The two-step calibration method of He, Gao, and Bai (Measurement, 2013) identifies the kinematic parameters of a dual-actuated mechanical servo press with parallel topology in two stages — active chains from moving-platform measurements, then passive chains from slider measurements — and cut the press's root-mean-square slider position error by 82%. Its experimental validation, however, rested on four simplifying assumptions: perfectly synchronized bilateral crank inputs, a perfectly symmetric bilateral structure (only left-side parameters identified, moving-platform x-coordinate fixed at zero), purely kinematic error sources (no joint friction, backlash, or clearance), and a planar 2-degree-of-freedom prototype. The open problem is to extend the methodology beyond these assumptions: simultaneous identification of both sides' parameters under structural asymmetry, inclusion of actuator non-synchronization in the error model, incorporation of non-kinematic errors, and experimental demonstration beyond the planar prototype, including on higher-degree-of-freedom multi-actuated presses. The problem remains open, with mainly local engineering significance.

Background

A mechanical servo press is a stamping (metal-forming) press whose slide, or ram, is driven by servo motors rather than a fixed-speed flywheel drive, so the slide's motion profile over the stroke is programmable; the working region near bottom dead center (BDC, the lowest point of the slide stroke) demands high positional accuracy. For very large press capacities a single motor cannot supply the torque, so presses are built multi-actuated: several servo motors drive one shared linkage mechanism. In the press family studied here, the mechanism has parallel topology — several linkage chains connect the frame to a common moving platform — and the actuation is redundant, meaning more actuators than the mechanism has degrees of freedom (DoF, the number of independent coordinates needed to specify its configuration). The press decomposes into active chains (the motor-driven parallelogram linkages that position a moving platform) and passive chains (unactuated force-amplifier linkages that transmit platform motion to the slider). He, Gao, and Zhang (2013) analyzed this architecture and explicitly studied a non-synchronous working mode, confirming that the two cranks can run out of phase in practice.

Kinematic calibration is the standard remedy for manufacturing and assembly tolerances: the actual geometric parameters (joint mounting positions on the frame, link lengths) deviate from design values, and calibration identifies the true values from measurements — here taken with a laser tracker, a portable interferometric instrument that measures the three-dimensional coordinates of a moving retroreflector with sub-millimeter accuracy. The identification solves a first-order error model obtained by totally differentiating the mechanism's loop-closure constraint equations (the algebraic equations expressing that each closed kinematic loop must close); the resulting sensitivity matrix relates input-angle errors and geometric parameter deviations to output (platform and slider) position errors, and linearly dependent parameters must be excluded to keep the calibration matrix full rank. He, Gao, and Bai (2013) proposed a two-step version of this for the multi-actuated press: first identify the active-chain parameters from moving-platform measurements, then identify the passive-chain parameters from slider measurements using the already-calibrated platform positions. On a dual-actuated planar 2-DoF prototype, the root-mean-square (RMS) platform error fell from 4.769 mm to 0.9954 mm and the slider error from 10.908 mm to 1.968 mm, an 82% improvement, with error sensitivity to geometric parameters shown to decrease near the working area.

The validation rested on idealizations the authors themselves flagged. The two servo motors were assumed perfectly synchronized, so the crank angles φ11\varphi_{11} and φ21\varphi_{21} were treated as identical at each commanded configuration and any non-synchronous discrepancy was left as an unmodelled input absorbed by the least-squares fit. The bilateral structure was assumed perfectly symmetric, so only left-side parameters were identified, the x-coordinate of the moving-platform center was fixed at zero, and both full-pose information and right-side geometry were discarded. Only kinematic error sources were modeled: joint backlash (the lost motion when contact between mating joint surfaces reverses direction), joint clearance (unwanted free play arising from the small gap between a joint pin and its mating bearing), friction, and link compliance were excluded. Finally, the demonstration was confined to a planar 2-DoF press. Generic remedies for pieces of this exist in neighboring literatures — calibration of non-fully-symmetric parallel robots (e.g., Shen et al., 2021), non-geometric error modeling and compensation in robot calibration (e.g., Luo et al., 2026), and synchronization control of multi-motor servo drives with backlash and friction (e.g., Zhao et al., 2016) — but none has been combined into, or validated for, the two-step active/passive-chain calibration of multi-actuated servo presses.

Problem Statement

Extend the two-step kinematic calibration methodology for multi-actuated mechanical servo presses with parallel topology so that it no longer relies on the idealizations under which it was validated, and specify the experimental validation required to establish the extended methodology. Concretely, relative to the 2013 baseline of He, Gao, and Bai: (1) formulate and identify an error model in which the geometric parameters of both bilateral chains (left and right active and passive chains) are identified simultaneously, without assuming perfect left–right structural symmetry or fixing the moving-platform x-coordinate to zero, resolving the identifiability and rank-deficiency problems that this full identification entails; (2) incorporate the non-synchronous error between the bilateral crank inputs (the discrepancy between φ11\varphi_{11} and φ21\varphi_{21}, whether from encoder offsets, controller timing mismatch, or motor dynamic differences) into the error model, so that actuator desynchronization is measured or estimated rather than silently absorbed into the geometric parameter estimates; (3) extend the error model beyond purely kinematic sources to include non-kinematic errors such as joint clearance, backlash, and friction (and link compliance where relevant), establishing which of these are identifiable from press pose measurements and how they enter the error budget; and (4) specify the experimental validation that would demonstrate the extended methodology beyond the original planar dual-actuated 2-DoF prototype, on multi-actuated servo press hardware of the same parallel-topology family, with extension to higher-degree-of-freedom press architectures as an optional further target. The common objective uniting these components is an extended calibration methodology, applicable to press hardware of the same family, that recovers the manufactured press's true geometry and dominant error sources — and hence delivers the slider positioning accuracy required for precision stamping near bottom dead center — without the symmetry, synchronization, and purely-kinematic idealizations of the 2013 demonstration.

References

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