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Abstract

Due to sliding force arising from the closed chain mechanism among the adhering points of a climbing caterpillar robot (CCR), a sliding phenomenon will happen at the adhering points, e.g., the vacuum pads or claws holding the surface. This sliding force makes the attachment of the climbing robot unsteady and reducesthe motion efficiency. According to the new bionic research on the soft-body structure of caterpillars, some flexible structures made of natural rubber bars are applied in CCRs correspondingly as an improvement to the old rigid mechanical design of the robotic structure. This paper firstly establishes the static model of the sliding forces, the distortion of flexible bars and the driving torques of joints. Then, a method to reduce the sliding force by exerting a compensating angle to an active joint of the CCR is presented. The analyses and experimental results indicate that the flexible structure and the compensating angle method can reduce the sliding forces remarkably.

Keywords

Flexible structure climbing robot natural rubber

1. Introduction

Because of their low-cost and small and dexterous features,small climbing robots have good prospects in many fields such as anti-terrorism, post-disasterrescue, engineering tests, etc. As one kind of small climbing robot, bionic climbing robotshave attracted widespread attention because of their good terrain adaptability. Based on the modular concept, these robots are built up with several identical units in chain configurations. In general, one typical unit is composed of a rotary joint, an adhering pad or claws and mechanical interfaces. Using different numbers of units, the researchers can conveniently build various configurations and mimic the gait of different worms, e.g., inchworms and caterpillars. Another advantage of the modular caterpillar robot concept is the intrinsic simplicity of testing new gaits and control algorithms, due to its simple control system in which only one joint will be controlled for angular positioning [14 –19].

In our previous research work, we found that an inchworm robot suffered from low safety when climbing on vertical surfaces, because of its open chain climbing kinematicsand limitedattaching points. Compared with the inchworm robot, the caterpillar climbing robot (CCR), which features at least seven units, closed chain climbing kinematicsand more adhering points, is more stable when climbing on complex surfaces[1][6][7]. One type of this robot is described in [1], as shown in Fig. 1. However, a sliding phenomenon, which was introduced by the redundant actuating, happened when the CCR was climbing in the closed chain state [9].

Figure 1.

Seven-joint bionic climbing robot^[1]

This sliding phenomenon leads to climbing instability and lowers the power efficiency. The redundant actuating phenomenon also exists in a real caterpillar, when the motion wave of its body is transferred from its tail to its head, but does not weaken the safety of its climbing. The anatomic structure of the caterpillar reveals that its muscles and body fluid pressure can be self-adjusting to adapt to the motion procedure[8].

Besides the modular concept, the flexible structure is becoming increasingly popular in the climbing robots. [2]describes a crawling robot which can steadily crawl on rough vertical surfaces. To attach on a surface reliably, the claws are designed with micro-spine tips and the flexible structures are employed to cope with the random micro features of the wall. [3]introduces an aircraft which can land on a rough vertical surface. It also uses micro-spines embedded in an adhering mechanism with flexible structures to ensure a reliable landing. The holding ability of a flexible structure is verified by[4]in which the proposed gripper with a flexible structure can hold a load of up to 1000N. The low rigidity of the flexible structure enables the climbing robot to adapt to the complex terrain passively. [5]depicts a crawling robot which can crawl along on the bark of a tree branch. The robot is built in a whole flexible structure to adapt to the rugged surface.

Current research reveals that the application of a flexible structure in climbing robots can not only endow the robots with high adaptability to rugged environments, it can also enhance the reliability of their climbing movement. The results both from the climbing robots and the bionic research domainsencourage us to embed some kinds of flexible structures into our modular CCR to cope with the sliding phenomenon.

This paper is organized as follows. We firstly analyse the sliding phenomenon of a basic four-bar linkage and point out the possibility of reducing the sliding forces by taking advantage of a flexible structure. Then, the paper establishes the static model of a flexible link and reveals the role of the proposed flexible links in reducing the sliding force. Then, a new joint control method is presented to further reduce the sliding force. After that, a series of experiments verify the success of the proposed methods. Finally, conclusions are drawn.

2. Sliding phenomenon and flexible structure

A. Sliding Phenomenon of a Caterpillar Robot witha Rigid Structure

With regard to the modular concept, the mechanism of a caterpillar robot can be simplified as a serially connected linkage structure[1][7][8], as shown in Fig. 2. The gaitof a caterpillar robot features a changing motion chain, which repeats between the open chain states and the closed chain state while the motion wave is transferred from the tail to the head. When the head and tail parts of the robot adhere to a surface, its middle parts, which contain four joints as shown in Fig. 3, run in the closed chain state. Obviously, this is a four-bar linkage with double rockers. When transferring the motion wave along the robot body, all of the joints in the closed chain should rotate following the kinematics of the four-bar linkage, as expressed by Equation (1).

{\begin{cases} \vec{Z_{1}} = r_{1} \\ \vec{Z_{2}} = r_{2} e^{j θ_{2}} \\ \vec{Z_{3}} = r_{3} e^{j θ_{3}} \\ \vec{Z_{4}} = r_{4} e^{j θ_{4}} \\ \vec{Z_{2}} + \vec{Z_{3}} = \vec{Z_{1}} + \vec{Z_{4}} \end{cases}

(1)

Figure 2.

Multi-bar linkage model of a caterpillar

Figure 3.

Four-bar linkage model of the middle part of a caterpillar robot in closed chain state

where, rj (j=1,2,3,4) is the length of a bar, i.e., one module. For a rigid bar, the length is a constant. θ_i (i=1,2,3,4) is the joint angle.Z_i(i=1,2,3,4) is a vector from one joint to another.

Basically, the adhering points of the robot provide the supporting forces to drive the robot forward. For a climbing robot, the reliable adhering points without sliding are particularly important to hold the robot on anysurface. In theory, the sliding forces of the adhering points in closed chain states are inevitable. Considering the simplest four-bar linkage model shown in Fig. 3 as an example, if only one joint is actively driven, the sliding force exerting on one adhering point can be estimated by ADAMS. The result in Fig. 4 reveals the relation between the sliding force ( F _X ) and the angle of Joint 1 (θ₁, see Fig. 3).

Though the sliding force shown in Fig. 4 is acceptable for a climbing robot, the real sliding situationdiffers greatly from the theoretical results. In our experiments, the sliding force of a caterpillar robot can be as large as 13N. The key issue behind this phenomenon is the over-actuated fact in the modular caterpillar robot. Because of the similarity of the modules, every joint in the four-bar linkage is active. Due to the inevitable control error of the joints and the large stiffness of the modules, the real sliding forces on the adhering pointsare unpredictable. Theforce is determined by the stall torques of the joint motors. Therefore, such anover-actuated closed chain with rigid structure suffers low walking efficiency andundesirableclimbing reliability.

Figure 4.

The sliding force on an adhering point of a full actuating four-bar linkage

According to our previous research, the real sliding force F _SS can be deduced by Equation (2), shown in Fig. 5 [1].

Figure 5.

Static model of the close chain^[1]

F_{S S} = f (\frac{M + F_{y}^{(43)} x_{B} - m_{3} l^{2} k_{3}^{2} α_{3} - m_{3} g k_{3} x_{B}}{y_{B}})

(2)

where, M is the bending torque caused by the inevitable control error of the joint and large stiffness of the module. F_X⁽⁴³⁾and Fy⁽⁴³⁾are separately the forces acting on joint B along the X and Y axisfrom bar 4 to bar 3 as shown in the far left of Fig. 6.F_X⁽²³⁾ and F_y⁽²³⁾are the forces acting on joint A along the X and Y axis from bar 2 to bar 3. l is the length of the bars. φ₃, φ₄, φ₅ are separately the rotation angles of bars AB, CB and CD.k₃ is the ratio of the distance from Joint C to the instant barycentre and the length of the bar3. m₃ is the mass of the bar 3. x_B and y_B is the coordinate of the joint B.α₃ is the angular acceleration of the bar 3. g is the gravity acceleration.

Figure 6.

One step motion of a caterpillar[8]

Though the exact value of M is neither calculable nor controllable, Equation (2) points out that the sliding force can be adjusted by changing the moment on the bar. However, the stiffness of a rigid bar is too large to make this method practicable, since any bias of the joint angle from its ideal value may result anunpredictable inner force in the closed chain.

B. Inspiration of Bionics Research

According to the results of bionics research[8], when the caterpillar is climbing, some of its body sections between the adhered gastropods will shrink actively (the shadow parts in Fig. 6), and other body sections will deform passively. Though the body sections in the closed chain also work in the over-actuated situation, the soft feature of the passive section reduces the rigidity remarkably. Thus, the motion wave is transferred along its body harmoniously. The role of the soft body of acaterpillar reminds us that reducing the rigidity of the module may be a simple way to make the active adjustment of the sliding force possible.

The basic idea of the following sections is to add a small incrementon the ideal angle value of the joint determined by Equation (1), to reduce the sliding force by taking advantage of the low rigidity of the module with a flexible structure, which will be made of the rubber material. The bias of the joint angle value is defined as the Joint Compensation Angle (JCA) in this paper.

3. The flexible structure and sliding force adjustment method

Taking the flexible structure into consideration, we rebuild the four-bar linkagemodel of the closed chain as shown in Fig. 7. Here, the four joints are all actively driven, thesame as the real modular caterpillar robot. The joint angle is indicated by θ_i. Link r2, link r3 and link r4 are all flexible bars. Ki (i=2,3,4) is the bending rigidity of link i. The two black triangles are the fixed constraints, e.g., the adhering points. The JCA of joint i is formulated as below, $Δ θ_{i} = θ_{R i} - θ_{T i}, i \in [1, 4]$ (3)

Figure 7.

A basic four-bar mechanism with JCAs

where, θ_Riis the actual angle position. θ_Tiis the theoretical angle position.

Assuming that all of the flexible bars are the same, the moments acting on the three moving bars will be similar. Therefore, bar r2 is randomly selected to reveal the relations among the sliding force, the rigidity of the bar and the JCA.

A. Moment Arising from the CJA

Fig. 8 shows the static model of bar r2. Assuming that the JCAs of the two ends of bar r2 are the same, it can be divided into two cantilevers, in which subscript i stands for the initial position[10].

Figure 8.

Simplified model of somite link

Because the deformation caused by JCA can be considered as the cantilever deformation, the moment M acting on a rubber flexible bar is formulated as follows[11]. $M = \frac{2 G b^{3} l}{3 h (1 - \frac{2}{κ h} \cdot \tanh \frac{κ h}{2})} \cdot Δ θ$ (4)

where, Gis the shear modulus of the rubber; b, h, l are the width, the height and the length of the flexible bar; κ is a coefficient factor determined by κ=240/b; Δθ is the difference betweenthe JCAs of two ends.

B. Tensile Force Caused by CJA

The tensile force caused by the JCA is formulated as below[12], $F = \frac{A}{l [\frac{3}{4 E} (1 - \frac{2}{β l} \tanh \frac{β l}{2}) + \frac{1}{K}]} Δ l$ (5)

whereA is the sectional area of the flexible bar;K is the bulk modulus of the rubber;E is the Young's modulus; β is a coefficient factor determined by β²=48/b²;Δl is the variation along the bar length caused by the JCA.

Assuming that the moments adding on the two ends of the link are the same, Δθ₁=Δθ₂, soΔl=lγsin²(Δθ₁/2), where γ is the characteristic radius factor and equals to 0.7346 for the rubber bar.

C. Relation between the Sliding Force and JCA

When the robot crawls onahorizontal surface, the static model of the adhered bar is depicted in Fig. 9.

Figure 9.

Static model of the bar adhering to a surface

In Fig. 9, F _X and F _Y are the forces onthe adhering points from the surface, and the sliding force F_SS=-F_X. F _A is the force along the flexible bar resulting from the bend deformation. ΣMis the sum of bending torque. P is the force caused by ΣM. F _R is the sliding force without the JCA. θ₁ is the joint rotation angle. F _SS can be formulated by Equation (6).

F_{S S} = - \frac{Σ M}{l} \sin θ_{1} + F_{A} \cdot \cos θ_{1} + F_{R}

(6)

If the JCA is onlyapplied to Joint 3,Δθ₃ does not equal zero, the ΣM is determined by Equation (7).

Σ M = K_{2} Δ θ_{3} + (\frac{K_{3} \cdot K_{4}}{K_{3} + K_{4}}) Δ θ_{3}

(7)

According to Equation (6) and (7), F _SS can be reduced by adjusting the Δθ₃.

4. Experiments testing the sliding forces

According to equations (4), (5) and (7), there are two factors influencing the F _SS . The one is the Young's modulusE of the material and the otheris the JCA. For a rubber material, the hardness H_r is positively correlated to E. Therefore, H_r and the JCA are taken into considerationin the following experiments. The experiment platform shown in Fig. 10 is designed to detect the sliding force. A basic closed chain unit with four modules is installed on the platform. The left module is fixed and the right module is connected to a force sensor to measure its F _SS . The bend rigidityof each module can be changed by choosingrigid barsor rubber bars with different hardnessfor the links. The motion wave is transferred from the left (as shown in Fig. 10) to the right. During this process, the value of F _SS will be detected. The joint furthest to the left, e.g.,Joint 1,rotates in a constant angular velocity.

Figure 10.

Experiment platform for sliding force testing

A. Influence of the Flexible Bar on the Sliding Force

In the experiments, the rubber bars with three different hardness (20HA, 30HA, 40HA), are tested. For contrast, the rigid bars made ofaluminium are also tested. The experiment resultsare shown in Fig. 11, in which the black curve represents the sliding force when only the rigid bars are used.

Figure 11.

The influence of the body hardness Hr on the sliding force F_SS without JCA

In Fig. 11, we can see that when the caterpillar is composed of only rigid modules,the F _SS is very large – up to 12.34 N. This is because the inevitable biases of the joint angles from ideal values block the rotation of some joints. When flexible bars are chosen, the maximum value of F _SS is reduced to near 5N for the rubber bars with 40HA hardness, 3N for 30HA and 2.7N for 20HA, respectively. Obviously, the F _SS becomes smaller if the hardness of the rubber bar is smaller. However, the reduction of hardness is limited by some physical requirements. For example, an appropriate rigidity is necessaryto support the weight of the impending parts when the head or tail of the robot is in the open chain state. According to our experiments, the rubber bars with 40HA hardness value are acceptable.

B. Influence of JCA on the Sliding Force

To reduce the sliding forcesof a flexible caterpillar robot further, the JCA is applied on Joint 3 by artificially adding a bias to the ideal value defined in Equation (1). The value of JCA is changed from 0° to 8°, including 0°, 2°, 4°, 6° and 8°. The experiment results are shown in Fig. 12.

Figure 12.

The results of different JCAs

Fig. 12 shows that the JCA obviously influence F _SS. When the JCA changes from 0° to 4°, F _SS is reduced along with the increasing JCA. But when the JCA is bigger than 4°, F _SS tends to be larger again. When the JCA equals 4°, the curve of F _SS (the red curvein the Fig. 14) is also smoother than the others. The other phenomenon that can be seen in Fig. 12 is that along with the rotation of Joint 1, the JCA corresponding to the smallest F _SS is also changed. As a result, the JCA can be segmentally adjusted according to Table 1 to acquire smaller F _SS. The final experiment results are shown in Fig. 13 (the black curve).

Figure 13.

Smaller FSS with segmental JCA values

Figure 14.

A seven-joint climbing caterpillar robot

Table 1.

JCA and the value of θ₁

θ₁	JCA
0°-5°	4°
5°-27°	2°
27°- end	8°

5. Experiments of a caterpillar robot with a flexible structure

A caterpillar climbing robot shown in Fig.14 is assembled using some natural rubber bars whose hardness is 40HA. When crawling, the robot attachesto the ground with some pins. Every module of this robot is composed of the same parts. One of them is shown in Fig. 15. There is one drive motor, one natural rubber flexible bar, two fixing structures and other connecting pieces in one module, as shown in Fig. 18. For comparison, a rigid bar can be installed beside the rubber bar to form a rigid module. The mass of the rigid bar is only 4g. Compared with 92g(4.3%)for the somite, this will not change the character of the robot orlose the comparability.

Figure 15.

A rigidity bar installed

Figure 16.

One module of the robot

Figure 17.

The vibrating distance testing theory

Figure 18.

The vibrating distance testing results

It is hard to directlymeasure the sliding force of the seven-jointrobot, however, the side-sliding phenomenon caused by F _SS can be found. According to the analysis above, the head of the robotwill vibrate due to the F _SS, ifonly the tail module is attached to the surface during the crawling procedure. If the value of F _SSis bigger, the vibrating magnitude with belarger. Meanwhile, the power consumption of the robot will becomelarger. Therefore, in this part, the vibrating distance and the power will be detected separately to show the effect of reduction of the F _SS.

B. Vibrating Distance Testing Experiments

Fig. 17 shows a testing platform to measure the vibrating magnitude of the head of the caterpillar robot with only the most left module (the tail) attached to the surface. In these experiments, the pins on the other modules are removed. During the crawling procedure, a laser rangerdetectsthe vibrating distance. The testing results are shown in Fig. 18.

In Fig. 18, the Mean-Square Deviation (MSD) of the vibrating distance of the rigidity bar is 0.42mm and those of the flexible bar with or without JCA are 0.28mm and 0.31mm, respectively. This means that the vibrating distancewith the flexible bar is reduced. In addition, compared with that of the flexible bar without JCA, the vibrating distance with JCA is reduced further. These experimental results indirectly prove that the flexible bar can reduce the F _SS.

B. Power ConsumptionTesting Experiment

As we know, the sliding forcesare mainly derived from the driving torques of the motors, whicharepositively correlated to the current value of the power source of the robot. Therefore, the power consume, e.g., the current value of the power source, can be measured to observe the change of the sliding force. The voltage of the supply source is U_S=5V. Aresister with a small value of R=0.5Ω is serially connected in the circuit. By measuring the voltage U_R of the resister, the supply source current can be calculated by I= U_R /R. Furthermore, the output power Pof supply source is calculated as follows, $P = \frac{n \cdot Δ t \cdot U \cdot U_{R}}{T \cdot R}$ (8)

where T is the period time of one step, n is the sample number and Δt is the time between two samples.

In these experiments, the robot moves one step in 20s. The flexible barswith 40HA are usedand the JCAs arealso applied to the system according to Table 1. For contrast, the rigid robot without the JCA gait is also tested. The P curves of both the flexible and rigid robot are measured separately and are shown in Fig. 19.

Figure 19.

The measuring result of P – the blue curve is the P of the flexible robot and the redcurve is that of the rigid robot

In Fig. 19, the Pvalue of the flexible robot is much lower than that of the rigid onewith the same crawling time. During the whole crawling procedure, the average Pvalue of the rigid robot is up to 1.67W, and thatof the flexible robot is 0.94W, 43.46% reduced. These indicate that the flexible structure and the JCA method clearly reduce the sliding forces and the power consumption.

6. Conclusion

This paper builds up static models of the sliding forces on the adhering points of the caterpillar robot by taking the solid and flexible modules into consideration respectively. Inspired by the real caterpillar in nature, the flexible structure is considered in a simple way to reduce the sliding force. Besides its low rigidity, the flexible module also makes it possible to apply some special control methods to reduce the sliding force further. The Joint Compensation Angle (JCA) method is proposed to add an artificial bias of joint angle to counteract the sliding force. A series of experiments verify the successof the proposed methods. The sliding force is reduced remarkably when the segmental JCA values are applied on the testing platform being composed of soft rubber parts.

Future work will concentrate on building the static model of the whole flexible caterpillar robot. The relationship between the joint torque and the wave delivering force will also be analysed. Finally, the work will focus on revealing how the driving works of the active joints change the motion energy of the robot, by which design and control methods will be found to realizehigh climbing efficiency.

Footnotes

7. Acknowledgments

This paper is sponsored by the National Natural Science Foundationof China,Grant No. 51075015.

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