# Transform Trajectory

Generate trajectory between two homogeneous transforms

**Library:**Robotics System Toolbox / Utilities

## Description

The Transform Trajectory block generates an interpolated trajectory
between two homogenous transformation matrices. The block outputs the transform at the times
given by the **Time** input, which can be a scalar or vector.

The trajectory is computed using quaternion spherical linear interpolation (SLERP) for the
rotation and linear interpolation for the translation. This method finds the shortest path
between positions and rotations of the transformation. Select the **Use custom time
scaling** check box to compute the trajectory using a custom time scaling. The
block uses linear time scaling by default.

The initial and final values are held constant outside the time period defined in
**Time interval**.

## Ports

### Input

`Time`

— Time point along trajectory

scalar | vector

Time point along trajectory, specified as a scalar or vector. In general, when specified as a scalar, this value is synced with simulation time and is used to specify the time point for sampling the trajectory. The block outputs a vector of the trajectory variables at that instant in time. If the time is specified as a vector, the block outputs a matrix with each column corresponding to each element of the vector.

**Data Types: **`single`

| `double`

`T0`

— Initial transformation matrix

4-by-4 homogeneous transformation

Initial transformation matrix, specified as a 4-by-4 homogeneous transformation.
The function generates a trajectory that starts at the initial orientation,
**T0**, and goes to the final orientation,
**TF**.

**Example: **`trvec2tform([1 10 -1])`

#### Dependencies

To enable this parameter, set the **Waypoint source** to
`External`

.

**Data Types: **`single`

| `double`

`TF`

— Final transformation matrix

4-by-4 homogeneous transformation

Final transformation matrix, specified as a 4-by-4 homogeneous transformation. The
function generates a trajectory that starts at the initial orientation,
**T0**, and goes to the final orientation,
**TF**.

**Example: **`trvec2tform([1 10 -1])`

#### Dependencies

To enable this parameter, set the **Waypoint source** to
`External`

.

**Data Types: **`single`

| `double`

`TimeInterval`

— Start and end times for trajectory

two-element vector

Start and end times for the trajectory, specified as a two-element vector.

**Example: **`[0 10]`

#### Dependencies

To enable this parameter, set the **Waypoint source** to
`External`

.

**Data Types: **`single`

| `double`

`TSTime`

— Time scaling time points

scalar | *p*-element vector

Time scaling time points, specified as a scalar or n *p*-element
vector, where *p* is the number of points for time scaling. By
default, the time scaling is a linear time scaling spanning the
**TimeInterval**. Specify the actual time scaling values in
**TimeScaling**.

If the **Time** input is specified at a time not specified by
these points, interpolation is used to find the right scaling time.

#### Dependencies

To enable this parameter, select the **Use custom time
scaling** check box and set **Parameter source** to
`External`

.

To specify a scalar, the **Time** input must be a
scalar.

**Data Types: **`single`

| `double`

`TimeScaling`

— Time scaling vector and first two derivatives

three-element vector | 3-by-*p* matrix

Time scaling vector and its first two derivatives, specified as a three element
vector or a 3-by-*p* matrix, where *m* is the length
of **TSTime**. By default, the time scaling is a linear time scaling
spanning the **TimeInterval**.

For a nonlinear time scaling, specify the values of the time points in the first row. The second and third rows are the velocity and acceleration of the time points, respectively. For example, to follow the path with a linear velocity to the halfway point, and then jump to the end, the time-scaling would be:

s(1,:) = [0 0.25 0.5 1 1 1] % Position s(2,:) = [1 1 1 0 0 0] % Velocity s(3,:) = [0 0 0 0 0 0] % Acceleration

#### Dependencies

To enable this parameter, select the **Use custom time
scaling** check box and set **Parameter source** to
`External`

.

To specify a three-element vector, the **Time** and
**TSTime** inputs must be a scalar.

**Data Types: **`single`

| `double`

### Output

`tform`

— Homogeneous transformation matrices

4-by-4-by-*m* homogenous matrix array

Homogeneous transformation matrices, returned as a 4-by-4-by-*m*
homogenous matrix array, where *m* is the number of points input to
**Time**.

`vel`

— Transform velocities

6-by-*m* matrix

Transform velocities, returned as a 6-by-*m* matrix, where
*m* is the number of points input to **Time**.
Each row of the vector is the angular and linear velocity of the transform as
`[wx wy wz vx vy vz]`

. *w* represents an angular
velocity and *v* represents a linear velocity.

`alpha`

— Transform accelerations

6-by-*m* matrix

Transform velocities, returned as a 6-by-*m* matrix, where
*m* is the number of points input to **Time**.
Each row of the vector is the angular and linear acceleration of the transform as
`[alphax alphay alphaz ax ay az]`

. *alpha*
represents an angular acceleration and *a* represents a linear
acceleration.

## Parameters

`Waypoint source`

— Source for waypoints

`Internal`

(default) | `External`

Specify `External`

to specify the **Waypoints**
and **Time points** parameters as block inputs instead of block
parameters.

`Initial transform`

— Initial transformation matrix

`trvec2tform([1 10 -1])`

(default) | 4-by-4 homogeneous transformation

Initial transformation matrix, specified as a 4-by-4 homogeneous transformation. The
function generates a trajectory that starts at the **Initial
transform** and goes to the **Final transform**.

**Data Types: **`single`

| `double`

`Final transform`

— Final transformation matrix

`eul2tform([0 pi pi/2])`

(default) | 4-by-4 homogeneous transformation

Final transformation matrix, specified as a 4-by-4 homogeneous transformation. The
function generates a trajectory that starts at the **Initial
transform** and goes to the **Final transform**.

**Data Types: **`single`

| `double`

`Time interval`

— Start and end times for trajectory

`[2 3]`

| two-element vector

Start and end times for the trajectory, specified as a two-element vector in seconds.

**Data Types: **`single`

| `double`

`Use custom time scaling`

— Enable custom time scaling

`off`

(default) | `on`

Enable to specify custom time scaling for the trajectory using the
**Parameter Source**, **Time scaling time**, and
**Time scaling values** parameters.

`Parameter source`

— Source for waypoints

`Internal`

(default) | `External`

Specify `External`

to specify the **Time scaling
time** and **Time scaling values** parameters as block
inputs instead of block parameters.

#### Dependencies

To enable this parameter, select the **Use custom time scaling**
check box.

`Time scaling time`

— Time scaling time points

`2:0.1:3`

(default) | scalar | *p*-element vector

Time scaling time points, specified as a scalar or *p*-element
vector, where *p* is the number of points for time scaling. By default,
the time scaling is a linear time scaling spanning the **Time
interval**. Specify the actual time scaling values in **Time scaling
values**.

If the **Time** input is specified at a time not specified by these
points, interpolation is used to find the right scaling time.

#### Dependencies

To enable this parameter, select the **Use custom time scaling**
check box.

To specify a scalar, the **Time** input must be a scalar.

**Data Types: **`single`

| `double`

`Time scaling values`

— Time scaling vector and first two derivatives

`[0:0.1:1; ones(1,11); zeros(1,11)]`

(default) | three-element vector | 3-by-*m* matrix

Time scaling vector and its first two derivatives, specified as a three-element
vector or 3-by-*p* matrix, where *p* is the length of
**Time scaling time**. By default, the time scaling is a linear time
scaling spanning the **Time interval**.

For a nonlinear time scaling, specify the values of the time points in the first row. The second and third rows are the velocity and acceleration of the time points, respectively. For example, to follow the path with a linear velocity to the halfway point, and then jump to the end, the time-scaling would be:

s(1,:) = [0 0.25 0.5 1 1 1] % Position s(2,:) = [1 1 1 0 0 0] % Velocity s(3,:) = [0 0 0 0 0 0] % Acceleration

#### Dependencies

To enable this parameter, select the **Use custom time scaling**
checkbox.

To specify a three-element vector, the **Time** and
**TSTime** inputs must be a scalar.

**Data Types: **`single`

| `double`

`Simulate using`

— Type of simulation to run

`Interpreted execution`

(default) | `Code generation`

`Interpreted execution`

— Simulate model using the MATLAB^{®}interpreter. This option shortens startup time but has a slower simulation speed than`Code generation`

. In this mode, you can debug the source code of the block.`Code generation`

— Simulate model using generated C code. The first time you run a simulation, Simulink^{®}generates C code for the block. The C code is reused for subsequent simulations, as long as the model does not change. This option requires additional startup time, but the speed of the subsequent simulations is comparable to`Interpreted execution`

.

**Tunable: **No

## Model Examples

## References

[1] Lynch, Kevin M., and Frank C.
Park. *Modern Robotics: Mechanics, Planning, and Control*. Cambridge
University Press, 2017.

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

## See Also

### Blocks

### Functions

**Introduced in R2019a**

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