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}}\cRef = [q_0,\,q_1,\,q_2,\,\nu_0,\,\nu_1,\,\nu_2]\tp\cRef = [{\mathbf{p}}\tp\cRef,\,\tnu\tp\cRef]\tp, type = Vector6D, size = 3, default = [0.,0.,0., 0.,0.,0.]]:reference coordinates (position and rotation vector \tnu) 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,\,\nu_0,\,\nu_1,\,\nu_2]\tp\cIni = [{\mathbf{u}}\tp\cIni,\,\tnu\tp\cIni]\tp, type = Vector6D, size = 3, default = [0.,0.,0., 0.,0.,0.]]:initial displacement coordinates {\mathbf{u}} and rotation vector \tnu relative to reference coordinates
- initialVelocities [\dot {\mathbf{q}}\cIni = [\dot q_0,\,\dot q_1,\,\dot q_2,\,\dot \nu_0,\,\dot \nu_1,\,\dot \nu_2]\tp\cIni = [\dot {\mathbf{u}}\tp\cIni,\,\dot \tnu\tp\cIni]\tp, 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
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}}}\cRefglobal 3D position vector of node; {\mathbf{u}}\cRef=0Displacement
: \LU{0}{{\mathbf{u}}}\cConfig = [q_0,\,q_1,\,q_2]\cConfig\tpglobal 3D displacement vector of nodeVelocity
: \LU{0}{{\mathbf{v}}}\cConfig = [\dot q_0,\,\dot q_1,\,\dot q_2]\cConfig\tpglobal 3D velocity vector of nodeAcceleration
: \LU{0}{{\mathbf{a}}}\cConfig = [\ddot q_0,\,\ddot q_1,\,\ddot q_2]\cConfig\tpglobal 3D acceleration vector of nodeCoordinates
: {\mathbf{c}}\cConfig = [q_0,\,q_1,\,q_2, \,\nu_0,\,\nu_1,\,\nu_2]\tp\cConfigcoordinate vector of node, having 3 displacement coordinates and 3 Euler anglesCoordinates\_t
: \dot{\mathbf{c}}\cConfig = [\dot q_0,\,\dot q_1,\,\dot q_2, \,\dot \nu_0,\,\dot \nu_1,\,\dot \nu_2]\tp\cConfigvelocity coordinates vector of nodeRotationMatrix
: [A_{00},\,A_{01},\,A_{02},\,A_{10},\,\ldots,\,A_{21},\,A_{22}]\cConfig\tpvector 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) coordinatesRotation
: [\varphi_0,\,\varphi_1,\,\varphi_2]\tp\cConfigvector 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)), recomputed from rotation matrixAngularVelocity
: \LU{0}{\tomega}\cConfig = \LU{0}{[\omega_0,\,\omega_1,\,\omega_2]}\cConfig\tpglobal 3D angular velocity vector of nodeAngularVelocityLocal
: \LU{b}{\tomega}\cConfig = \LU{b}{[\omega_0,\,\omega_1,\,\omega_2]}\cConfig\tplocal (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]\tp and three rotation coordinates, which is the rotation vector
\tnu = \varphi {\mathbf{n}} = \tnu\cConfig + \tnu\cRef,
with the rotation angle \varphi and the rotation axis {\mathbf{n}}. All coordinates {\mathbf{c}}\cConfig 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 \LU{0b}{\Rot(\tnu)}\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(\tnu)}\cConfig \LU{b}{\pLoc}
Note that \Rot(\tnu) is defined in function `` RotationVector2RotationMatrix``, see Section :ref:`sec-rigidbodyutilities-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 \LU{b}{\tomega} and thus the matrix \LU{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 :ref:`sec-rigidbodyutilities-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