## 3-D Multibody System Statistics

The 3-D Multibody System Statistics represent aggregate statistics generated from all physical networks that are associated with blocks from the Simscape™ Multibody™ library.

The tool generates statistics separately according to topologically distinct physical networks and then aggregates them into a single statistic.

The individual statistics are:

• Bodies (total, excluding ground) — Total number of bodies present in a mechanical system. This number equals the sum of two types of bodies: rigid components and flexible bodies. For more information, see the statistic descriptions for these types of bodies.

• Rigidly connected components (excluding ground) — Number of rigid components present in a mechanical system. Rigid components are subsets of rigidly connected blocks that represent rigid bodies or rigid frame networks in a model. These subsets include blocks from the Body Elements library, including the Variable Mass sublibrary, as well as Rigid Transform blocks.

Rigid connections within a rigid component can include Rigid Transform blocks, but not Weld Joint blocks. Rigid Transform blocks provide rigid connections between blocks in the same rigid component. Weld Joint blocks, like all joint blocks, provide connections between blocks in different rigid components.

This statistic excludes from the count any rigid component that rigidly connects to the World Frame blocks.

• Flexible bodies — Number of flexible bodies present in a mechanical system. These correspond to individual blocks from the Flexible Bodies sublibrary.

Each individual flexible body block counts as one in this statistic. For example, two flexible body blocks directly connected through a frame line count as two separate flexible bodies.

Flexible bodies do not combine with any rigid component blocks. For example, if two flexible body blocks are connected through a Rigid Transform block, then the Rigid Transform block makes up a rigid component wedged between two separate flexible bodies.

• Joints (total) — Total number of joints present in a mechanical system. This number equals the sum of three types of joints: explicit tree, cut, and implicit 6-DOF joints. For more information, see the statistic descriptions for these types of joints.

• Explicit tree joints — Number of joints in a mechanical system that correspond to explicit joint blocks. This number excludes joints that are cut to open kinematic loops.

• Implicit 6-DOF tree joints — Number of 6-DOF joints in a mechanical system that do not correspond to explicit joint blocks. Simscape Multibody adds one implicit 6-DOF joint for each portion of the system that is disconnected from the ground body. Such implicit joints never create kinematic loops and are therefore never cut.

• Cut joints — Number of joints that are cut from a mechanical system in order to open all of its closed kinematic loops. The number of cut joints equals the number of closed loops present in the system.

• Constraints — Total number of constraints in a mechanical system. This number includes constraint elements stemming from explicit constraint blocks as well as those generated from belt-cable networks. It does not include constraints stemming from cut joints.

• Tree degrees of freedom (total) — Total number of degrees of freedom in a mechanical system, before the application of any constraint equations. This number equals the sum of the total number of uncut joint degrees of freedom and the total number of body degrees of freedom. For more information, see the statistic descriptions for Tree joint degrees of freedom and Flexible body degrees of freedom.

• Tree joint degrees of freedom — Number of uncut joint degrees of freedom in a mechanical system. This number equals the sum of all degrees of freedom that the uncut joints provide. It excludes degrees of freedom associated with cut joints.

• Flexible body degrees of freedom — Number of body degrees of freedom in a mechanical system. This number equals the sum of all degrees of freedom that the flexible bodies provide. Rigid components do not have any degrees of freedom and do not contribute to this statistic.

• Position constraint equations (total) — Number of scalar equations that impose position constraints on a mechanical system. Constraint equations arise from three types of blocks: constraints, joints, and belt-cable blocks. Joint blocks contribute constraint equations only if the joints are cut. The number of position constraint equations that a cut joint contributes equals six minus the number of degrees of freedom that joint provides.

• Position constraint equations (non-redundant) — Number of unique position constraint equations associated with a model. This number is smaller than or equal to the total number of position constraint equations. The difference between the two is the number of redundant position constraint equations, which are satisfied whenever the unique position constraint equations are satisfied. Simscape Multibody attempts to remove redundant equations to improve simulation performance.

• Mechanism degrees of freedom (minimum) — Lower bound on the number of degrees of freedom in a mechanical system. It equals the difference between the total number of (unconstrained) degrees of freedom and the number of non-redundant position constraint equations. The actual number of degrees of freedom can exceed this lower bound if Simscape Multibody fails to detect a position constraint equation.

Some position constraint equations become redundant only in certain configurations. If an equation becomes redundant during simulation, the actual number of degrees of freedom in a model can change. However, that number must still equal or exceed the lower bound that this statistic provides.

• State vector size — Number of scalar values in the state vector of a mechanical system.

• Average number of degrees of freedom in kinematic loops — Average number of degrees of freedom in the closed kinematic loops of a mechanical system. The average number is taken over all loops in the system. If the system has no kinematic loops, this number equals zero.