# derivative

Time derivative of UAV states

## Description

determines the time derivative of the state of the UAV guidance model using the current
state, control commands, and environmental inputs. Use the state and time derivative with
`stateDerivative`

= derivative(`uavGuidanceModel`

,`state`

,`control`

,`environment`

)`ode45`

to simulate the UAV.

## Examples

### Simulate A Multirotor Control Command

This example shows how to use the `multirotor`

guidance model to simulate the change in state of a UAV due to a command input.

Create the multirotor guidance model.

model = multirotor;

Create a state structure. Specify the location in world coordinates.

s = state(model); s(1:3) = [3;2;1];

Specify a control command, `u`

, that specified the roll and thrust of the multirotor.

u = control(model); u.Roll = pi/12; u.Thrust = 1;

Create a default environment without wind.

e = environment(model);

Compute the time derivative of the state given the current state, control command, and environment.

sdot = derivative(model,s,u,e);

Simulate the UAV state using `ode45`

integration. The `y`

field outputs the multirotor UAV states as a 13-by-*n *matrix.

simOut = ode45(@(~,x)derivative(model,x,u,e), [0 3], s); size(simOut.y)

`ans = `*1×2*
13 3536

Plot the change in roll angle based on the simulation output. The roll angle (the X Euler angle) is the 9th row of the `simOut.y`

output.

plot(simOut.y(9,:))

Plot the change in the Y and Z positions. With the specified thrust and roll angle, the multirotor should fly over and lose some altitude. A positive value for Z is expected as positive Z is down.

figure plot(simOut.y(2,:)); hold on plot(simOut.y(3,:)); legend('Y-position','Z-position') hold off

You can also plot the multirotor trajectory using `plotTransforms`

. Create the translation and rotation vectors from the simulated state. Downsample (every 300th element) and transpose the `simOut`

elements, and convert the Euler angles to quaternions. Specify the mesh as the `multirotor.stl`

file and the positive Z-direction as `"down"`

. The displayed view shows the UAV translating in the Y-direction and losing altitude.

translations = simOut.y(1:3,1:300:end)'; % xyz position rotations = eul2quat(simOut.y(7:9,1:300:end)'); % ZYX Euler plotTransforms(translations,rotations,... 'MeshFilePath','multirotor.stl','InertialZDirection',"down") view([90.00 -0.60])

### Simulate A Fixed-Wing Control Command

This example shows how to use the `fixedwing`

guidance model to simulate the change in state of a UAV due to a command input.

Create the fixed-wing guidance model.

model = fixedwing;

Set the air speed of the vehicle by modifying the structure from the `state`

function.

```
s = state(model);
s(4) = 5; % 5 m/s
```

Specify a control command, `u`

, that maintains the air speed and gives a roll angle of `pi/12`

.

u = control(model); u.RollAngle = pi/12; u.AirSpeed = 5;

Create a default environment without wind.

e = environment(model);

Compute the time derivative of the state given the current state, control command, and environment.

sdot = derivative(model,s,u,e);

Simulate the UAV state using `ode45`

integration. The `y`

field outputs the fixed-wing UAV states based on this simulation.

simOut = ode45(@(~,x)derivative(model,x,u,e), [0 50], s); size(simOut.y)

`ans = `*1×2*
8 904

Plot the change in roll angle based on the simulation output. The roll angle is the 7th row of the `simOut.y`

output.

plot(simOut.y(7,:))

You can also plot the fixed-wing trajectory using `plotTransforms`

. Create the translation and rotation vectors from the simulated state. Downsample (every 30th element) and transpose the `simOut`

elements, and convert the Euler angles to quaternions. Specify the mesh as the `fixedwing.stl`

file and the positive Z-direction as `"down"`

. The displayed view shows the UAV making a constant turn based on the constant roll angle.

downsample = 1:30:size(simOut.y,2); translations = simOut.y(1:3,downsample)'; % xyz-position rotations = eul2quat([simOut.y(5,downsample)',simOut.y(6,downsample)',simOut.y(7,downsample)']); % ZYX Euler plotTransforms(translations,rotations,... 'MeshFilePath','fixedwing.stl','InertialZDirection',"down") hold on plot3(simOut.y(1,:),-simOut.y(2,:),simOut.y(3,:),'--b') % full path xlim([-10.0 10.0]) ylim([-20.0 5.0]) zlim([-0.5 4.00]) view([-45 90]) hold off

## Input Arguments

`uavGuidanceModel`

— UAV guidance model

`fixedwing`

object | `multirotor`

object

UAV guidance model, specified as a `fixedwing`

or `multirotor`

object.

`state`

— State vector

eight-element vector | thirteen-element vector

State vector, specified as a eight-element or thirteen-element vector. The vector is always filled with zeros. Use this function to ensure you have the proper size for your state vector.

For fixed-wing UAVs, the state is an eight-element vector:

**North**- Position in north direction in meters.**East**- Position in east direction in meters.**Height**- Height above ground in meters.**AirSpeed**- Speed relative to wind in meters per second.**HeadingAngle**- Angle between ground velocity and north direction in radians.**FlightPathAngle**- Angle between ground velocity and north-east plane in radians.**RollAngle**- Angle of rotation along body*x*-axis in radians per second.**RollAngleRate**- Angular velocity of rotation along body*x*-axis in radians per second.

For multirotor UAVs, the state is a thirteen-element vector in this order:

**World Position**-`[x y z]`

in meters.**World Velocity**-`[vx vy vz]`

in meters per second.**Euler Angles (ZYX)**-`[psi theta phi]`

in radians.**Body Angular Rates**-`[p q r]`

in radians per second.**Thrust**-`F`

in Newtons.

`environment`

— Environmental input parameters

structure

Environmental input parameters, returned as a structure. To generate this structure,
use `environment`

.

For fixed-wing UAVs, the fields of the structure are `WindNorth`

,
`WindEast`

,`WindDown`

, and
`Gravity`

. Wind speeds are in meters per second, and negative speeds
point in the opposite direction. Gravity is in meters per second squared (default
`9.81`

).

For multirotor UAVs, the only element of the structure is `Gravity`

(default `9.81`

) in meters per second squared.

`control`

— Control commands for UAV

structure

Control commands for UAV, specified as a structure. To generate this structure, use
`control`

.

For multirotor UAVs, the guidance model is approximated as separate PD controllers for each command. The elements of the structure are control commands:

`Roll`

- Roll angle in radians.`Pitch`

- Pitch angle in radians.`YawRate`

- Yaw rate in radians per second. (D = 0. P only controller)`Thrust`

- Vertical thrust of the UAV in Newtons. (D = 0. P only controller)

For fixed-wing UAVs, the model assumes the UAV is flying under the coordinated-turn condition. The Guidance Model equations assume zero side-slip. The elements of the bus are:

`Height`

- Altitude above the ground in meters.`Airspeed`

- UAV speed relative to wind in meters per second.`RollAngle`

- Roll angle along body forward axis in radians. Because of the coordinated-turn condition, the heading angular rate is based on the roll angle.

## Output Arguments

`stateDerivative`

— Time derivative of state

vector

Time derivative of state, returned as a vector. The time derivative vector has the
same length as the input `state`

.

## Extended Capabilities

### C/C++ Code Generation

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

## Version History

**Introduced in R2018b**

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