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Parametric Excitation Walking for Four-linked Bipedal Robot
Parametric Excitation Walking for Four-linked Bipedal Robot Y. Harata ∗ F. Asano ∗∗ K. Taji ∗ Y. Uno ∗ ∗
Department of Mechanical Science and Engineering, Graduate School of Engineering, Nagoya University, Furo, Chikusa, Nagoya, Aichi 464-8603, Japan (e-mail: {y harata, taji, uno}@nuem.nagoya-u.ac.jp}). ∗∗ Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan (e-mail: [email protected])
Abstract: In the bipedal walking, mechanical energy needs to be restored because of energy lost by heel strike collisions. Harata et al. have applied parametric excitation method to a kneed bipedal robot, and have shown that sustainable gait is generated with only knee torque. In this paper, we propose the method that combines the parametric excitation method for swing-leg with that for support-leg to improve the gait efficiency. To do this, we first extend the three-link bipedal robot to four-link bipedal robot by adding a support-leg knee, and then apply the parametric excitation method to the support-leg. Consequently, parametric excitation method for swing-leg and support-leg restores energy twice a step, and gait of the proposed method grows in efficiency. In the method proposed by Harata et al., the robot has large shin masses to restore much energy and has large semicircular feet to decrease the energy lost by heel strike. These features are unfavorable because common bipedal robots do not have such features. By simulation, we show that the bipedal robot with small shin masses or small feet can walk sustainably because the proposed method increases the quantity of restored energy. For example, the ratio of shin mass to thigh mass is one and the foot radius is reduced to one fifth of the previous method. Keywords: Walking, Robotics, Parametric excitation, Nonlinear control, Feedback linearization 1. INTRODUCTION In dynamic walking, mechanical energy of a bipedal robot dissipates by heel strike. For sustainable walking, restoration of mechanical energy is requisite. In passive dynamic walking proposed by McGeer (1990), potential energy is transported to kinetic energy as walking down a slope. On a level ground, it is necessary to restore mechanical energy by a certain method. Goswami et al. (1997) proposed energy tracking control, in which ankle and hip torque were designed to make energy constant and showed that the energy tracking control made stable limit cycle. Asano et al. (2000) proposed virtual passive dynamic walking in which the virtual gravity was adequately designed by ankle and hip torque so as to restore kinetic energy lost by collision. Another approach to restore mechanical energy is based on parametric excitation. A children’s swing is an example of parametric excitation. When playing on the swing, a person bends and stretches knees to increase amplitude of vibration. In other words, up-and-down motion of mass increases total mechanical energy. Asano et al. (2005) applied parametric excitation principle to a bipedal robot with telescopic legs which made the swing-leg mass upand-down (swing-leg excitation), and showed that the robot can walk sustainably on level ground. Asano et al. (2007) applied parametric excitation to a real machine.
Asano and Luo (2005) also proposed the support-leg excitation. Support-leg excitation is the method that makes the center of mass (COM) of the bipedal robot up-anddown by pumping telescopic support-leg. Harata et al. (2009) applied the parametric excitation principle to a kneed bipedal robot. Bending and stretching a swing-leg knee has the same effect of pumping the telescopic leg, and hence, the mechanical energy is restored based on parametric excitation. In the method, a bipedal robot can walk without hip actuation. However, there are two problems. One is that the bipedal robot needs much larger shin mass than thigh mass. If shin mass is small, COM does not move up-and-down sufficiently by bending and stretching knee and hence, mechanical energy is not sufficiently restored. Another problem is that the bipedal robot has large semicircular feet to decrease energy dissipation by heel strike (Asano and Luo (2007)). In this paper, we increase the gait efficiency for the parametric excitation method. To do this, we extend the threelink bipedal robot to four-link bipedal robot by adding a support-leg knee, and then, apply the parametric excitation method to the support-leg. The proposed method combines swing-leg excitation with support-leg excitation, and restores energy twice a step.
The proposed method consists of two parts. The first half is the support-leg excitation part, that is, at the beginning of each step, support-leg is bent at a given angle and is stretched to a straight posture. Then the method proceeds to the swing-leg excitation part. In this last half, a swingleg is bent and stretched in similar to the method proposed in Harata et al. (2009), but a robot stops stretching the swing-leg at a given angle instead of a straight posture. Then the swing-leg is kept in a bent position until heel strike occurs and swing-leg and support-leg are switched. We first apply the proposed method to a four-link bipedal robot and show that twice energy restorations are achieved. Numerical simulation results show that the efficiency of parametric excitation walking is improved remarkably by incorporating a little bit of support-leg excitation. Therefore, the design parameters of a robot are also much improved, such that, the ratio of shin and thigh mass is almost one, and the radius of circular foot is reduced to one fifth of the previous result. This paper is organized as follows: In Section 2 we show the bipedal robot with semicircular feet. In Section 3, we propose a sustainable gait generation method for the four linked bipedal robot and show simulation results. Parameter effect in this approach is discussed in Section 4. Finally in Section 5, we conclude this paper. 2. MODEL OF FOUR LINKED KNEED BIPED ROBOT WITH SEMICIRCULAR FEET Fig. 1 illustrates a bipedal robot discussed in this paper. The robot has five point mass and semicircular feet whose centers are on each leg. The semicircular feet have been shown to have same effects of equivalent ankle torque and to decrease energy dissipation of heel strike (Asano and Luo (2007)). We assume that the knees are only actuated. The dynamic equation during single support phase takes the form ˙ θ˙ + g(θ) = SuK , M (θ)θ¨ + C(θ, θ) (1) T where θ = [ θ1 θ2 θ3 θ4 ] is the generalized coordinate vector, M is the inertia matrix, C is the Coriolis force and the centrifugal force, and g is the gravity vector. The control input matrix, SuK , is described later in detail (Section 3). The robot gait consists of the following two phases. • Single support phase: The support-leg rotates around the contact point between a semicircular foot and ground. • Double support phase: This phase occurs instantaneously, and the support-leg and the swing-leg are exchanged after the heel strike. 3. PARAMETRIC EXCITATION BASED GAIT GENERATION 3.1 Control Input Design In this section, we explain control design for a kneed bipedal robot shown in Fig. 1. Define
mH
g
θ2
m2 -θ3
a2
u K1 m1
a1
b2 l 2
u K2
b1 l1
-θ4
θ1 R Fig. 1. Model of planar kneed biped robot with semicircular feet θ1 θ − θ2 − h1 x= 1 (2) , θ3 θ3 − θ4 − h2 then θ is rewritten by 1 0 0 0 0 1 −1 0 0 −h1 θ= x+ =: Lx + h, (3) 0 0 1 0 0 0 0 1 −1 −h2 where h1 is the reference trajectory for the support-leg knee and h2 is the reference trajectory for the swing-leg knee. Then θ˙ and θ¨ are ˙ θ˙ = Lx˙ + h, (4) ¨ ¨ ¨ + h. θ = Lx (5) By using x and h, the dynamic equation (1) in the single support phase is rewritten as ¨ + CLx˙ + C h ˙ + g = SuK . ¨ + Mh M Lx (6) Since the proposed robot has only knee actuation (Fig. 1), the control input matrix S is 1 0 −1 0 S= . (7) 0 1 0 −1 Define K as [ ] 0 1 0 0 K= L−1 M −1 S = S T M −1 S. (8) 0 0 0 1 Since an inertia matrix, M , is positive definite, the matrix K is positive definite, because S has full column rank. Select the knee torque uK as ( ) KP 1 y1 + KD1 y˙ 1 uK = K −1 Z + , KP 2 y2 + KD2 y˙ 2 where Z, y1 and y2 are defined by ¨ + CLx˙ + C h ˙ + g), Z = S T M −1 (M h
(9)
(10)
y1 (t) = θ1 − θ2 − h1 (t), (11) y2 (t) = θ3 − θ4 − h2 (t), and KP i and KDi are constant gain for PD control input. Using (7)–(10), the dynamic equation (6) reduces to
Table 1. Physical parameters of kneed bipedal model in Fig. 1
[
0.60 0.40 0.30 0.20
m m m m
R m1 m2 mH
y¨1 − KD1 y˙ 1 − KP 1 y1 y¨2 − KD2 y˙ 2 − KP 2 y2
0.5 4.0 1.0 5.5
]
m kg kg kg
[ ] 0 = . 0
(12)
Thus, by choosing constants KP i , KDi adequately, knee angles converge to desired trajectories.
Relative angle [rad]
l1 l2 a1 a2
support swing
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2 0
0.2
3.2 Proposed Method In this subsection we explain reference trajectories of the knees. The proposed method combines the support-leg excitation with the swing-leg excitation as follows. At the beginning of each step (that is, just after the heel strike), the support-leg knee is bent at a given angle, A1 , and is stretched to a straight posture. In this interval, the energy of the robot is restored based on the parametric excitation. Then, the swing-leg knee is bent to an angle A2 and stretched to an angle A1 . This swing-leg motion makes COM of the swing-leg up-and-down and the mechanical energy of the bipedal robot increases. After that, the swing-leg knee is kept in a bent position until heel strike occurs. After heel strike, the robot switches legs.
0.4
0.6 Time [s]
0.8
1
1.2
Fig. 2. Reference trajectory for knees
3.3 Numerical Simulation Table 1 shows the physical parameters of the robot used in our numerical simulation. In our simulation, the control input uK defined by (9) is determined for the reference trajectories (13a) and (13b) with an amplitude A1 = 0.1rad, A2 = 0.7rad, and settling time Tset1 = 0.2s, Tset2 = 0.58s and bending delay δ = 0.22s.
Fig. 3 illustrates simulation results between 100s and 103s after walking starts. Here, (a) is angular positions, (b) is angular velocities, (c) is the total mechanical energy, (d) is knee torques, (e) is relative angles and (f) is foot clearance. Foot clearance is the height of swing-leg foot h1 (t) = (θ1 − θ2 )d from the ground, and the robot scuffs the ground as this value is smaller than zero. From Figs. 3(c) and (e), it A (t < 0) 1 ( ) is observed that the total mechanical energy is restored π = A1 − A1 sin3 t (0 ≤ t ≤ Tset1 /2) (13a) when the swing-leg knee is bent, and is reduced when a Tset1 knee is stretched. The difference between the increased 0 (otherwise) energy and the decreased energy is the quantity of total h2 (t) = (θ3 − θ4 )d energy restoration. In addition, the total mechanical en ergy increases when support-leg knee stretches. Thus, in 0 (t2 < 0) ( ) the proposed method, the total mechanical energy is re π t2 (0 ≤ t2 < T2 /2) stored twice a step by the parametric excitation principle. −A2 sin3 T2 ( ) Fig. 3(f) shows that foot clearance is positive except for = π 3 ˜ −A2 + A sin (t2 − T2 /2) (T2 /2 ≤ t2 ≤ T2 ) the double support phase, and hence, the bipedal avoids T2 scuffing the ground. Fig. 4 shows stick diagram of one step −A1 (otherwise) for stable gait In this figure, black solid lines are swing-legs (13b) and red dashed lines are support-legs. where t2 = t − δ, T2 = Tset2 − δ and A˜ = A2 − A1 , Ai For the comparison purpose, we show in Fig. 5 that the simulation results of swing-leg excitation without support(i = 1, 2) is desired amplitude of vibration, Tseti (i = 1, 2) leg excitation. In the simulation, we set the parameters, is the desired settling-time which is the period during bending and stretching a knee and δ(> 0) is bending delay. A1 = 0.0rad and A2 = 1.1rad, so as that walking speed These reference trajectories are smooth enough to make of the swing-leg excitation almost equal to that of the the knees to track the reference trajectories exactly, by combined method shown in Fig. 3. Here, (a) is angular positions, (b) is the total mechanical energy, (c) is input using input torque shown in (9) . torques. From Fig. 3 and Fig. 5, it is observed that energy To avoid the overlap between support-leg and swing-leg variation of the combined method is about two-fifth of that excitations, we set δ > Tset1 . We assume that knee is of the swing-leg excitation only, and then the combined stretched before heel strike, so that Tset2 is smaller than method is more efficient. On the other hand, the maximum step period. If Tset2 is larger than step period, the biped absolute torque of the proposed method is larger than that robot does not walk. Fig. 2 shows an example of reference of the swing-leg excitation method. This is because the support-leg lifts the total mass of the robot. trajectories of knees. We design the reference trajectory of a support-leg knee h1 and the reference trajectory of a swing-leg knee h2 as follows:
(a) Angular position [rad] 0.5
θ2
0
θ3 θ4
−0.5 105
Walking direction
θ1
105.5
106
106.5 107 Time [s]
107.5
108
(b) Angular velocity [rad/s] 2
dθ1/dt
0
dθ2/dt dθ3/dt
−2 −4 105 104
dθ4/dt 105.5
106
106.5 107 Time [s]
107.5
108
Fig. 4. Stick diagram of parametric excitation based walking
(c) Total mechanical energy [J]
(a) Angular position [rad] 0.5
θ1
102
θ3
98 105
θ2
0
100
−0.5
105.5
106
106.5 107 Time [s]
107.5
105
(d) Input torque [Nm] support swing
10
θ4
108
0
104
105.5
106
106.5 Time [s]
107
107.5
108
(b) Total mechanical energy [J]
102 100
−10 105
105.5
106
106.5 107 Time [s]
107.5
108
98 105
105.5
(e) Relative angle [rad] 0 −0.2 −0.4 −0.6 −0.8 105
θ1−θ2 θ3−θ4
106.5 107 Time [s] (c) Input torque [Nm]
107.5
108
support swing
10 0 −10
105.5
106
106.5 107 Time [s]
107.5
108
(f) Foot clearance [m] 0.04 0.02 0 105.5
106
106.5 107 Time [s]
105
105.5
106
106.5 Time [s]
107
107.5
108
Fig. 5. Swing leg excitation based walking
0.06
105
106
107.5
108
Fig. 3. Steady walking patterns of parametric excitation based walking 4. EFFECT OF PARAMETERS 4.1 Effect of Knee Bending Angle In this subsection we discuss the effect of desired swingleg amplitude of vibration, A2 , with desired support-leg
amplitude of vibration, A1 , for four values, 0.0, 0.05, 0.1 and 0.15rad. The rest parameters of the reference trajectories, h1 and h2 are set as the previous section. We evaluate walking speed and specific resistance. Specific resistance proposed by Gabrielli and Karman (1950) is defined by ∫ T− ˙ ˙ ˙ ˙ + |uK1 (θ1 − θ2 )| + |uK2 (θ3 − θ4 )|dt/T , (14) µ= 0 Mg gV and represents the energy efficiency. When a specific resistance value is small, the energy efficiency is efficient. In (14), 0+ and T − represent the time just after and before collision with the ground, respectively, Mg is the total mass of the bipedal robot and V is the average walking speed.
In this subsection, we show that the bipedal robot with small shin mass or small feet can walk sustainably. Here, simulations are performed for desired support-leg amplitude A1 = 0.0, 0.05, 0.1, 0.15rad, swing-leg amplitude of vibration, A2 = 0.9rad, and for value of the rest parameters of the reference trajectories are the same as the previous section. First, we evaluate the effect of the ratio of shin and thigh mass. Fig. 7 show the simulation results. Here, (a) is walking speed and (b) is the specific resistance with respect to shin mass. We plot the value for which a bipedal robot can walk sustainably. In Fig. 7, the blue circles are the results with a desired amplitude of support-leg A1 = 0rad, the green squares are the results with A1 = 0.05rad, the red crosses are the results with A1 = 0.1rad and the gray triangles are the results with A1 = 0.15rad. Fig. 7 shows that, when the shin mass becomes small less than 3.5kg, the biped robot can not walk by swing-leg excitation only, while the robot can walk by the proposed method. From Fig. 7(b), it is observed that waking efficiency is increased as the shin mass becomes small in the propose method while it is decreased for swing-leg excitation only in almost region. This simulation result implies that the shin mass can be reduced by the proposed method. Next, we change the foot radius. Fig. 8 shows the simulation results. Here, (a) is walking speed and (b) is the specific resistance with respect to foot radius. In Fig. 8, the blue circles are the results with a desired amplitude of support-leg A1 = 0rad, the green squares are the results with A1 = 0.05rad, the red crosses are the results with A1 = 0.1rad and the gray triangles are the results with A1 = 0.15rad. It is observed that, when foot radius of the robot is smaller than 0.4m, the biped can not walk by swing-leg excitation only. On the other hand, the proposed method can generate the biped walking. Fig. 8 shows that the bipedal walking grows in speed and efficiency as the
Restored energy [J]
A1=0 A1=0.05 A1=0.10 A1=0.15
2
1.5
1
0.5
0
0.5
0.75 0.7 Walking speed [m/s]
4.2 Effect of Physical Parameters
2.5
0.6
0.7 0.8 Amplitude A 2 [rad] (a) Restored energy
0.9
1
0.9
1
0.9
1
A1=0 A1=0.05 A1=0.10 A1=0.15
0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.5
0.6
0.7 0.8 Amplitude A 2 [rad] (b) Walking speed
0.12 0.11 Specific resistance [−]
Fig. 6 shows the simulation results. Here, (a) is restored energy, (b) is walking speed and (c) is the specific resistance with respect to the desired amplitude A2 of swingleg bending. All data plotted in the figures are those for which a bipedal robot can walk sustainably. In Fig. 6, the blue circles are the results with a desired amplitude of support-leg A1 = 0rad, the green squares are the results with A1 = 0.05rad, the red crosses are the results with A1 = 0.1rad and the gray triangles are the results with A1 = 0.15rad. From Fig. 6, it is observed that by introducing A1 (> 0) the bipedal robot can walk in the range that A2 is smaller than 0.85. This is because energy restoration becomes large as A1 increases (shown in Fig. 6 (a)). Figs. 6 (b) and (c) show that the biped walking becomes fast and efficient when A1 increases. The figure also shows that the energy restored by swing-leg excitation increases as A2 becomes large. It is observed that the efficiency of parametric excitation walking is improved remarkably by incorporating a little bit of support-leg excitation. For example, specific resistance of swing-leg excitation only is about 0.975 at walking speed about 0.5m/s. On the other hand, specific resistance of the proposed method is about 0.06 at the same walking speed.
0.1 0.09
A1=0 A1=0.05 A1=0.10 A1=0.15
0.08 0.07 0.06 0.05 0.04 0.03 0.5
0.6
0.7 0.8 Amplitude A 2 [rad] (c) Specific resistance
Fig. 6. Walking indices with respect to desired amplitude of swing-leg foot radius R increases. This result implies that the bipedal robot with small feet can walk sustainably by the proposed method. 5. CONCLUSION In this paper, we proposed combined swing-leg parametric excitation with support-leg parametric excitation. Therefore, the total mechanical energy was restored twice a step, and the bipedal robot could walk when A2 is small. In
0.75
0.9 Walking speed [m/s]
Walking speed [m/s]
0.7 0.65 0.6 0.55 0.5
A 1 =0.0 A 1 =0.05 A 1 =0.1 A 1 =0.15
0.45 0.4 0.35 1
1.5
2
2.5 3 3.5 4 Shin mass m 1 [kg]
A1=0.0 A1=0.05 A1=0.1 A1=0.15
0.8 0.7 0.6 0.5 0.4 0.3
4.5
5
0
0.1
(a) Walking speed
0.11 0.1
Specific resistance [−]
A1=0.0 A1=0.05 A1=0.1 A1=0.15
0.12 Specific resistance [−]
0.3 0.4 Foot radius [m] (a) Walking speed
0.5
0.6
0.2
0.5
0.6
0.16
0.13
0.09 0.08 0.07
0.14 0.12 0.1
A1=0.0 A1=0.05 A1=0.1 A1=0.15
0.08
0.06 0.05 0.04
0.2
0.06
1
1.5
2
2.5 3 3.5 4 Shin mass m1 [kg] (b) Specific resistance
4.5
5
Fig. 7. Walking indices with respect to shin mass addition, we showed that the bipedal robot with small shin mass or small feet could walk sustainably. In the future works, we will optimize the desired trajectories. We also analyze energy restoration and bifurcation of nonlinear system in parametric excitation. In addition, we measure human walking and search the COM. By comparison the parametric excitation walking with human walking, we verify that human walking based on the parametric excitation principle. ACKNOWLEDGEMENTS This research was partly supported by the Japanese Center of Excellence program ”Education and Research of MicroNano Mechanics” (2008-2013). REFERENCES T. McGeer. Passive dynamic walking. Int. J. of Robotics Research, volume 9, nomber 2, pages 62–82, 1990. A. Goswami, B. Espiau and A. Keramane. Limit cycles in a passive compass gait bipedal and passivity-mimicking control laws. J. of Autonomous Robots, volume 4, number 3, pages 273–286, 1997. F. Asano, M. Yamakita and K. Furuta. Virtual passive dynamic walking and energy-based control laws. IEEE/RSJ Int. Conf. on Intelligent Robotics and Systems, pages 1149–1154, 2000.
0
0.1
0.3 0.4 Foot radius [m] (b) Specific resistance
Fig. 8. Walking indices with respect to foot radius F. Asano, Z.W. Luo and S. Hyon. parametric excitation mechanisms for dynamic bipedal walking. IEEE Int. Conf. on Robotics and Automation, pages 611–617, 2005. F. Asano, T. Hayashi, Z.W. Luo, S. Hirano and A. Kato. parametric excitation approaches to efficient bipedal walking. IEEE/RSJ Int. Conf. on Intelligent Robotics and Systems, pages 2210–2216, 2007. Y. Harata, F. Asano, Z.W. Luo, K. Taji and Y. Uno. Biped gait generation based on parametric excitation by kneejoint actuation. Robotica, published online, 17 March 2009. F. Asano and Z.W. Luo. Parametcally excited dynamic bipedal walking based on up-and-down hip motion. the 6th SICE System Integration Division Conf., pages 907– 908, 2005 (in Japanese). F. Asano and Z.W. Luo. The effect of semicircular feet on energy dissipation by heel-strike in dynamic bipedal locomotion. IEEE Int. Conf. on Robotics and Automation, pages 3976–3981, 2007. G. Gabrielli and Th. von Karman. What price speed? Specific power required for propulsion of vehicles. Mechanical engineering, volume 72, pages 775–781, 1950.
Department of Mechanical Science and Engineering, Graduate School of Engineering, Nagoya University, Furo, Chikusa, Nagoya, Aichi 464-8603, Japan (e-mail: {y harata, taji, uno}@nuem.nagoya-u.ac.jp}). ∗∗ Japan Advanced Institute of Science and Technology, 1-1 Asahidai, Nomi, Ishikawa 923-1292, Japan (e-mail: [email protected])
Abstract: In the bipedal walking, mechanical energy needs to be restored because of energy lost by heel strike collisions. Harata et al. have applied parametric excitation method to a kneed bipedal robot, and have shown that sustainable gait is generated with only knee torque. In this paper, we propose the method that combines the parametric excitation method for swing-leg with that for support-leg to improve the gait efficiency. To do this, we first extend the three-link bipedal robot to four-link bipedal robot by adding a support-leg knee, and then apply the parametric excitation method to the support-leg. Consequently, parametric excitation method for swing-leg and support-leg restores energy twice a step, and gait of the proposed method grows in efficiency. In the method proposed by Harata et al., the robot has large shin masses to restore much energy and has large semicircular feet to decrease the energy lost by heel strike. These features are unfavorable because common bipedal robots do not have such features. By simulation, we show that the bipedal robot with small shin masses or small feet can walk sustainably because the proposed method increases the quantity of restored energy. For example, the ratio of shin mass to thigh mass is one and the foot radius is reduced to one fifth of the previous method. Keywords: Walking, Robotics, Parametric excitation, Nonlinear control, Feedback linearization 1. INTRODUCTION In dynamic walking, mechanical energy of a bipedal robot dissipates by heel strike. For sustainable walking, restoration of mechanical energy is requisite. In passive dynamic walking proposed by McGeer (1990), potential energy is transported to kinetic energy as walking down a slope. On a level ground, it is necessary to restore mechanical energy by a certain method. Goswami et al. (1997) proposed energy tracking control, in which ankle and hip torque were designed to make energy constant and showed that the energy tracking control made stable limit cycle. Asano et al. (2000) proposed virtual passive dynamic walking in which the virtual gravity was adequately designed by ankle and hip torque so as to restore kinetic energy lost by collision. Another approach to restore mechanical energy is based on parametric excitation. A children’s swing is an example of parametric excitation. When playing on the swing, a person bends and stretches knees to increase amplitude of vibration. In other words, up-and-down motion of mass increases total mechanical energy. Asano et al. (2005) applied parametric excitation principle to a bipedal robot with telescopic legs which made the swing-leg mass upand-down (swing-leg excitation), and showed that the robot can walk sustainably on level ground. Asano et al. (2007) applied parametric excitation to a real machine.
Asano and Luo (2005) also proposed the support-leg excitation. Support-leg excitation is the method that makes the center of mass (COM) of the bipedal robot up-anddown by pumping telescopic support-leg. Harata et al. (2009) applied the parametric excitation principle to a kneed bipedal robot. Bending and stretching a swing-leg knee has the same effect of pumping the telescopic leg, and hence, the mechanical energy is restored based on parametric excitation. In the method, a bipedal robot can walk without hip actuation. However, there are two problems. One is that the bipedal robot needs much larger shin mass than thigh mass. If shin mass is small, COM does not move up-and-down sufficiently by bending and stretching knee and hence, mechanical energy is not sufficiently restored. Another problem is that the bipedal robot has large semicircular feet to decrease energy dissipation by heel strike (Asano and Luo (2007)). In this paper, we increase the gait efficiency for the parametric excitation method. To do this, we extend the threelink bipedal robot to four-link bipedal robot by adding a support-leg knee, and then, apply the parametric excitation method to the support-leg. The proposed method combines swing-leg excitation with support-leg excitation, and restores energy twice a step.
The proposed method consists of two parts. The first half is the support-leg excitation part, that is, at the beginning of each step, support-leg is bent at a given angle and is stretched to a straight posture. Then the method proceeds to the swing-leg excitation part. In this last half, a swingleg is bent and stretched in similar to the method proposed in Harata et al. (2009), but a robot stops stretching the swing-leg at a given angle instead of a straight posture. Then the swing-leg is kept in a bent position until heel strike occurs and swing-leg and support-leg are switched. We first apply the proposed method to a four-link bipedal robot and show that twice energy restorations are achieved. Numerical simulation results show that the efficiency of parametric excitation walking is improved remarkably by incorporating a little bit of support-leg excitation. Therefore, the design parameters of a robot are also much improved, such that, the ratio of shin and thigh mass is almost one, and the radius of circular foot is reduced to one fifth of the previous result. This paper is organized as follows: In Section 2 we show the bipedal robot with semicircular feet. In Section 3, we propose a sustainable gait generation method for the four linked bipedal robot and show simulation results. Parameter effect in this approach is discussed in Section 4. Finally in Section 5, we conclude this paper. 2. MODEL OF FOUR LINKED KNEED BIPED ROBOT WITH SEMICIRCULAR FEET Fig. 1 illustrates a bipedal robot discussed in this paper. The robot has five point mass and semicircular feet whose centers are on each leg. The semicircular feet have been shown to have same effects of equivalent ankle torque and to decrease energy dissipation of heel strike (Asano and Luo (2007)). We assume that the knees are only actuated. The dynamic equation during single support phase takes the form ˙ θ˙ + g(θ) = SuK , M (θ)θ¨ + C(θ, θ) (1) T where θ = [ θ1 θ2 θ3 θ4 ] is the generalized coordinate vector, M is the inertia matrix, C is the Coriolis force and the centrifugal force, and g is the gravity vector. The control input matrix, SuK , is described later in detail (Section 3). The robot gait consists of the following two phases. • Single support phase: The support-leg rotates around the contact point between a semicircular foot and ground. • Double support phase: This phase occurs instantaneously, and the support-leg and the swing-leg are exchanged after the heel strike. 3. PARAMETRIC EXCITATION BASED GAIT GENERATION 3.1 Control Input Design In this section, we explain control design for a kneed bipedal robot shown in Fig. 1. Define
mH
g
θ2
m2 -θ3
a2
u K1 m1
a1
b2 l 2
u K2
b1 l1
-θ4
θ1 R Fig. 1. Model of planar kneed biped robot with semicircular feet θ1 θ − θ2 − h1 x= 1 (2) , θ3 θ3 − θ4 − h2 then θ is rewritten by 1 0 0 0 0 1 −1 0 0 −h1 θ= x+ =: Lx + h, (3) 0 0 1 0 0 0 0 1 −1 −h2 where h1 is the reference trajectory for the support-leg knee and h2 is the reference trajectory for the swing-leg knee. Then θ˙ and θ¨ are ˙ θ˙ = Lx˙ + h, (4) ¨ ¨ ¨ + h. θ = Lx (5) By using x and h, the dynamic equation (1) in the single support phase is rewritten as ¨ + CLx˙ + C h ˙ + g = SuK . ¨ + Mh M Lx (6) Since the proposed robot has only knee actuation (Fig. 1), the control input matrix S is 1 0 −1 0 S= . (7) 0 1 0 −1 Define K as [ ] 0 1 0 0 K= L−1 M −1 S = S T M −1 S. (8) 0 0 0 1 Since an inertia matrix, M , is positive definite, the matrix K is positive definite, because S has full column rank. Select the knee torque uK as ( ) KP 1 y1 + KD1 y˙ 1 uK = K −1 Z + , KP 2 y2 + KD2 y˙ 2 where Z, y1 and y2 are defined by ¨ + CLx˙ + C h ˙ + g), Z = S T M −1 (M h
(9)
(10)
y1 (t) = θ1 − θ2 − h1 (t), (11) y2 (t) = θ3 − θ4 − h2 (t), and KP i and KDi are constant gain for PD control input. Using (7)–(10), the dynamic equation (6) reduces to
Table 1. Physical parameters of kneed bipedal model in Fig. 1
[
0.60 0.40 0.30 0.20
m m m m
R m1 m2 mH
y¨1 − KD1 y˙ 1 − KP 1 y1 y¨2 − KD2 y˙ 2 − KP 2 y2
0.5 4.0 1.0 5.5
]
m kg kg kg
[ ] 0 = . 0
(12)
Thus, by choosing constants KP i , KDi adequately, knee angles converge to desired trajectories.
Relative angle [rad]
l1 l2 a1 a2
support swing
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2 0
0.2
3.2 Proposed Method In this subsection we explain reference trajectories of the knees. The proposed method combines the support-leg excitation with the swing-leg excitation as follows. At the beginning of each step (that is, just after the heel strike), the support-leg knee is bent at a given angle, A1 , and is stretched to a straight posture. In this interval, the energy of the robot is restored based on the parametric excitation. Then, the swing-leg knee is bent to an angle A2 and stretched to an angle A1 . This swing-leg motion makes COM of the swing-leg up-and-down and the mechanical energy of the bipedal robot increases. After that, the swing-leg knee is kept in a bent position until heel strike occurs. After heel strike, the robot switches legs.
0.4
0.6 Time [s]
0.8
1
1.2
Fig. 2. Reference trajectory for knees
3.3 Numerical Simulation Table 1 shows the physical parameters of the robot used in our numerical simulation. In our simulation, the control input uK defined by (9) is determined for the reference trajectories (13a) and (13b) with an amplitude A1 = 0.1rad, A2 = 0.7rad, and settling time Tset1 = 0.2s, Tset2 = 0.58s and bending delay δ = 0.22s.
Fig. 3 illustrates simulation results between 100s and 103s after walking starts. Here, (a) is angular positions, (b) is angular velocities, (c) is the total mechanical energy, (d) is knee torques, (e) is relative angles and (f) is foot clearance. Foot clearance is the height of swing-leg foot h1 (t) = (θ1 − θ2 )d from the ground, and the robot scuffs the ground as this value is smaller than zero. From Figs. 3(c) and (e), it A (t < 0) 1 ( ) is observed that the total mechanical energy is restored π = A1 − A1 sin3 t (0 ≤ t ≤ Tset1 /2) (13a) when the swing-leg knee is bent, and is reduced when a Tset1 knee is stretched. The difference between the increased 0 (otherwise) energy and the decreased energy is the quantity of total h2 (t) = (θ3 − θ4 )d energy restoration. In addition, the total mechanical en ergy increases when support-leg knee stretches. Thus, in 0 (t2 < 0) ( ) the proposed method, the total mechanical energy is re π t2 (0 ≤ t2 < T2 /2) stored twice a step by the parametric excitation principle. −A2 sin3 T2 ( ) Fig. 3(f) shows that foot clearance is positive except for = π 3 ˜ −A2 + A sin (t2 − T2 /2) (T2 /2 ≤ t2 ≤ T2 ) the double support phase, and hence, the bipedal avoids T2 scuffing the ground. Fig. 4 shows stick diagram of one step −A1 (otherwise) for stable gait In this figure, black solid lines are swing-legs (13b) and red dashed lines are support-legs. where t2 = t − δ, T2 = Tset2 − δ and A˜ = A2 − A1 , Ai For the comparison purpose, we show in Fig. 5 that the simulation results of swing-leg excitation without support(i = 1, 2) is desired amplitude of vibration, Tseti (i = 1, 2) leg excitation. In the simulation, we set the parameters, is the desired settling-time which is the period during bending and stretching a knee and δ(> 0) is bending delay. A1 = 0.0rad and A2 = 1.1rad, so as that walking speed These reference trajectories are smooth enough to make of the swing-leg excitation almost equal to that of the the knees to track the reference trajectories exactly, by combined method shown in Fig. 3. Here, (a) is angular positions, (b) is the total mechanical energy, (c) is input using input torque shown in (9) . torques. From Fig. 3 and Fig. 5, it is observed that energy To avoid the overlap between support-leg and swing-leg variation of the combined method is about two-fifth of that excitations, we set δ > Tset1 . We assume that knee is of the swing-leg excitation only, and then the combined stretched before heel strike, so that Tset2 is smaller than method is more efficient. On the other hand, the maximum step period. If Tset2 is larger than step period, the biped absolute torque of the proposed method is larger than that robot does not walk. Fig. 2 shows an example of reference of the swing-leg excitation method. This is because the support-leg lifts the total mass of the robot. trajectories of knees. We design the reference trajectory of a support-leg knee h1 and the reference trajectory of a swing-leg knee h2 as follows:
(a) Angular position [rad] 0.5
θ2
0
θ3 θ4
−0.5 105
Walking direction
θ1
105.5
106
106.5 107 Time [s]
107.5
108
(b) Angular velocity [rad/s] 2
dθ1/dt
0
dθ2/dt dθ3/dt
−2 −4 105 104
dθ4/dt 105.5
106
106.5 107 Time [s]
107.5
108
Fig. 4. Stick diagram of parametric excitation based walking
(c) Total mechanical energy [J]
(a) Angular position [rad] 0.5
θ1
102
θ3
98 105
θ2
0
100
−0.5
105.5
106
106.5 107 Time [s]
107.5
105
(d) Input torque [Nm] support swing
10
θ4
108
0
104
105.5
106
106.5 Time [s]
107
107.5
108
(b) Total mechanical energy [J]
102 100
−10 105
105.5
106
106.5 107 Time [s]
107.5
108
98 105
105.5
(e) Relative angle [rad] 0 −0.2 −0.4 −0.6 −0.8 105
θ1−θ2 θ3−θ4
106.5 107 Time [s] (c) Input torque [Nm]
107.5
108
support swing
10 0 −10
105.5
106
106.5 107 Time [s]
107.5
108
(f) Foot clearance [m] 0.04 0.02 0 105.5
106
106.5 107 Time [s]
105
105.5
106
106.5 Time [s]
107
107.5
108
Fig. 5. Swing leg excitation based walking
0.06
105
106
107.5
108
Fig. 3. Steady walking patterns of parametric excitation based walking 4. EFFECT OF PARAMETERS 4.1 Effect of Knee Bending Angle In this subsection we discuss the effect of desired swingleg amplitude of vibration, A2 , with desired support-leg
amplitude of vibration, A1 , for four values, 0.0, 0.05, 0.1 and 0.15rad. The rest parameters of the reference trajectories, h1 and h2 are set as the previous section. We evaluate walking speed and specific resistance. Specific resistance proposed by Gabrielli and Karman (1950) is defined by ∫ T− ˙ ˙ ˙ ˙ + |uK1 (θ1 − θ2 )| + |uK2 (θ3 − θ4 )|dt/T , (14) µ= 0 Mg gV and represents the energy efficiency. When a specific resistance value is small, the energy efficiency is efficient. In (14), 0+ and T − represent the time just after and before collision with the ground, respectively, Mg is the total mass of the bipedal robot and V is the average walking speed.
In this subsection, we show that the bipedal robot with small shin mass or small feet can walk sustainably. Here, simulations are performed for desired support-leg amplitude A1 = 0.0, 0.05, 0.1, 0.15rad, swing-leg amplitude of vibration, A2 = 0.9rad, and for value of the rest parameters of the reference trajectories are the same as the previous section. First, we evaluate the effect of the ratio of shin and thigh mass. Fig. 7 show the simulation results. Here, (a) is walking speed and (b) is the specific resistance with respect to shin mass. We plot the value for which a bipedal robot can walk sustainably. In Fig. 7, the blue circles are the results with a desired amplitude of support-leg A1 = 0rad, the green squares are the results with A1 = 0.05rad, the red crosses are the results with A1 = 0.1rad and the gray triangles are the results with A1 = 0.15rad. Fig. 7 shows that, when the shin mass becomes small less than 3.5kg, the biped robot can not walk by swing-leg excitation only, while the robot can walk by the proposed method. From Fig. 7(b), it is observed that waking efficiency is increased as the shin mass becomes small in the propose method while it is decreased for swing-leg excitation only in almost region. This simulation result implies that the shin mass can be reduced by the proposed method. Next, we change the foot radius. Fig. 8 shows the simulation results. Here, (a) is walking speed and (b) is the specific resistance with respect to foot radius. In Fig. 8, the blue circles are the results with a desired amplitude of support-leg A1 = 0rad, the green squares are the results with A1 = 0.05rad, the red crosses are the results with A1 = 0.1rad and the gray triangles are the results with A1 = 0.15rad. It is observed that, when foot radius of the robot is smaller than 0.4m, the biped can not walk by swing-leg excitation only. On the other hand, the proposed method can generate the biped walking. Fig. 8 shows that the bipedal walking grows in speed and efficiency as the
Restored energy [J]
A1=0 A1=0.05 A1=0.10 A1=0.15
2
1.5
1
0.5
0
0.5
0.75 0.7 Walking speed [m/s]
4.2 Effect of Physical Parameters
2.5
0.6
0.7 0.8 Amplitude A 2 [rad] (a) Restored energy
0.9
1
0.9
1
0.9
1
A1=0 A1=0.05 A1=0.10 A1=0.15
0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.5
0.6
0.7 0.8 Amplitude A 2 [rad] (b) Walking speed
0.12 0.11 Specific resistance [−]
Fig. 6 shows the simulation results. Here, (a) is restored energy, (b) is walking speed and (c) is the specific resistance with respect to the desired amplitude A2 of swingleg bending. All data plotted in the figures are those for which a bipedal robot can walk sustainably. In Fig. 6, the blue circles are the results with a desired amplitude of support-leg A1 = 0rad, the green squares are the results with A1 = 0.05rad, the red crosses are the results with A1 = 0.1rad and the gray triangles are the results with A1 = 0.15rad. From Fig. 6, it is observed that by introducing A1 (> 0) the bipedal robot can walk in the range that A2 is smaller than 0.85. This is because energy restoration becomes large as A1 increases (shown in Fig. 6 (a)). Figs. 6 (b) and (c) show that the biped walking becomes fast and efficient when A1 increases. The figure also shows that the energy restored by swing-leg excitation increases as A2 becomes large. It is observed that the efficiency of parametric excitation walking is improved remarkably by incorporating a little bit of support-leg excitation. For example, specific resistance of swing-leg excitation only is about 0.975 at walking speed about 0.5m/s. On the other hand, specific resistance of the proposed method is about 0.06 at the same walking speed.
0.1 0.09
A1=0 A1=0.05 A1=0.10 A1=0.15
0.08 0.07 0.06 0.05 0.04 0.03 0.5
0.6
0.7 0.8 Amplitude A 2 [rad] (c) Specific resistance
Fig. 6. Walking indices with respect to desired amplitude of swing-leg foot radius R increases. This result implies that the bipedal robot with small feet can walk sustainably by the proposed method. 5. CONCLUSION In this paper, we proposed combined swing-leg parametric excitation with support-leg parametric excitation. Therefore, the total mechanical energy was restored twice a step, and the bipedal robot could walk when A2 is small. In
0.75
0.9 Walking speed [m/s]
Walking speed [m/s]
0.7 0.65 0.6 0.55 0.5
A 1 =0.0 A 1 =0.05 A 1 =0.1 A 1 =0.15
0.45 0.4 0.35 1
1.5
2
2.5 3 3.5 4 Shin mass m 1 [kg]
A1=0.0 A1=0.05 A1=0.1 A1=0.15
0.8 0.7 0.6 0.5 0.4 0.3
4.5
5
0
0.1
(a) Walking speed
0.11 0.1
Specific resistance [−]
A1=0.0 A1=0.05 A1=0.1 A1=0.15
0.12 Specific resistance [−]
0.3 0.4 Foot radius [m] (a) Walking speed
0.5
0.6
0.2
0.5
0.6
0.16
0.13
0.09 0.08 0.07
0.14 0.12 0.1
A1=0.0 A1=0.05 A1=0.1 A1=0.15
0.08
0.06 0.05 0.04
0.2
0.06
1
1.5
2
2.5 3 3.5 4 Shin mass m1 [kg] (b) Specific resistance
4.5
5
Fig. 7. Walking indices with respect to shin mass addition, we showed that the bipedal robot with small shin mass or small feet could walk sustainably. In the future works, we will optimize the desired trajectories. We also analyze energy restoration and bifurcation of nonlinear system in parametric excitation. In addition, we measure human walking and search the COM. By comparison the parametric excitation walking with human walking, we verify that human walking based on the parametric excitation principle. ACKNOWLEDGEMENTS This research was partly supported by the Japanese Center of Excellence program ”Education and Research of MicroNano Mechanics” (2008-2013). REFERENCES T. McGeer. Passive dynamic walking. Int. J. of Robotics Research, volume 9, nomber 2, pages 62–82, 1990. A. Goswami, B. Espiau and A. Keramane. Limit cycles in a passive compass gait bipedal and passivity-mimicking control laws. J. of Autonomous Robots, volume 4, number 3, pages 273–286, 1997. F. Asano, M. Yamakita and K. Furuta. Virtual passive dynamic walking and energy-based control laws. IEEE/RSJ Int. Conf. on Intelligent Robotics and Systems, pages 1149–1154, 2000.
0
0.1
0.3 0.4 Foot radius [m] (b) Specific resistance
Fig. 8. Walking indices with respect to foot radius F. Asano, Z.W. Luo and S. Hyon. parametric excitation mechanisms for dynamic bipedal walking. IEEE Int. Conf. on Robotics and Automation, pages 611–617, 2005. F. Asano, T. Hayashi, Z.W. Luo, S. Hirano and A. Kato. parametric excitation approaches to efficient bipedal walking. IEEE/RSJ Int. Conf. on Intelligent Robotics and Systems, pages 2210–2216, 2007. Y. Harata, F. Asano, Z.W. Luo, K. Taji and Y. Uno. Biped gait generation based on parametric excitation by kneejoint actuation. Robotica, published online, 17 March 2009. F. Asano and Z.W. Luo. Parametcally excited dynamic bipedal walking based on up-and-down hip motion. the 6th SICE System Integration Division Conf., pages 907– 908, 2005 (in Japanese). F. Asano and Z.W. Luo. The effect of semicircular feet on energy dissipation by heel-strike in dynamic bipedal locomotion. IEEE Int. Conf. on Robotics and Automation, pages 3976–3981, 2007. G. Gabrielli and Th. von Karman. What price speed? Specific power required for propulsion of vehicles. Mechanical engineering, volume 72, pages 775–781, 1950.