2. 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.)
3. 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.
4. 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.
5. 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?
This exercise explores the relationship between
workspace and configuration space using the examples shown in
Figure FigEx2.
1. 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).)
2. 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.)
3. 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.
4. 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.
5. 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?