docs: comparing slam algorithms

docs/how-to-guides/creating-maps-for-autoware/open-source-slam/.pages

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   - Optimized-SC-F-LOAM: optimized-sc-f-loam
   - SC-A-LOAM: sc-a-loam
   - SC-LeGO-LOAM: sc-lego-loam
+  - Comparing Errors of SLAM Algorithms: compare-slam-algorithms

docs/how-to-guides/creating-maps-for-autoware/open-source-slam/compare-slam-algorithms/.pages

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+nav:
+  - index.md

docs/how-to-guides/creating-maps-for-autoware/open-source-slam/compare-slam-algorithms/index.md

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+# Comparing Open Source SLAM Algorithms
+
+In order to compare SLAM algorithms their trajectory errors can be
+compared in terms of positions and orientations, thus ground truth
+should be generated and GNSS/INS can be used for this purpose.
+
+## Setup
+
+We have used a mapping kit which contains Velodyne VLP16 LiDAR and
+Applanix LVX GNSS/INS sensors. With using this kit we could collect
+data for ground truth and SLAM algorithms simultaneously. Since ground
+truth is generated by GNSS/INS, it should be transformed to LiDAR frame.
+Additionally, average positional error in the postprocessed GNSS/INS data
+is 0.06 meters.
+
+### Transform
+
+To calculate error's of slam algorithms ground truth trajectory and
+slam trajectories must be in the same frame. Hence, whether ground
+truth or slam trajectory should rotate and translate to the other's
+frame.
+
+Tf tree of this transform is:
+
+<p align="center">
+<img src="images/maptfinal.drawio.png" width="191" height="628">
+</p>
+
+To perform necessary rotation and translation firstly transform matrices
+are calculated then transforms applied on GNSS/INS positions by using
+calculated transform matrices.
+
+Before transformation:
+
+<p align="center">
+<img src="images/gnss_hdl_path_no_Tf.png" width="628" height="442">
+</p>
+
+## Results
+
+RViz output of both of the ground truth and slam trajectories and
+the graphs of the positional errors in meters of the slam algorithms
+are depicted below.
+
+|                         SLAM Algorithm                          |                    Trajectory After Transformation                     |                             Position Error Graphs                              |        Average Position Error (meters)         |
+| :-------------------------------------------------------------: | :--------------------------------------------------------------------: | :----------------------------------------------------------------------------: | :--------------------------------------------: |
+|   [hdl_graph_slam](https://github.com/koide3/hdl_graph_slam)    |    <img src="images/hdl_vs_gnss_path.png" width="200" height="400">    | <img src="images/hdl_graph_slam_position_errors.png" width="628" height="314"> | x:11.903188 <br> y: 3.887582 <br> z: 13.905123 |
+| [lidarslam_ros2](https://github.com/rsasaki0109/lidarslam_ros2) | <img src="images/lidarslam_vs_gnss_path.png" width="200" height="400"> |   <img src="images/lidarslam_position_errors.png" width="628" height="314">    | x: 8.949974 <br> y: 2.818557 <br> z: 16.495759 |
+
+## TODO
+
+- Transform between GNSS/INS trajectory and SLAM trajectory
+  will be implemented using ros tf.
+- SLAM algorithms tested with IMU data.
+- Remaining SLAM algorithms will be tested.