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Roger Labbe 2015-02-16 11:06:59 -08:00
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10_Unscented_Kalman_Filter.ipynb
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"I explained at the beginning of the chapter that there are two places for nonlinearities to occur. The first is in the state transition function, and the second is in the measurement function. For the linear Kalman filter we use the equation\n",
"\n",
"$$ \\mathbf{x}^-=\\mathbf{Fx}$$\n",
"$$ \\bf{x}^-=\\bf{Fx}$$\n",
"\n",
"to propagate the state from one epoch to the next. As you remember F defines a set of linear equations that performs this calculation. However, suppose we are tracking a ball or missile in flight. Air drag causes the behavior to be nonlinear, and therefore there are no linear equations that will compute the state.\n",
"\n",
"The linear Kalman filter uses the measurement function in this equation\n",
"\n",
"$$\\hat{\\mathbf{x}} = \\mathbf{K}(\\mathbf{z-Hx}^-)$$\n",
"$$\\hat{\\bf{x}} = \\bf{K}(\\bf{z-Hx}^-)$$\n",
"\n",
"to incorporate the measurements into the state. The term $\\mathbf{Hx}$ converts $\\mathbf{x}$ to a measurement using the linear equations implied by \\mathbf{H}$.\n",
"to incorporate the measurements into the state. The term $\\bf{Hx}$ converts $\\bf{x}$ to a measurement using the linear equations implied by $\\bf{H}$.\n",
"\n",
"The UKF assumes that both the measurement function and state transition function are nonlinear, though in some problems only one or the other are nonlinear. If they are nonlinear we can not represent them with a matrix. Instead, we will define a function $h(x)$ for the measurement function and $f(x)$ for the state transition function. We will implement these as Python functions that take a state $x$ and return either a state $x$ for the state transition function, or a measurement vector $z$ for the measurement function. The functions will look something like:\n",
"\n",