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(sec-item-noderigidbodyrxyz)=
# NodeRigidBodyRxyz

A 3D rigid body node based on Euler / Tait-Bryan angles for rigid bodies or beams. All coordinates lead to second order differential equations; NOTE: this node has a singularity if the second rotation parameter reaches $\psi_1 = (2k-1) \pi/2$, with $k \in \Ncal$ or $-k \in \Ncal$.

## Interface

- Python names: `NodeRigidBodyRxyz` or `RigidRxyz`, and `VRigidRxyz` for its visualization
- Provides: `Position`, `Orientation`, `RigidBody`, `RotationRxyz`
- Node markers that can be attached: [](#sec-item-markernodeposition), [](#sec-item-markernoderigid), [](#sec-item-markernoderotationcoordinate)
- Objects that take this node: [](#sec-item-objectmasspoint), [](#sec-item-objectrigidbody), [](#sec-item-objectbeamgeometricallyexact)

## Parameters

The parameters of the item; in a dictionary, its type is 'RigidBodyRxyz':

| name | type | size | default | description |
|---|---|---|---|---|
| **name** | String |  | '' | node's unique name |
| **referenceCoordinates** | Vector6D | 6 | [0.,0.,0., 0.,0.,0.] | (symbol: $\qv\cRef = [q_0,\,q_1,\,q_2,\,\psi_0,\,\psi_1,\,\psi_2]\tp\cRef = [\pv\tp\cRef,\,\tpsi\tp\cRef]\tp$) reference coordinates (3 position and 3 xyz Euler angles) of node ==> e.g. ref. coordinates for finite elements or reference position of rigid body (e.g. for definition of joints) |
| **initialCoordinates** | Vector6D | 6 | [0.,0.,0., 0.,0.,0.] | (symbol: $\qv\cIni = [q_0,\,q_1,\,q_2,\,\psi_0,\,\psi_1,\,\psi_2]\tp\cIni = [\uv\tp\cIni,\,\tpsi\tp\cIni]\tp$) initial displacement coordinates: ux,uy,uz and 3 Euler angles (xyz) relative to reference coordinates |
| **initialVelocities** | Vector6D | 6 | [0.,0.,0., 0.,0.,0.] | (symbol: $\dot \qv\cIni = [\dot q_0,\,\dot q_1,\,\dot q_2,\,\dot \psi_0,\,\dot \psi_1,\,\dot \psi_2]\tp\cIni = [\dot \uv\tp\cIni,\,\dot \tpsi\tp\cIni]\tp$) initial velocity coordinate: time derivatives of ux,uy,uz and of 3 Euler angles (xyz) |
| **visualization** | VNodeRigidBodyRxyz |  |  | parameters for visualization of item |

## Visualization parameters

The parameters of `VNodeRigidBodyRxyz`, given as `visualization`:

| name | type | size | default | description |
|---|---|---|---|---|
| **show** | Bool |  | True | set true, if item is shown in visualization and false if it is not shown |
| **drawSize** | float |  | -1. | drawing size (diameter, dimensions of underlying cube, etc.) for item; size == -1.f means that default size is used |
| **color** | Float4 | 4 | [-1.,-1.,-1.,-1.] | Default RGBA color for nodes; 4th value is alpha-transparency; R=-1.f means, that default color is used |

## Drawing

Drawn as [all nodes](#sec-drawing-nodes); with `nodes.showBasis`, also its frame, of length `nodes.basisSize`.

Settings beyond those of all nodes: [`general.axesTiling`](#sec-vsettingsgeneral), [`nodes.basisSize`](#sec-vsettingsnodes), [`nodes.showBasis`](#sec-vsettingsnodes).

## Output variables

Available as `OutputVariableType` in sensors, `Get...Output()` and other functions:

```{tabularcolumns} \Y{0.25}\Y{0.25}\Y{0.5}
```

| output variable | symbol | description |
|---|---|---|
| Position | $\LU{0}{\pv}\cConfig = \LU{0}{[p_0,\,p_1,\,p_2]}\cConfig\tp= \LU{0}{\uv}\cConfig + \LU{0}{\pv}\cRef$ | global 3D position vector of node; $\uv\cRef=0$ |
| Displacement | $\LU{0}{\uv}\cConfig = [q_0,\,q_1,\,q_2]\cConfig\tp$ | global 3D displacement vector of node |
| Velocity | $\LU{0}{\vv}\cConfig = [\dot q_0,\,\dot q_1,\,\dot q_2]\cConfig\tp$ | global 3D velocity vector of node |
| Acceleration | $\LU{0}{\av}\cConfig = [\ddot q_0,\,\ddot q_1,\,\ddot q_2]\cConfig\tp$ | global 3D acceleration vector of node |
| CoordinatesTotal |  | displacement/rotation coordinates of node including reference configuration |
| Coordinates | $\cv\cConfig = [q_0,\,q_1,\,q_2, \,\psi_0,\,\psi_1,\,\psi_2]\tp\cConfig$ | coordinate vector of node, having 3 displacement coordinates and 3 Euler angles |
| Coordinates_t | $\dot\cv\cConfig = [\dot q_0,\,\dot q_1,\,\dot q_2, \,\dot \psi_0,\,\dot \psi_1,\,\dot \psi_2]\tp\cConfig$ | velocity coordinates vector of node |
| Coordinates_tt | $\ddot\cv\cConfig = [\ddot q_0,\,\ddot q_1,\,\ddot q_2, \,\ddot \psi_0,\,\ddot \psi_1,\,\ddot \psi_2]\tp\cConfig$ | acceleration coordinates vector of node |
| RotationMatrix | $[A_{00},\,A_{01},\,A_{02},\,A_{10},\,\ldots,\,A_{21},\,A_{22}]\cConfig\tp$ | vector with 9 components of the rotation matrix $\LU{0b}{\Rot}\cConfig$ in row-major format, in any configuration; the rotation matrix transforms local ($b$) to global (0) coordinates |
| Rotation | $[\varphi_0,\,\varphi_1,\,\varphi_2]\tp\cConfig = [\psi_0,\,\psi_1,\,\psi_2]\tp\cRef + [\psi_0,\,\psi_1,\,\psi_2]\tp\cConfig$ | vector with 3 components of the Euler / Tait-Bryan angles in xyz-sequence ($\LU{0b}{\Rot}\cConfig=:\Rot_0(\varphi_0) \cdot \Rot_1(\varphi_1) \cdot \Rot_2(\varphi_2)$) |
| AngularVelocity | $\LU{0}{\tomega}\cConfig = \LU{0}{[\omega_0,\,\omega_1,\,\omega_2]}\cConfig\tp$ | global 3D angular velocity vector of node |
| AngularVelocityLocal | $\LU{b}{\tomega}\cConfig = \LU{b}{[\omega_0,\,\omega_1,\,\omega_2]}\cConfig\tp$ | local (body-fixed) 3D angular velocity vector of node |
| AngularAcceleration | $\LU{0}{\talpha}\cConfig = \LU{0}{[\alpha_0,\,\alpha_1,\,\alpha_2]}\cConfig\tp$ | global 3D angular acceleration vector of node |

(description-noderigidbodyrxyz)=
## Detailed description

### Coordinates

| index | symbol | kind | meaning | frame |
|---|---|---|---|---|
| 0, 1, 2 | $q_0,\,q_1,\,q_2$ | {ref}`ODE2 <ODE2>` | displacement of the reference point of the body | global |
| 3, 4, 5 | $\psi_0,\,\psi_1,\,\psi_2$ | {ref}`ODE2 <ODE2>` | change of the Tait-Bryan angles against their reference values: consecutive rotations about the $x$-, $y$- and $z$-axis | - |

### Configuration

The position of the reference point and the angles $\ttheta$ are the sums of reference and current
coordinates,

$$
\pv\cConfig = \pv\cRef + \uv\cConfig, \quad \ttheta\cConfig = \tpsi\cRef + \tpsi\cConfig ,
$$

and the rotation matrix, which transforms a local (body-fixed) position $\pLocB$ into the global
frame, $\LU{0}{\pLoc}\cConfig = \LU{0b}{\Rot}\cConfig \LU{b}{\pLoc}$, is

$$
\LU{0b}{\Rot} = \LU{01}{\Rot_0}(\theta_0) \LU{12}{\Rot_1}(\theta_1) \LU{2b}{\Rot_2}(\theta_2) ,
$$

see [](#sec-symbolsitems) for the elementary rotation matrices $\Rot_0$, $\Rot_1$ and $\Rot_2$.

### Frame and interpretation

The displacement is given in the global frame, and the angles describe the rotation of the body
frame $b$ against the global frame.

### Action on the equations of motion

The velocity transformation relates the time derivatives of the angles to the angular velocity,

$$
\begin{aligned}
\LU{0}{\tomega} &= \LU{0}{\Gm} \dot \ttheta, \\
\LU{b}{\tomega} &= \LU{b}{\Gm} \dot \ttheta.
\end{aligned}
$$

All six coordinates lead to {ref}`ODE2 <ODE2>` equations: the first three are the residuals of the forces in
the global frame, the last three the torque equations projected with $\LU{b}{\Gm\tp}$ (body frame)
or $\LU{0}{\Gm\tp}$ (global frame), see the equations of motion of [](#sec-item-objectrigidbody).
There is no constraint.

### Singularity

$\Gm$ is singular for $\theta_1 = \pm \pi/2$ (and every multiple of $\pi$ added): there the rotations
about the first and the third axis coincide, and the equations cannot be solved. Use the node only
for motions that stay away from it, or `NodeRigidBodyEP` or `NodeRigidBodyRotVecLG`.

For creating a `NodeRigidBodyRxyz` together with a rigid body, use `CreateRigidBody`, see
[](#sec-mainsystemextensions-createrigidbody).

(miniexample-noderigidbodyrxyz)=
## Mini example


```python
#a rigid body spinning about its z-axis; the rotation coordinates are Tait-Bryan angles
node = mbs.AddNode(NodeRigidBodyRxyz(referenceCoordinates=[0.5,0.2,0.1, 0,0,0],
                                     initialVelocities=[0,0,0, 0,0,0.5*np.pi])) #angle rates
inertia = InertiaCuboid(density=1000, sideLengths=[0.4,0.2,0.1])
mbs.AddObject(ObjectRigidBody(nodeNumber=node, mass=inertia.Mass(),
                              inertia=inertia.GetInertia6D()))

mbs.Assemble()
mbs.SolveDynamic()

#the third rotation coordinate after 1 second:
exu.sys['testResult'] = mbs.GetNodeOutput(node, exu.OutputVariableType.Coordinates)[5] #pi/2
```


Examples (Ex) and TestModels (TM) that show this item, with weblink to github: [`connectorRigidBodySpringDamperTest.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/connectorRigidBodySpringDamperTest.py) (TM), [`heavyTop.py`](https://github.com/jgerstmayr/EXUDYN/blob/master/python/TestModels/heavyTop.py) (TM)