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Design and Validation of a Novel Cable-Driven Hyper-Redundant Robot Based on Decoupled Joints

Design and Validation of a Novel Cable-Driven Hyper-Redundant Robot Based on Decoupled Joints Hindawi Journal of Robotics Volume 2021, Article ID 5124816, 16 pages https://doi.org/10.1155/2021/5124816 Research Article Design and Validation of a Novel Cable-Driven Hyper-Redundant Robot Based on Decoupled Joints 1,2 1 1 1 1 Long Huang, Bei Liu , Lairong Yin , Peng Zeng, and Yuanhan Yang School of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China Hunan Provincial Key Laboratory of Intelligent Manufacturing Technology for High-performance Mechanical Equipment, Changsha University of Science and Technology, Changsha 410114, China Correspondence should be addressed to Bei Liu; 17871947856@163.com and Lairong Yin; yinlairong@csust.edu.cn Received 15 July 2021; Revised 26 August 2021; Accepted 9 September 2021; Published 2 November 2021 Academic Editor: Yaoyao Wang Copyright © 2021 Long Huang et al. .is 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. In most of the prior designs of conventional cable-driven hyper-redundant robots, the multiple degree-of-freedom (DOF) bending motion usually has bending coupling effects. It means that the rotation output of each DOF is controlled by multiple pairs of cable inputs. .e bending coupling effect will increase the complexity of the driving mechanism and the risk of slack in the driving cables. To address these problems, a novel 2-DOF decoupled joint is proposed by adjusting the axes distribution of the universal joints. Based on the decoupled joint, a 4-DOF hyper-redundant robot with two segments is developed. .e kinematic model of the robot is established, and the workspace is analyzed. To simplify the driving mechanism, a kinematic fitting approach is presented for both proximal and distal segments and the mapping between the actuator space and the joint space is significantly simplified. It also leads to the simplification of the driving mechanism and the control system. Furthermore, the cable-driven hyper-redundant robot prototype with multiple decoupled joints is established. .e experiments on the robot prototype verify the advantages of the design. endoscope driven by multiple cables, which is more dex- 1. Introduction terous than rigid endoscopes [23]. Dong et al. proposed a In recent years, due to the advantages of compact structure continuum robot with a low ratio between diameter and and flexible bending motion in the confined environment, length based on compliant joints, which can apply to the hyper-redundant robots have received high attention in the inspection and maintenance of aero engines [24]. Based on field of minimally invasive surgery, maintenance, and testing universal joints, Jin et al. designed cable-driven snake-like 4- [1–3]. Various kinds of cable-driven hyper-redundant ro- DOF surgical forceps [25]. Kim et al. designed a novel rolling bots have been reported by researchers [4–11]. Generally, the joint with a block mechanism to develop a snake-like robot cable-driven hyper-redundant robots are usually composed for minimally invasive surgery [26]. of several segments driven by external actuators and mul- For most of the prior cable-driven hyper-redundant robots, the bending motion in all directions is usually tiple cables. Every segment consists of several identical joints in serial. Joint types for the cable-driven hyper-redundant coupled. It means the rotation output of each DOF is robot mainly can be classified into the 1-DOF joint and 2- controlled by multiple pairs of cable inputs [27]. Besides, the DOF joint [12]. .e 1-DOF joints mainly include the rev- mapping between the actuator space and joint space of the olute joint [13, 14], the flexible beam [15, 16], and the cy- robot is complex, which causes the driving mechanism and lindrical rolling joint [17, 18], while the multi-DOF joints the control system difficult [28]. Some researchers have tried mainly include the universal joint [19, 20], the flexible to address the drawbacks of the coupling bending motion backbone [21, 22], and the spherical rolling joint. Based on through special joint structure design and further simplify the flexible backbone, Li et al. developed a 2-DOF flexible the mapping between actuator space and joint space [29, 30]. 2 Journal of Robotics Based on multiple cylindrical rolling joints, Kim et al. in the joint. For the universal joint with two nonintersecting designed a cable-driven hyper-redundant robot, which di- rotation axes located between the two disks [32], it can be minishes the bending coupling effect by enlarging the space also proved that the coupling effect exists in the joint when at for the passage of the center cable [17]. least one rotation axis is located at the middle place of two In this paper, a novel 2-DOF decoupled joint is proposed disks, which is similar to the universal joint, as shown in by adjusting the distribution of two rotation axes. For the Figure 1. decoupled joint, a pair of antagonistic cable inputs only Based on the above analysis, we tried to change the controls a 1-DOF rotational output, and the rotational distribution of the joint rotation axes to avoid the coupling output of each DOF is only determined by a single pair of effect. .erefore, this paper proposes a novel joint without antagonistic cable inputs. By connecting two 2-DOF seg- coupling effects, as shown in Figure 2. .e two axes in the ments in serial, a cable-driven hyper-redundant robot is joint are, respectively, coincident with the upper surface of presented. Each segment consists of multiple identical 2- the lower disk and the lower surface of the upper disk. Since DOF decoupled joints in serial. A kinematics linear fitting points A and B are located at the axis w , points A and B 2 2 1 2 2 approach is presented to simplify the mapping between achieve circular motion around points A and B when the 1 1 actuator space and joint space. Based on the linear fitting upper disk rotates around the axis w . .erefore, the length error analysis, each pair of antagonistic cables is driven by a of cable A and cable B will not change, while the length of motor through a circular pulley, which can simplify the cable C and cable D will also change. Since the points C and design of the driving mechanism. .e proposed robot is D are located at the axis w , the points C and D can be 2 2 2 2 verified by the bending motion experiments, the cable considered as the fixed points when the upper disk rotates tension test, and the load experiments. around the axis w . In consequence, the length of cableC and .e rest of this paper is organized as follows: Section 2 cable D will not change, while the length of cable A and cable introduces the challenges of the existing robots. Moreover, a B will change. .is indicates that the coupling effect does not novel cable-driven hyper-redundant robot is also introduced exist in the proposed joint with the special distribution of in Section 2. .e kinematics and the robot workspace are two axes positions. analyzed in Section 3. .e presented kinematics linear fitting and error analysis of the robot are discussed in Section 4. .e 2.1.2. Challenge of the Driving Mechanism Design. .e re- robot prototype through some experiments is verified in lationship between cable length and bending angles should Section 5. Section 6 presents the conclusion. be considered to design the driving mechanism of the cable- driven hyper-redundant robot [33, 34]. For most cable- 2. Robot Design driven hyper-redundant robots, the relationship between the length inputs of each cable and bending angles is a nonlinear 2.1. Challenges of the Existing Robots function. It means the tightened amount on one cable is not equal to the released amount on the antagonistic cable when 2.1.1. Coupling Effect. For most of the prior designs of cable- the robot bends to an arbitrary configuration. It is difficult driven hyper-redundant robots, the 2-DOF bending motion for cable-driven hyper-redundant robots to design a com- in each segment of the robot is coupled. .e rotational pact and simple driving mechanism. output of each DOF of the joint is controlled by multiple .e following driving mechanism design approaches are pairs of antagonistic cable inputs. Consequently, the cou- adopted. .e first approach is that each cable is driven by a pling effect will increase the complexity of the robot’s driving separate motor and a circular cable pulley. It is convenient to mechanism and the risk of slack in the driving cables. design the driving mechanism. However, this approach .e relationship between the cable length and the ro- increases the complexity of the control system, as shown in tation angles is determined by the joint types and their Figure 3(a). .e second approach is that noncircular cable structural parameters. For instance, the coupling effect of the pulleys are designed to realize a motor driving a pair of revolute joints is determined by the cable distribution circle cables based on the nonlinear function, as shown in radius, the number of cables, and the distribution of the Figure 3(b). .is method significantly reduces the number of rotation axes. Since the cable distribution circle radius and the motor, but the fabrication and assembly of noncircular the number of cables are confined by the practical factors, cable pulleys require high accuracy. .e third approach is the distribution of joint rotation axes is a significant factor to that a pair of cables is driven by a separate motor and a avoid the coupling effect. Figure 1 shows a conventional circular cable pulley, as shown in Figure 3(c). .is method universal joint with two intersecting rotation axes located at requires the releasing amount of one cable is equal to the the middle of two disks [31]. When the upper disk rotates tightening amount of the antagonistic cable, which can around rotation axes w through the releasing of cable A and simplify the driving mechanism. tightening of cable C, cable B and cable D must be tightened simultaneously to avoid slack, as shown in Figure 1(b). Similarly, the rotation around axis w requires the control of 2.2. A Novel Robot Design with the Decoupled Joints. .is all four cables. Otherwise, cable A and cable B will become section proposes a cable-driven hyper-redundant robot slack. It can be proved that the rotational output of each based on the multiple decoupled joints, as shown in Figure 4. DOF requires the control of multiple pairs of antagonistic .e robot is composed of a proximal segment, a distal cables regardless of the location of the two intersecting axes segment, and a driving mechanism. Each segment consists of Journal of Robotics 3 A C 2 2 B 2 Disk B Joint D C Disk A A 1 1 1 C B 1 B 1 1 Release Tighten Tighten Driving mechanism Driving mechanism (a) (b) Figure 1: .e traditional universal joint with two intersecting rotation axes. B 2 Upper disk B 2 Joint A B C Lower disk C A C 1 1 1 B w w 1 1 1 Release Release Tighten Tighten Driving mechanism Driving mechanism (a) (b) Figure 2: A novel joint without coupling effects. Tighten Release Tighten Release Tighten Release (a) (b) (c) Figure 3: Classic driving system types. h Proximal segment Distal segment 4 Journal of Robotics Driving mechanism Joint structure parameters linkage disk Bending joint Proximal segment Distal segment linkage disk linkage disk Figure 4: Cable-driven hyper-redundant robot prototype. Table 1: Parameters of the joint structure. six identical 2-DOF joints. Each joint contains two disks and one spatial linkage. .e cylindrical bulge surface on the disk Symbol Description Value and the cylindrical concave surface on the spatial linkage H .e distance between two axes 8 mm cooperate to form two rotating pairs. .e axis w and axis w 1 2 t Disk thickness 2.5 mm coincide, respectively, with the upper surface of the lower r Cable distribution circle radius 4.25 mm disk and the lower surface of the upper disk. Joint structure d Robot diameter 10 mm parameters are defined, as shown in Table 1. (θ, φ) Joint variables in the proximal segment (−π/18, π/18) .e 2-DOF bending motion of each segment is achieved (α, β) Joint variables in the distal segment (−π/18, π/18) by two motors controlling a pair of antagonistic cables through the circular pulleys. Cable A, cable B, cable C, and joint space, and task space [35]. .e following assumptions cable D control the 2-DOF bending motion of the proximal are made in this study. In this proposed robot, there is no gap segment, as shown in Figure 5(a), while cable E, cable F, between the cables and the cable holes. .e cables’ shear cable G, and cable H control the 2-DOF bending motion of strains and elongation are negligible. .e cable tension the distal segment, as shown in Figure 5(b). Since the two exerting on each joint is the same. rotation axes of each joint are coincident with the end Based on these assumptions [36], the joint kinematics is surfaces of corresponding disks, the distance of the two first established to analyze the decoupled effect in the rotation axes is always equal to h regardless of the robot proposed 2-DOF joint. .e relationship between the sum of configurations. In addition, a pair of antagonistic cable the cable length change and bending angles theoretically inputs only control the 1-DOF rotational output, and the validates that the cables in the robot will not become slack. rotational output of each DOF of the joint is only deter- Besides, the robot kinematics is established and the robot mined by a single pair of antagonistic cable inputs. With this workspace is analyzed. design, the mapping between actuator space and joint space can be eventually simplified. Besides, the driving mechanism design of the robot is illustrated in Section 4. 3.1.JointKinematics. Since the proximal segment and distal segment have the same bending motion, we consider a single joint in the proximal segment as an example to establish the 3. Kinematics joint kinematics, as shown in Figure 6. .e kinematics of the cable-driven hyper-redundant robot .e coordinate systems {O }, {O }, and {O } are i 1i i+1 requires establishing the mapping between actuator space, established, respectively, on the center of the upper surface Journal of Robotics 5 2 H E O G O w 4 4 F 2 2 E 2 E BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 2 H 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 C F B 1 1 F 1 H E H 1 4 F E 3 3 (a) (b) Figure 5: .e 2-DOF joint of the proximal and distal segment. i+1 i+1 1i i+1 C y 2 1i 1i 0i C y 1 0i 1 x (x ) i 0i Figure 6: Kinematics coordinate system of the single joint. of the lower disk, the lower surface of the upper disk, and the (1) T � rot x ,θ􏼁 trans z , h􏼁 rot y ,φ 􏼁 trans z , t 􏼁 . i+1 i 0i 0i 1i upper surface of the upper disk. Axis x coincides with the axis w , axis y coincides with the axis w , and axis y is To establish the relationship between the cable length 1 1i 2 i+1 parallel to the axis w . .e transformation from the coor- 2 and bending angles, it is assumed that the position vector of dinate system {O } to {O } is as follows. First, the coor- any point p in {O } on the upper surface of the lower disk is i i+1 1 i dinate system {O } rotates angleθ around x axis to obtain the represented by p , while the position vector of any point p i i p1 coordinate system {O }. Second, the coordinate system {O } in {O } on the lower surface of the upper disk is represented 0i 0i 1i 1i moves h along the z axis and then rotates angle φ around by p . According to the coordinate transformation, the 0i p2 the y axis to obtain the coordinate system {O }. .ird, the position vector of any point p in {O } on the lower surface of 0i 1i 2 i coordinate system {O } moves t along the z axis to obtain the upper disk is represented by p . .e transformation can 1i 1i p2 the coordinate system {O }. Hence, the homogeneous be described as i+1 transformation matrix from the coordinate system {O } to i i 1i i p � R p + p , (2) p2 1i p2 1i {O } can be obtained as i+1 6 Journal of Robotics i i where R is the rotation matrix from {O } to {O } and p is 3.2. Robot Kinematics. Based on the joint kinematics, the i 1i 1i 1i the position vector of {O } relative to {O }. .erefore, the kinematics of the cable-driven hyper-redundant robot is 1i i relationship between cable length and angles θ and φ can be established. .e mapping between the actuator space, joint expressed as space, and task space is also obtained [39]. .e base coor- 􏼌 􏼌 dinate system {O } is established at the center of the lower 􏼌 􏼌 i i 􏼌 􏼌 􏼌 􏼌 l � p − p , (3) 􏼌 p2 p1􏼌 surface of the base disk, as shown in Figure 8. Axis x is parallel to the axis w , and axis z is perpendicular to the 1 0 where p and p , respectively, represent the intersection p1 p2 lower surface of the base disk. According to the joint ki- points between the cables, the upper surface of the lower nematics, the establishment principle of the coordinate disk, and the lower surface of the upper disk. systems {O } − {O } in the proximal segment is the same as 1 n Taking the cable length in a single joint of the proximal the coordinate system {O } − {O } in the distal segment. m1 mn segment as an example, the coordinates of the points A , B , 1 1 .erefore, the mapping between actuator space and joint C , and D in {O } are represented by p � (r, 0, 0), 1 1 i A1 space is first established. We assume that each segment of the i i i p � (0, − r, 0), p � (−r, 0, 0), and p � (0, r, 0), B1 C1 D1 robot contains n identical joints. In the straight configu- while the coordinates of the points A , B , C , and D in {O } 2 2 2 2 1i ration, each cable length in the proximal and distal segment 1i 1i are represented by p � (r, 0, 0), p � (0, − r, 0), A2 B2 can be obtained as 1i 1i p � (−r, 0, 0), and p � (0, r, 0). .erefore, the rela- C2 D2 tionship between the cable length and angles θ and φ can be L � nh + (n + 1)t, expressed as 􏼨 (5) L � 2nh + 2(n + 1)t, 􏽱��������������������� � 􏼌 􏼌 􏼌 􏼌 􏼌i i 2 2 2 ⎧ ⎪ 􏼌 ⎪ 􏼌 􏼌 l � p − p � 2r − 2r cφ + h − 2rh · sφ ⎪ A 􏼌 􏼌 A2 A1 where L represents the initial length of each cable driving ⎪ 􏽱������������������������������� ⎪ 􏼌 􏼌 the proximal segment and L represents the initial length of ⎪ 􏼌 􏼌 i i 2 2 ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 l � p − p � (−r · cθ − h · sθ + r) + (r · sθ + h · cθ) ⎨ 􏼌 􏼌 B B2 B1 each cable driving the distal segment. 􏽱��������������������� � , ⎪ 􏼌 􏼌 􏼌 􏼌 ⎪ According to equation (2) and equation (3), the rela- i i 2 2 2 ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 l � p − p � 2r − 2r cφ + h + 2rh · sφ 􏼌 􏼌 ⎪ C C2 C1 tionship between each cable length and bending angles in the 􏽱������������������������������� ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 i i 2 2 arbitrary configurations can be derived as ⎩ 􏼌 􏼌 􏼌 􏼌 l � p − p � (r · cθ − h · sθ − r) + (−r · sθ + h · cθ) D 􏼌 􏼌 D2 D1 􏼌 􏼌 􏼌 􏼌 i i 􏼌 􏼌 ⎧ ⎪ ′ 􏼌 􏼌 L � (n + 1)t + n p − p , (4) ⎨ P 􏼌 􏼌 p2 p1 􏼌 􏼌 (6) ⎪ 􏼌 􏼌 j j ⎩ 􏼌 􏼌 ′ ′ 􏼌 􏼌 L � L + (n + 1)t + n p − p , where l , l , l , and l represent the cable length in the joint d P 􏼌 􏼌 A B C, D p4 p3 of the proximal segment, cθ � cosθ, sθ � sinθ, cφ � cosφ, and where L represents the cables’ length in the proximal sφ � sinφ. p segment and L represents the cables’ length in the distal When angles θ and φ are equal to zero, each cable length d i i segment. In the proximal segment, p and p are the in the joint is equal to h, and the proximal segment keeps a p1 p2 intersection description between the cable on any side and straight configuration. According to equation (4), the length the upper surface of the lower disk and the lower surface of of cable A and cable C only depends on the angle φ, while the the upper disk in the coordinate system {O }, respectively. length of cable B and cable D only depends on the angle θ. i j j In the distal segment, and p and p are the intersection Consequently, during the 2-DOF bending motion of the p3 p4 description between the cable on any side and the upper proximal segment, a pair of antagonistic cable inputs only surface of the lower disk and the lower surface of the upper controls the 1-DOF rotation output, and the rotation output disk in the coordinate system {O }, respectively. .erefore, of each DOF of the joint is only determined by a single pair the mapping between the actuator space and joint space can of antagonistic cable inputs. .e 2-DOF bending motion in be established by equation (6). According to the trans- the proximal segment is completely decoupled. Similarly, the formation shown in Figure 8, the mapping between joint 2-DOF decoupled effect of the distal segment is the same as space and task space can be established. Hence, the ho- the decoupled effect of the proximal segment. mogeneous transformation matrix from the coordinate According to the literature [37, 38], if the sum of the system {O } to the coordinate system {O } can be 0 mn cables length changes in each one pair of antagonistic cables written as is positive, the cables will not become slack. When the proposed joint bends from a straight configuration to an 0 0 i n n T � T × ( T) × T × ( T) . (7) m1 j+1 mn 1 i+1 arbitrary bending configuration around axes w and w , the 1 2 relationship between the sum of the cable length changes in In the proposed cable-driven hyper-redundant robot, each one pair of antagonistic cables and bending angles is the adjacent two axes in each joint are perpendicular to shown in Figure 7. .e sum of length changes in the an- different bending planes, which causes that the inverse ki- tagonistic cables is positive regardless of the bending angles. nematics is difficult to solve through the analytical method .is indicates that the cables will not become slack when the [40–42]. .e Newton–Raphson iterative method can be used proximal segment and the distal segment achieve, respec- to solve the inverse kinematics, but it is not the research tively, 2-DOF bending motion. focus in this paper. Journal of Robotics 7 -4 -3 ×10 ×10 6 5 ∆ ∆ 0 0 -π/18 -π/36 0 π/36 π/18 -π/18 -π/36 0 π/36 π/18 Bending angle φ (rad) Bending angle θ (rad) (a) (b) Figure 7: .e sum of the antagonistic cables’ length changes in any pair of cables. (a) Bending angle φ (rad). (b) Bending angle θ (rad). nonlinear function. However, the following kinematics x linear fitting and error analysis will show that the rela- 1 x tionship can be well fitted to a linear function in a certain 1 y range of joint variables, and the tightened amount of the n-1 cable on one side is almost equal to the released amount of n-1 m1 the antagonistic cable when the robot configuration changes. n x m2 Hence, any pair of antagonistic cables in the robot can be y x m1 j x mn driven by a motor and a circle cable pulley, as shown in m2 Figure 3(c), which not only simplifies the driving mecha- mn nism but also reduces the control complexity. .e following mn contents are the kinematic linear fitting and error analysis in Figure 8: Coordinate system of the cable-driven hyper-redundant two segments. Based on the results, the driving mechanisms robot. of the two segments are designed. 3.3. Workspace Analysis. .e workspace of the cable-driven 4.1. Kinematics Linear Fitting in the Proximal Segment. hyper-redundant robot is determined by the joint geometry, Since the 2-DOF bending motion in the proximal segment is bending angles, and the joint number [43]. Based on the decoupled, cable A and cable C are considered as an example robot kinematics, the robot workspace is obtained. to perform the kinematic linear fitting using the polynomial Figure 9(a) shows the workspace of the proximal segment, fitting method. .e error values between the original and the while Figure 9(b) shows nine bending configurations of the fitting function are analyzed by the percentage error model. proximal segment when the joint angles (θ, φ) are, re- .e percentage error e % (φ) is defined as spectively, (0, 0), (0, π/36), (0, −π/36), (π/36, 0), (−π/36, 0), l (π/36, −π/36), (−π/36, π/36), (−π/36, −π/36), and (π/36, 􏼌 􏼌 􏼌 􏼌 l(φ) − l (φ) 􏼌 􏼌 􏼌 􏼌 (8) π/36). Based on this, Figure 9(c) shows the workspace of the e %(φ) � 100 · 􏼌 􏼌, 􏼌 􏼌 􏼌 l(φ) 􏼌 robot, while Figure 9(d) shows multiple bending configu- rations of the robot when joint angles (θ, φ, α, β) are, re- where l (φ) represents the original function and l’ (φ) spectively, (0, 0, 0, 0), (0, 0, 0, π/36), (0, 0, 0, −π/36), (0, 0, represents the fitting function. .e fitting curve and error π/36, 0), (0, 0, −π/36, 0), (π/36, 0, 0, 0), (−π/36, 0, 0, 0), (0 values between the original function and the fitting function −π/36, 0, 0), and (0, π/36, 0, 0). According to the above of cable A and cable C are solved by the MATLAB curve analysis, the more the segment number is, the larger the fitting tool, as shown in Figure 10. .e fitting functions of workspace of the robot becomes. cable A and cable C are represented by l (φ) � −4.237φ + 8 and l (φ) � 4.237φ + 8, respectively, as shown in Figure 10(a). 4. Kinematics Linear Fitting Within the range of bending angles shown in Table 1, the Based on equation (4) and equation (6), the relationship maximum fitting error between the original function and between each cable length and bending angles is the fitting function is 0.025%, and the maximum angle error of l +∆l (mm) A C l +∆l (mm) B D -40 -20 X (mm) -50 -100 X (mm) 25 50 -50 -25 X (mm) -20 -10 0 10 X (mm) 8 Journal of Robotics (a) (b) (c) (d) Figure 9: .e workspace and bending configurations of the robot. (a) .e workspace of the proximal section. (b) Nine bending con- figurations of the proximal section. (c) .e robotic workspace. (d) Multiple bending configurations of the robot. 8.8 0.025 8.6 Cable A Cable C 0.02 8.4 Cable A Cable C 8.2 0.015 0.01 7.8 7.6 0.005 7.4 l ’(φ)=4.237φ+8 l ’(φ)=-4.237φ+8 c A 7.2 0 -π/18 -π/36 0 π/36 π/18 -π/18 -π/36 0 π/36 π/18 Bending angle φ (rad) Bending angle φ (rad) Original function Fitting function (a) (b) Figure 10: Cable A and cable C error between the original and fitted function. (a) Bending angle φ (rad). (b) Bending angle θ (rad). ′ ′ the end disk in the proximal segment is less than 0.15%, as represented by l (θ) � −4.237θ + 8 and l (θ) � 4.237θ + 8, B D shown in Figure 10(b). Based on the same kinematics fitting respectively. .erefore, the relationship between each cable method, the fitting functions of cable B and cable D are length and bending angles is linear through the special 10 20 -20 -10 -30 Y (mm) -50 -100 Y (mm) -60 -30 30 60 Y (mm) -20 -10 Y (mm) e length of the cable A and cable C (mm) Z (mm) Z (mm) Errors e % Z (mm) Z (mm) Journal of Robotics 9 distribution of two rotation axes. It means the mapping fitting method. Within the range of bending angles shown in between actuator space and joint space is simplified. Table 1, the kinematics linear fitting results of cable E and Moreover, the 2-DOF bending motion of the proximal cable G in each joint of the proximal segment are shown in segment can be driven by two motors and two circular Figure 11. pulleys. Based on the literature [29], the robot motion ac- Similarly, the MATLAB curve fitting tool is also used to curacy is satisfied. solve the fitting function. .e fitting functions of cable E and cable G are represented by l (θ,φ) � −2.996θ − 2.996φ + 8, l (θ,φ) � 2.996θ + 2.996φ + 8. .e percentage error e % is G l 4.2. Kinematics Linear Fitting in the Distal Segment. redefined as According to equation (2) and equation (3), when the 􏼌 􏼌 􏼌 􏼌 􏼌 ′ 􏼌 l(θ,φ) − l (θ,φ) proximal segment undergoes 2-DOF bending motion, the 􏼌 􏼌 􏼌 􏼌 e %(θ,φ) � 100 · , (11) 􏼌 􏼌 􏼌 􏼌 length of all eight cables will change. .is means that the 􏼌 􏼌 l(θ,φ) bending motion between the proximal segment and distal where l (θ, φ) represents the original function and l’(θ, φ) segment is coupled. .erefore, the kinematics linear fitting represents the fitting function. of cable E, cable G, cable F, and cable H in proximal and According to equation (11), the maximum fitting errors distal segments should be considered to design the driving between the original function and fitting function of cable E mechanism of the distal segment. and cable G are less than 0.04%, while the maximum fitting When the proximal segment keeps the straight config- errors of the cable F and cable H are less than 0.25%, as uration, and the distal segment keeps an arbitrary bending shown in Figure 12. .erefore, the relationship between the configuration, the relationship between the length of cable E, length of cable E, cable G, cable F, and cable H, and bending cable G, cable F, and cable H and bending angles α and β can angles can be approximately linear. .is indicates that the be expressed as 􏽱��������������������� mapping between actuator space and joint space in the distal 2 2 2 ⎪ ⎧ segment is also simplified. .erefore, the 2-DOF bending ⎪ l � 2r − 2r cα + h − 2rh · sα , ⎪ E1 􏽱������������������������������� motion of the distal segment can be achieved by two motors ⎪ 2 2 and two circle pulleys. Based on the literature [29], the robot l � (−r · cβ − h · sβ + r) + (r · sβ + h · cβ) , F1 (9) 􏽱��������������������� motion accuracy is satisfied. ⎪ 2 2 2 l � 2r − 2r cα + h + 2rh · sα , G1 􏽱������������������������������� 2 2 5. Experiment Validation l � (r · cβ − h · sβ − r) + (−r · sβ + h · cβ) , H1 In this section, a 4-DOF cable-driven hyper-redundant robot where l , l , l , and l represent the length of a single E1 F1 G1 H1 prototype is established to validate the robot design. .e joint in the distal segment, cα � cosα, sα � sinα, cβ � cosβ, proposed robot includes the proximal segment, the distal and sβ � sinβ. According to equation (9), the 2-DOF bending segment, and the driving mechanism, as shown in motions of the distal segment are decoupled when the Figure 13(a). .e total length of the proximal segment and the proximal segment does not achieve the 2-DOF bending distal segment is 131 mm. .e cables’ diameter is 0.4 mm. .e motion. In addition, the relationship between the cable driving mechanism of the robot prototype includes a guiding length of the distal segment and the bending angles α andβ is device, a motor driving device, and a cable tension adjusting also approximately linear. device, as shown in Figure 13(b). .e rated speed of the motor If the proximal segment achieves 2-DOF bending mo- is 10 r/min, and the rated torque is 70kgcm. .e range of the tion, the lengths of cable E, cable G, cable F, and cable H will force sensor is 0–10 kg with an accuracy of 0.03%. also change. Hence, the cable length change relationship in According to the kinematic linear fitting relationship in the proximal segment should be considered to achieve the Section 4, the driving mechanisms of the proximal and distal kinematics linear fitting of the distal segment. According to segments are the same. For the proximal segment, cable A equation (2) and equation (3), the relationship between the and cable C are the two ends of one cable that is driven by length of cable E, cable G, cable F, and cable H and bending motor 1 to control the proximal segment bending in the x z angles θ and φ can be calculated as 0 0 􏼌 􏼌 plane. Cable B and cable D are also the two ends of one cable 􏼌 􏼌 􏼌i i 􏼌 ⎧ ⎪ 􏼌 􏼌 l � p − p , ⎪ 􏼌 􏼌 that is driven by motor 2 to control the proximal segment E2 E2 E1 ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 bending in the y z plane. 0 0 ⎪ 􏼌i i 􏼌 ⎪ 􏼌 􏼌 l � p − p , ⎨ 􏼌 􏼌 F2 F2 F1 For the distal segment, cable E and cable G are the two (10) 􏼌 􏼌 ⎪ 􏼌 􏼌 􏼌i i 􏼌 ⎪ ends of one cable that is driven by motor 3 to control the distal 􏼌 􏼌 ⎪ l � p − p , 􏼌 􏼌 ⎪ G2 G2 G1 ⎪ segment bending in the x z plane. Cable F and cable H are m1 m1 ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 i i ⎩ 􏼌 􏼌 􏼌 the two ends of one cable that is driven by motor 4 to control l � p − p , 􏼌 􏼌 H2 H2 H1 the distal segment bending in the y z plane. .e two ends m1 m1 where l , l , l , and l represent the cable length of the of each cable are fixedly connected to the end disks of the E2 F2 G2 H2 single joint in the proximal segment. Since the 2-DOF proximal segment and distal segment through knotting. .e bending motion in the proximal segment has similar ki- middle of each cable passes through each joint disk and winds nematics, cable E and cable G are considered as an example around the guide device, driving device, and tension adjusting to perform the kinematic linear fitting using the polynomial device, as shown in Figure 13(b). Each cable tension is Bending angle φ (rad) -π/18 -π/36 π/36 π/18 -π/18 -π/36 π/36 π/18 π/18 π/36 -π/36 -π/18 -π/18 -π/36 π/36 π/18 Bending angle φ (rad) Bending angle φ (rad) Bending angle φ (rad) π/18 0 π/36 -π/18 -π/36 Bending angle φ (rad) 10 Journal of Robotics 9.5 8.5 Cable E Cable G 7.5 Cable G Cable E 6.5 Original function Fitting function Figure 11: Kinematics linear fitting analysis of cable E and cable G. 0.02 0.25 0.015 0.20 0.01 0.15 0.1 0.005 0.05 (a) (b) 0.035 0.030 0.030 0.025 0.025 0.02 0.02 0.015 0.015 0.01 0.01 0.005 0.005 (c) (d) Figure 12: Kinematics linear fitting error analysis of cables E, G, F, and H. adjusted by changing the position of the sliding block. .e 5.1. Free Bending Motion. In this section, the multi-DOF cable tension values are tested by the tension sensors. .e bending motions of the proximal segment following experiments include the free bending motion test, and distal segment have experimented, as shown in the cable tension test, and payload experiments. Figure 14. Bending angle θ (rad) Bending angle θ (rad) Bending angle θ (rad) -π/18 -π/36 0 π/36 π/18 π/18 π/36 -π/36 -π/18 -π/18 -π/36 π/36 π/18 -π/18 0 -π/36 π/36 π/18 Bending angle θ (rad) Bending angle θ (rad) π/18 π/36 0 -π/36 -π/18 e error the cable E (e %) e error the cable G (e %) e length of the cable E and cable G (mm) e error the cable G (e %) e error the cable F (e %) l Journal of Robotics 11 Motor 3 Motor 4 e distal segment e proximal segment Tension adjusting Driving device Tension pulley device Guiding device e driving mechanism of the proximal segment Tension adjusting Guiding device device Tension pulley Driving device e driving mechanism of the distal segment Figure 13: Cable-driven hyper-redundant robot prototype. When the distal segment keeps a straight configuration, When the proximal segment keeps a straight config- the bending motion of the proximal segment in the x z uration, the bending motion of the distal segment in the 0 0 plane requires the coordinated work of motor 1, motor 3, x z plane only requires motor 3 working to change the m1 m1 and motor 4. Motor 2 does not work to ensure that the cable length of cable E and cable G. .e other motors do not lengths of cable B and cable D are unchanged. .e bending work to ensure that the cable lengths of cable A, cable C, configuration outputs of the proximal segment in the x z cable E, cable G, cable F, and cable H are unchanged. .e 0 0 plane are only determined by the inputs of motor 1. Motor 3 bending configuration outputs of the proximal segment in and motor 4 are driven to keep the straight configuration of the x z plane are only determined by the inputs of m1 m1 the distal segment. .e bending configurations are shown in motor 3. Motor 1, motor 2, and motor 4 are not driven to Figures 14(a)–14(c). keep the straight configuration of the proximal segment. When the distal segment keeps a straight configuration, .e bending configurations are shown in Figures 14(h)– the bending motion of the proximal segment in the y z 14(j). 0 0 plane requires the coordinated work of motor 2, motor 3, Similarly, the bending motion of the distal segment in and motor 4. Motor 1 does not work to ensure that the the y z plane only requires motor 4 working to change m1 m1 lengths of cable A and cable B are unchanged. .e bending the length of cable F and cable H. Besides, to verify the configuration outputs of the proximal segment in the x z multi-DOF bending motion of the robot, we consider the 0 0 plane are only determined by the inputs of motor 2. Motor 3 bending configuration of the proximal segment in the x z 0 0 and motor 4 are driven to keep the straight configuration of plane and the bending configuration of the distal segment the distal segment. .e bending configurations are shown in in the x z plane as an example, as shown in m1 m1 Figures 14(d)–14(g). Figures 14(k)–14(n). 12 Journal of Robotics 0° +30° +60° (a) (b) (c) (d) (e) (f) (g) 0° -60° -30° (h) (i) (j) (k) (l) (m) (n) Figure 14: Robot bending motion experiments. 5.2. Cable Tension Test. During the multiple bending mo- the length of each cable driving the distal segment, the tions, the cable average tension curves are used to illustrate average tension of each cable driving the distal segment will the design rationalization of the driving mechanism for the increase. proximal segment and distal segment [44]. For the proximal When the proximal segment keeps a straight configu- segment, the bending configuration in the x z plane is ration and the distal segment keeps a bending configuration 0 0 determined by the angle θ, while the bending configuration in the x z plane and y z plane, the average cable m1 m1 m1 m1 in the y z plane is determined by the angle φ. When the tension of each cable varies with the joint angles α and β, as 0 0 distal segment keeps a straight configuration and the shown in Figure 16. For the proximal segment, the bending proximal segment keeps a bending configuration in the x z configuration in the x z plane is determined by the angle 0 0 m1 m1 plane and y z plane, the average cable tension of each cable β, while the bending configuration in the y z plane is 0 0 m1 m1 varies with the bending angles θ and φ, as shown in determined by the angle α. Figure 15. Within the range of the bending angles of [−π/18, π/18], Within the joint angle ranges of [−π/18, π/18], when when only the distal segment bends in the x z plane, the m1 m1 only the proximal segment bends in the x z plane, the average tension of the cable F and cable H in the distal 0 0 average tension of cable A and cable C in the proximal segment varies in the range of 10 N–12 N, as shown in segment varies in the range of 13 N–15 N, as shown in Figure 16(a). When only the distal segment bends in the Figure 15(a). When only the proximal segment bends in the y z plane, the average tension of cable E and cable G in m1 m1 y z plane, the average tension of cable B and cable D in the the distal segment varies in the range of 11 N–13 N, as shown 0 0 proximal segment changes within the range of 10 N–13 N, as in Figure 16(b). For the cables of the proximal segment, the shown in Figure 15(b). For the cables of the distal segment, average tension of cable A, cable B, cable C, and cable D the average tension of the cable E, cable G, cable F, and cable varies in the range of 12 N–15 N. .erefore, the phenom- H varies in the range of 8 N–10 N and 6 N–10 N. Since the 2- enon of the cables slack does not appear during the multi- DOF bending motion of the proximal segment will change DOF bending motion. .e results indicate that the driving Journal of Robotics 13 –π/18 –π/36 0 π/36 π/18 Bending angle φ (rad) Cables of the proximal segment Cables of the distal segment (a) –π/18 –π/36 0 π/36 π/18 Bending angle θ (rad) Cables of the proximal segment Cables of the distal segment (b) Figure 15: Cable tension during the bending motion of the proximal segment. (a) Bending angle φ (rad). (b) Bending angle θ (rad). –π/18 –π/36 0 π/36 π/18 Bending angle β (rad) Cables of the proximal segment Cables of the distal segment (a) Figure 16: Continued. e average cable tension (N) e average cable tension (N) e average cable tension (N) 14 Journal of Robotics –π/18 –π/36 0 π/36 π/18 Bending angle α (rad) Cables of the proximal segment Cables of the distal segment (b) Figure 16: Cable tension during the bending motion of the distal segment. (a) Bending angle β (rad). (b) Bending angle α (rad). Figure 17: .e 1 N payload experiments in the different positions of the robot. mechanism design of the proximal and distal segments is disturbances. Since there are eight cables in the proximal reasonable. segment and four cables in the distal segment, the load capacity of the proximal segment is stronger than the load capacity of the distal segment. Besides, when the load po- 5.3. Payload Experiments. When most of the prior cable- sition keeps moving away from the driving mechanism, the driven hyper-redundant robots are subjected to small ex- deformation of the terminal position of the robot becomes ternal disturbance, the robots easily appear in the S con- larger. During the payload experiments, it can be easily figuration and even other uneven configurations [45]. In this known that the inevitable clearance between the cables and section, a 1 N weight is loaded at different positions of the cable holes, and the assembly errors of the initial con- different bending configurations in the proposed robot, as figuration of each joint will aggravate the deformation of the shown in Figure 17. According to the observation, the robot under the external disturbance. In addition, the robot proposed robot does not appear in the S configuration, load capacity is also determined by the robot prototype which means the proposed robot can resist small external structure parameters. e average cable tension (N) Journal of Robotics 15 manipulation arms for laparoendoscopic single-site surgery 6. Conclusions and Future Works (LESS),” International Journal of Medical Robotics and In this paper, a novel 2-DOF decoupled joint is first pro- Computer Assisted Surgery, vol. 14, no. 1, Article ID e1838, posed. A 4-DOF cable-driven hyper-redundant robot pro- 2018. [5] K. Xu, J. Zhao, and M. Fu, “Development of the SJTU totype is developed based on the 2-DOF decoupled joint. .e unfoldable robotic system (SURS) for single port laparos- kinematics model is established and the workspace is ana- copy,” IEEE, vol. 20, no. 5, pp. 2133–2145, 2014. lyzed. .e kinematic fitting approach is presented for both [6] L. Huang, L. Yin, B. Liu, and Y. 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Design and Validation of a Novel Cable-Driven Hyper-Redundant Robot Based on Decoupled Joints

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Copyright © 2021 Long Huang 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.
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Hindawi Journal of Robotics Volume 2021, Article ID 5124816, 16 pages https://doi.org/10.1155/2021/5124816 Research Article Design and Validation of a Novel Cable-Driven Hyper-Redundant Robot Based on Decoupled Joints 1,2 1 1 1 1 Long Huang, Bei Liu , Lairong Yin , Peng Zeng, and Yuanhan Yang School of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China Hunan Provincial Key Laboratory of Intelligent Manufacturing Technology for High-performance Mechanical Equipment, Changsha University of Science and Technology, Changsha 410114, China Correspondence should be addressed to Bei Liu; 17871947856@163.com and Lairong Yin; yinlairong@csust.edu.cn Received 15 July 2021; Revised 26 August 2021; Accepted 9 September 2021; Published 2 November 2021 Academic Editor: Yaoyao Wang Copyright © 2021 Long Huang et al. .is 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. In most of the prior designs of conventional cable-driven hyper-redundant robots, the multiple degree-of-freedom (DOF) bending motion usually has bending coupling effects. It means that the rotation output of each DOF is controlled by multiple pairs of cable inputs. .e bending coupling effect will increase the complexity of the driving mechanism and the risk of slack in the driving cables. To address these problems, a novel 2-DOF decoupled joint is proposed by adjusting the axes distribution of the universal joints. Based on the decoupled joint, a 4-DOF hyper-redundant robot with two segments is developed. .e kinematic model of the robot is established, and the workspace is analyzed. To simplify the driving mechanism, a kinematic fitting approach is presented for both proximal and distal segments and the mapping between the actuator space and the joint space is significantly simplified. It also leads to the simplification of the driving mechanism and the control system. Furthermore, the cable-driven hyper-redundant robot prototype with multiple decoupled joints is established. .e experiments on the robot prototype verify the advantages of the design. endoscope driven by multiple cables, which is more dex- 1. Introduction terous than rigid endoscopes [23]. Dong et al. proposed a In recent years, due to the advantages of compact structure continuum robot with a low ratio between diameter and and flexible bending motion in the confined environment, length based on compliant joints, which can apply to the hyper-redundant robots have received high attention in the inspection and maintenance of aero engines [24]. Based on field of minimally invasive surgery, maintenance, and testing universal joints, Jin et al. designed cable-driven snake-like 4- [1–3]. Various kinds of cable-driven hyper-redundant ro- DOF surgical forceps [25]. Kim et al. designed a novel rolling bots have been reported by researchers [4–11]. Generally, the joint with a block mechanism to develop a snake-like robot cable-driven hyper-redundant robots are usually composed for minimally invasive surgery [26]. of several segments driven by external actuators and mul- For most of the prior cable-driven hyper-redundant robots, the bending motion in all directions is usually tiple cables. Every segment consists of several identical joints in serial. Joint types for the cable-driven hyper-redundant coupled. It means the rotation output of each DOF is robot mainly can be classified into the 1-DOF joint and 2- controlled by multiple pairs of cable inputs [27]. Besides, the DOF joint [12]. .e 1-DOF joints mainly include the rev- mapping between the actuator space and joint space of the olute joint [13, 14], the flexible beam [15, 16], and the cy- robot is complex, which causes the driving mechanism and lindrical rolling joint [17, 18], while the multi-DOF joints the control system difficult [28]. Some researchers have tried mainly include the universal joint [19, 20], the flexible to address the drawbacks of the coupling bending motion backbone [21, 22], and the spherical rolling joint. Based on through special joint structure design and further simplify the flexible backbone, Li et al. developed a 2-DOF flexible the mapping between actuator space and joint space [29, 30]. 2 Journal of Robotics Based on multiple cylindrical rolling joints, Kim et al. in the joint. For the universal joint with two nonintersecting designed a cable-driven hyper-redundant robot, which di- rotation axes located between the two disks [32], it can be minishes the bending coupling effect by enlarging the space also proved that the coupling effect exists in the joint when at for the passage of the center cable [17]. least one rotation axis is located at the middle place of two In this paper, a novel 2-DOF decoupled joint is proposed disks, which is similar to the universal joint, as shown in by adjusting the distribution of two rotation axes. For the Figure 1. decoupled joint, a pair of antagonistic cable inputs only Based on the above analysis, we tried to change the controls a 1-DOF rotational output, and the rotational distribution of the joint rotation axes to avoid the coupling output of each DOF is only determined by a single pair of effect. .erefore, this paper proposes a novel joint without antagonistic cable inputs. By connecting two 2-DOF seg- coupling effects, as shown in Figure 2. .e two axes in the ments in serial, a cable-driven hyper-redundant robot is joint are, respectively, coincident with the upper surface of presented. Each segment consists of multiple identical 2- the lower disk and the lower surface of the upper disk. Since DOF decoupled joints in serial. A kinematics linear fitting points A and B are located at the axis w , points A and B 2 2 1 2 2 approach is presented to simplify the mapping between achieve circular motion around points A and B when the 1 1 actuator space and joint space. Based on the linear fitting upper disk rotates around the axis w . .erefore, the length error analysis, each pair of antagonistic cables is driven by a of cable A and cable B will not change, while the length of motor through a circular pulley, which can simplify the cable C and cable D will also change. Since the points C and design of the driving mechanism. .e proposed robot is D are located at the axis w , the points C and D can be 2 2 2 2 verified by the bending motion experiments, the cable considered as the fixed points when the upper disk rotates tension test, and the load experiments. around the axis w . In consequence, the length of cableC and .e rest of this paper is organized as follows: Section 2 cable D will not change, while the length of cable A and cable introduces the challenges of the existing robots. Moreover, a B will change. .is indicates that the coupling effect does not novel cable-driven hyper-redundant robot is also introduced exist in the proposed joint with the special distribution of in Section 2. .e kinematics and the robot workspace are two axes positions. analyzed in Section 3. .e presented kinematics linear fitting and error analysis of the robot are discussed in Section 4. .e 2.1.2. Challenge of the Driving Mechanism Design. .e re- robot prototype through some experiments is verified in lationship between cable length and bending angles should Section 5. Section 6 presents the conclusion. be considered to design the driving mechanism of the cable- driven hyper-redundant robot [33, 34]. For most cable- 2. Robot Design driven hyper-redundant robots, the relationship between the length inputs of each cable and bending angles is a nonlinear 2.1. Challenges of the Existing Robots function. It means the tightened amount on one cable is not equal to the released amount on the antagonistic cable when 2.1.1. Coupling Effect. For most of the prior designs of cable- the robot bends to an arbitrary configuration. It is difficult driven hyper-redundant robots, the 2-DOF bending motion for cable-driven hyper-redundant robots to design a com- in each segment of the robot is coupled. .e rotational pact and simple driving mechanism. output of each DOF of the joint is controlled by multiple .e following driving mechanism design approaches are pairs of antagonistic cable inputs. Consequently, the cou- adopted. .e first approach is that each cable is driven by a pling effect will increase the complexity of the robot’s driving separate motor and a circular cable pulley. It is convenient to mechanism and the risk of slack in the driving cables. design the driving mechanism. However, this approach .e relationship between the cable length and the ro- increases the complexity of the control system, as shown in tation angles is determined by the joint types and their Figure 3(a). .e second approach is that noncircular cable structural parameters. For instance, the coupling effect of the pulleys are designed to realize a motor driving a pair of revolute joints is determined by the cable distribution circle cables based on the nonlinear function, as shown in radius, the number of cables, and the distribution of the Figure 3(b). .is method significantly reduces the number of rotation axes. Since the cable distribution circle radius and the motor, but the fabrication and assembly of noncircular the number of cables are confined by the practical factors, cable pulleys require high accuracy. .e third approach is the distribution of joint rotation axes is a significant factor to that a pair of cables is driven by a separate motor and a avoid the coupling effect. Figure 1 shows a conventional circular cable pulley, as shown in Figure 3(c). .is method universal joint with two intersecting rotation axes located at requires the releasing amount of one cable is equal to the the middle of two disks [31]. When the upper disk rotates tightening amount of the antagonistic cable, which can around rotation axes w through the releasing of cable A and simplify the driving mechanism. tightening of cable C, cable B and cable D must be tightened simultaneously to avoid slack, as shown in Figure 1(b). Similarly, the rotation around axis w requires the control of 2.2. A Novel Robot Design with the Decoupled Joints. .is all four cables. Otherwise, cable A and cable B will become section proposes a cable-driven hyper-redundant robot slack. It can be proved that the rotational output of each based on the multiple decoupled joints, as shown in Figure 4. DOF requires the control of multiple pairs of antagonistic .e robot is composed of a proximal segment, a distal cables regardless of the location of the two intersecting axes segment, and a driving mechanism. Each segment consists of Journal of Robotics 3 A C 2 2 B 2 Disk B Joint D C Disk A A 1 1 1 C B 1 B 1 1 Release Tighten Tighten Driving mechanism Driving mechanism (a) (b) Figure 1: .e traditional universal joint with two intersecting rotation axes. B 2 Upper disk B 2 Joint A B C Lower disk C A C 1 1 1 B w w 1 1 1 Release Release Tighten Tighten Driving mechanism Driving mechanism (a) (b) Figure 2: A novel joint without coupling effects. Tighten Release Tighten Release Tighten Release (a) (b) (c) Figure 3: Classic driving system types. h Proximal segment Distal segment 4 Journal of Robotics Driving mechanism Joint structure parameters linkage disk Bending joint Proximal segment Distal segment linkage disk linkage disk Figure 4: Cable-driven hyper-redundant robot prototype. Table 1: Parameters of the joint structure. six identical 2-DOF joints. Each joint contains two disks and one spatial linkage. .e cylindrical bulge surface on the disk Symbol Description Value and the cylindrical concave surface on the spatial linkage H .e distance between two axes 8 mm cooperate to form two rotating pairs. .e axis w and axis w 1 2 t Disk thickness 2.5 mm coincide, respectively, with the upper surface of the lower r Cable distribution circle radius 4.25 mm disk and the lower surface of the upper disk. Joint structure d Robot diameter 10 mm parameters are defined, as shown in Table 1. (θ, φ) Joint variables in the proximal segment (−π/18, π/18) .e 2-DOF bending motion of each segment is achieved (α, β) Joint variables in the distal segment (−π/18, π/18) by two motors controlling a pair of antagonistic cables through the circular pulleys. Cable A, cable B, cable C, and joint space, and task space [35]. .e following assumptions cable D control the 2-DOF bending motion of the proximal are made in this study. In this proposed robot, there is no gap segment, as shown in Figure 5(a), while cable E, cable F, between the cables and the cable holes. .e cables’ shear cable G, and cable H control the 2-DOF bending motion of strains and elongation are negligible. .e cable tension the distal segment, as shown in Figure 5(b). Since the two exerting on each joint is the same. rotation axes of each joint are coincident with the end Based on these assumptions [36], the joint kinematics is surfaces of corresponding disks, the distance of the two first established to analyze the decoupled effect in the rotation axes is always equal to h regardless of the robot proposed 2-DOF joint. .e relationship between the sum of configurations. In addition, a pair of antagonistic cable the cable length change and bending angles theoretically inputs only control the 1-DOF rotational output, and the validates that the cables in the robot will not become slack. rotational output of each DOF of the joint is only deter- Besides, the robot kinematics is established and the robot mined by a single pair of antagonistic cable inputs. With this workspace is analyzed. design, the mapping between actuator space and joint space can be eventually simplified. Besides, the driving mechanism design of the robot is illustrated in Section 4. 3.1.JointKinematics. Since the proximal segment and distal segment have the same bending motion, we consider a single joint in the proximal segment as an example to establish the 3. Kinematics joint kinematics, as shown in Figure 6. .e kinematics of the cable-driven hyper-redundant robot .e coordinate systems {O }, {O }, and {O } are i 1i i+1 requires establishing the mapping between actuator space, established, respectively, on the center of the upper surface Journal of Robotics 5 2 H E O G O w 4 4 F 2 2 E 2 E BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB 2 H 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 C F B 1 1 F 1 H E H 1 4 F E 3 3 (a) (b) Figure 5: .e 2-DOF joint of the proximal and distal segment. i+1 i+1 1i i+1 C y 2 1i 1i 0i C y 1 0i 1 x (x ) i 0i Figure 6: Kinematics coordinate system of the single joint. of the lower disk, the lower surface of the upper disk, and the (1) T � rot x ,θ􏼁 trans z , h􏼁 rot y ,φ 􏼁 trans z , t 􏼁 . i+1 i 0i 0i 1i upper surface of the upper disk. Axis x coincides with the axis w , axis y coincides with the axis w , and axis y is To establish the relationship between the cable length 1 1i 2 i+1 parallel to the axis w . .e transformation from the coor- 2 and bending angles, it is assumed that the position vector of dinate system {O } to {O } is as follows. First, the coor- any point p in {O } on the upper surface of the lower disk is i i+1 1 i dinate system {O } rotates angleθ around x axis to obtain the represented by p , while the position vector of any point p i i p1 coordinate system {O }. Second, the coordinate system {O } in {O } on the lower surface of the upper disk is represented 0i 0i 1i 1i moves h along the z axis and then rotates angle φ around by p . According to the coordinate transformation, the 0i p2 the y axis to obtain the coordinate system {O }. .ird, the position vector of any point p in {O } on the lower surface of 0i 1i 2 i coordinate system {O } moves t along the z axis to obtain the upper disk is represented by p . .e transformation can 1i 1i p2 the coordinate system {O }. Hence, the homogeneous be described as i+1 transformation matrix from the coordinate system {O } to i i 1i i p � R p + p , (2) p2 1i p2 1i {O } can be obtained as i+1 6 Journal of Robotics i i where R is the rotation matrix from {O } to {O } and p is 3.2. Robot Kinematics. Based on the joint kinematics, the i 1i 1i 1i the position vector of {O } relative to {O }. .erefore, the kinematics of the cable-driven hyper-redundant robot is 1i i relationship between cable length and angles θ and φ can be established. .e mapping between the actuator space, joint expressed as space, and task space is also obtained [39]. .e base coor- 􏼌 􏼌 dinate system {O } is established at the center of the lower 􏼌 􏼌 i i 􏼌 􏼌 􏼌 􏼌 l � p − p , (3) 􏼌 p2 p1􏼌 surface of the base disk, as shown in Figure 8. Axis x is parallel to the axis w , and axis z is perpendicular to the 1 0 where p and p , respectively, represent the intersection p1 p2 lower surface of the base disk. According to the joint ki- points between the cables, the upper surface of the lower nematics, the establishment principle of the coordinate disk, and the lower surface of the upper disk. systems {O } − {O } in the proximal segment is the same as 1 n Taking the cable length in a single joint of the proximal the coordinate system {O } − {O } in the distal segment. m1 mn segment as an example, the coordinates of the points A , B , 1 1 .erefore, the mapping between actuator space and joint C , and D in {O } are represented by p � (r, 0, 0), 1 1 i A1 space is first established. We assume that each segment of the i i i p � (0, − r, 0), p � (−r, 0, 0), and p � (0, r, 0), B1 C1 D1 robot contains n identical joints. In the straight configu- while the coordinates of the points A , B , C , and D in {O } 2 2 2 2 1i ration, each cable length in the proximal and distal segment 1i 1i are represented by p � (r, 0, 0), p � (0, − r, 0), A2 B2 can be obtained as 1i 1i p � (−r, 0, 0), and p � (0, r, 0). .erefore, the rela- C2 D2 tionship between the cable length and angles θ and φ can be L � nh + (n + 1)t, expressed as 􏼨 (5) L � 2nh + 2(n + 1)t, 􏽱��������������������� � 􏼌 􏼌 􏼌 􏼌 􏼌i i 2 2 2 ⎧ ⎪ 􏼌 ⎪ 􏼌 􏼌 l � p − p � 2r − 2r cφ + h − 2rh · sφ ⎪ A 􏼌 􏼌 A2 A1 where L represents the initial length of each cable driving ⎪ 􏽱������������������������������� ⎪ 􏼌 􏼌 the proximal segment and L represents the initial length of ⎪ 􏼌 􏼌 i i 2 2 ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 l � p − p � (−r · cθ − h · sθ + r) + (r · sθ + h · cθ) ⎨ 􏼌 􏼌 B B2 B1 each cable driving the distal segment. 􏽱��������������������� � , ⎪ 􏼌 􏼌 􏼌 􏼌 ⎪ According to equation (2) and equation (3), the rela- i i 2 2 2 ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 l � p − p � 2r − 2r cφ + h + 2rh · sφ 􏼌 􏼌 ⎪ C C2 C1 tionship between each cable length and bending angles in the 􏽱������������������������������� ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 i i 2 2 arbitrary configurations can be derived as ⎩ 􏼌 􏼌 􏼌 􏼌 l � p − p � (r · cθ − h · sθ − r) + (−r · sθ + h · cθ) D 􏼌 􏼌 D2 D1 􏼌 􏼌 􏼌 􏼌 i i 􏼌 􏼌 ⎧ ⎪ ′ 􏼌 􏼌 L � (n + 1)t + n p − p , (4) ⎨ P 􏼌 􏼌 p2 p1 􏼌 􏼌 (6) ⎪ 􏼌 􏼌 j j ⎩ 􏼌 􏼌 ′ ′ 􏼌 􏼌 L � L + (n + 1)t + n p − p , where l , l , l , and l represent the cable length in the joint d P 􏼌 􏼌 A B C, D p4 p3 of the proximal segment, cθ � cosθ, sθ � sinθ, cφ � cosφ, and where L represents the cables’ length in the proximal sφ � sinφ. p segment and L represents the cables’ length in the distal When angles θ and φ are equal to zero, each cable length d i i segment. In the proximal segment, p and p are the in the joint is equal to h, and the proximal segment keeps a p1 p2 intersection description between the cable on any side and straight configuration. According to equation (4), the length the upper surface of the lower disk and the lower surface of of cable A and cable C only depends on the angle φ, while the the upper disk in the coordinate system {O }, respectively. length of cable B and cable D only depends on the angle θ. i j j In the distal segment, and p and p are the intersection Consequently, during the 2-DOF bending motion of the p3 p4 description between the cable on any side and the upper proximal segment, a pair of antagonistic cable inputs only surface of the lower disk and the lower surface of the upper controls the 1-DOF rotation output, and the rotation output disk in the coordinate system {O }, respectively. .erefore, of each DOF of the joint is only determined by a single pair the mapping between the actuator space and joint space can of antagonistic cable inputs. .e 2-DOF bending motion in be established by equation (6). According to the trans- the proximal segment is completely decoupled. Similarly, the formation shown in Figure 8, the mapping between joint 2-DOF decoupled effect of the distal segment is the same as space and task space can be established. Hence, the ho- the decoupled effect of the proximal segment. mogeneous transformation matrix from the coordinate According to the literature [37, 38], if the sum of the system {O } to the coordinate system {O } can be 0 mn cables length changes in each one pair of antagonistic cables written as is positive, the cables will not become slack. When the proposed joint bends from a straight configuration to an 0 0 i n n T � T × ( T) × T × ( T) . (7) m1 j+1 mn 1 i+1 arbitrary bending configuration around axes w and w , the 1 2 relationship between the sum of the cable length changes in In the proposed cable-driven hyper-redundant robot, each one pair of antagonistic cables and bending angles is the adjacent two axes in each joint are perpendicular to shown in Figure 7. .e sum of length changes in the an- different bending planes, which causes that the inverse ki- tagonistic cables is positive regardless of the bending angles. nematics is difficult to solve through the analytical method .is indicates that the cables will not become slack when the [40–42]. .e Newton–Raphson iterative method can be used proximal segment and the distal segment achieve, respec- to solve the inverse kinematics, but it is not the research tively, 2-DOF bending motion. focus in this paper. Journal of Robotics 7 -4 -3 ×10 ×10 6 5 ∆ ∆ 0 0 -π/18 -π/36 0 π/36 π/18 -π/18 -π/36 0 π/36 π/18 Bending angle φ (rad) Bending angle θ (rad) (a) (b) Figure 7: .e sum of the antagonistic cables’ length changes in any pair of cables. (a) Bending angle φ (rad). (b) Bending angle θ (rad). nonlinear function. However, the following kinematics x linear fitting and error analysis will show that the rela- 1 x tionship can be well fitted to a linear function in a certain 1 y range of joint variables, and the tightened amount of the n-1 cable on one side is almost equal to the released amount of n-1 m1 the antagonistic cable when the robot configuration changes. n x m2 Hence, any pair of antagonistic cables in the robot can be y x m1 j x mn driven by a motor and a circle cable pulley, as shown in m2 Figure 3(c), which not only simplifies the driving mecha- mn nism but also reduces the control complexity. .e following mn contents are the kinematic linear fitting and error analysis in Figure 8: Coordinate system of the cable-driven hyper-redundant two segments. Based on the results, the driving mechanisms robot. of the two segments are designed. 3.3. Workspace Analysis. .e workspace of the cable-driven 4.1. Kinematics Linear Fitting in the Proximal Segment. hyper-redundant robot is determined by the joint geometry, Since the 2-DOF bending motion in the proximal segment is bending angles, and the joint number [43]. Based on the decoupled, cable A and cable C are considered as an example robot kinematics, the robot workspace is obtained. to perform the kinematic linear fitting using the polynomial Figure 9(a) shows the workspace of the proximal segment, fitting method. .e error values between the original and the while Figure 9(b) shows nine bending configurations of the fitting function are analyzed by the percentage error model. proximal segment when the joint angles (θ, φ) are, re- .e percentage error e % (φ) is defined as spectively, (0, 0), (0, π/36), (0, −π/36), (π/36, 0), (−π/36, 0), l (π/36, −π/36), (−π/36, π/36), (−π/36, −π/36), and (π/36, 􏼌 􏼌 􏼌 􏼌 l(φ) − l (φ) 􏼌 􏼌 􏼌 􏼌 (8) π/36). Based on this, Figure 9(c) shows the workspace of the e %(φ) � 100 · 􏼌 􏼌, 􏼌 􏼌 􏼌 l(φ) 􏼌 robot, while Figure 9(d) shows multiple bending configu- rations of the robot when joint angles (θ, φ, α, β) are, re- where l (φ) represents the original function and l’ (φ) spectively, (0, 0, 0, 0), (0, 0, 0, π/36), (0, 0, 0, −π/36), (0, 0, represents the fitting function. .e fitting curve and error π/36, 0), (0, 0, −π/36, 0), (π/36, 0, 0, 0), (−π/36, 0, 0, 0), (0 values between the original function and the fitting function −π/36, 0, 0), and (0, π/36, 0, 0). According to the above of cable A and cable C are solved by the MATLAB curve analysis, the more the segment number is, the larger the fitting tool, as shown in Figure 10. .e fitting functions of workspace of the robot becomes. cable A and cable C are represented by l (φ) � −4.237φ + 8 and l (φ) � 4.237φ + 8, respectively, as shown in Figure 10(a). 4. Kinematics Linear Fitting Within the range of bending angles shown in Table 1, the Based on equation (4) and equation (6), the relationship maximum fitting error between the original function and between each cable length and bending angles is the fitting function is 0.025%, and the maximum angle error of l +∆l (mm) A C l +∆l (mm) B D -40 -20 X (mm) -50 -100 X (mm) 25 50 -50 -25 X (mm) -20 -10 0 10 X (mm) 8 Journal of Robotics (a) (b) (c) (d) Figure 9: .e workspace and bending configurations of the robot. (a) .e workspace of the proximal section. (b) Nine bending con- figurations of the proximal section. (c) .e robotic workspace. (d) Multiple bending configurations of the robot. 8.8 0.025 8.6 Cable A Cable C 0.02 8.4 Cable A Cable C 8.2 0.015 0.01 7.8 7.6 0.005 7.4 l ’(φ)=4.237φ+8 l ’(φ)=-4.237φ+8 c A 7.2 0 -π/18 -π/36 0 π/36 π/18 -π/18 -π/36 0 π/36 π/18 Bending angle φ (rad) Bending angle φ (rad) Original function Fitting function (a) (b) Figure 10: Cable A and cable C error between the original and fitted function. (a) Bending angle φ (rad). (b) Bending angle θ (rad). ′ ′ the end disk in the proximal segment is less than 0.15%, as represented by l (θ) � −4.237θ + 8 and l (θ) � 4.237θ + 8, B D shown in Figure 10(b). Based on the same kinematics fitting respectively. .erefore, the relationship between each cable method, the fitting functions of cable B and cable D are length and bending angles is linear through the special 10 20 -20 -10 -30 Y (mm) -50 -100 Y (mm) -60 -30 30 60 Y (mm) -20 -10 Y (mm) e length of the cable A and cable C (mm) Z (mm) Z (mm) Errors e % Z (mm) Z (mm) Journal of Robotics 9 distribution of two rotation axes. It means the mapping fitting method. Within the range of bending angles shown in between actuator space and joint space is simplified. Table 1, the kinematics linear fitting results of cable E and Moreover, the 2-DOF bending motion of the proximal cable G in each joint of the proximal segment are shown in segment can be driven by two motors and two circular Figure 11. pulleys. Based on the literature [29], the robot motion ac- Similarly, the MATLAB curve fitting tool is also used to curacy is satisfied. solve the fitting function. .e fitting functions of cable E and cable G are represented by l (θ,φ) � −2.996θ − 2.996φ + 8, l (θ,φ) � 2.996θ + 2.996φ + 8. .e percentage error e % is G l 4.2. Kinematics Linear Fitting in the Distal Segment. redefined as According to equation (2) and equation (3), when the 􏼌 􏼌 􏼌 􏼌 􏼌 ′ 􏼌 l(θ,φ) − l (θ,φ) proximal segment undergoes 2-DOF bending motion, the 􏼌 􏼌 􏼌 􏼌 e %(θ,φ) � 100 · , (11) 􏼌 􏼌 􏼌 􏼌 length of all eight cables will change. .is means that the 􏼌 􏼌 l(θ,φ) bending motion between the proximal segment and distal where l (θ, φ) represents the original function and l’(θ, φ) segment is coupled. .erefore, the kinematics linear fitting represents the fitting function. of cable E, cable G, cable F, and cable H in proximal and According to equation (11), the maximum fitting errors distal segments should be considered to design the driving between the original function and fitting function of cable E mechanism of the distal segment. and cable G are less than 0.04%, while the maximum fitting When the proximal segment keeps the straight config- errors of the cable F and cable H are less than 0.25%, as uration, and the distal segment keeps an arbitrary bending shown in Figure 12. .erefore, the relationship between the configuration, the relationship between the length of cable E, length of cable E, cable G, cable F, and cable H, and bending cable G, cable F, and cable H and bending angles α and β can angles can be approximately linear. .is indicates that the be expressed as 􏽱��������������������� mapping between actuator space and joint space in the distal 2 2 2 ⎪ ⎧ segment is also simplified. .erefore, the 2-DOF bending ⎪ l � 2r − 2r cα + h − 2rh · sα , ⎪ E1 􏽱������������������������������� motion of the distal segment can be achieved by two motors ⎪ 2 2 and two circle pulleys. Based on the literature [29], the robot l � (−r · cβ − h · sβ + r) + (r · sβ + h · cβ) , F1 (9) 􏽱��������������������� motion accuracy is satisfied. ⎪ 2 2 2 l � 2r − 2r cα + h + 2rh · sα , G1 􏽱������������������������������� 2 2 5. Experiment Validation l � (r · cβ − h · sβ − r) + (−r · sβ + h · cβ) , H1 In this section, a 4-DOF cable-driven hyper-redundant robot where l , l , l , and l represent the length of a single E1 F1 G1 H1 prototype is established to validate the robot design. .e joint in the distal segment, cα � cosα, sα � sinα, cβ � cosβ, proposed robot includes the proximal segment, the distal and sβ � sinβ. According to equation (9), the 2-DOF bending segment, and the driving mechanism, as shown in motions of the distal segment are decoupled when the Figure 13(a). .e total length of the proximal segment and the proximal segment does not achieve the 2-DOF bending distal segment is 131 mm. .e cables’ diameter is 0.4 mm. .e motion. In addition, the relationship between the cable driving mechanism of the robot prototype includes a guiding length of the distal segment and the bending angles α andβ is device, a motor driving device, and a cable tension adjusting also approximately linear. device, as shown in Figure 13(b). .e rated speed of the motor If the proximal segment achieves 2-DOF bending mo- is 10 r/min, and the rated torque is 70kgcm. .e range of the tion, the lengths of cable E, cable G, cable F, and cable H will force sensor is 0–10 kg with an accuracy of 0.03%. also change. Hence, the cable length change relationship in According to the kinematic linear fitting relationship in the proximal segment should be considered to achieve the Section 4, the driving mechanisms of the proximal and distal kinematics linear fitting of the distal segment. According to segments are the same. For the proximal segment, cable A equation (2) and equation (3), the relationship between the and cable C are the two ends of one cable that is driven by length of cable E, cable G, cable F, and cable H and bending motor 1 to control the proximal segment bending in the x z angles θ and φ can be calculated as 0 0 􏼌 􏼌 plane. Cable B and cable D are also the two ends of one cable 􏼌 􏼌 􏼌i i 􏼌 ⎧ ⎪ 􏼌 􏼌 l � p − p , ⎪ 􏼌 􏼌 that is driven by motor 2 to control the proximal segment E2 E2 E1 ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 bending in the y z plane. 0 0 ⎪ 􏼌i i 􏼌 ⎪ 􏼌 􏼌 l � p − p , ⎨ 􏼌 􏼌 F2 F2 F1 For the distal segment, cable E and cable G are the two (10) 􏼌 􏼌 ⎪ 􏼌 􏼌 􏼌i i 􏼌 ⎪ ends of one cable that is driven by motor 3 to control the distal 􏼌 􏼌 ⎪ l � p − p , 􏼌 􏼌 ⎪ G2 G2 G1 ⎪ segment bending in the x z plane. Cable F and cable H are m1 m1 ⎪ 􏼌 􏼌 ⎪ 􏼌 􏼌 i i ⎩ 􏼌 􏼌 􏼌 the two ends of one cable that is driven by motor 4 to control l � p − p , 􏼌 􏼌 H2 H2 H1 the distal segment bending in the y z plane. .e two ends m1 m1 where l , l , l , and l represent the cable length of the of each cable are fixedly connected to the end disks of the E2 F2 G2 H2 single joint in the proximal segment. Since the 2-DOF proximal segment and distal segment through knotting. .e bending motion in the proximal segment has similar ki- middle of each cable passes through each joint disk and winds nematics, cable E and cable G are considered as an example around the guide device, driving device, and tension adjusting to perform the kinematic linear fitting using the polynomial device, as shown in Figure 13(b). Each cable tension is Bending angle φ (rad) -π/18 -π/36 π/36 π/18 -π/18 -π/36 π/36 π/18 π/18 π/36 -π/36 -π/18 -π/18 -π/36 π/36 π/18 Bending angle φ (rad) Bending angle φ (rad) Bending angle φ (rad) π/18 0 π/36 -π/18 -π/36 Bending angle φ (rad) 10 Journal of Robotics 9.5 8.5 Cable E Cable G 7.5 Cable G Cable E 6.5 Original function Fitting function Figure 11: Kinematics linear fitting analysis of cable E and cable G. 0.02 0.25 0.015 0.20 0.01 0.15 0.1 0.005 0.05 (a) (b) 0.035 0.030 0.030 0.025 0.025 0.02 0.02 0.015 0.015 0.01 0.01 0.005 0.005 (c) (d) Figure 12: Kinematics linear fitting error analysis of cables E, G, F, and H. adjusted by changing the position of the sliding block. .e 5.1. Free Bending Motion. In this section, the multi-DOF cable tension values are tested by the tension sensors. .e bending motions of the proximal segment following experiments include the free bending motion test, and distal segment have experimented, as shown in the cable tension test, and payload experiments. Figure 14. Bending angle θ (rad) Bending angle θ (rad) Bending angle θ (rad) -π/18 -π/36 0 π/36 π/18 π/18 π/36 -π/36 -π/18 -π/18 -π/36 π/36 π/18 -π/18 0 -π/36 π/36 π/18 Bending angle θ (rad) Bending angle θ (rad) π/18 π/36 0 -π/36 -π/18 e error the cable E (e %) e error the cable G (e %) e length of the cable E and cable G (mm) e error the cable G (e %) e error the cable F (e %) l Journal of Robotics 11 Motor 3 Motor 4 e distal segment e proximal segment Tension adjusting Driving device Tension pulley device Guiding device e driving mechanism of the proximal segment Tension adjusting Guiding device device Tension pulley Driving device e driving mechanism of the distal segment Figure 13: Cable-driven hyper-redundant robot prototype. When the distal segment keeps a straight configuration, When the proximal segment keeps a straight config- the bending motion of the proximal segment in the x z uration, the bending motion of the distal segment in the 0 0 plane requires the coordinated work of motor 1, motor 3, x z plane only requires motor 3 working to change the m1 m1 and motor 4. Motor 2 does not work to ensure that the cable length of cable E and cable G. .e other motors do not lengths of cable B and cable D are unchanged. .e bending work to ensure that the cable lengths of cable A, cable C, configuration outputs of the proximal segment in the x z cable E, cable G, cable F, and cable H are unchanged. .e 0 0 plane are only determined by the inputs of motor 1. Motor 3 bending configuration outputs of the proximal segment in and motor 4 are driven to keep the straight configuration of the x z plane are only determined by the inputs of m1 m1 the distal segment. .e bending configurations are shown in motor 3. Motor 1, motor 2, and motor 4 are not driven to Figures 14(a)–14(c). keep the straight configuration of the proximal segment. When the distal segment keeps a straight configuration, .e bending configurations are shown in Figures 14(h)– the bending motion of the proximal segment in the y z 14(j). 0 0 plane requires the coordinated work of motor 2, motor 3, Similarly, the bending motion of the distal segment in and motor 4. Motor 1 does not work to ensure that the the y z plane only requires motor 4 working to change m1 m1 lengths of cable A and cable B are unchanged. .e bending the length of cable F and cable H. Besides, to verify the configuration outputs of the proximal segment in the x z multi-DOF bending motion of the robot, we consider the 0 0 plane are only determined by the inputs of motor 2. Motor 3 bending configuration of the proximal segment in the x z 0 0 and motor 4 are driven to keep the straight configuration of plane and the bending configuration of the distal segment the distal segment. .e bending configurations are shown in in the x z plane as an example, as shown in m1 m1 Figures 14(d)–14(g). Figures 14(k)–14(n). 12 Journal of Robotics 0° +30° +60° (a) (b) (c) (d) (e) (f) (g) 0° -60° -30° (h) (i) (j) (k) (l) (m) (n) Figure 14: Robot bending motion experiments. 5.2. Cable Tension Test. During the multiple bending mo- the length of each cable driving the distal segment, the tions, the cable average tension curves are used to illustrate average tension of each cable driving the distal segment will the design rationalization of the driving mechanism for the increase. proximal segment and distal segment [44]. For the proximal When the proximal segment keeps a straight configu- segment, the bending configuration in the x z plane is ration and the distal segment keeps a bending configuration 0 0 determined by the angle θ, while the bending configuration in the x z plane and y z plane, the average cable m1 m1 m1 m1 in the y z plane is determined by the angle φ. When the tension of each cable varies with the joint angles α and β, as 0 0 distal segment keeps a straight configuration and the shown in Figure 16. For the proximal segment, the bending proximal segment keeps a bending configuration in the x z configuration in the x z plane is determined by the angle 0 0 m1 m1 plane and y z plane, the average cable tension of each cable β, while the bending configuration in the y z plane is 0 0 m1 m1 varies with the bending angles θ and φ, as shown in determined by the angle α. Figure 15. Within the range of the bending angles of [−π/18, π/18], Within the joint angle ranges of [−π/18, π/18], when when only the distal segment bends in the x z plane, the m1 m1 only the proximal segment bends in the x z plane, the average tension of the cable F and cable H in the distal 0 0 average tension of cable A and cable C in the proximal segment varies in the range of 10 N–12 N, as shown in segment varies in the range of 13 N–15 N, as shown in Figure 16(a). When only the distal segment bends in the Figure 15(a). When only the proximal segment bends in the y z plane, the average tension of cable E and cable G in m1 m1 y z plane, the average tension of cable B and cable D in the the distal segment varies in the range of 11 N–13 N, as shown 0 0 proximal segment changes within the range of 10 N–13 N, as in Figure 16(b). For the cables of the proximal segment, the shown in Figure 15(b). For the cables of the distal segment, average tension of cable A, cable B, cable C, and cable D the average tension of the cable E, cable G, cable F, and cable varies in the range of 12 N–15 N. .erefore, the phenom- H varies in the range of 8 N–10 N and 6 N–10 N. Since the 2- enon of the cables slack does not appear during the multi- DOF bending motion of the proximal segment will change DOF bending motion. .e results indicate that the driving Journal of Robotics 13 –π/18 –π/36 0 π/36 π/18 Bending angle φ (rad) Cables of the proximal segment Cables of the distal segment (a) –π/18 –π/36 0 π/36 π/18 Bending angle θ (rad) Cables of the proximal segment Cables of the distal segment (b) Figure 15: Cable tension during the bending motion of the proximal segment. (a) Bending angle φ (rad). (b) Bending angle θ (rad). –π/18 –π/36 0 π/36 π/18 Bending angle β (rad) Cables of the proximal segment Cables of the distal segment (a) Figure 16: Continued. e average cable tension (N) e average cable tension (N) e average cable tension (N) 14 Journal of Robotics –π/18 –π/36 0 π/36 π/18 Bending angle α (rad) Cables of the proximal segment Cables of the distal segment (b) Figure 16: Cable tension during the bending motion of the distal segment. (a) Bending angle β (rad). (b) Bending angle α (rad). Figure 17: .e 1 N payload experiments in the different positions of the robot. mechanism design of the proximal and distal segments is disturbances. Since there are eight cables in the proximal reasonable. segment and four cables in the distal segment, the load capacity of the proximal segment is stronger than the load capacity of the distal segment. Besides, when the load po- 5.3. Payload Experiments. When most of the prior cable- sition keeps moving away from the driving mechanism, the driven hyper-redundant robots are subjected to small ex- deformation of the terminal position of the robot becomes ternal disturbance, the robots easily appear in the S con- larger. During the payload experiments, it can be easily figuration and even other uneven configurations [45]. In this known that the inevitable clearance between the cables and section, a 1 N weight is loaded at different positions of the cable holes, and the assembly errors of the initial con- different bending configurations in the proposed robot, as figuration of each joint will aggravate the deformation of the shown in Figure 17. According to the observation, the robot under the external disturbance. 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Published: Nov 2, 2021

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