Model predictive allocation control for leg-wheel mobile roboton loose soil considering wheel dynamics

Takatsugu Oda1 Hiroki Yoshikawa1 Naoki Shibata1 Kenichiro Nonaka1 Kazuma Sekiguchi1

1Mechanical systems engineering, Tokyo city university, Japan, {knonaka,ksekiguc}@tcu.ac.jp

Abstract

For the planetary exploration rover, to cope with the layerof heterogeneous superficial deposits called regolith isimportant so as to achieve designed traction. In orderto consider effects of wheel motion, model predictiveallocation control is proposed. To cope with complexterramechanics in MPC (model predictive control), theidentification technique is introduced; the proposed MPCis formed as a linear optimization problem. The rovermodel and terramechanics are described using model-ica and simulate to evaluate the performance of theproposed method. The suppressed of superfluous slipand enhancement of traction performance is numericallyshown.Keywords: Terramechanics, Allocation control, Modelpredictive control, Model identification.

1 Introduction

Developments of autonomous robots are widely studiedto apply for various environments in which human can-not work. Especially for planetary exploration, explo-ration rovers are generally adopted, because these robotscan move planetary surface efficiently (Weisbin et al.,1997). However, on many planets, planetary surface iscovered with a layer of heterogeneous superficial de-posits called regolith; wheels might slip and not gener-ate desired forces on these soil. In addition, since thedeformable loose soil on which wheels passed are com-pressed, the terrain property (e.g. soil density, resistanceforce etc.) would change (Senatore and Sandu, 2011).Then, for autonomous robots equipped with drive wheel,wheel control technique is important to suppress the slipand wheel sinkage.

Based on the terramechanics which deal with the inter-action of the wheel and soil (Taheri et al., 2015), manyvehicle control method is studied. In order to enhancetraction and energy performance, dual-criteria objectivefunction for control allocation is proposed in (Iagnemmaand Dubowsky, 2004). For the traction control of therover, PI controller with slip ratio estimator (Yoshida andHamano, 2002) and robust adaptive fuzzy control strat-

egy (Zhengcai and Yang, 2014) are developed. In addi-tion, MPC (model predictive control) which could con-sider the dynamics and constraint explicitly is appliedto planetary rovers (Krenn et al., 2013). However, thismethod adopts the linearization technique to considercomplex model and does not consider muli-pass effect.

In this paper, the model predictive allocation controlfor multi-wheel rovers considering the terramechanicsincluding the multi-pass effect is proposed. The wheeldynamics is explicitly considered using MPC, dynami-cally infeasible allocated solution which static optimiza-tion method might have is avoided. To consider complexterramechanics model in linear MPC, black box identi-fication technique is introduced; for identification, therover model is operated in only linear region of terrame-chanics. Effectiveness of the model predictive allocationcontrol is verified through numerical simulation usingmodelica; suppression of superfluous slip and enhance-ment of traction performance is shown. Note that to fo-cus on the longitudinal motion, we assume lateral motionof the rover could be sufficiently small.

2 Vehicle and wheel model

2.1 Leg-wheel mobile robot

In this paper, a leg-wheel mobile robot consists of a bodyand limbs, wheels, as depicted in Fig. 1 is considered.By using redundancy of limbs equipped with six joint,the desired wheel arrangement is achieved (Yoshikawaet al., 2017). The wheels could drive independently andgenerate tire force based on terramechanics which ex-press wheel-soil interaction mechanisms.

2.2 Wheel model based on terramechanics

We introduce the wheel model based on terramechan-ics (Senatore and Sandu, 2011) into the robot model inorder to simulate the force and moment characteristics ofthe tire on a deformable terrain (Yoshikawa et al., 2017).Roughly, in this model, generated force of each wheelsare calculated by integrating normal σ and shear τx stressdistribution model which described in Fig. 2.

____________________________________________________________________________________________________________240 Proceedings of the 2nd Japanese Modelica Conference

May 17-18, 2018, Tokyo, Japan DOI

10.3384/ecp18148240

z

x

y

Limb 1Limb 2

Limb 3

Limb 4 Limb 5 Limb 6

Figure 1. Leg-wheel mobile robot with six joints of each limb.

θ

r

fθ

rθ

h

hλ

)(θτx

mθ

xv

)(θσ

W

RT

Figure 2. Normal and shear stress distribution concept of ter-ramechanics while rolling.

The maximum stress angle is empirically estimated asfollowing a linear function of the slip ratio κ and the en-try angle θf:

θm = (a0 +a1κ)θf, (1)

where a0 and a1 are constant parameter. The slip ratiois explained as follows with the wheel radious r, angularvelocity ω and translational velocity vx:

κ =rω − vx

max(rω,vx), (2)

The empirical normal stress distribution based onReece’s formula are expressed as follows:

σ(θ) ={

σf(θ) (θm ≤ θ < θf),σr(θ) (θr < θ ≤ θm),

(3)

σf(θ) =(ck′c +ρbk′ϕ

)[ rb(cosθ − cosθf)

]n, (4)

σr(θ) =(ck′c +ρbk′ϕ

)[rb(cos{θf −

θ −θr

θm −θr(θf −θm)}− cosθf)

]n

, (5)

where the Bekker-Reece sinkage exponent n is a linearfunction of slip ratio; n = n0 + n1|κ|. b is wheel widthand c is cohesion stress of the soil, ρ is soil density, k′cis cohesion related soils parameter, k′ϕ is angle internalfriction related soil parameter, respectively.

The empirical shear stress distribution introduced byJanosi and Hanamoto is widely used (Taheri et al., 2015):

τx =(c+σ(θ) tanϕ)(1− e− jx(θ)/kx), (6)jx(θ) =r[θf −θ − (1−κ)(sinθf − sinθ)], (7)

where jx is the shear displacement in longitudinal direc-tion and ϕ is angle of internal friction of the soil, kx isshear deformation modulus in the longitudinal direction,respectively.

In order to express the multi-pass effect, as which theeffect of repetitive loading of deformable soils is ex-pressed, some parameters are modified; the modified soildensity ρp, soil cohesion cp and shear deformation mod-ulus in the longitudinal direction kxp are formulated asfollows:

ρp = ρ(1+(1− e

−κ0k1 )k2 + k3np

), (8)

cp = c(1+(1− e

−κ0k1 )k2 + k3np

), (9)

kxp = kx(1− (1− e

−κ0k1 )k2 − k3np

), (10)

where κ0 is slip ratio of previous pass and np is numberof passes, k1,k2,k3 are constant parameters, respectively.

The vertical force Fz and the longitudinal force Fx,rolling resistance torque TR are calculated by integratingnormal σ and shear τx stress distribution as follows:

Fz =rb∫ θf

θr{τx(θ)sinθ +σ(θ)cosθ}dθ , (11)

Fx =rb∫ θf

θr{τx(θ)cosθ −σ(θ)sinθ}dθ , (12)

TR =r2b∫ θf

θrτx(θ)dθ . (13)

Note that firstly in order to accord calculated load Fz togiven load W , i.e. W =Fz, Eq. (11) is solved with respectto sinkage h, and then Eqs. (12)(13) are calculated withthe calculated sinkage.

3 Wheel model identificationBecause the original wheel model based on terramechan-ics Eqs. (11)–(13) are too complex to consider in con-troller design, we extract a more simple model. In thispaper, we adopt black-box identification technique; tiredynamics is expressed as a first order delay system andtire model is expressed as a static linear function, asshown in Fig. 3. Because of dependent properties, wemake and refer the LUT (look up table) for each parame-ter.

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DOI Proceedings of the 2nd Japanese Modelica Conference 10.3384/ecp18148240 May 17-18, 2018, Tokyo, Japan 241

Wheel dynamics is expressed as follows:

Gw(s) =K

Tls+1, (14)

where the gain K(Tw,W,κ0,np) and the time constantTl(Tw,W ) are expressed using the LUT. On the de-formable terrain, small slip ratio region called linear re-gion is mainly used; a linear tire model is adopted. Thetire model, which express relationship between slip ratioκ and longitudinal force Fx, is expressed as follows:

Fx = aκ +b, (15)

where a(W,κ0,np) and b(W ) are parameter using theLUT.

4 Control methods

4.1 Conposition and guidance control

In this paper, the proposed method consists of guidancecontroller and model predictive allocation control. In theguidance controller, the vehicle total traction force is cal-culated based on the vehicle position and longitudinalvelocity. Then, in the allocation controller, the vehicletotal traction force is allocated among the wheels.

In the guidance controller, to achieve the target vehi-cle position xr and velocity vx,r, a linear quadratic regu-lator is adopted. The vehicle total traction force Fx,all isdesinged as follows:

Fx,all =−KLQ

[x− xrvx − vx,r

], (16)

where KLQ is the optimal feedback coefficient matrix andgiven by solving the continuous time algebraic Riccatiequation.

4.2 Model predictive allocation control

Control allocation could be solved without consideringany dynamics, because it is essentially static optimalproblem. However, without considering the dynamicsof the wheel, optimal solution might not be achievable.To deal with this problem, a model predictive allocationcontrol is proposed.

In this control allocation, state ξ = [κ1, · · · ,κ6]T and

input u = [Tw,1, · · · ,Tw,6]T are introduced to formulate

the proposed method as follows:

minu(i|k)

N+1

∑i=1

(||ξ (i|k)−ξr(i)||2Q + ||u(i|k)||2R), (17a)

subject toξ (i+1|k) = Aξ (i|k)+Bu(i|k), (17b)

Fx,all(i|k) =6

∑i=1

Fx,i(κi(i|k), fz,i,κ0,i,np,i), (17c)

|ξ (i|k)| ≤ ξ , (17d)|u(i|k)| ≤ u. (17e)

Eq. (17a) express the index function and Q,R > 0 areweight matrix and N is predicive horizon. ξr is ref-erence state which calculated by assuming equal trac-tion force distribution. Eq. (17b) is the discretized state-space representation model of Eq. (14). Eq. (17c) is thecondition for achieve vehicle total traction force Fx,all.Eqs. (17d)(17e) are the region of state and input ex-pressed by bound of state ξ and input u. Model predic-tive allocation control Eq. (17) is solved every samplingtimes with the condition ξ (1|k) = ξ (k); the first elementof imput u(1|k) is adopted as applied input at t.

5 Numerical simulation

5.1 Conditions

In order to verify the effectiveness of the proposedmethod, we model the leg-wheel mobile robot withwheel model based on terramechanics using modelica(see more details in (Yoshikawa et al., 2017)), and ap-ply the model predictive allocation control.

Some environmental parameters including soil param-eters are set, so as to simulate robot behavior in lunarenvironment (Ishigami et al., 2007; Senatore and Sandu,2011). The vehicles are commanded to run at constantspeed 0.4 m/s. A static control allocation control tech-nique, which does not consider any dynamics, with PIwheel velocity controller is introduced as a comparingmethod; same guidance controller Eq. (16) is adopted inthe both (Prop. and Comp.) controller. Then, the effec-tiveness of considering wheel dynamics in control allo-cation is clearly evaluated.

5.2 Results and discussions

Simulation results are shown in Fig. 4; we especiallyshow the response of limb 1–3, because of symmetricleg position. As Fig. 4(a) indicate, both methods almostachieve the target velocity. The allocated force in com-pared method is not achieved because of discontinuouschange; on the other hands, using the proposed method,which considers the wheel dynamics, the wheel torqueschange continuously. Then, wheel torques of the pro-posed method are smaller than that of comparing methodand is generated efficiently, as depicted in Fig. 4(b). Re-sult of the acceleration, Fig. 4(c) shows increased verticalload of rear limb. This load shift leads to the bias of theslip ratio, as depicted in Fig. 4(d). Neither considering

____________________________________________________________________________________________________________242 Proceedings of the 2nd Japanese Modelica Conference

May 17-18, 2018, Tokyo, Japan DOI

10.3384/ecp18148240

Torque

1),(

),,,(

wl

p0w

+sWTT

nWTK κwTSlip ratio κ

)(),,( p0 WbnWa +κκ

p0 ,, nW κ

Longitudinal

forcexF

p0 ,, nW κ

Figure 3. Structure of identified wheel model.

dynamics or not, the low load wheels are commandedlarge slip ratio so as to equalize tire forces. The longitu-dinal force distribution is, however, different from loaddistribution, because of considering the muti-pass effectand the compressed soil property, as depicted in Fig. 4(e).This slip ratio distributions should be suppressed, be-cause slip ratio occurs the sinkage as Fig. 4(d)(f) shows.Then, the amounts of sinkage are reduced in the pro-posed controller by reducing the slip ratio distributionshown in Fig. 4(g). Moreover, from the point of viewof the effect of considering wheel dynamics, desired ac-celeration in LQR controller is achieved, because theproposed controller allocates facile longitudinal force;then, overshooting phenomenon of vehicle velocity is re-moved. It is why total slip ratio value is reduced.

6 Conclusion

In this paper, to consider wheel dynamics in allocationcontrol, model predictive control method is adopted. Toconsider the complex and nonlinear terramechanics in-cluding multi-pass effect in linear MPC, identified wheelmodel consists of wheel dynamics and tire model is in-troduced. Numerical simulation is conducted using mod-elica, the effectiveness of the proposed control method isclearly shown.

Future direction of this study is to consider the lateralmotion and to evaluate the traction efficiency.

ReferencesKarl Iagnemma and Steven Dubowsky. Traction control of

wheeled robotic vehicles in rough terrain with applicationto planetary rovers. The international Journal of roboticsresearch, 23(10-11):1029–1040, 2004.

Genya Ishigami, Akiko Miwa, Keiji Nagatani, and KazuyaYoshida. Terramechanics-based model for steering maneu-ver of planetary exploration rovers on loose soil. Journal ofField robotics, 24(3):233–250, 2007.

Rainer Krenn, Andreas Gibbesch, Giovanni Binet, and AlbertoBemporad. Model predictive traction and steering controlof planetary rovers. 2013.

C Senatore and C Sandu. Off-road tire modeling and the multi-pass effect for vehicle dynamics simulation. Journal of Ter-ramechanics, 48(4):265–276, 2011.

Sh Taheri, C Sandu, S Taheri, E Pinto, and D Gorsich. A tech-nical survey on terramechanics models for tire–terrain inter-action used in modeling and simulation of wheeled vehicles.Journal of Terramechanics, 57:1–22, 2015.

CR Weisbin, D Lavery, and G Rodriguez. Robotics technologyfor planetary missions into the 21st century. 1997.

Kazuya Yoshida and Hiroshi Hamano. Motion dynamics andcontrol of a planetary rover with slip-based traction model.In Unmanned Ground Vehicle Technology IV, volume 4715,pages 275–287. International Society for Optics and Pho-tonics, 2002.

Hiroki Yoshikawa, Takatsugu Oda, Kenichiro Nonaka, andKazuma Sekiguchi. Modeling and simulation of wheel driv-ing systems based on terramechanics for planetary explana-tion rover using modelica. In Proceedings of the 12th In-ternational Modelica Conference, number 132, pages 901–907. Linköping University Electronic Press, 2017.

Li Zhengcai and Wang Yang. Robust adaptive fuzzy controlfor planetary rovers while climbing up deformable slopeswith longitudinal slip. Advances in Aerospace Engineering,2014, 2014.

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0

0.1

0.2

0.3

0.4

0.5

10 15 20 25 30 35 40

Vel

oci

ty v

x [

m/s

]

Time t [s]

Prop.Comp.Target

(a) Vehicle longitudinal velocity vx.

0 5

10 15 20 25 30 35 40

10 15 20 25 30 35 40

To

rqu

e T

w [

N·m

]

Time t [s]Prop. (Limb1)Prop. (Limb2)Prop. (Limb3)

Comp. (Limb1)Comp. (Limb2)Comp. (Limb3)

(b) Wheel torque Tw.

300

350

400

450

500

10 15 20 25 30 35 40

Ver

tica

l fo

rce

f z [

N]

Time t [s]Prop. (Limb1)Prop. (Limb2)Prop. (Limb3)

Comp. (Limb1)Comp. (Limb2)Comp. (Limb3)

(c) Vertical force Fz.

−0.02

0

0.02

0.04

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10 15 20 25 30 35 40

Sli

p r

atio

κ [

−]

Time t [s]Prop. (Limb1)Prop. (Limb2)Prop. (Limb3)

Comp. (Limb1)Comp. (Limb2)Comp. (Limb3)

(d) Slip ratio κ .

−10

0

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20

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50

10 15 20 25 30 35 40

Fo

rce

f x [

N]

Time t [s]Prop. (Limb1)Prop. (Limb2)Prop. (Limb3)

Comp. (Limb1)Comp. (Limb2)Comp. (Limb3)

(e) Longitudinal force Fx.

0.003

0.0035

0.004

0.0045

10 15 20 25 30 35 40

Sin

kag

e h

[m

]

Time t [s]Prop. (Limb1)Prop. (Limb2)Prop. (Limb3)

Comp. (Limb1)Comp. (Limb2)Comp. (Limb3)

(f) Wheel sinkage h.

0

0.005

0.01

0.015

0.02

0.025

0.03

Comp. Prop.

RM

S e

rrro

of

κ [

−]

Limb 1Limb 2Limb 3

(g) RMS of slip ratio κ .

Figure 4. Simulation results.

____________________________________________________________________________________________________________244 Proceedings of the 2nd Japanese Modelica Conference

May 17-18, 2018, Tokyo, Japan DOI

10.3384/ecp18148240