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Inverse Kinematics of Concentric Tube Robots in the Presence of Environmental Constraints

Inverse Kinematics of Concentric Tube Robots in the Presence of Environmental Constraints Hindawi Applied Bionics and Biomechanics Volume 2021, Article ID 4107732, 12 pages https://doi.org/10.1155/2021/4107732 Research Article Inverse Kinematics of Concentric Tube Robots in the Presence of Environmental Constraints 1 1 2 Mohammad Jabari , Manizhe Zakeri , Farrokh Janabi-Sharifi , and Somayeh Norouzi-Ghazbi Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3 Department of Biomedical Engineering, Ryerson University, Toronto, Canada Correspondence should be addressed to Manizhe Zakeri; m.zakeri@tabrizu.ac.ir Received 3 May 2021; Revised 4 July 2021; Accepted 20 July 2021; Published 16 August 2021 Academic Editor: Yanxin Zhang Copyright © 2021 Mohammad Jabari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Inverse kinematics (IK) of concentric tube continuum robots (CTRs) is associated with two main problems. First, the robot model (e.g., the relationship between the configuration space parameters and the robot end-effector) is not linear. Second, multiple solutions for the IK are available. This paper presents a general approach to solve the IK of CTRs in the presence of constrained environments. It is assumed that the distal tube of the CTR is inserted into a cavity while its proximal end is placed inside a tube resembling the vessel enabling the entry to the organ cavity. The robot-tissue interaction at the beginning of the organ- cavity imposed displacement and force constraints to the IK problem to secure a safe interaction between the robot and tissue. The IK in CTRs has been carried out by treating the problem as an optimization problem. To find the optimized IK of the CTR, the cost function is defined to be the minimization of input force into the body cavity and the occupied area by the robot shaft body. The optimization results show that CTRs can keep the safe force range in interaction with tissue for the specified trajectories of the distal tube. Various simulation scenarios are conducted to validate the approach. Using the IK obtained from the presented approach, the tracking accuracy is achieved as 0.01 mm which is acceptable for the application. 1. Introduction resentation of the physical environment extracted from pre- procedure medical images. Forward and inverse kinematics Nowadays, continuum robots (CRs) have found widespread of concentric tube continuum robots (CTCRs) using various geometrical approach are investigated in [11, 12], respec- applications in medicine, especially in minimally invasive surgeries (MIS). Catheters are an example of CRs that are tively. Anor et al. [13] presented a novel systematic approach widely used in MIS. In recent years, comprehensive to optimize the design of CTCRs for neurosurgical proce- researches have been done on the design, manufacturing, dures. These procedures require that the robot approach and development of steerable catheters. specified target areas while navigating and operating within CRs could be modeled using two main approaches: (i) an anatomically constrained workspace. A particular advan- Cosserat rod theory and (ii) constant curvature [1–4]. The tage of this approach is that it identifies the need for either former provides a precise model of CRs using differential fixed-curvature versus variable-curvature sections. Due to equations [5–7] while the latter provides a simpler model the vast variety of applications in concentric tube robots, with less time-cost and still acceptable accuracy [8, 9]. Lyons the optimization of concentric tube robots has always been et al. [10] presented a new method of optimization in IK for alluring for researchers. For example, a new design of CTCR medical devices to plan configurations with anatomically which is well suited to MIS inside small body cavities such as constrained for reaching specified targets. In this paper, they the heart is presented in [14]. This paper presents a general- calculate start position and orientation and a geometric rep- ized pattern search to optimize concentric tube robots while 2 Applied Bionics and Biomechanics base base base Figure 1: The scheme of a concentric tube robot comprising three precurved tubes. they can reach the target point with minimal curvature and (iii) The curvature of each segment of the robot remains length. Runge et al. [15] used evolutionary algorithms such constant, thus, the central backbone is always as genetic algorithm to optimize a soft robot. Bodily et al. located within a plane [16] used a genetic algorithm to optimize the reachability, dex- Figure 1 shows a three-segment concentric tube contin- terity, and manipulability of a multisegment continuum robot. uum robot. Based on the following assumption, the constant Luo et al. [17] designed a concentric tube manipulator that is curvature approach is utilized to model the system. appropriate for surgical environments. They presented a new The parameters used to describe the model of a three- topology of the tubes for the concentric tube robot, which segment continuum robot are listed in Table 1. can increase the stable workspace because it allows the usage Equation (1) calculates the length change of each of larger tube curvatures and/or curve lengths. Davarpanah segment. et al. [18] optimized a concentric tube robot using a genetic algorithm which allows them to reach the specificdestination L = L + ΔL , ð1Þ with high accuracy. Lloyd et al. [19] presented a novel model i i0 i for optimizing a task-specific in millimeter-scale for magneti- th cally actuated soft continuum robots used in medical applica- where L is the length of the i segment in the pri- i0 tions. Finally, most similar to our work, Cheong et al. [20] mary situation of insertion, and ΔL is the decrease or th applied a computational method to find optimal designs of increase in the i segment length which is task- continuum robots while considering reachability constraints. dependent. Then, Eq. (2) shows the total length of This paper contributes to the field by finding an optimized the catheter. inverse kinematic solution for a three-segment CTCR so that n n for a given position, the robot configuration would occupy L = 〠 L = 〠 L + ΔL , ð2Þ ðÞ minimized space. The proposed algorithm is very beneficial i i0 i i=1 i=1 in MIS, where the robot needs to work in a tight space. More- over, the optimized solution meets a safety interaction criterion 2.1. Coordinate Systems. Figure 2 presents the relations at the base of the CR. In practice, when the robot inserts to an between actuator space, configuration space, and task organ cavity, its shaft body would remain in the colons or ves- space. The actuator space includes tube lengths, the sels connected to that cavity. So, it is important that at the base configuration space covers the robot bending angles of the robot, which is inside a tighter space, the CR does not and the insertion length, and finally, the task space is apply forces to the surrounding tissues that may hurt. relevant to the end effector’s position and orientation. In MIS applications, the robot is ideally expected to per- The coordinate systems for a single-segment continuum form its mission with occupying the least possible workspace. robot (see Figure 3) are described as follows. To address this issue, in this paper, an algorithm is proposed that helps the minimum number of segments of a three- (1) Base Coordinate System. The base coordinate system segment CR be involved in achieving a target point. More- represented by fbg = fx, y, zg over, the robot-environment contact is modeled in the form of a spring which allows to apply interaction-force/displace- (2) End Coordinate System. The coordinate system is ment constraints to the base of the robot which assures a safe located at the end point of the tube (i.e., the intersec- interaction. Finally, the research is concluded in Section 4. tion of the tube with its nearby tube). The coordinate system is shown by E = fx , y , z g, i =1,2,3 for a i i i i three-segment catheter 2. Forward and Inverse Kinematics 2.2. Kinematics of Multisegment Robot. For the slender bar In this section, the structure of the constant and variable model, the curvature is not a constant and it can be widely length of a three-segment CTCR is described. The modeling used [21, 22]. As mentioned previously, the constant curva- assumptions of the CR are summarized as follows. ture approach is used to model the kinematics of the robot (Figure 3) [23, 24]. (i) The bending motion of the robot is planar The general mapping between the task space and the con- (ii) The curvature of the robot backbone is supposed to figuration space can be obtained by a homogeneous transfor- be constant mation matrix (Eq. (3)), ϕ Applied Bionics and Biomechanics 3 Table 1: The list of the nomenclatures used in this paper. Symbol Definition R The radius of curvature in the primary backbone is defined in the bending plate (the plate of X Z ) 1 1 The bending angle in the primary backbone is defined in the bending plate (X Z plate) at the point O . 1 1 φ ∈½ 02π The orientation angle of the robot which can rotate in the xy plate φ , α Rotation and bending angles of the first part of the catheter prox prox φ , α Rotation and bending angles of the second part of the catheter med med φ , α Rotation and bending angles of the third part of the catheter dist dist The primary length of the primary backbone th The position of the i secondary backbone fixed in the lower platform th The position of the i secondary backbone fixed in the upper platform O The center of the equilateral triangle made by the A points in the platform offg x, y, z C The center of the equilateral triangle made by the B points in the platform offg x , y , z i 1 1 1 P =½ x  y  z The position vector stating the center coordinate of the upper platform of each segment according to the lower platform c c c Curvature v =½ k φ  l The configuration space vector n The number of robot segments 3×3 R ∈ R The rotation matrix from the end disk coordinate system to the base coordinate system 0t Actuator space Configuration space Task space T T T (l , l , l ) (k, 𝜑, l) (x, y, z) 1 2 3 Tendon lengths Arc parameters Tip position Figure 2: Mapping between joint space variables, configuration space variables, and workspace variables. B y Top O´ y 1 𝛼 z Bottom B B 3 2 (a) (b) Figure 3: The introduced parameters in the robot: (a) the geometric display of a robot segment and (b) the display from the above of the robot platform. 4 Applied Bionics and Biomechanics 2 3 cφ 1 −ckl ðÞ ðÞ 2 cφskl ðÞ c φ ckl − 1 +1 −sφcφ ckl − 1 ðÞ ðÞ ðÞ ðÞ 6 7 6 7 "# 6 2 7 sφ 1 −ckl sφcφ ckl − 1 c φ 1 −ckl +ckl ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 6 7 R p ot sφskl ðÞ 6 7 T = = , ð3Þ 6 7 6 7 6 7 skl ðÞ 6 7 −cφskðÞl −sφskðÞl ckðÞl 4 5 where R represents the orientation of the end effector, p catheter tip and the corresponding target position ðP ot ref is the end position of the robot [25], c and s stand for cos = ½x  y  z Þ. Also, the optimized parameter would ref ref ref and sin, respectively. Using the length and radius of the be the configuration parameters including bending and rota- robot curvature shown in Figure 3, the angle α can be cal- tion angles and insertion length of each segment. According culated using to the number of extended segments (e.g., when two or three tubes are inserted out from the overlapped tube), the catheter ~ ̂ tip can be denoted by either p or p : To form cost function, 2 3 α = = kl, ð4Þ we first rearrange Eq. (6) as follows where R, l, and k are the arc length, bending radius, and ~ P = P + R α p : 2:ref 1 ot prox 2 ð7Þ curvature, respectively. More details on how to obtain P = P + R α P + R ðÞ α R α P : Eqs. (3) and (4) could be found in [25, 26]. The forward 3:ref 1 ot prox 2 ot med ot prox 3 kinematic model of the three-segment CR can be obtained using Next, the cost functions of the two-segment and three- b b 0 1 2 segment catheter can be defined in the form of T = T T T T : ð5Þ 3 0 1 2 3 J = γ p − p + R α p ot prox 2:ref 1 2 2.3. Inverse Kinematic for a Multisegment Robot. Increased ð8Þ 2 2 degrees of freedom (DOF) bring more redundancy for the p − p + R α ~p + α + α , 2,ref 1 ot prox 2 prox dist robot configuration. As a result, there are no closed-form solutions for the multisegment robot inverse kinematic. J = γ p − p + R α p + R ðÞ α R α p∧ ot prox ot med ot prox 3:ref 1 2 3 Moreover, there would be multiple solutions for the prob- lem of finding the inverse kinematics. To obtain the for-  p − p + R α p + R ðÞ α R α p ot prox ot med ot prox 3:ref 1 2 3 ward kinematic model, two different cases need to be 2 2 2 + α + α + α , prox med dist considered for the bending angle of the distal segment ° ° ° ð9Þ [27] including θ <90 and 90 ≤ θ ≤ 180 . The posi- dist dist tion of the central backbone of the catheter in the second where γ is a weighting factor. and third segments is achieved using 2.5. Safe Interaction Conditions. According to Figure 4, if you −1 p = R α p − p , want the robot to be safely interacting with the texture, at ðÞ 2 ot prox 2 1 first, the texture should be modeled into a spring, then, the ð6Þ −1 −1 p ̂ = R α p − p R α − p , amount of the allowed displacement for being in contact with ðÞ ðÞ 3 ot med 3 1 ot prox 2 the robot base should be obtained. Allowed ranges of the interaction force vary across surgical tasks. The size and fra- T T where the points P = ½x  y  z , P = ½x  y  z , 0 0 0 0 1 1 1 1 gility of the tissue are important parameters affecting the T T P = ½x  y  z , and P = ½x  y  z  are the catheter 2 2 2 2 3 3 3 3 allowed force range. base coordinate system, the intersection of the first and In this paper, the following assumptions are made second segments, the intersection of the second and third according to the literature [28, 29]. segments, and the end-plate of the third segment, respec- tively. More details can be found in [27]. Vectors ~p and 2 (i) Diameter of the main coronary arteries is about p are local positions of the end-effector in the coordinate 4 mm [30] systems at the base of each tube. (ii) The maximum allowed force applicable to the texture 2.4. Optimization Problem. The optimization problem is is considered as 0.2 N. The texture stiffness, at the defined to minimize the tracking error between the entrance of the cavity, is considered to be k =100N/ Applied Bionics and Biomechanics 5 Base Catheter tip Figure 4: The safe interaction with the texture modeled by the spring. −40 −50 −80 Y (mm) X (mm) Figure 5: Configuration of the three-segment catheter with bending and rotation angles. Table 2: Three-segment continuum robot bending and rotation m which corresponds to 2 mm displacement of the tis- angles. sue which is acceptable (Eq. (10)) Angles Segment 1 Segment 2 Segment 3 ° ° ° 63:24 53:54 35:58 Bending angle (α) F ° ° ° Rotation angle (φ) 49:80 310:05 178:89 F = KΔx ⇒ Δx = =±2mm: ð10Þ T T 73:26 36:63 76:21 Length (mm) Δx equals to the required displacement of P = ðx, y, zÞ T b so that the catheter tip would reach to the target point. 3. Results and Discussion qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In this section, first, we simulated the kinematic model 2 2 2 Δx = ðÞ x − x +ðÞ y − y +ðÞ z − z , T new old new old obtained in the previous section using MATLAB. new old Figure 5 demonstrates the simulation results for the ð11Þ three-segment catheter. The robot configuration parameters corresponding to the target point of the catheter tip are listed To obtain Δx , first, the shape of catheter corresponding in Table 2. to the desired tip catheter is obtained, and then among the Figure 6 shows the mapping between the position of the multiple solutions, those solutions which are meeting the robot end-effector and the configuration space variables. At safety constraint at the point of P are accepted. each iteration, the bending and rotation angles change smoothly Z (mm) 6 Applied Bionics and Biomechanics −10 −20 −30 0 102030405060708090 100 Sampling points Figure 6: Changes of the robot’s final position according to the mapping of the configuration space variables. Table 3: The simulation conditions for workspace. within ½0 π and ½0 2π, respectively. Step-increment equals ° ° to 1:8 for α and 3:6 for φ. The number of the chosen sample Single- Two- Three- Parameters points is 100. The weighting factor γ is set to 50. segment segment segment The length of each 120 mm 60 mm 40 mm 3.1. Continuum Robot Workspace. In this section, the robot segment workspace for a single, two, and three-segment catheter is 0 π 0 π/2 0 π/3 α ½ ½ ½ obtained and demonstrated. The variation range of the con- φ 0 2π 0 2π 0 2π ½ ½ ½ figuration parameters are shown in Table 3. As illustrated in Figures 7–12, with the increase of the number of robot segments, the thickness of the clouds form- ing the workspace is increasing, which reflects an enlarged accessible space and increased precision of the three- segment catheter compared to single- or two-segment cathe- ter. Hence, one can conclude that multisegment catheters are more appropriate choices for applications in which the cath- eter is supposed to operate in confined spaces. 3.2. Continuum Robot with Variable Length: Insertion Optimization. In this section, we consider a continuum robot with variable length. The purpose is to optimize the robot seg- ment lengths while meeting the aforementioned constraints. The schematic of the proposed optimization algorithm is shown in Figure 13. In this diagram, first, concentric tube robot receives the desired point and then calculates the distance between the robot’s tip and the desired point and increases the tube’slength −100 −80 −60 −40 −20 0 20 40 60 80 100 based on the calculations. If the desired point is in the range of X (mm) tube 1, only segment 1 would be involved to help the end- Figure 7: One-segment catheter workspace in 2D. effector reach the target point. Otherwise, the second and the third segments would be inserted to the workspace. ter of 120 mm length (60 mm for each segment). The target 3.3. Optimization Example: A Catheter with Two Segments point is set to P ½−47:98  − 33:92 33:90. Using the pro- (No Insertion Is Allowed). We consider a two-segment cathe- posed algorithm, as it has been shown in Figures 14 and 15, Coordinate points of x, y, z axis (mm) Z (mm) Applied Bionics and Biomechanics 7 50 100 −50 −50 Y (mm) −100 −100 X (mm) Figure 8: One-segment catheter workspace in 3D. −100 −80 −60 −40 −20 0 20 40 60 80 100 X (mm) Figure 9: Two-segment catheter workspace in 2D. after 2000 iterations, the optimal parameters of the robot shows itself; as a result, it will be possible to reduce the itera- including the bending and rotation angle of each segment tion limit when the velocity of the process seems to be impor- can be observed in Tables 4 and 5. tant to us. According to Figure 14, the amount of the cost function 3.4. Optimization Example: A Catheter with Two and Three after about 40 first iterations is approximately equal J = Segments (Insertion Is Allowed). As it was mentioned, in the 8:03. variable-length catheter, considering the target point, first, We also consider the hypothetical target point in the the proximal segment length increases from its primary coordinate system P = ½40:85 77:24 26:97 of for the length considered to be zero to its final length, and this same optimization of the three-segment catheter configuration trend for other links reports in order to approach the target space with a general length of 120 mm and a separate length point. of 40 mm for each segment. The three-segment catheter will have the same length for As it can also be seen in Figure 15, the amount of the cost each segment which is 40 mm has been used, so as to simulate function, after about 60 first iterations, equals J =6:59. how to access the variable-length catheter. The simulation results show that the catheter accuracy to We consider the target point at P = approach the point in question is too much, also, a consider- able amount of optimization, in the first several iterations, ½34:85 17:11 62:22 . After 2000 iterations, and after Z (mm) Z (mm) 8 Applied Bionics and Biomechanics −100 −100 −50 Y (mm) X (mm) Figure 10: Two-segment catheter workspace in 3D. −100 −80 −60 −40 −20 0 20 40 60 80 100 X (mm) Figure 11: Three-segment catheter workspace in 2D. optimization using the genetic algorithm, and also using the space with variable length. In the simulations, we consider cost function of Eq. (8), the amount of the shown response first the position of the robot tip at the entrance of the cavity in the 2016 Matlab will be in the form of Table 6. According in the form of the coordinate system of B = ½0 0 10 .At to these optimal parameters, the order of error is 0:01 mm for first, the optimization results for the point in question are two segment catheter which is small in comparison to three shown in Table 7 without the safe interaction with the fragile segment catheter that has an error greater than 1mm. There- structures like texture. If dB < Δx , the obtained position fore, the variable-length catheter will use two-segment of its and the parameters related to this position are correct, if inner parts in order to approach the desirable point, and this not, another response is chosen. same result has been shown in the last row of the first col- According to these optimal parameters, the order of error umn. Also, the optimization amount of the two-segment cost is 0:01 mm for a two-segment catheter which is small in com- function has been obtained to be the number of J =1:1032 parison to three segment catheter that has an error greater that is an appropriate amount for tracking the target point. than 1mm. Regarding the error amount, for each segment, the desirable link will be the second link. Also, the amount 3.5. Optimization of the Three-Segment Catheter with of the cost function for the two-segment of the variable- Variable Length Having the Safe Interacting with Fragile length catheter will be equal to 1:4539. Structures like Texture. We choose first the hypothetical As you can see in Table 8, for the catheter safe interacting point of P = ½30:16 25:59 65:96 as the target point, for with fragile structures like texture, according to the Eqs. (8) optimization of the three-segment catheter configuration and (9), at first, we obtained the safe limitation constraint Z (mm) Z (mm) Applied Bionics and Biomechanics 9 −100 −50 100 −50 Y (mm) 100 −100 X (mm) Figure 12: Three-segment catheter workspace in 3D. Start Get the target point The distance from tip's catheter is calculated 2 2 2 dist = x + y + z No If dist < L Yes A) Find the optimal solution by solving the inverse kinematics code of a single-segment robot B) Find the optimal solution by solving the inverse kinematics code of a two-segment robot C) Find the optimal solution by solving the inverse kinematics code of a three-segment robot Compare the answers A, B, C and give the optimal answer If L < dist < L + L 1 1 2 No Yes A) Find the optimal solution by solving the inverse kinematics code of a two-segment robot B) Find the optimal solution by solving the inverse kinematics code of a three-segment robot Compare the answers A, B and give the optimal answer No If L + L < dist < L + L + L 1 2 1 2 3 A) Find the optimal solution by solving the inverse kinematics code of a three segment robot Finish Figure 13: Flow chart for finding an optimal inverse kinematics. Z (mm) 10 Applied Bionics and Biomechanics × 10 0 5 10152025 30 35 40 45 50 Iteration Figure 14: Genetic algorithm for best fitness target point in two-segment catheter. 0 5 10152025 30 35 40 45 50 55 60 65 70 Iteration Figure 15: Genetic algorithm for best fitness target point in three-segment catheter. Table 4: Optimized results for reaching target point. Table 6: Optimized results for three-segment catheter with variable length. Optimal Optimal Reaching Section Error bending angle rotation angle point Two-segment Tracked point Three-segment Tracked point number (mm) (degree) (degree) (mm) θ : 44:6093 θ : 44:6093 prox prox 1 99.80 195.42 X: φ : 8:2412 φ : 8:2412 prox prox X2: 34.837 X3: 17.4794 0.17 θ : 38:9934 X: -40.81 θ : 38:9934 med med Y2: 17.099 Y3: 32.5204 Y: φ : 122:5938 φ : 122:5938 Y: -33.64 med med Z2: 66.199 Z3: 61.4679 0.28 2 88.57 324.01 Z: 33.73 θ : 192:3173 dist J: 1.1032 Z: φ : 145:7034 dist Link-num: 2 0.17 Table 5: Optimized results for reaching target point. Table 7: Optimized results for a three-segment catheter with variable length without interaction with fragile structures like Optimal Optimal Reaching texture. Section Error bending angle rotation angle point number (mm) (degree) (degree) (mm) Two-segment Tracked point Three-segment Tracked point 1 98.34 53.36 X: θ : 49:5561 θ : 53:3323 prox prox X: 40.86 0.01 φ : 55:6513 2 99.33 151.98 φ : 49:0286 prox prox X2: 30.1286 X3: 50.6561 Y: 77.27 Y: θ : 41:6511 θ : 41:5474 med med Y2: 25.5773 Y3: 1.2243 Z: 26.97 0.03 3 43.78 73.15 φ : 284:3094 φ : 255:1113 med med Z2: 65.9110 Z3: 72.9058 Z:0 θ : 124:2572 dist J: 1.4539 φ : 284:6543 Link-num: 2 dist for interacting with fragile structures like texture. Consider- ing the two mentioned conditions, we take into consideration the maximum amount of the allowed force which is applied the population in the algorithm, in each of the robot seg- to the texture to be 0:2N, and also the stiffness to be 100, at ments, the correct link for approaching the chosen target the entrance of the cavity. According to Eq. (10), the dis- point and the angle amounts of the catheter are obtained in placement amount at the contact point with the texture will the case that the distance amount of the robot is less than be obtained as ±2 mm. Now, after required iterations from 2 mm. Cost Cost Applied Bionics and Biomechanics 11 error for most real-time surgeries. In the final part, regarding Table 8: Optimized results for three-segment catheter with variable length with interaction with fragile structures like texture. force imposed onto the texture, and also the stiffness, the allowed distance amount with the texture causes the safe Two-segment Tracked point Three-segment Tracked point interaction. The implementation of the secure interaction θ : 53:4391 prox with the texture and the trajectory results indicate the cathe- θ : 53:4391 prox φ : 53:6859 ter has an accurate application in the medical procedure. prox φ : 53:6859 prox θ : 47:2743 X2: 30.0826 X3: 24.9051 Future works focusing on the addition of more constraints, med θ : 47:2743 med φ : 269:4255 Y2: 25.5582 Y3: 22.4644 and optimization of the distinguished constraints of the cost med φ : 269:4255 med Z2: 65.8237 Z3: 61.0422 function, try to lessen the tracking errors and to improve the J: 2.8294 θ : 299:592 dist stability of the existing catheter. Also, tracking a certain path Link-num: 2 φ : 220:5672 dist concerning the safe interaction with the texture for the cath- dB: 1.1641 eter will be carried out. According to the obtained error amount from the hypo- Data Availability thetical target point and the tracked point in the three- segment catheter, it can be concluded that the three-segment The MATLAB code data used to support the findings of this catheter has an undesirable tracking for approaching the target study are available from the corresponding author upon point, and also according to the obtained points in the table request. second column, and the little error amount the catheter will use two segments from its inner parts for approaching to the desired point. Conflicts of Interest Also, the optimized cost function amount of the two- segment catheter is obtained to be 2.8294 which is an The authors declare that they have no conflicts of interest. appropriate amount for tracking the target point. 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The average distal end positioning error is less than IEEE International Conference on Systems, Man, and Cyber- 1% for each segment of the catheter, which is an acceptable netics (SMC), pp. 1929–1936, Toronto, Canada, 2020. 12 Applied Bionics and Biomechanics [24] R. J. Webster III and B. A. Jones, “Design and kinematic [9] T. da Veiga, J. H. Chandler, P. Lloyd et al., “Challenges of con- tinuum robots in clinical context: a review,” Progress in Bio- modeling of constant curvature continuum robots: a review,” medical Engineering, vol. 2, no. 3, article 032003, 2020. The International Journal of Robotics Research, vol. 29, no. 13, pp. 1661–1683, 2010. [10] L. A. Lyons, R. J. Webster, and R. Alterovitz, “Planning active cannula configurations through tubular anatomy,” in 2010 [25] Y. Tian, M. Luan, X. Gao, W. Wang, and L. 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Duggan, “Optimal design of continuum robots with reachability constraints,” IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 3902– 3909, 2021. [21] J. Liu, Y. Mei, and X. Dong, “Post-buckling behavior of a double-hinged rod under self-weight,” Acta Mechanica Solida Sinica, vol. 26, no. 2, pp. 197–204, 2013. [22] J. L. Liu, Y. Mei, R. Xia, and W. L. Zhu, “Large displacement of a static bending nanowire with surface effects,” Physica E: Low- dimensional systems and Nanostructures, vol. 44, no. 10, pp. 2050–2055, 2012. [23] B. A. Jones and I. D. Walker, “Kinematics for multisection continuum robots,” IEEE Transactions on Robotics, vol. 22, no. 1, pp. 43–55, 2006. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Bionics and Biomechanics Hindawi Publishing Corporation

Inverse Kinematics of Concentric Tube Robots in the Presence of Environmental Constraints

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Hindawi Applied Bionics and Biomechanics Volume 2021, Article ID 4107732, 12 pages https://doi.org/10.1155/2021/4107732 Research Article Inverse Kinematics of Concentric Tube Robots in the Presence of Environmental Constraints 1 1 2 Mohammad Jabari , Manizhe Zakeri , Farrokh Janabi-Sharifi , and Somayeh Norouzi-Ghazbi Faculty of Mechanical Engineering, University of Tabriz, Tabriz, Iran Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3 Department of Biomedical Engineering, Ryerson University, Toronto, Canada Correspondence should be addressed to Manizhe Zakeri; m.zakeri@tabrizu.ac.ir Received 3 May 2021; Revised 4 July 2021; Accepted 20 July 2021; Published 16 August 2021 Academic Editor: Yanxin Zhang Copyright © 2021 Mohammad Jabari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Inverse kinematics (IK) of concentric tube continuum robots (CTRs) is associated with two main problems. First, the robot model (e.g., the relationship between the configuration space parameters and the robot end-effector) is not linear. Second, multiple solutions for the IK are available. This paper presents a general approach to solve the IK of CTRs in the presence of constrained environments. It is assumed that the distal tube of the CTR is inserted into a cavity while its proximal end is placed inside a tube resembling the vessel enabling the entry to the organ cavity. The robot-tissue interaction at the beginning of the organ- cavity imposed displacement and force constraints to the IK problem to secure a safe interaction between the robot and tissue. The IK in CTRs has been carried out by treating the problem as an optimization problem. To find the optimized IK of the CTR, the cost function is defined to be the minimization of input force into the body cavity and the occupied area by the robot shaft body. The optimization results show that CTRs can keep the safe force range in interaction with tissue for the specified trajectories of the distal tube. Various simulation scenarios are conducted to validate the approach. Using the IK obtained from the presented approach, the tracking accuracy is achieved as 0.01 mm which is acceptable for the application. 1. Introduction resentation of the physical environment extracted from pre- procedure medical images. Forward and inverse kinematics Nowadays, continuum robots (CRs) have found widespread of concentric tube continuum robots (CTCRs) using various geometrical approach are investigated in [11, 12], respec- applications in medicine, especially in minimally invasive surgeries (MIS). Catheters are an example of CRs that are tively. Anor et al. [13] presented a novel systematic approach widely used in MIS. In recent years, comprehensive to optimize the design of CTCRs for neurosurgical proce- researches have been done on the design, manufacturing, dures. These procedures require that the robot approach and development of steerable catheters. specified target areas while navigating and operating within CRs could be modeled using two main approaches: (i) an anatomically constrained workspace. A particular advan- Cosserat rod theory and (ii) constant curvature [1–4]. The tage of this approach is that it identifies the need for either former provides a precise model of CRs using differential fixed-curvature versus variable-curvature sections. Due to equations [5–7] while the latter provides a simpler model the vast variety of applications in concentric tube robots, with less time-cost and still acceptable accuracy [8, 9]. Lyons the optimization of concentric tube robots has always been et al. [10] presented a new method of optimization in IK for alluring for researchers. For example, a new design of CTCR medical devices to plan configurations with anatomically which is well suited to MIS inside small body cavities such as constrained for reaching specified targets. In this paper, they the heart is presented in [14]. This paper presents a general- calculate start position and orientation and a geometric rep- ized pattern search to optimize concentric tube robots while 2 Applied Bionics and Biomechanics base base base Figure 1: The scheme of a concentric tube robot comprising three precurved tubes. they can reach the target point with minimal curvature and (iii) The curvature of each segment of the robot remains length. Runge et al. [15] used evolutionary algorithms such constant, thus, the central backbone is always as genetic algorithm to optimize a soft robot. Bodily et al. located within a plane [16] used a genetic algorithm to optimize the reachability, dex- Figure 1 shows a three-segment concentric tube contin- terity, and manipulability of a multisegment continuum robot. uum robot. Based on the following assumption, the constant Luo et al. [17] designed a concentric tube manipulator that is curvature approach is utilized to model the system. appropriate for surgical environments. They presented a new The parameters used to describe the model of a three- topology of the tubes for the concentric tube robot, which segment continuum robot are listed in Table 1. can increase the stable workspace because it allows the usage Equation (1) calculates the length change of each of larger tube curvatures and/or curve lengths. Davarpanah segment. et al. [18] optimized a concentric tube robot using a genetic algorithm which allows them to reach the specificdestination L = L + ΔL , ð1Þ with high accuracy. Lloyd et al. [19] presented a novel model i i0 i for optimizing a task-specific in millimeter-scale for magneti- th cally actuated soft continuum robots used in medical applica- where L is the length of the i segment in the pri- i0 tions. Finally, most similar to our work, Cheong et al. [20] mary situation of insertion, and ΔL is the decrease or th applied a computational method to find optimal designs of increase in the i segment length which is task- continuum robots while considering reachability constraints. dependent. Then, Eq. (2) shows the total length of This paper contributes to the field by finding an optimized the catheter. inverse kinematic solution for a three-segment CTCR so that n n for a given position, the robot configuration would occupy L = 〠 L = 〠 L + ΔL , ð2Þ ðÞ minimized space. The proposed algorithm is very beneficial i i0 i i=1 i=1 in MIS, where the robot needs to work in a tight space. More- over, the optimized solution meets a safety interaction criterion 2.1. Coordinate Systems. Figure 2 presents the relations at the base of the CR. In practice, when the robot inserts to an between actuator space, configuration space, and task organ cavity, its shaft body would remain in the colons or ves- space. The actuator space includes tube lengths, the sels connected to that cavity. So, it is important that at the base configuration space covers the robot bending angles of the robot, which is inside a tighter space, the CR does not and the insertion length, and finally, the task space is apply forces to the surrounding tissues that may hurt. relevant to the end effector’s position and orientation. In MIS applications, the robot is ideally expected to per- The coordinate systems for a single-segment continuum form its mission with occupying the least possible workspace. robot (see Figure 3) are described as follows. To address this issue, in this paper, an algorithm is proposed that helps the minimum number of segments of a three- (1) Base Coordinate System. The base coordinate system segment CR be involved in achieving a target point. More- represented by fbg = fx, y, zg over, the robot-environment contact is modeled in the form of a spring which allows to apply interaction-force/displace- (2) End Coordinate System. The coordinate system is ment constraints to the base of the robot which assures a safe located at the end point of the tube (i.e., the intersec- interaction. Finally, the research is concluded in Section 4. tion of the tube with its nearby tube). The coordinate system is shown by E = fx , y , z g, i =1,2,3 for a i i i i three-segment catheter 2. Forward and Inverse Kinematics 2.2. Kinematics of Multisegment Robot. For the slender bar In this section, the structure of the constant and variable model, the curvature is not a constant and it can be widely length of a three-segment CTCR is described. The modeling used [21, 22]. As mentioned previously, the constant curva- assumptions of the CR are summarized as follows. ture approach is used to model the kinematics of the robot (Figure 3) [23, 24]. (i) The bending motion of the robot is planar The general mapping between the task space and the con- (ii) The curvature of the robot backbone is supposed to figuration space can be obtained by a homogeneous transfor- be constant mation matrix (Eq. (3)), ϕ Applied Bionics and Biomechanics 3 Table 1: The list of the nomenclatures used in this paper. Symbol Definition R The radius of curvature in the primary backbone is defined in the bending plate (the plate of X Z ) 1 1 The bending angle in the primary backbone is defined in the bending plate (X Z plate) at the point O . 1 1 φ ∈½ 02π The orientation angle of the robot which can rotate in the xy plate φ , α Rotation and bending angles of the first part of the catheter prox prox φ , α Rotation and bending angles of the second part of the catheter med med φ , α Rotation and bending angles of the third part of the catheter dist dist The primary length of the primary backbone th The position of the i secondary backbone fixed in the lower platform th The position of the i secondary backbone fixed in the upper platform O The center of the equilateral triangle made by the A points in the platform offg x, y, z C The center of the equilateral triangle made by the B points in the platform offg x , y , z i 1 1 1 P =½ x  y  z The position vector stating the center coordinate of the upper platform of each segment according to the lower platform c c c Curvature v =½ k φ  l The configuration space vector n The number of robot segments 3×3 R ∈ R The rotation matrix from the end disk coordinate system to the base coordinate system 0t Actuator space Configuration space Task space T T T (l , l , l ) (k, 𝜑, l) (x, y, z) 1 2 3 Tendon lengths Arc parameters Tip position Figure 2: Mapping between joint space variables, configuration space variables, and workspace variables. B y Top O´ y 1 𝛼 z Bottom B B 3 2 (a) (b) Figure 3: The introduced parameters in the robot: (a) the geometric display of a robot segment and (b) the display from the above of the robot platform. 4 Applied Bionics and Biomechanics 2 3 cφ 1 −ckl ðÞ ðÞ 2 cφskl ðÞ c φ ckl − 1 +1 −sφcφ ckl − 1 ðÞ ðÞ ðÞ ðÞ 6 7 6 7 "# 6 2 7 sφ 1 −ckl sφcφ ckl − 1 c φ 1 −ckl +ckl ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 6 7 R p ot sφskl ðÞ 6 7 T = = , ð3Þ 6 7 6 7 6 7 skl ðÞ 6 7 −cφskðÞl −sφskðÞl ckðÞl 4 5 where R represents the orientation of the end effector, p catheter tip and the corresponding target position ðP ot ref is the end position of the robot [25], c and s stand for cos = ½x  y  z Þ. Also, the optimized parameter would ref ref ref and sin, respectively. Using the length and radius of the be the configuration parameters including bending and rota- robot curvature shown in Figure 3, the angle α can be cal- tion angles and insertion length of each segment. According culated using to the number of extended segments (e.g., when two or three tubes are inserted out from the overlapped tube), the catheter ~ ̂ tip can be denoted by either p or p : To form cost function, 2 3 α = = kl, ð4Þ we first rearrange Eq. (6) as follows where R, l, and k are the arc length, bending radius, and ~ P = P + R α p : 2:ref 1 ot prox 2 ð7Þ curvature, respectively. More details on how to obtain P = P + R α P + R ðÞ α R α P : Eqs. (3) and (4) could be found in [25, 26]. The forward 3:ref 1 ot prox 2 ot med ot prox 3 kinematic model of the three-segment CR can be obtained using Next, the cost functions of the two-segment and three- b b 0 1 2 segment catheter can be defined in the form of T = T T T T : ð5Þ 3 0 1 2 3 J = γ p − p + R α p ot prox 2:ref 1 2 2.3. Inverse Kinematic for a Multisegment Robot. Increased ð8Þ 2 2 degrees of freedom (DOF) bring more redundancy for the p − p + R α ~p + α + α , 2,ref 1 ot prox 2 prox dist robot configuration. As a result, there are no closed-form solutions for the multisegment robot inverse kinematic. J = γ p − p + R α p + R ðÞ α R α p∧ ot prox ot med ot prox 3:ref 1 2 3 Moreover, there would be multiple solutions for the prob- lem of finding the inverse kinematics. To obtain the for-  p − p + R α p + R ðÞ α R α p ot prox ot med ot prox 3:ref 1 2 3 ward kinematic model, two different cases need to be 2 2 2 + α + α + α , prox med dist considered for the bending angle of the distal segment ° ° ° ð9Þ [27] including θ <90 and 90 ≤ θ ≤ 180 . The posi- dist dist tion of the central backbone of the catheter in the second where γ is a weighting factor. and third segments is achieved using 2.5. Safe Interaction Conditions. According to Figure 4, if you −1 p = R α p − p , want the robot to be safely interacting with the texture, at ðÞ 2 ot prox 2 1 first, the texture should be modeled into a spring, then, the ð6Þ −1 −1 p ̂ = R α p − p R α − p , amount of the allowed displacement for being in contact with ðÞ ðÞ 3 ot med 3 1 ot prox 2 the robot base should be obtained. Allowed ranges of the interaction force vary across surgical tasks. The size and fra- T T where the points P = ½x  y  z , P = ½x  y  z , 0 0 0 0 1 1 1 1 gility of the tissue are important parameters affecting the T T P = ½x  y  z , and P = ½x  y  z  are the catheter 2 2 2 2 3 3 3 3 allowed force range. base coordinate system, the intersection of the first and In this paper, the following assumptions are made second segments, the intersection of the second and third according to the literature [28, 29]. segments, and the end-plate of the third segment, respec- tively. More details can be found in [27]. Vectors ~p and 2 (i) Diameter of the main coronary arteries is about p are local positions of the end-effector in the coordinate 4 mm [30] systems at the base of each tube. (ii) The maximum allowed force applicable to the texture 2.4. Optimization Problem. The optimization problem is is considered as 0.2 N. The texture stiffness, at the defined to minimize the tracking error between the entrance of the cavity, is considered to be k =100N/ Applied Bionics and Biomechanics 5 Base Catheter tip Figure 4: The safe interaction with the texture modeled by the spring. −40 −50 −80 Y (mm) X (mm) Figure 5: Configuration of the three-segment catheter with bending and rotation angles. Table 2: Three-segment continuum robot bending and rotation m which corresponds to 2 mm displacement of the tis- angles. sue which is acceptable (Eq. (10)) Angles Segment 1 Segment 2 Segment 3 ° ° ° 63:24 53:54 35:58 Bending angle (α) F ° ° ° Rotation angle (φ) 49:80 310:05 178:89 F = KΔx ⇒ Δx = =±2mm: ð10Þ T T 73:26 36:63 76:21 Length (mm) Δx equals to the required displacement of P = ðx, y, zÞ T b so that the catheter tip would reach to the target point. 3. Results and Discussion qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In this section, first, we simulated the kinematic model 2 2 2 Δx = ðÞ x − x +ðÞ y − y +ðÞ z − z , T new old new old obtained in the previous section using MATLAB. new old Figure 5 demonstrates the simulation results for the ð11Þ three-segment catheter. The robot configuration parameters corresponding to the target point of the catheter tip are listed To obtain Δx , first, the shape of catheter corresponding in Table 2. to the desired tip catheter is obtained, and then among the Figure 6 shows the mapping between the position of the multiple solutions, those solutions which are meeting the robot end-effector and the configuration space variables. At safety constraint at the point of P are accepted. each iteration, the bending and rotation angles change smoothly Z (mm) 6 Applied Bionics and Biomechanics −10 −20 −30 0 102030405060708090 100 Sampling points Figure 6: Changes of the robot’s final position according to the mapping of the configuration space variables. Table 3: The simulation conditions for workspace. within ½0 π and ½0 2π, respectively. Step-increment equals ° ° to 1:8 for α and 3:6 for φ. The number of the chosen sample Single- Two- Three- Parameters points is 100. The weighting factor γ is set to 50. segment segment segment The length of each 120 mm 60 mm 40 mm 3.1. Continuum Robot Workspace. In this section, the robot segment workspace for a single, two, and three-segment catheter is 0 π 0 π/2 0 π/3 α ½ ½ ½ obtained and demonstrated. The variation range of the con- φ 0 2π 0 2π 0 2π ½ ½ ½ figuration parameters are shown in Table 3. As illustrated in Figures 7–12, with the increase of the number of robot segments, the thickness of the clouds form- ing the workspace is increasing, which reflects an enlarged accessible space and increased precision of the three- segment catheter compared to single- or two-segment cathe- ter. Hence, one can conclude that multisegment catheters are more appropriate choices for applications in which the cath- eter is supposed to operate in confined spaces. 3.2. Continuum Robot with Variable Length: Insertion Optimization. In this section, we consider a continuum robot with variable length. The purpose is to optimize the robot seg- ment lengths while meeting the aforementioned constraints. The schematic of the proposed optimization algorithm is shown in Figure 13. In this diagram, first, concentric tube robot receives the desired point and then calculates the distance between the robot’s tip and the desired point and increases the tube’slength −100 −80 −60 −40 −20 0 20 40 60 80 100 based on the calculations. If the desired point is in the range of X (mm) tube 1, only segment 1 would be involved to help the end- Figure 7: One-segment catheter workspace in 2D. effector reach the target point. Otherwise, the second and the third segments would be inserted to the workspace. ter of 120 mm length (60 mm for each segment). The target 3.3. Optimization Example: A Catheter with Two Segments point is set to P ½−47:98  − 33:92 33:90. Using the pro- (No Insertion Is Allowed). We consider a two-segment cathe- posed algorithm, as it has been shown in Figures 14 and 15, Coordinate points of x, y, z axis (mm) Z (mm) Applied Bionics and Biomechanics 7 50 100 −50 −50 Y (mm) −100 −100 X (mm) Figure 8: One-segment catheter workspace in 3D. −100 −80 −60 −40 −20 0 20 40 60 80 100 X (mm) Figure 9: Two-segment catheter workspace in 2D. after 2000 iterations, the optimal parameters of the robot shows itself; as a result, it will be possible to reduce the itera- including the bending and rotation angle of each segment tion limit when the velocity of the process seems to be impor- can be observed in Tables 4 and 5. tant to us. According to Figure 14, the amount of the cost function 3.4. Optimization Example: A Catheter with Two and Three after about 40 first iterations is approximately equal J = Segments (Insertion Is Allowed). As it was mentioned, in the 8:03. variable-length catheter, considering the target point, first, We also consider the hypothetical target point in the the proximal segment length increases from its primary coordinate system P = ½40:85 77:24 26:97 of for the length considered to be zero to its final length, and this same optimization of the three-segment catheter configuration trend for other links reports in order to approach the target space with a general length of 120 mm and a separate length point. of 40 mm for each segment. The three-segment catheter will have the same length for As it can also be seen in Figure 15, the amount of the cost each segment which is 40 mm has been used, so as to simulate function, after about 60 first iterations, equals J =6:59. how to access the variable-length catheter. The simulation results show that the catheter accuracy to We consider the target point at P = approach the point in question is too much, also, a consider- able amount of optimization, in the first several iterations, ½34:85 17:11 62:22 . After 2000 iterations, and after Z (mm) Z (mm) 8 Applied Bionics and Biomechanics −100 −100 −50 Y (mm) X (mm) Figure 10: Two-segment catheter workspace in 3D. −100 −80 −60 −40 −20 0 20 40 60 80 100 X (mm) Figure 11: Three-segment catheter workspace in 2D. optimization using the genetic algorithm, and also using the space with variable length. In the simulations, we consider cost function of Eq. (8), the amount of the shown response first the position of the robot tip at the entrance of the cavity in the 2016 Matlab will be in the form of Table 6. According in the form of the coordinate system of B = ½0 0 10 .At to these optimal parameters, the order of error is 0:01 mm for first, the optimization results for the point in question are two segment catheter which is small in comparison to three shown in Table 7 without the safe interaction with the fragile segment catheter that has an error greater than 1mm. There- structures like texture. If dB < Δx , the obtained position fore, the variable-length catheter will use two-segment of its and the parameters related to this position are correct, if inner parts in order to approach the desirable point, and this not, another response is chosen. same result has been shown in the last row of the first col- According to these optimal parameters, the order of error umn. Also, the optimization amount of the two-segment cost is 0:01 mm for a two-segment catheter which is small in com- function has been obtained to be the number of J =1:1032 parison to three segment catheter that has an error greater that is an appropriate amount for tracking the target point. than 1mm. Regarding the error amount, for each segment, the desirable link will be the second link. Also, the amount 3.5. Optimization of the Three-Segment Catheter with of the cost function for the two-segment of the variable- Variable Length Having the Safe Interacting with Fragile length catheter will be equal to 1:4539. Structures like Texture. We choose first the hypothetical As you can see in Table 8, for the catheter safe interacting point of P = ½30:16 25:59 65:96 as the target point, for with fragile structures like texture, according to the Eqs. (8) optimization of the three-segment catheter configuration and (9), at first, we obtained the safe limitation constraint Z (mm) Z (mm) Applied Bionics and Biomechanics 9 −100 −50 100 −50 Y (mm) 100 −100 X (mm) Figure 12: Three-segment catheter workspace in 3D. Start Get the target point The distance from tip's catheter is calculated 2 2 2 dist = x + y + z No If dist < L Yes A) Find the optimal solution by solving the inverse kinematics code of a single-segment robot B) Find the optimal solution by solving the inverse kinematics code of a two-segment robot C) Find the optimal solution by solving the inverse kinematics code of a three-segment robot Compare the answers A, B, C and give the optimal answer If L < dist < L + L 1 1 2 No Yes A) Find the optimal solution by solving the inverse kinematics code of a two-segment robot B) Find the optimal solution by solving the inverse kinematics code of a three-segment robot Compare the answers A, B and give the optimal answer No If L + L < dist < L + L + L 1 2 1 2 3 A) Find the optimal solution by solving the inverse kinematics code of a three segment robot Finish Figure 13: Flow chart for finding an optimal inverse kinematics. Z (mm) 10 Applied Bionics and Biomechanics × 10 0 5 10152025 30 35 40 45 50 Iteration Figure 14: Genetic algorithm for best fitness target point in two-segment catheter. 0 5 10152025 30 35 40 45 50 55 60 65 70 Iteration Figure 15: Genetic algorithm for best fitness target point in three-segment catheter. Table 4: Optimized results for reaching target point. Table 6: Optimized results for three-segment catheter with variable length. Optimal Optimal Reaching Section Error bending angle rotation angle point Two-segment Tracked point Three-segment Tracked point number (mm) (degree) (degree) (mm) θ : 44:6093 θ : 44:6093 prox prox 1 99.80 195.42 X: φ : 8:2412 φ : 8:2412 prox prox X2: 34.837 X3: 17.4794 0.17 θ : 38:9934 X: -40.81 θ : 38:9934 med med Y2: 17.099 Y3: 32.5204 Y: φ : 122:5938 φ : 122:5938 Y: -33.64 med med Z2: 66.199 Z3: 61.4679 0.28 2 88.57 324.01 Z: 33.73 θ : 192:3173 dist J: 1.1032 Z: φ : 145:7034 dist Link-num: 2 0.17 Table 5: Optimized results for reaching target point. Table 7: Optimized results for a three-segment catheter with variable length without interaction with fragile structures like Optimal Optimal Reaching texture. Section Error bending angle rotation angle point number (mm) (degree) (degree) (mm) Two-segment Tracked point Three-segment Tracked point 1 98.34 53.36 X: θ : 49:5561 θ : 53:3323 prox prox X: 40.86 0.01 φ : 55:6513 2 99.33 151.98 φ : 49:0286 prox prox X2: 30.1286 X3: 50.6561 Y: 77.27 Y: θ : 41:6511 θ : 41:5474 med med Y2: 25.5773 Y3: 1.2243 Z: 26.97 0.03 3 43.78 73.15 φ : 284:3094 φ : 255:1113 med med Z2: 65.9110 Z3: 72.9058 Z:0 θ : 124:2572 dist J: 1.4539 φ : 284:6543 Link-num: 2 dist for interacting with fragile structures like texture. Consider- ing the two mentioned conditions, we take into consideration the maximum amount of the allowed force which is applied the population in the algorithm, in each of the robot seg- to the texture to be 0:2N, and also the stiffness to be 100, at ments, the correct link for approaching the chosen target the entrance of the cavity. According to Eq. (10), the dis- point and the angle amounts of the catheter are obtained in placement amount at the contact point with the texture will the case that the distance amount of the robot is less than be obtained as ±2 mm. Now, after required iterations from 2 mm. Cost Cost Applied Bionics and Biomechanics 11 error for most real-time surgeries. In the final part, regarding Table 8: Optimized results for three-segment catheter with variable length with interaction with fragile structures like texture. force imposed onto the texture, and also the stiffness, the allowed distance amount with the texture causes the safe Two-segment Tracked point Three-segment Tracked point interaction. The implementation of the secure interaction θ : 53:4391 prox with the texture and the trajectory results indicate the cathe- θ : 53:4391 prox φ : 53:6859 ter has an accurate application in the medical procedure. prox φ : 53:6859 prox θ : 47:2743 X2: 30.0826 X3: 24.9051 Future works focusing on the addition of more constraints, med θ : 47:2743 med φ : 269:4255 Y2: 25.5582 Y3: 22.4644 and optimization of the distinguished constraints of the cost med φ : 269:4255 med Z2: 65.8237 Z3: 61.0422 function, try to lessen the tracking errors and to improve the J: 2.8294 θ : 299:592 dist stability of the existing catheter. Also, tracking a certain path Link-num: 2 φ : 220:5672 dist concerning the safe interaction with the texture for the cath- dB: 1.1641 eter will be carried out. According to the obtained error amount from the hypo- Data Availability thetical target point and the tracked point in the three- segment catheter, it can be concluded that the three-segment The MATLAB code data used to support the findings of this catheter has an undesirable tracking for approaching the target study are available from the corresponding author upon point, and also according to the obtained points in the table request. second column, and the little error amount the catheter will use two segments from its inner parts for approaching to the desired point. Conflicts of Interest Also, the optimized cost function amount of the two- segment catheter is obtained to be 2.8294 which is an The authors declare that they have no conflicts of interest. appropriate amount for tracking the target point. 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Journal

Applied Bionics and BiomechanicsHindawi Publishing Corporation

Published: Aug 16, 2021

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