Updated LATEX equations in notebook
mcarfagno
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0cebd24dec
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b3593ab100
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# mpc_python
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Python implementation of mpc controller for path tracking.
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## About
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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:
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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} $
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s.t.
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$ x(0) = x0 $
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$ x_{j+1} = Ax_{j}+Bu_{j})$ for $t< j <t+T-1 $
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$ U_{MIN} < u_{j} < U_{MAX} $ for $t< j <t+T-1 $
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The vehicle dynamics are described by the differential drive model:
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* $\dot{x} = v\cos{\theta}$
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* $\dot{y} = v\sin{\theta}$
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* $\dot{\theta} = w$
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The state variables of the model are:
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* $x$ coordinate of the robot
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* $y$ coordinate of the robot
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* $\theta$ heading of the robot
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The inputs of the model are:
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* $v$ linear velocity of the robot
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* $w$ angular velocity of the robot
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## Demo
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To run the demo:
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```bash
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python3 mpc_demo/main.py
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```
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## Requirements
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```bash
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pip3 install --user --requirement requirements.txt
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```
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@ -6,7 +6,7 @@ Python implementation of mpc controller for path tracking.
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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:
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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} $
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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} $
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s.t.
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@ -0,0 +1,184 @@
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#! /usr/bin/env python
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import numpy as np
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import matplotlib.pyplot as plt
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from matplotlib import animation
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from utils import compute_path_from_wp
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from cvxpy_mpc import optimize
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import sys
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import time
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# Robot Starting position
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SIM_START_X=0
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SIM_START_Y=2
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SIM_START_H=0
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# Classes
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class MPC():
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def __init__(self):
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# State for the robot mathematical model [x,y,heading]
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self.state = [SIM_START_X, SIM_START_Y, SIM_START_H]
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# Sim step
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self.dt = 0.25
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# starting guess output
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N = 5 #number of state variables
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M = 2 #number of control variables
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T = 20 #Prediction Horizon
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self.opt_u = np.zeros((M,T))
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self.opt_u[0,:] = 1 #m/s
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self.opt_u[1,:] = np.radians(0) #rad/s
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# Interpolated Path to follow given waypoints
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self.path = compute_path_from_wp([0,10,12,2,4,14],[0,0,2,10,12,12])
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# Sim help vars
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self.sim_time=0
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self.x_history=[]
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self.y_history=[]
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self.h_history=[]
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self.v_history=[]
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self.w_history=[]
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#Initialise plot
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plt.style.use("ggplot")
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self.fig = plt.figure()
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plt.ion()
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plt.show()
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def run(self):
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'''
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'''
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while 1:
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if self.state is not None:
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if np.sqrt((self.state[0]-self.path[0,-1])**2+(self.state[1]-self.path[1,-1])**2)<1:
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print("Success! Goal Reached")
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return
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#optimization loop
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start=time.time()
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self.opt_u = optimize(self.state,
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self.opt_u,
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self.path)
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# print("CVXPY Optimization Time: {:.4f}s".format(time.time()-start))
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self.update_sim(self.opt_u[0,1],self.opt_u[1,1])
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self.plot_sim()
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def update_sim(self,lin_v,ang_v):
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'''
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Updates state.
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:param lin_v: float
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:param ang_v: float
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'''
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self.state[0] = self.state[0] +lin_v*np.cos(self.state[2])*self.dt
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self.state[1] = self.state[1] +lin_v*np.sin(self.state[2])*self.dt
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self.state[2] = self.state[2] +ang_v*self.dt
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def plot_sim(self):
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self.sim_time = self.sim_time+self.dt
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self.x_history.append(self.state[0])
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self.y_history.append(self.state[1])
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self.h_history.append(self.state[2])
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self.v_history.append(self.opt_u[0,1])
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self.w_history.append(self.opt_u[1,1])
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plt.clf()
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grid = plt.GridSpec(2, 3)
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plt.subplot(grid[0:2, 0:2])
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plt.title("MPC Simulation \n" + "Simulation elapsed time {}s".format(self.sim_time))
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plt.plot(self.path[0,:],self.path[1,:], c='tab:orange',
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marker=".",
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label="reference track")
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plt.plot(self.x_history, self.y_history, c='tab:blue',
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marker=".",
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alpha=0.5,
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label="vehicle trajectory")
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# plt.plot(self.x_history[-1], self.y_history[-1], c='tab:blue',
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# marker=".",
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# markersize=12,
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# label="vehicle position")
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# plt.arrow(self.x_history[-1],
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# self.y_history[-1],
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# np.cos(self.h_history[-1]),
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# np.sin(self.h_history[-1]),
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# color='tab:blue',
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# width=0.2,
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# head_length=0.5,
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# label="heading")
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plot_car(self.x_history[-1], self.y_history[-1], self.h_history[-1])
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plt.ylabel('map y')
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plt.yticks(np.arange(min(self.path[1,:])-1., max(self.path[1,:]+1.)+1, 1.0))
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plt.xlabel('map x')
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plt.xticks(np.arange(min(self.path[0,:])-1., max(self.path[0,:]+1.)+1, 1.0))
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#plt.legend()
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plt.subplot(grid[0, 2])
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#plt.title("Linear Velocity {} m/s".format(self.v_history[-1]))
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plt.plot(self.v_history,c='tab:orange')
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locs, _ = plt.xticks()
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plt.xticks(locs[1:], locs[1:]*self.dt)
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plt.ylabel('v(t) [m/s]')
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plt.xlabel('t [s]')
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plt.subplot(grid[1, 2])
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#plt.title("Angular Velocity {} m/s".format(self.w_history[-1]))
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plt.plot(np.degrees(self.w_history),c='tab:orange')
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plt.ylabel('w(t) [deg/s]')
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locs, _ = plt.xticks()
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plt.xticks(locs[1:], locs[1:]*self.dt)
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plt.xlabel('t [s]')
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plt.tight_layout()
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plt.draw()
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plt.pause(0.1)
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def plot_car(x, y, yaw):
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LENGTH = 1.0 # [m]
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WIDTH = 0.5 # [m]
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OFFSET = LENGTH/2 # [m]
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outline = np.array([[-OFFSET, (LENGTH - OFFSET), (LENGTH - OFFSET), -OFFSET, -OFFSET],
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[WIDTH / 2, WIDTH / 2, - WIDTH / 2, -WIDTH / 2, WIDTH / 2]])
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Rotm = np.array([[np.cos(yaw), np.sin(yaw)],
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[-np.sin(yaw), np.cos(yaw)]])
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outline = (outline.T.dot(Rotm)).T
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outline[0, :] += x
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outline[1, :] += y
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plt.plot(np.array(outline[0, :]).flatten(), np.array(outline[1, :]).flatten(), 'tab:blue')
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def do_sim():
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sim=MPC()
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try:
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sim.run()
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except Exception as e:
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sys.exit(e)
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if __name__ == '__main__':
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do_sim()
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@ -822,37 +822,49 @@
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"\n",
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"The objective function used minimize (drive the state to 0) is:\n",
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"\n",
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"$J(x(t),U) = \\sum^{t+T-1}_{j=t} x^T_{j|t}Qx_{j|t} + u^T_{j|t}Ru_{j|t}$\n",
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"\n",
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"s.t.\n",
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"\n",
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"$ x(0) = x0$\n",
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"\n",
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"$ x_{j+1|t} = Ax_{j|t}+Bu_{j|t})$ for $t<j<t+T-1 $\n",
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"$\n",
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"\\begin{equation}\n",
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"\\begin{aligned}\n",
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"\\min_{} \\quad & \\sum^{t+T-1}_{j=t} x^T_{j|t}Qx_{j|t} + u^T_{j|t}Ru_{j|t}\\\\\n",
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"\\textrm{s.t.} \\quad & x(0) = x0\\\\\n",
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" & x_{j+1|t} = Ax_{j|t}+Bu_{j|t}) \\quad \\textrm{for} t<j<t+T-1 \\\\\n",
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"\\end{aligned}\n",
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"\\end{equation}\n",
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"$\n",
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"\n",
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"Other linear constrains may be applied,for instance on the control variable:\n",
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"\n",
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"$ U_{MIN} < u_{j|t} < U_{MAX} $ for $t<j<t+T-1 $\n",
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"$ U_{MIN} < u_{j|t} < U_{MAX} \\quad \\textrm{for} t<j<t+T-1 $\n",
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"\n",
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"The objective fuction accounts for quadratic error on deviation from 0 of the state and the control inputs sequences. Q and R are the **weight matrices** and are used to tune the response.\n",
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"\n",
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"Because the goal is tracking a **reference signal** such as a trajectory, the objective function is rewritten as:\n",
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"\n",
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"$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}$\n",
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"$\n",
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"\\begin{equation}\n",
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"\\begin{aligned}\n",
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"\\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",
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"\\end{aligned}\n",
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"\\end{equation}\n",
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"$\n",
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"\n",
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"where the error w.r.t desired state is accounted for:\n",
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"\n",
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"$\\delta x = x_{j,t,ref} - x_{j,t}$\n",
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"$ \\delta x = x_{j,t,ref} - x_{j,t} $\n",
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"\n",
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"#### Non-Linear MPC Formulation\n",
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"\n",
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"In general cases, the objective function is any non-differentiable non-linear function of states and inputs over a finite horizon T. In this case the constrains include nonlinear dynamics of motion.\n",
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"\n",
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"$J(x(t),U) = \\sum^{t+T}_{j=t} C(x_{j|t},{j|t})$\n",
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"\n",
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"s.t.\n",
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"\n",
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"$ x_{j+1|t} = f(x_{j|t},u_{j|t})$ for $t<j<t+T-1 $\n",
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"$\n",
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"\\begin{equation}\n",
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"\\begin{aligned}\n",
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"\\min_{} \\quad & \\sum^{t+T}_{j=t} C(x_{j|t},{j|t})\\\\\n",
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"\\textrm{s.t.} \\quad & x(0) = x0\\\\\n",
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" & x_{j+1|t} = f(x_{j|t},u_{j|t}) \\quad \\textrm{for} t<j<t+T-1 \\\\\n",
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"\\end{aligned}\n",
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"\\end{equation}\n",
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"$\n",
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"\n",
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"Other nonlinear constrains may be applied:\n",
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"\n",
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@ -822,37 +822,49 @@
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"\n",
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"The objective function used minimize (drive the state to 0) is:\n",
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"\n",
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"$J(x(t),U) = \\sum^{t+T-1}_{j=t} x^T_{j|t}Qx_{j|t} + u^T_{j|t}Ru_{j|t}$\n",
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"\n",
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"s.t.\n",
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"\n",
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"$ x(0) = x0$\n",
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"\n",
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"$ x_{j+1|t} = Ax_{j|t}+Bu_{j|t})$ for $t<j<t+T-1 $\n",
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"$\n",
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"\\begin{equation}\n",
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"\\begin{aligned}\n",
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"\\min_{} \\quad & \\sum^{t+T-1}_{j=t} x^T_{j|t}Qx_{j|t} + u^T_{j|t}Ru_{j|t}\\\\\n",
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"\\textrm{s.t.} \\quad & x(0) = x0\\\\\n",
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" & x_{j+1|t} = Ax_{j|t}+Bu_{j|t}) \\quad \\textrm{for} t<j<t+T-1 \\\\\n",
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"\\end{aligned}\n",
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"\\end{equation}\n",
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"$\n",
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"\n",
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"Other linear constrains may be applied,for instance on the control variable:\n",
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"\n",
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"$ U_{MIN} < u_{j|t} < U_{MAX} $ for $t<j<t+T-1 $\n",
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"$ U_{MIN} < u_{j|t} < U_{MAX} \\quad \\textrm{for} t<j<t+T-1 $\n",
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"\n",
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"The objective fuction accounts for quadratic error on deviation from 0 of the state and the control inputs sequences. Q and R are the **weight matrices** and are used to tune the response.\n",
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"\n",
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"Because the goal is tracking a **reference signal** such as a trajectory, the objective function is rewritten as:\n",
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"\n",
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"$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}$\n",
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"$\n",
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"\\begin{equation}\n",
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"\\begin{aligned}\n",
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"\\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",
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"\\end{aligned}\n",
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"\\end{equation}\n",
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"$\n",
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"\n",
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"where the error w.r.t desired state is accounted for:\n",
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"\n",
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"$\\delta x = x_{j,t,ref} - x_{j,t}$\n",
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"$ \\delta x = x_{j,t,ref} - x_{j,t} $\n",
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"\n",
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"#### Non-Linear MPC Formulation\n",
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"\n",
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"In general cases, the objective function is any non-differentiable non-linear function of states and inputs over a finite horizon T. In this case the constrains include nonlinear dynamics of motion.\n",
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"\n",
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"$J(x(t),U) = \\sum^{t+T}_{j=t} C(x_{j|t},{j|t})$\n",
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"\n",
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"s.t.\n",
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"\n",
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"$ x_{j+1|t} = f(x_{j|t},u_{j|t})$ for $t<j<t+T-1 $\n",
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"$\n",
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"\\begin{equation}\n",
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"\\begin{aligned}\n",
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"\\min_{} \\quad & \\sum^{t+T}_{j=t} C(x_{j|t},{j|t})\\\\\n",
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"\\textrm{s.t.} \\quad & x(0) = x0\\\\\n",
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" & x_{j+1|t} = f(x_{j|t},u_{j|t}) \\quad \\textrm{for} t<j<t+T-1 \\\\\n",
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"\\end{aligned}\n",
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"\\end{equation}\n",
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"$\n",
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"\n",
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"Other nonlinear constrains may be applied:\n",
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"\n",
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@ -723,9 +723,9 @@
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.5.2"
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"version": "3.6.9"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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"nbformat_minor": 4
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}
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