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updated discussion section of report
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mBerdal committed Dec 12, 2020
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4 changes: 4 additions & 0 deletions distr_opt.py
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Expand Up @@ -192,6 +192,10 @@ def get_area(x_a, precomputed_intersections, com_radius, mission_space):
def close_dist_repell(x_a, X_B_a, k_2):
return np.sum(np.exp(-k_2*np.linalg.norm(X_B_a - x_a.reshape(2, 1), axis=0)))

@staticmethod
def potential(x_a, X_B_a):
return 0.35*np.sum(1/np.linalg.norm(X_B_a - x_a.reshape(2, 1), axis=0))

@staticmethod
def objective(x_a, X_B_a, precomputed_intersections, com_radius, k_1, k_2, mission_space):
return -DistrOpt.get_area(x_a, precomputed_intersections, com_radius, mission_space) + k_1*DistrOpt.close_dist_repell(x_a, X_B_a, k_2)
85 changes: 35 additions & 50 deletions main.py
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Expand Up @@ -10,7 +10,7 @@
from global_opt import GlobalOpt
from distr_opt import DistrOpt

def all_spawn(M, X, com_radius, min_dist, d_min, k_1, k_2, fignames = "trash"):
def all_spawn(M, X, com_radius, min_dist, d_min, k_1, k_2, fignames = "trash", plot_steps = False):
config_fig, axs = plt.subplots(1, 2)
config_fig.set_size_inches(12, 6)
xmin, ymin, xmax, ymax = M.bounds
Expand Down Expand Up @@ -49,23 +49,20 @@ def all_spawn(M, X, com_radius, min_dist, d_min, k_1, k_2, fignames = "trash"):
print("---X_star---")
print(X_star)

###HER###

x_traj_fig, ex_traj_axs = plt.subplots(2, int((G.X_traj.shape[2]-1)/2))
x_traj_fig.set_size_inches(12, 7)
ex_traj_axs = ex_traj_axs.flatten()

for i in np.arange(G.X_traj.shape[2]-1):
X = G.X_traj[:, :, i]
ex_traj_axs[i].set_title("Initial" if i == 0 else f"After iteration {i}")
plot_distr(X, M, com_radius, ex_traj_axs[i])

#########

config_fig.savefig(f"report/figs/{fignames}_distr.pdf", format="pdf", dpi=config_fig.dpi)
area_traj_fig.savefig(f"report/figs/{fignames}_area_traj.pdf", format="pdf", dpi=area_traj_fig.dpi)
step_traj_fig.savefig(f"report/figs/{fignames}_step_traj.pdf", format="pdf", dpi=step_traj_fig.dpi)
x_traj_fig.savefig(f"report/figs/{fignames}_x_traj.pdf", format="pdf", dpi=x_traj_fig.dpi)

if plot_steps:
x_traj_fig, ex_traj_axs = plt.subplots(2, int((G.X_traj.shape[2]-1)/2))
x_traj_fig.set_size_inches(12, 7)
ex_traj_axs = ex_traj_axs.flatten()

for i in np.arange(G.X_traj.shape[2]-1):
X = G.X_traj[:, :, i]
ex_traj_axs[i].set_title("Initial" if i == 0 else f"After iteration {i}")
plot_distr(X, M, com_radius, ex_traj_axs[i])
x_traj_fig.savefig(f"report/figs/{fignames}_x_traj.pdf", format="pdf", dpi=x_traj_fig.dpi)

plt.show()

Expand All @@ -85,24 +82,6 @@ def single_spawn(M, N_dots, com_radius, box_bounds, min_dist):


if __name__=="__main__":
box_bounds = 5




obstacles = [
Obstacle.get_hexagon(2*np.ones((2, )), box_bounds*0.3),
Obstacle.get_horizontal_wall(np.array([-4.99, 1]), 5)
]
obstacles = []

M = MissionSpace(
np.array([
[-box_bounds, -box_bounds],
[box_bounds, -box_bounds],
[box_bounds, box_bounds],
[-box_bounds, box_bounds]
]), obstacles=obstacles)

tinyworld = MissionSpace(np.array([
[-0.75, -0.75],
Expand All @@ -125,15 +104,6 @@ def single_spawn(M, N_dots, com_radius, box_bounds, min_dist):
[-5, 5]
]))

bigworld2 = MissionSpace(np.array([
[-5, -5],
[5, -5],
[5, 5],
[-5, 5]
]), [
Obstacle.get_centered_box(5, 4)
])

complexworld = MissionSpace(np.array([
[-5, -5],
[20, -5],
Expand All @@ -151,23 +121,38 @@ def single_spawn(M, N_dots, com_radius, box_bounds, min_dist):
)
])

#f, a = plt.subplots()
#complexworld.plot(a)
#plt.show()

N_dots = 6
complexworld2 = MissionSpace(np.array([
[-5, -5],
[5, -5],
[5, 5],
[-5, 5]
]), obstacles=[
Obstacle.get_hexagon(
np.array([3, 3]), 1
),
Obstacle.get_hexagon(
np.array([-4, 2]), 1
),
Obstacle.get_hexagon(
np.array([2, -2.5]), 2
),
Obstacle.get_box(
np.array([-2, -1]), 3
),
Obstacle.get_vertical_wall(
np.array([-4.5, -4]), 5
)
])
N_dots = 20
com_radius = 3
min_dist = 0.2
d_min = 0.1

X = np.array([
[-5 + 0.1 + (-1)**(i)*0.05 for i in np.arange(N_dots)],
[-5 + (N_dots - i)*0.21 for i in np.arange(N_dots)]
#[-5 + (i+1)*0.21 for i in np.arange(N_dots)]
])

print(X)

"""
Test for (k_1, k_2) = [(0, *), (1, 0.5), (1, 1), (1, 2), (2, 0.5), (2, 1), (2, 2)]
"""
Expand Down
9 changes: 9 additions & 0 deletions obstacle.py
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Original file line number Diff line number Diff line change
@@ -1,6 +1,15 @@
import numpy as np

class Obstacle():
@staticmethod
def get_box(bottom_left_corner, side_length):
return np.array([
[bottom_left_corner[0], bottom_left_corner[1]],
[bottom_left_corner[0] + side_length, bottom_left_corner[1]],
[bottom_left_corner[0] + side_length, bottom_left_corner[1] + side_length],
[bottom_left_corner[0], bottom_left_corner[1] + side_length]
])

@staticmethod
def get_centered_box(box_bounds, rel_size):
return np.array([
Expand Down
103 changes: 80 additions & 23 deletions report/background.tex
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@@ -1,4 +1,5 @@
\section{Background}\label[sec]{background}
\subsection{Potential, Limitations and Progress of the MAV}
Micro Areal Vehicles (MAVs) have great potential in contributing to indoor search and rescue missions. Their small size and weight
make them easy to transport and allows for rapid field deployment. Furthermore they are agile, allowing them to operate in
complex environments and accessing hard-to-get-to places.
Expand All @@ -14,49 +15,105 @@ \section{Background}\label[sec]{background}
small and efficient enough hardware components such as power supplies, sensors and processors allowing them to quickly process incoming data and convert this information into actions.
The authors of \cite{MAV_enabling} conclude that as of March 2017 no fully autonomous MAV system exist.

Although a fully autonomous MAV system is yet to be realised, the advancement in transistor density shown in \figref{trans_density}, and the continously dimishing price of microprocessors continue to
allow for faster, and thus more complex, on-board computations. This allows for enhanced autonomous capabilites of the aircraft \cite{MAV_enabling} as it will be able to perform ever
Although a fully autonomous MAV system is yet to be realized, the advancement in transistor density shown in \figref{trans_density}, and the continuously diminishing price of microprocessors continue to
allow for faster, and thus more complex, on-board computations. This allows for enhanced autonomous capabilities of the aircraft \cite{MAV_enabling} as it will be able to perform ever
more complex computations locally without the aid of other computation units or human interference.
\begin{figure}[H]
\centering
\includegraphics[width=.85\textwidth]{figs/lion-price.png}
\caption{Evolution of energy density and cost of Lithium-ion batteries. Source: \cite{lion}.}
\label[fig]{lion_price}
\end{figure}
\clearpage
\begin{figure}
\centering
\includegraphics[width=.85\textwidth]{figs/transistors-per-microprocessor.pdf}
\caption{Evolution of the number of transistors per microprocessor.}
\label[fig]{trans_density}
\end{figure}

Most MAVs utilize electrical propulsion systems. This is due to the ability to miniaturize electrical propulsion systems for MAV applications with minimal loss of efficiency. Furthermore electrical propulsion systems have faster response time than combustion based systems \cite{MAV_enabling}. Energy is typically stored on
LiPo batteries. \figref{lion_price} shows how the energy density and price of Lithium-ion batteries have evolved over the last decades. As the energy density of Lithium-ion batteries already have and is projected to continue to increase \cite{lion_density}, this will allow
for fitting more power consuming equipment on the MAVs, and increasing their endurance. Thus enabling MAVs to take on more complex and long lasting missions, resulting in an increasing autonomy.
\begin{figure}
\centering
\includegraphics[width=.85\textwidth]{figs/lion-price.png}
\caption{Evolution of energy density and cost of Lithium-ion batteries. Source: \cite{lion}.}
\label[fig]{lion_price}
\end{figure}

Not only hardware components have made leaps over the last decades. The advancement in autonomy of MAV systems has been a field of great scientific interest.
The Robotics \& Perception Group at the University of Zurich have performed excessive research into the field of visual-inertial based real-time navigation and mapping
by MAVs in GNSS denied environments \cite{svo2, svo1}. The goal of which is enabling MAVs to localize themselves in and map unknown environments without external aid.

In \cite{svo2} a fast and robust visual-odometry algorithm (SVO) is presented. The method is faster than previously presented methods, and is therefore especially usefull
for MAV operations as it should be able to run on a onboard computer with limited processing power. It is concluded that the method is especially usefull for state estimation onboard MAVs as the algorithm runs
In \cite{svo2} a fast and robust visual-odometry algorithm (SVO) is presented. The method is faster than previously presented methods, and is therefore especially useful
for MAV operations as it should be able to run on a onboard computer with limited processing power. It is concluded that the method is especially useful for state estimation onboard MAVs as the algorithm runs
at a high enough rate to provide accurate state estimates. Furthermore the use of depth-filters yields an accurate map of the environment with few outliers.

In \cite{svo1} Scaramuzza et al. present a system for autonomous mapping of unknown indoor and outdoor environments performed by a single MAV using SVO. The MAV is supplied with
a trajectory it is to follow, and using only on-board processing and sensing follows said trajectory and provides at the same time a real-time map of the environment.

A lot of research has also been put into distributed swarm control: the task of making a set of MAVs work together to fulfill some collective goal, and doing so
with only local information. In \cite{connectivity_subgradient} De Gennaro and Jadbabaie present a distributed method for optimizing the connectivity of a multi-agent network.
In their approach each agent knows only the state of its neighbours, where another agent is defined as a neighbour if it fulfills some proximity condition.
Their approach shows through simulations that the the swarm reaches a configuration that corresponds to a maxima of the second smallest eigenvalue of the Laplacian matrix.
Developments such as those in \cite{svo1} and \cite{svo2} have increased the autonomous capabilities of MAVs, and as time passes more efficient software solutions will continue to do so.

\subsection{The Deployment Problem \& Blanket Coverage}
Multi agent distributed control is the task of making a set of agents work together to fulfill some collective goal, and doing so
with only local information. One such task is that studied in this report, self-deployment: Make a swarm of robots "[...] deploy
themselves in an environment without central coordination"\cite{BAYINDIR2016292}. A subcategory of the deployment problem is that referred to
in literature as the blanket coverage problem. In \cite{BAYINDIR2016292} the blanket coverage problem is defined as the task of finding a static
configuration of agents such that a cost function is maximized over an area.

Cassandras and Zhong present in \cite{cassandras} a distributed method for a multi-agent network in which the goal is maximizing the joint detection probability of randomly occuring events in an unknown environment.
They also present an algorithm that preserves connecticity of the network of agents. They conclude that using their method, the swarm of agents reach a configuration corresponding to a
local maxima of the joint detection probability function. Furthermore they find that when connecticity preservation is imposed, the swarm settles at a configuration in which the joint detection probability is
Cassandras and Zhong present in \cite{cassandras} a distributed method for a multi-agent blanket coverage.
They approach the blanket coverage problem using a probability based objective function. With their approach the
goal of the set of agents is maximizing the joint detection probability of randomly occurring events in the environment.
They also present an algorithm that preserves connectivity of the network of agents. They conclude that using their method, the swarm of agents reach a configuration corresponding to a
local maxima of the joint detection probability function. Furthermore they find that when connectivity preservation is imposed, the swarm settles at a configuration in which the joint detection probability is
smaller than when connectivity of the network is not taken into account.

Note to self:
\begin{itemize}
\item CrazyFlie (and other small scale drones)
\item Increase in computational power
\item State of the art drones
\end{itemize}
In \cite{sun2014escaping} Cassandras and Zhong "[...] address the problem of multiple local optima
commonly arising in optimization problems for multi-agent systems[...]". Utilizing the same probability based objective function as in \cite{cassandras}, boosting functions are applied to encourage agents to explore poorly
covered regions of the area and escape local optima. Three families of boosting functions are defined and tested through simulations. It is shown that the objective value post boosting is no worse than before.

In \cite{pot_field} the Howard et.al. present another approach to the blanket coverage problem: virtual potential fields. It is assumed that
agents are able to determine the range and bearing to both nearby agents and obstacles. Drawing inspiration from
electrostatic fields, agents are seen on as point charges who exert repulsive forces on one another, causing them to spread. Furthermore
obstacles also exert forces on the agents so that agents do not collide with obstacles in the environment. They show through simulations
that the potential field approach can be used to deploy multi agent networks for the means of blanket coverage, and conclude that area
coverage emerges from a combination of "[...] purely local rules"\cite{pot_field}.

\clearpage
\subsection{Multilateration}\label[ssec]{trilat}
Multilateration is the process of determining the positions of unknown points in space by measurements of distances from known points \cite{trilat_website}. In order
to perform this task in two-dimensional space, at least three known points are needed.


Given $n\geq 3$ beacons located at positions $\mathbf{x}_{a}\in\mathbb{R}^{2},\; 0\leq a<n$ where not all points
lie on a single line the location
of an entity, denoted by $\mathbf{y}\in\mathbb{R}^{2}$, can be determined as follows:
\begin{enumerate}
\item The entity broadcasts signal and starts a timer at $t_{0}$.
\item Beacons at $\mathbf{x}_{a},\; 0\leq a<n$ receive broadcasted signal and immediately responds with a packet containing $\mathbf{x}_{a}$.
\item When receiving the packet from beacon at $\mathbf{x}_{a}$, the entity stores the time of reception in a variable $t_{1, a}$.
\item When at least 3 beacons have responded, the entity calculates the distance
from itself to beacon at $\mathbf{x}_{a}$: $d_{a} = \frac{1}{2}s(t_{1, a} - t_{0})$, where $s$ is the propagation speed of the signal. The factor $\frac{1}{2}$ is due to the signal traveling
two times the distance between the entity and the beacon placed at $\mathbf{x}_{a}$ (the ping travels from the entity to the agent, and the packet
sent by the agent travels back again).
\item Based on the distances, $d_{a}$, and the positions of the beacons the entity can
determine its position by calculating the point where circles centered at $\mathbf{x}_{a}$ with radii $d_{a}$ intersect.
\end{enumerate}
If sufficiently many beacons respond an ML (Maximum Likelihood) estimator of the position of the entity can be computed \cite{10.1145/381677.381693}.
Defining the error function:\begin{equation}
e_{a}(\mathbf{y}) = s(t_{1, a} - t_{0, a}) - \norm{\mathbf{x}_{a} - \mathbf{y}} = d_{a} - \norm{\mathbf{x}_{a} - \mathbf{y}}
\end{equation}
We obtain the estimate for the position of the entity by solving:
\begin{equation}
\mathbf{y}_{\mathrm{MMSE}} = \min_{\mathbf{y}} \mathbf{E}^{T}\mathbf{E},\quad\mathbf{E} = \begin{bmatrix}
e_{0}(\mathbf{y})\\
\vdots\\
e_{n-1}(\mathbf{y})
\end{bmatrix}
\end{equation}
The position estimation error is affected by the measurement errors, by the geometry relating sensors and target, and by the estimation algorithm \cite{trilat_error}.
As the pings sent by the entity that is to be located travel at large velocities (speed of light) the resolution of the internal clock of the entity sets a bound on the accuracy of the estimated position.
As was found in \cite{CRB_multilat}, multilateration of the unknown position of an entity is most accurate when the entity is placed nearby or within the convex hull of the beacons. Hence spreading
the beacons is desirable in order to obtain accurate position estimates.
\figref{trilat_example} shows how the position of an entity can be determined from the known positions of 3 agents.
\begin{figure}[H]
\centering
\includegraphics[width=.7\textwidth]{figs/trilateration_example.pdf}
\caption{Position, $\mathbf{y}$, of entity determined by trilateration using known positions of $n = 3$ beacons.}
\label[fig]{trilat_example}
\end{figure}
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