2 Single Integrator (2D) – Kinodynamic Compendium

2.1 Dynamics

Parameters
- state space \(\mathcal{X}\)
- action space \(\mathcal{U}\)
State: \(\mathbf{x}= \begin{pmatrix}x, y\end{pmatrix}^\top \in \mathcal{X}\) [m, m] (position of the robot in the world frame)
Action: \(\mathbf{u}= \begin{pmatrix} v_x, v_y \end{pmatrix}^\top \in \mathcal{U}\) [m/s, m/s] (speed of the robot in the world frame)
Dynamics: \[ \begin{aligned} \dot x &= v_x\\ \dot y &= v_y \\ \end{aligned} \]

2.2 Differential Flatness

Pick flat outputs \(\mathbf{z}(t) = (x(t), y(t))^\top\), i.e., the position of the robot. Then we can compute all necessary variables if \(\mathbf{z}(t)\) is at least C1-continuous. \[ \begin{aligned} \mathbf{x}(t) &= g_x(\mathbf{z}) = \left(x, y\right)\\ \mathbf{u}(t) &= g_u(\dot{\mathbf{z}}) = \left(\dot x, \dot y \right) \end{aligned} \]

2.3 Invariance

The dynamics are translation-invariant.

2.4 Controllers

The system is linear, so an LQR controller is most suitable.

2.5 Useful Parameters

2.5.1 integrator1_2d_v0

A basic version proposed at (Ortiz-Haro et al. (2024))

\[ \mathcal{U}= [-0.5, 0.5] m/s \times [-0.5, 0.5] ms/s \]