The Development of a Theoretical Model for Simulation and Localization Improvement with a Polaris Ranger 800 HD Side By Side ATV Using Command Signals Robert Kidd Spring 2011 Magna Cum Laude Bachelor of Science in Mechanical Engineering Advisor: Dr. Carl Crane
Abstract This thesis explains the dev elopment of a vehicle model used to predict vehicle location based on initial conditions and command efforts. This model is to be developed so that it can be derived from the command signals of the Joint Architecture for Unmanned Systems (JAUS) and can be implemented in the C++ programming language. The resulting model showed sufficient accuracy to be used to test controller design by mimicking the performance of the physical system w ithin a desired acc uracy. While not be ing accurate enough to replace a precision GPS unit, the model can be used to regulate and replace low quality GPS data for a limited time
Table of Contents Page Abstract ................................ ................................ ................................ ................................ ........... 2 Introduction ................................ ................................ ................................ ................................ ..... 4 Motivation for S tudy ................................ ................................ ................................ ................... 5 Experimental Procedure ................................ ................................ ................................ .................. 6 Results ................................ ................................ ................................ ................................ ............. 8 Conclusion ................................ ................................ ................................ ................................ .... 16 References ................................ ................................ ................................ ................................ ..... 17
Introduction Since the first uses of GPS, there have been inherent errors that limit accuracy. Many of these errors affect the signal sent to GPS receivers from satellites. These errors can accumulate from irregularities in the upper and lower atmosphere to solar radio noise to relativistic e ffects between clocks on the ground and within the satellites. In addition, these errors within the satellite signals can be compounded by the signals being reflected off the environment before reaching the receivers or by environmental structures that pre vent the signals from being received at all. The summation of these effects can cause GPS da ta to vary between tens of meters and several centi meters depending on the system and the error correction methods used. There are various methods to minimize these effects, allowing very sensitive measurements under specific, cont rolled conditions (Kleusberg & Langley 1990 ). Part Three of the Joint Architecture for Unmanned Systems specifies that the command signal, or wrench command, for a vehicle should consist of a desired propulsive, resistive, and rotational effort. Each of these values should represent a percentage of maximum effort. Thus each of these efforts has a maximum of value of 100. Both the propulsive and resistive efforts have a minimum value of zero, but the rotational effort has a minimum effort of 100 to account for both clockwise and counterclockwise rotation (2007b, pp.20 21) Among the p revious efforts to account for these errors is to integrate the GPS with an inertial navigation system (INS) to smooth GPS errors over short distances (Cramer 1997, p 6). The INS uses a series o f accelerometers fixed on the object to be measured or incorporated into the center of a gimbal three dimensional space (Cramer 1997, p. 4 5). Another method of er ror compensation is dead reckoning. This system uses current, past, and future speeds and dire ctions to predict the current and future position (Bowditch, 1995, p.113). Neither of these methods is accurate over long distances as they use many assumptions about the system including an ideal system that is constant in time no external influence, and a perfect mathematical model (Cramer, 1997, p. 6; Bowditch, 1995, p. 113). The model being created in this study resembles a dead reckoning model. The difference between the two is that a true dead reckoning model would use the measured position, heading and velocity to predict the future location while the model studied here will first generate the current heading and velocity and then use the last position estimate to calculate the current position.
Motivation for Study Despite the se factors, GPS remains a major method of localization due to its overall traveled, and the ease and speed involved in setting it up. However, whe n a GPS receiver is pla ced within a building or under dense vegetation, the receiver will receive no signal, too few signals to perform calculations with much confidence, or the direct, clean signal will be to o jumbled with erroneou s signals so that the accurate signal cannot be filtered out. When the desired application requires GPS receivers to enter these troublesome areas, the result can be complete failure. If a model can be developed with sufficient accuracy, the anticipated position of the receiver can be calculated by the model when the GPS signal is insufficient for accurate localization. One of the other specifications of this study was that no sensors were to be used with the localization model. Specifically, this meant that the localization method could not use the sp a model that could be implemented on any vehicle without needing extra sensors or additions to the current design. Essentially, the desired design would be as isolated from the syste m as possible so that it could detect any error within the current state of the vehicle. This specification developed from the Unmanned Systems (2009a, pp. 4 5). This spec ification implies that the model should know nothing about the under lying system so that the platform can be changed with no change needing to occur within the implementation of the model. While this may seem short sighted in that the model is vehicle dep endent, it should be able to be moved from one Polaris vehicle to the next as long as they are the same vehicle type. Thus, this platform independence m ust be maintained as much as possible.
Experimental Procedure The apparatus used consisted of a Pol aris Ranger that had been modified to be controlled remotely through the JAUS architecture that was equipped with a GPS receiver, a GPS base station to increase the accuracy of the GPS receiver, and a computer to control the Polaris. The trials were run i n an open field that was as flat as possible to maximize the accuracy of the collected data and avoid any of the previously mentioned errors. The field needed to be level to have accurate translation from the wrench command to the vehicle data. If the vehi cle was traveling up an incline, the wrench would be unnecessarily large to accommodate the sloping terrain To develop the model, the speed and GPS calculated latitude, longitude, and yaw of the Polaris were recorded at regular intervals al ong with the JAUS wrench command Once the data collection was started, the vehicle was driven along seven different paths. The first four paths were used to generate an initial model while the next three paths were used to test and improve the model. The first path va ried the propulsive effort of the wrench command while maintaining constant zero values for the rotational and resistive efforts. This data was used to generate a model for the acceleration of the vehicle. The second path maximized the rotational effort while maintaining a positive propulsive effort and a zero resistive effort. This established a baseline for the path of the vehicle when turning right at various speeds. The third path mimicked the second path, except it minimized the rotational effort, es tablishing a baseline for the path of the vehicle when turning left. The fourth path maintained constant speeds as much as possible this gave the minimum propulsive effort needed to overcome the initial inertial of the system and the behavior of the system in steady state conditions. Once sufficient data had been collected along the first and fourth paths to give satisfactory results, the propulsive effort was zeroed and the resistive effort was increased. This allowed the resistive portion of the model to be calculated. To test and refine the model, the fifth path varied both the propulsive effort and the rotational effort so that the vehicle traveled along an arbitrary path that included acceleration, constant speed, straight travel, left hand turns, right hand turns, deceleration due to zero resistive and propulsive efforts, and a loop. Primarily, this allowed the rotational portion of the model to be tested.
The sixth path was designed to test the propulsive portion of the vehicle model. To do this, the v ehicle was driven in straight line with while significantly varying the propulsive effort, including a zero propulsive efforts. Finally, the resistive model was tested using the seventh path. To generate this path, the vehicle was brought to a stop several times while travelling along a straight line.
Results The rotational portion of the model proved to be the simplest. The yaw of the current time step can be calcula ted as Here, ( ) repr esents the yaw of the vehicle measured in radians from true north at the current time or the previous time step depending on the subscript, (K) represents the rotational effort at the current time step, (K max ) represents the maximum rotational effort of 10 0, (v) represents the This model was moderately accurate for the both the full right and full left conditions. However, during testing with the variable rotation path, some creep inaccuracies showed up due to inaccuracies in the model. These errors were accounted for with compensators that depend on the total time the system has been running (t total ) and the time since the last zero rotational effort (t zero ) This gave the following result: For all times when the desired effort is between 50 and 50, this result has sufficient accuracy. However, outside of this window, the (t zero ) compensation must be altered to give The combination of these models produced a result that is accurate to within 5 degrees. The resulting model and path are included below along with the error between the theoretical and actual yaw values. The 10 degree window is not perfect, but for simple tasks, it will be considered sufficient. The goal application for this study involves passing under scattered trees. Thus, the distance travelled without reliable GPS will be considered minimal, less than 10 meters. Thus, a 5 degree error will result in a n additional localization uncertainty of at most 0.872 meters due to the rotational portion of the vehicle model. Assuming a 3 meter uncertainty before the interference and a 1 meter uncertainty for the rotational portion after rounding, the 4 meter uncert ainty is considered acceptable.
Figure 1: Latitude and longitude of the path used to find the rotational portion of the model. The end of the path is denoted by the blue point in the lower left. Figure 2: The actual and predicted yaw values showing the close rela tionship of the two. The most visible discrepancy can be seen just past 10 seconds. This is not the largest error, but it is the most noticeable. 29.6462 29.6463 29.6464 29.6465 29.6466 29.6467 29.6468 -82.3545 -82.3544 -82.3543 -82.3542 -82.3541 -82.354 -82.3539 Latitude (deg) Longitude (deg) Path 4 End Point -6 -5 -4 -3 -2 -1 0 1 0.0000 10.0000 20.0000 30.0000 40.0000 50.0000 Yaw position (rad) Time (s) Actual Theoretical Minus Creep
Figure 3: Plot showing the error between the actual and predicted yaw as a function o f time. This error is considered random. It is within the window between 5 and 5 degrees, so it is considered acceptable. The propulsive portion of the vehicle model proved to be much more difficult than originally anticipated. This difficulty arose bec ause the vehicle response was expected to behave similar to an overdamped step response. However, because the input for the vehicle acceleration was never maintained long enough for the system to reach steady state this model could not be used. In stead, t he model construction started with comparing the acceleration verses the time as a natural log function of the time. Unfortunately, there were substantial difficulties with this plan. The vehicle has a wind up that is not well defined. Neither the relationship between the propulsive effort and the lag nor the minimum propulsive effort needed to begin vehicle motion were found. Instead, a lag of 0.45 seconds and a minimum wrench effort for motion of 14 were assumed. From these assumptions and the dat a collected from the first testing path, a general model was created. Additional multiplication steps were included to force the system to stop the excel calculations if the propulsive effort dropped below 14. The equation developed was -5 -4 -3 -2 -1 0 1 2 3 4 5 0 10 20 30 40 50 Error in Heading (deg) Time (s)
In this equation, (a ) represents the velocity of the current time step. (t zero ) represents the time si nce the system received a propulsive effort greater than 14. Finally, (P) represents the propulsive effort. The velocity was calculated from this acceleration over the previous time step length. absolute value of x. If the term within the natural log function could not be evaluated, the velocity was set to zero. There was no discernable pattern in the error generated from this function. To model the response of the car when the propulsive effort is removed, the acceleration verses time graph was examined again. T he relationship between the acceleration and the time appeared to have similar shapes, but different starting conditions. A trend line was added to the deceleration the follows the highest velocity. After this, the time elapsed (t zero ) was increased so that the starting point of each graph lay on this new trend line. This new point will be called the adjusted time elapsed (t adj ). The data for all of these cases lay very close to the trend l ine, so this equation was used. Thus, to calculate the speed after the propulsive effort was removed, the elapsed time was adjusted and then the trend line equation that follows was used. This equation also showed no discernable error pattern that could be canceled. The combined acceleration and deceleration plot is shown below along with the error involved. The equations were combined by using the equation that was dictated b y the propulsive effort at that point. Figure 4: The theoretical and actual velocity of the system as a function of time. The data is less coincident than the rotational effort plot. 0 1 2 3 4 5 6 7 0 5 10 15 20 25 30 Velocity (m/s) Time (s) Theoretical Actual
Figure 5: The error in the velocity as a function of time. This error can cause substantial difficulties if the desired speed is very slow because the error does not change significantly with the velocity. Lastly, the response of the model due to the resistive effort was calculated. Its calculations were similar to the pro cess for the propulsive calculation except with minimal lag and no minimum value for the response. Thus, the equation for the deceleration due to a resistive effort was calculated to be In this equation, (R) represents the resistive effort and ( ) represents the initial velocity before breaking started. The only change that occurs to this equation is that an error is generated if a minimal resistive effort is implemented for more than 0.25 sec onds before a larger effort is used. If that is the case, this error can be accounted for by resetting the (t zero ) value to zero just before the effort becomes greater than 20. The results of this portion of the model are shown below. -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0 5 10 15 20 25 30 Velocity Error (m/s) Time (s)
Figure 6: Plot of t he actual and theoretical velocity of the system as a function of time. The theoretical values were only calculated during the portion of the plot when the resistive effort was greater than zero. Figure 7: Plot of the error in the velocity between the ac tual and theoretical values as a function of time. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 30 35 Velocity Error (m/s) Time (s)
Up to this point, the model appears to be very accurate. However, in these tests, it was rotational model depended on the speed, but the speed was assumed to be independent of the steering effort. To test whether this assumption was accurate, the models were tested simultaneously. To do this, the data from the fifth path was examined. The rotational model was implement ed as before except that instead of using the speed recorded by the sensors on the vehicle, the speed generated from the propulsive model was used. The results of this system wer e superimposed on the previous L at/ Lon plots of the fifth path, producing the following. Figure 8: The latitude and longitude of the vehicle predicted without using any sensors superimposed on the previous graph of the fifth path. The sensor assisted estimation remains faithful to the original, but the unassisted model strays sign ificantly. However, it should be noted that the general shape of the path is maintained, though it is distorted. As the above graph illustrates, the system is absolutely insufficient to predict the vehicle location as a whole. Although the final differen ce between the predicted and actual location is only 8 meters, the model strayed as much as 30 meters from the measured position during the 60 29.6461 29.6462 29.6463 29.6464 29.6465 29.6466 29.6467 29.6468 -82.3545 -82.3544 -82.3543 -82.3542 -82.3541 -82.3540 -82.3539 Latitude (deg) Longitude (deg) Sensor Assisted Lat Lon Estimation Actual Lat Lon Unassisted Lat Lon Estimation
second trial This is completely unacceptable for the initial design goals for localization. While here, it shoul d be noted that the sensor assisted model shows remarkable accuracy. Its va riation is at most 1.25 meters during the trial. This range would be acceptable for all purposes described except it used the speed sensor. In an effort to discover where the model went wrong, the predicted speed was compared to the measured speed, producing Figure 9. Figure 9: The measured and predicted speed of the vehicle while following the fifth test path. It is easily evident that there are substantial errors in the propuls ive model. First, when the effort and a decreased propulsive effort. The vehicle accelerated slowly at this point, but the model continued to accelerate rapidl y. The second erroneous section is between 17 and 35 seconds. Here, the model completely failed. After a section that produced an accurate acceleration period the vehicle performed a sharp turn at the 21 second mark. At that point, the propulsive command was constant, but the speed did not increase as expected. Instead, it decreased. Apparently, the speed depends on the rotational effort to some degree. Additionally it 0 1 2 3 4 5 6 0 10 20 30 40 50 60 70 Speed (m/s) Time (s) Speed Estimate Measured Speed
than predicted. At this point, the propulsive effort was decreased below the threshold for vehicle motion. Essentially, this propulsive effort was enough to slow the deceleration of the vehicle, but not to cause it to move. Once this period of deceleration was completed, the model began to Two things become apparent from these results. First, the sp eed generated from the propulsive effort depends on the rotational effort of the vehicle. Second, the deceleration of the vehicle when the propulsive effort is non zero is not accura tely predicted with this model. A major complication in the development o f a model that corrected for these two error sources is the lack of data. Because only one trial was done varying both the speed and the direction of the vehicle, there is limited data to describe the results and no method available to test the results. Na turally, the best solution would be to collect more data. However, the Polaris ATV used to develop this model was bei ng loaned for a limited time. It was returned after the first round of tests. Thus, without proper data, the current model must be used. To limit errors, it is advised that the vehicle be run so that it only turns when it is at a constant speed, that no propulsive effort be used when the vehicle is decelerating, and that the vehicle be run at a low speed. The low speed portion is a recommenda tion based on the high errors that seem to exist when the vehicle is traveling in excess of 3 m/s. Conclusion Through this study, a model was developed that could be used to predict the position of a Polaris Ranger 800 HD Side By Side ATV accurately. The accuracy is insufficient to replace a GPS unit, how ever when aided by a sensor to detect the speed of the vehicle accuracy is sufficient to track when the GPS system gives faulty data and supply a n reliable position estimation until the GPS d ata becomes more reliable. When unassisted, the model is not accurate t o replace a faulty GPS signal. For localization, i ts only use would come if the GPS signal were blocked entirely, such as in a tunnel or building. However, t he model is more than suffic ient to act as a simulation to test other components within the JAUS system without the need to drive the physical vehicle. Because the simulation is to serve as a test bed, there is no need for it to be perfectly accurate.
References Bowditch N. (1995 ). The American Practical Navigator (1995 ed.). National Imagry and Mapping Agency (Ed.). Bethesda, MD: National Imagry and Mapping Agency. Retrieved from http://www.irbs.com/bowditch/ Cramer, M. (1997). GPS/INS Integration. In D. Fritsch & D. Hobbie (Eds.), Photogrammetric Retrieved from http://www.ifp.uni stuttgart.de/publications/phowo97/ cramer. pdf Department of Defense Joint Architecture for Unmanned Systems (JAUS). June 2007. Reference Architecture Version 3.3 Part 1. Retrieved from ht tp://openjaus.com/media/Final% 20RA%20V3 3%20Part%201 .pdf Department of Defense Joint Architecture for Unmanned Systems (JAUS). June 2007. Reference Architecture Version 3.3 Part 3. Retrieved from http://openjaus.com/media/Final% 20RA%20V3 3%20Part%203.pdf Kleusberg, A., and Langley, R, B. (1990). The limitations of GPS. GPS World, March/April, Vol. 1, No. 2, pp. 50 52.
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