A Versatile Continuum Gripping Robot with a Concealable Gripper

Continuum robots with their inherent compliance provide the potential for crossing narrow unstructured workspace and safely grasping various objects. However, the display gripper increases the size of the robots, and therefore, it tends to get stuck in constrained environments. This paper proposes a versatile continuum grasping robot (CGR) with a concealable gripper. The CGR can capture large objects with respect to the robot’s scale using the continuum manipulator and can grasp various objects using the end concealable gripper especially in narrow and unstructured workspaces. To perform the cooperative operation of the concealable gripper and the continuum manipulator, a global kinematic model based on screw theory and a motion planning approach referred to as “multi-node synergy method” for the CGR are presented. The simulation and experimental results show that objects of different shapes and sizes can be captured by the same CGR even in complex and narrow environments. Finally, in the future, the CGR is expected to serve for satellite capture in harsh space environments such as high vacuum, strong radiation, and extreme temperatures.


Introduction
Working in a complex and tight space is challenging for both workers and rigid robots, especially gripping operations such as industrial pipe dredging, sampling for cave exploration, and satellite capture. The continuum robot is a kind of bionic ro bots whose continuum backbone can be continuously de formed [1][2][3]. The continuum robot does not contain rigid links and recognizable rotating joints, which are inspired by the soft bodies of creatures, such as elephant trunks, octopus arms, and snakes [4][5][6][7][8][9][10]. Compared with the discretejointed traditional rigid robots, the continuum robot has an increased selfadaptability and safe interaction capacity due to its inher ent compliance [11][12][13][14][15]. Compared with the soft robots made of shape memory polymers [16] and conductive polymers [9], continuum robots have obvious advantages in robust grasps and efficient controllability. Continuum robots are ideal for these grasping tasks where the target workspaces are located far away from the robot's access point and lot of obstacles exists between them [17][18][19][20].
How to improve the adaptability of the grasping robot to the shape and size of the target object and the adaptability to the complex environment has received the attention of many researchers. The relative published studies can be divided into 3 types. (a) Winding. The continuum grasping robots (CGRs) in this category include the Festo Bionic Motion Robot [21,22], inspired by an elephant trunk and octopus, as well as cellular pneumatic mesh manipulators [23]. They have good safe han dling and adaptability, allowing easy grasping and movement of fragile or irregularly shaped objects, which is not the case for conventional rigid robots. However, these CGRs have dif ficulty in grasping small objects with respect to the robot's scale because of the bending curvature limitations. (b) In com bination with end effectors. Similar to traditional robots, the continuum robot can also grasp objects using end effectors. Yang et al. [24] installed a linear motordriven gripper to a tendondriven CGR for capturing smallsized objects. How ever, the drive of the gripper containing the rigid element is also located at the end of the CGR, which not only increases the load on the CGR but also limits its flexibility in narrow spaces. Alfalahi et al. [25] designed a concentric tube continuum robot with a flexible gripper at the end for intranasal skull base surgery. The robot gripper drive line is inserted through the innermost tube of the concentric tube manipulator, and the actuator is placed at the rear. This provides an idea for a light weight design. Moreover, it is difficult for this concentric tube robot to capture objects of large size. The exposed gripers men tioned above are prone to getting stuck in a narrow position, and the forward direction could not be correctly selected by the passive compliance of the gripper during the forward process (Fig. 1A). On the other hand, the exposed grip in creases the risk of entanglement in the environment, which can stop it from moving forward. (c) Multifinger collabo ration. It simulates the human handgrasping objects through multifinger coordination. The hydraulic CGR developed by Galloway et al. [26] allows flexible capture and sampling in the deep sea by controlling its internal pressure. CGRs of the same principle also include the humanoid soft hand [27] and the 6finger gripper [28]. The presence of multiple fingers has good adaptability to objects of different sizes. However, a large size limits its use in a narrow space, and the load capacity of this kind of robot is limited. Therefore, it is necessary to develop a lightweight CGR that can capture objects of different sizes in narrow and complex spaces (pipe cleaning, space station garbage capture, and cave sampling). Moreover, a CGR with a concealable gripper provides a possibility for this (Fig. 1B).
In this paper, a CGR with a concealable gripper is proposed for grasping operations in complex and narrow spaces. Objects with various shapes and sizes can be captured by the cooperative operation of the concealable gripper or the continuum manip ulator. The concealable gripper can be completely retracted into the backbone, which reduces the CGR volume when moving in tight spaces and avoids interference when the continuum ma nipulator performs coiling operations. When using only the roddriven, the CGR has the potential to achieve a light weight and compact design. Kinematics modeling and a motion planning approach, which is referred to as a "multinode col laboration method, " are then introduced for the proposed robot. The novel motion planning method has advantageous charac teristics. More precisely, it is a programming method in a global workspace, which does not require any convex assumption and does not depend on a convex index function of the searching space. In addition, allowing direct reference to the posture fea tures of CGR makes the method userfriendly.

Design and analysis of CGR
The proposed CGR contains 4 main parts ( Fig. 2A). The end concealable gripper, the continuum manipulator, and the con cealable gripper are connected as 1 unit, while the end conceal able gripper and the continuum manipulator are connected to the drive station.
The continuum manipulator (Fig. 2B) is composed of a central backbone, 6 constrained disks, and 3 drive rods. The backbone, which cannot be stretched or compressed in the axial direction, allows bending and deflection movements. The 3 NiTi alloy rods for driving are rigidly connected at their tips to the end disc, and the part below the base disc is covered with a polyethylene decoupling sleeve.
The decoupling principle is similar to a bicycle brake sys tem, where the deformation of the sleeve area does not change the length of the rod in the nonsleeve area. The flexible arm expands the CGR's working range. The flexible arm allows the end effector to work in areas far from the drive station.
The 4 fingers of the concealable gripper are composed of 2 elastic sheets. The 2 elastic sheets are orthogonally arranged on the elastic palm by a gap fit (Fig. 2C). The open state of the concealable gripper can be elastically deformed into a closed state, under the constraint of the spiral support and the pull of the driving rod (Fig. 2D). The decoupling sleeve of the concealable gripper is fixed to the spiral support. In addition, the end of the spiral support is designed as a funnel, so that the elastic return force of elastic sheets helps the concealable gripper to unfold.
Three identical linear modules (Fig. 2E) are distributed on the rear drive station. The drive rod is always in tension under the antagonism of the pulling force of the driver and the restor ing force of the elastic sheets. Displacement and force sensors are installed on the linear module to obtain the displacement and drive force of the drive rod.
The driver module of concealable gripper is shown in Fig.  2F. A rigid sleeve is installed outside the connection point of the rigid rod and the flexible rod. The rigid sleeve is rigidly connected to the upper support plate and the connection point does not move beyond the steel tube so that the thrust of the linear motor can be transmitted to the concealable grip per. The rigid rod is designed as a stepped shaft to restrain the upper limit of the actuator movement. The diameter of the elastic palm is larger than the inner diameter of the decou pling sleeve, which can restrain the lower limit of the linear movement.
In addition, the following features should be mentioned. The concealable gripper is underactuated, and the opening and closing of its 4 fingers are achieved only by the axial movement of 1 driving rod. In addition, the concealable gripper can be retracted into the backbone, which reduces the size of the CGR and protects the flexible clamps. Smallsized objects can be grasped by the con cealable gripper, while large objects with respect to the CGR's scale can be captured by the continuum manipulator, which enriches the robot's application scenarios. (c) The compliant flexible arm allows the CGR to work into narrow and unstruc tured spaces far from the drive station. (d) Compared with the drive mode of fluid and rotating joints, the roddriven mode provides the potential for miniaturization and lightweight design of CGRs. Illustration of a CGR with a concealable gripper that operates in a narrow tunnel environment, which performs better than the one with an exposed gripper. (A) A CGR with a rigid gripper tries to pass the tunnel and reach the target (purple) region but gets stuck at the corner of the tunnel. (B) CGR with a concealable gripper passes the tunnel and reaches the target region by hiding its last section.

Kinematics modeling of the continuum manipulator
The configuration space and drive space of CGR are inconsistent because of the roddriven method. Therefore, the kinematic pa rameters of the continuum manipulator arm are divided into 3 spaces (Fig. 4A). The driving space D = [S 1 S 2 S 3 ] T characterizes the length of the driving rods. The configuration space Θ = [θ φ] T describes the shape of the manipulator, and the workspace W = [x y z] T reflects the end position of the manipulator in Cartesian space. The complete forward kinematics includes ④ and ①, and the complete inverse kinematics includes ② and ③. Numbers ① to ④ express the order of introduction of the mapping.
The kinematic modeling of the continuum manipulator is performed based on the assumption of a constant curvature. In other words, the bending shape of the backbone is approx imately an arc whose curvature is the same at all the positions during the movement [29,30]. A simplified motion model of the continuum manipulator is shown in Fig. 4B). ∑ 0 (o 0 − x 0 y 0 z 0 ) is the base frame, where O 0 is the center of the base disk. In addition, ∑ 1 (o 1x 1 y 1 z 1 ) is the end frame, which is fixed to the center O 1 of the base disk. Axes z 0 and z 1 are tangential to the backbone arc. Axes x 0 and x 1 are points to the first hole of the base disk and the end disk, respectively. The y axis can be determined by the righthand rule. The dihedral angle θ∈[0,π] is defined as the angle between the base disk and end disk. The deflection angle ϕ∈[0,2π] is defined as the angle between the plane where the continuum manipulator is bent and the positive direction of the x 0 axis. Assuming that the length of the back bone's arc can neither be compressed nor stretched, it is set to a fixed value s.

1) Mapping from configuration space to workspace
When the continuum manipulator is working (θ ≠ 0), the position and posture of the end frame ∑ 1 can be obtained by 4 basic transformations from ∑ 0 . The base frame ∑ 0 is first translated from O 0 to O 1 to obtain frame ∑ 0,1 . The latter is then rotated around its z axis by an angle ϕ to obtain frame ∑ 0,2 . Afterward, frame ∑ 0,2 is rotated around its y axis by an angle θ to obtain frame ∑ 0,3 . Finally, frame ∑ 0,3 is rotated around its z axis by an angle −ϕ to obtain frame ∑ 1 . The twist coordinates of the 4 decomposed motions can be expressed as: (1) The twists of the twist coordinates are given by: where the symbol ^ is the wedge operator [31], which maps a vector from ℜ 6 to the Lie algebra se(3).
By applying the product of the exponentials formula [32], the transformation from the base frame ∑ 0 to the end frame ∑ 1 can be expressed as: where L 1 = S √ 2(1 − cos ) ∕ , R 0 1 , and P 0 1 can be com puted as: 2) Mapping from joint space to drive space This section details the mapping from Θ = [θ φ] T to D = [S 1 S 2 S 3 ] T of the continuum manipulator. The ge om etric relationship between Θ and D is shown in Fig. 4C). Figure 4D presents the crosssectional view of the x 0 o 0 y 0 of Fig. 4C. It can be deduced from Fig. 4D that: where α i (i = 1,2,3) is the angle between vector �⃗ r i (vector from the center O 0 to the center of the ith hole) and the plane where the backbone is bent. The radius of curvature R i corresponding to the drive rod can be expressed as: The length of the drive rods is then given by: Therefore, the mapping from joint space to drive space for the continuum manipulator can be expressed as: 3) Mapping from drive space to joint space It can be deduced from Eq. 7 that: Solving the above equations can lead to the mapping from the driving space to the joint space:

4) Mapping from workspace to joint space
Considering the range of ϕ ∈ [0,2π], ϕ can be expressed by Eq. 11 according to the geometric relationship presented in Fig. 4C: The bending angle can be easily obtained from Fig. 4C: Therefore, the mapping from the workspace to the joint space can be expressed by Eqs. 10 and 11.

Kinematics modeling of concealable gripper
The underactuated concealable gripper consists of 4 collet fin gers. It is assumed that the movement of each collet finger is cos 2 cos + sin 2 cos sin (cos − 1) cos sin cos sin (cos − 1) sin 2 cos + cos 2 sin sin − cos sin − sin sin cos

1) Mapping between drive space and joint space
The length h, which is the driving space parameter, is de fined as the distance between the end disc center O 1 and virtual joint center C. When the concealable gripper is just completely closed, h is written as h 0 . The boundary angle γ b is defined as the rotation angle from the boundary CA i of the ith collet finger to the reference axis CO 1 . γ o is the opening angle between CO 1 and CO i,2 . γ u is the underlying angle between CO i,2 and CA i . The mapping of the drive space to the joint space is given by: where length c is the diameter of the inner wall of the spiral support.
Furthermore, the mapping from joint space to drive space can be obtained: where the change value Δh represents the pulling length of the drive rod, and the initial setting h 0 = √ b 2 − c 2 is the length of the drive rod when the concealable gripper is just completely closed.
The opening angle of the concealable gripper is expressed as:

2) Mapping between joint space and workspace
The position and posture of the end frame ∑ i,2 of the ith collet finger can be obtained by 5 basic transformations from frame ∑ 1 . (a) Frame ∑ 1 is first translated from O 1 to C to obtain frame ∑ 1,1 . (b) In order to obtain the point A i on different fingers, frame ∑ 1,1 is rotated around its x axis by finger angle η(η = i*π/2) to obtain frame ∑ i,1,2 . (c) Afterward, frame ∑ i,1,2 is translated from C to A i to obtain frame ∑ i,1,3 . (d) Frame ∑ i,1,3 is then rotated around its x axis by angle β′(β′ = πβ−γ b ) to obtain frame ∑ i,1,4 . (e) Finally, frame∑ i,1,4 is translated from A i to O i,2 to obtain frame ∑ i,2 . The corresponding twist coordinates of the 5 decomposed motions can be expressed as: The transformation from the base frame ∑ 1 to the end frame ∑ i,2 is given by: is the distance from O 1 to C. For the ith collet finger, the forward kinematics can be written as: The pose of the center of the 4 fingers in frame ∑ 1 can be expressed as: where I 3×3 ∈ ℜ 3 is the identity matrix.
T is similar to the third one of P 1 i,2 . The pose of the center of the 4 fingers is given by:

Motion planning of the CGR
The CGR allows the gripping and handling of objects, which usually requires the coordinated operation of the continuum manipulator and the concealable gripper. To this end, a novel motion planning approach, referred to as the "multinode synergy method" for the CGR is proposed (Fig. 5). In this method, the key path nodes for the concealable gripper are first planned in Cartesian space, according to the task requirements. The parameters of the key path nodes contain the positions X e (t 0 ),X e (t 1 ),⋯,X e (t n ) of the grip points (i.e., centers of the 4 finger tips of the concealable gripper) and the boundary angles Θ g (t 0 ),Θ g (t 1 ),⋯,Θ g (t n ). The joint space node Θ c (t 0 ),Θ c (t 1 ),⋯,Θ c (t n ) for the continuum manipulator can then be obtained by inverse kinematics. Afterward, the complete joint space node Θ(t 0 ),Θ(t 1 ),⋯,Θ(t n ) for the CGR, combined with the joint space nodes for the continuum manipulator and the boundary angles, can be expressed as Θ = Θ c Θ g . The joint angle Θ(t), angular velocity Θ (t), and angular acceleration Θ (t) are then obtained according to the interpolation function. Finally, the drive space trajectory planning D(t) for CGR is performed by the inverse kinemat ics from the joint space to the drive space.

Numerical simulation
The CGR is numerically simulated and analyzed based on the designed prototype and the proposed kinematics algo rithm. The parameters used in the simulations are presented in Table 1.

Workspace analysis
The reachable workspace is an important basis for judging the performance of GCR. The path of the first collect finger is shown in Fig. 6A according to the positive kinematic Eq. 16 and the parameter variation range in Table 1. The extension length of the concealable gripper along the z axis is the largest (22.79 mm) when the boundary angle is γ b = 0.77. The maxi mum open radius (33.98 mm) of the concealable gripper is taken when the boundary angle is γ b = 1.30.
The coworking space of the continuum manipulator and concealable gripper is shown in Fig. 6B. Note that the coop erative workspace is an eggshellshaped envelope area. The cyan eggshell surface is the reachable space for the tip of the continuum manipulator, while the area between the colored eggshell surface and the cyan eggshell surface is the reachable space for the center of the end concealable gripper. The limit range of the concealable gripper is x max = y max = 162.98 mm, x min = y min = −162.98 mm, z max = 222.80 mm, and z min = −22.79 mm. The workspace limit range of the continuum ma nipulator is x max = y max = 144.90 mm, x min = y min = −144.90 mm, z max = 200 mm, and z min = 0 mm.

Simulation of cooperative motion
Moving the target object is one of the common operations of the grasping robot. To verify the efficiency of the kinematics model, a numerical simulation analysis is performed on the grasping and moving tasks. Assuming that there is a static spherical target object having a radius of 20 mm in the reach able workspace, it is required to move the center of the sphere from point A [98 98 127] T (mm) to point B [70 120 127] T (mm) by the CGR. The node parameters for this motion planning are presented in Table 2. The angular velocity and angular acceleration param eters for each node are set to 0.
The CGR achieves the above tasks through 5 basic actions ( Resetting. This is the inverse process of approaching the target. The angles, angular velocities, and angular accelerations of the continuum manipulator when performing transport task based on a 5polynomial interpolation are shown in Fig. 7F to H). Moreover, the drive space results corresponding to the transport task are shown in Fig. 7I. The length of each drive rod changes continuously and smoothly throughout the oper ation, which can efficiently avoid the drive damage caused by sudden speed changes.

Experiment of cooperative motion
The driving space parameters in Fig. 7D are used as the driving input of the prototype, and the bolt located on the table is suc cessfully moved away from the table (Fig. 8). The kinematic model and motion planning method proposed in this paper are reasonable, which is proved by the consistency between the above experimental results and simulation results.

Experiment of application
This section first illustrate the adaptability of CGR using the concealable gripper to pinch various styles of objects and grasp the same object in multiple ways. Then, the continuum manipulator is used to wrap objects with a larger size, which are not easy to be pinched by the concealable gripper, to demonstrate the complementarity of the manipulator arm and the hidden gripper in the grasping operation. Finally, the CGR performs grabbing tasks in a simulated constrained environ ment, demonstrating its potential for applications in cave sam pling, pipeline cleaning, and space station maintenance.
Objects of different shapes and sizes could be pinched by the concealable gripper. The block bolts, linear wrenches, flat sheet metal, and three dimensional (3D) shaped disc storage buckets could be stably pinched by the concealable gripper (Fig.  9A). It can be deduced that the volume is not the decisive factor affecting the grasping, and objects with larger volumes can also be grasped as long as there are suitable points for grasping. In addition, the same object could be pinched in different ways. The plastic handle can be stably held by grabbing (Fig. 9B) the (i) tail, (ii) waist, (iii) head, and (iv) lateral waist using the concealable gripper. It is crucial for the target object grasping in uncertain scenes.
The largesized objects with respect to the CGR's scale could also be wrapped with a continuum manipulator. The inherent flexibility of the backbone allowed the continuum manipulator to conform to balloons that could not be grasped by the con cealable gripper upon contact (Fig. 9C, i). This does not imply that the grasped target object necessarily passes through its fitting continuum manipulator. The edge of the target object in contact with the continuum manipulator may also be grasped as long as the force closure condition is satisfied (Fig. 9C, ii). The continuum manipulator and the concealable gripper have complementary advantages when grasping a variety of target objects.
Grasping objects in complex and confined environments is particularly attractive through the CGR. Bolts in complex obsta cles (Fig. 9D, i) and towels blocking pipes (Fig. 9D, ii) could be pinched by the concealable gripper of the CGR, assisted by the continuous deformation of the continuum manipulator. The concealable gripper can be completely retracted into the pro tective backbone, which can help the CGR flexibly move through the narrow and complicated space, and has a protective effect on the concealable gripper. In addition, knives dropped in nar row wall gaps, retrieved by the concealable gripper (Fig. 9D, iii), and hoops under the sofa could be captured by the continuum manipulator (Fig. 9D, iv). Finally, potential applications for CGR's operations include postdisaster rescue, pipe cleaning, and cave sampling.

Conclusion
In this study, a CGR with a concealable gripper is first proposed for grasping operations in complex and narrow spaces. A global kine matic model based on the screw theory and a motion planning ap proach referred to as "multinode synergy method" for the CGR are then presented to perform the cooperative operation of the concealable gripper and the continuum manipulator. In addition, using rods to transmit power from the rear, the CGR has the potential to achieve a lightweight and compact design, and the fragile control system can be safely protected at the rear.
In future work, we aim at combining the adaptability of changing scenarios, the safety of interaction, and a higher load capacity, which is challenging and crucial for for flexible grasping, especially for the capture of space junk, pipeline maintenance, and cave explo ration. In addition, accurately obtaining the grasping force exerted by the continuum robot on the target object and the contact force when the robot body interacts with the environment,especially in narrow and complex environments, is of our interest. performed the experiment and wrote the manuscript. S. Zou and N.M. contributed significantly to the editing. R.F., H.L., and S.Q. contributed to the analysis of the simulation. Com peting interests: The authors declare that they have no com peting interests.

Data Availability
The data used to support the findings of this study are included within the article.