Fixing the first node in Graph-based SLAM - Robotics Stack Exchange most recent 30 from robotics.stackexchange.com 2019-12-07T20:30:51Z https://robotics.stackexchange.com/feeds/question/16240 https://creativecommons.org/licenses/by-sa/4.0/rdf https://robotics.stackexchange.com/q/16240 2 Fixing the first node in Graph-based SLAM C.O Park https://robotics.stackexchange.com/users/19219 2018-08-16T21:50:51Z 2018-08-23T16:44:12Z <p>The first node in the graph SLAM should be fixed. The famous <a href="https://ieeexplore.ieee.org/document/5681215/" rel="nofollow noreferrer">"a tutorial on graph based slam"</a> paper is showing that we can fix a node by adding identity matrix. Why adding identity to the Hessian of the specific node results in fixing that node? What is the theory behind it? Any good material to read? </p> <p><a href="https://i.stack.imgur.com/ZnNPO.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ZnNPO.png" alt="enter image description here"></a></p> https://robotics.stackexchange.com/questions/16240/-/16273#16273 1 Answer by al-dev for Fixing the first node in Graph-based SLAM al-dev https://robotics.stackexchange.com/users/5610 2018-08-23T16:44:12Z 2018-08-23T16:44:12Z <p>Not a definitive answer but here are my thoughts: Fixing node $k$ is equivalent to enforcing $\Delta x_{k} = 0$. Now we must show that adding $I$ to diagonal block $H_{kk}$ will result in $\Delta x_{k} = 0$. Let's name $H_{orig}$ the original matrix $H$ before adding $I$ to diagonal block $H_{kk}$, we have: $$(H_{orig}+\begin{bmatrix} 0 &amp; &amp; &amp; &amp; &amp; \\ &amp; ... &amp; &amp; &amp; &amp; \\ &amp; &amp; 0 &amp; &amp; &amp; \\ &amp; &amp; &amp; I &amp; &amp; \\ &amp; &amp; &amp; &amp; 0 &amp; \\ &amp; &amp; &amp; &amp; &amp; ... \\ &amp; &amp; &amp; &amp; &amp; &amp; 0 \end{bmatrix})\Delta x = -b \Leftrightarrow H_{orig}\Delta x + \Delta x_{k} = -b$$ Can we show that $H_{orig}\Delta x + \Delta x_{k} = -b \Rightarrow \Delta x_{k} = 0$ ?</p>