A Robotic System With a Bendable Catheter for Laparoendoscopic Single Site Surgery

Hu, Zhenxuan; Li, Jinhua; Liu, Xingchi; Wang, Shuxin

doi:10.1115/1.4065730

Abstract

Robotic laparoendoscopic single site surgery (LESS) is emerging as a hot research topic with the advancement of robotics. However, the movement of the rigid catheter inserted through a fixed incision into the patient's body cavity is constrained to a conical workspace, and the surgical instruments introduced through channels in the catheter can hardly perform necessary operations when the target tissue is close to the boundary of this workspace. In this paper, we present a novel robotic system for LESS with a bendable catheter. The diameter of the bendable catheter is 30 mm and the length is 22 mm. The bendable portion of the catheter provides two degrees-of-freedom (DOF) within the body cavity and can be bent up to 45 deg. The system consists of two continuum instruments with 6DOF and a three-dimensional endoscope with 5DOF. System design, kinematic analysis, and teleoperation algorithm are introduced in detail. The simulation shows that the catheter centerline can be oriented toward the target tissue over a large area, thus providing a better initial position for the surgical instruments and enlarging the workspace of the instruments. Preliminary experiments are performed to verify the feasibility and effectiveness of the proposed system. The results prove the applicability of the system in LESS.

1 Introduction

Laparoendoscopic single site surgery (LESS) has been of great interest to researchers for decades. Compared with open surgery, LESS has the advantages of faster recovery, less adhesions, and reduced risk of infections [1]. However, there are still some drawbacks that limit its application in clinic.

A rigid catheter must be inserted into the patient's body cavity through a fixed incision, thereby its movement is restricted. The tip of the catheter can only scan a conical space, which cannot directly face the target tissue located near the boundary of the workspace. This may cause inconvenience in the operation of surgical instruments introduced through the catheter. In addition, the surgical instruments are introduced through channels arranged symmetrically around the centerline of the catheter. When the target tissue is close to the centerline of the catheter, the overlapped workspace of the instruments can cover a large fraction of the tissue. However, if the target tissue is far from the centerline, only one instrument can manipulate the tissue in a convenient pose, or the overlapped workspace can only cover a small region of the tissue as shown in Fig. 1(a). Sometimes none of the instruments can even touch the tissue.

Fig. 1

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Comparison of structures and system overview: (a) rigid catheter, (b) bendable catheter, and (c) system overview of the proposed robot

As the instrument is the primary execution unit of the robotic surgical system, its performance significantly affects the quality of a surgery. In recent decades, a large number of robotic LESS instruments have been proposed to realize more efficient surgical procedures, and stiffness and accuracy are the main concerns in their design. Serial rigid links connected by joints [2–4] and manipulators featuring embedded motors [5] are common design solutions. Linkage mechanisms [6–9] or plate springs [10] are usually used to strengthen the joint torque of manipulators. However, these mechanisms inevitably occupy a significant amount of space inside the patient's body cavity, which limits their dexterity during surgical procedures in narrow cavities and ability to reach the target region.

The continuum mechanisms provide an alternative option for instrument design, whose inherent flexibility ensures easier insertion of the manipulators into unstructured environments [11]. Nitinol rods [12] or tubes [13] and springs [14] are the main components for designing continuum manipulators. Although many continuum robots developed to date have not yet been fitted with end-effectors [15] or their end-effectors are limited in some motions [16], the performance of continuum instruments in surgical application is still attractive. As the continuum structures are more flexible than rigid joints, they can be more easily inserted into tortuous channels and reach the areas inaccessible to rigid instruments.

Moreover, the multistage adjustment of the instrument pose within the body cavity must be guaranteed to make the end-effectors approach the target tissue. Traditionally, this is mainly realized by adjusting the manipulator through additional joints with a large range of motion. However, these joints may result in a loss of accuracy and load capacity, and the extra degrees-of-freedom (DOF) may present control difficulties. Furthermore, in a typical robotic LESS procedure, a surgeon operates two robotic surgical instruments simultaneously via a master console. When either instrument is being replaced, the other one remains stationary. When the target position is far from the catheter centerline, the new instrument needs to be repositioned under the supervision of the endoscope, which requires an adjustment of its pose during the insertion. In this progress, the view of the unchanged instrument will be lost, increasing the risk of interference between tissue and instrument.

An alternative approach to extend the ability of the end-effector to approach the target tissue is to adjust the catheter that guides the instruments and the endoscope. A bendable structure with a low slenderness ratio can be applied to the straight sheath of a LESS system, providing greater orientation control of the tip [17], as shown in Fig. 1(b). Therefore, the instruments and the endoscope can be adjusted simultaneously, which ensures that the relative position between instruments and the endoscope remains constant. Lee et al. presented a robotic platform for LESS with a 6DOF guide tube capable of reaching various surgical sites within the abdominal cavity [18]. Although their guide tube can provide additional DOF to help the instruments locating, its length is relatively large (15 cm for the proximal segment and 10 cm for the distal segment) and the instruments are not replaceable.

This study presents a new robotic LESS system with a bendable catheter, as shown in Fig. 1(c). The bendable catheter is designed to extend the ability of instruments to approach target tissue in the body cavity. System design, kinematic analysis, and teleoperation algorithm are introduced in detail. Experiments are performed to verify the ability of the proposed continuum instrument.

2 Methods

2.1 Design of the Bendable Catheter.

The bendable catheter was designed to be adjustable within the body cavity to extend the operating area of the surgical instruments. The catheter provides two bending DOF within the body cavity and can be bent to 45 deg. The length of the bendable portion was designed to be 22 mm, which is much shorter than Olympus gastroscopy (about 80–120 mm). As the size of skin incision to access the abdominal cavity has a maximum diameter of 30–35 mm [19], the outer diameter of the catheter was set to 30 mm, providing two instrument channels and one endoscope channel, as shown in Fig. 2(a). Moreover, the rigid portion of the catheter could provide roll, pitch, and yaw motions via a robotic arm, which is not the main focus of this paper and is therefore not described in detail.

Fig. 2

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Structure of bendable catheter and rolling joint: (a)assembly of the bendable catheter, (b) structure of rolling joint, and geometric relations for deriving equations of (c) groove and (d) tooth

The bendable structure comprises two rolling joints, which has the potential to move smoothly and stably, and is easily miniaturized [20]. The radius R of the rolling surface is 3 mm. To prevent sliding and twisting of the rolling surfaces, a pair of teeth and grooves are designed on both sides parallel to the joint rotation plane as shown in Fig. 2(b). The profile of the teeth and grooves is cycloidal. Based on the geometric relations shown in Fig. 2, the cycloid equation of grooves and teeth can be derived.

Figure 2(c) shows the coordinate systems established to derive the cycloid equation of the groove. The joint with grooves is fixed and the joint with teeth rolls relative to the fixed joint. The trajectory traveled by the tip of the tooth makes the contour of the groove. Then the coordinates of the rolling center O_rt in the coordinate system x-O_fg-y can be deduced from geometric relations

{\begin{matrix} x_{O_{r t}}^{O_{f g}} = 2 R sin α \\ y_{O_{r t}}^{O_{f g}} = R (2 cos α - 1) \end{matrix}

(1)

where $α$ is half the rolling angle.

Let the height of the tooth be r, the coordinates of the tooth tip in the coordinate system x-O_rt-y can be obtained

{\begin{matrix} x_{a}^{O_{r t}} = - (R + r) sin 2 α \\ y_{a}^{O_{r t}} = - (R + r) cos 2 α \end{matrix}

(2)

Then the coordinates of the tooth tip in x-O_fg-y are obtained

{\begin{matrix} x_{a}^{O_{f g}} = x_{O_{r t}}^{O_{f g}} + x_{a}^{O_{r t}} \\ y_{a}^{O_{f g}} = y_{O_{r t}}^{O_{f g}} + y_{a}^{O_{r t}} \end{matrix}

(3)

Equation (3) is also the cycloid equation for the profile of the groove.

To obtain the cycloid equation of the tooth, a similar procedure is performed. The joint with teeth is fixed and coordinate systems are established as shown in Fig. 2(d). The joint with grooves rolls relative to the fixed joint. The trajectory traveled by an edge point

(x_{b}^{O_{f t}}, y_{b}^{O_{f t}})

of the groove makes the contour of the tooth. The coordinates of the rolling center O_rg in the coordinate system x-O_ft-y and coordinates of the edge point are obtained

{\begin{matrix} \begin{matrix} x_{O_{r g}}^{O_{f t}} = 2 R sin α \\ y_{O_{r g}}^{O_{f t}} = R (2 cos α - 1) \\ x_{b}^{O_{r g}} = - R sin (2 α + α_{1}) \end{matrix} \\ y_{b}^{O_{r g}} = - R cos (2 α + α_{1}) \end{matrix}

(4)

The cycloid equation of the tooth can be obtained as

{\begin{matrix} x_{b}^{O_{f t}} = x_{O_{r g}}^{O_{f t}} + x_{b}^{O_{r g}} \\ y_{b}^{O_{f t}} = y_{O_{r g}}^{O_{f t}} + y_{b}^{O_{r g}} \end{matrix}

(5)

Since the profile of the groove is formed by the trajectory of the tip of the tooth and the profile of the tooth is formed by the trajectory of the points on the edge of the groove, there will always be three points that remain in contact during the movement. The joints are manufactured by three-dimensional printing. Since the bendable catheter is primarily utilized for reorientation rather than precise manipulation, the friction between the joints is not considered.

2.2 Manipulator and Endoscope.

To accommodate the bendable catheter, the manipulator was designed with a continuum mechanism. Torsion is an essential factor to consider in designing continuum mechanisms as it can potentially give rise to unstable elastic behaviors, as well as kinematic inaccuracies [21]. Therefore, an elastic bellow was selected as the elastic joint of the manipulator to provide high torsional resistance.

As shown in Fig. 1(c), the manipulator arm consists of two elastic joints. The extension length of the first joint varies with the forward and backward movement of the instrument. Therefore, the lengths of the joints are designed to be 110 mm and 22 mm. Similar designs of continuum mechanism are introduced in Ref. [22]. These designs focus on the overall morphology and path planning of the continuum robot, such as the follow the leader motion, but we are more concerned with the pose of the end-effector. Each elastic joint provides two bending DOF and has a bending angle of at least 180 deg. The inner diameter of the joints is 3 mm and the outer diameter is 7 mm.

The endoscope consists of a camera head with a diameter of 10 mm and a two-segment continuum positioning arm which provides 5DOF as depicted in Fig. 1(c). The camera head is composed of two video cameras with a resolution of 1920 × 1080 and a length of 20 mm. Each segment of the positioning arm provides two bending DOF with a maximum bending angle of 90 deg. Two bundles of optical fibers are integrated into the camera head to provide light sources for the endoscope.

2.3 End-Effector and Axial Compression Resistance Mechanism.

The instruments should be able to rotate at the distal end, so as to avoid rotation of the entire manipulator in the catheter, which would lead to transmission errors and require a compensation algorithm [23]. In this case, our end-effector was designed to provide end rotation and grasping motion, whose structure is shown in Fig. 1(c). The drive cables are wound around a rotation shaft and guided toward the center of the continuum manipulator by a pair of guide wheels. When the drive cables are pulled, the rotation frame fixed to the grasper will rotate simultaneously with the rotation shaft. The end-effector can provide a rotation angle of ±180 deg.

As the elastic joints may be compressed during pretightening and pulling of the drive cables, a compression resistance mechanism should be inserted along the joint axis. As shown in Fig. 3, four 0.5 mm holes were drilled in each joint of the compression resistance mechanism, through which the drive cables for the end rotation and grasp pass. According to Refs. [14] and [24], the addition of passive rigid joints does not significantly affect the bending motion of elastic joints.

Fig. 3

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Compression resistance mechanism

2.4 Kinematics.

Although there are more accurate models based on mechanics or other methods [25,26], we still utilize the most commonly used constant curvature model due to the fact that the load of the instrument is not a constant value and the difficulty in sensing the shape and forces of a continuum robot. The first elastic joint of the manipulator is set to partially retract into the catheter. The length of the first joint outside the catheter is variable along with the movement of the manipulator. This mode of motion improves the maneuverability of the instrument in a limited workspace [27].

Eight and four cable channels with a diameter of 0.55 mm are evenly distributed along the circumference on first and second elastic joint, which are numbered as shown in Fig. 4(a). The definition of terms involved in kinematic analysis are shown in Table 1 and Fig. 4(b). ${{\hat{x}}_{i b}, {\hat{y}}_{i b}, {\hat{z}}_{i b}}$ and ${{\hat{x}}_{i e}, {\hat{y}}_{i e}, {\hat{z}}_{i e}}$ denote the base and end coordinate systems of the $i th$ elastic joint and are attached to the base and end plane of the joint. Moreover, ${\hat{x}}_{i b}$ and ${\hat{x}}_{i e}$ axes point to cable channel ${11}$ ⁠, ${\hat{y}}_{i b}$ and ${\hat{y}}_{i e}$ axes point to cable channel ${13}$ ⁠. It is important to note that ${{\hat{x}}_{1 b}, {\hat{y}}_{1 b}, {\hat{z}}_{1 b}}$ is also the base coordinate system of the entire manipulator with end-effector.

Fig. 4

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Coordinate systems of elastic joints: (a) numbering of cable channels and (b) coordinate systems of an elastic joint

Table 1

Definition of terms in kinematic analysis

Term	Description
$O_{i}$	Bending center of the $i th$ elastic joint
$O_{i b}$	Origin of ${{\hat{x}}_{i b}, {\hat{y}}_{i b}, {\hat{z}}_{i b}}$ coordinate frame
$O_{i e}$	Origin of ${{\hat{x}}_{i e}, {\hat{y}}_{i e}, {\hat{z}}_{i e}}$ coordinate frame
$φ_{i}$	Rotation angle between the bending plane and ${\hat{x}}_{i b}$ axis. $φ_{i} \in [- π, π]$ ⁠.
$θ_{i}$	Bending angle of the $i th$ elastic joint. $θ_{1} \in [0, π / 2]$ $θ_{2} \in [0, π]$ ⁠.
$L_{i}$	Length of centerline of the $i th$ elastic joint. $L_{1} = [22 mm, 85 mm]$ and $L_{2} = 22 mm$ ⁠.
$l$	Distance from end plane of second elastic joint to the center of the grasper. $l = 45 mm$ ⁠.
$d$	Distance between two adjacent elastic joints. $d = 2 mm$ ⁠.
$γ$	Rotation angle of end-effector along ${\hat{z}}_{e}$ axis. $γ = [- π, π]$ ⁠.
$p_{i}$	Position vector of $O_{i e}$ expressed in coordinate frame ${{\hat{x}}_{i b}, {\hat{y}}_{i b}, {\hat{z}}_{i b}}$
$T_{φ_{i,} θ_{i}}$	Transformation matrix of the $i th$ elastic joint segment
${}^{b}T_{e}$	Transformation matrix from base coordinate system ${{\hat{x}}_{1 b}, {\hat{y}}_{1 b}, {\hat{z}}_{1 b}}$ to the end coordinate system ${{\hat{x}}_{e}, {\hat{y}}_{e}, {\hat{z}}_{e}}$ and expressed in ${{\hat{x}}_{1 b}, {\hat{y}}_{1 b}, {\hat{z}}_{1 b}}$

Term	Description
$O_{i}$	Bending center of the $i th$ elastic joint
$O_{i b}$	Origin of ${{\hat{x}}_{i b}, {\hat{y}}_{i b}, {\hat{z}}_{i b}}$ coordinate frame
$O_{i e}$	Origin of ${{\hat{x}}_{i e}, {\hat{y}}_{i e}, {\hat{z}}_{i e}}$ coordinate frame
$φ_{i}$	Rotation angle between the bending plane and ${\hat{x}}_{i b}$ axis. $φ_{i} \in [- π, π]$ ⁠.
$θ_{i}$	Bending angle of the $i th$ elastic joint. $θ_{1} \in [0, π / 2]$ $θ_{2} \in [0, π]$ ⁠.
$L_{i}$	Length of centerline of the $i th$ elastic joint. $L_{1} = [22 mm, 85 mm]$ and $L_{2} = 22 mm$ ⁠.
$l$	Distance from end plane of second elastic joint to the center of the grasper. $l = 45 mm$ ⁠.
$d$	Distance between two adjacent elastic joints. $d = 2 mm$ ⁠.
$γ$	Rotation angle of end-effector along ${\hat{z}}_{e}$ axis. $γ = [- π, π]$ ⁠.
$p_{i}$	Position vector of $O_{i e}$ expressed in coordinate frame ${{\hat{x}}_{i b}, {\hat{y}}_{i b}, {\hat{z}}_{i b}}$
$T_{φ_{i,} θ_{i}}$	Transformation matrix of the $i th$ elastic joint segment
${}^{b}T_{e}$	Transformation matrix from base coordinate system ${{\hat{x}}_{1 b}, {\hat{y}}_{1 b}, {\hat{z}}_{1 b}}$ to the end coordinate system ${{\hat{x}}_{e}, {\hat{y}}_{e}, {\hat{z}}_{e}}$ and expressed in ${{\hat{x}}_{1 b}, {\hat{y}}_{1 b}, {\hat{z}}_{1 b}}$

${{\hat{x}}_{i t}, {\hat{y}}_{i t}, {\hat{z}}_{i t}}$ denotes the coordinate system that shares the same origin with ${{\hat{x}}_{i b}, {\hat{y}}_{i b}, {\hat{z}}_{i b}}$ ⁠, ${\hat{x}}_{i t}$ axis is in the bending plane and points to the bending center $O_{i}$ ⁠, and ${\hat{y}}_{i t}$ axis is perpendicular to the bending plane.

${{\hat{x}}_{e}, {\hat{y}}_{e}, {\hat{z}}_{e}}$ denotes the coordinate system fixed on the end-effector.

The rotation matrix and position vector of an elastic joint can be derived separately from the geometric relations. The virtual centerline characterizes the length and shape of the elastic joint. When the normal vector of the bending plane is determined, the rotation matrix can be easily derived from Rodrigues' rotation formula

R_{φ_{i}, θ_{i}} = I cos θ_{i} + {\hat{u}}_{i} {\hat{u}}_{i}^{T} (1 - cos θ_{i}) + {\tilde{u}}_{i} sin θ_{i}

(6)

In the above equation,

{\hat{u}}_{i}

is the normal vector of the bending plane,

{\hat{u}}_{i} = {[u_{i x} u_{i y} u_{i z}]}^{T} = {[- sin φ_{i} cos φ_{i} 0]}^{T}

⁠. The coordinate system

{{\hat{x}}_{i e}, {\hat{y}}_{i e}, {\hat{z}}_{i e}}

is obtained after the coordinate system

{{\hat{x}}_{i b}, {\hat{y}}_{i b}, {\hat{z}}_{i b}}

rotates around

{\hat{u}}_{i}

⁠. Actually,

{\hat{u}}_{i}

is equivalent to

{\hat{y}}_{i t}

⁠, which can be derived by rotating coordinate system

{{\hat{x}}_{i b}, {\hat{y}}_{i b}, {\hat{z}}_{i b}}

around

{\hat{z}}_{i b}

⁠. In addition,

{\tilde{u}}_{i}

denotes the skew symmetric matrix

{\tilde{u}}_{i} = [\begin{matrix} 0 & - u_{i z} & u_{i y} \\ u_{i z} & 0 & - u_{i x} \\ - u_{i y} & u_{i x} & 0 \end{matrix}]

The position vector of the

i th

elastic joint segment can be derived from the geometric relations as

p_{i} = [\begin{matrix} p_{x i} \\ p_{y i} \\ p_{z i} \end{matrix}] = [\begin{matrix} \frac{L_{i}}{θ_{i}} (1 - cos θ_{i}) cos φ_{i} \\ \frac{L_{i}}{θ_{i}} (1 - cos θ_{i}) sin φ_{i} \\ \frac{L_{i}}{θ_{i}} sin θ_{i} \end{matrix}]

(7)

When

θ_{i} = 0

⁠, the rotation matrix

R_{φ_{i}, θ_{i}}

becomes a unit matrix

I

⁠, and the position vector

p_{i} = {[\begin{matrix} 0 & 0 & L_{i} \end{matrix}]}^{T}

⁠. Then, the transformation matrix of the

i th

elastic joint is obtained

T_{φ_{i}, θ_{i}} = [\begin{matrix} R_{φ_{i}, θ_{i}} & p_{i} \\ 0 & 1 \end{matrix}]

(8)

Define the transformation matrix

T_{(\hat{z}, d, ψ)}

as a translation

d

along the

\hat{z}

axis and a rotation

ψ

around the

\hat{z}

axis. Then the forward kinematics of the manipulator with end-effector is obtained

{}^{b}T_{e} = T_{φ_{1}, θ_{1}} T_{({\hat{z}}_{1 e}, d, 0)} T_{φ_{2}, θ_{2}} T_{({\hat{z}}_{3 e}, l, γ)}

(9)

The angular velocity of the end-effector, expressed in coordinate system

{{\hat{x}}_{1 b}, {\hat{y}}_{1 b}, {\hat{z}}_{1 b}}

⁠, is obtained by summing the angular velocity components of the manipulator and end-effector along each axis

ω = {\hat{z}}_{1 b} φ_{1}^{'} + {\hat{y}}_{1 t} θ_{1}^{'} - {\hat{z}}_{1 e} φ_{1}^{'} + {\hat{z}}_{2 b} φ_{2}^{'} + {\hat{y}}_{2 t} θ_{2}^{'} - {\hat{z}}_{2 e} φ_{2}^{'} + {\hat{z}}_{e} γ'

(10)

where $\begin{matrix} φ_{1}^{'} & θ_{1}^{'} & L_{1}^{'} & φ_{2}^{'} & θ_{2}^{'} \end{matrix}$ and $γ^{'}$ are the values of the angular velocity of each joint parameter, $\begin{matrix} {\hat{z}}_{1 b} & {\hat{y}}_{1 t} & {\hat{z}}_{1 e} & {\hat{z}}_{2 b} & {\hat{y}}_{2 t} & {\hat{z}}_{2 e} \end{matrix}$ and ${\hat{z}}_{e}$ are coordinate axes expressed in coordinate system ${{\hat{x}}_{1 b}, {\hat{y}}_{1 b}, {\hat{z}}_{1 b}}$ ⁠.

Then the Jacobian matrix of the angular velocity can be derived by

J_{w} = {\hat{z}}_{1 b} + {\hat{y}}_{1 b} - {\hat{z}}_{1 e} + {\hat{z}}_{2 b} + {\hat{y}}_{2 t} - {\hat{z}}_{2 e} + {\hat{z}}_{e}

(11)

Taking the partial derivatives of the joint parameters separately yields the Jacobian matrix of linear velocity

J_{v}

⁠. Then the instantaneous kinematics is then given by

\dot{x} = [\begin{matrix} υ \\ ω \end{matrix}] = J ξ' = [\begin{matrix} J_{v} \\ J_{w} \end{matrix}] ξ'

(12)

where $υ$ and $ω$ are linear and angular velocities, and $ξ' = [\begin{matrix} φ_{1}^{'} & θ_{1}^{'} & L_{1}^{'} & φ_{2}^{'} & θ_{2}^{'} & γ \end{matrix}']^{T}$ ⁠.

As shown in Fig. 5(a), the overlapped workspace of the instruments can cover a workspace of 50 mm × 50 mm × 50 mm at any orientation of the proposed bendable catheter, which is sufficient for a typical cholecystectomy [28]. Furthermore, the bendable portion of the catheter also enlarges the workspace of the robotic system, as indicated in Fig. 5(b).

Fig. 5

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Overlapped workspace of the instruments: (a) workspace without bendable catheter and (b) enlarged workspace with the aid of the bendable catheter

2.5 Teleoperation.

The desired pose information of end-effector is generated by a master console with two handles. Each handle sends the position and orientation information to the controller of the corresponding instrument in the form of elements in the homogeneous matrix. The desired pose matrix

T_{d}

generated by master console is compared with the matrix

T_{c}

describing the current pose of the end-effector, so as to obtain the deviations of position and orientation. Specifically, the orientation deviation is derived from axis-angle representation. Then the deviations of position and orientation can be obtained as follows:

{\begin{matrix} Δ p = ‖ p_{d} - p_{c} ‖ \\ Δ θ = ‖ acos \frac{trace (R_{m}) - 1}{2} ‖ \end{matrix}

(13)

where $T_{d} = [\begin{matrix} R_{d} & p_{d} \\ 0 & 1 \end{matrix}]$ ⁠, $T_{c} = [\begin{matrix} R_{c} & p_{c} \\ 0 & 1 \end{matrix}]$ ⁠, and $R_{m} = R_{d} R_{c}^{- 1}$ ⁠.

Next, the deviations of position and orientation are compared with predefined thresholds to determine whether the end-effector pose is close enough to the desired pose. Thresholds of both

Δ p

and

Δ θ

are set as 0.01. The iterative program of inverse kinematics will be performed if the deviations exceed the defined thresholds. To achieve the desired configuration velocity, a singularity-robust algorithm which is inspired by Nakamura and Hanafusa [29] is used

ξ' = {\begin{matrix} J^{T} {(J J^{T})}^{- 1} \dot{x} (d_{t} \geq ε) \\ J^{T} {(J J^{T} + λ I)}^{- 1} \dot{x} (d_{t} < ε) \end{matrix}

(14)

In the above equation

{\begin{matrix} d_{t} = \det (J J^{T}) \\ λ = k {(1 - \frac{d_{t}}{ε})}^{2} \end{matrix}

(15)

where $k = 0.05$ and $ε = 0.01$ ⁠.

During a surgery, the connection between the handle and the corresponding instrument may be frequently disconnected and reconnected to optimize the position of the handle [8]. However, this may cause the console to transmit pose information that exceeds the workspace of the instrument. In this case, the iterative program cannot solve the inverse kinematics and will keep iterating. To avoid infinite iterations, a protection algorithm shall be integrated into the iterative program.

The maximum number of iterations is empirically set to 20 to determine whether the program is stuck in an infinite loop. The iteration will be stopped if the number of iterations exceeds the maximum number. Although the iteration can be stopped, the iterative program cannot find a solution and the instrument will remain motionless in its current position. In this case, the handle of the master console for the corresponding instrument should be manipulated to move from outside the workspace of the instrument to inside the workspace. During this process, the instrument will begin to follow the handle once the handle reaches the edge of the instrument's workspace. At this time, if the posture of the instrument is not consistent with that of the handle, the instrument may move suddenly and sharply to follow the posture of the handle. The instrument will not move smoothly with the handle until the posture of the instrument is consistent with the handle.

Since the abrupt motion can lead to operational risks, a continuous mapping algorithm is designed to ensure that the instrument can still move continuously when the pose information generated by the handle exceeds the workspace of the corresponding instrument. The flowchart of this algorithm is shown in Fig. 6 and the definition of terms is shown in Table 2.

Fig. 6

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Flowchart of the continuous mapping algorithm

Table 2

Definition of terms in continuous mapping algorithm

Term	Description
$R_{n}$	The $n th$ desired posture matrix of the instrument
$R_{a}$	The actual posture matrix of the instrument at the current moment
$R_{t}$	Posture matrix obtained by inverse kinematics calculation
$p_{n}$	The $n th$ desired position vector of the instrument
$p_{a}$	The actual position vector of the instrument at the current moment
$p_{t}$	Position vector of the instrument obtained by inverse kinematics calculation
$q_{t}$	Joint parameter of the instrument obtained by inverse kinematics calculation
$q_{a}$	The actual joint parameter of the instrument at the current moment
$R_{o z} (α)$	Rotation matrix after rotation by an angle $α$ around the z-axis
$φ_{i}^{n}$	The $n th$ desired rotation angle between the bending plane and ${\hat{x}}_{i b}$ axis of the $i th$ elastic joint
$θ_{i}^{n}$	The $n th$ desired bending angle of the $i th$ elastic joint
$L_{i}^{n}$	The $n th$ desired length of centerline of the $i th$ elastic joint
$γ^{n}$	The $n th$ desired rotation angle of the end-effector along ${\hat{z}}_{e}$ axis
$φ_{1}^{a}$	The actual rotation angle between the bending plane and ${\hat{x}}_{1 b}$ axis of the first elastic joint at the current moment
$θ_{1}^{a}$	The actual bending angle of the first elastic joint at the current moment
$L_{1}^{a}$	The actual length of centerline of the first elastic joint at the current moment

Term	Description
$R_{n}$	The $n th$ desired posture matrix of the instrument
$R_{a}$	The actual posture matrix of the instrument at the current moment
$R_{t}$	Posture matrix obtained by inverse kinematics calculation
$p_{n}$	The $n th$ desired position vector of the instrument
$p_{a}$	The actual position vector of the instrument at the current moment
$p_{t}$	Position vector of the instrument obtained by inverse kinematics calculation
$q_{t}$	Joint parameter of the instrument obtained by inverse kinematics calculation
$q_{a}$	The actual joint parameter of the instrument at the current moment
$R_{o z} (α)$	Rotation matrix after rotation by an angle $α$ around the z-axis
$φ_{i}^{n}$	The $n th$ desired rotation angle between the bending plane and ${\hat{x}}_{i b}$ axis of the $i th$ elastic joint
$θ_{i}^{n}$	The $n th$ desired bending angle of the $i th$ elastic joint
$L_{i}^{n}$	The $n th$ desired length of centerline of the $i th$ elastic joint
$γ^{n}$	The $n th$ desired rotation angle of the end-effector along ${\hat{z}}_{e}$ axis
$φ_{1}^{a}$	The actual rotation angle between the bending plane and ${\hat{x}}_{1 b}$ axis of the first elastic joint at the current moment
$θ_{1}^{a}$	The actual bending angle of the first elastic joint at the current moment
$L_{1}^{a}$	The actual length of centerline of the first elastic joint at the current moment

When the pose information generated by master console is outside the workspace of the instrument, the number of iterations of the inverse kinematics program will exceed the maximum number allowed, and the continuous mapping algorithm will be activated. At this point, the instrument maps only the orientation, not the position. This allows the posture of the instrument to remain aligned with that of the handle and to move continuously and smoothly throughout the entire surgical procedure. When the handle of master console moves from outside the workspace of the corresponding instrument to the edge, the instrument moves with the handle without any abrupt motion. The continuous mapping algorithm ensures that the mapping of the instrument and the handle is continuous when the handle frequently disconnects and establishes the connection with the instrument to optimize its position. Figure 7 shows the pose of the handle and instrument when the instrument crosses the edge of the workspace.

Fig. 7

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Position and orientation of the handle and instrument when the instrument crosses the edge of the workspace. ①②③ represent different moments. The curved trajectory of the instrument from moment ① to ③ is produced by bending the second elastic joint of the instrument in order to map the direction of the handle.

2.6 Experimental Method.

Three sets of experiments were conducted to verify the trajectory tracking accuracy, load capacity and master-slave operation capability of the robot.

To validate the ability of trajectory tracking, two sets of tests were conducted. The instrument was driven to follow a circular trajectory with a radius of 30 mm. The tip of the end-effector was driven to move along the radius from the center of the circle to the circumference and then along the circumference. The bendable catheter was straight in one of the tests, and was bent to 45 deg in the horizontal plane in the other test. An electromagnetic spatial measurement system (Aurora^®, Northern Digital Inc., Waterloo, ON, Canada) was used as the measurement equipment.

Weight lifting experiments were conducted to verify the load capacity of the instrument. A 200 g weight was selected for the experiments, since it would basically satisfy the requirements of LESS operation [16]. The weight needs to be lifted and hung onto three pegs in different locations.

Two sets of peg transfer tests were designed to simulate operations of transferring and positioning an object in surgery. A ring was moved from the center peg to each of the six surrounding pegs. Point positioning tests were conducted to simulate accurate operation in surgery. A needle of 0.5 mm diameter was held by the instrument. The needle was first located to the center point of the positioning plane, and then moved sequentially to eight surrounding points before returning to the center point, for a total of nine points. The time of moving the peg and locating the point were recorded.

2.7 Statistical Analysis.

In the trajectory tracking experiments, the actual trajectories were compared with the desired trajectories. The mean error and root-mean-square error were employed to evaluate the accuracy of the trajectories. In the peg transfer and point positioning experiments, the maximum and minimum values of operation time and the mean operation time with standard deviation were employed to evaluate the effectiveness of the operation.

3 Results and Discussion

3.1 Trajectory Tracking.

The experimental setup, trajectory, and curves of x/y coordinates with respect to time are shown in Fig. 8. The mean trajectory tracking errors in the two tests are 1.32 mm and 2.49 mm, respectively. The root-mean-square errors are 2.24 mm and 4.03 mm, respectively.

Fig. 8

View large Download slide

Experimental setup and results of trajectory tracking tests: (a) experimental setup, (b) and (c) trajectory and curves of x/y coordinates with respect to time when bendable catheter is straightened, (d) and (e) trajectory and curves of x/y coordinates with respect to time when bendable catheter is bent to 45 deg in horizontal plane

When the bendable catheter is bent to 45 deg, the trajectory tracking error of the instrument is slightly larger than that when the bendable catheter is straight. This is because the instrument channel is compressed, preventing the instrument from moving smoothly. It may be solved by inserting a tube with a smooth surface and high compression resistance, such as a braided tube with a protective coating, into the instrument channel.

3.2 Load Capacity.

As shown in Fig. 9, the instrument successfully lifted the weight. Due to the slack of drive cables, the tip position of the end-effector shows a noticeable deviation. Retightening the drives cables can effectively reduce the deviation. Furthermore, replacing the drive cables by Nitinol rods may be another effective way to improve the load capacity of the instrument.

Fig. 9

View large Download slide

Snapshots of load tests

3.3 Peg Transfer and Point Positioning.

It took a maximum of 24 s, a minimum of 13 s, and a mean of 16.5±3.6 s to move the ring to a peg in the straight state of the catheter, and a maximum of 26 s, a minimum of 11 s, and a mean of 18.7±5.3 s in the bent state. Snapshots of the peg transfer experiments are shown in Fig. 10(a). In the point positioning experiment, the maximum time to locate a point was 18 s, the minimum time was 5 s, and the average time was 8.2±3.6 s. Snapshots are shown in Fig. 10(b).

Fig. 10

View large Download slide

Snapshots of peg transfer tests and point positioning: (a) peg transfer and (b) point positioning

4 Conclusions

For robot-assisted LESS with rigid catheters, the limited movement of the catheter puts restrictions on the posture of the instruments, resulting in reduced instrument flexibility. In order to improve the ability of the instrument to reach the target region, we proposed a novel robotic system with a bendable catheter for LESS, which makes it easier to adjust the initial direction of the instruments. The system design is introduced in detail and the kinematic analysis of instrument is performed. Experimental results prove the feasibility and effectiveness of the proposed system in LESS. There are still some limitations with our design, which will be addressed in our future work: (1) A shorter end-effector is required. The relatively long end-effector reduces the variety of instrument poses. A shorter end-effector will allow more flexible instrument poses in dual-instrument operations. (2) Before withdrawing or replacing one instrument, the catheter needs to be adjusted to a straight state, during which the other instrument will also be influenced. In future work, a smoother method of instrument replacement needs to be proposed.

Funding Data

National Key Research and Development Program of China (Grant No. 2022YFB4700801; Funder ID: 10.13039/501100004517).

Data Availability Statement

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

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A Robotic System With a Bendable Catheter for Laparoendoscopic Single Site Surgery

Abstract

1 Introduction

2 Methods

2.1 Design of the Bendable Catheter.

2.2 Manipulator and Endoscope.

2.3 End-Effector and Axial Compression Resistance Mechanism.

2.4 Kinematics.

2.5 Teleoperation.

2.6 Experimental Method.

2.7 Statistical Analysis.

3 Results and Discussion

3.1 Trajectory Tracking.

3.2 Load Capacity.

3.3 Peg Transfer and Point Positioning.

4 Conclusions

Funding Data

Data Availability Statement

References

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A Robotic System With a Bendable Catheter for Laparoendoscopic Single Site Surgery

Abstract

1 Introduction

2 Methods

2.1 Design of the Bendable Catheter.

2.2 Manipulator and Endoscope.

2.3 End-Effector and Axial Compression Resistance Mechanism.

2.4 Kinematics.

2.5 Teleoperation.

2.6 Experimental Method.

2.7 Statistical Analysis.

3 Results and Discussion

3.1 Trajectory Tracking.

3.2 Load Capacity.

3.3 Peg Transfer and Point Positioning.

4 Conclusions

Funding Data

Data Availability Statement

References

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