Computing the $x$-$y$ position of a vehicle for constant acceleration

Question

I want to very simplistically plot the $x$-$y$ position of a vehicle in 2D that can accelerate/brake and drive a constant speed while turning etc.

The basic equations for determining the position for constant speed seem to be

$$x_{n+1} = x_n + vt\cos(\theta)$$ $$y_{n+1} = y_n + vt\sin(\theta)$$

where $\theta$ is the turning angle given in radians.

Am I correct in assuming that since $$v_{n+1} = v_n + at$$ for a constant acceleration we have:

$$x_{n+1} = x_n + (v_n+at)t\cos(\theta)$$ $$y_{n+1} = y_n + (v_n+at)t\sin(\theta)$$ ?

Answer 1

So I did a little C# demo to check that my math is correct first.

The math behind each simulation step is as follows (assuming non-zero steering angle). Below $\Delta t$ is the time step, $v_0$ the initial speed, $\vec{p}_0$ is the initial position, $\varphi_0$ the initial orientation angle, and $\ell$ is the wheelbase.

1. Define the step steering angle $\theta$, as well as the acceleration $\dot{v}$.
2. Define the turn radius $R = \ell / \tan \theta$.
3. Get speed change $\Delta v = \dot{v} \Delta t$.
4. Get distance change $\Delta s = v_0 \Delta t + \tfrac{1}{2} \dot{v} \Delta t^2$.
5. Get orientation angle change $\Delta \varphi = \Delta s / R$
6. Get new orientation angle $\varphi_1 = \varphi_0 + \Delta \varphi$
7. Get initial normal direction $\hat{n}_0 = \pmatrix{-\sin (\varphi_0) & \cos (\varphi_0)}$
8. Get final normal direction $\hat{n}_1 = \pmatrix{-\sin (\varphi_1) & \cos (\varphi_1)}$
9. Get position change $\Delta \vec{p} = R\, (\hat{n}_0 - \hat{n}_1)$
10. New position $\vec{p}_1 = \vec{p}_0 + \Delta \vec{p}$
11. New speed $v_1 = v_0 + \Delta v$
12. Get tangent direction $\hat{e}_1 = \pmatrix{ \cos(\varphi_1) & \sin(\varphi_1)}$
13. Get new velocity vector $\vec{v}_1 = v_1 \hat{e}_1$

_{NOTE: All angles are in radians. The car position is defined by the center of the rear axle since a car turns about the rear wheels.}

The key to this and the answer to your question is #9. It takes the current position, moves to the center of rotation (pivot point), and back into the new position by traversing an arc of length $\Delta s$.

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The kinematics of a 2D car is described by the following picture

where $R$ is the radius of turn (and $1/R$ is the turn curvature). In pink is the arc the car tracks (the local coordinate of the car shown mid-rear axle) and it has an arc length of $\Delta s$. This arc sweeps an angle of $\Delta \varphi = \Delta s/R$. The angle $\theta$ describing the big blue triangle is the steering angle, due to similar triangles in the steering geometry.

The initial and final position and orientation of the car coordinate system is shown at the ends of the pink arc, and you see they share the common pivot point which is always $R$ distance away, along the $\hat{n}$ direction.

Answer 2

Your ($v_n$ + at) is a final velocity. With constant acceleration, you can divide that by two to get the average velocity. In general, with constant acceleration: x = $x_o$ + $v_{ox}$t + (1/2)${(a_x)(t^2})$ There would be a similar equation for y (or z).

Answer 3

There are two cases I will address. In your question, you say "constant acceleration", which is trivial to model since you only need to plug the kinematic equations into your software and plot with known values. The next is non-constant acceleration, which needs a numerical solution.

The simplest numerical model for this would be Newton's Method. Essentially we discretize the coordinates (since computers can only approximate continuous systems). Hence \begin{equation} \frac{d\vec{v}}{dt} = \vec{a}(t) \rightarrow \vec{v}_{n+1} = \vec{v}_{n} + \vec{a}_{n}\Delta t \end{equation} And to go from velocity to position \begin{equation} \vec{r}_{n+1} = \vec{r}_{n} + \vec{v}_n\Delta t \end{equation} Each of these vector equations are really two one-dimensional equations (assuming the equations are not coupled, which could still be solved numerically) in both the $x$ and $y$ directions. For example, \begin{equation} x_{n+1} = x_{n} + v_{x,n}\Delta t \end{equation} You can now take these equations and plug it into any software (python, mathematica; probably not a calculator though) and specify the acceleration (for both the $x$ and $y$ directions), how big you want the time steps, and how many time steps you want to take. Then let your computer step the system forward in time with the equations above, save that data to a list or array, and plot the position against time.