# Documentation

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## 3-D Multibody System Statistics

This node represents aggregate statistics generated from all physical networks that are associated with blocks from Simscape™ Multibody™ Second Generation library.

Each statistic is generated separately from each topologically distinct physical network of these blocks and then aggregated to appear as a single statistic.

The individual statistics are:

• Number of rigidly connected components (excluding ground) — This statistic provides the 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 generally include blocks from the Body Elements library 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 block.

• Number of joints (total) — This statistic provides the 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 joints.

The kinematic graph provides a practical means to understand the topology of a model. This graph is a connected, undirected diagram in which each vertex corresponds to a rigid component and each edge corresponds to a joint. The total number of joints equals the total number of edges present in this graph.

The kinematic tree is a spanning tree of the kinematic graph in which each closed loop is opened by cutting one of its edges. If the kinematic graph contains no closed loops, it is identical to the kinematic tree.

• Number of explicit tree joints — This statistic provides the number of joints in the kinematic tree of a mechanical system that correspond to explicit joint blocks. Each tree joint corresponds to an edge in the kinematic tree. The number of explicit tree joints excludes joints cut from the kinematic graph to generate the kinematic tree.

For more information about kinematic graphs and trees, see the statistic description for Number of joints (total).

• Number of implicit 6-DOF tree joints — This statistic provides the number of 6-DOF joints in the kinematic tree of a mechanical system that do not correspond to explicit joint blocks. Simscape Multibody adds implicit 6-DOF joints when the kinematic graph of a model is not fully connected. These implicit joints connect previously disconnected portions of the graph to the ground body, adding the edges required to fully connect the graph. Implicit joints are always tree joints and do not create loops.

For more information about kinematic graphs and trees, see the statistic description for Number of joints (total).

• Number of cut joints — This statistic provides the number of joints that are cut from the kinematic graph of a mechanical system to generate the associated kinematic tree. The number of cut joints equals the number of closed loops present in the kinematic graph.

For more information about kinematic graphs and trees, see the statistic description for Number of joints (total).

• Number of constraints — This statistic provides the total number of constraint blocks in a mechanical system.

• Number of tree degrees of freedom — This statistic provides the total number of degrees of freedom in the kinematic tree of a mechanical system. This number equals the sum of all degrees of freedom that the tree joints provide. It excludes degrees of freedom associated with cut joints.

For more information about kinematic graphs and trees, see the statistic description for Number of joints (total).

• Number of position constraint equations (total) — This statistic provides the number of scalar equations that impose position constraints on a mechanical system. Constraint equations arise from two types of blocks: Constraints and Joints. Joint blocks contribute constraint equations only if the joints are cut in the kinematic tree. The number of position constraint equations that a cut joint contributes equals six minus the number of degrees of freedom that joint provides.

For more information about kinematic graphs and trees, see the statistic description for Number of joints (total).

• Number of position constraint equations (non-redundant) — This statistic provides the 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.

• Number of mechanism degrees of freedom (minimum) — This statistic provides a lower bound on the number of degrees of freedom in a mechanical system. It equals the difference between the number of tree 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 — This statistic provides the number of scalar values in the state vector of a mechanical system.

• Average kinematic loop length — This statistic provides the average number of edges—or, equivalently, vertices—in the closed loops of a kinematic graph. The average number is taken over all loops in the graph. If the graph has no kinematic loops, this number equals zero.

For more information about kinematic graphs and trees, see the statistic description for Number of joints (total).