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Powered Walking Based on the Passive Dynamic Principles: A Virtual Slope Walking Approach Zhibin Li, Ka Deng, Mingguo Zhao† Abstract— The presented work is motivated to implement the principles of passive dynamic walking (PDW) on the powered bipeds. The Virtual Slope Walking (VSW), as one PWD control approach, is chosen to be mathematically analyzed and realized to achieve fast dynamic walking. Our method of formulating the fixed point of the limit cycle gait is presented, and the walking pattern generation is designed in the polar coordinate for the better consistency with the model. The algorithms are implemented on a planar robot Stepper-2D, which achieved the walking speed from 0.69m/s upto 1.03m/s. The self-stabilized walking has the best robustness at the speed of 0.81m/s, where the robot was able to walk on the level ground, step over random obstacles smaller than 4% of the leg length, and step over a big obstacle of 7.2% of the leg length.

I. I NTRODUCTION In this paper, the motivation of investigating the passive dynamic walking (PDW) originates from the undeniable fact of foot landing impacts in dynamic walking. Though the zero moment point (ZMP) approach has rigorous control theory for the gait generation, however, it lacks the modeling of foot-ground impact nor at least considering the influence of impacts in the gait generation stage. As a result, this inadequacy makes the fast/dynamic walking more challenging and difficult to be realized in the ZMP framework. Dynamic walking on flat or uneven terrains remains challenging for most active powered bipeds nowadays. Although some advanced humanoids are able to perform such tasks, it is achieved through strong engineering approaches such as foothold re-planning, walking pattern generation, and precise joint position/torque tracking. Consequently, their power consumption is far beyond that of humans in addition to their limited adaption to the terrain variations. The powered bipeds built based on the passive dynamic paradigm, such as the Cornell bipeds [1] and the Delft’s ‘Flame’ and ‘TUlip’ robots [2], shed a light on the answers to these shortcomings found in the ZMP controlled bipeds. The human way of walking is more comparable to a ‘fall, capture balance, and fall’ process. During this, only a small amount of energy is required to compensate for the energy loss during foot-ground interaction, namely the heel strike impact and frictions. However, one major limitation of the passive walkers is their dependency on the power source, often a slope where the gravity supplies the positive work. Zhibin Li is with the Department of Advanced Robotics, Istituto Italiano di Tecnologia, via Morego 30, 16163, Genova, Italy. Ka Deng and Mingguo Zhao are with the Department of Automation, Tsinghua University, 100084 Beijing, P.R.China. Corresponding author: Mingguo Zhao. Email: [email protected], [email protected],

[email protected]

Based on the analysis of the dynamics of the passive dynamic walker [3], McGeer proposed solutions for the passive dynamic walking on level ground, including the push-off of the pre-swing leg, supplying torque on the joints and adjusting the leg length [4]. These suggestions were leading the research directions, though in-depth discussion was not yet presented. Exerting torque is the most direct way as demonstrated by Goswami [5], Spong [6]and Asano [7] et al. where level-ground walking was realized by continuously applying the torque on hips and ankles to track the references that are pre-designed based on the mechanical energy or the replacement of the gravity. Enabling the passive dynamic walkers to walk on level ground demands the energy supplement to compensate for the energy lost in collision, and meanwhile must retain the exploitation of the natural dynamics of the system. Push-off is a straightforward way to inject kinetic energy. McGeer, Kuo [8] and Collins [1], [9] both followed this idea and proved the feasibility in the theoretical investigation and the realization of real robots. The walking of the robot Mike was realized via actuating the swing leg to supply kinetic energy [10]. Adjusting the leg length is also a power injection method from the potential energy aspect. Minakata [11] and Asano [12] et al. studied the function of lifting the swing leg during one step which is named as the Parametric Excitation. Compared to the tracking control strategies, these energyinjecting methods are easier to implement and contribute for the walking speed and efficiency. From the aforementioned works, it can be ruled out that the underlaying principle of the passive dynamics based walkers is to supply the complementary energy into the system to sustain a stable limit cycle walking. In this paper, we chose the virtual slope walking (VSW) approach as a starting point to implement PDW for fast walking. The VSW based approach adjusts the length of the stance during walking to implement an asymmetric posture and mimic the downhill walking on the level ground. The input potential energy balances the kinetic energy loss due to the heel strike at every step, so the biped walking is sustainable. By this study, we wish to offer some insights to abridge the gap between two control realms: the ZMP based walkers and the passive dynamic walkers. Our control concept complies with the passive falling motion under the gravity, so the active control only spends very little effort during the passive behavior to save energy; and only injects energy into system when it is necessary. This paper reports our new improvements on top of the previous work in [13], and contributes as follows: elaborate

Preprint for research circulation only, please find official paper from IEEE Xplore better the basics of VSW by a systematic formulation based on rigid body dynamics, derive a set of control parameters for gait generation, and further enhance the algorithmic implementation by resolving inverse kinematics in polar coordinate. The improved dynamic gait generation has better foot ground clearance and requires least effort on parameter tuning for obtaining a various range of stable gaits. The paper is structured as follows. Section II presents the principles, modeling and formulation of the virtual slope walking. Section III explains the algorithmic implementation of the gait generation. Section IV demonstrates and analyzes the experiments of the dynamic and robust limit cycle walking on the flat and the rugged ground. We conclude the paper in Section V. II. P RINCIPLES OF V IRTUAL S LOPE WALKING

G

F

The maximum leg extension length is set as re =r0 −0.002 m to avoid knees singularity. The impact is plastic and the angular momentum conservation law is respected during the instantaneous exchange of support legs. The linear velocity along the new stance leg is reduced to zero due to impact, and the velocity component perpendicular the new stance leg is preserved. In the transition from nth step to (n+1)th step, we have ωfn re cos φ = rs ω0n+1 = re σω0n+1 So the angular velocity around the new point foot is cos φ n ωf (2) σ To have a stable limit cycle, we need to find the fixed point which consists of a state variable or a set of states that a stable limit cycle gait possesses at the beginning or the end of the successive gait transitions. The states are the same for every new step therefore it is called the fixed point. Simply speaking, to satisfy the fixed point criterion, the initial angular velocity ω0n+1 at the (n + 1)th step should be the same as that of the (n + 1)th step, namely ω0n . ω0n+1 =

ω0n+1 = ω0n Fig. 1: Energy compensation by stance leg extension The robot model consists of a point mass which represents the overall center of mass (COM) of the robot, and the massless telescopic leg which connects the COM to the point foot. In Fig. 1, the half transparent brown ramp represents the imaginary slope where the traditional passive walkers would step down. To equally create such behavior on the level ground, the frontal stance leg should be shorter than the rear one to create a ‘Virtual Slope’. To generate cyclic gait, the current stance leg needs to extend at each step and the coming stance leg needs to shorten, as the two successive steps shown in Fig. 1. Thus, the potential energy supplied by the extension of the stance leg is equal to the energy loss at touch down. This is similar to the ramp-walking where the robot passively walks down the slope. Therefore, this approach is termed as ‘Virtual Slope Walking’. A. Modeling and Formulation of Virtual Slope Walking The variables used in the derivation are listed beneath. r0 : the full leg length; re : the maximum extended leg length; rs : the minimum stance leg length; φ: the maximum inter-leg angle at the change of support leg; σ: ratio of stance leg over extended leg rs /re ; θ0 : the initial angle between stance leg and vertical line in a new step; θf : the final angle between stance leg and vertical line at the end of a step; ω0 : the initial angular velocity around the stance point foot; ωf : the final angular velocity around the stance point foot; Ek (t0 ): the initial kinetic energy; Ek (tf ): the final kinetic energy.

(1)

(3)

Substitute (2) into (3) ωfn =

σ ωn cos φ 0

(4)

The kinetic energy at the beginning/end of the nth step is 1 2 n 1 mrs ω0 = mσ 2 re2 ω0n 2 2 1 Ekn (tf ) = mre2 ωfn 2 Combining (4) and (5) yields Ekn (t0 ) =

Ekn (t0 )/Ekn (tf ) = cos2 φ.

(5a) (5b)

(6)

Therefore, the energy loss is only determined by the interleg angle φ. So given an initial state Ek (t0 ) of a stable limit cycle, the positive work W should satisfy   1 W = − 1 Ek (t0 ). (7) cos2 φ Given the gait parameters φ and σ, the initial angle of the stance with respect to the vertical axis is θ0 = − arctan

cos φ − σ σ − cos φ = arctan , sin φ sin φ

(8)

where cos φ < σ < 1. When σ = cos φ, the new stance leg strikes the level ground perpendicularly. So for flat ground walking, we set cos φ < σ < 1 to ensure a negative θ0 . The parameter σ results in an asymmetrical swept angle θ between the stance leg and vertical axis. The energy compensation can be understood in two means and both are correct and consistent with each other. 1) In the Cartesian coordinate, the positive work that compensates for the energy loss in heel strike comes

Preprint for research circulation only, please find official paper from IEEE Xplore from the active leg extension which increases the potential energy. The most efficient energy injection is the instantaneous leg extension at θ=0 so all the active work converts to the potential energy. 2) In the polar coordinate, the positive work is done by the gravitational torque due to the asymmetrical swept angle θ. The most efficient energy injection is the instantaneous leg extension at θ=0 to minimize the negative work and maximize the positive work. Assume a set of φ and σ of a stable limit cycle has the initial angular velocity ω0 . The initial angle θ0 is determined by φ and σ according to (8). Define θe the angle where the instantaneous leg extension occurs at te instant. The initial kinetic energy at t0 is 1 1 2 2 (9) mrs ω0 = mre2 σ 2 ω02 2 2 The kinetic energy at te− just before the leg extension is Ek (t0 ) =

Ek (t− e )

⇒ ωe2−

1 = mre2 σ 2 ωe2− 2 = Ek (t0 ) − mgrs (cos θe − cos θ0 ) 1 = mre2 σ 2 ω02 + mgre σ (cos θ0 − cos θe ) 2 g cos θ0 − cos θe = ω02 + 2 re σ

⇒ ωe+ = σ 2 ωe−

1 2 4 2 1 mre σ ωe− − mre2 σ 2 ωe2− 2 2 1 2 4 2 = mre (σ − σ )ωe2− < 0 2 The change of potential energy is

(10)

e0

 ef

 ef

 ,t mg

x (1)

(11)

(12)

(13)

(14)

So, the net mechanical energy is injected by ∆E = ∆Ep + ∆Ek (15) 1 = mgre (1 − σ) cos θe + mre2 (σ 4 − σ 2 )ωe2− 2 The positive work converts to the increased mechanical energy which balances the energy loss, it satisfies (7). So 1 ∆E = W = ( 2 − 1)Ek (t0 ) cos φ 1 = mgre (1 − σ) cos θe + mre2 (σ 4 − σ 2 )ωe2− 2 (16)

 f ,t f

0 ,t0

z y

∆Ek =

∆Ep = mgre (1 − σ) cos θe > 0

 G  G

 0

e0

The kinetic energy after the leg extension changes by ∆Ek = Ek (te+ ) − Ek (te− ) 1 1 = mre2 ωe2+ − mre2 σ 2 ωe2− 2 2 Substitute (11) into (12), we have

B. Equivalence of Instantaneous Leg Extension

mg

The angular velocities ωe− and ωe+ before and after the leg extension obey the angular momentum conservation. mrs2 ωe− = mre2 ωe+

Substitute (9) and (10) into (16) to cancel out intermediate variable ωe− , we obtain the formula for ω0 which is the fixed point for a stable limit cycle.   2 cos2 φ (1 − σ 3 ) cos θe + (σ 3 − σ) cos θ0 2 2 ω0 = ωc , (17) σ 2 (1 − σ 2 cos2 φ) q where ωc = rge , θ0 is a function of φ and σ as in (8). As (17) shows, the fixed point (θ0 , ω0 ) in virtual slope walking is determined by three parameters, namely φ, σ and θe , and θe =0 corresponds to a maximum value of ω0 . The parameter ωc is the natural frequency of the system itself. It means that given the same gait parameters φ and σ, the limit cycle gait frequency will be different for a different leg length. The shorter the leg is, the higher the gait frequency becomes.

mg

G  0

(2)

G  0 (3)

Fig. 2: Mechanical work done by gravitational torque, W1 > W2 > W3 . (1) Instantaneous leg extension at θe0 ;(2) Continuous leg extension between θe0 and θef ; (3) Instantaneous leg extension at θef . This section proves that the mechanical energy supplied by an arbitrary continuous leg extension between θe0 and θef is equivalent to an instantaneous leg extension at θe where θe0 < θe < θef . Except the leg extension period, the virtual leg r has the boundary constraints of r˙ = 0 at θ0 and θf respectively, so the corresponding kinetic energy of the translational motion along the virtual leg is zero. Hence, the leg extension changes no kinetic energy along virtual leg, but only the lever arm of gravity. Hence, the increased rotational kinetic energy around pivot comes from the net work done by the gravitational torque, Z θf (18) W = Ek (tf ) − Ek (t0 ) = τ (θ)dθ, θ0

where τ (θ) = mgr(θ) sin θ. The beneath inequality holds at three angular sections in the 2nd scenario of Fig. 2. Note the different leg lengths during different angles are as follows   r1 = r2 = r3 , τ1 = τ2 = τ3 , θ0 ≤ θ ≤ θe0 r1 > r2 > r3 , τ1 > τ2 > τ3 , θe0 < θ < θef .  r1 = r2 = r3 , τ1 = τ2 = τ3 , θef ≤ θ ≤ θf Therefore, we have Z θf Z τ1 (θ)dθ > θ0

θf

θ0

Z

θf

τ2 (θ)dθ >

τ3 (θ)dθ.

(19)

θ0

The work done by the gravity has the inequality constraint such that W1 > W2 > W3 .

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C. Self-stabilization of Virtual Slope Walking Substitute (10) into the increased mechanical energy ∆E in (16) to remove the intermediate variable ωe− , we obtain the equation of ∆E in terms of the initial angular velocity ωn (t0 ) at the nth step. t0 and tf denote the time instant at the beginning and the end of a new step. 1 ∆E = mgre (1 − σ) cos θe + mre2 (σ 4 − σ 2 ) 2   g cos θ0 − cos θe 2 · ωn (t0 ) + 2 re σ

(20)

(21)

For a stable limit cycle, recall (6), we have Ekn+1 (t0 ) equal to Ekn (t0 ). Therefore the following condition holds, Ekn+1 (t0 ) = Ekn (tf ) cos2 φ = (Ekn (t0 ) + ∆E) cos2 φ

Heel strike

4.6 4.4 4.2 4 3.8 3.6 −0.4

−0.3

−0.2

−0.1

0

0.1

0.2

(22)

Substitute (20) and the expressions of Ekn (t0 ), Ekn+1 (t0 ) according to (9) into (22), yields 1 2 2 2 mr σ ωn+1 (t0 ) 2 e 1 2 2 2 mr σ ωn (t0 ) + mgre (1 − σ) cos θe = 2 e   1 2 4 g cos θ0 − cos θe 2 2 + mre (σ − σ ) ωn (t0 ) + 2 cos2 φ 2 re σ  g 1 1 2 ⇒ ωn+1 (t0 ) = σ 2 ωn2 (t0 ) + 2 ( 2 − ) cos θe re σ σ   g cos θ0 − cos θe +(σ 2 − 1) 2 cos2 φ re σ (23) p Recall ωc = g/re , we have 2 ωn+1 (t0 ) = σ 2 cos2 φωn2 (t0 ) + 2 cos2 φωc2   1 1 1 · ( 2 − ) cos θe + (σ − ) (cos θ0 − cos θe ) σ σ σ (24)

0.3

0.4

0.5

0.6

θ [rad]

Fig. 3: Phase portrait of θ and θ˙ in a stable limit cycle. Define Ω as ω 2 , thus the single variable Ω becomes the indicator of the rotational kinetic energy of the system. Ωn+1 (t0 ) = σ 2 cos2 φΩn (t0 ) + 2 cos2 φΩc   1 1 1 · ( 2 − ) cos θe + (σ − ) (cos θ0 − cos θe ) σ σ σ (25) Equation (25) is the ‘stride function’ according to McGeer’s definition [3]. Denote the stride function as f, thus f(Ωn ) = σ 2 cos2 φΩn (t0 ) + 2 cos2 φΩc   1 1 1 · ( 2 − ) cos θe + (σ − ) (cos θ0 − cos θe ) . σ σ σ (26) Given the fixed point in (17), the projection of the state has the relation below based on the stride function in (26). Ωf = f(Ωf ).

The kinetic energy at the end of the nth step tf is Ekn (tf ) = Ekn (t0 ) + ∆E

5 4.8

θ˙ [rad/s]

According to (19), there exists an instantaneous extension angle ∃ θe ∈ (θe0 , θef ) which results in the same energy compensation as the continuous leg extension from θe0 to θef with boundary velocity r=0. ˙ The exact solution of θe depends on the continuous leg extension trajectory which could have a variety of possible realizations. This suggests that as long as the resulted states at the Poincar´e sections are identical, there are mathematically infinite state trajectories that lead through a funnel between two successive Poincar´e sections. Since every continuous leg extension corresponds to an instantaneous leg extension of equivalence, the instantaneous version is used hereafter for a simpler theoretical analysis, while in experiments we implement the continuous leg extension to suit the power limit of the actuation.

(27)

Fig. 3 shows an example of the phase portrait of our derived limit cycle gait with an existing fixed point. The stabilization of VSW is quantitatively measured by the eigenvalues of the Jacobian matrix of the stride function on fixed point. The maximum eigenvalue of Jacobian matrix indicates whether the system will regain the stable cyclic walking after a small disturbance around the fixed point. The basin of attraction (BOA) implies a subset of state space where Ω finally converges to the fixed point Ωf , thus a steady state walking pattern will be achieved. According to the former research [13], [14], the eigenvalues of the Jacobian matrix of the stride function at the fixed point reflects the level of disturbance rejection. If all the eigenvalues are within the unit circle, the disturbance reduces at every step. In this case, the states of system on Poincar´e section will return to the fixed point. Therefore, it is claimed that the fixed point is stable. Otherwise the fixed point is critical stable or even unstable. The maximum eigenvalue of the Jacobian matrix of f at the fixed point can be expressed by its first derivative df λ= = σ 2 cos2 φ. (28) dΩ Ω=Ωf Since σ < 1 and cos φ < 1, so the eigenvalue λ < 1, hence the system is asymptotically stable. A designed walking gait satisfying (28) will eventually converge to the fixed point at the rate of σ 2 cos2 φ per step. To sum up, the VSW gait is asymptotically stable and selfstabilized as long as the existence conditions of the fixed point as in (17), (26) and (27) are satisfied.

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100

100

θhip stance

θhip stance

 t = Tc ln

θf + Tc ωf θ0 + Tc ω0

 ,

(29)

q where Tc = gr¯ . The initial state (θ0 , ω0 ) are known now, thus the final state of the step (θf , ωf ) can also be obtained analytically by θf = θ0 +φ and (4). So the whole step period Ts can be calculated by (29). The leg extension control by angle is difficult to be implemented in the real system because of low resolution measurement. In our reported experiments, we set θef =θf , thus tef =Ts . Hence, this provides an alternative of controlling one parameter, the initial timing of the leg extension te0 , to be tuned instead of tuning θe0 . During stance, the angle and length of the virtual leg, pointing from hip to ankle, are t θf − θ0 (1 − cos π ), θst = θ0 + 2 Ts  r , 0 ≤ t < t  e0  s t−te0 s rs + re −r rst = 2 (1 − cos π tef −te0 ), te0 ≤ t ≤ tef   r ,t < t ≤ T e

ef

s

(30)

knee θswing

0

0.05

0.1

0.15

0.2

0.25

θknee stance

Joint trajectory [deg]

θhip swing

50

−50 0

θhip swing 50

0

−50 0

0.3

knee θswing

0.05

0.1

Time [s]

0.15

0.2

0.25

(b) te0 = 25%Ts

100

100

θhip stance

θhip stance

θknee stance θknee swing

0

0.05

0.1

0.15

0.2

Time [s]

(c) te0 = 50%Ts

0.25

θknee stance

Joint trajectory [deg]

hip θswing

50

−50 0

0.3

Time [s]

(a) te0 = 0%Ts

Joint trajectory [deg]

Equations (17) and (18) in Section II show that the leg extension occurs when θ ≥ 0 is more efficient to deliver energy into the system because the negative work done by the gravity torque is minimized. However, the measurement of this inclination angle with respect to the gravity requires the combined measurements of inertia sensor and joint angles which are noisy in our platform. So in the implementation, the leg extension is controlled by time te within [t0 , tf ] as an alternative of the angle θe within [θ0 , θf ]. Define the following parameters and variables for the gait pattern generation. Ts : the period of one step; te0 : the start time of leg extension; tef : the end time of leg extension; ts0 : the time of minimum swing leg length; tsf : the time of maximum swing leg length; rst : the stance leg length; rsw : the swing leg length; rs0 : the minimum swing leg length; θst : the stance leg angle with respect to hip; θsw : the swing leg angle with respect to hip. Given two gait parameters σ and φ, the initial angle θ0 is firstly determined by (8). According to (17), the initial velocity ω0 is uniquely determined by one control parameter θe . Choosing a value of θe ∈ [θ0 , θf ], we can obtain two neighbor angles θe0 = θe − ∆ and θef = θe + ∆ for continuous leg extension with an equivalent energy injection. Below is the generic formula to approximate the transition time between two successive states based on the linearization around the average leg length r¯ = rst2+re during stance

Joint trajectory [deg]

θknee stance

0.3

hip θswing

50

θknee swing

0

−50 0

0.05

0.1

0.15

0.2

0.25

0.3

Time [s]

(d) te0 = 75%Ts

Fig. 4: Joint trajectories with different leg extension time.

(a) te0 = 0%Ts

(b) te0 = 25%Ts

(c) te0 = 50%Ts

(d) te0 = 75%Ts

Fig. 5: Animation of gaits with different leg extension time.

During the swing phase, the virtual leg is controlled as θ0 − θf t θsw = θf + (1 − cos π ), 2 Ts  rs0 −re t  ,  re + 2 (1 − cos π ts0 ), 0 ≤ t < ts0 t−ts0 rs −rs0 rsw = rs0 + 2 (1 − cos π Ts −ts ), ts0 ≤ t ≤ tsf  0  rs , tsf < t ≤ Ts (31) and the minimum leg length is set as rs0 = 0.85 rst to obtain a good foot-ground clearance. Given the virtual leg length and angle, the inverse kinematics is solved considering the COM of the simplified model coincides with the hip of the real robot. In the end of a step, the joint trajectories of the stance and swing legs exchange. Since the boundary angles are the same, a smooth gait pattern can be obtained. Fig. 4 shows four sets of joint trajectories with different leg extension time te0 , and Fig. 5 shows the intuitive stick-

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(a) Free walking on the level ground

(b) Walking over random obstacle test

(c) Stepping over a large stair obstacle

Fig. 6: Snapshots of dynamic walking with 0.05s time interval.

figure animation. It should be noted that the real dynamic simulation will differ from this animation because the real robot has passive rotation around the pivot point, which is under-actuated. IV. E XPERIMENTS The experimental testbed is the Stepper-2D biped with a boom connected to its torso to constrain the motion in the sagittal plane. The robot has actuated hip and knee in each leg. Each joint is driven by a digital servo ‘Dynamixel RX28’ with series port communication for sending angular references. The mass and dimension of each segment are: trunk 390g, 118mm; thigh 30g, 125 mm; shank 165g, 125mm. The control system is built in Matlab Simulink RTW module to achieve the real-time control. The walking pattern generator is implemented using the algorithm presented in Section III. The control loop runs at 10 ms. The experiments show that measured walking speed of the robot can be achieved from 0.69 m/s upto 1.03 m/s. In this paper, two types of experiments were selected to be reported: free walking and walking over obstacles. In Fig. 6, the order starts from left to right and from top to bottom. Fig. 6a shows the snapshots of the highest walking speed without obstacle disturbance, the corresponding data are in Fig. 7. In Fig. 6b, the rubber plates with the thickness less than 4% of leg length were randomly distributed and overlapped on the ground. The purpose was to test the self-stability of the walking algorithm under the small and continuous disturbances. As can be seen in Fig. 6b, the robot was able to walk over all the obstacles without falling down.

In order to know the maximum disturbance that could be rejected by the self-stability property of the algorithm, a stair stepping experiment was carried out with no detection of the unexpected heel-strike for modulating any gait parameters. The proposed method produces the most robust walking at the real speed of 0.81m/s. As shown in Fig. 6c, the rubber plates, piped up at the height of 1.8cm (7.2% of leg length), were placed as an obstacle on the walking path, and the robot successfully walked up and down the stair. The corresponding data are in Fig. 8. Fig. 7 and Fig. 8 show the experimental data of measured walking distance and velocity, body pitch and height, and joint reference trajectories respectively for free walking (Fig. 6a) and walking over an obstacle (Fig. 6c). The body pitch is defined as positive when the robot leans forward, and the body height variation is shown by the relative movement around the nominal torso height. In Fig. 7, the ideal walking speed is 4.29 leg/s while the real measured speed is 4.12 leg/s (1.03 m/s), due to the unavoidable slipping. The torso inclination has about 4◦ magnitude of fluctuation. The torso height produces an up-down movement with 2cm variation, which implies a compass like gait. In Fig. 8, the instant at which the robot stepped on the stair obstacle is marked by yellow lines. Right after stepping on the obstacle, the measured torso height increased about 2 cm, which was close to the real height of the obstacle. V. C ONCLUSION The concept of implementing the passive dynamic walking principles in controlling the powered bipeds was demonstrated successfully by our study. Compared to the ZMP controlled bipedal robots, the Stepper-2D robot achieved fast walking with under-actuated point feet . The Stepper-2D robot also exhibited a fairly promising feature of self-stabilization by rejecting a large disturbance such as a stair-like obstacle at the height of 7.2% leg length, requiring neither the external collision detection nor the measurement of the change of the body inclination. The level of stability can be analyzed and understood simply based on two gait parameters φ and σ. The eigenvalue has simple formula as λ = σ 2 cos2 φ, and λ < 1 indicates the asymptotic stability of the limit cycle gait. R EFERENCES [1] S. H. Collins, A. Ruina, R. Tedrake, and M. Wisse, “Efficient bipedal robots based on passive-dynamic walkers,” Science, vol. 18, no. 307, pp. 1082–1085, February 2005. [2] D. Hobbelen, T. de Boer, and M. Wisse, “System overview of bipedal robots flame and tulip: Tailor-made for limit cycle walking,” in IEEE/RSJ International Conference on Intelligent Robots and Systems, 2008, pp. 2486–2491. [3] T. McGeer, “Passive dynamic walking,” International Journal of Robotics Research (Special Issue on Legged Locomotion), vol. 9, no. 2, pp. 62–82, 1990. [4] T. McGeer, “Stability and control of two-dimensional biped walking,” Center for Systems Science, Simon Fraser University, Burnaby, BC, Canada, Technical Report, vol. 1, 1988. [5] A. Goswami, B. Espiau, and A. Keramane, “Limit cycles in a passive compass gait biped and passivity-mimicking control laws,” Autonomous Robots, vol. 4, pp. 273–286, 1997.

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1

2

3

30

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.00

1

2

3

5

10

15

20

25

Walking velocity [m/s]

Walking velocity [m/s]

Walking distance [m] 30

6 5 4 3 2 1 00

Walking distance [m]

35 30 25 20 15 10 5 0 0 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

5

10

15 20 Time [s]

25

(a) Walking distance and velocity

Body pitch [deg] 16

17

16

17

18

19

20

18

19

20

Hip height [m]

Hip height [m]

7

8

4 5 Time [s]

6

7

8

Time [s]

5 0 5 102 0.05 0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.032

15.4

15.6

15.8

15.4

Time [s]

15.6

15.8

70 60 50 40 30 20 10 0 10 202.0 100 80 60 40 20 02.0

16.0

Left Right

15.2

Hip angle [deg]

Left Right

15.2

3

4

3

4

Time [s]

5

6

7

5

6

7

(b) Body posture and height

Knee angle [deg]

Hip angle [deg] Knee angle [deg]

6

10

(b) Body posture and height 60 50 40 30 20 10 0 10 20 30 15.0 100 80 60 40 20 0 15.0

5

(a) Walking distance and velocity

Body pitch [deg]

6 4 2 0 2 4 6 15 0.03 0.02 0.01 0.00 0.01 0.02 0.03 15

4

16.0

Left Right

2.2

2.4

2.6

2.8

3.0

Left Right

2.2

2.4

Time [s]

2.6

2.8

3.0

(c) Joint angular references

(c) Joint angular references

Fig. 7: Free walking at maximum speed 1.03 m/s. Gait parameters φ = 60◦ , σ = 0.83, te0 = 0.7Ts .

Fig. 8: Robust walking at speed 0.81 m/s. The occurrence of disturbance is highlighted by yellow line. Gait parameters φ = 50◦ , σ = 0.88, te0 = 0.8Ts .

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