Institutionen för systemteknik
Department of Electrical Engineering
Examensarbete
MALLS - Mobile Automatic Launch and Landing
Station for VTOL UAVs
Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping
av
Andreas Gising
LITH-ISY-EX--08/4190--SE
Linköping 2008
Department of Electrical Engineering Linköpings tekniska högskola
Linköpings universitet Linköpings universitet
MALLS - Mobile Automatic Launch and Landing
Station for VTOL UAVs
Examensarbete utfört i Reglerteknik
vid Tekniska högskolan i Linköping
av
Andreas Gising
LITH-ISY-EX--08/4190--SE
Handledare: Daniel Petersson, Janne Harju
isy, Linköpings universitet
Magnus Sethson
CybAero AB
Examinator: Martin Enqvist
isy, Linköpings universitet
Avdelning, Institution
Division, Department
Division of Automatic Control Department of Electrical Engineering Linköpings universitet
SE-581 83 Linköping, Sweden
Datum Date 2008-12-18 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-15980 ISBN — ISRN LITH-ISY-EX--08/4190--SE
Serietitel och serienummer
Title of series, numbering
ISSN
—
Titel
Title MALLS - Mobile Automatic Launch and Landing Station for VTOL UAVs
Författare
Author
Andreas Gising
Sammanfattning
Abstract
The market for vertical takeoff and landing unmanned aerial vehicles, VTOL UAVs, is growing rapidly. To reciprocate the demand of VTOL UAVs in offshore applica-tions, CybAero has developed a novel concept for landing on moving objects called MALLS, Mobile Automatic Launch and Landing Station.
MALLS can tilt its helipad and is supposed to align to either the horizontal plane with an operator adjusted offset or to the helicopter skids. Doing so, elim-inates the gyroscopic forces otherwise induced in the rotordisc as the helicopter is forced to change attitude when the skids align to the ground during landing or when standing on a jolting boat with the rotor spun up.
This master’s thesis project is an attempt to get the concept of MALLS closer to a quarter scale implementation. The main focus lies on the development of the measurement methods for achieving the references needed by MALLS, the hori-zontal plane and the plane of the helicopter skids. The control of MALLS is also discussed.
The measurement methods developed have been proved by tested implementa-tions or simulaimplementa-tions. The theories behind them contain among other things signal filtering, Kalman filtering, sensor fusion and search algorithms.
The project have led to that the MALLS prototype can align its helipad to the horizontal plane and that a method for measuring the relative attitude between the helipad and the helicopter skids have been developed. Also suggestions for future improvements are presented.
Nyckelord
Keywords MALLS, VTOL UAV, helicopter, helipad, sensor fusion, TDOA, Kalman filter, positioning.
Abstract
The market for vertical takeoff and landing unmanned aerial vehicles, VTOL UAVs, is growing rapidly. To reciprocate the demand of VTOL UAVs in off-shore applications, CybAero has developed a novel concept for landing on moving objects called MALLS, Mobile Automatic Launch and Landing Station.
MALLS can tilt its helipad and is supposed to align to either the horizontal plane with an operator adjusted offset or to the helicopter skids. Doing so, elim-inates the gyroscopic forces otherwise induced in the rotordisc as the helicopter is forced to change attitude when the skids align to the ground during landing or when standing on a jolting boat with the rotor spun up.
This master’s thesis project is an attempt to get the concept of MALLS closer to a quarter scale implementation. The main focus lies on the development of the measurement methods for achieving the references needed by MALLS, the hori-zontal plane and the plane of the helicopter skids. The control of MALLS is also discussed.
The measurement methods developed have been proved by tested implementa-tions or simulaimplementa-tions. The theories behind them contain among other things signal filtering, Kalman filtering, sensor fusion and search algorithms.
The project have led to that the MALLS prototype can align its helipad to the horizontal plane and that a method for measuring the relative attitude between the helipad and the helicopter skids have been developed. Also suggestions for future improvements are presented.
Sammanfattning
Marknaden för VTOL UAV:er växer snabbt, förkortningarna står för Vertical Ta-keoff and Landing Unmanned Aerial Vehicles, alltså vertikalt startande och lan-dande obemannade luftburna farkoster. För att möta marknadens önskemål om VTOL UAV-tillämpningar till havs, har CybAero utvecklat ett koncept för ver-tikal landning på föremål i rörelse. De har valt att kalla konceptet för MALLS, Mobile Automatic Launch and Landing Station.
MALLS kan vinkla sin helikopterplatta och tanken är att helikopterplattans lutning ska justeras mot antingen horisontalplanet eller det plan som helikopterme-darna utgör. Genom att göra det elimineras de gyroskopiska krafter som annars induceras i rotordisken under landning då helikopterns medar tvingas parallella med den sluttande marken eller då helikoptern står med roterande rotorblad på en krängande båt.
vi
Det här examensarbetet är ett försök att driva MALLS-projektet ett steg närm-re en fungerande kvartskaleimplementation. Huvudfokus ligger på att utveckla mätmetoder för de referenssignaler som MALLS behöver för att styras, vilka är horisontalplanet samt det plan som helikoptermedarna utgör. Även styrning av systemet diskuteras.
De mätmetoder som har utvecklats har verifierats i praktiska tester eller si-muleringar. Metoderna bygger på flera olika teorier så som signalbehandling, kal-manfiltrering, sensorfusion och sökalgoritmer.
Projektet har främst resulterat i att MALLS automatiskt kan justera sin heli-kopterplatta mot horisontalplanet och att mätmetoden för att mäta in helikopter-medarnas plan har utvecklats. Även förslag till framtida förbättringar presenteras.
Acknowledgments
Throughout this master’s thesis work I have been supported by many friends, supervisors and colleagues. I would especially like to thank my supervisors at CybAero - Magnus and Johan for their trust, belief and support for my ideas, Thomas for bracing me and creating a great atmosphere, my girlfriend Lena for the patience and support, my supervisors at ISY - Daniel and Janne for help with mathematics and proofreading and my examiner Martin for interesting viewpoints and discussions.
I would also like to thank my fellow master’s thesis workers Henrik and Björn for a great time working together with the MALLS project.
Contents
1 Introduction 1 1.1 Background . . . 1 1.2 Purpose . . . 1 1.3 Goal . . . 2 1.4 Challenges . . . 2 1.5 Limitations . . . 2 1.6 Thesis outline . . . 2 2 System overview 5 2.1 Measurement signals . . . 5 2.1.1 IMU . . . 6 2.1.2 Ultrasonic sensors . . . 6 2.2 System modes . . . 7 2.2.1 Simulation mode . . . 7 2.2.2 Launch mode . . . 7 2.2.3 Landing mode . . . 7 2.3 Control . . . 8 3 Modeling MALLSP 9 3.1 Moment model . . . 9 3.2 Angular model . . . 114 Measuring the attitude 13 4.1 Positioning via ranging and sensor fusion . . . 13
4.2 Restrictions of ultrasonic sound . . . 14
4.3 Sensor fusion . . . 14 4.3.1 TOA . . . 15 4.3.2 TDOA . . . 15 4.4 Estimation criterion . . . 16 4.5 Sensor arrangement . . . 18 4.6 Performance . . . 19 4.7 Hardware . . . 20
4.8 Sample rate and synchronization . . . 22
4.8.1 Sample rate . . . 22
4.8.2 Synchronization algorithm . . . 23 ix
x Contents
5 Measuring the vector of gravity 27
5.1 Angular rate measurements . . . 27
5.2 Acceleration measurements . . . 28
5.3 The Kalman filter . . . 29
5.3.1 Measurement models . . . 30
5.3.2 Tuning the Kalman filter . . . 31
5.4 Performance . . . 32
6 Discussion and conclusions 35 6.1 Today’s system . . . 35 6.2 Goal achievement . . . 36 6.3 Future system . . . 37 6.3.1 Hardware . . . 37 6.3.2 Control loop . . . 37 6.3.3 Computational power . . . 38 Bibliography 41
Chapter 1
Introduction
1.1
Background
The market for unmanned aerial vehicles, UAVs, is growing rapidly. An UAV can, without risking human safety, perform different tasks in hazardous environments such as toxics, war, boring endurance missions, risk of bad weather and so on. UAVs have huge benefits for both civil and defense applications.
CybAero is a Swedish company that develops and manufactures helicopter UAVs. Today CybAero is interested in developing an UAV-system being able to perform landings and takeoffs offshore, primarily on smaller ships but also on other kinds of vehicles.
For this application CybAero has developed a concept for landing at sea, based on a landing platform with a tilting helipad called MALLS, Mobile Automatic Launch and Landing Station. MALLS is supposed to control the attitude of the helipad such that the helicopter does not need to compensate for the changing attitude of the landing surface. If the helicopter rotordisc is forced to change attitude when it is spun up, strong gyroscopic forces are induced and they can make the helicopter crash. Landing a helicopter in a slope is also difficult because of the gyroscopic forces induced in the rotordisc as it is forced to change its attitude when the helicopter skids align to the ground.
1.2
Purpose
The purpose of this project is to verify that the concept of MALLS is a functional solution for helicopter landings on moving objects. In order to do this, the concept must be transformed to a operative system, enclosing all algorithms, measurement methods, sensors, control loops and implementation.
The system will be made as a quarter scale prototype and the expectation is that it will bring difficulties and issues regarding MALLS to surface, reducing development costs and time for the full scale implementation.
2 Introduction
1.3
Goal
The main objective with the project is to perform a helicopter landing on a moving object, proving the concept of MALLS. There is a long way to reach the main objective, and to facilitate the progress of the work, the following milestones have been stated:
• Develop a method for measuring the horizontal plane.
• Develop an unwired method for measuring the relative attitude between the
helipad and the helicopter.
• Control MALLS, given the reference signals.
• Implement and evaluate the horizontal plane measurements. • Implement and evaluate the relative attitude measurements. • Design a controller for the whole system.
• Integrate all subsystems to a functional quarter scale system, trim and
eval-uate it.
1.4
Challenges
The main challenge in the project is to develop the unwired measuring solution for the relative attitude between the helipad and the helicopter skids. The system’s influence on the helicopter must be minimal, radio communication is permitted and camera solutions are left out because of light condition issues. No references describing a corresponding problem have been found, so it will be developed from scratch.
1.5
Limitations
This project had a very clear goal, to develop and implement MALLS. How to reach the goal was never restricted, so the project has had few limitations. How-ever, the project is limited to the prototype of MALLS and its mechanical design, because it will not be rebuilt during the project. Further on, the developed mea-surement methods must not rely on radio communication and they will primary be designed and tested to fit the quarter scale implementation, not the full scale implementation. At last, neither the control of the helicopter nor is the device holding the helicopter fixed to the helipad after it has landed a part of this report.
1.6
Thesis outline
The thesis focuses on the measurement strategies and how to filter the measure-ment data to obtain the necessary references to control MALLS. A short intro-duction to the chapters follows.
1.6 Thesis outline 3
The second and third chapter cover the system in general and modeling of the platform. The fourth and fifth chapter cover the measurement methods regarding sensor arrangement, sensor fusion, signal filtering, performance and more. Fur-thermore, in the sixth chapter the results are presented and the conclusions are summarized in a proposal to a future system.
Chapter 2
System overview
One of the big issues when landing a helicopter on a boat is that the landing area changes its attitude. The problem is that the helicopter has a rotating mass, the rotordisc, and when a rotating mass is forced to change its attitude, gyroscopic forces act upon the mass, trying to maintain the angular momentum. This is the scenario when a helicopter lands in a slope or when it is standing on an object that changes its attitude. If the gyroscopic forces are too big the helicopter will tip and crash, causing monetary and possibly personal damage.
CybAero has developed and patented a concept for helicopter landings on boats and other moving objects. The patent proposes to use a system with a tilting helipad called MALLS, Mobile Automatic Launch and Landing Station, which can compensate for the moving object’s roll and pitch angles. MALLS keeps the attitude of its helipad fixed even though the object it is mounted on changes its attitude.
MALLS is a system under construction and to facilitate the development there is a prototype called MALLSP. MALLSP is made in quarter scale and has the extra feature of physically simulating sea movements. The simulating part of MALLSP is constructed from two square frames, fitted into each other where one can simulate pitch and the other roll. The stabilizing part of MALLSP consists of a helipad which also can be pitched and rolled and it is mounted in the center of the simulating frames. MALLSP is controlled by four angular servos via a controller card and a personal computer. The control software is written in C-code.
A photograph of MALLSP is shown i Figure 2.1 and a conceptual sketch is shown in Figure 3.1.
2.1
Measurement signals
MALLS depends on two different reference signals of which one is the horizontal plane and the other is the relative angle between the helipad and the helicopter skids. The horizon reference is computed from the measurements of an inertial measurement unit further discussed in Chapter 5, and the helicopter skid reference is proposed to be computed from ultrasonic range measurements via sensor fusion
6 System overview
Figure 2.1. A photograph of MALLSP showing two exterior frames for simulating movements of a boat and the stabilized plate in the center where the helipad is to be mounted.
further discussed in Chapter 4.
2.1.1
IMU
IMU is short for inertial measurement unit and it is used for monitoring the move-ments of the body it is mounted on. It measures angular rates and accelerations in three dimensions. An IMU is mounted right under the stabilized helipad, mon-itoring the movements of the simulated ship.
By merging the measurements from the IMU in a Kalman filter, the roll and pitch angles relative to the horizontal plane can be observed and the helipad can be controlled to keep its relative angle towards the horizontal plane fixed. The setup of the IMU and the signal filtering is further discussed in Chapter 4.
2.1.2
Ultrasonic sensors
Ultrasonic sensors are popular sensors for robotic projects because they are small, easy to use and cheap. Sound is considered ultrasonic if its frequency is above the human hearing spectrum, an approximate limit is often drawn at 20 kHz. Ultra-sonic sensors can act both as loudspeakers and microphones, converting electrical signals to ultrasonic sound or vice versa. Sometimes the ultrasonic sensors are optimized for either transmitting or receiving and then they can also be called ultrasonic transmitter and receiver respectively. The sensors can measure the dis-tance to nearby objects by transmitting an ultrasonic pulse and measure the time
2.2 System modes 7
until the pulse is received again. The transport delay via the speed of sound cor-responds to the distance that the pulse has traveled.
Via sensor fusion it is possible to estimate the position of a target if the dis-tances from the target to different points are known. Furthermore, if three posi-tions on an object are known, the orientation of the object can be calculated. In this application ultrasonic sensors are proposed for measuring distances between targets on the helicopter and points on the platform to be able to calculate the relative attitude between the helicopter and the helipad.
Sensor fusion and the sensor setup is described in more detail in Chapter 5.
2.2
System modes
MALLS always has to contribute to safe launch and landing conditions for the helicopter. To be able to perform a safe launch MALLS has to fix its helipad’s relative attitude to the horizon and to be able to perform a safe landing MALLS has to control the helipad’s attitude towards the helicopter skids. This leads to two main operation modes of MALLS, launch mode and landing mode. MALLSP also has an extra additive mode called simulation mode used for simulating sea movements.
2.2.1
Simulation mode
The simulation mode is very simple, it is just supposed to excite the helipad of MALLSP into something reminding of a boat at sea, i.e. two different sinusoidal signals, one for roll and one for pitch. The simulated motions are actuated via the exterior frames. Simulation mode is an additive mode which can be turned on or off independently of other modes.
2.2.2
Launch mode
In the launch mode MALLS is controlled to keep the helipad’s relative angle to-wards the horizon fixed. Even if the helipad could be perfectly controlled to be parallel to the horizon it is not the optimal reference. In case of relative wind, due to ship cruising and winds, an offset has to be adjusted by an operator. A helicopter hovering in winds will lean and therefore the platform should lean too. When launch mode is enabled the IMU will be used for calculating the reference signals.
The IMU is further discussed in Chapter 5.
2.2.3
Landing mode
In the landing mode MALLS is controlled to minimize the relative attitude between the helipad and the helicopter skids. An ultrasonic system, measuring range, is proposed for calculating the reference signals. Its operative range is at least 0-2 meters and it is robust in sea conditions regarding wind, fog, salty water and bright sunshine in contrast to a camera or other optical systems.
8 System overview
During a landing, the launch mode will be active until the helicopter is in operational range for the ultrasonic measurements. Thereafter the landing mode will be activated and control the helipad to minimize its relative angles to the helicopter skids. Directly after the helicopter has landed, the active mode has to be toggled back to launch mode again to prevent failure because the landing mode might be unstable when the helicopter is standing on the helipad. As long as the helicopter stands on the helipad the relative angle between them is unaffected from the helipad movements which means that an infinite small measurement error could make the helipad tilt to its maximum value causing the helicopter to crash.
The ultrasonic measurements are further discussed in Chapter 4.
2.3
Control
The current mechanical design of MALLSP and the chosen servos are not optimal for control. MALLSP was built in a rush to prove the concept, and therefore the design was focused on simplicity and construction speed instead of optimal control performance. MALLSP is controlled by four angular servos, two for the simulating frames and two for the stabilizing platform. The two servos for the helipad are high performing Kearfott servos [1]. They are quick and strong, have good resolution and small backlash. As for most angular servos they also have an embedded control loop which actuates the angle input as fast as possible.
When the launch mode is enabled the control of the stabilized helipad has no feedback because the IMU is mounted right under it, on the simulated boat. Of course, feedback control is the better option, but physical circumstances regarding this very prototype unfortunately does not allow the IMU to be mounted on the stabilizing platform. A Kalman filter uses measurements from the IMU to estimate the roll and pitch angles of the simulated boat where the angle estimates are used to calculate the control signals for the Kearfott servos. Due to the lack of feedback tuning is not possible, nor is diagnosis of the system’s performance.
Chapter 3
Modeling MALLSP
Two models of MALLSP have been configured via physical modeling of which one is for moment input and the other for angular input.
Statements and theorems in this chapter come from Physics Handbook [4], and the reference will not be indicated in the text. Figure 3.1 shows a conceptual sketch of MALLSP. Sim roll Sim pitch Stem Stern Roll Pitch s3 s4 s2 s1 d5 d6 l1 l2 l3 l4
Figure 3.1. A conceptual sketch of MALLSP. The exterior frames are used to physically simulate the movements of a boat, the center disc is the stabilized helipad. The angular servos are named s1-s4, the straight arrows represent servo controlled rods, and bent arrows represent angles.
3.1
Moment model
The model is built step by step, starting from three primary equations, the relation between torque and angular acceleration, Steiner’s parallel axis theorem and the relation between moment, force and lever arm length.
10 Modeling MALLSP
The relation between torque and angular acceleration is a result of Newton’s second law, adapted to rotating bodies
X
∀i
Mi= J ¨ϕ (3.1)
where M is the applied torque, J is the moment of inertia and ¨ϕis the angular ac-celeration. Steiner’s parallel axis theorem describes the moment of inertia around
z1 starting from the moment of inertia around the body’s center of mass
Jz1 = Jz2+ md
2 (3.2)
were z1is parallel to z2, separated with distance d and where z2intersects with the
body’s center of mass. The relation between moment, force and lever arm length is
M = F d (3.3)
where M is the moment, F is the force applied orthogonally to the lever arm at distance d from the center of rotation.
Moment of inertia for one square hollow rod
The moment of inertia for a rectangular parallelepiped with mass m, length l along the y-direction, base a along both the x- and the z-directions and where the origin is defined in the body’s center of mass is
Jx= Jz= m 12(a 2+ l2) (3.4) Jy= m 122a 2 (3.5)
Subtracting a smaller rectangular parallelepiped with same length l but base b results in the hollow square rod
Jx= Jz= 1 12 mal(a 2+ l2) − m bl(b2+ l2) (3.6) Jy= 1 12 2a 2mal− 2b2mbl (3.7)
where index mxl is for the mass of a square rod with base x and length l.
Moment of inertia for a frame of square hollow rods
Combining the moment of inertia for a square hollow rod (3.6) - (3.7), and Steiner’s theorem (3.2), gives the moment of inertia for a whole frame of square hollow rods. Under the assumption that the frames are neither rolled nor pitched (ϕ = θ = 0), the moments of inertia for the outer and the inner frames in the roll and pitch
3.2 Angular model 11 directions are Jo,ϕ= (3.8) 2 12 mal2(a 2+ l2 2) − mbl2(b 2+ l2 2) + mal12a 2− mbl 12b 2 +l2 2 4(mal1− mbl1) Ji,ϕ= (3.9) 2 12 mal4(a 2+ l2 4) − mbl4(b 2+ l2 4) + mal32a 2− mbl 32b 2 +l2 4 4(mal3− mbl3) Ji,θ = (3.10) 2 12 mal3(a 2+ l2 3) − mbl3(b 2+ l2 3) + mal42a 2− mbl 42b 2 +l2 3 4(mal4− mbl4)
where index i and o denote the inner and outer frame and ϕ and θ denote the roll and pitch directions respectively. The outer frame can not be pitched due to the mechanical design and thus its moment of inertia in the pitch direction is left out.
Moment of inertia for circular disk
The moment of inertia for a circular disk with mass mc and radius rc is
Jc,ϕ= Jc,θ= mc
rc2
4 (3.11)
Moment model for MALLSP
The moment model of MALLSP is summarized in four equations, one for each servo, and they are a result of the above equations. The parameters M1-M4 are
the applied torques, d1-d4are servo arm lengths and d5and d6are lever arm lengths
according to Figure 3.1. The lever arm of the disc is its radius rc, l5corresponds to
the altitude of the circular disc relative the simulating frames seen in Figure 2.1. The angles ϕ and θ are the roll and pitch angles respectively, and their reference is the inner simulating frame. The angles ϕs and θs are the simulated roll and
pitch angles respectively, and their reference is the floor.
M1= d1 rc (Jc,ϕ) ¨ϕ (3.12) M2= d2 rc (Jc,θ) ¨θ (3.13) M3= d3 d5 Jo,ϕ+ Ji,ϕ+ Jc,ϕ+ mcl 2 5 ¨ϕs (3.14) M4= d4 d6 Ji,θ+ Jc,θ+ mcl 2 5 ¨θs (3.15)
The relation between the lever arm lengths scales the torque from the servo to the moving frame or disc via the principals of torque, force and lever arm length (3.3).
3.2
Angular model
The angular model is very straight forward and only trigonometry is used to transform the servo angles to platform angles. One assumption is made which is
12 Modeling MALLSP
that the rods presented by straight arrows in Figure 3.1 does not change orientation even though they do because of the different lengths of the lever arms, but the small servo outputs makes the effect negligible.
ϕ= arcsin d1 rc sin (s1) (3.16) θ= arcsin d2 rc sin (s2) (3.17) ϕs= arcsin d3 d5sin (s3) (3.18) θs= arcsin d4 d6sin (s4) (3.19) where si are the servo output angles.
Chapter 4
Measuring the attitude
During landing, MALLS has to measure the relative attitude between the helipad and the helicopter skids and minimize it. When a helicopter lands in a slope, the rotor disc is forced to change its attitude as the skids are forced parallel to the slope. Changing attitude of a rotating body induces gyroscopic forces trying to maintain the angular momentum of the body. If the relative attitude is not small enough during landing the gyroscopic forces acting on the rotor disc will make the helicopter crash.
This chapter describes a proposed method for measuring the relative attitude between the helipad and the plane of the helicopter skids. The theory in this chapter, if nothing else is stated, comes from a compendium about sensor fusion, written by Fredrik Gustafsson [6].
4.1
Positioning via ranging and sensor fusion
Via sensor fusion it is possible to merge several measurements to get an estimate of something that is not directly measured. Merging the distances from three different points to the same target, the target’s position can be calculated. In this application the idea is to use sensor fusion of range measurements to determine several positions on the helicopter relative the helipad. When three or more targets on the helicopter have been positioned, the helicopter can both be positioned and orientated.
The challenging part here is how to measure the distance between a fixed point on the helipad and a fixed point on the helicopter. Most range finding systems rely on bounces, which makes it difficult to know what distance that actually has been measured. For example, laser is an accurate distance measurement method, but it relies on bounces and it is very difficult to know exactly which point the laser beam has been reflected on. The proposed solution is to use ultrasonic sound by separating the loudspeaker from the microphone and measure the time for a pulse to travel from the loudspeaker to the microphone. If the helicopter and the helipad are synchronized in time it is possible to measure the distance from an exact point of the helicopter to an exact point on the platform by placing
14 Measuring the attitude
the transmitter on the helicopter and the receiver on the helipad and measure the time for the ultrasonic pulse to travel between them. Expanding the single receiver to a hole array of receivers enables several range measurements to the same transmitter which can be used to position the transmitter via sensor fusion. If two transmitters are added to the helicopter three points can be positioned, and because their positions relative the helicopter are known the helicopter can be positioned and orientated.
The slow propagation of sound makes realtime issues such as sample rate and synchronization less critical than if using laser in the same way because the light travels faster than sound with a factor of a million.
4.2
Restrictions of ultrasonic sound
Ultrasonic waves propagate in the air with the speed of sound. The velocity de-pends on temperature, humidity and pressure, its approximate speed in indoor environment is 343 meters per second relative the air. There is a clear reason to get concerned when putting the system into practice under a hovering heli-copter. The turbulence will result in some unpredictable stochastic noise and the downward flow will make the sound travel faster relative the sensors and thereby add a systematic bias error. Further on, there is a possibility that the helicopter transmits ultrasonic sounds itself, which could afflict the measurements. These concerns have been examined and tested in practice by mounting a RC helicopter on a tripod and measuring the distance between it and the ground whilst simu-lating hovering state. The results showed that these defects were small enough to proceed with the ultrasonic sensors.
Another potential problem is that the ultrasonic sound possibly could echo for a time, resulting in one or more pulses bouncing between the ground and the rotor disc. Naturally, the probability for this effect to take place should increase as the helicopter gets closer to the helipad. The obvious remedy for the phenomenon is to decrease rate of the transmitted pulses although it directly would effect the control loop. This has not been fully tested and examined.
4.3
Sensor fusion
Sensor fusion concerns methods for merging measurements to obtain information about something that is not directly measured or to get more reliable data from noisy measurements. In this application sensor fusion is used to merge several range measurements from the same transmitter but to different receivers, in order to obtain an estimate of the transmitter position. Two different approaches can be used, time of arrival, TOA, and time difference of arrival, TDOA. We will see later that TOA is easier and more straightforward but requires absolute distances, where TDOA is more complex and uses more computational power but only relies on the difference in distance between the transmitter and the receivers.
Let us start discussing the theories in two dimensions where it becomes easier to understand the ideas.
4.3 Sensor fusion 15
4.3.1
TOA
If perfect synchronization between the transmitter and the receiver is available, the transport delay for ultrasonic sound to travel between them can be measured. The transport delay corresponds directly to the distance between the sensors via the speed of sound. If the distance between a transmitter with unknown position and a receiver with known position is r, the transmitter’s position is restricted to be anywhere on a circle with radius r and center point in the receiver’s position. The circle is called a TOA circle.
The equation for a circle with radius r and center point in p follows.
r=q(x − px)2+ (y − py)2 (4.1)
To delimit the solutions, the intersection of different TOA circles are consid-ered. When using three or more receivers the intersection of their TOA circles is one single point which is the solution to the system of equations containing all the TOA circles, see Figure 4.1(a).
r1 = q (x − p1x)2+ (y − p1y)2 r2 = q (x − p2x)2+ (y − p2y)2 .. . rn = q(x − pnx)2+ (y − pny)2 (4.2)
This is a straightforward and accurate way of positioning the transmitter, but it requires synchronization, meaning that the receiver must know when the received pulse was sent. In this application it is not trivial to synchronize the transmitters on the helicopter and the receivers on the helipad. One possible way of syn-chronization could be to send an infrared signal simultaneously with the sound, assuming it would arrive to the helipad instantly. Unfortunately it would bring an additional source of error, more complex implementation and would probably not work in bright sunshine because infrared light transmitted by the sun would interfere with the signal.
The TOA measurement equation for two dimensions is
yiT OA =
1
v
q
(xx− pix)2+ (xy− piy)2 (4.3)
where yi is the measured time for the pulse to travel from the transmitter x to the
receiver i, located at position pi, with the speed of sound, v.
4.3.2
TDOA
Time difference of arrival, TDOA, only depends on the difference of the transport delay between the transmitter x and the receivers pi and pj. This approach does
16 Measuring the attitude −8 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6
TOA circles, with measurement noise
(a) TOA circles from three receivers. The intersection of the three circles is the true transmitter position, marked in red and arrowed.
−8 −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6
TOA circles, with measurement noise
(b) TOA circles from three receivers with measurement noise. The arrowed intersection is no longer one single point and the solution of the transmit-ter position is ambiguous.
Figure 4.1. TOA circles from one transmitter and three receivers.
the receivers themselves, which is very suitable for this application. Furthermore, the system’s influence on the helicopter is minimal. TDOA is the chosen algorithm for this project.
Comparing the time of arrival for one pair of sensors, pi and pj, leads to an
expression describing the difference in distance to the transmitter, rij = ri− rj.
The value of rij corresponds to transmitter positions along a hyperbola, a TDOA
hyperbola.
The hyperbolas are characterized by their asymptotes and depend on the re-lation between rij and the receivers displacement. The asymptote angles give the
directions to far away points of the hyperbola. Further on in this report the hy-perbola angles will refer to a hyhy-perbola shape rather than a actual direction of the asymptotes. Note that the greatest value possible of the parameter rij is the
receiver displacement itself and it corresponds to asymptote angles of ±0◦. See
Figure 4.2(a) for an example of a hyperbola and its asymptotes.
In the same manner as for TOA measurements, using three or more receiver pairs, the intersection of the TDOA hyperbolas is a single point, the true position of the transmitter. The TDOA measurement equation for two dimensions is
yijT DOA = 1 v q (xx− pix)2+ (xy− piy)2− q (xx− pjx)2+ (xy− pjy)2 (4.4) where i < j, pi and pj are the receiver locations and yij is the time difference
between the sensor i and j receiving the pulse and v is the speed of sound.
4.4
Estimation criterion
The measurements suffers from noise, both from the turbulent environment and the time jitter in the hardware of the sensors. The noise is considered as an additive Gaussian term to the time stamps, and since yijT DOA is the difference of
4.4 Estimation criterion 17 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
TDOA hyperbola and its asymptotes, no measurement noise
(a) TDOA hyperbola and its asymp-totes when the receiver displacement is 2 units and the measured distance dif-ference is 1.225 units. The asymptotes are ±52◦. −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2
3 TDOA hyperbolas, with measurement noise
(b) TDOA hyperbola bands. The same measurement error creates bands of different width depending on asymp-tote angles and distance from hyper-bola focus. The asyptotes are ±84◦,
±52◦, ±16◦.
Figure 4.2. TDOA hyperbolas.
two time stamps the standard deviation of the noise is multiplied with the factor of√2. The measurement model for the TDOA algorithm in three dimensions is
hT DOA(x, pij) =
1
v
q
(xx− pix)2+ (xy− piy)2+ (xz− piz)2 −
1 v q (xx− pjx)2+ (xy− pjy)2+ (xz− pjz)2 + e v (4.5)
where the distribution of the Gaussian noise e is
e∼ N(0,√2σ) (4.6)
and σ is the standard deviation of a range measurement performed under a hov-ering helicopter.
When adding noise to the measurements, the solution get ambiguous. Both TOA circles and TDOA hyperbolas become bands with a width proportional to the noise variance. In the case of TDOA, the width of the band increases as the hyperbola asymptote angles decreases and when traveling away from the hyper-bola focus. Both these effects are a result of geometry; if the hyperhyper-bola asymptote angles are small or if the transmitter is far away from the receivers, a change in position will give a very small change of the value rij, see Figure 4.2(b).
The problem now is to find the position that minimizes a given loss function of the estimate error. The non-linear least squares criterion is used as loss function.
ˆ x= arg min x k yij− h(x, pij) k 2= arg min x yij− h(x, pij))T(y ij− h(x, pij) (4.7)
To find the position x that minimizes the loss function, Gauss-Newton’s algorithm is used. It iterates as follows.
ˆ
xk = ˆxk−1+ µk HT(ˆxk−1, pij)H(ˆxk−1, pij)
−1
HT(ˆxk−1, pij)(yij− h(ˆxk−1, pij))
18 Measuring the attitude
where H(x) = ▽xh(x) and µk is the step-size parameter. µk can be chosen as a
fix number in the range of 0 < µk<1 or optimized for each iteration via another
algorithm, reducing calculation costs [10]. In this application a small fixed step size is proposed because of the limitations of the ultrasonic sound update rate and the precise initial guess via the last estimate permits the extra calculation time.
Due to the sensor arrangement where all the receivers are placed in the same plane there is not a unique solution to the problem. The true position and its mirror point with negative z-component gives the same value for the loss function. Anyway, this does not matter due to the method shown in Section 4.5.
4.5
Sensor arrangement
This section describes a method to compensate for the bad effects of distributing all the receivers in the same plane, and it is a result of this master’s thesis work.
The sensors in the array on the helipad are unfortunately restricted to be placed more or less in the same plane due to the physical scope of the application. Simulations of positioning in a noisy environment using an array structured in a plane, shows that the variance of z suffers from the geometry. The variance of
z is significantly greater than the variance of x and y which is a result of the hyperbola bands discussed in Section 4.4, the effect is easily seen in Figure 4.5(a). The hyperbolas adding information about xzhave a wide band in the region around
the transmitter, which depends on the hyperbola asymptote’s small angles. In order to estimate an accurate relative angle between the helicopter and
Figure 4.3. To be able to relax the z-component from the calculations the two trans-mitters are placed above each other.
the helipad, the z-component is proposed to be omitted. To accomplish this the transmitters are placed above each other according to Figure 4.3. Knowing the transmitter displacement and which transmitter is active, the relative angle is a trigonometric function of the estimated x- and y-components of the transmitters. The transmitter displacement acts as a hypotenuse and the difference between the two transmitters’s x-components (or y-components) acts as the opposite cathethus in a right triangle, see Figure 4.4. The pitch and roll angles of the helipad relative
4.6 Performance 19
the plane of the helicopter skids are
ϕ= − arcsin ∆x ∆z (4.9) θ= − arcsin ∆y ∆z (4.10) ∆x ∆z
Figure 4.4. The angles are calculated from right triangles where the hypothenuses are known.
If the z-estimate is inaccurate when positioning using TDOA, it is desirable to have some hyperbolas with asymptotic angles close to ±90◦ because the x- and y-values of those hyperbolas are almost independent of the z-value which can make the x- and y-estimate accurate anyway. This is the case when the transmitter is somewhere in between the two receivers, i.e. when there are receivers on both sides of the transmitter which is actually qualified from the transmitter arrange-ment. Both transmitters can be centered under the rotordisc which means that two receivers always can be on each side of the transmitter because the center of the helicopter cannot descend on the edge of the helipad. Furthermore, under the center of the rotordisc the noise level is minimal and both transmitters will suffer from the same noise levels.
Note that neither the helicopter roll, pitch nor yaw angle can be observed be-cause the system does not provide enough information for that. What is observed is instead the helipad’s relative roll and pitch angles towards the plane of the heli-copter skids. The heliheli-copter angles could be observed if an additional transmitter was used and the IMU was mounted on the helipad, but the helicopter angles are not relevant to control the helipad.
4.6
Performance
Shielding objects on the helicopter and other circumstances will most likely prevent the ultrasonic pulses from reaching all the receivers. Therefore the system must not rely on all of them. In the noise-free case the sensor fusion requires measurements from at least four sensors (now discussing three dimensions), although simulations shows that in order to have a reliable estimate when adding noise, at least five
20 Measuring the attitude
sensors must be used.
The performance is demonstrated via simulations in Matlab & Simulink [3]. In the simulation environment thirteen receivers are arranged on a 1x1 meter square helipad. The two transmitters are placed straight above each other, the lower one at an altitude of 5 centimeters and the upper one at an altitude of 30 centimeters. During simulation, the true distance between each receiver and each transmitter is calculated, then noise is added according to experiments and finally the differences in distance are calculated. The standard deviation of the noise is 2.8 millimeters, the step-size µ is chosen to 0.3 and 100 iterations are performed in the algorithm. Figures 4.5 and 4.6 show the estimated position for 70 simulations for the lower transmitter. The transmitter estimates are marked with a red ring, its true position with an arrowed black star, used sensors with a blue cross, unused sensors with a blue dot, the axes units are millimeters. Figure 4.7 shows the unfiltered angle error gained when using the estimated transmitter positions to calculate the relative angle. The true relative angle is zero degrees.
−500 0 500 −500 0 500 −10 0 10 20 30 40 50 60 70 X Y
(a) Using all receivers, units in millimeters.
−500 0 500 −500 0 500 −10 0 10 20 30 40 50 60 70 X Y
(b) Only using the six closest receivers, units in millimeters.
Figure 4.5. Estimated positions for the lower transmitter. Note the different scales between the x-, y- and z-axes.
4.7
Hardware
In order to control the sensor array and keep all the sensors synchronized, sensors supporting the I2C interface [8] are proposed.
The ultrasonic transducer SRF02 [2] is an ultrasonic sensor mounted on a small chip with some logics. It features to send an ultrasonic burst on its own without a reception cycle, and the ability to perform a reception cycle without the preceding burst. It also supports the I2C standard and can be programmed to 16 unique
slave addresses.
The I2C standard features a general call, meaning all the sensors can be called
simultaneous with the same instruction. When the sensors are called to perform a reception cycle they start a counter and as they detect an incoming pulse they
4.7 Hardware 21 −500 −400 −300 −200 −100 0 100 200 300 400 500 −500 −400 −300 −200 −100 0 100 200 300 400 500 X Y
(a) Using all receivers, units in millimeters.
−500 −400 −300 −200 −100 0 100 200 300 400 500 −500 −400 −300 −200 −100 0 100 200 300 400 500 X Y
(b) Only using the six closest receivers, units in millimeters.
Figure 4.6. Same plots as fig 4.5, but viewed from above.
0 10 20 30 40 50 60 70 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 Simultion Degrees Pitch error Roll error
(a) Pitch and roll errors when using all 13 sen-sors. The pitch error has mean -0.02 degrees and standard deviation 0.77, the roll error has mean -0.016 and standard deviation 0.55 de-grees. 0 10 20 30 40 50 60 70 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 Simulation Degrees Pitch error Roll error
(b) Pitch and roll errors when using only the six closest sensors. The pitch error has mean 0.19 degrees and standard deviation 0.99, the roll error has mean -0.08 and standard devia-tion 0.77 degrees.
Figure 4.7. Angle errors when using different number of sensors.
store the counter value in a local register. There is only one register for storing the result, so the SRF02 only stores the time stamp of the first incoming pulse. The result from each receiver in the array can then be read, one by one, from a microcontroller without realtime issues. See Figure 4.8(a) for a picture of the SRF02 sensor.
The transmitters are characterized by two things, their central frequency and their beam pattern. From the beam pattern the beam angle can be defined, which most often is defined as the angle where the damping is 6dB. In this application it is desirable to have a beam angle of 180◦to make sure that the transmitted signal
covers the hole helipad, especially when the helicopter’s altitude is small.
The beam angles of ultrasonic transmitters varies from about 15◦ to 70◦, and
the SRF02 has a beam angle of 55◦. To obtain the desired beam angle of 180◦
a hardware modification is proposed. Attaching five sensors onto a sphere, all facing its center, a virtual transmitter with a beam angle of approximately 180◦
22 Measuring the attitude
(a) The SRF02 ultrasonic sensor. (b) Beam pattern for the SRF02.
Figure 4.8. The ultrasonic transducer SRF02.
can be achieved. The virtual transmitter’s position is in the sphere center. The arrangement leads to an offset error, but since the results of the measurements are subtracted from each other using the TDOA approach, the offset influence is liminal. The arrangement also brings a risk for a negative super-positioning phenomenon, if two or more signals from different transmitters are sent to the same receiver the signal can be cancelled completely. This must be tested and examined before the system is implemented.
4.8
Sample rate and synchronization
4.8.1
Sample rate
When the SRF02 is programmed to perform a reception cycle it is locked in re-ception mode for 70 milliseconds. It becomes an issue when measuring two trans-mitters with a delay of 70 milliseconds and comparing the results without using a dynamic model for the helicopter motion. For example, if the helicopter moves in 0.5 meters per second and the two transmitters are sampled with a delay of 70 milliseconds, the helicopter will travel 3.5 centimeters during the sample delay. Not taking this into account when calculating the relative angle will lead to an angle error of 8.0 degrees if the transmitters are separated 25 centimeters.
The maximum continuous sample rate for measuring both transmitters is 7 hertz and cannot be affected without modifying the hardware. However it is pos-sible to reduce the time elapsed between receiving the first and the second pulse. If the two transmitters are sending one pulse each, separated with a short delay, the reception cycle on the helipad can be synchronized to start in between of the two pulses, reducing the time jitter between the two measurements.
4.8 Sample rate and synchronization 23
Figure 4.9. Three of the five sensors placed on a sphere facing its center, making a virtual sensor in the sphere center achieving a beam angle of approximately 180◦.
4.8.2
Synchronization algorithm
On the helipad, a new reception cycle is started every 70 milliseconds. Assume that the first transmitter is sending a pulse at time t and the second transmitter at time t + δ. Then if the helipad can tune its reception cycles to start between of the two arriving pulses, it can make the pulse from the first transmitter arrive to the platform at δ
2 seconds before a new reception cycle and the pulse from the
second transmitter arrive at δ
2 seconds after a new reception cycle.
Decreasing the time δ to 10 milliseconds and assuming the same flight condi-tions as above, the helicopter will travel only 0.5 centimeters between the mea-surements and the angle error will be 1.1 degrees instead of 8.0 degrees. This is an improvement of 86%.
A synchronization algorithm can tune the reception start to appear in between of the arriving pulses. When the system is synchronized, it is also possible to identify the two transmitters which is required in the used sensor arrangement. A synchronization algorithm is proposed according to the flowchart shown in Fig-ure 4.11. To ease understanding of the synchronization algorithm, three different cases of synchronization are presented in Figure 4.10. There are three timelines where vertical bars represent the receivers starting a new reception cycle, the solid circles represent the arriving pulse from the first transmitter and the circles rep-resent the arriving pulse from the second transmitter.
The grade of synchronization is divided in three states: Out of sync, poorly
tuned and well tuned. In the two first states the reception start should be delayed
such as the next reception start will appear in between of the two arriving pulses. It is only in the third state that angle measurements can take place, because in the two first states the adjustment of the reception start interferes with the mea-surements. This means the system must be considered to be in the state well
24 Measuring the attitude
Figure 4.10. Three timelines showing different states of synchronization between the transmitters sending pulses and the receivers starting new reception cycles. Starting from the topmost the states are out of sync, poorly tuned and well tuned.
relative angle. The proposed limit between poorly tuned and well tuned is when one pulse arrives closer than δ
4 seconds to a reception start. Note that the tuning
is affected as the helicopter altitude change due to that the traveling time of the sound changes. The algorithm is supposed to be used with one or two receiver’s with centered positions on the helipad.
4.8 Sample rate and synchronization 25
Start
Start reception cycle and counter
Counter = 70ms ? Collect results Start new reception cycle
and restart counter
Received a pulse? Out of sync,
correct in next loop
Counter = 70ms ? Collect results Start new reception cycle
and restart counter
Received a pulse? Out of sync,
Counter − = δ
2+ t
Sync is OK, tune
Need tuning?
Counter + = δ
2− t
System is well tuned
No Yes No Yes No Yes No Yes Yes No
Figure 4.11. Synchronization-algorithm flow chart. The answer to the last dialog box, "Need tuning?", is "Yes" if one of the two pulses arrived closer than δ
4 seconds to the
reception start. The parameter t denotes value of the smallest collected result during the two cycles.
Chapter 5
Measuring the vector of
gravity
When the helicopter is standing on MALLS with its rotor spun up, MALLS has to keep the helipad’s relative angle to the horizon fixed. The angle offset from the horizontal plane mainly depends on the relative wind, making the helicopter lean when hovering. During landing the helipad naturally gets the offset from the ultrasonic measurements, but during launch the offset must be set by an operator. To estimate the horizontal plane the vector of gravity is used as a reference. It is estimated using an IMU and a Kalman filter [7]. When the vector of gravity is determined in the local coordinate system of the helipad, its roll and pitch angles can be surveyed.
5.1
Angular rate measurements
The IMU measures the angular rate in three orthogonal directions; x, y and z. Integrating the angular rate over time gives a value of how much the angle has changed since the integration started. To get hold of the current angle, the offset angle at time t0must be known according to the main clause of integral calculus,
Equation 5.1. This can be achieved by calibrating the IMU as it is held towards a known reference angle. The calibrating procedure will be returned to later.
α(t) = α(t0) +
t
Z
t0
˙α(t) dt (5.1)
The angular rate measurements also suffers from an unknown non-constant bias, such as the angular rate measurements drifts in time. The value of the angular rate obtained from the IMU is the sum of the non-constant bias and the true angular rate.
˙αIM U(t) = ˙αtrue(t) + ˙αbias(t) (5.2)
28 Measuring the vector of gravity
Assuming there is an estimate of the bias, the effect of the bias can be reduced. When integrating the angular rate to maintain the angle, simply subtract the bias estimate. ˆ αtrue(t) = ˆαtrue(t0) + t Z t0 ( ˙αIM U(t) − ˆ˙αbias(t)) dt = ˆ αtrue(t0) + t Z t0
( ˙αtrue(t) + ˙αbias(t) − ˆ˙αbias(t)
| {z }
≈0
) dt (5.3)
Now, let us get back to α(t0) and the calibrating issue. MALLS is a system
developed to operate on a ship, and ships offshore move constantly leading to that there are no known reference angles to use for calibration. This is where the gravity comes into use. If the vector of gravity can be estimated, it can be used as a virtual angle reference. The integral will calculate the angular change and an estimate of the vector of gravity will continuously tune the angle offset. With this approach continuous calibration is achieved and therefore the system is stable in time.
The vector of gravity is approximated using the acceleration measurements.
5.2
Acceleration measurements
In 1915, Einstein published the General Theory of Relativity, which postulated that the uniform acceleration and the gravitational field are equivalent [4]. The postulate yields for the IMU which registers the field of gravity as an acceleration. This means that the if the IMU is held still, its roll and pitch angles relative the horizon plane are trigonometric functions of the acceleration components in the
x-, y-, and z-directions.
Roll: ϕ = arctan ax az Pitch: θ = arctan ay az (5.4)
5.3 The Kalman filter 29
X
Y
Z
Figure 5.1. Coordinate system of MALLS when mounted on a ship.
However, the IMU is never held still, so the roll and pitch angles at best become noisy estimates of the true angles. It is expected that the IMU will experience some vibrations, both from the ship and from the platform itself. These disturbances have high frequencies and their impact can be reduced using a lowpass filter. Other disturbances, like the low frequency accelerations from a jolting boat, are inevitable and will make the estimates suffer anyway.
The angle estimates can be considered as virtual measurements of the angles and they are expected to have large covariance due to the false statement that the IMU is held still.
5.3
The Kalman filter
Under the observability conditions the Kalman filter can be used to produce state estimates for a state space model. When the noise characteristics are Gaussian, the Kalman filter is the best estimator among all linear and non-linear estimators in the mean-square error sense [5].
In this application, the noise of the virtual angle measurement cannot be con-sidered to be Gaussian since it is a result of a trigonometric function, but a Gaus-sian distribution is still a good approximation. Two identical Kalman filters are set up, one for roll and one for pitch. The filters observe the angle via the angular rate measurements and the virtual angle measurement. The angular rate is used to follow the dynamic changes of the angle and the noisy virtual angle to contin-uously tune the offset, replacing the calibration procedure.
30 Measuring the vector of gravity
The Kalman filter algorithm is implemented in two steps, time update and measurement update. The index k|k − 1 is denoted at time k given measurements
from time k− 1.
Time update:
ˆ
xk|k−1 = Aˆxk−1|k−1+ Buk−1
Pk|k−1 = APk−1|k−1AT + Q (5.5)
where ˆxk|k−1 is the predicted state, Pk|k−1 is the predicted covariance of the
pre-dicted state and Q is the covariance matrix of the process noise.
Measurement update: ˜ yk = C ˆxk|k−1− yk Sk = CPk|k−1CT + R Kk = Pk|k−1CTSk−1 ˆ xk|k = ˆxk|k−1− Kkyk˜ Pk|k = Pk|k−1− KkCPk|k−1 (5.6)
where ˜ykis the innovation, Skis the innovation covariance, Kkthe optimal Kalman
gain, ˆxk|k the updated state-estimate, Pk|k the updated covariance of the state
estimate and R is the covariance of the measured signal y.
The inversion of the innovation covariance can be circumvented via square root implementation [5], or by making the system output scalar.
5.3.1
Measurement models
The measurement model is a bit unevenly, so it is introduced in three steps. What is making the model different is that one of the actual system outputs is used as system input in the model and the other system output is a signal manipulated via trigonometry and false statements making it very noisy.
Ordinary model
The most obvious measurement model was never used in practice. It consists of three states, the true angle αtrue, the measured angular rate ˙αIM Uand the angular
rate bias ˙αbias. There are two system outputs and no system inputs, the sample
time is Ts. The state space matrices become
x= αtrue ˙αIM U ˙αbias , A= 1 Ts −Ts 0 1 0 0 0 1 , B= 0 0 0 , C=1 0 0 0 1 0
The second state is directly known from the second output and the system does not provide any additional information about it, so why estimate what is known? If the second state is removed the implementation gets more simple and requires less computational force. Removing states directly known from measurements results in reduced-order observers [9].
5.3 The Kalman filter 31
Reduced-order model
Removing the second state and the second output, the system is reduced to two states. However, the angular rate measurement is gone as well, and the first state depends on it. The signal can be recovered without adding states by letting it act as an input. The new state space matrices then become
x= αtrue ˙αbias , A=1 −Ts 0 1 , B= Ts 0 , C= 1 0,
where u = ˙αIM U. The reduced system has one less state, one less output and
therefor the inversion of the innovation covariance, Sk, becomes scalar due to the
system’s single output.
Reduced-order with lowpass filtering model
The single system output is the virtual measurement discussed earlier, but the Kalman filter does not know that yet. Since the Kalman filter is a linear estimator, it is difficult to include the tangent function, but the filter should at least know that the system output is a low pass filtered version of the true angle. Adding a new state describing the measured signal, αlp, the state space model finally looks
like this x= αtrue αlp ˙αbias , A= 1 0 −Ts (1 − a) a 0 0 0 1 , B= Ts 0 0 , C= 0 1 0
where a is the filter constant used in the low pass filtering of the noisy virtual angle measurement β.
yk= (1 − a) · βk−1+ a · yk−1 (5.7)
where 0 ≤ a ≤ 1.
The inversion of the innovation covariance is still scalar due to the system’s single output.
5.3.2
Tuning the Kalman filter
When tuning the Kalman filter it is important to remember that the virtual angle measurement is a trigonometric function of the acceleration measurements which are calculated under the false condition that the IMU is held still. The variance of this signal will be significantly greater than the variance of the angular rate which is a raw measurement.
The tuning compromise is a tradeoff between filter robustness and filter speed, i.e. how much vibrations and shocks the system should be able to handle without losing its orientation against how fast the filter should follow measurements. Of course, the final tuning of the filter depends on the unique application.
32 Measuring the vector of gravity
5.4
Performance
The demonstration of the performance is a bit vague. The mechanical design of MALLSP makes the whole system suffer. There are backlashes in attachment points and in the servo gearboxes which in combination with the fast servo outputs make the helipad shake. When the control signals to the stabilized helipad are actuated, the simulating frames are affected as well mainly due to the backlashes of the simulating servos. It is as if the helipad were standing on a balancing device, like a ball. Unfortunately, the IMU registers the movements of the simulating frames and sends new control inputs to the servos closing an unwanted devastating feedback loop.
The conclusion of the poor design is that the same set of control signals make the platform behave a bit differently each time it is repeated. This issue combined with that the true angles of the helipad cannot be measured in realtime make it difficult to benchmark different filters.
Some performance have been tested though, mostly to compare the reduced-order two state model with the reduced-reduced-order three state model. During the tests the simulation mode has been enabled and the launch mode has been alternated to be switched on or off. Figures 5.2 and 5.3 are plots from the tests, and there are no doubts that the control of the stabilized helipad affects the simulating platform and the IMU.
0 200 400 600 800 1000 1200 1400 1600 1800 2000 −15 −10 −5 0 5 10 15 Sample Roll
Filter test, helipad stabilization disabled
Two state model Three state model
0 200 400 600 800 1000 1200 1400 1600 1800 2000 −15 −10 −5 0 5 10 15 Sample Pitch
Two state model Three state model
Figure 5.2. Estimated angles during test runs where the stabilization of the helipad was disabled.
Due to the issues mentioned above it is not fair to compare the estimated angles between the two models since the are generated from two different runs,
5.4 Performance 33 0 200 400 600 800 1000 1200 1400 1600 1800 2000 −20 −10 0 10 20 Sample Roll
Filter test, helipad stabilization enabled
0 200 400 600 800 1000 1200 1400 1600 1800 2000 −15 −10 −5 0 5 10 15 Sample Pitch
Two state model Three state model
Two state model Three state model
Figure 5.3. Estimated angles during test runs where the stabilization of the helipad was enabled.
nor is it possible to calculate the estimate errors because the true angles are not known. However, the three state model seems to generate slightly larger angles which might be a sign of faster filter speed. The main conclusion of the tests is that the control of the helipad affects the simulating frames and the IMU.
In this experiment the simulated roll angle is a sinus of frequency about 0.5 hertz, and the simulated pitch angle a sinus of frequency about 0.2 hertz. The sample time is approximately 10 milliseconds.
Chapter 6
Discussion and conclusions
This chapter is a summary of the project’s results and the conclusions drawn re-garding MALLS and MALLSP. The project have run for almost five months and the path has changed direction several times during the lapse of time. Different setbacks have interfered with the project goals, resulting in interesting problems and innovative solutions, but unfortunately all the goals are not fulfilled.
The project has lead to good understanding of the system and many conclu-sions. With today’s facts, the project would have been planned differently, where the main issue is the mechanical design further discussed in Section 6.3.
6.1
Today’s system
Today MALLSP can simulate sea movements, measure the horizontal plane and control the helipad to be somewhat parallel to the horizontal plane. However, the control smoothness is not satisfying, mainly due to that the embedded feedback control inside the servos cannot be influenced.
No helicopter landing has been performed on MALLSP yet. Under good flight conditions it would be possible to perform a landing with the launch mode and the simulation mode enabled. The control is good enough, but the actual helicopter landing surface supposed to be fitted onto the stabilized disc is not manufactured. No endurance tests have been performed and to set up the whole system outdoors is an extensive work. The test result does not outweigh the effort, time and risk to perform the test, because today’s MALLSP suffers from teething troubles and it will be rebuilt anyway. The important conclusion drawn from the implementation is that when MALLSP is seen in action it is persuasive that the concept of a tilting helipad is of great help when performing a vertical landing on a boat.
A method for measuring the relative attitude between the helipad and the helicopter skids has been developed. It relies on ultrasonic range measurements and sensor fusion. Setbacks, mainly regarding the development environment for a microcontroller, delayed the development which lead to that it was never imple-mented. However, the system works very well in simulation environments.
