State vector (navigation)

In navigation, a state vector is a set of data describing exactly where an object is located in space, and how it is moving.

Mathematical representation

A state vector typically will contain seven elements: three position coordinates, three velocity terms, and the time at which these values were valid. Mathematically, in order to describe positions in a N-dimensional space (${\mathbb{ℝ}}^N$) then a state vector $\text{x}$ belongs to ${\mathbb{ℝ}}^{2N}$: $$\mathbf{x}(t)=\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\\ v_1(t)\\ v_2(t)\\ v_3(t)\end{array}\right]$$ or simply $$\mathbf{x}(t)=\left[\begin{array}{c}\mathbf{r}(t)\\ \mathbf{v}(t)\end{array}\right]$$ where $\mathbf{r}={\left[\begin{array}{ccc}{x}_1& x_2& x_3\end{array}\right]}^{\mathsf{T}}$ is the position vector and $\mathbf{v}=\dot{\mathbf{r}}={\left[\begin{array}{ccc}{v}_1& v_2& v_3\end{array}\right]}^{\mathsf{T}}$ is the velocity vector. Since there is freedom to choose coordinate systems for position, a state vector may also be expressed in a variety of coordinate systems (e.g. the North east down coordinate system).

See also

• Orbital state vectors