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# Documentation

## Glossary

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actuator

An actuator is a machine component that converts a Simulink® signal into SimMechanics™ force, torque, or motion signals.

• You can configure a body actuator to apply forces/torques to a body either as an explicit function of time or through feedback forces/torques.

• You can configure a joint actuator to apply forces/torques between the bodies connected on either side of the joint.

• You can configure a driver actuator to apply relative motion between the bodies connected on either side of the driver.

Two specialized SimMechanics actuators set joint initial conditions and apply stiction to a joint, respectively.

ActuatorWhat the Actuator Does
Body ActuatorApplies forces to a body
Driver ActuatorApplies motion to a time-dependent constraint
Joint ActuatorApplies forces or motions to a joint
Joint Initial Condition ActuatorSets a joint's initial conditions
Joint Stiction ActuatorApplies static and kinetic friction to joint motion

A SimMechanics Actuator block has an open round SimMechanics connector port for connecting with a Body, Joint, or Driver block and an angle bracket > Simulink inport for connecting with normal Simulink blocks, such as Source blocks for generating force or torque signals.

The adjoining CS of a Body coordinate system (CS) is the CS on the neighboring body or ground directly connected to the original Body CS by a Joint, Constraint, or Driver.

See also body, Body CS, coordinate system (CS), grounded CS, and World.

assembled configuration

A machine is in its assembled configuration once it passes from its home configuration through its initial configuration and its disassembled joints are then assembled. The assembly of joints can change body and joint configurations.

assembled joint

An assembled joint restricts the Body coordinate systems (CSs) on the two bodies at either end of the joint.

• For an assembled prismatic joint, the two Body CS origins must lie along the prismatic axis. The two Bodies translate relatively along the same axis.

For an assembled joint with multiple prismatic primitives, the two Body CS origins must lie in the plane or space defined by the directions of the prismatic axes.

• For an assembled revolute joint, the two Body CS origins must be collocated. The two Bodies rotate relatively about the same axis.

For an assembled joint with multiple revolute primitives, the two Body CS origins must be collocated.

• For an assembled spherical joint, the two Body CS origins must be collocated at the spherical primitive's pivot point. The two Bodies pivot relatively about this common origin.

You specify an assembly tolerance for assembled joints, the maximum dislocation distance allowed between all pairs of assembled Body CS origins and the maximum angle of misalignment between all pairs of assembled Body motion axes. If the distance dislocations and/or axis misalignments in an assembled joint grow larger than the assembly tolerance, the simulation stops with an error.

See also assembly tolerance, Body CS, collocation, disassembled joint, joint, and primitive joint.

assembly

An assembly represents a mechanical system in computer-aided design (CAD). An assembly includes parts (bodies with full geometric, mass, and inertia tensor information), as well as constraints (sometimes called mates) restricting the degrees of freedom of the parts.

Every assembly has a fundamental root attached to the assembly coordinate origin. Assemblies can also have one or more subassemblies branching off the main assembly at a single point.

Assembly specifications typically also include design tolerances and how subassemblies move and how they are connected to the main assembly.

assembly tolerance

The assembly tolerance is a range that determines how closely an assembled joint must be collocated and aligned. An assembled joint is connected on either side to Body coordinate systems (CSs) on two Bodies and restricts the relative configurations and motions of those Body CSs.

The assembly tolerances set the maximum dislocation of Body CS origins and maximum misalignment of motion axes allowed in assembled joints during the simulation.

• For assembled prismatic primitives, each pair of Body CS origins must lie in the subspace defined by the prismatic axes. Each pair of Bodies translates along these common axes.

• For assembled revolute primitives, each pair of Body CS origins must be collocated and their respective rotational axes aligned. Each pair of Bodies rotates about these common axes.

• For an assembled spherical primitive, the pair of Body CS origins must be collocated. The two Bodies pivot about this common origin.

A SimMechanics simulation attempts to assemble all joints in your machine at the start of simulation, including initially disassembled joints. If it cannot, the simulation stops with an error.

If the two Body CSs separate or the joint axes misalign in a way that makes their connecting assembled joint primitives no longer respect the assembly tolerances, the simulation stops with an error.

associativity

Associativity is a persistent and session-independent parallel relationship among certain components of a CAD assembly, Physical Modeling XML files exported from it, and SimMechanics models generated from the XML. This relationship preserves the identities and parallelisms of certain CAD components (parts, geometric and kinematic relationships, subassembly hierarchy) and the corresponding components of the SimMechanics model. These SimMechanics model components include:

• Body and Ground blocks

• Joint blocks

• Coordinate systems connected to Joints

• Subsystem hierarchies

CAD assembly and SimMechanics model components related in this way are associated. If they are not so related, they are not associated or nonassociated.

axis-angle rotation

An axis-angle rotation mathematically represents a three-dimensional spherical rotation as a rotation axis vector n = (nx,ny,nz) of unit length (n·n = nx2 + ny2 + nz2 = 1) and a rotation angle θ. Define the rotation axis by the vector n; rotate about that axis by θ using the right-hand rule. The n axis is sometimes called the eigenaxis.

The rotation axis direction is equivalent to specifying two independent angles; θ is the third independent angle making up the rotation.

In VRML, you represent body rotations by a vector signal [nx ny nz θ].

base (base body)

The base body is the body from which a joint is directed. The joint directionality runs from base to follower body.

Joint directionality sets the direction and the positive sign of all joint motion and force-torque data.

body

A body is the basic element of a mechanical system. It is characterized by its

• Mass properties (mass and inertia tensor)

• Position and orientation in space

• Attached Body coordinate systems

Bodies are connected to one another by joints, constraints, or drivers. Bodies carry no degrees of freedom.

You can attach to a Body block any number of Body coordinate systems (CSs). All SimMechanics Bodies automatically maintain a minimum of one Body CS at the body's center of gravity (CG). The Body block has distinctive axis triad CS ports , instead of the open, round connector ports , to indicate the attached Body CSs.

Body CS

A Body CS is a local coordinate system (CS) attached to a body, carried along with that body's motion. In general, bodies accelerate as they move, and therefore Body CSs define noninertial reference frames.

You can attach any number of Body CSs to a Body block, and you can choose where to place the Body CS origins and how to orient the Body CS axes. The Body block has distinctive axis triad Body CS ports instead of the open, round connector ports, to give you access to these Body CSs for connecting Joint, Sensor, and Actuator blocks.

Every Body block has an automatic, minimum Body CS at its center of gravity (CG). By default, it also has two other Body CSs for connection to adjacent Joints. Once you set the origin and axis orientation of each Body CS during Body configuration, the Body CSs are interpreted as fixed rigidly in that body during the simulation.

center of gravity (CG)

The center of gravity or center of mass of an extended body is the point in space about which the entire body balances in a uniform gravitational field. For translational dynamics, the body's entire mass can be considered as if concentrated at this point.

Every Body block has an automatic, minimum Body coordinate system (CS) with its origin at the CG — the CG CS. This origin point and the Body CS coordinate axes remain fixed rigidly in the body during the simulation.

See also body, Body CS, degree of freedom (DoF), inertia tensor, kinematics, and primitive joint.

CG
closed loop machine

A machine diagram contains one or more closed loops if, beginning at a starting point, you can trace a path through the machine back to the starting point without jumping out of or cutting the diagram. The number of closed loops is equal to the minimum number of cuttings needed to convert the diagram into a tree or open machine.

collocation

Collocation is the coincidence of two points in space, within assembly tolerances.

composite joint

A composite joint is a joint compounded from more than one joint primitive and thus representing more than one degree of freedom. The joint primitives constituting a composite joint are the primitives of that joint.

A spherical primitive represents three rotational degrees of freedom, but is treated as a primitive.

See also constrained joint, degree of freedom (DoF), joint, and primitive joint.

Computer-aided design systems or platforms provide an environment to design machines, with full geometric information about parts (bodies) and their spatial relationships, as well as the degrees of freedom and mass properties of the parts.

A CAD representation of a machine is an assembly.

See also assembly, associativity, body, constraint, degree of freedom (DoF), inertia tensor, mass, part, and STL.

connection line

You connect each SimMechanics block to another by using SimMechanics connection lines. These lines function only with SimMechanics blocks. They do not carry signals, unlike normal Simulink lines, and cannot be branched. You cannot link connection lines directly to Simulink lines.

By default, connection lines appear red and dashed if they are not anchored at both ends to a connector port . Once you anchor them, the lines become black and solid.

However, if you have selected a nonnull choice in Sample Time Display from your model's Format menu, the connection line displays instead with whatever color chosen to indicate the sample time.

connector port

A connector port is an anchor for a connection line. Each SimMechanics block has one or more open round SimMechanics connector ports for connecting to other SimMechanics blocks. You must connect these round ports only to other SimMechanics round ports. When an open connector port is attached to a connection line, the Port changes to solid .

A Connection Port block is provided in the SimMechanics library to create a round SimMechanics connector port for an entire subsystem on that subsystem's boundary.

constrained joint

A constrained joint is a composite joint with one or more internal constraints restricting the joint's primitives.

An example is the Screw block, which has a prismatic and a revolute primitive with their motions in fixed ratio. Only one of these degrees of freedom is independent.

constraint

A constraint is a restriction among degrees of freedom imposed independently of any applied forces/torques. A constraint removes one or more independent degrees of freedom, unless that constraint is redundant and restricts degrees of freedom that otherwise could not move anyway. Constraints can also create inconsistencies with the applied forces/torques that lead to simulation errors.

• Constraints are kinematic: they must involve only coordinates and/or velocities. Higher derivatives of coordinates (accelerations, etc.) are determined by the Newtonian force and torque laws and cannot be independently constrained.

• Constraints are holonomic (integrable into a form involving only coordinates) or nonholonomic (not integrable; that is, irreducibly involving velocities).

• Constraints specify kinematic relationships that are explicit functions of time (rheonomic) or not (scleronomic).

SimMechanics Constraints represent scleronomic constraints, and Drivers represent rheonomic constraints. SimMechanics Constraint and Driver blocks are attached to pairs of Body blocks.

• In SimMechanics models with closed loops, one Joint, Constraint, or Driver per loop is internally cut and replaced by an invisible scleronomic or rheonomic constraint.

• Constraints are redundant if they independently impose the same restrictions on a machine.

In computer-aided design (CAD) assemblies, a constraint restricts one or more degrees of freedom of the assembly parts. (CAD constraints are sometimes called mates.) When a CAD assembly is converted to a SimMechanics model, such restricted degrees of freedom are translated into specific joints. (SimMechanics bodies have no degrees of freedom.)

convex hull

The convex hull of a set of points in space is the surface of minimum area with convex (outward) curvature that passes through all the points in the set. In three dimensions, this set must contain at least four distinct, non-coplanar points to make a closed surface with nonzero enclosed volume.

The convex hull is an option for visualizing a SimMechanics body. The set of points is all the Body coordinate system (CS) origins configured in that Body block. The visualization of an entire machine is the set of the convex hulls of all its bodies. A SimMechanics convex hull excludes the body's center of gravity CS.

If a Body has fewer than four distinct, non-coplanar Body CSs, its convex hull is a lower-dimensional figure:

• Three distinct Body CSs produce a triangle without volume.

• Two distinct Body CSs produce a line without area.

• One Body CS produces a point without length.

coordinate system (CS)

A coordinate system is a geometric object defined, in a particular reference frame, by a choice of origin and orientation of coordinate axes, assumed orthogonal and Cartesian (rectangular). An observer attached to that CS measures distances from that origin and directions relative to those axes.

SimMechanics software supports two CS types:

• World: global or absolute inertial CS at rest

The assembly origin of a CAD assembly translated into a model becomes the World CS origin.

• Local:

• Grounded CS

• Body CS, including the center of gravity (CG) CS

Constraint points on CAD parts in a CAD assembly translated into a model become Body CS origins on the bodies representing the parts.

Local coordinate systems are sometimes called working frames.

degree of freedom (DoF)

A degree of freedom is a single coordinate of relative motion between two bodies. Such a coordinate is free only if it can respond without constraint or imposed motion to externally applied forces or torques. For translational motion, a DoF is a linear coordinate along a single direction. For rotational motion, a DoF is an angular coordinate about a single, fixed axis.

A prismatic joint primitive represents a single translational DoF. A revolute joint primitive represents a single rotational DoF. A spherical joint primitive represents three rotational DoFs in angle-axis form. A weld joint primitive represents zero DoFs.

directionality

The directionality of a joint, constraint, or driver is its direction of forward motion.

The joint directionality is set by the order of the joint's connected bodies and the direction of the joint axis vector. One body is the base body, the other the follower body. The joint direction runs from base to follower, up to the sign of the joint axis vector. Reversing the base-follower order or the joint axis vector direction reverses the forward direction of the joint.

Joint directionality sets the direction and the positive sign of all joint motion and force-torque data.

Directionality of constraints and drivers is similar, except there is no joint axis, only the base-follower sequence.

See also base (base body), body, follower (follower body), joint, and right-hand rule.

disassembled joint

A disassembled joint is a joint that need not respect the assembly tolerances of your machine.

• For a disassembled prismatic primitive, the Body coordinate system (CS) origins do not have to lie on the prismatic axis.

• For a disassembled revolute primitive, the Body CS origins do not have to be collocated.

• For a disassembled spherical primitive, the Body CS origins do not have to be collocated.

A SimMechanics simulation attempts to assemble all disassembled joints in your machine at the start of simulation. If it cannot, the simulation stops with an error.

You can use disassembled joints only in a closed loop, with no more than one per loop.

See also assembled joint, assembly tolerance, closed loop system, collocation, and topology.

DoF
driver

A driver is a constraint that restricts degrees of freedom as an explicit function of time (a rheonomic constraint) and independently of any applied forces/torques. A driver removes one or more independent degrees of freedom, unless that driver is inconsistent with the applied forces/torques and forces a simulation error.

You specify the driver function of time in a dialog box in terms of an input Simulink signal from a Driver Actuator.

SimMechanics Driver blocks are attached to pairs of Body blocks.

dynamics

Dynamics is distinguished from kinematics by explicit specification of applied forces/torques and body mass properties.

A forward dynamic analysis of a mechanical system specifies

• The topology of how bodies are connected

• The degrees of freedom (DoFs) and constraints among DoFs

• All the forces/torques applied to the bodies

• The mass properties (masses and inertia tensors) of the bodies

• The initial condition of all DoFs:

• Initial linear coordinates and velocities

• Initial angular coordinates and velocities

The analysis then solves Newton's laws to find the system's motion for all later times.

Inverse dynamics is the same, except that the system's motion is specified and the forces/torques necessary to produce this motion are determined.

See also constraint, degree of freedom (DoF), inertia tensor, kinematics, mass, and topology.

equivalent ellipsoid

The equivalent ellipsoid of a body is the homogeneous solid ellipsoid, centered at the body's center of gravity, with the same principal moments of inertia and principal axes as the body. A homogeneous solid ellipsoid is the simplest body with three distinct principal moments.

Every body has a unique equivalent ellipsoid, but a given homogeneous ellipsoid corresponds to an infinite number of other, more complicated, bodies. The rotational dynamics of a body depend only on its equivalent ellipsoid (which determines its principal moments and principal axes), not on its detailed shape.

The equivalent ellipsoid is an option for visualizing a SimMechanics body.

Euler angles

Euler angles mathematically represents a three-dimensional spherical rotation as a product of three successive independent rotations about three independent axes by three independent (Euler) angles. Follow the Euler angle convention by

1. Rotating about one axis (which rotates the other two).

2. Then rotating about a second axis (rotated from its original direction) not identical to the first.

3. Lastly, rotating about another axis not identical to the second.

There are 3*2*2 = 12 possible Euler angle rotation sequences.

fixed part

A fixed part of a computer-aided design assembly or subassembly is a part that is welded to the assembly or subassembly root. A fixed part cannot move relative to the root.

follower (follower body)

The follower body is the body to which a joint is directed. The joint directionality runs from base to follower body.

Joint directionality sets the direction and the positive sign of all joint motion and force-torque data.

fundamental root

The fundamental root is a point in a computer-aided assembly that does not move, usually coincident with the assembly origin. All translational and rotational motion of parts in the assembly reference this unmoving point.

The computer-aided assembly origin is recreated as the World coordinate system origin in a CAD-generated model.

ground

A ground or ground point is a point fixed at rest in the absolute or global inertial World reference frame.

Each ground has an associated grounded coordinate system (CS). The grounded CS's origin is identical to the ground point.

grounded CS

A grounded CS is a local CS attached to a ground point. It is at rest in World, but its origin is wherever the ground point is and in general shifted with respect to the World CS origin. The coordinate axes of a grounded CS are always parallel to the World CS axes.

The World coordinate axes are defined so that:

+x points right

+y points up (gravity in -y direction)

+z points out of the screen, in three dimensions

You automatically create a Grounded CS whenever you set up a Ground block.

home configuration

The bodies of a machine are in their home configuration when they are positioned and oriented purely according to the positions and orientations entered into the Body dialogs. This configuration assumes zero body velocities.

inertia tensor

The inertia or moment of inertia tensor of an extended rigid body describes its internal mass distribution and the body's angular acceleration in response to an applied torque.

Let V be the body's volume and ρ(r) its mass density, a function of vector position r within the body. Then the components of the inertia tensor I are:

${I}_{ij}=\underset{V}{\int }dV\left[{\delta }_{ij}{|r|}^{2}-{r}_{i}{r}_{j}\right]\rho \left(r\right)$

The indices i, j range over 1, 2, 3, or x, y, z. This tensor is a real, symmetric 3-by-3 matrix or equivalent MATLAB® expression.

The inertia tensor of a SimMechanics body is always evaluated in that body's center of gravity coordinate system (CG CS). That is, the origin is set to the body's CG and the coordinate axes are the CG CS axes.

Because the CG CS of a Body block is fixed rigidly in the body during simulation, the values of the inertia tensor components do not change as the body rotates.

initial condition actuator

An initial condition actuator sets a system's degrees of freedom nondynamically to prepare a system for dynamical integration, in a way consistent with all constraints.

The initial conditions are applied to a SimMechanics joint primitive.

initial configuration

A machine is in its initial configuration once all initial condition actuators have been applied to its joints. This step can change the positions and orientations of the machine's bodies, as well as apply nonzero initial velocities.

joint

A joint is a machine component that represents one or more mechanical degrees of freedom between two bodies. Joint blocks connect two Body blocks in a SimMechanics schematic. A Joint has no mass properties such as a mass or an inertia tensor, and the reaction force and torque are equal and opposite on its two connected Bodies.

A joint primitive represents one translational or rotational degree of freedom or one spherical (three rotational degrees of freedom in angle-axis form). Prismatic and revolute primitives have motion axis vectors. A weld primitive has no degrees of freedom.

A primitive joint contains one joint primitive. A composite joint contains more than one joint primitive.

Joints have a directionality set by their base-to-follower Body order and the direction of the joint primitive axis. The sign of all motion and force-torque data is determined by this directionality.

kinematics

A kinematic analysis of a mechanical system specifies topology, degrees of freedom (DoFs), motions, and constraints, without specification of applied forces/torques or the mass properties of the bodies.

The machine state at some time is the set of all

• Instantaneous positions and orientations

• Instantaneous velocities

of all bodies in the system, for both linear (translational) and angular (rotational) DoFs of the bodies.

Specification of applied forces/torques and solution of the system's motion as a function of time are given by the system's dynamics.

local CS

A local coordinate system (CS) is attached to either a Ground or a Body:

• A Grounded CS is automatically defined when you represent a ground point by a Ground block and is always at rest in the World reference frame. The origin of this Grounded CS is the same point as the ground point and not in general the same as the World CS origin.

• You define one or more Body CSs when you configure the properties of a Body. A Body CS is fixed rigidly in the body and carried along with that body's motion.

To indicate an attached coordinate system, a Body block has an axis triad CS port in place of the open, round connector port .

See also body, Body CS, coordinate system (CS), grounded CS, reference frame (RF), and World.

machine

In a SimMechanics model, a machine is a complete, connected block diagram representing one mechanical system. It is topologically isolated from any other machine in your model and has at least one ground and exactly one Machine Environment block.

A SimMechanics model has one or more machines.

machine precision constraint

A machine precision constraint implements motion restrictions on constrained degrees of freedom to the precision of your computer processor's arithmetic. It is the most robust, computationally intensive, and slowest-simulating constraint.

The precision to which the constraint is maintained depends on scale or the physical system of units.

mass

The mass is the proportionality between a force on a body and the resulting translational acceleration of that body.

Let V be the body's volume and ρ(r) its mass density, a function of position r within the body. Then the mass m is:

$m=\underset{V}{\int }dV\rho \left(r\right)$

The mass is a real, positive scalar or equivalent MATLAB expression.

A body's mass is insensitive to choice of reference frame, coordinate system origin, or coordinate axes orientation.

massless connector

A massless connector is a machine component equivalent to two joints whose respective primitive axes are spatially separated by a fixed distance. You can specify the gap distance and the axis of separation. The space between the degrees of freedom is filled by a rigid connector of zero mass.

You cannot actuate or sense a massless connector.

open machine

You can disconnect an open machine diagram into two separate diagrams by cutting no more than one joint.

Such machines can be divided into two types:

• An open chain is a series of bodies connected by joints and topologically equivalent to a line.

• An open tree is a series of bodies connected by joints in which at least one body has more than two joints connected to it. Bodies with more than two connected joints define branch points in the tree. A tree can be disconnected into multiple chains by cutting the branch points.

The end body of a chain is a body with only one connected joint.

part

A part represents a body in a computer-aided design (CAD) assembly. In CAD representations, a part typically includes full geometric, as well as mass and inertia tensor, information about a body.

Body degrees of freedom are restricted in CAD by constraints (sometimes called mates).

After translation into a SimMechanics model, parts are represented by bodies.

physical tree

You obtain the physical tree representation from a full machine diagram by removing actuators and sensors and cutting each closed loop once. The physical tree retains bodies, joints, constraints, and drivers.

primitive joint

A primitive joint expresses one degree of freedom (DoF) or coordinate of motion, if this DoF is a translation along one direction (prismatic joint) or a rotation about one fixed axis (revolute joint).

A SimMechanics spherical joint (three DoFs: two rotations to specify directional axis, one rotation about that axis) is also treated as a primitive joint.

These three types of primitive joints are the joint primitives from which composite joints are built.

A weld primitive has no degrees of freedom.

principal axes

The inertia tensor of a body is real and symmetric and thus can be diagonalized, with three real eigenvalues and three orthogonal eigenvectors. The principal axes of a body are these eigenvectors.

principal inertial moments

The inertia tensor of a body is real, symmetric, and diagonalizable, with three real eigenvalues and three orthogonal eigenvectors. The principal inertial moments or principal moments of inertia of a body are these eigenvalues, the diagonal values when the tensor is diagonalized.

The principal moments of a real body satisfy the triangle inequalities: the sum of any two moments is greater than or equal to the third moment.

If two of the three principal moments are equal, the body has some symmetry and is dynamically equivalent to a symmetric top. If all three principal moments are equal, the body is dynamically equivalent to a sphere.

quaternion

A quaternion mathematically represents a three-dimensional spherical rotation as a four-component row vector of unit length:

$q=\left[{n}_{x}\mathrm{sin}\left(\theta /2\right),{n}_{y}\mathrm{sin}\left(\theta /2\right),{n}_{z}\mathrm{sin}\left(\theta /2\right),\mathrm{cos}\left(\theta /2\right)\right]=\left[{q}_{\text{v}},{q}_{\text{s}}\right]$

with q*q = 1. The vector n = (nx,ny,nz) is a three-component vector of unit length: n·n = 1. The unit vector n specifies the axis of rotation. The rotation angle about that axis is θ and follows the right-hand rule.

The axis-angle representation of the rotation is just [ n θ ].

reference frame (RF)

A reference frame (RF) is the state of motion of an observer.

An inertial RF is a member of a set of all RFs moving uniformly with respect to one another, without relative acceleration. This set defines inertial space.

An RF is necessary but not sufficient to define a coordinate system (CS). A CS requires an origin point and an oriented set of three orthogonal axes.

RF
right-hand rule

The right-hand rule is the standard convention for determining the sign of a rotation: point your right thumb into the positive rotation axis and curl your fingers into the forward rotational direction.

root body

A root body is a component of a SimMechanics model translated from a computer-aided design (CAD) assembly. In the translated model, a block sequence Ground — Root Weld — Root Body or Root Body — Root Weld — Fixed Body represents the CAD assembly's fundamental or subassembly root, respectively.

In CAD assemblies, the fundamental or subassembly root represents a fixed point relative to which all part motion or subassembly part motion is measured. The fundamental root is usually the same as the assembly origin.

rotation matrix

A rotation matrix mathematically represents a three-dimensional spherical rotation as a 3-by-3 real, orthogonal matrix R: RTR = RRT = I, where I is the 3-by-3 identity and RT is the transpose of R.

$R=\left(\begin{array}{ccc}{R}_{11}& {R}_{12}& {R}_{13}\\ {R}_{21}& {R}_{22}& {R}_{23}\\ {R}_{31}& {R}_{32}& {R}_{33}\end{array}\right)=\left(\begin{array}{ccc}{R}_{\text{xx}}& {R}_{\text{xy}}& {R}_{\text{xz}}\\ {R}_{\text{yx}}& {R}_{\text{yy}}& {R}_{\text{yz}}\\ {R}_{\text{zx}}& {R}_{\text{zy}}& {R}_{\text{zz}}\end{array}\right)$

In general, R requires three independent angles to specify the rotation fully. There are many ways to represent the three independent angles. Here are two:

• You can form three independent rotation matrices R1, R2, R3, each representing a single independent rotation. Then compose the full rotation matrix R with respect to fixed coordinate axes (like World) as a product of these three: R = R3*R2*R1. The three angles are Euler angles.

• You can represent R in terms of an axis-angle rotation n = (nx,ny,nz) and θ with n·n = 1. The three independent angles are θ and the two needed to orient n. Form the antisymmetric matrix:

$J=\left(\begin{array}{ccc}0& -{n}_{z}& {n}_{y}\\ {n}_{z}& 0& -{n}_{x}\\ -{n}_{y}& {n}_{x}& 0\end{array}\right)$

Then Rodrigues' formula simplifies R:

sensor

A sensor is a machine component that measures the motion of, or forces/torques acting on, a body or joint. A sensor can also measure the reaction forces in a constraint or driver constraining a pair of bodies.

A SimMechanics Sensor block has an open round SimMechanics connector port for connecting with a Body or Joint block and an angle bracket > Simulink outport for connecting with normal Simulink blocks, such as a Sinks block like Scope.

See also actuator, body, connector port, constraint, driver, joint, and primitive joint.

spanning tree

You obtain the spanning tree representation from a full machine diagram by removing everything except bodies and joints and cutting each closed loop once.

stabilizing constraint

A stabilizing constraint implements motion restrictions on constrained degrees of freedom by modifying the dynamics of a system so that the constraint manifold is attractive, without changing the constrained solution. This constraint solver type is computationally the least intensive, the least robust, and the fastest-simulating.

The precision to which the constraint is maintained depends on scale or the physical system of units.

stiction actuator

A stiction actuator is a machine component that applies discontinuous friction forces to a joint primitive according to the relative velocity of one body with the other body.

If this relative velocity drops below a specified threshold, the relative motion ceases and the bodies or joints become locked rigidly to one another by static friction.

Above that threshold, the bodies or joints move relative to one another with kinetic friction.

STL

Stereolithographic (STL) format is an open file format for specifying the three-dimensional surface geometry or shape of a body. STL format specifies surface geometry by linked triangles whose edges and vertices are oriented by the right-hand rule.

SimMechanics visualization uses STL format files for custom body geometries and supports both ASCII and binary STL variants.

See also body, computer-aided design (CAD), convex hull, equivalent ellipsoid, part, right-hand rule, VRML and 3D Systems, creator of STL format, at www.3dsystems.com.

subassembly

A subassembly represents of a subset of machine parts and constraints (mates) in computer-aided design (CAD).

A subassembly is attached to its parent CAD assembly at a single branching point.

A subassembly is either flexible or rigid. That is, its parts either can move with respect to one another, or they cannot.

subassembly root

A subassembly root is a point in a computer-aided design (CAD) subassembly that does not move relative to the assembly point off of which it branches. All translational and rotational motion of parts in the subassembly reference this unmoving point.

tolerancing constraint

A tolerancing constraint implements motion restrictions on constrained degrees of freedom only up to a specified accuracy and/or precision.

This accuracy/precision is independent of any accuracy/precision limits on the solver used to integrate the system's motion, although constraints cannot be maintained to greater accuracy than the accuracy of the solver.

The precision to which the constraint is maintained depends on scale or the physical system of units.

Tolerancing constraints are moderately robust and moderately intensive and execute at moderate speed. They are less intensive than machine precision constraints, but computationally more intensive than stabilizing constraints.

Tolerancing constraints are most useful in realistic simulation of constraint slippage ("slop" or "play").

topology

The topology of a machine diagram is the global connectivity of all its elements. A diagram's elements are its bodies, and its connections are its joints, constraints, and drivers. Two topologies are equivalent if you can transform one diagram into another by continuous deformations and without cutting connections or joining elements.

An open machine diagram has no closed loops.

• An open chain is topologically equivalent to a line; and each body is connected to only two other bodies, if the body is internal, or one other body if it is at an end.

• An open tree has one or more branch points. A branch point is where an internal body is connected to more than two joints. A tree can be disconnected into multiple chains by cutting at the branch points.

A closed loop machine diagram has one or more closed loops. The number of closed loops is equal to the minimum number of joints, minus one, that must be cut to dissociate a diagram into two disconnected diagrams.

An actual diagram can have one of these primitive topologies or can be built from multiple primitive topologies.

See also body, closed loop machine, constraint, driver, joint, machine, open machine, and spanning tree.

VRML

Virtual Reality Modeling Language (VRML) is an open, Web-oriented ISO standard for defining three-dimensional virtual worlds and bodies in multimedia and the Internet.

In VRML, body rotations are represented in the axis-angle form. The SimMechanics RotationMatrix2VR block converts rotation matrices to the equivalent axis-angle forms.

World

In the SimMechanics environment, World is a kinematic and geometric construct defining both the absolute inertial reference frame (RF) and absolute coordinate system (CS) in that RF. World has a fixed origin and fixed coordinate axes that cannot be changed.

The World coordinate axes are defined so that:

+x points right

+y points up (gravity in -y direction)

+z points out of the screen, in three dimensions

The assembly origin of a CAD assembly translated into a model becomes the World CS origin.