SCHOOL of ENGINEERING Page i of v STUDENT NAME: TUTOR NAME: Andrea Paoli PROGRAMME: BEng Electrical Engineering M ELE1002M ODULE CODE: MODULE TITLE: INTRODUCTION TO ROBOTICS SUBJECT: POSES AND TRAJECTORIES COURSEWORK TITLE: ASSIGNMENT 1 OF 2 COURSEWORK WEIGHTING (%): 30% Issue Date: 12/12/2016 Due Date: 30/01/2017 Feedback Date: 10/02/2017 PERFORMANCE CRITERIA: TARGETED LEARNING OUTCOMES 1. Explain how things can be described in 2D using poses 2. Explain how things can be described in 3D using poses 3. Apply rotations and translations to describe a pose and transform a pose into another in 2D and 3D 4. Define and analyse a mono and multi-dimensional trajectories 5. Explain how poses can be measured 6. Use Matlab to solve problems related to poses definition and analysis Important Information – Please Read Before Completing Your Work All students should submit their work by the date specified using the procedures specified in the Student Handbook. An assessment that has been handed in after this deadline will be marked initially as if it had been handed in on time, but the Board of Examiners will normally apply a lateness penalty. Your attention is drawn to the Section on Academic Misconduct in the Student's Handbook. All work will be considered as individual unless collaboration is specifically requested, in which case this should be explicitly acknowledged by the student within their submitted material. Any queries that you may have on the requirements of this assessment should be e-mailed to [email protected]. No queries will be answered after respective submission dates. You must ensure you retain a copy of your completed work prior to submission.SCHOOL of ENGINEERING Page ii of v COURSEWORK BRIEF: Write your assignment using Word, Pages or Open Office. Justify your replies explaining the methodology you applied. Any time you refer to Matlab code, append and comment it. Whenever your Matlab output is a graphical picture, save it as a png file and insert it in your file. Submit the hard copy to the School Office. Exercise 1 Consider two reference frames ሼܣሽ and ሼܤሽ. Reference frame ሼܣሽ is the world reference frame. Reference frame ሼܤሽ is obtained by ሼܣሽ by means of a translational component ሺ4; 10ሻ and a rotation of /ߨ4. 1. Depict the two reference frames. 2. Write the homogeneous transform ஺ܶ஻. Explain how you get this result. 3. Consider a point ܲ described in frame ሼܣሽ by ஺ܲ ൌ ሾ2 6ሿ ். Evaluate the coordinates ஻ܲ of point ܲ described in frame ሼܤሽ. Explain how you get this result. 4. Use Matlab to check your replies 1, 2 and 3. Exercise 2 Consider two reference frames ሼܣሽ and ሼܤሽ. Reference frame ሼܣሽ is the world reference frame. Reference frame ሼܤሽ is obtained by ሼܣሽ by means of a translational component ሺെ1; െ1; െ2ሻ and a rotational component described by the following Euler's angles: ቀగ ସ ; గ ଷ ; െ గ ସቁ. 1. Write the homogeneous transform ஺ܶ஻. Explain how you get this result. 2. Consider a point ܲ described in frame ሼܣሽ by ஺ܲ ൌ ሾ0 1 0ሿ ். Evaluate the coordinates ஻ܲ of point ܲ described in frame ሼܤሽ. Explain how you get this result. 3. Use Matlab to check your replies 1 and 2. Exercise 3 Consider the following scenario. Reference frame ሼܹሽ is the world reference frame. Reference frame ሼܥሽ is attached to a fixed camera and has a translational component ሺ0; 0; 10ሻ and a rotational component with respect to ሼܹሽ described by the following Euler's angles: ቀ0; గ ସ ; െ గ ସቁ. Reference frame ሼܴሽ is attached to a mobile robot and has a translational component ሺ5; െ5; െ10ሻ and a rotational component with respect to ሼܥሽ described by the following Euler's angle: ቀ0; 0; െ గ ଷቁ.SCHOOL of ENGINEERING Page iii of v 1. Compute the homogeneous transform ௐܶோ. Explain how you get this result. 2. Consider a target point described in frame ሼܥሽ by ஼ܲ ൌ ሾ5 5 0ሿ ். Evaluate the coordinates ௐܲ and ோܲ of point ܲ described in frames ሼܹሽ and ሼܴሽ respectively. Explain how you get this result. 3. Use Matlab to depict reference frames ሼܹሽ, ሼܥሽ and ሼܴሽ. 4. Use Matlab to check answers 1 and 2. Exercise 4 Consider a 3rd order polynomial trajectory with the following boundaries conditions: ଴ݐൌ 0, ݐଵ ൌ 10, ݏሺ଴ݐሻ ൌ ݏሶሺ଴ݐሻ ൌ 0, ݏሺݐଵሻ ൌ െ2, ݏሶሺݐଵሻ ൌ െ1. Write the polynomial function that satisfies these boundaries conditions. Explain how you get this result. Exercise 5 Consider a rotational trajectory that starts in the configuration described by Euler's angles ሺ0; 0; 0ሻ and ends in the configuration described by Euler's angles ቀగ ସ ; 0; െ గ ଺ቁ. Using a linear interpolation, write the rotation matrix corresponding to the orientation when 70% of the way along the path. Explain how you get this result. Exercise 6 Describe the working principle of gyroscopes, accelerometers and magnetometers. Also, explain how an Inertial Measurement Unit (IMU) works Exercise 7 Write a Matlab script to rotate and display a cube. The coordinates of the corners (or vertices) of the cube are given in the world coordinate frame by the columns of the matrix ܸ: ܸ ൌ ൥െ െ െ1 1 1 1 1 െ െ1 1 1 1 െ1 െ െ1 1 1 1 1 1 െ െ1 1 1 1 െ1 1 1 െ1൩, Another matrix ( ܧedges) describes each of the edges that needs to be drawn. For example if a column of this matrix is [2, 6] this means that an edge should be drawn between vertex 2 and vertex 6, that is, between the points given by columns 2 and 6 of the matrix ܸ: ܧൌ ቂ1 2 3 4 5 6 7 8 1 2 3 4 2 3 4 1 6 7 8 5 5 6 7 8ቃ. This approach, a set of vertices and a set of edges, is a common way to represent mesh objects in a graphics system.SCHOOL of ENGINEERING Page iv of v We wish to view the cube from a virtual camera with a coordinate frame ሼܥሽ whose pose is given by a translational component ሺ4; 5; 6ሻ and a rotational component described by the following Euler's angle: ቀగ ଶ ; 0; െ గ ସቁ. Plot the cube as it appears from ሼܥሽ using the MATLAB plot3 function to draw 3D lines. Comment your code and depict the graphical output.SCHOOL of ENGINEERING Page v of v MARKING CRITERIA: COURSEWORK WILL BE MARKED ACCORDING TO THE FOLLOWING UNIVERSITY CRITERIA. 90-100%: a range of marks consistent with a first where the work is exceptional in all areas; 80-89%: a range of marks consistent with a first where the work is exceptional in most areas. 70-79%: a range of marks consistent with a first. Work which shows excellent content, organisation and presentation, reasoning and originality; evidence of independent reading and thinking and a clear and authoritative grasp of theoretical positions; ability to sustain an argument, to think analytically and/or critically and to synthesise material effectively. 60-69%: a range of marks consistent with an upper second. Well-organised and lucid coverage of the main points in an answer; intelligent interpretation and confident use of evidence, examples and references; clear evidence of critical judgement in selecting, ordering and analysing content; demonstrates some ability to synthesise material and to construct responses, which reveal insight and may offer some originality. 50-59%: a range of marks consistent with lower second; shows a grasp of the main issues and uses relevant materials in a generally business-like approach, restricted evidence of additional reading; possible unevenness in structure of answers and failure to understand the more subtle points: some critical analysis and a modest degree of insight should be present. 40-49%: a range of marks which is consistent with third class; demonstrates limited understanding with no enrichment of the basic course material presented in classes; superficial lines of argument and muddled presentation; little or no attempt to relate issues to a broader framework; lower end of the range equates to a minimum or threshold pass. 35-39%: achieves many of the learning outcomes required for a mark of 40% but falls short in one or more areas. 30-34%: a fail; may achieve some learning outcomes but falls short in most areas; shows considerable lack of understanding of basic course material and little evidence of research. 0-29%: a fail; basic factual errors of considerable magnitude showing little understanding of basic course material; falls substantially short of the learning outcomes for compensation. MARKING SCHEME: ALL EXERCISES HAVE AN EQUAL MARKING WEIGHT. BEGIN YOUR WORK ON THE FOLLOWING PAGE IF YOU ARE WORD PROCESSING YOUR COURSEWORK (DELETE PAGE IF HAND-WRITING)SCHOOL of ENGINEERING Page 1 of 1