# Documentation

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## Constraining and Driving Degrees of Freedom

The Simscape™ Multibody™ Constraints & Drivers library provides a set of blocks to model constraints on the relative motions of two bodies. You model the constraint by connecting the appropriate Constraint or Driver block between the two bodies. As with joints, the blocks each have a base and follower connector port, with the body connected to the follower port viewed as moving relative to the body connected to the base port. For example, the following model constrains Body2 to move along a track that is parallel to the track of Body1.

### Types of Mechanical Constraints

Constraint and Driver blocks enable you to model time-independent constraints or time-dependent drivers.

• Constraint and unactuated Driver blocks model scleronomic (time-independent) constraints.

• Actuated Driver blocks (see Actuating a Driver) model rheonomic (time-dependent) constraints.

Scleronomic constraints lack explicit time dependence; that is, their time dependence appears only implicitly through the coordinates x. Rheonomic constraints have explicit time dependence as well, in addition to implicit time dependence through the x.

Holonomic constraint functions depend only on body positions, not velocities:

`$f\left({x}_{B},{x}_{F};t\right)=0$`

Constraints of the form

`$g\left({x}_{B},{\stackrel{˙}{x}}_{B},{x}_{F},{\stackrel{˙}{x}}_{F};t\right)=0$`

can sometimes be integrated into a form dependent only on positions; but if not, they are nonholonomic. For example,

• The one-dimensional rolling of a wheel of radius R along a line (the x-axis) imposes a holonomic constraint, x = Rθ.

• The two-dimensional rolling of a sphere of radius R on a plane (the xy-plane) imposes a nonholonomic constraint, ds = R·dθ, with ds2 = dx2 + dy2. This constraint is nonholonomic because there is not enough information to solve the constraint independently of the dynamics.

### What Constraints and Drivers Do

Constrained and driven bodies are still free to respond to externally imposed forces/torques, but only in a way consistent with the constraints.

Constraints and drivers can only remove degrees of freedom from a machine. Constraints and unactuated Drivers prevent the machine from moving in certain ways. Unactuated Drivers hold the constrained degrees of freedom between the connected pair of bodies in their initial state. Actuated Drivers externally impose a relative motion between pairs of bodies, starting with the bodies' initial state. See Counting Model Degrees of Freedom.

This section discusses modeling constraints and drivers in a general way.

The section ends with two examples, Constraint Example: Gear Constraint and Driver Example: Angle Driver.

See the reference pages for information on the specific constraint that a Constraint or Driver block imposes.

### Directionality of Constraints and Drivers

Like joints, constraints and drivers have directionality. The sequence of base to follower body determines the directionality of the constraint or driver. The directionality determines how the sign of Driver Actuator signals affects the motion of the follower relative to the base and the sign of signals output by constraint and driver sensors.

### Solving Constraints

A Simscape Multibody simulation uses a constraint solver to find the motion, given the model's Constraint and Driver blocks. You can specify both the constraint solver type and the constraint tolerances used to find the constraint solution. See Maintaining Constraints for more information.

#### Mitigating Constraint Singularities

Some constraints, whether time-independent (Constraints) or time-dependent (Drivers), can become singular when the constrained bodies take on certain relative configurations; for example, if the two body axes line up when the Bodies are connected by an Angle Driver. The simulation slows down as a constraint becomes singular.

If you find a constrained model running slowly, consider selecting the Use robust singularity handling option in the Constraints tab of your machine's Machine Environment block dialog. See Handling Motion Singularities.

### Restrictions on Constraint and Driver Blocks

The following restrictions apply to the use of Constraint and Driver blocks in a model:

• Constraint and Driver blocks can appear only in closed loops. A closed loop cannot contain more than one Constraint or Driver block.

• A Constraint or Driver must connect exactly two Bodies.

### Constraint Example: Gear Constraint

The `mech_gears` model illustrates the Gear Constraint. Open the Body and Gear Constraint blocks.

Body1 and Body2 have their CG positions 2 meters apart. CS1 and CS2 on Body1 are collocated with the Body1 CG, and similarly, CS1 and CS2 on Body2 are collocated with the Body2 CG.

The Gear Constraint between them has two pitch circles. One is centered on the CS2 at the base Body, which is Body1, and has radius 1.5 meters. The other is centered on CS1 at the follower Body, which is Body2, and has radius 0.5 meters. The distance between CS2 on Body1 and CS1 on Body2 is 2 meters. The sum of the pitch circle radii equals this distance, as it must.

#### Visualizing the Gear Motion

When you run a simulation, the 3-D visualization window opens automatically, displaying your multibody model as a system of convex hulls. Start the simulation and watch the CG CS axis triads spin around. The CG triad at Body2 rotates three times faster than the CG triad at Body1, because the pitch circle centered on Body2 is three times smaller.

You can see the same behavior in the Scope. The upper plot shows the motion of Revolute2, and the lower plot the motion of Revolute1. Note that angular motion is mapped to the interval (-180o, +180o) degrees.

The Gear Constraint is inside a closed loop formed by

Ground_1–Revolute1–Body1–Gear Constraint–Body–Revolute2–Ground_2

Although Ground_1 and Ground_2 are distinct blocks, they represent different points on the same immobile ground at rest in World. So the blocks form a loop.

### Driver Example: Angle Driver

The following two models illustrate the Angle Driver, both without and with a Driver Actuator.

#### The Angle Driver Without a Driver Actuator

The first is `mech_angle_unact`. Open the Body2 block.

The bodies form a double pendulum of two rods. The Body Sensor is connected to Body2 at CS3 = CS2 and measures all three components of Body2's angular velocity vector with respect to the ground.

The Angle Driver is connected between Body2 and Ground_2. Because the Angle Driver is not actuated in this model, it acts during the simulation as a time-independent constraint to hold the angle between Body2 and Ground_2 constant at its initial value.

#### Visualizing the Angle Driver Motion

When you run a simulation, the 3-D visualization window opens up, displaying your model as a system of convex hulls.

Start the simulation. The upper body swings like a pendulum, but the lower body maintains its horizontal orientation with respect to the horizontal ground. The Scope measures Body2's angular velocity with respect to ground, and this remains at zero.

#### The Angle Driver With a Driver Actuator

The second model is `mech_angle_act`. Open the Driver Actuator block.

The Driver Actuator drives the Angle Driver block. Here, the Actuator accepts a constant angular velocity signal from the Simulink® blocks. The Actuator also requires the angle itself and the angular acceleration, together with the angular velocity, in a vector signal format. The Angle Driver's angle signal is added to the angle's initial value.

The Body Sensor again measures three components of Body2's angular velocity with respect to the ground. Constant1 drives the angle at 15o/second. While the simulation is running, this angle changes at the constant rate. At the same time, the assembly and the constant length of the two pendulum rods must be maintained by Simulink, while both rods are subject to gravity. As the two axes line up, the mutual constraint between the bodies enforced the Driver becomes singular. The simulation slows down.

As in the Gear Constraint model, the two Ground blocks in these models represent points on the same immobile ground at rest in World, so the Angle Driver is part of a closed loop.