Merge branch 'main' of https://github.com/DojoSim/Dojo.jl into main

Author: simon-lc

docs/src/system_identification.md

@@ -1,2 +1,45 @@
 # System Identification
 
+A [real-world dataset](https://github.com/DAIRLab/contact-nets/tree/main/data) is used to learn the geometric and friction properties of a block being tossed onto a flat surface. Gradient-based optimization is employed to regress parameters and real-to-sim validation is performed. The ground truth system is shown in orange and the learned system in blue.
+
+## Geometry 
+
+```@raw html
+<img src="../../examples/animations/box_learning.gif" width="300"/>
+```
+
+The eight locations of the block's corners relative to its center of mass are learned.
+
+## Friction 
+
+```@raw html
+<img src="../../examples/animations/cone_learning.gif" width="200"/>
+```
+
+A friction coefficient, describing a friction cone, is learned for all of the contact points.
+
+## Real-To-Sim 
+
+```@raw html
+<img src="../../examples/animations/box_toss.gif" width="300"/>
+```
+
+The system parameters are learned to within a ``\pm 5 \%`` error from their ground-truth values. These parameters are then compared to the ground-truth system in simulation.
+
+## Learning 
+
+The cost function: 
+
+``\mathcal{L}(\mathcal{D}, \theta) = \sum_{Z \in \mathcal{D}} L(Z, \theta) = \sum_{Z \in \mathcal{D}} \frac{1}{2} ||s(z_{-}, z, \theta) - z_{+}||_W^2``,
+
+is used where ``\mathcal{D}`` is a dataset of trajectories containing tuples ``(z_{-}, z, z_{+})`` of state sequences, with system parameters ``\theta \in \mathbf{R}^p``, and where ``s : \mathbf{Z} \times \mathbf{Z} \times \mathbf{R}^p \rightarrow \mathbf{Z}`` represents the simulator.
+
+A quasi-Newton method is employed to optimize the cost function and uses gradients:
+
+``\frac{\partial L}{\partial \theta} = {\frac{\partial s}{\partial \theta}}^T W \Big(s(z_{-}, z, \theta) - z_{+} \Big)``,
+
+and the following Gauss-Newton approximation: 
+
+``\frac{\partial^2 L}{\partial \theta^2} \approx {\frac{\partial s}{\partial \theta}}^T W \frac{\partial s}{\partial \theta}``,
+
+of the cost function Hessian, which only relies on Jacobians from Dojo.