Problem question Problem 6. Localization using range only information. A simple two dimensional robot localization problem using range information can be formulated as an optimization problem. Suppose a mobile robot is navigating in an environment and it observes 4 landmarks. The (x,y) position of the 4 landmarks are known. The robot can only observe the range to the 4 landmarks. The range means the distance between the landmark and the robot. The problem is to estimate the robot position using the ranges and the landmark positions. (1) Derive the model of the localization problems with 4 landmarks; (2) solve the problem; (3) discuss the conditions when the problem can or cannot be solved; (4) write a report. pROBLEM DESCRIPTION There is a robot in a 2 dimensional space which can observe 4 landmarks. The position of these landmarks as (x,y) are known. It is also given that the robot can only observe the range to the given four landmarks. Here, range means the distance between the landmark and robot. Using the ranges and the position of the landmarks, estimate the position of the robot in the given 2-D space. The given 2-D space is shown in figure, with the position of the landmarks. Methematical Model problem analysis There are four landmarks and their ranges are known. In addition, we also know robot can observe landmarks. However, we don’t know the position of robot. In order to make sure that robot can observe each landmark, landmarks must be in robot’s observing range. This means each distance between robot and landmark should be within that range. Therefore, all areas which can satisfy this constraint are where the robot is possibly in. Furthermore, the position which can satisfy range constrain from all four landmarks at same time is the most optimum answer. objective function Calculate the distance between the robot and landmark by the distance formula. We can sum up all the equations for the ranges of each landmark to get our objective function as follows: f= + + + This formula is our objective function to express four landmarks can be observed at the same time. Minimizing the value of f will direct us to the position of robot.  Explanation of the Approach There are 4 parts in Matlab process. First is to set positions of four landmarks and robot. Second is to calculate the real distance (range) between each landmark and robot. Third is to add some noise to cover robot’s position. And the last part is to estimate the robot position and to plot it. Set the landmarks and robot position Firstly, set four landmarks which are [ 2.5,1.5 ], [ 4,1.5 ], [ 2.5,2.5 ] and [ 4,2 ]. Then, set the position of the robot as [3, 0]. Here, robot position is initially taken into consideration for the calculation of the range. Calculate the real distance (Range) Use loop to turn positions of four landmark into equations ‘’, ‘’, and ‘’. Each range is the real distance between landmark and robot. Adding noise In order to simulate a realistic environment, use ‘randn’ to add some random noise in the process. The equation of noise is ‘noiseInMeasurements= sdev_rangeMeasurement*randn(size(RealRanges))’. Therefore, the distance which is measured is ‘MeasuredRanges = Ranges + noiseInMeasurements’. Estimate the robot position Finally, combine 4 differences between the real distance and estimated range. By fminunc function, to minimize the new value of combined difference, estimate the robot position.  Results and discussion For the given localization problem, it estimates the position of the robot using the range only information. We have also added some noise to the approach to solve it in a realistic way. Thus, the robot position is estimated along with the noise, so it may differ slightly from the original position of the robot. Conditions when the problem can or cannot be solved: ⦁ For the localization problems like the given one, with four landmarks set distant apart from each other, can be solved and the position of the robot can be determined easily. ⦁ Graphically close landmarks: If the given landmarks are very close to each other as shown in figure, there can be several problems which can complicate the localization of the robot position. The problems that can occur are: ⦁ One landmark may interfere in between the position or range of the other landmark further complicating the computation of the robot position. ⦁ If the landmark are too close, and comparatively the robot is too far, then two or more landmarks can act as a single point landmark. ⦁ Polarized landmarks: The landmarks may be grouped into two or more and polarized at the end parts as shown in figure. With these types of the landmarks also, the same type of problems may occur as discussed above. Moreover, if the distance between robot and landmarks is too far the four polarized landmarks may act as just two landmarks, as shown in figure below, which decreases the accuracy of the position of the robot. ⦁ If the landmarks are reduced, then also it will become difficult to estimate the position of the robot as the approximation is done in more complex manner. In the other way, more the landmarks more will be the accuracy in finding out the position of the robot. ⦁ The uncertainty about the position will increase if the robot is moving around in the space. ⦁ For any case, if we know the inaccuracy of the encoders and detectors, we can work out to remove error and increase the accuracy.