ObjectJointRollingDisc

A joint representing a rolling rigid disc (marker 1) on a flat surface (marker 0, ground body) in global \(x\)-\(y\) plane. The contraint is based on an idealized rolling formulation with no slip. The contraints works for discs as long as the disc axis and the plane normal vector are not parallel. It must be assured that the disc has contact to ground in the initial configuration (adjust z-position of body accordingly). The ground body can be a rigid body which is moving. In this case, the flat surface is assumed to be in the \(x\)-\(y\)-plane at \(z=0\). Note that the rolling body must have the reference point at the center of the disc. NOTE: the cases of normal other than \(z\)-direction, wheel axis other than \(x\)-axis and moving ground body needs to be tested further, check your results!

Simpler: mbs.CreateRollingDisc adds this item, with what it needs, in one call.

Interface

Python names: ObjectJointRollingDisc or RollingDiscJoint, and VRollingDiscJoint for its visualization
Markers it acts on: those providing Position and Orientation: MarkerBodyRigid, MarkerNodeRigid, MarkerSuperElementRigid, MarkerKinematicTreeRigid

Parameters

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

name	type	size	default	description
name	String		‘’	constraints’s unique name
markerNumbers	ArrayMarkerIndex	2	[ invalid (-1), invalid (-1) ]	(symbol: \([m0,m1]\tp\)) list of markers used in connector; \(m0\) represents the ground and \(m1\) represents the rolling body, which has its reference point (=local position [0,0,0]) at the disc center point
constrainedAxes	ArrayIndex	3	[1,1,1]	(symbol: \(\jv=[j_0,\,\ldots,\,j_2]\)) flags, which determine which constraints are active, in which \(j_0\) represents lateral motion, \(j_1\) longitudinal (forward/backward) motion and \(j_2\) represents the normal (contact) direction
activeConnector	Bool		True	flag, which determines, if the connector is active; used to deactivate (temporarily) a connector or constraint
discRadius	PReal		0	defines the disc radius; must be given: the default is only a placeholder
discAxis	Vector3D	3	[1,0,0]	(symbol: \(\LU{m1}{\wv_{1}}, \;\; \vert \LU{m1}{\wv_{1}}\vert = 1\)) axis of disc defined in marker \(m1\) frame
planeNormal	Vector3D	3	[0,0,1]	(symbol: \(\LU{m0}{\vv_{PN}}\)) normal to the contact / rolling plane defined in marker \(m0\) coordinates
visualization	VObjectJointRollingDisc			parameters for visualization of item

Visualization parameters

The parameters of VObjectJointRollingDisc, 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
discWidth	float		0.1	width of disc for drawing
color	Float4	4	[-1.,-1.,-1.,-1.]	RGBA connector color; if R==-1, use default color

Drawing

Drawn as all joints; the disc of width discWidth, with 4 times general.cylinderTiling segments, and its axis.

Settings beyond those of all joints: general.cylinderTiling.

Output variables

Available as OutputVariableType in sensors, Get...Output() and other functions:

output variable	symbol	description
Position	\(\LU{0}{\pv}_{G}\)	current global position of contact point between rolling disc and ground
Velocity	\(\LU{0}{\vv}_{trail}\)	current velocity of the trail (according to motion of the contact point along the trail!) in global coordinates; this is not the velocity of the contact point; needs further testing for general case of relative moving bodies
ForceLocal	\(\LU{J1}{\fv} = \LU{0}{[f_0,\, f_1,\, f_2]\tp}= [-\zv^T \LU{0}{\wv_{lat}}, \, -\zv^T \LU{0}{\wv_2}, \, -\zv^T \LU{0}{\vv_{PN}}]\tp\)	contact forces acting on disc, in special \(J1\) joint coordinates, \(f_0\) being the lateral force (parallel to ground plane), \(f_1\) being the longitudinal force and \(f_2\) being the normal force
RotationMatrix	\(\LU{0,J1}{\Am} = [\LU{0}{\wv_{lat}},\, \LU{0}{\wv}_2,\, \LU{0}{\vv_{PN}}]\)	transformation matrix of special joint coordinates \(J1\) to global coordinates

Detailed description

Definition of quantities

intermediate variables	symbol	description
marker m0 position	\(\LU{0}{\pv}_{m0}\)	current global position of marker \(m0\); needed only if body \(m0\) is not a ground body
marker m0 orientation	\(\LU{0,m0}{\Rot}\)	current rotation matrix provided by marker m0 (assumed to be rigid body)
marker m0 velocity	\(\LU{0}{\vv}_{m0}\)	current global velocity which is provided by marker m0 (assumed to be rigid body)
marker m0 angular velocity	\(\LU{0}{\tomega}_{m0}\)	current angular velocity vector provided by marker m0 (assumed to be rigid body)
marker m1 position	\(\LU{0}{\pv}_{m1}\)	center of disc
marker m1 orientation	\(\LU{0,m1}{\Rot}\)	current rotation matrix provided by marker m1
marker m1 velocity	\(\LU{0}{\vv}_{m1}\)	accordingly
marker m1 angular velocity	\(\LU{0}{\tomega}_{m1}\)	current angular velocity vector provided by marker m1
ground normal vector	\(\LU{0}{\vv_{PN}} = \LU{0,m0}{\Am} \LU{m0}{\vv_{PN}}\)	normalized normal vector to the ground plane, moving with marker \(m0\)
ground position B	\(\LU{0}{\pv}_{B}\)	disc center point projected on ground in plane normal (\(z\)-direction, \(z=0\))
ground position C	\(\LU{0}{\pv}_{C}\)	contact point of disc with ground in global coordinates
ground velocity C	\(\LU{0}{\vv}_{Cm1}\)	velocity of disc (marker 1) at ground contact point (must be zero if ground does not move)
ground velocity C	\(\LU{0}{\vv}_{Cm2}\)	velocity of ground (marker 0) at ground contact point (is always zero if ground does not move)
wheel axis vector	\(\LU{0}{\wv_1} =\LU{0,m1}{\Rot} \LU{m1}{\wv_{1}} \)	normalized disc axis vector
longitudinal vector	\(\LU{0}{\wv_2}\)	vector in longitudinal (motion) direction
lateral vector	\(\LU{0}{\wv_{lat}} = \LU{0}{\vv_{PN}} \times \LU{0}{\wv}_2\)	vector in lateral direction, parallel to ground plane
contact point vector	\(\LU{0}{\wv_3}\)	normalized vector from disc center point in direction of contact point C
\(D1\) transformation matrix	\(\LU{0,D1}{\Am} = [\LU{0}{\wv_1},\, \LU{0}{\wv_2},\, \LU{0}{\wv_3}]\)	transformation of special disc coordinates \(D1\) to global coordinates
\(J1\) transformation matrix	\(\LU{0,J1}{\Am} = [\LU{0}{\wv_{lat}},\, \LU{0}{\wv}_2,\, \LU{0}{\vv_{PN}}]\)	transformation of special joint \(J1\) coordinates to global coordinates
algebraic variables	\(\zv=[\lambda_0,\,\lambda_1,\,\lambda_2]\tp\)	vector of algebraic variables (Lagrange multipliers) according to the algebraic equations

Geometric relations

The main geometrical setup is shown in the following figure: First, the contact point \(\LU{0}{\pv}_{C}\) must be computed. With the helper vector,

\[ \LU{0}{\xv} = \LU{0}{\wv}_1 \times \LU{0}{\vv_{PN}} \]

we obtain a disc coordinate system, representing the longitudinal direction,

\[ \LU{0}{\wv}_2 = \frac{1}{|\LU{0}{\xv}|} \LU{0}{\xv} \]

and the vector to the contact point,

\[ \LU{0}{\wv}_3 = \LU{0}{\wv}_1 \times \LU{0}{\wv}_2 \]

The contact point \(C\) can be computed from

\[ \LU{0}{\pv}_{C} = \LU{0}{\pv}_{m1} + r \cdot \LU{0}{\wv}_3 \]

The velocity of the contact point at the disc is computed from,

\[ \LU{0}{\vv}_{Cm1} = \LU{0}{\vv}_{m1} + \LU{0}{\tomega}_{m1} \times (r\cdot \LU{0}{\wv}_3) \]

If marker 0 body is (moving) rigid body instead of a ground body, the contact point \(C\) is reconstructed in body of marker 0,

\[ \LU{m0}{\pv}_{C} = \LU{m0,0}{\Rot} (\LU{0}{\pv}_{C} - \LU{0}{\pv}_{m0}) \]

The velocity of the contact point at the marker 0 body reads

\[ \LU{0}{\vv}_{Cm0} = \LU{0}{\vv}_{m0} + \LU{0}{\tomega}_{m0} \times \left( \LU{0,m0}{\Rot} \LU{m0}{\pv}_{C} \right) \]

Connector constraint equations

Constraints for activeConnector = True:

The non-holonomic, index 2 constraints for the tangential and normal contact follow from (an index 3 formulation would be possible, but is not implemented yet because of mixing different jacobians)

\[ \LU{J1,0}{\Am} \left(\vr{\LU{0}{\vv}_{Cm1,x}}{\LU{0}{\vv}_{Cm1,y}}{\LU{0}{\vv}_{Cm1,z}} - \vr{\LU{0}{\vv}_{Cm0,x}}{\LU{0}{\vv}_{Cm0,y}}{\LU{0}{\vv}_{Cm0,z}} \right) = \Null \]

In case that activeConnector = False, the Lagrange multipliers are set to zero:

\[ \zv = \Null \]

Note that since version 1.8.27 the constraints can be turned on/off separately with constrainedAxes=[b0,b1,b2], in which b0 represents the flag for lateral motion, b1 switches the constraint for forward motion and b2 for motion in plane normal direction.

Mini example

#a disc rolling without slip on the ground plane: the constraint of ideal rolling
r = 0.2
inertia = InertiaCylinder(density=1000, length=0.05, outerRadius=r, axis=0)
node = mbs.AddNode(NodeRigidBodyEP(referenceCoordinates=[0,0,r]+eulerParameters0,
                   initialVelocities=[0,-2,0]+list(AngularVelocity2EulerParameters_t([2/r,0,0], eulerParameters0))))
disc = mbs.AddObject(ObjectRigidBody(nodeNumber=node, mass=inertia.Mass(), inertia=inertia.GetInertia6D()))
mbs.AddLoad(LoadMassProportional(markerNumber=mbs.AddMarker(MarkerBodyMass(bodyNumber=disc)), loadVector=[0,0,-9.81]))
mGround = mbs.AddMarker(MarkerBodyRigid(bodyNumber=oGround, localPosition=[0,0,0]))
mDisc = mbs.AddMarker(MarkerBodyRigid(bodyNumber=disc, localPosition=[0,0,0]))
mbs.AddObject(ObjectJointRollingDisc(markerNumbers=[mGround, mDisc], discRadius=r, discAxis=[1,0,0],
                                     planeNormal=[0,0,1]))

mbs.Assemble()
mbs.SolveDynamic()

#it rolls on with the initial velocity: y = -2*t at t=1
exu.sys['testResult'] = mbs.GetNodeOutput(node, exu.OutputVariableType.Position)[1] #-2

Relevant Examples (Ex) and TestModels (TM) with weblink to github: bicycleIftommBenchmark.py (Ex), reinforcementLearningRobot.py (Ex), rollingCoinTest.py (TM), rollingDiscTangentialForces.py (TM), rotatingTableTest.py (TM), createFunctionsTest.py (TM), createRollingDiscTest.py (TM)