# Project MP1 & MP2 Solution

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## Description

# MP1:(PDF inside the MP1 folder has more detailed project descriptions/assignment details)

# Problem1:(multilaterate)

– You are to implement the function named multilaterate(distances) in the provided template python code in the ﬁle multilaterate.py. The function multilaterate(distances) takes a list of lists as its input. The list should have four landmark locations with associated (x,y,z,d) in formation in which x,y,z are the location of the landmark in 3D and d is the distance of the landmark to the unknown point to be localized. You should test your program to ensure it works with diﬀerent input points correctly. Your program will be tested on several diﬀerent setups and should provide reasonable output (i.e. for each coordinate of (x,y,z), either the absolute or relative error is less than 10^−6). multilaterate.py takes a single argument: the dataﬁle name. Your program should then print out the location that is computed. A skeleton multilaterate.py is provided.

# Problem2:(Kalman Filter)

– as the title suggests, this project had us implement Kalman filters

# MP2

Shortest Path Roadmap algorithm for a translating point.

– You will be implementing the Shortest Path Roadmap algorithm for a point robot. The robot resides in a region with the lower left corner being (0,0) and the upper right corner being (10,10). There are multiple polygonal obstacles in the region. An example of the obstacles is provided in the ﬁle env 01.txt. In the ﬁle, each line represents a list of clockwise arranged x-y coordinates that deﬁne a polygonal obstacle. To see a visualization of the environment, you may run the following code:

python visualize.py env 01.txt

– You are also given a skeleton ﬁle spr.py to work with. The current code takes in as arguments a ﬁle describing the polygonal obstacles, and coordinates for the start and goal conﬁgurations. For example, you should be able to run the command

python spr.py env 01.txt 1.0 2.0 3.0 4.0

– You are to implement the Shortest Path Roadmap algorithm following the steps listed below. Do not change code (function signature) that are provided to you for grading purpose. If you do, you may receive 0 credit.

1. Compute the reﬂex vertices [20 points]. You are to implement the function

findReflexVertices(polygons) to identify all reﬂex vertices. The vertices should be returned as a list (see spr.py for more details). You may assume that there are only polygons (no lines or points) .

2. Compute roadmap edges and their lengths [30 points]. You are to check for each pair of reﬂex vertices whether the edge between them should be part of the shortest path roadmap. As the valid edges are identiﬁed, you are to build the roadmap (graph). To store the map, ﬁrst assign each reﬂex vertex a label (e.g., 1,2,3,…). You can then represent the roadmap as adjacency lists. As the outcome, you should provide two dictionaries. The ﬁrst dictionary should map a vertex label to vertex coordinates, e.g., {1: [5.2,6.7], 2: [9.2,2.3], …} In the above dictionary, vertex 1 has a coordinate of (5.2,6.7) and vertex 2 has a coordinate of (9.2,2.3). The second dictionary should map a vertex label to a list of vertices that are adjacent to that vertex and also store the lengths of the edges, e.g., {1: [[2, 5.95], [3, 4.72]], 2: [[1, 5.95], [5, 3.52]], …} In the above dictionary, vertex 1 is adjacent to vertices 2 and 3. Vertex 2 is adjacent to verices 1 and 5. The distance between vertices 1 and 2 is 5.95 and the distance between vertices 1 and 3 is 4.72. The distance between vertices 2 and 5 is 3.52. Your code should go in the function

computeSPRoadmap(polygons, reflexVertices) The argument reflexVertices is the list of reﬂex vertices returned from the function

findReflexVertices()

3. Putting things together [20 points]. You are to add the start now and the goal to the roadmap graph (i.e., the adjacency list) and perform the search. You need to implement the function

updateRoadmap(polygons, vertexMap, adjListMap, x1, y1, x2, y2) to add the start and goal vertices to the roadmap so later the uniform-cost search function could complete the search. Your function should return a list of vertex labels as the path. You may assume the start and goal vertices are not inside of a polygon.

4. Implement a uniform-cost search algorithm [30 points]. You are to implement the uniform-cost search algorithm that we have covered in class to work on the adjacency list from the previous step. Note that this is a standard uniform-cost search algorithm that should work with any edge-weighted graph. Your code should go in the function

uniformCostSearch(adjListMap, start, goal) where adjListMap has the same structure as the adjacency list mentioned in the previous task, I.e., it has the format {1: [[2, 5.95], [3, 4.72]], 2: [[1, 5.95], [5,3.52]], …} The arguments start and goal are the vertex labels of the start and goal.

5. Visualization bonus [10 points]. Through modifying the visualization code supplied in visualize.py, draw the roadmap (using green lines, including the start and the goal) and the path that you computed (using red lines).

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