started working on optimization based controller
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@ -2,9 +2,12 @@ from casadi import *
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# look at: https://github.com/casadi/casadi/blob/master/docs/examples/python/vdp_indirect_multiple_shooting.py
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T = 3.0
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N = 30
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class OpenLoopSolver:
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def __init__(self, N=60, T=6.0):
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self.T = T
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self.N = N
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def solve(self, x0):
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x = SX.sym('x')
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y = SX.sym('y')
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theta = SX.sym('theta')
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@ -12,24 +15,27 @@ state = vertcat(x, y, theta)
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r = 0.03
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R = 0.05
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d = 0.02
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#r = SX.sym('r')
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#R = SX.sym('R')
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#d = SX.sym('d')
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omegar = SX.sym('omegar')
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omegal = SX.sym('omegal')
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control = vertcat(omegar, omegal)
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f1 = (r/2 * cos(theta) - r*d/(2*R) * sin(theta)) * omegar + (r/2 * cos(theta) + r*d/(2*R) * sin(theta)) * omegal
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f2 = (r/2 * sin(theta) + r*d/(2*R) * cos(theta)) * omegar + (r/2 * sin(theta) - r*d/(2*R) * cos(theta)) * omegal
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# model equation
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f1 = (r / 2 * cos(theta) - r * d / (2 * R) * sin(theta)) * omegar + (r / 2 * cos(theta) + r * d / (2 * R) * sin(
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theta)) * omegal
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f2 = (r / 2 * sin(theta) + r * d / (2 * R) * cos(theta)) * omegar + (r / 2 * sin(theta) - r * d / (2 * R) * cos(
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theta)) * omegal
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f3 = r / (2 * R) * omegar - r / (2 * R) * omegal
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xdot = vertcat(f1, f2, f3)
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f = Function('f', [x, y, theta, omegar, omegal], [f1, f2, f3])
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print("f = {}".format(f))
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# cost functional
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L = x ** 2 + y ** 2 + 1e-2 * theta ** 2 + 1e-4 * (omegar ** 2 + omegal ** 2)
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# Fixed step Runge-Kutta 4 integrator
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M = 4 # RK4 steps per interval
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DT = T/N/M
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DT = self.T / self.N / M
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print("DT = {}".format(DT))
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f = Function('f', [state, control], [xdot, L])
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X0 = MX.sym('X0', 3)
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@ -46,7 +52,7 @@ if runge_kutta:
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X = X + DT / 6 * (k1 + 2 * k2 + 2 * k3 + k4)
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Q = Q + DT / 6 * (k1_q + 2 * k2_q + 2 * k3_q + k4_q)
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else:
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DT = T/N
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DT = self.T / self.N
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k1, k1_q = f(X, U)
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X = X + DT * k1
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Q = Q + DT * k1_q
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@ -69,8 +75,8 @@ lbg = []
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ubg = []
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# Formulate the NLP
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Xk = MX([1.1, 1.1, 0.0])
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for k in range(N):
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Xk = MX(x0)
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for k in range(self.N):
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# New NLP variable for the control
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U1k = MX.sym('U1_' + str(k), 2)
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# U2k = MX.sym('U2_' + str(k))
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@ -91,25 +97,25 @@ for k in range(N):
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# Create an NLP solver
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prob = {'f': J, 'x': vertcat(*w), 'g': vertcat(*g)}
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solver = nlpsol('solver', 'ipopt', prob);
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self.solver = nlpsol('solver', 'ipopt', prob)
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# Solve the NLP
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sol = solver(x0=w0, lbx=lbw, ubx=ubw, lbg=lbg, ubg=ubg)
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sol = self.solver(x0=w0, lbx=lbw, ubx=ubw, lbg=lbg, ubg=ubg)
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w_opt = sol['x']
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# Plot the solution
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u_opt = w_opt
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x_opt = [[1.1, 1.1, -0.0]]
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for k in range(N):
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x_opt = [x0]
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for k in range(self.N):
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Fk = F(x0=x_opt[-1], p=u_opt[2*k:2*k+2])
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x_opt += [Fk['xf'].full()]
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x1_opt = [r[0] for r in x_opt]
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x2_opt = [r[1] for r in x_opt]
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x3_opt = [r[2] for r in x_opt]
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tgrid = [T/N*k for k in range(N+1)]
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tgrid = [self.T/self.N*k for k in range(self.N+1)]
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import matplotlib.pyplot as plt
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plt.figure(1)
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plt.figure(2)
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plt.clf()
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plt.plot(tgrid, x1_opt, '--')
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plt.plot(tgrid, x2_opt, '-')
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@ -119,17 +125,18 @@ plt.xlabel('t')
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plt.legend(['x1','x2','x3','u'])
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plt.grid()
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#plt.show()
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#return
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# alternative solution using multiple shooting (way faster!)
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opti = Opti() # Optimization problem
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# ---- decision variables ---------
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X = opti.variable(3,N+1) # state trajectory
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Q = opti.variable(1,N+1) # state trajectory
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X = opti.variable(3,self.N+1) # state trajectory
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Q = opti.variable(1,self.N+1) # state trajectory
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posx = X[0,:]
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posy = X[1,:]
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angle = X[2,:]
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U = opti.variable(2,N) # control trajectory (throttle)
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U = opti.variable(2,self.N) # control trajectory (throttle)
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#T = opti.variable() # final time
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# ---- objective ---------
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@ -138,8 +145,8 @@ U = opti.variable(2,N) # control trajectory (throttle)
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# ---- dynamic constraints --------
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#f = lambda x,u: vertcat(f1, f2, f3) # dx/dt = f(x,u)
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dt = T/N # length of a control interval
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for k in range(N): # loop over control intervals
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dt = self.T/self.N # length of a control interval
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for k in range(self.N): # loop over control intervals
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# Runge-Kutta 4 integration
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k1, k1_q = f(X[:,k], U[:,k])
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k2, k2_q = f(X[:,k]+dt/2*k1, U[:,k])
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@ -149,7 +156,7 @@ for k in range(N): # loop over control intervals
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q_next = Q[:,k] + dt/6*(k1_q + 2 * k2_q + 2 * k3_q + k4_q)
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opti.subject_to(X[:,k+1]==x_next) # close the gaps
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opti.subject_to(Q[:,k+1]==q_next) # close the gaps
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opti.minimize(Q[:,N])
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opti.minimize(Q[:,self.N])
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# ---- path constraints -----------
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#limit = lambda pos: 1-sin(2*pi*pos)/2
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@ -157,9 +164,9 @@ opti.minimize(Q[:,N])
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opti.subject_to(opti.bounded(-10,U,10)) # control is limited
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# ---- boundary conditions --------
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opti.subject_to(posx[0]==1.10) # start at position 0 ...
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opti.subject_to(posy[0]==1.10) # ... from stand-still
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opti.subject_to(angle[0]==0.0) # finish line at position 1
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opti.subject_to(posx[0]==x0[0]) # start at position 0 ...
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opti.subject_to(posy[0]==x0[1]) # ... from stand-still
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opti.subject_to(angle[0]==x0[2]) # finish line at position 1
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#opti.subject_to(speed[-1]==0) # .. with speed 0
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opti.subject_to(Q[:,0]==0.0)
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@ -168,11 +175,12 @@ opti.subject_to(Q[:,0]==0.0)
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#opti.subject_to(X[2,:]<=4) # Time must be positive
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#opti.subject_to(X[2,:]>=-2) # Time must be positive
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r = 0.25
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p = (0.5, 0.5)
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for k in range(N):
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opti.subject_to((X[0,k]-p[0])**2 + (X[1,k]-p[1])**2 > r**2)
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pass
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# avoid obstacle
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#r = 0.25
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#p = (0.5, 0.5)
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#for k in range(self.N):
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# opti.subject_to((X[0,k]-p[0])**2 + (X[1,k]-p[1])**2 > r**2)
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# pass
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# ---- initial values for solver ---
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@ -195,7 +203,7 @@ plot(sol.value(posx),label="posx")
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plot(sol.value(posy),label="posy")
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plot(sol.value(angle),label="angle")
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plt.figure()
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plt.figure(3)
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plot(sol.value(posx), sol.value(posy))
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ax = plt.gca()
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circle = plt.Circle(p, r)
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@ -20,6 +20,8 @@ import matplotlib.animation as anim
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import time
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from casadi_opt import OpenLoopSolver
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from marker_pos_angle.msg import id_pos_angle
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class Robot:
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@ -59,7 +61,7 @@ def f_ode(t, x, u):
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class RemoteController:
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def __init__(self):
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self.robots = [Robot(3)]
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self.robots = [Robot(5)]
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self.robot_ids = {}
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for r in self.robots:
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@ -69,7 +71,7 @@ class RemoteController:
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self.rc_socket = socket.socket()
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try:
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pass
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self.rc_socket.connect(('192.168.1.101', 1234)) # connect to robot
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self.rc_socket.connect(('192.168.1.103', 1234)) # connect to robot
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except socket.error:
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print("could not connect to socket")
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@ -117,6 +119,8 @@ class RemoteController:
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plt.xlabel('x-position')
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plt.ylabel('y-position')
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self.ols = OpenLoopSolver()
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def ani(self):
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self.ani = anim.FuncAnimation(self.fig, init_func=self.ani_init, func=self.ani_update, interval=10, blit=True)
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plt.ion()
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@ -208,7 +212,8 @@ class RemoteController:
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keyboard_control = False
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keyboard_control_speed_test = False
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pid = True
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pid = False
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open_loop_solve = True
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if keyboard_control: # keyboard controller
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events = pygame.event.get()
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@ -330,6 +335,15 @@ class RemoteController:
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time.sleep(dt)
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elif open_loop_solve:
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# open loop controller
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events = pygame.event.get()
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for event in events:
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if event.type == pygame.KEYDOWN:
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if event.key == pygame.K_UP:
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self.ols.solve(self.xms[-1])
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def main(args):
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rospy.init_node('controller_node', anonymous=True)
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