Numerical Computation-Based Position Estimation for QR Code Object Marker: Mathematical Model and Simulation

Abstract

Providing position and orientation estimations from a two-dimensional (2D) image is challenging, as such images lack depth information between the target and the automation system. This paper proposes a numerical-based monocular positioning method to determine the position and orientation of a single quick response (QR) code object marker. The three-dimensional (3D) positional information can be extracted from the underdetermined system using the QR code’s four vertices as positioning points. This method uses the fundamental principles of the pinhole imaging theory and similar triangular rules to correspond the QR code’s corner points in a 3D environment to the 2D image. The numerical-based model developed with suitable guessing parameters and correct updating rules successfully determines the QR code marker’s position. At the same time, an inversed rotation matrix determines the QR code marker’s orientation. Then, the MATLAB platform simulates the proposed positioning model to identify the maximum rotation angles detectable at various locations using a single QR code image with the known QR code’s size and the camera’s focal length. The simulation results show that the proposed numerical model can measure the position and orientation of the tilted QR code marker within 30 iterations with great accuracy. Additionally, it can achieve no more than a two-degree angle calculation error and less than a five millimeter distance difference. Overall, more than 77.28% of the coordinate plane simulated shows a converged result. The simulation results are verified using the input value, and the method is also capable of experimental verification using a monocular camera system and QR code as the landmark.

Keywords

• numerical method
• 3D positioning
• pose estimation
• pinhole imaging
• machine vision
• QR code marker