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IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 20, NO. 3, JUNE 2004

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A Novel Gait Generation for Biped Walking Robots Based on Mechanical Energy Constraint Fumihiko Asano, Masaki Yamakita, Norihiro Kamamichi, and Zhi-Wei Luo Abstract—This paper proposes novel energy-based gait generation and control methods for biped robots based on an analysis of passive dynamic walking. First, we discuss the essence of dynamic walking using a passive walker on a gentle slope from the mechanical energy point of view. Second, we propose a simple and effective gait-generation method, which imitates the energy behavior in every walking cycle considering the zero-moment point condition and other factors of the active walker. The control strategy is formed by taking into account the features of mechanical energy dissipation and restoration. Following the proposed method, the robot can exhibit a natural and reasonable walk on a level ground without any gait planning and design in advance. The effectiveness of the method is examined through numerical simulations and experiments.

Fig. 1.

Experimental walking machine and its ideal model. TABLE I PARAMETERS OF THE EXPERIMENTAL MACHINE

Index Terms—Biped robots, gait generation, legged locomotion, mechanical energy, passive dynamic walking.

I. INTRODUCTION The control problem of biped locomotion has been studied and experimentally demonstrated by many researchers. Among them, the study of Kajita et al. [10] is especially excellent. The linear inverted pendulum mode (LIPM) proposed by them has been used by many researchers and successfully applied to actual walking machines. In this method, the desired trajectories are specified by a potential energy-conserving orbit, and the complexity of the biped dynamics can be reduced. Recently, this method has been extended to a three-dimensional (3-D) version, and its experimental result using a 12–degree-of-freedom (DOF) walking machine has been reported [11]. On the other hand, McGeer experimentally studied a simple unpowered walking machine, which is well known as passive dynamic walking [9], and nowadays its concept is familiar. After McGeer’s work, Goswami et al. proposed an energy-based control law as an application of passive walking to the active one on a level ground [3], [4]. Spong has also studied and proposed an energy-shaping control method based on passive dynamic walking [15]. In their methods, however, stability of the zero-moment point (ZMP) was not investigated enough. The control strategy based on passive dynamic walking has the property of automatic gait generation. It realizes dynamics-based control without any approximations such as linearization or disregard of leg mass. This type of study will be done widely in the future. The importance of transition impacts in legged locomotion has been mentioned and analyzed by Hurmuzlu et al. [7], [8]. Dissipation of the mechanical energy at the transition instant strongly affects the gait sta-

bility. In passive dynamic walking, this phenomenon determines the shape of following gait. LIPM does not take into account the dissipation mechanism enough and is a synthesis method, considering mainly the single support phase. Recently, Grizzle et al. studied the stability of a bipedal gait from a hybrid system point of view using an extended Poincaré’s section method [6]. Their synthesis method provides a reduction form of gait-stability analysis. In this paper, we propose novel gait generation and control laws based on the mechanical energy of the walking robots. First, we analyze the mechanism of passive dynamic walking anew and consider its essence from the mechanical energy point of view. From the simulation results, we can see that the mechanical energy has a simple pattern, including dissipation and restoration. Second, we introduce a simple control law called energy-constraint control (ECC) for the active walkers, which imitates the energy mechanism of passive dynamic walking. By the effect of the proposed control with an energy-constraint condition, a stable walking pattern can be generated easily without any gait planning and design in advance. Furthermore, we consider an energy-feedback control law using a reference energy trajectory, and we analyze the gait stability. The validity of our method is investigated through numerical simulations and experiments. Finally, an extension of ECC to a kneed biped is considered, and the successful realization of dynamic walking is demonstrated via numerical simulations. II. COMPASS-GAIT BIPED ROBOT

Manuscript received December 18, 2002; revised April 8, 2003. This paper was recommended for publication by Associate Editor H. Arai and Editor A. De Luca upon evaluation of the reviewers’ comments. This work was supported in part by the Ministry of Education, Science, and Culture of Japan under Grant-in-Aid for COE Research #09CE2004. This paper was presented in part at the IEEE/RSJ International Conference on Intelligent Robots and Systems, Lausanne, Switzerland, September 30-October 4, 2002. F. Asano and Z. W. Luo are with the Bio-Mimetic Control Research Center, The Institute of Physical and Chemical Research (RIKEN), Nagoya 463-0003, Japan (e-mail: [email protected]; [email protected]). M. Yamakita and N. Kamamichi are with the Tokyo Institute of Technology, Tokyo 152-8552, Japan (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TRA.2004.824685

In this paper, a simplest planar 2-link full-actuated walking model, called the compass-gait walker, is chosen as the control object. Fig. 1 (left) shows the experimental walking machine, which was designed as a nearly ideal compass-gait biped model. This robot has three dc motors with encoders in the hip block to reduce the weight of the legs. The ankle joints are driven by the motors via timing belts. Table I lists the values of the robot parameters. Fig. 1 (right) shows the simplest ideal compass-gait model of the experimental machine, where mH , m kg and l = a + b m are the hip mass, leg mass, and leg length, respectively. Its dynamic equation is given by

1042-296X/04$20.00 © 2004 IEEE

M ( ) + C ( ; _ )_ + g ( ) =

(1)

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Fig. 2. Mechanism of gravity in passive dynamic walking.

where = [ 1 2 ]T is the angle vector of the robot’s configuration, and the details of the matrices are as follows:

mH 12 + ml2 + ma2 M () = 0mbl cos(1 0 2 )

C ( ; _ ) = g ( ) =

0

0mbl cos(1 0 2 )

mb2 0mbl sin(1 0 2 )_2

mbl sin(1 0 2 )_1 0(mH l + ml + ma) sin 1 g mb sin 2

Fig. 3. Virtual transformed torque inputs and energy during passive walking. (a) Virtual transformed torque inputs. (b) E .

0

0mb cos(2 0 )g cos :

and the control torque vector is

= Su =

1

u1 u2

1

01

0

=

u1 + u2 0 u2 :

(2)

The transition is assumed to be inelastic and without slipping. With the assumption and based on the law of conservation of angular momentum, we can derive the following compact equation between the preimpact and postimpact angular velocities:

Q+ ()_ where

Q + ( ) = Q 0 ( ) =

+

=

Q0 ()_

0

(3)

mH l2 + ma2 + ml(l 0 b cos 2) mb(b 0 l cos 2) 0mbl cos 2 mb2 2 (mH l + 2mal) cos 2 0 mab 0mab 0mab 0

and is the hip angle at the transition instant given by

0 0 = 1 0 2 2

=

2+ 0 1+ > 0: 2

For further detail derivations, the reader should refer to [5]. III. PASSIVE-WALKING MECHANISM REVISITED It has been considered that passive dynamic walking suggests the essence of dynamic walking. Therefore, it is important to analyze the nature of dynamic walking using a passive walker as a reference of the automatic gait generator. The impulsive transition feature, without a double-support phase, can be intuitively regarded as a vigor for fast and energy-effective walking. In order to get the vigor, the walking machine must store the mechanical energy, and the impulsive and inelastic collision with the ground makes the energy dissipate discontinuously. This section exploits the essence of dynamic biped walking from the mechanical energy point of view. Let us define virtual total mechanical energy E under the gravity condition of Fig. 2 (right) as follows: T E := 1 _ M ( )_ + P ( ; ) (4) 2

where the potential energy is given by

P ( ; ) = (mH l + ml + ma) cos(1 0 )g cos

The virtual transformed torque u1 and u2 are given as the transformed effect of the horizontal gravity element g sin . Fig. 3 shows the simulation results of passive dynamic walking on a gentle slope whose angle is 0.005 rad. Fig. 3(a) and (b) show the evolution of transformed torque and virtual energy E , respectively. From (a), we can see that both u1 and u2 are almost constant-like, and thus, ZMP is also kept within a narrow range, as will be shown in (10). This property is effective in the virtual passive walking on a level ground from the postural stability point of view [1]. It is apparent from (b) that the mechanical energy is dissipated at the transition instant and restored during the swing phase. This energy mechanism can be considered as an essential characteristic of dynamic walking and the factor of vigor. In the case of steady walking, since the potential energy at every double-support phase is the same, the restored energy is only the kinetic one. Fig. 4 shows the trajectories of a steady passive dynamic walking pattern with respect to time, where = 0:03 rad. In general, we can state the following. CH1) The total mechanical energy of the robot E increases monotonically during the swing phase. CH2) _1 > 0 always holds. CH3) There exists an instant when _1 0 _2 = 0. CH1 and CH2 always hold, regardless of physical and initial conditions, but CH3 does not always hold, as it depends on physical parameters and slope angle. We can confirm CH2 and CH3 from Fig. 4. It is also clear that CH1 is satisfied from Fig. 3(b). From the results, the essence of dynamic walking based on passive walking can be summarized as follows. E1) The walking pattern is generated automatically, including impulsive transitions, and converges to a steady limit cycle. E2) The total energy is restored during the swing phase monotonically, and is dissipated at every transition instant impulsively. E2 is considered to be an important characteristic in gait generation, and is the basic concept of our method. The details will be described in Section IV. IV. ENERGY-CONSTRAINT CONTROL In our previous work, we have proposed virtual passive walking, considering an artificial gravity condition called virtual gravity [1]. This


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Fig. 5. Mechanism of the foot part and ZMP.

Fig. 6.

Reaction moment at the heel-strike instant.

From (6) and the target energy-constraint condition of (7), we then get T _ S u = _1 u1 + _1 Fig. 4.

Steady passive-walking cycle.

imitates the gravity mechanism in the original passive walking. A virtual gravity toward the walking direction is a driving force for the robot, and the stable limit cycle can be generated automatically without any gait design in advance. The validity was also confirmed experimentally [17]. The determination of a virtual gravity is, however, equivalent to that of control inputs. So there is no freedom to control other factors, i.e., ZMP. From the analysis of the passive walking, we can deduce some natural and effective gait-generation strategy. Based on the nature of monotonic increase of energy, we can formulate a simple control strategy for the compass-gait biped. A. Control Law The total mechanical energy of the robot can be expressed as 1 T E = _ M ( )_ + P ( ) 2

(5)

where P is the potential energy. The power input to the system is the time rate of the total energy, that is T T (6) E_ = _ = _ S u: Suppose now that we use a simple control law which imitates the characteristic CH1. Let > 0 be a positive constant and consider the following condition: E_ = :

(7)

This means that the robot’s energy increases monotonically with a constant rate. We call this control method ECC. In this method, the walking speed becomes faster with respect to (w.r.t.) the increase of , in other words, the parameter corresponds to the slope angle in passive dynamic walking. Let us consider the following output function: T H ( ) := E_ 0 = _ 0 (8) and the target control constraint can be rewritten as H ( ) = 0. Therefore, the ECC is, in other words, the output zeroing control.

0 _2

u2 =

(9)

an indeterminate equation. Therefore, the control problem yields how to solve this equation for the control inputs u1 and u2 in real time. The property of this control strategy is that the control inputs can be determined easily by only adjusting the feedforward parameter , and T this value can be determined by considering E_ = _ in virtual passive dynamic walking. The control policy is, however, how to determine in real time, but this problem is still open and must be investigated in the future. B. Relation Between ZMP and Reaction Moment The actual walking machine has foot parts and a problem of reaction moment is generated. The geometrical specifications of the stance leg and foot part is shown in Fig. 5. In this paper, the ZMP is calculated by the following approach. We assume: 1) the mass and volume of the foot part can be ignored; 2) the tip of the stance leg is always connected with the ground. Under these assumptions, we can calculate the ZMP in the coordinate shown in Fig. 5 as ZMP =

0 Ru1n

(10)

where u1 is the ankle torque acting not on the foot link but on the leg link, and Rn is the vertical element of the reaction force which is given by Rn =

0(mH l + ml + ma) _12 cos 1 + 1 sin 1 0mb _22 cos 2 + 2 sin 2 + (mH + 2m)g:

From Fig. 5, it is obvious that the ZMP is always shifted behind the ankle joint, however, at the transition instant, the robot is critically affected by the reaction moment from the floor, as shown in Fig. 6. Considering the reaction-moment effect, we can reform the ZMP equation for the simplest model as follows: u1 + urm ZMP = 0 Rn

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where urm > 0 is the equivalent torque of the reaction moment, and due to this, the ZMP is shifted further backward. This phenomenon can be considered to obstruct the transition. Since the actual walking machines generally have foot parts with toe and heel, such problems sometimes arise. From the aforementioned point of view, we conclude that the ZMP should be shifted forward on the ankle joint just after the transition instant to cancel the reaction moment. Based on the observation, in the following, we consider an intuitive ZMP manipulation algorithm using the freedom of solution of the indeterminate (9).

carefully so as not to destroy the original limit cycle. The combination between and u+ is most important for generating a stable limit cycle. Although a precise impedance-control law considering the impulsive collision effect has been studied by Park et al. [13], [14], this paper does not treat the problem as they do. In this paper, the authors would like to emphasize manipulability of the ZMP using the freedom of solution of the indeterminate equation, rather than strict analysis of reaction moment. We would like to yield another opportunity to study the detailed ZMP manipulation.

C. Principal Ankle-Torque Control

D. Principal Hip-Torque Control

From a practical point of view, as mentioned above, the two most important control factors of dynamic biped walking are mechanical-energy restoration and ZMP control. To keep the energy-constraint condition, we should reconsider the solution algorithm. First, we must realize energy restoration (i.e., gait generation), and second, the ZMP condition must be guaranteed without destroying the energy-restoration condition. Based on the considerations, we first discuss the following approach: 1) determine the value of ; 2) determine the ankle torque u1 ; 3) by substituting and u1 into (9), we can solve it for u2 . In order to shift the ZMP, let us consider the following simple ankletorque control:

As mentioned before, we must switch the controller (i.e., solution algorithm) to avoid the singularity CH3 in a cycle. As a new method, we propose the following new strategy: 1) determine the value of ; 2) determine the hip torque u2 ; 3) by substituting and u2 into (9), we can solve it for u1 . In this case, u1 is obtained as follows:

u1 =

u0 < 0; if s T u+ > 0; if s > T

0 _1 u6 : _1 0 _2

(11)

Therefore, we can determine the hip torque uniquely according to and u1 . However, u2 has a singularity at _1 0 _2 = 0, which was mentioned before as CH3. This condition must be taken into account, and we propose a switching control law, described later. Before it, we consider a more reasonable switching algorithm from u0 to u+ . In general, for the most part of a cycle from the beginning, the condition _1 0 _2 > 0 holds (see Fig. 4). Hence, the sign of u2 is identical with that of 0 _1 u1 . If u1 = u0 , this sign is positive because of , _1 > 0 and u0 < 0. At the beginning of a cycle, 0 _1 u+ increases monotonically because of 1 < 0 (see Fig. 4) and

d 0 _1 u+ = 01 u+ > 0: dt This does not link with a special choice of the control parameters. Therefore, if < _1 u+ holds at the beginning, it is reasonable to switch when 0 _1 u+ = 0 so as to keep u2 given by (11) always positive under the condition _1 0 _2 > 0. According to this method, the hip torque can always contribute the energy restoration because of _1 0 _2 u2 > 0. The switching algorithm of u1 is summarized as follows:

u1 =

u0 < 0; if _1 u+ u+ > 0; if > _1 u+ .

0 _1 0 _2 u2 : _1

(12)

Note that here we use the assumption of CH2. In this paper, we consider u2 in the following form:

u2 = _1 0 _2 :

(13)

Assuming > 0, this leads to the following inequality:

where s is a virtual time that is reset at every transition instant, and s_ = 1:0. This comes from the fact that u1 must be negative to shift the ZMP forward, and if u1 > 0, then the ZMP moves behind the ankle joint. In this case, u2 is obtained as follows:

u2 =

u1 =

The value of u+ can be determined empirically, based on the simulation results of virtual passive dynamic walking. u0 must be determined

_1 0 _2 u2 = _1 0 _2

2

0:

(14)

From (6) and (9), it is clear that the hip torque u2 can always contribute to energy restoration during a cycle, and u2 = 0 when _1 0 _2 = 0, this is reasonable in the energy consumption. E. Switching Control In order to control the ZMP as well as to avoid the singularity, we must consider a switching algorithm from principal ankle- to hip-torque control. As a simple method, we choose the switching timing when 1 = rad in trial. At this instant, we reset so that the control inputs become continuous according to the following relationship:

0 _ sw u+ u2 = _1sw 0 _2sw = sw 1 sw _1 0 _2 from which we can calculate as follows:

=

0 _1sw u+

_1sw 0 _2sw

2

where the superscript “sw” stands for the switching instant. The obtained is used during its cycle and reset at every switching instant. Fig. 7 shows the simulation results of the dynamic walking by the proposed switching control from principal ankle- to hip-torque control. The control parameters are chosen as = 0:07 J/s, u+ = 0:15, and u0 = 00:05 N 1 m. Fig. 7(a)–(c) show the evolution of the control inputs, mechanical energy, and ZMP, respectively. By the effect of the principal ankle-torque control, the ZMP is shifted forward on the ankle joint without destroying the energy-restoration condition. From Fig. 7(b), we can see that the hip torque becomes very large as the ZMP is shifted forward because of u1 = u0 , but this does not affect the ZMP, and the postural stability of the foot is maintained. It should be mentioned that the possibility of arbitrary ZMP manipulation under automatic gait generation is a very important property for controlling a biped robot to realize stable dynamic walking.


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Fig. 8. Simulation results of dynamic walking by energy-tracking control where = 10:0, = 10:0, and E = 22:0 J. (a) Control inputs. (b) Total mechanical energy.

Fig. 7. Simulation results of dynamic walking by ECC. (a) Control inputs. (b) Total mechanical energy. (c) ZMP.

F. Discussion Here, we compare our method with Goswami et al.’s approach [3], [4], the energy-tracking control. The proposed energy-control approach in [3], [4] is formulated as T (15) E_ = _ S u = 0(E 0 Ed ) where Ed J (constant) is the reference energy, and positive scalar > 0 is the feedback gain. A solution of (15) by constant torque ratio > 0, which gives the condition u1 = u2 , is obtained as Su =

0 0+11

0

(E E d ) ( + 1)_1 _2

0

(16) Fig. 9. Simulation results of dynamic walking by ECC where and = 10:0. (a) Control inputs. (b) Total mechanical energy.

and in our case, a solution by constant torque ratio is given by Su =

+1

01

( + 1)_1

0 _2

:

(17)

Figs. 8 and 9 show the simulation results of active dynamic walking on a level by the torque given by (16) and (17) without manipulating the ZMP actively (we here do not treat the ZMP manipulation problem). The two cases are equal in walking speed. From the simulation results, we can see that in the case of ECC, the energy consumption is better than those of energy-tracking control. Furthermore, in our approach, the maximum ankle torque is about three times smaller than that of Goswami’s approach, and this yields a better ZMP condition creation. These results imply that the mechanical energy of a walking robot should be restored monotonically, not be converged to a constant value, to generate a natural gait. V. EXPERIMENTAL RESULTS In order to confirm the validity of the proposed method, we carried out an actual walking experiment using our developed machine, intro-

= 0:34 J/s

duced in Fig. 1. The AT-class PC (Pentium III 1.0 GHz) running Windows 98 receives information from all encoders and outputs the control signals to the servomotors. To implement the control law, we used RT MA T X [12] for real-time computation with the sampling period 1.0 ms. Since the proposed methods are called model-matching control, they are not robust for uncertainty. In this research, we use the model-following control of the motion generated by the virtual internal model (VIM) which is a reference model in the computer. Every postimpact condition of VIM is reset to that of the actual machine. By using the VIM, the uncertainties of identification, which is a crucial factor in the case of model-matching control, can be compensated. The dynamics of VIM is given by +C ^ ( d ) M d ^ ( d ; _ d )_ d + g^( d ) = d ; p p d = T _ d p

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Fig. 11.

Eigenvalues and maximum singular values of

rF w.r.t. .

In the case of steady walking, the relation F (x3 ) = x3 holds, and x3 is the equilibrium point of state at just after the transition instant. For xk around the limit cycle, the mapping function a small perturbation x F can be expressed in terms of Taylor series expansion as Fig. 10.

3 F (xk ) = F (x + x xk )

Experimental results. (a) Angular positions. (b) Angular velocities.

x3 + (rF )xxk

(20)

where and the control input for the actual robot is given by

rF :=

^ ( d )u + C ^ ( d ; _ d )_ d + g =M ^( d ) u = d

0 K D _ 0 _ d 0 K P ( 0 d ) 0 K I

(

0 d )dt:

The VIM started walking from the following initial condition: _ d (0) = [ 0:68

0:62 ]

T;

d (0) = [

00:14

0:14 ]

T

and its state was calculated and updated in real time. At every transition instant, the angular positions of VIM were reset to that of the actual machine. The ankle joint of the swing leg is controlled by a proportional-integral-derivative (PID) controller in order to keep the foot posture horizontal. The experimental results are shown in Fig. 10. The tuning parameters are chosen as = 0:075 J/s, u+ = 0:15, u0 = 00:05 N 1 m, and = 0:05 rad empirically. Fig. 10(a) and (b) show the evolution of angular positions and velocities of the actual machine and VIM, respectively. The actual angular velocities are calculated by differentiation thorough a filter whose transfer function is 70=(s + 70). VI. STABILITY ANALYSIS AND ENERGY-FEEDBACK CONTROL Equation (15) implies that the walking system becomes robust through the reference-energy tracking. In other words, this control expands the basin of attraction of a limit cycle. However, our method (17) is called feedforward control, which gives only the energy change ratio without any information to attract the trajectories. Based on the observations, in this section, we first analyze the stability of the walking cycle, and then consider an energy-feedback control law in order to increase the robustness of the walking system. The Poincaré return map is denoted below as F x k+1 = F (xk )

(18)

where the discrete state xk is chosen as x k :=

2+ [k ] 1+ [k ] _1+ [k ] _2+ [k ]

0

@F F (x) (21) @x x x =x is the Jacobian (gradient) w.r.t. x3 . By performing numerical simulations, F can be calculated approximately. Fig. 11 (left) shows the eigenvalues of F (see the = 0:0 case) in the case that the control input is given by (17), where = 10:0 and = 0:30 J/s (ZMP control

r

r

is not considered). Since all eigenvalues are in the unit circle, the discrete walking system is stable, however, it is not robust enough against disturbances and initial conditions as well as the original passive dynamic walking. Next, let us consider an energy-feedback control using a reference energy trajectory. Consider the following control:

T

E_ = _ S u = E_ d + (Ed

that is, relative hip joint angle and angular velocities just after the k th impact. The function F consists of (1) and (3), but cannot be expressed analytically. Therefore, we must compute F by numerical simulation following an approximation algorithm.

(22)

that is, determine the control input so that the closed energy system yields d (E dt

0 Ed (s)) = 0 (E 0 Ed (s))

(23)

where > 0 is the feedback gain. The original ECC can be recognized by choosing E_ d =

(24)

and = 0 in (22). By integrating (24) w.r.t. time, we can obtain the reference energy Ed using virtual time s as Ed (s) = E0 + s

(25)

where E0 J is the energy value when s = 0 s. A solution of (22) using constant torque ratio yields Su = =

(19)

0 E)

+1

01

+1

01

0 E) 0 _2 + (E0 + s 0 E ) : ( + 1)_1 0 _2

E_ d + (Ed (s) ( + 1)_1

(26)

Fig. 12 shows the simulation result of step period of active dynamic walking by energy-feedback control w.r.t. the value of . The adjust parameters are chosen as E0 = 21:86 J, = 0:30 J/s, and = 10:0, respectively. The robot starts from an initial condition close to steady value. From the results, we can confirm the effectiveness of the control law. On the other hand, Fig. 11 (left) shows the eigenvalues of rF w.r.t.


Fig. 12.

Convergence of step period.

Fig. 13.

Response of step period against external force.

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Fig. 14. Kneed biped model.

. We can see that the system is stable, however, the robust performance

cannot be decided clearly. Fig. 11 (right) shows the maximum singular values of rF versus . Magnitude of the maximum singular value imxk k because of x xk , and xk+1 = (rF )x plies convergence speed of kx from the graph we can confirm that the robust performance is improved according to . In order to investigate the robustness against external forces, we also performed numerical simulation of active dynamic walking in the presence of external forces. Fig. 13 shows the simulation result of step period. The robot starts from the steady initial condition, and is pushed by a small force at the hip position to the walking direction. From the result, we can see that the recovery to a steady state is dramatically improved by the control effect. Although autonomy is destroyed by applying this method, we can improve the robustness of the walking system. We should consider other robust control laws which do not destroy the autonomy of the closed walking system. VII. CONTROL OF KNEED BIPED ROBOT This section extends the dynamic walking control for a kneed biped model. A. The Kneed Model Fig. 14 shows the model of a kneed biped walking robot. The robot consists of three links; the swing leg has one upper and lower part (thigh and shank), and the stance leg is assumed to be locked at the knee joint, that is, regarded as a one-link rigid bar. For further details, the reader should refer to [2] and [18].

In the case of a kneed biped, the time derivative of the total mechanical energy is given by

0 _2

u2 + _3

0 _2

u3 :

(27)

Therefore, during the three-link phase, the indeterminate equation E_ = is written as _1 u1 + (_1

Thus, we must determine the other two conditions which are independent of (28). In this paper, the following control objectives are considered: 1) equation (28) is always satisfied; 2) knee joints are controlled and lifted enough to avoid foot scuffing by constant torques, and are mechanically locked (active knee lock). In order to achieve the second control objective, the knee torque u3 should always be determined first during each phase. The knee joint is driven by a constant feedforward torque until the active knee-lock instant during the first stage (see Table II). In order to consider the control algorithm, we divide the walking cycle into five phases, as shown in Fig. 15. Postimpact Phase: We concluded that the ZMP should be shifted forward of the ankle joint just after the transition instant, so in the first stage, the ankle torque u1 should be concretely determined, as well as u3 , in advance. Then we can solve (28) for u2 by the information u1 , u3 and as u2 =

B. The Solution Algorithms

T E_ = _ S u = _1 u1 + _1

TABLE II ORDER OF PRIORITY FOR CONTROL INPUTS

0 _2 )u2 + (_3 0 _2 )u3 = :

(28)

0 _1 u1 0 _1

_3

0 _2

0 _2

u3

:

(29)

The ZMP control interval is chosen as s T s, and the switching time T is chosen empirically. This control algorithm will be changed to principal knee-torque control when s = T s as described below. Three-Link Phase I: Until the active knee-lock instant, principal knee-torque control must be applied continuously. In other words, u3 is given in advance. Then the indeterminate equation in this phase is given by _1 u1 + _1

0 _2

u2 =

0 (_3 0 _2 )u3 :

(30)

572

Fig. 15.


Typical walking step and divided phases.

Thus, we can determine u1 and u2 based on the values of and u3 . To solve (30), we propose a solution using a constant torque ratio > 0. By substituting u1 = u2 into (30), we obtain u2 as u2 =

0

0 _ u ( + 1)_ 0 _

_3

2

1

3

(31)

2

where is determined according to torque limits of the joints. Then the control input can be written as follows: _ ; ; =

0 0

(_ _ )u (+1)_ _

0

u3

+1

01

:

(32)

Therefore, we can choose the parameters u3 , , and to determine the control input . If the control limits are the same for u1 and u2 , they are optimized by a standard constant-optimization method with the constraints u3 and . Modified-Compass Phase: The knee joint is locked mechanically when _2 = _3 (active knee-lock algorithm [2]). After this instant, u3 = 0 N 1 m until active knee-lock off. Since during this phase, the robot seems to be a compass-gait biped with a short swing leg, we call this walking motion “modified compass gait.” By the effect of the knee-lock control, the robot can avoid foot scuffing during the swing phase. In this phase, the indeterminate equation is given by _1 u1 + _1

0 _

2

u2 = :

Fig. 16. Dynamic walking with knees by ECC. TABLE III FEEDFORWARD VALUES OF u

AND

u

(33)

By substituting u1 = u2 into (33), we can solve it for u2 , and is obtained as +1 _ ; ; = (+1)_ 0_ (34) 01 :

the control input is finally determined according to the information of other control inputs, “4” denotes not so important, and “0” denotes mechanical lock, that is, u3 = 0.

This is equivalent to (32) with u3 = 0. Three-Link Phase II: The knee-joint constraint will be systematrad. 1 is reasonable to be used as ically locked off when 1 = a trigger (switching flag) because it increases monotonically during every cycle. We must choose carefully so that the passive knee lock will be done before the heel-strike collision. In this phase, the control input is determined by the same algorithm of the three-link phase I. Compass Phase: In this phase, the control input is calculated by the same algorithm of the modified-compass phase as mentioned before. Table II lists the order of priority for control inputs in each phase. “ ” denotes the control input should be determined first, “2” denotes

In order to investigate the validity of the proposed control law, we demonstrate a dynamic walking simulation (see Fig. 16). The algorithm of the feedforward control inputs u1 and u3 is shown in Table III. The control parameters are chosen as = 5:0 J/s, = 3:8, u10 = 03:0, u31 = 00:30, u32 = 00:40 N 1 m and = 0:0 rad. The control interval is 1.0 ms. From the simulation results, we can see that the total mechanical energy is restored monotonically during every cycle without losing the energy-constraint condition, and the ZMP is shifted forward of the ankle joint by the effect of switching control of u1 (Fig. 17). By the freedom of the control strategy (control inputs), we can design more variation of walking patterns than virtual gravity algorithms.

0

C. Numerical Simulations


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that some effective control laws based on mechanical energy will be proposed. A control strategy partially using reference trajectories might be also effective in making the walking system robust. In addition, the effect of the upper part of the body should also be investigated to extend our method to high-DOF humanoid robots. REFERENCES

Fig. 17. Simulation results of dynamic walking with knees by ECC. (a) Angular positions. (b) Control inputs. (c) Total mechanical energy. (d) ZMP.

VIII. CONCLUSIONS In this paper, as a clue to elucidate a natural and energy-effective walking pattern-generation method, we have considered passive dynamic walking as a reference model. The control-design technique used in this study was shown to be effective in generating a walking pattern, and its validity has been proved by numerical simulations and experiments. The experimental results showed that it is possible to generate a dynamic gait automatically with only a simple constraint condition. The effect of ZMP manipulation, however, has not been confirmed yet. More detailed experimental case studies must be done in the future. The authors think that an energy-restoration concept is worth taking into consideration to generate a natural walking pattern, and will be used widely in the future, as well as a potential energy-conserving orbit. Throughout this paper, we can conclude that it can be considered that a clue of dynamics-based control is not to track desired trajectories, but to move along mechanical energy flow involuntarily. This approach must become stronger and seems to provide a new field of robotic gait synthesis in the future. A problem yet to be solved is to realize robust walking control NOT depending on time which can enable active walking on a rough floor, with unsuitable initial conditions and mass balance. The authors hope

[1] F. Asano, M. Hashimoto, N. Kamamichi, and M. Yamakita, “Extended virtual passive dynamic walking and virtual passivity-mimicking control laws,” in Proc. IEEE Int. Conf. Robotics and Automation, vol. 3, 2001, pp. 3084–3089. [2] F. Asano and M. Yamakita, “Virtual gravity and coupling control for robotic gait synthesis,” IEEE Trans. Syst., Man, Cybern. A, vol. 31, pp. 737–745, June 2001. [3] A. Goswami, B. Espiau, and A. Keramane, “Limit cycles and their stability in a passive bipedal gait,” in Proc. IEEE Int. Conf. Robotics and Automation, 1996, pp. 246–251. , “Limit cycles in a passive compass gait biped and passivity-mim[4] icking control laws,” Auton. Robots, vol. 4, no. 3, pp. 273–286, 1997. [5] A. Goswami, B. Thuilot, and B. Espiau. (1996) Compass-Like Biped Robot Part I: Stability and Bifurcation of Passive Gaits, Res. Rep. INRIA 2613. [Online]. Available: http://www.inria.fr/rrrt/rr-2996.html [6] J. W. Grizzle, G. Abba, and F. Plestan, “Asymptotically stable walking for biped robots: Analysis via systems with impulse effects,” IEEE Trans. Automat. Contr., vol. 46, pp. 51–64, Jan. 2001. [7] Y. Hurmuzlu and G. D. Moskowitz, “Role of impact in the stability of bipedal locomotion,” Int. J. Dynam., Stability Syst., vol. 1, no. 3, pp. 217–234, 1986. [8] Y. Hurmuzlu and T. H. Chang, “Rigid body collisions of a special class of planar kinematic chains,” IEEE Trans. Syst., Man, Cybern., vol. 22, pp. 964–971, May 1992. [9] T. McGeer, “Passive dynamic walking,” Int. J. Robot. Res., vol. 9, no. 2, pp. 62–82, 1990. [10] S. Kajita, T. Yamaura, and A. Kobayashi, “Dynamic walking control of a biped robot along a potential conserving orbit,” IEEE Trans. Robot. Automat., vol. 8, pp. 431–438, Aug. 1992. [11] S. Kajita, O. Matsumoto, and M. Saigo, “Real-time 3-D walking pattern generation for a biped robot with telescopic legs,” in Proc. IEEE Int. Conf. Robotics and Automation, 2001, pp. 2299–2306. [12] M. Koga, Numerical Computation With MATX (in Japanese). Tokyo, Japan: Tokyo Denki Univ. Press, 2000. [13] J. H. Park, “Impedance control for biped robot locomotion,” IEEE Trans. Robot. Automat., vol. 17, pp. 870–882, Dec. 2001. [14] J. H. Park and K. D. Kim, “Biped robot walking using gravity-compensated inverted pendulum mode and computed torque control,” in Proc. IEEE Int. Conf. Robotics and Automation, 1998, pp. 3528–3533. [15] M. W. Spong, “Passivity-based control of the compass gait biped,” in Proc. World Congr. IFAC, 1999, pp. 19–23. [16] M. Vukobratovic´ and Y. Stepanenko, “On the stability of anthropomorphic systems,” Math. Biosci., vol. 15, pp. 1–37, 1972. [17] M. Yamakita, N. Kamamichi, and F. Asano, “Virtual coupling control for dynamic bipedal walking,” in Proc. IEEE/RSJ Int. Conf. Intelligent Robots and Systems, 2001, pp. 233–238. [18] M. Yamakita and F. Asano, “Extended passive velocity field control with variable velocity fields for a kneed biped,” Adv. Robot., vol. 15, no. 2, pp. 139–168, 2001.