WHEN IS MORE NOT BETTER? Karen T. Sutherland Department of Computer Science University of Wisconsin La Crosse La Crosse, WI 56401

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1 WHEN IS MORE NOT BETTER? Karen T. Sutherland Department of Computer Science University of Wisconsin La Crosse La Crosse, WI ABSTRACT It is often assumed that when sensing in autonomous robot navigation, the more measurements taken, the better. In many cases, this is indeed true. However, we have found that in an outdoor, unstructured environment where the number of possible measurements one can take is limited and error is not normally distributed, additional measurements may cause more harm than good. These rogue measurements are not interpreted as outliers due to the fact that there is no observed central tendency in the measurements as a whole. This paper describes ongoing work in determining how a qualitative rating can be used with sparse data to weight measurements and how these weighted measurements can best be combined with map data to produce a good estimate of location. Experiments have been run using U. S. Geographical Survey Digital Elevation Map (DEM) data from mountainous terrain in eastern Utah. INTRODUCTION Finding one s own location in the environment is a critical skill for a robot navigator, especially in an outdoor unstructured environment. 10 Localization errors outdoors tend to have more severe consequences than those indoors. In an indoor environment, proximity sensors can warn of impending disaster. Outdoors, it is possible, with a small error in localization measure, to send oneself off a cliff. Problems which occurred with nonconvergence of measurements taken to self locate in such an environment caused us to look more closely at the assumptions under which we were expecting that convergence. The result was a realization that merely taking more measurements is not always a wise move. It should first be made clear what is meant by an unstructured environment. Any indoor envi-

2 ronment, regardless of clutter, is, by our definition, structured. An outdoor environment containing buildings, roads, light beacons and other such items 3 is, by our definition, structured. An outdoor, unstructured environment has no man made objects to use for landmarks. It is also the type of environment in which the Global Positioning System (GPS) and instruments such as a compass or barometric altimeter may fail to give valid location readings. 9 Landmarks are naturally occurring physical terrain features such as mountain peaks or ridgelines. Due to the large distance between viewpoint and feature, proximity sensors are useless. Measurements are taken from camera images. The foundation for the assumption that more measurements are better lies in the Law of Large Numbers. 4 If we let X k be a sequence of mutually independent random variables with a common distribution and if the expectation = E(X k ) exists, then for every > 0 as n! 1 X 1 + ::: + X n P robability? n >! 0; In other words, the probability that the average will differ from the expectation by less than an arbitrarily prescribed tends to one. The Law of Large Numbers is more general than the often quoted Central Limit Theorem because it holds even when the random variables X k have no finite variance. Maximum likelihood estimators, such as the Kalman filter, 7 are often used in robot navigation. However, it should be noted that all of these estimators assume that numerous measurements are taken. Indeed, the Kalman filter was developed for applications in signal processing, in which many measurements are taken. Other problems with using the Kalman filter in this type of environment are that it assumes both that a good starting estimate is available and that errors in the data are Gaussian. Furthermore, it is most often the Extended Kalman filter (EKF) which is used. 1,6,8 Whereas, under the given assumptions, the Kalman filter is guaranteed to converge, the EKF has no such guarantee of convergence. 2 It is possible, of course, for a maximum likelihood estimator to work well when a sequence of measurements does not take on values at random, as random variables should, or when there is no common distribution. In fact, it follows from the Central Limit Theorem that even when random variables each have an arbitrary distribution, a great many of them display a distribution which closely resembles the normal. 5 However, the issue with which this paper is concerned is the fact that this all depends on a limit which is approached for very large n. The number of measurements taken is critical. UNIQUE ERRORS Errors in localization in this type of environment are unique due to the type of landmarks available, the magnitude of the distances involved and the accuracy of available maps.

Figure 1: When the number of pixels in the image is constant, the larger the actual area covered by the image, the greater the error due to

Errors in discretization of the image become worse as distance to landmarks increases.

The scale of the mountain view is such that one pixel represents about 4 meters, while one pixel along the right door jam in the indoor scene

Although an outdoor navigator can, in general, allow more room for maneuvering than if navigating indoors, it is usually not at this level of

Much more significant is the error in the map.

3 Figure 1: When the number of pixels in the image is constant, the larger the actual area covered by the image, the greater the error due to discretization. One pixel represents 4 meters on the left and 0.2 inches on the right. Errors in discretization of the image become worse as distance to landmarks increases. Figure 1 shows two typical images used in navigation tasks, one outdoors and the other indoors. The scale of the mountain view is such that one pixel represents about 4 meters, while one pixel along the right door jam in the indoor scene represents approximately 0.2 inches. Although an outdoor navigator can, in general, allow more room for maneuvering than if navigating indoors, it is usually not at this level of magnitude. Figure 2: A rendering of the DEM data showing the smooth contour which results from the 30 meter square averaging. Much more significant is the error in the map. Whereas maps of structured environments can, in general, be made quite exact, the USGS DEM data is made up of measurements averaged over a grid of cells of 30 meters square. Most of the USGS DEM s were produced by running paper contour maps through a drum scanner, extracting the contour lines and interpolating to estimate elevations. This averaging produces a type of inverse discretization which is, for the outdoor example shown in Figure 1, approximately ten times that which is caused by discretization in the image. The smoothing which results from this operation can be seen in the

4 rendering in Figure 2. Features which may be very sharp in the actual view are smoothed out in the map data and the resultant rendering. Since these are the very features which would be taken from the view to use as landmarks, the smoothing significantly affects accuracy. Errors in elevation deception, as shown in the diagram in Figure 3, where a subpeak or subridge in the image is mistaken for a peak or ridgeline on the map, are not common indoors or in a structured outdoor environment. An observer located between B and C will see peak or ridge 1 as the highest point while an observer located between A and B will see peak or ridge 2 as the highest point. Not until the observer moves beyond point A will the true summit be observed. Errors such as these add large inaccuracies to localization measurements. Figure 3: Misidentifying a subridge or subpeak as the true landmark is a common error in navigation in unstructured environments. The true summit will be seen only by an observer located to the left of point A. Most importantly, the number of landmarks from which to take readings is extremely limited in this environment. The contour map in Figure 4 shows one area approximately 21 kilometers by 14 kilometers. The navigator s path is shown by the heavy black line. The black dots are located at the only visible, distinguishable mountain peaks which could be used for landmarks. Only four to five of them were visible at any one time. A navigator in such an environment cannot simply add more beacons from which to take readings. Figure 4: The number of distinguishable landmarks is small at any one time.

5 INTERACTIVE MEASUREMENTS We have developed a strategy for measuring in environments with few natural landmarks which analyzes the map data in the area of each landmark in addition to the measurements which are taken. These interactive measurements are meshed together to produce a more robust result. The example presented here uses mountain peaks as point landmarks. As in the method described by Sutherland and Thompson, 9 self localization is determined by measuring the angle between pairs of three point landmarks which have been identified on the map. The strategy focuses on errors in elevation deception, as described in the previous section, since these were the errors which could most easily be addressed. Although misidentifying a subpeak for a nearby given peak is a distinct possibility, we assume that there is no misidentification of one landmark peak for another. We know which three peaks we are using and we know which is which. The navigator is moving toward a goal, measuring as it moves. The process is as follows: Step 1: Sweep peaks and record elevations. The first step in this process is to search the map data around each landmark peak for possible nearby subpeaks. These searches are in the form of a sweep around the peak while measuring the elevations at each 30 meter interval. Although the map data has been averaged, any average elevation close to the elevation of the peak is a candidate for a mismatch. The sweep is done only half way around each peak on the side on which the navigator is located. The radius of the sweep is arbitrarily set. It follows that the shallower the inclination up to the peak, the higher the probability of a mismatch. Any distinguishable point with elevation close to that of the peak will later be tagged for analysis. Step 2: Estimate navigator location. The angles between landmark pairs as well as the angles of elevation from the navigator to each of the three landmarks are then measured. This results in an initial estimate of location. Step 3: Check elevations. An arbitrary error threshold 1 is set for the angle of elevation to each peak. Each elevation E i at location i recorded for each of the three peaks in Step 1 is tested as follows: If arctan( E p?e i D(p;i) < + 1 tag E i, where E p is the elevation of the landmark peak and D(p; i) is the distance from the landmark peak to the point at which elevation E i was measured. We now have a pool of the tagged elevations for each peak which are candidates for subpeak misidentification. Step 4: Cull candidates. The candidates found in Step 3 are culled using the initial estimate of location. An arbitrary error threshold 2 is set for the angle measured from navigator heading to the landmark peak with vertex at the estimated navigator location. The angle i from navigator heading to each candidate i is then measured.

6 If i <? 2 or i > + 2, candidate i is culled from the pool. The pool of tagged elevations for each peak now consists of locations which could be misidentified as the landmark peak and which lie within the line of view from the navigator to the peak. A probability of occlusion, based on the elevation of the subpeak, is associated with each candidate in the pool. The navigator now moves. The direction and distance of this move is calculated as a linear programming problem with the following goals: 1. The move is toward the goal. 2. The move removes candidates with high probability of occlusion in each landmark pool from the line of view. 3. The move is of minimal distance. At this point, the process repeats. However, the sweep needn t be complete. Indeed, it is often unnecessary to sweep at all on a subsequent move. Likewise, the elevation checks are not completely repeated, but only augmented whenever additional sweeping is required. Candidates which were culled in an earlier pass may move into a line of view and candidates which were previously in a line of view may be culled. Consequently, the process involves for most moves only reprocessing the candidates for each peak and reassigning probabilities. # Subpeaks Peak 1 Peak 2 Peak 3 Initial pool Culled pool Table 1: Candidates for misidentification for one set of landmark triples. Initial experiments have been done in simulation using USGS DEM data from mountainous terrain in eastern Utah. Runs using this strategy have been compared to runs through the same terrain but with new measurements taken at evenly distributed points on the path. Table 1 shows the initial and culled pools of candidates for one such run. This small number of candidates for a mismatch was typical in most runs. Significant improvement has been noted when the strategy is used. This has, in most cases, been due to not moving past good measuring locations while not staying at locations where the probability of elevation deception is high. In the few cases when no improvement in localization occurred, the runs which used the strategy did no worse than those which did not. Our current work focuses on testing the strategy in an actual outdoor environment in the bluff area of southwestern Wisconsin.

7 Advantages of this strategy: Rather than measurements being taken at specific time intervals or after specific distances traveled, they are only taken when the navigator has moved to a position where the measurement has a chance of adding to the accuracy of the localization. Numerous measurements incorporating the very same elevation deception errors are not being averaged into the calculations of location. The necessary elevations and angles can be calculated quickly. After the first pass, incremental calculations are minimal. The moving and recalculating also address the discretization error in the image. Moving the minimal distance to remove the candidates for occlusion from the line of view prevents the navigator from missing a good measurement situation. The navigator is always moving toward the goal. Drawback of the strategy: The radius of the area scanned around each landmark peak must be preset, as must both 1 and 2. If set too small, occluding subpeaks will not be picked up. If set too large, determining probabilities of occlusion and removing from the pool of candidates with high probabilities of occlusion on a move will be difficult if not impossible to do. Work is ongoing on how best to determine these values. CONCLUSION We have shown that, when used in simulation, a strategy which combines measurement analysis of the view with measurement analysis of the map data as opposed to simply taking random measurements can significantly improve localization in outdoor, unstructured environments with few landmarks. This method has served to compensate for the inability to make numerous useful measurements by taking advantage of the few good measurements which can be made. At the present time, this strategy is being tested in an actual outdoor environment, using both peak and ridgeline landmarks.

8 REFERENCES [1] Nicholas Ayache and Oliver D. Faugeras. Maintaining representations of the environment of a mobile robot. IEEE Transactions on Robotics and Automation, 5(6): , December [2] Daniel L. Boley, Karen T. Sutherland, and Erik S. Steinmetz. Robot localization from landmarks using recursive total least squares. In Proceedings of the IEEE International Conference on Robotics and Automation, volume 4, pages IEEE, April [3] David Dai and Daryl Lawton. Range-free qualitative navigation. In Proceedings of the IEEE International Conference on Robotics and Automation, volume 1, pages IEEE, May [4] William Feller. An Introduction to Probability Theory and its Applications. John Wiley and Sons, [5] Arthur Gelb. Applied Optimal Estimation. The MIT Press, [6] P. Hébert, S. Betgé-Brezetz, and R. Chatila. Reasoning with Uncertainty in Robotics, chapter Probabilistic Map Learning: Necessity and Difficulties. Springer, [7] R. E. Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engineering, pages 35 45, [8] Akio Kosaka and Avinash C. Kak. Fast vision-guided mobile robot navigation using modelbased reasoning and prediction of uncertainties. CVGIP: Image Understanding, 56(3): , November [9] Karen T. Sutherland and William B. Thompson. Localizing in unstructured environments: Dealing with the errors. IEEE Transactions on Robotics and Automation, pages , December [10] Karen T. Sutherland and William B. Thompson. Pursuing projections: Keeping a robot on path. In Proceedings of the IEEE International Conference on Robotics and Automation, volume 4, pages IEEE, May 1994.

Exploiting Composite Features in Robot Navigation

EXPLOITING COMPOSITE FEATURES IN ROBOT NAVIGATION 69 Exploiting Composite Features in Robot Navigation Jennifer Davison, Kelly Hasler Faculty Sponsor: Karen T. Sutherland, Department of Computer Science