import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle, Rectangle
from casadi import *
class TrajOptResolver():
"""
A class that resolves conflicts using trajectoy optimization.
"""
def __init__(self, num_robots, robot_radius, starts, goals, circle_obstacles, rectangle_obstacles,
rob_dist_weight, obs_dist_weight, control_weight, time_weight):
self.num_robots = num_robots
self.starts = starts
self.goals = goals
self.circle_obs = circle_obstacles
self.rect_obs = rectangle_obstacles
self.rob_dist_weight = rob_dist_weight
self.obs_dist_weight = obs_dist_weight
self.control_weight =control_weight
self.time_weight = time_weight
self.robot_radius = MX(robot_radius)
def dist(self, robot_position, circle):
"""
Returns the distance between a robot and a circle
params:
robot_position [x,y]
circle [x,y,radius]
"""
return sumsqr(robot_position - transpose(circle[:2]))
def apply_quadratic_barrier(self, d_max, d, c):
"""
Applies a quadratic barrier to some given distance. The quadratic barrier
is a soft barrier function. We are using it for now to avoid any issues with
invalid initial solutions, which hard barrier functions cannot handle.
params:
d (float): distance to the obstacle
c (float): controls the steepness of curve.
higher c --> gets more expensive faster as you move toward obs
d_max (float): The threshold distance at which the barrier starts to apply
"""
return c*fmax(0, d_max-d)**2
def log_normal_barrier(self, sigma, d, c):
return c*fmax(0, 2-(d/sigma))**2.5
def problem_setup(self, N, x_range, y_range):
"""
Problem setup for the multi-robot collision resolution traj opt problem
inputs:
- N (int): number of control intervals
- x_range (tuple): range of x values
- y_range (tuple): range of y values
outputs:
- problem (dict): dictionary containing the optimization problem
and the decision variables
"""
opti = Opti() # Optimization problem
# ---- decision variables --------- #
X = opti.variable(self.num_robots*3, N+1) # state trajectory (x,y,heading)
pos = X[:self.num_robots*2,:] # position is the first two values
x = pos[0::2,:]
y = pos[1::2,:]
heading = X[self.num_robots*2:,:] # heading is the last value
U = opti.variable(self.num_robots*2, N) # control trajectory (v, omega)
vel = U[0::2,:]
omega = U[1::2,:]
T = opti.variable() # final time
# ---- obstacle setup ------------ #
circle_obs = DM(self.circle_obs) # make the obstacles casadi objects
# ------ Obstacle dist cost ------ #
# TODO:: Include rectangular obstacles
dist_to_other_obstacles = 0
for r in range(self.num_robots):
for k in range(N):
for c in range(circle_obs.shape[0]):
circle = circle_obs[c, :]
d = self.dist(pos[2*r : 2*(r+1), k], circle)
dist_to_other_obstacles += self.apply_quadratic_barrier(2*(self.robot_radius + circle[2]), d, 5)
# ------ Robot dist cost ------ #
dist_to_other_robots = 0
for k in range(N):
for r1 in range(self.num_robots):
for r2 in range(self.num_robots):
if r1 != r2:
d = sumsqr(pos[2*r1 : 2*(r1+1), k] - pos[2*r2 : 2*(r2+1), k])
dist_to_other_robots += self.apply_quadratic_barrier(2*self.robot_radius, d, 1)
# ---- dynamics constraints ---- #
dt = T/N # length of a control interval
pi = [3.14159]*self.num_robots
pi = np.array(pi)
pi = DM(pi)
for k in range(N): # loop over control intervals
dxdt = vel[:,k] * cos(heading[:,k])
dydt = vel[:,k] * sin(heading[:,k])
dthetadt = omega[:,k]
opti.subject_to(x[:,k+1]==x[:,k] + dt*dxdt)
opti.subject_to(y[:,k+1]==y[:,k] + dt*dydt)
opti.subject_to(heading[:,k+1]==fmod(heading[:,k] + dt*dthetadt, 2*pi))
# ------ Control panalty ------ #
# Calculate the sum of squared differences between consecutive heading angles
heading_diff_penalty = 0
for k in range(N-1):
heading_diff_penalty += sumsqr(fmod(heading[:,k+1] - heading[:,k] + pi, 2*pi) - pi)
# ------ cost function ------ #
opti.minimize(self.rob_dist_weight*dist_to_other_robots
+ self.obs_dist_weight*dist_to_other_obstacles
+ self.time_weight*T
+ self.control_weight*heading_diff_penalty)
# ------ control constraints ------ #
for k in range(N):
for r in range(self.num_robots):
opti.subject_to(sumsqr(vel[r,k]) <= 0.2**2)
opti.subject_to(sumsqr(omega[r,k]) <= 0.2**2)
# ------ bound x, y, and time ------ #
opti.subject_to(opti.bounded(x_range[0],x,x_range[1]))
opti.subject_to(opti.bounded(y_range[0],y,y_range[1]))
opti.subject_to(opti.bounded(0,T,100))
# ------ initial conditions ------ #
for r in range(self.num_robots):
opti.subject_to(heading[r, 0]==self.starts[r][2])
opti.subject_to(pos[2*r : 2*(r+1), 0]==self.starts[r][0:2])
opti.subject_to(pos[2*r : 2*(r+1), -1]==self.goals[r])
return {'opti':opti, 'X':X, 'T':T}
def solve_optimization_problem(self, problem, initial_guesses=None, solver_options=None):
opti = problem['opti']
if initial_guesses:
for param, value in initial_guesses.items():
print(f"param = {param}")
print(f"value = {value}")
opti.set_initial(problem[param], value)
# Set numerical backend, with options if provided
if solver_options:
opti.solver('ipopt', solver_options)
else:
opti.solver('ipopt')
try:
sol = opti.solve() # actual solve
status = 'succeeded'
except:
sol = None
status = 'failed'
results = {
'status' : status,
'solution' : sol,
}
if sol:
for var_name, var in problem.items():
if var_name != 'opti':
results[var_name] = sol.value(var)
return results
def solve(self, N, x_range, y_range, initial_guesses):
"""
Setup and solve a multi-robot traj opt problem
input:
- N (int): the number of control intervals
- x_range (tuple):
- y_range (tuple):
"""
problem = self.problem_setup(N, x_range, y_range)
results = self.solve_optimization_problem(problem, initial_guesses)
X = results['X']
sol = results['solution']
# Extract the values that we want from the optimizer's solution
pos = X[:self.num_robots*2,:]
x_vals = pos[0::2,:]
y_vals = pos[1::2,:]
theta_vals = X[self.num_robots*2:,:]
return sol,pos, x_vals, y_vals, theta_vals
def get_local_controls(self, controls):
"""
Get the local controls for the robots in the conflict
"""
l = self.num_robots
final_trajs = [None]*l
for c in self.conflicts:
# Get the robots involved in the conflict
robots = [self.all_robots[r.label] for r in c]
# Solve the trajectory optimization problem
initial_guess = None
sol, x_opt, vels, omegas, xs,ys = self.solve(20, initial_guess)
pos_vals = np.array(sol.value(x_opt))
# Update the controls for the robots
for r, vel, omega, x,y in zip(robots, vels, omegas, xs,ys):
controls[r.label] = [vel, omega]
final_trajs[r.label] = [x,y]
return controls, final_trajs
def plot_paths(self, x_opt):
fig, ax = plt.subplots()
# Plot obstacles
for obstacle in self.circle_obs:
# if len(obstacle) == 2: # Circle
ax.add_patch(Circle(obstacle, obstacle[2], color='red'))
# elif len(obstacle) == 4: # Rectangle
# ax.add_patch(Rectangle((obstacle[0], obstacle[1]), obstacle[2], obstacle[3], color='red'))
if self.num_robots > 20:
colors = plt.cm.hsv(np.linspace(0.2, 1.0, self.num_robots))
elif self.num_robots > 10:
colors = plt.cm.tab20(np.linspace(0, 1, self.num_robots))
else:
colors = plt.cm.tab10(np.linspace(0, 1, self.num_robots))
# Plot robot paths
for r,color in zip(range(self.num_robots),colors):
ax.plot(x_opt[r*2, :], x_opt[r*2+1, :], label=f'Robot {r+1}', color=color)
ax.scatter(x_opt[r*2, :], x_opt[r*2+1, :], color=color, s=10 )
ax.scatter(self.starts[r][0], self.starts[r][1], s=85,color=color)
ax.scatter(self.goals[r][0], self.goals[r][1], s=85,facecolors='none', edgecolors=color)
ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.legend()
ax.set_aspect('equal', 'box')
plt.ylim(0,640)
plt.xlim(0,480)
plt.title('Robot Paths')
plt.grid(False)
plt.show()