ObjectRigidBody2D

A 2D rigid body which is attached to a rigid body 2D node. The body obtains coordinates, position, velocity, etc. from the underlying 2D node.

Additional information for ObjectRigidBody2D:

  • This Object has/provides the following types = Body, SingleNoded
  • Requested Node type = Position2D + Orientation2D + Position + Orientation
  • Short name for Python = RigidBody2D
  • Short name for Python visualization object = VRigidBody2D

The item ObjectRigidBody2D with type = ‘RigidBody2D’ has the following parameters:

  • name [type = String, default = ‘’]:
    objects’s unique name
  • physicsMass [\(m\), type = UReal, default = 0.]:
    mass [SI:kg] of rigid body
  • physicsInertia [\(J\), type = UReal, default = 0.]:
    inertia [SI:kgm\(^2\)] of rigid body w.r.t. reference point; this is equal to the center of mass, if physicsCenterOfMass = 0
  • physicsCenterOfMass [\(\LU{b}{{\mathbf{b}}_{COM}}\), type = Vector2D, size = 2, default = [0.,0.]]:
    local position of COM relative to the body’s reference point; if the vector of the COM is [0,0], the computation will not consider additional terms for the COM and it is faster
  • nodeNumber [\(n_0\), type = NodeIndex, default = invalid (-1)]:
    node number (type NodeIndex) for 2D rigid body node
  • visualization [type = VObjectRigidBody2D]:
    parameters for visualization of item

The item VObjectRigidBody2D has the following parameters:

  • show [type = Bool, default = True]:
    set true, if item is shown in visualization and false if it is not shown
  • graphicsDataUserFunction [type = PyFunctionGraphicsData, default = 0]:
    A Python function which returns a bodyGraphicsData object, which is a list of graphics data in a dictionary computed by the user function; the graphics elements need to be defined in the local body coordinates and are transformed by mbs to global coordinates
  • graphicsData [type = BodyGraphicsData]:
    Structure contains data for body visualization; data is defined in special list / dictionary structure

DESCRIPTION of ObjectRigidBody2D

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

  • Position: \(\LU{0}{{\mathbf{p}}}\cConfig(\pLocB) = \LU{0}{\pRef}\cConfig + \LU{0}{\pRef}\cRef + \LU{0b}{\Rot}\pLocB\)
    global position vector of body-fixed point given by local position vector \(\pLocB\)
  • Displacement: \(\LU{0}{{\mathbf{u}}}\cConfig + \LU{0b}{\Rot}\pLocB\)
    global displacement vector of body-fixed point given by local position vector \(\pLocB\)
  • Velocity: \(\LU{0}{{\mathbf{v}}}\cConfig(\pLocB) = \LU{0}{\dot{\mathbf{u}}}\cConfig + \LU{0b}{\Rot}(\LU{b}{\tomega} \times \pLocB\cConfig)\)
    global velocity vector of body-fixed point given by local position vector \(\pLocB\)
  • VelocityLocal: \(\LU{b}{{\mathbf{v}}}\cConfig(\pLocB) = \LU{b0}{\Rot} \LU{0}{{\mathbf{v}}}\cConfig(\pLocB)\)
    local (body-fixed) velocity vector of body-fixed point given by local position vector \(\pLocB\)
  • RotationMatrix: \(\mathrm{vec}(\LU{0b}{\Rot})=[A_{00},\,A_{01},\,A_{02},\,A_{10},\,\ldots,\,A_{21},\,A_{22}]\cConfig\tp\)
    vector with 9 components of the rotation matrix (row-major format)
  • Rotation: \(\theta_{0\mathrm{config}}\)
    scalar rotation angle of body
  • AngularVelocity: \(\LU{0}{\tomega}\cConfig\)
    angular velocity of body
  • AngularVelocityLocal: \(\LU{b}{\tomega}\cConfig\)
    local (body-fixed) 3D velocity vector of node
  • Acceleration: \(\LU{0}{{\mathbf{a}}}\cConfig(\pLocB) = \LU{0}{\ddot{\mathbf{u}}} + \LU{0}{\talpha} \times (\LU{0b}{\Rot} \pLocB) + \LU{0}{\tomega} \times ( \LU{0}{\tomega} \times(\LU{0b}{\Rot} \pLocB))\)
    global acceleration vector of body-fixed point given by local position vector \(\pLocB\)
  • AccelerationLocal: \(\LU{b}{{\mathbf{a}}}\cConfig(\pLocB) = \LU{b0}{\Rot} \LU{0}{{\mathbf{a}}}\cConfig(\pLocB)\)
    local (body-fixed) acceleration vector of body-fixed point given by local position vector \(\pLocB\)
  • AngularAcceleration: \(\LU{0}{\talpha}\cConfig\)
    angular acceleration vector of body
  • AngularAccelerationLocal: \(\LU{b}{\talpha}\cConfig = \LU{b0}{\Rot} \LU{0}{\talpha}\cConfig\)
    local angular acceleration vector of body

Definition of quantities

intermediate variables
symbol
description
reference position
\(\pRefG\cConfig + \pRefG\cRef = \LU{0}{{\mathbf{p}}}(n_0)\cConfig\)
reference point, only equal to the position of COM if \(\LU{b}{{\mathbf{b}}_{COM}}=\Null\); provided by node \(n_0\) in any configuration (except reference)
reference point displacement
\(\LU{0}{{\mathbf{u}}}\cConfig =\pRefG\cConfig = [q_0,\;q_1,\;0]\cConfig\tp = \LU{0}{{\mathbf{u}}}(n_0)\cConfig\)
displacement of reference point which is provided by node \(n_0\) in any configuration; NOTE that for configurations other than reference, it is follows that \(\pRefG\cRef - \pRefG\cConfig\)
reference point velocity
\(\LU{0}{{\mathbf{v}}}\cConfig = [\dot q_0,\;\dot q_1,\;0]\cConfig\tp = \LU{0}{{\mathbf{v}}}(n_0)\cConfig\)
velocity of reference point which is provided by node \(n_0\) in any configuration
body rotation
\(\LU{0}{\theta}_{0\mathrm{config}} = \theta_0(n_0)\cConfig = \psi_0(n_0)\cRef + \psi_0(n_0)\cConfig\)
rotation of body as provided by node \(n_0\) in any configuration
body rotation matrix
\(\LU{0b}{\Rot}\cConfig = \LU{0b}{\Rot}(n_0)\cConfig\)
rotation matrix which transforms local to global coordinates as given by node
local position
\(\pLocB = [\LU{b}{b_0},\,\LU{b}{b_1},\,0]\tp\)
local position as used by markers or sensors
body angular velocity
\(\LU{0}{\tomega}\cConfig = \LU{0}{[\omega_0(n_0),\,0,\,0]}\cConfig\tp\)
rotation of body as provided by node \(n_0\) in any configuration
(generalized) coordinates
\({\mathbf{c}}\cConfig = [q_0,q_1,\;\psi_0]\tp\)
generalized coordinates of body (= coordinates of node)
generalized forces
\(\LU{0}{{\mathbf{f}}} = [f_0,\;f_1,\;\tau_2]\tp\)
generalized forces applied to body
applied forces
\(\LU{0}{{\mathbf{f}}}_a = [f_0,\;f_1,\;0]\tp\)
applied forces (loads, connectors, joint reaction forces, …)
applied torques
\(\LU{0}{\ttau}_a = [0,\;0,\;\tau_2]\tp\)
applied torques (loads, connectors, joint reaction forces, …)

Equations of motion

The equations of motion in case that physicsCenterOfMass=\(\Null\) read:

\[\mr{m}{0}{0} {0}{m}{0} {0}{0}{J} \vr{\ddot q_0}{\ddot q_1}{\ddot \psi_0} = \vr{f_0}{f_1}{\tau_2} = {\mathbf{f}}.\]

if physicsCenterOfMass is nonzero, we resort to (not that \(J\) represents the moment of inertia related to the reference point!):

\[\mr{m}{0}{G_x} {0}{m}{G_y} {G_x}{G_y}{J} \vr{\ddot q_0}{\ddot q_1}{\ddot \psi_0} = \vr{m \dot \psi_0^2 b_x }{m \dot \psi_0^2 b_y}{0} + \vr{f_0}{f_1}{\tau_2} = {\mathbf{f}}.\]

where we use the relations caused by the non-zero center of mass

\[\vp{G_x}{G_y} = m \vp{b_y}{-b_x} \quad \mathrm{and} \quad \vp{b_x}{b_y} = \LU{0}{{\mathbf{b}}_{COM}}\]

Position-based markers can measure position \({\mathbf{p}}\cConfig(\pLocB)\) depending on the local position \(\pLocB\). The position jacobian depends on the local position \(\pLocB\) and is defined as,

\[\LU{0}{{\mathbf{J}}_{pos}} = \partial \LU{0}{{\mathbf{p}}}\cConfig(\pLocB)\cCur / \partial {\mathbf{c}}\cCur = \mr{1}{0}{-\sin(\theta)\LU{b}{b_0} - \cos(\theta)\LU{b}{b_1}} {0}{1}{\cos(\theta)\LU{b}{b_0}-\sin(\theta)\LU{b}{b_1}} {0}{0}{0}\]

which transforms the action of global forces \(\LU{0}{{\mathbf{f}}}\) of position-based markers on the coordinates \({\mathbf{c}}\),

\[{\mathbf{Q}} = \LU{0}{{\mathbf{J}}_{pos}\tp} \LU{0}{{\mathbf{f}}}_a\]

Note that a LoadCoordinate on coordinate 2 of the node would add a torque \(\tau_2\) on the RHS. The rotation jacobian, which is computed from angular velocity, reads

\[\LU{0}{{\mathbf{J}}_{rot}} = \partial \LU{0}{\tomega}\cCur / \partial \dot {\mathbf{c}}\cCur = \mr{0}{0}{0} {0}{0}{0} {0}{0}{1}\]

and transforms the action of global torques \(\LU{0}{\ttau}\) of orientation-based markers on the coordinates \({\mathbf{c}}\),

\[{\mathbf{Q}} = \LU{0}{{\mathbf{J}}_{rot}\tp} \, \LU{0}{\ttau}_a\]

Userfunction: graphicsDataUserFunction(mbs, itemNumber)

A user function, which is called by the visualization thread in order to draw user-defined objects. The function can be used to generate any BodyGraphicsData, see Section GraphicsData. Use exudyn.graphics functions, see Section Module: graphics, to create more complicated objects. Note that graphicsDataUserFunction needs to copy lots of data and is therefore inefficient and only designed to enable simpler tests, but not large scale problems.

For an example for graphicsDataUserFunction see ObjectGround, Section ObjectGround.

arguments / return
type or size
description
mbs
MainSystem
provides reference to mbs, which can be used in user function to access all data of the object
itemNumber
int
integer number of the object in mbs, allowing easy access
returnValue
BodyGraphicsData
list of GraphicsData dictionaries, see Section GraphicsData

MINI EXAMPLE for ObjectRigidBody2D

 1node = mbs.AddNode(NodeRigidBody2D(referenceCoordinates = [1,1,0.25*np.pi],
 2                                   initialCoordinates=[0.5,0,0],
 3                                   initialVelocities=[0.5,0,0.75*np.pi]))
 4mbs.AddObject(RigidBody2D(nodeNumber = node, physicsMass=1, physicsInertia=2))
 5
 6#assemble and solve system for default parameters
 7mbs.Assemble()
 8mbs.SolveDynamic()
 9
10#check result
11exudynTestGlobals.testResult = mbs.GetNodeOutput(node, exu.OutputVariableType.Position)[0]
12exudynTestGlobals.testResult+= mbs.GetNodeOutput(node, exu.OutputVariableType.Coordinates)[2]
13#final x-coordinate of position shall be 2, angle theta shall be np.pi

Relevant Examples and TestModels with weblink:

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