# stateTransitionJacobian

## Description

returns the Jacobian matrix for the state transition function of the
`jac`

= stateTransitionJacobian(`model`

,`filter`

,`dt`

,`varargin`

)`model`

object inherited from the `positioning.INSMotionModel`

abstract class.

**Note**

Implementing this method is optional for a subclass of the `positioning.INSMotionModel`

abstract class. If you do not implement this
method, the subclass uses a Jacobian matrix calculated by numerical
differentiation.

## Examples

### Customize Motion Model Used with `insEKF`

Customize a 1-D constant velocity motion model used with an `insEKF`

object. Customize the motion model by inheriting from the `positioning.INSMotionModel`

interface class and implement the `modelstates`

and `stateTranistion`

methods. You can also optionally implement the `stateTransitionJacobian`

method. These sections provide an overview of how the `ConstantVelocityMotion`

class implements the `positioning.INSMotionModel`

methods, but for more details on their implementation, see the attached `ConstantVelocityMotion.m`

file.

**Implement modelstates method**

To model 1-D constant velocity motion, you need to return only the 1-D position and velocity state as a structure. When you add a `ConstantVelocityMotion`

object to an `insEKF`

filter object, the filter adds the `Position`

and `Velocity`

components to the state vector of the filter.

**Implement stateTransition method**

The `stateTransition`

method returns the derivatives of the state defined by the motion model as a structure. The derivative of the `Position`

is the `Velocity`

, and the derivative of the `Velocity`

is `0`

.

**Implement stateTransitionJacobian method**

The `stateTransitionJacobian`

method returns the partial derivatives of `stateTransition`

method, with respect to the state vector of the filter, as a structure. All the partial derivatives are 0, except the partial derivative of the derivative of the `Position`

component, which is the `Velocity`

, with respect to the `Velocity`

state, is `1`

.

**Create and add inherited object**

Create a `ConstantVelocityMotion`

object.

cvModel = ConstantVelocityMotion

cvModel = ConstantVelocityMotion with no properties.

Create an `insEKF`

object with the created `cvModel`

object.

filter = insEKF(insAccelerometer,cvModel)

filter = insEKF with properties: State: [5x1 double] StateCovariance: [5x5 double] AdditiveProcessNoise: [5x5 double] MotionModel: [1x1 ConstantVelocityMotion] Sensors: {[1x1 insAccelerometer]} SensorNames: {'Accelerometer'} ReferenceFrame: 'NED'

The filter state contains the `Position`

and `Velocity`

components.

stateinfo(filter)

`ans = `*struct with fields:*
Position: 1
Velocity: 2
Accelerometer_Bias: [3 4 5]

**Show customized ConstantVelocityMotion class**

`type ConstantVelocityMotion.m`

classdef ConstantVelocityMotion < positioning.INSMotionModel % CONSTANTVELOCITYMOTION Constant velocity motion in 1-D % Copyright 2021 The MathWorks, Inc. methods function m = modelstates(~,~) % Return the state of motion model (added to the state of the % filter) as a structure. % Since the motion is 1-D constant velocity motion, % retrun only 1-D position and velocity state. m = struct('Position',0,'Velocity',0); end function sdot = stateTransition(~,filter,~, varargin) % Return the derivative of each state with respect to time as a % structure. % Deriviative of position = velocity. % Deriviative of velocity = 0 because this model assumes constant % velocity. % Find the current estimated velocity currentVelocityEstimate = stateparts(filter,'Velocity'); % Return the derivatives sdot = struct( ... 'Position',currentVelocityEstimate, ... 'Velocity',0); end function dfdx = stateTransitionJacobian(~,filter,~,varargin) % Return the Jacobian of the stateTransition method with % respect to the state vector. The output is a structure with the % same fields as stateTransition but the value of each field is a % vector containing the derivative of that state relative to % all other states. % First, figure out the number of state components in the filter % and the corresponding indices N = numel(filter.State); idx = stateinfo(filter); % Compute the N partial derivatives of Position with respect to % the N states. The partial derivative of the derivative of the % Position stateTransition function with respect to Velocity is % just 1. All others are 0. dpdx = zeros(1,N); dpdx(1,idx.Velocity) = 1; % Compute the N partial derivatives of Velocity with respect to % the N states. In this case all the partial derivatives are 0. dvdx = zeros(1,N); % Return the partial derivatives as a structure. dfdx = struct('Position',dpdx,'Velocity',dvdx); end end end

## Input Arguments

`model`

— Motion model used with INS filter

object inherited from `positioning.INSMotionModel`

class

Motion model used with an INS filter, specified as an object inherited from the `positioning.INSMotionModel`

abstract class.

`filter`

— INS filter

`insEKF`

object

INS filter, specified as an `insEKF`

object.

`dt`

— Filter time step

positive scalar

Filter time step, specified as a positive scalar.

**Data Types: **`single`

| `double`

## Output Arguments

`jac`

— Jacobian matrix for state transition equation

*S*-by-*N* real-valued matrix

Jacobian matrix for the state transition equation, returned as an
*S*-by-*N* real-valued matrix. *S*
is the number of fields in the returned structure of the `modelstates`

method of the motion model, and *N* is the dimension of the state
maintained in the `State`

property of the
`filter`

.

## Version History

**Introduced in R2022a**

## See Also

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