On the Use of Tension Transition Zones for Kinematic and Compliance Performance Analysis of Wire-Actuated Continuum Robots

The design of distal dexterity devices for minimally invasive surgery (MIS) has been the focus of intense research over the past three decades. While some designs use embedded actuation within the robot structure (e.g., Refs. [1–4]), the vast majority of surgical robotic instrument designs use wire-rope (tendon) actuation (for example, Refs. [5–7] as commercial designs and Refs. [8–12] as academic works). Due to size, sterilization, and cost constraints, the actuation and joint-level sensing for tendon-actuated distal dexterity device are placed extracorporeally. This means that the force and motion transmission losses due to the actuation tendons experiencing cumulative friction along their routing pathways and due to wire compliance cannot be easily accounted for. The many works attempting to model these losses (e.g., Refs. [13–17]) have not propagated to industrial devices due to complexity and cost constraints. Therefore, these motion and force transmission losses are considered in this paper as joint-level uncertainties leading to uncertainty in the end-effector position and orientation (pose).

This paper is concerned with the simulation-informed design of wire-actuated distal dexterity devices while considering that actuation tendons can only transmit tension forces. We ignore designs employing Bowden cable/conduit pairs that may also transmit compressive forces because, when the wires pass through the small distal dexterity device, applying full constraint along the wire-rope path is impractical. Therefore, this paper aims to inform medical instrument/robot designers about the proper procedure for modifying the traditional kinematic performance measures and compliance to bind the expected end-effector pose uncertainty due to some assumed joint-level uncertainty contributed by wire extension/compliance and motion transmission losses.

We use continuum robots as a generic architecture for illustrating the concepts in this paper. These robots, as illustrated in Fig. 1 have a fixed-length central backbone surrounded by actuation tendons that are circumferentially distributed equidistantly at a known pitch circle. While the illustration shows a bending central backbone, there are alternate design embodiments that use a constrained central backbone made of disks with rolling cams [8,18,19], sliding cylinders/disks/spheres [10,20–22], or cross revolute joints [11]. The analysis and concepts presented in this paper may be applied to each of these architectures, and we use the architecture in Fig. 1 to illustrate these concepts. Of course, the instantaneous kinematics derivation for each of these architectures will vary. Still, the overarching theme of accounting for zones of the workspace where wires may go slack persists through all these design variations.

Previous works on manipulability measures for wire-actuated devices mainly focus on wire-actuated parallel mechanisms/robots [23–25]. Shen et al. [23] discussed the need for considering force constraints, i.e., wire tension should always be greater than zero, in the kinematic analysis of the wire-actuated mechanisms. They also derived the manipulability measures using almost the same methodologies as a conventional rigid-link manipulator. This chapter discusses how we partially revise the definitions of dexterity measures for conventional rigid-link manipulators by considering the fact that not all wires actively contribute to performing the commanded movement.

The most relevant works to this work are prior research addressing wrench closure for wire-actuated robots. Kurtz and Hayward [26] suggested modified performance measures taking into account the wire tensions in wire-actuated parallel robots. They expressed the bounds on the dexterity measures as a function of the maximal wire force in the direction where the Jacobian nullspace is smallest. In Ref. [27], the force closure conditions of a cable-driven serial chain were considered. Configurations of wire routing were considered where force closure may be maintained. In contrast to these works, Jin et al. considered the effect of loss of wrench closure on force transmissibility performance measures in parallel mechanisms.

Our work differs from these in that we consider practical considerations for wire-actuated surgical wrists that do not necessarily assume that actuation redundancy is a practical way to maintain force/wrench closure. This assumption stems from market-driven competitive cost constraints for these surgical devices.

In light of the aforementioned works, the contribution of this paper is in introducing the simple notion of tension transition zones for robots without reliance on actuation redundancy. We show how this concept is easily applied to continuum robots with tendon actuation, and we also show how to use this concept to consider the portion of the Jacobian corresponding to the tendons that can remain in tension. We show how this approach produces more conservative (larger) predictions of end-effector pose uncertainty compared to the traditional approach, where the full Jacobian is (incorrectly) used. Finally, we also consider the effects of the tension transition zones on the computation of the configuration-space compliance of continuum robots.

For the case study presented in this paper, we consider a continuum robot comprised of a flexible central backbone, $nd$ spacer disks (for the continuum architectures considered in this work and presented in Fig. 1, we assume $nd=6$⁠), and $nw$ actuation wires that are circumferentially distributed on a pitch circle with radius $r$⁠. Figure 1 provides a visual representation of the key parameters and key coordinate frames used to define the kinematics of the continuum segment. In the following subsections, we follow the modeling approach in Refs. [28,29] and present these results for completeness.

In this section, we discuss the notion of wire tension zones and wire tension maps that take into account the continuum segment bending directions subject to the limitations of using the actuation wires only as tension elements.

Using individual wire tension zones, we define the wire tension map for a given continuum robot design configurations. A wire tension map considers the intersection of wire tension zones for all actuation wires. It can be used to understand the actuation limitations of a given design configuration and to inform the computation of dexterity. To illustrate these concepts, we next consider two design configurations with three or four wires for actuation.

Using a full circle to represent $δ∈[0,2π]$⁠, we can draw wire tension zones as circular sectors. Figure 2 presents the wire tension zones for a three-wire continuum segment. Zone $Δ1$ is shown on the base disk, and the other zones are offset for clarity.

The wire tension zones in Fig. 2 are drawn for $δ˙=0$⁠. We note that different values for $δ˙≠0$ correspond with a phase shift of these tension zones.

Figure 3 shows this phenomenon for the second wire. In this figure, the shaded areas denote wire tension zones (ranges along the horizontal axis) where $q˙2<0$⁠.

Figure 4 shows plots of the joint speeds $q˙k,k=1,2,3$ for sample combinations of $δ˙=0$⁠, $δ˙>0$⁠, and $δ˙<0$⁠. Shaded zones on the x-axis denote the radial bending directions corresponding to the wire tension zones. As stated above, these tension zones undergo a rotation (phase shift). Therefore, for the rest of the paper, we will proceed with the tension zones depicted in Fig. 2.

By overlaying the individual wire tension zones, the wire tension map shown in Fig. 5 is obtained. Note that for the three-wire design configuration, there are zones of bending directions where only one wire is in tension. These zones alternate with other zones where two wires are in tension. In this figure, tension zones where two wires are in tension are denoted by the circled notations

When considering the transitions that occur when the bending direction passes between zones of overlap to no overlap, one needs to consider the scenario where two wires initially in tension transition into only one wire in tension. Figure 6 shows a time strip of the wire tension transitions that occur as we slowly sweep the continuum segment bending direction in a clockwise direction. The arrow denotes the direction of bending of the continuum segment (the projections of the continuum segment on the base disk). Actuation wires denoted by hollow circles depict wires outside their tension zones for that particular bending direction.

When the bending plane falls within the patterned zones shown in Fig. 6, only one wire can be in tension to promote the bending of the continuum segment actively. Considering the case of bending in $δ=0deg$⁠, we note that a filled dot shows wire 1 because it is the only wire that can promote bending by pulling. Also, backbones 2 and 3 are shown as hollow circles because they lie outside their respective tension zones. As $δ$ increases and passes $30deg$⁠, the arrow exits the shaded zone wires 1 and 3, which can promote bending by tension (e.g., see $δ=66deg$ configuration). As $δ$ increases to $90deg$⁠, the arrow enters the shaded zone, and again, only wire 3 can promote bending by pulling. Using this logic, the figure continues depicting these tension transitions as the bending direction continues, sweeping a full circle in the clockwise direction.

Repeating the same process for the four-wire configuration leads to the wire tension map shown in Fig. 7. For clarity, only the wire tension zone for wire 1 is drawn within the circle, and the others are offset. While in the three-wire configuration, there are zones where adjacent wire tension zones do not overlap, in the four-wire design configuration, the individual wire tension zones always overlap. This means that the zones where only one wire can pull the continuum segment to bend it end up collapsing to the four lines, as shown in Fig. 7 for the bending directions passing through each wire. In all other zones, the four-wire segment will always have two wires pulling the segment to achieve a desired bending angle.

Considering the wire tension transitions as the bending direction is swept a full circle in the clockwise direction, we obtain the result shown in Fig. 8. The figure shows that with the exception of bending directions $δ=0deg$⁠, $90deg$⁠, $180deg$⁠, $270deg$⁠, the four-wire configuration has two wires capable of pulling to share the load while bending the continuum segment.

Having established the notions of wire tension maps, we next discuss how the modeling in Sec. 2 (which is for push-pull designs) needs to be modified to account for wire rope (pull only) actuation. We then present a simulation study to compare the three-wire and four-wire design configurations in terms of dexterity, stiffness, and expected tip deflections. This subsection discusses the key modifications needed to account for pull-only actuation.

The original configuration-to-joint-space mapping $Jqψ$ in Eq. (9) does not take into account whether a wire is pulling or pushing. Therefore, it can yield joint speeds that cause slack in specific cables. Since such slack wires cannot promote active bending, we must remove them from consideration when we carry out the kinematics, statics, and dexterity analysis.

By using the wire tension transition maps in Figs. 6 and 8, it is possible to discern which wires do not promote active bending by pulling. Alternately, one could use Eq. (13) to test each wire for $ψ˙$ within the unit sphere and numerically construct the wire tension zones by definition. Once these wires are identified, a reduced/modified Jacobian$J~qψ$ is obtained from $Jqψ$ by eliminating the rows that correspond to the wires that cannot promote bending while pulling. The instantaneous direct kinematics Jacobian $Jξq$ is then obtained by using $J~qψ$ instead of $Jqψ$ in Eq. (12).

In this section, a preliminary model of the compliance of the continuum wrist will be presented. The starting point for derivation will be a simplified statics model, which assumes a constant curvature bending profile and negligible frictional effects. This simplifying assumption in the case of bounding end-effector pose uncertainty due to compliance for a particular design is justified because friction can help with reducing the device deflection, and ignoring it results in a conservative estimate for the end-effector pose uncertainty. Additionally, since most surgical devices are small and lightweight, we assume that the potential energy of the wrist is dominated by elastic bending, with gravitational energy ignored.

The elastic potential energy of a wire-actuated continuum wrist can be defined by Euler-Bernoulli beam bending as $U=EYIθ~22L$ where $EY$ and $I$ refer to Young’s modulus and cross-sectional area moment of inertia of the central backbone, respectively. The length of the bending segment is referred to as $L$⁠, and the magnitude of bending is referred to as $θ~=θL−θ0$⁠. This follows the notation presented in Sec. 2 where $θL$ is the angle from $x^1$ to $z^2$ and $θ0=π2$⁠.

This second derivative term is the Hessian of the bending energy and can be found from the gradient of Eq. (24).

In the preceding expression, the term $JξψTΔwg$ refers to a perturbation of the generalized forces applied to the tip of the continuum segment as defined in configuration space. This can be redefined as a perturbation of a generalized configuration space force, $Δf$⁠. Following Hooke’s Law, the term in between brackets represents the configuration space stiffness, $Kψ$⁠. The inverse of this matrix results in the configuration space compliance matrix (⁠$Cψ=Kψ−1$⁠), which when multiplied by configuration space forces returns a configuration space displacement.

It must be noted that a force in configuration space is two-dimensional (⁠$f∈R2$⁠), while a full task space wrench is six-dimensional (⁠$wg∈R6$⁠). When transforming from the lower dimensional configuration space to the higher dimensional task space, only a subspace within the task space can be fully defined. Therefore, the transformed stiffness matrix in task space (Eq. (30a)) is simply a minimum-norm solution of the stiffness matrix, and additional constraints would need to be added for a more exact result.

To illustrate the use of the modeling approach and the reduced Jacobians based on wire tension zones, we conduct simulation studies using the traditional method with the full and reduced Jacobians. We compare the expected performance of the same continuum segment with two design alternatives using three or four wires.

To compare performance, we use kinematic conditioning and end-effector pose uncertainty using the traditional and reduced Jacobian, and we also compare the three-wire and four-wire designs.

To compare the effect of using the reduced Jacobian on the expected performance of the three-wire and four-wire architecture, we used the local kinematic conditioning index (KCI) defined as the inverse condition number³ and the minimal singular value $σmin$ of both the translational and orientational portions of $Jξq$⁠. Referring to Eq. (12), we compare the KCI and $σmin$ when using $Jqψ$ versus when replacing it with the reduced Jacobian $J~qψ$ to produce $J~ξq$⁠. We then split the resulting Jacobian into the top three rows (translational Jacobian $Jξqv$ or $J~ξqv$⁠) and bottom three rows (orientational Jacobian $Jξqω$ or $J~ξqω$⁠). After producing these matrices, we compute their KCI and the minimal singular values.

Figures 9 presents the KCI across a portion of the configuration space (of $θL∈[0deg,5deg,10deg,…,85deg]$ and $δ∈[0deg,2.5deg,5deg,…,360deg]$⁠). Figures 9(a) and 9(b) show the surface plots of the kinematic isotropy determined using the traditional Jacobian (push-pull actuation) and reduced Jacobian (pull-only actuation) for the three-wire design alternative, respectively. Note that the abrupt jumps in Fig. 9(b) occur when the segment of a three-wire design transitions from having only one wire in tension to having two wires in tension. Such transitions occur at $δ=π/6,5π/6,$ and $3π/2$ as shown in Figs. 5 and 6. Unlike the three-wire design, the four-wire alternative does not suffer from such transitions because there are always at least two wires in tension. The KCI plot for the four-wire design using the traditional Jacobian results in an identical surface plot as in Figs. 9(a) and 9(d).

Figure 10 repeats the analysis for $σmin$⁠, which is associated with the least dexterous motion direction given by its corresponding singular vector. Again, the results are split for translation and orientation. In Figs. 10(a) and 10(d), we used the traditional Jacobian $Jξq$ while in Figs. 10(b) and 10(e) we used the reduced Jacobian $J~ξq$⁠.

Both Figs. 9(b), 9(e) and 10(b), 10(e) show that the KCI and $σmin$ using the reduced Jacobian exhibit a periodic behavior with a period equal to the separation angle between the wires (⁠$120deg$ versus $90deg$ for the three- and four-wire designs, respectively). As the continuum segment bends, its variations in its KCI and $σmin$ as a function of $δ$ increased as expected since the moment arm of the actuation wire about the bending axis of the continuum changes as a function of the cosine of the angle of the wire’s radial angle relative to the bending axis.

A more easily interpretable performance measure to a designer is the expected end-effector backlash/position uncertainty for a given joint-level position uncertainty. We, therefore, used the same translation and orientation Jacobians to compute the pose uncertainty using Eq. (19). For both three-wire and four-wire design alternatives, we determined uncertainty in end-effector pose across the same portion of the configuration space. We also compared the expected end-effector pose uncertainty using both the traditional (push-pull actuation) and reduced (pull-only actuation) Jacobians. The values used for $|Δq|$ (joint-level uncertainty) ranged from 0.2 mm to 1 mm with increments of 0.2 mm.

Figure 11 presents the comparative results of the backlash-induced uncertainty in the end-effector position $Δxgv$⁠. The plots shown in Fig. 11 represent the maximum uncertainty in the position of the end-disk for the joint-level uncertainty of $|Δq|=0.4mm$⁠. Note that the valleys in Figs. 11(a) and 11(b) correspond with the transitions from only one wire being in tension to two wires in tension (refer Figs. 6 and 8 for depictions of different wire states). The maximum uncertainty in the end-effector position for three-wire and four-wire designs are compared in Fig. 11(d). The three-wire design reports an increase in end-effector uncertainty by 20-50% depending upon the segment configuration. Figure 11(c) shows how sensitive positional uncertainty is to different amounts of joint-level uncertainty. Based on the surface plots in Fig. 11(c), we can say that the three-wire design is about 39% more sensitive to the joint-level uncertainty than the four-wire design alternative.

Similar observations are made for orientation uncertainty as shown in Fig. 12. Figure 12 presents the comparative results of the backlash-induced uncertainty in the end-effector orientation $Δxgω$⁠. The plots shown in Fig. 12 represent the maximum uncertainty in the orientation of the end-disk for the joint-level uncertainty of $|Δq|=0.4mm$⁠. Note that the valleys in Figs. 12(a) and 12(b) correspond with the transitions from only one wire being in tension to two wires in tension (refer Figs. 6 and 8 for depictions of different wire states). The maximum uncertainty in the end-effector orientation for three-wire and four-wire designs is compared in Fig. 12(d). The three-wire design reports an increase in end-effector uncertainty by 25–45% depending upon the segment configuration. Figure 12(c) shows how sensitive orientational uncertainty is to different amounts of joint-level uncertainty. Based on the surface plots in Fig. 12(c), we can say that the three-wire design is about 45% more sensitive to the joint-level uncertainty than the four-wire design alternative.

In this subsection, we consider the effect of using the traditional versus reduced Jacobian for bounding the end-effector uncertainty due to compliance. We also consider whether a three-wire design can achieve similar performance by changing wire stiffness. In these simulations, the configuration space was swept within reasonable ranges $θL=[89deg,84deg,…,45deg]$ and $δ=[0deg,15deg,…,360deg]$ for both three-wire and four-wire configurations. The wrench applied to the end-effector (⁠$wg=[−5N,0,0,0,0,0]T$⁠) is always in the same direction with respect to the world coordinate frame. The simplified statics model using Eq. (23) was used for this with the constraint that each joint-level force is in tension, $τi≤0$⁠. The model was solved as a constrained optimization problem using the interior point method in matlab. Additionally, the assumption here is that no preload is applied to the actuating wires unless explicitly reported, in which case the joint-level force are constrained to the bounds $τi≤fmin$⁠, where $fmin$ refers to the minimal wire preload.

In Fig. 13 and Table 1, the compliance computed across the workspace is reported as the magnitude of positional deflection (Fig. 13(a)) and angular deflection (Fig. 13(b)). Throughout the workspace, the deflections between the three-wire and four-wire designs are compared to show the relative increase in compliance. With no pretension, the three-wire designs report increases in deflection by about 85–105% depending on the segment configuration compared with the four-wire design. The average increase throughout the workspace for linear and angular deflections is 95.9% for the three-wire design.

In Figs. 13(e)–13(h), the deflections are computed using the traditional (non-reduced) joint-configuration space Jacobian, which considers stiffness in slack actuation lines. For the case of no pretension (Figs. 13(e) and 13(f)), the traditional Jacobian greatly under-predicts the observed deflection. However, by comparing Fig. 13(c) to Fig. 13(g) and Fig. 13(d) to Fig. 13(h) we conclude that the use of pretension decreases predicted deflections greatly and that deflections predicted using the reduced Jacobian match the traditional Jacobian. In the case of pretension, each tendon is enforced to be in tension affect the stiffness of the device, and thus no reduction of the Jacobian occurs.

Given the differences in compliance under the three-wire and four-wire design alternatives shown in Fig. 13, there may exist design flexibility to explore. Specifically, the wire stiffness, $kw$⁠, may be increased to increase joint-level stiffness, affecting the robot’s stiffness. While we assumed a generic median cable stiffness of 75 N/mm, additional simulations were performed to evaluate compliance across the workspace for joint-level stiffnesses of 55 N/mm, 65 N/mm, 85 N/mm, 95 N/mm, and 105 N/mm. Results for both three-wire and four-wire at these varying stiffness ranges were included in Figs. 14(a)–14(d), while means and standard deviations of compliance across the workspace are reported in Table 2. For each stiffness range, it was found that the three-wire design results in average positional deflections, which are around 90–99% greater than the four-wire design alternative. Additionally, it can be seen that similar deflection ranges can be achieved between three-wire and four-wire designs with different wire stiffness. In Table 2, for example, a three-wire design with wires of 105 N/mm stiffness can achieve better positional deflections than a four-wire design with wires of 55 N/mm stiffness.

This paper presented the notions of wire tension maps and zones for wire-actuated continuum robots and wrists used for surgery. We put forth the notion of using the reduced Jacobian according to the tension transition maps to account for the wires capable of inducing motion/force via pulling. We also used the three-wire and four-wire design configurations as case studies for a simulation study that compared kinematics-based end-effector uncertainty and compliance-induced end-effector uncertainty.

Our simulation study showed several things: (1) the use of the traditional Jacobian overestimates the expected KCI of the robot since it assumes that all joints can contribute to motion using push-pull actuation; (2) the minimal singular value when computed using the reduced Jacobian is larger than when using the traditional Jacobian, which again demonstrates that the use of the traditional Jacobian would underestimate the noise amplification from joint space to task space; (3) the minimal singular value for the three-wire design is larger than the four-wire design, which means that the three-wire design would suffer larger joint space uncertainty amplification; (4) the addition of an extra wire can help improve the KCI; (5) the use of the traditional Jacobian underestimates the compliance-induced tip uncertainty compared to the reduced Jacobian and the use of wire pretension helps attenuate these deflections. The comparisons above also looked at the three-wire versus four-wire designs as case studies and showed performance improvement with an increase in the number of actuation wires. However, this increase may not be practical within the context of surgical instruments due to the associated loss of available cross-sectional area and the additional cost of an actuator. Also, we showed that the use of pretension and an increase in wire stiffness can result in a three-wire configuration with similar performance to the four-wire configuration. The key point is to alert designers to the proper use of wire tension maps and the reduced Jacobian when comparing design alternatives.

This work was partly supported by funds from J&J Ethicon—Robotics and Digital Solutions, under SRA00000096AM1 and Vanderbilt University funds.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$k$ =

index of the secondary backbones (wires/cables), $(k=1,2,…nw)$ where $nw$ is the number of actuation wire ropes (cables).

$s$ =

arc-length parameter of the central curve that passes through the center of the discs. At the base disk $s=0$ and at the end disk $s=L$

$r$ =

pitch radius of the holes on each disk through which secondary backbones (cables) are passed

$L$ =

length of the central curve

$q$ =

a joint space vector, $[l1l2⋯lk⋯lnw]T∈Rnw$⁠. Here $lk$ is the length of the $kth$ wire

$lk$ =

length of the $kth$ secondary backbone (cable) measured from the base disk to the end disk

$nw$ =

a number of actuation wires

${B}$ =

a Cartesian coordinate system⁴ defined at the base of the continuum segment.

${1}$ =

a Cartesian coordinate system whose origin coincides with the origin of the frame, ${B}$⁠, and the orientation is given by the rotation of $(−δ)$ about $z^b$⁠. The $xz$ plane of this frame is the bending plane of the continuum segment.

$Jqψ$ =

a Jacobian matrix that maps configuration space velocity, $ψ˙$⁠, to joint space velocity, $q˙$

$Jξψ$ =

a Jacobian matrix that maps configuration space velocity, $ψ˙$⁠, to the task space twist, $bξg/b$

$Jξq$ =

a Jacobian matrix that maps joint space velocity, $q˙$⁠, to the task space twist, $bξg/b$

$A†$ =

denotes Moore–Penrose inverse of the matrix $A$

$β$ =

division angle defined as, $β=2πn$ where n is the number of secondary backbones (cables)

$δ$ =

a right-handed rotation angle measured from $x^1$ about $z^b$ to the bending plane

$δk$ =

a right-handed rotation angle measured from the bending plane about $z^1$ to a line passing through the central curve and the $kth$ secondary backbone at $s=0$

${2}$ =

a Cartesian coordinate system whose origin coincides with the center of the end disk and the orientation is given by the rotation of $(θ0−θL)$ about $y^1$

${G}$ =

a Cartesian coordinate system whose origin coincides with the origin of the frame, ${2}$⁠, and the orientation is given by the rotation of $δ$ about $z^2$

$ψ$ =

a configuration space vector, $[θδ]T∈R2$

$θ(s)$ =

an angle of the central curve tangent in the bending plane at some arc length $s$ along the curve. The $θ(s=L)$ and $θ(s=0)$ are denoted by $θL$ and $θ0$ respectively. Also, the value of $θ0$ is constant and is equal to $π2$⁠.

$ρ$ =

radius of curvature. Considering the constant curvature bending model, it is defined as $Lθ0−θL$