicable. The error can be minimised through recalibration [10].
2.3.2 FSR sensor
A Force Sensitive Resistor sensor, FSR sensor, works like a resistor where the re-sistance is proportional to the force applied onto the sensor. The rere-sistance can be translated to force by using a curve such as the one in Figure 2.5.
Figure 2.5. Present resistance due to applied force. Curve in logarithmic scale [11].
To make the applied force more easily controlled when used in a system, the resis-tance of the FSR sensor can be converted to voltage using a voltage divider, see Figure 2.6. The FSR sensor is connected to a measuring resistor in the circuit and the output voltage is then given by equation 2.3 [11]
Vout =
RMV+
(RM + RF SR) (2.3)
2.4. SLIP SENSORS
Figure 2.6. A FSR Voltage Divider [11]
2.4
Slip sensors
The term slip in relation to robot grippers is used to describe the situation where the frictional forces are less than the gravitational force on a lifted object. There are many ways to measure and calculate slip rate such as distance measurements using ultrasound sensors, cameras, or light emission of different frequencies. Other methods are relating vibrations created in the gripper during slip to the rate of slip through different dielectric membranes. There are also methods using mechanical rollers built in to the gripper in order to measure object displacement and displace-ment rate [14].
2.4.1 Mechanical roller
The most common construction for slip measurements using a mechanical roller is shown in Figure 2.7. The construction consists of a hollow roller with internal ball bearings fitted on a leaf spring. The leaf spring is attached to the back of the gripper surface and the roller makes contact with the lifted object through a cut-out in the gripper surface. The leaf spring makes sure that the roller stays in contact with the surface of the object and also allows the surfaces of the gripper to stay in contact with the object being lifted. When the object starts to slip, it is detected by the subsequent rotation of the roller. The rotation is then measured by a potentiometric sensor or a photoelectric sensor through readouts of slits in the roller itself or an extra disk. Effectively this is an angle measurement that in the code together with the dimension of the hardware will be interpreted as a slip distance [12].
Figure 2.7. Cross sectional view of mechanical roller used in a sensory system for measuring slip [12]
CHAPTER 2. THEORY There have been several works done on grippers with slip sensors using similar constructions as the mechanical roller described above. In these grippers the rollers have also been used as the claws of the grippers, though it restricts the possibility to have other sensors built into the claw construction [13]-[15]. By taking the rate of the slippage, i.e., the velocity of the rotation of the roller, into account the grip force can be regulated in one single step. This will reduce the total slip time of the object compared to incrementally increasing the force until slippage is eliminated [16].
2.5
Control Theory
2.5.1 PID Controller
A Proportional-Integral-Derivative (PID) controller, is a commonly used controller in feedback systems for industries. The proportional part effects the speed of the system but does nothing to decrease static errors in the system. The integral part of the controller deals with static error by adding up the previous error of the system over time through an integrator, but may affect the stability of the system. The derivative part of the controller tries to improve stability by predicting future errors of the system. Together these three parts aim to increase speed and stability and while decreasing static error. The equation for signal regulation can be seen in equation 2.4 and 2.5. u(t) = Kpe(t) + KI Z t 0 e(τ)dτ + KD d dte(t) (2.4) e(t) = r(t) − y(t) (2.5)
In equation 2.4 u(t) is the control signal and e(t) is the static error that consists of the differential between the reference signal, r(t), and the output signal, y(t), shown in equation 2.5. The constants Kp, KI and KD are the corresponding coefficients
of the three different parts of the controller, P, I and D[17].
In robotic gripper applications a PID controller can be used to control the force exerted on an object to achieve lift. The requirement to lift an object is that the gravitational force of the object is equal to the exerted force of the gripper multiplied with the coefficient of friction [7]. The exerted force can be registered through a force sensor in the gripper and by using it as the output signal of the system the er-ror can be calculated with equation 2.5, where the required force is the reference[18]. This would require that the weight of the object and the coefficient of friction are known. When working with objects with unknown weights the required force is replaced with a force that dependents on the input from a slip sensor, which is then set as the reference in equation 2.5 [13].
2.5. CONTROL THEORY
2.5.2 Cascade Controller
When dealing with two or more systems in series with each other a cascade regulator might be realisable. The main building block of the regulator is a measurable intermediate signal between the systems. An example of this can be seen in Figure 2.8, where z is the aforementioned intermediate signal.
Figure 2.8. Figure of two systems in series with each other [17]. Made in
Power-point[19]
The idea is to use a regulator to control z as if it was the input of the system. This can be done using one external feedback loop for calculating the reference signal
zref using the output y, and one internal to actually regulate z. This can be seen
in Figure 2.9, where the internal loop is inside the dotted box.
Figure 2.9. Figure of a cascade regulated system with the inner feedback loop inside the dotted box [17]. Made in Powerpoint[19]
One thing to note about cascade regulation is that the performance of the system as a whole is dependent on the internal feedback loop. The faster the internal loop is the better the system will perform in regards to speed, time delay and stability [17].
2.5.3 Filtering
When handling signals, either analog or digital, the quality is often far from ideal. Signals often contain random variations of different, but often high, frequencies. This leads to the need of filtering the signal before computation to ensure more reliable and less random results. Commonly used filters are averaging filters or low pass filters.
An averaging filter takes several data points in to account and uses the mean of these as the signal to be accessed. In computer applications a recursive version of
CHAPTER 2. THEORY an averaging filter is often used, a moving average filter. The formula for a moving average filter is shown in equation 2.6.
xi = xi−1+ 1
n(xi− xi−n) (2.6)
In equation 2.6 xi, and xi−1 are the current and previous average and xi, xi−n and
n are the current value, the first measured value and the amount of data points.
This method places equal emphasis on each data point and leads to a data point in the past having the same influence as a current data point. In a system where the signal is stable and not trending in any direction this may be desirable, but in most situations it is not. In this case another moving average filter is more suitable, namely the Exponentially Weighted Moving Average filter (EWMA or EMA). The general formula for a EWMA filter can be seen in equation 2.7 and is also identical to the discrete first order low pass filter.
xi = αxi−1+ (1 − α)xi (2.7)
In equation 2.7 xi is still the current average, xi−1 is the previous and xi is the
current value. The main differences are the weighting factor α and that there is no longer any need to track the amount of data points. The filter, through α, also enables the user to chose how much bias the filtered signal should have towards previous measurements, which may be useful when dealing with a noisy signal [20].
Chapter 3
Demonstrator
3.1
Hardware and electronics
3.1.1 Crane
The crane can be seen in Figure 3.1 and consists of a wooden box with four metal rods and one lead screw connected with a flexible coupling to a 12V DC motor. A platform which the gripper is attached to moves up and down due to the rotation of the lead screw and a nut in the platform. The wooden top plate and the acrylic frame together with the four rods stabilize the platform during the up and down motion.
Figure 3.1. Image of the crane and gripper platform. Made in Solid Edge ST10, and rendered in KeyShot [23]
On top off and beneath the gripper platform two micro switches are placed that are pushed down when the platform reaches the top or bottom of the rods. This will make automation of the platform movement easier.
CHAPTER 3. DEMONSTRATOR
3.1.2 Robot Gripper
The gripper, that is presented in Figure 3.2, uses two parallel plates for picking up objects. The plates are connected through rails on the bottom of the gripper platform to the racks on top of the platform. The traversal motion of the plates is then accomplished by using a rack and pinion drive with a DC motor installed on the top side of the platform.
One of the plates has a rectangular cut-out where the Slip sensor is placed, see Figure 3.2. The FSR sensor is mounted on the opposite plate and has a rubber dome attached to its active surface to distribute the force evenly over the whole area. Because of the dome shape a second plate, which can be pushed horizontally towards the sensor, was put between the sensor and the object to ensure that the direct contact between the claw and the lifted object is sufficient.
Figure 3.2. Image of the gripper with the two plates attached through the rails in the gripper platform to the racks on the opposite side. On one of the plates is the Slip sensor mounted and on the other the FSR sensor. Made in Solid Edge ST10 and rendered in KeyShot[23]
3.1.3 Slip sensor
The mechanical roller style Slip sensor can be seen in Figure 3.3. The main struc-ture consists of a plate at the back for attaching the two tension springs, the holder and the roller itself. A potentiometer is fitted into the roller in such a way that its rotation is locked in with the rotation of the roller. On the opposite side of the potentiometer is a deep grove ball bearing mounted into the roller, providing free rotation on that side.
The final construction of the Slip sensor ended up using a rotational potentiometer with approximately 315° of rotation where the rotation was picked up through an analog port on the Arduino. Together with the roller, with a diameter of 22mm, this enables a measurement interval of around 60mm using equation 3.1.
ls=
nr
360°πdr (3.1)
3.1. HARDWARE AND ELECTRONICS
The dr in the equation is the outer diameter of the roller and nr is the available
rotation in degrees.
Figure 3.3. Picture of the Slip sensor with its mounting structure attached. Made in Solid Edge ST10 and rendered in KeyShot[23]
Due to the force requirement for rotation double sided tape had to be attached onto the surface of the roller to retain the contact between the roller and the lifted object.
3.1.4 Circuit
The circuitry of the project consists of seven main systems which can be seen in Figure 3.4. The brain of the whole project, the Arduino Uno, can be seen in the top. It was decided that the Arduino will not get its power from the external 12V power supply due to the convenience of it being powered by a Universal Serial Bus (USB) port of a laptop and at the same time enabling easy tuning and monitoring.
CHAPTER 3. DEMONSTRATOR One of the seven systems that is not mentioned previously in Theory is the Liq-uid Crystal Display (LCD) and its control Inter-Integrated Circuit (I2C) module. The theory behind the inner workings of these will not be mentioned due to them not being an integral part of the project and was simply added just for ease of demonstration.
3.1.5 Motor driver
The chosen motor driver for this project was the Adafruit TB6612, shown in Figure 3.5, which is mainly used for stepper motors. Given the fact that a stepper motor was first intended to be used for grip modulation the driver satisfied the require-ments for the intended motor. The driver is suitable for controlling two DC motors using Pulse Width Modulation (PWM) and reversing voltage polarity which is what the final configuration of the gripper ended up being.
Figure 3.5. Picture of the Adafruit TB6612 motor driver[22]
The driver runs on a 2,7V to 5V logic voltage and has a separated motor voltage suitable for operation between 4,5V and 13,5V. As previously mentioned the driver has two channels each suitable for 1,2A of continuous current and peak current at 3A for about 20 milliseconds [22].
3.1.6 FSR
The FSR is connected, as previously mentioned in Theory 2.3.2, to a measuring resistor, in this case a 12 kΩ resistor. The measuring resistor is connected to an analog port on the Arduino which enables a 1024 segmented measuring interval between 0 and 5V for the voltage across the resistor.
3.2
Software
Since the two main inputs for the program on the Get a grip robot are the FSR and the Slip sensor these are also the main focus in the code, that can be found in Appendix B.1. The triggers for changes in motor voltages and the different states of the program are all controlled by these two sensory inputs. The logic of the code adheres to the flowchart that can be seen in Figure 3.6.
3.2. SOFTWARE
Figure 3.6. Flowchart of the algorithm for the Get a Grip project. Made in
draw.io[24]
3.2.1 Control System
Since the goal of this project is to lift objects with unknown weights it requires as mentioned in Theory 2.5.1 input from a slip sensor to set a force reference value using a PID regulator. The force reference regulator takes an analog voltage input, which together with equation 3.1 is used to segment the total voltage input over the total available slip distance that is measurable for the sensor. This in turn makes
CHAPTER 3. DEMONSTRATOR each voltage value equivalent to a certain distance in millimeters. This calculated value is then processed by a low pass filter and subtracted with the previous value to check if any slippage has occurred since the last run of the main loop. The dif-ference is compared to a redif-ference, set to zero to avoid slip, and then a new force reference is calculated by the controller.
To achieve the calculated force reference another PID regulator is used to con-trol the exerted force in the gripper, given by the FSR sensor embedded into the gripper. By taking the analog voltage over a 12 kΩ resistance as input and using equation 2.3 the resistance of the FSR is calculated. Through previous tests, using set weights to assess how the resistance of the FSR changes under load, a force is extracted. The test results of the FSR calibrations can be seen in Appendix A.1. This force is then compared to the force reference according to equation 2.5 and the motor voltages is adjusted accordingly.
3.2.2 Crane
The software that the crane is operated with works like a basic switch case algorithm. In the initialization phase it runs the platform motor downwards until the bottom micro switch is triggered. It then waits for the exerted force in the gripper to reach the initial force reference before turning the platform motor on and making the platform rise.
To finish off the lifting sequence it runs the platform motor upwards until the upper micro switch is triggered and then turns it off. To repeat the sequence the Arduino reset button is pressed.
3.3
Testing
In this section the tests that were performed on the sensors and the construction as a whole are presented. These tests aim to answer, or at least partly answer, the research questions posed in this report.
3.3.1 Current Regulation
To answer the first research question How consistent and accurate is the relationship
between the torque and the motor current in the gripper? a built in FSR sensor was
used to register the applied force in the gripper. The FSR sensor was calibrated by doing three sets of tests putting known weights of 100-700 grams to examine the corresponding resistance for each weight. With the average values of the results from the tests an expression using a curve fitting method was evaluated to obtain a relationship between the force applied and the resistance in the FSR sensor. With the calibrated sensor three test cycles on the current regulation method were made. In each test cycle the force was set to have values 1-4 N, aiming for equal
3.3. TESTING
force point in each test cycle, and the resulting current was documented. Between each cycle a break of five minutes was taken, so the motor could cool slightly.
3.3.2 Force Reference Regulation
To partly answer the second research question ”Which PID regulator gives optimal
performance of the system in regards to overshoot, rise time and reaction time?”
several tests were performed on the Force Reference regulator. These tests aimed to guide the tuning of the force reference regulator that takes input from the slip sensor. The tests consisted of manually rotating the slip sensor 180° as smoothly as possible aiming for full rotation in about two seconds. The tests were then performed using a low pass filter and different P, I and D parameters in equation 2.4, where these parameters are the same as the corresponding coefficients. Finally the resulting force reference value were plotted using Excel [25].
3.3.3 Force Regulation
To partly answer the second research question ”Which PID regulator gives optimal
performance of the system in regards to overshoot, rise time and reaction time?”
several tests were performed on the Force regulator. These tests aimed to guide the tuning of the force regulation that takes input from the FSR. The tests consisted of measuring the force displayed by the FSR over time when going from a force reference of 1 N to 2 N. The test was then performed using different P, I and D parameters. Finally the results were plotted using Excel [25].
3.3.4 System Performance
To answer the second research question ”Which PID regulator gives optimal
perfor-mance of the system in regards to overshoot, rise time and reaction time?” tests on
the complete system and lifting sequence were performed. The results of the two previous tests on the Force Reference Regulator and the Force Regulator were used as a starting point when determining the final PID parameters for the complete system.
The tests consisted of lifting a box with different weights and tuning the parameters of the two regulators in order to maximize the lifted weight without exerting more force than necessary. The complete sequence of the test can be seen in Figure 3.6 and is explained in Demonstrator 3.2.
Chapter 4
Result
4.1
Current Regulation
The results of the current regulation tests can be seen in Figure 4.1, 4.2, and 4.3. The red lines are the linear fittings of the test data of each test and the exact equation of these can be seen just above the graphs.
Figure 4.1. Graph showing exerted force as a function of motor current. The blue circles are the measured values from test 1 and the red line is the linear adaptation to these points. The equation of the linear adaptation is shown above the graph. Made in Matrix Laboratory (Matlab)[26]
CHAPTER 4. RESULT
Figure 4.2. Graph showing exerted force as a function of motor current. The blue circles are the measured values from test 2 and the red line is the linear adaptation to these points. The equation of the linear adaptation is shown above the graph. Made in Matlab[26]
Figure 4.3. Graph showing exerted force as a function of motor current. The blue circles are the measured values from test 3 and the red line is the linear adaptation to these points. The equation of the linear adaptation is shown above the graph. Made in Matlab[26]
4.2. FORCE REFERENCE REGULATION
4.2
Force Reference Regulation
The results of the force reference testing can be seen in Figure 4.4, Figure 4.5, and in Appendix A.2. The PID parameters seen above the graph in Figure 4.5 are the chosen parameters after testing. A low pass filter was applied to the input signal since this resulted in a smoother output.
A couple of basic tests, with separated control parameters, were performed in order to get a better understanding of the characteristics of the sensor readings. These tests, together with more comparisons between filtered and unfiltered results, can be found in Appendix A.2.
Figure 4.4. Graph comparing the force reference over time with and without pre-filtering. The blue line is the force reference when using a moving average low pass filter and the orange line is without the filter. Made in Excel[25]
CHAPTER 4. RESULT
4.3
Force Regulation
The results of the force regulation testing can be seen in Figure 4.6 , Figure 4.7, Figure 4.8 and Appendix A.3. Figure 4.6 shows how the integral coefficient can effect stability and Figure 4.7 shows how proportional amplification effects system stability. The PID parameters shown in Figure 4.8 are the chosen parameters after testing.
Figure 4.6. Graph of the exerted force plotted over time. Made in Excel[25]
Figure 4.7. Graph comparing exerted force between different controllers plotted over time. The blue line uses a proportional control coefficient of 60 and the orange 70. Made in Excel[25]
Figure 4.8. Graph of the exerted force plotted over time. Made in Excel[25]
4.4. SYSTEM PERFORMANCE
4.4
System Performance
The results of the system performance testing can be see in Figure 4.9, Figure 4.10, and Appendix A.4. Figure 4.9 shows how the exerted force in the gripper tracks the force reference when lifting an object weighing 250 grams. This was the maximum achieved weight lifted during the testing of the complete system.
An object weighing 300 grams was also tested, with results in Figure 4.10, to show what happens to the exerted force when the motor current of the gripper DC motor surpassed the maximum allowed current through the motor driver.
Figure 4.9. Graph showing how the exerted force tracks the force reference during a lift sequence with an object weighing 250 grams. Made in Excel[25]
Figure 4.10. Graph showing how the exerted force tracks the force reference during a lift sequence with an object weighing 300 grams. Made in Excel[25]
Chapter 5
Discussion and conclusions
5.1
Discussion
5.1.1 Current Regulation
The results shown in the Current Regulation tests were quite conclusive. They showed that the consistency and accuracy of the motor current is not, at least in this application, suitable for force regulation. The same exerted force seems to require different motor current in the different tests. This leads to different curve adaptations and therefore no usable expression for the PID regulator. Even though it is not conclusive, since the amount of tests where quite low, there also seems to be a downward trend on the slope of the curve further into testing. This may be due to increasing motor temperatures during testing.
There are a couple of conceivable reasons why using motor current as an indicator for exerted force might be difficult in our construction.
The first being that the friction in the plate slots, where the two gripper plates slide, is a bit to high or not consistent throughout the whole motion. This could be due to material choice resulting in a slip stick situation, where the rails stick to the side of the slots and when the motor torque is increased they make a leap before sticking to the sidewalls again. Different claw layouts might be more suitable when using motor current for force modulation.
The second being if the regulation towards the force reference was initiated above or below the reference itself. The mechanics of the the gripper seemed to have an affect on how well the gripper was able to hold and or achieve a certain force. This may also relate to the inconsistent friction between the slots and rails brought up earlier.
One realization made during testing was that at around 4 N of exerted force in the gripper the motor current reached the maximum continues current that the mo-tor driver could handle. This means that in this configuration the maximum force exerted by the gripper is around 4 N.
CHAPTER 5. DISCUSSION AND CONCLUSIONS
The combined results and realizations of this test was one of the main reasons why the following tests, and the project as a whole, used the FSR as input for the Force Regulator.
5.1.2 Force Reference Regulation
The results in the Force Reference Regulation tests showed that there was a need for prefiltering the signal but also that there were quite a lot of dead spots in the measurable interval of the potentiometer. This led to the reference regulation being quite jumpy. It is also worth adding that the way the tests where performed was probably not optimal, but the force requirements at different parts of the rotation varied a lot which limited options regarding testing methods.
As predicted the derivative part of the controller had a negative effect on the force reference regulation, leading to a very unstable output. The proportional part of the controller also had a negative effect on the smoothness of the output, but was deemed a necessary trade-off to enable higher reference setting at lower slip distance. Throughout the tests it was observed that a higher relative integral coefficient, com-pared to the proportional coefficient, lead to smoother reference setting.
The regulator shown in Figure 4.5 was deemed a sufficient starting point for optimisation of the controller.
5.1.3 Force Regulation
The results in the Force Regulation tests were quite straight forward. We were able to achieve a fast initial response, but struggled to eliminate the static error without effecting stability. Since we wanted a controller without overshoot the controller shown in Figure 4.8 was deemed a sufficient starting point for optimisation. The derivative part of the controller was kept at zero due to issues with overshoot and stability. The integral coefficient also seemed to affect the stability if set too high, though the low value might on the other hand be the reason why the static error was hard to eliminate. This might not be a huge problem in regards to performance because insufficient grip force will just lead to the reference increasing and the ex-erted force being increased that way.
Worth mentioning is that the FSR was damage during reconstruction of the claw. This led to the resistance at each given load being cut in half. Two tests were per-formed, the two last graphs seen in Appendix A.1, after damaging the sensor and the FSR seemed to stabilize around a new set of constants. Therefore this should not affect the accuracy in any significant way.
5.2. CONCLUSION
5.1.4 System Performance
The results in the System Performance tests yielded a maximum liftable weight of 250 grams using Force Regulator parameters P = 80, I = 3 and D = 0 and the Force Reference Regulator parameters from separate testing, shown in Figure 4.5.
The reason why both the proportional and integral parameters were increased compared to the result shown in Result 4.3 was due to initial static error. At the start of the test when gripping was initialized the motor torque was too low to get the exerted force close enough to the force reference. This is probably due to the friction in the rails of the claw discussed in Discussion 5.1.1. Unfortunately this adjustment led to some instability at the initial parts of gripping, resulting in over-shoot followed by overcompensation, that can be seen in Figure 4.9. It was decided that this was a better compromise than allowing a high initial static error since this led to higher current drawn by the motor and therefore lower lifting capacity. The static error still remained relatively high.
The main limiter of the system seems to be the motor driver and therefore the limit of the control system is still unknown. Using a driver with higher maximum operational current would allow for further testing.
5.2
Conclusion
To conclude the results and answer the questions posed:
• Motor current as an input for controlling exerted gripper force in the gripper configuration presented in this project is most likely not an accurate or con-sistent method. This being due to the unpredictable friction between the rails and the slots in the gripper and also increasing temperatures in the motor. • A potentiometer might not be optimal for measuring slip due to the varying
force needed to turn it at different parts of the rotational interval. It also has some deadspots that could lead to the output of the regulator being quite jumpy.
• Filtering the signal from the Slip sensor before using it for computation im-proved smoothness of the output noticeably in testing.
• Both PID regulators exhibit good performance during separate testing and characteristics seem to carry over when the two are put together.
• The optimal PID parameters, if the internal friction in the system was lower, are the ones produced in the separate regulator tests. Due to system charac-teristics the slightly higher parameters produced during system performance testing were deemed necessary.
CHAPTER 5. DISCUSSION AND CONCLUSIONS It is worth noting that there is no accurate way of knowing that the force exerted in the gripper is the same as the readings from the FSR. Even though calibration was performed before testing the condition changes between testing and actual use. In the gripper it is difficult to ensure that the pressure on the sensor is applied orthogonal on to the sensor.
This will probably not affect the performance of the gripper negatively since the main input of the system is whether the object is slipping or not. The only thing affected is the precision of the force being displayed on screen.
Chapter 6
Recommendations and Future work
One of the main parts that needs improvement in the Get a grip robot is the Slip sensor. Due to lack of availability in some parts of the construction the sensory system is not as precise as it could have been.
Firstly the potentiometer requires a bit to much force to rotate. This led to having to use double sided tape in order to keep contact between the lifted object and the roller, which in turn effected the natural slipping sequence. A lower resistance encoder might be more suitable.
Secondly the springs that where used to help the roller to stay in contact with the lifted object while, at the same time, enabling the object to make contact with the gripper surface were a bit to stiff. This resulted in the sensor not fully retracting which in our case did not pose a big problem. But if you want to lift more delicate objects you want the pressure being spread out on as large surface as possible. One thing that could be investigated in future work is how different gripper ge-ometry and mechanics effect the reliability of using motor current as an indicator of exerted force in the gripper. A design with less internal friction using pivots and sprockets might give more conclusive results. An external force gauge could also be used to both test the accuracy of the FSR and the motor current method.
Another thing that could be investigated in future work is additional controllers for both force reference regulation and force regulation. The force reference regula-tor could, for example, have an extra set of control parameters used when the slip speed is high in order to quickly increase the reference fed to the force regulator. The force regulator could have an extra set of control parameter for situations when the exerted force is further away from the reference, but also one when the exerted force is higher than the reference in order to safely decrease the force.
Instead of using PID regulators fuzzy controllers could also be investigated for use in this application. A multifinger gripper setup investigated by three researchers at International Islamic University Malaysia might be a good starting point for further work [13].
CHAPTER 6. RECOMMENDATIONS AND FUTURE WORK
Lastly a setup with the same sensory combination would be interesting to see in either a robotic prosthetic arm or just mounted to an industrial multipurpose robot arm.
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Appendix A
Test Results
A.1
FSR Calibration
Figure A.1. Graph of the resistance in the FSR plotted over the applied force. The blue circles are the values in resistance taken over three separate tests and the red line is the curve adaptation. Tests performed 2019-04-04. Made in Matlab[26]
APPENDIX A. TEST RESULTS
Figure A.2. Graph of the resistance in the FSR plotted over the applied force. The blue circles are the average values in resistance taken over three separate tests and the red line is the curve adaptation. Tests performed 2019-04-22. Made in Matlab[26]
Figure A.3. Graph of the resistance in the FSR plotted over the applied force. The blue circles are the average values in resistance taken over three separate tests and the red line is the curve adaptation. Tests performed 2019-04-29. Made in Matlab[26]
A.2. FORCE REFERENCE REGULATION
A.2
Force Reference Regulation
Figure A.4. Graph of the force reference over time using only a proportional regu-lator and no prefiltering. Made in Excel[25]
Figure A.5. Graph of the force reference over time using only a integral regulator and no prefiltering. Made in Excel[25]
Figure A.6. Graph of the force reference over time using only a derivative regulator and no prefiltering. Made in Excel[25]
APPENDIX A. TEST RESULTS
Figure A.7. Graph of the force reference over time using only a proportional regu-lator and a EMWA with α = 0, 2. Made in Excel[25]
Figure A.8. Graph of the force reference over time using only a integral regulator and a EMWA with α = 0, 2. Made in Excel[25]
Figure A.9. Graph of the force reference over time using only a derivative regulator and a EMWA with α = 0, 2. Made in Excel[25]
A.2. FORCE REFERENCE REGULATION
Figure A.10. Graph of the force reference over time using a proportional integral regulator and a EMWA with α = 0, 2. Made in Excel[25]
Figure A.11. Graph of the force reference over time using a integral derivative regulator and a EMWA with α = 0, 2. Made in Excel[25]
Figure A.12. Graph of the force reference over time using a proportional integral derivative regulator and a EMWA with α = 0, 2. Made in Excel[25]
APPENDIX A. TEST RESULTS
A.3
Force Regulation
Figure A.13. Graph of the exerted force over time using a proportional regulator. Made in Excel[25]
Figure A.14. Graph of the exerted force over time using a proportional integral regulator. Made in Excel[25]
Figure A.15. Graph of the exerted force over time using a proportional integral regulator. Made in Excel[25]
A.3. FORCE REGULATION
Figure A.16. Graph of the exerted force over time using a proportional integral derivative regulator. Made in Excel[25]
Figure A.17. Graph of the exerted force over time using a proportional integral derivative regulator. Made in Excel[25]
Figure A.18. Graph of the exerted force over time using a proportional integral regulator. Made in Excel[25]
APPENDIX A. TEST RESULTS
Figure A.19. Graph of the exerted force over time using a proportional integral regulator. Made in Excel[25]
Figure A.20. Graph of the exerted force over time using a proportional integral regulator. Made in Excel[25]
Figure A.21. Graph of the exerted force over time using a proportional integral derivative regulator. Made in Excel[25]
A.4. SYSTEM PERFORMANCE
Figure A.22. Graph of the exerted force over time using a proportional integral regulator. Made in Excel[25]
Figure A.23. Graph of the exerted force over time using a proportional integral derivative regulator. Made in Excel[25]
A.4
System Performance
Figure A.24. Graph showing how the exerted force tracks the force reference during a lift sequence with an object weighing 100 grams. Made in Excel[25]
APPENDIX A. TEST RESULTS
Figure A.25. Graph showing how the exerted force tracks the force reference during a lift sequence with an object weighing 150 grams. Made in Excel[25]
Figure A.26. Graph showing how the exerted force tracks the force reference during a lift sequence with an object weighing 200 grams. Made in Excel[25]
