Frame A and point lie on the plane :
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Note that unit vector | Mathinline |
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| body | --uriencoded--z_%7B\mathrm%7BA%7D%7D |
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is pointing out of the screen. The coordinates of point are (2.1, 0.5, 0) when expressed in frame A. Frame B lies in the same plane and has the orientation shown below:
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(a) Find the rotation matrix | Mathinline |
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| body | --uriencoded--%7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D |
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that relates the orientation of frames A and B. Use any strategy to check your answer and explain why it is correct.
(b) For part (b) only, suppose the origins of frames A and B are coincident. Use your answer from part (a) and the expression | Mathinline |
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| body | --uriencoded--p_%7B\mathrm%7BA%7D%7D = %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \ p_%7B\mathrm%7BB%7D%7D |
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to compute | Mathinline |
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| body | --uriencoded--p_%7B\mathrm%7BB%7D%7D |
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. Recall that rotation matrices are orthogonal, which means that | Mathinline |
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| body | --uriencoded--\left( %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \right)%5e%7B-1%7D = \left( %7B%7D%5e%7B\mathrm%7BA%7D%7D\!R_%7B\mathrm%7BB%7D%7D \right)%5e%7B\operatorname%7BT%7D%7D |
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.
(c) Now suppose the origin of frame B is at | Mathinline |
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| body | --uriencoded--3.7 x_%7B\mathrm%7BA%7D%7D + 1.1 y_%7B\mathrm%7BA%7D%7D + 0 z_%7B\mathrm%7BA%7D%7D |
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relative to the origin of frame A. What is the transformation matrix | Mathinline |
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| body | --uriencoded--%7B%7D%5e%7B\mathrm%7BA%7D%7DT_%7B\mathrm%7BB%7D%7D |
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that relates frames A and B?
(d) Use your answer from part (c) and the expression below to compute | Mathinline |
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| body | --uriencoded--p_%7B\mathrm%7BB%7D%7D |
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:
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\begin{Bmatrix}p_{\mathrm{A}}\\1\end{Bmatrix} = {}^{\mathrm{A}}T_{\mathrm{B}} \begin{Bmatrix}p_{\mathrm{B}}\\1\end{Bmatrix} |
(e) Sketch frame A, frame B, and point to confirm that your answer to part (d) is correctIn this problem, we will use the OpenSim API to check the solution to problem 7.3 Introduction to transformation matrices. (If you haven't completed that problem yet, please start there.)
(a) Download and install OpenSim.
(b) Read the Common Scripting Commands and Scripting in the GUI pages in the OpenSim documentation.
(c) Define frame A and point p in a model. You can do this by writing a script that creates a new model, or you could add components to an existing model. For simplicity, the instructions here will add components to the arm26 model. First, open arm26.osim. Next, type the following in the ScriptingShell Window, filling in the missing information as indicated:
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| language | py |
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| theme | Midnight |
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| linenumbers | true |
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model = getCurrentModel()
sphere = modeling.Sphere(0.02)
transA = modeling.Transform( modeling.Vec3(a,b,c) ) #fill in a,b,c
pofA = modeling.PhysicalOffsetFrame(model.getGround(), transA)
pofA.setName('pofA')
model.addComponent(pofA)
pofA.attachGeometry(sphere)
m = modeling.Marker('p', pofA, modeling.Vec3(a,b,c) ) #fill in a,b,c
model.addMarker(m)
model.finalizeConnections()
model.print('step1.osim') |
(d) Close the arm26 model and open step1.osim. You will see a sphere at the origin of frame A and a marker at point p. Next, we will add frame B. Type the following in the ScriptingShell Window, filling in the missing information as indicated:
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| title | Solution (only visible by instructors; please contact us to request access) |
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| Include Page |
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| S7.3 Introduction to transformation matrices |
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| S7.3 Introduction to transformation matrices |
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