Body

Rigid body with mass, inertia tensor and one frame connector (12 potential states)

Library

Modelica/Mechanics/MultiBody/Parts

Description

Rigid body with mass and inertia tensor. All parameter vectors have to be resolved in frame_a. The inertia tensor has to be defined with respect to a coordinate system that is parallel to frame_a with the origin at the center of mass of the body.

By default, this component is visualized by a cylinder located between frame_a and the center of mass and by a sphere that has its center at the center of mass. If the cylinder length is smaller as the radius of the sphere, e.g., since frame_a is located at the center of mass, the cylinder is not displayed. Note, that the animation may be switched off via parameter animation = false.

States of Body Components

Every body has potential states. If possible a tool will select the states of joints and not the states of bodies because this is usually the most efficient choice. In this case the position, orientation, velocity and angular velocity of frame_a of the body will be computed by the component that is connected to frame_a. However, if a body is moving freely in space, variables of the body have to be used as states. The potential states of the body are:

• The position vector frame_a.r_0 from the origin of the world frame to the origin of frame_a of the body, resolved in the world frame and the absolute velocity v_0 of the origin of frame_a, resolved in the world frame (= der(frame_a.r_0)).
• If parameter useQuaternions in the "Advanced" menu is true (this is the default), then 4 quaternions are potential states. Additionally, the coordinates of the absolute angular velocity vector of the body are 3 potential states.
If useQuaternions in the "Advanced" menu is false, then 3 angles and the derivatives of these angles are potential states. The orientation of frame_a is computed by rotating the world frame along the axes defined in parameter vector "sequence_angleStates" (default = {1,2,3}, i.e., the Cardan angle sequence) around the angles used as potential states. For example, the default is to rotate the x-axis of the world frame around angles[1], the new y-axis around angles[2] and the new z-axis around angles[3], arriving at frame_a.

The quaternions have the slight disadvantage that there is a non-linear constraint equation between the 4 quaternions. Therefore, at least one non-linear equation has to be solved during simulation. A tool might, however, analytically solve this simple constraint equation. Using the 3 angles as states has the disadvantage that there is a singular configuration in which a division by zero will occur. If it is possible to determine in advance for an application class that this singular configuration is outside of the operating region, the 3 angles might be used as potential states by setting useQuaternions = false.

In text books about 3-dimensional mechanics often 3 angles and the angular velocity are used as states. This is not the case here, since 3 angles and their derivatives are used as potential states (if useQuaternions = false). The reason is that for real-time simulation the discretization formula of the integrator might be "inlined" and solved together with the body equations. By appropriate symbolic transformation the performance is drastically increased if angles and their derivatives are used as states, instead of angles and the angular velocity.

Whether or not variables of the body are used as states is usually automatically selected by the Modelica translator. If parameter enforceStates is set to true in the "Advanced" menu, then body variables are forced to be used as states according to the setting of parameters "useQuaternions" and "sequence_angleStates".

Parameters

NameLabelDescriptionData TypeValid Values

mo_animation

animation

= true, if animation shall be enabled (show cylinder and sphere)

Scalar

true
false

mo_r_CM

r_CM

Vector from frame_a to center of mass, resolved in frame_a

Vector of size 3

mo_m

m

Mass of rigid body

Scalar

mo_I_11

I_11

(1,1) element of inertia tensor

Scalar

mo_I_22

I_22

(2,2) element of inertia tensor

Scalar

mo_I_33

I_33

(3,3) element of inertia tensor

Scalar

mo_I_21

I_21

(2,1) element of inertia tensor

Scalar

mo_I_31

I_31

(3,1) element of inertia tensor

Scalar

mo_I_32

I_32

(3,2) element of inertia tensor

Scalar

mo_I

I

inertia tensor

Matrix of size 3x3

mo_R_start

R_start

Orientation object from world frame to frame_a at initial time

FromModelica('Modelica.Mechanics.MultiBody.Frames.Orientation')

mo_z_a_start

z_a_start

Initial values of angular acceleration z_a = der(w_a), i.e., time derivative of angular velocity resolved in frame_a

Vector of size 3

mo_Q_start

Q_start

Quaternion orientation object from world frame to frame_a at initial time

Vector of size 4

mo_phi_start

phi_start

Potential angle states at initial time

Vector of size 3

NameLabelDescriptionData TypeValid Values

mo_angles_fixed

angles_fixed

= true, if angles_start are used as initial values, else as guess values

Scalar

true
false

mo_angles_start

angles_start

Initial values of angles to rotate frame_a around 'sequence_start' axes into frame_b

Vector of size 3

mo_sequence_start

sequence_start

Sequence of rotations to rotate frame_a into frame_b at initial time

Vector of size 3

mo_w_0_fixed

w_0_fixed

= true, if w_0_start are used as initial values, else as guess values

Scalar

true
false

mo_w_0_start

w_0_start

Initial or guess values of angular velocity of frame_a resolved in world frame

Vector of size 3

mo_z_0_fixed

z_0_fixed

= true, if z_0_start are used as initial values, else as guess values

Scalar

true
false

mo_z_0_start

z_0_start

Initial values of angular acceleration z_0 = der(w_0)

Vector of size 3

NameLabelDescriptionData TypeValid Values

mo_sphereDiameter

sphereDiameter

Diameter of sphere

Scalar

mo_sphereColor

sphereColor

Color of sphere

Vector of size 3

mo_cylinderDiameter

cylinderDiameter

Diameter of cylinder

Scalar

mo_cylinderColor

cylinderColor

Color of cylinder

Vector of size 3

mo_specularCoefficient

specularCoefficient

Reflection of ambient light (= 0: light is completely absorbed)

Scalar

NameLabelDescriptionData TypeValid Values

mo_enforceStates

enforceStates

= true, if absolute variables of body object shall be used as states (StateSelect.always)

Scalar

true
false

mo_useQuaternions

useQuaternions

= true, if quaternions shall be used as potential states otherwise use 3 angles as potential states

Scalar

true
false

mo_sequence_angleStates

sequence_angleStates

Sequence of rotations to rotate world frame into frame_a around the 3 angles used as potential states

Vector of size 3

NameLabelDescriptionData TypeValid Values

mo_r_0

r_0

r_0

Structure

mo_r_0/fixed

fixed

Cell of vectors of size 3

true
false

mo_r_0/start

start

Cell of vectors of size 3

mo_v_0

v_0

v_0

Structure

mo_v_0/fixed

fixed

Cell of vectors of size 3

true
false

mo_v_0/start

start

Cell of vectors of size 3

mo_a_0

a_0

a_0

Structure

mo_a_0/fixed

fixed

Cell of vectors of size 3

true
false

mo_a_0/start

start

Cell of vectors of size 3

mo_w_a

w_a

w_a

Structure

mo_w_a/fixed

fixed

Cell of vectors of size 3

true
false

mo_w_a/start

start

Cell of vectors of size 3

mo_z_a

z_a

z_a

Structure

mo_z_a/fixed

fixed

Cell of vectors of size 3

true
false

mo_z_a/start

start

Cell of vectors of size 3

mo_g_0

g_0

g_0

Structure

mo_g_0/fixed

fixed

Cell of vectors of size 3

true
false

mo_g_0/start

start

Cell of vectors of size 3

Ports

NameTypeDescriptionIO TypeNumber

frame_a

implicit

Coordinate system fixed at body

input

1