Motion Planning & Trajectory Optimization

Robotics

Motion Planning in robotics focuses on devising efficient paths for a robot from its current state to a goal, while Trajectory Optimization refines these paths for smooth and obstacle-free motion, crucial for precise robotic movements in various applications

Motion Planning

Potential Fields

Here I will use potential fields to get a motion plan for the 2-DoF environment shown below. Here the drone’s position is $θ = [𝑥, 𝑦]^{𝑇}$ .

Implementing the potential fields approach:

• Setting $θ_{𝑠 𝑡 𝑎 𝑟 𝑡} = [0, 0]^{𝑇} a n d θ_{𝑔 𝑜 𝑎 l} = [1, 1]^{𝑇}$

• The first obstacle has $c e n t e r 𝑐_{1} = [0.3, 0.5]^{𝑇} a n d r a d i u s 𝑟_{1} = 0.125$ .

• The second obstacle has center $c e n t e r 𝑐_{2} = [0.7, 0.5]^{𝑇} a n d r a d i u s 𝑟_{2} = 0.225$ .

NOTE: Always start with a low learning rate $α$ in your g

Implementations

Python

Code

import numpy as np
import matplotlib
matplotlib.use('Agg')  # Set the backend to a non-interactive backend
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation, PillowWriter

# Parameters
theta_start = np.array([0.0, 0.0])
theta_goal = np.array([1.0, 1.0])
centers = np.array([[0.3, 0.5], [0.7, 0.5]])
radii = np.array([0.125, 0.225])

# Function to compute potential field
def field(theta, theta_goal, centers, radii):
    U = 0.5 * np.linalg.norm(theta_goal - theta)**2
    for idx in range(len(radii)):
        center = centers[idx]
        radius = radii[idx]
        dist = np.linalg.norm(center - theta)
        if dist < radius:
            U += 0.5 * (1/dist - 1/radius)**2
    return U

# Initialize figure
fig, ax = plt.subplots()
ax.grid(True)

# Drawing circles
for idx in range(len(radii)):
    circle = plt.Circle(centers[idx], radii[idx], color=[0.5, 0.5, 0.5], fill=False)
    ax.add_patch(circle)

# Start and goal points
ax.plot(theta_start[0], theta_start[1], 'ko', markerfacecolor='k')
ax.plot(theta_goal[0], theta_goal[1], 'ko', markerfacecolor='k')
ax.axis('equal')

# Parameters for gradient descent
theta = theta_start
delta = 0.01  # Adjusted delta value
learning_rate = 0.01  # Adjusted learning rate
path = [theta_start]  # To store the path

# Animation update function
def update(frame):
    global theta
    if np.linalg.norm(theta - theta_goal) < 0.1:
        return

    U = field(theta, theta_goal, centers, radii)
    U1 = field(theta + np.array([delta, 0.0]), theta_goal, centers, radii)
    U2 = field(theta + np.array([0.0, delta]), theta_goal, centers, radii)
    Ugrad = np.array([U1 - U, U2 - U]) / delta
    theta -= learning_rate * Ugrad
    path.append(theta.copy())
    ax.plot([p[0] for p in path], [p[1] for p in path], 'o', color=[1, 0.5, 0], markerfacecolor=[1, 0.5, 0])

# Create animation
ani = FuncAnimation(fig, update, frames=np.arange(300), repeat=False)

# Save to GIF using PillowWriter
gif_path = 'Potential_Field.gif'
ani.save(gif_path, writer=PillowWriter(fps=30))

MATLAB


clear
close all
% Parameters
theta_start = [0; 0];
theta_goal = [1; 1];
centers = [.3, .7; .5, .5];
radii = [.125, .225];
% Create figure
figure
grid on
hold on
for idx = 1:length(radii)
 viscircles(centers(:,idx)', radii(idx), 'Color', [0.5, 0.5, 0.5]);
end
plot(theta_start(1), theta_start(2), 'ko', 'MarkerFaceColor', 'k')
plot(theta_goal(1), theta_goal(2), 'ko', 'MarkerFaceColor', 'k')
axis equal
% Gradient descent down potential field
theta = theta_start;
delta = 0.01;
learning_rate = 0.01;
for idx = 1:1000
 if norm(theta - theta_goal) < 0.1
 break
 end
 U = field(theta, theta_goal, centers, radii);
 U1 = field(theta + [delta; 0], theta_goal, centers, radii);
 U2 = field(theta + [0; delta], theta_goal, centers, radii);
 Ugrad = [U1 - U; U2 - U] / delta;
 theta = theta - learning_rate * Ugrad;
 plot(theta(1), theta(2), 'o', 'color', [1, 0.5, 0], ...
 'MarkerFaceColor', [1, 0.5, 0])
end
% Find potential field at position theta
function U = field(theta, theta_goal, centers, radii)
 U = 0.5 * norm(theta_goal - theta)^2;
 for idx = 1:length(radii)
 center = centers(:, idx);
 radius = radii(idx);
 dist = norm(center - theta);
 if dist < radius
 U = U + 0.5 * (1/dist - 1/radius)^2;
 end
 end
end

Result:

Animated GIF

Modifying the position of the obstacles so that a valid plan from 𝜃𝑠𝑡𝑎𝑟𝑡 to 𝜃𝑔𝑜𝑎𝑙 exists but the potential fields planner fails (i.e., gets stuck). Below is a simulation that shows the obstacles and the failed motion plan.

There are many possible solutions. For instance, I am setting $𝑐_{1} = [0.4, 0.5]^{𝑇}$ .

Motion Plan Failure

Implementations

Python

Code

import numpy as np
import matplotlib
matplotlib.use('Agg')  # Set the backend to a non-interactive backend
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation, PillowWriter

# Parameters
theta_start = np.array([0.0, 0.0])
theta_goal = np.array([1.0, 1.0])
centers = np.array([[0.4, 0.5], [0.5, 0.5]])
radii = np.array([0.125, 0.225])

# Function to compute potential field
def field(theta, theta_goal, centers, radii):
    U = 0.5 * np.linalg.norm(theta_goal - theta)**2
    for idx in range(len(radii)):
        center = centers[idx]
        radius = radii[idx]
        dist = np.linalg.norm(center - theta)
        if dist < radius:
            U += 0.5 * (1/dist - 1/radius)**2
    return U

# Initialize figure
fig, ax = plt.subplots()
ax.grid(True)

# Drawing circles
for idx in range(len(radii)):
    circle = plt.Circle(centers[idx], radii[idx], color=[0.5, 0.5, 0.5], fill=False)
    ax.add_patch(circle)

# Start and goal points
ax.plot(theta_start[0], theta_start[1], 'ko', markerfacecolor='k')
ax.plot(theta_goal[0], theta_goal[1], 'ko', markerfacecolor='k')
ax.axis('equal')

# Parameters for gradient descent
theta = theta_start
delta = 0.01  # Adjusted delta value
learning_rate = 0.01  # Adjusted learning rate
path = [theta_start]  # To store the path

# Animation update function
def update(frame):
    global theta
    if np.linalg.norm(theta - theta_goal) < 0.1:
        return

    U = field(theta, theta_goal, centers, radii)
    U1 = field(theta + np.array([delta, 0.0]), theta_goal, centers, radii)
    U2 = field(theta + np.array([0.0, delta]), theta_goal, centers, radii)
    Ugrad = np.array([U1 - U, U2 - U]) / delta
    theta -= learning_rate * Ugrad
    path.append(theta.copy())
    ax.plot([p[0] for p in path], [p[1] for p in path], 'o', color=[1, 0.5, 0], markerfacecolor=[1, 0.5, 0])

# Create animation
ani = FuncAnimation(fig, update, frames=np.arange(300), repeat=False)

# Save to GIF using PillowWriter
gif_path = 'Potential_Field_fail.gif'
ani.save(gif_path, writer=PillowWriter(fps=30))

MATLAB

clear
close all

% Start and goal environments
theta_start= [0; 0];
theta_goal = [1; 1];

% Modified obstacle positions and sizes
obs_c21 = [0.3; 0.5];  % Shifted first obstacle closer to the second
obs_r21 = 0.15;        % Slightly increased radius
obs_c22 = [0.8; 0.5];  % Shifted second obstacle closer to the first
obs_r22 = 0.25;        % Slightly increased radius

% Visualize the environment
figure
grid on
hold on
axis([0, 1, 0, 1])
axis equal
viscircles(obs_c21', obs_r21, 'Color', [0.5, 0.5, 0.5]);
viscircles(obs_c22', obs_r22, 'Color', [0.5, 0.5, 0.5]);
plot(0, 0, 'ko', 'MarkerFaceColor', 'k');
plot(1, 1, 'ko', 'MarkerFaceColor', 'k');

% Setting the variables
alpha= 0.01;
epsilon = 0.1;
delta= 0.01;

% Initial trajectory
theta(:,1) = theta_start;
t=1;
del_Unet=1;

while norm(del_Unet)> epsilon
    del_Ux= U_theta(theta(:,t) + [delta;0]);
    del_Uy= U_theta(theta(:,t)+ [0;delta]);
    del_U= U_theta(theta(:,t));

    del_Unet=[del_Ux-del_U; del_Uy-del_U]/delta;

    theta(:,t+1)= theta(:,t)- alpha*del_Unet;
    t=t+1;
end


grid on
hold on
axis equal
plot(theta(1,:), theta(2,:), 'o-',...
    'Color', [1, 0.5, 0], 'LineWidth', 2);
    

function U = U_theta(theta) 
    beta=2;
    gamma=1;
    theta_goal = [1; 1];
    % Modified obstacle positions and sizes
    obs_c21 = [0.3; 0.5];  % Update obstacle parameters
    obs_r21 = 0.15;
    obs_c22 = [0.6; 0.5];
    obs_r22 = 0.25;

    Urep1=0;
    Urep2=0;

    Uatt=0.5*beta*norm(theta_goal-theta)^2;    

    if norm(obs_c21-theta)<= obs_r21
        Urep1=0.5* gamma*((1/norm(obs_c21-theta))- (1/obs_r21))^2;
    end
    if norm(obs_c22-theta)<= obs_r22
        Urep2= 0.5*gamma*((1/norm(obs_c22-theta))- (1/obs_r22))^2;
    end

    Urep= Urep1+ Urep2;
    disp(Urep);
    U= Uatt + Urep;
end

Result:

Animated GIF

This failure occurs because the robot uses gradient descent to move towards decreasing potential energy, and the robot gets trapped in a local minimum.

In this modification, the obstacles are placed closer to each other, potentially creating a narrow corridor that the drone might not be able to navigate due to the combined repulsive forces. This setup can result in the planner getting stuck in a local minimum, where the drone is unable to progress towards the goal. Thus, we place a local minimim in between the start and goal. Gradient descent can only climb “down” so if it’s surrounded by high gradients, it can’t escape.

Trajectory Optimization

(MATLAB Implementations provided below)

Here we are using trajectory optimization to perform motion planning in 2-DoF environments. As before, the mobile robot’s position is $θ = [𝑥, 𝑦]^{𝑇}$ .

Implementing the trajectory optimization algorithm Below. My code should be able to work with an arbitrary number of waypoints and circular obstacles.

Thus setting the initial trajectory $ξ^{0}$ as:

$ξ^{0} = (\begin{array}{cc} l i n s p a c e (θ_{s t a r t} (1), θ_{g o a l} (1), k) \\ l i n s p a c e (θ_{s t a r t} (1), θ_{g o a l} (1), k) \end{array})$

Environment 1: One obstacle with $c e n t e r 𝑐_{1} = [0.2, 0.35]^{𝑇} a n d r a d i u s 𝑟_{1} = 0.2$ . A second obstacle with $c e n t e r 𝑐_{2} = [0.5, 0.3]^{𝑇} a n d r a d i u s 𝑟_{2} = 0.2$ . A third obstacle with $c e n t e r 𝑐_{3} = [0.7, 0.5]^{𝑇} a n d r a d i u s 𝑟_{3} = 0.2$ . Setting $𝑘 = 20$ .

MATLAB Code

clear
close all

% start and goal
theta_start = [0;0];
theta_goal = [1;1];
centers = [0.2 0.5, 0.7; 0.35 0.3, 0.5];
radii = [0.2, 0.2, 0.2];

% initial trajectory
n = 2;
k = 20;
xi_0 = [linspace(theta_start(1), theta_goal(1), k);...
 linspace(theta_start(2), theta_goal(2), k)];
xi_0_vec = reshape(xi_0, [], 1);

% start and goal equality constraints
A = [eye(n) zeros(n, n*(k-1));...
 zeros(n, n*(k-1)), eye(n)];
B = [theta_start; theta_goal];

% nonlinear optimization
options = optimoptions('fmincon','Display','final',...
 'Algorithm','sqp','MaxFunctionEvaluations',1e5);
xi_star_vec = fmincon(@(xi) cost(xi, centers, radii), xi_0_vec, ...
 [], [], A, B, [], [], [], options);
xi_star = reshape(xi_star_vec, 2, []);

% plot result
figure
grid on
hold on
axis equal
for idx = 1:length(radii)
 viscircles(centers(:, idx)', radii(idx), 'Color', [0.5, 0.5, ...
 0.5]);
end
plot(xi_0(1,:), xi_0(2,:), 'o-', 'Color', [0.3, 0.3, ...
 0.3], 'LineWidth', 3);
plot(xi_star(1,:), xi_star(2,:), 'o-', 'Color', [1, 0.5, ...
 0], 'LineWidth', 3);
% cost function to minimize

function C = cost(xi, centers, radii)
 xi = reshape(xi, 2, []);
 C = 0;
 for idx = 2:length(xi)
 theta_curr = xi(:, idx);
 theta_prev = xi(:, idx - 1);
 C = C + norm(theta_curr - theta_prev)^2;
 for jdx = 1:length(radii)
 center = centers(:, jdx);
 radius = radii(jdx);
 if norm(theta_curr - center) < radius
 C = C + 0.5*(1/norm(theta_curr - center) - 1/radius)^2;
 end
 end
 end
end

Result:

Environment 2: One obstacle with $c e n t e r 𝑐_{1} = [0.5, 0.3]^{𝑇} a n d r a d i u s 𝑟_{1} = 0.3$ . A second obstacle with $c e n t e r 𝑐_{2} = [0.5, 0.7]^{𝑇} a n d r a d i u s 𝑟_{2} = 0.2$ . Setting $𝑘 = 15$ waypoints.

MATLAB Code

clear
close all

% start and goal
theta_start = [0;0];
theta_goal = [1;1];
centers = [0.5 0.5; 0.3 0.7];
radii = [0.3, 0.2];

% initial trajectory
n = 2;
k = 15;
xi_0 = [linspace(theta_start(1), theta_goal(1), k);...
 linspace(theta_start(2), theta_goal(2), k)];
xi_0_vec = reshape(xi_0, [], 1);

% start and goal equality constraints
A = [eye(n) zeros(n, n*(k-1));...
 zeros(n, n*(k-1)), eye(n)];
B = [theta_start; theta_goal];

% nonlinear optimization
options = optimoptions('fmincon','Display','final',...
 'Algorithm','sqp','MaxFunctionEvaluations',1e5);
xi_star_vec = fmincon(@(xi) cost(xi, centers, radii), xi_0_vec, ...
 [], [], A, B, [], [], [], options);
xi_star = reshape(xi_star_vec, 2, []);

% plot result
figure
grid on
hold on
axis equal
for idx = 1:length(radii)
 viscircles(centers(:, idx)', radii(idx), 'Color', [0.5, 0.5, ...
 0.5]);
end
plot(xi_0(1,:), xi_0(2,:), 'o-', 'Color', [0.3, 0.3, ...
 0.3], 'LineWidth', 3);
plot(xi_star(1,:), xi_star(2,:), 'o-', 'Color', [1, 0.5, ...
 0], 'LineWidth', 3);

% cost function to minimize
function C = cost(xi, centers, radii)
 xi = reshape(xi, 2, []);
 C = 0;
 for idx = 2:length(xi)
 theta_curr = xi(:, idx);
 theta_prev = xi(:, idx - 1);
 C = C + norm(theta_curr - theta_prev)^2;
 for jdx = 1:length(radii)
 center = centers(:, jdx);
 radius = radii(jdx);
 if norm(theta_curr - center) < radius
 C = C + 0.5*(1/norm(theta_curr - center) - 1/radius)^2;
 end
 end
 end
end

Result:

Environment 3: One obstacle with $c e n t e r 𝑐_{1} = [0.55, 0.5]^{𝑇} a n d r a d i u s 𝑟_{1} = 0.3$ . Trajectory should have $𝑘 = 10$ waypoints.

MATLAB Code

clear
close all

% start and goal
theta_start = [0;0];
theta_goal = [1;1];
centers = [0.55; 0.5];
radii = 0.3;

% initial trajectory
n = 2;
k = 10;
xi_0 = [linspace(theta_start(1), theta_goal(1), k);...
 linspace(theta_start(2), theta_goal(2), k)];
xi_0_vec = reshape(xi_0, [], 1);

% start and goal equality constraints
A = [eye(n) zeros(n, n*(k-1));...
 zeros(n, n*(k-1)), eye(n)];
B = [theta_start; theta_goal];

% nonlinear optimization
options = optimoptions('fmincon','Display','final',...
 'Algorithm','sqp','MaxFunctionEvaluations',1e5);
xi_star_vec = fmincon(@(xi) cost(xi, centers, radii), xi_0_vec, ...
 [], [], A, B, [], [], [], options);
xi_star = reshape(xi_star_vec, 2, []);

% plot result
figure
grid on
hold on
axis equal
for idx = 1:length(radii)
 viscircles(centers(:, idx)', radii(idx), 'Color', [0.5, 0.5, ...
 0.5]);
end
plot(xi_0(1,:), xi_0(2,:), 'o-', 'Color', [0.3, 0.3, ...
 0.3], 'LineWidth', 3);
plot(xi_star(1,:), xi_star(2,:), 'o-', 'Color', [1, 0.5, ...
 0], 'LineWidth', 3);

% cost function to minimize
function C = cost(xi, centers, radii)
 xi = reshape(xi, 2, []);
 C = 0;
 for idx = 2:length(xi)
 theta_curr = xi(:, idx);
 theta_prev = xi(:, idx - 1);
 C = C + norm(theta_curr - theta_prev)^2;
 for jdx = 1:length(radii)
 center = centers(:, jdx);
 radius = radii(jdx);
 if norm(theta_curr - center) < radius
 C = C + 0.5*(1/norm(theta_curr - center) - 1/radius)^2;
 end
 end
 end
end

Result:

Legend: Here, in all the three environments the gray line is the initial trajectory $ξ^{0}$ , while the orange line is the optimal trajectory $ξ$ .

Special Scenario

Consider an environment with two obstacles:

$θ_{s t a r t} = [0, 0]^{𝑇} a n d θ_{g o a l} = [1, 1]^{𝑇}$ .
First obstacle with $c e n t e r 𝑐_{1} = [0.4, 0.6]^{𝑇} a n d r a d i u s 𝑟_{1} = 0.2$ .
Second obstacle with $c e n t e r 𝑐_{2} = [0.6, 0.4]^{𝑇} a n d r a d i u s 𝑟_{2} = 0.2$ .
The trajectory $ξ$ should have $k = 20$ waypoints.

Modifying the initial trajectory $ξ^{0}$ so that the optimal trajectory goes around both obstacles.

Solution:

For this environment an initial trajectory $ξ^{0}$ that moves straight from start to goal gets stuck. The nonlinear optimizer cannot find a way to modify this initial trajectory to decrease the cost (i.e., we are stuck in a local minima), and so the final answer simply jumps across the obstacles. We can fix this problem by choosing an initial trajectory that moves either above or below the obstacles.

For example, we can try:

$ξ^{0} = (\begin{array}{cc} 0 & 0 & 0 & . & . & . & 0 \\ 0 & 1 / (k - 1) & 2 / (k - 1) & . & . & . & 1 \end{array})$

Where $ξ^{0} \in R^{𝑛 \times 𝑘}$

This results in an optimal trajectory $ξ$ that goes above both obstacles. See the figure below. If we select an initial trajectory that causes the robot to converge to a path below both obstacles, that is also fine.

Result:

--- title: "Motion Planning & Trajectory Optimization" description: "Motion Planning in robotics focuses on devising efficient paths for a robot from its current state to a goal, while Trajectory Optimization refines these paths for smooth and obstacle-free motion, crucial for precise robotic movements in various applications" categories: [Robotics] image: Trajectory_Cover_GIF.gif format: html: code-fold: show code-overflow: wrap code-tools: true --- # Motion Planning ## Potential Fields Here I will use potential fields to get a motion plan for the 2-DoF environment shown below. Here the drone’s position is $\normalsize \theta = [𝑥, 𝑦]^𝑇$. Implementing the potential fields approach: • Setting $\normalsize \theta_{𝑠𝑡𝑎𝑟𝑡} = [0, 0]^𝑇 \; and \; \theta_{𝑔𝑜𝑎l} = [1, 1]^𝑇$ • The first obstacle has $\normalsize center \; 𝑐_1 = [0.3, 0.5]^𝑇 \; and \; radius \; 𝑟_1 = 0.125$. • The second obstacle has center $\normalsize center \; 𝑐_2 = [0.7, 0.5]^𝑇 \; and \; radius \; 𝑟_2 = 0.225$. NOTE: Always start with a low learning rate $\normalsize \alpha$ in your g ### Implementations #### Python ```{python} import numpy as np import matplotlib matplotlib.use('Agg') # Set the backend to a non-interactive backend import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation, PillowWriter # Parameters theta_start = np.array([0.0, 0.0]) theta_goal = np.array([1.0, 1.0]) centers = np.array([[0.3, 0.5], [0.7, 0.5]]) radii = np.array([0.125, 0.225]) # Function to compute potential field def field(theta, theta_goal, centers, radii): U = 0.5 * np.linalg.norm(theta_goal - theta)**2 for idx in range(len(radii)): center = centers[idx] radius = radii[idx] dist = np.linalg.norm(center - theta) if dist < radius: U += 0.5 * (1/dist - 1/radius)**2 return U # Initialize figure fig, ax = plt.subplots() ax.grid(True) # Drawing circles for idx in range(len(radii)): circle = plt.Circle(centers[idx], radii[idx], color=[0.5, 0.5, 0.5], fill=False) ax.add_patch(circle) # Start and goal points ax.plot(theta_start[0], theta_start[1], 'ko', markerfacecolor='k') ax.plot(theta_goal[0], theta_goal[1], 'ko', markerfacecolor='k') ax.axis('equal') # Parameters for gradient descent theta = theta_start delta = 0.01 # Adjusted delta value learning_rate = 0.01 # Adjusted learning rate path = [theta_start] # To store the path # Animation update function def update(frame): global theta if np.linalg.norm(theta - theta_goal) < 0.1: return U = field(theta, theta_goal, centers, radii) U1 = field(theta + np.array([delta, 0.0]), theta_goal, centers, radii) U2 = field(theta + np.array([0.0, delta]), theta_goal, centers, radii) Ugrad = np.array([U1 - U, U2 - U]) / delta theta -= learning_rate * Ugrad path.append(theta.copy()) ax.plot([p[0] for p in path], [p[1] for p in path], 'o', color=[1, 0.5, 0], markerfacecolor=[1, 0.5, 0]) # Create animation ani = FuncAnimation(fig, update, frames=np.arange(300), repeat=False) # Save to GIF using PillowWriter gif_path = 'Potential_Field.gif' ani.save(gif_path, writer=PillowWriter(fps=30)) ``` **MATLAB** <details> ```matlab clear close all % Parameters theta_start = [0; 0]; theta_goal = [1; 1]; centers = [.3, .7; .5, .5]; radii = [.125, .225]; % Create figure figure grid on hold on for idx = 1:length(radii) viscircles(centers(:,idx)', radii(idx), 'Color', [0.5, 0.5, 0.5]); end plot(theta_start(1), theta_start(2), 'ko', 'MarkerFaceColor', 'k') plot(theta_goal(1), theta_goal(2), 'ko', 'MarkerFaceColor', 'k') axis equal % Gradient descent down potential field theta = theta_start; delta = 0.01; learning_rate = 0.01; for idx = 1:1000 if norm(theta - theta_goal) < 0.1 break end U = field(theta, theta_goal, centers, radii); U1 = field(theta + [delta; 0], theta_goal, centers, radii); U2 = field(theta + [0; delta], theta_goal, centers, radii); Ugrad = [U1 - U; U2 - U] / delta; theta = theta - learning_rate * Ugrad; plot(theta(1), theta(2), 'o', 'color', [1, 0.5, 0], ... 'MarkerFaceColor', [1, 0.5, 0]) end % Find potential field at position theta function U = field(theta, theta_goal, centers, radii) U = 0.5 * norm(theta_goal - theta)^2; for idx = 1:length(radii) center = centers(:, idx); radius = radii(idx); dist = norm(center - theta); if dist < radius U = U + 0.5 * (1/dist - 1/radius)^2; end end end ``` </details> #### Result: <img src="Potential_Field.gif" alt="Animated GIF" loop width="50%" height="50%"> * Modifying the position of the obstacles so that a valid plan from 𝜃𝑠𝑡𝑎𝑟𝑡 to 𝜃𝑔𝑜𝑎𝑙 exists but the potential fields planner fails (i.e., gets stuck). Below is a simulation that shows the obstacles and the failed motion plan. There are many possible solutions. For instance, I am setting $\normalsize 𝑐_1 = [0.4, 0.5]^𝑇$. ## Motion Plan Failure ### Implementations #### Python ```{python} import numpy as np import matplotlib matplotlib.use('Agg') # Set the backend to a non-interactive backend import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation, PillowWriter # Parameters theta_start = np.array([0.0, 0.0]) theta_goal = np.array([1.0, 1.0]) centers = np.array([[0.4, 0.5], [0.5, 0.5]]) radii = np.array([0.125, 0.225]) # Function to compute potential field def field(theta, theta_goal, centers, radii): U = 0.5 * np.linalg.norm(theta_goal - theta)**2 for idx in range(len(radii)): center = centers[idx] radius = radii[idx] dist = np.linalg.norm(center - theta) if dist < radius: U += 0.5 * (1/dist - 1/radius)**2 return U # Initialize figure fig, ax = plt.subplots() ax.grid(True) # Drawing circles for idx in range(len(radii)): circle = plt.Circle(centers[idx], radii[idx], color=[0.5, 0.5, 0.5], fill=False) ax.add_patch(circle) # Start and goal points ax.plot(theta_start[0], theta_start[1], 'ko', markerfacecolor='k') ax.plot(theta_goal[0], theta_goal[1], 'ko', markerfacecolor='k') ax.axis('equal') # Parameters for gradient descent theta = theta_start delta = 0.01 # Adjusted delta value learning_rate = 0.01 # Adjusted learning rate path = [theta_start] # To store the path # Animation update function def update(frame): global theta if np.linalg.norm(theta - theta_goal) < 0.1: return U = field(theta, theta_goal, centers, radii) U1 = field(theta + np.array([delta, 0.0]), theta_goal, centers, radii) U2 = field(theta + np.array([0.0, delta]), theta_goal, centers, radii) Ugrad = np.array([U1 - U, U2 - U]) / delta theta -= learning_rate * Ugrad path.append(theta.copy()) ax.plot([p[0] for p in path], [p[1] for p in path], 'o', color=[1, 0.5, 0], markerfacecolor=[1, 0.5, 0]) # Create animation ani = FuncAnimation(fig, update, frames=np.arange(300), repeat=False) # Save to GIF using PillowWriter gif_path = 'Potential_Field_fail.gif' ani.save(gif_path, writer=PillowWriter(fps=30)) ``` **MATLAB** <details> ```matlab clear close all % Start and goal environments theta_start= [0; 0]; theta_goal = [1; 1]; % Modified obstacle positions and sizes obs_c21 = [0.3; 0.5]; % Shifted first obstacle closer to the second obs_r21 = 0.15; % Slightly increased radius obs_c22 = [0.8; 0.5]; % Shifted second obstacle closer to the first obs_r22 = 0.25; % Slightly increased radius % Visualize the environment figure grid on hold on axis([0, 1, 0, 1]) axis equal viscircles(obs_c21', obs_r21, 'Color', [0.5, 0.5, 0.5]); viscircles(obs_c22', obs_r22, 'Color', [0.5, 0.5, 0.5]); plot(0, 0, 'ko', 'MarkerFaceColor', 'k'); plot(1, 1, 'ko', 'MarkerFaceColor', 'k'); % Setting the variables alpha= 0.01; epsilon = 0.1; delta= 0.01; % Initial trajectory theta(:,1) = theta_start; t=1; del_Unet=1; while norm(del_Unet)> epsilon del_Ux= U_theta(theta(:,t) + [delta;0]); del_Uy= U_theta(theta(:,t)+ [0;delta]); del_U= U_theta(theta(:,t)); del_Unet=[del_Ux-del_U; del_Uy-del_U]/delta; theta(:,t+1)= theta(:,t)- alpha*del_Unet; t=t+1; end grid on hold on axis equal plot(theta(1,:), theta(2,:), 'o-',... 'Color', [1, 0.5, 0], 'LineWidth', 2); function U = U_theta(theta) beta=2; gamma=1; theta_goal = [1; 1]; % Modified obstacle positions and sizes obs_c21 = [0.3; 0.5]; % Update obstacle parameters obs_r21 = 0.15; obs_c22 = [0.6; 0.5]; obs_r22 = 0.25; Urep1=0; Urep2=0; Uatt=0.5*beta*norm(theta_goal-theta)^2; if norm(obs_c21-theta)<= obs_r21 Urep1=0.5* gamma*((1/norm(obs_c21-theta))- (1/obs_r21))^2; end if norm(obs_c22-theta)<= obs_r22 Urep2= 0.5*gamma*((1/norm(obs_c22-theta))- (1/obs_r22))^2; end Urep= Urep1+ Urep2; disp(Urep); U= Uatt + Urep; end ``` </details> #### Result: <img src="Potential_Field_fail.gif" alt="Animated GIF" loop width="50%" height="50%"> This failure occurs because the robot uses gradient descent to move towards decreasing potential energy, and the robot gets trapped in a **local minimum**. In this modification, the obstacles are placed closer to each other, potentially creating a narrow corridor that the drone might not be able to navigate due to the combined repulsive forces. This setup can result in the planner getting stuck in a local minimum, where the drone is unable to progress towards the goal. Thus, we place a local minimim in between the start and goal. **Gradient descent** can only climb "down" so if it's surrounded by high gradients, it can't escape. ## Trajectory Optimization **(MATLAB Implementations provided below)** Here we are using trajectory optimization to perform motion planning in 2-DoF environments. As before, the mobile robot’s position is $\normalsize \theta = [𝑥, 𝑦]^𝑇$. Implementing the trajectory optimization algorithm Below. My code should be able to work with an arbitrary number of waypoints and circular obstacles. Thus setting the initial trajectory $\normalsize \xi^0$ as: $$\normalsize \xi^0 = \left(\begin{array}{cc} linspace(\theta_{start}(1), \; \theta_{goal}(1), \; k)\\ linspace(\theta_{start}(1), \; \theta_{goal}(1), \; k) \end{array}\right) $$ * **Environment 1: One obstacle with $\normalsize center \; 𝑐_1 = [0.2, 0.35]^𝑇 \; and \; radius \; 𝑟_1 = 0.2$. A second obstacle with $\normalsize center \; 𝑐_2 = [0.5, 0.3]^𝑇 \; and \; radius \; 𝑟_2 = 0.2$. A third obstacle with $\normalsize center \; 𝑐_3 = [0.7, 0.5]^𝑇 \; and \; radius \; 𝑟_3 = 0.2$. Setting $\normalsize 𝑘 = 20$.** <details open> <summary>MATLAB Code</summary> ```matlab clear close all % start and goal theta_start = [0;0]; theta_goal = [1;1]; centers = [0.2 0.5, 0.7; 0.35 0.3, 0.5]; radii = [0.2, 0.2, 0.2]; % initial trajectory n = 2; k = 20; xi_0 = [linspace(theta_start(1), theta_goal(1), k);... linspace(theta_start(2), theta_goal(2), k)]; xi_0_vec = reshape(xi_0, [], 1); % start and goal equality constraints A = [eye(n) zeros(n, n*(k-1));... zeros(n, n*(k-1)), eye(n)]; B = [theta_start; theta_goal]; % nonlinear optimization options = optimoptions('fmincon','Display','final',... 'Algorithm','sqp','MaxFunctionEvaluations',1e5); xi_star_vec = fmincon(@(xi) cost(xi, centers, radii), xi_0_vec, ... [], [], A, B, [], [], [], options); xi_star = reshape(xi_star_vec, 2, []); % plot result figure grid on hold on axis equal for idx = 1:length(radii) viscircles(centers(:, idx)', radii(idx), 'Color', [0.5, 0.5, ... 0.5]); end plot(xi_0(1,:), xi_0(2,:), 'o-', 'Color', [0.3, 0.3, ... 0.3], 'LineWidth', 3); plot(xi_star(1,:), xi_star(2,:), 'o-', 'Color', [1, 0.5, ... 0], 'LineWidth', 3); % cost function to minimize function C = cost(xi, centers, radii) xi = reshape(xi, 2, []); C = 0; for idx = 2:length(xi) theta_curr = xi(:, idx); theta_prev = xi(:, idx - 1); C = C + norm(theta_curr - theta_prev)^2; for jdx = 1:length(radii) center = centers(:, jdx); radius = radii(jdx); if norm(theta_curr - center) < radius C = C + 0.5*(1/norm(theta_curr - center) - 1/radius)^2; end end end end ``` </details> #### Result: ![](Images/Traj_Env_1.png){fig-align="center" width=50%} * **Environment 2: One obstacle with $\normalsize center \; 𝑐_1 = [0.5, 0.3]^𝑇 \; and \; radius \; 𝑟_1 = 0.3$. A second obstacle with $\normalsize center \; 𝑐_2 = [0.5, 0.7]^𝑇 \; and \; radius \; 𝑟_2 = 0.2$. Setting $\normalsize 𝑘 = 15$ waypoints.** <details open> <summary>MATLAB Code</summary> ```matlab clear close all % start and goal theta_start = [0;0]; theta_goal = [1;1]; centers = [0.5 0.5; 0.3 0.7]; radii = [0.3, 0.2]; % initial trajectory n = 2; k = 15; xi_0 = [linspace(theta_start(1), theta_goal(1), k);... linspace(theta_start(2), theta_goal(2), k)]; xi_0_vec = reshape(xi_0, [], 1); % start and goal equality constraints A = [eye(n) zeros(n, n*(k-1));... zeros(n, n*(k-1)), eye(n)]; B = [theta_start; theta_goal]; % nonlinear optimization options = optimoptions('fmincon','Display','final',... 'Algorithm','sqp','MaxFunctionEvaluations',1e5); xi_star_vec = fmincon(@(xi) cost(xi, centers, radii), xi_0_vec, ... [], [], A, B, [], [], [], options); xi_star = reshape(xi_star_vec, 2, []); % plot result figure grid on hold on axis equal for idx = 1:length(radii) viscircles(centers(:, idx)', radii(idx), 'Color', [0.5, 0.5, ... 0.5]); end plot(xi_0(1,:), xi_0(2,:), 'o-', 'Color', [0.3, 0.3, ... 0.3], 'LineWidth', 3); plot(xi_star(1,:), xi_star(2,:), 'o-', 'Color', [1, 0.5, ... 0], 'LineWidth', 3); % cost function to minimize function C = cost(xi, centers, radii) xi = reshape(xi, 2, []); C = 0; for idx = 2:length(xi) theta_curr = xi(:, idx); theta_prev = xi(:, idx - 1); C = C + norm(theta_curr - theta_prev)^2; for jdx = 1:length(radii) center = centers(:, jdx); radius = radii(jdx); if norm(theta_curr - center) < radius C = C + 0.5*(1/norm(theta_curr - center) - 1/radius)^2; end end end end ``` </details> #### Result: ![](Images/Traj_Env_2.png){fig-align="center" width=50%} * **Environment 3: One obstacle with $\normalsize center \; 𝑐_1 = [0.55, 0.5]^𝑇 \; and \; radius \; 𝑟_1 = 0.3$. Trajectory should have $\normalsize 𝑘 = 10$ waypoints.** <details open> <summary>MATLAB Code</summary> ```matlab clear close all % start and goal theta_start = [0;0]; theta_goal = [1;1]; centers = [0.55; 0.5]; radii = 0.3; % initial trajectory n = 2; k = 10; xi_0 = [linspace(theta_start(1), theta_goal(1), k);... linspace(theta_start(2), theta_goal(2), k)]; xi_0_vec = reshape(xi_0, [], 1); % start and goal equality constraints A = [eye(n) zeros(n, n*(k-1));... zeros(n, n*(k-1)), eye(n)]; B = [theta_start; theta_goal]; % nonlinear optimization options = optimoptions('fmincon','Display','final',... 'Algorithm','sqp','MaxFunctionEvaluations',1e5); xi_star_vec = fmincon(@(xi) cost(xi, centers, radii), xi_0_vec, ... [], [], A, B, [], [], [], options); xi_star = reshape(xi_star_vec, 2, []); % plot result figure grid on hold on axis equal for idx = 1:length(radii) viscircles(centers(:, idx)', radii(idx), 'Color', [0.5, 0.5, ... 0.5]); end plot(xi_0(1,:), xi_0(2,:), 'o-', 'Color', [0.3, 0.3, ... 0.3], 'LineWidth', 3); plot(xi_star(1,:), xi_star(2,:), 'o-', 'Color', [1, 0.5, ... 0], 'LineWidth', 3); % cost function to minimize function C = cost(xi, centers, radii) xi = reshape(xi, 2, []); C = 0; for idx = 2:length(xi) theta_curr = xi(:, idx); theta_prev = xi(:, idx - 1); C = C + norm(theta_curr - theta_prev)^2; for jdx = 1:length(radii) center = centers(:, jdx); radius = radii(jdx); if norm(theta_curr - center) < radius C = C + 0.5*(1/norm(theta_curr - center) - 1/radius)^2; end end end end ``` </details> #### Result: ![](Images/Traj_Env_3.png){fig-align="center" width=50%} *Legend: Here, in all the three environments the gray line is the initial trajectory $\normalsize \xi^0$, while the orange line is the optimal trajectory $\normalsize \xi$.* # Special Scenario **Consider an environment with two obstacles:** * $\normalsize \theta_{start} = [0, 0]^𝑇 \; and \; \theta_{goal} = [1, 1]^𝑇$. * First obstacle with $\normalsize center \; 𝑐_1 = [0.4, 0.6]^𝑇 \; and \; radius \; 𝑟_1 = 0.2$. * Second obstacle with $\normalsize center \; 𝑐_2 = [0.6, 0.4]^𝑇 \; and \; radius \; 𝑟_2 = 0.2$. * The trajectory $\normalsize \xi$ should have $\normalsize k = 20$ waypoints. Modifying the initial trajectory $\normalsize \xi^0$ so that the optimal trajectory goes around both obstacles. **Solution:** For this environment an **initial trajectory $\normalsize \xi^0$** that moves straight from start to goal gets stuck. The nonlinear optimizer cannot find a way to modify this initial trajectory to **decrease the cost** (i.e., we are stuck in a local minima), and so the final answer simply jumps across the obstacles. We can fix this problem by choosing an initial trajectory that moves either above or below the obstacles. For example, we can try: $$\normalsize \xi^0 = \left(\begin{array}{cc} 0 && 0 && 0 && . && . && . && 0\\ 0 && 1/(k-1) && 2/(k-1) && . && . && . && 1 \end{array}\right) $$ Where $\normalsize \xi^0 ∈ R^{𝑛×𝑘}$ This results in an optimal trajectory $\normalsize \xi$ that goes above both obstacles. See the figure below. If we select an initial trajectory that causes the robot to converge to a path below both obstacles, that is also fine. #### Result: ![](Images/Traj_Env_4_special.png){fig-align="center" width=50%} ```{=html} <script> const tooltipTriggerList = document.querySelectorAll('[data-bs-toggle="tooltip"]') const tooltipList = [...tooltipTriggerList].map(tooltipTriggerEl => new bootstrap.Tooltip(tooltipTriggerEl)) </script> <style> div#quarto-sidebar-glass { display: none !important; } ul.navbar-nav.navbar-nav-scroll { -webkit-flex-direction: row !important; } /* #quarto-sidebar { padding: 5px; } #quarto-sidebar > * { padding: 5px; } div.sidebar-menu-container > * { padding: 5px 5px 5px 5px; } #quarto-margin-sidebar { padding: 40px; } */ </style> ```