A simple method for calibrating force plates and force treadmills using an instrumented pole

Abstract

We propose a new method for calibrating force plates to reduce errors in center of pressure locations, forces, and moments. These errors may be caused by imperfect mounting of force plates to the ground or by installation of a treadmill atop a force plate, which may introduce distorting loads. The method, termed the Post-Installation Least-Squares (PILS) calibration, combines features of several previous methods into a simple procedure. It requires a motion capture system and an instrumented pole for applying reference loads. Reference loads may be manually applied to the force plate in arbitrary locations and directions. The instrumented pole measures applied load magnitudes through a single-axis load cell, and load directions through motion capture markers. Reference data and imperfect force plate signals are then combined to form a linear calibration matrix that simultaneously minimizes mean square errors in all forces and moments. We applied the procedure to standard laboratory force plates, as well as a custom-built, split-belt force treadmill. We also collected an independent set of verification data for testing. The proposed calibration procedure was found to reduce force errors by over 20%, and moment errors by over 60%. Center of pressure errors were also reduced by 63% for standard force plates and 91% for the force treadmill. The instrumented pole is advantageous because it allows for fast and arbitrary load application without needing a precise fixture for aligning loads. The linear calibration matrix is simpler than nonlinear correction equations and more compatible with standard data acquisition software, yet it yields error reductions comparable to more complex methods.

Introduction

Accurate measurements of ground reaction forces from force plates are important in many areas of biomechanics research. For example, inverse dynamics analysis is highly sensitive to errors in force magnitude and center of pressure (COP) location [1], [2]. Force plates are also used to assess foot placement and its variability during human walking (e. g., [3]), again with a high sensitivity to error. Some errors may be due to misalignment of the force plates with respect to the motion capture coordinate system. Others may be caused by distortions in load-sensing elements due to imperfect mounting conditions [4]; a typical specification requires the mounting surface to be flat within 0.05 mm to ensure low distortions (AMTI, Watertown, MA). Instrumented treadmills, in which treadmills are mounted atop force sensing elements, are becoming increasingly prevalent in biomechanical analysis [5], [6], [7], [8], [9]. These tend to exacerbate accuracy concerns, not only because the mounting of the treadmill introduces additional distortions, but also because the compliance and dynamics of the mechanical structures reduce the fidelity of load measurements. These factors can lead to substantially large errors compared to original manufacturer specifications. The errors can potentially be reduced, however, through proper calibration performed after mounting the force plate and any optional structures, such as a treadmill.

Errors may be quantified by comparing force plate measurements such as center of pressure (COP) against external reference values such as those obtained from a motion capture system [10], [11]. The reference COP location may easily be obtained from motion capture markers fixed to a rigid rod or pole, whose tip may be pressed against the force plate from arbitrary orientations and in arbitrary locations. Force magnitude errors may be assessed with a pole that is instrumented with a load cell [12].

Calibration methods adjust inaccurate force plate measurements based on their mismatch with reference values. Correction equations, whether linear or nonlinear in form, are characterized by unknown parameters or coefficients whose values are derived from observed errors, typically through optimization techniques. A particularly simple method is to apply purely vertical loads to the force plate in known locations. Verkerke et. al. [13] demonstrated that COP errors could thus be reduced substantially with a set of 10 coefficients in a polynomial correction formula. Rabuffetti et al. [14] showed that the loads need not be applied solely in the vertical direction, so long as the motion of the pole about the point of contact is recorded. It is further possible to correct not only COP, but also the other force and moment data sensed by the force plate [15]. This requires that loads be applied along other axes. Black et al. [15] developed a custom-built test fixture for that purpose, capable of applying static loads along precise, externally-referenced axes. They optimized 36 nonlinear correction coefficients to reduce COP errors.

The various merits of these procedures might be combined into a single procedure. The application of loads to the pole in arbitrary directions [14] could be extended to provide reference data for not only COP location but also the applied forces, using an instrumented pole [12]. Doing so would allow for calibration of force and moment measurements about all axes as with [15], but without requiring a specialized fixture. Interesingly, the combination of the advantages from [12], [13], [14], [15] would not require an increase in complexity. A procedure for calibrating all force plate axes could use fairly arbitrary loads applied in a number of locations, followed by a simple least-squares optimization of the appropriate correction coefficients.

We propose a simplified procedure that utilizes an instrumented pole and a least-squares optimization of a linear model to quickly calibrate force plates along all axes and over the full range of expected loading conditions. The corrections can reduce the effects of misalignment and distortion, and improve the accuracy of force, moment, and COP measurements. We term the proposed procedure the “Post-Installation Least-Squares” (PILS) Procedure, as it is intended to correct for errors introduced during installation of force plates. We demonstrate the PILS Procedure in two common applications: calibration of standard ground-embedded force plates, and calibration of a custom-built, split-belt, instrumented treadmill.