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mcarfagno 2020-04-07 16:45:11 +01:00
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.ipynb_checkpoints/README-checkpoint.md Normal file
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# mpc_python
Python implementation of mpc controller for path tracking.
## About
The MPC is a model predictive path following controller which does follow a predefined reference path Xref and Yref by solving an optimization problem. The resulting optimization problem is shown in the following equation:
MIN $ J(x(t),U) = \sum^{t+T-1}_{j=t} (x_{j,ref} - x_{j})^T_{j}Q(x_{j,ref} - x_{j}) + u^T_{j}Ru_{j} $
s.t.
$ x(0) = x0 $
$ x_{j+1} = Ax_{j}+Bu_{j})$ for $t< j <t+T-1 $
$ U_{MIN} < u_{j} < U_{MAX} $ for $t< j <t+T-1 $
The vehicle dynamics are described by the differential drive model:
* $\dot{x} = v\cos{\theta}$
* $\dot{y} = v\sin{\theta}$
* $\dot{\theta} = w$
The state variables of the model are:
* $x$ coordinate of the robot
* $y$ coordinate of the robot
* $\theta$ heading of the robot
The inputs of the model are:
* $v$ linear velocity of the robot
* $w$ angular velocity of the robot
## Demo
![](img/demo.gif)
To run the demo:
```bash
python3 mpc_demo/main.py
```
## Requirements
```bash
pip3 install --user --requirement requirements.txt
```

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README.md
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@ -6,7 +6,7 @@ Python implementation of mpc controller for path tracking.
The MPC is a model predictive path following controller which does follow a predefined reference path Xref and Yref by solving an optimization problem. The resulting optimization problem is shown in the following equation: The MPC is a model predictive path following controller which does follow a predefined reference path Xref and Yref by solving an optimization problem. The resulting optimization problem is shown in the following equation:
MIN $ J(x(t),U) = \sum^{t+T-1}_{j=t} (x_{j,ref} - x_{j})^T_{j}Q(x_{j,ref} - x_{j}) + u^T_{j}Ru_{j} $ min $ J(x(t),U) = \sum^{t+T-1}_{j=t} (x_{j,ref} - x_{j})^T_{j}Q(x_{j,ref} - x_{j}) + u^T_{j}Ru_{j} $
s.t. s.t.

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mpc_demo/.ipynb_checkpoints/main-checkpoint.py Executable file
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#! /usr/bin/env python
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import animation
from utils import compute_path_from_wp
from cvxpy_mpc import optimize
import sys
import time
# Robot Starting position
SIM_START_X=0
SIM_START_Y=2
SIM_START_H=0
# Classes
class MPC():
def __init__(self):
# State for the robot mathematical model [x,y,heading]
self.state = [SIM_START_X, SIM_START_Y, SIM_START_H]
# Sim step
self.dt = 0.25
# starting guess output
N = 5 #number of state variables
M = 2 #number of control variables
T = 20 #Prediction Horizon
self.opt_u = np.zeros((M,T))
self.opt_u[0,:] = 1 #m/s
self.opt_u[1,:] = np.radians(0) #rad/s
# Interpolated Path to follow given waypoints
self.path = compute_path_from_wp([0,10,12,2,4,14],[0,0,2,10,12,12])
# Sim help vars
self.sim_time=0
self.x_history=[]
self.y_history=[]
self.h_history=[]
self.v_history=[]
self.w_history=[]
#Initialise plot
plt.style.use("ggplot")
self.fig = plt.figure()
plt.ion()
plt.show()
def run(self):
'''
'''
while 1:
if self.state is not None:
if np.sqrt((self.state[0]-self.path[0,-1])**2+(self.state[1]-self.path[1,-1])**2)<1:
print("Success! Goal Reached")
return
#optimization loop
start=time.time()
self.opt_u = optimize(self.state,
self.opt_u,
self.path)
# print("CVXPY Optimization Time: {:.4f}s".format(time.time()-start))
self.update_sim(self.opt_u[0,1],self.opt_u[1,1])
self.plot_sim()
def update_sim(self,lin_v,ang_v):
'''
Updates state.
:param lin_v: float
:param ang_v: float
'''
self.state[0] = self.state[0] +lin_v*np.cos(self.state[2])*self.dt
self.state[1] = self.state[1] +lin_v*np.sin(self.state[2])*self.dt
self.state[2] = self.state[2] +ang_v*self.dt
def plot_sim(self):
self.sim_time = self.sim_time+self.dt
self.x_history.append(self.state[0])
self.y_history.append(self.state[1])
self.h_history.append(self.state[2])
self.v_history.append(self.opt_u[0,1])
self.w_history.append(self.opt_u[1,1])
plt.clf()
grid = plt.GridSpec(2, 3)
plt.subplot(grid[0:2, 0:2])
plt.title("MPC Simulation \n" + "Simulation elapsed time {}s".format(self.sim_time))
plt.plot(self.path[0,:],self.path[1,:], c='tab:orange',
marker=".",
label="reference track")
plt.plot(self.x_history, self.y_history, c='tab:blue',
marker=".",
alpha=0.5,
label="vehicle trajectory")
# plt.plot(self.x_history[-1], self.y_history[-1], c='tab:blue',
# marker=".",
# markersize=12,
# label="vehicle position")
# plt.arrow(self.x_history[-1],
# self.y_history[-1],
# np.cos(self.h_history[-1]),
# np.sin(self.h_history[-1]),
# color='tab:blue',
# width=0.2,
# head_length=0.5,
# label="heading")
plot_car(self.x_history[-1], self.y_history[-1], self.h_history[-1])
plt.ylabel('map y')
plt.yticks(np.arange(min(self.path[1,:])-1., max(self.path[1,:]+1.)+1, 1.0))
plt.xlabel('map x')
plt.xticks(np.arange(min(self.path[0,:])-1., max(self.path[0,:]+1.)+1, 1.0))
#plt.legend()
plt.subplot(grid[0, 2])
#plt.title("Linear Velocity {} m/s".format(self.v_history[-1]))
plt.plot(self.v_history,c='tab:orange')
locs, _ = plt.xticks()
plt.xticks(locs[1:], locs[1:]*self.dt)
plt.ylabel('v(t) [m/s]')
plt.xlabel('t [s]')
plt.subplot(grid[1, 2])
#plt.title("Angular Velocity {} m/s".format(self.w_history[-1]))
plt.plot(np.degrees(self.w_history),c='tab:orange')
plt.ylabel('w(t) [deg/s]')
locs, _ = plt.xticks()
plt.xticks(locs[1:], locs[1:]*self.dt)
plt.xlabel('t [s]')
plt.tight_layout()
plt.draw()
plt.pause(0.1)
def plot_car(x, y, yaw):
LENGTH = 1.0 # [m]
WIDTH = 0.5 # [m]
OFFSET = LENGTH/2 # [m]
outline = np.array([[-OFFSET, (LENGTH - OFFSET), (LENGTH - OFFSET), -OFFSET, -OFFSET],
[WIDTH / 2, WIDTH / 2, - WIDTH / 2, -WIDTH / 2, WIDTH / 2]])
Rotm = np.array([[np.cos(yaw), np.sin(yaw)],
[-np.sin(yaw), np.cos(yaw)]])
outline = (outline.T.dot(Rotm)).T
outline[0, :] += x
outline[1, :] += y
plt.plot(np.array(outline[0, :]).flatten(), np.array(outline[1, :]).flatten(), 'tab:blue')
def do_sim():
sim=MPC()
try:
sim.run()
except Exception as e:
sys.exit(e)
if __name__ == '__main__':
do_sim()

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notebook/.ipynb_checkpoints/MPC_cvxpy-checkpoint.ipynb
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@ -822,37 +822,49 @@
"\n", "\n",
"The objective function used minimize (drive the state to 0) is:\n", "The objective function used minimize (drive the state to 0) is:\n",
"\n", "\n",
"$J(x(t),U) = \\sum^{t+T-1}_{j=t} x^T_{j|t}Qx_{j|t} + u^T_{j|t}Ru_{j|t}$\n", "$\n",
"\n", "\\begin{equation}\n",
"s.t.\n", "\\begin{aligned}\n",
"\n", "\\min_{} \\quad & \\sum^{t+T-1}_{j=t} x^T_{j|t}Qx_{j|t} + u^T_{j|t}Ru_{j|t}\\\\\n",
"$ x(0) = x0$\n", "\\textrm{s.t.} \\quad & x(0) = x0\\\\\n",
"\n", " & x_{j+1|t} = Ax_{j|t}+Bu_{j|t}) \\quad \\textrm{for} t<j<t+T-1 \\\\\n",
"$ x_{j+1|t} = Ax_{j|t}+Bu_{j|t})$ for $t<j<t+T-1 $\n", "\\end{aligned}\n",
"\\end{equation}\n",
"$\n",
"\n", "\n",
"Other linear constrains may be applied,for instance on the control variable:\n", "Other linear constrains may be applied,for instance on the control variable:\n",
"\n", "\n",
"$ U_{MIN} < u_{j|t} < U_{MAX} $ for $t<j<t+T-1 $\n", "$ U_{MIN} < u_{j|t} < U_{MAX} \\quad \\textrm{for} t<j<t+T-1 $\n",
"\n", "\n",
"The objective fuction accounts for quadratic error on deviation from 0 of the state and the control inputs sequences. Q and R are the **weight matrices** and are used to tune the response.\n", "The objective fuction accounts for quadratic error on deviation from 0 of the state and the control inputs sequences. Q and R are the **weight matrices** and are used to tune the response.\n",
"\n", "\n",
"Because the goal is tracking a **reference signal** such as a trajectory, the objective function is rewritten as:\n", "Because the goal is tracking a **reference signal** such as a trajectory, the objective function is rewritten as:\n",
"\n", "\n",
"$J(x(t),U) = \\sum^{t+T-1}_{j=t} \\delta x^T_{j|t}Q\\delta x_{j|t} + u^T_{j|t}Ru_{j|t}$\n", "$\n",
"\\begin{equation}\n",
"\\begin{aligned}\n",
"\\min_{} \\quad & \\sum^{t+T-1}_{j=t} \\delta x^T_{j|t}Q\\delta x_{j|t} + u^T_{j|t}Ru_{j|t}\n",
"\\end{aligned}\n",
"\\end{equation}\n",
"$\n",
"\n", "\n",
"where the error w.r.t desired state is accounted for:\n", "where the error w.r.t desired state is accounted for:\n",
"\n", "\n",
"$\\delta x = x_{j,t,ref} - x_{j,t}$\n", "$ \\delta x = x_{j,t,ref} - x_{j,t} $\n",
"\n", "\n",
"#### Non-Linear MPC Formulation\n", "#### Non-Linear MPC Formulation\n",
"\n", "\n",
"In general cases, the objective function is any non-differentiable non-linear function of states and inputs over a finite horizon T. In this case the constrains include nonlinear dynamics of motion.\n", "In general cases, the objective function is any non-differentiable non-linear function of states and inputs over a finite horizon T. In this case the constrains include nonlinear dynamics of motion.\n",
"\n", "\n",
"$J(x(t),U) = \\sum^{t+T}_{j=t} C(x_{j|t},{j|t})$\n", "$\n",
"\n", "\\begin{equation}\n",
"s.t.\n", "\\begin{aligned}\n",
"\n", "\\min_{} \\quad & \\sum^{t+T}_{j=t} C(x_{j|t},{j|t})\\\\\n",
"$ x_{j+1|t} = f(x_{j|t},u_{j|t})$ for $t<j<t+T-1 $\n", "\\textrm{s.t.} \\quad & x(0) = x0\\\\\n",
" & x_{j+1|t} = f(x_{j|t},u_{j|t}) \\quad \\textrm{for} t<j<t+T-1 \\\\\n",
"\\end{aligned}\n",
"\\end{equation}\n",
"$\n",
"\n", "\n",
"Other nonlinear constrains may be applied:\n", "Other nonlinear constrains may be applied:\n",
"\n", "\n",

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notebook/MPC_cvxpy.ipynb
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@ -822,37 +822,49 @@
"\n", "\n",
"The objective function used minimize (drive the state to 0) is:\n", "The objective function used minimize (drive the state to 0) is:\n",
"\n", "\n",
"$J(x(t),U) = \\sum^{t+T-1}_{j=t} x^T_{j|t}Qx_{j|t} + u^T_{j|t}Ru_{j|t}$\n", "$\n",
"\n", "\\begin{equation}\n",
"s.t.\n", "\\begin{aligned}\n",
"\n", "\\min_{} \\quad & \\sum^{t+T-1}_{j=t} x^T_{j|t}Qx_{j|t} + u^T_{j|t}Ru_{j|t}\\\\\n",
"$ x(0) = x0$\n", "\\textrm{s.t.} \\quad & x(0) = x0\\\\\n",
"\n", " & x_{j+1|t} = Ax_{j|t}+Bu_{j|t}) \\quad \\textrm{for} t<j<t+T-1 \\\\\n",
"$ x_{j+1|t} = Ax_{j|t}+Bu_{j|t})$ for $t<j<t+T-1 $\n", "\\end{aligned}\n",
"\\end{equation}\n",
"$\n",
"\n", "\n",
"Other linear constrains may be applied,for instance on the control variable:\n", "Other linear constrains may be applied,for instance on the control variable:\n",
"\n", "\n",
"$ U_{MIN} < u_{j|t} < U_{MAX} $ for $t<j<t+T-1 $\n", "$ U_{MIN} < u_{j|t} < U_{MAX} \\quad \\textrm{for} t<j<t+T-1 $\n",
"\n", "\n",
"The objective fuction accounts for quadratic error on deviation from 0 of the state and the control inputs sequences. Q and R are the **weight matrices** and are used to tune the response.\n", "The objective fuction accounts for quadratic error on deviation from 0 of the state and the control inputs sequences. Q and R are the **weight matrices** and are used to tune the response.\n",
"\n", "\n",
"Because the goal is tracking a **reference signal** such as a trajectory, the objective function is rewritten as:\n", "Because the goal is tracking a **reference signal** such as a trajectory, the objective function is rewritten as:\n",
"\n", "\n",
"$J(x(t),U) = \\sum^{t+T-1}_{j=t} \\delta x^T_{j|t}Q\\delta x_{j|t} + u^T_{j|t}Ru_{j|t}$\n", "$\n",
"\\begin{equation}\n",
"\\begin{aligned}\n",
"\\min_{} \\quad & \\sum^{t+T-1}_{j=t} \\delta x^T_{j|t}Q\\delta x_{j|t} + u^T_{j|t}Ru_{j|t}\n",
"\\end{aligned}\n",
"\\end{equation}\n",
"$\n",
"\n", "\n",
"where the error w.r.t desired state is accounted for:\n", "where the error w.r.t desired state is accounted for:\n",
"\n", "\n",
"$\\delta x = x_{j,t,ref} - x_{j,t}$\n", "$ \\delta x = x_{j,t,ref} - x_{j,t} $\n",
"\n", "\n",
"#### Non-Linear MPC Formulation\n", "#### Non-Linear MPC Formulation\n",
"\n", "\n",
"In general cases, the objective function is any non-differentiable non-linear function of states and inputs over a finite horizon T. In this case the constrains include nonlinear dynamics of motion.\n", "In general cases, the objective function is any non-differentiable non-linear function of states and inputs over a finite horizon T. In this case the constrains include nonlinear dynamics of motion.\n",
"\n", "\n",
"$J(x(t),U) = \\sum^{t+T}_{j=t} C(x_{j|t},{j|t})$\n", "$\n",
"\n", "\\begin{equation}\n",
"s.t.\n", "\\begin{aligned}\n",
"\n", "\\min_{} \\quad & \\sum^{t+T}_{j=t} C(x_{j|t},{j|t})\\\\\n",
"$ x_{j+1|t} = f(x_{j|t},u_{j|t})$ for $t<j<t+T-1 $\n", "\\textrm{s.t.} \\quad & x(0) = x0\\\\\n",
" & x_{j+1|t} = f(x_{j|t},u_{j|t}) \\quad \\textrm{for} t<j<t+T-1 \\\\\n",
"\\end{aligned}\n",
"\\end{equation}\n",
"$\n",
"\n", "\n",
"Other nonlinear constrains may be applied:\n", "Other nonlinear constrains may be applied:\n",
"\n", "\n",

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notebook/MPC_scipy.ipynb
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@ -723,9 +723,9 @@
"name": "python", "name": "python",
"nbconvert_exporter": "python", "nbconvert_exporter": "python",
"pygments_lexer": "ipython3", "pygments_lexer": "ipython3",
"version": "3.5.2" "version": "3.6.9"
} }
}, },
"nbformat": 4, "nbformat": 4,
"nbformat_minor": 2 "nbformat_minor": 4
} }