MarkerBodyRigid

A rigid-body (position+orientation) body-marker attached to a local (body-fixed) position \(\pLocB = [b_0,\; b_1,\; b_2]\) (\(x\), \(y\), and \(z\) coordinates) of the body. It provides position and orientation (rotation), as well as the according derivatives. It can be used for most connectors, joints or loads where either position, position and orientation, or orientation are required.

Interface

Provides: Position, Orientation
Bodies it can be placed on: ObjectGround, ObjectMass1D, ObjectRotationalMass1D, ObjectRigidBody, ObjectRigidBody2D, ObjectFFRF, ObjectFFRFreducedOrder, ObjectANCFCable2D, ObjectALEANCFCable2D, ObjectANCFBeam, ObjectBeamGeometricallyExact2D, ObjectBeamGeometricallyExact
Connectors, constraints and loads that can use it: ObjectConnectorSpringDamper, ObjectConnectorCartesianSpringDamper, ObjectConnectorRigidBodySpringDamper, ObjectConnectorLinearSpringDamper, ObjectConnectorTorsionalSpringDamper, ObjectConnectorGravity, ObjectConnectorHydraulicActuatorSimple, ObjectConnectorDistance, ObjectConnectorRollingDiscPenalty, ObjectContactConvexRoll, ObjectContactSphereSphere, ObjectContactSphereTorus, ObjectContactSphereTriangle, ObjectContactCurveCircles, ObjectJointGeneric, ObjectJointRevoluteZ, ObjectJointPrismaticX, ObjectJointSpherical, ObjectJointRollingDisc, ObjectJointRevolute2D, ObjectJointPrismatic2D, LoadForceVector, LoadTorqueVector

Parameters

The parameters of the item; in a dictionary, its type is ‘BodyRigid’:

name	type	size	default	description
name	String		‘’	marker’s unique name
bodyNumber	ObjectIndex		invalid (-1)	body number to which marker is attached to
localPosition	Vector3D	3	None (zero)	(symbol: \(\pLocB\)) local body position of marker; e.g. local (body-fixed) position where force is applied to; the translation of localHT
localHT	HomogeneousTransformation		None (identity)	(symbol: \(\LU{b}{\Hm}_{m}\)) the frame of the marker in the body frame, as homogeneous transformation: its translation is localPosition, its rotation \(\LU{bm}{\Rot}\) turns the marker frame against the body; a 4x4 matrix, its 16 values row by row or an exu.HT; None: not given; given together with localPosition, both must agree
visualization	VMarkerBodyRigid			parameters for visualization of item

Visualization parameters

The parameters of VMarkerBodyRigid, 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

Drawing

Drawn as all markers.

Detailed description

Marker quantities

quantity	symbol	as computed
position	\(\LU{0}{\pv}_m\)	the output variable `Position` of the body at the local position \(\pLocB\)
velocity	\(\LU{0}{\vv}_m\)	the output variable `Velocity` of the body at \(\pLocB\)
rotation matrix	\(\LU{0m}{\Rot} = \LU{0b}{\Rot} \LU{bm}{\Rot}\)	the rotation \(\LU{0b}{\Rot}\) of the body at \(\pLocB\) (for a rigid body the rotation of the body), turned by the rotation \(\LU{bm}{\Rot}\) of `localHT`
angular velocity	\(\LU{m}{\tomega} = \LU{bm}{\Rot}\tp \LU{b}{\tomega}\)	the angular velocity of the body at \(\pLocB\), in the marker frame

Position, velocity and rotation matrix are global quantities; the angular velocity is local. \(\pLocB\) is given in the body frame, from the reference point of the body. The marker frame is localHT = \([\LU{bm}{\Rot}\;\pLocB;\;\Null\tp\;1]\) in the body frame: localPosition is its translation, and its rotation is the unit matrix unless localHT is given. The Jacobians are global and do not depend on \(\LU{bm}{\Rot}\).

Jacobians

\[ \LU{0}{\Jm_{pos}} = \frac{\partial \LU{0}{\vv}_m}{\partial \dot\qv} , \quad \LU{0}{\Jm_{rot}} = \frac{\partial \LU{0}{\tomega}_m}{\partial \dot\qv} , \]

the derivatives of the global velocity and angular velocity with respect to the velocity coordinates of the body (its access functions TranslationalVelocity_qt and AngularVelocity_qt). For ObjectRigidBody they are computed from its node directly, with the velocity transformation \(\LU{0}{\Gm}\) of the rotation parameters in \(\LU{0}{\Jm_{rot}}\).

Mini example

#position and orientation of a rigid body: a torque on it, held by a rigid body spring-damper
inertia = InertiaCuboid(density=1000, sideLengths=[0.4,0.2,0.1])
node = mbs.AddNode(NodeRigidBodyEP(referenceCoordinates=[1,0,0]+eulerParameters0))
body = mbs.AddObject(ObjectRigidBody(nodeNumber=node, mass=inertia.Mass(),
                                     inertia=inertia.GetInertia6D()))
mBody = mbs.AddMarker(MarkerBodyRigid(bodyNumber=body, localPosition=[0,0,0]))
mGround = mbs.AddMarker(MarkerBodyRigid(bodyNumber=oGround, localPosition=[1,0,0]))
mbs.AddObject(ObjectConnectorRigidBodySpringDamper(markerNumbers=[mGround, mBody],
                                                   stiffness=np.diag([1e4,1e4,1e4, 100,100,100]),
                                                   damping=np.zeros((6,6))))
mbs.AddLoad(LoadTorqueVector(markerNumber=mBody, loadVector=[0,0,1]))

mbs.Assemble()
mbs.SolveStatic()

#rotation about z: M/k_rot, for the small angle
exu.sys['testResult'] = mbs.GetNodeOutput(node, exu.OutputVariableType.Rotation)[2] #0.01

Examples (Ex) and TestModels (TM) that show this item, with weblink to github: tutorialSpringDamperCreate.py (Ex), tutorialRigidBody.py (Ex), tutorialRigidBodyCreate.py (Ex)