NodeRigidBodyRxyz.rst

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 that 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.

Additional information for NodeRigidBodyRxyz:

• This Node has/provides the following types = Position, Orientation, RigidBody, RotationRxyz
• Short name for Python = RigidRxyz
• Short name for Python visualization object = VRigidRxyz

The item NodeRigidBodyRxyz with type = 'RigidBodyRxyz' has the following parameters:

• name [type = String, default = '']:

  node's unique name
• referenceCoordinates [{\mathbf{q}}\cRef = [q_0,\,q_1,\,q_2,\,\psi_0,\,\psi_1,\,\psi_2]\tp\cRef = [{\mathbf{p}}\tp\cRef,\,\tpsi\tp\cRef]\tp, type = Vector6D, size = 6, default = [0.,0.,0., 0.,0.,0.]]:

  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 [{\mathbf{q}}\cIni = [q_0,\,q_1,\,q_2,\,\psi_0,\,\psi_1,\,\psi_2]\tp\cIni = [{\mathbf{u}}\tp\cIni,\,\tpsi\tp\cIni]\tp, type = Vector6D, size = 6, default = [0.,0.,0., 0.,0.,0.]]:

  initial displacement coordinates: ux,uy,uz and 3 Euler angles (xyz) relative to reference coordinates
• initialVelocities [\dot {\mathbf{q}}\cIni = [\dot q_0,\,\dot q_1,\,\dot q_2,\,\dot \psi_0,\,\dot \psi_1,\,\dot \psi_2]\tp\cIni = [\dot {\mathbf{u}}\tp\cIni,\,\dot \tpsi\tp\cIni]\tp, type = Vector6D, size = 6, default = [0.,0.,0., 0.,0.,0.]]:

  initial velocity coordinate: time derivatives of ux,uy,uz and of 3 Euler angles (xyz)
• visualization [type = VNodeRigidBodyRxyz]:

  parameters for visualization of item

The item VNodeRigidBodyRxyz has the following parameters:

• show [type = Bool, default = True]:

  set true, if item is shown in visualization and false if it is not shown
• drawSize [type = float, default = -1.]:

  drawing size (diameter, dimensions of underlying cube, etc.) for item; size == -1.f means that default size is used
• color [type = Float4, size = 4, default = [-1.,-1.,-1.,-1.]]:

  Default RGBA color for nodes; 4th value is alpha-transparency; R=-1.f means, that default color is used

---

DESCRIPTION of NodeRigidBodyRxyz

The following output variables are available as OutputVariableType in sensors, Get...Output() and other functions:

• Position: \LU{0}{{\mathbf{p}}}\cConfig = \LU{0}{[p_0,\,p_1,\,p_2]}\cConfig\tp= \LU{0}{{\mathbf{u}}}\cConfig + \LU{0}{{\mathbf{p}}}\cRef

  global 3D position vector of node; {\mathbf{u}}\cRef=0
• Displacement: \LU{0}{{\mathbf{u}}}\cConfig = [q_0,\,q_1,\,q_2]\cConfig\tp

  global 3D displacement vector of node
• Velocity: \LU{0}{{\mathbf{v}}}\cConfig = [\dot q_0,\,\dot q_1,\,\dot q_2]\cConfig\tp

  global 3D velocity vector of node
• Acceleration: \LU{0}{{\mathbf{a}}}\cConfig = [\ddot q_0,\,\ddot q_1,\,\ddot q_2]\cConfig\tp

  global 3D acceleration vector of node
• Coordinates: {\mathbf{c}}\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{\mathbf{c}}\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{\mathbf{c}}\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

Detailed information: The node has 3 displacement coordinates [q_0,\,q_1,\,q_2]\tp and 3 rotation coordinates [\psi_0,\,\psi_1,\,\psi_2]\tp for consecutive rotations around the 0, 1 and 2-axis (x, y and z). All coordinates {\mathbf{c}}\cConfig lead to second order differential equations. The rotation matrix \LU{0b}{\Rot}\cConfig transforms a local (body-fixed) 3D position \pLocB = \LU{b}{[b_0,\,b_1,\,b_2]}\tp to global 3D positions,

\LU{0}{\pLoc}\cConfig = \LU{0b}{\Rot}\cConfig \LU{b}{\pLoc}

Note that the Euler angles \ttheta\cCur are computed as sum of current coordinates plus reference coordinates,

\ttheta\cCur = \tpsi\cCur + \tpsi\cRef.

The rotation matrix is defined as function of the rotation parameters \ttheta=[\theta_0,\,\theta_1,\,\theta_2]\tp

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

see Section :ref:`sec-symbolsitems` for definition of rotation matrices \Rot_0, \Rot_1 and \Rot_2.

The derivatives of the angular velocity vectors w.r.t.the rotation velocity coordinates \dot \ttheta=[\dot \theta_0,\,\dot \theta_1,\,\dot \theta_2]\tp lead to the {\mathbf{G}} matrices, as used in the equations of motion for rigid bodies,

\LU{0}{\tomega} &=& \LU{0}{{\mathbf{G}}} \dot \ttheta, \\
\LU{b}{\tomega} &=& \LU{b}{{\mathbf{G}}} \dot \ttheta.

For creating a NodeRigidBodyRxyz together with a rigid body, there is a rigidBodyUtilities function AddRigidBody, see Section :ref:`sec-rigidbodyutilities-addrigidbody`, which simplifies the setup of a rigid body significantely!

Relevant Examples and TestModels with weblink:

performanceMultiThreadingNG.py (Examples/), explicitLieGroupIntegratorPythonTest.py (TestModels/), explicitLieGroupIntegratorTest.py (TestModels/), explicitLieGroupMBSTest.py (TestModels/), heavyTop.py (TestModels/), connectorRigidBodySpringDamperTest.py (TestModels/)

The web version may not be complete. For details, consider also the Exudyn PDF documentation : theDoc.pdf