NodeRigidBodyRotVecLG

A 3D rigid body node based on rotation vector and Lie group methods for rigid bodies; the node has 3 displacement coordinates and three rotation coordinates and can be used in combination with explicit Lie Group time integration methods.

Authors: Gerstmayr Johannes, Holzinger Stefan

Additional information for NodeRigidBodyRotVecLG:

• This Node has/provides the following types = Position, Orientation, RigidBody, RotationRotationVector
• Short name for Python = RigidRotVecLG
• Short name for Python visualization object = VRigidRotVecLG

The item NodeRigidBodyRotVecLG with type = ‘RigidBodyRotVecLG’ has the following parameters:

• name [type = String, default = ‘’]:

  node’s unique name
• referenceCoordinates [$\mathbf{q}{}_{\mathrm{ref}}=[q_0,q_1,q_2,{\nu }_0,{\nu }_1,{\nu }_2]{}^{\mathrm{T}}{}_{\mathrm{ref}}=[\mathbf{p}{}^{\mathrm{T}}{}_{\mathrm{ref}},\mathit{\nu }{}^{\mathrm{T}}{}_{\mathrm{ref}}]{}^{\mathrm{T}}$, type = Vector6D, size = 3, default = [0.,0.,0., 0.,0.,0.]]:

  reference coordinates (position and rotation vector $\mathit{\nu }$) of node ==> e.g. ref. coordinates for finite elements or reference position of rigid body (e.g. for definition of joints)
• initialCoordinates [$\mathbf{q}{}_{\mathrm{ini}}=[q_0,q_1,q_2,{\nu }_0,{\nu }_1,{\nu }_2]{}^{\mathrm{T}}{}_{\mathrm{ini}}=[\mathbf{u}{}^{\mathrm{T}}{}_{\mathrm{ini}},\mathit{\nu }{}^{\mathrm{T}}{}_{\mathrm{ini}}]{}^{\mathrm{T}}$, type = Vector6D, size = 3, default = [0.,0.,0., 0.,0.,0.]]:

  initial displacement coordinates $\mathbf{u}$ and rotation vector $\mathit{\nu }$ relative to reference coordinates
• initialVelocities [$\dot{\mathbf{q}}{}_{\mathrm{ini}}=[{\dot{q}}_0,{\dot{q}}_1,{\dot{q}}_2,{\dot{\nu }}_0,{\dot{\nu }}_1,{\dot{\nu }}_2]{}^{\mathrm{T}}{}_{\mathrm{ini}}=[\dot{\mathbf{u}}{}^{\mathrm{T}}{}_{\mathrm{ini}},\dot{\mathit{\nu }}{}^{\mathrm{T}}{}_{\mathrm{ini}}]{}^{\mathrm{T}}$, type = Vector6D, size = 3, default = [0.,0.,0., 0.,0.,0.]]:

  initial velocity coordinate: time derivatives of displacement and angular velocity vector
• visualization [type = VNodeRigidBodyRotVecLG]:

  parameters for visualization of item

The item VNodeRigidBodyRotVecLG 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

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DESCRIPTION of NodeRigidBodyRotVecLG

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

• Position: ${}^0\mathbf{p}{}_{\mathrm{config}}={}^0[p_0,p_1,p_2]{}_{\mathrm{config}}{}^{\mathrm{T}}={}^0\mathbf{u}{}_{\mathrm{config}}+{}^0\mathbf{p}{}_{\mathrm{ref}}$

  global 3D position vector of node; $\mathbf{u}{}_{\mathrm{ref}}=0$
• Displacement: ${}^0\mathbf{u}{}_{\mathrm{config}}=[q_0,q_1,q_2]{}_{\mathrm{config}}{}^{\mathrm{T}}$

  global 3D displacement vector of node
• Velocity: ${}^0\mathbf{v}{}_{\mathrm{config}}=[{\dot{q}}_0,{\dot{q}}_1,{\dot{q}}_2]{}_{\mathrm{config}}{}^{\mathrm{T}}$

  global 3D velocity vector of node
• Acceleration: ${}^0\mathbf{a}{}_{\mathrm{config}}=[{\ddot{q}}_0,{\ddot{q}}_1,{\ddot{q}}_2]{}_{\mathrm{config}}{}^{\mathrm{T}}$

  global 3D acceleration vector of node
• Coordinates: $\mathbf{c}{}_{\mathrm{config}}=[q_0,q_1,q_2,{\nu }_0,{\nu }_1,{\nu }_2]{}^{\mathrm{T}}{}_{\mathrm{config}}$

  coordinate vector of node, having 3 displacement coordinates and 3 Euler angles
• Coordinates\_t: $\dot{\mathbf{c}}{}_{\mathrm{config}}=[{\dot{q}}_0,{\dot{q}}_1,{\dot{q}}_2,{\dot{\nu }}_0,{\dot{\nu }}_1,{\dot{\nu }}_2]{}^{\mathrm{T}}{}_{\mathrm{config}}$

  velocity coordinates vector of node
• RotationMatrix: $[A_{00},A_{01},A_{02},A_{10},\dots ,A_{21},A_{22}]{}_{\mathrm{config}}{}^{\mathrm{T}}$

  vector with 9 components of the rotation matrix ${}^{0b}\mathbf{A}{}_{\mathrm{config}}$ in row-major format, in any configuration; the rotation matrix transforms local ($b$) to global (0) coordinates
• Rotation: $[{\phi }_0,{\phi }_1,{\phi }_2]{}^{\mathrm{T}}{}_{\mathrm{config}}$

  vector with 3 components of the Euler/Tait-Bryan angles in xyz-sequence (${}^{0b}\mathbf{A}{}_{\mathrm{config}}=:{\mathbf{A}}_0({\phi }_0)\cdot {\mathbf{A}}_1({\phi }_1)\cdot {\mathbf{A}}_2({\phi }_2)$), recomputed from rotation matrix
• AngularVelocity: ${}^0\mathit{\omega }{}_{\mathrm{config}}={}^0[{\omega }_0,{\omega }_1,{\omega }_2]{}_{\mathrm{config}}{}^{\mathrm{T}}$

  global 3D angular velocity vector of node
• AngularVelocityLocal: ${}^b\mathit{\omega }{}_{\mathrm{config}}={}^b[{\omega }_0,{\omega }_1,{\omega }_2]{}_{\mathrm{config}}{}^{\mathrm{T}}$

  local (body-fixed) 3D angular velocity vector of node

Detailed information: For a detailed description on the rigid body dynamics formulation using this node, see Holzinger and Gerstmayr .

The node has 3 displacement coordinates $[q_0,q_1,q_2]{}^{\mathrm{T}}$ and three rotation coordinates, which is the rotation vector

$$\mathit{\nu }=\phi \mathbf{n}=\mathit{\nu }{}_{\mathrm{config}}+\mathit{\nu }{}_{\mathrm{ref}},$$

with the rotation angle $\phi$ and the rotation axis $\mathbf{n}$. All coordinates $\mathbf{c}{}_{\mathrm{config}}$ lead to second order differential equations, However the rotation vector cannot be used as a conventional parameterization. It must be computed within a nonlinear update, using appropriate Lie group methods. The first 3 equations are residuals of translational forces in global coordinates, while the last 3 equations are residual of local (body-fixed) torques, compare the equations of motion of the rigid body.

The rotation matrix ${}^{0b}\mathbf{A}(\mathit{\nu }){}_{\mathrm{config}}$ transforms a local (body-fixed) 3D position ${}^b\mathbf{v}={}^b[b_0,b_1,b_2]{}^{\mathrm{T}}$ to global 3D positions,

$${}^0\mathbf{b}{}_{\mathrm{config}}={}^{0b}\mathbf{A}(\mathit{\nu }){}_{\mathrm{config}}{}^b\mathbf{b}$$

Note that $\mathbf{A}(\mathit{\nu })$ is defined in function `` RotationVector2RotationMatrix``, see Section Function: RotationVector2RotationMatrix.

A Lie group integrator must be used with this node, which is why the is used, the rotation parameter velocities are identical to the local angular velocity ${}^b\mathit{\omega }$ and thus the matrix ${}^b\mathbf{G}$ becomes the identity matrix.

Note, that the node automatically switches to Lie group integration of its rotational coordinates, both in explicit integration as well as for implicit time integration. This node avoids typical singularities of rotations and is therefore perfectly suited for arbitrary motion. Furthermore, nonlinearities are reduced, which may improve implicit time integration performance.

For creating a NodeRigidBodyRotVecLG together with a rigid body, there is a rigidBodyUtilities function AddRigidBody, see Section Function: AddRigidBody, which simplifies the setup of a rigid body significantely!

Relevant Examples and TestModels with weblink:

explicitLieGroupIntegratorPythonTest.py (TestModels/), explicitLieGroupIntegratorTest.py (TestModels/), explicitLieGroupMBSTest.py (TestModels/)

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