Effects of Constraints on Bipedal Balance Control - IEEE Xplore

Proceedings of the 2006 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 2006

ThA12.3

Effects of Constraints on Bipedal Balance Control C. Yang and Q. Wu 

Abstract— The satisfaction of constraints between the feet and the ground is important during balance control of bipedal standing. In this work, the effects of such constraints on balance control of a simplified two-dimensional biped are studied. The bounds imposed on the control torque due to the constraints are first determined. Such control bounds have significant effects on designing balance control laws and can be used to predict the type of falls. It is also found that the angular velocity plays a crucial role in satisfying the constraints. For the biped under study, the friction constraint determines the critical angular velocity, above which the constraints are violated, but the Center of Pressure (COP) constraint dictates the bounds of the control torque in a large region around upright position.

C

I. INTRODUCTION

of balance is an essential component of bipedal movements. Stabilization of bipedal models during standing has attracted much attention in the past two decades. Various control strategies have been developed, which include adaptive control [1], sliding mode control [2], neural network control [3-5], and fuzzy control [6]. In much of the previous work, bipedal feet were assumed to be fixed on the ground once they contacted the ground, i.e., the constraints between the feet and the ground are assumed to be satisfied automatically, and the controllers were designed only considering the motion regulation. However, the satisfaction of the constraints between the feet and the ground imposes bounds to the control torques [7-10]. Without the considerations of the control bounds, the controllers may cause the violation of the constraints. The effects of constraints on bipedal locomotion have not been investigated rigorously. Research on such effects is important for understand the mechanics of bipedal balance control and the interactions between the biped and the environment, which is crucial for developing bipedal robots. ONTROL

ZMP coincide [11]. Thus, the tip-over constraint is equivalent to the COP constraint as both feet are on the same level ground. The effects of the gravity and friction constraints have been studied in our previous work [10]. In this paper, we consider the COP constraint and the interactions among the gravity, friction and COP constraints. The bipedal model is simplified as an inverted pendulum with a rigid foot-link. The biped is assumed to move in a sagittal plane. The first objective of this work is to determine the control bounds when satisfying the above three constraints. The second objective is to explore the effects of the constraints on bipedal standing. Questions to be answered are: what are the physical indications of the control bounds? In addition to the bounds imposed to the control torque, are there any other conditions imposed to the bipedal systems during balance control of standing? How do the constraints interact? Which constraint is more dominant? The paper is organized as follows. Section 2 describes dynamic equations of the bipedal model and the constraint inequalities. Since the effects of the gravity and friction constraints have been studied in our previous work [10], the effects of the COP constraint is presented in section 3. Section 4 discusses the effects of all constraints, followed by conclusions in Section 5. II. DYNAMIC EQUATIONS AND CONSTRAINTS The simplified two-link bipedal model includes a foot link, which provides a base of support on the ground and an inverted pendulum representing the legs, trunk, arms and head as shown in Fig. 1. Y L

T

There are four constraints between the feet and the ground during bipedal standing, namely, the gravity constraint, i.e., the feet do not lift from the ground; the friction constraint, i.e., the feet do not slide; the Center of Pressure (COP) constraint, i.e., the center of pressure is within the contact surface between the feet and the ground, and the tip-over constraint, i.e., the biped does not rotate around the toe or the heel. The tip-over constraint can be characterized by the zero moment point (ZMP) when both feet are on the same level ground. For this case, it has been proven that COP and Manuscript received September 15, 2005. Caixia Yang (e-mail: [email protected]) and Qiong Wu (corresponding author, phone: 1-204-4748843; fax: 1-204-2757507; e-mail: [email protected]) are with the Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, MB, R3T 5V6, Canada

1-4244-0210-7/06/$20.00 ©2006 IEEE

r

toe Fgy

mg

W

mfg

b

Fgx

heel

Xcop

X

c a Lf

Fig. 1. Simplified bipedal model The foot position is assumed to be bilaterally symmetric and stationary, and the biped moves in the sagittal plane. The dynamic equations are developed using Euler-Lagrangian Equation and are shown below:

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mgr sin T  ( I  mr 2 )T mrT cos T  mrT 2 sin T

W Fgx

(m f  m) g  mrTsin T  mrT 2 cos T

Fgy x cop

Lf a

(1a)

'4

(1b)

I  mr 2

(1c)

bFgx  W  cm f g

where Fgx and Fgy are the horizontal and vertical ground reaction forces. W , T , T and T are the ankle torque (counter clockwise as "+"), angular displacement, velocity and acceleration of the body (clockwise as "+"), respectively. The parameters a , b , c , x cop , r , L , L f , m f and m are the horizontal distance between the ankle

and the heel, ankle height, horizontal distance between the mass center of the foot and the ankle, distance between the center of pressure and the toe, distance between the center of mass of the body and the ankle, length of pendulum, length of the foot, mass of the foot and mass of the body, respectively. The origin of the fixed coordinate system is located at the toe. The x-axis is horizontal pointing from the toe to the heel, and the y-axis is upward. Note that with a positive angle, T , the biped leans posteriorly (clockwise rotation) and with a negative T , the biped leans anteriorly (counter clockwise rotation). Three constraints can be written as: (2a) Fgy t 0 Fgx d PFgy

(2b)

0 d x cop d L f

(2c)

In this section, we will determine the range of the ankle torque, which guarantee the satisfaction of the COP constraint. Some insights into the effect of COP constraint on the bipedal system are also discussed. The COP constraint requires that the COP reside within the boundary of support. Inequality (2c) can be divided into two parts: (3a) x cop d L f Substituting equation (1d) into considering that Fgy t 0 , we have

'4

mr (a sin T  b cosT )  ( I  mr 2 )

(5c)

'5

I  mr  mr[((L

(5d)

aFgy  bFgx  W  cm f g t 0

( L f  a ) Fgy  bFgx  W  cm f g t 0

(3),

2

f

 a ) sin T  b cosT ]

For a general bipedal model, it is not unreasonable to assume that the distance between the ankle and the mass center of the body is greater than the length of the foot, i.e., r ! L f , thus r ! a . As  90 o d T d 90 o , mrb cos T t 0 , mrasinT  mr 2 and mr ( L f  a ) sin T d mr ( L f  a) d mr 2 .

Thus ' 4  0 and ' 5 ! 0 . From inequalities (5a) and (5b), we have the bounds of control torque (6a) W cop  toe d W d W cop  heel where W coptoe and W copheel are ankle torques when the COP is at the toe and at the heel, respectively. They can be written as: W cop  heel

1 {mrmgr sin T ( a sin T  b cos T )  ( I  mr 2 ) '4 [ mr T ( a cos T  b sin T )  ( 2

W cop toe

Lf mf 2

1 {mrmgr sin T [( L f  a ) sin T  b cos T ] '5  ( I  mr 2 ) mr T 2 [( L  a ) cos T  b sin T ]  ( I  mr 2 )[

Lf mf 2

(6b)

 am ) g ]}

f

(6c)

 ( L f  a )m]g}

should be held. By examining W cop  heel  W cop  toe

and

1 { ' 5 { mrmrg sin T ( a sin T  b cos T ) ' 4'5

Lfmf  ( I  mr 2 )[ mr T 2 ( a cos T  b sin T )  (  am ) g ]} 2  ' 4 { mrmrg sin T [( L f  a ) sin T  b cos T ]  ( I  mr 2 ) mr T 2 [( L f  a ) cos T  b sin T ]

(3b) inequality



To satisfy the COP constraint, condition W cop  heel ! W cop toe

CENTER OF PRESSURE CONSTRAINT AND ITS EFFECT

x cop t 0



(5a)

where

The effects of the friction and gravity constraints on balance control of bipedal standing have been investigated in our previous work [10]. The results directly related to this studied are summarized in Appendix A. III.

mr a sin T  b cos T mgr sin T  mrT 2 I  mr 2 a cos T  b sin T  a m f  m g  cm f g

mr[(L f  a) sinT  b cosT ] '5 Wt mgr sinT  mrT 2 2 (5b) I  mr I  mr2 [(L f  a) cosT  b sinT ]  ( L f  a)(m f  m) g  cmf g

(1d)

Fgy

Wt

 ( I  mr 2 )[

Lfmf 2

 ( L f  a ) m ] g }}

(4a)

with ' 4 ' 5  0 , the following condition must be satisfied: T  T (7a)

(4b)

Where

Substituting equations (1b) and (1c) into inequality (4) and combining with (1a), we have

cr1

Tcr1

­mrb(m f  m) cosT ½ g °° °° ® mrcmf sinT ¾ (7b) mrM cosT  mrb ° 2 2 2 ° ¯° M m f  m  m r sin T ¿°



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and M

I  mr 2 .

Inequality (7a) indicates that the angular velocity of the biped plays an important role in keeping the COP within the base of support. If the angular velocity is higher than the critical value, Tcr1 , the condition W cop  heel ! W cop toe will be violated. So regardless of the ankle torque, the COP constraint will be violated. IV. EFFECTS OF ALL CONSTRAINTS In this section, we will determine the bounds of the control torque satisfying all constraints and investigate their effects on bipedal standing. For simplicity, the angle of the biped is restricted within the critical angle rT * which are defined in (A7) and (A8) in the Appendix A. The gravity constraint is not in effect since it is always satisfied when the friction constraint is satisfied [10]. The control torque should be: max(W slipanterior,W coptoe ) d W d min(W slip posterior,W copheel )

(8)

where W slip  posterior and W slip  anterior are the ankle torque at the on-set that the foot-link slips posteriorly and anteriorly, respectively, and they are defined in equations (A12), (A13) in the Appendix A. W cop  heel and W cop toe are defined in equations (6b) and (6c). Inequality (8) is valid under the condition that the upper bound of the ankle torque is higher than the lower bound, which requires

T  min(Tcr1 , Tcr 2 )

(9)

Where Tcr1 and Tcr 2 are critical angular velocities determined by the COP and the friction constraints, respectively, and are defined in equations (7b) and (A11). Due to the complexity of the mathematical forms of the torques, the numerical results are explored. The parameters from previous literature [7] are used here and are listed in Table I. A computer program is developed to determine the minimum between W slip posterior and W copheel , and the maximum between

W slipanterior and W coptoe . Next min(W slip  posterior , W cop heel ) and max(W slip  anterior , W coptoe ) are compared to assure the validity of the control T  min(T , T ) , W cr1

cr 2

bounds.

slip posterior

Note that as  W slipanterior ! 0 , and

W copheel  W coptoe ! 0 . The regions in the phase plane satisfying the three constraints are shown in Fig. 2. Note that with the friction coefficient of 0.5, the critical angles, rT * defined in (A7) and (A8), are r 63o . Curves a and b are the critical angular velocities associated with the COP constraint and the friction constraint, respectively. We first found that, with the parameters listed in Table I, it is always valid that Tcr1 ! Tcr 2 . Thus, for the biped under study, the critical angular velocity is determined by the

friction constraint. If the angular velocity is higher than Tcr 2 , the constraints cannot be satisfied regardless of the control torque. This result indicates the importance to keep the angular velocity of the biped below the critical value. Validity of the bounds, i.e., the upper control bound is higher than the lower control bound in (8), indicates the satisfaction of the constraints, and the possibility of designing a control law to satisfy the constraints. The regions with valid control bounds are shown in Fig. 2 as regions 1-5 in white color, while the regions with invalid control bounds are shown in gray color. Regions 1-5 are defined by different control bounds as shown in Table II. The results show that if the states of the biped fall in the white regions, it is possible to design a controller to satisfy all three constraints, whereas in the gray regions, regardless of the control torque, the constraints will be violated. Thus stabilization of biped during standing is impossible. It is interesting to see that region 2 is the largest and most important region since the biped will reside in region 2 ultimately. Simulation results show that the control bounds are determined by the COP constraint when the angular velocity is below the critical angular velocity Tcr 2 which is determined by the friction constraint. This result shows that the COP constraint is the most dominant among the three constraints. The ranges of the control torque versus the angle and the angular velocity of the biped are shown in Fig. 3. The smooth grey surfaces are the lower control bounds and the grid surfaces are the upper control bounds. Since the sign of the angular velocity has no effects on the control bounds, the ranges of the control torque with positive angular velocity are presented. Figures 3a and 3b show the ranges of the torque with the angle from  63o to 63 o and the angular velocities from 0 to 1.5rad / s and from 1.5 to 2rad / s , respectively. Figures 3c and 3d show the angle from  40 o to 40 o and the angular velocities from 2 to 2.5rad / s and from 2.5 to 3rad / s , respectively. As the angular velocity is below 2.8rad/s as shown in Fig. 3a-3d, the upper bound of the control torque, determined by the constraint that the COP remains in front of the heel, is not sensitive to the changes in the angle and angular velocity of the biped, except when the angle, T , is below  20o and the angular velocity, T , is below 1.5rad/s as shown in Fig. 3a. However, the lower bound of the control torque is sensitive to the changes in both the angle and angular velocity of the biped as shown in Fig.3. Such a lower bound is determined by the constraint that the COP remains behind the toe, which increases with both the angle and the angular velocity. Such an increase in the lower bound reduces the range of the control torque, which makes the control design more challenging. Figure 3 shows the effects of the angle and angular velocity of the biped on balance control of bipedal standing. Low angular velocities not only ensure the satisfaction of the constraints, but also enlarge the range of the control torque, which makes the balance control feasible. On the other hand, large angle of the biped reduces the

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range of the ankle torque, which makes the control design challenging. V.

DISCUSSIONS

During balance control of bipedal standing, the constraints between the feet and the ground must be maintained. The satisfaction of these constraints imposes bounds on the control torque, which is crucial for designing balance control. In this work, the effects of three important constraints, namely the gravity constraint, the friction constraint and the COP constraint, on balance control of a planar biped during standing are investigated. Together with our previous work [10], analytical solutions to the bounds of the controlled ankle torque satisfying each individual constraint are determined. The control bounds satisfying all three constraints are also obtained using a numerical method. Regions in the phase plane satisfying all constraints are determined. More importantly, regions that can not satisfy the constraints are also identified. If the states of the biped fall in such regions, regardless of the control torque, the constraints will be violated and stabilization cannot be achieved. The changes in the range of the control torque versus the states are also obtained. A large range of the control torque is desirable since it makes the control design feasible. Furthermore, the results of control bounds show explicitly the specific constraint causing such bounds, which, in turn, predicts the potential movement of the foot once the constraint is violated. For example, in Region 2, with the ankle torque higher than the upper control bound, i.e., W ! W copheel , the COP (ZMP) will move behind the heel and the foot-link will tip over about the heel, while with a ankle torque below the lower control bound in the same region, i.e., W  W coptoe , the COP (ZMP) will move ahead of the toe and the foot-link will turn over about the toe. The solutions to the control bounds are important not only for designing balance control laws, but also for preparing protective measures for the bipedal robots. Through this work, we found that the satisfaction of the constraints imposes not only the bounds to the control torque, but also conditions to the angular velocity of the biped. The critical angular velocity has been determined analytically. It is further found that the angular velocity dictates the nature of the bounds to the control torque, i.e., as the angular velocity varies, the bounds of the control torque are resulted from different constraints. More importantly, in a region around the upright position within the critical angles, if the angular velocity is above the critical one, regardless of the control torque, the constraints cannot be satisfied. This region is important since the biped will reside in it ultimately. Thus, keeping the angular velocity below the critical value is crucial in balance control. Comparing the friction constraint and the COP constraint in the region T * d T d T * , the critical angular velocity for the biped under study is determined by the friction constraint, indicating the importance of the friction constraint. However, as the angular velocity is below the critical value,

the bounds of the control torque are determined only by the COP constraint in the largest and most important region 2. Therefore, the COP constraint is the most dominant among three constraints when the angular velocity is kept below the critical value. This finding shows the importance of the foot design in satisfying all constraints in balance control of bipedal standing. The work, presented in this paper, is a comprehensive investigation of the effects of the constraints on the bipedal standing. The bounds of the control toque have significant impact on designing balance control strategies. Furthermore, our results also reveal interesting physical insights into the effects of constraints on bipedal standing and into the interactions among the constraints and between the biped and the environment. Such insights allow us to understand the mechanics of bipedal balance control, which has positive impact on developing better bipedal walking robots. As for the future work, we are considering more thorough parametric analysis when all constraints are considered. REFERENCES [1]

Chew, C. M. Pratt, G.A. “Adaptation to load variations of a planar biped: Height control using robust adaptive control,” Robotics and Autonomous Systems 35(1), pp. 1-22, Apr. 30, 2001 [2] Mu, X. and Wu, Q. “Development of a complete dynamic model of a planar five-link biped and sliding mode control of its locomotion during the double support phase,” International Journal of Control 77(8), pp. 789-799, May 2004 [3] Kuperstein , M. and Wang, J. “Neural Controller for Addaptive Movements with Unforeseen Payloads,” IEEE Trans. Neural Networks 1(1), pp. 137-142, 1990. [4] Narendra , K. S. and Parthasarathy, K. “Identification and control of dynamic systems using neural networks,” IEEE Trans. Neural Networks, pp. 4-27, Mar. 1990. [5] Bersini, H. and Gorrini, V. “A Simplification of the BackpropagationThrough-Time Algorithm for Optimal Neurocontrol,” IEEE, Trans. On Neural Networks 8(2), pp. 437-441, Mar. 1997. [6] Zhi Liu and Chunwen Li “Fuzzy Neural Networks Quadratic Stabilization Output Feedback Control for Biped Robots via H_infiniti Approach” IEEE Trans. On Systems, Man, And Cybernetics—Part B: Cybernetics 33(1), Feb. 2003. [7] Pai, Y. C. and Patton, J. “Center of mass velocity-position predictions for balance control,” Journal of Biomechanics 30(4), pp. 347-354, 1997. [8] Pai, Y.C. and Iqbal, K., “Simulated movement termination for balance recovery: can movement strategies be sought to maintain stability in the presence of slipping or forced sliding?” Journal of Biomechanics 32, pp.779-786, 1999. [9] Iqbal, K. and Pai, Y.C., “Predicted region of stability for balance recovery: motion at the knee joint can improve termination of forward movement,” Journal of Biomechanics 33, pp. 1619-1627, 2000. [10] Yang, C and Wu, Q., “Effects of gravity and friction constraints on bipedal balance control,” Proceedings of 2005 IEEE International Conference on Control Applications, pp. 1093-1098, 2005. [11] Vukobratovic, M. and Borovac, B., “Zero-Moment Point-Thirty Years of its Life,” International Journal of Humanoid Robotics 1(1), pp.157173, 2004

Appendix A

In our previous work [10], it was proven that the friction constraint is more important than the gravity constraint, i.e., once the friction constraint is satisfied, the gravity constraint will be automatically satisfied. In this section, only the

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results of friction constraint are summarized here. The friction constraint, shown in inequality (2b), can be divided into two parts as follows: (A1) Fgx t  PFgy (A2)

Fgx d PFgy

The satisfaction of inequality (A1) indicates that slipping posteriorly of the foot (slipping along the positive x-axis as shown in Fig. 1) is prevented. Similarly, the satisfaction of inequality (A2) indicates that slipping anteriorly (slipping along the negative x-axis) is prevented. The procedure for determining the bounds of the ankle torque without slipping is that the angular velocity, T , is first solved from equation (1a), and is then substituted into equations (1b) and (1c). Considering inequalities (A1) and (A2), we have: '2 (A3) (W  mgrsinT ) d P(mf  m)g  mrT2 (sinT  P cosT ) I  mr2 '3 (W  mgrsinT ) t mrT2 (P cosT  sinT )  P(mf  m)g (A4) I  mr2 where ' 2 mr cos T  P sin T (A5) mr cos T  P sin T

(A6) In order to solve the control torque, W , we need to divide '2 both sides of inequality (A3) by and inequality I  mr 2 '3 (A4) by . Such divisions lead to two critical angles I  mr 2 at ' 2 0 and ' 3 0 , which are calculated below: '3

1

T cr1

T * tan 1 ( ) P

T cr 2

T *

W slip  anterior

T 2 d

Tcr2 2 (T *) P

( m f  m) g

mr ( P cos T  sin T )

1 P 2

mr

( m f  m) g

I  mr [mrT 2 ( P cos T  sin T ) '3 (A13)

In region T *  T  T * , the biped is around the upright position, from inequalities (A3) and (A4), we have (A14) W slip  anterior d W d W slip  posterior The inequality (A14) requires that W slip posterior ! W slip anterior . By examining W slip  posterior  W slip  anterior , we found that the following condition must be satisfied:

T  Tcr 2 (T )

(A15)

where Tcr 2 is given in equation (A11). Inequality (A15) shows that as the biped is around the upright position (within r T * ), the angular velocity must be below Tcr 2 . Otherwise, regardless of the control torque, the friction constraint will be violated. Under the condition of (A15), inequality (A14) shows the bounds for the ankle torque in which the friction constraint will be satisfied.

(A10)

cos(T * ) Tcr2 2 (T *)

mr where the critical angular velocity, Tcr 2 (T ) , is defined as:

Tcr 2 (T )

mgr sin T 

 P ( m f  m) g ]

(A8)

cos T * mr P ( m f  m) g

(A12)

2

P As the biped is at the critical angles, inequalities (A3) and (A4) lead to ( m f  m) g P ( m f  m) g P T 2 d mr ( P cos T  sin T ) mr 1 P 2 (A9) ( m f  m) g

I  mr 2 [ P ( m f  m) g '2

 mrT 2 (sin T  P cos T )]

(A7)

1  tan 1 ( )

mgr sin T 

W slip  posterior

( m f  m) g

cos T (A11) mr As the biped is not at the critical angles, torques W slip posterior and W slip anterior can be determined, based on

inequalities (A3) and (A4), respectively. Thus we have 2514

TABLE I. PHYSICAL PARAMETERS USED IN THE NUMERICAL ANALYSIS

Body height

H = 1.78 m

Body mass

mass = 80 kg

Feet mass

mf = 2*0.0145*mass = 2.32 kg

Pendulum mass

m = mass - mf = 77.68 kg

Length of ankle-tocenter of mass

r = 0.575*H = 1.02 m

Feet length

Lf = 0.152*H = 0.27 m

Horizontal ankle-toheel distance

a = 0.19*Lf = 0.05 m

Vertical ankle height

b = 0.039*H = 0.07 m

Horizontal ankle-tocenter of foot

c = 0.5*Lf – a = 0.085 m

Pendulum length

L = H – b = 1.71 m

Coefficient of friction

P = 0.5

Gravity acceleration

g = 9.80 m/s2

TABLE II. THE CONTROL BOUNDS SATISFYING CONSTRAINTS Regions

Control bounds

Region1

W coptoe  W  W slip  posterior

Region2

W coptoe  W  W copheel

Region3

W slip anterior  W  W cop heel

Region4

W coptoe  W  W slip  posterior

Region5

W slip anterior  W  W cop heel

(b)

(c)

Fig. 2. Regions in the phase plane

(d)

(a)

Fig. 3. Ranges of the control torques

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