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nominal_trajectory_with_obstacles.py
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nominal_trajectory_with_obstacles.py
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"""
IMPORTS
"""
import casadi
import math as m
import matplotlib.pyplot as plt
import numpy as np
"""
FUNCTIONS
"""
def env_setup():
plt.xlim(0,10)
plt.ylim(0,10)
obs = np.array([[3,4,0.5],[7,8,0.5],[1,8,0.3]])
for i in range(obs.shape[0]):
rad = obs[i,2]; x = obs[i,0]; y = obs[i,1]
c = plt.Circle((x,y),radius = rad,fc='b')
plt.gca().add_patch(c)
return obs
def robot_pose_plot():
start = np.array([2,2,0])
goal = np.array([2,6,m.pi/2])
return (start,goal)
def plot_trajec(U_tot,T):
start,goal = robot_pose_plot()
x_start = start[0]
y_start = start[1]
x_goal = goal[0]
y_goal = goal[1]
env_setup()
plt.plot(x_start,y_start,'^g')
plt.plot(x_goal,y_goal,'^k')
X_tot = np.zeros((3,T))
for i in range(T):
if i==0:
X_tot[:,i] = start[:]
else:
x = X_tot[0,i-1]; y=X_tot[1,i-1];theta = X_tot[2,i-1]
x_new = x + 0.1*U_tot[0,i-1]*m.cos(theta)
y_new = y + 0.1*U_tot[0,i-1]*m.sin(theta)
theta_new = theta + 0.1*U_tot[0,i-1]*m.tan(U_tot[1,i-1])
X_tot[0,i] = x_new; X_tot[1,i] = y_new ; X_tot[2,i]=theta_new
plt.plot(X_tot[0,:],X_tot[1,:])
plt.show()
def robot_pose():
"""
Through this we can access start and end from anywhere
"""
start = casadi.DM([[2],[2],[0]])
goal = casadi.DM([[2],[6],[m.pi/2]])
return(start,goal)
def robot_params():
L = 1
X_max = casadi.DM([[10],[10]]) #works both with DM and not DM
return(L,X_max)
def update(X,U):
"""
X contains [x,y,theta] in a column
U contains [v,s] in a column
"""
x = X[0]; y = X[1] ; theta = X[2]
v = U[0]; s = U[1]
dt = 0.1 #update time
L,_ = robot_params()
#kinematics
x_new = x + v*casadi.cos(theta)*dt
y_new = y + v*casadi.sin(theta)*dt
theta_new = theta + ((v*casadi.tan(s))/L)*dt
#lets make it into a casadi thing
X_new = casadi.blockcat([[x_new],[y_new],[theta_new]])
return X_new
def find_X_tot(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
X_tot = casadi.MX(3,T)
start,goal = robot_pose()
for i in range(T):
if i == 0:
X_tot[:,i] = start[:]
else:
X_tot[:,i] = update(X_tot[:,i-1],U_tot[:,i-1])
return X_tot
def obstacle_cost(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
X_tot = find_X_tot(U_var,T)
obs = env_setup()
obs = casadi.DM(obs)
cost = 0
for i in range(T):
x = X_tot[0,i]; y = X_tot[1,i]
x0 = obs[0,0];y0 = obs[0,1];r0 = obs[0,2]
x1 = obs[1,0];y1 = obs[1,1];r1 = obs[1,2]
x2 = obs[2,0];y2 = obs[2,1];r2 = obs[2,2]
cost = cost + r0*5000*casadi.exp(-((x-x0)**2+(y-y0)**2 - r0**2))
cost = cost + r1*5000*casadi.exp(-((x-x1)**2+(y-y1)**2 - r1**2))
cost = cost + r2*5000*casadi.exp(-((x-x2)**2+(y-y2)**2 - r2**2))
return cost
def total_cost(U_var,T):
Q = casadi.DM([[20,0,0],[0,20,0],[0,0,3]])
R = casadi.DM([[20,0],[0,200]])
Qf = 40000*casadi.DM([[5,0,0],[0,5,0],[0,0,10]])
cost = 0
#find X_tot
U_tot = casadi.reshape(U_var,2,T)
X_tot = find_X_tot(U_var,T)
start,goal = robot_pose()
for i in range(T):
x = goal - X_tot[:,i]; u = U_tot[:,i]
cost = cost + casadi.mtimes(casadi.mtimes(x.T,Q),x) + casadi.mtimes(casadi.mtimes(u.T,R),u)
terminal_cost = casadi.mtimes(casadi.mtimes((goal - X_tot[:,i]).T,Qf),goal - X_tot[:,i])
obs_cost = obstacle_cost(U_var,T)
cost = cost + terminal_cost + obs_cost
return cost
def boundary_constraints(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
X_tot = find_X_tot(U_var,T)
c = casadi.MX(2,T)
_,X_max = robot_params()
for i in range(T):
c[:,i] = X_max - X_tot[0:2,i]
c = casadi.reshape(c,2*T,1)
return c
def X_min_constaints(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
X_tot = find_X_tot(U_var,T)
X_var = casadi.reshape(X_tot,3*T,1)
return X_var
def steering_constraint(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
st_max = m.pi/2
c = casadi.MX(1,T)
for i in range(T):
c[:,i] = st_max - U_tot[1,i]
return c
def steering_constraint2(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
st_max = m.pi/2
c = casadi.MX(1,T)
for i in range(T):
c[:,i] = st_max + U_tot[1,i]
return c
def velocity_constraint(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
v_max = 10
c = casadi.MX(1,T)
for i in range(T):
c[:,i] = v_max - U_tot[0,i]
return c
def velocity_constraint2(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
v_max = 10
c = casadi.MX(1,T)
for i in range(T):
c[:,i] = U_tot[0,i] + v_max
# c[:,i] = U_tot[0,i]
return c
def acceleration_limit(U_var,T):
U_tot = casadi.reshape(U_var,2,T)
a_max = 0.05
v_max = 0.05
c = casadi.MX(2,T-1)
for i in range(T-1):
c[0,i] = a_max**2 - casadi.mtimes((U_tot[0,i+1]-U_tot[0,i]).T,U_tot[0,i+1]-U_tot[0,i])
c[1,i] = v_max**2 - casadi.mtimes((U_tot[1,i+1]-U_tot[1,i]).T,U_tot[1,i+1]-U_tot[1,i])
# c[:,T] = casadi.DM([[0.1],[0.1]])
c = casadi.reshape(c,2*(T-1),1)
return c
def nominal_trajectory():
opti = casadi.Opti()
T = 100
U_var = opti.variable(2*T,1)
X_tot = find_X_tot(U_var,T)
#objective
opti.minimize(total_cost(U_var,T))
#subject to
opti.subject_to(boundary_constraints(U_var,T)>0)
opti.subject_to(X_min_constaints(U_var,T)>0)
opti.subject_to(steering_constraint(U_var,T)>0)
opti.subject_to(steering_constraint2(U_var,T)>0)
opti.subject_to(velocity_constraint2(U_var,T)>0)
opti.subject_to(velocity_constraint(U_var,T)>0)
opti.subject_to(acceleration_limit(U_var,T)>0)
opti.solver('ipopt')
sol = opti.solve()
U = sol.value(U_var)
U_tot = casadi.reshape(U,2,T)
print(U_tot)
plot_trajec(U_tot,T)
nominal_trajectory()
# env_setup()
# plt.show()