ECE 7360 : FISP

1. ECE 7360 : FISP Robust Autonomous Robot Localization Using Interval Analysis by Madhumita Chandrasekharan

2. ECE 7360 : FISP Contents • Introduction • Formulation of the Problem • Elementary Localization Tests • Interval Localization Tests • Recursive Set Inversion • Test Cases • Conclusion

3. ECE 7360 : FISP Introduction • The aim is the autonomous localization of a robot from • distance measurements provided by a belt of onboard sensors. • The environment is assumed to be two-dimensional and a map • of its landmarks is available to the robot. • The classical localization methods have many limitations that • are taken care of in the interval localization presented here. • The method proposed here: • - is based on bounded-error set estimation. • - bypasses the data-association step • - handles the problem as nonlinear in a global way. • - robust to outliers.

4. ECE 7360 : FISP Classical Localization : Limitations • Each data point provided by a sensor must be associated with a given landmark. • Methods are based on linearization, which makes these methods inherently local. • Lack of robustness to outliers, for example to sensor malfunctions or outdated maps.

5. ECE 7360 : FISP Formulation of the problem • Computations involves two frames: • World frame W • Frame with origin c = (xc, yc) R • A point m will be represented in the world frame with • respect to the frame R as

6. ECE 7360 : FISP • The parameters to be estimated are : • Coordinate xc • Coordinate yc • Angle q • Configuration vector : • These parameters form the configuration vector given by • P = (xc , yc , q )T • Given some initial search box[po] in configuration space • robot localization can be formulated as the task of • characterizing the set S = { p  [po] |t(p) holds true} • where t(p) is a suitable test or a combination of tests.

7. ECE 7360 : FISP Measurements Ns = number of ultrasonic sensors. s = position of the ith sensor. q = orientation. g = half-aperture di = distance to the closest reflecting landmark C = emission cone = (s, q , g ) To take measurement inaccuracy into account , each data point di is associated with an interval [di] = [di (1 - ai ) , di (1+ ai )] where ai is the known relative measurement accuracy of sensor i.

8. ECE 7360 : FISP Prior Information • Two types of prior information are considered • Map of the environment. • M = { [aj,bj]|j = 1,….,nw} • where nw is the number of oriented segments which • describe the landmarks. • Set described by polygons to which p is known a • priori to belong.

9. ECE 7360 : FISP Localization Tests • Data–association Test • This test is used to estimate the robot configuration • from range data provided by ultrasonic sensors. • It checks whether a given configuration is consistent • with the data, given their imprecision. • In order to achieve this information from the robot frame • R is translated to the world frame W. • The algorithm for testing a given configuration is based • on the notion of remoteness of a segment from a sensor. • The condition where a given segment may not be found • as it lies in the shadow of another is also accounted for.

10. ECE 7360 : FISP Localization Tests In_room test It is assumed that the map partitions the world in 2 sets. The interior The exterior

11. ECE 7360 : FISP Localization Tests The following test enables to eliminate configurations for which it would not be in the interior. Test in_room (m):

12. ECE 7360 : FISP Localization Tests Leg_in test This test provides a necessary condition for p to be constant with the ith measurement. where ci is the point at a distance equal to the lower bound of [di]

13. ECE 7360 : FISP Interval Tests Boolean Intervals A Boolean interval is an element of IB = {0, [0,1], 1}, where 0 stands for false. 1 stands for true. [0,1] stands for intermediate.

14. ECE 7360 : FISP Interval Tests Inclusion Tests Let IRn be the set of all n-dimensional vectors of real intervals. An inclusion test for the test t : Rn -> IB such that for any [x],t ([x])  t[]([x]) In other words,

15. ECE 7360 : FISP Interval Tests Interval Extensions for the localization tests. A natural interval extension of each elementary data-association test datai is built Where ri is the remoteness of the map from sensor i. If t is the Boolean result of a test and y and z are two real numbers, then The interval counterpart ofc(t,y,z) is given by

16. ECE 7360 : FISP Interval Tests Interval Extensions for the localization tests. Interval test in_room[]([m]) : [m] = box enclosing the set m([p]) for a given interval configuration p. c[m] = center of [m] The extension of extending leg_in is obtained by substituting in_room[] for in_room.

17. ECE 7360 : FISP Interval Tests Combining localization tests. The three elementary designs will be combined into a global test.

18. ECE 7360 : FISP Recursive Set Inversion The set S = { p  [po] | t(p) = 1} can also be written as t-1[po] =1 . Characterizing S can be viewed as a problem of set inversion which can be solved in an approximated way using the SIVIA ( Set Inversion Via Interval Analysis). Here a recursive version of SIVIA is used which reduces the amount of testing required in the solution set S in an union of boxes in configuration space, Ŝ. This is done with the help of the notion of masked tests.

19. ECE 7360 : FISP Recursive Set Inversion • If t[]([po]) = 1 • [ po] is in the solution set S and is stored in Ŝ. • If t[]([po]) = 0 • [ po] has a void intersection with S and is dropped • altogether from further consideration. • If t[]([po]) = [0,1] and if the width of [ po] is larger than • some specified e, then [ po] is bisected ,leading to two • child subboxes L[p] and R[p], and the test t[](.) is • recursively applied to each of them.Any box of width • less than e is considered small enough and • incorporated in Ŝ.

20. ECE 7360 : FISP Recursive Set Inversion Masked tests : If the value of an elementary inclusion test over a box[p] is either true or false, this result remains valid for any of the subbox of [p]. Hence its no longer necessary to evaluate these over the children boxes unless the prior test results have uncertain values. The results of these tests will be stored in a mask [m] attached to the parent box. The resulting masked test, which is also in charge of updating the value of [m] will be denoted by t[] ([p], [m]).

21. ECE 7360 : FISP Recursive Set Inversion Masked Sivia The masked tests described are incorporated into Sivia with the help of recursive function CLASSIFY.

22. ECE 7360 : FISP Test Cases • Interval based localization is illustrated on three test cases. • These tests are based on simulations but the test cases are realistic and based on the robot with the following characteristics : • The robot has 24 ultrasonic sensors on its • periphery. • Each sensor has an emission angle of 0.2 rad. • The initial test domain in configuration space • is [-12m, 12m] X [-12m,12m] X [0, 2p] • The precision parameter e is taken as 0.04.

23. ECE 7360 : FISP Test Cases The robot is located in a room described by the following Figure and the map available to the robot matches this environment.

24. ECE 7360 : FISP Test Cases First Test Case The following diagram shows the emission diagram of the 24 sensors. Computing time for the various test cases is shown in the table.

25. ECE 7360 : FISP Test Cases First Test Case The result obtained by using the complete algorithm described in the previous sections is shown below.

26. ECE 7360 : FISP Test Cases Second Test Case The second test case has the same room and map but the actual unknown configuration is (xc , yc , q ) = (1, -7.5, p) The emission diagram is shown below.

27. ECE 7360 : FISP Test Cases Second Test Case The MASKSIVIA finds the set of boxes that are selected.

28. ECE 7360 : FISP Test Cases Second Test Case The figure indicates that due to local symmetries there are two radically different types of possible configurations, each of which corresponds to a different association of segments of the map with distances measured by the sensors.

29. ECE 7360 : FISP Test Cases Third Test Case The problem with outliers and outdated maps are taken into account. • Changes in this are • The previous pillar is moved and a second one added. • Two of the distances have been taken equal to twice • their previous values. • The actual configuration is the same as the first case.

30. ECE 7360 : FISP Test Cases Third Test Case Emission cones 1 and 6 are inconsistent because of the presence of obstacles that are closer to the sensors. Emission cones 2 to 5 correspond to the two misplaced pillars.

31. ECE 7360 : FISP Relevance to Current Research • A laser range sensor is used for localization. • The test area is divided into 160 segments. • The parameters to be estimated are{ (xi , yj q ) | i = 1:11, • j =1:15, q = 1:72} • The data obtained at each point is available as the map of • the area with obstacles. • The algorithm perceived for localization has three tests. • Hence the above method can be adopted to this current • research.

32. ECE 7360 : FISP Conclusion • Autonomous robot localization is well amenable to solution via interval analysis because the number of estimated parameters is small. • It is not necessary to consider all possible associations between sensor data and landmarks, nor the choices of outliers and the data points. This avoids combinatorial explosion. • The results obtained are global and hence any configuration compatible with prior information and measurement cannot be missed. • The results by this method are extremely robust, and the estimator can handle a majority of outliers.

33. ECE 7360 : FISP Conclusion • The computing times seem acceptable for static localization. • This method is flexible and additional information on the physics of the problem can easily be incorporated. • This can be implemented for other types of sensors such as rotating laser range finders, as well as multi-sensor data fusion.

34. ECE 7360 : FISP Reference Robust Autonomous Robot Localization Using Interval Analysis. MICHEL KIEFFER, LUC JAULIN, ERIC WALTER AND DOMINIQUE MEIZEL. Link to the paper http://www.istia.univ-angers.fr/~jaulin/publications.html