Motion Models

All motion models below are simplifications of a real system. The Unity car is more complex than all of them, but the Unity drone is quite similar to the Dynamic Point model. You decide if and how you use them in the algorithms you create.

Kinematic Point model

The kinematic point model has a velocity bound, but is free to instantly change velocity within that bound. It is a reasonable model for movements on time scales such that accelerations and rotations can be assumed to happen instantly. Such as agents moving (sufficiently) large distances over long times.

$\begin{array}{ccc} \dot{x} & = & u_{x} \\ \dot{y} & = & u_{y} \end{array}$ $\begin{array}{ccc} \dot x& = &u_x\\ \dot y& = &u_y \end{array}$

with $| | (u_{x}, u_{y}) | | \leq v_{m a x}$ $||(u_x, u_y)|| \leq v_{max}$

Dynamic Point model

The dynamic point model has bounds on both velocity and acceleration, and provides a reasonable model for quadcopters, hovercraft and spaceships.

$\begin{array}{ccc} \ddot{x} & = & u_{x} \\ \ddot{y} & = & u_{y} \end{array}$ $\begin{array}{ccc} \ddot x& = &u_x\\ \ddot y& = &u_y \end{array}$

with $| | (u_{x}, u_{y}) | | \leq a_{m a x} and | | (\dot{x}, \dot{y}) | | \leq v_{m a x}$ $||(u_x, u_y)|| \leq a_{max} \mbox{ and } ||(\dot x, \dot y)|| \leq v_{max}$

Differential Drive model

The differential drive model has an orientation $θ$ $\theta$ , and can not move sideways without turning first. It has bounds on velocity as well as angular velocity. It is a reasonable model for tracked vehicles, such as tanks, as well as wheelchairs and many wheeled robots.

$\begin{array}{ccc} \dot{x} & = & v \cos ϕ \\ \dot{y} & = & v \sin ϕ \\ \dot{θ} & = & ω \end{array}$ $\begin{array}{ccc} \dot x& = &v\cos\phi \\ \dot y& = &v\sin\phi \\ \dot \theta & = & \omega \end{array}$

with $| | v | | \leq v_{m a x}, | | ω | | \leq ω_{m a x}$ $||v|| \leq v_{max}, ~|| \omega|| \leq \omega_{max}$

Kinematic Car model

The kinematic car has orientation $θ$ $\theta$ as well as a front wheel steering angle $ϕ$ $\phi$ . It is a reasonable model for cars traveling without skidding.

$\begin{array}{ccc} \dot{x} & = & v \cos θ \\ \dot{y} & = & v \sin θ \\ \dot{θ} & = & \frac{v}{L} \tan ϕ \end{array}$ $\begin{array}{ccc} \dot x& = &v\cos\theta \\ \dot y& = &v\sin\theta \\ \dot \theta & = &\frac{v}{L}\tan \phi \end{array}$

with $| | v | | \leq v_{m a x}, | | ϕ | | \leq ϕ_{max}$ $||v|| \leq v_{max},~|| \phi || \leq \phi_{\max}$ .