Exercise 25.8 [confspace-exercise]
This exercise explores the relationship between workspace and configuration space using the examples shown in Figure FigEx2.
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Consider the robot configurations shown in Figure FigEx2(a) through (c), ignoring the obstacle shown in each of the diagrams. Draw the corresponding arm configurations in configuration space. (Hint: Each arm configuration maps to a single point in configuration space, as illustrated in Figure FigArm1(b).)
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Draw the configuration space for each of the workspace diagrams in Figure FigEx2(a)–(c). (Hint: The configuration spaces share with the one shown in Figure FigEx2(a) the region that corresponds to self-collision, but differences arise from the lack of enclosing obstacles and the different locations of the obstacles in these individual figures.)
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For each of the black dots in Figure FigEx2(e)–(f), draw the corresponding configurations of the robot arm in workspace. Please ignore the shaded regions in this exercise.
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The configuration spaces shown in Figure FigEx2(e)–(f) have all been generated by a single workspace obstacle (dark shading), plus the constraints arising from the self-collision constraint (light shading). Draw, for each diagram, the workspace obstacle that corresponds to the darkly shaded area.
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Figure FigEx2(d) illustrates that a single planar obstacle can decompose the workspace into two disconnected regions. What is the maximum number of disconnected regions that can be created by inserting a planar obstacle into an obstacle-free, connected workspace, for a 2DOF robot? Give an example, and argue why no larger number of disconnected regions can be created. How about a non-planar obstacle?
| $\quad\quad\quad\quad\quad\quad$ | $\quad\quad\quad\quad\quad\quad$ | $\quad\quad\quad\quad\quad\quad$ |
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| (a) | (b) | (c) |
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| (d) | (e) | (f) |