State vector

From QAOP Wiki

The state vector of a system describes the configuration of all of the bodies in the system i.e. where the members of a system are located.

The state vector for a system of particles

The state vector for a system of ${n}$ particles is defined as the column vector containing the positions of all of the particles

$\mathbf{q} = \begin{bmatrix} P_1 \\ P_2 \\ \vdots \\ P_n \end{bmatrix}$

where $P_i$ is a vector containing the position of the $i$ th particle.

The time derivative of the state vector for a system of particles contains the velocities of all of the particles.

$\dot{\mathbf{q}} = \begin{bmatrix} \mathbf{v}_1 \\ \mathbf{v}_2 \\ \vdots \\ \mathbf{v}_n \end{bmatrix}$

where $\mathbf{v}_i$ is a vector containing the velocity of the $i$ th particle.

The state vector for a system of rigid bodies

The state vector for a system of ${n}$ rigid bodies is defined as the column vector containing the positions and orientations of all of the rigid bodies.

$\mathbf{q} = \begin{bmatrix} P_1 \\ q_1 \\ P_2 \\ q_2 \\ \vdots \\ P_N \\ q_n \end{bmatrix}$

where $P_i$ is the position of the $i$ th rigid body and and $q_i$ is a quaternion representing the orientation of the $i$ th rigid body.

The derivative of the state vector for a system of rigid bodies contains the linear and angular velocities of the rigid bodies.

$\dot{\mathbf{q}} = \begin{bmatrix} \mathbf{v}_1 \\ \mathbf{\omega}_1 \\ \mathbf{v}_2 \\ \mathbf{\omega}_2 \\ \vdots \\ \mathbf{v}_n \\ \mathbf{\omega}_n \end{bmatrix}$

State vector

From QAOP Wiki

The state vector for a system of particles

The state vector for a system of rigid bodies

Views

Personal tools

Navigation

Search

Toolbox