MASTER'S THESIS. Laser Guided Vehicle Java and MATLAB for Control

Transcription

1 MASTER'S THESIS 2005:185 CIV Laser Guided Vehicle Java and MATLAB for Control Bengt Gunnarsson Luleå University of Technology MSc Programmes in Engineering Department of Computer Science and Electrical Engineering Division of Robotics and Automation 2005:185 CIV - ISSN: ISRN: LTU-EX--05/185--SE

2 Laser Guided Vehicle; Java and MATLAB for Control Bengt Gunnarsson

3 Abstract This study addresses the generic problem of navigating a robot between obstacles. The experimental case tested here is driving through doorways, which can be seen as a special case of a junction between corridors. The robot used was the MICA wheelchair. On-board there is a scanning laser tilted downwards, a fiber optic gyro and odometric encoders. From the laser scan walls and the door are found using the Hough/Radon transform. The results with the control law used gave a standard deviation of 1.4 cm for the lateral error and 2.2 degrees in angle. This can be improved by tuning of control law. The programming was mainly done in Java. Java methods can be integrated in a MATLAB script. MATLAB and Java together was found to be an excellent test-bed for designing and testing algorithms. All computations was done in real-time. A long term goal in robotics is to fuse together many sensors into one platform and make a multipurpose vehicle. Keywords: Java, MATLAB, Tilted scanning laser, Hough transform, Corridor junctions, Doorway, Real-time control.

4 Contents 1 Introduction Background Some Goals Notations The MICA Wheelchair The Mobile Robot The Scanning Laser The Implementation Languages Computer and Communication Transformation and Rotation Extracting Geometric Parameters from the Laser Scan The Hough/Radon Transform Zooming in the Hough/Radon Transform Extracting Valuable Information from the Hough/Radon Transform Geometry, Object Labeling and Control; A Prestudy Geometry and Object Labeling Kinematic Motion Models Control Law Navigating Through Junctions The Kalman Filter Extensions to Junctions Control Law Navigating Through Doors Laser Measurements The Innovation Process i

5 6.3 Time Analysis Error when Driving Through Doors Evaluation and Discussion Evaluation Discussion Future work A Appendix: Tolerances for Passing Between Two Obstacles 35 A.1 Representation of the Door A.2 Cylinder A.3 Wall A.4 One wall, One Cylinder ii

6 Chapter 1 Introduction 1.1 Background A lot of work has been put into improving the system on the MICA (Mobile Internet Connected Assistant) wheelchair work the past few years. One system is fully implemented and developed, the Lazerway TM. The new system has the SICK LMS200 laser on-board. The real test has now been to make every component on the robot work together. Not only just work together, but improve the performance aswell. One of the challenges is to make all sensors act together with good synchronization and a small time delay. A lot of the focus has been on making sensors work independently of each other on the robot, and a system for retrieving the data. Then of course the testing phase, to make sure that each sensor works properly. The things left to do is to test all sensors together and make a complete algorithm for navigating with the sensors. The most intuitive starting point is to make the robot go through corridors, and through doorways. 1.2 Some Goals One long term goal behind this MSc. thesis is modeling of system integration when lasers is the key sensors. Topics listed at the start was Drive through a corridor. Be able to drive through a doorway. Analyze the repeatability of the trajectory. Drive around a corner in the corridor, or junctions of multiple corners.

7 1. Introduction 2 Another goal is to get the robot to be able to avoid driving into crap in the corridor. Analyze Laser alignment error. Error analyze, compare test and modeling. Time error behavior. Time synchronization and time delay. Measuring errors and calibration errors. Scene interpretation error. Dead reckoning error. 1.3 Notations Symbols and abbreviations. MICA W-LAN GPS α tilt θ R θ L γ L,γ R d R d L α v v s v R,v L ω L l X X p C R p F U Mobile Internet Connected Assistant. Wireless Local Area Network. Global Positioning System. Angle which the SICK laser is tilted towards the floor Angle from the Hough transform representing the right walls angular position relative the robot. Same as θ R but derived from the left wall. Angle of robot relative to the corridor. Distance to the right wall, derived from the Hough transform. Same as d R, but for the left wall. Angle of the steering wheel. Velocity of the robot. Velocity of the robots steer wheel. Velocity of right respectively left drive wheel. Angular velocity of the robot. Length to the steer wheel from the center of rotation. Distance between two drive wheels. State vector. Predicted state vector. Covariance matrix. Predicted covariance. System matrix. Input vector.

8 Chapter 2 The MICA Wheelchair A short introduction of the MICA wheelchair follows; The robot is built on an ordinary wheelchair chassis of make Boden Rehab. The communications between the encoders, gyro and control unit is built on a CAN-bus. The SICK LMS200 laser communicates on a serial interface. The communication between the robot and the user is based on W-LAN, IEEE b. A PC104- computer on the robot handles the communications with the sensors. The PC104 is an ordinary computer but has a aluminium frame and has a smaller format than an ordinary PC. It is built to be used in the industry where the demands for durability is much higher. The operating system used is Linux Red Hat The Mobile Robot The robot used is the MICA wheelchair see figure 2.1. The MICA wheelchair should here be seen as a test-bed for various equipment. The sensors used in this study is SICK LMS200 range scanning laser. The original odometric wheel encoders. KVH-ECORE 1000 fiber optic rate gyro. The position data updating frequency is 20 Hz. The position update rate is faster than the laser updating. The position can be calculated seperately with dead reckoning. However, every 0.2 s the position estimate should get a correction based on the laser data.

9 2. The MICA Wheelchair 4 Figure 2.1: The MICA Wheelchair. 2.2 The Scanning Laser The SICK LMS200 is a range scanning laser. It is possible to change a number of parameters in the laser, dependig of the application. The maximum measured range is limited to 80 meters, the possible settings are 8 or 80 meters. It can use a angular resolution of 0.5 over 180 or 0.25 over 100. The setting used in this work is 0.5 over 180. The update interval of the laser is 5 Hz. The laser gives a new scan every 0.2 s. A good starting point would be to use every laser scan, this means that every loop with calculations, updating the position should not take more than 0.2 s.

10 2. The MICA Wheelchair The Implementation Languages The implementation is done in Mathworks MATLAB Matlab has support for Java. The parts that are computationally expensive could with benefit be made in Java. Matlab is also a great tool for matrix calculations. It also gives great possibilities for plotting data, for making graphs in both two and three dimensions etc. The server software on the PC104 on the wheelchair is written in C++. The server communicates with Matlab through Java methods. New Java methods are easily implemented. To make a new module that is based on a new algorithm is easy to do. One of the disadvantages with Matlab is that it is slow since it is interpreting the code, especially when using plot functions. To speed up calculations it is possible make the algorithms in Java. If that is not sufficient there is possible to go another step and compile the java code to machine code. If this still is not enough there is something wrong/inefficient with the algorithm or more computer power would be necessary. 2.4 Computer and Communication The computer on the MICA wheelchair is a PC104 with a 266 MHz Pentium processor. It has 64 Mb RAM and has three gigabyte of harddisk space. The encoders, gyro and control unit are connected by CAN-bus. The SICK LMS200 laser is connected on a serial interface. The PC104 has a PCMCIA slot where a standard IEEE b WLAN card is attached. The processing of data from the sensors can be made on another computer connected by WLAN. One of the advatages is that the processing, calculations can be made on a much faster computer than the one on the robot. One of the disadvatages is that it takes some time to send data back and forth between the user and the robot. 2.5 Transformation and Rotation The SICK LMS200 laser is mounted on the back support of the seat on the MICA wheelchair. The laser unit is slightly misaligned. By using a rigid surface, part of the robots framework, a transformation matrix could be

11 2. The MICA Wheelchair 6 derived and used for correction. See figure 2.2. The α tilt can be calculated Figure 2.2: The laser is misaligned on the robot in several directions. The tilt angle is often disturbed since it is very easy to change it. This means that the tilt is not fixed to 20.3 o. by α tilt = arcsin ( heightlaser length lasertofloor ) (2.1) Here the range from laser to the floor is calculated by analyzing a scan. The height of the laser with its current placing is 1.25 m, see figure 2.3. The tilt angle can be changed. These adjustment parameters is relative to the robot. However, the robot in its turn can be tilted or is tilted in several directions. This is because of the wheels that are air-filled, and its hard to keep the inflation/load constant. Just by driving it from indoor to outdoor on a cold day will probably misalign the laser, as well as a flat tire. When taking into consideration that the laser is not mounted straight and that it is tilted. The following misalignment angles was found 20.3 o around the X-axis. 1.7 o around the Y-axis. 2.0 o around the Z-axis.

12 2. The MICA Wheelchair 7 Figure 2.3: The Laser is put 35 cm from the COR - Center Of Rotation along the Y-axis of the robot. The Laser is also located 1.25 meters above the floor. This results in the three rotational matrices below cos ( ) 1.7 π 180 R X (1.7 o ) = R Y (α tilt ) = R Z (2.0 o ) = sin ( 1.7 π sin ( 1.7 π ) 1 ( 0 0 cos 1.7 π cos (α tilt ) sin (α tilt ) 0 sin (α tilt ) cos (α tilt ) cos ( 2.0 π 180 sin ( 2.0 π 180 ) ) (2.2) ) ( ) sin 2.0 π ) cos ( ) 2.0 π (2.3) (2.4) These three rotational matrices could of course be multiplied together to one correction matrix. These matrices should not be derived manually, since they change almost from day to day.

13 2. The MICA Wheelchair 8 Figure 2.4: Example of a corrected scan, but not translated, as seen only in the range of four meters the scan is around 0.2 m wrong in the X-axis because of the yaw, pitch and roll offset. The calibration matrices can be found by looking at the floor. And then rotating in a clear room/corridor. Using the Hough/Radon lines found will be sufficient to adjust the laser in all directions.

14 Chapter 3 Extracting Geometric Parameters from the Laser Scan The best way to extract linear segments in a laser scan is to use the Hough/Radon transform. Walls and other planar surfaces such as doors becomes easy to find. If it then is know where the object is that is tracked. Then it is possible to make the Hough/Radon transform work faster. 3.1 The Hough/Radon Transform The Hough/Radon transform is based on the equation for a line. Consider a line at distance d and angle θ of the normal. On this line there is a number of points (x i, y i ). Then d = x i cos (θ) + y i sin (θ) (3.1) Where x and y is taken directly from the scans. The Hough/Radon transform counts all the points at an certain angle θ and at a distance d and in an predefined stripe from that distance, in this case 1 cm. d restricted to an interval [0, 8] and θ is also restricted to an interval [0, 2 π]. Systematically go through these intervals in a loop and build up an matrix, where the highest peaks represents an distance and an angle where there is a straight line. Also, the floor is always on the same place in the scan, and depends of tilt of the laser. This makes it easy to remove some irrelevant data from the scan in a early stage of the algorithm.

15 3. Extracting Geometric Parameters from the Laser Scan 10 Figure 3.1: The Hough/Radon transform counts all the points at an certain angle θ and at a distance d and in an predefined stripe from that distance, in this case 1 cm 3.2 Zooming in the Hough/Radon Transform To be able to zoom in the Hough/Radon transform is needed. Since, the laser can measure up to 80 meters and if reasonable resolution is wanted, the Hough/Radon matrix could be unnecessary large. If a length resolution of five centimeters in length and 0.5 degrees of the angular resolution. Then the Hough/Radon matrix will consist of elements. If the length resolution is made smaller, the time needed to extract the Hough/Radon matrix still will be rather long. Longer than the 0.2 s time that it takes to get a new laser scan. The computation time of the Hough/Radon transform can be pretty long in worst case. The solution was first of all to write the Hough/Radon transform in Java. With the inputs, the laser scan and an array with lengths and angles. Where suspected objects are located. This window could be used to extract objects at a certain distance and angle.

16 3. Extracting Geometric Parameters from the Laser Scan 11 Figure 3.2: The Hough/Radon transform for the measurements in figure 2.4. Where the three tops represents two walls and the floor. The rightmost top belongs to the leftmost since the is some wrap-around because of the fact that sin (2 π) = sin (0). If a large window is used, and if two(or more) peaks are suppose to be in that area. Then remove the first peaks laser scan points, and then start over with the remaining laser scan points. Benefits of Zooming the Hough/Radon Transform The Hough/Radon transform is modified to be able to check any distances and angles. It is possible to use R p, the predicted covariances in a Kalman filter R p = H C p H T (3.2) For each of the parameters in the Kalman filter to reduce the search area in the Hough/Radon matrix. This windowing function could easily reduce the calculation time up to ten times. This is a part of figure 3.2 same resolution,

17 3. Extracting Geometric Parameters from the Laser Scan 12 Figure 3.3: This is the a part of figure 3.2 and it is the leftmost top. It is just 10 o to 10 o and the distances is 1.1 to 1.3 meters. but only 20 degrees and 2 dm instead of 0 to 8 meters and 360 degrees. If it is known where a wall is within say 20 degrees and 2 dm there is no need to compute the whole Hough/Radon area. That is the the main disadvantage is that no new surfaces will be found this way. A complementing function for finding new objects has be made. 3.3 Extracting Valuable Information from the Hough/Radon Transform To find the right and left wall of a corridor is based on the fact that they are parallel to each other. γ L γ R = 180 (3.3)

18 3. Extracting Geometric Parameters from the Laser Scan 13 Although, it is required that the robot starts with two walls left and right defined. The deviation from the zero angle of the corridor respectively the deviation from the middle point can be calculated by θ = γ L + γ R 2 and (3.4) y = d L d R (3.5) 2 It is of course so that a filter of some sort has to be applied to take care of any spuriouses, preferably a Kalman filter.

19 Chapter 4 Geometry, Object Labeling and Control; A Prestudy This chapter will describe the basic geometry and kinematic models to be used to control the robot. We start up with the simplest case, driving in a corridor. At this point there will be no obstacles present in the corridor. 4.1 Geometry and Object Labeling Basically the goal is to find a feedback law from laser measurements in such a way that the robot drives automatically between objects. The simplest case is a corridor with right and left walls. The approach to the more general case is the following; We consider the following steps: 1. A scene/workplace with several objects. 2. Local range sensing in the forward direction of the moving robot. 3. A decision on how to locally pass between the objects. 4. Based on this decision a labeling of the objects with respect to Right(R) or Left(L) of the planned trajectory. 5. Polygons of straight lines can be used to describe the Right(R) and Left(L) envelops. Indoors, the Range Weighted Hough/Radon transform gives a natural input in many cases.

20 4. Geometry, Object Labeling and Control; A Prestudy 15 Figure 4.1: Robot at pose (x,y,θ) in a corridor. The steering wheel(dog) is directed α towards the aiming point(rabbit) in the middle of the corridor. 6. The distance transform gives a general method for planning the trajectory. This is not studied in this report. 7. The simplest case above is a corridor. The suggested control system should have some generality. If we, say, want the robot to turn around in the corridor the input to the control system should just be a relabeling R L and L R of the walls. 4.2 Kinematic Motion Models The MICA wheelchair has motors on both of its front wheels. The back caster wheels are only there as support wheels. This means that the robot

21 4. Geometry, Object Labeling and Control; A Prestudy 16 can rotate around its own axis, when one wheel drives forward and the other wheel drives backwards. Next we are to describe the basic control law of the robot. Thus consider a robot with a real or virtual steering wheel at distance L in front of the two fixed wheels. The steering wheel angle is α and the velocity of the steering wheel is v s. Then the robots velocity will be and the angular velocity of the robot will be v = v s cosα (4.1) ω = ( ) vs cosα (4.2) L For a robot with motors on both fixed front wheels we have v = (v R + v L ) 2 (4.3) ω = (v R v L ) (4.4) l where l is the distance between the two drive wheels. The two sets of equations are just alternative descriptions. [2] gives more details. 4.3 Control Law When driving in a corridor, laser measurements of the walls will generate very clear peaks in the Hough transform. If the robot is suppose to travel in the middle of the corridor, then the walls should be located at the same distance and be parallel to the direction of travel. This can be used to design a control law. The control systems task will be to control the robot based on the two most significant peaks in the Hough transform. The peaks should be located at the same distance, and have the angles γ R = 90 o and γ L = 90 o. To further improve the accuracy the least square line fitting is applied to the points in the wall stripe returned by the Hough transform.

22 4. Geometry, Object Labeling and Control; A Prestudy 17 The resulting approximation is d L,d R and γ L, γ R shown in figure 4.1 above. The robots position and orientation relative to the corridor is given by y = d R d L 2 (4.5) θ = γ L + γ R (4.6) 2 This control law was tested in an earlier master thesis here in Lulea[7]. The steering along the middle line in the corridor is made by a control law used in Mobile Robot Navigation using the Range-weighted Hough transform [2]. Another control law that have been used is the dog-rabbit control law.

23 Chapter 5 Navigating Through Junctions This chapter will describe the basic equations needed to navigate along a corridor and through junctions of corridors. Passing through doors i also considered. The parameters are essentially as in the Hough/Radon transform. 5.1 The Kalman Filter Consider a corridor with two walls and a Kalman filter with four states X k = [ d L γ L d R γ R ] (5.1) where γ L and d L is the angle for the wall respectively the distance. The angles are measured counter-clockwise from the robots frame, compare figure 4.1 above. The process model is updated as X k+1 = F X k + G k U k (5.2) where X k is the current position data. The control during the interval τ s. U k = [ v ω ] τ (5.3) where v is the velocity and ω is the angular velocity. Depending on the types of robots v and ω are given by equations 4.1 to 4.4 above. The input matrix G equals G = sin(γ L ) sin(γ R ) (5.4)

24 5. Navigating Through Junctions 19 Figure 5.1: Corridor with a change of direction and/or width. When the laser sees this combination of two corridors, we need to extend the state in equation 5.1 to eight variables. and F is the system matrix which is a four by four unit matrix F = (5.5) The filter is updated every 200 ms, i.e. τ = 0.2s. Observe that if we have a perfect corridor the estimation of d L and d R will have a strong (negative) correlation, since d L + d R is equal to the width of the corridor. There will also be a similar strong correlation between γ L and γ R.

25 5. Navigating Through Junctions 20 Figure 5.2: A 90 o turn of a corridor. Is also a simplification of normal orthogonal corridor junctions. 5.2 Extensions to Junctions figure 4.1 and the equations above are for estimating the position of the peaks in the Radon/Hough transform. In this subsection we describe how this basic equation can be extended to junctions, open doors etc. The simplest case is in figure 5.1 when the corridor changes direction and/or width. If we use the Kalman filter described above the new walls will not be accepted Inside the association gate. Depending on the control law/rule the robot will stop or run into the new (left) wall. To handle such cases the state vector in equation 5.1 should be extended with new walls (described by distances d and angles γ) as they are found by the laser. We leave the explicit equations for G and F to the reader.

26 5. Navigating Through Junctions 21 Figure 5.3: This is an example of an open door in the right side of the corridor. This is the case tested in the next chapter. Note, in this case are the two line segments to the left not connected - there is a gap. For both the case in figure 5.1 and in figure 5.2 the laser can observe two walls left of the planned trajectory. The state vector is then X = [ d L1 γ L1 d L2 γ L2 ; d R γ R ] (5.6a) When the robot moves on there will also be an extension of the X for a second wall to the right giving X = [ d L1 γ L1 d L2 γ L2 ; d R1 γ R1 d R2 γ R2 ] (5.6b) The case with the open door in figure 5.3 can be described in the same way as equation 5.6 above. The main difference is that the two line segments L 1 and L 2 are not connected. Note that the angle γ L2 will in this case initially be negative.

27 5. Navigating Through Junctions Control Law There is a range of possibilities for designing control laws to our problem of navigating between objects. The choices stood between a dog-rabbit control law and more smooth one that is the one used. A Dog-Rabbit control law Figure 5.4: The navigation between two objects is described by a pair of reference points and a steer line in between. The reference points put on the two walls/objects, and the steer-line in between the objects. The properties of the control between the points depends on the robot and can be tuned by the gains k 1 and k 2 in equation 5.8 has large disadvantages in that it is slow and cuts corners or after tuning produces a large overshoot. It is not preferred in this case since we want one control law for all cases studied. Several geometrical cases are given in Appendix 1. Expressed in laser coordinates, the angle to the point in between the two objects θ = γ R + γ L (5.7) 2 The robot will then always strive to go to the line that goes orthogonal to a line between the reference points. Where the angle of the steer wheel will be α = arctan ( k 1 dr d L 2 If we interpret figure 4.1 we get that + k 2 sin (θ) ) θ (5.8) k 1 = 1 d (5.9)

28 5. Navigating Through Junctions 23 k 2 = L d (5.10) This control law is also useful in other cases, since it is possible to put the reference points on any object. For example a corner or on a door opening.

29 Chapter 6 Navigating Through Doors This chapter is on navigating through doors. The navigation through doors is essentially the same as navigating in junctions. There are some differences. In a junction the new corridor is most often orthogonal to the corridor from which the robot are leaving. The open door into the hall could be open at different angles. Another difference is the fact that a doorway is narrower than a corridor, so the error has to be low when navigating through a door. 6.1 Laser Measurements The case studied is illustrated in figure 6.1. The corresponding laser scan is plotted in figure 6.2. The thickness of the door can not be ignored. In this case the door is six centimeters thick. If the door thickness is ignored and the door is open at a large angle the crossing point of the two Hough/Radon lines will be moved E = 0.06 (6.1) sin (θ door ) If the door is even thicker the error will increase faster if not taken into consideration. Although, if the door is open at 90 degrees the error will be zero. 6.2 The Innovation Process The first step in this evaluation of the result is to analyze the innovation process. This is some kind of testing the internal consistency of the model and measurements.

30 6. Navigating Through Doors 25 Figure 6.1: Picture taken from the position of the laser.the corresponding laser scan is plotted in figure 6.2 The innovation is defined as the difference between the predicted values and the measured values. In figure 6.3 is the distance innovation plot for navigating through a door. In figure 6.4 is the angle innovation plot for navigating through a door. At t 31s the Kalman filter loses track of the door, and wall, since the robot has passed into the room. Consequently the three times the sigma limits are exceeded. At t 20s the robot makes a rather hard turn and the limits for the wall increases. The innovation is down on approximately one cm in distance. This is the resolution of the laser. That is an satisfying innovation. The innovation for the angle is approximately one degree. This is also a satisfying innovation.

31 6. Navigating Through Doors 26 Figure 6.2: Laser scan taken in a corridor with a door open. 6.3 Time Analysis A lot of effort have to go into making all algorithms time effective. Many small sources of time consuming code will at the end give a noneffective algorithm. The Hough/Radon transform - An extremely time consuming way of processing data, although there is only 361 points. There is other ways of extracting linear elements from laser range scan data, but no-one as robust as the Hough/Radon transform. Acquire the position data - As a laser scan is received, matching encoder and gyro data must be acquired. Position data gets a time stamp and are put in an buffer. Searching in this buffer is at the moment slow, so the buffer

32 6. Navigating Through Doors 27 Figure 6.3: Range innovation, Green: innovation of wall. Red: innovation of door. The innovation for the door is leaving the 3-σ boundary 0.8s before the one for the wall. should be small. It should not be larger than that it fits all position data packets between two laser-scans. Updating the Kalman filter. - This might be the fastest operation done, but there is a lot of matrix multiplying. If the Kalman filter is expanded to contain more states this must be taken into more consideration, and probably be implemented in Java. Plotting - When testing algorithms, it is nice to have some plots to look at. MATLAB is an excellent tool to use for this purpose. However, there are some disadvantages, MATLAB is a bit slow. Real-time plotting should be

33 6. Navigating Through Doors 28 Figure 6.4: Angle innovation, Cyan: innovation of wall. Magenta: innovation of door. The innovation for the door is leaving the 3-σ boundary 0.8s before the one for the wall. avoided, although it is wanted. 6.4 Error when Driving Through Doors A number of test was made to evaluate this algorithm. First of all to find k 1 and k 2 in the control law, equation 5.8. This might not be the most effective control law for driving through tight spots, since it produces a overshoot. Although, the overshoot is possible to reduce to be approximately ten centimeters. In figure 6.5 it is easy to see the overshoot in x. However, the doorframe is not placed at the peak of the overshoot but at y = 1.5 where

34 6. Navigating Through Doors 29 the overshoot is in at only a few centimeters. In figure 6.6 the last two meters is zoomed from figure 6.5, where it is easier to se that the error in x at y = 1.5 is just a few centimeters. The gain parameters in the control law k 1 and k 2, that gives a satisfying result is k 1 = 10.7 and k 2 = 1.4 where the k 1 parameter controls how much gain should be put in on getting nearer the centerline see figure 5.4. The k 2 parameter controls how much the angle error affect the steering. In figure 6.5 it is not totally obvious how the robot drove to get to the end position. A clearer picture can be achieved the state space plot (x,y,θ) in figure 6.7. There it is possible to se that the robot first turns almost 90 degrees before turning back 90 degrees again and then goes through the door. The robot was also placed off center from the goal line, but had almost the right direction. It drove in a sort of S-curve to get to the goal position. There is some things worth mentioning in figure 6.6. At y = 1.5 something happens. That is the place where the robot starts to go through the door and the curve starts to get noisy. The interpretation is 1. The Hough lines is shorter here, the Hough lines might not be based on the wall or the door and might therefore not be totally correct. 2. There is a small door-step that disturbs the robot when passing through the door. There is also possible that both of these error sources act together and create a greater error. What will happen to the error if the true position and orientation of the laser is not according to the model. If the center of rotation is not correct. What effect will these errors have on the robot when driving through a doorway. One of the other error sources is that the odometric encoders are expected to add approximately one percent of the travelled distance.

35 6. Navigating Through Doors 30 Figure 6.5: Trajectory when the robot is passing through the door: Plotting two dimensional x,y. The door frame is passed at y = 1.5 m and x = 0.

36 6. Navigating Through Doors 31 Figure 6.6: Approximately the last two meters in figure 6.5. There is a zoom-in along the x-axis.

37 6. Navigating Through Doors 32 Figure 6.7: Going through the door: Plotting three dimensional state variables (x,y, θ). The door frame is passed at y = 1.5 and x = 0. Observe the initial turning of the robot.

38 Chapter 7 Evaluation and Discussion 7.1 Evaluation The algorithm for driving through a door works very well. Although, the control law could be better, the control law used in these experiments produces an overshoot. However the overshoot is not to big that it is not possible to control the robot. The result driving through the doorway gave a standard deviation of 1.4 cm and 2.2 degrees based on 10 runs. These results have the right magnitude, and are good enough for driving through the door. 7.2 Discussion The main goal in this master thesis has been to make the robot, the MICA wheelchair, move around in an indoor environment. Using a test-bed based on Java and MATLAB. Driving through a doorway and through junction are maybe the most basic moves an indoor-robot must cope with. The fact that moving through a doorframe can be described as driving through junctions makes it possible to eliminate some scenarios. This makes it possible to use to same control law for both driving through doors and through junctions. There is still a lot more to be done in this field. Although, a Java and MATLAB test-bed is big step into making it easier to test and evaluate new algorithms. 7.3 Future work Some of the goals set at the very beginning of this study has not been reached. Some of the future work to be done is

39 7. Evaluation and Discussion 34 Get the robot to be able to avoid driving into crap in the corridor. Analyze Laser alignment error. Error analyze, compare test and modelling. Time error behavior. Time synchronization and time delay. Measuring errors and calibration errors. Scene interpretation error. Dead reckoning error. Some of these goals has been mentioned shortly in the report but further studies is needed.

40 Appendix A Tolerances for Passing Between Two Obstacles A.1 Representation of the Door A door can be represented as two cylinders which in two dimensions will be two circles. A.2 Cylinder There is two conditions, equations A.1 and A.3 that must be fulfilled to get maximum angle at which the robot can can pass between the cylinders. These conditions are written for one cylinder but it is easy to repeat them for the other cylinder. ) ( ( Lw 2 + ξ 0 sin (α) > r + W ) (A.1) 2 Where And α = arccos ( r + W 2 L w2 + r + ξ 0 L > W > L w ) (A.2) Or if this condition is fulfilled ( ) (L ) 2 ( Lw W 2 + r + ξ 0 + x > ) 2 cos ( arctan ( L 2 W 2 ) ± θ ) (A.3)

41 A. Appendix: Tolerances for Passing Between Two Obstacles 36 Figure A.1: Robot between two cylinders. Figure A.2: Robot next to one cylinder. Could be a part of figure A.1

42 A. Appendix: Tolerances for Passing Between Two Obstacles 37 Where ( x = r r cos arcsin ( )) ηpos r And η pos is the position of a corner of the robot in the η-direction. (A.4) A.3 Wall The condition for two walls ( ) (L ) 2 ( ) Lw W 2 ( ( L ) ) 2 + ξ 2 0 > + cos arctan W ± θ (A.5) This condition is similar to equation A.3. If r, x 0. When the radius in a cylinder is large the cylinder could be considered a wall, since x is so small that it could be disregarded. A.4 One wall, One Cylinder One wall and one cylinder requires that both the conditions discussed above is combined and fulfilled.

43 A. Appendix: Tolerances for Passing Between Two Obstacles 38 Figure A.3: Filled line(green): two cylinders. Dashed line(blue): two walls. Radius of cylinders 0.05 to 20 meters. As seen the larger the radius of the cylinders becomes they more and more becomes similar walls. W = 0.6, L = 1.5 and L w = 1.0.

44 A. Appendix: Tolerances for Passing Between Two Obstacles 39 Figure A.4: Filled line(green): one cylinder, one wall. Dashed line(blue): two walls. Radius of cylinders 0.05 to 20 meters. As seen the larger the radius of the cylinders becomes they more and more becomes similar walls. Here we also can se that it can benefit to drive nearer the cylinder if the cylinder is small. W = 0.6, L = 1.5 and L w = 1.0.

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