$\mathbf{F}$$\=⟨f_x\,f_y\,f_z⟩\,$is the reaction force vector,

$\mathbf{r}\={\mathbf{r}}_b\mathbf{-}{\mathbf{r}}_a\=⟨\left(x_b-x_a\right)\,\left(y_b-y_a\right)\,\left(z_b-z_a\right)⟩$, is the relative displacement vector,

${\mathit{r}}_0$, is the undeformed distance to frame_b from frame_a expressed along the inboard frame (frame_a), and

$\mathbf{v}\=\frac{d}{\mathrm{dt}}\mathit{r}\,$is the relative velocity vector.

$\mathbf{M} \=⟨{\mathrm{\tau }}_x\,{\mathrm{\tau }}_y\,{\mathrm{\tau }}_z⟩\,$is the reaction torque vector,

$\mathbf{\theta }$$\={\left[{\mathrm{\θ}}_1\,{\mathrm{\θ}}_2\,{\mathrm{\θ}}_3\right]}^T\,$are the Euler angles - defined by the Rotation Sequence [1,2,3] and calculated from the relative rotation matrix of frame_b with respect to frame_a (see Euler Angle Sensor),

${\mathbf{\theta }}_0$, designates the undeformed rotation of frame_b with respect to frame_a, and

$\mathbf{\omega }$$\=⟨{\mathrm{\ω}}_x\,{\mathrm{\ω}}_y\,{\mathrm{\ω}}_z⟩\,$is the relative angular velocity vector.

State Value

Description

1

ON state.

The state of the Gripper when $\mathrm{active}\mathbf{\=}\mathbf{true}$ at initialization or $\mathrm{T\_\_on/off}$ seconds have passed since $\mathrm{active}$ became true during simulation. In this state the reaction forces and torques are applied as described above.

2

ON2OFF state.

The state of the Gripper when less than $\mathrm{T\_\_on/off}$ seconds have passed since $\mathrm{active}$ became false during simulation. In this state the reaction forces and torques are multiplied by an internal variable $\mathrm{onoff}$ that decreases linearly from 1 to 0 over $\mathrm{T\_\_on/off}$ seconds. Changing the value of $\mathrm{active}$ while in this state has no effect.

3

OFF state.

The state of the Gripper when $\mathrm{active}\mathbf{\=}\mathbf{false}$ at initialization or $\mathrm{T\_\_on/off}$ seconds have passed since $\mathrm{active}$ became false during simulation. In this state there are no reaction forces and torques applied.

4

OFF2ON state.

The state of the Gripper when less than $\mathrm{T\_\_on/off}$ seconds have passed since $\mathrm{active}$ became true during simulation. In this state the reaction forces and torques are multiplied by an internal variable $\mathrm{onoff}$ that increases linearly from 0 to 1 over $\mathrm{T\_\_on/off}$ seconds. Changing the value of $\mathrm{active}$ while in this state has no effect.

Name

Description

Modelica ID

${\mathrm{frame}}_a$

Inboard frame

frame_a

${\mathrm{frame}}_b$

Outboard frame

frame_b

$\mathrm{active}$

Boolean signal to activate/deactivate the joint

active

Case

Use Torque Reactions

In Place

Description

1

The Gripper is activated and deactivated by an input boolean signal and only force reactions (due to translational springs) are applied.

2

Unlike case #1, the initial translational offset between frame_a and frame_b is used as the undeformed lengths of the translational springs. These undeformed lengths are updated any time the Gripper is not active and is then made active.

3

The same as case #1 and torque reactions (due to angular springs) are applied. When these options are selected the joint rotational stiffness and damping must be specified.

4

The same as case #2 and torque reactions (due to angular springs) are applied. When these options are selected the joint rotational stiffness and damping must be specified. The initial rotational offset between frame_a and frame_b is used as the undeformed angles of the angular springs. The undeformed angles are updated any time the Gripper is not active and is then made active.

Symbol

Default

Units

Description

Modelica ID

$\mathrm{T\_\_on/off}$

0.1

[s]

Time constant for activation/deactivation

T

$\mathrm{Linear} \mathrm{Reactions}$

All

With this enumeration, the directions in which the gripper applies reaction forces is controlled. There are seven choices:

TransDir

$\mathrm{K\_\_s}$

1e6

[N m]

Linear stiffness in all three directions.

cT

$\mathrm{K\_\_d}$

1e3

[N s/m]

Linear damping coefficient in all three directions.

dT

$\mathrm{Use} \mathrm{Torque} \mathrm{Reactions}$

When checked (true) torque reactions are also included.

useMoment

$\mathrm{K\_\_\θ}$

1e4

[N m/rad]

Angular stiffness for all three rotations ([1,2,3] Euler angles).

cR

$\mathrm{K\_\_\ω}$

1e2

[N m s/rad]

Angular damping coefficient in all three directions.

dR

$\mathrm{In} \mathrm{Place}$

When checked (true) the joint will use the displacement (translational and rotational) between frame_a and frame_b at the initial or activation time as the spring offsets.

useInPlace

$\mathrm{Angular} \mathrm{Of}\mathrm{fsets}$

Keep Current

Options for how to select angle offsets when using $\mathrm{In} \mathrm{Place}$.

setAngles