initial commit LTV_MPC
matssteinweg
parent
d0479e511c
commit
76df3ce814
329
MPC.py
329
MPC.py
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@ -1,31 +1,32 @@
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import numpy as np
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import cvxpy as cp
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import osqp
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import scipy as sp
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from scipy import sparse
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##################
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# MPC Controller #
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##################
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class MPC:
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def __init__(self, model, T, Q, R, Qf, StateConstraints, InputConstraints,
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Reference):
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def __init__(self, model, N, Q, R, QN, StateConstraints, InputConstraints):
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"""
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Constructor for the Model Predictive Controller.
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:param model: bicycle model object to be controlled
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:param T: time horizon | int
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:param Q: state cost matrix
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:param R: input cost matrix
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:param Qf: final state cost matrix
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:param QN: final state cost matrix
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:param StateConstraints: dictionary of state constraints
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:param InputConstraints: dictionary of input constraints
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:param Reference: reference values for state variables
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"""
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# Parameters
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self.T = T # horizon
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self.N = N # horizon
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self.Q = Q # weight matrix state vector
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self.R = R # weight matrix input vector
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self.Qf = Qf # weight matrix terminal
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self.QN = QN # weight matrix terminal
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# Model
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self.model = model
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@ -34,11 +35,7 @@ class MPC:
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self.state_constraints = StateConstraints
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self.input_constraints = InputConstraints
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# Reference
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self.reference = Reference
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# Current control and prediction
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self.current_control = None
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self.current_prediction = None
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# Initialize Optimization Problem
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@ -50,111 +47,71 @@ class MPC:
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time step.
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"""
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# Instantiate optimization variables
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self.x = cp.Variable((self.model.n_states+1, self.T + 1))
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self.u = cp.Variable((2, self.T))
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# number of input and state variables
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nx = self.model.n_states
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nu = 1
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# Instantiate optimization parameters
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self.kappa = cp.Parameter(self.T+1)
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self.x_0 = cp.Parameter(self.model.n_states+1, 1)
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self.A = cp.Parameter(self.model.A.shape)
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self.B = cp.Parameter(self.model.B.shape)
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# system matrices
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self.A = cp.Parameter(shape=(nx, nx*self.N))
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self.B = cp.Parameter(shape=(nx, nu*self.N))
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self.A.value = np.zeros(self.A.shape)
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self.B.value = np.zeros(self.B.shape)
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# Initialize cost
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cost = 0
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# reference values
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xr = np.array([0., 0., -1.0])
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self.ur = cp.Parameter((nu, self.N))
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self.ur.value = np.zeros(self.ur.shape)
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# Initialize constraints
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constraints = [self.x[:, 0] == self.x_0]
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# constraints
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umin = self.input_constraints['umin']
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umax = self.input_constraints['umax']
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xmin = self.state_constraints['xmin']
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xmax = self.state_constraints['xmax']
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for t in range(self.T):
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# initial state
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self.x_init = cp.Parameter(self.model.n_states)
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# set dynamic constraints
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constraints += [self.x[:-1, t + 1] == self.A[:-1, :]
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@ self.x[:, t] + self.B[:-1, :] @ self.u[:, t],
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self.x[-1, t + 1] == self.kappa[t+1]]
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# set input constraints
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inputs = ['D', 'delta']
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for input_name, constraint in self.input_constraints.items():
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input_id = inputs.index(input_name)
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if constraint[0] is not None:
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constraints.append(-self.u[input_id, t] <= -constraint[0])
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if constraint[1] is not None:
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constraints.append(self.u[input_id, t] <= constraint[1])
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# Set state constraints
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for state_name, constraint in self.state_constraints.items():
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state_id = self.model.spatial_state.list_states(). \
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index(state_name)
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if constraint[0] is not None:
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constraints.append(-self.x[state_id, t] <= -constraint[0])
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if constraint[1] is not None:
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constraints.append(self.x[state_id, t] <= constraint[1])
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# update cost function for states
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for state_name, state_reference in self.reference.items():
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state_id = self.model.spatial_state.list_states(). \
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index(state_name)
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cost += cp.norm(self.x[state_id, t] - state_reference, 2) * self.Q[
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state_id, state_id]
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# update cost function for inputs
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cost += cp.norm(self.u[0, t], 2) * self.R[0, 0]
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cost += cp.norm(self.u[1, t], 2) * self.R[1, 1]
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# set state constraints
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for state_name, constraint in self.state_constraints.items():
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state_id = self.model.spatial_state.list_states(). \
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index(state_name)
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if constraint[0] is not None:
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constraints.append(-self.x[state_id, self.T] <= -constraint[0])
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if constraint[1] is not None:
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constraints.append(self.x[state_id, self.T] <= constraint[1])
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# update cost function for states
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for state_name, state_reference in self.reference.items():
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state_id = self.model.spatial_state.list_states(). \
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index(state_name)
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cost += cp.norm(self.x[state_id, self.T] - state_reference, 2) * \
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self.Qf[state_id, state_id]
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# sums problem objectives and concatenates constraints.
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problem = cp.Problem(cp.Minimize(cost), constraints)
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# Define problem
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self.u = cp.Variable((nu, self.N))
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self.x = cp.Variable((nx, self.N + 1))
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objective = 0
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constraints = [self.x[:, 0] == self.x_init]
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for n in range(self.N):
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objective += cp.quad_form(self.x[:, n] - xr, self.Q) + cp.quad_form(self.u[:, n] - self.ur[:, n], self.R)
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constraints += [self.x[:, n + 1] == self.A[:, n*nx:n*nx+nx] * self.x[:, n]
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+ self.B[:, n*nu] * (self.u[:, n] - self.ur[:, n])]
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constraints += [umin <= self.u[:, n], self.u[:, n] <= umax]
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objective += cp.quad_form(self.x[:, self.N] - xr, self.QN)
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constraints += [xmin <= self.x[:, self.N], self.x[:, self.N] <= xmax]
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problem = cp.Problem(cp.Minimize(objective), constraints)
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return problem
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def get_control(self):
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def get_control(self, v):
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"""
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Get control signal given the current position of the car. Solves a
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finite time optimization problem based on the linearized car model.
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"""
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# get current waypoint curvature
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kappa_ref = [wp.kappa for wp in self.model.reference_path.waypoints
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[self.model.wp_id:self.model.wp_id+self.T+1]]
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nx = self.model.n_states
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nu = 1
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# Instantiate optimization parameters
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self.kappa.value = kappa_ref
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self.x_0.value = np.array(self.model.spatial_state[:] + [kappa_ref[0]])
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self.A.value = self.model.A
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self.B.value = self.model.B
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for n in range(self.N):
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current_waypoint = self.model.reference_path.waypoints[self.model.wp_id+n]
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next_waypoint = self.model.reference_path.waypoints[
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self.model.wp_id + n + 1]
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delta_s = next_waypoint - current_waypoint
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kappa_r = current_waypoint.kappa
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self.A.value[:, n*nx:n*nx+nx], self.B.value[:, n*nu:n*nu+nu] = self.model.linearize(v, kappa_r, delta_s)
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self.ur.value[:, n] = kappa_r
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# Solve optimization problem
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self.problem.solve(solver=cp.ECOS, warm_start=True)
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self.x_init.value = np.array(self.model.spatial_state[:])
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self.problem.solve(solver=cp.OSQP, verbose=True)
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# Store computed control signals and associated prediction
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try:
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self.current_control = self.u.value
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self.current_prediction = self.update_prediction(self.x.value)
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except TypeError:
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print('No solution found!')
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exit(1)
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self.current_prediction = self.update_prediction(self.x.value)
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delta = np.arctan(self.u.value[0, 0] * self.model.l)
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# RCH - get first control signal
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D_0 = self.u.value[0, 0]
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delta_0 = self.u.value[1, 0]
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return D_0, delta_0
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return delta
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def update_prediction(self, spatial_state_prediction):
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"""
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@ -168,12 +125,180 @@ class MPC:
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x_pred, y_pred = [], []
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# get current waypoint ID
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wp_id_ = np.copy(self.model.wp_id)
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print('#########################')
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for t in range(self.T):
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associated_waypoint = self.model.reference_path.waypoints[wp_id_+t]
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for n in range(self.N):
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associated_waypoint = self.model.reference_path.waypoints[self.model.wp_id+n]
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predicted_temporal_state = self.model.s2t(associated_waypoint,
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spatial_state_prediction[:, t])
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spatial_state_prediction[:, n])
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print('delta: ', np.arctan(self.u.value[0, n] * self.model.l))
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print('e_y: ', spatial_state_prediction[0, n])
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print('e_psi: ', spatial_state_prediction[1, n])
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print('t: ', spatial_state_prediction[2, n])
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print('+++++++++++++++++++++++')
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x_pred.append(predicted_temporal_state.x)
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y_pred.append(predicted_temporal_state.y)
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return x_pred, y_pred
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class MPC_OSQP:
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def __init__(self, model, N, Q, R, QN, StateConstraints, InputConstraints):
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"""
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Constructor for the Model Predictive Controller.
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:param model: bicycle model object to be controlled
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:param T: time horizon | int
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:param Q: state cost matrix
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:param R: input cost matrix
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:param QN: final state cost matrix
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:param StateConstraints: dictionary of state constraints
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:param InputConstraints: dictionary of input constraints
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:param Reference: reference values for state variables
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"""
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# Parameters
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self.N = N # horizon
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self.Q = Q # weight matrix state vector
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self.R = R # weight matrix input vector
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self.QN = QN # weight matrix terminal
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# Model
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self.model = model
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# Constraints
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self.state_constraints = StateConstraints
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self.input_constraints = InputConstraints
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# Current control and prediction
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self.current_prediction = None
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# Initialize Optimization Problem
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self.optimizer = osqp.OSQP()
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def _init_problem(self, v):
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"""
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Initialize optimization problem for current time step.
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"""
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# Number of state and input variables
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nx = self.model.n_states
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nu = 1
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# Constraints
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umin = self.input_constraints['umin']
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umax = self.input_constraints['umax']
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xmin = self.state_constraints['xmin']
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xmax = self.state_constraints['xmax']
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# LTV System Matrices
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A = np.zeros((nx * (self.N + 1), nx * (self.N + 1)))
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B = np.zeros((nx * (self.N + 1), nu * (self.N)))
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# Reference vector for state and input variables
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ur = np.zeros(self.N)
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xr = np.array([0.0, 0.0, -1.0])
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# Offset for equality constraint (due to B * (u - ur))
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uq = np.zeros(self.N * nx)
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# Iterate over horizon
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for n in range(self.N):
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# Get information about current waypoint
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current_waypoint = self.model.reference_path.waypoints[
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self.model.wp_id + n]
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next_waypoint = self.model.reference_path.waypoints[
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self.model.wp_id + n + 1]
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delta_s = next_waypoint - current_waypoint
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kappa_r = current_waypoint.kappa
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# Compute LTV matrices
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A_lin, B_lin = self.model.linearize(v, kappa_r, delta_s)
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A[nx + n * nx:n * nx + 2 * nx, n * nx:n * nx + nx] = A_lin
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B[nx + n * nx:n * nx + 2 * nx, n * nu:n * nu + nu] = B_lin
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# Set kappa_r to reference for input signal
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ur[n] = kappa_r
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# Compute equality constraint offset (B*ur)
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uq[n * nx:n * nx + nx] = B_lin[:, 0] * kappa_r
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# Get equality matrix
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Ax = sparse.kron(sparse.eye(self.N + 1),
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-sparse.eye(nx)) + sparse.csc_matrix(A)
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Bu = sparse.csc_matrix(B)
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Aeq = sparse.hstack([Ax, Bu])
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# Get inequality matrix
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Aineq = sparse.eye((self.N + 1) * nx + self.N * nu)
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# Combine constraint matrices
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A = sparse.vstack([Aeq, Aineq], format='csc')
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# Get upper and lower bound vectors for equality constraints
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lineq = np.hstack([np.kron(np.ones(self.N + 1), xmin),
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np.kron(np.ones(self.N), umin)])
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uineq = np.hstack([np.kron(np.ones(self.N + 1), xmax),
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np.kron(np.ones(self.N), umax)])
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# Get upper and lower bound vectors for inequality constraints
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x0 = np.array(self.model.spatial_state[:])
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leq = np.hstack([-x0, uq])
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ueq = leq
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# Combine upper and lower bound vectors
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l = np.hstack([leq, lineq])
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u = np.hstack([ueq, uineq])
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# Set cost matrices
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P = sparse.block_diag([sparse.kron(sparse.eye(self.N), self.Q), self.QN,
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sparse.kron(sparse.eye(self.N), self.R)], format='csc')
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q = np.hstack(
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[np.kron(np.ones(self.N), -self.Q.dot(xr)), -self.QN.dot(xr),
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-self.R.A[0, 0] * ur])
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# Initialize optimizer
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self.optimizer = osqp.OSQP()
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self.optimizer.setup(P=P, q=q, A=A, l=l, u=u, verbose=False)
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def get_control(self, v):
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"""
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Get control signal given the current position of the car. Solves a
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finite time optimization problem based on the linearized car model.
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"""
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# Number of state variables
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nx = self.model.n_states
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# Initialize optimization problem
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self._init_problem(v)
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# Solve optimization problem
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dec = self.optimizer.solve()
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x = np.reshape(dec.x[:(self.N+1)*nx], (self.N+1, nx))
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u = np.arctan(dec.x[-self.N] * self.model.l)
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self.current_prediction = self.update_prediction(u, x)
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return u
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def update_prediction(self, u, spatial_state_prediction):
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"""
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Transform the predicted states to predicted x and y coordinates.
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Mainly for visualization purposes.
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:param spatial_state_prediction: list of predicted state variables
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:return: lists of predicted x and y coordinates
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"""
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# containers for x and y coordinates of predicted states
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x_pred, y_pred = [], []
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# get current waypoint ID
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print('#########################')
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for n in range(self.N):
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associated_waypoint = self.model.reference_path.waypoints[self.model.wp_id+n]
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predicted_temporal_state = self.model.s2t(associated_waypoint,
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spatial_state_prediction[n, :])
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print('delta: ', u)
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print('e_y: ', spatial_state_prediction[n, 0])
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print('e_psi: ', spatial_state_prediction[n, 1])
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print('t: ', spatial_state_prediction[n, 2])
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print('+++++++++++++++++++++++')
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x_pred.append(predicted_temporal_state.x)
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y_pred.append(predicted_temporal_state.y)
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@ -1,17 +1,17 @@
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from map import Map
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import numpy as np
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from reference_path import ReferencePath
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from spatial_bicycle_models import SimpleBicycleModel, ExtendedBicycleModel
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from spatial_bicycle_models import BicycleModel
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import matplotlib.pyplot as plt
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from MPC import MPC
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from time import time
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from MPC import MPC, MPC_OSQP
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from scipy import sparse
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import time
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if __name__ == '__main__':
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# Select Simulation Mode | 'Race' or 'Q'
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sim_mode = 'Race'
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# Select Model Type | 'Simple' or 'Extended'
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model_type = 'Simple'
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# Create Map
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if sim_mode == 'Race':
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@ -45,46 +45,23 @@ if __name__ == '__main__':
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# Initial state
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e_y_0 = 0.0
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e_psi_0 = 0.0
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v_x_0 = 0.1
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v_y_0 = 0
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omega_0 = 0
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t_0 = 0
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t_0 = 0.0
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v = 1.0
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if model_type == 'Extended':
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car = ExtendedBicycleModel(reference_path=reference_path,
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e_y=e_y_0, e_psi=e_psi_0, v_x=v_x_0, v_y=v_y_0,
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omega=omega_0, t=t_0)
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elif model_type == 'Simple':
|
||||
car = SimpleBicycleModel(reference_path=reference_path,
|
||||
e_y=e_y_0, e_psi=e_psi_0, v=v_x_0)
|
||||
else:
|
||||
car = None
|
||||
print('Invalid Model Type!')
|
||||
exit(1)
|
||||
car = BicycleModel(reference_path=reference_path,
|
||||
e_y=e_y_0, e_psi=e_psi_0, t=t_0)
|
||||
|
||||
##############
|
||||
# Controller #
|
||||
##############
|
||||
|
||||
if model_type == 'Extended':
|
||||
Q = np.diag([1, 0, 0, 0, 0, 0])
|
||||
Qf = Q
|
||||
R = np.diag([0, 0])
|
||||
Reference = {'e_y': 0, 'e_psi': 0, 'v_x': 1.0, 'v_y': 0, 'omega': 0, 't':0}
|
||||
elif model_type == 'Simple':
|
||||
Reference = {'e_y': 0, 'e_psi': 0, 'v': 4.0}
|
||||
Q = np.diag([0.0005, 0.05, 0.5])
|
||||
Qf = Q
|
||||
R = np.diag([0, 0])
|
||||
else:
|
||||
Q, Qf, R, Reference = None, None, None, None
|
||||
print('Invalid Model Type!')
|
||||
exit(1)
|
||||
|
||||
T = 5
|
||||
StateConstraints = {'e_y': (-0.2, 0.2), 'v': (0, 4)}
|
||||
InputConstraints = {'D': (-1, 1), 'delta': (-0.44, 0.44)}
|
||||
mpc = MPC(car, T, Q, R, Qf, StateConstraints, InputConstraints, Reference)
|
||||
N = 20
|
||||
Q = sparse.diags([0.01, 0.0, 0.4])
|
||||
R = sparse.diags([0.01])
|
||||
QN = Q
|
||||
InputConstraints = {'umin': np.array([-np.tan(0.44)/car.l]), 'umax': np.array([np.tan(0.44)/car.l])}
|
||||
StateConstraints = {'xmin': np.array([-0.2, -np.inf, 0]), 'xmax': np.array([0.2, np.inf, np.inf])}
|
||||
mpc = MPC_OSQP(car, N, Q, R, QN, StateConstraints, InputConstraints)
|
||||
|
||||
##############
|
||||
# Simulation #
|
||||
|
|
@ -93,26 +70,22 @@ if __name__ == '__main__':
|
|||
# logging containers
|
||||
x_log = [car.temporal_state.x]
|
||||
y_log = [car.temporal_state.y]
|
||||
psi_log = [car.temporal_state.psi]
|
||||
v_log = [car.temporal_state.v_x]
|
||||
D_log = []
|
||||
delta_log = []
|
||||
|
||||
# iterate over waypoints
|
||||
for wp_id in range(len(car.reference_path.waypoints)-T-1):
|
||||
for wp_id in range(len(car.reference_path.waypoints)-mpc.N-1):
|
||||
|
||||
# get control signals
|
||||
D, delta = mpc.get_control()
|
||||
start = time.time()
|
||||
delta = mpc.get_control(v)
|
||||
end = time.time()
|
||||
u = np.array([v, delta])
|
||||
|
||||
# drive car
|
||||
car.drive(D, delta)
|
||||
car.drive(u)
|
||||
|
||||
# log current state
|
||||
# log
|
||||
x_log.append(car.temporal_state.x)
|
||||
y_log.append(car.temporal_state.y)
|
||||
v_log.append(car.temporal_state.v_x)
|
||||
D_log.append(D)
|
||||
delta_log.append(delta)
|
||||
|
||||
###################
|
||||
# Plot Simulation #
|
||||
|
|
@ -121,19 +94,13 @@ if __name__ == '__main__':
|
|||
car.reference_path.show()
|
||||
|
||||
# plot car trajectory and velocity
|
||||
plt.scatter(x_log, y_log, c='g', s=15)
|
||||
plt.scatter(x_log[:-1], y_log[:-1], c='g', s=15)
|
||||
|
||||
# plot mpc prediction
|
||||
if mpc.current_prediction is not None:
|
||||
x_pred = mpc.current_prediction[0]
|
||||
y_pred = mpc.current_prediction[1]
|
||||
plt.scatter(x_pred, y_pred, c='b', s=10)
|
||||
plt.scatter(mpc.current_prediction[0], mpc.current_prediction[1], c='b', s=5)
|
||||
|
||||
plt.title('MPC Simulation: Position: {:.2f} m, {:.2f} m, Velocity: '
|
||||
'{:.2f} m/s'.format(car.temporal_state.x,
|
||||
car.temporal_state.y, car.temporal_state.v_x))
|
||||
plt.title('MPC Simulation: Position: {:.2f} m, {:.2f} m'.
|
||||
format(car.temporal_state.x, car.temporal_state.y))
|
||||
plt.xticks([])
|
||||
plt.yticks([])
|
||||
plt.pause(0.0000001)
|
||||
|
||||
plt.pause(0.00000001)
|
||||
plt.close()
|
||||
|
|
|
|||
|
|
@ -7,7 +7,7 @@ from abc import ABC, abstractmethod
|
|||
#########################
|
||||
|
||||
class TemporalState:
|
||||
def __init__(self, x, y, psi, v_x, v_y):
|
||||
def __init__(self, x, y, psi):
|
||||
"""
|
||||
Temporal State Vector containing car pose (x, y, psi) and velocity
|
||||
:param x: x position in global coordinate system | [m]
|
||||
|
|
@ -19,8 +19,6 @@ class TemporalState:
|
|||
self.x = x
|
||||
self.y = y
|
||||
self.psi = psi
|
||||
self.v_x = v_x
|
||||
self.v_y = v_y
|
||||
|
||||
|
||||
########################
|
||||
|
|
@ -39,6 +37,9 @@ class SpatialState(ABC):
|
|||
def __getitem__(self, item):
|
||||
return list(vars(self).values())[item]
|
||||
|
||||
def __setitem__(self, key, value):
|
||||
vars(self)[list(vars(self).keys())[key]] = value
|
||||
|
||||
def __len__(self):
|
||||
return len(vars(self))
|
||||
|
||||
|
|
@ -60,40 +61,18 @@ class SpatialState(ABC):
|
|||
|
||||
|
||||
class SimpleSpatialState(SpatialState):
|
||||
def __init__(self, e_y, e_psi, v):
|
||||
def __init__(self, e_y, e_psi, t):
|
||||
"""
|
||||
Simplified Spatial State Vector containing orthogonal deviation from
|
||||
reference path (e_y), difference in orientation (e_psi) and velocity
|
||||
:param e_y: orthogonal deviation from center-line | [m]
|
||||
:param e_psi: yaw angle relative to path | [rad]
|
||||
:param v: absolute velocity | [m/s]
|
||||
:param t: time | [s]
|
||||
"""
|
||||
super(SimpleSpatialState, self).__init__()
|
||||
|
||||
self.e_y = e_y
|
||||
self.e_psi = e_psi
|
||||
self.v = v
|
||||
|
||||
|
||||
class ExtendedSpatialState(SpatialState):
|
||||
def __init__(self, e_y, e_psi, v_x, v_y, omega, t):
|
||||
"""
|
||||
Extended Spatial State Vector containing separate velocities in x and
|
||||
y direction, angular velocity and time
|
||||
:param e_y: orthogonal deviation from center-line | [m]
|
||||
:param e_psi: yaw angle relative to path | [rad]
|
||||
:param v_x: velocity in x direction (car frame) | [m/s]
|
||||
:param v_y: velocity in y direction (car frame) | [m/s]
|
||||
:param omega: anglular velocity of the car | [rad/s]
|
||||
:param t: simulation time| [s]
|
||||
"""
|
||||
super(ExtendedSpatialState, self).__init__()
|
||||
|
||||
self.e_y = e_y
|
||||
self.e_psi = e_psi
|
||||
self.v_x = v_x
|
||||
self.v_y = v_y
|
||||
self.omega = omega
|
||||
self.t = t
|
||||
|
||||
|
||||
|
|
@ -161,45 +140,60 @@ class SpatialBicycleModel(ABC):
|
|||
|
||||
return x, y, psi
|
||||
|
||||
def drive(self, D, delta):
|
||||
def drive(self, input, state=None, kappa=None, delta_s=None):
|
||||
"""
|
||||
Update states of spatial bicycle model. Model drive to the next
|
||||
waypoint on the reference path.
|
||||
:param D: acceleration command | [-1, 1]
|
||||
:param delta: angular velocity | [rad]
|
||||
Drive.
|
||||
:param state: state vector for which to compute derivatives
|
||||
:param input: input vector
|
||||
:param kappa: curvature of corresponding waypoint
|
||||
:return: numpy array with spatial derivatives for all state variables
|
||||
"""
|
||||
|
||||
# Get spatial derivatives
|
||||
spatial_derivatives = self.get_spatial_derivatives(D, delta)
|
||||
if state is None and kappa is None and delta_s is None:
|
||||
state = np.array(self.spatial_state[:])
|
||||
# Get delta_s | distance to next waypoint
|
||||
next_waypoint = self.reference_path.waypoints[self.wp_id + 1]
|
||||
delta_s = next_waypoint - self.current_waypoint
|
||||
# Get current curvature
|
||||
kappa = self.current_waypoint.kappa
|
||||
|
||||
# Get delta_s | distance to next waypoint
|
||||
next_waypoint = self.reference_path.waypoints[self.wp_id+1]
|
||||
delta_s = next_waypoint - self.current_waypoint
|
||||
spatial_derivatives = self.get_spatial_derivatives(state, input, kappa)
|
||||
|
||||
# Update spatial state (Forward Euler Approximation)
|
||||
self.spatial_state += spatial_derivatives * delta_s
|
||||
# Update spatial state (Forward Euler Approximation)
|
||||
self.spatial_state += spatial_derivatives * delta_s
|
||||
|
||||
# Assert that unique projections of car pose onto path exists
|
||||
assert self.spatial_state.e_y < (1 / (self.current_waypoint.kappa +
|
||||
self.eps))
|
||||
# Assert that unique projections of car pose onto path exists
|
||||
#assert self.spatial_state.e_y < (1 / (self.current_waypoint.kappa +
|
||||
# self.eps))
|
||||
|
||||
# Increase waypoint ID
|
||||
self.wp_id += 1
|
||||
# Increase waypoint ID
|
||||
self.wp_id += 1
|
||||
|
||||
# Update current waypoint
|
||||
self.current_waypoint = self.reference_path.waypoints[self.wp_id]
|
||||
# Update current waypoint
|
||||
self.current_waypoint = self.reference_path.waypoints[self.wp_id]
|
||||
|
||||
# Update temporal_state to match spatial state
|
||||
self.temporal_state = self.s2t()
|
||||
# Update temporal_state to match spatial state
|
||||
self.temporal_state = self.s2t()
|
||||
|
||||
# Update s | total driven distance along path
|
||||
self.s += delta_s
|
||||
# Update s | total driven distance along path
|
||||
self.s += delta_s
|
||||
|
||||
# Linearize model around new operating point
|
||||
self.A, self.B = self.linearize()
|
||||
# Linearize model around new operating point
|
||||
# self.A, self.B = self.linearize()
|
||||
|
||||
else:
|
||||
|
||||
spatial_derivatives = self.get_spatial_derivatives(state, input,
|
||||
kappa)
|
||||
|
||||
# Update spatial state (Forward Euler Approximation)
|
||||
state += spatial_derivatives * delta_s
|
||||
|
||||
return state
|
||||
|
||||
@abstractmethod
|
||||
def get_spatial_derivatives(self, D, delta):
|
||||
def get_spatial_derivatives(self, state, input, kappa):
|
||||
pass
|
||||
|
||||
@abstractmethod
|
||||
|
|
@ -207,12 +201,12 @@ class SpatialBicycleModel(ABC):
|
|||
pass
|
||||
|
||||
|
||||
########################
|
||||
# Simple Bicycle Model #
|
||||
########################
|
||||
#################
|
||||
# Bicycle Model #
|
||||
#################
|
||||
|
||||
class SimpleBicycleModel(SpatialBicycleModel):
|
||||
def __init__(self, reference_path, e_y, e_psi, v):
|
||||
class BicycleModel(SpatialBicycleModel):
|
||||
def __init__(self, reference_path, e_y, e_psi, t):
|
||||
"""
|
||||
Simplified Spatial Bicycle Model. Spatial Reformulation of Kinematic
|
||||
Bicycle Model. Uses Simplified Spatial State.
|
||||
|
|
@ -223,24 +217,22 @@ class SimpleBicycleModel(SpatialBicycleModel):
|
|||
"""
|
||||
|
||||
# Initialize base class
|
||||
super(SimpleBicycleModel, self).__init__(reference_path)
|
||||
super(BicycleModel, self).__init__(reference_path)
|
||||
|
||||
# Constants
|
||||
self.C1 = 0.5
|
||||
self.C2 = 17.06
|
||||
self.Cm1 = 12.0
|
||||
self.Cm2 = 2.17
|
||||
self.Cr2 = 0.1
|
||||
self.Cr0 = 0.6
|
||||
self.l = 0.06
|
||||
|
||||
# Initialize spatial state
|
||||
self.spatial_state = SimpleSpatialState(e_y, e_psi, v)
|
||||
self.spatial_state = SimpleSpatialState(e_y, e_psi, t)
|
||||
|
||||
# Number of spatial state variables
|
||||
self.n_states = len(self.spatial_state)
|
||||
|
||||
# Initialize temporal state
|
||||
self.temporal_state = self.s2t()
|
||||
|
||||
# Compute linear system matrices | Used for MPC
|
||||
self.A, self.B = self.linearize()
|
||||
# self.A, self.B = self.linearize()
|
||||
|
||||
def s2t(self, reference_waypoint=None, reference_state=None):
|
||||
"""
|
||||
|
|
@ -253,547 +245,85 @@ class SimpleBicycleModel(SpatialBicycleModel):
|
|||
|
||||
if reference_state is None and reference_waypoint is None:
|
||||
# Get pose information from base class implementation
|
||||
x, y, psi = super(SimpleBicycleModel, self).s2t()
|
||||
x, y, psi = super(BicycleModel, self).s2t()
|
||||
# Compute simplified velocities
|
||||
v_x = self.spatial_state.v
|
||||
v_y = 0
|
||||
else:
|
||||
# Get pose information from base class implementation
|
||||
x, y, psi = super(SimpleBicycleModel, self).s2t(reference_waypoint,
|
||||
x, y, psi = super(BicycleModel, self).s2t(reference_waypoint,
|
||||
reference_state)
|
||||
v_x = reference_state[2]
|
||||
v_y = 0
|
||||
|
||||
return TemporalState(x, y, psi, v_x, v_y)
|
||||
return TemporalState(x, y, psi)
|
||||
|
||||
def get_temporal_derivatives(self, D, delta):
|
||||
def get_temporal_derivatives(self, state, input, kappa):
|
||||
"""
|
||||
Compute relevant temporal derivatives needed for state update.
|
||||
:param D: duty-cycle of DC motor | [-1, 1]
|
||||
:param delta: steering command | [rad]
|
||||
:param state: state vector for which to compute derivatives
|
||||
:param input: input vector
|
||||
:param kappa: curvature of corresponding waypoint
|
||||
:return: temporal derivatives of distance, angle and velocity
|
||||
"""
|
||||
|
||||
# Compute velocity components | Approximation for small delta
|
||||
v_x = self.spatial_state.v
|
||||
v_y = self.spatial_state.v * delta * self.C1
|
||||
|
||||
# Compute velocity along waypoint direction
|
||||
v_sigma = v_x * np.cos(self.spatial_state.e_psi) - v_y * np.sin(
|
||||
self.spatial_state.e_psi)
|
||||
e_y, e_psi, t = state
|
||||
v, delta = input
|
||||
|
||||
# Compute velocity along path
|
||||
s_dot = 1 / (1 - (self.spatial_state.e_y * self.current_waypoint.kappa)) * v_sigma
|
||||
s_dot = 1 / (1 - (e_y * kappa)) * v * np.cos(e_psi)
|
||||
|
||||
# Compute yaw angle rate of change
|
||||
psi_dot = self.spatial_state.v * delta * self.C2
|
||||
psi_dot = v / self.l * np.tan(delta)
|
||||
|
||||
# Compute acceleration
|
||||
v_dot = (self.Cm1 - self.Cm2 * self.spatial_state.v) * D - self.Cr2 * (
|
||||
self.spatial_state.v ** 2) - self.Cr0 - (
|
||||
self.spatial_state.v * delta) ** 2 * self.C2 * self.C1 ** 2
|
||||
return s_dot, psi_dot
|
||||
|
||||
return s_dot, psi_dot, v_dot
|
||||
|
||||
def get_spatial_derivatives(self, D, delta):
|
||||
def get_spatial_derivatives(self, state, input, kappa):
|
||||
"""
|
||||
Compute spatial derivatives of all state variables for update.
|
||||
:param D: duty-cycle of DC motor | [-1, 1]
|
||||
:param delta: steering angle | [rad]
|
||||
:param state: state vector for which to compute derivatives
|
||||
:param input: input vector
|
||||
:param kappa: curvature of corresponding waypoint
|
||||
:return: numpy array with spatial derivatives for all state variables
|
||||
"""
|
||||
|
||||
e_y, e_psi, t = state
|
||||
v, delta = input
|
||||
|
||||
# Compute temporal derivatives
|
||||
s_dot, psi_dot, v_dot = self.get_temporal_derivatives(D, delta)
|
||||
s_dot, psi_dot = self.get_temporal_derivatives(state, input, kappa)
|
||||
|
||||
# Compute spatial derivatives
|
||||
d_e_y_d_s = (self.spatial_state.v * np.sin(self.spatial_state.e_psi)
|
||||
+ self.spatial_state.v * delta * self.C1 * np.cos(
|
||||
self.spatial_state.e_psi)) / s_dot
|
||||
d_e_psi_d_s = psi_dot / s_dot - self.current_waypoint.kappa
|
||||
d_v_d_s = v_dot / s_dot
|
||||
d_e_y_d_s = v * np.sin(e_psi) / s_dot
|
||||
d_e_psi_d_s = psi_dot / s_dot - kappa
|
||||
d_t_d_s = 1 / s_dot
|
||||
|
||||
return np.array([d_e_y_d_s, d_e_psi_d_s, d_v_d_s])
|
||||
return np.array([d_e_y_d_s, d_e_psi_d_s, d_t_d_s])
|
||||
|
||||
def linearize(self, D=0, delta=0):
|
||||
def linearize(self, v=None, kappa_r=None, delta_s=None):
|
||||
"""
|
||||
Linearize the system equations around the current state and waypoint.
|
||||
:param delta: reference steering angle | [rad]
|
||||
:param D: reference duty-cycle of DC-motor | [-1, 1]
|
||||
"""
|
||||
:param kappa_r: kappa of waypoint around which to linearize
|
||||
"""
|
||||
|
||||
# Get current state | operating point to linearize around
|
||||
e_y = self.spatial_state.e_y
|
||||
e_psi = self.spatial_state.e_psi
|
||||
v = self.spatial_state.v
|
||||
|
||||
# Get curvature of current waypoint
|
||||
kappa = self.reference_path.waypoints[self.wp_id].kappa
|
||||
|
||||
# Get delta_s
|
||||
next_waypoint = self.reference_path.waypoints[self.wp_id+1]
|
||||
delta_s = next_waypoint - self.current_waypoint
|
||||
|
||||
##############################
|
||||
# Helper Partial Derivatives #
|
||||
##############################
|
||||
|
||||
# Compute velocity components
|
||||
v_x = v
|
||||
v_y = v * delta * self.C1
|
||||
|
||||
# Compute partial derivatives of s_dot w.r.t. each state variable,
|
||||
# input variable and kappa
|
||||
s_dot = 1 / (1 - e_y*kappa) * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi))
|
||||
d_s_dot_d_e_y = kappa / (1-e_y*kappa)**2 * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi))
|
||||
d_s_dot_d_e_psi = 1 / (1 - e_y*kappa) * (-v_x * np.sin(e_psi) - v_y * np.cos(e_psi))
|
||||
d_s_dot_d_v = 1 / (1 - e_y*kappa) * (np.cos(e_psi) - delta * self.C1 * np.sin(e_psi))
|
||||
# d_s_dot_d_D = 0
|
||||
d_s_dot_d_delta = 1 / (1 - e_y*kappa) * (- v * self.C1 * np.sin(e_psi))
|
||||
d_s_dot_d_kappa = e_y / (1-e_y*kappa)**2 * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi))
|
||||
|
||||
# Compute partial derivatives of v_psi w.r.t. each state variable,
|
||||
# input variable and kappa
|
||||
v_psi = (v_x * np.sin(e_psi) + v_y * np.cos(e_psi))
|
||||
# d_v_psi_d_e_y = 0
|
||||
d_v_psi_d_e_psi = v_x * np.cos(e_psi) - v_y * np.sin(e_psi)
|
||||
d_v_psi_d_v = np.sin(e_psi) + self.C1 * delta * np.cos(e_psi)
|
||||
# d_v_psi_d_D = 0
|
||||
d_v_psi_d_delta = self.C1 * v * np.cos(e_psi)
|
||||
# d_v_psi_d_kappa = 0
|
||||
|
||||
# Compute partial derivatives of psi_dot w.r.t. each state variable,
|
||||
# input variable and kappa
|
||||
psi_dot = v * delta * self.C2
|
||||
# d_psi_dot_d_e_y = 0
|
||||
# d_psi_dot_d_e_psi = 0
|
||||
d_psi_dot_d_v = delta * self.C2
|
||||
# d_psi_dot_d_D = 0
|
||||
d_psi_dot_d_delta = v * self.C2
|
||||
# d_psi_dot_d_kappa = 0
|
||||
|
||||
# Compute partial derivatives of v_dot w.r.t. each state variable,
|
||||
# input variable and kappa
|
||||
v_dot = (self.Cm1 - self.Cm2 * v) * D - self.Cr2 * (v ** 2) - self.Cr0 \
|
||||
- (v * delta) ** 2 * self.C2 * (self.C1 ** 2)
|
||||
# d_v_dot_d_e_y = 0
|
||||
# d_v_dot_d_e_psi = 0
|
||||
d_v_dot_d_v = -self.Cm2 * D - 2 * self.Cr2 * v - 2 * v * (delta ** 2) \
|
||||
* self.C2 * (self.C1 ** 2)
|
||||
d_v_dot_d_D = self.Cm1 - self.Cm2 * v
|
||||
d_v_dot_d_delta = -2 * (v ** 2) * delta * self.C2 * self.C1 ** 2
|
||||
# d_v_dot_d_kappa = 0
|
||||
|
||||
#############################
|
||||
# State Partial Derivatives #
|
||||
#############################
|
||||
|
||||
# Use pre-computed helper derivatives to compute spatial derivatives of
|
||||
# all state variables using Quotient Rule
|
||||
|
||||
# Compute partial derivatives of e_y w.r.t. each state variable,
|
||||
# input variable and kappa
|
||||
# e_y = v_psi / s_dot
|
||||
d_e_y_d_e_y = - d_s_dot_d_e_y * v_psi / (s_dot**2)
|
||||
d_e_y_d_e_psi = (d_v_psi_d_e_psi * s_dot - d_s_dot_d_e_psi * v_psi) / (s_dot**2)
|
||||
d_e_y_d_v = (d_v_psi_d_v * s_dot - d_s_dot_d_v * v_psi) / (s_dot**2)
|
||||
d_e_y_d_D = 0
|
||||
d_e_y_d_delta = (d_v_psi_d_delta * s_dot - d_s_dot_d_delta * v_psi) / (s_dot**2)
|
||||
d_e_y_d_kappa = - d_s_dot_d_kappa * v_psi / (s_dot**2)
|
||||
|
||||
# Compute partial derivatives of e_psi w.r.t. each state variable,
|
||||
# input variable and kappa
|
||||
# e_psi = psi_dot / s_dot - kappa
|
||||
d_e_psi_d_e_y = - d_s_dot_d_e_y * psi_dot / (s_dot**2)
|
||||
d_e_psi_d_e_psi = - d_s_dot_d_e_psi * psi_dot / (s_dot**2)
|
||||
d_e_psi_d_v = (d_psi_dot_d_v * s_dot - d_s_dot_d_v * psi_dot) / (s_dot**2)
|
||||
d_e_psi_d_D = 0
|
||||
d_e_psi_d_delta = (d_psi_dot_d_delta * s_dot - d_s_dot_d_delta * psi_dot) / (s_dot**2)
|
||||
d_e_psi_d_kappa = - d_s_dot_d_kappa * psi_dot / (s_dot**2) - 1
|
||||
|
||||
# Compute partial derivatives of v w.r.t. each state variable,
|
||||
# input variable and kappa
|
||||
# v = v_dot / s_dot
|
||||
d_v_d_e_y = - d_s_dot_d_e_y * v_dot / (s_dot**2)
|
||||
d_v_d_e_psi = - d_s_dot_d_e_psi * v_dot / (s_dot**2)
|
||||
d_v_d_v = (d_v_dot_d_v * s_dot - d_s_dot_d_v * v_dot) / (s_dot**2)
|
||||
d_v_d_D = d_v_dot_d_D * s_dot / (s_dot**2)
|
||||
d_v_d_delta = (d_v_dot_d_delta * s_dot - d_s_dot_d_delta * v_dot) / (s_dot**2)
|
||||
d_v_d_kappa = - d_s_dot_d_kappa * v_dot / (s_dot**2)
|
||||
|
||||
#############
|
||||
# Jacobians #
|
||||
#############
|
||||
|
||||
# Construct Jacobian Matrix
|
||||
a_1 = np.array([d_e_y_d_e_y, d_e_y_d_e_psi, d_e_y_d_v, d_e_y_d_kappa])
|
||||
a_2 = np.array([d_e_psi_d_e_y, d_e_psi_d_e_psi, d_e_psi_d_v, d_e_psi_d_kappa])
|
||||
a_3 = np.array([d_v_d_e_y, d_v_d_e_psi, d_v_d_v, d_v_d_kappa])
|
||||
|
||||
b_1 = np.array([d_e_y_d_D, d_e_y_d_delta])
|
||||
b_2 = np.array([d_e_psi_d_D, d_e_psi_d_delta])
|
||||
b_3 = np.array([d_v_d_D, d_v_d_delta])
|
||||
|
||||
# Add extra row for kappa | Allows for updating kappa during MPC
|
||||
# optimization
|
||||
a_4 = np.array([0, 0, 0, 0])
|
||||
b_4 = np.array([0, 0])
|
||||
|
||||
Ja = np.stack((a_1, a_2, a_3, a_4), axis=0)
|
||||
Jb = np.stack((b_1, b_2, b_3, b_4), axis=0)
|
||||
# Get linearization state
|
||||
if kappa_r is None and delta_s is None:
|
||||
# Get curvature of linearization waypoint
|
||||
kappa_r = self.reference_path.waypoints[self.wp_id].kappa
|
||||
# Get delta_s
|
||||
next_waypoint = self.reference_path.waypoints[self.wp_id + 1]
|
||||
delta_s = next_waypoint - self.current_waypoint
|
||||
|
||||
###################
|
||||
# System Matrices #
|
||||
###################
|
||||
|
||||
# Construct system matrices from Jacobians. Multiply by sampling
|
||||
# distance delta_s + add identity matrix (Forward Euler Approximation)
|
||||
A = Ja * delta_s + np.identity(Ja.shape[1])
|
||||
B = Jb * delta_s
|
||||
# Construct Jacobian Matrix
|
||||
a_1 = np.array([1, delta_s, 0])
|
||||
a_2 = np.array([-kappa_r**2*delta_s, 1, 0])
|
||||
a_3 = np.array([-kappa_r/v*delta_s, 0, 0])
|
||||
|
||||
b_1 = np.array([0, ])
|
||||
b_2 = np.array([delta_s, ])
|
||||
b_3 = np.array([0, ])
|
||||
|
||||
A = np.stack((a_1, a_2, a_3), axis=0)
|
||||
B = np.stack((b_1, b_2, b_3), axis=0)
|
||||
|
||||
return A, B
|
||||
|
||||
|
||||
##########################
|
||||
# Extended Bicycle Model #
|
||||
##########################
|
||||
|
||||
class ExtendedBicycleModel(SpatialBicycleModel):
|
||||
def __init__(self, reference_path, e_y, e_psi, v_x, v_y, omega, t):
|
||||
"""
|
||||
Construct spatial bicycle model.
|
||||
:param e_y: initial deviation from reference path | [m]
|
||||
:param e_psi: initial heading offset from reference path | [rad]
|
||||
:param v: initial velocity | [m/s]
|
||||
:param reference_path: reference path model is supposed to follow
|
||||
"""
|
||||
super(ExtendedBicycleModel, self).__init__(reference_path)
|
||||
|
||||
# Constants
|
||||
self.m = 0.041
|
||||
self.Iz = 27.8e-6
|
||||
self.lf = 0.029
|
||||
self.lr = 0.033
|
||||
|
||||
self.Cm1 = 0.287
|
||||
self.Cm2 = 0.0545
|
||||
self.Cr2 = 0.0518
|
||||
self.Cr0 = 0.00035
|
||||
|
||||
self.Br = 3.3852
|
||||
self.Cr = 1.2691
|
||||
self.Dr = 0.1737
|
||||
self.Bf = 2.579
|
||||
self.Cf = 1.2
|
||||
self.Df = 0.192
|
||||
|
||||
# Spatial state
|
||||
self.spatial_state = ExtendedSpatialState(e_y, e_psi, v_x, v_y, omega, t)
|
||||
|
||||
# Temporal state
|
||||
self.temporal_state = self.s2t()
|
||||
|
||||
# Linear System Matrices
|
||||
self.A, self.B = self.linearize()
|
||||
|
||||
def s2t(self, reference_waypoint=None, predicted_state=None):
|
||||
"""
|
||||
Convert spatial state to temporal state
|
||||
:return temporal state equivalent to self.spatial_state
|
||||
"""
|
||||
|
||||
# compute velocity information
|
||||
if predicted_state is None and reference_waypoint is None:
|
||||
# get information from base class
|
||||
x, y, psi = super(ExtendedBicycleModel, self).s2t()
|
||||
v_x = self.spatial_state.v_x
|
||||
v_y = self.spatial_state.v_y
|
||||
else:
|
||||
# get information from base class
|
||||
x, y, psi = super(ExtendedBicycleModel, self).s2t(reference_waypoint,
|
||||
predicted_state)
|
||||
v_x = predicted_state[2]
|
||||
v_y = predicted_state[3]
|
||||
|
||||
return TemporalState(x, y, psi, v_x, v_y)
|
||||
|
||||
def get_forces(self, delta, D):
|
||||
"""
|
||||
Compute forces required for temporal derivatives of v_x and v_y
|
||||
:param delta:
|
||||
:param D:
|
||||
:return:
|
||||
"""
|
||||
|
||||
F_rx = (self.Cm1 - self.Cm2 * self.spatial_state.v_x) * D - self.Cr0 - self.Cr2 * self.spatial_state.v_x ** 2
|
||||
|
||||
alpha_f = - np.arctan2(self.spatial_state.omega*self.lf + self.spatial_state.v_y, self.spatial_state.v_x) + delta
|
||||
F_fy = self.Df * np.sin(self.Cf*np.arctan(self.Bf*alpha_f))
|
||||
|
||||
alpha_r = np.arctan2(self.spatial_state.omega*self.lr - self.spatial_state.v_y, self.spatial_state.v_x)
|
||||
F_ry = self.Dr * np.sin(self.Cr * np.arctan(self.Br*alpha_r))
|
||||
|
||||
return F_rx, F_fy, F_ry, alpha_f, alpha_r
|
||||
|
||||
def get_temporal_derivatives(self, delta, F_rx, F_fy, F_ry):
|
||||
"""
|
||||
Compute temporal derivatives needed for state update.
|
||||
:param delta: steering command
|
||||
:param D: duty-cycle of DC motor
|
||||
:return: temporal derivatives of distance, angle and velocity
|
||||
"""
|
||||
|
||||
# velocity along path
|
||||
s_dot = 1 / (1 - (self.spatial_state.e_y * self.current_waypoint.kappa)) \
|
||||
* (self.spatial_state.v_x * np.cos(self.spatial_state.e_psi)
|
||||
+ self.spatial_state.v_y * np.sin(self.spatial_state.e_psi))
|
||||
|
||||
# velocity in x and y direction
|
||||
v_x_dot = (F_rx - F_fy * np.sin(delta) + self.m * self.spatial_state.
|
||||
v_y * self.spatial_state.omega) / self.m
|
||||
v_y_dot = (F_ry + F_fy * np.cos(delta) - self.m * self.spatial_state.
|
||||
v_x * self.spatial_state.omega) / self.m
|
||||
|
||||
# omega dot
|
||||
omega_dot = (F_fy * self.lf * np.cos(delta) - F_ry * self.lr) / self.Iz
|
||||
|
||||
return s_dot, v_x_dot, v_y_dot, omega_dot
|
||||
|
||||
def get_spatial_derivatives(self, delta, D):
|
||||
"""
|
||||
Compute spatial derivatives of all state variables for update.
|
||||
:param delta: steering angle
|
||||
:param psi_dot: heading rate of change
|
||||
:param s_dot: velocity along path
|
||||
:param v_dot: acceleration
|
||||
:return: spatial derivatives for all state variables
|
||||
"""
|
||||
|
||||
# get required forces
|
||||
F_rx, F_fy, F_ry, _, _ = self.get_forces(delta, D)
|
||||
|
||||
# Compute state derivatives
|
||||
s_dot, v_x_dot, v_y_dot, omega_dot = \
|
||||
self.get_temporal_derivatives(delta, F_rx, F_fy, F_ry)
|
||||
|
||||
|
||||
d_e_y = (self.spatial_state.v_x * np.sin(self.spatial_state.e_psi)
|
||||
+ self.spatial_state.v_y * np.cos(self.spatial_state.e_psi)) \
|
||||
/ (s_dot + self.eps)
|
||||
d_e_psi = (self.spatial_state.omega / (s_dot + self.eps) - self.current_waypoint.kappa)
|
||||
|
||||
d_v_x = v_x_dot / (s_dot + self.eps)
|
||||
d_v_y = v_y_dot / (s_dot + self.eps)
|
||||
d_omega = omega_dot / (s_dot + self.eps)
|
||||
d_t = 1 / (s_dot + self.eps)
|
||||
|
||||
return np.array([d_e_y, d_e_psi, d_v_x, d_v_y, d_omega, d_t])
|
||||
|
||||
def linearize(self, delta=0, D=0):
|
||||
"""
|
||||
Linearize the system equations around the current state and waypoint.
|
||||
:param delta: reference steering angle
|
||||
:param D: reference dutycycle
|
||||
"""
|
||||
|
||||
# get current state
|
||||
e_y = self.spatial_state.e_y
|
||||
e_psi = self.spatial_state.e_psi
|
||||
v_x = self.spatial_state.v_x
|
||||
v_y = self.spatial_state.v_y
|
||||
omega = self.spatial_state.omega
|
||||
t = self.spatial_state.t
|
||||
|
||||
# get information about current waypoint
|
||||
kappa = self.reference_path.waypoints[self.wp_id].kappa
|
||||
|
||||
# get delta_s
|
||||
next_waypoint = self.reference_path.waypoints[self.wp_id + 1]
|
||||
delta_s = next_waypoint - self.current_waypoint
|
||||
|
||||
# get temporal derivatives
|
||||
F_rx, F_fy, F_ry, alpha_f, alpha_r = self.get_forces(delta, D)
|
||||
s_dot, v_x_dot, v_y_dot, omega_dot = self.\
|
||||
get_temporal_derivatives(delta, F_rx, F_fy, F_ry)
|
||||
|
||||
##############################
|
||||
# Forces Partial Derivatives #
|
||||
##############################
|
||||
|
||||
d_alpha_f_d_v_x = 1 / (1 + ((omega * self.lf + v_y) / v_x)**2) * (omega * self.lf + v_y) / (v_x**2)
|
||||
d_alpha_f_d_v_y = - 1 / (1 + ((omega * self.lf + v_y) / v_x)**2) / v_x
|
||||
d_alpha_f_d_omega = - 1 / (1 + ((omega * self.lf + v_y) / v_x)**2) * (self.lf / v_x)
|
||||
d_alpha_f_d_delta = 1
|
||||
|
||||
d_alpha_r_d_v_x = - 1 / (1 + ((omega * self.lr - v_y) / v_x)**2) * (omega * self.lr - v_y) / (v_x**2)
|
||||
d_alpha_r_d_v_y = - 1 / (1 + ((omega * self.lr - v_y) / v_x)**2) / v_x
|
||||
d_alpha_r_d_omega = 1 / (1 + ((omega * self.lr - v_y) / v_x)**2) * (self.lr * v_x)
|
||||
|
||||
d_F_fy_d_v_x = self.Df * np.cos(self.Cf * np.arctan(self.Bf * alpha_f)) * self.Cf / (1 + (self.Bf * alpha_f)**2) * self.Bf * d_alpha_f_d_v_x
|
||||
d_F_fy_d_v_y = self.Df * np.cos(self.Cf * np.arctan(self.Bf * alpha_f)) * self.Cf / (1 + (self.Bf * alpha_f)**2) * self.Bf * d_alpha_f_d_v_y
|
||||
d_F_fy_d_omega = self.Df * np.cos(self.Cf * np.arctan(self.Bf * alpha_f)) * self.Cf / (1 + (self.Bf * alpha_f)**2) * self.Bf * d_alpha_f_d_omega
|
||||
d_F_fy_d_delta = self.Df * np.cos(self.Cf * np.arctan(self.Bf * alpha_f)) * self.Cf / (1 + (self.Bf * alpha_f)**2) * self.Bf * d_alpha_f_d_delta
|
||||
|
||||
d_F_ry_d_v_x = self.Dr * np.cos(self.Cr * np.arctan(self.Br * alpha_r)) * self.Cr / (1 + (self.Br * alpha_r)**2) * self.Br * d_alpha_r_d_v_x
|
||||
d_F_ry_d_v_y = self.Dr * np.cos(self.Cr * np.arctan(self.Br * alpha_r)) * self.Cr / (1 + (self.Br * alpha_r)**2) * self.Br * d_alpha_r_d_v_y
|
||||
d_F_ry_d_omega = self.Dr * np.cos(self.Cr * np.arctan(self.Br * alpha_r)) * self.Cr / (1 + (self.Br * alpha_r)**2) * self.Br * d_alpha_r_d_omega
|
||||
|
||||
d_F_rx_d_v_x = - self.Cm2 * D - 2 * self.Cr2 * v_x
|
||||
d_F_rx_d_D = self.Cm1 - self.Cm2 * v_x
|
||||
|
||||
##############################
|
||||
# Helper Partial Derivatives #
|
||||
##############################
|
||||
|
||||
d_s_dot_d_e_y = kappa / (1-e_y*kappa)**2 * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi))
|
||||
d_s_dot_d_e_psi = 1 / (1 - e_y*kappa) * (-v_x * np.sin(e_psi) - v_y * np.cos(e_psi))
|
||||
d_s_dot_d_v_x = 1 / (1 - e_y*kappa) * np.cos(e_psi)
|
||||
d_s_dot_d_v_y = -1 / (1 - e_y*kappa) * np.sin(e_psi)
|
||||
d_s_dot_d_omega = 0
|
||||
d_s_dot_d_t = 0
|
||||
d_s_dot_d_delta = 0
|
||||
d_s_dot_d_D = 0
|
||||
d_s_dot_d_kappa = e_y / (1-e_y*kappa)**2 * (v_x * np.cos(e_psi) - v_y * np.sin(e_psi))
|
||||
# Check
|
||||
|
||||
c_1 = (v_x * np.sin(e_psi) + v_y * np.cos(e_psi))
|
||||
d_c_1_d_e_y = 0
|
||||
d_c_1_d_e_psi = v_x * np.cos(e_psi) - v_y * np.sin(e_psi)
|
||||
d_c_1_d_v_x = np.sin(e_psi)
|
||||
d_c_1_d_v_y = np.cos(e_psi)
|
||||
d_c_1_d_omega = 0
|
||||
d_c_1_d_t = 0
|
||||
d_c_1_d_delta = 0
|
||||
d_c_1_d_D = 0
|
||||
d_c_1_d_kappa = 0
|
||||
# Check
|
||||
|
||||
d_v_x_dot_d_e_y = 0
|
||||
d_v_x_dot_d_e_psi = 0
|
||||
d_v_x_dot_d_v_x = (d_F_rx_d_v_x - d_F_fy_d_v_x * np.sin(delta)) / self.m
|
||||
d_v_x_dot_d_v_y = - (d_F_fy_d_v_y * np.sin(delta) + self.m * omega) / self.m
|
||||
d_v_x_dot_d_omega = - (d_F_fy_d_omega * np.sin(delta) + self.m * v_y) / self.m
|
||||
d_v_x_dot_d_t = 0
|
||||
d_v_x_dot_d_delta = - (F_fy * np.cos(delta) + d_F_fy_d_delta * np.sin(delta)) / self.m
|
||||
d_v_x_dot_d_D = d_F_rx_d_D / self.m
|
||||
d_v_x_dot_d_kappa = 0
|
||||
|
||||
d_v_y_dot_d_e_y = 0
|
||||
d_v_y_dot_d_e_psi = 0
|
||||
d_v_y_dot_d_v_x = (d_F_ry_d_v_x + d_F_fy_d_v_x * np.cos(delta) - self.m * omega) / self.m
|
||||
d_v_y_dot_d_v_y = (d_F_ry_d_v_y + d_F_fy_d_v_y * np.cos(delta)) / self.m
|
||||
d_v_y_dot_d_omega = (d_F_ry_d_omega + d_F_fy_d_omega * np.cos(delta) - self.m * v_x) / self.m
|
||||
d_v_y_dot_d_t = 0
|
||||
d_v_y_dot_d_delta = d_F_fy_d_delta * np.cos(delta) / self.m
|
||||
d_v_y_dot_d_D = 0
|
||||
d_v_y_dot_d_kappa = 0
|
||||
|
||||
d_omega_dot_d_e_y = 0
|
||||
d_omega_dot_d_e_psi = 0
|
||||
d_omega_dot_d_v_x = (d_F_fy_d_v_x * self.lf * np.cos(delta) - d_F_ry_d_v_x * self.lr) / self.Iz
|
||||
d_omega_dot_d_v_y = (d_F_fy_d_v_y * self.lf * np.cos(delta) - d_F_fy_d_v_y * self.lr) / self.Iz
|
||||
d_omega_dot_d_omega = (d_F_fy_d_omega * self.lf * np.cos(delta) - d_F_fy_d_omega * self.lr) / self.Iz
|
||||
d_omega_dot_d_t = 0
|
||||
d_omega_dot_d_delta = (- F_fy * np.sin(delta) + d_F_fy_d_delta * np.cos(delta)) / self.Iz
|
||||
d_omega_dot_d_D = 0
|
||||
d_omega_dot_d_kappa = 0
|
||||
|
||||
#######################
|
||||
# Partial Derivatives #
|
||||
#######################
|
||||
|
||||
# derivatives for E_Y
|
||||
d_e_y_d_e_y = -c_1 * d_s_dot_d_e_y / (s_dot**2)
|
||||
d_e_y_d_e_psi = (d_c_1_d_e_psi * s_dot - d_s_dot_d_e_psi * c_1) / (s_dot**2)
|
||||
d_e_y_d_v_x = (d_c_1_d_v_x * s_dot - d_s_dot_d_v_x * c_1) / (s_dot**2)
|
||||
d_e_y_d_v_y = (d_c_1_d_v_y * s_dot - d_s_dot_d_v_y * c_1) / (s_dot**2)
|
||||
d_e_y_d_omega = (d_c_1_d_omega * s_dot - d_s_dot_d_omega * c_1) / (s_dot**2)
|
||||
d_e_y_d_t = 0
|
||||
d_e_y_d_D = 0
|
||||
d_e_y_d_delta = (d_c_1_d_delta * s_dot - d_s_dot_d_delta * c_1) / (s_dot**2)
|
||||
d_e_y_d_kappa = -d_s_dot_d_kappa * c_1 / (s_dot**2)
|
||||
|
||||
# derivatives for E_PSI
|
||||
d_e_psi_d_e_y = - omega * d_s_dot_d_e_y / (s_dot**2)
|
||||
d_e_psi_d_e_psi = - omega * d_s_dot_d_e_psi / (s_dot**2)
|
||||
d_e_psi_d_v_x = (- omega * d_s_dot_d_v_x) / (s_dot**2)
|
||||
d_e_psi_d_v_y = (- omega * d_s_dot_d_v_y) / (s_dot**2)
|
||||
d_e_psi_d_omega = (s_dot - omega * d_s_dot_d_omega) / (s_dot**2)
|
||||
d_e_psi_d_t = 0
|
||||
d_e_psi_d_delta = (- omega * d_s_dot_d_delta) / (s_dot**2)
|
||||
d_e_psi_d_D = (- omega * d_s_dot_d_D) / (s_dot**2)
|
||||
d_e_psi_d_kappa = -d_s_dot_d_kappa * omega / (s_dot**2) - 1
|
||||
|
||||
# derivatives for V_X
|
||||
d_v_x_d_e_y = - d_s_dot_d_e_y * v_x_dot / (s_dot**2)
|
||||
d_v_x_d_e_psi = - d_s_dot_d_e_psi * v_x_dot / (s_dot**2)
|
||||
d_v_x_d_v_x = (d_v_x_dot_d_v_x * s_dot - d_s_dot_d_v_x * v_x_dot) / (s_dot**2)
|
||||
d_v_x_d_v_y = (d_v_x_dot_d_v_y * s_dot - d_s_dot_d_v_y * v_x_dot) / (s_dot**2)
|
||||
d_v_x_d_omega = (d_v_x_dot_d_omega * s_dot - d_s_dot_d_omega * v_x_dot) / (s_dot**2)
|
||||
d_v_x_d_t = 0
|
||||
d_v_x_d_delta = (d_v_x_dot_d_delta * s_dot - d_s_dot_d_delta * v_x_dot) / (s_dot**2)
|
||||
d_v_x_d_D = d_v_x_dot_d_D * s_dot / (s_dot**2)
|
||||
d_v_x_d_kappa = -d_s_dot_d_kappa * v_x_dot / (s_dot**2)
|
||||
|
||||
# derivatives for V_Y
|
||||
d_v_y_d_e_y = - d_s_dot_d_e_y * v_y_dot / (s_dot ** 2)
|
||||
d_v_y_d_e_psi = - d_s_dot_d_e_psi * v_y_dot / (s_dot ** 2)
|
||||
d_v_y_d_v_x = (d_v_y_dot_d_v_x * s_dot - d_s_dot_d_v_x * v_y_dot) / (
|
||||
s_dot ** 2)
|
||||
d_v_y_d_v_y = (d_v_y_dot_d_v_y * s_dot - d_s_dot_d_v_y * v_y_dot) / (
|
||||
s_dot ** 2)
|
||||
d_v_y_d_omega = (d_v_y_dot_d_omega * s_dot - d_s_dot_d_omega * v_y_dot) / (
|
||||
s_dot ** 2)
|
||||
d_v_y_d_t = 0
|
||||
d_v_y_d_delta = (d_v_y_dot_d_delta * s_dot - d_s_dot_d_delta * v_y_dot) / (
|
||||
s_dot ** 2)
|
||||
d_v_y_d_D = d_v_y_dot_d_D * s_dot / (s_dot ** 2)
|
||||
d_v_y_d_kappa = -d_s_dot_d_kappa * v_y_dot / (s_dot ** 2)
|
||||
|
||||
# derivatives for Omega
|
||||
d_omega_d_e_y = (d_omega_dot_d_e_y * s_dot - omega_dot * d_s_dot_d_e_y) / (s_dot**2)
|
||||
d_omega_d_e_psi = (d_omega_dot_d_e_psi * s_dot - omega_dot * d_s_dot_d_e_psi) / (s_dot**2)
|
||||
d_omega_d_v_x = (d_omega_dot_d_v_x * s_dot - omega_dot * d_s_dot_d_v_x) / (s_dot**2)
|
||||
d_omega_d_v_y = (d_omega_dot_d_v_y * s_dot - omega_dot * d_s_dot_d_v_y) / (s_dot**2)
|
||||
d_omega_d_omega = (d_omega_dot_d_omega * s_dot - omega_dot * d_s_dot_d_omega) / (s_dot**2)
|
||||
d_omega_d_t = (d_omega_dot_d_t * s_dot - omega_dot * d_s_dot_d_t) / (s_dot**2)
|
||||
d_omega_d_delta = (d_omega_dot_d_delta * s_dot - omega_dot * d_s_dot_d_delta) / (s_dot**2)
|
||||
d_omega_d_D = (d_omega_dot_d_D * s_dot - omega_dot * d_s_dot_d_D) / (s_dot**2)
|
||||
d_omega_d_kappa = (d_omega_dot_d_kappa * s_dot - omega_dot * d_s_dot_d_kappa) / (s_dot**2)
|
||||
|
||||
# derivatives for T
|
||||
d_t_d_e_y = - d_s_dot_d_e_y / (s_dot**2)
|
||||
d_t_d_e_psi = - d_s_dot_d_e_psi / (s_dot ** 2)
|
||||
d_t_d_v_x = - d_s_dot_d_v_x / (s_dot ** 2)
|
||||
d_t_d_v_y = - d_s_dot_d_v_y / (s_dot ** 2)
|
||||
d_t_d_omega = - d_s_dot_d_omega / (s_dot ** 2)
|
||||
d_t_d_t = 0
|
||||
d_t_d_delta = - d_s_dot_d_delta / (s_dot ** 2)
|
||||
d_t_d_D = 0
|
||||
d_t_d_kappa = - d_s_dot_d_kappa / (s_dot ** 2)
|
||||
|
||||
a_1 = np.array([d_e_y_d_e_y, d_e_y_d_e_psi, d_e_y_d_v_x, d_e_y_d_v_y, d_e_y_d_omega, d_e_y_d_t, d_e_y_d_kappa])
|
||||
a_2 = np.array([d_e_psi_d_e_y, d_e_psi_d_e_psi, d_e_psi_d_v_x, d_e_psi_d_v_y, d_e_psi_d_omega, d_e_psi_d_t, d_e_psi_d_kappa])
|
||||
a_3 = np.array([d_v_x_d_e_y, d_v_x_d_e_psi, d_v_x_d_v_x, d_v_x_d_v_y, d_v_x_d_omega, d_v_x_d_t, d_v_x_d_kappa])
|
||||
a_4 = np.array([d_v_y_d_e_y, d_v_y_d_e_psi, d_v_y_d_v_x, d_v_y_d_v_y, d_v_y_d_omega, d_v_y_d_t, d_v_y_d_kappa])
|
||||
a_5 = np.array([d_omega_d_e_y, d_omega_d_e_psi, d_omega_d_v_x, d_omega_d_v_y, d_omega_d_omega, d_omega_d_t, d_omega_d_kappa])
|
||||
a_6 = np.array([d_t_d_e_y, d_t_d_e_psi, d_t_d_v_x, d_t_d_v_y, d_t_d_omega, d_t_d_t, d_t_d_kappa])
|
||||
a_7 = np.array([0, 0, 0, 0, 0, 0, 1])
|
||||
A = np.stack((a_1, a_2, a_3, a_4, a_5, a_6, a_7), axis=0) * delta_s
|
||||
A[0, 0] += 1
|
||||
A[1, 1] += 1
|
||||
A[2, 2] += 1
|
||||
A[3, 3] += 1
|
||||
A[4, 4] += 1
|
||||
A[5, 5] += 1
|
||||
b_1 = np.array([d_e_y_d_D, d_e_y_d_delta])
|
||||
b_2 = np.array([d_e_psi_d_D, d_e_psi_d_delta])
|
||||
b_3 = np.array([d_v_x_d_D, d_v_x_d_delta])
|
||||
b_4 = np.array([d_v_y_d_D, d_v_y_d_delta])
|
||||
b_5 = np.array([d_omega_d_D, d_omega_d_delta])
|
||||
b_6 = np.array([d_t_d_D, d_t_d_delta])
|
||||
b_7 = np.array([0, 0])
|
||||
B = np.stack((b_1, b_2, b_3, b_4, b_5, b_6, b_7), axis=0) * delta_s
|
||||
|
||||
# set system matrices
|
||||
return A, B
|
||||
|
|
|
|||
Loading…
Reference in New Issue