I've posted a question asking about how to go about this problem in general here:
Trajectory Optimization for "Rocket" using scipy.optimize.minimize
Ideally I would like to just minimize the final time, but I could not get the optimizer to append the time to the variable that can be adjusted properly, so I decided that I would just try and minimize u^2 instead for now.
Here is the code:
# Code
t_f = 1.0
t = np.linspace(0., t_f, num = 10) # Time array for 1 second into the future with 0.01 increment
u = np.zeros(t.size) + 650
print(u)
g = -650
initial_position = 0
initial_velocity = 0
final_position = 100
final_velocity = 100
def car_dynamics(x):
# Create time vector
# t = np.linspace(0., t_f, num = 100) # Time array for 1 second into the future with 0.01 increment
# Integrate over entire time to find v as a function of t
a = x + g
v = int.cumtrapz(a, t, initial = 0) + initial_velocity
# Integrate v(t) to get s(t)
s = int.cumtrapz(v, t, initial = 0) + initial_position
return s, v
def constraint1(x): # Final state constraints (Boundary conditions)
s, v = car_dynamics(x)
print('c1', s[0] - initial_position)
return s[0] - initial_position
def constraint2(x): # Initial state constraints (initial conditions)
s, v = car_dynamics(x)
print('c2', v[0] - initial_velocity)
return v[0] - initial_velocity
def constraint3(x):
s, v = car_dynamics(x)
print('c3', s[-1] - final_position)
return s[-1] - final_position
def constraint4(x):
s, v = car_dynamics(x)
print('c4', v[-1] - final_velocity)
return v[-1] - final_velocity
def constraint5(x):
return x - 1000
def objective(x):
u2 = np.square(x)
return np.sum(u2)
cons = [{'type':'eq', 'fun':constraint1},
{'type':'eq', 'fun':constraint2},
{'type':'eq', 'fun':constraint3},
{'type':'eq', 'fun':constraint4}]
# {'type':'ineq', 'fun':constraint5}]
result = minimize(objective, u, constraints = cons, method = 'SLSQP', options={'eps':500, 'maxiter':1000, 'ftol':0.001, 'disp':True})
print(result)
The code runs but the optimizer fails. Here is the error from the output.
c1 0.0
c2 0.0
c3 -100.0
c4 -100.0
c1 0.0
c2 0.0
c3 -100.0
c4 -100.0
c1 0.0
c1 0.0
c1 0.0
c1 0.0
c1 0.0
c1 0.0
c1 0.0
c1 0.0
c1 0.0
c1 0.0
c1 0.0
c2 0.0
c2 0.0
c2 0.0
c2 0.0
c2 0.0
c2 0.0
c2 0.0
c2 0.0
c2 0.0
c2 0.0
c2 0.0
c3 -100.0
c3 -73.76543209876543
c3 -50.617283950617285
c3 -56.79012345679013
c3 -62.96296296296296
c3 -69.1358024691358
c3 -75.30864197530863
c3 -81.4814814814815
c3 -87.65432098765432
c3 -93.82716049382715
c3 -98.45679012345678
c4 -100.0
c4 -72.22222222222223
c4 -44.44444444444445
c4 -44.44444444444445
c4 -44.44444444444445
c4 -44.444444444444436
c4 -44.44444444444445
c4 -44.44444444444448
c4 -44.44444444444445
c4 -44.44444444444442
c4 -72.22222222222221
Singular matrix C in LSQ subproblem (Exit mode 6)
Current function value: 4225000.0
Iterations: 1
Function evaluations: 12
Gradient evaluations: 1
fun: 4225000.0
jac: array([1800., 1800., 1800., 1800., 1800., 1800., 1800., 1800., 1800.,
1800.])
message: 'Singular matrix C in LSQ subproblem'
nfev: 12
nit: 1
njev: 1
status: 6
success: False
x: array([650., 650., 650., 650., 650., 650., 650., 650., 650., 650.])
It seems the constraints aren't being met in some amount of iterations. Should I switch my objective function to contain the final velocity and final position? I've tried different step sizes and what not with the same exit code.
Is there a better way to use this function for what I am trying to get? I am trying to get the control vector u(t) over the entire interval from t0 to t_f, that way I can send these commands to a rocket for optimal control. For now I've simplified the optimization to a single axis, just to learn how to use the function. But as you can see I have not succeeded.
Similar examples would be extremely helpful, and I am open to other methods of optimization, as long as they are numerical, and relatively fast as I plan to implement this as a Model Predictive Controller in real-time eventually.
