Compact Optical Single-axis Joint Torque Sensor Using Redundant Photo-Reflectors and Quadratic-Programming Calibration

Hyun-Bin Kim; Byeong-Il Ham; Kyung-Soo Kim

cs.ROMar 17, 2026

Compact Optical Single-axis Joint Torque Sensor Using Redundant Photo-Reflectors and Quadratic-Programming Calibration

Authors: Hyun-Bin Kim, Byeong-Il Ham, Kyung-Soo Kim

Organizations: MSC (Mechatronics, Systems and Control) lab in the KAIST(Korea Advanced Institute of Science and Technology which is in the Daehak-Ro 291, Daejeon, South Korea

Abstract

This study proposes a non-contact photo-reflector-based joint torque sensor for precise joint-level torque control and safe physical interaction. Current-sensor-based torque estimation in many collaborative robots suffers from poor low-torque accuracy due to gearbox stiction/friction and current-torque nonlinearity, especially near static conditions. The proposed sensor optically measures micro-deformation of an elastic structure and employs a redundant array of photo-reflectors arranged in four directions to improve sensitivity and signal-to-noise ratio. We further present a quadratic-programming-based calibration method that exploits redundancy to suppress noise and enhance resolution compared to least-squares calibration. The sensor is implemented in a compact form factor (96 mm diameter, 12 mm thickness). Experiments demonstrate a maximum error of 0.083%FS and an RMS error of 0.0266 Nm for z-axis torque measurement. Calibration tests show that the proposed calibration achieves a 3 sigma resolution of 0.0224 Nm at 1 kHz without filtering, corresponding to a 2.14 times improvement over the least-squares baseline. Temperature chamber characterization and rational fitting based compensation mitigate zero drift induced by MCU self heating and motor heat. Motor-level validation via torque control and admittance control confirms improved low torque tracking and disturbance robustness relative to current-sensor-based control.

Figures & tables

Fig. 1: Configuration and operating principle of the proposed sensor. When an external torque is applied, the reflective surface rotates, altering the amount of light reflected toward the photo-reflector; this change in reflected intensity causes the output voltage to increase or decrease accordingly.

Fig. 2: Assembly of the printed circuit board (PCB) and the elastic structure, and parameters of the elastic beam.

Fig. 3: Schematic illustration of the elastic structure. Here, $\Delta r_{t}$ denotes the bending angle induced by $T_{z}$ , and $F_{A}$ represents the force applied due to $T_{z}$ .

Parameter	$l$	$b$	$h$	$r_{i}$	$l_{s}$
Value (mm)	21	4	7	19	30

TABLE I: Value of Parameters

Property	Value	Unit
Dimension	$\phi$ 96 $\times$ 12	mm
Weight	114.5	g
Load capacity	$F_{xy}$ :7700, $F_{z}$ :3180, $T_{xy}$ :62, $T_{z}$ :80	N,N,N $\cdot$ m,N $\cdot$ m
Communication	CAN-FD	-
Electronic bandwidth	Effective bandwidth (rise-time-limited) $\sim$ 5 [ 23 ]	kHz
Resonant frequency	3.407	kHz

TABLE II: Specifications of the proposed torque sensor

Fig. 4: Assembly overview: (a) photograph of the assembled PCB and elastic structure; (b) PCB layout showing the sensor location; (c) FEA of the elastic structure.

Fig. 5: Calibration setup: (a) photograph of the calibration setup using the ATI MINI85, the proposed sensor, and a lever arm; (b) schematic diagram of the communication interfaces and experimental devices used for the calibration setup.

Fig. 6: Calibration experiment: (a) measured responses obtained by applying test weights five times in the negative and positive directions; (b) ADC outputs of each sensor as a function of the applied torque.

Fig. 7: Nonlinearity and hysteresis experiments: (a) upper- and lower-bound average curve and the corresponding linear fit used for nonlinearity evaluation; (b) magnified view of (a); (c) nonlinearity deviation as a function of the applied torque; (d) hysteresis deviation as a function of the applied torque.

Fig. 8: Crosstalk test and resolution experiments: (a) test setup for the crosstalk experiments, where weights were applied to generate moments about the $x$ - and $y$ -axes; (b) measured responses under the applied $x$ - and $y$ -axis torques; (c) sensor voltage variation under an $x$ -axis torque; (d) sensor voltage variation under a $y$ -axis torque; (e) noise level after calibration using the least-squares method; (f) noise level after calibration using the proposed method.

	Max Percentage Error(FS%)	RMS Error(N $\cdot$ m)	Nonlinearity(FS%)	Hysteresis(FS%)
z-axis Torque	0.0830	0.0266	0.111	0.139

TABLE III: Error Analysis, Nonlinearity and Hysteresis

Repeatability (FS%)	Crosstalk $T_{x}$ (FS%)	Crosstalk $T_{y}$ (FS%)
0.0849	0.148	0.126

TABLE IV: REPEATABILITY TEST AND CROSSTALK ANALYSIS

Fig. 9: Temperature compensation experiment: (a) temperature chamber and the sensor placement inside the chamber; (b) temperature-dependent drift and the corresponding rational fitting curve; (c) drift at zero external torque, before and after temperature compensation; (d) drift under an external torque of 10 N $\cdot$ m, before and after temperature compensation.

Sensor	MINI85	DYJN-104	Proposed Sensor	Proposed Sensor
			(Uncompensated)	(Compensated)
Zero Torque RMS Error(N $\cdot$ m)	0.0074	0.0442	0.0123	0.0091
10 N $\cdot$ m Torque RMS Error(N $\cdot$ m)	0.0061	0.0314	0.0326	0.0224

TABLE V: RMS Error of one hour drift

Fig. 10: Motor application experiment setup: (a) photograph of the application setup integrating the motor, the proposed sensor, and the load cell; (b) schematic diagram of the communication interfaces and experimental devices used for the motor application experiments.

Fig. 11: Step-response experiments. The light green shaded region indicates the results obtained using only the motor current sensor (without the torque sensor), whereas the gray shaded region indicates the results obtained using feedback from the proposed sensor: (a) 0.05 N $\cdot$ m; (b) 0.1 N $\cdot$ m; (c) 0.2 N $\cdot$ m; (d) disturbance-rejection test while regulating 0.5 N $\cdot$ m under an external disturbance.

Experiment	0.05 N $\cdot$ m	0.1 N $\cdot$ m	0.2 N $\cdot$ m	0.5 N $\cdot$ m with Disturbance
Current Sensor
RMS Error(N $\cdot$ m)	0.0329	0.0566	0.0685	0.1893
Torque Sensor
RMS Error(N $\cdot$ m)	0.0026	0.0026	0.0025	0.0058

TABLE VI: Step-Reference Tracking Test Analysis

Fig. 12: Sinusoidal and square-wave experiments: (a) sinusoidal test with an amplitude of 0.05 N $\cdot$ m at 0.2 Hz (current-sensor-only control); (b) square-wave test with an amplitude of 0.1 N $\cdot$ m at 1 Hz (current-sensor-only control); (c) sinusoidal test with an amplitude of 10 N $\cdot$ m at 0.2 Hz (current-sensor-only control); (d) proposed-sensor-based control for the test in (a); (e) proposed-sensor-based control for the test in (b); (f) proposed-sensor-based control for the test in (c).

Experiment	0.05 N $\cdot$ m Sinusoidal	0.1 N $\cdot$ m Square	10 N $\cdot$ m Sinusoidal
Current Sensor
RMS Error(N $\cdot$ m)	0.0685	0.0924	1.2022
Peak Error(N $\cdot$ m)	0.0505	0.1426	2.2611
Torque Sensor
RMS Error(N $\cdot$ m)	0.0205	0.0436	0.1520
Peak Error(N $\cdot$ m)	0.0301	0.0797	0.4244

TABLE VII: Sinusoidal and Square-Wave Tracking Performance

Fig. 13: Admittance control experiment: (a) low-torque, low-speed interaction, exhibiting mild damping behavior; (b) high-torque, high-speed interaction, exhibiting pronounced damping behavior.

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Sep 27, 2026J/K move · Enter open · S save

A Disk-Shaped Magnetoelastic Torque Sensor for Robotic Joints Using Permanent Magnetization

Bruno Brajon, Enrico Gasparin, Nicole Yazigy, Lisa Salamin+4

Melexis Technologies SA, Bevaix, Switzerland · CSEM SA, Neuchˆatel, Switzerland

Open paper AskPDF arXiv

Direct torque sensing is a growing need in the robotics community to enable precise control and interactions where torque estimation from motor current is not sufficient. This paper presents a novel disk-shaped magnetoelastic torque sensor with a compact axial envelope of about 1 cm, suitable for integration in robotic joints. A four-magnetometer architecture is used to measure the field modulated by the stress affecting a narrow magnetized region while rejecting the effects of parasitic cross forces. A custom-designed magnetic shield enhances the torque sensitivity while reducing the external stray fields by a factor 7x. The device measures the torque with an accuracy of 1.34 %FS relative to the 50 Nm full scale (FS). The paper details the development of the sensor through the mechanical design, the magnetization procedure, and the experimental validation. The results demonstrate the potential of the proposed sensor for robotic applications.

Abstract

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