Beyond Constant Curvature: A New Mechanics Model for Unidirectional Notched-Tube Continuum Wrists

Abstract

This paper presents a new mechanics model for unidirectional notched-tube continuum wrists, a class of mechanisms frequently used to implement distal steering in needle-sized surgical robotic instruments. Existing kinematic models available for these devices are based on the simplifying assumption that, during actuation, all the notches undergo the same amount of deflection, so that the shape of a wrist can be approximated by an arc of constant curvature. This approach is analytically attractive, but, as we show in this paper, it can sometimes fail to provide good tracking accuracy. In this article, we provide a new model that relaxes the assumption above, and we report experimental evidence showing its superior accuracy. We model wrist deflection using Castigliano’s second theorem, with the addition of a capstan friction term that accounts for frictional losses on the actuation tendon. Because notched-tube wrists are typically made of Nickel-Titanium (Nitinol), which has nonlinear stress-strain characteristics, we use a technique to obtain a local linearized approximation of the material modulus, suitable for use in the deflection model. The result of our modeling is a system of nonlinear equations that can be solved numerically to predict the wrist configuration based on the applied actuation force. Experimental results on physical specimens show that this improved model provides a more accurate estimate of wrist kinematics than prior models assuming constant curvature bending.

1. Introduction

The quest for minimal invasiveness in surgery has catalyzed technical innovation in micromanufacturing, smart materials, and continuum robotics, enabling the creation of surgical robotic systems which grant unprecedented access to, and dexterity within, confined areas in the body. Such innovations promise improved therapeutic outcomes, reduced morbidity, and enhanced cosmesis thanks to the realization of highly-dexterous, needle-sized arms capable of entering the body through small incisions or natural orifices [1–6]. Within this area of research, notched-tube wrists are emerging as an appealing approach to provide distal steering in tiny surgical instruments. They are fabricated by cutting a pattern of notches into the body of a thin-walled superelastic Nickel-Titanium (Nitinol) tube. Removing material from the body of the tube creates a compliant section that can be articulated by applying a bending moment, e.g., by pulling one or more tendons attached to the tip of the tube. Numerous wrist designs have been proposed in the literature in the past decade, with tube diameters ranging between 0.5 and 10 mm (see e.g. [7–20]).

Among these, York and Swaney et al. [10,12] proposed a notched-tube wrist where the notches are created asymmetrically with respect to the centerline of the tube, as shown in Fig. 1. Compared to other wrist designs, this solution has two main benefits: (i) cutting asymmetric notches effectively offsets the neutral bending plane of the tube, enabling bending in tighter radii; (ii) actuation only requires a single tendon, which makes the wrist simple to operate. A downside of asymmetric notches is that they only permit bending in a single direction, but this limitation can be overcome by axially rotating the entire tube. Since their appearance in [12], asymmetric notched-tube wrists have rapidly been adopted in the development of numerous medical devices, including instruments for endoscopic ear procedures [21–24] and brain surgery [25,26], a set of miniaturized instruments for the da Vinci robot [27,28], a robotic radiofrequency (RF) probe for the treatment of liver tumors [29], a robotic guidewire for vascular surgery [30], and a pair of forceps for robot-assisted vocal folds surgery [31].

Fig. 1.

Notched-tube wrists, typically manufactured from superelastic Nickel-Titanium (Nitinol), are articulated by pulling a tendon attached to the tip, causing the wrist to bend. When tendon friction and material nonlinearity are accounted for, wrist kinematics can deviate from constant curvature assumptions, resulting in significant tracking error for large deflections.

While asymmetric notched-tube wrists are conceptually simple to manufacture and operate, their mechanics have previously been modeled under the assumption that, during actuation, all the notches in these devices undergo the same amount of deformation; with this assumption, it is possible to approximate the shape of a wrist to that of an arc of constant curvature [10, 12, 26, 30]. While a constant curvature model is convenient and can work in some applications (for a general overview of the kinematics of constant curvature continuum robots, interested readers are referred to [32]), one can observe practical devices taking on variable curvature shapes (see Fig. 2). In this paper, we present a model capable of describing the non-constant curvature shapes that asymmetric notched-tube wrists can sometimes take on.

Fig. 2.

Actuation of two different notched-tube wrists. In wrist (a), which features five identical notches, the most proximal notch bends first (due to tendon friction, which makes the moment highest at the base of the device), while the most distal notch is the last one to reach the hard stop. Wrist (b), which features variable notch depths with the tip notch being deepest, exhibits the opposite behavior, i.e., the most distal notch is the one that closes first since it is the least stiff of the notches (tip-first bending). In this paper, we propose a mechanics model capable of explaining these different behaviors.

To accurately model wrist deflection, we analyze how tendon friction and material nonlinearities affect the loading conditions applied by the tendon at each notch. We note that the importance of accounting for these two factors was already established in prior modeling studies describing similar devices: in [13], Moses and colleagues describe a large-lumen continuum robot consisting of two nested Nitinol tubes with alternating notches cut in the sides, and they present a physics-based dynamic model for this manipulator where tendon friction is modeled as Coulomb friction. This work was later extended by Gao et al. [11], who developed a more advanced model for the same manipulator, based on Cosserat rod theory. In this work, the authors propose a method to linearize the stress-strain curve of Nitinol, i.e., determine an equivalent Young’s modulus, to model the material behavior. We note that the work described in [11, 13] is not directly applicable to the devices we describe in this paper due to fundamental differences in the construction and in the mechanics of the flexural elements.

For asymmetric notched-tube wrists, prior studies have investigated how friction and material nonlinearities affect a wrist overall deflection, but still under the limiting assumption of constant curvature [10, 12, 26, 30]. One notable exception is the work by Legrand et al. [33], who recently proposed a model that relaxes this assumption. The model we propose in this paper parallels the one described in [33], but it was derived using different assumptions, which enabled us to obtain a simpler closed-form formulation without sacrificing descriptive power. The key contribution of this paper is a system of recursive equations that describe the bending moment experienced by each individual notch in a wrist, and hence the corresponding deflection. We report on experiments that demonstrate the accuracy of the proposed model, and we further compare its descriptive power to that of prior models. On a more general level, this paper contributes a deeper understanding of the mechanics of asymmetric notched-tube joints: This manuscript is the first to present an in-depth analysis of the behavior of each individual notch in a wrist, and to elucidate how the combined role of tendon friction and material non-linearity affects wrist deflection.

2. Review of Prior Modeling

In this section, we briefly review the modeling work carried out in prior research. We begin with a review of the wrist kinematics proposed in [10, 12] and then move on to a description of existing statics models [10, 12, 26, 30, 33, 34], which provide a foundation for the model proposed in this manuscript.

Let us consider a generic notched-tube wrist, with dimensions as noted in Fig. 3. Throughout this paper, we will use the convention of numbering notches from 1 (most proximal) to n (most distal). When a pulling force F_p is applied on the actuation tendon, a moment M will be created at the tip of the tube, which will make the wrist bend. A key modeling assumption is that the tendon used for actuation is attached to the opposite side of the wrist spine, as illustrated in Fig. 3.

Fig. 3.

A notched-tube wrist is an open kinematic chain consisting of a sequence of interleaving notches and uncut sections. Throughout this paper, we use c_j to indicate the length of the j^th uncut section, while h_j and g_j respectively denote the length and depth of the j^th notch. Wrist actuation is performed by applying a pulling force F_p on the tendon. Kinematic models provide a mapping between the tendon displacement Δl and the tube pose T_wrist ∈ SE(3). A consequence of assuming a frictionless tendon in prior models is that each notch undergoes the same amount of deflection θ_j.

2.1. Wrist Kinematics

The goal of kinematic modeling is to find a homogeneous transformation matrix T_wrist ∈ SE(3) between the base of the wrist and the tip, given the total amount of tendon displacement Δl (refer to Figure 3). This transformation can be calculated with the following product:

$$T_{\text{wrist }}=\prod _{j=1}^n{T}_{\text{notch },j}{T}_{\text{uncut },j}$$ (1)

where T_notch,j and T_uncut,j are the transformations across the j^th notched and uncut sections, respectively. We assume that uncut sections do not bend, therefore T_uncut,j is a simple translation along the z axis by a distance c_j equal to the length of the j^th uncut section. For the notched sections, it is convenient to assume that bending occurs in the shape of a constant curvature arc. Using well-established kinematic relations for constant-curvature continuum robots [32], T_notch,j can be calculated as:

$$T_{\text{notch },j}=\left[\begin{array}{cccc}\text{cos}\left({\kappa }_j{s}_j\right)& 0& \text{sin}\left({\kappa }_j{s}_j\right)& \frac{1-\text{cos}\left({\kappa }_j{s}_j\right)}{{\kappa }_j}\\ 0& 1& 0& 0\\ -\text{sin}\left({\kappa }_j{s}_j\right)& 0& \text{cos}\left({\kappa }_j{s}_j\right)& \frac{\text{sin}\left({\kappa }_j{s}_j\right)}{{\kappa }_j}\\ 0& 0& 0& 1\end{array}\right]$$ (2)

where k_j and s_j are the curvature and arc length as measured with respect to the centerline of the tube (refer to Fig. 4). Note that θ_j = s_j k_j, where θ_j denotes the bending angle of the j^th notch.

Fig. 4.

Detail of a single notch. The bending angle of the j^th notch is denoted with θ_j, the curvature is k_j, and the arc length is s_j. The distance between the centerline of the tube and the neutral bending plane is ${\overline{y}}_j$.

To complete the kinematic modeling, it is necessary to calculate the relation between the tendon displacement Δl and the arc parameters k_j and s_j. From [10, 12], these relations are:

$${\kappa }_j\approx \frac{\Delta l}{h_j\left(r_i+{\overline{y}}_j\right)-\Delta l{\overline{y}}_j},s_j=\frac{h_j}{1+{\overline{y}}_j{\kappa }_j}$$ (3)

where r_i is the inner radius of the tube and ${\overline{y}}_j$ is the location of the neutral bending plane with respect to the centerline of the tube, as illustrated in Fig. 4. The value of ${\overline{y}}_j$ can be calculated using the equations in [10, 12].

The kinematic model described above implicitly assumes that, during actuation, all the notches undergo the same amount of deflection, i.e., for a wrist comprised of n geometrically identical notches, each notch undergoes the same amount of deflection, i.e., θ₁ = θ₂ = ⋯ = θ_n. As we have shown earlier in Fig. 2, this assumption does not always describe the shape of these devices accurately. In this paper, we relax this assumption and explicitly model tendon friction and material nonlinearities to describe these non-constant curvature behaviors.

2.2. Wrist Statics

In [10, 12], York and Swaney et al. proposed a statics model to determine the actuation force necessary for wrist actuation. Briefly, the authors first calculate the total amount of elastic energy U stored in the wrist, then apply Castigliano’s First Theorem to calculate the bending moment M acting on the wrist:

$$M=F_{p}L=\frac{\partial U}{\partial \theta }$$ (4)

where θ is the total deflection experienced by the wrist, i.e., $\theta ={\sum }_{j=1}^n{\theta }_j$, and L is the moment arm length. When the tendon is attached to the opposite side of the spine, as shown in Fig. 4, the moment arm has length

$$L=r_i+{\overline{y}}_j$$ (5)

where r_i is the inner tube radius. Note how the statics model described above assumes a frictionless tendon. In practice, friction between the actuation tendon and the inner wall of the tube can be non-negligible. In [10, 12], these frictional losses were accounted for by introducing an empirical correction factor on the right-hand side of Eq. (4). In this paper, we develop a theory that explains the presence of these frictional losses, and we use the resulting model to describe the (varying) loading conditions at each individual notch.

The influence of tendon friction on wrist deflection was also considered by Chitalia et al. in their work [30], whose objective was to develop a miniaturized robotic manipulator for vascular procedures. The design proposed in this work involves two bi-directional notched-tube joints, which are created by interleaving asymmetric notches with opposite orientations. To model wrist deflection, the authors assume constant curvature bending, and they propose a linear relation between the tendon pulling force and the wrist curvature. Frictional losses on the actuation tendon are accounted for using an exponential term, similarly to what we propose in this paper. In more recent work, the same group developed a robotic manipulator that combines a bi-directional wrist and a unidirectional wrist arranged sequentially on the same Nitinol tube [26]. In this work, the authors maintain the assumption that all the notches undergo the same amount of deflection.

More recently, Legrand et al. [33] were the first to report a statics model for asymmetric notched wrists that does not make the assumption of constant curvature bending. In their model, frictional losses along the actuation tendon are assumed to create different loading conditions at each notch, and friction is modeled using the capstan friction model. In this paper, we use a similar formulation, however, as we shall see in section 3.2, a key difference between this work and [33] is that here we assume friction to occur at the notch edges, whereas Legrand and colleagues assume friction to occur between the tendon and the uncut tube sections. Because of this different modeling assumption, in this work we are able to derive a simpler closed-form expression to describe the bending of each individual notch.

In separate work, Eastwood et al. [34] developed a kineto-static model that uses Castigliano’s Second Theorem to estimate the deflection of a wrist subject to external forces. Their work does not account for tendon friction, which is the focus of this article.

3. Wrist Mechanics

We now study the mechanics of notched tube wrists, and formulate a model that describes the deflection of each individual notch. Our analysis is structured as follows: we first focus on the mechanics of notched sections and use Castigliano’s Second Theorem to derive a formulation for the notch deflection θ_j based on the local bending moment M_j experienced at each notch. We then propose a model for the frictional forces occurring between the actuation tendon and the notch edges, and study how these losses affect the local moments M_j. The result of our analysis is a set of recursive equations that can be solved to calculate the wrist kinematics.

Throughout our analysis, we consider the same generic notched wrist illustrated earlier in Fig. 3. To denote the wrist dimensions, we use the nomenclature introduced in the previous section. For convenience, relevant variables are summarized in Table 1.

Table 1.

Wrist Parameters

r i	Inner Radius of Wrist
h j	Height of notch
c j	Length of uncut section
g j	Depth of cut for notch
θ j	Angle of notch deflection
${\overline{y}}_j$	Distance to neutral bending plane
n	Number of notches

3.1. Notch Deflection via Castigliano’s Second Theorem

Recall from basic mechanics [35] that the total strain energy U stored in a cantilever beam of length L subject to a bending moment M is given by:

$$U={\int }_L\frac{M^2}{2EI}dz$$ (6)

where E is the elastic modulus, and I is the second moment of area of the beam about its neutral axis. This relation is valid so long as we operate within the linear (elastic) region of the stress-strain curve for the material that the beam is made of. In section 3.3, we will extend our analysis to consider the case of materials with a non-linear stress-strain curve, as is the case with Nitinol.

If we assume that M, E, and I are uniform along the length of the beam, Eq. (6) simplifies to:

$$U=\frac{M^{2}L}{2EI}$$ (7)

Having obtained a relation for the strain energy U, we can now use Castigliano’s Second Theorem to determine the angular deflection of the beam:

$$\theta =\frac{\partial U}{\partial M}=\frac{ML}{EI}.$$ (8)

These results for a cantilever beam can be applied to calculate the deflection of each notched section in a wrist. For a notched-tube joint with n notches, the bending angle of the j^th notch θ_j is given by:

$${\theta }_j=\frac{M_j{h}_j}{E_j{I}_j}$$ (9)

where M_j is the local bending moment, E_j is the local elastic modulus, I_j is the second moment of area, and h_j is the notch height. Note that this equation is general in the sense that it allows for variations in notch geometry and material.

3.2. Modeling the Frictional Losses on the Actuation Tendon

We now study tendon friction and evaluate how it affects the loading conditions at each notch. Without loss of generality, let us consider the 3-notch wrist shown in Fig. 5. Pulling on the tendon with a force F_p exerts a force F₃ at the distal tip of the wrist. If there were no losses, then F_p would be equal to F₃. However, as it was found in prior work in [10, 12, 26, 30], F₃ will generally be smaller than F_p due to the frictional contact occurring between the wire and the corners of the notched sections. This implies that, as the tendon is pulled to deflect the wrist from an initially straight configuration, friction will progressively reduce the tension on the tendon towards the distal end of the wrist. To model tendon friction, here we make the following two assumptions: (i) friction occurring at the notch edges dominates frictional effects that may occur elsewhere along the tendon; (ii) friction occurring between the tendon and the notch edges can be modeled using a capstan loss model. As the wire goes through a generic j^th notch, its tension will be subjected to two successive capstan loss terms, each for one of the notch edges (see Fig. 5), and each equal to $e^{-\mu {\theta }_j/2}$, where θ_j is the notch angle, and μ is a static friction term. Based on these considerations, we can model the force experienced at each notch, F_j, as a function of the pulling force, F_p and the angle of each notch:

$$F_j=F_p\prod _{k=1}^j{e}^{-\mu {\theta }_k}.$$ (10)

Note from Eq. (10) that, due to the product of decaying exponentials, the effective force experienced by each notch monotonically decreases towards the distal end, which enables us to account for the non-constant curvature effects induced by tendon friction.

Fig. 5.

Diagram of the forces and moments being experienced at each notch due the pulling force, and the capstan loss model being applied at each corner of the notches.

We are now ready to calculate the bending moment experienced by each notch, which is simply given by:

$$M_j=F_j\left(r_i+{\overline{y}}_j\right).$$ (11)

In the equation above, the term $r_i+{\overline{y}}_j$ represents the moment arm. Recall from section 2.2 that this expression for the moment arm length is valid so long the actuation tendon is attached at the tip of the tube on the opposite side of the spine.

Finally, we can substitute M_j in Eq. (9) to obtain an expression that relates the bending angle of the j^th notch to the pulling force F_p, yielding the key modeling result of this paper:

$${\theta }_j=\frac{\left(r_i+{\overline{y}}_j\right)h_j}{E_j{I}_j}{F}_p\prod _{k=1}^j{e}^{-\mu {\theta }_k},j=1,\dots ,n.$$ (12)

The system of transcendental equations above - note that θ_j appears both on the left-hand side as a linear term and on the right-hand side as an argument to one of the exponentials - can be solved numerically to obtain the notch deflections for a given actuation force F_p. Note also how the right-hand side of Eq. (12) contains E_j, i.e., the elastic modulus of the material. To account for the fact that Nitinol has a nonlinear stress-strain curve, we replace E_j with an effective (secant) modulus E_{eff ,j}, as we describe in the following.

3.3. Handling Material Nonlinearities

The stress-strain curve of Nitinol has a distinctive non-linear behavior that is convenient to approximate using a piecewise formulation:

$$\sigma (\epsilon )=\{\begin{array}{ll}\epsilon E_{lin}\hfill & \epsilon \le {\epsilon }_l\hfill \\ {\sigma }_l+\left(\epsilon -{\epsilon }_l\right)E_{se}\hfill & {\epsilon }_l<\epsilon \le {\epsilon }_p\hfill \\ {\sigma }_p+E_p{e}^{-\beta /\left(\epsilon -{\epsilon }_p\right)}\hfill & {\epsilon }_p<\epsilon \hfill \end{array}$$ (13)

where E_lin, E_se, and E_p are the effective moduli in the linear, super-elastic, and plastic regions (respectively); σ_l, σ_p are the lower and upper plateau stresses; ε_l, ε_p are the lower and upper plateau strains; and β is a heuristic parameter. This formulation was adapted from a similar piecewise formulation presented in [10] and modified to include the effects of strain hardening during the detwinned martensite phase as the material approaches plastic deformation [36]. It should be noted that this latter term was included only for completeness, as in principle it is not desirable for the material to enter this region during wrist operation. An illustrative stress-strain curve of Nitinol, showing the three modeled regions, is shown in Fig. 6.

Fig. 6.

Illustrative stress-strain curve of Nitinol as per Eq. (13).

From Eq. (13) and Fig. 6, it is evident that so long we operate within the elastic region of the stress-strain curve, it is possible to estimate the deflection θ_j by simply replacing E_j with E_lin in Eq. (12). However, as soon as the amount of strain created on the notch exceeds ε_l, i.e., as soon as the material enters the superelastic regime, the linear modulus E_lin no longer adequately captures the mechanics of the notch, and we need to find an approximation for E_j that ensures consistency with the assumptions of the model. For this purpose, we use the effective (secant) modulus E_eff,j. To calculate E_eff,j, we use a gradient descent approach. We begin by setting $E_{\text{eff},j}^{[0]}=E_{lin}$, and then use the following update law:

$$E_{\text{eff},j}^{[k]}=E_{\text{eff},j}^{[k-1]}-\eta \left(E_{\text{eff},j}^{[k-1]}-\frac{{\sigma }_j}{{\epsilon }_j}\right)$$ (14)

where is η the descent rate, and σ_j and ϵ_j respectively are the stress and strain created on the j^th notched section. The execution of the algorithm is illustrated in Fig. 7. At each step k of the algorithm, the values of σ_j and ε_j are updated to reflect the update in the effective elastic modulus. To perform these updates, we use the relations derived in [10], which we report below for completeness:

$${\kappa }_j=\frac{{\theta }_j}{h_j-{\overline{y}}_j{\theta }_j}$$ (15)

$${\epsilon }_j=\frac{\kappa \left(r_o-{\overline{y}}_j\right)}{1+{\overline{y}}_j\kappa }.$$ (16)

The deflection angle θ_j used in the relations above is calculated by evaluating Eq. (12) with a modulus value of E^[k−1]. This process is repeated until the modulus converges (|E^[k] − E^[k−1]| < ϵ = 10⁻⁴).

Fig. 7.

Iterative secant modulus approximation of superelastic material behavior: (top) modulus approximation for a tube in the superelastic range, (bottom) convergence plot.

4. Experimental Verification

Here we describe the experiments that we performed to verify the model proposed in the previous section. We created three different wrists designs, with notch geometries as listed in Table 2. Nitinol tubing was procured from Johnson Matthey (West Chester, PA, USA), and notches were created by laser cutting. Laser cutting was performed by Pulse Systems (Concord, CA, USA) using a femtosecond laser and on-axis cutting. Wrists A and B feature a uniform notch geometry, and they were designed to achieve a maximum deflection of 150° and 90° , respectively. These two wrists were designed to test the main hypothesis of this paper; namely, that uniform-notched wrists undergo non-constant curvature due to the presence of tendon friction, as illustrated earlier in Fig. 1(a) (i.e., proximal notches close first). Wrist C was instead designed to compensate for these frictional losses by having increasingly deeper notches towards the distal end, with the goal of replicating the tip-first bending behavior observed previously in Fig. 1(b). All the wrists above were equipped with a 0.1 mm actuation tendon made of Nitinol.

Table 2.

Geometric design parameters of the wrists used for experimental verification. Each wrist has an inner diameter of 1.4 mm and an outer diameter of 1.62 mm.

	Wrist A	Wrist B	Wrist C
h (mm)	0.8	0.5	1.0
c (mm)	1.2	1.5	1.0
g (mm)	1.4	1.4	[1.36, 1.39, 1.42, 1.45]
n	5	5	4

4.1. Experimental Setup and Procedure

The experimental setup is shown in Fig. 8. Each wrist was mounted on a collet and the actuation tendon was pulled by means of a linear slider (Velmex, Bloomfield, NY, USA). Throughout each experiment, the tension on the tendon was monitored with an FT-17 force sensor (Alberobotics, Genoa, Italy).

Fig. 8.

Experimental setup. Notched-tube wrists were mounted in a collet, and a linear slider was used to pull the actuation tendon. Tension on the tendon was measured by means of a force sensor, while the wrist deflection was recorded with a digital single-lens reflex camera.

Experiments were carried out as follows: starting from an initially straight configuration, each wrist was progressively bent by pulling the tendon in increments of 0.25 mm until all the notches were observed to be fully closed. At each step, we took a photograph with a digital single-lens reflex camera outfitted with a macro objective (Nikkor 40mm f/2.8G, Nikon Corporation, Tokyo, Japan) to document wrist deflection. Each wrist was bend-cycled a total of five times. After the experiment, each photograph was processed in MATLAB (MathWorks Inc., Natick, MA, USA) to obtain the wrist shape and measure the bending angle of each notch. This was a semi-automatic process, which first involved the identification of all the notch edges through the manual digitization of the corner points on the notches. The resolution of each photograph was 4928 × 3624 pixels, with a pixel-per-mm ratio of 0.01 mm/pixel. The baseline accuracy of angle measurements was determined by repeated measurements of a reference object displaying a 1° angle. We repeated the measurement 20 times, and we observed a Root-Mean-Square Error (RMSE) of 0.11°.

4.2. Identification of Material Properties

For reasonable (non-plastic) strains, the mechanics model in Eq. (12) is dependent on three different material properties of Nitinol, namely the linear elastic modulus E_lin, the superelastic modulus E_se, and the lower strain plateau ε_lp, plus the coefficient of friction μ of the actuation tendon. To identify these parameters, we ran a global optimization routine in MATLAB seeking to solve the following problem:

$$\underset{E_{lin},E_{se},{\epsilon }_l,\mu }{\text{min}}\Vert e_A\Vert +\Vert e_B\Vert +\Vert e_C\Vert$$ (17)

where e_A, e_B, and e_C, are (n + 1)-dimensional vectors containing the model RMSEs of the individual notches (i.e., the error made in the prediction of the bending angle of each notch) and total wrist deflection for Wrist A, Wrist B, and Wrist C respectively, and the ∥·∥ operator denotes the Euclidean norm. We used the GlobalSearch function of MATLAB with fmincon to find the set of parameters E_lin, E_se, ε_l, μ that would converge on the global minimum to Eq. (17). GlobalSearch was run with 2800 starting points but otherwise default conditions. The results of the material identification routine produced the parameters listed in Table 3. We note that the optimized linear modulus is lower than values presented in similar studies (24.3 GPa in [26], 60 GPa in [14]). This discrepancy could be explained by several potential causes, including variations in manufacturing and annealing, the presence of stress concentrations at each notch root, and the effects of Brazier flattening, which would manifest as an apparent reduction in modulus. It is also possible that the estimate of the linear modulus is implicitly capturing some unmodeled effects, and we plan to investigate this topic further in future work.

Table 3.

Material parameters determined by global search optimization.

Elin	ESe	εl	μ
10 GPa	3 GPa	0.028	0.13

4.3. Model Verification

Using the material properties identified in the previous section, we generated the plots in Figure 9, which show a comparison between experimental data and model predictions. The model was found to accurately predict the deflection of each notch. The RMSEs of the model with respect to each individual notch are reported in Table 4. The largest observed error was 2.47° (notch 1 in wrist C), while the RMSE was around or below 1° for most other notches.

Fig. 9.

Experimental Results. (Top Row) Model output for the three different wrist designs considered in the experiments. Wrists A and B were designed to exhibit proximal-first bending, while wrist C displayed tip-first bending. Note how the model correctly captures the notch closing sequence in all cases. (Bottom Row) Comparison of model output versus experimental data.

Table 4.

Model RMSE (degrees) for each notch for wrists A, B, and C.

Notch	Wrist A	Wrist B	Wrist C
1	1.80	0.43	2.47
2	1.32	0.41	1.49
3	1.01	0.39	1.15
4	0.84	0.57	0.79
5	0.89	0.57	—

Fig. 10 shows that the model accurately predicts the shape of each wrist over their entire range of motion. The model was able to estimate the overall wrist deflection with percent RMSEs of 2.5%, 1.8%, and 2.6% on Wrists A, B, and C, respectively. We observed a slightly larger tracking error for Wrist A as the proximal notches began to close fully. We believe this can be attributed to the larger tracking error of notch 1 at those configurations (see Table 4): based on basic geometric intuition, the prediction of the wrist shape is more sensitive to errors on the proximal notches than the distal ones.

Fig. 10.

Comparison between the wrist shapes observed experimentally and the wrist shapes predicted by the model. The lines in the figure above represent the wrist center line. This data shows that the proposed model describes wrist deflection well across the entire range of motion.

4.4. Comparison with Simpler Models

To illustrate the benefits of our model, we report results comparing its tracking accuracy to that of three simpler models, i.e., (i) one where we assume linear material properties (but still account for tendon friction); (ii) one where we assume linear material properties and also consider the tendon to be frictionless (note that under this set of assumptions, wrist deflection can be treated as a pure elastic beam bending problem; i.e. notch angles can be simply calculated with Eq. (9)); and finally, (iii) the pure kinematic model described earlier in Section 2.1. To ensure a fair comparison with our model, we repeated the material identification procedure described in section 4.2 to identify a value of elastic modulus that yields the best tracking accuracy for models (i) and (ii).

Results are summarized in Fig. 11. It can be observed that the shape of the wrist estimated by our matches the experimental wrist shape more accurately than all other models.

Fig. 11.

Comparison of the accuracy of the proposed model versus three other models using simplifying assumptions. This figure shows three sample wrist shapes observed during experimentation, along with the wrist shapes predicted by the four different models. The model we propose in this paper predicts the wrist shape almost exactly, while models assuming simplifying assumptions generally provide a higher tracking error.

Fig. 12 summarizes the tip tracking RMSEs afforded by each of the four models under consideration. These results further demonstrate the higher accuracy of our model (average RMSE: 0.33 mm) compared to the other three simpler models (average RMSEs: 0.92 mm, 0.90 mm, and 5.45 mm, for models (i), (ii), and (iii) respectively).

Fig. 12.

Tip tracking accuracy (Root-Mean-Square Error) of three different models calculated on Wrists A, B, and C.

5. Discussion

The aim of the model presented in this paper is to elucidate how the combined effects of tendon friction and material super-elasticity determine the bending behavior of unidirectional notched-tube wrists. The accuracy of our model was verified through experiments on three different Nitinol wrists, each designed in such a way to display a different curvature profile during bending. The model was shown to be able to predict the disparate loading conditions at each notch, and therefore accurately capture the actual shape of each wrist.

By observing the Force/Deflection plots for Wrists A and B in Fig. 9, the effects of tendon friction manifest as differences in slope for each notch under loading. We observe that, for more distal notches, parasitic capstan friction effects ultimately reduce the effective moment seen by each notch, thereby reducing deflection for a given tendon force. This explains why, in wrists with uniform notch geometry, proximal notches tend to close first. Because proximal notches close first, they also enter the super-elastic regime earlier (refer to Fig. 9), which further exacerbates the difference in bending angle with distal notches.

The fidelity of our model is especially well illustrated in the Force/Deflection plots for Wrist C in Fig. 9, where the effects of tendon friction and material nonlinearity on the resulting kinematics are abundantly clear. For an input force of 1 N, notches 1 and 2 remain in the linear elastic regime, whereas notches 3 and 4 have transitioned to superelastic, as indicated by the abrupt increase in slope. Our model matches the experimental data extremely well, and a simpler model relying on linear elastic material assumptions would not be able to capture this phenomenon.

A potential source of error in the model is the assumption that the entire notch will transition from linear-elasticity to super-elasticity once the strain in the notch reaches a specific threshold. This is best demonstrated by the small error we see in accurately predicting the transition for Wrist C which has a varying backbone thickness. For the first two notches, we underestimate the strain threshold for the transition, while overestimating the strain threshold for the most proximal notch. For beam bending, the entire beam does not experience the same amount of strain. Fibers closer to the neutral bending plane will experience strains approaching zero while fibers closer to the outer edge of the beam will experience the maximum amount of strain. Our model assumes that the entire beam is undergoing the same amount of strain. When our model transitions from the linear elastic region to the superelastic region, in reality, only a portion of the notch is actually transitioning due to the varying strain across the cross-section of the tube. In future work, we will investigate whether the tracking performance of our model can be improved further by accounting for strain variation throughout the notch cross-section and its resulting effect on the notch modulus. We note that an approach to model the effects of strain variation across the notch cross-section was recently proposed in [26]. During experimentation, it was also observed that the tested samples never entered the plastic regime despite undergoing significant curvature. In future work, we will further assess the validity of our model by testing it against samples undergoing such extreme curvature such that the local notch strain exceeds the upper plateau strain ε_p.

While the modeling work described in this paper parallels the one described in [33], it presents some important differences. One important distinction, as it was described earlier in section 2.2, is the way in which tendon friction is modeled, with this work assuming friction to occur at the notch edges, while the authors in [33] assume friction to occur between the tendon and the uncut tube sections. Because of this different modeling assumption, in this work we are able to express the deflection of each notch as a function of the bending angles of all the notches proximal to it, leading to a simple closed-form expression, i.e., Eq. (12). Another fundamental difference is in the way in which we approximate the linear modulus (section 3.3). In this paper, we use a general stress-strain curve, i.e., Eq. (13), which is parameterized by only three variables (the linear modulus E_lin, the transition strain ε_l, and the superelastic modulus E_se) that can be either prescribed or fit experimentally. The secant modulus (i.e., the effective linear modulus for the current strain) is then approximated through gradient descent. Conversely, in [33] the entire stress-strain curve must be measured experimentally a priori before the model can be run. As such, the model in [33] can only be run when the full experimental curve is known, whereas our model can be run with only the three fit parameters known.

Finally, we note that tendon thickness is not a factor that was studied in this work. We used a thin (0.1 mm) Nitinol tendon in our experiments, and we assumed that, except for friction, this wire does not contribute significantly to the mechanics of the wrist. This assumption may not be true for thicker wires, which generally present a higher bending stiffness, which may play into the mechanics of the device. In extreme cases, the tendon could prevent notches from fully closing. The tendon thickness is not used as a parameter for the model but it is implicitly accounted for in the material property identification for the tube.

6. Conclusion

As the use of notched-tube wrists in robotic minimally invasive surgeries increases, it is important to have accurate models for the simulation and actuation of these devices. Currently available models for unidirectional notched-tube wrists assume that, during actuation, all the notches undergo the same amount of deflection, so that the wrist shape can be described by an arc of constant curvature. In this paper, we presented a mechanics-based model which relaxes this assumption in order to determine the deflection of the notches individually. We modeled the frictional losses using a capstan friction model and accounted for the non-linear material properties using a secant modulus. The result of our modeling is a system of transcendental equations that can be numerically solved to calculate the deflection of each notch in a wrist based on the actuation force. Experimental data was gathered using three wrists with varying geometries, and the proposed model was shown to outperform prior models in accurately predicting the wrist deflection.