example for optimal control of robot
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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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x = SX.sym('x')
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y = SX.sym('y')
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theta = SX.sym('theta')
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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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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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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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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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U = MX.sym('U', 2)
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X = X0
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Q = 0
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runge_kutta = True
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if runge_kutta:
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for j in range(M):
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k1, k1_q = f(X, U)
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k2, k2_q = f(X + DT/2 * k1, U)
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k3, k3_q = f(X + DT/2 * k2, U)
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k4, k4_q = f(X + DT * k3, U)
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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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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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F = Function('F', [X0, U], [X, Q],['x0','p'],['xf','qf'])
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#F_euler = Function('F_euler', [X0, U], [Xeuler, Qeuler], ['x0', 'p'], ['xf', 'qf'])
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Fk = F(x0=[0.2,0.3, 0.0],p=[-1.1, 1.1])
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print(Fk['xf'])
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print(Fk['qf'])
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# Start with an empty NLP
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w=[]
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w0 = []
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lbw = []
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ubw = []
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J = 0
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g=[]
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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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# 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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w += [U1k]
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lbw += [-10, -10]
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ubw += [10, 10]
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w0 += [0, 0]
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# Integrate till the end of the interval
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Fk = F(x0=Xk, p=U1k)
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Xk = Fk['xf']
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J=J+Fk['qf']
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# Add inequality constraint
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#g += [Xk[1]]
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#lbg += [-.0]
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#ubg += [inf]
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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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# Solve the NLP
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sol = 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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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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import matplotlib.pyplot as plt
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plt.figure(1)
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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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plt.plot(tgrid, x3_opt, '*')
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#plt.step(tgrid, vertcat(DM.nan(1), u_opt), '-.')
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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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# 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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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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#T = opti.variable() # final time
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# ---- objective ---------
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#opti.minimize(T) # race in minimal time
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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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# 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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k3, k3_q = f(X[:,k]+dt/2*k2, U[:,k])
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k4, k4_q = f(X[:,k]+dt*k3, U[:,k])
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x_next = X[:,k] + dt/6*(k1+2*k2+2*k3+k4)
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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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# ---- path constraints -----------
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#limit = lambda pos: 1-sin(2*pi*pos)/2
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#opti.subject_to(speed<=limit(pos)) # track speed limit
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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(speed[-1]==0) # .. with speed 0
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opti.subject_to(Q[:,0]==0.0)
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# ---- misc. constraints ----------
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#opti.subject_to(X[1,:]>=0) # Time must be positive
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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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# ---- initial values for solver ---
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#opti.set_initial(speed, 1)
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#opti.set_initial(T, 1)
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# ---- solve NLP ------
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opti.solver("ipopt") # set numerical backend
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sol = opti.solve() # actual solve
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#x0 = sol.value(opti.x)
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#lam_g0 = sol.value(opti.lam_g)
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#opti.set_initial(opti.lam_g, lam_g0)
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#opti.set_initial(opti.x, x0)
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#opti.solve()
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from pylab import plot, step, figure, legend, show, spy
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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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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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ax.add_artist(circle)
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#plot(limit(sol.value(pos)),'r--',label="speed limit")
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#step(range(N),sol.value(U),'k',label="throttle")
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legend(loc="upper left")
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plt.show()
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pass
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# linearization
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# A = zeros(states.shape[0], states.shape[0])
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# for i in range(f.shape[0]):
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# for j in range(states.shape[0]):
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# A[i,j] = diff(f[i,0], states[j])
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# Alin = A.subs([(theta,0), (omegar,0), (omegal,0)])
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# print("A = {}".format(Alin))
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# B = zeros(f.shape[0], controls.shape[0])
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# for i in range(f.shape[0]):
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# for j in range(controls.shape[0]):
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# B[i,j] = diff(f[i,0], controls[j])
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# print("B = {}".format(B))
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# dfdtheta = diff(f, theta)
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#print(dfdtheta.doit())
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# takeaway: linearization is not helpful, because the linearized system is not stabilizable
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# -> alternative: use nonlinear control method
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