Updated robot localization example.
Put into a function, added parameters so we can run the alg with different landmarks and sigmas.
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Roger Labbe committed
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@@ -269,10 +269,6 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"**Author's note: this is still being heavily edited and added to - there is a lot of duplicate material, incorrect code, and so on. The examples are just a brain dump, not fully formulated and not ready to be used.**\n",
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"\n",
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"**The Unscented Kalman filter (UKF) chapter is much further along. The UKF is almost always better performing than the Extended Kalman filter, and is much easier to implement, so if you have an urgent need for nonlinear Kalman filter I'll point you towards that chapter for now.**\n",
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"\n",
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"At this point in the book we have developed the theory for the linear Kalman filter. Then, in the last two chapters we broached the topic of using Kalman filters for nonlinear problems. I chose to start off with the Unscented Kalman filter, which probably felt like quite a departure from the linear Kalman filter math. It was also a departure from the historical development of the Kalman filter.\n",
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"\n",
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"Shortly after Dr. Kálmán published his paper on the Kalman filter people recognized that his theory was limited to linear problems and immediately began work on extending the theory to work with nonlinear problems. As I have said elsewhere the only equation we really know how to solve is $\\mathbf{Ax} = \\mathbf{b}$. That is a slight exaggeration. Many of us have spent years at school learning sophisticated tricks and techniques to handle equations that are not linear. Yet after all that work the vast majority of equations that arise from simple physical systems remain intractable. Certainly there is no way to find general analytic solutions to the Kalman filter equations for nonlinear systems. \n",
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@@ -316,7 +312,7 @@
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"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x6e9b3c8>"
|
||||
"<matplotlib.figure.Figure at 0x510d2b0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -776,9 +772,9 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x8047390>"
|
||||
"<matplotlib.figure.Figure at 0x808c710>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -786,9 +782,9 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x80d7240>"
|
||||
"<matplotlib.figure.Figure at 0x593d5f8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -796,9 +792,9 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAwMAAAEtCAYAAABUG7xMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7fdd470>"
|
||||
"<matplotlib.figure.Figure at 0x58c5668>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -901,7 +897,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAQEAAAAmBAMAAAA2HDP0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nUqOHbKQnT/0tv+59MpOT+AmTXbj1s0kQCQAAAABJRU5ErkJggg==\n",
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAQEAAAAmBAMAAAA2HDP0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
|
||||
"text/latex": [
|
||||
"$$\\left[\\begin{matrix}\\frac{x_{pos}}{\\sqrt{x_{alt}^{2} + x_{pos}^{2}}} & 0 & \\frac{x_{alt}}{\\sqrt{x_{alt}^{2} + x_{pos}^{2}}}\\end{matrix}\\right]$$"
|
||||
],
|
||||
@@ -1050,7 +1046,7 @@
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x80b8978>"
|
||||
"<matplotlib.figure.Figure at 0x8067a90>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -1483,7 +1479,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 17,
|
||||
"execution_count": 37,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
@@ -1492,13 +1488,14 @@
|
||||
"from filterpy.kalman import ExtendedKalmanFilter as EKF\n",
|
||||
"from numpy import dot, array, sqrt\n",
|
||||
"class RobotEKF(EKF):\n",
|
||||
" def __init__(self, dt, wheelbase):\n",
|
||||
" def __init__(self, dt, wheelbase, sigma_vel, sigma_steer):\n",
|
||||
" EKF.__init__(self, 3, 2, 2)\n",
|
||||
" self.dt = dt\n",
|
||||
" self.wheelbase = wheelbase\n",
|
||||
" self.sigma_vel = sigma_vel\n",
|
||||
" self.sigma_steer = sigma_steer\n",
|
||||
"\n",
|
||||
" a, x, y, v, w, theta, time = symbols('a, x, y, v, w, theta, t')\n",
|
||||
"\n",
|
||||
" d = v*time\n",
|
||||
" beta = (d/w)*sympy.tan(a)\n",
|
||||
" r = w/sympy.tan(a)\n",
|
||||
@@ -1525,7 +1522,7 @@
|
||||
" V = array(self.V_j.evalf(subs=self.subs)).astype(float)\n",
|
||||
"\n",
|
||||
" # covariance of motion noise in control space\n",
|
||||
" M = array([[0.1*u[0]**2, 0], [0, sigma_steer**2]])\n",
|
||||
" M = array([[self.sigma_vel*u[0]**2, 0], [0, self.sigma_steer**2]])\n",
|
||||
"\n",
|
||||
" self.P = dot(F, self.P).dot(F.T) + dot(V, M).dot(V.T)\n",
|
||||
"\n",
|
||||
@@ -1547,7 +1544,7 @@
|
||||
" coshb = cos(h + b)\n",
|
||||
" return x + array([[-r*sinh + r*sinhb],\n",
|
||||
" [r*cosh - r*coshb],\n",
|
||||
" [b]])\n"
|
||||
" [b]])"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -1580,210 +1577,214 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The rest of the code runs the simulation, and shouldn't need to much comment by now. I create a variable `map` that contains the coordinates of the landmarks. I update the simulated robot position 10 times a second, but run the EKF only once. This is for two reasons. First, we are not using Runge Kutta to integrate the differental equations of motion, so a narrow time step allows our simulation to be more accurate. Second, it is fairly normal in embedded systems to have limited processing speed. This forces you to run your Kalman filter only as frequently as absolutely needed."
|
||||
"The rest of the code runs the simulation and plots the results, and shouldn't need too much comment by now. I create a variable `landmarks` that contains the coordinates of the landmarks. I update the simulated robot position 10 times a second, but run the EKF only once. This is for two reasons. First, we are not using Runge Kutta to integrate the differental equations of motion, so a narrow time step allows our simulation to be more accurate. Second, it is fairly normal in embedded systems to have limited processing speed. This forces you to run your Kalman filter only as frequently as absolutely needed."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 19,
|
||||
"execution_count": 113,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x805b5f8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from filterpy.common import plot_covariance_ellipse\n",
|
||||
"from math import sqrt, tan, cos, sin, atan2\n",
|
||||
"dt = 1.0\n",
|
||||
"ekf = RobotEKF(dt, wheelbase=0.5)\n",
|
||||
"sigma_steer = np.radians(1)\n",
|
||||
"sigma_h = np.radians(1)\n",
|
||||
"sigma_r = 0.1\n",
|
||||
"#np.random.seed(1234)\n",
|
||||
"\n",
|
||||
"m = array([[5, 10],\n",
|
||||
" [10, 5],\n",
|
||||
" [15, 15]])\n",
|
||||
"def run_localization(landmarks, sigma_vel, sigma_steer, sigma_range, sigma_bearing):\n",
|
||||
" ekf = RobotEKF(dt, wheelbase=0.5, sigma_vel=sigma_vel, sigma_steer=sigma_steer)\n",
|
||||
" ekf.x = array([[2, 6, .3]]).T\n",
|
||||
" ekf.P = np.diag([.1, .1, .1])\n",
|
||||
" ekf.R = np.diag([sigma_range**2, sigma_bearing**2])\n",
|
||||
"\n",
|
||||
"ekf.x = array([[2, 6, .3]]).T\n",
|
||||
"u = array([1.1, .01])\n",
|
||||
"ekf.P = np.diag([.1, .1, .1])\n",
|
||||
"ekf.R = np.diag([sigma_r**2, sigma_h**2])\n",
|
||||
"c = [0, 1, 2]\n",
|
||||
" sim_pos = ekf.x.copy() # simulated position\n",
|
||||
" u = array([1.1, .01]) # steering command (vel, steering angle radians)\n",
|
||||
"\n",
|
||||
"xp = ekf.x.copy()\n",
|
||||
" plt.scatter(landmarks[:, 0], landmarks[:, 1], marker='s', s=60)\n",
|
||||
" for i in range(200):\n",
|
||||
" sim_pos = ekf.move(sim_pos, u, dt/10.) # simulate robot\n",
|
||||
" plt.plot(sim_pos[0], sim_pos[1], ',', color='g')\n",
|
||||
"\n",
|
||||
"plt.scatter(m[:, 0], m[:, 1], marker='s', s=60)\n",
|
||||
"for i in range(150):\n",
|
||||
" xp = ekf.move(xp, u, dt/10.) # simulate robot\n",
|
||||
" plt.plot(xp[0], xp[1], ',', color='g')\n",
|
||||
" if i % 10 == 0:\n",
|
||||
" ekf.predict(u=u)\n",
|
||||
"\n",
|
||||
" if i % 10 == 0:\n",
|
||||
" ekf.predict(u=u)\n",
|
||||
" plot_covariance_ellipse((ekf.x[0,0], ekf.x[1,0]), ekf.P[0:2, 0:2], std=6,\n",
|
||||
" facecolor='b', alpha=0.08)\n",
|
||||
"\n",
|
||||
" plot_covariance_ellipse((ekf.x[0,0], ekf.x[1,0]), ekf.P[0:2, 0:2], std=10,\n",
|
||||
" facecolor='b', alpha=0.08)\n",
|
||||
" x, y= sim_pos[0, 0], sim_pos[1, 0]\n",
|
||||
" for lmark in landmarks:\n",
|
||||
" d = np.sqrt((lmark[0] - x)**2 + (lmark[1] - y)**2) \n",
|
||||
" a = atan2(lmark[1] - y, lmark[0] - x) - sim_pos[2, 0]\n",
|
||||
" z = np.array([[d + randn()*sigma_range], [a + randn()*sigma_bearing]])\n",
|
||||
"\n",
|
||||
" for lmark in m:\n",
|
||||
" d = np.sqrt((lmark[0] - xp[0, 0])**2 + (lmark[1] - xp[1, 0])**2) + randn()*sigma_r\n",
|
||||
" a = atan2(lmark[1] - xp[1, 0], lmark[0] - xp[0, 0]) - xp[2, 0] + randn()*sigma_h\n",
|
||||
" z = np.array([[d], [a]])\n",
|
||||
" ekf.update(z, HJacobian=H_of, Hx=Hx, residual=residual,\n",
|
||||
" args=(lmark), hx_args=(lmark))\n",
|
||||
"\n",
|
||||
" ekf.update(z, HJacobian=H_of, Hx=Hx, residual=residual,\n",
|
||||
" args=(lmark), hx_args=(lmark))\n",
|
||||
"\n",
|
||||
" plot_covariance_ellipse((ekf.x[0,0], ekf.x[1,0]), ekf.P[0:2, 0:2], std=10,\n",
|
||||
" facecolor='g', alpha=0.4)\n",
|
||||
"\n",
|
||||
" #plt.plot(ekf.x[0], ekf.x[1], 'x', color='r')\n",
|
||||
"\n",
|
||||
"plt.axis('equal')\n",
|
||||
"plt.show()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Example: A falling Ball"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"**author's note: ignore this section for now. **\n",
|
||||
"\n",
|
||||
"In the **Designing Kalman Filters** chapter I first considered tracking a ball in a vacuum, and then in the atmosphere. The Kalman filter performed very well for vacuum, but diverged from the ball's path in the atmosphere. Let us look at the output; to avoid littering this chapter with code from that chapter I have placed it all in the file `ekf_internal.py'."
|
||||
" plot_covariance_ellipse((ekf.x[0,0], ekf.x[1,0]), ekf.P[0:2, 0:2], std=6,\n",
|
||||
" facecolor='g', alpha=0.4)\n",
|
||||
" plt.axis('equal')\n",
|
||||
" #plt.xlim((0,20))\n",
|
||||
" plt.show()\n",
|
||||
" return ekf"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 20,
|
||||
"execution_count": 110,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xa0b1550>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[ 0.01969482 0.02094075 0.00154535]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"landmarks = array([[5, 10], [10, 5], [15, 15], [20, 5]])\n",
|
||||
"\n",
|
||||
"ekf = run_localization(landmarks, sigma_vel=0.1, sigma_steer=np.radians(1),\n",
|
||||
" sigma_range=0.3, sigma_bearing=0.1)\n",
|
||||
"print(ekf.P.diagonal())"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"I have plotted the landmarks as solid squares. The path of the robot is drawn with a dashed line, which is admittedly hard to see. The covariance after the predict step is drawn in a very light shade of blue, and the covariance after all of the landmark measurements are incorporated are shown in green. To make them visible at this scale I have set the ellipse boundary at 10$\\sigma$.\n",
|
||||
"\n",
|
||||
"From this we can see that there is a lot of uncertainty added by our motion model, and that most of the error in in the direction of motion. We can see that from the shape of the blue ellipses. After a few steps we can see that the filter incorporates the landmark measurements. \n",
|
||||
"\n",
|
||||
"We can see the fantastic effect that multiple landmarks has on our uncertainty by only using the first two landmarks."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 109,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": true
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x806fd30>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[ 0.02126405 0.04551132 0.00022729]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"ekf = run_localization(landmarks[0:2], sigma_vel=1.e-10, sigma_steer=1.e-10,\n",
|
||||
" sigma_range=1.4, sigma_bearing=.05)\n",
|
||||
"print(ekf.P.diagonal())"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We can see that the covariance gets smaller as it passes through the landmarks but quickly expands once past them. Let's see what happens with only one landmark"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 114,
|
||||
"metadata": {
|
||||
"collapsed": false,
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0xafeb7b8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[ 0.26908765 0.8039095 0.00355399]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"ekf = run_localization(landmarks[0:1], sigma_vel=1.e-10, sigma_steer=1.e-10,\n",
|
||||
" sigma_range=1.4, sigma_bearing=.05)\n",
|
||||
"print(ekf.P.diagonal())"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"As you probably suspected, only one landmark produces a very bad covariance. What is worse, the filter starts to diverge from the robot's path. On the other hand, a large number of landmarks allows us to make very accurate estimates."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 119,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x803bb00>"
|
||||
"<matplotlib.figure.Figure at 0x5834ac8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[ 0.00883285 0.00866094 0.00077284]\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import ekf_internal\n",
|
||||
"ekf_internal.plot_ball()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We can artificially force the Kalman filter to track the ball by making $Q$ large. That would cause the filter to mistrust its prediction, and scale the kalman gain $K$ to strongly favor the measurments. However, this is not a valid approach. If the Kalman filter is correctly predicting the process we should not 'lie' to the filter by telling it there are process errors that do not exist. We may get away with that for some problems, in some conditions, but in general the Kalman filter's performance will be substandard.\n",
|
||||
"landmarks = array([[5, 10], [10, 5], [15, 15], [20, 5], [15, 10], \n",
|
||||
" [10,14], [23, 14], [25, 25], [10, 20]])\n",
|
||||
"\n",
|
||||
"Recall from the **Designing Kalman Filters** chapter that the acceleration is\n",
|
||||
"\n",
|
||||
"$$a_x = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_x \\\\\n",
|
||||
"a_y = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_y- g\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"These equations will be *very* unpleasant to work with while we develop this subject, so for now I will retreat to a simpler one dimensional problem using this simplified equation for acceleration that does not take the nonlinearity of the drag coefficient into account:\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"$$\\begin{aligned}\n",
|
||||
"\\ddot{y} &= \\frac{0.0034ge^{-y/20000}\\dot{y}^2}{2\\beta} - g \\\\\n",
|
||||
"\\ddot{x} &= \\frac{0.0034ge^{-x/20000}\\dot{x}^2}{2\\beta}\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"Here $\\beta$ is the ballistic coefficient, where a high number indicates a low drag."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This is still nonlinear, so we need to linearize this equation at the current state point. If our state is position and velocity, we need an equation for some arbitrarily small change in $\\mathbf{x}$, like so:\n",
|
||||
"\n",
|
||||
"$$ \\begin{bmatrix}\\Delta \\dot{x} \\\\ \\Delta \\ddot{x} \\\\ \\Delta \\dot{y} \\\\ \\Delta \\ddot{y}\\end{bmatrix} = \n",
|
||||
"\\large\\begin{bmatrix}\n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial x} & \n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial \\dot{x}} & \n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial y} & \n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial \\dot{y}} \\\\ \n",
|
||||
"\\frac{\\partial \\ddot{x}}{\\partial x} & \n",
|
||||
"\\frac{\\partial \\ddot{x}}{\\partial \\dot{x}}& \n",
|
||||
"\\frac{\\partial \\ddot{x}}{\\partial y}& \n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial \\dot{y}}\\\\\n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial x} & \n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial \\dot{x}} & \n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial y} & \n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial \\dot{y}} \\\\ \n",
|
||||
"\\frac{\\partial \\ddot{y}}{\\partial x} & \n",
|
||||
"\\frac{\\partial \\ddot{y}}{\\partial \\dot{x}}& \n",
|
||||
"\\frac{\\partial \\ddot{y}}{\\partial y}& \n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial \\dot{y}}\n",
|
||||
"\\end{bmatrix}\\normalsize\n",
|
||||
"\\begin{bmatrix}\\Delta x \\\\ \\Delta \\dot{x} \\\\ \\Delta \\dot{y} \\\\ \\Delta \\ddot{y}\\end{bmatrix}$$\n",
|
||||
"\n",
|
||||
"The equations do not contain both an x and a y, so any partial derivative with both in it must be equal to zero. We also know that $\\large\\frac{\\partial \\dot{x}}{\\partial x}\\normalsize = 0$ and that $\\large\\frac{\\partial \\dot{x}}{\\partial \\dot{x}}\\normalsize = 1$, so our matrix ends up being\n",
|
||||
"\n",
|
||||
"$$\\mathbf{F} = \\begin{bmatrix}0&1&0&0 \\\\\n",
|
||||
"\\frac{0.0034e^{-x/22000}\\dot{x}^2g}{44000\\beta}&0&0&0\n",
|
||||
"\\end{bmatrix}$$\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\\begin{aligned}\\ddot{x} &= -\\frac{1}{2}C_d\\rho A \\dot{x}\\\\\n",
|
||||
"\\ddot{y} &= -\\frac{1}{2}C_d\\rho A \\dot{y}-g\\end{aligned}$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 21,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sympy.abc import *\n",
|
||||
"from sympy import *\n",
|
||||
"\n",
|
||||
"init_printing(pretty_print=True, use_latex='mathjax')\n",
|
||||
"\n",
|
||||
"x1 = (0.0034*g*exp(-x/22000)*((x)**2))/(2*b) - g\n",
|
||||
"\n",
|
||||
"x2 = (a*g*exp(-x/c)*(Derivative(x)**2))/(2*b) - g\n",
|
||||
"\n",
|
||||
"#pprint(x1)\n",
|
||||
"#pprint(Derivative(x)*Derivative(x,n=2))\n",
|
||||
"#pprint(diff(x2, x))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"** orphan text\n",
|
||||
"This approach has many issues. First, of course, is the fact that the linearization does not produce an exact answer. More importantly, we are not linearizing the actual path, but our filter's estimation of the path. We linearize the estimation because it is statistically likely to be correct; but of course it is not required to be. So if the filter's output is bad that will cause us to linearize an incorrect estimate, which will almost certainly lead to an even worse estimate. In these cases the filter will quickly diverge. This is where the 'black art' of Kalman filter comes in. We are trying to linearize an estimate, and there is no guarantee that the filter will be stable. A vast amount of the literature on Kalman filters is devoted to this problem. Another issue is that we need to linearize the system using analytic methods. It may be difficult or impossible to find an analytic solution to some problems. In other cases we may be able to find the linearization, but the computation is very expensive. **\n"
|
||||
"ekf = run_localization(landmarks, sigma_vel=0.1, sigma_steer=np.radians(1),\n",
|
||||
" sigma_range=0.3, sigma_bearing=0.1)\n",
|
||||
"print(ekf.P.diagonal())"
|
||||
]
|
||||
}
|
||||
],
|
||||
@@ -1803,7 +1804,7 @@
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.4.1"
|
||||
"version": "3.4.3"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
|
||||
@@ -16,7 +16,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"execution_count": 1,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -30,7 +30,7 @@
|
||||
"@import url('http://fonts.googleapis.com/css?family=Arimo');\n",
|
||||
"\n",
|
||||
" div.cell{\n",
|
||||
" width: 850px;\n",
|
||||
" width: 900px;\n",
|
||||
" margin-left: 0% !important;\n",
|
||||
" margin-right: auto;\n",
|
||||
" }\n",
|
||||
@@ -117,9 +117,6 @@
|
||||
" font-family: 'Arimo',verdana,arial,sans-serif;\n",
|
||||
" line-height: 125%;\n",
|
||||
" font-size: 120%;\n",
|
||||
" width:740px;\n",
|
||||
" margin-left:auto;\n",
|
||||
" margin-right:auto;\n",
|
||||
" text-align:justify;\n",
|
||||
" text-justify:inter-word;\n",
|
||||
" }\n",
|
||||
@@ -240,7 +237,10 @@
|
||||
" },\n",
|
||||
" displayAlign: 'center', // Change this to 'center' to center equations.\n",
|
||||
" \"HTML-CSS\": {\n",
|
||||
" scale:85,\n",
|
||||
" scale:100,\n",
|
||||
" availableFonts: [\"Neo-Euler\"],\n",
|
||||
" preferredFont: \"Neo-Euler\",\n",
|
||||
" webFont: \"Neo-Euler\",\n",
|
||||
" styles: {'.MathJax_Display': {\"margin\": 4}}\n",
|
||||
" }\n",
|
||||
" });\n",
|
||||
@@ -250,7 +250,7 @@
|
||||
"<IPython.core.display.HTML object>"
|
||||
]
|
||||
},
|
||||
"execution_count": 3,
|
||||
"execution_count": 1,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@@ -368,16 +368,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAt0AAAEPCAYAAACeKXhLAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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 truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x7f50438>"
|
||||
"<matplotlib.figure.Figure at 0x6ee6160>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -488,19 +488,19 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAATBAMAAAC0B+rjAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAIokQdkQymVRmzbur3ZYT\n5QUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABhSURBVAgdY2BgYGAEYiBgQqFMILwwCKUAotgSChuB\nVAXjBC4BBu4JnAJMBxh8gdrYLzBcB1JcGxheMkgyyDUwLGGYyLCdgSGWQYG7AKitcqcaSDvjBBDJ\nwCkApqC2sYA4ADr+DeweQRu1AAAAAElFTkSuQmCC\n",
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAASBAMAAAB/WzlGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdolEMpnNqyK7EO/d\nVGbrAyG9AAAAX0lEQVQIHWNgYGBgBGIgYEKhjCE8ARDFEppeAaQqFyh0NzBwFjAbsBYwyC9gYmA6\nwHAUqI1zAsNHhoUM6y8wJDJcYShlYLBhUGAPAGqLm6sC0s7xAEQy8DWAKag1YEsBkk8P0x6j9zMA\nAAAASUVORK5CYII=\n",
|
||||
"text/plain": [
|
||||
"\\phi"
|
||||
]
|
||||
},
|
||||
"execution_count": 7,
|
||||
"execution_count": 3,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@@ -523,14 +523,14 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAwBAMAAAB+l06WAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZjKJ\nRFRer8KoAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOklEQVQ4EWNggAMhOAuJYfIJiQNjqoRhE2Vg\nH76irBk/s2C+H3r0fyzgA328YQ+xGs2yXCCfGSxmGuINk2MWALJ4QDymBIbzF6DCrA+ADLAomwID\nRwNUVBNEq4MIjgkMbL9BDCCIAxG3QQTjJ7go0wSQwAEQAQSc0DTKs4CBgXtD8GOwIMP9AxAaZGwc\n8wJGAzB3CkSQoYqBgWUBlwFPAojP6sDAwAtyHNDYSqDDWMHGlwEFKgQYGNgLGBimAkUZgaoY+AIY\njJnV/wItBXJ+MVgx2D8AMrYyMBQzMH1hYFgD5IgzLGZwBdLMc86EAQ2KN2CYBOTdZDjAEgCkGYGB\nBxS1D+A6AORxR3mdBlIwwPaJUwHEZl4AImGA+89TMJPLACYCpiWEwBQ4HBES70GWAB2NEAGxoL6H\nCAIAAx9b8PUdS4gAAAAASUVORK5CYII=\n",
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAwBAMAAAB+l06WAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZjKJ\nRFRer8KoAAABNklEQVQ4EWNggAMhOAuJYfIJiQNjqoRhE2VgH76irBk/s2C+H3r0fyzgA328YQ+x\nGs2yXDjfNMQbxmYWALLYQTymBIbzF6DCrA+ADB4Qh02BgaMBxAACTRABFuWYwMD2G8QDgjgQcRdE\nMH6CizJNAAkUgAgg4ISmUZ4FDAzcoWLJYEGG+wcgNMjYFIUDFgZg7hSIIEMVAwNLAu8F1gQQn9WB\ngYEX5DigsfUKPAw8QJqBoQyIKwSAzi9gYJgKdBiLA1CAL4DBmFn9L9BSIOc7gzqD/gMgYysDQzED\n0xcGhjVAjjjDa4ZUIM0850wY0KB4A4ZJQN5NhgNsAUCaERh4QFH7AK4DQB53lN9pIAUDbJ84FUBs\n5gUgEga4/zwFM7kMYCJgWkIITIFDDCHxHmQJNMwRoozIWgFe5Vu10Z1KpwAAAABJRU5ErkJggg==\n",
|
||||
"text/plain": [
|
||||
" 1 \n",
|
||||
"───────\n",
|
||||
@@ -538,7 +538,7 @@
|
||||
"2⋅╲╱ φ "
|
||||
]
|
||||
},
|
||||
"execution_count": 6,
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@@ -556,20 +556,20 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAIsAAAAcBAMAAACjVFE3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nCW5MapLtTWzvpPkP4lmKXRXJfWcAAAAASUVORK5CYII=\n",
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAIsAAAAcBAMAAACjVFE3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
|
||||
"text/plain": [
|
||||
" ⎛ 2 ⎞\n",
|
||||
"(φ - 1)⋅⎝φ + 1⎠"
|
||||
]
|
||||
},
|
||||
"execution_count": 7,
|
||||
"execution_count": 5,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@@ -587,14 +587,14 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"execution_count": 6,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ4AAABLCAMAAAB3JhEVAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nMB2G8BXjPe1/W0n2gP6pJLOpfwGec844NmVfmwAAAABJRU5ErkJggg==\n",
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAZ4AAABLCAMAAAB3JhEVAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nc844NmVfmwAAAABJRU5ErkJggg==\n",
|
||||
"text/plain": [
|
||||
"⎡ x - xₐ y - yₐ ⎤\n",
|
||||
"⎢ ────────────────────────── 0 ────────────────────────── 0⎥\n",
|
||||
@@ -609,7 +609,7 @@
|
||||
"⎣╲╱ (x - x_b) + (y - y_b) ╲╱ (x - x_b) + (y - y_b) ⎦"
|
||||
]
|
||||
},
|
||||
"execution_count": 8,
|
||||
"execution_count": 6,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
@@ -677,7 +677,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"execution_count": 7,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -707,7 +707,7 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"execution_count": 8,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
@@ -743,16 +743,16 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 11,
|
||||
"execution_count": 9,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x872c860>"
|
||||
"<matplotlib.figure.Figure at 0x81e45f8>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
@@ -838,6 +838,141 @@
|
||||
"We start by assuming some function $\\mathbf f$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Example: A falling Ball"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"**author's note: ignore this section for now. **\n",
|
||||
"\n",
|
||||
"In the **Designing Kalman Filters** chapter I first considered tracking a ball in a vacuum, and then in the atmosphere. The Kalman filter performed very well for vacuum, but diverged from the ball's path in the atmosphere. Let us look at the output; to avoid littering this chapter with code from that chapter I have placed it all in the file `ekf_internal.py'."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x81ac748>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"import ekf_internal\n",
|
||||
"ekf_internal.plot_ball()"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We can artificially force the Kalman filter to track the ball by making $Q$ large. That would cause the filter to mistrust its prediction, and scale the kalman gain $K$ to strongly favor the measurments. However, this is not a valid approach. If the Kalman filter is correctly predicting the process we should not 'lie' to the filter by telling it there are process errors that do not exist. We may get away with that for some problems, in some conditions, but in general the Kalman filter's performance will be substandard.\n",
|
||||
"\n",
|
||||
"Recall from the **Designing Kalman Filters** chapter that the acceleration is\n",
|
||||
"\n",
|
||||
"$$a_x = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_x \\\\\n",
|
||||
"a_y = (0.0039 + \\frac{0.0058}{1+\\exp{[(v-35)/5]}})*v*v_y- g\n",
|
||||
"$$\n",
|
||||
"\n",
|
||||
"These equations will be *very* unpleasant to work with while we develop this subject, so for now I will retreat to a simpler one dimensional problem using this simplified equation for acceleration that does not take the nonlinearity of the drag coefficient into account:\n",
|
||||
"\n",
|
||||
"\n",
|
||||
"$$\\begin{aligned}\n",
|
||||
"\\ddot{y} &= \\frac{0.0034ge^{-y/20000}\\dot{y}^2}{2\\beta} - g \\\\\n",
|
||||
"\\ddot{x} &= \\frac{0.0034ge^{-x/20000}\\dot{x}^2}{2\\beta}\n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n",
|
||||
"Here $\\beta$ is the ballistic coefficient, where a high number indicates a low drag."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"This is still nonlinear, so we need to linearize this equation at the current state point. If our state is position and velocity, we need an equation for some arbitrarily small change in $\\mathbf{x}$, like so:\n",
|
||||
"\n",
|
||||
"$$ \\begin{bmatrix}\\Delta \\dot{x} \\\\ \\Delta \\ddot{x} \\\\ \\Delta \\dot{y} \\\\ \\Delta \\ddot{y}\\end{bmatrix} = \n",
|
||||
"\\large\\begin{bmatrix}\n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial x} & \n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial \\dot{x}} & \n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial y} & \n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial \\dot{y}} \\\\ \n",
|
||||
"\\frac{\\partial \\ddot{x}}{\\partial x} & \n",
|
||||
"\\frac{\\partial \\ddot{x}}{\\partial \\dot{x}}& \n",
|
||||
"\\frac{\\partial \\ddot{x}}{\\partial y}& \n",
|
||||
"\\frac{\\partial \\dot{x}}{\\partial \\dot{y}}\\\\\n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial x} & \n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial \\dot{x}} & \n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial y} & \n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial \\dot{y}} \\\\ \n",
|
||||
"\\frac{\\partial \\ddot{y}}{\\partial x} & \n",
|
||||
"\\frac{\\partial \\ddot{y}}{\\partial \\dot{x}}& \n",
|
||||
"\\frac{\\partial \\ddot{y}}{\\partial y}& \n",
|
||||
"\\frac{\\partial \\dot{y}}{\\partial \\dot{y}}\n",
|
||||
"\\end{bmatrix}\\normalsize\n",
|
||||
"\\begin{bmatrix}\\Delta x \\\\ \\Delta \\dot{x} \\\\ \\Delta \\dot{y} \\\\ \\Delta \\ddot{y}\\end{bmatrix}$$\n",
|
||||
"\n",
|
||||
"The equations do not contain both an x and a y, so any partial derivative with both in it must be equal to zero. We also know that $\\large\\frac{\\partial \\dot{x}}{\\partial x}\\normalsize = 0$ and that $\\large\\frac{\\partial \\dot{x}}{\\partial \\dot{x}}\\normalsize = 1$, so our matrix ends up being\n",
|
||||
"\n",
|
||||
"$$\\mathbf{F} = \\begin{bmatrix}0&1&0&0 \\\\\n",
|
||||
"\\frac{0.0034e^{-x/22000}\\dot{x}^2g}{44000\\beta}&0&0&0\n",
|
||||
"\\end{bmatrix}$$\n",
|
||||
"\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"$$\\begin{aligned}\\ddot{x} &= -\\frac{1}{2}C_d\\rho A \\dot{x}\\\\\n",
|
||||
"\\ddot{y} &= -\\frac{1}{2}C_d\\rho A \\dot{y}-g\\end{aligned}$$"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 11,
|
||||
"metadata": {
|
||||
"collapsed": true
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"from sympy.abc import *\n",
|
||||
"from sympy import *\n",
|
||||
"\n",
|
||||
"init_printing(pretty_print=True, use_latex='mathjax')\n",
|
||||
"\n",
|
||||
"x1 = (0.0034*g*exp(-x/22000)*((x)**2))/(2*b) - g\n",
|
||||
"\n",
|
||||
"x2 = (a*g*exp(-x/c)*(Derivative(x)**2))/(2*b) - g\n",
|
||||
"\n",
|
||||
"#pprint(x1)\n",
|
||||
"#pprint(Derivative(x)*Derivative(x,n=2))\n",
|
||||
"#pprint(diff(x2, x))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"** orphan text\n",
|
||||
"This approach has many issues. First, of course, is the fact that the linearization does not produce an exact answer. More importantly, we are not linearizing the actual path, but our filter's estimation of the path. We linearize the estimation because it is statistically likely to be correct; but of course it is not required to be. So if the filter's output is bad that will cause us to linearize an incorrect estimate, which will almost certainly lead to an even worse estimate. In these cases the filter will quickly diverge. This is where the 'black art' of Kalman filter comes in. We are trying to linearize an estimate, and there is no guarantee that the filter will be stable. A vast amount of the literature on Kalman filters is devoted to this problem. Another issue is that we need to linearize the system using analytic methods. It may be difficult or impossible to find an analytic solution to some problems. In other cases we may be able to find the linearization, but the computation is very expensive. **\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
@@ -869,7 +1004,7 @@
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.4.3"
|
||||
"version": "3.4.1"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
|
||||
+4
-4
@@ -40,8 +40,8 @@ def residual(a,b):
|
||||
return y
|
||||
|
||||
|
||||
sigma_r = 0.1
|
||||
sigma_h = np.radians(1)
|
||||
sigma_r = 1
|
||||
sigma_h = .1#np.radians(1)
|
||||
sigma_steer = np.radians(1)
|
||||
|
||||
#only partway through. predict is using old control and movement code. computation of m uses
|
||||
@@ -130,8 +130,8 @@ def H_of(x, p):
|
||||
dist = np.sqrt(hyp)
|
||||
|
||||
H = array(
|
||||
[[-(px - x[0, 0]) / dist, -(py - x[1, 0]) / dist, 0],
|
||||
[ (py - x[1, 0]) / hyp, -(px - x[0, 0]) / hyp, -1]])
|
||||
[[(-px + x[0, 0]) / dist, (-py + x[1, 0]) / dist, 0.],
|
||||
[ -(-py + x[1, 0]) / hyp, -(px - x[0, 0]) / hyp, -1.]])
|
||||
return H
|
||||
|
||||
|
||||
|
||||
Reference in new issue
Block a user