Trajectory adaptation of biomimetic equilibrium point for ...

Autonomous Robots (2021) 45:155–174https://doi.org/10.1007/s10514-020-09955-4

Trajectory adaptation of biomimetic equilibrium point for stablelocomotion of a large-size hexapod robot

Chen Chen1 · Fusheng Zha1 ·Wei Guo1 · Zhibin Li2 · Lining Sun1 · Junyi Shi1

Received: 11 September 2019 / Accepted: 5 November 2020 / Published online: 22 November 2020© Springer Science+Business Media, LLC, part of Springer Nature 2020

AbstractThis paper proposes a control scheme inspired by the biological equilibrium point hypothesis (EPH) to enhance the motionstability of large-size legged robots. To achieve stable walking performances of a large-size hexapod robot on different outdoorterrains, we established a compliant-leg model and developed an approach for adapting the trajectory of the equilibrium pointvia contact force optimization. The compliant-leg model represents well the physical property between motion state of therobot legs and the contact forces. The adaptation approach modifies the trajectory of the equilibrium point from the forceequilibrium of the system, and deformation counteraction. Several real field experiments of a large-size hexapod robot walkingon different terrains were carried out to validate the effectiveness and feasibility of the control scheme, which demonstratedthat the biologically inspired EPH can be applied to design a simple linear controller for a large-size, heavy-duty hexapodrobot to improve the stability and adaptability of the motion in unknown outdoor environments.

Keywords Equilibrium point hypothesis · Compliant-leg model · Contact force optimization · Deformation counteraction ·Large-size hexapod robot

1 Introduction

On discontinuous terrain surfaces, legged robots can achievesignificant mobility advantages in such a challenging envi-ronment compared to wheeled types (Semini et al. 2016;Gehring et al. 2016). But by far, legged robots are still notas agile as legged animals on this planet. Stable and robustmotion control of different kinds of legged robots, especiallylarge-size legged robots, remains an important research topicto address (Li et al. 2016; Hodoshima et al. 2007; Irawan andNonami 2011; Zhuang et al. 2017). Particularly for a large-size legged robot, the deformations of robot structure, ie body

Electronic supplementary material The online version of this article(https://doi.org/10.1007/s10514-020-09955-4) containssupplementary material, which is available to authorized users.

B Wei [email protected]

Zhibin [email protected]

1 Harbin Institute of Technology, No.92 West Dazhi Street,Harbin, Heilongjiang Province, China

2 School of Informatics, The University of Edinburgh,Edinburgh, UK

and legs, as well as the significant foot-ground impacts arenot negligible and can largely downgrade the walking perfor-mance, ie motion stability and body attitude, especially ontough terrains. Compared to small-size legged robots, large-size robots have difficulty in recovering the body posture dueto very large inertia, and the resulted instability may furtherlead to irreversible damage to the robot. Therefore, the capa-bility of enhancing a stable motion is of crucial importancefor large-size legged robots.

The core of stable motion control of legged robots is howto control the motions of the legs properly (Caron et al.2016). This is hard because of the mechanical complex-ity, redundant degrees of freedom, unknown terramechanics,and so on. But even though the biomechanical propertiesof limbs are complex, a wide variety of activities can stillbe taken by the legged creatures without much knowl-edge known beforehand about the environments. Inspired bythe motion dexterity of legged creatures, and motivated bythe understanding and further applications of the biologicalmotion mechanisms, many proposals have been presentedby the neuroscientists through long term observations andexperiments on different species. Among all the proposals,mechanical reductionism and equilibrium point hypothesismay be most directly related to the legged robot control.

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156 Autonomous Robots (2021) 45:155–174

In the field of mechanical reductionism, scholars believean assumption that motor actions of limbs can be reduced tomechanical programming. In line with this assumption, theyargue that the central nervous system (CNS) forms inter-nal models of limbs, namely the inverse dynamics models(Shadmehr andMussa-Ivaldi 1994; Thoroughman and Shad-mehr 2000; Buchanan et al. 2005; Erdemir et al. 2007; Russoet al. 2014). The EMG activities and torques underlying thecontrol process of limbs are in continuously programmingand computing by the CNS using the inverse dynamics mod-els. Based on the mechanical reductionism, many controllersusing inverse dynamics have been developed for bionic limbcontrolling. Villard et al. (1993) designed a controller basedon the inverse dynamics model of the quadruped robot RAL-PHY to achieve stable locomotion. In their controller, theinverse dynamics model was used for the single leg controllevel to get the joint torque needed. Li et al. (2015) pro-posed a controller based on a constrained dynamics modelfor a quadruped robot with flexible joints. In this controller,a dynamic force distribution approach and a fuzzy-basedadaptive control method of the joints were proposed to sup-press uncertainties of the perturbing forces and the dynamicsmodel. Roy et al. (2010) proposed an energy efficient con-troller for the hexapod locomotion based on the analysisof the dynamics model of a hexapod robot. The energyconsumption model was established based on the dynam-ics model. The effectiveness of the controller was verifiedthrough simulation experiments. A similar idea of energyconsumption optimization based on the inverse dynamicsmodel was also employed by Mahapatra et al. (2015) indesigning their controller for the hexapod locomotion. Torealize better dynamic quadrupedal locomotion of a robotcalled StarlETH, (Hutter et al. 2014) proposed an operationalspace controller based on hierarchical task optimization. Theprojected dynamics model of floating base systems was usedto reduce the optimization dimensionality.

Although the inverse dynamics model is widely employedby scholars in designing their controllers, but it may not bethe ultimate key to solve the robot motion control problem.At least two limitations need to be further breached. The firstis the complexity of modeling. Modern robots are develop-ing rapidly. New structures, like legged robots with flexiblejoints, and redundantmechanical degrees of freedom (DOFs)severely challenge the dynamics modeling process. The sec-ond is themodel accuracy.Adynamicmotion process usuallyshows strong nonlinear characteristics. For a robot motioncontroller based on dynamics modeling, model parametersmay be time-varying and need to be estimated with highaccuracy to get a good control performance (Wensing et al.2017; Wu et al. 2010; Tournois et al. 2017; Abdellatif andHeimann 2009). But the reality is that some unpredictablequantities, like joint frictions or the disturbances from theenvironment, are hard to be estimated. Therefore additional

compensations for model uncertainties (Grotjahn et al. 2004;Wang et al. 2018), or interactive models with environments(Ding et al. 2013) are always needed. The two limitationstogether lead to a great amount of nonlinear computations.The expensive computations are not friendly with the design-ing of real-time controllers. Besides, whether the CNS needsto know the exactmodel of themusculoskeletal system is stilla debatable question, because the control process seems to bequite time-consuming and difficult for a biological system.

In contrast to the inverse dynamics control, the equilib-rium point hypothesis (EPH) pioneered by Feldman (1966)seems more in line with biological motions. Triggered bythe observations of human motor behaviors, Feldman didsystematically theoretical and experimental efforts, and pro-posed that the limb motion was controlled in the way ofshifting the limb from one equilibrium position to another.In other words, motor control could be realized by adjustinga succession of equilibrium points (EPs) overtime (Feld-man 2009). He proposed a model to describe the functionof the muscle-reflex system. In this model, the muscle wasabstracted into a springwith adjustable resting length. There-fore, muscle force only related to the spring stiffness, and thedifference between the actual length/position and the equilib-rium length/position (Asratyan and Feldman 1965). Duringthe EP transition process, no knowledge about the environ-ment or the internal model of the musculoskeletal system isneeded. The functional tuning of the CNS remains constant.This hypothesis has been putting forward by many schol-ars for its compelling feature that motion and posture areintegrated into a single mechanism (Bizzi et al. 1991; Shad-mehr and Arbib 1992; Suzuki and Yamazaki 2005; Latash2010; Ambike et al. 2016). Obviously, compared with theinverse dynamics control, the EPH can avoid the uncertainparameters and complex nonlinear computations. Therefore,it is more attractive when an effective real-time controller isneeded in real-world deployment.

Regardless of the various hypotheses of EPH for motorcontrol in biological systems (Gomi and Kawato 1996;Popescu and Rymer 2000; Hinder and Milner 2003), manyEP controllers have been developed and implemented on dif-ferent robots successfully due to the original tolerance forthe unpredicted external disturbances. Taking advantages ofthe muscle-spring model, Gu and Ballard (2006) developeda two-phase EP controller for controlling simulated humanmovements. In this controller, given a task inCartesian space,the EP solutions were computed in joint space based on theidea of minimizing gravity potential energy. Motion exe-cutions were taken charged by the muscle-spring model.According to EPH, the force exerted on a limb depends on thedifference between the actual positions andEPs, and the stiff-ness and damping parameters about the EPs. Based on this,Mukaibo et al. (2004) developed a two-linked manipulatorwith a double-actuator joint to generate human-like move-

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Autonomous Robots (2021) 45:155–174 157

ments. One of the actuators was used for position control,and the other was used for the joint stiffness control. Using asimple control scheme, this mechanism could allow a robotlimb to achieve self-stability in unknown environments. Asimilar idea inspired by the EPH was also employed by Kimet al. (2013) and Park et al. (2006) in designing their redun-dant actuationmanipulators. Jain andKemp (2009) proposeda control scheme based on the EPH for a 7 DoF anthropo-morphic armwith series elastic joint actuators. In this controlscheme, two controllers to generate the Cartesian space EPsfor the arm’s end effector were developed. Then using theinverse kinematics, the joint EPs could be obtained. Gengand Gan (2009) developed a control structure for a planarbiped walking robot. In this control structure, a local jointEP controller was designed to realize the control of a singlejoint. Stable walking was achieved without any computationof the robot dynamics. Shi et al. (2018) proposed an EP con-troller for a quadruped robot. In this controller, by solvinga QP problem to achieve desired equilibrium contact forces,the foot-tip EPs were indirectly obtained. The effectivenessof the controller was verified through a series of experiments.

Because the EPH was originated from the studies ofhumans, so the applicability of EP controllers for control-ling limb-like manipulators is very desirable (Metta et al.2000). Nevertheless, although controllers inspired by theEPH show significant advantages in many aspects, EP con-trollers designed for legged robots, especially for large-sizelegged robots, are rarely seen. Legged robots are base-floating systems with redundant DoFs. Both single-leg con-trol and inter-leg coordination should be taken into accountwhen a good controller is wanted to be designed. Further-more, unlike manipulators that operate in unchanged specificenvironments, legged systems usually work in unstructuredenvironments. The complex interaction between the feet andterrains will significantly affect the motion states of therobots. Therefore, how to obtain stable EP trajectories is themain problem currently limiting the development of EP con-trollers for legged systems.

Inspired by the advantages of the currently developedEP controllers, and motivated by solving the stable walk-ing problem of a large-size hexapod robot, a simple controlscheme based on an EP trajectory modification method isproposed in this paper. Based on the compliant-leg modelestablished in this control scheme, no nonlinear dynamicscomputations are employed when designing the whole con-trol logic. The whole control scheme is totally linear, andeasy to be realized. An EP trajectory modification method isintroduced in the control scheme. Different from the existingstable control methods for small legged robots, the deforma-tions of the large-size hexapod robotwhich cannot be ignoredare significantly considered while designing the EP trajec-tory modification method. Through the work of the controlscheme, stable hexapod locomotions can be realized just by

adjusting the EP trajectories over time without any knowl-edge about the environment. Therefore good versatility ofthe control scheme can be ensured.

This paper is organized as follows: in the second part, thecontrol problems of the large-size hexapod robot and the con-trol scheme are introduced briefly. In the third part, an EPtrajectory modification method via contact-force optimiza-tion is introduced in detail. In the fourth part, the experimentresults to demonstrate the effectiveness of the control schemeare discussed.

2 The hexapod robot system and the controlscheme inspired by EPH

2.1 The problem statement of the large-sizehexapod robot

The large-size hexapod robot developed for large load car-rying transportation in challenging outdoor environments isdepicted in Fig. 1. The robot is designed in the size of over 4m× 2.5 m× 2 mwhile the weight of it is 2.5 t. The left threelegs are marked as leg 1, leg 2 and leg 3 while the right threelegs are marked as leg 4, leg 5 and leg 6. Three electricallyactuated joints are assembled on each leg of the robot, whichare the horizontal joint (H-joint), the vertical joint (V-joint),and the swing joint (S-joint). Various sensors, such as aninertial measurement unit (IMU), 3D contact-force sensors,joint-displacement sensors, are equipped on this robot. TheIMU is used to obtain the robot body attitude values duringwalking (the attitude angles, attitude angular velocities andattitude angular accelerations). The 3D contact-force sensorequipped on each foot of the robot is used to detect the contactforces of each leg. The joint-displacement sensors are used todetect the real positions of the joints to realize close loop con-trol at joint level, andkinematics computation (the kinematicsmodel of the robot employed in this paper can be found in theauthors’ previous work (Chen et al. 2019)).With themultiplesensors fitted, it is convenient to do biomimetic locomotionanalyses and realize different kinds of robot motion controlschemes.

For such a large-size legged robot walking on unpre-dictable terrains, the motion stability is of crucial importanceand needs to be primarily considered when designing themotion control system. Compared with the existing small-size legged robots, the great mass and inertia of the large-sizehexapod robot, and the unpredictable terrain conditions,together will cause two main extraordinary problems thatare rarely seen on the small-size robots. The two problemswhich will significantly influence the motion performancesof the robot are briefly introduced below.

1. The deformations of the robot: the mass of a small-size legged robot is usually small and therefore it is usually

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Leg 1Leg 2

Leg 3

Leg 4Leg 5

Leg 6

IMU

V-joint

H-joint S-joint

3D force sensor

Fig. 1 The whole structure of the large-size hexapod robot with eigh-teen electrically actuated joints

Fig. 2 The deterioration ofmotion caused by the structure deformationsand the large contact impacts

treated as a pure rigid structure, namely the deformations ofthe robot are usually neglected. But for our large-size hexa-pod robot with a great mass, the rigid component assumptionis no longer applicable and the deformations of the robot willgreatly influence the motion altitude and attitude, especiallywhen a commonly used triangle gait form with only threelegs supporting the robot body is employed.

2. The great contact impacts between the robot’s feet andthe unpredictable terrains: the terrains which legged robotsmove on are usually uneven and unpredictable. Due to theunpredictable terrain conditions, and the robot’s large massproperty, great contact impacts between the robot’s feet andthe terrains will occur. The contact impacts will significantlybreak up the system force equilibriumwhich plays an impor-tant roll in ensuring the balance walking of the robot, androbot body posture deviations will be seen.

Without proper compensation and control applied, thesetwo problems together downgrade the walking performance,as shown in Fig. 2, where the blue solid line is the desiredbody posture, the red dash line represents the actual bodyposture while the green double dot dash line shows the defor-mation of robot leg.

M

Fig. 3 The abstracted compliant-leg model of the large-size hexapodrobot

2.2 The problem formulation and the controlscheme

As discussed in Sect. 2.1, to solve these two problems ofthe large-size hexapod robot, the deformations of the robotcannot be neglected, namely the robot cannot be treated asan ideal rigid structure. Furthermore, the foot forces must becontrolled in a compliant way to reduce the contact impactsand ensure the system force equilibrium. To achieve this goal,not only the motion of the legs but also the foot forces mustbe taken into account at the same time in the design of thecontrol system. Therefore, a proper model which can bridgethe gap between the motion of the legs and the foot forces isvery essential to be established.

To establish a propermodel which can be used for the con-trol process of the large-size hexapod robot, the abstractedspring-muscle model of the EPH may be a good example. Inthe spring-muscle model, the muscle force can be controlledand the desired dynamics characteristics can be obtained byonly appropriately modifying the EPs which are usually rep-resented by the equilibrium positions of the end effector, orthe stiffness coefficient. Inspired by the spring-musclemodel,in this paper, a simple compliant-leg model which can reflectthe deformations of the robot is established. Imitating thespring-muscle model of the EPH, the EP trajectory is definedas a sequence of the robot foot positions.

The deformations of the robot mainly consists of twoparts: the deformation of the robot body, and the deforma-tions of the legs. To get a simple model which can be used,all the deformations are treated to be the deformations of thelegs. To imitate the situation of deformation, the robot legis abstracted into a mass-spring-damper model, as shown inFig. 3. It is well to be reminded that only the vertical defor-mation of the leg is taken into account because the tangentialdeformation of the leg is really small and has little influenceon the moving performance of the robot.

Based on the mass-spring-damper model, the relationshipbetween the foot force and the deformation of the leg can be

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achieved, as shown in Eq. (1).

C Fiz = f lagi · C fi = f lagi · (Kiz · ΔC Piz + Ciz · ΔC Piz)(1)

where f lagi represents the status flag of leg i , if leg i isin stance phase then f lagi = 1, otherwise f lagi = 0.C Fiz represents the vertical foot force of leg i . C fi representsthe equivalent foot force of leg i based on the mass-spring-damper model. ΔC Piz represents the vertical foot positionerror of leg i caused by the leg deformation. Kiz and Ciz

represent the equivalent stiffness and damping coefficients,respectively.

Because of the average low moving speed, the motionprocess of the large-size hexapod robot can be assumed to bequasi-static. Thus, the damping component can be neglected,and Eq. (1) can be further reduced to Eq. (2).

C Fiz = f lagi · C fi = f lagi · Kiz · ΔC Piz (2)

The stiffness coefficient Kiz is usually easy to get throughthe method of calibration (the calibration method is intro-duced in “Appendix A”). So the key point to solve theproblems discussed in Sect. 2.1 is how to control the footforces in a compliant way to compensate the deformationsof the robot and keep the system force equilibrium based onthe compliant-leg model. To solve this question, an EP tra-jectory modification method via contact-force optimizationis developed in this paper, which will be introduced in detailin Sect. 3.

In this method, by establishing the relationship betweenthe desired motion state and the contact foot forces, ΔC Pizunder the desired motion state can be obtained. Furthermore,based on the compliant-leg model, desired contact forces canbe calculated. Thenwith the implementation of an impedancemodel with force tracking, the final EP trajectory modifica-tion value to ensure system force equilibrium, and counterthe robot’s deformations, can be obtained through a com-pliant way. The brief structure of the whole control schemebased on the EP trajectory modification method is shown inFig. 4. Meanings of the parameters in Fig. 4 are explained inTable 1.

3 The EP trajectory modification viacontact-force optimization

For a hexapod robot, the support foot, or namely the foot instance phase, is the only medium in contact with the externalenvironment and drive the robot to move. Thus, interactivecontact forces are of crucial importance for the achievementof the system equilibrium. Nevertheless, the contact forcesare always affected by the unknown terrains. The unpre-

CiFP

iJiJ

iJCi finalPC

iP UJ

CiF

CiF

C CCiPC

ta

CizP

Fig. 4 The brief structure of the control scheme based on the EP mod-ification method in Cartesian space

Table 1 Important parameters of the EP control scheme

Parameter Quantity

C Pi The original foot position vector of leg iCat The desired motion acceleration of the robotCω The actual angular velocity of the robotCα The actual angular acceleration of the robotC Fi The desired foot force vector of leg iC F ′

i The actual foot force vector of leg i

ΔC PiF The force equilibrium EP trajectory

modification value

ΔC Piz The vertical deformation of leg i under

system forces

Ji The desired joint position vector of leg i

J ′i The actual joint position vector of leg i

ΔJi The joint position deviation

UJ The control quantity of the joint motorC Pi_ f inal The final desired EP trajectory

dicted interactive dynamic relationship between a robot’sfeet and environment, and the complexity of the terrame-chanics, together make it hard to control the contact forcesdirectly. A bad contact-force distribution may lead a seriesof consequences, such as foot sliding, unbalanced momen-tum on the torso, or even the lost of stability. Aiming tosolve this problem and get an equilibrium system locomo-tion, a contact-force optimization method is set up to modifythe stance-phase EP trajectories in real time. Through themodificationmethod, desired contact forces can be indirectlyachieved.

To get an equilibrium motion, the mechanical state of therobot during moving should be analyzed. Figure 5 shows ageneral motion state of the hexapod robot, in which C rep-

resents the robot body coordinate. CV = [CVx ,

CVy,CVz

]T

and Cat = [Catx , Caty, Catz

]Trepresent themotion velocity

and acceleration of C . Cω = [Cωx ,

Cωy,Cωz

]Tand Cα =

[Cαx ,

Cαy,Cαz

]Trepresent the angular velocity and angular

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CG

VxC

atxCVz

C

atzC

atyCVyC

xC

xC

zC

zC

yC

yC

FixFiz

Fiy

C

Fig. 5 Ageneralmotion state and themotion parameters of the hexapodrobot

acceleration of C , respectively. CG = [CGx ,

CGy,CGz

]T

represents the position of center of mass (COM) in the robotbody coordinate frame, namely C coordinate frame. Thus,the acceleration of COM can be written as

Ca = Cat + Cω × (Cω × CG) + Cα × CG (3)

where Ca represents the acceleration of COM. Then the totalforces and moments acting on COM can be written as

W ={

C FW = −mCa − m[0 0 g

]TCMW = −I Cα − Cω × I Cω

(4)

where I represents the inertia tensor at COM. W representsthe set of the desired virtual force vectors C FW and momentvectors CMW acting on COM.

Treating the foot contacting the terrain as a point andneglecting the interactive moments acting on the foot, themotion equilibrium equation of the hexapod robot can beobtained as shown in Eq. (5),

AF + W = 0 (5)

with

A =[E . . . EC1 . . . C6

](6)

F = [C F1T . . . C F6T

]T(7)

C Fi =[C Fix ,

C Fiy,C Fiz

]T(8)

W = [C FW T CMW

T]T

(9)

Ci =⎡

⎣0 CGz − C Piz C Piy − CGy

C Piz − CGz 0 CGx − C PixCGy − C Piy C Pix − CGx 0

⎤

⎦ (10)

where E represents the 3 × 3 identity matrix. F representsthe set of the desired contact force vectors which is mappedfrom the foot tip to the COM. C Fi represents the desiredcontact force vector of leg i , which is mapped from the foottip to the COM in the three-dimensional Cartesian space.

Equation (5) significantly bridges the gap between the sys-tem’s desiredmotion state and the robot contact forces,wherethe desired forces to ensure system force equilibrium mustmeet the demand of Eq. (5). Nevertheless, the order of matrixA is 6 × 18, and for a hexapod robot, the number of feet inthe stance phase must not be less than three. Therefore, theachievement of a unique solution of the desired foot forceswhich can meet the demand of Eq. (5) has become a redun-dancy problem. To solve this problem, additional constraintsare needed.

For controllers based on inverse dynamics, solving thisproblem is usually transformed into solving a quadratic pro-gramming (QP) problem. A nonlinear optimization goalbased on the inverse dynamics model of the robot, likeminimization of the energy consumption, is usually estab-lished. Then equality constraint like Eq. (5), and inequalityconstraints are additional defined to make the redundancyproblem match the standard form of a QP problem. By solv-ing the QP problem, a unique solution of the desired contactforces which can meet the demand of system force equilib-rium can be obtained.

Although the inverse dynamics controllers can solve thisredundancy problem, as discussed in the introduction sec-tion, the inverse dynamics modeling is usually complex andfull of nonlinear computations. When the robot’s configura-tion is complex, this problem becomes worse. In the authors’previous work (Chen et al. 2019), a linear foot force distribu-tion algorithm is attempted to solve the redundancy problem.But in that algorithm, an obvious defect can be found throughmathematical analysis. The foot force distribution algorithmcannot ensure system force equilibrium for all the supportforms of the hexapod robot. For example, when support formC showed in Fig. 4 in Ref. (Chen et al. 2019) is employedas the current gait form, none of the constraints for the threesituations showed in Fig. 7 in Ref. (Chen et al. 2019) can beused. Similar problems can be found in other support forms.The system force equilibrium cannot be ensured mathemat-ically for all the support forms.

To reduce the complexity of computation, and get a uniquesolution of the desired contact forces which can ensure sys-

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tem force equilibrium for all the support forms of the hexapodrobot, in this paper, constraints based on deformation com-patibility equation, foot-sliding resistance and internal-forceelimination are proposed to solve the redundancy problem.

The deformation compatibility equation is set up basedon the compliant-leg model, namely Eq. (2). Assuming thatunder the function of forces andmoments acting on the body,the robot body makes tiny movements, and there is no posi-tion change of the robot foot with respect to the ground, thenthe position change of the robot foot with respect to the robotbody can be written in the form shown in Eq. (11).

ΔC Pi = RWC Pi − C Pi + ΔQ (11)

with

RW ≈⎡

⎣1 ΔβΔγ − Δα Δβ + ΔαΔγ

Δα ΔαΔβΔγ + 1 ΔαΔβ − Δγ

−Δβ Δγ 1

⎤

⎦ (12)

where ΔC Pi =[ΔC Pix ,ΔC Piy,ΔC Piz

]Trepresents the

position change of the robot foot i . Δα, Δβ and Δγ rep-resent tiny yaw, pitch and roll angular changes of the robotbody, respectively. ΔQ = [

ΔQx ,ΔQy,ΔQz]T represents

the three tiny displacements of the robot body.Thus, based onEq. (2), (11) and (12),ΔC Piz and the defor-

mation compatibility equation can be achieved, as shown inEqs. (13) and (14), respectively.

ΔC Piz = −ΔβC Pix + Δγ C Piy + ΔQz (13)C Fiz = −( f lagi Kiz

C Pix )Δβ + ( f lagi KizC Piy)Δγ

+( f lagi Kiz)ΔQz (14)

Walking straight forward is the most important movingstate of the hexapod robot. It is mainly driven by the tangen-tial contact forces along the robot moving direction, namelyC Fix . To keep a stable walking of the robot, foot-slidingresistance is taken into account for the calculation of C Fix .To reduce the risk of foot sliding, the maximum ratio of thetangential force to the normal force should be as small aspossible. According to Gardner et al. (1990), this goal canbe reached by making the ratio of every support leg equalsto the ratio of the total tangential forces to the total normalforces. Therefore, C Fix can be calculated using Eq. (15).

C Fix = μC Fiz (15)

with

μ = C FWx/C FWz (16)

where C FWx represents the sumof all C Fix . C FWz representsthe sum of all C Fiz .

To calculate the other component of the tangential force,namely C Fiy , internal-force elimination is taken into accountto improve the durability of the robot’s mechanical com-ponents. Just like the normal contact force, to distinguishwhether the foot is in stance phase or not, C Fiy is defined as

C Fiy = f lagi · C fiy (17)

where C fiy represents the alternative contact force of leg ialong the y direction.

The two legs assembled in the middle of the robot body,namely leg 2 and leg 5, have little effect on the yaw anglechange of the body, the yaw angle change is mainly drivenby the other four legs. So in order to reduce the internal forcebetween legs, a constraint is set up, as shown in Eq. (18).

{C f1y = C f4yC f3y = C f6y

(18)

The constraint for the middle two contact forces is set upas shown in Eq. (19).

⎧⎪⎨

⎪⎩

C F2y = f lag2f lag1 + f lag2 + f lag3

C FWy

C F5y = f lag5f lag4 + f lag5 + f lag6

C FWy

(19)

Thus, based on Eq. (5) to Eq. (19), a system of linearequations can be achieved, as shown in Eq. (20).

B · x

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

f lag1 f lag3 f lag4 f lag6 0 0 00 0 0 0 b25 b26 b27b31 b32 b33 b34 b35 b36 b370 0 0 0 b45 b46 b47b51 b52 b53 b54 b55 b56 b571 0 −1 0 0 0 00 1 0 −1 0 0 0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

C f1yC f3yC f4yC f6yΔβ

Δγ

ΔQz

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎣

−C FWy − C F2y − C F5y−C FWz

−CMWx +(C P2z−CGz)C F2y+(C P5z−CGz)

C F5y−CMWy

−CMWx −(C P2x −CGx )C F2y−(C P5x −CGx )

C F5y00

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎦

= b

(20)

where the unknown coefficients of matrix B are explained inTable 2.

It can be verified that for all gait types of a hexapod robotlisted in Ref. (Chen et al. 2019), Eq. (21) can be satisfied.Namely, a unique solution of Eq. (20) can be achieved.Whenthe solution x ofEq. (20) is obtained, desired contact forces ofall legs in stance phase can be finally calculated according to

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Table2

The

unknow

ncoefficientsof

matrixB

Parameter

Value

Parameter

Value

b 25

−6 ∑ i=1flag

iKizCP i

xb 2

6

6 ∑ i=1flag iKizCP i

y

b 27

6 ∑ i=1flag iKiz

b 31

−flag 1

(CP 1

z−

CG

z)

b 32

−flag 3

(CP 3

z−

CG

z)b 3

3−flag 4

(CP 4

z−

CG

z)

b 34

−flag 6

(CP 6

z−

CG

z)b 3

5−


x(C

P iy−

CG

y)

b 36


y(C

P iy−

CG

y)

b 37

6 ∑ i=1flag iKiz

(CP i

y−

CG

y)

b 45

−6 ∑ i=1flag iKizCP i

x[μ

(CP i

z−

CG

z)−

(CP i

x−

CG

x)]

b 46


y[μ

(CP i

z−

CG

z)−

(CP i

x−

CG

x)]

b 47

6 ∑ i=1flag iKiz

[μ(C

P iz−

CG

z)−

(CP i

x−

CG

x)]

b 51

flag 1

(CP 1

x−

CG

x)

b 52

flag 3

(CP 3

x−

CG

x)

b 53

flag 4

(CP 4

x−

CG

x)

b 54

flag 6

(CP 6

x−

CG

x)

b 55

6 ∑ i=1flag iK

(iz)

CP i

xμ

(CP i

y−

CG

y)

b 56

−6 ∑ i=1flag iKizCP i

yμ

(CP i

y−

CG

y)

b 57

−6 ∑ i=1flag iKiz

μ(C

P iy−

CG

y)

123


Eqs. (14), (15), (17) and (19). The system force equilibriumcan be ensured for all gait types during walking.

rank[B] = rank[B, b] = 7 (21)

Although the desired contact forces for the system equi-librium can be obtained through the method proposed above,it is still hard to get a good contact force tracking perfor-mance through a direct way. Direct force close-loop controlmethod may cause the system unstable due to the sensitiv-ity and noises of the foot force sensors equipped. To solvethis problem, a position-based impedance controlmodelwithforce tracking is used, as shown in Eq. (22).

ΔFi = MdΔC Pi F + CdΔ

C Pi F + KdΔC PiF (22)

where ΔFi represents foot force deviation of leg i betweenthe actual foot force C F ′

i and the desired foot force C Fi .ΔC PiF represents the EP trajectory modification value,namely foot positionmodification value.Md ,Cd and Kd rep-resent the user-defined impedance coefficients, respectively.

The EP trajectory modification method via contact-forceoptimization proposed here is more like biological behav-iors of the legged creatures or humans. Legged creatures orhumans can optimize contact forces according to the motionstate wanted. This optimization is finally achieved by chang-ing foot motion trajectories. For example, if a human stepson a soft terrain and a large contact force is needed to keepthe body balance, deeper foot positionwill be taken. Throughthis control method, not only the system equilibrium but alsoa compliance interaction between the robot foot and terraincan be obtained.

To make the whole control logic clearly in detail, Fig. 4 isextended intoFig. 6,where the color circled numbers, namelythe green circled 1 and the red circled 2, represent the signaltunnels that pass the same signal. Eq.(n) in Fig. 6 representsequation (n).

4 Experiments and result discussions

Toverify the effectiveness of the control scheme in improvingthe walking performance and ensuring the walking stability,several experiments were carried out on the real large-sizehexapod robot. First, a common rigid flat terrain walkingexperiment was carried out to demonstrate the effectivenessof the control scheme in compensating the robot’s deforma-tions. Second, an artificial soft terrain walking experimentwas carried out to demonstrate the advantages of the con-trol scheme in terrain deformation counteraction and keepingthe stable walking attitude. Third, a natural field walkingexperiment was carried out to verify the practical engineer-ing application performance of the control scheme.

Cta

CiFP

iJiJ

iJCi finalPC

iP UJ

CiF C

iF

C C

CiP

CizP

Ca

CWF

CWM

zQ

C5yFC

2yF

CizP

CixF

CizF

Cyf

CyfC

yf Cyf

C1yF C

3yF C4yF C

6yF

iF

Fig. 6 The detailed EP control scheme

The hexapod robot is equipped with an industrial PC con-troller from Beckhoff with an i5 CPU and a real-time controlsystem inside. Fast data exchanges can be ensured by usingthe Ethernet. The motion control program is written in thecombination of the C++ language and PLC language onTwinCAT 3 platform.

It is well to be noticed that only triangle gait form wasemployed in the three experiments (about the stable hexapodgait forms, see the authors’ previous work (Zha et al. 2019)).The triangle gait form is most commonly used for hexapodrobots. A hexapod robotwalkingwith a triangle gait form canbemore sensitive to the deformations of the robot and the ter-rain changes because only three legs are supporting the robotbody at the same time during walking. Besides, although thewalking stability may be increased with the increase in thenumber of support legs, this will also cause the problem ofwalking decay (Irawan and Nonami 2011). Therefore, thetriangle gait form is more proper to be employed to exam thewalking performance.

To demonstrate the walking improvement of the robotusing the EP control scheme better, during every experiment,walking only with the kinematics control scheme (WKC),and walking with a typical inverse dynamics control scheme(WID) were employed as comparisons with walking with theEP control scheme (WEP). In the former two experiments, nospecial contact detection was employed in the three controlschemes. In the third experiment, to ensure a safe walking, asimple contact detection was employed which will be briefly

123


iJ

iJ

iJC

iP UJ

(a)

iJ iJCi finalPC

iP UJ

CiP

Cta

CiFP

iJ

CiF

CiF

C Cut

iJ

ut

iJ

iJ

(b)

Fig. 7 The brief structures of the kinematics control scheme and theinverse dynamics control scheme: a the kinematics control scheme, andb the inverse dynamics control scheme

introduced in Sect. 4.3. The control schemes of WKC andWID are shown in Fig. 7.

It is well to be noticed that for the WID control schemein Fig. 7b, only the force equilibrium control part of WEPin Fig. 4 is replaced by a force distribution QP solver basedon inverse dynamics. Through this way, the influence on thewalking performance caused by the parameter changes willbe reduced. The force distribution QP solver was employedfrom reference (Wang et al. 2020). This is a typical inversedynamics solver. Similar solvers can be found in other works,such as reference (Roy et al. 2010) and reference (Mahapatraet al. 2015). Ji_ori, Ji_ori, and Ji_ori in Fig. 7b represent theoriginal joint position, joint velocity, and joint accelerationvector of leg i , respectively. The inverse dynamics modelingprocess of the large-size hexapod robot can be seen in theauthors’ previous work (Wang et al. 2018).

Some global parameters used in the experiments areshown in Table 3.

4.1 The rigid flat terrain walking experiment

The walking parameters used in this experiment is shown inTable 4. In Table 4, the duty factor represents the ratio ofthe support time to the cycle time. The cycle time representsthe time period for a leg to complete a swing and a supportphase. The motion velocity of the robot can be calculatedusing Eq. (23).

Vx = Sxλ · T (23)

The walking process is shown in Fig. 8. The terrain is themost commonly seen rigid cement concrete pavement. Dur-ing walking, the deformation of the terrain can be neglected.The triangular support regions bounded by the yellow linesshow the changes of the support legs during walking. Thecomparative experiment result of the robot body attitudechanges is shown in Fig. 9.

During the experiment, the desired pitch and roll angleswere set the same to 0 degree while the desired body heightwas set to 1380 mm. Based on the previous experience inlaboratory commissioning tests of the robot, a stable motionrange for the robot’s pitch and roll angles is set to ±1.5degrees to evaluate themotion performances, just as the rangebetween the twoblack double dot-dash lines shown inFig. 9b.

It can be analyzed from Fig. 9 that although the robotcan move within the stable motion range, the pitch angle,roll angle, and body height of WKC fluctuate significantly.The fluctuations seem at a low-value level, but for a large-size robot, macroscopic posture changes during the robotwalking can be seen. Because the deformation of the rigidterrain can be nearly neglected, therefore without any reg-ulation of the EP trajectories, the body attitude fluctuationsare mainly caused by the deformations of the robot body andlegs. This phenomenon matches the deformation assumptionof the large-size robot and can provide a realistic basis forthe abstracted compliant-leg model, namely Eq. (2).

Compared with WKC, the walking performances weresignificantly improved by WID and WEP. The max abso-lute tracking error of the pitch angle was reduced from 0.86degree (WKC) to 0.41degree (WID), and0.37degree (WEP),respectively. Themax absolute tracking error of the roll anglewas reduced from1.08 degrees (WKC) to 0.58 degree (WID),

Table 3 Global parameters for the experiments

Parameter Quantity Value

Kiz = [K1z, K2z, K3z, K4z, K5z, K6z

]Vertical stiffness coefficient of the leg [146, 325, 177, 149, 327, 175] kN m−1

Kd Stiffness coefficient of the impedance model 120 kNm−1

Cd Damping coefficient of the impedance model 20 kNm−1 s

Md Inertia coefficient of the impedance model 3000 kg

123


Table 4 Walking parameters


Sx Step length 400 mm

T Cycle time 8 s

λ Duty factor 0.5

h Step height 120 mm

and 0.49 degree (WEP), respectively. The standard deviationof the pitch angle was reduced from 0.28 degree (WKC) to0.12 degree (WID), and 0.08 degree (WEP), respectively. Thestandard deviation of the roll angle was reduced from 0.37degree (WKC) to 0.18 degree (WID), and 0.13 degree(WEP),respectively. The max absolute tracking error of the bodyheight was reduced from 41.13 mm (WKC) to 38.53 mm(WID), and 14.92 mm (WEP), respectively. The standarddeviation of the body height was reduced from 9.85 mm(WKC) to 4.81 mm (WID), and 4.01 mm (WEP), respec-tively. (The body height measuring method is introduced inAppendix B).

The disparate control performance mainly results fromthe different system states brought by the different controlschemes. In contrast to the common system of WKC, therobot system equilibrium can be more easily obtained usingthe WID and WEP control schemes. Taking the robot rollangle variations for instance, the roll angle fluctuation stateis mainly determined by the foot force relationship of the leftand right supporting legs. Therefore, Fz5, namely verticalfoot force of leg 5, is set in comparison with Fz1 + Fz3,namely the sum of the vertical contact forces of leg 1 and leg3, as shown in Fig. 10.

Figure 10 shows the vertical foot force comparisons ofWKC, WID, and WEP during the same three stance phases,respectively. During a steady stance phase, if a stable rollangle is desired to be achieved, Fz1 + Fz3 ≈ Fz5 should be

(a)

(b)

(c)

Fig. 9 The robot’s body attitude comparisons as a result of the rigidflat terrain walking experiment: a the body pitch angle comparison, bthe body roll angle comparison and c the body height comparison

Fig. 8 The snapshots of therigid flat terrain walking process

123


(a)

(b)

(c)

Fig. 10 The vertical foot force comparisons: a the vertical foot forcecomparison of WKC, b the vertical foot force comparison of WID, andc the vertical foot force comparison of WEP

satisfied. But as shown in Fig. 10a, when the common controlscheme, namely WKC is employed, the foot force deviationbetween Fz1 + Fz3 and Fz5 is very obviously seen. The footforce deviation is mainly caused by the deformations of therobot body and legs. Leg 5 is the only support leg on oneside of the robot body. Compared with leg 1 and leg 3, thelarge payload will easily cause more deformation of leg 5.The force equilibrium of the robot can hardly be achieved ifno regulation of the system is introduced.

In contrast to the force relationship of WKC, force equi-librium of the robot system can be nearly satisfied under thefunction ofWID andWEP, as shown in Fig. 10b, c. The forceratio of Fz1 + Fz3 to Fz5 can almost maintain unchanged andthe sum of the three legs’ vertical contact forces is almoststeady at 25 kN during the stance phases. In other words,the sum of the supporting legs’ vertical contact forces almostequals to the gravity of the robot. During any common stance

phase of each walking cycle, Fz1 + Fz3 ≈ Fz5 can be satis-fied, which means the robot is force balanced in the coronalplane. Because of the balanced contact forces during thestance phase, the amplitude fluctuation of the robot’s rollangle is small. System equilibrium can be ensured.

Although both the WID and WEP system can ensure sys-tem force equilibrium during the flat terrain walking, there isstill a significant gap between them in regulating body height.As an instance, Fig. 11 shows the vertical foot position com-parison of leg 5 during two support phases under the threecontrol schemes to demonstrate the advantage of the WEPsystem in robot body height regulation. (The actual verti-cal foot position computation is introduced in “AppendixB”, Eq. (27).) Because no contact detection algorithm isemployed in the three control schemes, obvious foot posi-tion changes caused by the foot force fluctuations can beseen in Fig. 11 when leg status switches. These foot positionchanges will cause the robot body height changes, which canbe seen from Fig. 9c. During the flat terrain walking, thedesired body height is 1380 mm. Ideally, if the desired robotbody height is wanted to be ensured, the vertical foot positionof all the support legs with respect to the body coordinate Cshould be -1380 mm. But it can be seen from Fig. 11 thatboth the WKC and WID system cannot meet this demand.Take a steady period of the first support phase for instance toanalyze the foot position improvement quantitativly, namelythe grey area from t1 = 9.2 s to t2 = 12.2 s in Fig. 11. Theaverage vertical foot positions of leg 5 under WKC, WID,and WEP systems during this period are −1339.10mm,−1346.91mm, and −1380.42mm, respectively. Comparedwith WKC and WID, due to the deformation compensation,the vertical foot position of WEP can almost track the idealvalue with little deviation. This is why the robot body heightof WEP can be regulated much better than the other twocontrol schemes.

This different control performance is mainly caused bythe deformations of the robot. Although the foot position ofleg 5 of WID can be regulated through the regulation of footforces, it can still not compensate the robot deformationsto ensure the desired body height. As a comparison, underthe function of the EP control scheme, the leg deformation,namely ΔC Piz , can be computed in advance, and compen-sated through a feed-forward way. Therefore, the robot bodyheight can be ensured. It can be inferred that with the growthof the robot’smass and payload, the deformations of the robotwill increase.When facing this situation, the advantage of theEP control scheme proposed will be more significant.

4.2 The artificial soft terrain walking experiment

The results of the rigid flat terrain walking experimentverified the effectiveness of the EP control scheme in coun-teracting the large-size hexapod robot’s deformations. But

123


Fig. 11 The vertical foot position comparison of leg 5 under the threecontrol schemes




T Cycle time 10 s

λ Duty factor 0.5


in fact, the terrains on which the legged robots always walkcannot be as flat or rigid as desired, and they are usually softwith obvious deformations andunevenwith topographicfluc-tuations. Unknown disturbances caused by the deformationand the roughness of the terrain will lead to a more unstablemotion performance. To verify the effectiveness of the EPcontrol scheme in improving the robot’s walking stability onthe uneven soft terrain, an artificial soft terrainwalking exper-iment was carried out. The walking parameters are shown inTable 5.

The artificial soft terrain which was constructed of EPEplates and plywood plates is shown in Fig. 12. Each pieceof the soft EPE plates is 1m × 1m × 20mm while each ofthe hard plywood plates is 0.7m × 0.5m × 22mm (Fig. 12shows some acceptable manufacturing errors in thickness).The basement of this terrain was constructed with two lay-ers of the blue EPE plates shown in Fig. 12. Several yellowEPE plates were randomly placed on and besides the base-ment. The highest part of the terrainwas constructedwith fivelayers of the EPE plates. Because the EPE plate is soft andcan produce noticeable deformationwhen being compressed,therefore it is very suitable to be employed to simulate thedeformation of the terrain. The plywood plates are placedto simulate the hard parts of the terrain to make the experi-ment more natural. The same comparative control schemes,namely WKC and WID showed in Fig. 7, were employed to

Fig. 12 The artificial soft terrain constructedofEPEplates andplywoodplates

make a comparison with the EP control scheme proposed.The walking process is shown in Fig. 13.

During walking, the soft terrain was easily compressedwith obvious deformation, as shown in Fig. 14. Comparedwith the rigid terrain walking, the deformations of the ter-rain and the robot itself will together lead to a more unstablemotion if no compensation is employed into the control sys-tem. This bad situation can be seen through the attitudechanges of the robot, as shown in Fig. 15.

It can be seen fromFig. 15 that comparedwith the rigid flatterrain walking experiment, the robot’s pitch and roll anglesfluctuate even more sever during walking with WKC. Dur-ing some periods of walking, the pitch and roll angles moveout of the stable range. This phenomenon matches the guessthat the deformations of the terrain and the robot itself willseriously influence the walking stability. As a comparisonwith the walking performance of WKC, the robot achievedmore stable walking performances underWID andWEP sys-tems. The fluctuations of the robot’s pitch and roll anglesreduced significantly. The max absolute tracking error ofthe pitch angle was reduced from 1.83 degrees (WKC) to0.75 degree (WID), and 0.65 degree (WEP), respectively.The standard deviation of the pitch angle was reduced from0.79 degree (WKC) to 0.32 degree (WID), and 0.15 degree(WEP), respectively. The max absolute tracking error of theroll angle was reduced from 2.21 degrees (WKC) to 1.24degrees (WID), and 1.19 degrees (WEP), respectively. Thestandard deviation of the roll angle was reduced from 0.86degree (WKC) to 0.36degree (WID), and0.22degree (WEP),respectively. The pitch and roll angles were controlled withinthe stable range during the motion processes of WID andWEP.

Similar to the rigid flat terrainwalking experiment, Fig. 16shows the vertical foot force comparisons of WKC, WID,and WEP during the same five stance phases as an instanceto explain the roll angle improvement. It can be seen from

123


Fig. 13 The snapshots ofwalking on the artificial softterrain surfaces

Fig. 14 The deformation of the artificial soft terrain

(a)

(b)

Fig. 15 The robot’s body attitude comparisons as a result of the artificialsoft terrain walking experiment: a the body pitch angle comparison andb the body roll angle comparison.

Fig. 16a that compared with the rigid flat terrain walking,the foot forces of the support legs fluctuated more severduring walking on the uneven artificial soft terrain withWKC system. The foot force balance was significantly bro-

ken. This is why big roll angle fluctuation can be seen inFig. 15b. When WKC was replaced by WID and WEP,obvious improvements of the vertical foot force relation-ship can be seen from Fig. 16b, c. Although compared withthe rigid flat terrain walking, Fz1 + Fz3 ≈ Fz5 cannot beideally ensured under both WID and WEP systems due tothe deformation and uneven characteristic of the artificialsoft terrain, the fluctuations of the vertical foot forces weresignificantly reduced. Compared with WKC, the deviationsbetween Fz1 + Fz3 and Fz5 were obviously reduced whenWID and WEP were employed. Therefore, the motion per-formances were improved.

It can be seen from the experiment results shown inFigs. 15 and 16 that comparedwithWID, thewalking perfor-mance ofWEP is improved, even though the improvement isnot very big. But in the EP control scheme proposed, no non-linear inverse dynamics model was employed. As discussedin the introduction part, the inverse dynamics modeling pro-cess is usually quite complex. Besides, compared with thecomplex QP solver commonly employed by the traditionalinverse dynamics controllers, the computation process of thefoot forces ensuring system force equilibrium in the EP con-trol scheme proposed is quite simple. The whole controlscheme is totally linear. Therefore, achieveing the same sta-ble control performance, the EP control scheme proposed ismore simple to be employed.

4.3 The natural field walking experiment

From the theoretical analysis and the experiment result anal-yses above, the advantages of the EP control scheme inwalking posture regulation and simple computation havebeen demonstrated. To finally test the applicability of theEP control scheme in ensuring stable motion of the robot ina real natural environment, a natural field walking experi-ment was carried out. In this experiment, only the EP controlscheme was employed. The natural field is shown in Fig. 17.The natural field is constructed with several randomly placedholes dug on an irregular slope topography. The soil of thefield is soft and can lead to obvious foot sinkages of the robotduring walking. The depth of the holes is 200 mm. Due to

123


(a)

(b)

(c)

Fig. 16 The vertical foot force comparisons: a the vertical foot forcecomparison of WKC, b the vertical foot force comparison of WID, andc the vertical foot force comparison of WEP

the depth of the holes, to ensure a safe walking process, asimple contact detection was employed. A rest time periodof 3 seconds is set between each triangle step. During the resttime, the robot will maintain steady with six legs supportingthe body. Due to the function of the EP control scheme, thedesired foot forces of the legs will be computed. If one swingleg steps into the hole at the leg status switching time, thedesired foot force won’t be reached. Then under the functionof the impedance model in the EP control scheme, a deeperstep will be taken until the desired foot force is reached. Dur-ing the reaching time, the other legs will keep their positionsunchanged. Once the desired foot force is reached, the robotwill take the next triangle step to move forward. The walkingparameters during this experiment are shown in Table 6.

It is well to be noticed that in this experiment, the slopeangle of the field is natural and uneven. During walking,

Fig. 17 The real field test irregularities with random holes




T Cycle time 10 s

λ Duty factor 0.5


the body of the robot was always parallel to the slope. So thedesiredpitch angle in this experimentwas as sameas the slopeangle. The slope angle was obtained using a macro terrainrecognition method in the authors’ previous work (Zha et al.2019). The walking process of the robot is shown in Fig. 18.

As shown in Fig. 18, the body of the robot is leaning dur-ing the walking process because of the time-varying slopeangle. The actual pitch and roll angles of the robot during thenatural field walking is shown in Fig. 19. The range betweenthe black double dot-dash lines is the stable motion range. Itcan be seen that during the walking process, the pitch and rollangles are almost within the stable motion ranges. The pitchangle can almost track the desired value, namely, the slopeangle detected. The max absolute tracking errors of the pitchand roll angles are 1.57 degrees and 1.95 degrees, respec-tively. The tracking errors beyond the stable motion rangeare caused by the sudden foot slippage due to the unstablelanding angle of the foot in the hole, and the foot sinkagedue to the soft soil in the hole. The foot landing angle of therobot employed in this article cannot be controlled becausethe ankle joint is a passive joint. Furthermore, in a real natu-ral environment, the detailed terrain roughness informationis hard to be obtained unless specific sensors are equipped onthe robot. Excluding the uncontrolled factors, the robot canachieve a stable motion under the function of the EP controlscheme proposed.

Figure 20 shows the vertical foot positions of leg 1, leg 3and leg 5with respect to the robot body coordinateC . Duringone stance phase of this experiment, leg 1, leg 3 and leg 5 sup-ported the robot body at the same time and drove the robot to

123


Fig. 18 The snapshots of thenatural field walking process

(a)

(b)

Fig. 19 The robot’s body attitude changes as a result of the natural fieldwalking experiment: a the body pitch angle change and b the body rollangle change

move forward. Because of the EP control scheme proposed,the vertical foot positions of the three legs are not the sameand keep changing during the walking process. Obvious foottrajectory modifications, namely the EP trajectory modifica-tions can be seen in Fig. 20. The EP trajectory modificationsenable the robot to adapt to the terrain changes, and thereforethe walking stability can be ensured.

Figure 21 shows the vertical foot force comparison of Fz5and Fz1 + Fz3 during three stance phases. It can be seen thatdue to the uneven terrain conditions, Fz1+Fz3 ≈ Fz5 is hardto be totally ensured. The deviation between Fz1 + Fz3 andFz5 leads to the fluctuation of the body’s roll angle. But thefoot force deviation is acceptable because the roll angle isalmost controlled within the stable range.

Fig. 20 The vertical foot positions of leg 1, leg 3 and leg 5 with respectto the robot body coordinate C

Fig. 21 The vertical foot force comparison of WEP

5 Conclusion and future works

In this paper, the importance of the motion stability and thecontrol problems of a large-size hexapod robot have beendemonstrated. Themain theoretical contribution of this paper

123


is the proposal of a simple control scheme inspired by thebiological EPH. The control scheme is mainly based onan EP trajectory modification method which is designed toachieve the system equilibrium of the robot, and deformationcounteraction. In this control scheme, no complex dynam-ics computation is employed. A compliant-leg model whichis similar to the muscle-spring model in the EPH has beenused and based on this abstracted model, the deformationswhich will influence the walking stability of the large-sizehexapod robot have been significantly considered. Differentexperiments based on the real large-size hexapod robot havebeen carried out. The effectiveness of the control scheme inensuring the walking stability of the robot has been verifiedthrough the analyses of the experiment results.

Different from the existing inverse dynamics controllersdeveloped for commonly seen small-size robots, the EP con-trol scheme proposed in this paper shows two significantimprovements. The first one is deformation counteractionwhich is of crucial importance in ensuring the body alti-tude of the large-size hexapod robot. It can be inferred thatwith the increase in the mass and payload of a large-sizehexapod robot, the advantage of deformation counteractionwill become more significant. Furthermore, the deformationcounteraction characteristic may allow the robot designersto use low-cost materials to build their robots with largermechanical errors. The second improvement is the simplic-ity of the control scheme. The control scheme is totallylinear without any nonlinear computations. Regardless ofthe kinematics differences, the control scheme can be easilyemployed by other robot designers. Due to the good versatil-ity,more applications of the control scheme on other hexapodrobots can be expected.

The EP trajectory modification method proposed in thispaper is more like a feed-forward control method. Althoughthe posture fluctuations of the large-size hexapod robot can bereduced to ensure stable locomotion, complete posture track-ing can not be realized only using this method. Therefore, tofurther improve the posture tracking ability, and environmentadaptability of the large-size hexapod robot, more trajectorymodification methods based on the feedback informationsof the sensors will be introduced into the control scheme inthe future. Besides, more natural terrain moving experimentswill be carried out in the future.

Acknowledgements The authors disclosed receipt of the followingfinancial support for the research, authorship, and/or publicationof this article: authors gratefully thank the support of the NaturalScience Foundation of China (No. 61773139), the Shenzhen Sci-ence and Technology Program (Nos. KQTD2016 11215134654) andthe Shenzhen special fund for future industrial development (No.JCYJ20160425150757025).

Appendix A

To calibrate the stiffness coefficient of the robot leg, namelyKiz in Eq. (2), a simple method was used.

First, we initialized the robot posture with six legs sup-porting the robot body on a rigid flat terrain, and set a desiredbody altitude d. Then we employed a plumb line with oneend fixed on the robot body, and the other end perpendicularto the ground. By measuring the length of the plumb line,we obtained the actual body altitude d ′ under the gravity ofthe robot. At last, we measured the vertical foot force C Fiz ′of leg i by reading the feedback data from the 3D contact-force sensor equipped. Based on C Fiz ′ and the body altitudedeviation d − d ′, Kiz was obtained, as shown in Eq. (24).The schematic diagram of the calibration process is shownin Fig. 22.

Kiz =C Fiz ′

d − d ′ (24)

Appendix B

In the authors’ previous work (Zha et al. 2019), the macroterrain can be nearly abstracted as a support plane which isconstructed with all the support feet. The general equation ofthis plane can be expressed in Eq. (25), which has the sameform as the Eq. (1) in reference (Zha et al. 2019).

Ax + By + Dz + 1 = 0 (25)

The coefficients of Eq. (25), namely A, B, and D can becomputedusing the least squaremethod, as shown inEq. (26).

d d

izK

Fig. 22 The schematic diagram of the calibration process

123


⎡

⎣ABD

⎤

⎦ =⎡

⎣a11 a12 a13a21 a22 a23a31 a32 a33

⎤

⎦

−1 ⎡

⎣−∑ C Pix ′−∑ C Piy ′−∑ C Piz∗

⎤

⎦ (26)

with

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

a11 = ∑ C Pix ′2

a22 = ∑ C Piy ′2

a33 = ∑ C Piz∗2a12 = a21 = ∑ C Pix ′ · C Piy ′a13 = a31 = ∑ C Pix ′ · C Piz∗a23 = a32 = ∑ C Piy ′ · C Piz∗C Piz∗ = C Piz ′ + C Fiz ′/Kiz

(27)

where C Pix ′, C Piy ′, and C Piz ′ represent the computation-based foot positions of the support leg i along the x , y andz directions in the robot body coordinate C , respectively.C Piz∗ represents the actual foot position of the support leg ialong the z direction in the robot body coordinate C . C Fiz ′represents the actual vertical foot force of the support legi achieved from the feedback data of the 3D contact forcesensor. Kiz represents the vertical stiffness coefficient of thesupport leg i .

It is well to be noticed that during the computation ofEq. (26), the three foot positions along different directionsare not obtained through the same way. Just as discussed inSect. 2.2, the tangential deformation of the support leg i is toosmall and can be neglected. Therefore,C Pix ′ andC Piy ′ whichare obtained through the forward kinematics computationcan be used precisely to represent the actual foot positions(the kinematics modeling process of the large-size hexapodrobot employed in this paper can be found in the authors’previous work (Chen et al. 2019)). But for the vertical footposition, because of the obvious vertical deformation of leg i ,the forward kinematics computation along the z direction isnot accurate enough. In other words, C Piz ′ obtained directlythrough the forward kinematics computation cannot be useddirectly to represent the actual foot position.Therefore,C Piz∗with the vertical deformation considered is used to representthe actual vertical foot position.

Based on the A, B, and D computed from Eq. (26), theperpendicular distance from the origin of the body coordi-nate C to the support plane, namely the actual body heightd ′ during hexapod locomotion, can be obtained through thecomputation of Eq. (28).

d ′ = 1√A2 + B2 + D2

(28)

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Chen Chen received his M.Sc.degree in Automotive Engineer-ing from Harbin Institute of Tech-nology in 2015. He is now work-ing toward his Ph.D. degree inMechatronics Engineering in HarbinInstitute of Technology. His researchinterests include the autonomouscontrol, dynamic analysis and con-trol system designs of mobile robots,especially legged types.

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Fusheng Zha received the asso-ciate degree in mechatronics engi-neering from Wuhu Radio & TVUniversity in 1997. He receivedthe M.Sc. degree in mechanicaldesign and theory from LanzhouUniversity of Technology in 2005.He received the Ph.D. degree inmechatronics engineering fromHarbin Institute of Technology in2012. His research interests includeCPG control, quadruped robot andneural network.

Wei Guo received the B.Sc. andM.Sc. degrees in Control Scienceand Engineering from Harbin Insti-tute of Technology in 1985 and1994, respectively. She receivedthe Ph.D. degree in mechanicalengineering from Harbin Instituteof Technology in 2002. Her researchinterests include industrial robottechnology and legged robot con-trol.

Zhibin Li received his B.E. degreefrom Harbin Institute of Technol-ogy in 2007 and joint Ph.D. degreefrom Italian Institute of Technol-ogy and University of Genova in2012. He is now a Lecturer (Assis-tant Professor) in the School ofInformatics at the University ofEdinburgh. His research interestsinclude deep reinforcement learn-ing of dynamic locomotion of leggedrobots, and multi-contact motionplanning of multi-limbed robots.

Lining Sun received the B.Sc.degree in mechanical engineeringfrom Harbin Institute of Technol-ogy in 1985, where he receivedthe M.Sc. degree in mechanicalengineering in 1988 and the Ph.D.degree in mechatronics engineer-ing in 1993. His research inter-ests include industrial robot tech-nology, humanoid arm and robotcontrol.

Junyi Shi received his B.E. degreein Mechanical Design, Manufac-turing and Automation in LanzhouUniversity of Technology in 2018.He is currently working towardshis M.Sc. Degree in MechanicalEngineering in Harbin Institute ofTechnology. His research interestsinclude simultaneous localizationand mapping, place recognition,and legged robots.

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