High-speed and energy-efficient biped locomotion based on Virtual ...

Auton Robot (2011) 30: 199–216 DOI 10.1007/s10514-010-9201-4

High-speed and energy-efficient biped locomotion based on Virtual Slope Walking Hao Dong · Mingguo Zhao · Naiyao Zhang

Received: 31 March 2009 / Accepted: 23 July 2010 / Published online: 7 December 2010 © Springer Science+Business Media, LLC 2010

Abstract In our previous work, we have presented results on Virtual Slope Walking, that is when a robot walks on level ground down a virtual slope by leg length modulation, based on the potential energy restoration in Passive Dynamic Walking. In this paper, we introduce the model of Virtual Slope Walking with Trajectory Leg Extension (TLE) and equivalent Instantaneous Leg Extension (ILE) under the Equivalent Definition. The analytic solution of the model’s fixed point is obtained to analyze the essence of Virtual Slope Walking. We systematically investigate the characteristics and illustrate the effect of model parameters: the length-shortening ratio β, the equivalent extension angle θII∗ , and the inter-leg angle ϕ0 . We examine the energy efficiency and walking speed to demonstrate that Virtual Slope Walking is effective in generating high speed and energyefficient walking. The high energy efficiency of the proposed model is theoretically confirmed. And the fast walking is validated by the experiments of a planar biped robot Stepper-2D, which achieves a sufficiently fast relative speed of 4.48 leg/s. Keywords Biped locomotion · High speed · Energy efficiency · Virtual Slope Walking Abbreviations m the mass of the body; r the length of the stance leg; H. Dong · M. Zhao () · N. Zhang Department of Automation, Tsinghua University, Beijing 100084, P.R. China e-mail: [email protected] H. Dong e-mail: [email protected] N. Zhang e-mail: [email protected]

rs , re θ ω ϕ g Ec Ed ET T Ts , Te

the length of the stance leg before and after leg extension; the clockwise angle of the stance leg with respect to the vertical line; the angular velocity of the stance leg; the inter-leg angle; gravitational acceleration; the complementary energy in one step; the dissipated energy in one step; the total mechanical energy; step period; the start and end time of stance leg extension, respectively.

1 Introduction A Passive Dynamic Walker can walk down a shallow slope without any control or actuation, originally demonstrated by McGeer via simulations and experiments (McGeer 1990). Since then, McGeer’s finding is confirmed with various passive walking studies. Goswami et al. studies the characteristics of passive compass-gaits using Poincaré return map (Goswami et al. 1998). Coleman analyzes the dynamic behavior of a rimless wheel (Coleman 1998). Garcia et al. analytically studied the essential of passive dynamic walking using the simplest walking model (Garcia et al. 1998). Since the passive walker performs natural and highly energy-efficient gaits, its concept has been used as a starting point for designing actuated walkers that are able to walk on level ground (Collins et al. 2005). The slope in Passive Dynamic Walking is a fundamental property, and can be replaced by a variety of other energy sources. As McGeer stated in (McGeer 1988), there are several options for adding

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power and control to the Passive Dynamic Walker for level walking based on the kinematic energy compliment. Examples include torque application between legs, which is realized in the delft pneumatic biped Denise (Wisse 2004); torque on the stance leg-ankle joint, which is realized in Meta (Hobbelen and Wisse 2008); impulse application on the trailing leg as it leaves the ground, which is realized in the Cornell Biped (Collins et al. 2005). These applications resulted in highly energy-efficient gaits, but limited walking speed. On the other hand, from the viewpoint of equivalent gravity effect, Goswami et al. (1997) proposed active control algorithm using Passive Dynamic Walking as a reference model, leading to the work of Spong (1999) and Asano et al. (2004). And the biped Lucy (Verrelst et al. 2005), actuated by pleated pneumatic artificial muscles, demonstrates successful active walking by incorporating the Passive Dynamic Walking model with variable compliance (Vanderborght et al. 2008). Asano et al. introduced “virtual gravity field” toward the horizontal direction as a driving force for levelground walking robot (Asano et al. 2005). These modelbased control applications are dependent on the precision of the model, and lose naturalness of Passive Dynamic Walking by gaining controllability. Moreover, Asano and Luo (2008) propose the level walking by pumping the swing leg from the viewpoint of potential energy restoration, demonstrating a potential realization of energy efficient and high speed walking. Furthermore, Honjo et al. (2008) and Harata et al. (2009) introduce the parametric excitation for potential energy restoration by extending and shortening the stance leg in powered walking. This presents a more energy-efficient gait than just pumping the swing leg. Besides, the active locomotion derives directly from Passive Dynamic Walking. A stable walking can also be realized using nonlinear oscillators under the viewpoint of limit cycle principle (Aoi and Tsuchiya 2005), which is another essence of Passive Dynamic Walking. The concept of Passive Dynamic Walking comes from that of stable walking and results from the balance between the restored energy from the slope and dissipated energy at heelstrike. A question then arises as how to produce such an energy balance mechanism for powered walking on level ground. Previous studies demonstrate several successful realizations of energy-efficient walking (Collins and Ruina 2005). However, these energy-efficient walkers achieve the lowest energy supplement by decreasing the walking speed. It is still a challenge for generating fast biped locomotion without losing energy efficiency under the corresponding speed. This question leads us to conclusion that if the balance between the restored and dissipated energy can be flexibly adjusted, a large range of walking speed can be generated. Then a biped locomotion with both high speed and its corresponding energy efficiency can be obtained. Based

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on this idea, we proposed Virtual Slope Walking by introducing the leg length modulation and realized a fast and energy-efficient biped walking gait (Zhao et al. 2008; Dong et al. 2009) from the potential energy restoration in Passive Dynamic Walking. The swing leg is shortened relative to the stance leg prior to the heelstrike, and then the effect would be like taking a downhill step, which we named Virtual Slope. By actively extending the stance leg and shortening the swing leg, a balance between the complementary energy and the dissipated energy is realized in Virtual Slope Walking. The energy restoration can be easily adjusted in Virtual Slope Walking, producing a large range of walking speed. In this paper, we analytically study the model of Virtual Slope Walking and demonstrate that a fast and energyefficient gait can be generated based on the potential energy restoration in Virtual Slope Walking. Since the Trajectory Leg Extension (TLE) is so complicated that it is hard to analytically study the characteristics of Virtual Slope Walking, we introduce the Equivalent Definition: Trajectory Leg Extension (TLE) can be equivalent to the Instantaneous Leg Extension (ILE). Based on ILE, we further systematically investigate the characteristics of Virtual Slope Walking from an analytic solution of the fixed point. It is shown from the analyzing results that both high speed and energy efficiency can be easily obtained in Virtual Slope Walking. This validity is confirmed by the theoretical results and successful walking experiments of a planar biped robot Stepper-2D, demonstrating a sufficiently fast dynamic walking. The remainder of this paper is organized as follows. In Sect. 2, Virtual Slope Walking is introduced. In Sect. 3, the model of Virtual Slope Walking with TLE and equivalent ILE is presented. In Sect. 4, we analyze the characteristics of Virtual Slope Walking. Section 5 presents the walking experiments of Stepper-2D and Sect. 6 concludes and presents a future work.

2 Principle of Virtual Slope Walking In Passive Dynamic Walking, a robot can descend a gentle slope with no energy input other than gravity, and have no active control, as shown in Fig. 1. The lost gravitational potential energy while the robot walks downhill is transformed into the walking kinetic energy and gets dissipated at heelstrike (McGeer 1990). If the system state has to appropriate initial condition, a stable gait can then be synthesized if the complementary gravitational potential energy Es balances with the dissipated energy Ed , asymptotically. We suppose that the robot leg length can be shortened boundlessly, and the swing leg can be swung around arbitrarily quickly to the position with constant inter-leg angle

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at heelstrike. In level walking, the swing leg is shortened by a constant ratio during each step. In this way, the center of mass experiences a virtual slope as shown in Fig. 2(a). Just like Passive Dynamic Walking, a stable gait can be achieved continuously under an appropriate initial conditions. However, in practical walking, the leg length of the robot cannot be shortened boundlessly. So after heelstrike, it is required that the stance leg shortened in the previous walking step should be actively extended during the following swing phase with an amount of potential energy Ec added into the system, as shown in Fig. 2(b). A stable gait can also be synthesized on level ground with the complementary energy Ec balances with the dissipated energy Ed asymptotically. Virtual Slope Walking is defined as a combined process of actively extending the stance leg and actively swinging and shortening the swing leg.

Fig. 1 Passive Dynamic Walking

Fig. 2 Virtual Slope Walking Fig. 3 Model of Virtual Slope Walking

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3 Model of Virtual Slope Walking 3.1 Model description A diagram of our walking model is shown in Fig. 3. We define that a walking step starts when the new stance leg (light line) has just made contact with the ground, see I. The stance leg swings to the position at θ = θII for the beginning of extension, see II. The stance leg extension ends at θ = θIII , see III. The swing leg (heavier line) is shortened and swung to the position with constant inter-leg angle before heelstrike, and hits the ground in the bottom right picture, see IV. The details of the model and the basic assumptions are listed below. Mass: The model has two straight, massless legs and a point mass body at the hip. This model is based on the basic assumption that humans have compact bodies and light legs. The forces the legs exert on the upper body act through the center of mass, thus apply very little rotational moment on the upper body. Leg: The two legs are modeled as a telescopic actuator with a point foot. The stance leg length is extended from rs to re , and the swing leg length is shortened from re to rs . The length-shortening ratio is defined as β = rs /re .

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Actuation: The stance leg is actuated for extension, while swing phase is an unactuated inverted pendulum. The swing leg is actuated for shortening and swinging to the position with a constant inter-leg angle before heelstrike. Stance leg extension: The stance leg extension is realized by introducing a smooth trajectory of leg length r from θ = θII to θ = θIII , namely TLE. The trajectory is assumed to satisfy the velocity-constraint conditions r˙ (θ = θII ) = r˙ (θ = θIII ) = 0, guaranteeing that the complementary energy is transformed into the gravitational potential energy and rotatory kinetic energy without radial kinetic energy. Heelstrike: the impact of the swing leg with the ground is assumed to be fully inelastic (no slip, no bounce). This implies that during the instantaneous transition stage: – the robot configuration remains unchanged; – the angular momentum of the robot about the swing-foot contact point is conserved, which leads to a discontinuous change on the velocity of the point mass body; – double stance is assumed to occur instantaneously. When the swing leg hits the ground and sticks, the previous stance leg lifts up; – the inter-leg angle ϕ at heelstrike is kept constant, which means that the swing leg can be swung around arbitrarily quickly to the position with constant inter-leg angle before heelstrike, fixed impact posture provides dimensionreduction in the state space and asymptotic stability of the fixed point since the swing leg is modeled as a massless leg.

The governing equations of the robot consist of nonlinear differential equations for the swing phase and algebraic equations for the transition of heelstrike. (1) Swing phase from I to II: Using the Lagrangian Equations, the second-order differential equations of motion are given below for the stance leg with constant length rs : g sin θ (t) rs

(1)

Rescaling the time as by defining dimensionless time √ τ = g/re t, (1) can be rewritten as θ¨ (τ ) =

1 sin θ (τ ) β

where F is the force that the legs exert on the center of mass during the stance leg extension. (3) Swing phase from III to IV: Similarly to (2), the equations of motion for the stance leg with constant length re are θ¨ (τ ) = sin θ (τ )

(4)

(4) Heelstrike transition from IV to I of the subsequent step: The heelstrike from step n to the subsequent step n + 1 occurs when the geometric collision condition  θI (n + 1) = −(ϕ0 − θIV (n)) (5) β cos θI (n + 1) = cos θIV (n) is met. Here the ‘I’ and ‘IV’ subscripts denote the instants I and IV, respectively and ϕ0 is the constant of the inter-leg angle at heelstrike. Equation (5) also reflects a change of names for the two legs. The swing leg becomes the new stance leg, and vice versa. From the conservation of angular momentum about the swing-foot contact point at heelstrike, we obtain the following transition equation: ωI (n + 1) =

cos ϕ0 ωIV (n) β

(6)

Equations (2)–(6) construct the dynamic equations of hybrid system of the model in Virtual Slope Walking.

3.2 Governing equations

θ¨ (t) =

Using the Lagrangian Equations, the equations of motion are  re 2 ˙ θ¨ (τ ) = r(τ ) sin θ (τ ) − r(τ ) r˙ (τ )θ (τ ) (3) F r¨ (τ ) = r(τ )θ˙ 2 (τ ) − re cos θ (τ ) + re mg

(2)

To simplify notation, we will refer to dimensionless time τ as the time variable. (2) Stance leg extension phase from II to III: The stance leg acts as an inverted pendulum with variable length r(τ ).

3.3 Numerical simulations of TLE Incorporating the smooth and velocity-constraint conditions into the model assumptions for TLE, we introduce the trajectory of stance leg length r(τ ) for the realization of leg extension as follows: ⎧ ⎪ ⎨ rs θ < θII r(τ ) = 1−β r sinπ (τ −Tr /2) + 1+β r (7) ⎪ Tr 2 e ⎩ 2 e θII ≤ θ ≤ θIII , τ ∈ [0, Tr ] where θII is the start position of leg extension and Tr is the dimensionless time period of the leg extension phase. By applying (7) and numerically integrating the dynamic equations (2)–(6), the simulation results for TLE of Virtual Slope Walking are presented in Fig. 4. As shown in Fig. 4(a) and (b), the stable cyclic walking motion is generated by the effect of TLE in Virtual Slope Walking. It is also indicated in Fig. 4(c) and (d) that the total mechanical energy ET is restored during the stance leg

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Fig. 5 ILE of Virtual Slope Walking. II III: instant of TLE phase II∗ III∗ : instant of ILE phase

ILE provides the possibility to analytically study the essence of Virtual Slope Walking. As shown in Fig. 5, ILE is the simplest special case of ∗ . The stance leg extends instantaneously TLE when θII∗ = θIII ∗ at θII , which is defined as the equivalent extension angle. While the swing phase from I to II∗ and III∗ to IV is the same unactuated inverted pendulum as that of TLE. The heelstrike condition of these two phases is also the same. The stance leg extends instantaneously from II∗ to III∗ , which implies that during the instantaneous transition stage: – the robot configuration remains unchanged, except the stance leg length; – the angular momentum of the robot about the support point on the ground of its stance leg is conserved. This conservation law leads to a discontinuous change on the velocity of the point mass body.

Fig. 4 Simulation results for TLE of stable Virtual Slope Walking: θII = 0◦ , Tr = 0.2, β = 0.8, ϕ0 = 45◦

extension phase and dissipated at heelstrike. This verifies the effect of the energy balance mechanism in Virtual Slope Walking. Since the simulation results are generally based on the specific trajectories of stance leg length and high-order nonlinear differential equations, it is hard to investigate the model analytically for the general and theoretical results. We introduce an equivalent leg extension to solve this problem in the following section. 3.4 Equivalent Instantaneous Leg Extension (ILE) TLE can be considered as an integral process of complementary energy, which is always analyzed by numerical simulation. From the viewpoint of complementary energy, the leg extensions with different trajectories can be considered as equivalent if they have the same energy restoration processes. There exists the simplest leg extension case of TLE, namely ILE (Zhao et al. 2009), which can equivalently replace any TLE with the same energy restoration process.

From the conservation of angular momentum about the stance-foot contact point in the ILE, we obtain the following transition equation: ∗ ωIII = β 2 ωII∗

(8)

ILE (Fig. 5) is equivalent to TLE (Fig. 3) under the following definition. Equivalent definition: Given parameters (m, β, ϕ0 ) for a periodical gait, if the stance-leg angle θ and angular velocity ω in ILE at the start (I) and end (IV) are equal to those of TLE, the two leg extensions are equivalent. Let ET (i) and ET∗ (i) be the total mechanical energy of the system for TLE and ILE respectively, where i denotes the instant of the leg extension phase. According to the Equivalent Definition, if ILE has the same stance-leg angle θ and angular velocity ω as TLE at instants I and IV, resulting in the same kinematic and potential energy respectively, we can obtain  ∗ ET (I) = ET (I) (9) ET∗ (IV) = ET (IV) From the energy conservation of the unactuated inverted pendulum swing phase from Fig. I to Fig. II∗ and from

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Fig. III∗ to Fig. IV in ILE, we have  ∗ ∗ ET (II ) = ET∗ (I) ET∗ (III∗ ) = ET∗ (IV)

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(10)

Also from the energy conservation of the unactuated inverted pendulum swing phase from Fig. I to Fig. II and from Fig. III to Fig. IV in TLE, we obtain  ET (II) = ET (I) (11) ET (III) = ET (IV) From (9), (10), and (11), we have the following energy equalities:  ∗ ET (II) = ET (II) (12) ET∗ (III) = ET (III) Substituting the stance leg angle and angular velocity for the energy expression into (12), the energy equality can be represented as ⎧1 ∗2 2 ∗ ⎪ ⎨ 2 mgωII β re + mgβre cos θII (13) = 12 mgωII2 β 2 re + mgβre cos θII ⎪ ⎩1 1 ∗2 ∗ 2 2 mgωIII re + mgre cos θII = 2 mgωIII re + mgre cos θIII Note that kinetic energy is represented with dimensionless time τ . From (8) and (13), θII∗ can be represented as

 1 2 4 2 3 ∗ 2 (ωIII − β ωII ) + cos θIII − β cos θII θII = ± arccos 1 − β3 (14) Consequently, we can conclude that given TLE, there exists two equivalent ILE with symmetric extension angle θII∗ , which can be determined using (14). ILE provides the possibility to analyze the essence of Virtual Slope Walking in an analytic way. We will verify the equivalence of the dynamic behavior between these two extensions and study the characteristics of Virtual Slope Walking in the following section.

Our Poincaré section is at the start of a step, namely Panel I in Fig. 5, just after heelstrike. Based on the Poincaré mapping method, the dynamic behavior of the walking is determined by the system state in the Poincaré section, but not every instant between the sections. Consequently, based on the Equivalent Definition, two equivalent extensions have the same state at the Poincaré section, such that they have the equivalent dynamic behavior. Then we can use ILE to analyze TLE. Given the system state at instant I, the Poincaré map f determines the state just after the next heelstrike. Note that in the geometric collision condition equation (5), the stanceleg angle θI is constant with inter-leg angle ϕ0 . The heelstrike transition reduces this problem from 2D state space {θI , ωI } to a one-dimensional map f, only consisting of angular velocity ωI . While the system has only one independent initial condition, we need to specify ωI (n) at the start of walking step n to fully determine the subsequent motion at steps n + 1, n + 2, . . . , so that ωI (n + 1) can be obtained from the Poincaré mapping of ωI (n). Applying the conservation of energy two swing stages at step n (from I to II∗ and from III∗ to IV as shown in Fig. 5), we have the following energy conservation equations: 1 mgωI2 (n)β 2 re + mgβre cos θI 2 1 = mgωII∗2 (n)β 2 re + mgβre cos θII∗ 2 1 ∗2 mgωIII (n)re + mgre cos θII∗ 2 1 2 = mgωIV (n)re + mgre cos θIV 2

(15)

From the heelstrike transition equations (5) and (6), legextension transition equation (8), and (15), we obtain the walking map as follows:

 1 ωI2 (n + 1) = β 2 cos2 ϕ0 ωI2 (n) + 2 cos2 ϕ0 cos θII∗ 2 − β β 

1 −β (16) − cos θI β

4 Characteristics of Virtual Slope Walking 4.1 Poincaré section and return map The general procedure for the study of this model is based on interpreting a step as a Poincaré map, or, as McGeer termed it, a ‘stride function’ (McGeer 1990). Gait limit cycles are fixed points of this function. This walking map can be analytically expressed for ILE. We use this analytic representation to study the model characteristics for analytical probing and graphical visualization.

To simplify the definition of the map f, q is taken to be the square of the angular velocity. q = ωI2

(17)

Since the stance leg angle θI is constant, q indicates that the kinetic energy represents the total mechanical energy of the system. This state variable is different from that of Passive Dynamic Walking. By substitution of variables, the Poincaré map f is linear in q:

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Fig. 6 Typical gait cycles from numerical simulation results. (a) Angular velocity, (b) system state in phase space. The light solid and heavy dashed lines represent the trajectories of TLE and ILE respectively. The transition instants match the plots of Figs. 3 and 5. At

a gait cycle, heelstrike returns the system to its initial conditions. The leg extension trajectory from II to III is sinusoid connecting θII and θIII . θII = 0◦ , θIII = 15.8◦ , θII∗ = 9.6◦ , β = 0.8, ϕ0 = 45◦

f(q) = β 2 cos2 ϕ0 q

ergy source that can replace the slope in Passive Dynamic Walking for energy restoration.



+ 2 cos2 ϕ0 cos θII∗



1 1 −β − β − cos θI β β2



(18) Let q replace the state variable ωI . Using numerical simulations, typical plots of the cyclic walking motions resulting for TLE and its equivalent ILE (only with positive equivalent extension angle) are shown in Fig. 6. It can be seen from Fig. 6(a) that the angular velocity of ILE is the same as that of TLE at the Poincaré section, producing the same fixed point of the walking map, namely the equivalent cyclic walking motion, while the trajectories between the Poincaré sections are different. As shown in Fig. 6(b), a walking step starts at instant I, and the stance leg swings to instant II as an unactuated inverted pendulum, following the arrows in the phase trajectory. From II to III, the stance leg is extended following a smooth trajectory. Then, the stance leg swings like an unactuated inverted pendulum from III to IV. Heelstrike occurs at the phase trajectory from IV to I, resulting in a discontinuous change of the stance leg angular velocity and the exchange of the two legs. TLE can be equivalently replaced by ILE as shown in Fig. 6(b). The phase trajectory from II∗ to III∗ corresponds to ILE, with a discontinuous change of the stance leg angular velocity and unchanged stance leg angle. It is seen in Fig. 6 that these two phases exhibit the same dynamic behavior, verifying the Equivalent Definition in Sect. 3.4. The dynamic behavior in Fig. 6(b) is different from that of Passive Dynamic Walking. There exist two mechanical energy orbits during the swing phase in Virtual Slope Walking, while only one orbit in Passive Dynamic Walking. It is indicated that the effect of the stance leg extension is the en-

4.2 Existence of a fixed point A state q f is a fixed point of f if f(q f ) = q f . From (18), the fixed point of f is qf =

2 cos2 ϕ0 [cos θII∗ (1 − β 3 ) − cos θI (β − β 3 )] β 2 (1 − cos2 ϕ0 β 2 )

(19)

The stance-leg angle θI at Poincaré section can be obtained from (5) as follows: θI = − arctan

β − cos ϕ0 sin ϕ0

(20)

Three sufficient conditions have to be satisfied for the existence of a fixed point: (1) By definition of q, the sign of q is always positive; this requirement is qf > 0

(21)

From (19) and (21), this inequality can be rewritten as a criterion relating the length-shortening ratio β, the equivalent extension angle θII∗ , and the inter-leg angle ϕ0 : cos θII∗ >

β − β3 cos θI 1 − β3

(22)

where θI is the function of β and ϕ0 in (20). (2) For an existing fixed point, the robot must be able to pass the apex at mid-stance repeatedly after each heelstrike, i.e., that the robot will not fall backwards. The

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lower boundary of the angular velocity can be determined analytically as follows. It is required that the angular velocity of the apex at the mid-stance ωmid > 0, namely qmid > 0. By conservation of energy, the initial state of the system is given in two conditions: 1 mgqβ 2 re + mgβre cos θI 2 1 = mgqmid β 2 re + mgβre cos 0◦ 2 ⎧1 mgqβ 2 re + mgβre cos θI ⎪ ⎪ ⎪2 ⎪ ⎨ = 1 mgq ∗ β 2 re + mgβre cos θ ∗ II II 2 ⎪ 1 mgq ∗ re + mgre cos θ ∗ ⎪ III II 2 ⎪ ⎪ ⎩ = 12 mgqmid re + mgre cos 0◦

θII∗ ≥ 0

(23)

θII∗ < 0

(24)

(3) Geometric restrictions are listed below: – the sign of the stance leg angle θI should be negative, this requires (26)

– the stance leg angular velocity just after heelstrike should be greater than zero: from (6), we obtain π ϕ0 < 2

(27)

– the stance leg should be extended before heelstrike, this requires θII∗
(25) ∗ ∗ ⎩ 2 1−cos θII + cos θII −cos θI θ ∗ < 0 4 II β β

β > cos ϕ0

Since β and cos ϕ0 are always less than unity, the eigenvalue of the Jacobian J is within the unit circle. The fixed point is always asymptotically stable. Note that this stability principle is basically equivalent to that of a rimless wheel (Coleman 1998). Also, the fixed point of a passive walker is always asymptotically stable if the inter-leg angle is kept constant at heelstrike (Ikemata et al. 2006).

(28)

4.3 Stability analysis Once the fixed point is obtained, we would like to characterize it as stable or unstable. Only the stable fixed point produces successful walking. The eigenvalues of the Jacobian J of the Poincaré map f govern the stability of the fixed point. If all eigenvalues are within the unit circle, meaning that a small perturbation will decay with time, then the gait cycle is asymptotically stable. Considering the one-dimensional map f in (18), the eigenvalues of the Jacobian J can be represented as the first derivative of the map f at the fixed point:  df  λ= = β 2 cos2 ϕ0 (29) dq q=q f

This walking model is not a conservative holonomic system, since energy is gained in stance leg extension and lost at heelstrike. As discussed indirectly in Hurmuzlu and Moskowitz (1986), the dissipative collision provides the possibility of asymptotic stability. How asymptotic stability is approached can be seen by considering the balance between the restored and dissipated energy of the system. The complementary energy Ec is a net change of the total mechanical energy in stance-leg extension phase, which can be represented as 1 ∗2 (n)re + mgre cos θII∗ Ec (n) = mgωIII 2 1 − mgωII∗2 (n)β 2 re − mgβre cos θII∗ 2 = mgre (1 − β) cos θII∗  1  − mg 1 − β 2 β 2 re ωII∗2 (n) 2

(30)

From (15) and (30), we can obtain Ec as follows:   1 Ec (n) = − mgre β 2 1 − β 2 q(n) 2     + mgre 1 − β 3 cos θII∗ − β 1 − β 2 cos θI (31) Given the model parameters m, re , β, θII∗ , and ϕ0 , the constant term of (30) indicates the potential energy restoration, meaning that the restored potential energy is kept constant for each step. The variable term of (30) indicates that during the stance leg extension, there exists dissipation of the kinetic energy, since β is always less than unity. We can conclude that stance leg extension provides the change of both potential and kinetic energies simultaneously. And for ILE, the potential energy is restored while the kinetic energy is dissipated, resulting in the complementary energy Ec from the combined effect. Also, it is shown in (31) that the energy gained by the stance leg extension is a linear function of the state variable q. From (18) and (31), we can obtain the relationship of Ec (n + 1) and Ec (n) as follows: Ec (n + 1) = β 2 cos2 ϕ0 Ec (n) + mgre sin2 ϕ0     × 1 − β 3 cos θII∗ − β 1 − β 2 cos θI

(32)

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Then Ec (n) can be represented as

4.4.2 Energy efficiency

 n   Ec (n) = E f + β 2 cos2 ϕ0 1 − β 2  1 cos2 ϕ0 f 2 × − mgre β q(0) + E 2 sin2 ϕ0 Ef =

mgre sin2 ϕ0 [(1 − β 3 ) cos θII∗ − β(1 − β 2 ) cos θI ] 1 − β 2 cos2 ϕ0 (33)

where q(0) is the initial system state. It can be concluded from (33) that the complementary energy Ec will approach E f asymptotically, since the absolute value of β 2 cos2 ϕ0 is always less than unity. The dissipated energy Ed is a net change of the total mechanical energy at heelstrike, which is equal to the dissipated kinetic energy because the potential energy does not change at the instant of heelstrike. So Ed can be represented as 1 1 2 Ed (n) = mgre ωIV (n) − mgβ 2 re ωI2 (n + 1) 2 2 1 2 = mgre β 2 sin2 ϕ0 ωIV (n) 2

ct = =

=

(35)

(36)

Then Ed (n) can be represented as n  Ed (n) = E f + β 2 cos2 ϕ0  1 × mgre β 4 sin2 ϕ0 q(0) − β 2 cos2 ϕ0 E f 2

Used energy Weight · Distance traveled

(37)

where q(0) is the initial system state. The changing process of Ed (n) in (37) is the same as that in (33), meaning that the dissipated energy Ed and complementary energy Ec will be balanced asymptotically. Once the gait is in a periodical state, Ec is equal to Ed , producing the fixed point of the Poincaré map. Such energy restoration and dissipation mechanism provides the asymptotic stability of the fixed point.

(38)

From (19) and (31), we obtain ct as follows:

(34)

Given the model parameters m, re , β, θII∗ , and ϕ0 , Ed is also a linear function of the state variable q. From (18) and (35), we can obtain the relationship between Ed (n + 1) and Ed (n) as follows: Ed (n + 1) = β 2 cos2 ϕ0 Ed (n) + mgre sin2 ϕ0     × 1 − β 3 cos θII∗ − β 1 − β 2 cos θI

ct =

mg ·



re2

Ec 2 + β re2 − 2βre2 cos ϕ0

− 1 mgre (β 2 − β 4 )q f 2 mg · re2 + β 2 re2 − 2βre2 cos ϕ0 +

From (15) and (34), we can obtain Ed as follows: 1 Ed (n) = mgre β 4 sin2 ϕ0 q(n) + mgre sin2 ϕ0 2     × 1 − β 3 cos θII∗ − β 1 − β 2 cos θI

Since the complementary energy Ec from stance leg extension totally compensates for the dissipated energy Ed at heelstrike without unnecessary loss, it leads to the energyefficient walking, just as that in Passive Dynamic Walking. The specific cost of transport ct is a measure of energy efficiency of locomotion (Collins et al. 2005). The specific cost of transport ct of the fixed point is defined as

mgre [(1 − β 3 ) cos θII∗ − (β − β 3 ) cos θI ]  mg · re2 + β 2 re2 − 2βre2 cos ϕ0

1 − cos2 ϕ0 (1 − β 3 ) cos θII∗ − (β − β 3 ) cos θI  · 1 − cos2 ϕ0 β 2 1 + β 2 − 2β cos ϕ0

(39)

The main question of how to attain energy-efficient biped locomotion in Virtual Slope Walking can be answered by studying how to decrease ct by adjusting the model parameters β, θII∗ , and ϕ0 . Figure 7 shows the analysis of the specific cost ct changing with β, θII∗ , and ϕ0 based on (39). As shown in Fig. 7(a), the specific cost ct decreases monotonically with the increase in β. The minimum ct = 0.0146 occurs at β = 0.99, ϕ0 = 45◦ , producing maximum efficiency. This is because an increase in β causes a net decrease in the extended leg length, resulting in a decrease in the complementary energy Ec . It is indicated from Fig. 7(b) that ct decreases as the stance leg is extended far away from the vertical line. This comes from the decrease in the vertical projection of the extended leg length, also resulting in a decrease in the complementary energy Ec . Figure 7(c) shows that ct decreases as ϕ0 increases. However, the change in ϕ0 has little effect on ct . It can be concluded from Fig. 7 that β plays a major role in improving the energy efficiency, while θII∗ and ϕ0 have little effect. The minimum specific cost ct in Virtual Slope Walking is much smaller than that of previous works. Several values of specific cost in related studies on dynamic walkers are listed in Table 1. We can confirm that Virtual Slope Walking is a relatively small value compared with others. It is also important that the energy efficiency can be modulated by adjusting the controllable parameters β, θII∗ , and ϕ0 based on (39).

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Fig. 7 The specific cost ct with regard to model parameters: (a) β; (b) θII∗ ; (c) ϕ0

Table 1 Comparison of energy efficiency Type of dynamic walker

Specific cost ct

Human

0.05

Honda’s Asimo

1.6

Cornell Biped (Collins et al. 2005)

0.055

Passive Dynamic Walking (McGeer 1990)

sin γ (γ = 0.005–0.09)

Virtual Passive Dynamic Walking (Asano et al. 2005)

tan γ (γ = 0.005–0.09)

Parametric excitation with elasticity (Asano and Luo 2008)

0.031

Virtual Slope Walking

1−cos2 ϕ0 1−cos2 ϕ0 β 2

·

(1−β 3 ) cos θII∗ −(β−β 3 ) cos θI

√

1+β 2 −2β cos ϕ0

(0.0146–0.281)

4.5 Influence of model parameters on the gait Referring to the analytical expression of the fixed point q f in (19) and (20), q f is determined by three parameters: the length-shortening ratio β, the equivalent extension angle θII∗ , and the inter-leg angle ϕ0 . The influence of each parameter

on the fixed point is presented individually in the following section. Also, we will use two criterion functions to illustrate the effect of the parameters. Let T be the steady-step period, which is obtained from numerical simulation. The walk√ ing speed is evaluated by the Froude number Fr = v/ ¯ gre (Hobbelen and Wisse 2007), where the average walking speed v¯ is defined as  Distance traveled re 1 + β 2 − 2β cos ϕ0 = (40) v¯ = T T Then, the Froude number Fr can be represented as   1 + β 2 − 2β cos ϕ0 re 1 + β 2 − 2β cos ϕ0 = (41) Fr = √ gre T τ0 where τ0 is the steady-step period (corresponding to the dimensionless time τ ). The performance of walking speed with respect to these three parameters is then presented as follows. 4.5.1 Effect of the length-shortening ratio β The fixed point and walking speed evaluations are shown as a function of β in Fig. 8 with four values of ϕ0 . Figure 8(a)

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209

Fig. 8 The fixed point and Walking Speed Evaluations with regard to the length-shortening ratio β. (a) Fixed point. (b) Step period. (c) Froude number Fr

shows q f decreases with an increase in β. An increase in β causes a net decrease in the extended leg length, resulting in a decrease in the complementary energy Ec . On the other hand, the dissipated energy Ed at heelstrike can be represented as the function of β and ϕ0 : 1 Ed = mgre q tan2 ϕ0 β 2 2

(42)

where Ed increases as β increases. Consequently, the total mechanical energy ET is lowered, and q f decreases. From Fig. 8(b) and (c), we can see also that the walking speed is decreasing as β increases. The main conclusion from these graphs is that a decrease in β leads to a greater fixed point q f and faster walking. However, there is a lower limit of β shown in (26). And β is also restricted by the physical parameters of the real robot. 4.5.2 Effect of the equivalent extension angle θII∗ The fixed point and walking speed evaluations are shown as a function of θII∗ in Fig. 9 with two values of β and ϕ0 , respectively. Figure 9(a) shows a second-order relationship between q f and θII∗ . As θII∗ approaches zero from both sides,

q f increases and reaches the maximum value at θII∗ = 0◦ . The vertical projection of the extended leg length increases as θII∗ approaches zero, and more potential energy is complemented. As a consequence, the total energy ET increases, and q f increases. The vertical projection of the extended leg length reaches its maximum at θII∗ = 0◦ . From Fig. 9(b) and (c), we can also see that the walking speed varied the same as that of the fixed point with respect to θII∗ . It can be concluded from this graph that extending the stance leg more closer to the mid-stance will result in a greater fixed point q f and faster walking. From (14), we can extend this conclusion to TLE that the extension phase being close to the mid-stance also produces fast walking. 4.5.3 Effect of the inter-leg angle ϕ0 The fixed point and walking speed evaluations are shown as a function of ϕ0 in Fig. 10 with four values of β. As shown in Fig. 10(a), q f decreases with an increase in ϕ0 . The dissipated energy Ed increases as ϕ0 increases in (42). As a consequence, the total energy ET decreases, and q f decreases. From Fig. 10(b) and (c), though the step length is larger as ϕ0 increases, the walking is slower with a larger ϕ0 .

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Fig. 9 The fixed point and Walking Speed Evaluations with regard to equivalent extension angle θII∗ . (a) Fixed point. (b) Step period. (c) Froude number Fr

Therefore, we can conclude from this graph that a larger ϕ0 results in a smaller q f and a slower walking. From the above analysis, it can be concluded that the fixed point is parameterized by β, θII∗ , and ϕ0 . The expected fixed point can be easily determined by adjusting these three controllable parameters, which is a significant improvement for Passive Dynamic Walking. The walking speed can be sufficiently fast with adjusted parameters β, θII∗ , and ϕ0 , which will be confirmed in the experiment results. In conclusion, the analysis of this section derive from the analytical expression of the fixed point in the walking map, but do not depend on the numerical simulations.

5 Experiments 5.1 Planar biped robot Stepper-2D We use the planar biped robot Stepper-2D as the test bed of Virtual Slope Walking. As shown in Fig. 11, Stepper-2D is mounted on a boom to constrain the body motion in the sagittal plane. The boom has three orthogonal DOF and the

length is six times more than the height of the robot, so its effect on the robot sagittal movement can be ignored. Stepper2D is 368 mm in height and 780 g in weight. The details of the body parameters are shown in Table 2. The leg with a point foot is actuated in the hip and knee joint by digital servo motors. The telescopic leg motion is realized by bending and unbending the knee joint. The swing leg motion is realized by hip motor actuation. All digital servo motors are controlled by a computer through the serial bus. 5.2 Virtual Slope Walking gait generation The gait generation begins with the creation of swing and stance leg length trajectories based on the TLE of Virtual Slope Walking. We use three key angles to generate the leg length trajectories, as shown in Fig. 12 with the positive sign. We first define the virtual leg as the line joining the hip point and the foot point of the actual leg as shown in Fig. 12. Let ϕ be the angle from the virtual stance leg to the virtual swing leg, i.e. the inter-leg angle, determining the swing leg motion. Let αst be the angle from the virtual stance leg to the

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211

Fig. 10 The fixed point and Walking Speed Evaluations with regard to inter-leg angle ϕ0 . (a) Fixed point. (b) Step period. (c) Froude number Fr

Fig. 12 Key angles for gait generation

Fig. 11 Planar biped Stepper-2D with point foot

Table 2 Body parameters of Stepper-2D Mass (g)

Length (mm)

Trunk

390

118

Thigh

30

125

Shank

165

125

Total

780

368

thigh, determining the stance leg length. Let αsw be the angle from the virtual swing leg to the thigh, determining the

swing leg length. Since the exact shape of the trajectories is of little concern in Virtual Slope Walking, we simply use smooth sinusoids to generate the trajectories of ϕ, αst , and αsw , which also satisfy the velocity-constraint conditions in stance leg extension. As shown in Fig. 13(a), αst varies from α1 to 0 in the period of Ts to Te , resulting in the stance leg extension with β = cos α1 . Before Ts and after Te , αst remains invariant, resulting in the constant stance leg length. Here, αsw varies from 0 to α2 in the first half of step, resulting in the swing leg shortening and avoiding contacting with the ground. And, in the second half of the step, αsw varies from α2 to α1 for heelstrike. Note that ϕ varies from

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Fig. 13 Trajectories for the key angles. (a) αst , and αsw (b) ϕ α1 = 15◦ , α2 = 30◦ , ϕ0 = 35◦ , T = 0.3 s, Ts = 0.5T , Te = 0.9T

Fig. 14 Definition of joint angles

Fig. 15 Stick diagrams for α1 = 15◦ , α2 = 30◦ , ϕ0 = 35◦ , T = 0.3 s, Ts = 0.5T , Te = 0.9T

ϕ0 to −ϕ0 in the whole period, resulting in a fixed posture at heelstrike with inter-leg angle ϕ0 , as shown in Fig. 13(b). In practical walking, it is required to transform the trajectories of ϕ, αst , and αsw to the trajectories of the joint angles. The definition of the joint angles with the positive sign is shown in Fig. 14. Since ϕ is equally divided between the two hip joints. Then we have the following transformation equations: ⎧ Ship1 = − 12 ϕ + αst ⎪ ⎪ ⎪ ⎨ Ship2 = 12 ϕ + αsw ⎪ ⎪ Sknee1 = 2αst ⎪ ⎩ Sknee2 = 2αsw

(43)

The Virtual Slope Walking motion is shown in Fig. 15 using a stick diagram, presenting the process of actively extending the stance leg and swinging and shortening the swing leg. 5.3 Numerical analysis of realistic gait generation The theoretical results in the previous section are based on the analytical expression of the fixed point in ILE. The main precondition is that there always exists an ILE, which is

equivalent to any given TLE under the Equivalent Definition. Given the gait generation algorithm that produces the stance leg length trajectory for a real walking experiment, the corresponding equivalent instantaneous extension angle θII∗ can be obtained from numerical simulations. Thus, the realistic gait generation can be analyzed by the results of ILE. The trajectories of the stance leg extension are parameterized by α1 , ϕ0 , Ts , Te , and T . α1 simply determines the length-shortening ratio β as β = cos α1 . And Ts , Te determine the timing of leg extension. Applying the real trajectory in Fig. 13(a) and numerically integrating the dynamic equations (2)–(6), we can obtain the start and end positions and velocity of the system during the stance leg extension phase, which correspond to Ts and Te . Consequently, the equivalent extension angle θII∗ can be determined by (14), meaning that the equivalent ILE is obtained. The equivalent extension angle θII∗ obtained in the numerical simulations is shown as a function of Ts and Te in Fig. 16 with two values of α1 and ϕ0 , respectively. As shown in Fig. 16, for any fixed start time Ts , the equivalent extension angle θII∗ will be far from the mid-stance as the end time Te increases, and also for any fixed end time Te , θII∗ will be far from the mid-stance as the start time Ts

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213

Fig. 16 The equivalent extension angle with regard to the start and end time of stance leg extension in realistic gait generation

increases. The corresponding Ts and Te of the minimum θII∗ in each figure of Fig. 16 suggests that the start and end positions of the leg extension lying on either side of the midstance will produce the minimum θII∗ , resulting in a maximum walking speed. Besides Ts and Te , the effect of α1 and ϕ0 in the realistic gait generation is the same as that of β and ϕ0 in ILE. Consequently, the theoretical results in ILE can be extended to TLE in realistic gait generation of Virtual Slope Walking, which can be applied as a guide for parameters tuning in the experiments. 5.4 Experimental results The fixed point and walking speed can be determined by the gait parameters α1 , ϕ0 , Ts , Te , and T . Furthermore, high speed walking on Stepper-2D can be achieved by tuning these parameters based on the presented analysis. We first determine the boundary value of T and ϕ0 under the hardware limit of the servo motors. Then, the maximum α1 can be determined by the existence criterion of the fixed point and the hardware limit of the servo motors. After that, Ts , Te can be tuned to extend the stance leg close to the mid-stance. Finally, α2 is adjusted to avoid foot scuffing at mid-stance. After some hand-tuning, we get the maximum walking speed with the joint trajectories shown in Fig. 17.

Using (14) and numerical simulation with the parameters shown in Fig. 17, we obtain the equivalent extension angle |θII∗ | = 5.1◦ , indicating that the stance leg is extended close to the mid-stance and a fast walking is generated correspondingly. Stepper-2D walks 10.539 m in 9.41 s (1 circuit with the radius of 159.5 cm), resulting in the speed of 1.12 m/s. The relative speed is 4.48 leg/s, and the Froude number is 0.716, which is the fastest relative speed among known biped robots. Figure 18 presents the image sequences of three walking steps. All the videos of the walking experiments of Stepper-2D are available on our website http://www.au. tsinghua.edu.cn/robotlab/rwg/Robots/Stepper_2D.htm. Several values of relative walking speed in related studies on dynamic walkers are listed in Table 3. We have shown that Stepper-2D achieves sufficiently fast walking compared with other robots.

6 Conclusions and future work In this paper, we analytically studied Virtual Slope Walking by introducing ILE under the Equivalent Definition. Based on the analytic solution of the fixed point, we investigate the characteristics of Virtual Slope Walking by use of existence criterions, stability, energy balance and efficiency, and effect

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Fig. 17 Trajectories of the joint angles in the maximum Walking Speed Experiment. (a) ship1 , and ship2 (b) sknee1 , and sknee2 α1 = 34.2◦ , β = 0.8281, α2 = 54.2◦ , ϕ0 = 45◦ , T = 0.2 s, Ts = 0.5T , Te = 0.7T

Fig. 18 Image sequences of Walking Experiment

Table 3 Comparison of Relative Walking Speed Type of dynamic walker

Relative Walking Speed (leg/s)

Human Olympic Record

4–5

Stepper-2D (Zhao et al. 2009)

4.48

RunBot (Geng et al. 2006)

3.48

Honda’s Asimo

2.49

Rabbit (Chevallereau et al. 2003)

1.5

Flamingo (Pratt et al. 2001)

1.4

of model parameters. We conclude from the analysis that the fixed point is always asymptotically stable and can be controlled flexibly by adjusting the model parameters: β, θII∗ , and ϕ0 . Thus, the energy efficiency and walking speed also can be adjusted flexibly by these parameters in a large range. The theoretical value of the energy efficiency from (39) validates that Virtual Slope Walking achieves sufficiently highly energy-efficient walking compared with others. And, Virtual Slope Walking is also effective in generating high speed walking, which is then confirmed by the experiments conducted on the Stepper-2D. The model we presented in this paper mainly focuses on the energy restoration from the center of mass of the robot, which plays a major role in the energy efficient and high-speed walking during Virtual Slope Walking. However,

there also exists the effect from the leg mass for the energy restoration in Virtual Slope Walking. We will introduce a compass-like biped model to investigate the effect of the leg mass in Virtual Slope Walking in the future. We find that the exact trajectories of the telescopic leg motion are of little concern in Stepper-2D’s experimental results. As long as the energy balance condition and existence criterions are satisfied, the robot can generate stable walking with various shapes of smooth trajectory. Our long-term goal of this research is to prove that Virtual Slope Walking can generate a fast and energy-efficient gait in a simple way without an elaborate trajectory control algorithm. Furthermore, we plan to investigate the energy efficiency of Virtual Slope Walking in a walking experiment of Stepper-2D by introducing more sensors to measure the actual specific cost of transport. Disturbance rejection is another important aspect for Virtual Slope Walking in future work. We will introduce the real-time sensing of the ground perturbation, as described in Kajita and Tani (1997), and study the sensorbased powered walking from the viewpoint of kinematic energy complement. Acknowledgements This work was supported in part by the National Nature Science Foundation of China (No. 60875065) and Open Project Foundation of National Robotics Technology and System Key Lab of China (No. SKLRS200718).

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Hao Dong is a Ph.D. candidate in Robotics and Intelligent Control Lab, Department of Automation, Tsinghua University. He was born in Inner Mongolia, China, in 1984. He received the M.E. Bachelor Degree in Department of Precision Instruments and Mechanology of Tsinghua University in 2004. Currently, he is working on the stability analysis of Virtual Slope Walking.

Mingguo Zhao is an Associate Professor of Department of Automation, Tsinghua University. He received his B.E., M.Sc. and Ph.D. degrees from Harbin Institute of Technology in 1995, 1997 and 2001 respectively. From 2001 to 2003, he held a postdoctoral position in Department of Precision Instrument, Tsinghua University. His research interests include biped locomotion control and robot self-localization.

216

Auton Robot (2011) 30: 199–216 Naiyao Zhang graduated from the Department of Electrical Engineering of Tsinghua University in 1970. Since then, he has been engaged in the teaching and research work in the Department of Automation, Tsinghua University. He has been in Stuttgart University, Germany as a visiting scholar from 1992 to 1993. His current research interests include: hierarchical fuzzy systems and control, intelligent control of robot soccer, etc.