3D Vehicle Collision Probability Using Minkowski Difference Projections
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Solution Overview
Problem
Existing methods for determining collision probability in three-dimensional space are computationally intensive and prone to excessive overestimation, making real-time trajectory planning for aerial vehicles challenging, especially when dealing with complex traffic situations involving uncertainty in sensor data and unknown intentions of other traffic participants.
Innovation Solution
A method and device that approximate a vehicle and an object using geometric bodies, calculate Minkowski differences with indeterminacies, and standardize these differences to determine collision probability efficiently, using sensor data and potentially external information, with a control unit to compute collision probabilities in real-time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional collision probability methods are used in three-dimensional space, then measurement precision is improved, but computing time increases excessively
Solution Approach 1:
The three-dimensional space is segmented into multiple two-dimensional projection planes (e.g., XY, XZ, YZ planes). The collision probability calculation is divided into separate calculations for each plane, allowing parallel processing and reducing the overall computational complexity from O(n³) to O(n²) while maintaining accuracy through the combination of results from all planes.
Solution Approach 2:
The problem is transformed from a direct three-dimensional collision probability calculation into multiple two-dimensional projection calculations. By projecting the three-dimensional positions and uncertainty regions onto two-dimensional planes, the method leverages efficient 2D algorithms while reconstructing the 3D collision probability from the projected results, achieving a balance between precision and computational efficiency.
2Loss of time
If simplifying assumptions are made for collision probability estimation, then computing time is reduced, but overestimation of collision probability occurs
Solution Approach 1:
Instead of calculating the exact three-dimensional collision probability directly (excessive action), the method performs partial calculations in two-dimensional projection planes and combines them. This partial approach avoids the computational burden of full 3D integration while ensuring that the combined results do not excessively overestimate the collision probability, as each plane's contribution is weighted appropriately.
Solution Approach 2:
The method creates two-dimensional copies (projections) of the three-dimensional uncertainty regions and vehicles. By calculating collision probabilities in these simplified 2D copies and then combining them, the approach avoids the complexity of direct 3D calculation while maintaining accuracy through the mathematical relationship between the projections and the original 3D space.
3Device complexity
If three-dimensional collision probability calculation is performed without special treatment, then device complexity is reduced, but calculation accuracy deteriorates
Solution Approach 1:
The method introduces a dimensional transformation approach where three-dimensional collision probability is calculated by performing multiple two-dimensional calculations in projection planes. This special treatment of dimensions maintains relatively simple device architecture while achieving accurate results through the mathematical combination of projected plane calculations, avoiding the need for complex direct 3D integration algorithms.
Data Source
AI summary
Determining a collision probability of a vehicle with an object. A three-dimensional position and orientation of the vehicle and the object is detected. The vehicle is approximated by at least one first geometrical element, that includes a part of the vehicle. The same is done for the object. A first error and a second error is determined in relation to the geometrical element for the vehicle or object. A Minkowski difference is formed for each combination of the at least one first geometrical element with the at least one second geometrical element. A third error is standardized via a transformation. The transformation serves to calculate calculation elements. The collision probability of the vehicle with the object is determined on the basis a spatial union of the calculation elements.