# Spherical Joint

Joint with one spherical primitive

Joints

## Description

This block represents a joint with three rotational degrees of freedom. One spherical primitive provides the three rotational degrees of freedom. The base and follower frame origins remain coincident during simulation.

Joint Degrees of Freedom

The joint block represents motion between the base and follower frames as a single time-varying transformation. The spherical primitive (S) applies this transformation, which causes the follower frame to rotate with respect to the base frame about an arbitrary 3-D axis. This joint primitive is not susceptible to gimbal lock.

Joint Transformation

A set of optional state targets guide assembly for each joint primitive. Targets include position and velocity. A priority level sets the relative importance of the state targets. If two targets are incompatible, the priority level determines which of the targets to satisfy.

Internal mechanics parameters account for energy storage and dissipation at each joint primitive. Springs act as energy storage elements, resisting any attempt to displace the joint primitive from its equilibrium position. Joint dampers act as energy dissipation elements. Springs and dampers are strictly linear.

In all but lead screw and constant velocity primitives, joint limits serve to curb the range of motion between frames. A joint primitive can have a lower bound, an upper bound, both, or, in the default state, neither. To enforce the bounds, the joint adds to each a spring-damper. The stiffer the spring, the harder the stop, or bounce, if oscillations arise. The stronger the damper, the deeper the viscous losses that gradually lessen contact oscillations or, in overdamped primitives, keep them from forming altogether.

Each joint primitive has a set of optional actuation and sensing ports. Actuation ports accept physical signal inputs that drive the joint primitives. These inputs can be forces and torques or a desired joint trajectory. Sensing ports provide physical signal outputs that measure joint primitive motion as well as actuation forces and torques. Actuation modes and sensing types vary with joint primitive.

## Parameters

### Spherical Primitive: State Targets

Specify the desired initial states of the spherical joint primitive and their relative priority levels. States that you can target include position and velocity. Use the priority level to help the assembly algorithm decide which of the state targets in a model to more precisely satisfy should conflicts between them arise.

Even in the absence of state target conflicts, the true initial states may differ from those specified here. Such discrepancies can occur due to kinematic constraints arising from other parts of the model. If a state target cannot be satisfied precisely, it is satisfied approximately. Discrepancies are noted in Simscape Variable Viewer (in the Apps gallery, click ).

Specify Position Target

Check to specify the desired rotation of the follower frame relative to the base frame at the start of simulation.

Priority

Select state target priority. This is the importance level assigned to the state target. If all state targets cannot be simultaneously satisfied, the priority level determines which targets to satisfy first and how closely to satisfy them. This option applies to both position and velocity state targets.

Priority LevelDescription
`High (desired)`Satisfy state target precisely
`Low (approximate)`Satisfy state target approximately

Note

During assembly, high-priority targets behave as exact guides. Low-priority targets behave as rough guides.

Value

Select a method to specify the joint primitive state target.

MethodDescription
`None`Constrain the base and follower frames to share the same orientation.
`Aligned Axes`Set frame rotation by aligning two follower frame axes with two base frame axes.
`Standard Axis`Specify frame rotation as an angle about a standard axis (x, y, or z).
`Arbitrary Axis`Specify frame rotation as an angle about a general [x, y, z] axis.
`Rotation Sequence`Specify frame rotation as a sequence of three elementary rotations.
`Rotation Matrix`Specify frame rotation as a right-handed orthogonal rotation matrix.
`Aligned Axes`

Select two pairs of base-follower frame axes.

ParameterDescription
Pair 1First pair of base-follower frame axes to align.
Pair 2Second pair of base-follower frame axes to align. Axis choices depend on Pair 1 axis selections.
`Standard Axis`

Select a standard rotation axis, resolved in the base frame, and specify the follower frame rotation angle.

ParameterDescription
AxisStandard rotation axis (X, Y, or Z) resolved in the base frame.
AngleFollower frame rotation angle about the rotation axis with respect to the base frame.
`Arbitrary Axis`

Select a general 3-D rotation axis, resolved in the base frame, and specify the follower frame rotation angle.

ParameterDescription
AxisGeneral rotation axis [X Y Z] resolved in the base frame.
AngleFollower frame rotation angle about the rotation axis with respect to the base frame.
`Rotation Sequence`

Specify a sequence of three elementary rotations about the selected permutation of x, y, and z axes. These rotation sequences are also known as Euler and Tait-Bryan sequences. The rotations are those of the follower frame relative to the frame selected in the Rotate About parameter.

If you set the Rotate About parameter to `Follower Frame`, the follower frame rotates about its own axes. These axes change orientation with each successive rotation. If you set the Rotate About parameter to `Base Frame`, the follower frame rotates about the fixed base frame axes.

ParameterDescription
SequenceSequence of axes about which to apply the elementary rotations.
AnglesThree-element vector with elementary rotation angles about the axes specified in the Sequence parameter.
`Rotation Matrix`

Specify the 3×3 transformation matrix of a proper rotation between the base and follower frames. The matrix must be orthogonal and have determinant +1. The default matrix is ```[1 0 0; 0 1 0; 0 0 1]```.

Specify Velocity Target

Check to specify the desired rotational velocity of the follower frame relative to the base frame at the start of simulation.

Value

Enter the relative rotational velocity of the follower frame against the base frame, as projected on the axes of the selected Resolution Frame (by default `Follower`). This parameter requires a three-element vector with the [x y z] components of the resolved relative velocity.

Resolution Frame

Select the frame in which to resolve the components of the velocity target. The resolution frame is not a measurement frame—the specified velocity is always that of the follower frame relative to the base frame. The resolution frame merely provides an alternate set of axes with respect to which to interpret the relative velocity components. The default setting is `Follower`.

### Spherical Primitive: Internal Mechanics

Specify the spherical primitive internal mechanics. This includes linear spring and damping forces, accounting for energy storage and dissipation, respectively. To ignore internal mechanics, keep spring stiffness and damping coefficient values at the default value of `0`.

Equilibrium Position

Select a method to specify the spring equilibrium position. The equilibrium position is the rotation angle between base and follower port frames at which the spring torque is zero.

MethodDescription
`None`Constrain the base and follower frames to share the same orientation.
`Aligned Axes`Set frame rotation by aligning two follower frame axes with two base frame axes.
`Standard Axis`Specify frame rotation as an angle about a standard axis (x, y, or z).
`Arbitrary Axis`Specify frame rotation as an angle about a general [x, y, z] axis.
`Rotation Sequence`Specify frame rotation as a sequence of three elementary rotations.
`Rotation Matrix`Specify frame rotation as a right-handed orthogonal rotation matrix.
`Aligned Axes`

Select two pairs of base-follower frame axes.

ParameterDescription
Pair 1First pair of base-follower frame axes to align.
Pair 2Second pair of base-follower frame axes to align. Axis choices depend on Pair 1 axis selections.
`Standard Axis`

Select a standard rotation axis, resolved in the base frame, and specify the follower frame rotation angle.

ParameterDescription
AxisStandard rotation axis (X, Y, or Z) resolved in the base frame.
AngleFollower frame rotation angle about the rotation axis with respect to the base frame.
`Arbitrary Axis`

Select a general 3-D rotation axis, resolved in the base frame, and specify the follower frame rotation angle.

ParameterDescription
AxisGeneral rotation axis [X Y Z] resolved in the base frame.
AngleFollower frame rotation angle about the rotation axis with respect to the base frame.
`Rotation Sequence`

Specify a sequence of three elementary rotations about the selected permutation of x, y, and z axes. These rotation sequences are also known as Euler and Tait-Bryan sequences. The rotations are those of the follower frame relative to the frame selected in the Rotate About parameter.

If you set the Rotate About parameter to `Follower Frame`, the follower frame rotates about its own axes. These axes change orientation with each successive rotation. If you set the Rotate About parameter to `Base Frame`, the follower frame rotates about the fixed base frame axes.

ParameterDescription
SequenceSequence of axes about which to apply the elementary rotations.
AnglesThree-element vector with elementary rotation angles about the axes specified in the Sequence parameter.
`Rotation Matrix`

Specify the 3×3 transformation matrix of a proper rotation between the base and follower frames. The matrix must be orthogonal and have determinant +1. The default matrix is ```[1 0 0; 0 1 0; 0 0 1]```.

Spring Stiffness

Enter the linear spring constant. This is the torque required to displace the joint primitive by a unit angle. The term linear refers to the mathematical form of the spring equation. The default is `0`. Select a physical unit. The default is `N*m/deg`.

Damping Coefficient

Enter the linear damping coefficient. This is the torque required to maintain a constant joint primitive angular velocity between base and follower frames. The default is `0`. Select a physical unit. The default is `N*m/(deg/s)`.

### Spherical Primitive: Limits

Limit the range of motion of the joint primitive. Joint limits use spring-dampers to resist travel past the bounds of the range. A joint primitive can have a lower bound, an upper bound, both, or, in the default state, neither. The stiffer the spring, the harder the stop, or bounce, if oscillations arise. The stronger the damper, the larger the viscous losses that gradually lessen contact oscillations or, in overdamped primitives, keep them from forming altogether.

Specify Lower Limit

Select to add a lower bound to the range of motion of the joint primitive.

Specify Upper Limit

Select to add an upper bound to the range of motion of the joint primitive.

Value

Location past which to resist joint travel. The location is the offset from base to follower, as measured in the base frame, at which contact begins. It is a distance along an axis in prismatic primitives, an angle about an axis in revolute primitives, and an angle between two axes in spherical primitives.

Spring Stiffness

Resistance of the contact spring to displacement past the joint limit. The spring is linear and its stiffness is constant. The larger the value, the harder the stop. The proportion of spring to damper forces determines whether the stop is underdamped and prone to oscillations on contact.

Damping Coefficient

Resistance of the contact damper to motion past the joint limit. The damper is linear and its coefficient is constant. The larger the value, the greater the viscous losses that gradually lessen contact oscillations, if any arise. The proportion of spring to damper forces determines whether the stop is underdamped and prone to oscillations on contact.

Transition Region

Region over which to raise the spring-damper force to its full value. The region is a distance along an axis in prismatic primitives, an angle about an axis in revolute primitives, and an angle between two axes in spherical primitives.

The smaller the region, the sharper the onset of contact and the smaller the time-step required of the solver. In the trade-off between simulation accuracy and simulation speed, reducing the transition region improves accuracy while expanding it improves speed.

### Spherical Primitive: Actuation

Specify actuation options for the spherical joint primitive. Actuation modes include Torque only. Selecting a torque input adds the corresponding physical signal port to the block. Use this port to specify the actuation torque signal.

Torque

Select a source for the actuation torque. The default setting is `None`.

Actuation Torque SettingDescription
`None`Apply no actuation torque.
`Provided by Input`Apply an actuation torque based on a physical signal. The signal specifies the torque acting on the follower frame with respect to the base frame. An equal and opposite torque acts on the base frame. Selecting this option exposes additional parameters.
Torque (X), Torque (Y), Torque (Z)

Select in order to actuate the spherical joint primitive about each standard Cartesian axis (X, Y, Z) separately. The block exposes the corresponding physical signal ports. Use these ports to specify the actuation torque signals. The signals must be scalar values.

Torque (XYZ)

Select in order to actuate the spherical joint primitive about an arbitrary axis [X Y Z]. The block exposes the corresponding physical signal port. Use this port to specify the actuation torque signal. The signal must be a 3-D vector.

Frame

Select the frame to resolve the actuation torque signal in. The axes of this frame establish the directions of the X, Y, and Z torque components. The default setting is `Base`.

### Spherical Primitive: Sensing

Select the motion variables to sense in the spherical joint primitive. The block adds the corresponding physical signal ports. Use these ports to output the numerical values of the motion variables.

The block measures each motion variable for the follower frame with respect to the base frame. It resolves that variable in the resolution frame that you select from the Frame drop-down list.

Motion VariablesDescription
PositionQuaternion describing follower frame rotation with respect to base frame. The quaternion coefficients are $\left[\mathrm{cos}\left(\frac{\theta }{2}\right),{n}_{x}\mathrm{sin}\left(\frac{\theta }{2}\right),{n}_{y}\mathrm{sin}\left(\frac{\theta }{2}\right),{n}_{z}\mathrm{sin}\left(\frac{\theta }{2}\right)\right]$. The measurement is the same in all measurement frames.
Velocity (X), Velocity (Y), Velocity (Z)Angular velocity components about X, Y, and Z axes.
Velocity3–D angular velocity vector with components about X, Y, and Z axes.
Acceleration (X), Acceleration (Y), Acceleration (Z)Angular acceleration components about X, Y, and Z axes.
Acceleration3–D angular acceleration vector with components about X, Y, and Z axes.
Frame

Select the frame to resolve the measurement in. The axes of this frame establish the directions of X, Y, and Z vector components. The default setting is `Base`.

### Mode Configuration

Specify the mode of the joint. The joint mode can be normal or disengaged throughout the simulation, or you can provide an input signal to change the mode during the simulation.

Mode

Select one of the following options to specify the mode of the joint. The default setting is `Normal`.

MethodDescription
`Normal`The joint behaves normally throughout the simulation.
`Disengaged`The joint is disengaged throughout the simulation.
`Provided by Input`This option exposes the mode port that you can connect to an input signal to change the joint mode during the simulation. The joint mode is normal when the input signal is `0` and disengaged when the input signal is `-1`. The joint mode can be changed many times during the simulation.

### Composite Force/Torque Sensing

Select the composite forces and torques to sense. Their measurements encompass all joint primitives and are specific to none. They come in two kinds: constraint and total.

Constraint measurements give the resistance against motion on the locked axes of the joint. In prismatic joints, for instance, which forbid translation on the xy plane, that resistance balances all perturbations in the x and y directions. Total measurements give the sum over all forces and torques due to actuation inputs, internal springs and dampers, joint position limits, and the kinematic constraints that limit the degrees of freedom of the joint.

Direction

Vector to sense from the action-reaction pair between the base and follower frames. The pair arises from Newton's third law of motion which, for a joint block, requires that a force or torque on the follower frame accompany an equal and opposite force or torque on the base frame. Indicate whether to sense that exerted by the base frame on the follower frame or that exerted by the follower frame on the base frame.

Resolution Frame

Frame on which to resolve the vector components of a measurement. Frames with different orientations give different vector components for the same measurement. Indicate whether to get those components from the axes of the base frame or from the axes of the follower frame. The choice matters only in joints with rotational degrees of freedom.

Constraint Force

Dynamic variable to measure. Constraint forces counter translation on the locked axes of the joint while allowing it on the free axes of its primitives. Select to output the constraint force vector through port fc.

Constraint Torque

Dynamic variable to measure. Constraint torques counter rotation on the locked axes of the joint while allowing it on the free axes of its primitives. Select to output the constraint torque vector through port tc.

Total Force

Dynamic variable to measure. The total force is a sum across all joint primitives over all sources—actuation inputs, internal springs and dampers, joint position limits, and kinematic constraints. Select to output the total force vector through port ft.

Total Torque

Dynamic variable to measure. The total torque is a sum across all joint primitives over all sources—actuation inputs, internal springs and dampers, joint position limits, and kinematic constraints. Select to output the total torque vector through port tt.

## Ports

This block has two frame ports. It also has optional physical signal ports for specifying actuation inputs and sensing dynamical variables such as forces, torques, and motion. You expose an optional port by selecting the sensing check box corresponding to that port.

### Frame Ports

• B — Base frame

• F — Follower frame

### Actuation Ports

The spherical joint primitive provides the following actuation ports:

• t — Actuation torque vector [tx, ty, tz] acting on the spherical joint primitive

• tx, ty, tz — X, Y, and Z components of the actuation torque acting on the spherical joint primitive

### Sensing Ports

The spherical primitive provides the following sensing ports:

• Q — Orientation of the spherical joint primitive in quaternion form

• wx, wy, wz — X, Y, and Z angular velocity components of the spherical joint primitive

• w — Angular velocity [wx, wy, wz] of the spherical joint primitive

• bx, by, bz — X, Y, and Z angular acceleration components of the spherical joint primitive

• b — Angular acceleration [bx, by, bz] of the spherical joint primitive

• tll — Torque due to contact with the lower limit of the spherical joint primitive, given as the signed magnitude of the torque vector

• tul — Torque due to contact with the upper limit of the spherical joint primitive, given as the signed magnitude of the torque vector

The following sensing ports provide the composite forces and torques acting on the joint:

• fc — Constraint force

• tc — Constraint torque

• ft — Total force

• tt — Total torque

### Mode Port

Mode configuration provides the following port:

• mode — Value of the mode of the joint. If the input is equal to `0`, the joint behaves normally. If the input is equal to `-1`, the joint behaves as disengaged.

## Version History

Introduced in R2012a