Assignment 3: C-Space and C-Obstacle
Contact: asathyam@cs.umd.edu February 2020
1 C-Space Basics
1.(10 points) Derive the C-Space for 3D rigid body.
a. Consider the GL(3) group first and impose constraints to obtain O(3)
and SO(3) groups. Identify redundant equations while imposing constraints.
b. What is the shape of the groups? Show that the antipodal points in the SO(3) group represent the same rotation. Elucidate your answer with diagrams.
2.(5 points) Determine the C-space for: a. Planar robot with a 2R arm.
b. Spherical Pendulum
2 C-Obstacle
3.(20 points) Assume that a planar robot R is modeled by an oriented line segment of length 4 that can move freely in the plane XY plane. R’s config- uration is described by the coordinates of the center point of R and the angle θ (with respect to the X axis). Note that since R is an oriented line segment, the orientations θ and θ + π are different.Assume that an obstacle O, a square centered at (0,2) with side length 2 is present in the workspace.
a. Draw the C-obstacles corresponding to O when R is only allowed to translate at fixed orientations θ = 0, π/4, and π/2. Draw three C-obstacles, one for each orientation.
b. How would the C-obstacle corresponding to O look like in the 3-D C-space of R(that is, when R translates and rotates). Discuss whether the C-obstacle is connected (i.e., made of one single piece)? Does it contain holes? Is it convex? At which orientations its cross-section undergoes qualitative changes? Is there
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a repeating pattern along the orientation axis?
c. If we constrain the center-point of R to remain on the x-axis, and R translates and rotates with this constraint. What is the configuration space of R? Draw (approximately) the C-obstacle corresponding to O. Discuss whether the C-Obstacle is connected in this case.
3 Convexity in workspace and C-Space
4.(10 points) Consider a robot A consisting of a single rigid body that only translates and static obstacle Bin a 3-D workspace W. A’s configuration is rep- resented by (x,y,z) which are the coordinates of a reference point in A relative to a global coordinate frame in W. If both A and B are convex subsets of W, is the C-obstacle corresponding to Bin the C-Space of A convex? [Hint: Consider asetS⊂Rn. SisconvexifforanytwopointsXandYofSdescribedbytheir n coordinates, the point λX + (l-λ)Y (with 0 < λ < 1) is in S.]
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