Research ArticleThe Gait Design and Trajectory Planning of a Gecko-InspiredClimbing Robot

Xuepeng Li ,1 Wei Wang ,1 Shilin Wu ,2 Peihua Zhu ,3 Fei Zhao ,1

and Linqing Wang 1

1Robotics Institute, Beihang University, Beijing 100191, China2School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China3FAW-Volkswagen Automotive Co., Ltd., Changchun, Jilin 130000, China

Correspondence should be addressed to Xuepeng Li; xpli613@163.com

Received 9 November 2017; Revised 22 January 2018; Accepted 5 March 2018; Published 22 April 2018

Academic Editor: QingSong He

Copyright © 2018 Xuepeng Li 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.

Inspired by the dynamic gait adopted by gecko, we had put forward GPL (Gecko-inspired mechanism with a Pendular waist andLinear legs) model with one passive waist and four active linear legs. To further develop dynamic gait and reduce energyconsumption of climbing robot based on the GPL model, the gait design and trajectory planning are addressed in this paper.According to kinematics and dynamics of GPL, the trot gait and continuity analysis are executed. The effects of structuralparameters on the supporting forces are analyzed. Moreover, the trajectory of the waist is optimized based on system energyconsumption. Finally, a bioinspired robot is developed and the prototype experiment results show that the larger body lengthratio, a certain elasticity of the waist joint, and the optimized trajectory contribute to a decrease in the supporting forces andreduction in system energy consumption, especially negative forces on supporting feet. Further, the results in our experimentspartly explain the reasonability of quadruped reptile’s kinesiology during dynamic gait.

1. Introduction

Wall-climbing robots can move and work on a vertical wallto complete various tasks, which have attracted much atten-tion of researchers around the world and have wide applica-tion fields including antiterrorism, postdisaster rescue,engineering test, and maintenance and inspection for haz-ardous environment [1–4]. Compared to the wheeled robotsand caterpillar robots, the multilegged climbing robots have adistinct advantage of strong adaptability to unknown envi-ronment and uneven wall. Certainly, we must own the factthat they are limited by low velocity and large energy con-sumption. Here, it becomes very important to address gaitdesign and trajectory planning of multilegged climbingrobots, especially for bioinspired climbing robots.

Generally, gaits of multilegged climbing robot couldbe classified into two kinds by leg raise sequence andstability: the quasistatic gait and dynamic gait. As the

name implies, the quasistatic gait shows that the wholerobot system maintains static balance during the climbingprocess, which is easy to ensure the behavior of robot[5]. Many previous works have been carried out to pro-mote development of climbing robots, such as the Climb-ing Mini-Whegs, the Waalbot II [6], the RiSE [7, 8], theStintov, and the StickyBot [9, 10], characterized withpivot joints on the legs. These climbing robots can beapplied in many challenging environments such as tree,glasses, cabinets, or concrete surfaces. However, theirworks are limited to relatively low velocities due to thequasistatic gait.

With the development of biomimetics, the dynamic gaitwhich is closer to actual biologic locomotion has beenapplied to improve vertical climbing efficiency and reducepower consumption. To describe the dynamic gait, somemodels are abstracted from the biologic locomotion pattern,including the spring-mass (SM) model [11], the ubiquitous

HindawiApplied Bionics and BiomechanicsVolume 2018, Article ID 2648502, 13 pageshttps://doi.org/10.1155/2018/2648502

Spring-Loaded Inverted Pendulum (SLIP) model, theLateral-Leg Spring (LLS) model [12], and the F-G model[13]. As expected, the climbing robots adopting dynamic gaitperformed an excellent performance from the perspective ofhigh speed and low energy cost [14, 15]. However, there arestill inevitable questions about the stability of climbingmotion. With the DynoClimber and ROCR as an example,the lack of supporting feet makes both of them easily rotatearound the vertical axis parallel with the climbing surfaceand leads to the weak stability during pendulum.

Trajectory planning for multilegged climbing robotsrefers to two major steps. Firstly, the trajectory solution fora required motion task could be obtained based on the for-ward and inverse kinematics. Furthermore, it is to ensure thatthe final path satisfies the needs of the desired conditions,such as minimum path, time, and energy consumption.Wang et al. investigated optimal attaching and detaching tra-jectory for wall climbing which guarantee reliability of theclimbing robot [16]. Chen et al. operated the leg trajectoriesof the leg-wheel hybrid robot in a periodic manner on eachstep during stair climbing in view of a series of similar geom-etry constraints and limitation of the joint power density[17]. Akinfiev and Armada analyzed the influence of gravityon trajectory planning for climbing robots [18]. These workshave improved the climbing performance of robots to acertain extent.

In our previous work, we have proposed the GPL(Gecko inspired mechanism with a Pendular waist andLinear legs) model and verified its feasibility [19]. TheGPL model inspired by the F-G model consists of tworigid bodies (upper part and lower part with tail), fourlinear legs with spring buffers, and a passive waist joint.And it aims at explaining the relationships of the loco-motion dynamics, the variables of the movement, andthe parameters of the mechanism for sprawl quadrupedclimbing animals. To further develop dynamic gait andreduce energy consumption of climbing robots based onthe GPL model, the gait design and trajectory planningwill be addressed in this paper.

The organization of this paper is as follows: firstly, kine-matics and dynamics of GPL are derived systematically andthe conditions of gait reuse are obtained in the second sec-tion. And the singularity and feasible region of the waist areanalyzed in detail. Then, the effects of structural parameterson the supporting forces are explored, including body lengthratio, driving angle, and elasticity of the waist joint. More-over, the trajectory of the waist is optimized based on systemenergy consumption. To testify our analysis, a bioinspiredrobot is developed and the prototype experiments are per-formed. Finally, conclusions are drawn and the future workis described.

2. Gait Design and Continuity Analysis

2.1. Kinematics and Dynamics of GPL. As illustrated inFigure 1, the GPL model abstracted from the gecko’s mor-phology and kinesiology is composed of four linear legs Lii = 1 – 4 , an upper part rigid body P1, a lower part rigidbody P2, and a passive revolute joint called waist P. Here,

the tail of the bionic prototype is very light and ignored tosimplify the dynamic analysis. According to bionic research[25], the driving angle β between the driving force along thelegs and the long axis of the body is almost constant duringa moving cycle. Here, β keeps a constant value (10°). Thefront legs L1, L3 and rear legs L2, L4 are fixed on the upperand lower rigid body, respectively. Namely, the leg and thecorresponding rigid body have the same angular velocity.In order to facilitate the analysis and design, waist P fixedon the upper part P1 is treated as a reference point of loco-motion for the model. The lower body can revolve about thewaist freely. To imitate gecko’s wrist, one passive rotary jointis designed at the end of each leg, which allows the leg torotate on the climbing surface. The GPL’s locomotion canbe realized by the legs’ extending or contracting motion,cooperating with alternation of feet’s attachment and wring.It notes that the waist has a certain elasticity by linking tworigid bodies with two linear springs.

The trot gait inspired by gecko’s climbing is adopted forthe GPL model, as shown in Figure 2. One stride of the trotgait during a stable climbing movement can be divided into

Q2

L1 (t) L3 (t)

L2 (t)L4 (t)

P1

P2 (m2, I2)

P (Px, Py)

d3

d4

Q3

Q4

Q1(m1, I1)

X1

X2

𝛽

b

O X

Y

O′La

Lb

θ1

θ1

z1

z2

s1 s3

s2s4

𝛽

F1y

F1x

F1

F2y

F2x

F2

F2

𝛼2

𝛼1

Figure 1: The GPL and its dynamic model derived from gecko.The red dotted line indicates the current adherent mechanism.The black dots on the foot mean the corresponding diagonal legsare attached with the climbing surface, while the gray dots meanthe corresponding diagonal legs are swinging. Here, La and Lb arethe horizontal and vertical distances between the diagonal stancefeet, respectively. Li t and xP denote the length of linear legsLi and the position of P, which can be controlled. Let mj,Xj, andI j j = 1, 2 be mass, the position of the center, and the inertiamoment tensor matrix of Pj in coordinate O , respectively. Qj,dj, θi, and αj denote the leg’s fixed point on Pj, the distance fromxP to Qj, the angle between leg and the vertical line, and the anglebetween corresponding leg and line PQj, respectively. Β and αj areconstant values. zj denotes the distance from xP to the center of Pj.

2 Applied Bionics and Biomechanics

two symmetrical phases. In one phase, the left front leg andthe right rear leg begin to establish attachment with theclimbing surface. Then, the left front leg contracts and theright rear leg extends, pulling the body upward, with otherlegs executing opposite motion. Meanwhile, the waist swingsas a pendulum about the stance foot. Next, the other one exe-cutes the symmetrical motion.

In view of the symmetry of gait, we choose the procedureof one phase for analysis and apply the result to the next sym-metrical phase, as shown by red dotted lines in Figure 1.Thus, the GPL model can be treated as a planar five-barmechanism RPRPR with two degrees of freedom duringone phase.

Let O and O′ be a coordinate frame O − XY fixed onthe right rear at O and left front foot at O′, respectively. Thewaist P rotates with the upper part around the stance foot O′.The pose of GPL can be expressed by by either qL = L1L2

T ,xP = Px Py

T , or X j = Pxj Pyj T as below (details can befound in [20]).

xp =Px

Φ L2 −Φ L1 − 2LaPx + C2 Lb

,

xp = JqL,

X j =Px

Py

+sin θj − β zj

−1 j+1 cos θ j − β zj,

1

where J is a 2 × 2 Jacobian matrix.

J = 1Lb × Px + La × Py

mPyn Lb − Py

−mPx n La + Px

2

Based on (1) and (2), when given the length qL of lin-ear legs, the position xP of the waist can be solved and viceversa, which lays the foundation for trajectory planning.

For further dynamic analysis of GPL, driving forces onthe linear legs are analyzed, which significantly determinethe control and performance of the robot. According toAutumn et al.’s research [21], energy loss during a rapidlocomotion was generally attributed to the decelerationforces produced by the supporting feet. In that case, gravityand the gecko’s legs decelerate the climbing within each step,resulting in velocity fluctuation. Therefore, it is worth ana-lyzing the supporting forces on foot.

Let F j be the active force applied on the linear legs. Let EK

and EP be the kinetic energy and potential energy of the tem-plate; it can be calculated as follows:

Ek qL, qL = 12〠

j

1I2j θj +mj Pxj + Pyj ,

Ep qL = Eps + Epg,3

where

Eps =12 kwσ

2w + 〠

j

1kj Lj − Lsj

2 ,

Epg = 〠j

1mjgPy

4

Here, the Eps and Epg denote the elastic potential energyand the gravitational potential energy of the template, respec-tively. kj is the stiffness coefficient of the axil legs Lj. kw andσw are the stiffness coefficient and torsional angle of the pen-dular waist, respectively.

According to Lagrange dynamical equation, driving forceτ = F1F2

T , generated by the stretchout and drawback of thelegs, can be calculated as follows:

O XY

S/2

(d)

O′

O XY

(e)

O′

O X

Y

(f)

O XY

O′

(g)

𝛽

𝛽

OL2 (tm+)

L4 (tm+)

L3 (tm+)

L1 (tm+)

(a) (c)(b)

L2 (t0)

L1 (t0)L3 (t0)

L4 (t0) 𝛽

b

O XY

O′

𝜃1

𝜃4 𝜃4T

𝜃1T 𝜃3T

𝜃2T

𝜃1T

𝜃4T

𝛼3𝛼1T xpT

𝛼3T

𝛼2T

𝜃3T

𝜃2T

xpT𝛼3T

𝛼2T𝛼4T𝛼1T𝛼4T

𝛼2

𝛼1

𝛼4

𝜃3

𝜃2

𝛾1

𝛾2𝛾2

𝛾1

𝛾2

xp

𝛽

T

L1 (tm−)L3 (tm−)

𝛽

𝛽

O XY

O′

γ1

L4 (tm−)

L2 (tm−) X

Y

O

O′ O′

L4 (T/2)L2 (T/2)

L1 (T/2) L3 (T/2)

L3 (te−)

L4 (te−)

L1 (te−)

L2 (te+)

L4 (te+)

L3 (te+)L1 (te+)

L2 (te−)L2 (T)

L4(T)

L3 (T)L1 (T)

S/2

Figure 2: The trot gait inspired by gecko’s climbing for the GPL model. Here, red dotted lines denote that the corresponding mechanismsare executing the active movement. (a)–(c) show the gait movement principle during half one gait. (a) The initial state of trot gait. L1 and L2are adherent; L3 and L4 are swing. (b) L1 contracts and L2 extends; L3 and L4 are swing; and the waist swings as a pendulum and movesforward half stride S/2. (c) The role transformation of corresponding adherent and swinging diagonal feet. (d)–(e) The two processes aresimilar with (a) and (b). And the waist moves forward the other half stride S/2. (f) The role transformation status. (g) The initial statefor the next gait.

3Applied Bionics and Biomechanics

τ = ddt

∂T∂qL

−∂T∂qL

,

T = Ek qL, qL + Ep qL

5

Therefore, the joint power of the robot and system energyconsumption in actual application of the climbing robot canbe obtained as follows:

Pj t = Ra

k2Mη2N2

G

F2j t +

Lj t F j t

η,

E t = 〠n

j=1

tm

t0

Pj t dt  j = 1, 26

2.2. Condition of Gait Reuse. Generally, the gait designed forthe robot has obvious periodicity in view of motion stabilityand control complexity. When a symmetrical gait is adopted,the trajectory planning for a step could be reused in the entireregular motion trajectory, especially linear trajectory or circletrajectory. Unlike the other quadruped robots, there is only acontrollable local degree of freedom on the foot for the GPLmodel. This means that when two diagonal legs are attachedwith the climbing surface, the remaining legs are in an approx-imately free state, leading to the uncontrollable poses of thecorresponding feet. In other words, the floating feet poses can-not reach the necessary conditions needed for the trot gaitreuse when the GPL model is in the transitional state fromone phase to the other one in one stride (S). In this paper,the conditions of GPL gait reuse are analyzed and the gait con-tinuity is ensured by taking linear trajectory as an example.

Figures 2(a)–(c) describe the gait movement principleduring half one gait. When t = t0, the climbing robot is inthe initial state, in which the diagonal legs with the blackdot on the foot are attached with the climbing surfaceand the diagonal legs with the blank dot on the foot areswinging; see Figure 2(a). Namely, the left front leg andthe right rear leg begin to establish attachment with theclimbing surface. Then, the left leg contracts and the rightleg extends, pulling the body upward. When t = tm—, theclimbing robot is in the transitional state; see Figure 2(b).It is obvious that the position of waist P moves from xPto xPT , and the distance in vertical direction is just half astride. When t = tm+, the other diagonal mechanismindicated by red dotted lines starts to execute the activemovement, as shown in Figure 2(c). Now, the currentstate becomes the initial state of the next half one gait. Inthe same way, the right front leg and the left rear legbegin to establish attachment with the climbing surfaceand execute a similar movement. So, the state transition isrealized from the former half one gait to the next one. Ofcourse, the following conditions need to be satisfied.

Set the initial time t = t0 and the terminal time t = tmduring half a gait. Let xP and xPT , Li t0 and Li tm ,and θi and θiT (i = 1 – 4) be the position of the waist,length of linear legs, and angle between leg and the corre-sponding vertical line in the initial and transitional states ofgait, respectively. Let γj j = 1, 2 be the angle between line

PQj and PQj+2, keeping a constant value. Let S be the verticalstride of one gait.

To realize the gait continuity, the following constraintsneed to be satisfied when the following phase repeats the tra-jectory of the previous phase in one stride.

(a) Structure parameter constraints

It is mainly considered that the prototype designshould keep bilateral symmetry. Thus,

αj = αj+2,dj = dj+2  j = 1, 2

7

(b) Leg pose constraints

These constraints include two aspects: feet angle andleg length. Here, the leg lengths can be actively con-trolled by motors.

θj = θ j+2 T ,Lj t0 = L j+2 tm   j = 1, 2

8

(c) Waist position constraints about central symmetry

This is determined by the trot gait adopted by theGPL model. Thus,

xp + xpT = −La2 9

Based on the geometric constraints, there exists afixed relationship as follows:

θj + αj + θj+2 + αj+2 = γj,θ jT + αj + θ j+2 T + α j+2 T = γj  j = 1, 2

10

According to (1), (7), (8), (9), and (10), the condi-tions of gait reuse and continuity for the GPL modelcan be obtained as follows.

xp = f1 θ1, θ2 ,xpT = f1 θ1T , θ2T ,

f1 θ1T , θ2T + f1 θ1, θ2 = −La2 ,

11

where

θjT = 2βj − θj 12

From the above, it can be seen that the position ofthe waist has specific limitations to realize the gaitcontinuity during the climbing process, rather thanany position.

4 Applied Bionics and Biomechanics

3. Trajectory Planning of GPL

3.1. Singularity and Feasible Region of the Waist. For thetrajectory planning, singularity and workspace are bothinevitable problems. Let ∣J∣ denote the determinant of theJacobian matrix J of the GPL model. When ∣J∣→∞ is satis-fied, the singularity of the GPL model occurs.

∣J∣→∞⇔Lb × Px + La × Py = 0 13

Obviously, the singularity of the robot will occur whenthe position of the waist is located on the diagonal anomalyline from (13). In this case, the driving forces on each sup-porting foot will be infinite theoretically. In order to addressthis issue, the waist trajectory should avoid intersection withthe diagonal anomaly line. According to (5) and (6), it sug-gests that the farther the waist trail keeps away from the diag-onal line linking the supporting feet, the smaller themaximum force on the axial legs. This rule should be consid-ered during trajectory planning of the waist for the robot.

As we know, trajectory planning must be restricted to thecorresponding workspace. Here, the waist xP is selected as thereference point for trajectory planning. The feasible region ofthe waist can be obtained according to previous analysis. LetW ∈ R2 denote the feasible region of the waist. Let Px min andPx max be the minimum and maximal values of Px for a cer-tain Py within the feasible region of the waist, respectively.

From (1), the relation between Px and Py can be derivedas follows:

Px = −d2sin α2sec θ2 − Pytan θ2 14

Given a large range Py min, Py max of Py beyond thefeasible region, Px min and Px max can be confirmed by thebelow constraints.

Minimize→Px min = min Px ,Px max = min −Px ,

subject toLj min ≤ Lj ≤ Lj max  j = 1, 2 ,Py = L2cos θ2 + d2cos θ2 + α2 ,Py min ≤ Py ≤ Py max

15

Based on (1) and (15), a series of points including(Px min, Py) and (Px max, Py) constitute the boundary of Wduring half a gait. In the same way, the feasible regionof the waist during the following half a gait can be deter-mined, which is symmetric with the previous feasible regionabout the central axis and has a translational distance in theclimbing direction.

3.2. Effect of Structural Parameters. As mentioned above, thefeasible region of the waist directly affects trajectory planningof the climbing robot. Both driving forces of active legs andthe feasible region of the waist are related to the structuralparameters of the robot, including body length ratio, drivingangle, and waist spring coefficient. Therefore, the structuralparameters should be reasonably selected, which contributes

to a decrease in energy consumption and improvement in thestability of the climbing robot.

3.2.1. The Effect of Body Length Ratio on Supporting Forces ofClimbing Robot. To avoid the disturbance of unreasonablemotion path to structural parameters, it is necessary to plana smooth motion trail of the waist to highlight the effect ofstructural parameters on the climbing performance of therobot. In terms of control weaknesses of the single curvefunction, an acceleration function composed by piecewisecontinuous function is adopted, which combines advantagesof trapezoidal function and sine function. In this case, theacceleration curve function transits smoothly. Thus, the iner-tia force of the system could be reduced, which effectivelyimproves the climbing speed and stability.

Let T be the working time during half a gait. Here, let LIjand LT j j = 1, 2 be the initial and terminal lengths of legs Lj,respectively. According to kinematic analysis of the robot, LIjand LT j can be solved based on the start and terminal points(xP and xPT) of waist trajectory at initial and terminal times.

To ensure the smooth path and no impact velocity of theGPL model, the following boundary conditions are listed.Namely, the corresponding velocity and acceleration of legsLj are both equal to zero at initial and terminal times.

Lj t0 = LIj,

Lj t0 = 0,

Lj t0 = 0,Lj tm = LT j,

Lj tm = 0,

Lj tm = 0

16

Define the trajectory interpolation functions s τ , and lets τ be the independent variable. Thus,

Lj t = LIj + LT j − LIj s τ ,

Lj t = LT j − LIj sτ

T,

Lj t = LT j − LIj sτ

T2 ,

17

where s τ meets the following constraints:

0 ≤ s τ ≤ 1,0 ≤ τ ≤ 1,

τ = t − t0T

,

T = tm − t0

18

Give priority to acceleration, and set s τ as follows:

5Applied Bionics and Biomechanics

s τ =

k sin 4π τ, 0 ≤ τ ≤18

k, 18 ≤ τ ≤

38 ,

k sin 4πτ − π , 38 ≤ τ ≤

58 ,

−k, 58 ≤ τ ≤

78 ,

k sin 4πτ − 3π , 78 ≤ τ ≤ 1

19

Integrating (19) with respect to time and combiningthe initial and terminal conditions, the value of k can bedetermined as follows:

s τ =tm

t0

1

0s τ dτ,

s 0 = 0,s 1 = 1

20

According to (16), (17), (19), and (20) and structuralparameters of the climbing robot (Table 1), the planned tra-jectory of the waist can be obtained. Combining the kine-matic analysis of robot, the whole controlling function oflegs can be confirmed.

To study the effect of the length ratio of front and rearbody on driving forces of the robot, the different body lengthratios λ = 8/2, λ = 6/4, λ = 5/5, and λ = 4/6 are set to calculatethe driving forces according to the planned trajectory of waistand kinematic analysis. According to Figure 3, the support-ing forces on front and rear feet present a significant decreas-ing trend as the increasing body length ratio. It is worthmentioning that the supporting forces on rear foot display asmaller negative value with the larger body length ratio,which contributes to a reduction in negative work of therobot. It suggests that the waist close to the bottom of thelower body can effectively optimize driving forces and systemenergy consumption.

3.2.2. The Effect of Driving Angle on Start and Terminal Pointsof Waist Trajectory. According to bionic research [22, 23], thedriving angle β of reptile gecko usually keeps a certain value.From (11), it can be shown that β has an obvious influenceon the start and termination point space of the waist. Toexplore the effect of the driving angle on the start and terminalpoints of the waist, the driving angle range [8, 11] is selected tocompute the start and termination point space and the feasibleregion of the waist based on the following dimension con-straints and structural parameters (Table 1).

L1min

L2min≤

L1 t

L2 t≤

L1max

L2max21

From Figure 4, the range of the waist feasible regionshows a slightly increasing trend with driving angle β. Asmentioned above, trajectory planning should avoid the sin-gular line due to waist joint on singular position, causing

infinite force. Therefore, the start and termination pointsof the waist joint should be at the same side of the singularline. In addition, the motion trajectory must be within thewaist feasible region in terms of structural constraint.Namely, the start and termination points of the waist alsoneed to be located in the feasible region. Here, the optimaldriving angle is 9°.

3.2.3. The Effect of Waist Spring Coefficient on SupportingForces of Climbing Robot. As we know, the reverse drivingforces produce negative work, increasing system energyconsumption. The driving forces are associated with thewaist spring coefficient according to (3) and (5). Hence,the waist spring coefficient of the robot is analyzed toobtain a reasonable value, which contributes to a decreasein negative driving forces.

To optimize the energy consumption, we need to reducethermal energy during robot climbing. It means that thenegative work done by the supporting legs should beavoided. For a given waist trajectory or a climbing dynamicgait, the negative work will be as small as possible when thefollowing (22) is satisfied. According to the equation, we canobtain the range of kw (0.503–0.616) to ensure the positivedriving forces.

Table 1: Structural parameters of the climbing robot.

Parameters La (mm) d1 (mm) L1min (mm) L1max (mm)

Value 230 168 103 168

m1 (g) Lb (mm) d2 (mm) L2min (mm) L2max (mm)

243 420 113 85 150

m2 (g) NG η (%) Ra (Ω) KM (Nm/A)

223 2 77 4 10.2

1 2Time (s) Time (s)

𝜆 = 4/6𝜆 = 5/5

𝜆 = 6/4𝜆 = 8/2

03

4

5

6

7

8

9

10

11

12

0 1 2−4

−3

−2

−1

0

1

2

3

4

5

Supp

ortin

g fo

rce o

n re

ar fo

ot (N

)

Supp

ortin

g fo

rce o

n re

ar fo

ot (N

)Figure 3: Supporting forces of the GPL model with different bodylength ratios.

6 Applied Bionics and Biomechanics

F1 > 0,F2 > 0

22

Given the range of kw 0 – 1N · mm , the system energyconsumption is calculated based on the above planning tra-jectory of the waist as well as (5) and (6). Figure 5(a) showsthat the supporting forces on front and rear feet decreaseand increase with kw, respectively. There exist the largest neg-ative and positive driving forces in the case of kw = 1N · mmthan the ones in other cases. From Figure 5(b), it can be seenthat the lesser negative forces active legs produce, the lesserthe system energy consumption and the better the movementperformance of the robot. Here, the reasonable spring coeffi-cient can improve athletic performance while an oversizevalue of the spring coefficient may generate a negative effecton climbing motion of the robot. We can also find that thesystem energy consumption can obtain the smallest valuewhen kw = 0 5N · mm. This is in accord with the above anal-ysis based on (22). Hence, the value of kw = 0 5N · mm isselected in this paper.

3.3. Trajectory Optimization Based on Energy Consumption.During practical applications, there are high requirementsof energy supply to be put forward for the robot system. Gen-erally, the load-bearing capability of multilegged climbingrobots is limited, which cannot carry greater weight energystorage units. Thus, it is significant to reasonably controlmechanical energy consumption and improve climbing effi-ciency of the robot during the design process of the climbingrobot. In this section, we focus on trajectory optimizationbased on system energy consumption.

Here, the waist joint coordinates can be expressed by them-rank polynomial through the Hermite interpolationmethod, as follows:

xp t = aj0 + aj1 t − t0 +⋯ + ajm t − t0m, j = 1,

yp t = aj0 + aj1 t − t0 +⋯ + ajm t − t0m, j = 2

23

The above equation can be written in matrix form, asfollows:

q A, t =xp t

yp t= Ah t , 24

where A is the multinomial coefficient matrix, which needs tobe calculated.

A =a10 ⋯ a1m

a20 ⋯ a2m∈ R2× m+1 ,

h t = 1 t − t0 ⋯ t − t0m T

25

As in Section 3.2, the driving angle and body length ratiohave been discussed and the optimal values have beenobtained. The driving force curves of supporting feet at everystart and termination points of the waist joint can be accom-plished according to (1), (2), and (13). Considering the min-imum driving force as the optimization goal, the best startpoint q A, t0 and termination point q A, tm of the waist

joint are selected. To ensure the smooth motion track of therobot, it is necessary to guarantee the angular velocity andangular acceleration to be continuous. Therefore, the follow-ing constraint equations need to be satisfied:

q A, tk =xp tk

yp tk=

x1k

x2k= Ah tk ,

q A, tk =xp tk

yp tk=

x1 k+1

x2 k+1= Ah tk ,

q A, tk =xp tk

yp tk=

x1 k+2

x2 k+2= Ah tk   k = 0,m ,

26

where xij is the boundary condition, which denotes the posi-tion, velocity, and acceleration of the waist at the start andterminal points.

Combining the above equation constraints, the boundaryconditions can be obtained as follows:

Ceqj A = 0, j = 1 – 6 27

In addition, the waist joint coordinates are limited to thedimension constraint relation as in (15). Thus, there are fourinequality constraints as follows:

Ci A ≤ 0, i = 1 – 4 28

In terms of constraint condition for waist joint coordi-nate x or y, there are six equation constraints. And thereare twelve equations and four inequalities for the waist jointcoordinate in x and y directions. Therefore, at least sevenundetermined coefficients are needed to solve the solution.Here, m = 6.

According to the above theoretical analysis, the systemenergy consumption is mainly caused by active joints. So,the power consumption of the robot at any time can be rep-resented by multinomial coefficient matrices A and h t .

P A, t = 〠2

j=1Pj t = Dq A, t + Hq A, t + Sq A, t + C

29

The optimization problem of system energy consump-tion can be described as

Minimize→A

tm

t0

P A, t dt,

subject toCeqj A = 0,Ci A ≤ 0

30

So the trajectory optimization of the waist can be trans-formed into a nonlinear programming problem in the non-linear equality and inequality constraints. According to thestructural parameters and gait conditions, the multinomial

7Applied Bionics and Biomechanics

coefficient matrices A can be solved and the optimal trajec-tory of the waist can be obtained, as shown in Figure 6.

4. Experiments and Results

4.1. Prototype Design Based on the GPL Model. The robotbased on the GPL model is composed of two rigid partsand four identical leg modules. According to Figure 7(a),there is no difference between mechanical configurations ofthe upper and lower parts in addition to the size parameter.And both parts connect to each other by the passive waist,which is a rotational joint in practice. Here, the position ofthe waist joint can be adjusted along the central axis of the

upper part. The linear leg modules fixed on the rigid partsare adopted to realize the desired motion of the climbingrobot, and they keep a certain angle with the body axis. Inorder to facilitate parameter adjustment, the angle and posi-tion of the rotating joint can be regulated manually. FromFigure 7(b), the linear leg module consists of spines, bufferunits, a linear guide with a rack, and a linear slider fixed onthe rigid part. Here, the spine is fixed on the cylindrical sliderthat can slide in a linear track. The track named as springshells is divided into two segmentations by a cylindricalslider, which plays a buffer role in two directions. To acquirethe accurate motion of the robot, the linear leg modules aredirectly driven by a rotary motor with a rack and pinion

−230 −180 −130 −80 −300

50

100

150

200

250

300

350

400

y (m

m)

Waist feasible region

The trace of startand terminal points

of the waist

Singular line

Center line

X (mm)𝛽 = 8°𝛽 = 9°

𝛽 = 10°𝛽 = 11°

Figure 4: The start and termination point space and the feasible region of the waist.

0 0.5 1t (s)

−6

−4

−2

0

2

4

6

8

F1 (N

)

F2 (N

)

−4

−2

0

kw = 0kw = 0.1kw = 0.2kw = 0.3

kw = 0.4kw = 0.5kw = 0.6

kw = 0.8kw = 0.9kw = 1

kw = 0.7

2

4

6

8

10

0 0.5 1t (s)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1t (s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

P (W

)

kw = 0kw = 0.1kw = 0.2kw = 0.3

kw = 0.4kw = 0.5kw = 0.6

kw = 0.8kw = 0.9kw = 1

kw = 0.7

(b)

Figure 5: The effect of the waist spring coefficient. (a) The supporting forces on front and rear feet with different waist spring coefficients. (b)The system energy consumption with different waist spring coefficients.

8 Applied Bionics and Biomechanics

mechanism. The corresponding linear speed of linear legscan be regulated and controlled by servo motors.

In view of the overturning torque in the whole climbingprocess, it is necessary to reduce the torque to avoid dropdamage. As we know, the overturning torque becomes largeras the thickness of the robot increases. So, the thickness of therobot is limited to 14.8mm and the motor is placed on therigid parts. Moreover, a thin flat tail acting as a support isdesigned to obtain a more stable climbing movement. In fact,the tail is ignored in dynamics of the GPL model due to itsmicro weight. In addition, the controller and battery arelocated in the upper part, which have the effect of counterweight by adjusting their corresponding position.

4.2. Actual Climbing Experiments and Force Measurement onSupporting Feet. In order to verify the designed gait and con-dition of gait reuse, the test platform with linen is establishedand the actual climbing experiment is conducted. Accordingto Figure 8(a), the robot adopts a symmetrical gait andmoves smoothly during a whole gait circle, which showsan intuitive understanding about gait reuse and continuity.It suggests that the theoretical analysis is validated and itprovides the basis for the trajectory planning of the robot.It should be noted that the climbing speed is 66mm/s inthe dynamic analysis for robot climbing. Actually, the

speed of the prototype is limited by failure of the spine’sattachment/detachment to the cloth-covered climbing sur-face in the climbing experiment, which will be improved.Therefore, the speed of the prototype is reduced to 50mm/sto maintain a more stable climbing in the test. Meanwhile,the force-measuring platform is built to explore the effect ofstructural parameters on supporting forces of the robot, asshown in Figure 8(b).

In the supporting force measurement experiment, theplanning trajectory of the waist in Section 3.2.1 is applied.And the ranges of the waist spring coefficient and bodylength ratio are in accord with the above analysis. FromFigure 9, we can obviously find that the supporting forceson front and rear feet have the same decreasing trend withthe increase in λ. Meanwhile, the ranges of supportingforces vary greatly with λ under a given waist spring coef-ficient kw. This indicates that the larger body length ratiocontributes to a decrease in the driving forces, which is con-sistent with theoretical analysis. Based on (6), the systemenergy consumption is directly related to the supportingforces, which is expected to be as small as possible. More-over, the waist spring coefficient has a significant effect onsupporting forces, especially forces on rear feet. FromFigures 9(a) and 9(b), there exist negative values on therear-supporting foot when kw is 0N·mm and 0.2N·mm.

0 T/2 ( + )T/2 ( − ) T

O′ O′

O′ O′

L3L3

L1L1

L4L4

L2L2

P P PP

O O

O O

Figure 6: The optimal trajectory of waist during the whole gait. Here, the blue line and blue circle dot denote the optimal trajectory and thestart and terminal points during the half gait, respectively.

Tail

Spine

Linear leg

WaistLower part

Battery and controller

Upper part

(a) The configuration of the climbing prototype

Spine

Buffer system

Servo motor

Linear leg

Adjustable part

Gear

Linear sliderLinear rack

Cylindrical sliderSpring shells

(b) Design of the leg module

Figure 7: Prototype design based on the GPL model.

9Applied Bionics and Biomechanics

According to Figures 9(c) and 9(d), the negative forcesdecrease gradually with the increase in kw while the overallvalues of supporting forces become larger. Based on previous

analysis, the negative forces on the rear-supporting footshould be avoided to reduce energy consumption in the pro-cess of climbing. In addition, there exist obvious fluctuations

S

h

TT/20

1 2 3 4 5

(a)

Force sensorThe vertical wall

The climbingrobot

Force sensor

Signal amplifier

Signal collector

(b)

Figure 8: Actual climbing experiments on cloth-covered climbing surface and force measurement on supporting feet. (a) The sequence ofclimbing gait. Here, the red dotted line indicates the current adherent mechanism and the yellow circles indicate the adherent feet. (b) Thetest facilities of climbing robot and supporting force experiment.

0 0.5 1 1.5 256789

1011121314

Time (s)

F1 (N

)

0 0.5 1 1.5 2−3−2−1

0123456

Time (s)

F2 (N

)

𝜆 = 4/6𝜆 = 5/5

𝜆 = 6/4𝜆 = 8/2

(a)

0 0.5 1 1.5 27

8

9

10

11

12

13

14

Time (s)

F1 (N

)

0 0.5 1 1.5 2−1

0

1

2

3

4

5

6

7

Time (s)

F2 (N

)

𝜆 = 4/6𝜆 = 5/5

𝜆 = 6/4𝜆 = 8/2

(b)

𝜆 = 4/6𝜆 = 5/5

𝜆 = 6/4𝜆 = 8/2

0 0.5 1 1.5 27

8

9

10

11

12

13

14

Time (s)

F1 (N

)

0 0.5 1 1.5 20

1

2

3

4

5

6

7

Time (s)

F2 (N

)

(c)

𝜆 = 4/6𝜆 = 5/5

𝜆 = 6/4𝜆 = 8/2

0 0.5 1 1.5 210

12

14

16

18

20

22

24

26

Time (s)

F1 (N

)

0 0.5 1 1.5 22

4

6

8

10

12

14

16

18

Time (s)

F2 (N

)

(d)

Figure 9: The supporting forces on front and rear feet with different waist spring coefficients and body length ratios: (a) kw = 0N · mm; (b)kw = 0 2N · mm; (c) kw = 0 5N · mm; (d) kw = 1N · mm.

10 Applied Bionics and Biomechanics

of supporting forces in Figures 9(a), 9(b), and 9(d). Thismeans that the stability and structural strength of theclimbing robot are easily damaged from the changingstresses to the actuators and manipulator structure. There-fore, 0.5N·mmmay be a more reasonable value of kw in viewof the whole positive amounts and smaller range of support-ing forces on both feet.

4.3. Energy Consumption Experiments. To verify the effec-tiveness of the optimized trajectory in Section 3.3, theenergy consumption experiments are carried out, as shownin Figure 10. According to bionic research on gecko, thesinusoidal curve fitting the trajectory of the gecko’s waist isapplied to contrast with optimized trajectory [24, 26]. Andthe current sensors are adopted to obtain the instantaneouscurrent values of motors. The instantaneous input power ofthe motor is the product of the instantaneous armature cur-rent and the instantaneous armature voltage. Thus, the totalinput power of the robot can be obtained as follows. Here,Uj t is a constant 5V.

P t = 〠n

j=1

tm

t0U j t × ij t 31

According to Figure 11, the power based on optimizedtrajectory is less than that based on bionic trajectory of thegecko. The former is 7.39W and the latter is 9.79W, in whichthe energy consumption based on optimized trajectory saves24.5% than the latter one does. Obviously, it suggests that theoptimized trajectory is effective, enhancing the theoreticalanalysis in this paper. Moreover, the results in our experi-ments may partly explain the reasonability of the quadrupedreptile’s kinesiology during dynamic gait.

5. Conclusion and Future Work

Benefitting from dynamic gait, the GPL model inspired bygecko has been proposed in our previous work. In order tofurther develop dynamic gait and reduce energy consump-tion of the climbing robot based on the GPL model, the gaitdesign and trajectory planning are analyzed in this paper.In view of the special construction of the GPL model, the trotgait is adopted and the conditions of gait reuse are presented.According to the kinematics and dynamics of the GPLmodel, the structural parameters have a significant effect onthe trajectory planning of the robot. The prototype experi-ment results show that the larger body length ratio and a cer-tain elasticity of the waist joint contribute to a decrease in the

P(t) = 𝛴 ⟆

S

10 cmh

tTT/20 2T3T/2

n

j=1tmt0 Uj (t) × ij (t)

⑦① ② ③ ④ ⑤ ⑥

Figure 10: Energy consumption experiments of the climbing robot.

Time (s)

P (W

)

Bionic trajectoryof the gecko

Optimized trajectory

Former halfphaseLatter halfphase

0

2

4

6

8

10

12

0 0.5 1 1.5 2 2.5 3 3.5 4

(a)

P (W

)

012345678

Time (s)0 0.5 1 1.5 2 2.5 3 3.5 4

Bionic trajectoryof the gecko

Optimized trajectory

Former halfphaseLatter halfphase

(b)

Figure 11: Energy consumption of the climbing robot with different trajectories. (a) The total input power of the robot with unfaltering data;(b) the total input power of robot with faltering data.

11Applied Bionics and Biomechanics

supporting forces and reduction in system energy consump-tion, especially negative forces on supporting feet. Moreover,it suggests that the driving angle plays an important role onobtaining a reasonable performance of gait continuity. Fromthe perspective of gecko, the pendular waist with certain elas-ticity is beneficial to storing kinetic energy in fluctuation andreducing energy consumption in the process of climbing.Therefore, the optimal trajectory of the waist is plannedbased on system energy consumption. The energy consump-tion experiments about optimal trajectory and gecko’s sinu-soidal curve are conducted, where a similar trend and lessvalue of power suggest that the optimized trajectory is effec-tive. Further, the results in our experiments partly explain thereasonability of the quadruped reptile’s kinesiology duringdynamic gait.

In the future work, the deformation of the structurewill be considered in dynamics of the GPL model. Andthe flexible material and foot structure with multipledegrees of freedom will be included in the future design ofclimbing robots.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this manuscript.

Acknowledgments

This study is supported by the National Natural ScienceFoundation of China (no. 51475018), Beijing Natural ScienceFoundation (no. 3162018), and Innovation Practice Founda-tion of BUAA for Graduates (no. YCSJ-01-201709).

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13Applied Bionics and Biomechanics

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