Gait generation and control for biped robots with underactuation degree one

Automatica 47 (2011) 1605–1616

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Gait generation and control for biped robots with underactuation degree one✩ Yong Hu, Gangfeng Yan, Zhiyun Lin ∗ Asus Intelligent Systems Lab, College of Electrical Engineering, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, PR China

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Article history: Received 23 April 2010 Received in revised form 28 January 2011 Accepted 8 February 2011 Available online 25 May 2011 Keywords: Biped walking Periodic motion Stability Underactuated systems Time scaling

abstract The paper develops a unified feedback control law for n degree-of-freedom biped robots with one degree of underactuation so as to generate periodic orbits on different slopes. The periodic orbits on different slopes are produced from an original periodic orbit, which is either a natural passive limit cycle on a specific slope or a stable periodic walking gait on level ground generated with active control. First, inspired by the controlled symmetries approach, a general result on gait generation on different slopes based on a periodic orbit on a specific slope is obtained. Second, the time-scaling control approach is integrated to reproduce geometrically same periodic orbits for biped robots with one degree of underactuation. The degree of underactuation is compensated by one degree-of-freedom in the temporal evolution that scales the original periodic orbit. Necessary and sufficient conditions are investigated for the existence and stability properties of periodic orbits on different slopes with the proposed control law. Finally, the proposed approach is illustrated by two kinds of underactuated biped robots: one has a passive gait on a specific ground slope and the other does not have a natural passive gait. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction The primary objective of the paper is to develop a feedback control strategy that translates periodic orbits existing on certain shallow slopes or level ground to other slopes and to find conditions for the existence and stability properties of such orbits. Our approach integrates the controlled symmetries method and the time-scaling control approach to extend walking on slopes for n degree-offreedom biped robots with one degree of underactuation and it addresses both passive gaits and active gaits in a unified framework. The related work of passive dynamics has added insights into the design of anthropomorphic robots since the pioneer work of McGeer (1990). Various analysis and prototypes that demonstrate efficient human-like walking have been developed based on passive dynamics (Garcia, Chatterjee, Ruina, & Coleman, 1998; Kajita, Yamaura, & Kobayashi, 1992; Spong, 1999). The techniques have also been extended to three-dimensional biped robots. For example, the first three-dimensional, kneed, two-legged, passivedynamic walking machine powered by gravity down a shallow slope is developed and described in Collins, Wisse, and Ruina (2001). Real three-dimensional robots based on passive dynamics,

✩ The work was supported by National Natural Science Foundation of China under Grant 60875074. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Warren E. Dixon under the direction of Editor Andrew R. Teel. ∗ Corresponding author. Tel.: +86 571 87951637; fax: +86 571 87952152. E-mail addresses: [email protected] (Y. Hu), [email protected] (G. Yan), [email protected] (Z. Lin).

0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.04.018

which can walk on level ground with small active power sources substituting for gravity, are presented in Collins, Ruina, Tedrake, and Wisse (2005). Passive walking appears very natural and is helpful to explain the efficiency of human bipedal locomotion. Nevertheless, stable passive gaits exist only for a limited class of biped robots with specific structures and parameters, and are extremely sensitive to slope angles. In order to achieve a wide range of stable walking and provide a large basin of attraction, the controlled symmetries method is proposed in Spong and Bullo (2005). It is shown that for n degreeof-freedom robots with full actuation, a passive gait that exists on a specific slope can be realized on other slopes via potential energy shaping. In addition, a method called hybrid Routhian reduction is developed to analyze and design a feedback control law for biped robots, which generates three-dimensional walking on the flat ground from two-dimensional planar gaits (Ames & Gregg, 2007; Ames, Gregg, Wendel, & Sastry, 2007). The idea is further improved in Gregg and Spong (2008), where sagittal plane dynamics of a full actuated three-dimensional biped robot is decoupled from its unstable yaw and lean dynamics in a rigorous manner. Recently, an anthropomorphic walking model is investigated in Sinnet and Ames (2009a,b). Although the gaits investigated in these works contain some underactuated phases, it still requires full actuation of the biped robots. Hence, these approaches are limited to a subclass of biped robots and cannot be extended directly to underactuated biped robots without further modifications. Moreover, for most biped robots, there does not exist a natural passive walking gait (Chevallereau et al., 2003; Geppert, 2004; Hirose & Takenaka, 2001; Ishida, 2004; Kaneko et al., 2004). Instead, periodic walking gaits are realized by active control.

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Y. Hu et al. / Automatica 47 (2011) 1605–1616

In order to achieve a more human-like active gait and not to consider too many constraints to ensure full actuation in designing simpler and cheaper robots, some recent works begin to focus on feedback control of underactuated biped robots. A relevant work that successfully addresses the challenge of underactuation is hybrid zero dynamics. Based on hybrid zero dynamics, planar three-link and five-link biped walking robots with a torso and one degree of underactuation are investigated in Grizzle, Abba, and Plestan (2001) and Westervelt, Grizzle, and Koditschek (2003), where the control design involves the judicious choice of a set of holonomic constraints that are asymptotically imposed on the robot via partial feedback linearization and optimal parameters to tune the hybrid zero dynamics for achieving stable walking with low energy consumption. The approach produces a closedloop system with provable stability properties by analyzing a scalar Poincaré map. Recently, some results on extending toward three-dimensional underactuated biped robots are presented in Chevallereau, Grizzle, and Shih (2009). The method is effective for underactuated biped robots, but it must explicitly take into account the complex natural dynamics, so the desired holonomic constraints cannot be obtained simply and do not have a unified formula for different slopes. Moreover, the desired holonomic constraints are chosen over a pre-assigned, finitely parameterized family of constraints. It is adequate when the control goal is to generate a gait with certain stability and energetic properties. However, when the goal is to exactly achieve a gait of interest such as a natural passive gait, it is unlikely to reproduce the specific gait by only choosing the holonomic constraints from the preassigned, finitely parameterized family of constraints. To overcome the limitation of the hybrid zero dynamics (HZD) control approach for this objective, sample-based HZD control is developed by Westervelt, Morris, and Farrell (2006, 2007b), which samples a given gait and then uses the sampled quantities to define the holonomic constraints of an HZD controller. Thus, a given gait is reproduced. In the paper, we also address the feedback control design for underactuated biped robots with underactuation degree one but focus on translating a passive gait or an active gait existing on a specific slope to other slopes. We expect to derive a simple unified feedback control law for different slopes and theoretically prove the existence and stability properties of the resulting periodic orbits. The idea is inspired by the controlled symmetries approach, but integrates the time-scaling control method to address the challenge of underactuation. The time-scaling control method is used for path following of robotic manipulators (Arai, Tanie, & Shiroma, 1998; Dahl & Nielsen, 1990; Szadeczky-Kardoss & Kiss, 2006) and is also employed to track a cyclic reference trajectory for an underactuated five-link planar biped robot (Chevallereau, 2003). In Chevallereau (2003), a new variable called virtual time is considered as a supplementary control input to compensate for the reduction of number of actuators, and temporal convergence of the resulting trajectory is then transformed to the convergence of the derivative of virtual time to the constant value 1. Instead of carefully designing a cyclic reference trajectory and making the derivative of virtual time converge to 1 like Chevallereau (2003), in the paper, we concentrate on a stable gait existing on a specific slope or level ground and design a feedback control law such that the resulting periodic orbits on other slopes are the same as those orbits generated by controlled symmetries for full actuated bipeds in geometry. Thus, the derivative of virtual time is expected to be periodic rather than converging to 1. Compared to the sample-based HZD control approach by Westervelt et al. (2006, 2007b), our approach does not need to explicitly know the periodic gait of interest for reproducing it on different slopes, while the sample-based HZD control approach does need it so that full-state information of the gait can be sampled to define the holonomic

constraints. Moreover, in practice, since it may not be possible to solve exactly the map from time t to the free variable θ of the holonomic constraints for a given gait, cubic spline interpolation is used and thus the reproduced gait is an approximation. However, our approach is able to exactly translate a natural passive gait or an actively controlled walking gait to other slopes. In the paper, we present necessary and sufficient conditions on whether there exists a solution of several steps or a global solution when the ground slope changes to another. Moreover, we show that the resultant solution is periodic if the initial condition of virtual time is properly selected. Also, stability properties of the reproduced walking gaits on different slopes are presented. The proposed control law is simple and suitable for different slopes without redesigning. The conditions for existence and stability are also very easy to check. Furthermore, for planar robots with underactuation degree one, some properties of several key parameters related to stability are analyzed. Finally, examples of underactuated biped robots with passive gaits and active gaits are given to illustrate the efficiency of our method with a series of numerical simulations. 2. Preliminaries and system models In this section, we consider a general model of biped robots with n degrees of freedom. Let q be a vector of generalized coordinates in the configuration space Q , which is of n-dimension, and let u be a vector of forces and torques, which is of m-dimension. The mathematical model for biped robots we consider is that of a controlled Lagrangian system subject to instantaneous impacts (Goswami, Espiau, & Keramane, 1997; Goswami, Thuilot, & Espiau, 1998; Westervelt, Grizzle, Chevallereau, Choi, & Morris, 2007a; Westervelt et al., 2007b), that is, d ∂L dt ∂ q˙



−

∂L = Bu for q ̸∈ S ∂q

q+ = q− q˙ + = ∆q q˙ −

−

whenever q

(1)

∈S

where L is the Lagrangian of the system, B ∈ Rn×m describes the effects of actuators on the generalized coordinates, the superscripts − and + denote the state value before and after instantaneous impacts, respectively, ∆q is the n-by-n impact matrix depending on q, and the set S represents the switching surface, e.g., for a walking robot, states at which the foot of the swing leg hits the ground and a new step begins. The Lagrangian L is given by

L(q, q˙ ) = K (q, q˙ ) − V (q) =

1 2

q˙ T D(q)˙q − V (q)

where K (q, q˙ ) is the total kinetic energy, V (q) is the total potential energy, and D(q) is the inertia matrix. The controlled Lagrangian equation yields the equation of motion D(q)¨q + C (q, q˙ )˙q + G(q) = Bu, where C (q, q˙ ) is the matrix of Coriolis and centrifugal terms and G(q) = ∂∂Vq is the gradient of the potential energy field. In the paper, we assume the following. Assumption A.1. The system has underactuation degree one, that is, Rank(B) = m = n − 1. Assumption A.2. The kinetic energy K and the impact equations of the system are invariant under a group action Φ : A × Q → Q , i.e., for all a ∈ A,

Y. Hu et al. / Automatica 47 (2011) 1605–1616

d ∂ La

K (Φa (q), Tq Φa (˙q)) = K (q, q˙ ) Tq Φa (∆q q˙ ) = ∆Φa (q) Tq Φa (˙q)

dt ∂ q˙

where Tq Φa denotes the tangent function to Φa mapping Tq Q (linear space of tangent vectors at q ∈ Q ) to TΦa (q) Q (linear space of tangent vectors at Φa (q) ∈ Q ).



Specifically, we are interested in the symmetry properties in the Lagrangian dynamics of biped robots with respect to the action of changing the ground slope. The action of changing the ground slope defines a group action of SO(2) in the planar case and SO(3) in the general case. It is shown in Spong and Bullo (2005) that Assumption A.2 is satisfied for biped robots under the action of changing the ground slope. Let Sa denote the switching surface after applying the action of changing the ground slope. Under Assumption A.2, if the system is fully actuated (i.e., Rank(B) = n), the following result is presented in Spong and Bullo (2005). Theorem 2.1 (Spong & Bullo, 2005). If q(t ) is a solution of (1) with u = 0, then Φa−1 (q(t )) is a solution of D(q)¨q + C (q, q˙ )˙q + G(q) = Bu



+

for q ̸∈ Sa

−

q =q q˙ + = ∆Φa (q) q˙ −

whenever q− ∈ Sa

(2)

(3)

By this result, a control law that compensates the gravitational torque acting on the biped is obtained to make passive gait slope invariant. In other words, the approach allows for the construction of control laws that translate stable periodic orbits for passive bipeds to stable periodic orbits for fully actuated bipeds walking on arbitrary slopes. However, it cannot directly address underactuated systems without further modifications. It should be emphasized that the ground contact force is not invariant under rotations of the world frame though the impact map and the kinetic energy are invariant. Therefore, in order that the translated gait is valid on a different slope, the ground contact force constraint has to be satisfied. This constraint should be taken into account in all the results throughout the paper. 3. Generalization of controlled symmetries Theorem 2.1 provides a scheme to translate limit cycles existing for u = 0 to any other ground slopes. Nevertheless, stable passive walking gaits exist only for a limited class of biped robots. For most biped robots, stable walking gaits are achieved by applying certain control efforts. i.e., u ̸= 0. In this section, we will show how a walking gait existing on a specific ground slope under certain control effort is translated to other ground slopes. The result is a direct generalization of controlled symmetries. Theorem 3.1. Suppose Assumption A.2 holds. If q(t ) is a solution of (1) with u = u0 (q, q˙ ), then Φa−1 (q(t )) is a solution of the system D(q)¨q + C (q, q˙ )˙q + G(Φa (q)) = Bu



+

for q ̸∈ Sa

−

q =q q˙ + = ∆Φa (q) q˙ −

whenever q− ∈ Sa

(4)

q+ = q− q˙ + = ∆Φa (q) q˙ −

−

whenever q

(6)

∈ Sa

where La (q, q˙ ) = L(Φa (q), Tq Φa (˙q)). We can also write the left hand side of the Lagrangian equation (6) as d ∂ La dt ∂ q˙

−

∂ La d ∂ Ka ∂ Ka ∂ Va = − + ∂q dt ∂ q˙ ∂q ∂q

where Ka = K (Φa (q), Tq Φa (˙q)) and Va = V (Φa (q)). Considering the invariance property of the kinetic energy under the action Φa in Assumption A.2, Eq. (6) is equivalent to D(q)¨q + C (q, q˙ )˙q + G(Φa (q)) = Bu0 (Φa (q), Tq Φa (˙q)). Again by Assumption A.2, the impact equation is also invariant under the action Φa . Thus, the conclusion follows.  Remark 3.1. Theorem 3.1 tells us that any limit cycle of the system (1) with u = u0 (q, q˙ ) can be reproduced by the system (4) with the control law (5). In a particular case, when the matrix B is of full rank, the following control law (7)

can be directly applied to the original system, which is a generalization of the potential energy shaping control (3) in Spong and Bullo (2005). In other words, the potential shaping control law (3) is simply obtained by letting u0 (Φa (q), Tq Φa (˙q)) = 0 in (7). From Theorem 3.1, the following corollary about the basin of attraction is presented without a proof. It can also be seen in Spong and Bullo (2005). Corollary 3.1. If (q0 , q˙ 0 ) lies in the basin of attraction of a limit cycle for the system (1) with u = u0 (q, q˙ ), then (Φa−1 (q0 ), Tq Φa−1 (˙q0 )) lies in the basin of attraction of the limit cycle reproduced by the system (4) with (5). 4. Time-scaling control In the paper, the biped we consider has one degree of underactuation. In order to reproduce stable periodic orbits that exist on a specific slope or level ground on any other ground slope for underactuated bipeds with one degree of underactuation, we invoke the time-scaling control approach and then apply the results obtained in the previous section to solve the problem. The time-scaling control approach is first presented in Chevallereau (2003) for the tracking of a cyclic reference trajectory in geometry for which the idea is the same as path following in mobile robot control. 4.1. Control law We introduce a new variable τ = g (t ) as in Chevallereau (2003) where g : R → R is a smooth function and satisfies g (0) = 0. Next, we define s(τ ) = q(t ).

(8)

Then it can be obtained that q˙ (t ) = s˙(τ )τ˙

with the following control law u = u0 (Φa (q), Tq Φa (˙q)).

∂ La = Bu0 (Φa (q), Tq Φa (˙q)) for q ̸∈ Sa ∂q

u = B−1 (G(q) − G(Φa (q))) + u0 (Φa (q), Tq Φa (˙q))

with the following control law u = B−1 (G(q) − G(Φa (q))).

−

1607

(5)

Proof. If q(t ) is a solution trajectory to (1) with u = u0 (q, q˙ ), then Φa−1 (q(t )) is a solution of the system

q¨ (t ) = s˙(τ )τ¨ + s¨(τ )τ˙ 2

(9)

where s˙ and s¨ represent the first and second derivative of s with respect to τ , and τ˙ and τ¨ are the first and second derivative of τ with respect to t.

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Y. Hu et al. / Automatica 47 (2011) 1605–1616

q˙ + = ∆Φa (q) q˙ − .

Substituting (8) and (9) into (1) leads to



τ˙ 2 D(s)¨s + C (s, s˙)˙s + τ¨ D(s)˙s + G(s) = Bu.

(10)

Under Assumption A.1, we can find B⊥ ∈ R1×n such that B⊥ B = 0, which is called the left-annihilator of B. Also, we can find a pseudoinverse B+ of B satisfying B+ B = In−1 . Left-multiplying B⊥ and B+ to (10), respectively, one obtains

 

 + τ¨ D(s)˙s + G(s) = 0,     B+ τ˙ 2 D(s)¨s + C (s, s˙)˙s + τ¨ D(s)˙s + G(s) = u. B⊥ τ˙ 2 D(s)¨s + C (s, s˙)˙s

(20)

According to (8) and (9), Eq. (20) can be written as



(21) s˙+ τ˙ + = ∆Φa (s) s˙− τ˙ − . + − Comparing (19) and (21), it follows that τ˙ = τ˙ , which means τ˙ is continuous at the moment of impact and so is ζ . Combining this continuity observation together with (13), (14) and (18), it is obtained that under the hypothesis τ˙ ̸= 0, the overall closed-loop system is as follows: when s ̸∈ Sa ,



α(s)

dζ dτ

+ 2β(s)ζ + ξ (s) = 0

(22a)

D(s)¨s + C (s, s˙)˙s + G(Φa (s)) = Bu0 (Φa (s), Tg Φa (˙s))

(22b)

and when s ∈ Sa , Now we consider the following time-scaling control law:

  B τ˙ 2 G(Φa (s)) − G(s) τ¨ = B⊥ D(s)˙s  u = B+ τ¨ D(s)˙s − τ˙ 2 G(Φa (s)) + G(s)  + τ˙ 2 Bu0 (Φa (s), Tg Φa (˙s))

ζ + = ζ −,

s+ = s− ,

s˙+ = ∆Φa (s) s˙− .

(22c)

⊥

(11a)

(11b)

where Φa (·) is an action of changing the ground slope, Tq Φa (·) is the tangent function to Φa , and u0 (·, ·) is an active control such that the system (1) with u = u0 (q, q˙ ) has a limit cycle. If a passive limit cycle is considered, then u0 (·, ·) = 0 in that case. Considering the control law (11) and taking into account the fact   B⊥ B+

that

τ˙

2

has full rank, it is obtained that



D(s)¨s + C (s, s˙)˙s + G(Φa (s)) − Bu0 (Φa (s), Tg Φa (˙s))



= 0. (12)

(13)

κ(τ , τ0 ) =

for s ̸∈ Sa . And for s ∈ Sa , s =s , s˙+ = ∆Φa (s) s˙− .

(14)

α(s) := B D(s)˙s ̸= 0.

(15)

This expression is indeed the angular momentum (Chevallereau, 2003). For a desired limit cycle, we can check whether it crosses zero or not. The control law (11) actually introduces an auxiliary dynamic system of τ . Define β(s) := −B⊥ G(Φa (s)) and ξ (s) := B⊥ G(s). Then (11a) can be written as

α(s)τ¨ + β(s)τ˙ 2 + ξ (s) = 0. Applying the change of variables, ζ =

dτ

=

d

(16) 1 2

τ˙ 2 , we have

 τ˙ 2 dt τ˙ τ¨ = = τ¨ . dt dτ τ˙

τ0

dζ dτ

2

+ 2β(s)ζ + ξ (s) = 0.

τ0

α(Φa−1 (q(σ )))

 dσ ,

ξ (Φa−1 (q(σ2 ))) α( Φa−1 (q(σ2 ))) ∫ σ2 2β(Φa−1 (q(σ1 )))

× exp

α(Φa−1 (q(σ1 )))

 dσ1 dσ2 .

For the problem of translating a periodic orbit for bipeds, the periodic solution is known a priori. So for a given action of changing the ground slope Φa , the functions α(·), β(·), ξ (·), δ(·, ·), and κ(·, ·) can also be known. We define Ck :=

max

kT ≤τ ≤(k+1)T

κ(τ , kT )

for each step k. When τ = kT , it is clear that κ(τ , kT ) = 0. Hence, the following must hold: Ck ≥ 0 for any k. Note that α(Φa−1 (q(τ ))), β(Φa−1 (q(τ ))), and ξ (Φa−1 (q(τ ))) are periodic with period T (here T is the period of the solution for bipeds). Thus, it follows that 0≤τ ≤T

(17)

(18)

Moreover, at the moment of impact (s ∈ Sa ), s˙+ = ∆Φa (s) s˙− .

β(Φa−1 (q(σ )))

Ck = C0 = max κ(τ , 0).

1

Then, Eq. (16) becomes

α(s)

τ

τ

τ0

⊥

dζ

∫

−

On the other hand, notice that (11a) is not singular as long as

(23)

where

 ∫ δ(τ , τ0 ) = exp −

D(s)¨s + C (s, s˙)˙s + G(Φa (s)) = Bu0 (Φa (s), Tg Φa (˙s))

+

In this subsection, we show how the solution of (22) is related to a walking gait on certain slope. Let q(t ) be a periodic solution (with period T ) of (1) with u = u0 (q, q˙ ), that is, it is a limit cycle of the system on a specific slope. Then by Theorem 3.1, it follows that Φa−1 (q(τ )) is a solution of (22b)–(22c). However, in order to make Φa−1 (q(τ )) be a solution of the system after applying the proposed control law (11), the hypothesis τ˙ ̸= 0 has to be satisfied. Next, we are going to find out the conditions such that a solution exists or at least a local solution exists. Note that when α(Φa−1 (q(τ ))) ̸= 0, we can derive the explicit solution for (22a), which is given as

ζ (τ ) = δ 2 (τ , τ0 )ζ (τ0 ) − δ 2 (τ , τ0 )κ(τ , τ0 )

When τ˙ ̸= 0, Eq. (12) is equivalent to



4.2. Walking gaits

(19)

Similarly, we can get the following impact equation in terms of q:

The periodic solution q(t ) of the biped robot with control u = u0 (q, q˙ ) may exist for all t ∈ [0, ∞) on a specific slope or a level ground, but the solution of the biped robot after applying the proposed time-scaling control law (11) may or may not exist for all time. We say that a solution of the biped robot after applying the time-scaling control law (11) defined in the time interval [0, t ∗ ) is a forward-motion solution if τ˙ (t ) > 0 for all t ∈ [0, t ∗ ). Next, we present necessary and sufficient conditions for the existence of a forward-motion solution and provide the explicit solution formula in terms of the periodic solution for the biped with control u = u0 (q, q˙ ).

Y. Hu et al. / Automatica 47 (2011) 1605–1616

Theorem 4.1. Let K > 0 be an integer. Suppose that q(t ) is a periodic solution with period T of (1) with u = u0 (q, q˙ ) and satisfies α(Φa−1 (q(t ))) ̸= 0. There exists a forward-motion solution defined in [0, g −1 ((K + 1)T )), namely, Φa−1 (q(g (t ))), for the system (1) with the control law (11) if and only if

τ˙ (g −1 (kT )) >



2C0 ,

(24)

for k = 0, 1, . . . , K . Proof. (⇐H) Note that ζ (τ ) = 21 τ˙ 2 . So from the condition (24), we get that ζ (kT ) > C0 for k = 0, 1, . . . , K . Moreover, according to (23) and the fact C0 = Ck , we obtain that for τ ∈ [kT , (k + 1)T )



=

u

ξ (Φa

−1

1609

u0 (q, q˙ ). Note that α(Φa−1 (q(τ ))), β(Φa−1 (q(τ ))), and (q(τ ))) are periodic with period T , so we know that

δ((k + 1)T , kT ) = δ(T , 0), κ((k + 1)T , kT ) = κ(T , 0). For simplicity, we denote δ¯ = δ(T , 0) and κ¯ = κ(T , 0). Also, we use ζ (k) to represent the value ζ (kT ) in the following. Then from (23), we can have a discrete time model for the evolution of ζ , that is,

  ζ (k + 1) = δ¯ 2 ζ (k) − κ¯ .

(26)

 2 ζ (τ ) > δ (τ , kT ) Ck − κ(τ , kT ) .

If δ¯ ̸= 1, the unique equilibrium point of (26) can be derived as

By the definition of Ck , the right hand side of the above inequality is positive and so is ζ (τ ) for all τ ∈ [kT , (k + 1)T ). This is true for k = 0, 1, . . . , K . Therefore, ζ (τ ) > 0 for τ ∈ [0, (K + 1)T ). This condition ensures that τ˙ does not cross zero for t ∈ [0, g −1 ((K + 1)T )). Furthermore, since τ˙ (0) > 0, it follows that τ˙ = g˙ (t ) is always positive for t ∈ [0, g −1 ((K + 1)T )). Thus, it is inferred that Φa−1 (q(τ )) is a forward-motion solution of (12) for t ∈ [0, g −1 ((K +1)T )). Substituting τ = g (t ) results in the solution formula Φa−1 (q(g (t ))). (H⇒) For any k = 0, 1, . . . , K and τ ∈ [kT , (k + 1)T ), we have

ζ∗ =

  ζ (τ ) = δ 2 (τ , kT ) ζ (kT ) − κ(τ , kT ) .

(25)

Since δ 2 (τ , kT ) > 0 and moreover since there exists a forwardmotion solution defined in [0, g −1 ((K + 1)T )), which implies τ˙ > 0 and ζ (τ ) = 12 τ˙ 2 > 0 during this time interval, it then follows from (25) that√ζ (kT ) − √κ(τ , kT ) > 0. It in turn implies that τ˙ (g −1 (kT )) > 2Ck = 2C0 for k = 0, 1, . . . , K .  The theorem states that we can translate a periodic walking orbit existing on a specific slope or level ground to other slopes by our proposed control law. However, the ground contact force constraint has to be checked for the walking gait in the new slope. Certain number of steps can be achieved on other slopes by selecting appropriate initial τ˙ (0) to satisfy (24). The resulting gait is exactly the same as the desired gait in geometry though the configuration state q of each step does not precisely track the desired gait in time and moreover, the velocity is different from the desired walking unless τ˙ = 1. If condition (24) holds for all k, then the forward-motion solution exists globally, which is stated in the following corollary. Corollary 4.1. Suppose that q(t ) is a periodic solution with period T of (1) with u = u0 (q, q˙ ) and satisfies α(Φa−1 (q(t ))) ̸= 0. There exists a global forward-motion solution, namely, Φa−1 (q(g (t ))) with t ∈ [0, ∞), for √ the system (1) with the control law (11) if and only if τ˙ (g −1 (kT )) > 2C0 for any integer k. The result can be deduced from Theorem 4.1 directly. In Theorem 4.1 and Corollary 4.1, for the existence of a solution, it requires to check τ˙ at the beginning of every step, namely, the moment when the foot of the swing leg hits the ground and a new step begins. However, if we can make τ˙ (t ) periodic with τ˙ (0) = τ˙ (T ′ ) = · · · = τ˙ (kT ′ ) where 0, T ′ , 2T ′ , . . . are the time instants a new step begins, then we only need to find an appropriate initial √ condition τ˙ (0) and check whether it satisfies τ˙ (0) > 2C0 , which would lead to the existence of a global forward-motion solution. In next subsection, we are going to address this issue. 4.3. Periodicity Consider the discrete time sequence, 0, T , 2T , . . . at which the impact occurs for the biped robot with the control effort

δ¯ 2 κ¯ . δ¯ 2 − 1

(27)

Next, we present our main result showing that the walking gait generated by our proposed control is periodic, which tracks the desired gait in geometry. Theorem 4.2. Suppose q(t ) is a periodic solution with period T of (1) with u = u0 (q, q˙ ) such that δ¯ ̸= 1 and α(Φa−1 (q(t ))) ̸= 0. If

ζ ∗ > C0 ,

(28)

then a global forward-motion solution exists for the closed-loop system with the control law (11), namely, Φa−1 (q(g (t ))) with g (0) = √ 0 and g˙ (0) = 2ζ ∗ . Moreover, the solution is periodic with period T′ =

T

∫ 0

1 g˙ (g −1 (σ ))

dσ .

Proof. Let ζ (τ ) be a solution of (23) with initial condition ζ (0) = ζ ∗ . Then it can be easily seen that ζ (τ ) is periodic with period T , that is, ζ (τ ) = ζ (τ + T ). Thus,

τ˙ (g −1 (kT )) = τ˙ (0) =



2ζ ∗ >



2C0 .

Then, by Corollary 4.1, it follows √ that with the initial conditions g (0) = 0 and τ˙ (0) = g˙ (0) = 2ζ ∗ , Φa−1 (q(g (t ))) is a global forward-motion solution. Next, we will show the solution is periodic. Recall that τ = g (t ), so we can write dτ dt

= g˙ (t ) = g˙ (g −1 (τ )).

Since τ˙ = g˙ (t ) > 0 for t ∈ [0, +∞), we can also write dt dτ

=

1 g˙ (g −1 (τ ))

.

(29)

Since ζ = 12 τ˙ 2 , then from the periodicity of ζ with respect to τ , it also holds that g˙ (g −1 (τ )) is periodic with respect to τ and its period is T . That is, τ +T

∫ τ

1 g˙ (g −1 (σ ))

dσ =

T

∫ 0

for any τ . We denote T ′ =

1 g˙ (g −1 (σ ))

T

dσ

1 d 0 g˙ (g −1 (σ ))

σ . Integrating both sides

of (29) from τ to τ + T results in t + T ′ = g −1 (τ + T ). Thus,

Φa−1 (q(g (t + T ′ ))) = Φa−1 (q(τ + T ))

= Φa−1 (q(τ )) = Φa−1 (q(g (t )))

(30)

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Y. Hu et al. / Automatica 47 (2011) 1605–1616

 ∂ Q  = δ¯ 2 ∂ζ x=x∗ ,ζ =ζ ∗

and g˙ (t + T ′ )˙q(g (t + T ′ )) = g˙ (t )˙q(g (t )). This proves the periodicity of the solution.

and the linearized model of (32) is



 ∂ P   ∂x  ∗ ∗ x(k + 1)  x=x ,ζ =ζ = ζ (k + 1)  ∂ Q  ∂x  ∗ ∗ 

Remark 4.1. In Theorem 4.2, the condition ζ > C0 is also a necessary condition to ensure the existence of the global forward√ motion solution Φa−1 (q(g (t ))) with g (0) = 0 and g˙ (0) = 2ζ ∗ . In other words, if ζ ∗ ≤ C0 , g˙ (t ) may not be strictly positive, which means the biped robot may fall backward. ∗

that though we can select the initial condition√τ˙ (0) = √ Notice 2ζ ∗ such that τ˙ (kT ′ ) periodically revisits the value 2ζ ∗ , if the discrete time system (26) is unstable, then with a small perturbation, τ˙ (t ) may diverge and thus the closed-loop system is unstable.

In this subsection, we derive a necessary and sufficient condition for the stability properties of the biped robots when applying our proposed control law. We show that the asymptotical stability of the reproduced gait is equivalent to the asymptotical stability of the original gait and δ¯ < 1. Theorem 4.3. Suppose the conditions in Theorem 4.2 hold. If the periodic solution q(t ) of system (1) with u = u0 (q, q˙ ) is asymptotically stable and δ¯ < 1, then the periodic solution Φa−1 (q(g (t ))) with √ g (0) = 0 and g˙ (0) = 2ζ ∗ of the system with the control law (11) is asymptotically stable. Proof. Suppose that the periodic solution q(t ) of system (1) with u = u0 (q, q˙ ) is asymptotically stable. Then from Theorem 3.1 and Corollary 3.1, we know that Φa−1 (q(t )) is an asymptotically stable periodic solution of system (4) under the control law (5). Let S be the (2n − 1)-dimensional surface of section at which the instantaneous impact occurs. Denote x∗ the intersection point of the periodic solution and S . Then we can find a Poincaré map in the neighborhood U of x∗ , i.e., P : U → S . So it follows that the discrete time system x(k + 1) = P (x(k)),

x∈S

(31)

is asymptotically stable with respect to the fixed point x∗ . Although the vector x is 2n-dimensional, the solution generated by the above discrete time system with an initial state in S is restricted in the (2n − 1)-dimensional surface S . Now consider the closed-loop system (22). Let

S = {(x, ζ ) : x ∈ S } ′

be the 2n-dimensional surface. It is known that (x∗ , ζ ∗ ) with ζ ∗ defined in (27) is the intersection point of S ′ and the periodic solution of the closed-loop system (22). Then we can find a Poincaré map in the neighborhood U′ of (x∗ , ζ ∗ ), i.e., P ′ : U′ → S ′ , and the evolution in discrete time is x(k + 1) = P′ ζ (k + 1)

]

x(k) ζ (k)

[

]

,

[ ] x

ζ

∈ S′.

(32)

Note that the dynamics (22b) for x does not rely on ζ and is exactly the same as the system (4) under the control law (5). Hence, the Poincaré map P ′ : U′ → S ′ must be of the form

[

 0[

]  x(k)  ζ (k)  δ¯ 2

]

x=x ,ζ =ζ

which is of lower triangular form. Thus, by the condition δ¯ < 1 and by the conclusion that (31) is asymptotically stable, it follows that (32) is asymptotically stable with respect to the fixed point (x∗ , ζ ∗ ). Consequently, the periodic solution of the closed-loop system (22) is asymptotically stable.  4.5. Some properties of δ¯ and κ¯ for planar robots

4.4. Stability

[

[

P ( x) . Q (x, ζ )

]

Moreover, notice that (26) actually is the map Q (x, ζ ) of the component ζ for x = x∗ since it is obtained on the periodic solution. Therefore,

Consider the action of changing the ground slope and let γ0 denote the variation of the slope angle. For planar biped robots, we have the following special result. Theorem 4.4. For planar bipeds, the parameter δ¯ is a constant invariant to the action of changing the ground slope and κ¯ is a sine function of γ0 , i.e.,

κ¯ = c1 sin(γ0 + c2 ) where c1 and c2 are constants. Proof. Consider a general n-link planar biped robot. Let q = (q1 , q2 , . . . , qn ) be the generalized coordinates such that only q1 represents an absolute angle relative to the ground and the others qr = (q2 , . . . , qn ) are all relative angles between links. Recall that

 ∫ δ¯ = exp −

T

β(Φa−1 (q(σ )))

 dσ

α(Φa−1 (q(σ ))) ∫ ξ (Φa−1 (q(σ2 ))) 0

κ¯ =

T

∫ 0

α(Φa−1 (q(σ2 )))

σ2

exp

0

2β(Φa−1 (q(σ1 )))

α(Φa−1 (q(σ1 )))

 dσ1 dσ2

where

α(Φa−1 (q(τ ))) = B⊥ D(Φa−1 (q(τ )))Tq Φa (˙q(τ )), β(Φa−1 (q(τ ))) = −B⊥ G(q(τ )), ξ (Φa−1 (q(τ ))) = B⊥ G(Φa−1 (q(τ ))). For the action of changing the ground slope, we can get Φa (q) = (q1 + γ0 , qr ) and Tq Φa (˙q) = q˙ . It is noticed that the kinetic energy K is independent of the choice of world coordinate frame (Spong, Hutchinson, & Vidyasagar, 2004) and thus D(q) is independent of the action of changing the ground slope, namely, D(Φa−1 (q)) = D(q). As a result, from the formula of α(·), it is known that its value is not affected by the action of changing the ground slope. Also, notice that β(·) is not affected by the action of changing the ground slope from its formula as well. So the value of δ¯ remains constant, which can be seen from its formula above. For a planar biped robot, the potential energy V can be written as V (q) = g0 (qr ) cos(q1 ) where g0 (qr ) is a function of qr . Thus, we can express

∂V B⊥ G(q) = B⊥ = g1 (qr ) sin(q1 ) + g2 (qr ) cos(q1 ) ∂q where g1 (qr ) and g1 (qr ) are functions of qr . Consider a periodic solution q(t ). Then we can re-write ξ as a function of τ and γ0 , that is,

Y. Hu et al. / Automatica 47 (2011) 1605–1616

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Fig. 1. Mathematical model of a compass biped robot.

ξ (Φa−1 (q(τ ))) = C1 (τ ) sin(γ0 ) + C2 (τ ) cos(γ0 ) where C1 (τ ) and C2 (τ ) are functions of τ . Then by checking the formula of κ¯ above, it is obtained that κ¯ is a sine function of γ0 . If another set of generalized coordinates qa = (qa1 , qa2 , . . . , qan ) is used, the same conclusion can still be obtained by applying the following invertible coordinate transformation qa = Tr

Fig. 2. When γ0 varies from −20° to 20°, the values of δ¯ and κ¯ are shown in the top, and the values of ζ ∗ and C0 are shown in the bottom.

[ ] q1 qr

where Tr is a n-by-n full-rank matrix.



5. Examples and simulations In this section, we apply our results to two examples of planar underactuated bipeds: one is a compass biped robot, which has a passive gait on the ground slope γ ∗ = 3°, and the other is a fivelink biped, which does not own any passive gait on any slope. We are going to demonstrate how these two types of bipeds produce periodic walking gaits on different slopes in the same framework using our proposed approach. Simulations are also provided. In what follows, we let FN denote the normal component of the ground reaction force which is along the direction perpendicular to the walking surface and let FT denote the tangential component of the ground reaction force which is along the direction parallel to the walking surface. Also, we denote the absolute value of the ratio FT /FN as µ.

Fig. 3. Periodic solutions over real time t for γ0 = −0.9°, −0.5°, 0°, 3°, and 5°.

zero dynamics control approach. For this robot, the control matrix B is

[

]

1 B= . −1

First, we consider a compass biped. The structure and physical parameters of the robot are given in Fig. 1. The robot has two degrees of freedom. We denote q = (q1 , q2 ) the configuration state of the robot, where q1 is the angle of the stance leg about the vertical and q2 is the angle of the swing leg about the vertical. The parameter γ in Fig. 1 is the slope angle of the ground. The value of γ is positive or negative for downslope and upslope, respectively. We consider two cases for the compass biped robot: (i) It has only hip actuation; (ii) It has only ankle actuation. In both cases, the robot has one degree of underactuation. In the following, we show that how our proposed control law translates an asymptotic stable passive gait existing on the slope γ ∗ = 3° to other slopes γ = γ ∗ + γ0 where γ0 represents the variation of the slope angle.

¯ κ, For the parameters given in Fig. 1, the values of δ, ¯ ζ ∗ are calculated when γ0 varies in the range [−20°, 20°] and are shown in Fig. 2. From the figure, it can be seen that when γ0 varies, the value of δ¯ remains constant as Theorem 4.4 stated. Also, from the numerical calculation drawn in Fig. 2, we can observe that δ¯ is strictly less than 1 for this case and ζ ∗ > C0 when γ0 > −0.97°. It means, the limit of stability is a slope of γ = 2.03°. When γ > 2.03°, asymptotic stable walking gaits exist with the control law (11) by Theorem 4.3. For γ0 = −0.9°, −0.5°, 0°, 3°, and 5°, the periodic solutions of the closed-loop system over real time t are shown in Fig. 3; the phase portraits of τ˙ and τ¨ are depicted in Fig. 4; and the periodic solutions of the closed-loop system over virtual time τ are depicted in Fig. 5 where the geometry shapes are exactly the same as the periodic solutions generated by the method of controlled symmetries. Table 1 gives the minimum values of vertical component FN and the maximum values of the ratio µ over the periodic orbits on these slopes.

5.1.1. Hip actuation The compass biped robot with only one actuated joint in the hip is studied in Westervelt et al. (2007b) using sample-based hybrid

5.1.2. Ankle actuation The compass biped robot with ankle actuation is studied in Holm and Spong (2008). We take this example as another

5.1. Compass bipeds

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Y. Hu et al. / Automatica 47 (2011) 1605–1616

Fig. 4. Phase diagrams of τ¨ and τ˙ for γ0 = −0.9°, −0.5°, 0°, 3°, and 5°.

Fig. 6. When γ0 varies from −20° to 20°, the values of δ¯ and κ¯ are shown in the top, and the values of ζ ∗ and C0 are shown in the bottom.

Fig. 5. Periodic solutions over virtual time τ for γ0 = −0.9°, −0.5°, 0°, 3°, and 5°. Fig. 7. Periodic solutions over real time t for γ0 = −0.5°, −0.3°, 0°, 1°, and 2°. Table 1 The minimum values of FN and the maximum values of µ over the periodic orbit on different slopes for hip actuation.

γ0 (°)

Min(FN ) (N)

Max(µ)

−0.9 −0.5

139.1 133.5 126.4 83.8 55.3

0.2941 0.2856 0.2744 0.1894 0.4749

0 3 5

illustration for our proposed control law. For this robot, the control matrix is

[ ] B=

1 , 0

which also satisfies our Assumption A.1. Through numerical calculations, the values of δ¯ , κ, ¯ ζ ∗ , and C0 with respect to γ0 ∈ [−20°, 20°] are drawn in Fig. 6. From the figure, it follows from Theorem 4.2 that a global forward-motion solution exists when γ0 < 2.4°. It means, for any slope with its angle γ < 5.4°, a periodic gait can be achieved with the control law (11). But valid ground slopes ensuring the existence of valid periodic gaits involve considerations of the ground reaction force constraint and actuator limits. As an illustration, for γ0 = −0.5°, −0.3°, 0°, 1°, and 2°, the periodic solutions of the closedloop system over real time t are shown in Fig. 7; the phase diagrams of τ˙ and τ¨ are depicted in Fig. 8; and the periodic solutions of the closed-loop system over virtual time τ are plotted in Fig. 9,

Fig. 8. Phase diagrams of τ¨ and τ˙ for γ0 = −0.5°, −0.3°, 0°, 1°, and 2°.

respectively. Table 2 provides the minimum values of vertical component FN and the maximum values of the ratio µ over the periodic orbits on these slopes for ankle actuation. However, these periodic solutions are not stable due to the fact δ¯ > 1 observed in the figure, according to Theorem 4.3. It might be possible to add additional control efforts to stabilize the periodic solutions, but it is beyond the scope of the paper.

Y. Hu et al. / Automatica 47 (2011) 1605–1616

Fig. 9. Periodic solutions over virtual time τ for γ0 = −0.5°, −0.3°, 0°, 1°, and 2°. Table 2 The minimum values of FN and the maximum values of µ over the periodic orbits on different slopes for ankle actuation.

γ0 (°)

Min(FN ) (N)

Max(µ)

−0.5 −0.3

101.4 111.4 126.4 159.2 168.9

0.7048 0.5082 0.2744 0.2268 0.5053

0 1 2

1613

Fig. 11. Two steps of walking gaits. (The stance leg is red, the swing leg is blue, and the torso is black. The black dots represent the center of mass of the robot.) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

and Westervelt et al. (2003), which involves partial feedback linearization and holonomic constraints for actuated joints. The mathematical model of the five-link biped has the same form as (1) with the generalized coordinates q = (q1 , q2 , . . . , q5 ) (see Fig. 10). Denote qa = (q2 , q3 , q4 , q5 ), the vector of actuated coordinates, and denote θ = q1 + 0.5q2 as shown in Fig. 10. Then we introduce the following output function y = h(q) = qa − hd (θ )

(33)

where hd (·) is a function constructed based on Bézier polynomials of degree 6, for which 20 parameters, denoted by α , are yet to be determined. The 20 free parameters α are selected to satisfy a set of physical constraints and to minimize the cost function J (α) =

1 Ls

Ts

∫

(u∗ (t ))T u∗ (t )dt

0

where Ls is the step width, Ts is the duration of each step, and

 ∗

u =

 −1   ∂ 2 hd (θ ) 2 ∂ h(q) −1 ∂ h(q) −1 D (q)B θ˙ + D (q)(C q˙ + G) ∂q ∂θ 2 ∂q

is the torque required to ensure the virtual constraint y = qa − hd (θ ) = 0 all the time provided that initially it is satisfied. The virtual constraint surface y = 0 corresponds to a periodic walking gait of the biped. Moreover, in order to make the resultant walking gait asymptotically stable, the following control law is proposed, Fig. 10. Mathematical model of a five-link biped robot.

 u = u0 (q, q˙ ) = u − ∗

5.2. Five-link bipeds Second, we consider the example of five-link underactuated biped robot shown in Fig. 10, which has one degree of underactuation. The parameter values of the robot used in this paper are the same as the ones of RABBIT in Westervelt et al. (2007a). For this robot, there does not exist a stable passive gait on any slope. Hence, we first use the hybrid zero dynamics method to design a control law for the purpose of producing a stable periodic walking orbit on level ground. Then we apply our control strategy so that it can reproduce a stable periodic walking orbit on other slopes. 5.2.1. Stable walking orbits on level ground To produce a stable walking orbit on level ground, we adopt the hybrid zero dynamics method appeared in Chevallereau and Aoustin (2001), Morris and Grizzle (2005), Westervelt et al. (2007a)

 −1 ∂ h(q) −1 D (q)B (Kd y˙ + Kp y) ∂q

(34)

with properly chosen parameters Kd and Kp such that y¨ + Kd y˙ + Kp y = 0 is asymptotically stable. Let the average walking speed be 0.7 m/s and the absolute value of the ratio of the horizontal and vertical components of ground reaction force µ be less than 0.2. By numerical calculations, two steps of walking gaits on level ground corresponding to the virtual constraint surface y = 0 are demonstrated in Fig. 11, while the periodic torso orbit is presented in Fig. 12. 5.2.2. Stable walking gaits on sloped environments The five-link biped robot has one degree of underactuation and the control matrix is

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Y. Hu et al. / Automatica 47 (2011) 1605–1616

Fig. 12. The periodic torso orbit in the phase plane.

Fig. 13. When γ0 varies from −20° to 20°, the values of δ¯ and κ¯ are shown in the top, and the values of ζ ∗ and C0 are shown in the bottom.



0 1  B = 0 0 0

0 0 1 0 0

0 0 0 1 0

Fig. 14. Periodic gaits on a 1° upslope and a 5° downslope.

Fig. 15. Top: the simulated response on 1° upslope with respect to real time t. Bottom: the simulated response on 1° upslope with respect to virtual time τ .



0 0  0 . 0 1

We apply our approach to extend the periodic walking gait obtained on level ground to a slope environment by the action of changing the ground slope. Let γ0 be the variation of the slope angle. The value of γ0 is either positive or negative corresponding to downslope or upslope. By varying γ0 in the range [−20°, 20°], it is calculated that δ¯ = 0.8149, which is strictly less than 1, and ζ ∗ > C0 when γ0 > −1.04° (see Fig. 13). Thus, it follows from Theorem 4.3 that the control law (11) results in a stable periodic walking orbit on slopes of γ0 > −1.04°. It means that for downslopes the robot can have a stable walking gait using this method, but the maximal upslope it can have such a stable gait is 1.04°. A larger upslope is infeasible for the robot because the ankle of the robot is not actuated. However, for a different walking gait designed on level ground, this approach may lead to a different maximal upslope on which the robot is able to have a stable walking gait. Hence, if we take this factor into account in optimizing the walking gait on level ground, the biped may generate stable walking gaits on relatively large upslopes by our approach. We simulate two situations of 1° upslope and 5° downslope for which our control law (11) is applied with u0 given in (34). The stick

diagrams of the resulting periodic gaits are demonstrated in Fig. 14. In the simulation, the initial state is not on the periodic orbit. In particular, q1 (0) has a 10° deviation away from the periodic orbit. The initial value τ˙ (0) is selected to be 1. It is shown in Figs. 15 and 16 that for both situations, the trajectory of the torso asymptotically converges to a periodic orbit, where the top subfigure shows the trajectory with respect to real time t, and the bottom subfigure shows the trajectory with respect to virtual time τ . The periodic orbits with respect to τ shown in the bottom of Figs. 15 and 16 are the same as the one existing on level ground. Moreover, the phase diagram of τ˙ and τ¨ is plotted in Fig. 17 for γ0 = −1°, 0° and 5°. It is noticed that if γ0 > 0°, then τ˙ < 1, which means the real time is faster than the virtual time, and if γ0 < 0°, then τ˙ > 1, which means the real time is slower than the virtual time. If γ0 = 0°, then τ = t , τ˙ = 1, i.e., the real time is the same as the virtual time. The evolutions of the virtual time with respect to the real time for γ0 = −1° and γ0 = 5° are depicted in Fig. 18, which illustrate the observation. In addition, the evolutions of τ˙ are drawn in Fig. 19 for the two situations as well. It is shown that τ˙ converges to a periodic orbit. Table 3 lists the minimum values of vertical component FN and the maximum values of the ratio µ over the periodic orbits on different slopes. The simulation results verify that our proposed control law reproduces stable periodic orbits on slope environments as we theoretically analyzed.

Y. Hu et al. / Automatica 47 (2011) 1605–1616

Fig. 16. Top: the simulated response on 5° downslope with respect to real time t. Bottom: the simulated response on 5° downslope with respect to virtual time τ .

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Fig. 19. The evolution of τ˙ asymptotically converges to a periodic trajectory (top: 1° upslope; bottom: 5° downslope). Table 3 The minimum values of FN and the maximum values of µ over the periodic orbits on different slopes.

γ0 (°)

Min(FN ) (N)

Max(µ)

−1 −0.5

238.8 225.3 211.6 102.4 24.0

0.2147 0.2021 0.1999 0.3412 0.6531

0 3 5

Fig. 17. Phase diagram of τ¨ and τ˙ for γ0 = −1°, 0°, and 5°.

the issue of underactuation which makes controlled symmetries unapplicable, the time-scaling control approach is introduced to hold the same geometry shape of the walking gait as the original one on a specific slope or level ground. The degree of underactuation is compensated by one degree-of-freedom in the temporal evolution. In the paper, necessary and sufficient conditions are investigated for the existence of a periodic solution on other slopes when applying the proposed time-scaling control law and for the stability properties of these periodic solutions. A shortcoming of our approach inherited from controlled symmetries is that the posture of the robot does not change when the slope changes. This limits the gaits that can be achieved on arbitrary slopes. References

Fig. 18. The evolution of τ with respect to the real time t. (top: 1° upslope; bottom: 5° downslope).

6. Conclusion The paper investigates the problem of walking on slopes for n degree-of-freedom biped robots with one degree of underactuation. The approach is to translate a walking gait existing only on certain shallow slopes or level ground to other slopes. To address

Ames, A. D., & Gregg, R. D. (2007). Stably extending two-dimensional bipedal walking to three dimensions. In Proceedings of American control conference. New York, USA (pp. 2848–2854). Ames, A. D., Gregg, R. D., Wendel, E. D. B., & Sastry, S. (2007). On the geometric reduction of controlled three-dimensional bipedal robotic walkers. Lecture Notes in Control and Information Sciences, 366, 183–196. Arai, H., Tanie, K., & Shiroma, N. (1998). Time-scaling control of an underactuated manipulator. Journal of Robotic Systems, 15(9), 525–536. Chevallereau, C. (2003). Time-scaling control for an underactuated biped robot. IEEE Transactions on Robotics and Automation, 19(2), 362–368. Chevallereau, C., Abba, G., Aoustin, Y., Plestan, F., Westervelt, E. R., de Wit, C. C., et al. (2003). Rabbit: a testbed for advanced control theory. IEEE Control Systems Magazine, 23(5), 57–79. Chevallereau, C., & Aoustin, Y. (2001). Optimal reference trajectories for walking and running of a biped robot. Robotica, 19(5), 557–569. Chevallereau, C., Grizzle, J. W., & Shih, C. (2009). Asymptotically stable walking of a five-link underactuated 3D bipedal robot. IEEE Transactions on Robotics, 25(1), 37–50. Collins, S., Ruina, A., Tedrake, R., & Wisse, M. (2005). Efficient bipedal robots based on passive-dynamic walkers. Science, 307(5712), 1082–1085. Collins, S. H., Wisse, M., & Ruina, A. (2001). A three-dimensional passive-dynamic walking robot with two legs and knees. The International Journal of Robotics Research, 20(7), 607–615. Dahl, O., & Nielsen, L. (1990). Torque-limited path following by online trajectory time scaling. IEEE Transactions on Robotics and Automation, 6(5), 554–561.

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Y. Hu et al. / Automatica 47 (2011) 1605–1616

Garcia, M., Chatterjee, A., Ruina, A., & Coleman, M. (1998). The simplest walking model: stability, complexity, and scaling. Journal of Biomechanical Engineering, 120(2), 281–288. Geppert, L. (2004). QRIO, the robot that could. IEEE Spectrum, 41(5), 34–37. Goswami, A., Espiau, B., & Keramane, A. (1997). Limit cycles in a passive compass gait biped and passivity-mimicking control laws. Autonomous Robots, 4(3), 273–286. Goswami, A., Thuilot, B., & Espiau, B. (1998). A study of the passive gait of a compasslike biped robot: symmetry and chaos. The International Journal of Robotics Research, 17(12), 1282–1311. Gregg, R. D., & Spong, M. W. (2008). Reduction-based control with application to three-dimensional bipedal walking robots. In Proceedings of American control conference. Seattle, WA, USA (pp. 880–887). Grizzle, J. W., Abba, G., & Plestan, F. (2001). Asymptotically stable walking for biped robots: analysis viasystems with impulse effects. IEEE Transactions on Automatic Control, 46(1), 51–64. Hirose, R., & Takenaka, T. (2001). Development of the humanoid robot ASIMO. Honda R&D Technical Review, 13(1), 1–6. Holm, J. K., & Spong, M. W. (2008). Kinetic energy shaping for gait regulation of underactuated bipeds. In Proceedings of IEEE international conference on control applications. San Antonio, USA (pp. 1232–1238). Ishida, T. (2004). Development of a small biped entertainment robot QRIO. In Proceedings of the 2004 international symposium on micro-nanomechatronics and human science. Nagoya, Japan (pp. 23–28). Kajita, S., Yamaura, T., & Kobayashi, A. (1992). Dynamic walking control of a biped robot along a potential energyconserving orbit. IEEE Transactions on Robotics and Automation, 8(4), 431–438. Kaneko, K., Kanehiro, F., Kajita, S., Hirukawa, H., Kawasaki, T., & Hirata, M. et al. (2004). Humanoid robot HRP-2. In Proceedings of 2004 IEEE international conference on robotics and automation. New Orleans, USA (pp. 1083–1090). McGeer, T. (1990). Passive dynamic walking. The International Journal of Robotics Research, 9(2), 62–82. Morris, B., & Grizzle, J. W. (2005). A restricted Poincare map for determining exponentially stable periodic orbits in systems with impulse effects: application to bipedal robots. In Proceedings of 44th IEEE conference on decision and control and 2005 European control conference. Seville, Spain (pp. 4199–4206). Sinnet, R. W., & Ames, A. D. (2009a). 2D bipedal walking with knees and feet: a hybrid control approach. In Proceedings of 48th IEEE conference on decision and control and 28th Chinese control conference. Shanghai, China (pp. 3200–3207). Sinnet, R. W., & Ames, A. D. (2009b). 3D bipedal walking with knees and feet: a hybrid control approach. In Proceedings of 48th IEEE conference on decision and control and 28th Chinese control conference. Shanghai, China (pp. 3208–3213). Spong, M. W. (1999). Passivity based control of the compass gait biped. In Proceedings of IFAC world congress. Beijing, China (pp. 19–23). Spong, M. W., & Bullo, F. (2005). Controlled symmetries and passive walking. IEEE Transactions on Automatic Control, 50(7), 1025–1031. Spong, M. W., Hutchinson, S., & Vidyasagar, M. (2004). Robot dynamics and control. John Wiley & Sons. Szadeczky-Kardoss, E., & Kiss, B. (2006). Tracking error based on-line trajectory time scaling. In Proceedings of international conference on intelligent engineering systems. London, UK (pp. 80–85).

Westervelt, E. R., Grizzle, J. W., Chevallereau, C., Choi, J. H., & Morris, B. (2007a). Feedback control of dynamic bipedal robot locomotion. CRC Press. Westervelt, E. R., Grizzle, J. W., & Koditschek, D. E. (2003). Hybrid zero dynamics of planar biped walkers. IEEE Transactions on Automatic Control, 48(1), 42–56. Westervelt, E. R., Morris, B., & Farrell, K. D. (2006). Sample-based hzd control for robustness and slope invariance of planar passive bipedal gaits. In Proceedings of 14th mediteraneen conference on control and automation. Westervelt, E. R., Morris, B., & Farrell, K. D. (2007). Analysis results and tools for the control of planar bipedal gaits using hybrid zero dynamics. Autonomous Robots, 23(2), 131–145.

Yong Hu received the B.S. degree in electrical engineering and automation from Nanjing University of Aeronautics and Astronautics, China in 2005 and received the Master degree in control theory and control engineering from the same university in 2008. He is currently working toward the Ph.D. degree in electrical engineering at Zhejiang University, China. His current research interests include nonlinear dynamic control and biped robots.

Gangfeng Yan received the B.S. and M.S. degrees in Control Theory and Control Engineering from Zhejiang University, China, in 1981 and 1984, respectively. He is currently a professor in the Department of Systems Science and Engineering, Zhejiang University. His research interests include hybrid systems, neural networks, and cooperative control.

Zhiyun Lin received his Ph.D. degree in electrical and computer engineering from University of Toronto, Canada, in 2005. From 2005 to 2007, he was a postdoctoral researcher in the Department of Electrical and Computer Engineering at University of Toronto, Canada. In 2007 he joined the College of Electrical Engineering at Zhejiang University, China, as a Professor. His recent research is concentrated on large scale networked multi-agent systems, wireless sensor networks, switched and hybrid systems, and feedback control of biped robots.