|
FhSim
3.1.0
Marine systems simulation
|
|
|
FhSim
3.1.0
Marine systems simulation
|
6DOF rigid body with gravity
Collaboration diagram for 6D Body:Simulation of a rigid body in six degrees of freedom with gravity. Implemented in class C6DBody.
\begin{eqnarray*} \mathbf{\dot{x}}_{1} &=&\mathbf{R}_{b}^{n}\mathbf{x}_{3}, \\ \mathbf{\dot{x}}_{2} &=&\mathbf{T}_{\mathbf{\Theta }}\left( \mathbf{x}_{2}\right) \mathbf{x}_{4}, \\ \left[ \begin{array}{c} \mathbf{\dot{x}}_{3} \\ \mathbf{\dot{x}}_{4} \end{array} \right] &=&\left( \mathbf{M}_{b}^{RB}\right) ^{-1}\left( \mathbf{\sum \tau }^{b} \right), \\ \end{eqnarray*}
where the states are designated
\begin{eqnarray*} \mathbf{x} &=&\left[ \begin{array}{cccc} \mathbf{x}_{1}^{T} & \mathbf{x}_{2}^{T} & \mathbf{x}_{3}^{T} & \mathbf{x}_{4}^{T}\end{array}\right] ^{T}, \\ \mathbf{x}_{1} &\mathbf{=}&\left[ \begin{array}{ccc} x_{bn}^{n} & y_{bn}^{n} & z_{bn}^{n}\end{array}\right] ^{T}, \\ \mathbf{x}_{2} &\mathbf{=}&\left[ \begin{array}{ccc} \varphi & \theta & \psi\end{array}\right] ^{T}, \\ \mathbf{x}_{3} &\mathbf{=}&\left[ \begin{array}{ccc} u_{bn}^{b} & v_{bn}^{b} & w_{bn}^{b}\end{array}\right] ^{T}, \\ \mathbf{x}_{4} &\mathbf{=}&\left[ \begin{array}{ccc} p_{bn}^{b} & q_{bn}^{b} & r_{bn}^{b}\end{array} \right] ^{T}, \end{eqnarray*}
and \( \mathbf{M}_{b}^{RB}\in \mathbf{R}^{6\times 6} \) is the rigid body mass matrix of the body. In this SimObject, the mass matrix is taken to be diagonal, and it is calculated based on the mass and the specified inertia radi. \(\mathbf{\tau}^{b}\), are the sum of the forces and moments acting on the body. The Coriolis force terms are neglected due to an assumption of low speeds and small mass of the system.
The orientation is the quaternion \(\mathbf{q}\) (state Quater, scalar first), with \(\mathbf{\dot{q}} = \frac{1}{2}\mathbf{q}\otimes(0,\mathbf{x}_{4}) + \frac{1}{2\tau}(1-|\mathbf{q}|^2)\mathbf{q}\), where the angular velocity \(\mathbf{x}_{4}\) is in the body frame and \(\tau\) = 0.01 s drives the norm of \(\mathbf{q}\) towards one.
The analytic state Jacobian is complete for the ODE above, including the rotation of gravity and of the global Force and Moment into the body frame. The SimObject has no port Jacobians, so the dependence of Force and Moment on other states is not supplied.
| Name | Width | Description |
|---|---|---|
| Force | 3 | Force acting on the body at the centre of gravity [N]. |
| Moment | 3 | Moment around the center of gravity [Nm]. |
| Name | Width | Description |
|---|---|---|
| Pos | 3 | Position of the body center of gravity. |
| Vel | 3 | Linear velocity of the body in the global frame (LocalVel rotated by Quater). |
| Quater | 4 | Quaternion orientation of the body. |
| LocalVel | 3 | Linear velocity of the body in the body-fixed frame. |
| AngVel | 3 | Angular velocity of the body in the body-fixed frame. |
| Name | Width | Description |
|---|---|---|
| Mass | 1 | Mass of the object in kg. |
| InertiaRadi | 3 | Inertia ?. |
| Material | 1 | The visualization material (Default: Simple/Black). |
| Mesh | 1 | The visualization mesh (geometry) (Default: fhSphere.mesh). |
| MeshScale | 3 | Mesh scale in visualization (Default: [1,1,1]. |
| MeshOrigin | 3 | Mesh origin relative to body center of gravity in visualization (Default: [0,0,0]. |
| Size | 3 | Body size, used with MeshScale to set visualization size (Default: [1,1,1]). |
| Name | Width | Description |
|---|---|---|
| Pos | 3 | 3D position [m]. |
| Quater | 4 | Quaternion orientation. |
| LocalVel | 3 | Linear velocity [m/s]. |
| AngVel | 3 | Angular velocity [rad/s]. |
This SimObject is referred to as Body/6DBody
1.9.8