Body

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 acoordinate system that is parallel to frame_a with theorigin at the center of mass of the body.

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

States of Body Components

Every body has potential states. If possible a tool will selectthe states of joints and not the states of bodies because this isusually the most efficient choice. In this case the position, orientation,velocity and angular velocity of frame_a of the body will be computedby the component that is connected to frame_a. However, if a body is movingfreely in space, variables of the body have to be used as states. The potentialstates 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 anon-linear constraint equation between the 4 quaternions.Therefore, at least one non-linear equation has to be solvedduring simulation. A tool might, however, analytically solve thissimple constraint equation. Using the 3 angles as states has thedisadvantage that there is a singular configuration in which adivision by zero will occur. If it is possible to determine in advancefor an application class that this singular configuration is outsideof the operating region, the 3 angles might be used as potentialstates by setting useQuaternions = false.

In text books about 3-dimensional mechanics often 3 angles and theangular velocity are used as states. This is not the case here, since3 angles and their derivatives are used as potential states(if useQuaternions = false). The reasonis that for real-time simulation the discretization formula of theintegrator might be "inlined" and solved together with the body equations.By appropriate symbolic transformation the performance isdrastically increased if angles and theirderivatives are used as states, instead of angles and the angularvelocity.

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