Impact of Generic Tendon Routing on Tension Loss of Tendon-Driven Continuum Manipulators with Planar Deformation

A. Problem Statement

To completely consider the effect of GTR on tension loss and deformation behavior of a generic variable-curvature TD-CM with planar bending shown in Fig. 1, we develop a new modeling framework that has the following new features:

Remark 1: As shown in Fig. 1, for a TD-CM with a GTR, the arc length along a routed tendon s_c is not necessarily equal to the midline arc length s (i.e.,$s_c\ne s$). Thus, the developed models in our previous studies (i.e., [7], [8] and [18]) with this assumption are not valid for TD-CMs with a GTR and may cause modeling inaccuracies. Therefore, unlike our previous studies, the proposed modeling framework takes into account this difference and generalizes our previous models in which tendons were limited to a parallel or straight routing. This feature enables expanding the design and work spaces of TD-CMs with GTR by accurately modeling their tension loss and deformation behavior.

Remark 2: The proposed modeling framework is derived based on a new mathematical formulation that uses three physically-intuitive generic functions to describe internal distributed forces on a GTR. It also introduces new concepts of “tension loss factor” and “projection factor” to provide insights of tendon-tension transmission loss in TD-CMs. This framework can be easily incorporated in the Cosserat rod theory (e.g., [5], [24]), which makes it more generic, physically-intuitive, and extendable.

Assumption 1: In this study, we assume a TD-CM with a fully-constrained GTR that causes planar deformation behavior. We also neglect tendons’ elongation and backlash effects.

B. Distributed Forces along a GTR

To develop a more accurate model and consider the difference between tendon arc length s_c and TD-CM’s midline arc length s, the kinematics of a GTR is thoroughly derived in this section. As shown in Fig. 1, the position vector ${\mathit{p}}_c(s)$ of the GTR is first expressed as follows:

where p(s) denotes the position vector of TD-CM’s midline and ${\mathit{R}}_c(s)$ denotes a vector from origin of the body frame $\{x-y\}$ to the corresponding point on the routed tendon. Using this definition, the arc length s_c along the routed tendon can be expressed as:

where $\parallel \cdot \parallel$ denotes the two norm throughout this paper. Using (2), we can define the differential $d{s}_c$ as:

where ${\mathit{p}}_{cx}^{\prime }(s)$ and ${\mathit{p}}_{cy}^{\prime }(s)$ represent the partial derivative of position ${\mathit{p}}_c(s)$ in the x and y directions with respect to s, respectively. Of note, $(\cdot {)}^{\prime }$ denotes the differentiation with respect to s throughout this paper. As shown in the zoomed region B of Fig. 1, to model distributed forces along a GTR, the directions associated with the tendon tension T(s), distributed normal force ${\mathit{f}}_n$ and distributed friction ${\mathit{f}}_f$ need to be obtained. Here, ${\mathit{f}}_f\left({\kappa }_c,T\right)$ and ${\mathit{f}}_n\left({\kappa }_c,T\right)$ are functions of tendon curvature ${\kappa }_c(s)$ and tendon tension profile T(s). During actuation, a tendon slides along the curved channel in which tendon tension T(s) and distributed friction ${\mathit{f}}_f\left({\kappa }_c,T\right)$ are both in the tangent direction ${\mathit{t}}_{\text{tendon}}(s)$ while distributed normal force ${\mathit{f}}_n\left({\kappa }_c,T\right)$ is in the direction of tendon curvature ${\mathit{n}}_{\text{tendon}}(s)$. The tangent direction of the curve ${\mathit{p}}_c(s)$ can be described using arc length parameterization s_c as follows:

By the chain rule and using (2) and (3), we can write: $\frac{d}{d{s}_c}=\frac{d}{ds}\cdot \frac{ds}{d{s}_c}=\frac{d}{ds}\cdot \frac{1}{d{s}_c/ds}=\frac{d}{ds}\cdot \frac{1}{\Vert {\mathit{p}}_c^{\prime }(s)\Vert }$. Hence, (4) can be expressed as [5]:

Also, tendon tension T(s) along the arc length parameter s is a function of tendon wrap angle θ(s) (shown in Fig. 1) [8] and using (5) can be written as:

where ${\mathit{t}}_{\text{tendon}}(s)$ denotes the direction vector of tendon tension. T_base is the input tension at the base of the TD-CM. As for the exponential term, $\lambda =-sign\left(T^{\prime }(s)\right)$ (where $T^{\prime }(s)=[T(s+ds)-T(s)]/ds$ [8]) indicates the sliding direction of the curved tendon along the channel, while µ is the frictional coefficient of the sliding motion. Here we use a Coulomb friction model and the tendon loading/motion is quasi-static (e.g., [6], [16], [25]). Also, $\theta (s)={\int }_0^s\Vert {\mathit{p}}_c^{\prime }(\xi )\Vert \cdot {\kappa }_c(\xi )d\xi$ represents the tendon wrap angle shown in the zoomed region A of Fig. 1. Of note, the tendon curvature κ_c is non-negative [26].

Remark 3: In (6), the ”tension loss factor” ${\mathcal{F}}_{t.l.}^{\star }$ captures the tendon-tension transmission loss phenomenon of TD-CMs with GTR. Of note, in contrast with the well-known frictionless Cosserat-rod models (e.g., [5], [11]), this equation can take into account the effects of distributed friction forces on tension loss and deformation behavior of TD-CMs.

Remark 4: Unlike previous studies (e.g., [15], [17], [16]), in the implementation of (6), there is no need for a priori known information or experimental measurements on tendon’s configuration (e.g., wrap angle or tendon curvature distribution) or TD-CM’s configuration (e.g., robot’s shape). The only known variable in this equation is the input actuating tension T_base which includes the applied tendon tension history and the other variables are not required to be known a priori.

Using (3), (5), and the Frenet–Serret Formulas [26], the further differentiations of p_c(s) can be obtained as follows:

Throughout the paper, $(\dots )\times (\dots )$ denotes the cross product of two vectors. Using (7), parameter ${\kappa }_c(s)$ in (6) can be expressed as:

The direction of tendon curvature (i.e., ${\mathit{n}}_{\text{tendon}}$) can also be obtained using (7):

As shown in the zoomed region B of Fig. 1, internal distributed forces are functions of tendon’s curvature, and tension distribution:

Derivation of (10) and (11) can be found in Appendix A. Substituting (5), (6), (8), and (9) into (10) and (11), the distributed normal force ${\mathit{f}}_n\left({\kappa }_c,T\right)$ can be written as:

The distributed friction force ${\mathit{f}}_f\left({\kappa }_c,T\right)$ can also be derived as:

In (12), ${\mathcal{P}}^{\star }{}_n$ is defined as the “projection factor” that can “project” tendon tension to the distributed normal force. Similarly, in (13), ${\mathcal{P}}^{\star }{}_{\mathit{f}}$ is defined as the “projection factor” that can “project” tendon tension to distributed friction force. Therefore, we can use a general physically-intuitive form to express the internal distributed forces as follow:

where ${\mathcal{P}}^{\star }$ represents ${\mathcal{P}}^{\star }{}_n$ or ${\mathcal{P}}^{\star }{}_f$.

Remark 5: Equation (13) gives the analytical form of distributed friction force along a routed tendon of TD-CMs, which either has been ignored or not been investigated by all the previous studies of both frictionless Cosserat-rod models (e.g., [5], [11]) and tendon-sheath systems (e.g., [15], [16]).

Remark 6: Equation (14) generalizes and provides a physically-intuitive understanding of internal distributed forces. When tendon tension is transmitted from the base of TD-CMs to a point on the curved routed tendon, it first experiences an exponential tension loss and then is “projected” to the normal and tangent directions of the tendon curve. The “projection factor” is characterized by the tendon configurations (e.g., tendon position and curvature) and the friction coefficient.

C. Distributed Moment on the TD-CM’s Midline

The distributed moment that a routed tendon applies to the TD-CM’s midline consists of the contributions from the anchor point force (see Fig. 1) and the described internal distributed forces. The anchor point moment is the cross product of the arm vector and the anchor point force vector:

Distributed moment can also be expressed as the integral of the cross product of the arm vector and the distributed forces as follows:

where ${\mathit{f}}_t\left({\kappa }_c,T\right)$ represents the sum of the internal distributed normal force ${\mathit{f}}_n\left({\kappa }_c,T\right)$ and the distributed friction force ${\mathit{f}}_f\left({\kappa }_c,T\right)$ along the GTR. Similar to (16), external distributed moment can be written as:

where ${\mathit{f}}_e(\xi )$ denotes the distributed forces along the TD-CM’s body and ${\mathit{p}}_s(\xi )$ is the position vector of a point on the TD-CM’s surface (shown in Fig. 1). Now, using the static equilibrium equation, we can relate the GTR effects as well as the internal and external distributed loads to the deformation behavior of TD-CMs [8]:

where E is the Young’s modulus and I(s) is the second moment of area of the TD-CM’s cross sections. ${\omega }_b(s)$ is the angular rate of change of the TD-CM’s configuration [8].

To solve such highly nonlinear model expressed in (18) with exponential terms in the integral calculation, we have developed an iterative solution algorithm. This algorithm requires the solution of the friction-less Cosserat-rod model (e.g., [11], [5]) in the first iteration. The obtained solution will then be used as an initial guess for iteratively solving the proposed friction-based model. More details of sovling equation (18) can be found in references [7], [8] and [18].

To capture the tension loss due to the friction and particularly hysteresis behavior of tendon-tension transmission in TD-CMs, we propose a procedure summarized in Algorithm 1. In this algorithm, we particularly need to obtain the direction of motion/friction of tendons to calculate the sign of $\lambda =-sign\left(T^{\prime }(s)\right)$ during the tendons’ actuation phases and identify the three pulling, transition, and releasing phases observed in TD-CMs experiencing tension loss [6], [16]. As shown in lines 1–5 of Algorithm 1, during the pulling phase, we first apply tendon tension T_base quasi-statically at TD-CM’s base from an initial zero to a maximum tendon tension T_{base max}. During this phase, the entire tendon moves towards proximal/base side of the manipulator and tension distribution decreases since the friction force accumulates along the length (i.e., $\lambda (s)=+1$) [6], [16]. Using the known λ(s) and for each applied T_base, we can calculate the tension at the TD-CM’s tip T_tip by solving equilibrium equation (18). Obviously, among these calculated T_tip, the maximum output T_{tip_max} is associated with the maximum applied T_base.

After the pulling phase, we start releasing the tendon in which the TD-CM first experiences a transition phase [6], [16]. In this phase, part of tendon moves from base toward the distal/tip end of manipulator and more specifically to a “sign transition point of λ(s)” while tension distribution is increased–due to the accumulated friction force in the opposite direction of tendon’s motion. Of note, beyond the “sign transition point of λ(s)”, there is no tendon motion and no change is observed in the tension distribution (i.e., λ(s) is partially −1 and partially +1) [6], [16]. Therefore, the maximum measured output tension at the TD-CM tip (i.e., T_{tip_max}) stays constant until the input tension at the base is below an unknown certain value that needs to be calculated. To obtain the whole transition phase, we need to find an input tension $T_{\text{base\_switch}}$ that can lead to a switch from a transition phase to a releasing phase where the entire tendon moves towards the distal end and tension distribution increases due to the accumulation of friction force along the entire length (i.e., $\lambda (s)=-1$). Therefore, as summarized in lines 6–15 of Algorithm 1, during releasing tendon, if input tension is greater than $T_{\text{base\_switch}}$, TD-CM will be in the transition phase in which output tension at the tip stays constant; otherwise, TD-CM is in the releasing phase. Using known λ(s) and T_base, the output T_tip can be calculated by solving (18). Of note, by setting $\lambda (s)=-1$ and using the T_{tip_max} as the input, equation (18) can be solved backward to calculate $T_{\text{base\_switch}}$.

A. Simulations

To evaluate the capability of the proposed model, we conducted simulations on a conical-shape TD-CM with 8 different GTR paths (i.e., path 1∼8), shown in Fig. 2 and Fig. 3. The TD-CM geometry is defined using the parameters shown in Fig. 1 i.e.,: L = 400 mm, R_outer,0 = 30 mm, and r_outer,0 = 15 mm. To investigate the GTR effects on tension loss of TD-CMs, the simulations of the considered eight GTR paths (i.e., path 1∼8) are performed with identical input tensions (i.e., T_base = 60 N and T_base = 40 N in Fig. 2 and Fig. 3, respectively). In general, as shown in Fig. 2 and Fig. 3, we have considered two different classes of GTRs. Fig. 2(a) illustrates a single curved tendon passing through a converging sinusoidal wave path (i.e., path 1∼4) from TD-CM base to the tip. Among these GTR paths, path 1∼3 are defined based on equation (19), while path 4 is defined based on equation (20) and the parameters detailed in Table I.

Moreover, in Fig. 3(a), a multi-segment GTR path of straight-sine-straight (i.e., path 5∼8) is considered and expressed by (21) and the parameters detailed in Table I.

Algorithm 1:

Solving the proposed model in pulling, transition, and releasing phases

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Fig. 2(a) and Fig. 3(a) compare the TD-CM deformation behavior as well as the distributed forces (i.e., f_n and f_f) along the routed tendon between these GTR paths. Fig. 2(b) and Fig. 3(b) illustrate the tendon tension distribution profile along the arc length parameter s. Fig. 2(c) and Fig. 3(c) show the tendon curvature distribution and wrap angle along the arc length parameter s. Of note, to solely investigate the effects of GTR and remove the influence of other factors, the distributed gravitational force is not considered in these simulations. However, in the performed experimental studies, the TD-CMs are under the influence of distributed gravitational loads and this effect has been considered in the modeling procedure.

B. Experiments

As shown in Fig. 4, to experimentally evaluate the performance of the proposed friction-based model, we molded two TD-CMs (i.e., TD-CM I and TD-CM II) with different GTR paths (i.e., path I and path II, respectively). The dimensions of these robots are identical and defined using the parameters shown in Fig. 1, i.e., L = 250 mm, R_outer,0 = 20 mm and r_outer,0 = 13 mm. As shown in Fig. 4(a), the diameter of the inner hole 1 of TD-CM I is 14 mm and its depth is 20 mm. The diameter of the inner hole 2 is 14 mm and its depth is 170 mm. The diameter of the inner hole 1 of TD-CM II is 14 mm with 70 mm length and the diameter of the inner hole 2 is 14 mm with 95 mm length (Fig. 4(b)). The GTR path I (shown in Fig. 4(a)) of TD-CM I and path II (shown in Fig. 4(b)) of TD-CM II are both defined based on (21). Details of parameters and experimental setup are shown in Fig. 4. More information about this setup can be found in [8].

During the experimental process, we first pulled the tendon using the linear actuation mechanism to continuously generate input tendon tensions at the TD-CM base (i.e., T_base = 0 N∼45 N with 5 N increment). In the pulling phase, we simultaneously measured the corresponding tension at the TD-CM tip using the aforementioned load cell. Similarly, we released the tendon and measured the tension at the tip of TD-CM (i.e., releasing/unbending phase) at the end of the bending phase. Of note, we repeated all experiments three times. Fig. 5 shows the tendon-tension transmission loss of the GTR in path I of TD-CM I from experiments, the proposed model, and the friction-less model output results during both bending and unbending phases. Similarly, Fig. 6 compares the tension transmission loss of the GTR in TD-CM II with path 2 from experiments, proposed model prediction, and friction-less model prediction during both bending and unbending phases.