https://mmt.github.io/smark-notes/posts/ Recent content in Posts on sMark Notes [DRAFT] Hugo -- gohugo.io en-us © Copyright Mark M. Tobenkin 2020 Mon, 04 Jan 2021 10:38:42 -0800 https://mmt.github.io/smark-notes/posts/navigation/so3_trapz/ Mon, 04 Jan 2021 10:38:42 -0800 https://mmt.github.io/smark-notes/posts/navigation/so3_trapz/ This post is about numerical integration on $SO(3)$, i.e. how to approximate the solution of: $$ \frac{d}{dt} C(t) = C(t)[\omega(t)]_\times $$ given only $C(t_0) \in SO(3)$ and samples of $\omega(\cdot)$. https://mmt.github.io/smark-notes/posts/navigation/bortz/ Sun, 03 Jan 2021 20:00:21 -0800 https://mmt.github.io/smark-notes/posts/navigation/bortz/ This post gives a derivation of the Bortz Equation, which is useful for deriving integration schemes for rotation matrices. https://mmt.github.io/smark-notes/posts/geometry/so3_deriv/ Sun, 03 Jan 2021 20:00:07 -0800 https://mmt.github.io/smark-notes/posts/geometry/so3_deriv/ This post is about finding the directional derivative: $$ \left . https://mmt.github.io/smark-notes/posts/navigation/so3_integrals/ Sun, 03 Jan 2021 11:48:07 -0800 https://mmt.github.io/smark-notes/posts/navigation/so3_integrals/ This post is about a family of integrals that are important when computing state transition matrices in certain navigation problems as well as some other applications, namely: $$ C_n(v, T) : = \int_0^{T} \int_0^{s_n} \ldots \int_0^{s_2}\exp(s_1 [v]_\times) ds_1 \; ds_2 \;\ldots \; ds_n \qquad \; n \in \{1, 2, \ldots \}. https://mmt.github.io/smark-notes/posts/geometry/cross_product_power_series/ Thu, 31 Dec 2020 23:24:05 -0800 https://mmt.github.io/smark-notes/posts/geometry/cross_product_power_series/ This post is about power series of the form: $$ \alpha_0 I + \sum_{n=0}^\infty \alpha_n [v]_\times^n. https://mmt.github.io/smark-notes/posts/systems/transition_matrices/ Mon, 28 Dec 2020 23:38:22 -0800 https://mmt.github.io/smark-notes/posts/systems/transition_matrices/ This post has a few tricks for deriving state transitions matrices. https://mmt.github.io/smark-notes/posts/systems/parameter_sensitivity/ Mon, 28 Dec 2020 10:45:34 -0800 https://mmt.github.io/smark-notes/posts/systems/parameter_sensitivity/ We can find a general expression for the derivative: $$ \left . https://mmt.github.io/smark-notes/posts/geometry/cross_product/ Sun, 27 Dec 2020 12:55:13 -0800 https://mmt.github.io/smark-notes/posts/geometry/cross_product/ This note reviews important properties of what you might call the cross product matrix.