[SOLVED] Com s 476/576 midterm exam
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body of radius r operating in a 2D rectangular workspace, as previously described in
Homework 3.As before, the robot is modeled as a rigid body with a circular base of radius r. The
robot can move in two ways:
• Translation T(d): The robot can move along the direction specified by its current
orientation, where d ∈ R is the distance parameter. A positive distance d > 0
indicates forward movement, while a negative distance d < 0 indicates backward
movement.• Rotation R(θ): The robot can rotate in place by an angle θ around its center,
where θ ∈ R is the rotation parameter.The workspace, where the robot operates, is a rectangular area defined by the set
[0, Xmax] × [0, Ymax], with the bottom-left corner at (0, 0) and the top-right corner at
(Xmax, Ymax). The world contains a set O of circular obstacles, each defined by a tuple
o = (xo, yo, ro) ∈ R
2 × R>0, where (xo, yo) is the position of the obstacle’s center and ro
is its radius.The robot initial configuration is qI = (xI , yI , θI ). Unlike in Problem 3 of Homework
3, the robot’s initial position (xI , yI ) is not restricted to grid points; it can be located
anywhere within the workspace. Additionally, collisions must be considered at all times
during the robot’s movement.The goal configuration qG = (xG, yG, θG) includes both position and orientation, and
the robot is considered to have reached the goal only when its configuration exactly
matches qG, not just when its center is at (xG, yG).The objective of this problem is to determine a sequence of robot’s center positions
p = (x0, y0)(x1, y1). . .(xn, yn), where (x0, y0) = (xI , yI ) and (xn, yn) = (xG, yG), along
a path τ : [0, 1] → Cfree such that when projecting τ onto the xy-plane, it forms a
sequence of line segments l1l2 . . . ln with each segment li connecting (xi−1, yi−1) and
(xi
, yi). Here, Cfree = C Cobs, where C = R
2 × S is the configuration space and Cobs
is the configuration space obstacle as defined in class. In other words, the sequence of
line segments corresponding to p must ensure that the robot remains entirely within the
workspace and avoid collisions with any obstacle through its motion.1. (15 points for COM S 4760, 10 points for COM S 5760) Can the problem of computing the sequence of positions p be formulated as a discrete planning problem?
Explain your reasoning by addressing the following points:• Provide a formal, mathematical definition of the free configuration space
C
p
free. Here, the superscript p indicates that the focus is solely on computing
the sequence p of positions rather than the full path τ . So C
p
free is not necessarily the same as Cfree defined earlier. If your definition of C
p
free differs from
Cfree, explain the reasons for this difference.• Specify whether C
p
free is countable, and if so, whether they are finite.
• Describe how to obtain the sequence p = (x0, y0)(x1, y1). . .(xn, yn) once a
path τ
p
: [0, 1] → Cp
free is found, assuming that when projecting τ
p onto the
xy-plane, it corresponds to a sequence of line segments.2. (15 points for COM S 4760, 10 points for COM S 5760) Solve the planning problem
using RRT:
Use the single-tree search outlined in Section 5.5.3 to compute a path p within
C
p
free, starting from q
p
I
and ending at q
p
G. Here, q
p
I
and q
p
G are the projection of qI
and qG, respectively, on the xy-plane.You should check periodically if the tree can be connected to q
p
G. This can be
easily done by setting α(i) as q
p
G with a certain probability pr.For example, the
book recommends pr = 0.01. You should have this as a parameter in your code.
Once q
p
G is successfully added to the tree, quit the loop and compute the path p
from q
p
I
to q
p
G.Implementation Requirements: You are required to implemented a class
named CircularMobileRobotPathPlanning with the following methods in a file
called midterm.py.
• constructor:
__init__(
self,
robot_radius: float,
Xmax: float,
Ymax: float,
circular_obstacles: list
)
– robot radius is the radius r of the robot.
– Xmax and Ymax define the boundaries of the workspace, corresponding to
Xmax and Ymax, respectively.– circular obstacles is a list of circular obstacles, each specified as a
tuple (xo, yo, ro), where xo, yo, ro represent the obstacle’s center
coordinates xo, yo and radius ro, respectively.
• plan method:
plan(
self,
2
qI: tuple,
qG: tuple
)Here, qI and qG are tuples of three elements representing the initial and
goal configurations, i.e., qI = (xI, yI, tI) and qG = (xG, yG, tG) where
xI, yI, tI, xG, yG, tG are floats corresponding to xI , yI , θI , xG, yG, θG,
respectively.This method should return a tuple (G, p) where:
– G is a graph with a draw(self, ax) function that draws the graph on
the specified axis ax
– p is the computed path p = (x0, y0)(x1, y1). . .(xn, yn), represented as
a list of tuples [p[0], p[1], …, p[n]], where each p[i] is a tuple
(xi, yi), corresponding to the position (xi
, yi) in p.An example main.py file is provided, along with a file draw cspace.py containing
the plotting functionalities. Your implementation should be compatible with these
files, so that when running
python main.py
a plot similar to that in Figure 1 is generated. Additionally, your implementation
will be tested with different obstacle configurations and varying values of r, Xmax,
Ymax, qI , qG.Figure 1: The result of RRT planning with p = 0.1, showing the tree and the path from
xI to xG.3. (COM S 5760 only, 10 points) Consider a scenario where two robots b1 and b2,
each with a radius r, operates within this 2D rectangular workspace.In addition to finding individual paths for each robot, we must ensure that their
paths are collision-free, meaning that the robots must not collide with each other
at any point during their execution.Discuss the potential coordination challenges that arise in multi-robot motion
planning by addressing the following points:
• Provide a formal, mathematical definition of the configuration space C2 for
this multi-robot motion planning problem, where the subscript 2 indicates
the setup involving two robots.• Provide a formal, mathematical definition of the free configuration space
Cfree,2 ⊆ C2 that excludes the configurations where any robot collides with
circular obstacles, workspace boundaries, or the other robot.• Identify assumptions that enable the formulation of a discrete motion planning problem for this two-robot setup. Describe the resulting state space X,
the set of actions U, the action space U(x) for each state x ∈ X, and the
state transition function f : X × U → X.Unlike in Problem 3 of Homework 3, you are explicitly NOT allowed to
assumed that collisions—whether with the workspace boundaries, circular
obstacles, or the other robots—can only occur when the robot’s center is
precisely at a grid point. Collisions must be checked continuously throughout
the robot’s execution, including when the robots move between grid points.Note: A path for a robot may be represented either as a full path τ : [0, 1] →
Cfree,1, where Cfree,1 ⊂ R
2 × S
2
is the free configuration space for a scenario where
there is a single robot within this 2D rectangular workspace, or as a sequence of
robot’s center positions, as described before. When answering the above points,
specify how you will represent the paths.