An Inverted Pendulum Based Approach to Biped Trajectory Generation with Swing Leg Dynamics

Kemalettin Erbatur#1, Utku Seven#2

#Mechatronics Program, Sabanci University Orhanli-Tuzla, 34956 Istanbul, Turkey

#1erbatur@sabanciuniv.edu #2utkuseven@su.sabanciuniv.edu

Abstract— Reference trajectory generation is one of the key problems in biped walking robot research. The linear inverted pendulum model (LIPM) is employed widely as a useful model which simplifies trajectory generation task. Many reference generation algorithms use the Zero Moment Point (ZMP) Criterion for the LIPM in order to achieve stable walking trajectories. However, LIMP ignores the dynamics of the swing leg. This can lead to tracking problems, especially when the legs are heavy. This paper uses a two-mass LIPM and proposes a fifth order state space description for the dynamics of the robot body and the swing leg in the swing phase. The body center of mass (CMB) reference trajectory is obtained for given foot placement references and the desired ZMP trajectory. An inverse kinematics based position controller is then employed for locomotion. The walking performances with the one-mass and one-mass-two-mass switching linear inverted pendulum models are finally compared via 3D full-dynamics simulations of a 12 degrees of freedom (DOF) biped robot. The results indicate that the proposed model switching between one-mass and two-mass models is useful in improving the stability of the walk.

I. INTRODUCTION

Humanoid design is currently among the most exciting research topics in the field of robotics and there are many successful projects reported [1-7]. The motivation of the research is the suitability of the biped structure for tasks in the human environment as substitutes of human.

However, the control of a biped humanoid is a difficult task due to the many degrees of freedom involved and the non-linear and hard to stabilize dynamics [8-9].

Walking reference trajectory generation is an important problem. Reference generation techniques which employ LIPM as a simplified biped robot model are reported [10-12]. An intuitive demand for the biped robot reference generation is that the reference trajectory should be a stable one, in the sense that it should not lead to unrecoverable falling motion. The ZMP Criterion [8] is widely employed in the stability analysis of biped robot walk. LIPM based reference generation methods obtained by applying the ZMP criterion in the design process are reported too [13, 18]. In this approach the ZMP during a stepping motion is kept in the supporting foot sole for the stability, while the robot center of mass follows the Linear Inverted Pendulum path.

Not considered in the LIPM is however the dynamics of the swing leg. LIPM assumes that the robot can be modelled as a point mass body and massless legs. The swing leg is however

creating a disturbance in the swing phase, especially when the legs are heavy. [19] proposes a two-mass model in order to alleviate the problems caused by the swing leg. They work on the reference generation of a six DOF robot in the saggital plane and so the equations of motion analytically. A solution for the disturbing gravity effect on the swing leg is proposed in [20] too. [7] proposes a gravity compensation technique for the full body robot motion.

This paper suggests the use of a LIPM which switches between one-mass and two-mass models for the double and single support phases, respectively. The one-mass model is described by a third order state-space realization [21] and a fifth order state-space system is used to describe the dynamics of the two-mass model in both realizations, the CMB is one of the state variables. Simulating the state-space model under state feedback to reach desired ZMP and swing foot trajectories, and recording the CMB history provides the CMB reference. The joint trajectories are then obtained by inverse kinematics and PID controllers are employed independently at joint level for locomotion.

The reference generation and control algorithm is tested with a 3-D full dynamic simulator on the model of a 12 DOF biped robot. Results indicate better performance of the one-mass-two-mass switching LIPM over the one-mass LIPM.

The next section describes the biped robot model used in the simulations. Section III presents ZMP and swing leg trajectory generation. The one-mass and two-mass Linear Inverted Pendulum models, their state-space descriptions and the off-line state feedback simulation for CMB reference generation is discussed in Section IV. The simulation results are given in Section V and lastly the conclusion is presented

II. THE BIPED MODEL

The biped model used in this paper consists of two 6-DOF legs and a trunk connecting them. Three joint axes are positioned at the hip. Two joints are at the ankle and one at the knee. Link sizes and masses of the biped are given in Table I.

A view of the animation window is shown in Fig. 1. The full-dynamics 3-D simulation scheme is similar to the one in [22]. The ground contact is modelled by an adaptive penalty based method. The details of the simulation algorithm and contact modeling can be found in [23]. This model used in the simulation studies presented in Section V.

978-1-4244-1862-6/07/$25.00 © 2007 IEEE HUMANOIDS’07216

TABLE I MASSES AND LENGTHS OF THE ROBOT LINKS

Link Dimensions (LxWxH) [m] Mass [kg] Trunk 0.2 x 0.4 x 0.5 50 Thigh 0.27 x 0.1 x 0.1 12 Calf 0.22 x 0.05 x 0.1 0.5 Foot 0.25 x 0.12 x 0.1 5.5

Fig. 1 A snapshot from the animation window.

III. ZMP AND SWING FOOT REFERENCES

As mentioned in Section I, ZMP and swing foot references are required for the CMB reference generation simulations. The Zero Moment Point criterion states that this point should be under in the supporting polygon of a robot for that robot to be stable. ZMP and swing foot reference generation algorithm employed in this paper is as presented in [17]. Mainly, the procedure can be summarized as follows:

1) Specify the support foot locations (step size B and foot-to-foot distance A2 ) as in Fig. 2.

2) Specify the distance travelled by the ZMP under the support foot ( b2 ) in the single support phase (Fig.3).

3) Specify a walking period ( T2 in Fig. 3). 4) Generate x-direction ZMP reference ref

xp (walking direction) as a “staircase of ramps” (Fig. 3) with the following expression

( )�∞

=−−+−=

1)2()

2(2

k

refx kTtubBTt

Tbp

where u is the unit step function. 5) Generate the y-direction ZMP reference ref

yp with the assumption that the foot switching occurs instantaneously. This corresponds to the square wave shaped reference curve in Fig. 4:

( ) ( ) ( )�∞

=−−+=

112

k

krefy kTtuAtAup

6) Smooth the staircase and square wave shapes to introduce a smooth switching between single support phases (that is, a switching with a double support

phase). The resulting ZMP references are shown in Figures 5 and 6 for the simulation studies presented in this paper. These are obtained from the curves in Figures 3 and 4 by the smoothing algorithm in [17].

7) y-direction foot reference positions are constant. The foot x-direction position references in Fig. 5 are chosen as smooth functions based on sinusoidal building blocks. They are determined by the desired support foot locations (Fig. 2) and the step period T2 . The swing foot z-direction references are chosen such that the respective foot is above ground level when the foot x-direction reference is moving forward in Fig. 5.

Fig. 2 Desired foot locations.

Fig. 3 ZMP x-direction reference with moving ZMP under the support foot.

Fig. 4 ZMP y-direction reference before smoothing.

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Fig. 5 ZMP reference after smoothing and swing foot references, x-direction.

Fig. 6 ZMP y-direction reference after smoothing.

IV. THE TWO-MASS MODEL AND CMB TRAJECTORY GENERATION

In this section, firstly the one-mass LIPM equations which are used as the base of many reference generation algorithms are revisited. The generalization to the two-mass case will follow. Finally, the robot body center of mass position reference generation via off-line state feedback simulation will be discussed. Figure 7 shows the one-mass LIPM in the x-z-plane. In this figure, xc is the x-directional position, zc is the z-directional position and xc�� is the x-directional acceleration of the center of mass of the robot. g is the gravitational acceleration constant (9.81 2/ sm ) and M is the body weight.

The assumption that the height of the mass is constant decouples the x and y direction motion equations. The ZMP x-component xp can be obtained from the equation

)( xxzx pcMgccM −=�� (1) as

xz

xx cgccp ��−= . (2)

Note that when this model serves as a simplified walking robot dynamics description, M stands for the combined mass of the trunk and the legs and c is the location of the overall

center of mass, usually abbreviated as COM. However, using the CMB instead of the real COM is also common because of practical reasons.

With (2), by applying the Laplace transform we obtain )()( 2

xxx ccLsP ��ω−= (3)

Where ω is defined as gcz . This equation can be used to obtain the xc trajectory from the given desired xp trajectory. These approaches followed by a number of studies. However [21] points out that typical ZMP measurement is a noisy one due to the force sensors in the measurement system and adds a low pass filter into the equation:

)(1

1)( 2xxx ccL

ssP ��ω

τ−

+= . (4)

In this equation τ is the filter time constant. Such a filter is typically used in real applications and (4) does indeed reflect a realistic situation. [21] uses this model to obtain the state-space description

x

B

x

x

x

A

x

x

x

cccp

ccp

dtd

��

������

�� ��� ��� �

��

�

�

���

�

�−+

���

�

�

���

�

�

���

�

�

���

�

�−=

���

�

�

���

�

�

10

000100011 2 τωττ

,

[ ] � xD

x

x

x

C

x cccp

p ��

������

0001 +���

�

�

���

�

�=

(5)

and employs (5) in feedback control for online stabilization. In this paper, however, we use this equation for online trajectory generation as follows:

Firstly the continuous state-space realization (A,B,C,D) is discretized into a discrete realization (F,G,H,J) for computer implementation. A state feedback gain K is obtained by pole placement techniques for stable closed loop pole locations. This gain is used in the block diagram shown in Fig. 8 where

xN and uN are the feedforward gain and state reference gain matrices computed as

Fig. 7.One -mass LIPM

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Fig. 8 The state-feedback controller

��

���

���

���

�=�

�

���

�−

101

JGHF

NN

u

x . (6)

The derivation of (6) is given in [24]. r in Fig. 8 is the ZMP x-direction reference. The trajectories of the state variable xcis stored and used as the robot body center of mass position reference. Since xc and foot references (Fig. 5.) are expressed in the same coordinate frame (world coordinate frame), inverse kinematics can be employed to obtain the corresponding joint variable references. PID independent joint controllers are then applied for locomotion control purpose. The robot can indeed walk with this kind of reference trajectory as simulation results in Section V indicate too.

However there is a drawback in the single support phases. The weight of the swing leg is not considered in one-mass LIPM and it disturbs the motion of the center of mass during walking. This effect can be identified from front and back, right and left deviations from the planned COM trajectory. In this paper (as also in [19]), a two-mass LIPM is proposed to alleviate this problem in reference trajectory generation. For the two-mass LIPM, the x-y-plane configuration is shown in Fig. 9. In this figure defined are two center of mass locations which actually differ from, COM, the overall center of mass of robot.

Fig. 9 Two-mass LIPM.

the robot. xc now stands for the robot body (or trunk) center of mass, CMB and xl is the swing leg center of mass, CML. m is the mass of the swing leg. The ZMP equation obtained from Fig. 9 is as,

gll

mMml

mMm

gcc

mMMc

mMMp z

xxz

xxx����

)()()()( +−

++

+−

+= . (7)

where xl is the x-directional position of center of mass of

swing leg, xl�� is the x-directional acceleration of center of

mass of swing leg and zl is the z-directional position of center of mass of swing leg. (7) can be put into the form

)()( 22 γω xxxxx lrrlcRRcLsP ���� −+−= . (8) where

RmM

M =+ )(

, rmM

m =+ )(

, 2ω=gcz , 2γ=

glz . (9)

The definitions of ω and τ are as for (3). Similar to the case with equation (3), a low pass filter is added to this equation to obtain

)(1

1)( 22 γωτ xxxxx lrrlcRRc

ssP ���� −+−

+= . (10)

With the selection of new state variables as xp , xc , xc� , xl

and xl� (9) can be realized in state-space as

��

���

�

������

�

�

������

�

� −−

+

������

�

�

������

�

�

������

�

�

������

�

�−

=

������

�

�

������

�

�

x

x

x

x

x

x

x

x

x

x

x

x

lc

rR

llccprR

llccp

dtd

����

�

�

�

�

10000100

00000100000000000100001 22 τγτωτττ

. (11)

By setting the output vector y as [ ]Txx lp , the output

equation is obtained as,

��

���

���

���

�+

������

�

�

������

�

�

��

���

�=

x

x

D

x

x

x

x

x

B

lc

llccp

y ����

���

�

�

��� ���� ��0000

0100000001

. (12)

Again this system discretized into a discrete state space realization (F,G,H,J) and a suitable state feedback gain K is obtained by pole placement. A suitable set for a feed forward gain and state reference gain can be found from (6).

The simulation runs again in accordance with the block diagram in Fig. 8 for the two-mass case. However the reference vector is now accommodating the ZMP x-direction reference and the swing leg center of mass x-direction reference xl . The leg center of mass locations are computed from the current CMB value and swing foot references at every computation cycle by computing the link center of mass coordinates. The history of xc is stored to be employed as the robot body center of mass reference in the walking time. Inverse kinematics and PID controllers are also employed as in the one-mass case.

The one-mass and two-mass models are used during the trajectory generation simulation alternatingly: When the robot

219

is in the double support phase, the one-mass model is used and the two-mass model is used in the single support phase. Continuity is addressed by reinitializing the starting conditions every time the model switching occurs.

All the discussions above are for the x-direction. However, for the y direction, the deviations are very similar. The whole approach is also applied for the y-directional case and references are obtained for this direction too. As a result these two results are combined to realize the reference for the CMB.

V. SIMULATION RESULTS

In this section, a walking trajectory with the parameters listed in Table II is considered. Two cases are simulated:

1) CMB trajectory generated with a one-mass model [18]for double and single support phases with the ZMP and swing foot references in Section III.

2) CMB trajectory generated by the one-mass-two-mass switching model with the ZMP and swing foot references in Section III.

The choices for the body and leg mass and heights are given in Table III The leg and body heights are obtained from typical walking simulations by the trajectory generation in [18] and they are further tuned by trial and error for good walking performance. Fig. 10 shows the first case (the one-mass case). Deviations from the planned COM trajectory are observed in the swing phases in both x and y directions. The tracking in the double support phase is more successful than in the single support phase. The CMB reference and actual trajectories for the second case (one-mass-two-mass model alternating case) are shown in Fig. 11. The results shown in the Figures 10 and 11 are both obtained with the inverse kinematics based open loop position controller without any feedback from the body inclination angles, body position and ground interaction sensors during the walk. Therefore, their performances are purely determined by the performances of the reference generation algorithms. However, it is not straightforward to decide which of the two trajectories shown in Figures 10 and 11 reflects a better walking performance. It can be observed that the swing phase CMB tracking performance is better with the one-mass-two-mass switching reference generation model, in that the x-direction tracking error is much less than the one with the one–mass model. This indicates that the effect of the swing foot weight is indeed decreased by the reference generation algorithm. On the other hand it is also observed that the y-direction overshoots at the peaks of the curves are larger with the one-mass-two-mass switching model. Our last observation is that the behaviour at these peaks is more oscillatory with the one-mass model. The animations corresponding to these curves also indicate that the walk with the proposed method is smoother in the y-direction.

The observations above indicate that the one-mass-two-mass reference generation algorithm produces slightly better walking results when compared with the one-mass LIPM based reference generation. Our interpretation of this result is that even with the two mass model, the LIMP is an approximation of complex dynamics and improving this model adds only marginally to the walking performance. Our

bottom-line is that on-line ZMP feedback control methods, even with simpler models, can contribute much better to the walking performance than improved LIP models can do.

VI. CONCLUSIONS

A biped walking robot trajectory reference generation algorithm based on ZMP considerations and on the switching between one-mass and two-mass Linear Inverted Pendulum Models is presented in this paper. Simulation studies indicate a slightly better performance of this model when compared with the one-mass LIPM case. This indicates that the vital point in walking algorithms lies more in on-line ZMP feedback control than in an improved model. Author’s work on building a full-body robot for experimental validation of the results is in continuation.

TABLE II SOME OF THE IMPORTANT SIMULATION PARAMETERS

Parameter Value Step height 0.02 m Step period 3 s Foot to foot y-direction distance 0.16 m Foot to foot y-direction ZMP reference distance 0.2 m Step size 0.2 m ZMP motion under the support foot 0.08 m

TABLE III SOME OF THE IMPORTANT SIMULATION PARAMETERS

One-Mass Model One-Mass-Two-Mass Model Parameter Value Parameter Value

zc 0.77 m zc 0.77 m

M 86 kg M 68 kg

zl 0.31 m

m 10 kg

Fig. 10 CMB position and CMB reference position projections on the x-y-plane (ground plane) with the one-mass LIPM.

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Fig. 11 CMB position and CMB reference position projections on the x-y-plane (ground plane) with the one-mass/ two-mass switching LIPM.

ACKNOWLEDGMENT

This work has been supported by TUBITAK, the Scientific and Technological Research Council of Turkey (research project and grant no: 106E040).

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