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Separation method of bending and torsion in shape sensing based on FBG sensors array

Xinhua Yi, Xiangyan Chen, Hongchao Fan, Fei Shi, Xiaomin Cheng, and Jinwu Qian

Open Access Open Access

Abstract

This paper presents a theoretical method for separating bending and torsion of shape sensing sensor to improve sensing accuracy during its deformation. We design a kind of shape sensing sensor by encapsulating three fibers on the surface of a flexible rod and forming a triangular FBG sensors array. According to the configuration of FBG sensors array, we derive the relationship between bending curvature and bending strain, and set up a function about the packaging angle of FBG sensor and strain induced by torsion under different twist angles. Combined with the influence of bending and torsion on strain, we establish a nonlinear matrix equation resolving three unknown parameters including maximum strain, bending direction and wavelength shift induced by torsion and temperature. The three parameters are sufficient to separate bending and torsion, and acquire two scalar functions including curvature and torsion, which could describe 3D shape of rod according to Frenet-Serret formulas. Experimental results show that the relative average error of measurement about maximum strain, bending direction is respectively 2.65% and 0.86% when shape-sensing sensor is bent into an arc with a radius of 260 mm. The separating method also applied to 2D shape and 3D shape of reconstruction, and the absolute spatial position maximum error is respectively 3.79mm and 11.10mm when shape-sensing sensor with length 500mm is bent into arc shape with a radius 260mm and helical curve. The experiment results verify the feasibility of separating method, which would provide effective parameters for precise 3D reconstruction model of shape sensing sensor.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In recent years, shape sensing has offered significant advantages and demonstrated great promise in terms of technological advances and extensive clinical application, which has been wildly applied in the research of continuum robots including otolaryngology, abdominal surgery, and intravascular interventions, particularly for endoscopy [1–3]. Shape sensing can provide the location and trajectory tracking of medical devices in the interventional operations, which is very useful to track any deviation from the planned trajectory to minimize positioning error and procedural complication during the subsequent manipulation [4]. Shape sensing can reconstruct 3D shape and sense large deflection of continuum dexterous manipulators, as we did in our earlier study [5]. We applied it to the endoscopy and provided a real time visualization for endoscopist to decrease pain and danger to the patient. Because continuum robots have a fundamentally different structure than conventional manipulators composed of discrete rigid links connected by joints [6–8], model-based research on kinematics and mechanics remains challenging to implement accurate intraoperative shape sensing of continuum robots [9–11]. Considering the size limitation of continuum robot as medical tools, and not to affect the mechanical properties of medical tools, shape sensing sensor should not be too large in size, especially to meet with very small diameter requirement. FBG based shape reconstruction techniques could be a good choice since FBG-based sensing techniques are capable of providing shape estimation without requiring the kinematics-based modeling, and it can help to achieve precise and reliable motion control of continuum robots used in these surgical procedures requiring accurate and real-time shape sensing [12–13].

At present, various manufacturing techniques of FBG shape sensing sensor explored in the literatures are related to the choice of substrate material, geometry, number of fibers. The commonly used fiber configurations based on FBG sensors array include orthogonal and triangular configuration by investigating the applications of FBG-based shape reconstruction presented in the researches [14–16]. The obvious difference about orthogonal and triangular configuration is the solution to curvature and torsion, in addition the different number and layout of FBGs in the cross-section. The solution about orthogonal configuration applies the calculation formula of right triangles directly to obtain maximum strain and bending direction, which is not very robust and susceptible to be affected by the ambient temperature, packaging error. Although some orthogonal configuration of four fibers supported the temperature compensation and curvature-based shape measurement, but suffered from poor resolution to determine the twist angle. FBG sensors array with a triangular configuration could establish a nonlinear matrix equation and use an optimization method to get solution, which supports temperature compensation, removes noise and allows for measurement of torsion. The triangular configuration has been validated to achieve better accuracy for shape estimation in comparison to the orthogonal configuration.

FBG-based methods of shape reconstruction mainly focus on using curvature-strain model, which is associated with the strain derived from wavelength shift [17–18]. According to Frenet-Serret theory [19–20], torsion and curvature of rod are important parameters indirectly used to describe geometric properties of rod when some forces or force-couples act upon it. In 2D space, the shape of rod can be described mathematically by a scalar variable, which is a curvature function along central axis of rod. However, two scalar functions including curvature and torsion along central axis of rod are required to describe the shape of rod in 3D space. Obviously, it is not enough to reconstruct accurately the shape of rod only using curvature-strain model for 3D rod if not taking into account the torsion. If curvature and torsion functions of rod could be acquired using shape sensing sensor, Frenet-Serret formulas can be employed in reconstructing shape of rod. Here, we need to set up a method to separate curvature and torsion by strain information measured from FBG sensors array. Charles el.at used four-core optical fiber to sense bend and twist [21], of which, the three outer cores are used to measure twist and bend, and the central core serves as a reference for axial strain and temperature changes since it does not respond to bending or twist if located on the fiber's centerline. Actually, it is very difficult to locate the central core on the fiber’s centerline, and then affect the measurement. Ran Xu el.at developed a novel helically wrapped FBG sensor and use force-curvature-strain model on Cosserat rod theory to provide simultaneous curvature, torsion, and force measurement [22].

The contribution of this paper is to develop a separation method of bending and torsion, which uses strain-curvature and strain-torsion model to establish a nonlinear matrix equation and use non-linear least squares analysis solving three parameters including maximum strain, bending direction, wavelength shift induced by torsion and temperature. The first two parameters are employed to acquire curvature and torsion at the detection point, the last parameter is used to distinguish the effect of temperature on wavelength. The discrete curvature and torsion are used to reconstruct 2D and 3D curve based on discrete Frenet-Serret theory. Section II presents FBG sensors array design, derives the relationship between strain and bending, and figures out the mathematical model about strain distribution induced by twist in different twist rate and different helical-angle. The shape reconstruction model is also described in Section II. Results and errors of using the separation method to reconstruction 2D and 3D curve are provided in section III as shape sensing sensor is deformed in a series of known curves, and conclusions appear in section IV.

2. Principal of shape sensing sensor

Shape sensing technology is based on wavelength division multiplexing (WDM) using the basic network topology, and provides a means of measuring distributed wavelength shift of FBGs. A FBG has unique characteristics to perform as a strain sensor because the deformation of the optical fiber leads to a change in the period of the microstructure and, consequently, of the Bragg wavelength, so we can use the wavelength shift to monitor distribution strain along the direction of shape sensing sensor. In general, the strain is mainly induced by twist and bend for the elongated rod, if we can distinguish the axial twist and bending through the strain information, shape sensing can be performed by monitoring distributed strain and convert measured distributed strain to three-dimensional distributed position.

FBG sensors array as an important part of shape sensing sensor is composing of substrate, optical fiber, coating layer and encapsulation layer. The shape memory alloy based on NiTi material is chosen as a substrate due to its stability, practicability and superior thermo-mechanic performance. Three acrylate coated SMF-28e optical fibers with the diameter 250 µm are bonded around SMA wire using AB adhesive at every 120 degrees, and the diameter of SMA wire is about 0.75 mm. On every fiber, there are five Bragg Gratings with the length 5 mm and the spacing between two neighbor gratings is about 100mm±0.1 mm. The packaging diameter of FBG sensor is 1.5 mm. The triangular configuration of FBG sensors array is showed in Fig. 1. To satisfy wavelength shift scope of every FBG for changing of temperature and strain, and guarantee the wavelength not to overlap between two neighboring FBGs, FBG wavelength shift is set in range of about 8 nm. The center wavelength of five gratings in one optical fiber is respectively set to 1524 nm, 1532 nm, 1540 nm, 1548 nm, 1556 nm, and the actual deviation of central wavelength is about ±0.3 nm for every grating.

 figure: Fig. 1.

Fig. 1. Diagram of FBG sensors array.

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For shape sensing sensor, one of the key techniques is how to use strain information to judge the states of twist and bend of measured rod when it is deformed under certain force or torque. The following analysis can help us to understand it. Under bend, each of FBG sensor experiences alternating state of tension or compression along the center axis of substrate. The center axial of substrate, which is precisely located in the center of sensors, experiences no strain. The three outer FBGs exhibit sinusoidal strain responses in the same cross section, and each phase difference is 120 degrees. By comparing the amplitude and phase of these three strain curves, we can determine the applied bend radius and its direction relative to the reference direction. When the substrate of shape sensing sensor is twisted in the direction of the helix, the tensile strain is applied to three outer gratings. Under twist, all of the three FBGs sensors experience a common-mode strain in the same cross section.

2.1 The curvature-strain model

One of factors affecting the shape of rod is bending direction and bending radius, the following question is how to acquire the relation between strain and bending. Figure 2(a) shows distribution of triple FBGs in one cross section. When bending curvature of the cross section is ${c_b}$, FBG-a, FBG-b and FBG-c are respectively under tension or compression, and the strain value can be calculated by bending radius. The maximum strain of the cross section is,

$${\varepsilon _{bm}} = \frac{{{r_f} + {r_s}}}{{{r_b}}} = ({r_f} + {r_s}) \cdot {c_b}$$
where, ${r_f}$, ${d_f}$ is respectively the radius and diameter of fiber, ${r_s},{d_s}$ is respectively the radius and diameter of substrate, ${r_b}$ is bending radius of the section and ${\varepsilon _{bm}}$ is the maximum bending strain of the section.

 figure: Fig. 2.

Fig. 2. (a) Distribution of triple FBGs in one cross-section. (b) Diagram of FBG bending strain.

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The strain of every FBG can be derived according to the diagram of FBG sensors as Fig. 2(b) shows. If the radial direction of FBG-b is set to reference direction, so the strain of FBG-a, FBG-b and FBG-c can be obtained,

$$\left\{ \begin{array}{l} {\varepsilon_b} = \textrm{-1} \cdot {\varepsilon_{bm}}\cos \beta \\ {\varepsilon_a} = \textrm{-1} \cdot {\varepsilon_{bm}}\cos \alpha ,\,\,\alpha = \beta - \frac{2}{3}\pi \\ {\varepsilon_c} = \textrm{-1} \cdot {\varepsilon_{bm}}\cos \psi ,\,\,\psi = \beta \textrm{ + }\frac{2}{3}\pi \end{array} \right.$$
where, ${\mathrm{\beta} } \in ({ - {{\mathrm{\pi}} }\sim {{\mathrm{\pi}} }} ),\; \alpha ,\beta ,\Psi$ is respectively the angle about radial direction of FBG-a, FBG-b and FBG-c relative to bending direction, ${{{\mathrm{\varepsilon}} }_a},{{{\mathrm{\varepsilon}} }_b},{{{\mathrm{\varepsilon}} }_c}$ is respectively the strain that FBG-a, FBG-b, FBG-c induced by bending.

Ideally, the strain of each FBG sensor induced by bending is offset from the others’ by 120° in phase, and normalized waveforms of the instantaneous strain in the same section with bending angle increasing is shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Normalized waveforms of strain for FBG sensors in the different bending direction.

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2.2 The torsion-strain model

The torsion always exists for a 3D space curve, and the model of pure torsion has been described in many literatures for a flexible shaft of uniform circular cross section along its length [23]. In the paper we concerned that the strain induced by twist which is different from shear strain. To better measure the strain suffered from twist, FBG sensor should always be bonded to the surface of substrate helically as Fig. 4(a) shows. The deformation of grating is showed in Fig. 4(b), and the geometric relationship between the length of FBG and shear strain can be written as,

$$l^{\prime} = \sqrt {{{(\textrm{2}\pi r + \textrm{2}\pi r \cdot \tan \alpha \cdot \tan \gamma )}^2} + {{(\textrm{2}\pi r \cdot \tan \alpha )}^2}}$$
where, ${\mathrm {\gamma} }$ is shear strain, ${\mathrm {\alpha} }$ is packaging angle relative to the circumferential direction, $l^{\prime}$ is the deformation length, and r is the distance between the central axis of substrate and the central axis of fiber.

 figure: Fig. 4.

Fig. 4. (a) Packaging diagram of FBG Sensor. (b) Geometric relationship about FBG deformation and twist angle.

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The strain under twist can be expressed as,

$${\varepsilon _t} = \frac{{l^{\prime} - l}}{l} = \frac{{l^{\prime}}}{l}\textrm{ - 1} = \sqrt {\textrm{1 + si}{\textrm{n}^2}\alpha \textrm{ta}{\textrm{n}^2}\gamma + \sin (2\alpha )\tan \gamma } \textrm{ - 1}$$
In addition, the shear strain is at its maximum value at the outer surface of the substrate, we are able to express the relationship between shear strain and angle of twist,
$$\gamma \approx \tan (\gamma ) = \frac{{\phi r}}{s}$$
where, $\textrm{s}$ is the length of shaft, $\emptyset $ is respectively twist angle.

Substituting twist angle for shear strain, the strain can be rewritten as,

$${\varepsilon _t} = \sqrt {\textrm{1 + si}{\textrm{n}^2}\alpha \frac{{{\phi ^2}{r^2}}}{{{s^2}}} + \sin (2\alpha )\frac{{\phi r}}{s}} \textrm{ - 1}$$
From Eq. (4), we can get that the strain induced in the FBG sensor is not only related to the twist angle, but also related to the helical-angle of the sensor bonded on the shaft. In order to increase the sensitivity of the measuring system, the optimal direction of the FBG sensor can be derived by partial derivative of strain function with respect to variable packaging angle.
$$\frac{{d{\varepsilon _t}}}{{d\alpha }} = \frac{1}{2}{\textrm{(1 + si}{\textrm{n}^2}\alpha \frac{{{\phi ^2}{r^2}}}{{{s^2}}} + \sin (2\alpha )\frac{{\phi r}}{s})^{ - {\textstyle{1 \over 2}}}}.(\sin (2\alpha )\frac{{{\phi ^2}{r^2}}}{{{s^2}}} + 2\cos (2\alpha )\frac{{\phi r}}{s})$$
Let $\frac{{d{\varepsilon _t}}}{{d\alpha }} = 0$, the optimized helical-angle is
$$\tan (2\alpha ) = - \frac{{2s}}{{r\phi }}$$
In Eq. (4), the relation between the strain and twist angle is not linearity when the helical angle is constant, which will make it difficult to acquire the torsion through strain. In order to better understand the effect of the helical angle and the twist angle change on strain, we construct a three-dimensional surface map of three parameters according to Eq. (4) as Fig. 5(a) shows. We set the scope of helical angle is from 0 to 1.57 rad and twist rate is between −0.00628 rad/mm and 0.00628 rad/mm. According to simulation results, Fig. 5(b) presents some special strain curves in different twist rate and helical angle respectively at 0, 30, 36, 45, 60 and 90 degrees. Obviously, no strain is induced in the twist when the helical angle is 0 degree. When the helical angle is 90 degree, the FBG sensor is always suffered from tension strain whether the twist direction is clockwise or counterclockwise, and the change of strain with torsion rate is non-linear but has a good symmetry. When the helical angle is 30, 36 and 45 degrees, the twist direction is consistent with the state of the tensile or compressive strain of the FBG sensor, this means that when the twist direction is clockwise, the FBG sensor is subjected to tensile strain, otherwise compressive strain. The change of strain has a good linear relationship with the change of the torsion rate, and the best linear relationship and the best sensitivity respectively come from the helical angles of 30 and 45 degrees. While the helical angle is 60 degrees, there is a good linear relationship between a compressive strain and torsion rate, but a nonlinear relation between a tensile strain and torsion rate. Considering that the purpose of separating torsion and bending is to more accurately reconstruct the shape of shape sensing sensor, moreover, torsion deformation is analyzed by strain induced by torsion, so the sensitivity and linearity between torsion and strain directly affect the response speed and precision of shape reconstruction. Combining the requirement of shape reconstruction with simulation results, 30 degree is chosen as the optimized helical angle, in which direction there is the largest sensitivity of FBG sensors and a good linear relation between strain and twist rate.

 figure: Fig. 5.

Fig. 5. (a) Strain distribution induced by twist in different twist rate and helical-angle. (b) Strain induced by twist in different twist rate and helical angle at 0, 30, 36, 45, 60 and 90 degrees respectively.

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2.3 Extracting bending strain and twist strain

Since FBG sensors are sensitive to temperature and strain, so it is difficult to distinguish the coupling effect of the temperature and strain by measuring the wavelength shift of FBGs, this cross-sensitivity seriously affect the application in shape sensing. The triangular configuration can help us to distinguish the effect of temperature on wavelength.

The general strain suffered by bend and twist for FBG sensor is

$$\varepsilon^{\prime} = {\varepsilon_\textrm{b}} + {\varepsilon _\textrm{t}}$$
where, $\varepsilon ^{\prime},\;{\varepsilon _b},\;{\varepsilon _t}$ are respectively total strain, bending strain and torsional strain.

The wavelength shift induced by strain and temperature can be defined as,

$$\begin{aligned} \Delta \lambda ^{\prime} &= {k_\varepsilon } \cdot \varepsilon ^{\prime} \\ \Delta \lambda ^{\prime\prime} &= {k_T} \cdot \Delta T \end{aligned}$$
where $\Delta {\lambda^{\prime}},\Delta {\lambda ^{\prime\prime}}$ are wavelength shift induced by strain and temperature respectively, ${k_\varepsilon },{k_T}$ are calibrating coefficients about wavelength shift vs strain, wavelength shift vs temperature.

The wavelength shift induced can be written as,

$$\left[ {\begin{array}{c} {\Delta {\lambda_1}}\\ {\Delta {\lambda_2}}\\ {\Delta {\lambda_3}} \end{array}} \right] = \left[ {\begin{array}{c} {\Delta {\lambda_1}^\prime }\\ {\Delta {\lambda_2}^\prime }\\ {\Delta {\lambda_3}^\prime } \end{array}} \right] + \left[ {\begin{array}{c} {\Delta {\lambda_1}^{\prime \prime }}\\ {\Delta {\lambda_2}^{\prime \prime }}\\ {\Delta {\lambda_3}^{\prime \prime }} \end{array}} \right] = \left[ {\begin{array}{c} {{k_{\varepsilon 1}}({\varepsilon_{b1}} + {\varepsilon_{t1}})}\\ {{k_{\varepsilon 2}}({\varepsilon_{b2}} + {\varepsilon_{t2}})}\\ {{k_{\varepsilon 3}}({\varepsilon_{b3}} + {\varepsilon_{t3}})} \end{array}} \right] + \left[ {\begin{array}{c} {{k_{T1}} \cdot \Delta {T_1}}\\ {{k_{T2}} \cdot \Delta {T_2}}\\ {{k_{T3}} \cdot \Delta {T_3}} \end{array}} \right]$$
where $\Delta {\lambda _1},\;\Delta {\lambda _2},\;\Delta {\lambda _3}$ are respectively wavelength shift of three FBG sensors in the same cross section.

According to the triangular configuration of FBG sensors array, the strain under twist is theoretically the same for the triple FBG sensors, and temperature change is approximately consistent in the same section. To facilitate the theoretical analysis, the parameters for strain coefficient are set to be approximately equal for every FBG sensor, as well as for temperature coefficient, and these parameters can be acquired by experimental calibration. Equation (9) can be expressed as,

$$\left[ {\begin{array}{c} {\Delta {\lambda_{1}}}\\ {\Delta {\lambda_{2}}}\\ {\Delta {\lambda_{3}}} \end{array}} \right] = \left[ {\begin{array}{c} {{k_\varepsilon } \cdot {\varepsilon_{bm}} \cdot \cos \left( {\beta - {\textstyle{2 \over 3}}\pi } \right))}\\ {{k_\varepsilon } \cdot {\varepsilon_{bm}} \cdot \cos (\beta )}\\ {{k_\varepsilon } \cdot {\varepsilon_{bm}} \cdot (\beta + {\textstyle{4 \over 3}}\pi )} \end{array}} \right] + \left[ {\begin{array}{c} {{k_\varepsilon } \cdot {\varepsilon_t}}\\ {{k_\varepsilon } \cdot {\varepsilon_t}}\\ {{k_\varepsilon } \cdot {\varepsilon_t}} \end{array}} \right] + \left[ {\begin{array}{c} {{k_T} \cdot \Delta T}\\ {{k_T} \cdot \Delta T}\\ {{k_T} \cdot \Delta T} \end{array}} \right]$$
We use the parameter $\Delta {\lambda _x}$ to represent ${k_\varepsilon } \cdot {\varepsilon _t} + {k_T} \cdot \Delta T$, and Eq. (10) can be simplified as,
$$\left[ {\begin{array}{c} {\Delta {\lambda_{1}}}\\ {\Delta {\lambda_{2}}}\\ {\Delta {\lambda_{3}}} \end{array}} \right] = \left[ {\begin{array}{c} {{f_1}(x)}\\ {{f_2}(x)}\\ {{f_3}(x)} \end{array}} \right] = \left[ {\begin{array}{c} {{k_\varepsilon } \cdot {\varepsilon_{bm}} \cdot \cos \left( {\beta - {\textstyle{2 \over 3}}\pi } \right))}\\ {{k_\varepsilon } \cdot {\varepsilon_{bm}} \cdot \cos (\beta )}\\ {{k_\varepsilon } \cdot {\varepsilon_{bm}} \cdot \cos (\beta + {\textstyle{{2} \over 3}}\pi )} \end{array}} \right] + \left[ {\begin{array}{c} {\Delta {\lambda_{\tau t}}}\\ {\Delta {\lambda_{\tau t}}}\\ {\Delta {\lambda_{\tau t}}} \end{array}} \right]$$
where, $\Delta {\lambda _{\tau t}}$ is wavelength shift induced by twist and temperature, and $x = \{{{\varepsilon_{bm}},\;\beta ,\;\Delta {\lambda_{\tau t}}} \}$.

In the matrix equation, the parameters ${\varepsilon _{bm}}$, $\beta $ and $\Delta {\lambda _{\tau t}}$ are unknown parameters, $\Delta {\lambda _1},\Delta {\lambda _2}$ and $\Delta {\lambda _3}$ are known parameters acquired by FBG sensors array, and ${k_\varepsilon }$ is the strain coefficient that can be acquired by experimental calibration. To solve the nonlinear matrix equation, the form of non-linear least squares analysis is employed to fit a set of three observations with the model that is non-linear in three unknown parameters. It is desired to find unknown parameters such that the model function fits best the given data in the least squares sense, that is the sum of squares,

$$\mathop {\min }\limits_x ||{r(x)} ||_2^2 = \mathop {\min }\limits_x (({r_1}{(x)^2} + {r_2}{(x)^2} + {r_3}{(x)^2})$$
where, the residual $r(x )$ is given by,
$$r(x = \{ {\varepsilon _{bm}},\beta ,\Delta {\lambda _{\tau \textrm{t}}}\} ) = \left[ {\begin{array}{c} {{r_1}(x)}\\ {{r_1}(x)}\\ {{r_3}(x)} \end{array}} \right] = \left[ {\begin{array}{c} {\Delta {\lambda_1} - {f_1}(x)}\\ {\Delta {\lambda_2} - {f_2}(x)}\\ {\Delta {\lambda_3} - {f_3}(x)} \end{array}} \right]$$
The optimal solution of the parameters $x = \{{{\varepsilon_{bm}},\beta ,\Delta {\lambda_{\tau t}}} \}$ can be obtained when the minimum value of $\mathop {\min }\nolimits_x \Vert r(x )\Vert_2^2$ occurs.

The above method has extracted the bending information including the maximum strain, bending direction and the wavelength shift ${\Delta }{{\lambda }_x}$ induced by temperature and torsion in one cross-section. The following method can help us to distinguish the effects of temperature and twist on wavelength shift respectively. To better calculate the torsion, a more general approach starts by setting up a set of discrete points $\{{{{\boldsymbol {o}}_i}} \}$ along the length of continuous central axis $\textbf{u}(s )$. Defining two local coordinate systems at each point, a set of unit vectors $\{{{\textbf{u}_i},{\textbf{v}_i},{\textbf{w}_i}} \}$ is as fixed coordinate system at every cross-section and $\{{{\textbf{T}_i},{\textbf{N}_i},{\textbf{B}_i}} \}$ as moving coordinate system which is attached to the curve and describes the shape of the curve independent of any parameterization as Fig. 6 shows. The unit tangent vector ${\textbf{T}_i}$, the unit normal vector ${\textbf{N}_i}$ and the unit bi-normal vector ${\textbf{B}_i}$ are three mutually perpendicular vectors used to describe a discrete curve in three dimensions. The curvature vector ${{\boldsymbol {\kappa} }_i}$ is consistent with the normal vector ${\textbf{N}_i}$. In the more general case, curvature vector must be perpendicular to tangent vector when the curvature vector is not equal to zero.

 figure: Fig. 6.

Fig. 6. Model of twist

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In an untwisted rod, the torsion is interpreted as rotation of curvature vector. However, if rod is twisted, this twist will cause an additional rotation of the curvature vector, In general then it can be expressed as,

$$\tau = \frac{{d\beta }}{{ds}}$$
where, ${\tau }$ is the torsion of torsion, ${\beta}$ is the bending angle relative to the reference direction, and $\textrm{s}$ is the length of central axis between two adjacent cross-sections.

The discrete torsion formulas ${{\tau }_i}$ in two adjacent moving coordinate systems can be defined as,

$${\tau _i} = \frac{{{\beta _\textrm{i}} - {\beta _{i - 1}}}}{{{s_i}}}\textrm{ (i} = 1,2, \cdots ,n; {\beta _\textrm{0}} = 0)$$
Here, we define the torsion direction as clockwise if the sign of torsion is positive and counterclockwise otherwise. The discrete strain ${\varepsilon _{ti}}$ under torsion in Eq. (4) can be discretized as,
$${\varepsilon _{ti}} = \sqrt {\textrm{1 + si}{\textrm{n}^2}\alpha \cdot {\tau _i}^\textrm{2} \cdot {r^2} + \sin (2\alpha ) \cdot {\tau _i} \cdot r} - 1$$
According to the simulation results in section 2.2, the optimized helical angle ${\alpha }$ is 30 degrees. In our experiment, we try to bond the FBG sensors on the substrate at a helical angle of 30 degree using different processing methods, including a laser method employed to cut three helical grooves on the substrate. However, it is very difficult to finish it when length to diameter ratio of the substrate is relatively large with the length of 1 m and the diameter of 0.75 mm, and so in this experiment we use an angle that is easy to package FBG sensors on the substrate along the axis direction, as well as ${\alpha } = {90^0}$. According to the Eq. (14), let ${\alpha } = {90^0}$, the discrete strain ${\varepsilon _{ti}}$ induced by twist can be rewritten as,
$${\varepsilon _{ti}} = \sqrt {{r^2} \cdot {\tau _i}^2 + 1} - 1 (\textrm{i} = 1,2, \cdots, n)$$
According to the Eq. (8), the wavelength shift induced by bending, torsion and temperature can be defined respectively as,
$$\Delta {\lambda ^{\prime}_i} = \Delta {\lambda ^{\prime}_{bi}} + \Delta {\lambda ^{\prime}_{\tau i}} = {k_\varepsilon } \cdot {\varepsilon _{bi}} + {k_\varepsilon } \cdot {\varepsilon _{ti}}$$
$$\Delta {\lambda ^{\prime\prime}_i} = \Delta {\lambda _i} - \Delta {\lambda ^{\prime}_i}$$
where, $\Delta \lambda_{\tau i}^{\prime},\;\Delta \lambda _{bi}^{\prime}$ is respectively the wavelength shift induced by bending and twist in the discrete node, $\Delta {\lambda ^{\prime}},\Delta {\lambda ^{\prime\prime}}$ is respectively the wavelength shift induced by strain and temperature in the discrete node.

The above theoretical method helps us distinguish the effects of bending, twisting and temperature changes on wavelengths. Temperature separation can especially eliminate the effect of temperature cross-sensitivity. The discrete bending and torsion provide enough information for 3D reconstruction of shape sensing sensor.

2.4 Shape sensing model

Using wavelength shift of FBG sensors array, we can derive curvature and torsion at the location of FBG sensors. The following question is how to use curvature and torsion to reconstruct the shape of rod. For a continual flexible rod, it can be simplified to be a curve, and every node in the curve can be considered as a micro arch segment and with the Frenet–Serret frame to describe the moving coordinate frame, curvature vector ${N_i}$ and tangent vector ${T_i}$ are employed to set up osculating plane as Fig. 7 shows. When the micro-arch segment approach the point, the bend of the arch segment can be derived in the osculating plane.

 figure: Fig. 7.

Fig. 7. Shape reconstruction model of shape sensing sensor

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For the neighboring moving coordinate about $\{{{o_{i - 1}}} \}$ and $\{{{o_i}} \}$, the rotation matrix can be defined as, firstly, the $\{{{o_{i - 1}}} \}$ rotate ${{\theta }_{i - 1}}$ around the binomial vector ${B_{i - 1}} = {T_{i - 1}} \times {N_{i - 1}}$, and then rotate angle ${\emptyset _i}$ around the new curvature vector ${T_{i - 1}}$, and the rotation matrix about $\{{{o_i}} \}$ and $\{{{o_{i - 1}}} \}$ can be described by,

$$\left[ {\begin{array}{c} {{T_i}}\\ {{N_i}}\\ {{B_i}} \end{array}} \right] = \left[ {\begin{array}{ccc} {\cos ({\theta_{i\textrm{ - 1}}})} &{ - \sin ({\theta_{i\textrm{ - 1}}})} &0\\ {\sin ({\theta_{i\textrm{ - 1}}})} &{\cos ({\theta_{i\textrm{ - 1}}})} &0\\ 0 &0 &1 \end{array}} \right]\left[ {\begin{array}{ccc} 1 &0 &0\\ 0 &{\cos ({\phi_i})} &{ - \sin ({\phi_i})}\\ 0 &{\sin ({\phi_i}} &{\cos ({\phi_i})} \end{array}} \right]\left[ {\begin{array}{c} {{T_{i\textrm{ - 1}}}}\\ {{N_{i\textrm{ - 1}}}}\\ {{B_{i\textrm{ - 1}}}} \end{array}} \right]$$
So the rotation matrix between $\{{{o_i}} \}$ and $\{{{o_{i - 1}}} \}$ can be obtained,
$$\textbf{R}_i^{i\textrm{ - 1}} = \left[ {\begin{array}{ccc} {\cos ({\theta_{i\textrm{ - 1}}})} &{ - \sin ({\theta_{i\textrm{ - 1}}}).\cos ({\phi_i})} &{ - \sin ({\theta_{i\textrm{ - 1}}}).\sin ({\phi_i})}\\ {\sin ({\theta_{i\textrm{ - 1}}})} &{\sin ({\theta_{i\textrm{ - 1}}}).\cos ({\phi_i})} &{ - \cos ({\theta_{i\textrm{ - 1}}}).\sin ({\phi_i})}\\ 0 &{\sin ({\phi_i})} &{\cos ({\phi_i})} \end{array}} \right]$$
The rotation angle ${{\theta }_{i - 1}}$ can be expressed as,
$${\theta _{i - 1}} = {\kappa _{i - 1}}.d{s_{i - 1}}$$
where, ${\kappa _{i - 1}}$ is the curvature of arch length in the ${o_{i - 1}}$ coordinate frame, $d{s_{i - 1}}$ is the length of micro-arch segment in the ${o_{i - 1}}$ coordinate frame.

The twist angle ${\emptyset _i}$ is the angle about osculating plane $O{P_i}$ deviating from the osculating plane $O{P_{i - 1}}$, it can be expressed as,

$${\phi _i} = {\beta _{i + 1}}\textrm{ - }{\beta _i}$$
where, ${\beta _i}$ is the bending angle relative to the reference direction in the discrete node.

The relative position of $\{{{o_i}} \}$ coordinate system in $\{{{o_{i - 1}}} \}$ coordinate system is

$$\textbf{t}_i^{i\textrm{ - 1}} = {\left[ {\begin{array}{ccc} {{d_T}} &{{d_N}} &{{d_B}} \end{array}} \right]^T} = {\left[ {\begin{array}{ccc} {\frac{1}{{{k_{i\textrm{ - 1}}}}}.\sin ({\theta_{i\textrm{ - 1}}})} &{\frac{1}{{{k_{i\textrm{ - 1}}}}}.(1 - \cos ({\theta_{i\textrm{ - 1}}}))} &0 \end{array}} \right]^T}$$
Moreover, the transformation matrix of neighbor Frenet-Serret coordinate system can be determined as,
$$\textbf{T}_i^{i\textrm{ - 1}} = \left[ {\begin{array}{cc} {\textbf{R}_i^{i\textrm{ - 1}}} &{\textbf{t}_i^{i\textrm{ - 1}}}\\ 0 &1 \end{array}} \right]$$
Every original point ${o_i}$ of $\{{{o_i}} \}$ local coordinate system in the global coordinate system $\{o \}$ can be written as,
$$\left\{ {\begin{array}{ll} {{o_i} = \prod\limits_{k = 0}^{k = i} {T_k^{k\textrm{ - 1}}} \times {{\left[ {\begin{array}{cccc} 0 &0 &0 &1 \end{array}} \right]}^T}} &{\textrm{i} > 0}\\ {{o_i} = \left[ {\begin{array}{ccc} 0 &0 &0 \end{array}} \right]} &{\textrm{i} = 0} \end{array}} \right.$$

3. Experiment and discussion

To testify validity, reliability and repeatability of the separating method, we set up a serial of experiments including experimental calibration, 2D and 3D shape reconstruction. The shape sensing experimental system is set up as Fig. 8 shows, which mainly composes of four components. The first component is to calibrate the coefficient between bending curvature and wavelength shift, and create circle curve with different radius for testifying accuracy of 2D curve shape reconstruction as shown in the bottom left of Fig. 8. The second component is employed to calibrate the coefficient between torsion and wavelength shift, and generate helical curve for verifying accuracy of 3D shape reconstruction as the upper left of Fig. 8 shows. The third component is FBG wavelength demodulator (MOI, si-425) shown in upper right of Fig. 8, which is used to monitor the wavelength shift of FBG sensors array. The last component is software system, which integrates wavelength acquisition, separating algorithm and shape reconstruction as shown in the bottom right of Fig. 8. The description of the components are in details in Fig. 9, Fig. 15 and Fig. 16 respectively.

 figure: Fig. 8.

Fig. 8. Experiment setup for shape sensing

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 figure: Fig. 9.

Fig. 9. Calibration platform

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In order to analyze the relationship between strain and wavelength shift, we set up a calibration platform by drilling many holes in an aluminum plate with a thickness of 10 mm to form a series of circles as Fig. 9 shows. Positioning pins are to constrain the sensor of shape sensing and keep every FBG sensor in the same bending radius. We use a linear fitting method to obtain the strain coefficient between strain and wavelength shift, which is ${k_\varepsilon } = 0.328 \pm 0.015nm/\mu \varepsilon$ as Fig. 10 shows. The negative sign of strain represents compression strain, and the positive sign is tension strain in the horizontal axis.

 figure: Fig. 10.

Fig. 10. Wavelength shift vs strain

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The first experiment is a pure bending experiment as Fig. 9 shows, which means the wavelength shift induced by twist does not exist in theory. The bending radius and bending direction is respectively ${r_b} = 260$mm and ${\beta } = {60^0}$, which has been illustrated in Fig. 2 and the theoretical maximum strain is 3.85${{\mathrm{\mu}} \varepsilon }$. To verify the repeatability and reliability of separating method, we continuously bend the sensor of shape sensing ten times from a straight line into an arc with a radius of 260 mm, and record wavelength shift of three FBG sensors in one cross section. We bring experimental results in Table 1 into the Eq. (14) and get the solution of three parameters $\{{{\varepsilon_{bm}},\;\beta ,\;\Delta {\lambda_{\tau t}}} \}$. The range of measurement for the maximum strain is between 3.74${{\mathrm{\mu}} \varepsilon}$ and 3.75${{\mathrm{\mu}} \varepsilon }$ shown in Fig. 11(a), and the relative average error of the measurement is 2.65%. Figure 11(b) shows the theoretical angle and measured angle about bending direction relative to reference direction, the range of measured angle is from 59.42 degrees to 59.53 degrees, and the relative average error of measurement is 0.86%. The measured strain induced by twist vary between −0.09${{\mathrm{\mu}} \varepsilon }$ and −0.1${{\mathrm{\mu}} \varepsilon }$, the theoretical value is zero.

 figure: Fig. 11.

Fig. 11. (a) Measured and theoretical value for max strain. (b) Measured and theoretical value for bending direction.

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Tables Icon

Table 1. Wavelength shift induced by bending with 260mm radius for ten times.

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Reasons for deviation between measured values and theoretical values are analyzed as follows. Firstly, the triple FBG sensors are theoretically distributed at 120-degree angle in one cross-section, but actual location is not consistent with theoretical location because of the packaging error. Packaging error would seriously affect the measurement, so high-precision encapsulation method and correction algorithm could solve this problem. Secondly, FBGs are bonded on the substrate using AB adhesive, the strain transferred from the substrate through the bonding layer to FBGs smaller than that on the substrate because of transmission loss. The substrate strain sensed by FBG sensors is underestimated and thus required to be corrected in the future research.

In fact, it is very difficult to testify the feasibility about the separating method of bending and torsion strain by direct measurement especially for 3D curve, so we deign two experiments to indirectly verify the feasibility. One is the shape reconstruction of 2D curve and the other is 3D curve. Here, 3D curve is interpreted as a curve with bending and torsion, 2D curve as a curve with only bending. The shape sensing sensor used in this experiment is described in Fig. 1, the effective length of the shape sensing sensor is 500 mm with five detection locations, and adjacent detection location is spaced at 100 mm.

In the experiment of 2D curve, the shape sensing sensor is bent into an arc shape with the radius 260 mm. According to the separating method, we acquire two sets of data about maximum strain and bending direction in five detection location, and substitute respectively the data into Eq. (1) and Eq. (15) to obtain two sets of curvature and torsion, which are {0.003800, 0.003837, 0.003836, 0.003835, 0.003836, 0.003834}(1/mm) and {0, 8.0e-5, −9.7e-5, −4.1e-5, 2.1e-4, −1.1e-4}(rad/mm). To better observe the changes in the curvature and torsion along central axis, we use piecewise cubic Hermite interpolation method to acquire the curve of torsion and curvature as Fig. 12, the maximum deviation of curvature and torsion is 0.000037 1/mm and 2.1e-4 rad/mm respectively, the actual shape, the theoretical shape and measured shape are shown in Fig. 13. The maximum average position error in x, y, z direction respectively is 2.22 mm, −1.32 mm, 2.77 mm, and the absolute spatial position maximum error is 3.79 mm, which is defined as the square root of the square sum of the deviation between theoretical position and measured position in x, y, z direction. Figure 14 shows the position errors gradually increase at the location of the central axis 100 mm, 200 mm, 300 mm, 400 mm, and 500 mm respectively.

 figure: Fig. 12.

Fig. 12. (a) Curve of curvature along central axis. (b) Curve of torsion along central axis.

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 figure: Fig. 13.

Fig. 13. (a) Actual shape of shape sensing sensor. (b) The theoretical and measured shape of shape sensing sensor.

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 figure: Fig. 14.

Fig. 14. The measurement error in x, y, z direction and absolute position error.

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The experimental platform of 3D curve is designed as Fig. 15 shows, which is to generate approximate helical curve. It is composed of three stepper motors and controllers. The left stepper motor is fixed on the operation platform, and apply full constraints at the proximal end of shape sensing sensor. The middle stepper motor is to control twist angle at the distal end of shape sensing sensor and can translate freely along the axial direction by ball screw because the right motor is on the non-self-locking state. Figure 16 shows the shape of generating helix curve while twist angle gradually increases from zero to 100 degrees at the distal end of shape sensing sensor. After ten repetitions of an experiment, we acquire the range of maximum errors, which is 4.09∼6.53 mm, 1.77∼5.25 mm and 5.07∼7.28 mm in X, Y and Z direction respectively. The absolute spatial position maximum error is 6.75∼11.10 mm. The maximum relative position error is 1.68%∼2.76%. Figure 17 demonstrates the best-measured shape and theoretical shape, and Fig. 18 shows the position errors gradually increase at the location of the central axis 100 mm, 200 mm, 300 mm, 400 mm and 500 mm respectively.

 figure: Fig. 15.

Fig. 15. The experimental platform for generating helical curve and wavelength shift measuring of FBG Sensors array.

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 figure: Fig. 16.

Fig. 16. Helical curve generating.

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 figure: Fig. 17.

Fig. 17. The theoretical and measured shape of 3D helical curve.

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 figure: Fig. 18.

Fig. 18. The measurement error in x, y, z direction and absolute position error.

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4. Conclusion

In this research, we employee a triangular FBG sensors array bonding on the surface of the elongated SMA rod to design a kind of shape sensing sensor, which configuration can distinguish bend and twist better than the perpendicular FBG sensors array in our earlier research. We analyze the effect about bending radius and bending direction on wavelength shift of triple FBGs under pure bending, and twist angle on wavelength shift of triple FBGs under pure twist. Considering FBG sensor is sensitive to temperature, twist and bend, we set up a nonlinear matrix equation to solve three parameters including maximum bending strain, bending direction and wavelength shift induced by twist and temperature, use the change of bending direction of adjacent detecting location to define torsion and acquire the wavelength shift induced by temperature. This method is so robust for measurement error correction to obtain the optimal solution. An accurate shape reconstruction model of elongated rod has been set up using bending angle associated with the curvature and twist angle associated with the torsion according to Frenet-Serret formulas. Experimental results show that the relative average error of measurement about maximum strain, bending direction are respectively 2.65% and 0.86% under pure bend. The separating method also is employed to reconstruct 2D and 3D shape. The experimental results demonstrate that the absolute spatial position maximum error of shape recontrcution is 3.79 mm when the rod with length 500 mm is bent into an arc shape with the radius 260 mm in 2D space, and the absolute spatial position maximum error of shape reconstruction is 6.75 mm when the rod with 500 mm is deformed into helical curve in 3D space. The experiment verify the feasibility of separating method, which would provide effective parameters for precise 3D reconstruction model of flexible rod using curvature and torsion. Future works will investigate the factors that affecting the accuracy of reconstruction and combine reconstruction method with the dynamics model of shape sensing sensor to improve further the accuracy of shape sensing.

Funding

Zhejiang Province Public Welfare Technology Application Research Project (2015C31155); Science and Technology Commission of Shanghai Municipality (18JC1410402); Department of Education of Zhejiang Province (Y201533757); Natural Science Foundation of Ningbo (2018A610190).

Acknowledgments

The experimental platform of 3D curve was provided by Part Rolling Key Laboratory of Zhejiang Province, Ningbo University, Ningbo Zhejiang 315211, China.

Disclosures

The authors declare no conflicts of interest.

References

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Figures (18)
Fig. 1.
Fig. 1. Diagram of FBG sensors array.

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Fig. 2.
Fig. 2. (a) Distribution of triple FBGs in one cross-section. (b) Diagram of FBG bending strain.

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Fig. 3.
Fig. 3. Normalized waveforms of strain for FBG sensors in the different bending direction.

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Fig. 4.
Fig. 4. (a) Packaging diagram of FBG Sensor. (b) Geometric relationship about FBG deformation and twist angle.

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Fig. 5.
Fig. 5. (a) Strain distribution induced by twist in different twist rate and helical-angle. (b) Strain induced by twist in different twist rate and helical angle at 0, 30, 36, 45, 60 and 90 degrees respectively.

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Fig. 6.
Fig. 6. Model of twist

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Fig. 7.
Fig. 7. Shape reconstruction model of shape sensing sensor

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Fig. 8.
Fig. 8. Experiment setup for shape sensing

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Fig. 9.
Fig. 9. Calibration platform

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Fig. 10.
Fig. 10. Wavelength shift vs strain

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Fig. 11.
Fig. 11. (a) Measured and theoretical value for max strain. (b) Measured and theoretical value for bending direction.

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Fig. 12.
Fig. 12. (a) Curve of curvature along central axis. (b) Curve of torsion along central axis.

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Fig. 13.
Fig. 13. (a) Actual shape of shape sensing sensor. (b) The theoretical and measured shape of shape sensing sensor.

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Fig. 14.
Fig. 14. The measurement error in x, y, z direction and absolute position error.

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Fig. 15.
Fig. 15. The experimental platform for generating helical curve and wavelength shift measuring of FBG Sensors array.

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Fig. 16.
Fig. 16. Helical curve generating.

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Fig. 17.
Fig. 17. The theoretical and measured shape of 3D helical curve.

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Fig. 18.
Fig. 18. The measurement error in x, y, z direction and absolute position error.

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Tables (1)

Tables Icon

Table 1. Wavelength shift induced by bending with 260mm radius for ten times.

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Equations (28)

Equations on this page are rendered with MathJax. Learn more.

ε b m = r f + r s r b = ( r f + r s ) ⋅ c b
{ ε b = -1 ⋅ ε b m cos ⁡ β ε a = -1 ⋅ ε b m cos ⁡ α , α = β − 2 3 π ε c = -1 ⋅ ε b m cos ⁡ ψ , ψ = β  +  2 3 π
l ′ = ( 2 π r + 2 π r ⋅ tan ⁡ α ⋅ tan ⁡ γ ) 2 + ( 2 π r ⋅ tan ⁡ α ) 2
ε t = l ′ − l l = l ′ l  - 1 = 1 + si n 2 α ta n 2 γ + sin ⁡ ( 2 α ) tan ⁡ γ  - 1
γ ≈ tan ⁡ ( γ ) = ϕ r s
ε t = 1 + si n 2 α ϕ 2 r 2 s 2 + sin ⁡ ( 2 α ) ϕ r s  - 1
d ε t d α = 1 2 (1 + si n 2 α ϕ 2 r 2 s 2 + sin ⁡ ( 2 α ) ϕ r s ) − 1 2 . ( sin ⁡ ( 2 α ) ϕ 2 r 2 s 2 + 2 cos ⁡ ( 2 α ) ϕ r s )
tan ⁡ ( 2 α ) = − 2 s r ϕ
ε ′ = ε b + ε t
Δ λ ′ = k ε ⋅ ε ′ Δ λ ′ ′ = k T ⋅ Δ T
[ Δ λ 1 Δ λ 2 Δ λ 3 ] = [ Δ λ 1 ′ Δ λ 2 ′ Δ λ 3 ′ ] + [ Δ λ 1 ′ ′ Δ λ 2 ′ ′ Δ λ 3 ′ ′ ] = [ k ε 1 ( ε b 1 + ε t 1 ) k ε 2 ( ε b 2 + ε t 2 ) k ε 3 ( ε b 3 + ε t 3 ) ] + [ k T 1 ⋅ Δ T 1 k T 2 ⋅ Δ T 2 k T 3 ⋅ Δ T 3 ]
[ Δ λ 1 Δ λ 2 Δ λ 3 ] = [ k ε ⋅ ε b m ⋅ cos ⁡ ( β − 2 3 π ) ) k ε ⋅ ε b m ⋅ cos ⁡ ( β ) k ε ⋅ ε b m ⋅ ( β + 4 3 π ) ] + [ k ε ⋅ ε t k ε ⋅ ε t k ε ⋅ ε t ] + [ k T ⋅ Δ T k T ⋅ Δ T k T ⋅ Δ T ]
[ Δ λ 1 Δ λ 2 Δ λ 3 ] = [ f 1 ( x ) f 2 ( x ) f 3 ( x ) ] = [ k ε ⋅ ε b m ⋅ cos ⁡ ( β − 2 3 π ) ) k ε ⋅ ε b m ⋅ cos ⁡ ( β ) k ε ⋅ ε b m ⋅ cos ⁡ ( β + 2 3 π ) ] + [ Δ λ τ t Δ λ τ t Δ λ τ t ]
min x ⁡ | | r ( x ) | | 2 2 = min x ⁡ ( ( r 1 ( x ) 2 + r 2 ( x ) 2 + r 3 ( x ) 2 )
r ( x = { ε b m , β , Δ λ τ t } ) = [ r 1 ( x ) r 1 ( x ) r 3 ( x ) ] = [ Δ λ 1 − f 1 ( x ) Δ λ 2 − f 2 ( x ) Δ λ 3 − f 3 ( x ) ]
τ = d β d s
τ i = β i − β i − 1 s i  (i = 1 , 2 , ⋯ , n ; β 0 = 0 )
ε t i = 1 + si n 2 α ⋅ τ i 2 ⋅ r 2 + sin ⁡ ( 2 α ) ⋅ τ i ⋅ r − 1
ε t i = r 2 ⋅ τ i 2 + 1 − 1 ( i = 1 , 2 , ⋯ , n )
Δ λ i ′ = Δ λ b i ′ + Δ λ τ i ′ = k ε ⋅ ε b i + k ε ⋅ ε t i
Δ λ i ′ ′ = Δ λ i − Δ λ i ′
[ T i N i B i ] = [ cos ⁡ ( θ i  - 1 ) − sin ⁡ ( θ i  - 1 ) 0 sin ⁡ ( θ i  - 1 ) cos ⁡ ( θ i  - 1 ) 0 0 0 1 ] [ 1 0 0 0 cos ⁡ ( ϕ i ) − sin ⁡ ( ϕ i ) 0 sin ⁡ ( ϕ i cos ⁡ ( ϕ i ) ] [ T i  - 1 N i  - 1 B i  - 1 ]
R i i  - 1 = [ cos ⁡ ( θ i  - 1 ) − sin ⁡ ( θ i  - 1 ) . cos ⁡ ( ϕ i ) − sin ⁡ ( θ i  - 1 ) . sin ⁡ ( ϕ i ) sin ⁡ ( θ i  - 1 ) sin ⁡ ( θ i  - 1 ) . cos ⁡ ( ϕ i ) − cos ⁡ ( θ i  - 1 ) . sin ⁡ ( ϕ i ) 0 sin ⁡ ( ϕ i ) cos ⁡ ( ϕ i ) ]
θ i − 1 = κ i − 1 . d s i − 1
ϕ i = β i + 1  -  β i
t i i  - 1 = [ d T d N d B ] T = [ 1 k i  - 1 . sin ⁡ ( θ i  - 1 ) 1 k i  - 1 . ( 1 − cos ⁡ ( θ i  - 1 ) ) 0 ] T
T i i  - 1 = [ R i i  - 1 t i i  - 1 0 1 ]
{ o i = ∏ k = 0 k = i T k k  - 1 × [ 0 0 0 1 ] T i > 0 o i = [ 0 0 0 ] i = 0
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