# Differentially flat system - kinematic car¶

This example demonstrates the use of the flatsys module for generating trajectories for differentially flat systems. The example is drawn from Chapter 8 of FBS2e.

## Code¶

  1# kincar-flatsys.py - differentially flat systems example
2# RMM, 3 Jul 2019
3#
4# This example demonstrates the use of the flatsys module for generating
5# trajectories for differnetially flat systems by computing a trajectory for a
6# kinematic (bicycle) model of a car changing lanes.
7
8import os
9import numpy as np
10import matplotlib.pyplot as plt
11import control as ct
12import control.flatsys as fs
13import control.optimal as obc
14
15#
16# System model and utility functions
17#
18
19# Function to take states, inputs and return the flat flag
20def vehicle_flat_forward(x, u, params={}):
21    # Get the parameter values
22    b = params.get('wheelbase', 3.)
23
24    # Create a list of arrays to store the flat output and its derivatives
25    zflag = [np.zeros(3), np.zeros(3)]
26
27    # Flat output is the x, y position of the rear wheels
28    zflag = x
29    zflag = x
30
31    # First derivatives of the flat output
32    zflag = u * np.cos(x)  # dx/dt
33    zflag = u * np.sin(x)  # dy/dt
34
35    # First derivative of the angle
36    thdot = (u/b) * np.tan(u)
37
38    # Second derivatives of the flat output (setting vdot = 0)
39    zflag = -u * thdot * np.sin(x)
40    zflag =  u * thdot * np.cos(x)
41
42    return zflag
43
44
45# Function to take the flat flag and return states, inputs
46def vehicle_flat_reverse(zflag, params={}):
47    # Get the parameter values
48    b = params.get('wheelbase', 3.)
49
50    # Create a vector to store the state and inputs
51    x = np.zeros(3)
52    u = np.zeros(2)
53
54    # Given the flat variables, solve for the state
55    x = zflag  # x position
56    x = zflag  # y position
57    x = np.arctan2(zflag, zflag)  # tan(theta) = ydot/xdot
58
59    # And next solve for the inputs
60    u = zflag * np.cos(x) + zflag * np.sin(x)
61    thdot_v = zflag * np.cos(x) - zflag * np.sin(x)
62    u = np.arctan2(thdot_v, u**2 / b)
63
64    return x, u
65
66# Function to compute the RHS of the system dynamics
67def vehicle_update(t, x, u, params):
68    b = params.get('wheelbase', 3.)             # get parameter values
69    dx = np.array([
70        np.cos(x) * u,
71        np.sin(x) * u,
72        (u/b) * np.tan(u)
73    ])
74    return dx
75
76# Plot the trajectory in xy coordinates
77def plot_results(t, x, ud, rescale=True):
78    plt.subplot(4, 1, 2)
79    plt.plot(x, x)
80    plt.xlabel('x [m]')
81    plt.ylabel('y [m]')
82    if rescale:
83        plt.axis([x0, xf, x0-1, xf+1])
84
85    # Time traces of the state and input
86    plt.subplot(2, 4, 5)
87    plt.plot(t, x)
88    plt.ylabel('y [m]')
89
90    plt.subplot(2, 4, 6)
91    plt.plot(t, x)
93
94    plt.subplot(2, 4, 7)
95    plt.plot(t, ud)
96    plt.xlabel('Time t [sec]')
97    plt.ylabel('v [m/s]')
98    if rescale:
99        plt.axis([0, Tf, u0 - 1, uf + 1])
100
101    plt.subplot(2, 4, 8)
102    plt.plot(t, ud)
103    plt.xlabel('Ttime t [sec]')
104    plt.ylabel('$\delta$ [rad]')
105    plt.tight_layout()
106
107#
108# Approach 1: point to point solution, no cost or constraints
109#
110
111# Create differentially flat input/output system
112vehicle_flat = fs.FlatSystem(
113    vehicle_flat_forward, vehicle_flat_reverse, vehicle_update,
114    inputs=('v', 'delta'), outputs=('x', 'y'),
115    states=('x', 'y', 'theta'))
116
117# Define the endpoints of the trajectory
118x0 = [0., -2., 0.]; u0 = [10., 0.]
119xf = [40., 2., 0.]; uf = [10., 0.]
120Tf = 4
121
122# Define a set of basis functions to use for the trajectories
123poly = fs.PolyFamily(6)
124
125# Find a trajectory between the initial condition and the final condition
126traj1 = fs.point_to_point(vehicle_flat, Tf, x0, u0, xf, uf, basis=poly)
127
128# Create the desired trajectory between the initial and final condition
129T = np.linspace(0, Tf, 500)
130xd, ud = traj1.eval(T)
131
132# Simulation the open system dynamics with the full input
133t, y, x = ct.input_output_response(
134    vehicle_flat, T, ud, x0, return_x=True)
135
136# Plot the open loop system dynamics
137plt.figure(1)
138plt.suptitle("Open loop trajectory for kinematic car lane change")
139plot_results(t, x, ud)
140
141#
142# Approach #2: add cost function to make lane change quicker
143#
144
145# Define timepoints for evaluation plus basis function to use
146timepts = np.linspace(0, Tf, 10)
147basis = fs.PolyFamily(8)
148
149# Define the cost function (penalize lateral error and steering)
151    vehicle_flat, np.diag([0, 0.1, 0]), np.diag([0.1, 1]), x0=xf, u0=uf)
152
153# Solve for an optimal solution
154traj2 = fs.point_to_point(
155    vehicle_flat, timepts, x0, u0, xf, uf, cost=traj_cost, basis=basis,
156)
157xd, ud = traj2.eval(T)
158
159plt.figure(2)
160plt.suptitle("Lane change with lateral error + steering penalties")
161plot_results(T, xd, ud)
162
163#
164# Approach #3: optimal cost with trajectory constraints
165#
166# Resolve the problem with constraints on the inputs
167#
168
169# Constraint the input values
170constraints = [
171    obc.input_range_constraint(vehicle_flat, [8, -0.1], [12, 0.1]) ]
172
173# TEST: Change the basis to use B-splines
174basis = fs.BSplineFamily([0, Tf/2, Tf], 6)
175
176# Solve for an optimal solution
177traj3 = fs.point_to_point(
178    vehicle_flat, timepts, x0, u0, xf, uf, cost=traj_cost,
179    constraints=constraints, basis=basis,
180)
181xd, ud = traj3.eval(T)
182
183plt.figure(3)
184plt.suptitle("Lane change with penalty + steering constraints")
185plot_results(T, xd, ud)
186
187# Show the results unless we are running in batch mode
188if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
189    plt.show()
190
191
192#
193# Approach #4: optimal trajectory, final cost with trajectory constraints
194#
195# Resolve the problem with constraints on the inputs and also replacing the
196# point to point problem with one using a terminal cost to set the final
197# state.
198#
199
200# Define the cost function (mainly penalize steering angle)
202    vehicle_flat, None, np.diag([0.1, 10]), x0=xf, u0=uf)
203
204# Set terminal cost to bring us close to xf
205terminal_cost = obc.quadratic_cost(vehicle_flat, 1e3 * np.eye(3), None, x0=xf)
206
207# Change the basis to use B-splines
208basis = fs.BSplineFamily([0, Tf/2, Tf], [4, 6], vars=2)
209
210# Use a straight line as an initial guess for the trajectory
211initial_guess = np.array(
212    [x0[i] + (xf[i] - x0[i]) * timepts/Tf for i in (0, 1)])
213
214# Solve for an optimal solution
215traj4 = fs.solve_flat_ocp(
216    vehicle_flat, timepts, x0, u0, cost=traj_cost, constraints=constraints,
217    terminal_cost=terminal_cost, basis=basis, initial_guess=initial_guess,
218    # minimize_kwargs={'method': 'trust-constr'},
219)
220xd, ud = traj4.eval(T)
221
222plt.figure(4)
223plt.suptitle("Lane change with terminal cost + steering constraints")
224plot_results(T, xd, ud, rescale=False)  # TODO: remove rescale
225
226# Show the results unless we are running in batch mode
227if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
228    plt.show()