**26**

*Apr*

### fbg-based Force-Torque Sensor – Fundamentals of the sensor

The presented force-torque sensor makes use of fiber Bragg gratings (FBG) as optical strain sensors. These devices are written into the core of optical single-mode fibers. The FBG consists of a periodic modulation of the refractive index in the core of the fiber. The length of a FBG is usually several millimeters. From the periodic modulation a narrow optical reflection-band at the Bragg-wavelength ?_{B} is induced. Since the wavelength of this reflection band is depending on temperature and strain, both quantities may be measured relatively to a calibration value by measuring the wavelength of the optical reflection. The wavelength shift upon strain is approximately 1.2 pm per µm/m. Using the small bandwidth of a single FBG, several FBGs may be multiplexed in a single fiber by choosing the Bragg- wavelength differently for each sensor. In this work, we make use of this wavelength division multiplexing capability.

## Mechanical transducer structure

Force torque sensors are built from a mechanical transducer structure which is installed at the position where forces and torques are to be determined. The installation breaks the flow of forces and guides it through the transducer structure, leading to a position dependent stress tensor ?(r). The stress leads to strains ?(r) and a deformation or displacement field **u**(**r**) of the structure. The strains are then conventionally measured by resistive strain gauges. To determine all six mechanical degrees of freedom namely the forces **F** and the torques **M** requires at least six linearly independent measurements for their reconstruction.

A transducer structure which fulfills the independency requirement is the Steward Platform (Kang 2011), consisting of a top plate a bottom plate and six legs connecting them. When the strains of the six legs are measured, the forces and torques can be reconstructed. This necessitates, however, that the strain in each leg is homogeneous along the length of the strain sensor, otherwise it will lead to some sort of integral value. A special problem arises when fiber Bragg gratings are used as strain sensors. If an inhomogeneous strain field is applied to the sensor, the spectral response will change in wavelength and in shape, leading to measurement error.

The transducer structure is mounted at two positions; conventionally the top and bottom side. If the two connections to the force/torque sensor are modeled as very rigid, their deformation will be zero. If additionally a linear materials law (Hook`s law) is assumed then an increase in force or torque will lead to a proportional increase in stresses which in turn leads to a proportional increase in the corresponding strain and displacement field, which is just the property of a linear spring. This allows introducing a simplified model of a force torque sensor as depicted in Figure 2.

The figure demonstrates how the transducer structure may be thought of as consisting of very rigid upper plate which is connected to a very rigid lower plate by a linear spring. Between the top plate and the bottom plate, the FBG sensors are placed in the model. Due to a deformation of the spring, the top plate will move relative to the bottom plate, thereby straining the FBGs.

The displacement of the top plate has all degrees of freedom of a rigid body which are three translational motions **U** and three rotatory motions ?. Summarizing them in a vector **R** = {**U**^{T}, ?^{T}}^{T}, and summarizing the forces and torques in the vector **f** = {**F**^{T}, **M**^{T}}^{T} the model is as follows:

**R = K **x **f**

where K is the spring´s stiffness matrix. In general the transducer´s stiffness matrix is difficult to obtain in an analytical way. This is only possible for very simple structures. For more complicated structures a finite element simulation may be employed.

## Direct measurement of displacement

The approach followed in this work is to directly measure the displacement of the rigid body motion of the top plate by fiber Bragg gratings. These gratings are broad between top and bottom plate and are thus not bonded to the structure in the region of the sensor. This ensures a homogeneous strain field along the FBG as no structural strains from the transducer structure may couple to the fiber. The displacement field of the top plate is given by

**u**(**r**) = **U** + **r** x ?

Positioning of the fiber Bragg gratings in the displacement field must ensure the linear independency of all sensor responses. With the above assumption of a homogeneous strain distribution the *i*th fiber Bragg grating measures a longitudinal strain ?_{zz.i} Assuming only small motions of the top plate, the strain of the *i*th sensor, fixed at position **r** is as follows.

where **e*** _{i}* is a vector parallel to the

*i*lh fiber Bragg grating of unity length,

*l*is the length of the fiber that is braced between top and bottom plate. By writing all strains as a vector, XXXXX the response of the force torque sensor may be summarized in the compliance matrix C (Svinin and Uchiyama 1995) leading to

_{i}

f = C x ?

## Stiffness matrix

The compliance matrix may either be obtained from the calculated stiffness matrix of measured directly by applying the individual loads of **f** and thus obtaining the inverse of C column by column. The inversion poses a problem, if C is badly conditioned leading to high sensitivity to calibration errors. Therefore, a well-conditioned sensor has to be a design goal. Since the design goal may also include having different sensitivities or maximum loads of **f** and, it is convenient to introduce the normalized compliance matrix

where N? = diag{?* _{i}*} and N

*= diag{*

_{f}**f**

*}. With the normalized compliance matrix a measure for the conditioning of the sensor may be determined from the condition number of C*

_{i}*which is the quotient of its largest and smallest singular value. For a well-conditioned sensor it should be close to 1.*

_{N}## Minimum bending radius

The construction of a small volume fiber optic force/torque sensor comes with the severe limitation of minimal bending radius. A simple electrical wire, as the ones that may be used for connecting resistive strain gauges, can easily be bent around a diameter of one millimeter or so. In contrast, an optical glass fiber will break within a few seconds (Lodi et al. 1997) at that bending diameter. A second issue, bending loss, causing attenuation of the optical signal, is of minor importance. At bending diameters that allow reasonable lifetimes (~5 mm), attenuation of commercially available sensor fibers is insignificant, considering the lengths of the fiber bent at that diameter.