Intuition The one core ideaThe Extended Kalman Filter is just the ordinary Kalman Filter with one repair: wherever the real world curves, we replace the curve by its tangent line at our current best guess so the straight-line filter math still works. Everything on this page — vectors, derivatives, slopes, matrices — exists to build the meaning of that one sentence: a curve, zoomed in, is a line.
This page assumes you have seen nothing . We name every symbol the parent note throws at you, draw the picture behind it, and say why the EKF cannot live without it. Read top to bottom; each item is a brick for the next.
x state is the shortest list of numbers that fully describes what your vehicle is doing right now . We stack that list into a column and call it x
Plain words: "everything I need to know to predict the next instant."Picture: a single arrow-tipped dot living in a room whose axes are the quantities (position, velocity, angle...). See the figure below.Why the topic needs it: the filter's entire job is to track this dot as new sensor data arrives.
For example a 2D target is x = [ x , y ] ⊤ ⊤ [ x , y ] ⊤
Definition Measurement vectorz measurement is what your sensor actually reports. It is not the state — it is some function of the state, often a distorted one.
Plain words: "the number the radar screen shows," which may be range, or an angle, not the raw x , y Picture: a second, smaller room; an arrow (the function h Why: the whole difficulty of the EKF is that the map from state to measurement is curved .
Why can't we just read the state directly? ::: Sensors report indirect, nonlinear quantities (a range $\sqrt{x^2+y^2}$, a bearing angle) — never the clean state itself.
f h
f ( x ) next state (how things move).h ( x ) measurement it would produce.Picture: two black boxes. Feed a dot in, a dot comes out — but the output dot may be somewhere curved-looking, not a straight scaling of the input.Why: the parent note writes x k = f ( x k − 1 , u ) + w z k = h ( x k ) + v are the model of your system.
The subscript k k x k − 1 x k u control input — the commands you sent (throttle, steering) that also move the state.
A map is linear if doubling the input doubles the output and inputs add cleanly: straight lines stay straight, no bending. In symbols a linear map is just multiplication by a fixed matrix, y = M x
Picture: a grid of squares gets stretched/rotated but stays a grid of straight lines.Nonlinear means the grid bends — the picture below shows exactly this.
Intuition Why nonlinearity breaks the ordinary Kalman FilterA bell-curve of uncertainty (a Gaussian ) pushed through a straight map comes out another bell curve. Pushed through a bent map it comes out lop-sided and skewed — no longer a bell curve. The Kalman Filter only knows how to carry bell curves. So we must first straighten the bend locally .
What stays Gaussian through a linear map but not a nonlinear one? ::: The uncertainty cloud (a Gaussian); a curve skews it into a non-Gaussian shape.
d x d g derivative of a function at a point is the steepness of the curve right there — "how many units does the output rise for one tiny unit of input."
Plain words: "the slope of the ramp you feel if you stand at that exact spot."Picture: zoom into any smooth curve far enough and it looks straight; the derivative is the slope of that straight zoom-in. See figure.Why the EKF needs it: the derivative is the tangent line's slope, and the tangent line is our straightened stand-in for the curve.
This is the single most important formula on the page — the EKF applies it at every time step. Deeper background lives in Taylor Series & Linearization .
Common mistake The tangent is only goodnearby
Far from x ^
Definition Partial derivative∂ x ∂ g x and y partial derivative with respect to x only x
Picture: stand on a hillside; the partial in x y Why: states have many components, so "the slope" is really a collection of one-direction slopes.
The curly ∂ d
What does $\partial r/\partial x$ mean in words? ::: How much the range $r$ changes when only $x$ moves, with $y$ held still.
Definition Jacobian matrixJ Jacobian .
Row i = "how output g i Column j = "how input x j Picture: the best flat sheet (linear map) tangent to the bent grid at your point — the multi-input version of "the tangent line."Why: the Kalman equations need a constant matrix . The Jacobian is the best constant matrix approximating the curve locally — so it is exactly what F H
J = ∂ x 1 ∂ g 1 ∂ x 1 ∂ g 2 ∂ x 2 ∂ g 1 ∂ x 2 ∂ g 2
In the EKF the two Jacobians earn their own letters: F = ∂ f / ∂ x H = ∂ h / ∂ x Jacobian Matrix & Multivariable Calculus .
Definition Covariance matrixP P how uncertain you are, and in which directions the uncertainty is correlated.
Plain words: "the fuzziness of the state-dot" — a small P P Picture: an ellipse (a squashed circle) drawn around the state-dot; long axis = the direction you're least sure of.Why: the filter tracks not just where it thinks the vehicle is, but how much to trust that — and that trust ellipse must be carried forward too.
Intuition Why covariance transforms byF P F ⊤ not by f slope the map has — i.e. by the Jacobian F F F ⊤ f
The Q R w v Covariance Propagation .
atan2 ( y , x ) arctan ( y / x ) 18 0 ∘ atan2 x y
Picture: a full 36 0 ∘ Why: radar bearing measurements need the right quadrant, and angle differences must be wrapped to ( − π , π ] 35 9 ∘ − 1 ∘ − 2 ∘ 35 8 ∘ atan2 & Angle Wrapping .
State vector x and measurement z
Tangent line = Taylor first order
Covariance P as an ellipse
This map feeds directly into the parent, the EKF topic note . If the tangent-line idea still feels shaky, the gentler linear case lives in Kalman Filter (linear) , and when linearization is too crude the alternative is Unscented Kalman Filter (UKF) . The big-picture role sits in State Estimation in GNC .
Say each answer aloud before revealing.
What is a state vector x
What does ⊤
Difference between z x z x
What do f h f h
Why does nonlinearity break the ordinary KF? ::: It skews a Gaussian into a non-Gaussian shape the KF can't carry.
What is a derivative in one phrase? ::: The local slope — output rise per tiny input nudge.
Write the first-order Taylor line. ::: g ( x ) ≈ g ( x ^ ) + g ′ ( x ^ ) ( x − x ^ )
What is a partial derivative? ::: Slope when you nudge one input and freeze the rest.
What is the Jacobian? ::: The grid of all partial derivatives — the best local linear (constant-matrix) approximation.
What are F H f h
What does P
Why does P F P F ⊤ f ( P )
What does atan2 arctan 36 0 ∘
