# polynomialTrajectory

## Description

The `polynomialTrajectory`

System object™ generates trajectories using a specified piecewise polynomial.

You can create a piecewise-polynomial structure using trajectory
generators like `minjerkpolytraj`

, and
`minsnappolytraj`

, as
well as any custom trajectory generator. You can then pass the structure to the
`polynomialTrajectory`

System object to create a trajectory interface for scenario simulation using the `uavScenario`

object.
You can use the `polynomialTrajectory`

object to specify the trajectory for
`uavPlatform`

motion.

To generate a trajectory from a piecewise polynomial:

Create the

`polynomialTrajectory`

object and set its properties.Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

## Creation

### Description

### Input Arguments

`pp`

— Piecewise polynomial

structure

Piecewise polynomial, specified as a structure that defines the polynomial for each section of the piecewise trajectory.

**Data Types: **`struct`

## Properties

Unless otherwise indicated, properties are *nontunable*, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
`release`

function unlocks them.

If a property is *tunable*, you can change its value at
any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

`SampleRate`

— Sample rate of trajectory (Hz)

`100`

(default) | positive scalar

Sample rate of the trajectory in Hz, specified as a positive scalar.

**Tunable: **Yes

**Data Types: **`double`

`SamplesPerFrame`

— Number of samples per output frame

`1`

(default) | positive integer

Number of samples per output frame, specified as a positive integer.

**Data Types: **`double`

`Orientation`

— Orientation at each waypoint

*N*-element `quaternion`

column vector | 3-by-3-by-*N* array of real numbers

Orientation at each waypoint, specified as an *N*-element `quaternion`

column vector
or 3-by-3-by-*N* array of real numbers. *N* is the
number of waypoints.

Each `quaternion`

must have a norm of `1`

. Each
3-by-3 rotation matrix must be an orthonormal matrix. Each quaternion or rotation matrix
is a frame rotation from the local navigation coordinate system to the current body
coordinate system at the corresponding waypoint.

If you do not specify this property, then the object sets yaw to the direction of travel at each waypoint, and pitch and roll are subject to the values of the AutoPitch and AutoBank properties, respectively.

**Data Types: **`double`

`Waypoints`

— Positions in navigation coordinate system (m)

*N*-by-3 matrix

This property is read-only.

Positions in the navigation coordinate system, in meters, specified as an
*N*-by-3 matrix. The columns of the matrix correspond to the first,
second, and third axes, respectively. The rows of the matrix, *N*,
correspond to individual waypoints.

The object infers this value from the piecewise polynomial
`pp`

.

**Data Types: **`double`

`TimeOfArrival`

— Timestamp at each waypoint (s)

*N*-element column vector of nonnegative increasing
numbers

This property is read-only.

Timestamp at each waypoint, in seconds, specified as an *N*-element
column vector. The number of samples, *N*, is the same as the number of
samples (rows) in Waypoints property.
The value of each element of the vector must be greater than the value of the previous
element.

The object infers this value from the piecewise polynomial
`pp`

.

**Data Types: **`double`

`AutoPitch`

— Align pitch angle with direction of motion

`false`

or `0`

(default) | `true`

or `1`

Align the pitch angle with the direction of motion, specified as a logical
`0`

(`false`

) or `1`

(`true`

). When specified as `true`

, the pitch angle
automatically aligns with the direction of motion. If specified as
`false`

, the object sets the pitch angle to `0`

(level orientation).

#### Dependencies

To set this property, you must not specify the Orientation property.

**Data Types: **`logical`

`AutoBank`

— Align roll angle to counteract centripetal force

`false`

or `0`

(default) | `true`

or `1`

Align the roll angle to counteract the centripetal force, specified as a logical
`0`

(`false`

) or `1`

(`true`

). When specified as `true`

, the roll angle
automatically counteracts the centripetal force. If specified as
`false`

, the object sets the roll angle to `0`

(flat
orientation).

#### Dependencies

To set this property, you must not specify the Orientation property.

**Data Types: **`logical`

`ReferenceFrame`

— Reference frame of trajectory

`"NED"`

(default) | `"ENU"`

Reference frame of the trajectory, specified as `"NED"`

(North-East-Down) or `"ENU"`

(East-North-Up).

**Data Types: **`char`

| `string`

`Velocities`

— Velocity in navigation coordinate system at each waypoint (m/s)

*N*-by-3 matrix

This property is read-only.

Velocity in the navigation coordinate system at each waypoint, in meters per second,
specified as an *N*-by-3 matrix. The columns of the matrix correspond
to the first, second, and third axes, respectively. The number of samples,
*N*, is the same as the number of samples (rows) in Waypoints
property.

The object infers this value from the derivative of the piecewise polynomial
`pp`

.

**Data Types: **`double`

`Course`

— Horizontal direction of travel (degrees)

*N*-element real vector

This property is read-only.

Horizontal direction of travel, in degrees, specified as an
*N*-element real vector. The number of samples, *N*,
is the same as the number of samples (rows) in Waypoints
property.

The object infers this value from the derivative of the piecewise polynomial
`pp`

.

**Data Types: **`double`

`GroundSpeed`

— Groundspeed at each waypoint (m/s)

*N*-element real vector

This property is read-only.

Groundspeed at each waypoint, in meters per second, specified as an
*N*-element real vector. The number of samples, *N*,
is the same as the number of samples (rows) in Waypoints
property.

The object infers this value from the derivative of the piecewise polynomial
`pp`

.

**Data Types: **`double`

`ClimbRate`

— Climb rate at each waypoint (m/s)

*N*-element real vector

This property is read-only.

Climb Rate at each waypoint, in meters per second, specified as an
*N*-element real vector. The number of samples, *N*,
is the same as the number of samples (rows) in Waypoints
property.

The object infers this value from the derivative of the piecewise polynomial
`pp`

.

**Data Types: **`double`

## Usage

### Description

`[`

outputs a frame of trajectory data based on specified creation arguments and properties.
The trajectory returns `position`

,`orientation`

,`velocity`

,`acceleration`

,`angularVelocity`

] = trajectory()`NaN`

for positions and orientations outside the
range of the time of arrival.

### Output Arguments

`position`

— Position in local navigation coordinate system

*M*-by-3 matrix

Position in the local navigation coordinate system, returned as an
*M*-by-3 matrix in meters.

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

`orientation`

— Orientation in local navigation coordinate system

*M*-element `quaternion`

column vector | 3-by-3-by-*M* real array

Orientation in the local navigation coordinate system, returned as an
*M*-element `quaternion`

column
vector or a 3-by-3-by-*M* real array.

Each quaternion or 3-by-3 rotation matrix is a frame rotation from the local navigation coordinate system to the current body coordinate system at the corresponding sample.

*M* is specified by the SamplesPerFrame
property.

**Note**

If the consecutive roots of the velocity polynomial are within
1e^{-3} seconds of each other, indicating a sudden change
in direction, then the calculated orientation between these roots will be adjusted
to maintain continuity.

**Data Types: **`double`

`velocity`

— Velocity in local navigation coordinate system

*M*-by-3 matrix

Velocity in the local navigation coordinate system, returned as an
*M*-by-3 matrix, in meters per second.

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

`acceleration`

— Acceleration in local navigation coordinate system

*M*-by-3 matrix

Acceleration in the local navigation coordinate system, returned as an
*M*-by-3 matrix, in meters per second squared.

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

`angularVelocity`

— Angular velocity in local navigation coordinate system

*M*-by-3 matrix

Angular velocity in the local navigation coordinate system, returned as an
*M*-by-3 matrix, in radians per second.

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

## Object Functions

To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named `obj`

, use
this syntax:

release(obj)

### Specific to `polynomialTrajectory`

`lookupPose` | Obtain pose information for certain time |

`waypointInfo` | Get waypoint information table |

### Common to All System Objects

## Examples

### Generate Trajectory from Piecewise Polynomial Using `polynomialTrajectory`

Use the `minjerkpolytraj`

function to generate the piecewise polynomial and the time samples for the specified waypoints of a trajectory.

waypoints = [0 20 20 0 0; 0 0 5 5 0; 0 5 10 5 0]; timePoints = cumsum([0 10 1.25*pi 10 1.25*pi]); numSamples = 100; [~,~,~,~,pp,~,tsamples] = minjerkpolytraj(waypoints,timePoints,numSamples);

Use the `polynomialTrajectory`

System object to generate a trajectory from the piecewise polynomial that a multirotor must follow. Specify the sample rate of the trajectory and the orientation at each waypoint.

eulerAngles = [0 0 0; 0 0 0; 180 0 0; 180 0 0; 0 0 0]; q = quaternion(eulerAngles,"eulerd","ZYX","frame"); traj = polynomialTrajectory(pp,SampleRate=100,Orientation=q);

Inspect the waypoints, times of arrival, and orientation by using `waypointInfo`

.

waypointInfo(traj)

`ans=`*5×3 table*
TimeOfArrival Waypoints Orientation
_____________ ________________________________________ ________________
0 0 0 0 {1x1 quaternion}
10 20 0 5 {1x1 quaternion}
13.927 20 5 10 {1x1 quaternion}
23.927 0 5 5 {1x1 quaternion}
27.854 6.9321e-14 -1.0658e-13 -8.9706e-14 {1x1 quaternion}

Obtain pose information one buffer frame at a time.

[pos,orient,vel,acc,angvel] = traj(); i = 1; spf = traj.SamplesPerFrame; while ~isDone(traj) idx = (i+1):(i+spf); [pos(idx,:),orient(idx,:), ... vel(idx,:),acc(idx,:),angvel(idx,:)] = traj(); i = i + spf; end

Get the yaw angle from the orientation.

eulOrientation = quat2eul(orient); yawAngle = eulOrientation(:,1);

Plot the generated positions and orientations, as well as the specified waypoints.

plot3(pos(:,1),pos(:,2),pos(:,3), ... waypoints(1,:),waypoints(2,:),waypoints(3,:),"--o") hold on % Plot the yaw using quiver. quiverIdx = 1:100:length(pos); quiver3(pos(quiverIdx,1),pos(quiverIdx,2),pos(quiverIdx,3), ... cos(yawAngle(quiverIdx)),sin(yawAngle(quiverIdx)), ... zeros(numel(quiverIdx),1)) title("Position") xlabel("X (m)") ylabel("Y (m)") zlabel("Z (m)") legend({"Position","Waypoints","Orientation"}) axis equal hold off

### Obtain Pose Information of Polynomial Trajectory at Certain Time

Use the `minsnappolytraj`

function to generate the piecewise polynomial and the time samples for the specified waypoints of a trajectory.

waypoints = [0 20 20 0 0; 0 0 5 5 0; 0 5 10 5 0]; timePoints = linspace(0,30,5); numSamples = 100; [~,~,~,~,~,pp,~,~] = minsnappolytraj(waypoints,timePoints,numSamples);

Use the `polynomialTrajectory`

System object to generate a trajectory from the piecewise polynomial. Specify the sample rate of the trajectory.

traj = polynomialTrajectory(pp,SampleRate=100);

Inspect the waypoints and times of arrival by using `waypointInfo`

.

waypointInfo(traj)

`ans=`*5×2 table*
TimeOfArrival Waypoints
_____________ ________________________________________
0 0 0 0
7.5 20 0 5
15 20 5 10
22.5 0 5 5
30 2.4897e-13 -2.7471e-12 -2.6352e-12

Obtain the time of arrival between the second and fourth waypoint. Create timestamps to sample the trajectory.

t0 = traj.TimeOfArrival(2); tf = traj.TimeOfArrival(4); sampleTimes = linspace(t0,tf,1000);

Obtain the position, orientation, velocity, and acceleration information at the sampled timestamps using the `lookupPose`

object function.

[pos,orient,vel,accel,~] = lookupPose(traj,sampleTimes);

Get the yaw angle from the orientation.

eulOrientation = quat2eul(orient); yawAngle = eulOrientation(:,1);

Plot the generated positions and orientations, as well as the specified waypoints.

plot3(pos(:,1),pos(:,2),pos(:,3), ... waypoints(1,:),waypoints(2,:),waypoints(3,:),"--o") hold on % Plot the yaw using quiver. quiverIdx = 1:100:length(pos); quiver3(pos(quiverIdx,1),pos(quiverIdx,2),pos(quiverIdx,3), ... cos(yawAngle(quiverIdx)),sin(yawAngle(quiverIdx)), ... zeros(numel(quiverIdx),1)) title("Position and Orientation") xlabel("X (m)") ylabel("Y (m)") zlabel("Z (m)") legend({"Position","Waypoints","Orientation"}) axis equal hold off

Plot the velocity profiles.

figure subplot(3,1,1) plot(sampleTimes,vel(:,1)) title("Velocity Profile") ylabel("v_x (m/s)") subplot(3,1,2) plot(sampleTimes,vel(:,2)) ylabel("v_y (m/s)") subplot(3,1,3) plot(sampleTimes,vel(:,3)) ylabel("v_z (m/s)") xlabel("Time (sec)")

Plot the acceleration profiles.

figure subplot(3,1,1) plot(sampleTimes,accel(:,1)) title("Acceleration Profile") ylabel("a_x (m/s^2)") subplot(3,1,2) plot(sampleTimes,accel(:,2)) ylabel("a_y (m/s^2)") subplot(3,1,3) plot(sampleTimes,accel(:,3)) ylabel("a_z (m/s^2)") xlabel("Time (sec)")

## Extended Capabilities

### C/C++ Code Generation

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

## Version History

**Introduced in R2023a**

## MATLAB 명령

다음 MATLAB 명령에 해당하는 링크를 클릭했습니다.

명령을 실행하려면 MATLAB 명령 창에 입력하십시오. 웹 브라우저는 MATLAB 명령을 지원하지 않습니다.

Select a Web Site

Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .

You can also select a web site from the following list:

## How to Get Best Site Performance

Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.

### Americas

- América Latina (Español)
- Canada (English)
- United States (English)

### Europe

- Belgium (English)
- Denmark (English)
- Deutschland (Deutsch)
- España (Español)
- Finland (English)
- France (Français)
- Ireland (English)
- Italia (Italiano)
- Luxembourg (English)

- Netherlands (English)
- Norway (English)
- Österreich (Deutsch)
- Portugal (English)
- Sweden (English)
- Switzerland
- United Kingdom (English)