Kinematics is the study of motion without considering the forces that cause that motion. For a point mass moving along a single dimension (the $x$-axis), its entire motion is defined by its position, velocity, and acceleration.
The motion of the point mass is governed by two fundamental differential relationships that link position ($x$), velocity ($v$), and acceleration ($a$). These equations define the instantaneous rate of change for the position and velocity.
Velocity ($v$) is the first time derivative of position ($p$). It tells us how quickly the position is changing:
$$v = \frac{dp}{dt} = \dot{p}$$
The dot over $p$ notates the time differentiation.
Acceleration ($a$) is the first time derivative of velocity ($v$), or the second time derivative of position. It tells us how quickly the velocity is changing:
$$a = \frac{dv}{dt} = \dot{v}$$
Acceleration can therefore also be expressed as the second time derivative of position:
$$a = \frac{d^2p}{dt^2} = \ddot{p}$$
The double dot ($\ddot{p}$) signifies the second time differentiation.
While the differential equations define the instantaneous state, we use integration to calculate the position and velocity at a time $t$ based on the known acceleration and initial conditions.
We will assume a constant acceleration ($a$) over the interval $[t_0, t]$. This assumption is often necessary when discretizing motion for estimation algorithms like the Kalman Filter.
We start with the acceleration equation
$$\frac{dv}{dt} = a \implies dv = a \, dt$$
and integrate it with respect to time from the initial time $t_0$ to the current time $t$. We use $\tau$ as variable of integration to avoid a "notational collision".
$$\int_{v_0}^{v(t)} dv = \int_{t_0}^{t} a \, d\tau$$
where
Since $a$ is constant, we can pull it out of the integral:
$$v(t) - v_0 = a \int_{t_0}^{t} d \tau$$
$$v(t) - v_0 = a (t - t_0)$$
By rearranging the terms, we obtain the final equation for velocity $v$:
$$v(t) = v_0 + a(t - t_0)$$
| Term | Meaning |
|---|---|
| $v(t)$ | Velocity at time $t$. |
| $v_0$ | Initial velocity at time $t_0$. |
| $a$ | Constant acceleration. |
| $(t - t_0)$ | Elapsed time ($\Delta t$). |
Next, we substitute the integrated velocity equation $v(\tau) = v_0 + a(\tau - t_0)$ into the position equation ($\dot{p} = v$) and integrate with respect to time from $t_0$ to $t$ using the dummy variable $\tau$:$$\int_{p_0}^{p(t)} dp = \int_{t_0}^{t} v(\tau) \, d\tau$$
$$\int_{p_0}^{p(t)} dp = \int_{t_0}^{t} [v_0 + a(\tau - t_0)] \, d\tau$$
1. Change of Variables Let $\Delta\tau = (\tau - t_0)$. Since $t_0$ is a constant, the differential is $d(\Delta\tau) = d\tau$. We also change the limits of integration:
2. Substitute into the Integral $$p(t) - p_0 = \int_{0}^{(t - t_0)} (v_0 + a\Delta\tau) \, d(\Delta\tau)$$
3. Integrate term by term Now the integration is straightforward because we are integrating with respect to the elapsed time $\Delta\tau$:
$$p(t) - p_0 = \left[ v_0 \Delta\tau + \frac{1}{2} a (\Delta\tau)^2 \right]_{0}^{(t - t_0)}$$
4. Evaluate at the limits Substituting the upper limit $(t - t_0)$ and the lower limit $0$: $$p(t) - p_0 = \left( v_0(t - t_0) + \frac{1}{2} a(t - t_0)^2 \right) - (0 + 0)$$
5. Final Result By adding $p_0$ to both sides, we obtain the standard kinematic equation: $$p(t) = p_0 + v_0(t - t_0) + \frac{1}{2} a(t - t_0)^2$$
| Term | Meaning |
|---|---|
| $p(t)$ | Position at time $t$. |
| $p_0$ | Initial position at time $t_0$. |
| $v_0(t - t_0)$ | Displacement due to constant initial velocity. |
| $\frac{1}{2} a(t - t_0)^2$ | Displacement due to constant acceleration. |
These two integrated equations form the basis for predicting the next state in a linear motion model, which is the starting point for developing the transition matrix in a Kalman Filter.
Estimation algorithms like the Kalman Filter operate on data sampled at fixed time intervals. Therefore, the continuous equations of motion must be converted—or discretized—into a form that predicts the state from one discrete time step ($k-1$) to the next ($k$).
The fundamental equations of kinematics, such as $v = \frac{dp}{dt}$, are continuous. They require knowing the acceleration $a$ at every instant in time, $t$.
However, in real-world systems:
To bridge this gap, we discretize the equations by applying the integral form over a fixed, finite time interval, $\Delta t$.
The entire process of discretization relies on a crucial assumption about the behavior of the system input (acceleration) within the short interval $\Delta t = t_k - t_{k-1}$:
Assumption: The acceleration ($a_{k-1}$) is constant and known throughout the entire time interval $[t_{k-1}, t_k]$.
This simplifies the integrals and allows us to derive the linear matrix equations used later in the State-Space Model.
We use the integrated form of $\dot{v} = a$:
$$\int_{v_{k-1}}^{v_k} dv = \int_{t_{k-1}}^{t_k} a_{k-1} \, dt$$
Since $a_{k-1}$ is assumed constant over the interval, we pull it out and define the duration as $\Delta t = t_k - t_{k-1}$:$$v_k - v_{k-1} = a_{k-1} \int_{t_{k-1}}^{t_k} dt$$$$v_k - v_{k-1} = a_{k-1} (t_k - t_{k-1}) = a_{k-1} \Delta t$$
Final Discrete Velocity Equation:
$$v_k = v_{k-1} + a_{k-1} \Delta t$$
This equation shows that the velocity at time $k$ is the velocity at time $k-1$ plus the change in velocity caused by the constant acceleration $a_{k-1}$ over the time step $\Delta t$.
We use the integrated form of $\dot{p} = v$, substituting the discrete velocity solution:
$$\int_{p_{k-1}}^{p_k} dp = \int_{t_{k-1}}^{t_k} v(t) \, dt$$
We substitute $v(t) = v_{k-1} + a_{k-1} (t - t_{k-1})$ into the integral. For convenience, let $\tau = t - t_{k-1}$, so $dt = d\tau$, and the integration limits are from $0$ to $\Delta t$.
$$\int_{p_{k-1}}^{p_k} dp = \int_{0}^{\Delta t} [v_{k-1} + a_{k-1} \tau] \, d\tau$$
Integrating term by term:
$$p_k - p_{k-1} = v_{k-1} \int_{0}^{\Delta t} d\tau + a_{k-1} \int_{0}^{\Delta t} \tau \, d\tau$$
$$p_k - p_{k-1} = v_{k-1} [\tau]_{0}^{\Delta t} + a_{k-1} \left[ \frac{1}{2}\tau^2 \right]_{0}^{\Delta t}$$
$$p_k - p_{k-1} = v_{k-1} \Delta t + \frac{1}{2} a_{k-1} (\Delta t)^2$$
Final Discrete Position Equation:
$$p_k = p_{k-1} + v_{k-1} \Delta t + \frac{1}{2} a_{k-1} (\Delta t)^2$$
These two discrete equations, which link the state at $k-1$ directly to the state at $k$, are the State Transition Equations used to build the State-Space Model for the Kalman Filter's prediction step.