{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-06T14:01:22.000Z","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":"2026-04-06T00:00:00.000Z","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":5387,"title":"Triple function composition","description":"Given three functions f,g and h, create the composed function y=f(g(h)).\r\n\r\nExample \r\n\r\n f = @(x) x+1\r\n g = @(x) x/2\r\n h = @(x) x^2\r\n\r\nAnd x1=8; x2=10; x3=1;\r\n\r\n y(x1) = 33\r\n y(x2) = 51\r\n y(x3) = 1.5","description_html":"\u003cp\u003eGiven three functions f,g and h, create the composed function y=f(g(h)).\u003c/p\u003e\u003cp\u003eExample\u003c/p\u003e\u003cpre\u003e f = @(x) x+1\r\n g = @(x) x/2\r\n h = @(x) x^2\u003c/pre\u003e\u003cp\u003eAnd x1=8; x2=10; x3=1;\u003c/p\u003e\u003cpre\u003e y(x1) = 33\r\n y(x2) = 51\r\n y(x3) = 1.5\u003c/pre\u003e","function_template":"function y = compose3(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nf= @(x) x+1;\r\ng= @(x) x/2;\r\nh= @(x) x^2;\r\n\r\nx1 = 8 ; x2 = 10; x3=1;\r\n\r\ny1 = 33; y2 = 51; y3=1.5;\r\n\r\ny=compose3(f,g,h);\r\n\r\nassert(isequal(y(x1),y1))\r\nassert(isequal(y(x2),y2))\r\nassert(isequal(y(x3),y3))\r\n\r\n%%\r\nf= @(x) log(x);\r\ng= @(x) x+6;\r\nh= @(x) x/2;\r\n\r\nx1 = 8 ; x2 = 4.4; x3 = 6;\r\n\r\ny1 = 2.3026; y2 = 2.1041; y3=2.1972;\r\n\r\ny=compose3(f,g,h);\r\n\r\nassert(abs(y(x1)-y1)\u003c=1e-4)\r\nassert(abs(y(x2)-y2)\u003c=1e-4)\r\nassert(abs(y(x3)-y3)\u003c=1e-4)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":38414,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":50,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-25T15:27:41.000Z","updated_at":"2026-03-04T14:44:19.000Z","published_at":"2015-03-25T15:27:41.000Z","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\u003eGiven three functions f,g and h, create the composed function y=f(g(h)).\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\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ f = @(x) x+1\\n g = @(x) x/2\\n h = @(x) x^2]]\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\u003eAnd x1=8; x2=10; x3=1;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ y(x1) = 33\\n y(x2) = 51\\n y(x3) = 1.5]]\u003e\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\"}]}"},{"id":44837,"title":"Composing relative poses in 2D: problem 1","description":"We consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north. \r\n\r\nThere is a playing field with a reference frame denoted by {F}.  The rigid-body displacement from {O} to {F} is given by the homogenous transformation |TF| .\r\n\r\nThere are two robots (robot 1 and robot 2) on the playing field, with attached body-fixed coordinate frame {B} and {C} respectively with their origins at the centre of the robot. The x-axis of each robot's frame points in the robot's forward direction.  \r\n\r\nThe rigid-body displacement of robot 1 with respect to the world frame is estimated by GPS to be the homogenous transformation |TB| .  The displacement of robot 2 with respect to robot 1 is estimated by a novel radar sensor and is given by the homogenous transformation |TBC| .\r\n\r\nWhere is robot 2 with respect to the field?","description_html":"\u003cp\u003eWe consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.\u003c/p\u003e\u003cp\u003eThere is a playing field with a reference frame denoted by {F}.  The rigid-body displacement from {O} to {F} is given by the homogenous transformation \u003ctt\u003eTF\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eThere are two robots (robot 1 and robot 2) on the playing field, with attached body-fixed coordinate frame {B} and {C} respectively with their origins at the centre of the robot. The x-axis of each robot's frame points in the robot's forward direction.\u003c/p\u003e\u003cp\u003eThe rigid-body displacement of robot 1 with respect to the world frame is estimated by GPS to be the homogenous transformation \u003ctt\u003eTB\u003c/tt\u003e .  The displacement of robot 2 with respect to robot 1 is estimated by a novel radar sensor and is given by the homogenous transformation \u003ctt\u003eTBC\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eWhere is robot 2 with respect to the field?\u003c/p\u003e","function_template":"function TFC = user_function(TF, TB, TBC)\r\n  % Input:  TF a 3x3 homogeneous transformation matrix\r\n  %         TB a 3x3 homogeneous transformation matrix\r\n  %         TBC a 3x3 homogeneous transformation matrix\r\n  %         P a 2x1 vector representing the coordinate of a point\r\n  % Output: TFC a 3x3 homogeneous transformation matrix\r\n  %\r\n  TFC = ;\r\nend","test_suite":"th = 2*pi*rand;\r\nt = rand(2,1)*20-10;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTF = [R t; 0 0 1];\r\n\r\nth = 2*pi*rand;\r\nt = rand(2,1)*20-10;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTB = [R t; 0 0 1];\r\n\r\nth = 2*pi*rand;\r\nt = rand(2,1)*20-10;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTBC = [R t; 0 0 1];\r\n\r\n\r\n%%\r\nT = user_function(TF, TB, TBC)\r\n\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\nT = user_function(TF, TB, TBC)\r\nassert(isequal(T(3,:), [0 0 1]), 'The bottom row of the homogeneous transformation matrix is not correct')\r\n\r\n%% valid rotation matrix\r\nT = user_function(TF, TB, TBC)\r\nR = T(1:2,1:2);\r\nassert( abs(det(R)-1) \u003c 0.01, 'The determinant of the rotation submatrix is not correct')\r\n\r\n%% correct value\r\nT = user_function(TF, TB, TBC)\r\nTref = inv(TB*TBC)*TF*T;\r\nassert( norm(Tref-eye(3,3)) \u003c 1e-6, 'The homogeneous transform value is not correct')","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":13332,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":64,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":77,"created_at":"2019-01-20T08:11:47.000Z","updated_at":"2026-03-08T02:06:59.000Z","published_at":"2019-01-20T08:33:16.000Z","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\u003eWe consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.\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\u003eThere is a playing field with a reference frame denoted by {F}. The rigid-body displacement from {O} to {F} is given by the homogenous transformation\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\u003eTF\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\u003eThere are two robots (robot 1 and robot 2) on the playing field, with attached body-fixed coordinate frame {B} and {C} respectively with their origins at the centre of the robot. The x-axis of each robot's frame points in the robot's forward direction.\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\u003eThe rigid-body displacement of robot 1 with respect to the world frame is estimated by GPS to be the homogenous transformation\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\u003eTB\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e . The displacement of robot 2 with respect to robot 1 is estimated by a novel radar sensor and is given by the homogenous transformation\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\u003eTBC\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\u003eWhere is robot 2 with respect to the field?\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\"}]}"},{"id":44831,"title":"Composing relative poses in 2D: problem 2","description":"We consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north. \r\n\r\nThere is a playing field with a reference frame denoted by {F}.  The rigid-body displacement from {O} to {F} is given by the homogenous transformation |TF| .\r\n\r\nThere is a robot with an attached body-fixed coordinate frame {B} whose origin is in the centre of the robot, and whose x-axis points in the robot's forward direction.  The rigid-body displacement displacement of {B} relative to {F} is given by the homogenous transformation |TB| .\r\n\r\nThe robot carries a laser scanner which reports the range and bearing of landmarks with respect to its coordinate frame {S} with zero bearing angle corresponding to its x-axis.  The scanner is displaced from the centre of the robot and the displacement of frame {S} relative to {B} is given by the homogenous transformation |TS| .\r\n\r\nThere is a landmark with position described by a coordinate vector P with respect to the world coordinate frame {O}.\r\n\r\nWhat is the range and bearing of the landmark as observed by the laser scanner?","description_html":"\u003cp\u003eWe consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.\u003c/p\u003e\u003cp\u003eThere is a playing field with a reference frame denoted by {F}.  The rigid-body displacement from {O} to {F} is given by the homogenous transformation \u003ctt\u003eTF\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eThere is a robot with an attached body-fixed coordinate frame {B} whose origin is in the centre of the robot, and whose x-axis points in the robot's forward direction.  The rigid-body displacement displacement of {B} relative to {F} is given by the homogenous transformation \u003ctt\u003eTB\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eThe robot carries a laser scanner which reports the range and bearing of landmarks with respect to its coordinate frame {S} with zero bearing angle corresponding to its x-axis.  The scanner is displaced from the centre of the robot and the displacement of frame {S} relative to {B} is given by the homogenous transformation \u003ctt\u003eTS\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eThere is a landmark with position described by a coordinate vector P with respect to the world coordinate frame {O}.\u003c/p\u003e\u003cp\u003eWhat is the range and bearing of the landmark as observed by the laser scanner?\u003c/p\u003e","function_template":"function [bearing, range] = user_function(TF, TB, TS, P)\r\n  % Input:  TF a 3x3 homogeneous transformation matrix\r\n  %         TB a 3x3 homogeneous transformation matrix\r\n  %         TS a 3x3 homogeneous transformation matrix\r\n  %         P a 2x1 vector representing the coordinate of a point\r\n  % Output: bearing, a scalar angle\r\n  %         range, a scalar distance\r\n  bearing = ;\r\n  range = ;\r\nend","test_suite":"th = 2*pi*rand - pi;\r\nt = rand(2,1)*60-30;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTF = [R t; 0 0 1];\r\n\r\nth = 2*pi*rand - pi;\r\nt = rand(2,1)*20-10;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTB = [R t; 0 0 1];\r\n\r\nth = 30*pi/180;\r\nt = [0.5; -0.3];\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTS = [R t; 0 0 1];\r\n\r\nP = rand(2,1) * 500 - 250;\r\n\r\nPref = inv(TF * TB * TS) * [P; 1];\r\n[bearingRef,rangeRef] = cart2pol( Pref(1), Pref(2) );\r\n\r\n[bearing, range] = user_function(TF, TB, TS, P);\r\n\r\n%%\r\nassert(isscalar(bearing) \u0026 isreal(bearing), 'Bearing angle must be a real scalar')\r\n%%\r\nassert(isscalar(range) \u0026 isreal(range), 'Range must be a real scalar')\r\n%%\r\nassert(abs(bearing-bearingRef) \u003c 0.001, 'Bearing angle is not correct')\r\n%%\r\nassert(abs(range-rangeRef) \u003c 0.001, 'Range is not correct')\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":13332,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":53,"test_suite_updated_at":"2019-01-10T10:52:52.000Z","rescore_all_solutions":false,"group_id":77,"created_at":"2019-01-10T10:46:31.000Z","updated_at":"2026-03-08T02:54:18.000Z","published_at":"2019-01-10T10:52:52.000Z","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\u003eWe consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.\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\u003eThere is a playing field with a reference frame denoted by {F}. The rigid-body displacement from {O} to {F} is given by the homogenous transformation\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\u003eTF\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\u003eThere is a robot with an attached body-fixed coordinate frame {B} whose origin is in the centre of the robot, and whose x-axis points in the robot's forward direction. The rigid-body displacement displacement of {B} relative to {F} is given by the homogenous transformation\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\u003eTB\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\u003eThe robot carries a laser scanner which reports the range and bearing of landmarks with respect to its coordinate frame {S} with zero bearing angle corresponding to its x-axis. The scanner is displaced from the centre of the robot and the displacement of frame {S} relative to {B} is given by the homogenous transformation\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\u003eTS\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\u003eThere is a landmark with position described by a coordinate vector P with respect to the world coordinate frame {O}.\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\u003eWhat is the range and bearing of the landmark as observed by the laser scanner?\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":5387,"title":"Triple function composition","description":"Given three functions f,g and h, create the composed function y=f(g(h)).\r\n\r\nExample \r\n\r\n f = @(x) x+1\r\n g = @(x) x/2\r\n h = @(x) x^2\r\n\r\nAnd x1=8; x2=10; x3=1;\r\n\r\n y(x1) = 33\r\n y(x2) = 51\r\n y(x3) = 1.5","description_html":"\u003cp\u003eGiven three functions f,g and h, create the composed function y=f(g(h)).\u003c/p\u003e\u003cp\u003eExample\u003c/p\u003e\u003cpre\u003e f = @(x) x+1\r\n g = @(x) x/2\r\n h = @(x) x^2\u003c/pre\u003e\u003cp\u003eAnd x1=8; x2=10; x3=1;\u003c/p\u003e\u003cpre\u003e y(x1) = 33\r\n y(x2) = 51\r\n y(x3) = 1.5\u003c/pre\u003e","function_template":"function y = compose3(x)\r\n  y = x;\r\nend","test_suite":"%%\r\nf= @(x) x+1;\r\ng= @(x) x/2;\r\nh= @(x) x^2;\r\n\r\nx1 = 8 ; x2 = 10; x3=1;\r\n\r\ny1 = 33; y2 = 51; y3=1.5;\r\n\r\ny=compose3(f,g,h);\r\n\r\nassert(isequal(y(x1),y1))\r\nassert(isequal(y(x2),y2))\r\nassert(isequal(y(x3),y3))\r\n\r\n%%\r\nf= @(x) log(x);\r\ng= @(x) x+6;\r\nh= @(x) x/2;\r\n\r\nx1 = 8 ; x2 = 4.4; x3 = 6;\r\n\r\ny1 = 2.3026; y2 = 2.1041; y3=2.1972;\r\n\r\ny=compose3(f,g,h);\r\n\r\nassert(abs(y(x1)-y1)\u003c=1e-4)\r\nassert(abs(y(x2)-y2)\u003c=1e-4)\r\nassert(abs(y(x3)-y3)\u003c=1e-4)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":38414,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":50,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-25T15:27:41.000Z","updated_at":"2026-03-04T14:44:19.000Z","published_at":"2015-03-25T15:27:41.000Z","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\u003eGiven three functions f,g and h, create the composed function y=f(g(h)).\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\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ f = @(x) x+1\\n g = @(x) x/2\\n h = @(x) x^2]]\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\u003eAnd x1=8; x2=10; x3=1;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ y(x1) = 33\\n y(x2) = 51\\n y(x3) = 1.5]]\u003e\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\"}]}"},{"id":44837,"title":"Composing relative poses in 2D: problem 1","description":"We consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north. \r\n\r\nThere is a playing field with a reference frame denoted by {F}.  The rigid-body displacement from {O} to {F} is given by the homogenous transformation |TF| .\r\n\r\nThere are two robots (robot 1 and robot 2) on the playing field, with attached body-fixed coordinate frame {B} and {C} respectively with their origins at the centre of the robot. The x-axis of each robot's frame points in the robot's forward direction.  \r\n\r\nThe rigid-body displacement of robot 1 with respect to the world frame is estimated by GPS to be the homogenous transformation |TB| .  The displacement of robot 2 with respect to robot 1 is estimated by a novel radar sensor and is given by the homogenous transformation |TBC| .\r\n\r\nWhere is robot 2 with respect to the field?","description_html":"\u003cp\u003eWe consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.\u003c/p\u003e\u003cp\u003eThere is a playing field with a reference frame denoted by {F}.  The rigid-body displacement from {O} to {F} is given by the homogenous transformation \u003ctt\u003eTF\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eThere are two robots (robot 1 and robot 2) on the playing field, with attached body-fixed coordinate frame {B} and {C} respectively with their origins at the centre of the robot. The x-axis of each robot's frame points in the robot's forward direction.\u003c/p\u003e\u003cp\u003eThe rigid-body displacement of robot 1 with respect to the world frame is estimated by GPS to be the homogenous transformation \u003ctt\u003eTB\u003c/tt\u003e .  The displacement of robot 2 with respect to robot 1 is estimated by a novel radar sensor and is given by the homogenous transformation \u003ctt\u003eTBC\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eWhere is robot 2 with respect to the field?\u003c/p\u003e","function_template":"function TFC = user_function(TF, TB, TBC)\r\n  % Input:  TF a 3x3 homogeneous transformation matrix\r\n  %         TB a 3x3 homogeneous transformation matrix\r\n  %         TBC a 3x3 homogeneous transformation matrix\r\n  %         P a 2x1 vector representing the coordinate of a point\r\n  % Output: TFC a 3x3 homogeneous transformation matrix\r\n  %\r\n  TFC = ;\r\nend","test_suite":"th = 2*pi*rand;\r\nt = rand(2,1)*20-10;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTF = [R t; 0 0 1];\r\n\r\nth = 2*pi*rand;\r\nt = rand(2,1)*20-10;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTB = [R t; 0 0 1];\r\n\r\nth = 2*pi*rand;\r\nt = rand(2,1)*20-10;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTBC = [R t; 0 0 1];\r\n\r\n\r\n%%\r\nT = user_function(TF, TB, TBC)\r\n\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\nT = user_function(TF, TB, TBC)\r\nassert(isequal(T(3,:), [0 0 1]), 'The bottom row of the homogeneous transformation matrix is not correct')\r\n\r\n%% valid rotation matrix\r\nT = user_function(TF, TB, TBC)\r\nR = T(1:2,1:2);\r\nassert( abs(det(R)-1) \u003c 0.01, 'The determinant of the rotation submatrix is not correct')\r\n\r\n%% correct value\r\nT = user_function(TF, TB, TBC)\r\nTref = inv(TB*TBC)*TF*T;\r\nassert( norm(Tref-eye(3,3)) \u003c 1e-6, 'The homogeneous transform value is not correct')","published":true,"deleted":false,"likes_count":2,"comments_count":1,"created_by":13332,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":64,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":77,"created_at":"2019-01-20T08:11:47.000Z","updated_at":"2026-03-08T02:06:59.000Z","published_at":"2019-01-20T08:33:16.000Z","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\u003eWe consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.\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\u003eThere is a playing field with a reference frame denoted by {F}. The rigid-body displacement from {O} to {F} is given by the homogenous transformation\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\u003eTF\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\u003eThere are two robots (robot 1 and robot 2) on the playing field, with attached body-fixed coordinate frame {B} and {C} respectively with their origins at the centre of the robot. The x-axis of each robot's frame points in the robot's forward direction.\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\u003eThe rigid-body displacement of robot 1 with respect to the world frame is estimated by GPS to be the homogenous transformation\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\u003eTB\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e . The displacement of robot 2 with respect to robot 1 is estimated by a novel radar sensor and is given by the homogenous transformation\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\u003eTBC\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\u003eWhere is robot 2 with respect to the field?\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\"}]}"},{"id":44831,"title":"Composing relative poses in 2D: problem 2","description":"We consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north. \r\n\r\nThere is a playing field with a reference frame denoted by {F}.  The rigid-body displacement from {O} to {F} is given by the homogenous transformation |TF| .\r\n\r\nThere is a robot with an attached body-fixed coordinate frame {B} whose origin is in the centre of the robot, and whose x-axis points in the robot's forward direction.  The rigid-body displacement displacement of {B} relative to {F} is given by the homogenous transformation |TB| .\r\n\r\nThe robot carries a laser scanner which reports the range and bearing of landmarks with respect to its coordinate frame {S} with zero bearing angle corresponding to its x-axis.  The scanner is displaced from the centre of the robot and the displacement of frame {S} relative to {B} is given by the homogenous transformation |TS| .\r\n\r\nThere is a landmark with position described by a coordinate vector P with respect to the world coordinate frame {O}.\r\n\r\nWhat is the range and bearing of the landmark as observed by the laser scanner?","description_html":"\u003cp\u003eWe consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.\u003c/p\u003e\u003cp\u003eThere is a playing field with a reference frame denoted by {F}.  The rigid-body displacement from {O} to {F} is given by the homogenous transformation \u003ctt\u003eTF\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eThere is a robot with an attached body-fixed coordinate frame {B} whose origin is in the centre of the robot, and whose x-axis points in the robot's forward direction.  The rigid-body displacement displacement of {B} relative to {F} is given by the homogenous transformation \u003ctt\u003eTB\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eThe robot carries a laser scanner which reports the range and bearing of landmarks with respect to its coordinate frame {S} with zero bearing angle corresponding to its x-axis.  The scanner is displaced from the centre of the robot and the displacement of frame {S} relative to {B} is given by the homogenous transformation \u003ctt\u003eTS\u003c/tt\u003e .\u003c/p\u003e\u003cp\u003eThere is a landmark with position described by a coordinate vector P with respect to the world coordinate frame {O}.\u003c/p\u003e\u003cp\u003eWhat is the range and bearing of the landmark as observed by the laser scanner?\u003c/p\u003e","function_template":"function [bearing, range] = user_function(TF, TB, TS, P)\r\n  % Input:  TF a 3x3 homogeneous transformation matrix\r\n  %         TB a 3x3 homogeneous transformation matrix\r\n  %         TS a 3x3 homogeneous transformation matrix\r\n  %         P a 2x1 vector representing the coordinate of a point\r\n  % Output: bearing, a scalar angle\r\n  %         range, a scalar distance\r\n  bearing = ;\r\n  range = ;\r\nend","test_suite":"th = 2*pi*rand - pi;\r\nt = rand(2,1)*60-30;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTF = [R t; 0 0 1];\r\n\r\nth = 2*pi*rand - pi;\r\nt = rand(2,1)*20-10;\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTB = [R t; 0 0 1];\r\n\r\nth = 30*pi/180;\r\nt = [0.5; -0.3];\r\nR = [cos(th) -sin(th); sin(th) cos(th)];\r\nTS = [R t; 0 0 1];\r\n\r\nP = rand(2,1) * 500 - 250;\r\n\r\nPref = inv(TF * TB * TS) * [P; 1];\r\n[bearingRef,rangeRef] = cart2pol( Pref(1), Pref(2) );\r\n\r\n[bearing, range] = user_function(TF, TB, TS, P);\r\n\r\n%%\r\nassert(isscalar(bearing) \u0026 isreal(bearing), 'Bearing angle must be a real scalar')\r\n%%\r\nassert(isscalar(range) \u0026 isreal(range), 'Range must be a real scalar')\r\n%%\r\nassert(abs(bearing-bearingRef) \u003c 0.001, 'Bearing angle is not correct')\r\n%%\r\nassert(abs(range-rangeRef) \u003c 0.001, 'Range is not correct')\r\n\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":1,"created_by":13332,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":53,"test_suite_updated_at":"2019-01-10T10:52:52.000Z","rescore_all_solutions":false,"group_id":77,"created_at":"2019-01-10T10:46:31.000Z","updated_at":"2026-03-08T02:54:18.000Z","published_at":"2019-01-10T10:52:52.000Z","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\u003eWe consider a world reference frame denoted by {0} which has its x-axis pointing east and its y-axis pointing north.\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\u003eThere is a playing field with a reference frame denoted by {F}. 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