{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":44838,"title":"Pose from bearing angles in 2D","description":"A robot moving on the plane has a sensor that measures the bearing angle to two mapped landmarks, that is, the world frame coordinates of the landmarks are known and represented by position vectors |P1| and |P2| .  The bearing angles to the landmarks, with respect to the x-axis of the robot's coordinate frame are |th1| and |th2| respectively.  The robot's forward direction is parallel to its x-axis and its heading angle, with respect to magnetic north is |thb| .\r\n\r\nDetermine the pose of the robot expressed as a homogeneous transformation matrix with respect to the world coordinate frame.  In surveying this is known as a resection problem.","description_html":"\u003cp\u003eA robot moving on the plane has a sensor that measures the bearing angle to two mapped landmarks, that is, the world frame coordinates of the landmarks are known and represented by position vectors \u003ctt\u003eP1\u003c/tt\u003e and \u003ctt\u003eP2\u003c/tt\u003e .  The bearing angles to the landmarks, with respect to the x-axis of the robot's coordinate frame are \u003ctt\u003eth1\u003c/tt\u003e and \u003ctt\u003eth2\u003c/tt\u003e respectively.  The robot's forward direction is parallel to its x-axis and its heading angle, with respect to magnetic north is \u003ctt\u003ethb\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eDetermine the pose of the robot expressed as a homogeneous transformation matrix with respect to the world coordinate frame.  In surveying this is known as a resection problem.\u003c/p\u003e","function_template":"function T = user_function(P1, P2, th1, th2, thb)\r\n% Input:  P1 a 2x1 vector representing the coordinate of a point\r\n%         P2 a 2x1 vector representing the coordinate of a point\r\n%         th1 bearing, a scalar angle\r\n%         th2 bearing, a scalar angle\r\n%         thb heading, a scalar angle\r\n% Output: T a 3x3 homogeneous transformation matrix\r\n  T = ;\r\nend","test_suite":"%\r\nP1 = [10 20]';\r\nP2 = [20 20]';\r\nP = rand(2,1)*5 + [5 5]';\r\nthb = rand*0.2;\r\nx = P1 - P;\r\nth1 = atan2(x(2), x(1)) - thb;\r\nx = P2 - P;\r\nth2 = atan2(x(2), x(1)) - thb;\r\n\r\nT = user_function(P1, P2, th1, th2, thb);\r\n\r\n%% test size and complexity\r\nassert(all(size(T)==3), 'The matrix must be 3x3');\r\nassert(isreal(T), 'The matrix must be real, not complex');\r\n\r\n%% bottom row\r\nassert(isequal(T(3,:), [0 0 1]), 'The bottom row of the homogeneous transformation matrix is not correct')\r\n\r\n%% x coordinate\r\nassert(abs(T(1,3)-P(1))\u003c1e-4, 'The representation of the x-coordinate is not correct')\r\n\r\n%% y coordinate\r\nassert(abs(T(2,3)-P(2))\u003c1e-4, 'The representation of the y-coordinate is not correct')\r\n\r\n%% valid rotation matrix\r\nR = T(1:2,1:2);\r\nassert( abs(det(R)-1) \u003c 1e-4, 'The determinant of the rotation submatrix is not correct')\r\n\r\n%% correct rotation matrix\r\nR = T(1:2,1:2);\r\nassert( abs(atan2(R(2,1), R(1,1)) - thb) \u003c 1e-4, 'The rotation matrix is not correct, check your calculation of the heading SSW and whether you are using radians or degrees')\r\n\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":13332,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":77,"created_at":"2019-01-21T00:32:56.000Z","updated_at":"2026-02-15T06:52:16.000Z","published_at":"2019-01-21T00:37:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA robot moving on the plane has a sensor that measures the bearing angle to two mapped landmarks, that is, the world frame coordinates of the landmarks are known and represented by position vectors\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e . The bearing angles to the landmarks, with respect to the x-axis of the robot's coordinate frame are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eth1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eth2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e respectively. The robot's forward direction is parallel to its x-axis and its heading angle, with respect to magnetic north is\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ethb\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e .\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDetermine the pose of the robot expressed as a homogeneous transformation matrix with respect to the world coordinate frame. In surveying this is known as a resection problem.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":44838,"title":"Pose from bearing angles in 2D","description":"A robot moving on the plane has a sensor that measures the bearing angle to two mapped landmarks, that is, the world frame coordinates of the landmarks are known and represented by position vectors |P1| and |P2| .  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The robot's forward direction is parallel to its x-axis and its heading angle, with respect to magnetic north is \u003ctt\u003ethb\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eDetermine the pose of the robot expressed as a homogeneous transformation matrix with respect to the world coordinate frame.  In surveying this is known as a resection problem.\u003c/p\u003e","function_template":"function T = user_function(P1, P2, th1, th2, thb)\r\n% Input:  P1 a 2x1 vector representing the coordinate of a point\r\n%         P2 a 2x1 vector representing the coordinate of a point\r\n%         th1 bearing, a scalar angle\r\n%         th2 bearing, a scalar angle\r\n%         thb heading, a scalar angle\r\n% Output: T a 3x3 homogeneous transformation matrix\r\n  T = ;\r\nend","test_suite":"%\r\nP1 = [10 20]';\r\nP2 = [20 20]';\r\nP = rand(2,1)*5 + [5 5]';\r\nthb = rand*0.2;\r\nx = P1 - P;\r\nth1 = atan2(x(2), x(1)) - thb;\r\nx = P2 - P;\r\nth2 = atan2(x(2), x(1)) - thb;\r\n\r\nT = user_function(P1, P2, th1, th2, thb);\r\n\r\n%% test size and complexity\r\nassert(all(size(T)==3), 'The matrix must be 3x3');\r\nassert(isreal(T), 'The matrix must be real, not complex');\r\n\r\n%% bottom row\r\nassert(isequal(T(3,:), [0 0 1]), 'The bottom row of the homogeneous transformation matrix is not correct')\r\n\r\n%% x coordinate\r\nassert(abs(T(1,3)-P(1))\u003c1e-4, 'The representation of the x-coordinate is not correct')\r\n\r\n%% y coordinate\r\nassert(abs(T(2,3)-P(2))\u003c1e-4, 'The representation of the y-coordinate is not correct')\r\n\r\n%% valid rotation matrix\r\nR = T(1:2,1:2);\r\nassert( abs(det(R)-1) \u003c 1e-4, 'The determinant of the rotation submatrix is not correct')\r\n\r\n%% correct rotation matrix\r\nR = T(1:2,1:2);\r\nassert( abs(atan2(R(2,1), R(1,1)) - thb) \u003c 1e-4, 'The rotation matrix is not correct, check your calculation of the heading SSW and whether you are using radians or degrees')\r\n\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":13332,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":20,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":77,"created_at":"2019-01-21T00:32:56.000Z","updated_at":"2026-02-15T06:52:16.000Z","published_at":"2019-01-21T00:37:35.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA robot moving on the plane has a sensor that measures the bearing angle to two mapped landmarks, that is, the world frame coordinates of the landmarks are known and represented by position vectors\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eP2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e . The bearing angles to the landmarks, with respect to the x-axis of the robot's coordinate frame are\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eth1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eth2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e respectively. The robot's forward direction is parallel to its x-axis and its heading angle, with respect to magnetic north is\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ethb\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e .\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDetermine the pose of the robot expressed as a homogeneous transformation matrix with respect to the world coordinate frame. In surveying this is known as a resection problem.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray 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