Two-Dimensional-Time-To-Collision

This repository allows for fast computation of two-dimensional Time-To-Collision (2D-TTC). This is particularly useful for evaluating the traffic conflict risk at intersections, but for sure can also be used in the scenario of highways.

A document explaining my computation is provided here, where the core idea is $TTC=\frac{DTC}{|\boldsymbol{v}_{ij}|}$ and $DTC$ means a distance to collision, as shown in the figure below.

Note that this method follows the typical definition of TTC assuming constant velocity at the time moment of evaluation (a reference can be found here). This clearly differs from alternative definitions of 2D-TTC such as time advantage or predicted PET (https://doi.org/10.1016/j.aap.2010.03.021). Due to the drawback of the constant-velocity assumption, this method may not suit well for slow and conscious interactions.

If you use this software in your work, please cite it using the following metadata:

@software{jiao2023ttc,
author = {Jiao, Yiru},
month = mar,
title = {{A fast calculation of two-dimensional Time-to-Collision}},
url = {https://github.com/Yiru-Jiao/Two-Dimensional-Time-To-Collision},
year = {2023}
}

Libarary requirements

Any versions of pandas and numpy.

Usage

Use function TTC(samples, 'dataframe') or TTC(samples, 'values') to compute two-dimensional Time-To-Collision.

For example,

import sys
sys.path.append('') # add the path where you save this `.py` file
import TwoDimTTC

# To return a dataframe with the input vehicle pair samples, where 2D-TTC are saved in a new column named 'TTC'
samples = TwoDimTTC.TTC(samples, 'dataframe')

# To return a numpy array of 2D-TTC values
ttc = TwoDimTTC.TTC(samples, 'values')

Input

The first input is a pandas dataframe of vehicle pair samples, which should include the following columns. Note that the heading direction of a vehicle is not necessarily the direction of its velocity, especially when the vehicle is stopping or steering. The heading (hx, hy) is normalized to a unit vector in this function.

• x_i : x coordinate of the ego vehicle $i$ (usually assumed to be centroid)
• y_i : y coordinate of the ego vehicle $i$ (usually assumed to be centroid)
• vx_i : x coordinate of the velocity of the ego vehicle $i$
• vy_i : y coordinate of the velocity of the ego vehicle $i$
• hx_i : x coordinate of the heading direction of the ego vehicle $i$
• hy_i : y coordinate of the heading direction of the ego vehicle $i$
• length_i : length of the ego vehicle $i$
• width_i : width of the ego vehicle $i$
• x_j : x coordinate of another vehicle $j$ (usually assumed to be centroid)
• y_j : y coordinate of another vehicle $j$ (usually assumed to be centroid)
• vx_j : x coordinate of the velocity of another vehicle $j$
• vy_j : y coordinate of the velocity of another vehicle $j$
• hx_j : x coordinate of the heading direction of another vehicle $j$
• hy_j : y coordinate of the heading direction of another vehicle $j$
• length_j : length of another vehicle $j$
• width_j : width of another vehicle $j$

The second input allows outputing a dataframe with inputed samples plus a new column named 'TTC', or mere TTC values.

Output

If ttc==np.inf, the ego vehicle $i$ and another vehicle $j$ will never collide if they keep current speed.

A negative TTC means the boxes of the ego vehicle $i$ and another vehicle $j$ are overlapping. This is due to approximating the space occupied by a vehicle with a rectangular. In other words, ttc<0 in this computation means the collision between the two vehicles almost (or although seldom, already) occurred.

Note that mere TTC computation can give an extreme small positive value even when the vehivles are overlapping a bit. In order to improve the accuracy, please use function CurrentD(samples, 'dataframe') or CurrentD(samples, 'values') to further exclude overlapping vehicles. This function calculate current distance between the ego vehicle $i$ and another vehicle $j$, which indicate overlapping when the value is negative.

# Within pandas dataframe
samples = TwoDimTTC.TTC(samples, 'dataframe')
samples = TwoDimTTC.CurrentD(samples, 'dataframe')
samples.loc[(samples.CurrentD<0)&(samples.TTC<np.inf)&(samples.TTC>0),'TTC'] = -1

# Using numpy array of values
ttc = TwoDimTTC.TTC(samples, 'values')
current_dist = TwoDimTTC.CurrentD(samples, 'values')
ttc[(current_dist<0)&(ttc<np.inf)&(ttc>0)] = -1

Efficiency

Use function efficiency(samples, iterations) to test the computation efficiency.

For example,

print('Average time cost = {:.4f} second(s)'.format(TwoDimTTC.efficiency(samples, 10)

The following table shows approximately needed computation time (tested for 10 iterations of experiments).

number of vehicle pairs	computation time (s)
1e4	0.0357
1e5	0.4342
1e6	7.1657

Copyright

Copyright (c) 2022 Yiru Jiao. All rights reserved.

This work is licensed under the terms of the MIT license. For a copy, see https://opensource.org/licenses/MIT.