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Roger Labbe 2015-12-20 12:42:30 -08:00
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8
06-Multivariate-Kalman-Filters.ipynb
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@ -2177,8 +2177,8 @@
"metadata": {},
"source": [
"$$\\begin{aligned}\\mathbf x \n",
"&=\\mathbf{\\bar x}+\\mathbf K(z-\\mathbf{Hx)} \\\\\n",
"&= \\mathbf{\\bar x}+\\mathbf Ky\\end{aligned}$$\n",
"&=\\mathbf{\\overline x}+\\mathbf K(z-\\mathbf{Hx)} \\\\\n",
"&= \\mathbf{\\overline x}+\\mathbf Ky\\end{aligned}$$\n",
"\n",
"Here the residual $y$ is a scalar. Therefore it is multiplied into each element of $\\mathbf K$. Therefore we have\n",
"\n",
@ -2186,10 +2186,10 @@
"\n",
"Which gives this system of equations: \n",
"\n",
"$$x = \\bar x + yK_x\\\\\n",
"$$x = \\overline x + yK_x\\\\\n",
"\\dot x = \\overline{\\dot x} + yK_{\\dot x}$$\n",
"\n",
"The prediction $\\bar x$ was computed as $x + \\bar x \\Delta t$. If the prediction was perfect then the residual will be $y=0$ (ignoring noise in the measurement) and the velocity estimate will be unchanged. On the other hand, if the velocity estimate was very bad then the prediction will be very bad, and the residual will be large: $y >> 0$. In this case we update the velocity estimate with $yK_{\\dot x}$. $K_{\\dot x}$ is proportional to $COV(x,\\dot x)$. Therefore the velocity is updated by the error in the position times the a value proportional to the covariance between the position and velocity. The higher the correlation the larger the correction. \n",
"The prediction $\\overline x$ was computed as $x + \\overline x \\Delta t$. If the prediction was perfect then the residual will be $y=0$ (ignoring noise in the measurement) and the velocity estimate will be unchanged. On the other hand, if the velocity estimate was very bad then the prediction will be very bad, and the residual will be large: $y >> 0$. In this case we update the velocity estimate with $yK_{\\dot x}$. $K_{\\dot x}$ is proportional to $COV(x,\\dot x)$. Therefore the velocity is updated by the error in the position times the a value proportional to the covariance between the position and velocity. The higher the correlation the larger the correction. \n",
"\n",
"To bring this full circle, $COV(x,\\dot x)$ is the off-diagonal elements of $\\mathbf P$. Recall that those values were computed with $\\mathbf{FPF}^\\mathsf T$. So the covariance of position and velocity is computed during the predict step. The Kalman gain for the velocity is proportional to this covariance, and we adjust the velocity estimate based on how inaccurate it was during the last epoch times a value proportional to this covariance. \n",
"\n",

22
07-Kalman-Filter-Math.ipynb
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@ -1208,16 +1208,16 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"The Kalman filter predict equation is $\\mathbf{\\bar{x}} = \\mathbf{Fx} + \\mathbf{Bu}$. Hence the prediction is\n",
"The Kalman filter predict equation is $\\mathbf{\\overline x} = \\mathbf{Fx} + \\mathbf{Bu}$. Hence the prediction is\n",
"\n",
"$$\\mathbf{\\bar{x}} = \\begin{bmatrix}\n",
"$$\\mathbf{\\overline x} = \\begin{bmatrix}\n",
"1 & \\Delta t \\\\ 0 & 1\\end{bmatrix}\\begin{bmatrix}\n",
"x\\\\ \\dot x\\end{bmatrix}\n",
"$$\n",
"\n",
"which multiplies out to \n",
"\n",
"$$\\begin{aligned}\\bar{x} &= x + v\\Delta t \\\\\n",
"$$\\begin{aligned}\\overline x &= x + v\\Delta t \\\\\n",
"\\bar{\\dot x} &= \\dot x\\end{aligned}$$.\n",
"\n",
"This works for linear ordinary differential equations (ODEs), but does not work well for nonlinear equations. For example, consider trying to predict the position of a rapidly turning car. Cars turn by pivoting the front wheels, which cause the car to pivot around the rear axle. Therefore the path will be continuously varying and a linear prediction will necessarily produce an incorrect value. If the change in the system is small enough relative to $\\Delta t$ this can often produce adequate results, but that will rarely be the case with the nonlinear Kalman filters we will be studying in subsequent chapters. Another problem is that even trivial systems produce differential equations for which finding closed form solutions is difficult or impossible. \n",
@ -1570,14 +1570,14 @@
"\n",
"$$\n",
"\\begin{aligned}\n",
"\\mathbf{\\bar{x}} &= \\mathbf{F x} + \\mathbf{B u} \\\\\n",
"\\mathbf{\\overline x} &= \\mathbf{F x} + \\mathbf{B u} \\\\\n",
"\\mathbf{\\bar{P}} &= \\mathbf{FPF}^\\mathsf{T} + \\mathbf Q\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"For a univariate problem the state $\\mathbf x$ only has one variable, so it is a $1\\times 1$ matrix. Our motion $\\mathbf{u}$ is also a $1\\times 1$ matrix. Therefore, $\\mathbf{F}$ and $\\mathbf B$ must also be $1\\times 1$ matrices. That means that they are all scalars, and we can write\n",
"\n",
"$$\\bar{x} = Fx + Bu$$\n",
"$$\\overline x = Fx + Bu$$\n",
"\n",
"Here the variables are not bold, denoting that they are not matrices or vectors. \n",
"\n",
@ -1612,8 +1612,8 @@
"$$\n",
"\\begin{aligned}\n",
"\\mathbf{K}&= \\mathbf{\\bar{P}H}^\\mathsf{T} (\\mathbf{H\\bar{P}H}^\\mathsf{T} + \\mathbf R)^{-1} \\\\\n",
"\\textbf{y} &= \\mathbf z - \\mathbf{H \\bar{x}}\\\\\n",
"\\mathbf x&=\\mathbf{\\bar{x}} +\\mathbf{K\\textbf{y}} \\\\\n",
"\\textbf{y} &= \\mathbf z - \\mathbf{H \\overline x}\\\\\n",
"\\mathbf x&=\\mathbf{\\overline x} +\\mathbf{K\\textbf{y}} \\\\\n",
"\\mathbf P&= (\\mathbf{I}-\\mathbf{KH})\\mathbf{\\bar{P}}\n",
"\\end{aligned}\n",
"$$\n",
@ -1623,8 +1623,8 @@
"$$\n",
"\\begin{aligned}\n",
"K &=\\frac{\\bar{P}}{\\bar{P} + R} \\\\\n",
"y &= z - \\bar{x}\\\\\n",
"x &=\\bar{x}+Ky \\\\\n",
"y &= z - \\overline x\\\\\n",
"x &=\\overline x+Ky \\\\\n",
"P &= (1-K)\\bar{P}\n",
"\\end{aligned}\n",
"$$\n",
@ -1801,9 +1801,9 @@
"\n",
"Let's start with some definitions which should be familiar to you. First, we define the innovation as \n",
"\n",
"$$\\delta \\mathbf{\\bar{z}}= \\mathbf z - h(\\mathbf{\\bar{x}})$$\n",
"$$\\delta \\mathbf{\\bar{z}}= \\mathbf z - h(\\mathbf{\\overline x})$$\n",
"\n",
"where $\\mathbf z$ is the measurement, $h(\\bullet)$ is the measurement function, and $\\delta \\mathbf{\\bar{z}}$ is the innovation, which we abbreviate as $y$ in FilterPy. In other words, this is the equation $\\mathbf{y} = \\mathbf z - \\mathbf{H\\bar{x}}$ in the linear Kalman filter's update step.\n",
"where $\\mathbf z$ is the measurement, $h(\\bullet)$ is the measurement function, and $\\delta \\mathbf{\\bar{z}}$ is the innovation, which we abbreviate as $y$ in FilterPy. In other words, this is the equation $\\mathbf{y} = \\mathbf z - \\mathbf{H\\overline x}$ in the linear Kalman filter's update step.\n",
"\n",
"Next, the *measurement residual* is\n",
"\n",