Tan, Clévy and Chaillet (Micromachines, 2015) calibrated a 1-DoF nanopositioning stage with a static polynomial-plus-thermal model and an adaptive scheme that updates compensation by periodically revisiting an absolute reference point they call the KiloMeter-Zero (KMZ), improving positioning accuracy from the micrometer range to about 400 nm (static) and below 100 nm (adaptive) in five-day tests, and from 1400 nm to 200 nm on a 2-DoF microrobot. However, the KMZ drift measurement was taken with an external laser interferometer, the static model loses validity once ambient temperature leaves the training range, and the framework assumes quasi-static operation. The paper's explicit future work is the implementation of KMZ drift updates with integrated low-cost on-board sensors; the paper additionally notes, at dispersed points, three related issues: the optimal number and placement of temperature sensors at microscale; principled design of the training temperature range and training procedure so that static-model compensation survives later ambient excursions; and the quasi-static scope of the validated framework, with a combination of planning-layer calibration and inner-loop dynamic control proposed but not developed. This record states the integrated problem anchored in that future work: a sensor-integrated calibration and adaptation architecture that sustains the demonstrated 100-nanometer-class ISO 9283 positioning accuracy over long-term operation, including ambient-temperature excursions beyond the training range, without continuous external metrology. No published resolution of these issues in the microscale nanopositioning context is known to date.

Background

A nanopositioning stage is a precision motion platform — typically a flexure-based compliant mechanism (a structure that moves by elastic deformation of thin flexures rather than bearings) driven by piezoelectric stack actuators — producing displacements with nanometer resolution over travel ranges of tens to hundreds of micrometers. Such stages are the motion backbone of microrobotics: micromanipulation and microassembly, mechanical characterization of biological samples, and scanning-probe or photonics-alignment tasks all require controlling the relative position of a tool and a sample to tens of nanometers. Two error sources dominate. First, geometric errors: the mapping from commanded to actual position deviates from ideal because of manufacturing tolerances and actuator nonlinearities such as hysteresis and creep of piezoelectric ceramics. Second, thermal drift: slow displacement of the whole structure caused by ambient temperature variation and self-heating; at microscale, a fraction of a degree Celsius can shift the stage by hundreds of nanometers.

Positioning performance is quantified following ISO 9283:1998, the manipulator performance standard that defines positioning accuracy (deviation of the mean attained position from the commanded target at each pose, over repeated cycles) and repeatability (the scatter of attained positions around their mean). Because sensors that directly measure tool-sample relative position over a full workspace are difficult and costly to integrate, one strategy keeps the manufacturer's actuator-level closed loop (a PID, i.e. proportional-integral-derivative, controller on an internal capacitive sensor) and corrects the commanded input at the planning layer using a calibrated inverse model — the open-loop, or calibration, approach. This requires a model capturing both error sources.

Tan, Clévy and Chaillet (2015), extending their 2013 ICRA study of a single-axis stage under thermal disturbance, formulated such a calibration for a Physik Instrumente P-625.1CD stage (500 micrometer stroke) monitored by four K-type thermocouples and a laser interferometer (an optical displacement sensor resolving 0.3 nm) as the external reference. Their static model combines a polynomial geometric term with a first-order linear thermal term,

q=∑i=1n1aixn1−i+1+∑i=1n2biti+λ,q = \sum_{i=1}^{n_1} a_i x^{n_1-i+1} + \sum_{i=1}^{n_2} b_i t_i + \lambda,

where xx and qq are measured and commanded positions, tit_i are the readings of the n2n_2 temperature sensors, and the coefficients are identified by stepwise regression (a statistical variable-selection procedure that retains the most influential terms). Because thermal drift at microscale is largely position-independent — all points of the workspace shift nearly equally — they also proposed an adaptive model based on the KiloMeter-Zero (KMZ): an absolute reference point that the stage periodically revisits to measure the accumulated drift, whose value λKMZ\lambda_{KMZ} then updates the compensation over the whole stroke, a compromise between open-loop and closed-loop control at the planning layer. In five-day experiments the uncalibrated stage reached about 1 micrometer of error; the static model held roughly 400 nm while ambient temperature stayed inside the training range but degraded once it left it (after about 70 hours of the test); the KMZ adaptive model stayed below 100 nm throughout. The calibration also transferred to a 2-DoF microrobot, improving 2D accuracy from 1400 nm to 200 nm.

The demonstrated performance, however, depends on conditions that do not hold in deployment. The KMZ drift measurement was taken with the external interferometer; the static model's validity is tied to the temperature range seen during training, with no principled procedure for choosing that range; the framework assumes quasi-static operation; and even the temperature sensing feeding the static model is ad hoc — the authors note that several thermocouples are needed, that their minor reading differences matter at microscale, and that 'so far, nobody has succeeded in defining where the good locations are to be measured'. At the macroscale, machine-tool thermal-error compensation has developed systematic temperature-variable selection and sensor-placement methodology, from grouping-based selection in the late 1990s to digital-twin-based placement (using a simulated thermal replica of the machine) in 2025; but those methods target identification accuracy on machine tools, not the microscale regime in which the sensors themselves are intrusive relative to the stage and the effects to be identified are nanometers. The 2015 conclusions name the integration of low-cost, high-resolution, small-range on-board sensors into the stage as future work, and the paper's robustness discussion proposes combining manufacturer-side geometric calibration with user-side KMZ updates from internal low-cost sensors.

Problem Statement

The problem, as left open by Tan, Clévy and Chaillet (2015), is to make calibration-based (open-loop at the planning layer) thermal-drift compensation of micro-/nanopositioning stages deployable in real operation: to sustain, without continuous external metrology, the accuracy class the source paper demonstrated only with an interferometer as reference — maximum per-pose ISO 9283 positioning accuracy of order 100 nm over multi-day operation, versus about 1 micrometer uncalibrated and 400 nm for the static model within its training range. The source paper's explicit future work defines the core requirement: the design and experimental validation of an on-board, low-cost, high-resolution, small-range sensor integrated into the stage that implements periodic KiloMeter-Zero drift updates — the adaptive mechanism of the source paper, which there relied on an external laser interferometer — so that the demonstrated accuracy class is preserved over long-term operation. The sensor's 'low-cost', 'high-resolution' and 'small-range' qualifiers are relative to the external laser-interferometer metrology that the on-board update replaces: the sensor must occupy and cost substantially less than that metrology, resolve below the accuracy class stated above, and need span only the local drift about the KiloMeter-Zero rather than the full stage stroke.

Three related issues are discussed in the source paper and would be engaged by a deployable architecture, though the paper does not itself formulate them as part of its future work. Issues (a) and (b) are part of the target, whereas (c) is a related extension rather than part of it:

(a) a principled criterion and method for choosing the number and locations of temperature sensors on and around a microscale stage, where nanometer-level drifts, few usable sensing locations, and the physical intrusion of the sensors themselves constrain identification quality;

(b) a principled way to design the training procedure — including the temperature range and conditions — such that the static (polynomial-plus-linear-thermal) model's compensation performance under later ambient-temperature excursions is predictable, or a defensible criterion for when static calibration must be replaced by adaptive updating;

(c) an extension of the drift-compensating calibration beyond quasi-static positioning: the source paper proposes, but does not develop, combining planning-layer calibration-based accuracy control with inner-loop dynamic control of the transient part, the two being 'fully complementary'.

The target is a sensor-integrated architecture demonstrated experimentally on physical micro-/nanopositioning stage(s): sustained ISO 9283 positioning accuracy of the class demonstrated in the source paper (order 100 nm maximum per-pose accuracy) over at least one multi-day validation spanning ambient-temperature excursions beyond the training range, with compensation during operation driven only by the on-board sensors and the selected temperature sensors.

References

  1. Ning Tan, Cédric Clévy, Nicolas Chaillet, 'Calibration of Nanopositioning Stages', Micromachines 6(12):1856-1875, 2015. DOI: 10.3390/mi6121461. https://doi.org/10.3390/mi6121461 (source paper)
  2. Ning Tan, Cedric Clevy, Nicolas Chaillet, 'Calibration of single-axis nanopositioning cell subjected to thermal disturbance', 2013 IEEE International Conference on Robotics and Automation (ICRA), pp. 3660-3665, 2013. DOI: 10.1109/ICRA.2013.6631091. https://doi.org/10.1109/ICRA.2013.6631091
  3. Houari Bettahar, Olivier Lehmann, Cedric Clevy, Nadege Courjal, Philippe Lutz, '6-DoF Full Robotic Calibration Based on 1-D Interferometric Measurements for Microscale and Nanoscale Applications', IEEE Transactions on Automation Science and Engineering 19(1):348-359, 2022. DOI: 10.1109/TASE.2020.3031856. https://doi.org/10.1109/TASE.2020.3031856
  4. Jae Wan Kim, Jong-Ahn Kim, Sung Hoon Eom, 'Calibration of 3 DoF microstages with XYΘ sensors', Journal of Physics: Conference Series 1065:152001, 2018. DOI: 10.1088/1742-6596/1065/15/152001. https://doi.org/10.1088/1742-6596/1065/15/152001
  5. Yindi Cai, Zeguang Zhang, Dianpeng Zhang, Ruyue Wang, Kuang-Chao Fan, 'Micro/nanopositioning coplanar stage with real-time compensation of volumetric error', Measurement 229:114424, 2024. DOI: 10.1016/j.measurement.2024.114424. https://doi.org/10.1016/j.measurement.2024.114424
  6. Chih-Hao Lo, Jingxia Yuan, Jun Ni, 'Optimal temperature variable selection by grouping approach for thermal error modeling and compensation', International Journal of Machine Tools and Manufacture 39(9):1383-1396, 1999. DOI: 10.1016/S0890-6955(99)00009-7. https://doi.org/10.1016/S0890-6955(99)00009-7
  7. Sebastian Lang, Mario Zorzini, Stephan Scholze, Josef Mayr, Markus Bambach, 'Sensor placement utilizing a digital twin for thermal error compensation of machine tools', Journal of Manufacturing Systems 80:243-257, 2025. DOI: 10.1016/j.jmsy.2025.03.003. https://doi.org/10.1016/j.jmsy.2025.03.003
  8. Yuliang Wang, Huimin Wang, Shusheng Bi, 'Real time drift measurement for colloidal probe atomic force microscope: a visual sensing approach', AIP Advances 4(5):057130, 2014. DOI: 10.1063/1.4880242. https://doi.org/10.1063/1.4880242