walking pattern generation on inclined and uneven terrains ......walking pattern generation on...

Walking Pattern Generation on Inclined andUneven Terrains for Humanoid Robots

Young-Dae Hong and Jong-Hwan Kim

Department of Electrical Engineering, KAIST291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea

e-mail: {ydhong, johkim}@rit.kaist.ac.kr

Abstract. This paper introduces a walking pattern generation methodon an inclined terrain in both pitch and roll directions, and uneven ter-rain. The walking pattern generation method is based on a modifiablewalking pattern generator (MWPG) which allows a zero moment point(ZMP) variation in real-time. As a navigational command set, a 3-Dcommand state (CS) is defined, which consists of single and double sup-port times, sagittal and lateral step lengths, and foot height of the swingleg. In the single support phase, the primary dynamics of the humanoidrobot on the inclined terrain is modeled as a 3-D linear inverted pendu-lum model (LIPM) with constant center of mass (CoM) height, and thedynamic equation of the 3-D LIPM is derived to obtain the sagittal andlateral CoM motions. Using the sagittal and lateral CoM motions, thesagittal and lateral CoM trajectories are generated to satisfy the sagittaland lateral step lengths of the swing leg. The foot trajectories of theswing leg are generated according to the commanded sagittal and lateralstep lengths, and foot height. In the double support phase, the verticalCoM trajectory is generated to satisfy the foot height of the swing legfrom the single support phase. The walking pattern generation method isimplemented on a simulation model of the small-sized humanoid robot,HanSaRam-IX (HSR-IX) and the effectiveness is demonstrated throughcomputer simulations.

Key words: Humanoid robot, 3-D command state (CS), modifiablewalking pattern generator (MWPG), locomotion on inclined and uneventerrains, 3-D linear inverted pendulum model (LIPM).

1 Introduction

In the robotics society, a humanoid robot is a representative research topic. Alot of humanoid robots have been developed [1]–[3] and many walking patterngeneration methods for stable walking have been studied [4]–[7]. In most of walk-ing pattern generation methods, it is assumed that the terrain is flat. However,there exist not only flat but also inclined or uneven terrains in real environment.Therefore, recently, various walking pattern generation methods were developedfor walking on the inclined terrain [8]–[11] and uneven terrain [12]–[16].

2 Young-Dae Hong and Jong-Hwan Kim

This paper presents a novel walking pattern generation method on variousenvironments for humanoid robots. Particularly, the walking pattern generationmethod on an inclined terrain in both pitch and roll directions, and uneven ter-rain is introduced. In the previous researches on walking pattern generation onthe inclined terrain and uneven terrain, the humanoid robot was unable to mod-ify independently elements of a walking pattern, i.e. single and double supporttimes, sagittal and lateral step lengths, and foot height of the swing leg withoutany additional footstep for adjusting the center of mass (CoM) motion. Thus,in this paper, the walking pattern generation method is developed to solve thisproblem. the walking pattern generation method is based on the conventionalmodifiable walking pattern generator (MWPG) which allows the zero momentpoint (ZMP) variation in real-time by closed form functions [5], [6]. The conven-tional MWPG can be applied only on the flat terrain.

The walking pattern generation method in this paper extended the MWPGto independently modify the elements of the walking pattern on the inclinedterrain and uneven terrain. As a navigational command set, a 3-D commandstate (CS) is defined, which consists of the single and double support times,sagittal and lateral step lengths, and foot height of the swing leg. The CoMtrajectories in the single and double support phases are generated to satisfy the3-D CS. In the single support phase, the primary dynamics of the humanoidrobot on the inclined terrain is modeled as a 3-D linear inverted pendulummodel (LIPM) with constant CoM height, and the dynamic equation of the 3-D LIPM is derived to obtain the sagittal and lateral CoM motions. Using thesagittal and lateral CoM motions, the sagittal and lateral CoM trajectories aregenerated to satisfy the sagittal and lateral step lengths of the swing leg. The foottrajectories of the swing leg are generated according to the commanded sagittaland lateral step lengths, and foot height. In the double support phase, the verticalCoM trajectory is generated to satisfy the foot height of the swing leg from thesingle support phase. The walking pattern generation method is implementedon a simulation model of the small-sized humanoid robot, HanSaRam-IX (HSR-IX), developed at the Robot Intelligence Technology laboratory, KAIST and theeffectiveness is demonstrated through computer simulations.

This paper is organized as follows. Section 2 presents the walking patterngeneration method. The 3-D CS is presented, the CoM trajectory generationsin the single and double support phases are described, and the foot trajectorygeneration of the swing leg is also explained. In Section 3, the simulation resultsare presented and finally conclusions follow in Section 4.

2 Walking Pattern Generation

The walking of the humanoid robot consists of single and double support phases,and in the single support phase, the primary dynamics of the humanoid roboton the flat terrain is modeled as a 3-D LIPM [4]. In the 3-D LIPM, it is assumedthat the support leg is a weightless telescopic limb and the mass is concentrated

Walking Pattern Generation on Inclined and Uneven Terrains for Humanoid Robots 3

as a single point without vertical motion. Consequently, it is possible to decouplethe sagittal and lateral CoM motion equations.

In the conventional 3-D LIPM, it is assumed that the ZMP is fixed at thecontact point. Consequently, in the single support phase, the CoM motion ofthe 3-D LIPM is unmodifiable, which means that the humanoid robot is unableto modify independently the elements of the walking pattern, i.e. the single anddouble support times, the sagittal and lateral step lengths, and the foot directionof the swing leg. However, in the MWPG, by allowing the ZMP variation, theCoM position and velocity can be changed independently at any time during thesingle support phase [5]–[7]. Thus, the MWPG enables the humanoid robot tomodify the elements of the walking pattern independently by the ZMP functionswithout any extra footstep for adjusting the CoM motion. In this paper, formodifiable walking on the inclined terrain and uneven terrain as well as flatterrain, the conventional MWPG is extended.

2.1 3-D CS

As a navigational command set, the conventional CS is insufficient to walk onvarious environments [5]–[7]. Thus, the foot height of the swing leg should beconsidered in the CS. In this paper, the 3-D CS is defined as a novel navigationalcommand set as follows:

Definition 1. 3-D CS is defined as

3-D CS ≡ [T ssl/r T ds

l/r Sl/r Ll/r Hl/r]

where

T ssl/r: single support time;

T dsl/r: double support time;

Sl/r: sagittal step length of left/right leg;Ll/r: lateral step length of left/right leg;Hl/r: foot height of left/right leg.

2.2 CoM Trajectory Generation in Single Support Phase

Fig. 1 shows the 3-D LIPM on the inclined terrain in both pitch and roll direc-tions. The following equations present the sagittal and lateral CoM motions ofthe 3-D LIPM on the inclined terrain.

Sagittal CoM motion:[xf

vfTc

]=

[cosh( T

Tc) sinh( T

Tc)

sinh( TTc) cosh( T

Tc)

] [xi

viTc

]− 1

Tc

[∫ T

0sinh( T

Tc)p̄(t)dt∫ T

0cosh( T

Tc)p̄(t)dt

]

+

[−gT 2

c sin θv(−1 + cosh( TTc))

−gT 2c sin θv sinh(

TTc)

](1)


Z

cZz

hR

X

RY

RZ

v

}{R

Fig. 1. 3-D LIPM on inclined terrain.

Lateral CoM motion:[yf

wfTc

]=

[cosh( T

Tc) sinh( T

Tc)

sinh( TTc) cosh( T

Tc)

] [yi

wiTc

]− 1

Tc

[∫ T

0sinh( T

Tc)q̄(t)dt∫ T

0cosh( T

Tc)q̄(t)dt

]

+

[−gT 2

c sin θh(−1 + cosh( TTc))

−gT 2c sin θh sinh(

TTc)

](2)

with

Tc =

√Zc

g cos θv cos θh

where (xi, vi)/(xf , vf ) and (yi, wi)/(yf , wf ) represent the initial/final positionand velocity of the CoM in sagittal and lateral planes, respectively. T is theremaining single support time. p(t) and q(t) are the ZMP functions for sagittaland lateral CoM motions, respectively. p(t) = p(T − t) and q(t) = q(T − t).

Using the above sagittal and lateral CoMmotions (1) and (2), the sagittal andlateral CoM trajectories in the single support phase are generated to satisfy thesagittal and lateral step lengths. The CoM position and velocity in the sagittaland lateral planes are defined as a walking state (WS) of the 3-D LIPM and theWS is derived for the commanded CS [5]–[7]. Then, the sagittal and lateral CoMtrajectories satisfying the WS are generated by (1) and (2).

2.3 CoM Trajectory Generation in Double Support Phase

In this paper, the vertical CoM trajectory in the double support phase is gener-ated to satisfy the foot height of the swing leg from the single support phase,Hl/r

instead of using the constant CoM height. As shown in Fig. 2, the CoM heightmaintains the constant Zc during the single support phase, and then it movesto Zc +Hl/r during the double support phase by the cubic spline interpolation


rlH /

rlH /

x

z

cZ

Fig. 2. Vertical CoM motion in double support phase.

with Zc at t = 0 and Zc +Hl/r at t = T dsl/r as follows:

z(t) = −2Hl/r

T dsl/r

3 t3 + 3

Hl/r

T dsl/r

2 t2 + Zc. (3)

Note that the vertical CoM trajectory is defined with respect to the local co-ordinate frame attached on the support leg. In the double support phase, thesagittal and lateral CoM motions travel with constant velocity.

2.4 Foot Trajectory Generation

The sagittal and lateral foot trajectories of the swing leg, (xfoot, yfoot) aregenerated by the cubic spline interpolation with the sagittal and lateral steplengths at the previous footstep, (−Spre

r/l , − Lprer/l ) at t = 0 and (Sl/r, Ll/r) at

t = T ssl/r as follows:

xfoot(t) = −2Sl/r + Spre

r/l

T ssl/r

3 t3 + 3Sl/r + Spre

r/l

T ssl/r

2 t2 − Sprer/l (4)

yfoot(t) = −2Ll/r + Lpre

r/l

T ssl/r

3 t3 + 3Ll/r + Lpre

r/l

T ssl/r

2 t2 − Lprer/l . (5)

The vertical foot trajectory of the swing leg, zfoot is generated by a cycloidfunction. To satisfy the foot height Hl/r, the trajectory ∆zfoot is added, whichis generated by the cubic spline interpolation with the foot height of the swingleg at the previous footstep, −Hpre

r/l at t = 0 and Hl/r at t = T ssl/r as follows:

zfoot(t) = r

(1− cos

(2πt

T ssl/r

))+∆zfoot(t) (6)

with

∆zfoot(t) = −2Hl/r +Hpre

r/l

T ssl/r

3 t3 + 3Hl/r +Hpre

r/l

T ssl/r

2 t2 −Hprer/l


(a) (b) (c)

Fig. 3. (a) HSR-IX. (b) Simulation model. (c) Configuration.

where r denotes the radius of the cycloid circle.

3 Simulation Results

3.1 HanSaRam-IX

The proposed algorithm was implemented on the simulation model of the small-sized humanoid robot, HSR-IX in Fig. 3. The simulation model of HSR-IX wasmodeled by Webots which is the 3-D robotics simulation software and enablesusers to conduct the physical and dynamical simulation [17]. HSR has beenin continual development and research by the Robot Intelligence Technologylaboratory, KAIST [6]. Its height and weight are 52.8 cm and 5.5 kg, respectively.It has 26 DOFs which consist of 12 DC motors with harmonic drives in the lowerbody and 16 RC servo motors in the upper body (two servo motors in eachhand control). The on-board Pentium-III compatible PC, running RT-Linux,calculates the proposed algorithm every 5 msec in real-time. To measure groundreaction forces on the feet and the real ZMP trajectories while walking, fourforce sensing resisters are equipped on each foot.

3.2 Modifiable Walking on Inclined Terrain

To verify the modifiable walking on the inclined terrain in pitch and roll direc-tions, the 3-D CS list in Table 1 was used for the simulation, in which sagittaland lateral step lengths were independently changed while maintaining the samewalking period at each footstep. The 3-D CS can be predefined or provided bythe footstep planning algorithm [18], [19]. Fig. 4 shows the snapshot of the walk-ing simulation on the inclined terrain in pitch and roll directions. HSR-IX walkedstably on the inclined terrain in pitch and roll directions by modifying the sagit-tal and lateral step lengths of the swing leg according to the commanded 3-D


Table 1. Commanded 3-D CS list for walking simulation on inclined terrain in pitchand roll directions (time and length units were given in seconds and centimeters, re-spectively.)

Steps T ssl/r T ds

l/r Sl/r Ll/r Hl/r

1st (right foot) 0.8 0.4 4.0 -6.0 0.0

2nd (left foot) 0.8 0.4 4.0 6.0 0.0

3rd (right foot) 0.8 0.4 1.0 -10.0 0.0

4th (left foot) 0.8 0.4 4.0 6.0 0.0

5th (right foot) 0.8 0.4 4.0 -6.0 0.0

6th (left foot) 0.8 0.4 2.0 9.0 0.0

7th (right foot) 0.8 0.4 4.0 -6.0 0.0

8th (left foot) 0.8 0.4 4.0 6.0 0.0

9th (right foot) 0.8 0.4 4.0 -6.0 0.0

10th (left foot) 0.8 0.4 2.0 9.0 0.0

11th (left foot) 0.8 0.4 4.0 -6.0 0.0

12th (right foot) 0.8 0.4 4.0 6.0 0.0

13th (left foot) 0.8 0.4 0.0 -6.0 0.0

Fig. 4. Snapshot of the walking simulation on inclined terrain in pitch and roll direc-tions.

CS list at every footstep. Fig. 5 shows the generated walking pattern, CoM, andZMP trajectories when the terrain was inclined upward and leftside simultane-ously (θv = 15◦ and θh = 5◦). As the figure shows, the CoM trajectory whichwas shifted forward and left simultaneously was generated for stable walking.It can be shown that the ZMP trajectories in x-axis and y-axis follow the foottrajectories with a small variation. The small variation was mainly caused bythe dynamic difference between the robot and the 3-D LIPM. However, the ZMPtrajectories were within the upper and lower boundaries of foot trajectories. Ac-


10 0 10 20 30 40 50 10

5

0

5

10

15

20

25

x [cm]

y [

cm

]

CoM trajectory

Foot boundary

Foot position

0 5 10 15

0

20

40

60

80

100

Time [s]

xzm

p [

cm

]

ZMP trajectory

Foot trajectory

Upper boundary of foot trajectory

Lower boundary of foot trajectory

0 5 10 15 10

5

0

5

10

15

Time [s]

yzm

p [

cm

]

Fig. 5. Generated walking pattern, CoM, and ZMP trajectories when the terrain wasinclined upward and leftside simultaneously in the simulation (θv = 15◦ and θh = 5◦).

cordingly, the robot was able to walk stably on the inclined terrain in both pitchand roll directions.

3.3 Modifiable Walking on Uneven Terrain

In the simulation environment, there were five boards with different heights (0.5,1.0, and 1.5 cm) and sizes, and the 3-D CS list for the simulation was determinedaccording to the information about the terrain. Table 2 shows the commanded 3-D CS list, in which sagittal and lateral step lengths, and foot height of the swingleg were independently changed while maintaining the same walking period at


Table 2. Commanded 3-D CS list for walking simulation on uneven terrain (time andlength units were given in seconds and centimeters, respectively.)

Steps T ssl/r T ds

l/r Sl/r Ll/r Hl/r

1st (right foot) 0.8 0.4 7.0 -6.0 0.0

2nd (left foot) 0.8 0.4 7.0 6.0 1.0

3rd (right foot) 0.8 0.4 7.0 -6.0 -1.0

4th (left foot) 0.8 0.4 7.0 6.0 1.0

5th (right foot) 0.8 0.4 5.0 -11.0 -1.0

6th (left foot) 0.8 0.4 7.0 6.0 1.0

7th (right foot) 0.8 0.4 7.0 -6.0 0.5

8th (left foot) 0.8 0.4 8.0 9.0 -1.0

9th (right foot) 0.8 0.4 6.0 -9.0 0.5

10th (left foot) 0.8 0.4 7.0 6.0 -1.0

11th (right foot) 0.8 0.4 8.0 -6.0 0.0

12th (left foot) 0.8 0.4 7.0 6.0 0.0

13th (right foot) 0.8 0.4 0.0 -6.0 0.0

Fig. 6. Snapshot of the walking simulation on uneven terrain.

each footstep for walking on the uneven terrain. Fig. 6 shows the snapshot of thewalking simulation on the uneven terrain. HSR-IX walked stably on the uneventerrain by modifying the sagittal and lateral step lengths, and foot height ofthe swing leg according to the commanded 3-D CS list at every footstep. Fig.7 shows the generated CoM trajectories with respect to the global coordinateframe attached on the terrain. As shown in the figure, in the single supportphases, the sagittal and lateral CoM trajectories were generated to satisfy thesagittal and lateral step lengths in the 3-D CS list by (1) and (2), and the verticalCoM trajectory maintained the constant value. In the double support phases,the vertical CoM trajectory was generated to satisfy the foot heights of the swingleg from the single support phases by (3). In addition, the foot trajectories of the


Fig. 7. Generated sagittal, lateral and vertical CoM trajectories with respect to theglobal coordinate frame attached on the terrain. The thick and thin lines represent theCoM trajectories in the single and double support phases, respectively.

swing leg were generated by (4), (5), and (6). Fig. 8 shows the measured ZMPtrajectories in the walking simulation. It can be seen that the ZMP trajectoriesin the x-axis and y-axis followed the foot trajectories with a small variation.The small variation of the ZMP trajectories was mainly due to the dynamicdifference between HSR-IX and the 3-D LIPM. However, the ZMP trajectorieswere within the upper and lower boundaries of foot trajectories, which meansthat HSR-IX could walk on the uneven terrain while maintaining stability.

4 Conclusion

In this paper, the novel walking pattern generation method based on the MWPGwas introduced for modifiable walking on the inclined terrain in pitch and rolldirections, and uneven terrain. As a novel navigational command set, the 3-DCS was defined and then, the CoM trajectories in the single and double supportphases were generated to satisfy the 3-D CS, also the foot trajectory of the swingleg was generated. The effectiveness of the walking pattern generation methodwas verified through simulations using the simulation model of the small-sizedhumanoid robot, HSR-IX. Consequently, by using the walking pattern generation


0 5 10 15

0

50

100

150

Time [s]

xzm

p [

cm

]

ZMP trajectory

Foot trajectory

Upper boundary of foot trajectory

Lower boundary of foot trajectory

0 5 10 15 10

5

0

5

10

15

Time [s]

yzm

p [

cm

]

Fig. 8. Measured ZMP trajectories in the walking simulation on uneven terrain.

method, the humanoid robot could walk stably on the inclined terrain in bothpitch and roll directions, and uneven terrain following the commanded 3-D CSlist by modifying the sagittal and lateral step lengths, and foot height of theswing leg at every footstep.

5 Acknowledgments

This research was supported by Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded by the Ministry of Edu-cation, Science and Technology (2012-0000150).

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