Jacobian for constant-velocity motion
Compute the state Jacobian for a two-dimensional constant-velocity motion model for a one second update time.
state = [1,1,2,1].'; jacobian = constveljac(state)
jacobian = 4×4 1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1
Compute the state Jacobian for a two-dimensional constant-velocity motion model for a half-second update time.
state = [1;1;2;1];
Compute the state update Jacobian for 0.5 second.
jacobian = constveljac(state,0.5)
jacobian = 4×4 1.0000 0.5000 0 0 0 1.0000 0 0 0 0 1.0000 0.5000 0 0 0 1.0000
w— State noise
State noise, specified as a scalar or real-valued real valued N-by-1 vector. N is the number of motion dimensions. For example, N = 2 for the 2-D motion. If specified as a scalar, the scalar value is expanded to an N-by-1 vector.
jacobian— Constant-velocity motion Jacobian
Constant-velocity motion Jacobian, returned as a real-valued 2N-by-2N matrix. N is the number of spatial degrees of motion.
noisejacobian— Constant velocity motion noise Jacobian
Constant velocity motion noise Jacobian, returned as a real-valued 2N-by-N matrix. N is the number of spatial degrees of motion. The Jacobian is constructed from the partial derivatives of the state at the updated time step with respect to the noise components.
For a two-dimensional constant-velocity motion, the Jacobian matrix for a time step, T, is block diagonal:
The block for each spatial dimension has this form:
For each additional spatial dimension, add an identical block.