Double Integrator Model

This page describes the double integrator vehicle model, which abstracts the vehicle motion via fully decoupled longitudinal and lateral dynamics in a curvilinear coordinate system.

Currently, we do not support state conversion from the double integrator model to the kinematic/dynamic bicycle model or vice versa!

See also the vehicle model interface documentation for additional information.

---

State Vector

The state vector is defined as:

\[ \mathbf{x} = \begin{bmatrix} s \\ d \\ v_s \\ v_d \end{bmatrix} \]

where:

Symbol	Description
\(s, d\)	Position in a given curvilinear coordinate system.
\(v_s\)	Longitudinal velocity (along the reference path)
\(v_d\)	Lateral velocity (with respect to the reference path)

---

Control Inputs

The control input vector is:

\[ \mathbf{u} = \begin{bmatrix} a_s \\ a_d \end{bmatrix} \]

where:

Symbol	Description
\(a_s\)	Longitudinal acceleration
\(a_d\)	Lateral acceleration

Input bounds are derived from the vehicle parameters a_long_max, a_lat_max.

---

Continuous-Time Dynamics

The nominal (continuous-time) dynamics are governed by the following set of differential equations:

---

Longitudinal Motion

\[ \dot{v_s} = u_1 \]

---

Lateral Motion

\[ \dot{v_d} = u_2 \]

---

State Bounds

The velocity bounds are enforced as:

\[ v\_{x,\min} \le v_x \le v\_{x,\max}, \quad v\_{y,\min} \le v_y \le v\_{y,\max} \]

with typical values:

• \(v_x \in [0, 10]\) m/s
• \(v_y \in [-2, 2]\) m/s

---

Normalized Accelerations

Normalized longitudinal and lateral accelerations are computed as:

\[ a\_{\mathrm{long,norm}} = \frac{a_x}{a\_{\mathrm{long,max}}} \]

\[ a\_{\mathrm{lat,norm}} = \frac{a_y}{a\_{\mathrm{lat,max}}} \]

---

Time Discretization

In contrast to the nonlinear vehicle models, we use the matrix exponential to obtain the nominal discrete-time dynamics of the double integator model.

---