A Spherical Two-Degree-of-Freedom Angular Displacement Sensor

Multidimensional angular displacement detection technology is essential for enabling the high-precision control of advanced spherical motors and ball hinges. However, existing conventional methods based on optical or electromagnetic principles are limited in their capacity for obtaining measurements in real time or in their ability to combine high accuracy with a large measurement range. This article addresses these issues by proposing a novel spherical two-degree-of-freedom (two-DOF) angular displacement sensor that employs two distinct spherical electrode array configurations as the stator and rotor, respectively. The electrodes on the stator and rotor are separated by a short radial distance and represent capacitors with a capacitance that varies according to the area of overlap between the electrodes. Four two-phase orthogonal sinusoidal excitation signals of identical frequency and amplitude are applied along the two spatially orthogonal rotational angles (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta \text {)}$ </tex-math></inline-formula> of the stator electrode array, respectively. As a result, each electrode in the rotor electrode array captures induced sinusoidal signals of identical frequency, whose phase characteristics are directly correlated with the rotational angles of the rotor in both the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> directions. The induced signals from rotor electrodes are processed using a spatial phase combination method, which yields separate measurement signals <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${U}_{{o}\alpha }$ </tex-math></inline-formula>, containing only angular displacement information for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> direction, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${U}_{{o}\beta }$ </tex-math></inline-formula>, containing only angular displacement information for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> direction. Finally, the angular displacements of the rotor along the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> directions are determined independently by extracting the phases from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${U}_{{o}\alpha }$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${U}_{{o}\beta }$ </tex-math></inline-formula>, respectively. Accuracy measurements on an experimental platform demonstrate that the calibrated sensor obtains a peak-to-peak error of 0.012° in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta $ </tex-math></inline-formula> directions over an entire ±20° measurement range in each direction.