Symbology changes.
Working on finalizing the symbology used in this book. Added appendix to document what other books in the field use for the Kalman filter equations.
This commit is contained in:
Roger Labbe
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Appendix_Symbols_and_Notations.ipynb
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Appendix_Symbols_and_Notations.ipynb
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"source": [
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"Appendix I : Symbology"
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]
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"source": [
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"This is just notes at this point. \n",
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"\n",
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"### State\n",
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"\n",
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"$x$ (Brookner, Zarchan, Brown)\n",
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"\n",
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"$\\underline{x}$ Gelb)\n",
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"\n",
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"### State at step n\n",
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"\n",
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"$x_n$ (Brookner)\n",
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"\n",
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"$x_k$ (Brown, Zarchan)\n",
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"\n",
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"$\\underline{x}_k$ (Gelb)\n",
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"\n",
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"\n",
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"\n",
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"### Prediction\n",
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"\n",
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"$x^-$\n",
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"\n",
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"$x_{n,n-1}$ (Brookner) \n",
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"\n",
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"$x_{k+1,k}$\n",
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"\n",
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"\n",
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"## measurement\n",
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"\n",
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"\n",
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"$x^*$\n",
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"\n",
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"\n",
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"\n",
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"Y_n (Brookner)\n",
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"\n",
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"##control transition Matrix\n",
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"\n",
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"$G$ (Zarchan)\n",
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"\n",
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"\n",
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"Not used (Brookner)"
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"source": [
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"#Nomenclature\n",
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"\n",
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"\n",
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"## Equations\n",
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"### Brookner\n",
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"\n",
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"$$\n",
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"\\begin{aligned}\n",
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"X^*_{n+1,n} &= \\Phi X^*_{n,n} \\\\\n",
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"X^*_{n,n} &= X^*_{n,n-1} +H_n(Y_n - MX^*_{n,n-1}) \\\\\n",
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"H_n &= S^*_{n,n-1}M^T[R_n + MS^*_{n,n-1}M^T]^{-1} \\\\\n",
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"S^*_{n,n-1} &= \\Phi S^*_{n-1,n-1}\\Phi^T + Q_n \\\\\n",
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"S^*_{n-1,n-1} &= (I-H_{n-1}M)S^*_{n-1,n-2}\n",
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"\\end{aligned}$$\n",
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"\n",
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"### Gelb\n",
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"\n",
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"$$\n",
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"\\begin{aligned}\n",
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"\\underline{\\hat{x}}_k(-) &= \\Phi_{k-1} \\underline{\\hat{x}}_{k-1}(+) \\\\\n",
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"\\underline{\\hat{x}}_k(+) &= \\underline{\\hat{x}}_k(-) +K_k[Z_k - H_k\\underline{\\hat{x}}_k(-)] \\\\\n",
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"K_k &= P_k(-)H_k^T[H_kP_k(-)H_k^T + R_k]^{-1}\\\\\n",
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"P_k(+) &= \\Phi_{k-1} P_{k-1}(+)\\Phi_{k-1}^T + Q_{k-1} \\\\\n",
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"P_k(-) &= (I-K_kH_k)P_k(-)\n",
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"\\end{aligned}$$\n",
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"\n",
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"\n",
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"### Brown\n",
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"\n",
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"$$\n",
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"\\begin{aligned}\n",
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"\\hat{x}^-_{k+1} &= \\phi_{k}\\hat{x}_{k} \\\\\n",
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"\\hat{x}_k &= \\hat{x}^-_k +K_k[z_k - H_k\\hat{x}^-_k] \\\\\n",
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"K_k &= P^-_kH_k^T[H_kP^-_kH_k^T + R_k]^{-1}\\\\\n",
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"P^-_{k+1} &= \\phi_k P_k\\phi_k^T + Q_{k} \\\\\n",
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"P_k &= (I-K_kH_k)P^-_k\n",
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"\\end{aligned}$$\n",
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"## \n",
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"\n",
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"### Zarchan\n",
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"\n",
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"$$\n",
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"\\begin{aligned}\n",
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"\\hat{x}_{k} &= \\Phi_{k}\\hat{x}_{k-1} + G_ku_{k-1} + K_k[z_k - H\\Phi_{k}\\hat{x}_{k-1} - HG_ku_{k-1} ] \\\\\n",
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"M_{k} &= \\Phi_k P_{k-1}\\phi_k^T + Q_{k} \\\\\n",
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"K_k &= M_kH^T[HM_kH^T + R_k]^{-1}\\\\\n",
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"P_k &= (I-K_kH)M_k\n",
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"\\end{aligned}$$\n",
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"\n",
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"\n",
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"\n",
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"\n",
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"\n",
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File diff suppressed because one or more lines are too long
@ -86,7 +86,8 @@ Contents
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* [**Chapter N: Designing Nonlinear Kalman Filters**](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Designing_Nonlinear_Kalman_Filters.ipynb)
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Building on the material in chapters 8 and 9 we walk through the design of several nonlinear filters.
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*[**Appendix: Symbols and Notations**](http://nbviewer.ipython.org/github/rlabbe/Kalman-and-Bayesian-Filters-in-Python/blob/master/Appendix_Symbols_and_Notations.ipynb)
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Math symbology used by this book and by the notable books in the literature.
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Reading the book
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def show_residual_chart():
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plt.xlim([0.9,2.5])
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plt.ylim([0.5,2.5])
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plt.ylim([1.5,3.5])
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plt.scatter ([1,2,2],[1,2,1.3])
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plt.scatter ([2],[1.8],marker='o')
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plt.scatter ([1,2,2],[2,3,2.3])
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plt.scatter ([2],[2.8],marker='o')
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ax = plt.axes()
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ax.annotate('', xy=(2,2), xytext=(1,1),
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ax.annotate('', xy=(2,3), xytext=(1,2),
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arrowprops=dict(arrowstyle='->', ec='b',shrinkA=3, shrinkB=4))
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ax.annotate('prediction', xy=(2.04,2.), color='b')
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ax.annotate('measurement', xy=(2.05, 1.28))
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ax.annotate('prior measurement', xy=(1, 0.9))
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ax.annotate('residual', xy=(2.04,1.6), color='r')
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ax.annotate('new estimate', xy=(2,1.8),xytext=(2.1,1.8),
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ax.annotate('prediction', xy=(2.04,3.), color='b')
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ax.annotate('measurement', xy=(2.05, 2.28))
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ax.annotate('prior estimate', xy=(1, 1.9))
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ax.annotate('residual', xy=(2.04,2.6), color='r')
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ax.annotate('new estimate', xy=(2,2.8),xytext=(2.1,2.8),
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arrowprops=dict(arrowstyle='->', ec="k", shrinkA=3, shrinkB=4))
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ax.annotate('', xy=(2,2), xytext=(2,1.3),
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ax.annotate('', xy=(2,3), xytext=(2,2.3),
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arrowprops=dict(arrowstyle="-",
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ec="r",
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shrinkA=5, shrinkB=5))
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@ -124,4 +124,4 @@ def show_x_with_unobserved():
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if __name__ == "__main__":
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show_x_with_unobserved()
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show_residual_chart()
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@ -122,7 +122,7 @@
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" \n",
|
||||
"\n",
|
||||
"\n",
|
||||
"[**Appendix: Symbols and Notations**](not implemented)\n",
|
||||
"[**Appendix: Symbols and Notations**](http://nbviewer.ipython.org/urls/raw.github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python/master/Appendix_Symbols_and_Notations.ipynb)\n",
|
||||
"\n",
|
||||
"Symbols and notations used in this book. Comparison with notations used in the literature.\n",
|
||||
"\n",
|
||||
|
||||
Loading…
Reference in New Issue
Block a user