Waypoint trajectory generator

The `waypointTrajectory`

System object™ generates trajectories using specified waypoints. When you create the System object, you can optionally specify the time of arrival, velocity, and orientation at
each waypoint.

To generate a trajectory from waypoints:

Create the

`waypointTrajectory`

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

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

returns a
System object, `trajectory`

= waypointTrajectory`trajectory`

, that generates a trajectory based on default
stationary waypoints.

specifies the `trajectory`

= waypointTrajectory(`Waypoints`

,`TimeOfArrival`

)`Waypoints`

that the generated trajectory passes through
and the `TimeOfArrival`

at each waypoint.

sets each creation argument or property `trajectory`

= waypointTrajectory(`Waypoints`

,`TimeOfArrival`

,`Name,Value`

)`Name`

to the specified
`Value`

. Unspecified properties and creation arguments have default or
inferred values.

```
trajectory =
waypointTrajectory([10,10,0;20,20,0;20,20,10],[0,0.5,10])
```

creates a waypoint
trajectory System object, `trajectory`

, that starts at waypoint
`[10,10,0]`

, and then passes through `[20,20,0]`

after
0.5 seconds and `[20,20,10]`

after 10 seconds.Creation arguments are properties which are set during creation of the System object and cannot be modified later. If you do not explicitly set a creation argument value, the property value is inferred.

If you specify any creation argument, then you must specify both the Waypoints and TimeOfArrival creation
arguments. You can specify `Waypoints`

and
`TimeOfArrival`

as value-only arguments or name-value pairs.

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

**Tunable: **Yes

**Data Types: **`double`

`SamplesPerFrame`

— Number of samples per output frame`1`

(default) | positive scalar integerNumber of samples per output frame, specified as a positive scalar integer.

**Tunable: **Yes

**Data Types: **`double`

`Waypoints`

— Positions in the navigation coordinate system (m)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.

To set this property, you must also set valid values for the TimeOfArrival property.

**Data Types: **`double`

`TimeOfArrival`

— Time at each waypoint (s)Time corresponding to arrival at each waypoint in seconds, specified as an
*N*-element column vector. The first element of
`TimeOfArrival`

must be `0`

. The number of
samples, *N*, must be the same as the number of samples (rows) defined
by `Waypoints`

.

To set this property, you must also set valid values for the Waypoints property.

**Data Types: **`double`

`Velocities`

— Velocity in navigation coordinate system at each waypoint (m/s)Velocity in the navigation coordinate system at each way point 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*, must be the same as the number of samples (rows) defined by
`Waypoints`

.

If the velocity is specified as a non-zero value, the object automatically calculates the course of the trajectory. If the velocity is specified as zero, the object infers the course of the trajectory from adjacent waypoints.

To set this property, you must also set valid values for the Waypoints and TimeOfArrival properties.

**Data Types: **`double`

`Course`

— Horizontal direction of travel (degree)Horizontal direction of travel, specified as an *N*-element real
vector in degrees. The number of samples, *N*, must be the same as the
number of samples (rows) defined by `Waypoints`

. If neither
`Velocities`

nor `Course`

is specified, course
is inferred from the waypoints.

To set this property, the `Velocities`

property must not be
specified in object creation.

**Data Types: **`double`

`GroundSpeed`

— Groundspeed at each waypoint (m/s)Groundspeed at each waypoint, specified as an *N*-element real
vector in m/s. If the property is not specified, it is inferred from the waypoints. The
number of samples, *N*, must be the same as the number of samples
(rows) defined by `Waypoints`

.

To set this property, the `Velocities`

property must not be
specified at object creation.

**Data Types: **`double`

`Climbrate`

— Climbrate at each waypoint (m/s)Climbrate at each waypoint, specified as an *N*-element real vector
in degrees. The number of samples, *N*, must be the same as the number
of samples (rows) defined by `Waypoints`

. If neither
`Velocities`

nor `Course`

is specified,
climbrate is inferred from the waypoints.

To set this property, the `Velocities`

property must not be
specified at object creation.

**Data Types: **`double`

`Orientation`

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

column vector
or 3-by-3-by-*N* array of real numbers. The number of quaternions or
rotation matrices, *N*, must be the same as the number of samples
(rows) defined by `Waypoints`

.

If `Orientation`

is specified by quaternions, the underlying
class must be `double`

.

To set this property, you must also set valid values for the Waypoints and TimeOfArrival properties.

**Data Types: **`quaternion`

| `double`

`AutoPitch`

— Align pitch angle with direction of motion`false`

(default) | `true`

Align pitch angle with the direction of motion, specified as `true`

or `false`

. When specified as `true`

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

, the pitch angle is set to zero (level orientation).

To set this property, the `Orientation`

property must not be
specified at object creation.

`AutoBank`

— Align roll angle to counteract centripetal force`false`

(default) | `true`

Align roll angle to counteract the centripetal force, specified as
`true`

or `false`

. When specified as
`true`

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

, the roll angle is set to zero (flat
orientation).

To set this property, the `Orientation`

property must not be
specified at object creation.

`ReferenceFrame`

— Reference frame of trajectory`'NED'`

(default) | `'ENU'`

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

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

(East-North-Up).

`[`

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

,`orientation`

,`velocity`

,`acceleration`

,`angularVelocity`

] = trajectory()

`position`

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

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

`orientation`

— Orientation in local navigation coordinate systemOrientation in the local navigation coordinate system, returned as an
*M*-by-1 `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.

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

`velocity`

— Velocity in local navigation coordinate system (m/s)Velocity in the local navigation coordinate system in meters per second, returned
as an *M*-by-3 matrix.

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

`acceleration`

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

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

`angularVelocity`

— Angular velocity in local navigation coordinate system (rad/s)Angular velocity in the local navigation coordinate system in radians per second,
returned as an *M*-by-3 matrix.

*M* is specified by the SamplesPerFrame
property.

**Data Types: **`double`

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)

`waypointTrajectory`

`waypointInfo` | Get waypoint information table |

`lookupPose` | Obtain pose information for certain time |

`perturbations` | Perturbation defined on object |

`perturb` | Apply perturbations to object |

`waypointTrajectory`

trajectory = waypointTrajectory

trajectory = waypointTrajectory with properties: SampleRate: 100 SamplesPerFrame: 1 Waypoints: [2x3 double] TimeOfArrival: [2x1 double] Velocities: [2x3 double] Course: [2x1 double] GroundSpeed: [2x1 double] ClimbRate: [2x1 double] Orientation: [2x1 quaternion] AutoPitch: 0 AutoBank: 0 ReferenceFrame: 'NED'

Inspect the default waypoints and times of arrival by calling `waypointInfo`

. By default, the waypoints indicate a stationary position for one second.

waypointInfo(trajectory)

`ans=`*2×2 table*
TimeOfArrival Waypoints
_____________ ___________
0 0 0 0
1 0 0 0

Create a square trajectory and examine the relationship between waypoint constraints, sample rate, and the generated trajectory.

Create a square trajectory by defining the vertices of the square. Define the orientation at each waypoint as pointing in the direction of motion. Specify a 1 Hz sample rate and use the default `SamplesPerFrame`

of 1.

waypoints = [0,0,0; ... % Initial position 0,1,0; ... 1,1,0; ... 1,0,0; ... 0,0,0]; % Final position toa = 0:4; % time of arrival orientation = quaternion([0,0,0; ... 45,0,0; ... 135,0,0; ... 225,0,0; ... 0,0,0], ... 'eulerd','ZYX','frame'); trajectory = waypointTrajectory(waypoints, ... 'TimeOfArrival',toa, ... 'Orientation',orientation, ... 'SampleRate',1);

Create a figure and plot the initial position of the platform.

figure(1) plot(waypoints(1,1),waypoints(1,2),'b*') title('Position') axis([-1,2,-1,2]) axis square xlabel('X') ylabel('Y') grid on hold on

In a loop, step through the trajectory to output the current position and current orientation. Plot the current position and log the orientation. Use `pause`

to mimic real-time processing.

orientationLog = zeros(toa(end)*trajectory.SampleRate,1,'quaternion'); count = 1; while ~isDone(trajectory) [currentPosition,orientationLog(count)] = trajectory(); plot(currentPosition(1),currentPosition(2),'bo') pause(trajectory.SamplesPerFrame/trajectory.SampleRate) count = count + 1; end hold off

Convert the orientation quaternions to Euler angles for easy interpretation, and then plot orientation over time.

figure(2) eulerAngles = eulerd([orientation(1);orientationLog],'ZYX','frame'); plot(toa,eulerAngles(:,1),'ko', ... toa,eulerAngles(:,2),'bd', ... toa,eulerAngles(:,3),'r.'); title('Orientation Over Time') legend('Rotation around Z-axis','Rotation around Y-axis','Rotation around X-axis') xlabel('Time (seconds)') ylabel('Rotation (degrees)') grid on

So far, the trajectory object has only output the waypoints that were specified during construction. To interpolate between waypoints, increase the sample rate to a rate faster than the time of arrivals of the waypoints. Set the `trajectory`

sample rate to 100 Hz and call `reset`

.

trajectory.SampleRate = 100; reset(trajectory)

Create a figure and plot the initial position of the platform. In a loop, step through the trajectory to output the current position and current orientation. Plot the current position and log the orientation. Use `pause`

to mimic real-time processing.

figure(1) plot(waypoints(1,1),waypoints(1,2),'b*') title('Position') axis([-1,2,-1,2]) axis square xlabel('X') ylabel('Y') grid on hold on orientationLog = zeros(toa(end)*trajectory.SampleRate,1,'quaternion'); count = 1; while ~isDone(trajectory) [currentPosition,orientationLog(count)] = trajectory(); plot(currentPosition(1),currentPosition(2),'bo') pause(trajectory.SamplesPerFrame/trajectory.SampleRate) count = count + 1; end hold off

The trajectory output now appears circular. This is because the `waypointTrajectory`

System object™ minimizes the acceleration and angular velocity when interpolating, which results in smoother, more realistic motions in most scenarios.

Convert the orientation quaternions to Euler angles for easy interpretation, and then plot orientation over time. The orientation is also interpolated.

figure(2) eulerAngles = eulerd([orientation(1);orientationLog],'ZYX','frame'); t = 0:1/trajectory.SampleRate:4; plot(t,eulerAngles(:,1),'ko', ... t,eulerAngles(:,2),'bd', ... t,eulerAngles(:,3),'r.'); title('Orientation Over Time') legend('Rotation around Z-axis','Rotation around Y-axis','Rotation around X-axis') xlabel('Time (seconds)') ylabel('Rotation (degrees)') grid on

The `waypointTrajectory`

algorithm interpolates the waypoints to create a smooth trajectory. To return to the square trajectory, provide more waypoints, especially around sharp changes. To track corresponding times, waypoints, and orientation, specify all the trajectory info in a single matrix.

% Time, Waypoint, Orientation trajectoryInfo = [0, 0,0,0, 0,0,0; ... % Initial position 0.1, 0,0.1,0, 0,0,0; ... 0.9, 0,0.9,0, 0,0,0; ... 1, 0,1,0, 45,0,0; ... 1.1, 0.1,1,0, 90,0,0; ... 1.9, 0.9,1,0, 90,0,0; ... 2, 1,1,0, 135,0,0; ... 2.1, 1,0.9,0, 180,0,0; ... 2.9, 1,0.1,0, 180,0,0; ... 3, 1,0,0, 225,0,0; ... 3.1, 0.9,0,0, 270,0,0; ... 3.9, 0.1,0,0, 270,0,0; ... 4, 0,0,0, 270,0,0]; % Final position trajectory = waypointTrajectory(trajectoryInfo(:,2:4), ... 'TimeOfArrival',trajectoryInfo(:,1), ... 'Orientation',quaternion(trajectoryInfo(:,5:end),'eulerd','ZYX','frame'), ... 'SampleRate',100);

Create a figure and plot the initial position of the platform. In a loop, step through the trajectory to output the current position and current orientation. Plot the current position and log the orientation. Use `pause`

to mimic real-time processing.

figure(1) plot(waypoints(1,1),waypoints(1,2),'b*') title('Position') axis([-1,2,-1,2]) axis square xlabel('X') ylabel('Y') grid on hold on orientationLog = zeros(toa(end)*trajectory.SampleRate,1,'quaternion'); count = 1; while ~isDone(trajectory) [currentPosition,orientationLog(count)] = trajectory(); plot(currentPosition(1),currentPosition(2),'bo') pause(trajectory.SamplesPerFrame/trajectory.SampleRate) count = count+1; end hold off

The trajectory output now appears more square-like, especially around the vertices with waypoints.

Convert the orientation quaternions to Euler angles for easy interpretation, and then plot orientation over time.

figure(2) eulerAngles = eulerd([orientation(1);orientationLog],'ZYX','frame'); t = 0:1/trajectory.SampleRate:4; eulerAngles = plot(t,eulerAngles(:,1),'ko', ... t,eulerAngles(:,2),'bd', ... t,eulerAngles(:,3),'r.'); title('Orientation Over Time') legend('Rotation around Z-axis', ... 'Rotation around Y-axis', ... 'Rotation around X-axis', ... 'Location', 'SouthWest') xlabel('Time (seconds)') ylabel('Rotation (degrees)') grid on

This example shows how to create an arc trajectory using the `waypointTrajectory`

System object™. `waypointTrajectory`

creates a path through specified waypoints that minimizes acceleration and angular velocity. After creating an arc trajectory, you restrict the trajectory to be within preset bounds.

**Create an Arc Trajectory**

Define a constraints matrix consisting of waypoints, times of arrival, and orientation for an arc trajectory. The generated trajectory passes through the waypoints at the specified times with the specified orientation. The `waypointTrajectory`

System object requires orientation to be specified using quaternions or rotation matrices. Convert the Euler angles saved in the constrains matrix to quaternions when specifying the `Orientation`

property.

% Arrival, Waypoints, Orientation constraints = [0, 20,20,0, 90,0,0; 3, 50,20,0, 90,0,0; 4, 58,15.5,0, 162,0,0; 5.5, 59.5,0,0 180,0,0]; trajectory = waypointTrajectory(constraints(:,2:4), ... 'TimeOfArrival',constraints(:,1), ... 'Orientation',quaternion(constraints(:,5:7),'eulerd','ZYX','frame'));

Call `waypointInfo`

on `trajectory`

to return a table of your specified constraints. The creation properties `Waypoints`

, `TimeOfArrival`

, and `Orientation`

are variables of the table. The table is convenient for indexing while plotting.

tInfo = waypointInfo(trajectory)

tInfo = 4x3 table TimeOfArrival Waypoints Orientation _____________ ____________________ ________________ 0 20 20 0 {1x1 quaternion} 3 50 20 0 {1x1 quaternion} 4 58 15.5 0 {1x1 quaternion} 5.5 59.5 0 0 {1x1 quaternion}

The trajectory object outputs the current position, velocity, acceleration, and angular velocity at each call. Call `trajectory`

in a loop and plot the position over time. Cache the other outputs.

figure(1) plot(tInfo.Waypoints(1,1),tInfo.Waypoints(1,2),'b*') title('Position') axis([20,65,0,25]) xlabel('North') ylabel('East') grid on daspect([1 1 1]) hold on orient = zeros(tInfo.TimeOfArrival(end)*trajectory.SampleRate,1,'quaternion'); vel = zeros(tInfo.TimeOfArrival(end)*trajectory.SampleRate,3); acc = vel; angVel = vel; count = 1; while ~isDone(trajectory) [pos,orient(count),vel(count,:),acc(count,:),angVel(count,:)] = trajectory(); plot(pos(1),pos(2),'bo') pause(trajectory.SamplesPerFrame/trajectory.SampleRate) count = count + 1; end

Inspect the orientation, velocity, acceleration, and angular velocity over time. The `waypointTrajectory`

System object™ creates a path through the specified constraints that minimized acceleration and angular velocity.

figure(2) timeVector = 0:(1/trajectory.SampleRate):tInfo.TimeOfArrival(end); eulerAngles = eulerd([tInfo.Orientation{1};orient],'ZYX','frame'); plot(timeVector,eulerAngles(:,1), ... timeVector,eulerAngles(:,2), ... timeVector,eulerAngles(:,3)); title('Orientation Over Time') legend('Rotation around Z-axis', ... 'Rotation around Y-axis', ... 'Rotation around X-axis', ... 'Location','southwest') xlabel('Time (seconds)') ylabel('Rotation (degrees)') grid on figure(3) plot(timeVector(2:end),vel(:,1), ... timeVector(2:end),vel(:,2), ... timeVector(2:end),vel(:,3)); title('Velocity Over Time') legend('North','East','Down') xlabel('Time (seconds)') ylabel('Velocity (m/s)') grid on figure(4) plot(timeVector(2:end),acc(:,1), ... timeVector(2:end),acc(:,2), ... timeVector(2:end),acc(:,3)); title('Acceleration Over Time') legend('North','East','Down','Location','southwest') xlabel('Time (seconds)') ylabel('Acceleration (m/s^2)') grid on figure(5) plot(timeVector(2:end),angVel(:,1), ... timeVector(2:end),angVel(:,2), ... timeVector(2:end),angVel(:,3)); title('Angular Velocity Over Time') legend('North','East','Down') xlabel('Time (seconds)') ylabel('Angular Velocity (rad/s)') grid on

**Restrict Arc Trajectory Within Preset Bounds**

You can specify additional waypoints to create trajectories within given bounds. Create upper and lower bounds for the arc trajectory.

figure(1) xUpperBound = [(20:50)';50+10*sin(0:0.1:pi/2)';60*ones(11,1)]; yUpperBound = [20.5.*ones(31,1);10.5+10*cos(0:0.1:pi/2)';(10:-1:0)']; xLowerBound = [(20:49)';50+9*sin(0:0.1:pi/2)';59*ones(11,1)]; yLowerBound = [19.5.*ones(30,1);10.5+9*cos(0:0.1:pi/2)';(10:-1:0)']; plot(xUpperBound,yUpperBound,'r','LineWidth',2); plot(xLowerBound,yLowerBound,'r','LineWidth',2)

To create a trajectory within the bounds, add additional waypoints. Create a new `waypointTrajectory`

System object™, and then call it in a loop to plot the generated trajectory. Cache the orientation, velocity, acceleration, and angular velocity output from the `trajectory`

object.

% Time, Waypoint, Orientation constraints = [0, 20,20,0, 90,0,0; 1.5, 35,20,0, 90,0,0; 2.5 45,20,0, 90,0,0; 3, 50,20,0, 90,0,0; 3.3, 53,19.5,0, 108,0,0; 3.6, 55.5,18.25,0, 126,0,0; 3.9, 57.5,16,0, 144,0,0; 4.2, 59,14,0, 162,0,0; 4.5, 59.5,10,0 180,0,0; 5, 59.5,5,0 180,0,0; 5.5, 59.5,0,0 180,0,0]; trajectory = waypointTrajectory(constraints(:,2:4), ... 'TimeOfArrival',constraints(:,1), ... 'Orientation',quaternion(constraints(:,5:7),'eulerd','ZYX','frame')); tInfo = waypointInfo(trajectory); figure(1) plot(tInfo.Waypoints(1,1),tInfo.Waypoints(1,2),'b*') count = 1; while ~isDone(trajectory) [pos,orient(count),vel(count,:),acc(count,:),angVel(count,:)] = trajectory(); plot(pos(1),pos(2),'gd') pause(trajectory.SamplesPerFrame/trajectory.SampleRate) count = count + 1; end

The generated trajectory now fits within the specified boundaries. Visualize the orientation, velocity, acceleration, and angular velocity of the generated trajectory.

figure(2) timeVector = 0:(1/trajectory.SampleRate):tInfo.TimeOfArrival(end); eulerAngles = eulerd(orient,'ZYX','frame'); plot(timeVector(2:end),eulerAngles(:,1), ... timeVector(2:end),eulerAngles(:,2), ... timeVector(2:end),eulerAngles(:,3)); title('Orientation Over Time') legend('Rotation around Z-axis', ... 'Rotation around Y-axis', ... 'Rotation around X-axis', ... 'Location','southwest') xlabel('Time (seconds)') ylabel('Rotation (degrees)') grid on figure(3) plot(timeVector(2:end),vel(:,1), ... timeVector(2:end),vel(:,2), ... timeVector(2:end),vel(:,3)); title('Velocity Over Time') legend('North','East','Down') xlabel('Time (seconds)') ylabel('Velocity (m/s)') grid on figure(4) plot(timeVector(2:end),acc(:,1), ... timeVector(2:end),acc(:,2), ... timeVector(2:end),acc(:,3)); title('Acceleration Over Time') legend('North','East','Down') xlabel('Time (seconds)') ylabel('Acceleration (m/s^2)') grid on figure(5) plot(timeVector(2:end),angVel(:,1), ... timeVector(2:end),angVel(:,2), ... timeVector(2:end),angVel(:,3)); title('Angular Velocity Over Time') legend('North','East','Down') xlabel('Time (seconds)') ylabel('Angular Velocity (rad/s)') grid on

Note that while the generated trajectory now fits within the spatial boundaries, the acceleration and angular velocity of the trajectory are somewhat erratic. This is due to over-specifying waypoints.

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

The object function, `waypointInfo`

, does not
support code generation.

Usage notes and limitations:

See System Objects in MATLAB Code Generation (MATLAB Coder).

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