Documentation

constturnjac

Jacobian for constant turn-rate motion

Description

example

jacobian = constturnjac(state) returns the updated Jacobian, jacobian, for constant turn-rate Kalman filter motion model for a one-second step time. The state argument specifies the current state of the filter. Constant turn rate means that motion in the x-y plane follows a constant angular velocity and motion in the vertical z directions follows a constant velocity model.

example

jacobian = constturnjac(state,dt) specifies the time step, dt.

[jacobian,noisejacobian] = constturnjac(state,dt,w) also specifies noise, w, and returns the Jacobian, noisejacobian, of the state with respect to the noise.

Examples

collapse all

Compute the Jacobian for a constant turn-rate motion state. Assume the turn rate is 12 degrees/second. The time step is one second.

state = [500,0,0,100,12];
jacobian = constturnjac(state)
jacobian = 5×5

1.0000    0.9927         0   -0.1043   -0.8631
0    0.9781         0   -0.2079   -1.7072
0    0.1043    1.0000    0.9927   -0.1213
0    0.2079         0    0.9781   -0.3629
0         0         0         0    1.0000

Compute the Jacobian for a constant turn-rate motion state. Assume the turn rate is 12 degrees/second. The time step is 0.1 second.

state = [500,0,0,100,12];
jacobian = constturnjac(state,0.1)
jacobian = 5×5

1.0000    0.1000         0   -0.0010   -0.0087
0    0.9998         0   -0.0209   -0.1745
0    0.0010    1.0000    0.1000   -0.0001
0    0.0209         0    0.9998   -0.0037
0         0         0         0    1.0000

Input Arguments

collapse all

State vector for a constant turn-rate motion model in two or three spatial dimensions, specified as a real-valued vector.

• When specified as a 5-element vector, the state vector describes 2-D motion in the x-y plane. You can specify the state vector as a row or column vector. The components of the state vector are [x;vx;y;vy;omega] where x represents the x-coordinate and vx represents the velocity in the x-direction. y represents the y-coordinate and vy represents the velocity in the y-direction. omega represents the turn rate.

• When specified as a 7-element vector, the state vector describes 3-D motion. You can specify the state vector as a row or column vector. The components of the state vector are [x;vx;y;vy;omega;z;vz] where x represents the x-coordinate and vx represents the velocity in the x-direction. y represents the y-coordinate and vy represents the velocity in the y-direction. omega represents the turn rate. z represents the z-coordinate and vz represents the velocity in the z-direction.

Position coordinates are in meters. Velocity coordinates are in meters/second. Turn rate is in degrees/second.

Example: [5;0.1;4;-0.2;0.01]

Data Types: double

Time step interval of filter, specified as a positive scalar. Time units are in seconds.

Example: 0.5

Data Types: single | double

State noise, specified as a scalar or real-valued M-by-(D+1)-length vector. D is the number of motion dimensions. D is two for 2-D motion and D is three for 3-D motion. The vector components are [ax;ay;alpha] for 2-D motion or [ax;ay;alpha;az] for 3-D motion. ax, ay, and az are the linear acceleration noise values in the x-, y-, and z-axes, respectively, and alpha is the angular acceleration noise value. If specified as a scalar, the value expands to a (D+1) vector.

Data Types: single | double

Output Arguments

collapse all

Constant turn-rate motion Jacobian, returned as a real-valued 5-by-5 matrix or 7-by-7 matrix depending on the size of the state vector. The Jacobian is constructed from the partial derivatives of the state at the updated time step with respect to the state at the previous time step.

Constant turn-rate motion noise Jacobian, returned as a real-valued 5-by-(D+1) matrix where D is two for 2-D motion or a real-valued 7-by-(D+1) matrix where D is three for 3-D motion. The Jacobian is constructed from the partial derivatives of the state at the updated time step with respect to the noise components.