Logic motion algorithm and design of flexible limb of gecko-imitating robot

The tower climbing robot is supposed to climb up and down smoothly along the power transmission tower skeleton and stay firmly at the desired position, and install the safety cable at the appropriate position of the tower end, providing the basic life safety guarantee for the operator’s in climbing and operating for the maintenance of the tower connection of the high-voltage transmission line. Because of the non-uniform arrangement of the handles along the angular steel of the skeleton, the most critical problem to be solved in the climbing process is to find and eliminate pseudo targets and reliably grasp the real targets and move forward reliably. In this paper, we proposed and designed a set of searching and pseudo target elimination algorithms for robot flexible limbs with the chassis and intelligent control. The grasping climbing state sequence sets of robot flexible limbs were defined and constructed in Euclidean n-dimensional space with a real number, and a complete cycle map of the flexible limb behaviors was drawn. By using the norm sequence set of the distances between the centers of two adjacent handles on the same side of the skeleton, the machine vision and the specific W-type sensor layout, the pseudo target and real target can be discriminated effectively, so that the robot’s grasping and climbing can be established on a reliable mathematical basis. Flexible limbs consist of a built-in motor, screw expansion mechanism, flexible palm-fingers, and W type of sensor layout region. In the process of crawling, the screw mechanism makes macroscopically the limb approach the handle, meanwhile the palm-fingers justify microscopically to the center of the handle by the pressure-uniform algorithm, ensuring that the robot climbs, discriminates the pseudo target, and stays at any position of the skeleton. Simulation study and case analysis show that the proposed flexible limb and climbing-removing pseudo-target algorithm are of correctness, reliability, and practice.


Introduction
Robot climbing and its application have always been the focus of experts, scholars, and industrial engineers for several decades. The theory and technology of vertical-climbing robots are intrinsic and put into wide use, which has become the focus of research and application of industrial robots.
The design and use of safety cables have gone through two development stages: trial facility and generalization and specialization. In the climbing or moving robot field, several achievements have been gained. Liu et al. [1] developed a track-type inverted climbing robot called SpinyCrawler, using a spiny track with an opposed gripping mechanism with the ability of generating considerable adhesion to achieve stable inverted climbing. Kapoutsis et al. [2] created the plug-and-play algorithm for multi-robot applications. Gao et al. [3] presented a new climbing-wall detection robot mechanism consisting of two climbing modules. The two climbing modules were connected by an anti-overturning mechanism to provide a capacity of anti-overturning during overcoming of an obstacle with a detection mechanism which is installed at the bottom of the robot. Virgili-Llop et al. [4] carried out a convex programming-based guidance algorithm to capture a tumbling object. Villagrossi et al. [5]  cast iron deburring cell. Li et al. [6] proposed a motion planning algorithm with probabilistic guarantees for limbed robots with stochastic gripping forces. And the planners based on deterministic models with a worst-case uncertainty could be conservative and inflexible to consider the stochastic behavior of the contact, especially when a gripper is installed. Shirai et al. [7] presented a motion planning algorithm with probabilistic guarantees for limbed robots with stochastic gripping forces, with planners based on deterministic models with a worst-case uncertainty being conservative and inflexible to consider the stochastic behavior of the contact, especially when a gripper was installed. Brown et al. [8] examined how cockroaches vary their behavior as the slope was changed from horizontal to vertical, with special care to examine individual leg forces when possible. Gallentine et al. [9] presented new plant-inspired robotic tendril-bearing and intertwining stem hardware and corresponding novel attachment strategies for thin continuum robots. Amirhosseini et al. [10] presented the development of a small and lightweight differential hexapedal microrobot, inspired by a cockroach locomotion mechanism with detailed design, prototyping, and performance evaluation. Chen et al. [11] proposed a novel pneumatic soft actuator, which can perform bending in different directions under positive or negative air pressure with the actuators composed of multiple airbags designed. Fukui et al. [12] presented a ceiling-mounted mobile robot, HanGrawler, with high-speed mobility and the ability to freely select and adjust its route under a ceiling plate. The HanGrawler hangs from the holes of a perforated metal ceiling plate using newly developed mechanically constrained hanging mechanisms mounted on crawler-type traveling equipment. Hirasawa [13] described about improvements in the mobility of a stair-climbable mobile robot on a step field standardized by the National Institute of Standards and Technology being a simulated artificial rough terrain used for testing rescue robots. Chen et al. [14] established a theoretical model of the rotation flow, containing two important parameters, and verified it experimentally, with application of the computational fluid dynamics (CFD) method to illustrate the secondary flow relative to the blades, revealing that it gave rise to a nonlinear velocity distribution. Serebrennyi et al. [15] discussed the dynamics of a newly developed mobile robot with a vertical movement mechanism on magnetic caterpillar movers and, based on analysis of the current solutions in the field of vertical movement robots, proposed an original kinematic scheme of the movement of the vertical movement mechanism that allows implementing a number of critical advantages compared to current schemes. Chen et al. [16] presented a novel adhesion method that leverages capillary and lubrication effects to achieve simultaneous adhesion and sliding with design of a 47-mg adhesion pad and installation of it on a 1.4-g insect-scale quadrupedal robot to demonstrate locomotion on inverted and inclined surfaces. Seo et al. [17] proposed a novel mechanical design of a cable-driven parallel robot (CDPR), called a dual ascender robot (DAR), which consists of two ascenders and two structures for rope-measuring sensors. Choi et al. [18] proposed the modeling methodology and its experimental verification that could maximize the lifting shear force of the electroadhesive device to reach well over the human-finger grip force, say, ca. 8.9 kPa, which has not been achieved yet in this device system. Fan et al. [19] developed a novel style of a permanent-magnetic adsorption mechanism using an electromagnetic method and internal force compensation principle. Wu et al. [20] introduced an aircraft skin inspection robot which sticks to the surface of an aircraft by suction adhesion with the purpose of operating adsorption force to ensure a smooth and fast movement of the robot. Liu et al. [21] aimed at task requirements on multiple walls, and based on the above three bionic structures, a wall-climbing robot with the composed mode of "grabbing + adhesion + adsorption" is presented via the law of mechanism configuration synthesis. Chavdarov et al. [22] investigated the possibilities of climbing higher obstacles while maintaining the overall dimensions of a walking robot through design improvements and experiments. Yoo et al. [23] proposed a novel multi-wound differential pulley winch (MWDPW) component for assisting the ascending/descending operations of a wall climbing robot. The robotic platform enabling rope access in dangerous environments (ROPE RIDE) climbs vertical walls using a rope and embedded winch. Masoud and Ito et al. [24,25] developed a soft climbing robot made of silicone. An octopus-like behavior is realized by a simple mechanism utilizing the dynamics of the soft body, and the robot can grasp various objects of unknown shape. Austin et al. [26] outlined some of the specific motion planning challenges faced when attempting to plan for legged systems with dynamic gaits, with specific instances of these demonstrated by the dynamic climbing platform TAILS. Wardana et al. [27] proposed the design of a single-wheeled robot capable of climbing stairs. The robot is equipped with the proposed climbing mechanism, which enables it to climb stairs. Sun et al. [28] proposed a control framework to tackle the hybrid locomotion problem of wheeled-legged robots. It comes as a hierarchical structure with three layers: hybrid foot placement planning, center of mass (CoM) trajectory optimization, and whole-body control. Zhou et al. [29] proposed a convolutional neural network-based positioning scheme composed of a global bounding box detector and a local wheel detector.
The current study puts forward and designs a set of robot and intelligent control with the hardware system of a homing-culling algorithm, and the pseudo target defined in Euclidean n-dimensional space with a real number. A climbing state sequence, a two adjacent center distance sequence 1 3 of the handles on the same side of the angular steel, is constructed, efficiently and reliably identifying false targets and obstacle-overcoming with realization of the biomimetic gecko movement, discrimination of the pseudo target, and successful obstacle-overcoming.
2 Analysis and expression of the climbing logic of a transmission tower

Function analysis
The functions of the robot limbs mainly include the following: (1) Climb up, down, and stay in the desired position along the skeleton handle of the tower; (2) The working handle in the direction of detection and searching, namely, the target handle; (3) Flexible limb length. By changing the length of forelimb or hind limb, the center of the target handle and the center of the palm and finger can be conformed; (4) Grasp handles; (5) Provide and transmit power and movement to make the vehicle run; (6) Support bearing device to realize carrying capacity.

Climbing physical model and kinematic model
Driven by a stepping motor, the front and rear limbs rotate counterclockwise ( 180 • − 2 ) from a relative horizontal angle to handle 1 and handle 2, respectively, to push the vehicle chassis to move L/2 forward, or forward: In the XOZ plane, the displacement and velocity of any point on the vehicle are Meanwhile, the acceleration is

Crawling route
Based on gecko crawling bionics, a "3-1 climbing mode" is formed-3 points of positioning support and 1 point of power transmission to realize forward movement. Accordingly, under the given working conditions in Fig. 1, the movement mode in Fig. 3 is adopted to realize gecko climbing. The specific model is as follows: Press gecko limbs -forelimb 2 arms, hind limb 2 legs and feet, respectively defined as arm 1, arm 2, legs, and foot 1 and foot 2. The handrails corresponding to moments are handle 2, handle 4, handle 1, and handle 3, which are called points 2, 4, 1, and 3, respectively. Its working state is points 1, 2, and 3 in the state of holding the handrail, and arm 2 in the state of stretching forward in space, forming a (1,2,3) − 4 pattern. In the next instantaneous state, points 2, 3, and 4 are in the state of holding the handrail, while arm 1 extends forward in space, forming a pattern of (2,3,4) − 1. Follow the loop to achieve a forward climb. This is known as the three-point grab-and-one-point forward stretch mode. The progressive pattern sequence can be expressed.

The flexible limb and its design
Its core composition is a screw nut mechanism. Driven by a stepper motor, the middle sleeve with an end key drives the nut to turn, and the nut's outer cylinder undertakes bearing and turns relative to the shell. The rotating nut pushes the lead screw in a straight line. When the motor rotation is changed, so is the lead screw.

W-type design of the discrete -continuous sensing region
Ensure where b is the minimum center distance of two adjacent irregularly distributed handles on the same side. Take When the small cell slices are divided on the length of W, if the sensing element is arranged with length l 1 and the center of the sensing element is designated as the extraction location, then the detection density is where t is the average value of the total length on one bend of the sensing region.
Since the length of each sensing element l 1 is greater than the length of the segmented thin section l o , the cylindrical surface of the target handle in front can be detected in the entire w × m region, and the geometric center of each sensitive element in contact with the cylindrical bar is regarded as the position of the contact point of the handle, as shown in Fig. 5.

Grasping palm design
When the hand grip is designed, the grip force should be uniform and have automatic compensation function, so as to eliminate the limited distribution and certain error of the sensing element in the process of detection and seeking.
(a) Composition and layout of palm-fingers It is composed of palm 72, finger 77, palm and finger sensitive elements, rotating shaft 74, stepper motor 73, etc., as shown in Fig. 7; (b) Palm design A part of the retractable forelimb -the working face is made into a part of the cylinder, and the diameter can be chosen as the diameter of the handle, but the tolerance is positive deviation To adapt to the handle cylinder working face. (c) Means design Finger shape as a pawl -its working face diameter slightly larger than the handle diameter Take = 1.5 − 2mm . One end of the finger is fixedly connected with the shaft sleeve, and the bearing is matched with the shaft to rotate in two directions under the driving of the stepping motor, and the angular stroke is rotated to ensure that the limbs work reliably without interference during searching and grasping. (d) Layout and mechanical design of sensing elements There are at least two sensors on the palm region, three sensors on the finger region.
One end of the finger is fixedly connected with the shaft sleeve, and the bearing is matched with the shaft to rotate in two directions under the driving of the stepping motor, and the angular stroke is rotated to ensure that the limbs work reliably without interference during searching and grasping. (e) Layout and mechanical design of sensing elements There are at least two sensors on the palm region, three sensors on the finger region. If the press on the palm and fingers is The real press on the sensor i is Then, the infinitesimal adjustment will be down to the direction of leaving the sensor, so as uniform the presses on the palm and fingers, as well as eliminating the error raised during the homing.
During the grasping of the hand onto the handle, the fingers and palm come into union to hold the handle firmly with pressures, Take average of the five values which is very near the optimal values as prescribed. Then, calculate d p If the pressure in discussion, p i , is considerably greater than the miniature motor, it will drive the corresponding finger or part of the palm forward to the handle. If p i is smaller than the miniature motor, it will retract the finger back. In this case, pressure equalization is realized.

Climbing gesture
In the Euclidean n-dimensional space with the real number domain, the spatial position of the handle on the tower skeleton, the robot's homing and decertifying targets, and the climbing attitude will be defined. Based on the bionics of the gecko, the four limbs are supposed as the anterior limb 1 (arm 1), anterior limb 2 (arm 2), hind limb 1 (leg 1), and hind limb 2 (leg 2); see Fig. 2. In a moment, the state of the robot in climbing will be where S i -the ith climbing sequency, which may even be depicted in the following: where l -number of climbing steps; it is thus defined as the climbing step with respect to the forward guide with one step. l may have the same value as with n; fix(n 1 , n 2 , n 3 ) -code of grasping limb, 3; drive limb j -forward guide code, 1.
The flexible limbs are thus defined respectively in terms of the above descriptions (Fig. 3).

Movement state sequential set and expression
When a limb, say arm1, becomes the drive that makes the carrier forward, then the expression drive (limb1) is used for clarification from other limbs. The four limbs in drive will be formulated in terms of

3
The handles corresponding to the moment are handle 2, handle 4, handle 1, and handle 3, which are called working state points 2, 4, 1, and 3, respectively, as shown in Fig. 4. Set in the original position, the climbing robot limbs are defined by Eq. (3). Its working state is that points 1, 2, 3, and 4 are all in the grip state and ready to crawl forward, namely, attitude A. The first step of crawling forward is corresponding to attitude B. The limbs are rotated forward with the center of the handle as the center of rotation respectively forward 180° − β(drive (limb3) = leg1), drive (limb4) = leg2). In the second step, arm 1 takes hinge 1 as the center of rotation, leans forward, and grabs the handle of target 1 (see "Pseudo target elemination algorithm" for process details). In this process, working points 3, 4, and 2 are still in the grip state, see state C, form (2, 3, 4-1); arm 2 extends forward in space centering on hinge 3 and grabs target 1 handle. During this process, work points 1, 2, and 4 are in a grip state (drive (limb j ) = arm2)), forming a (1,2,4) − 3 pattern, see attitude D. In the next instantaneous state, the working state is point (1,3,4), which is in the state of holding the handrail, while the leg-foot 1 extends forward in space and grabs the handle of the original arm 1, forming a pattern of (1,3,4) − 2. See state E. In the following action, the working state points (1,2,3) are in the state of holding the handrail, while the second leg extends forward in the space and grabs the handle of the original second arm, forming a pattern of (1,2,3,) − 4. See state F. At this point, you go in a loop with L/2. This is known as the three-point grab-and-one-point forward stretch mode. Then, crawl into the next loop. The progressive 3-1 pattern sequence can be expressed as gecko climbing mode: The backward climbing is equivalent to the forelimb modification, while the forelimb modification is similar to the forward circulation mode and has a symmetrical form with Formula (10), so it is no longer redundantly expressed.

Homing by the sensor technology
The algorithm is as follows: First, set the target handle within the sector swept by the flexible limb: In the area swept by the limb, the target handle must be included, that is, the length of the sensing area should cover the target handle, as shown in Fig. 5.
The sensing elements are distributed as follows. Since each sensor length, l 1 , is designed greater than the fine district piece length l o , in some part of the entire domain of the w × m area the generatrix of the handle in front can be touched and detected, and each contact with the cylinder is supposed to be the geometric center of the sensitive element, thus ensuring the capture of one target with the location of the handle, as shown in Fig. 6.
The ith sensor in the "W" distribution of the sensing area touches the cylinder side of the handle, generating the spatial coordinate position:  In the unified coordinate system, we have Then, the spatial two points may be transformed into planetary problem, with the distance of the two points The transformation between the touch generatrix of the handle and the center of the handle will be Equation (14) expresses the target, handle center, and coordinates from the touched point on the handle generatrix coordinates (Fig. 7). So, in this way, a series of the center coordinates of the handles may be obtained (Figs. 8 and 9).

Homing by the machine vision
As shown in Fig. 10, when using machine vision, after the robot climbs one step, the target handle enters the visual field. At the same time, the visual system prompts the target center coordinates to the control system, and compares  Calculation results match the test data.

Pseudo target elimination algorithm
Due to the structural form of the tower frame, the distribution of handles on the same side of the triangle steel is irregular, that is, the distance b between the two adjacent handles is far less than the spacing of regular arrangement, as shown in Figs. 3 and 5. In this case, in the "climb up" mode, two adjacent handles that are not arranged in a regular pattern, the nearer one cannot be used and becomes the "fake target" or "pseudo target," while the farther one is the handle to be used and becomes the "real target." Therefore, the false targets must be eliminated, and the corresponding algorithm is as follows: In the Euclidean n-dimensional space with the real number domain, the center positions of the same side handles are composed into a set {C i }, and then the number set is composed of the distance norm sequence between the rotation centers of any adjacent handles on the same side, and is put into the database. The distance norm of any two adjacent handles is: then becomes a set of distance norm sequence d i,i−1 = C i − C i−1 , that is, So, the constituent elements of the distance norm sequence set are Take the minimum value and second minimum value of the elements: Now take the position of the first handle, C 1 x 1 , y 1 , z 1 , and the second minimum value of the elements the distance norm sequence set, S b2 , as the basic parameters of the climbing robot, then the general spatial locus of the flexible arm may be obtained And the specific locus of the touch point of the sensor on the flexible arm corresponding onto the handle and the specific locus of the palm center, both swept in space, respectively, may be depicted in terms of where R 1 and R 2 -respectively the radii between the track of the contact point between the sensor and the handle and the track of the palm grasping center relative to the common instantaneous center C s , as shown in Fig. 7.
As shown in Figs. 3 and 6, the distribution of the same side handles is non-uniform. Under normal circumstances, the distance between any two adjacent handles is with a certain value. However, at the turning point of the tower frame, the distance between the two adjacent handles would be b = 200 mm, which is much smaller than the normal distance of P = (450-500) × 2 mm. At this time, the robot should grasp a farther handle. In this way, the nearer handle in the abovementioned becomes a false target and must be eliminated in the control process. The algorithm adopted is to use Eqs. (16), (17), and (18) to find the minimum element of the distance norm sequence set, S b1 , at the same time, judge where Δ -distance between the two irregular adjacent handles, which is much smaller than that of the two regular adjacent handles (900-1000 mm).
If Eq. (22) is true, then the coordinates of the two handles forming S b1 will be Therefore, it can be determined that the farther handle is the real target, and the nearer handle is the pseudo target.
If the next target is settled, then the movement of the palm center approaches linearly the center of the handle and to make the two overlap, according to Eq. (14).

Discussion
The design of the flexible limb and the pseudo target discrimination algorithm have been successfully developed.

Coincidence algorithm of the handle center and the palm center
Norm calculation is adopted with Eq. (16), and the two centers overlap. It is a fetal and the most important step in climbing. A case study has shown that with CCD, the robot judges the position of the target roughly, and then sensing region touches the generatrix of the handle and provides a relatively accurate location. Finally, the palm self-adjusts its center with respect to that of the handle by the pressure equalization algorithm.
(23) C S b1 1 = C , C S b1 2 = C −1 , = 1, 2......n 6.2 Firm grasping the handle by the pressure equalization algorithm Figures 11 and 12 show the complete procedure of the behavior for firmly grasping the handle. Suppose the palm with its fingers may make rotation within the angle of , by rotation, the fingers go over 1∕3 , 1∕3 , and again 1∕3 , in turn. When the surface of the finger touches the handle, the sensors on the fingers as well as on the palm are subjected to different pressures. According to Eqs. (5) and (6), the motor built-in in the limb drives the screw move forward or backward, meanwhile another motor make the fingers run approximately, realizing the pressure equalization algorithm.

Design and simulation of the flexible limb for climbing and obstacle-overcoming
The so-called flexible limb functions supporting the chassis, making the robot forward, and supplying the combination of the revolution round axis of hinge by motor 1 and linear movement of the front part of the limb by motor 2. In this  case, the limb system is of 2 DOF, and needs to control precisely. There are two ways to climb forward with the flexible limbs. One is the clockwise revolution of the limb, meanwhile the limb length has to retract somewhere so the end of the limb may avoid the obstacle where r 1 radius of the end circle or the length of the part from the center of palm; L s the lead of the screw in the limb; S the start number of the screw; v the average velocity of the chassis.
Another one is the counterclockwise revolution of the limb. But first, a relief of the palm-fingers is done and then a clockwise revolution of the limb goes over the handle a little more and turns counterclockwise for the next handle. In the latter case, the trajectory is rather long and the angle to rotate through is Simulations are carried out for the positions and gestures of the limb in the first case, see Fig. 13. Indeed, there are several feasible solutions to climb and overcome the obstacle; here, the situation of the extreme position and gesture is depicted. A very possible way to overcome the obstacle is that the robot runs with the shortest route or with the shortest time between the two adjacent handles at points A and A′; meanwhile, the limb stretches to the longest magnitude at β and contracts to the shortest stage at α.
Simulations are also carried out for the position of the limb for the second case, see Fig. 14.
A comparison between the two cases are made for the optimal solution. Because of a longer route to go or a longer time to spend, or a larger space for the trajectory in the second case, the first solution is taken as the final decision.

Conclusion
A set of climbing robot with intelligent control realizing homing and pseudo-target eliminating algorithm was put forward in Euclidean n-dimensional space with the real number, and a catch climbing state sequence set was proposed and applied on the same side two adjacent rotary centers of the handles of the tower skeleton apart from the norm of the sequence number set. Taking the gecko climbing as the bionic object, the series of grasping and climbing states of the climbing robot is defined and constructed in the Euclidean n-dimensional space with the real number domain, and the corresponding handle, limb state, grasping, and guiding model are defined with the so-called three-point grasping and one-point forward stretching driving model: The number theory and database technology have been employed to define, store, and access the norm of center position distance of any two adjacent handles on the same side, the norm sequence number set, the minimum norm, and the secondary minimum norm, so that the climbing and homing of the robot, the elimination of false targets, and the intelligent control are based on reliable mathematics.
Third, using the two technology methods of sensing and machine vision, through the machine vision guidance and flexible arm design, the limbs sweep the sector, the sensing elements arranged in the induction area "W" style, and the  discrete induction detection is continuous and seamless, to ensure the limb sensing area and the target cylinder surface effective contact, to fulfill the search. The precise location of the center position of the handle is obtained by the coordinate transformation algorithm on the cylindrical surface.
The norm sequence number set of the center distance of any two adjacent handles on the same side of the skeleton has been originated and used effectively in discrimination of false targets and real targets, raised by regular and irregular arrangements of handles on the same side.
A pressure equalization algorithm has been created and adopted in grasping handles, laying no less than 5 distributed sensing elements on the working face of the palm-finger, adjusting the pressure between palm-fingers adaptively according to the pressure deviation, ensuring that the center of the palm-finger is coexisting with the center of the palmfinger, and grasping the handle reliably.
In the Euclidean n-dimensional space with a real number, the tower climbing robot's catch hold climbing state sequence number set has been defined and the corresponding database constructed. Combined with the gecko help handle creepy robot limbs, symmetrical design in "climbing down" mode of algorithms, both homing or eliminating pseudo targets, only set the related sequence database elements in reverse, the fore limbs replace the hind legs, then a full set of algorithms and corresponding database in "climbing up" mode can fully be used, to simplify the program, the robot structure improving the reliability of the system. Simulation study and case analysis show that the proposed flexible limb and climbing-removing pseudo-target algorithms are of correctness, reliability, and practice.