This paper focuses on the development of a model-based feedback controller to realize high versatility of fully actuated planar bipedal robotic walking. To conveniently define both symmetric and asymmetric walking patterns, we propose to use the left and the right legs for gait characterization. In addition to walking pattern tracking error, a biped's position tracking error in Cartesian space is included in the output function in order to enable high-level task planning and control such as multi-agent coordination. A feedback controller based on input–output linearization and proportional–derivative control is then synthesized to realize exponential tracking of the desired walking pattern as well as the desired global position trajectory. Sufficient stability conditions of the hybrid time-varying closed-loop system are developed based on the construction of multiple Lyapunov functions. In motion planning, a new method of walking pattern design is introduced, which decouples the planning of global motion and walking pattern. Finally, simulation results on a fully actuated planar biped show the effectiveness of the proposed walking strategy.

Introduction

There are a variety of approaches to realize bipedal robotic walking. One of the most frequently used methods is based on the zero moment point (ZMP) balance criterion [1,2]. Enforcing the ZMP criterion turns a biped into a fixed-based robot manipulator as it requires flat-footed walking, which results in limited walking speed.

The concept of viability has been introduced to walking control [3,4], and capturability has been proposed as a computationally feasible approximation of viability [58]. Because capturability is defined based on the viability of walking instead of the state of balance, the associated walking strategies may be less conservative than the ZMP-based approach.

Another approach to realizing bipedal robotic walking is based on nonlinear control theories and the concept of hybrid zero dynamics (HZD) [913]. With the HZD framework, the walking stabilization problem becomes a stabilization problem of the closed-loop control system. Orbitally, asymptotic stabilization of an underactuated walking biped with hybrid dynamics was first achieved based on feedback control under the assumption of finite time convergence during the continuous walking phase [9]. Later on, finite time stabilization was relaxed to sufficiently fast exponential stabilization, and the concepts of virtual constraints and HZD were introduced [1214]. Because the HZD framework only specifies a general structure of walking dynamics, it can be applied to a variety of biped models. In addition to provable closed-loop stability of the control system, another advantage of the HZD-based approach is the achievable high walking speed. Furthermore, the walking strategy based on HZD has been extended to rough terrain walking [15], three-dimensional walking [16], and neutrally stable walking [17]. Besides the HZD framework, orbital stabilization of underactuated walking has also been realized based on transverse linearization [1820].

For fully actuated walking, the HZD framework has been extended to achieve exponential tracking of the desired walking pattern as well as the desired walking speed [11,21,22]. However, because orbital stabilization cannot control a biped's global position trajectory in Cartesian space, only velocity tracking has been addressed. Therefore, satisfactory position tracking in Cartesian space is not guaranteed, and the walking versatility is limited. Controlling a biped's global position trajectory is desirable because it enables high-level task planning and control such as multi-agent coordination and obstacle avoidance.

Also, previous studies on orbital stabilization typically utilize the traditional support-swing gait characterization that describes a bipedal gait using the support and the swing legs. This gait characterization can be used to conveniently define symmetric gaits but not asymmetric gaits. Therefore, planning and control of asymmetric bipedal robotic gaits have not been fully investigated in previous studies. However, asymmetric gait tracking is potentially meaningful for related research areas such as motion planning and control of prosthetic devices and exoskeletons because asymmetric gaits are common in injured human walking.

In this study, exponential stabilization of fully actuated planar bipedal walking is realized with enhanced versatility as compared with the previous studies on orbital stabilization. The left–right gait characterization is utilized to describe a bipedal gait so as to conveniently define both symmetric and asymmetric gaits. Although this type of gait characterization has been previously adopted [23], planning and control of asymmetric walking have not been fully studied. Another benefit of the left–right gait characterization is that the states representing joint positions become continuous and well defined upon a swing-foot landing. In control design, input–output linearization is utilized to synthesize a controller to achieve exponential tracking of the desired global motion and walking pattern, and the closed-loop stability conditions are derived based on the construction of multiple Lyapunov functions [24]. In motion planning, a new method of walking pattern design is presented, which not only guarantees that the planned motion respects the impact events but also enables decoupled planning of the desired global position trajectory and the desired walking pattern. The proposed walking strategy is validated through simulated walking of a planar biped model with three revolute joints. Simulation results show that the proposed control design can achieve exponential tracking of the desired global position trajectory in Cartesian space as well as the desired walking pattern in the configuration space. A comparison with previous studies on orbital stabilization is also presented to validate the improved versatility.

In Sec. 2, the left–right gait characterization and the full-order hybrid walking dynamics are presented along with the tracking error of the desired position trajectory in Cartesian space and the desired walking pattern. A model-based feedback control law based on input–output linearization is presented in Sec. 3. In Sec. 4, the closed-loop stability is analyzed based on the construction of multiple Lyapunov functions. A new method of walking pattern design is proposed in Sec. 5, which enables decoupled planning of global motion and walking pattern. In Sec. 6, improved versatility of the proposed walking strategy as compared with previous studies is illustrated through simulations, and the effects of control gains on the closed-loop stability are analyzed.

Preliminary results of control design and stability analysis presented in this paper were initially reported in Ref. [25]. The present paper includes important results of motion planning and fully developed theorems on stability analysis, which were omitted from Ref. [25]. Also, a comparison with previous work on versatility is presented through simulations.

Problem Formulation

The objective of this study is to achieve exponentially stable walking with high versatility so that a fully actuated planar bipedal robot can exponentially track the desired walking pattern, both symmetric and asymmetric, as well as the desired global position trajectory in Cartesian space. In order to reach this goal, a bipedal gait is characterized by the left and the right legs instead of the traditional characterization by the support and the swing legs, the full-order hybrid walking dynamics are modeled under the left-right gait characterization, and the expressions of the walking pattern tracking error and the global position tracking error are derived.

Gait Characterization.

The traditional gait characterization describes a bipedal gait using the support and the swing legs [9,20,21]. Under the support-swing gait characterization, the states that represent positions and velocities of the support and the swing legs are always discontinuous at the end of a step because of the role switching of the support and the swing legs. Here, we utilize the left–right gait characterization to describe a bipedal gait. There are two main advantages of using the left and the right legs for gait characterization. First, the states that represent joint positions will be continuous and well defined upon an impact [23,26,27]. Although the states that represent joint velocities may still experience sudden jumps due to landing impacts, the definition of these states is at least consistent. Second, the left–right gait characterization enables planning and control of asymmetric gaits, which will be detailed in Secs. 3 and 5.

With the left–right gait characterization, the full-order walking dynamics are presented in Secs. 2.22.4.

Hybrid Walking Dynamics With Impulse Effects.

In this study, a fully actuated planar biped with three revolute joints, identical legs, and massless thin feet is considered (see Fig. 1). There are two actuators at the hip and one at each ankle, and three of them are active at any moment except for the one at the swing ankle. Assume that the swing foot always lands flat and remains in static, full contact with the walking surface until the next landing occurs. Under this assumption, the biped is considered fully actuated during continuous phases because it has three degrees-of-freedom during continuous phases and three independent active actuators. Also, assume that the landing impact is a rigid-body contact with an impulse effect and that the double-support phase is instantaneous [9]. Without loss of generality, suppose that the swing leg length is adjustable. Otherwise, when the swing leg passes the support leg, the planar biped with a compass gait can only avoid scuffing the walking surface exactly at the vertical configuration of both legs [9].

Let Q3 be the configuration space of the biped when its support foot is in static, full contact with the walking surface and the joint position limits are satisfied. As shown in Fig. 1, let q denote the joint position vector of the bipedal robot 
q=[q1,q2,q3]TQ
(1)

where q1, q2, and q3 represent the joint angles of the left leg, the right leg, and the trunk with respect to the world coordinate frame OwXwZw, respectively.

Let u denote the joint torque vector 
u=[u1,u2,u3,u4]T4
(2)

where u1, u2, u3, and u4 are defined as in Fig. 1.

Based on the left–right gait characterization introduced in Sec. 2.1, a complete walking cycle (or, a stride) consists of two successive steps, which include two single-support phases (SSP) and two swing-foot landings. The two SSPs are the left-in-support SSP and the right-in-support SSP. The two swing-foot landings are the left-to-right-support landing and the right-to-left-support landing. System dynamics during the two SSPs are continuous time, but the swing-foot landings are modeled as impulse effects.

When the swing foot hits the walking surface, a landing impact occurs. This switching event can be defined by a switching surface Si(q,q˙) (i ∈ {L, R}) as 
{left-to-right-support:SL(q,q˙):={(q,q˙)TQ:hL(q)=0,dhLdqq˙<0}right-to-left-support:SR(q,q˙):={(q,q˙)TQ:hR(q)=0,dhRdqq˙<0}
(3)

where the subscript i ∈ {L, R} indicates whether the left (L) or the right (R) leg is in support, hL(q):=lcos(q1)lcos(q2) is the swing foot height when the left leg is in support, and hR(q):=lcos(q2)lcos(q1) is the swing foot height when the right leg is in support (l is the leg length).

Under the assumption that the support foot is in static, full contact with the walking surface, that a double-support phase is instantaneous, and that the landing impact is a rigid-body contact with an infinitesimally small period, the hybrid walking dynamics can be compactly expressed as 
{ΣL:{ML(q)q¨+hL(q,q˙)=BLuL,if(q,q˙)SL(q,q˙)[q+;q˙+]=ΔL(q,q˙),if(q,q˙)SL(q,q˙)ΣR:{MR(q)q¨+hR(q,q˙)=BRuR,if(q,q˙)SR(q,q˙)[q+;q˙+]=ΔR(q,q˙),if(q,q˙)SR(q,q˙)
(4)
where Mi:Q3×3 is the inertia matrix, hi:TQ3 is the sum of the Coriolis, centrifugal, and gravitational terms, Bi3×3 is a nonsingular input matrix, ui3 is the torque vector of the active joints, and the reset map Δi:TQ6 represents landing impact dynamics. The expression of Δi(q,q˙) can be obtained as 
Δi(q,q˙):=[qIi(q)q˙],i{L,R}
(5)

where the derivation of Ii:Q3×3 is given in Ref. [28]. Equation (5) indicates that the states representing joint velocities may experience a sudden jump at the switching event, but the states representing joint positions are always continuous because of the left–right gait characterization.

Tracking Error of the Desired Global Position Trajectory.

To accomplish complex tasks such as multi-agent coordination and obstacle avoidance, it is necessary for a biped to follow the desired travel path with the desired motion in Cartesian space, which can be formulated as a contouring control problem in general. By constructing an orthogonal global task coordinate frame along the desired travel path, the contour error and the motion along the desired travel path can be separately represented in two sets of coordinates, based on which contour error minimization and desired motion following along the contour can be decoupled into a stabilization problem and a trajectory tracking problem [29]. In this study, we want to solve this contouring control problem for the fully actuated planar biped model in Fig. 1, which reduces to the problem of position trajectory tracking along the Xw-axis because the Xw-axis is the only feasible travel path for the planar biped. The complete problem of contouring control will be considered in our future work on three-dimensional walking of fully actuated bipedal robots.

Suppose that the walking direction aligns with the positive direction of the Xw-axis of the world coordinate frame (see Fig. 1). Let (xl, zl) and (xr, zr) be the left foot and the right foot positions in the world coordinate frame, respectively. Let the biped's hip position s along the Xw-axis represent its global position in Cartesian space. With reference to Fig. 1, s can be expressed as 
s={xl+lsin(q1)(left-in-support)xr+lsin(q2)(right-in-support)
(6)

Let sd(t) denote the desired trajectory of s along the Xw-axis. Then, the tracking error of the desired global position trajectory sd(t) can be expressed as s − sd(t).

Let 
qst:={q1(left-in-support)q2(right-in-support)
be the support-leg angle. Under the assumption that the support foot position is known, the desired position trajectory of the support leg can be obtained from sd(t) as 
qstd(t):={q1d(t):=sin1(sd(t)xll)(left-in-support)q2d(t):=sin1(sd(t)xrl)(right-in-support)
(7)

Then, the tracking error of qst − qstd(t) can be used to indicate the global position tracking error s − sd(t). If a control law is synthesized such that qst − qstd(t) is driven to zero exponentially, then the exponential tracking of the desired global position trajectory sd(t) is realized.

Tracking Error of the Desired Walking Pattern.

A walking pattern is defined as the relative evolution of a biped's joint positions with respect to a reference (or, encoding) variable in a complete walking cycle [9]. Tracking a preplanned walking pattern is desirable partly because the joint motion can be synchronized with respect to the reference variable. Furthermore, a proper walking pattern design can be utilized to construct the hybrid invariance of the desired motion, which will be detailed in Sec. 5.

Let 
s¯={sxl(left-in-support)sxr(right-in-support)
(8)
be the relative position of the hip with respect to the support foot along the Xw-axis. Note that s¯(t) increases monotonically in t during forward walking.
With s¯ chosen as the reference variable, the desired walking pattern is introduced as 
gi(s¯,qsw,q3)=0,i{L,R}
where qsw:={q2(left-in-support)q1(right-in-support) represents the swing leg position and gi(s¯,qsw,q3) is defined as 
{left-in-support:gL(s¯,qsw,q3):=[q2ϕ1L(s¯)q3ϕ2L(s¯)]right-in-support:gR(s¯,qsw,q3):=[q1ϕ1R(s¯)q3ϕ2R(s¯)]
(9)

where the functions ϕji(s¯)(i{L,R},j{1,2}) will be determined with a new method of walking pattern design in Sec. 5.

Hence, the tracking error of the desired walking pattern is simply gi(s¯,qsw,q3). If gi(s¯,qsw,q3) is exponentially driven to zero with a controller design, then exponential tracking of the desired walking pattern is realized.

Now, we will use an example to illustrate why the left–right gait characterization can be utilized to conveniently define an asymmetric gait. From Eqs. (6) and (8), one has 
s¯=lsin(qst)
(10)
From Eq. (10), a function of qst is introduced to represent ϕji(s¯) as 
ϕ̃ji(qst):=ϕji(s¯)=ϕji(lsin(qst))
(11)
for i ∈ {L, R} and j ∈ {1, 2}. Thus, under the left–right gait characterization, the desired walking pattern of the swing-leg angle qsw with respect to the support-leg angle qst can be expressed as 
{left-in-support:q2ϕ̃1L(q1)=0right-in-support:q1ϕ̃1R(q2)=0
(12)

Such a walking pattern is illustrated in Fig. 2. From the solid lines in Fig. 2, one can see that an asymmetric gait can be conveniently defined by differentiating the left and the right legs because ϕ̃1L(qst) and ϕ̃1R(qst) can be chosen independently. However, as shown by the dashed line in Fig. 2, the traditional walking characterization based on the support and the swing legs can at most represent a symmetric walking pattern where ϕ̃1L(qst)=ϕ̃1R(qst)=ϕ̃1(qst).

Feedback Control Based on Input–Output Linearization

The control objective of this study is to realize exponential tracking of the desired walking pattern in the configuration space as well as the desired position trajectory in Cartesian space. As the first step of our ongoing research, it is assumed that there are no modeling errors or disturbances, and input–output linearization is utilized to synthesize the needed controller to achieve the control objectives. Specifically, the swing leg and the trunk are driven to exponentially converge to the desired walking pattern encoded by the support leg angle, and the support leg is driven to realize exponential tracking of the desired motion in Cartesian space.

With this goal in mind and from Eqs. (7), (9), and (11), the output function is designed as the tracking error of the desired global position in Cartesian space and the desired walking pattern 
{left-in-support:yL:=qqLd(t,q1)right-in-support:yR:=qqRd(t,q2)
(13)
where 
qLd(t,q1):=[q1d(t)ϕ̃1L(q1)ϕ̃2L(q1)]andqRd(t,q2):=[ϕ̃1R(q2)q2d(t)ϕ̃2R(q2)]

Note that the output function yi (i ∈ {L, R}) can be designed as three-dimensional because the biped has three independent active actuators. By exponentially driving the output function yi (i ∈ {L, R}) to zero, exponential tracking of the desired global motion sd(t) and the desired walking pattern gi(s¯,qsw,q3)=0 (i ∈ {L, R}) can be realized simultaneously.

From Eq. (13), one obtains 
y¨i=Pi(qst)q¨zi(t,qst,q˙st),i{L,R}
(14)
during continuous phases, where 
PL:=[100dϕ̃1Ldq110dϕ̃2Ldq101],zL:=[q¨1d(t)d2ϕ̃1Ldq12q˙12d2ϕ̃2Ldq12q˙12]PR:=[1dϕ̃1Rdq200100dϕ̃2Rdq21],zR:=[d2ϕ̃1Rdq22q˙22q¨2d(t)d2ϕ̃2Rdq22q˙22]
(15)

From Eq. (15), it can be known that Pi(qst) is always invertible.

From Eqs. (4) and (14), one has 
y¨i=Ni(q)uiLi(t,q,q˙),i{L,R}
(16)

where Ni=PiMi1Bi is proved to be invertible and Li=PiMi1hi+zi.

Therefore, the feedback control law based on input–output linearization is defined as 
ui=Ni1(vi+Li)
(17)
which results in a linear system 
y¨i=vi,i{L,R}
(18)
If vi is chosen as a proportional–derivative (PD) controller 
vi=KPiyiKDiy˙i,i{L,R}
(19)
where KPi3×3 and KDi3×3 are both nonsingular diagonal matrices, one then obtains a linear system 
x˙=Aix,i{L,R}
(20)
with 
x:=[x1,x2,x3,x4,x5,x6]T:=[yiy˙i]χ
(21)
and 
Ai:=[03×3I3×3KPiKDi]6×6
(22)

where χ is the full state space, 03×33×3 is a zero matrix, and I3×33×3 is an identity matrix.

If KPi and KDi are chosen such that Ai is Hurwitz, then there exists a real positive-definite-symmetric matrix Wi such that Vi(x)=xTWix is a Lyapunov function candidate for the continuous-phase dynamics in Eq. (20) and there exist positive constants c1i, c2i, and c3i (i ∈ {L, R}) such that Vi(x) satisfies 
c1ix2Vi(x)c2ix2andV˙i(x)c3iVi(x)
(23)
for all x during continuous phases [30].

Stability Analysis

Based on previous analysis, the closed-loop walking dynamics can be compactly written as 
{ΣL:{x˙=ALx,if(t,x)SLR(t,x)x+=ΔLR(t,x),if(t,x)SLR(t,x)ΣR:{x˙=ARx,if(t,x)SRL(t,x)x+=ΔRL(t,x),if(t,x)SRL(t,x)
(24)

where the expressions of ΔLR and ΔRL can be derived from Δi and yi and the expressions of the switching surfaces SLR(t, x) and SRL(t, x) can be obtained from Si and yi.

Without loss of generality, suppose that the walking process begins with the left leg in support and the right leg in the air at t = t0. Without loss of generality, suppose t0 = 0. Let TLk and TRk,k{1,2,} represent the actual moments of the kth left-to-right-support and the kth right-to-left-support impacts, respectively. Without loss of generality, assume TR0=t0. Let τLk and τRk,k{1,2,} denote the desired moments of the kth left-to-right-support impact assuming x(t)=0t>TRk1 and the kth right-to-left-support impact assuming x(t)=0t>TLk, respectively.

Properties of TLk and TRk are summarized in Theorem 1, which is introduced based on Lemma 2 in Ref. [31].

Theorem 1. Consider the fully actuated walking system in Eq. (24). Let the following conditions hold:

  • (A1) There is no beating effect at impacts.

  • (A2) The desired global position trajectory sd(t) is continuously differentiable and monotonically increasing for t > 0.

  • (A3) The function ϕ̃ji(qst)(i{L,R},j{1,2}) in Eq. (11) is continuously differentiable in qst.

Then, there exists a small neighborhood U of the point (τik,0),i{L,R},k{1,2,} such that Tik(τik,pik(τik)) is a unique continuously differentiable function in U, where pLk(t) is the solution of x˙=ALx with initial condition pLk(TRk1+)=x(TRk1+) for t(TRk1,+) and pRk(t) is the solution of x˙=ARx with initial condition pRk(TLk+)=x(TLk+) for t(TLk,+). Also, Tik has the following properties:

  • (P1) TLk(τLk,0)=τLk,TRk(τRk,0)=τRk; and

  • (P2) there exists a positive number Lτ such that |Tik(τ,w)Tik(τ,u)|Lτwu,(τ,w),(τ,u)U.

Proof. By the definitions of Tik and τik(i{L,R},k{1,2,}), the property (P1) holds. From the conditions (A1)–(A3) and Eqs. (3), (13), and (24), it can be known that the functions that define the continuous dynamics, the reset maps, and the switching surfaces are all continuously differentiable in t and x. Then, by the implicit function theorem, the property (P2) holds.

We are now ready to present the main theorem.

Theorem 2. Let the conditions (A1)–(A3) hold. Assume that ΔLR(τLk,0)=0 and ΔRL(τRk,0)=0 hold for any k ∈ {1, 2,…}. Then, there exist positive-definite diagonal matrices KPi and KDi (i ∈ {L, R}) and a positive number δ such that the hybrid time-varying closed-loop system in Eq. (24) is locally exponentially stable for all x(0)Bδ(0):={xχ:x<δ}.

Proof. Without loss of generality, suppose that the walking process begins with the left-in-support continuous phase.

Let VL(x) and VR(x) be the Lyapunov functions associated with the left-in-support and the right-in-support phases, respectively. When the walking process begins with the left-in-support continuous phase, the Kth step (K ∈{1, 3, 5…}) is a left-in-support phase and the Kth switching is a left-to-right-support switching. Let VR|K+ and VL|K+1+ (K ∈ {1, 3, 5,…}) denote the values of Lyapunov functions right after the Kth and the (K + 1)th switchings, respectively. By stability analysis via multiple Lyapunov functions [24], the overall system is exponentially stable if VL(x) and VR(x) are exponentially decreasing in the left-in-support and the right-in-support phases, respectively, and if {VR|1+,VR|3+,VR|5+} and {VL|2+,VL|4+,VL|6+} are both strictly decreasing sequences.

As explained in Sec. 3, if KPi and KDi (i ∈ {L, R}) are chosen such that Ai is Hurwitz, then the continuous-phase subsystems are exponentially stabilized. Therefore, the remaining task is to derive stability conditions to guarantee that the sequences {VR|1+,VR|3+,VR|5+} and {VL|2+,VL|4+,VL|6+} are both strictly decreasing. This requirement can be rewritten as 
VR|K+2+<VR|K+andVL|K+3+<VL|K+1+
(25)
where K is defined as 
K{1,3,5,}
for the rest of this proof.

First, we prove that there exist positive-definite diagonal matrices KPi and KDi (i ∈ {L, R}) and a positive number δR such that VR|K+2+<VR|K+ for any x(0)BδR(0).

From Eq. (23), one has 
VR(x)ec3R(tTK)VR|K+
(26)
during the continuous-phase right after the Kth impact and 
VL(x)ec3L(tTK+1)VL|K+1+
(27)
during the continuous-phase right after the (K + 1)th impact.
Because of the assumption ΔRL(τK+1,0)=0, one has 
x|K+1+=ΔRL(TK+1,x|K+1)ΔRL(TK+1,x|K+1)ΔRL(τK+1,x|K+1)+ΔRL(τK+1,x|K+1)ΔRL(τK+1,0)
(28)

where TK+1 is the moment of the actual (K + 1)th impact, x|K+1 and x|K+1+ represent the values of x right before and after the (K + 1)th impact, respectively, and τK+1 is the moment of the desired (K + 1)th impact assuming x(t) = ∀t > TK.

Because the reset map ΔRL(t,x) is continuously differentiable in t and x, it is locally Lipschitz continuous in t and x. Hence, there exists rR*>0 such that for any x(0)BrR*(0), one has 
ΔRL(τK+1,x|K+1)ΔRL(τK+1,0)LΔRxx|K+1
(29)
and 
ΔRL(TK+1,x|K+1)ΔRL(τK+1,x|K+1)LΔRt|TK+1τK+1|
(30)
for some Lipschitz constants LΔRx and LΔRt.
Define 
x̃R(t):=eAR(tTK)x|K+,t>TK
(31)
By Theorem 1, there exist hR*>0 and Lτ > 0 such that 
|TK+1τK+1|=|TK+1(τK+1,x̃(τK+1))TK+1(τK+1,0)|Lτx̃R(τK+1)
(32)
for any x(0)BhR*(0).
From Eqs. (28)(30), and (32), one has 
x|K+1+2LΔR(x|K+12+x̃R(τK+1)2)
(33)

where LΔR:=max(2LΔRx2,2LΔRt2Lτ2).

Similarly 
x|K+2+2LΔL(x|K+22x̃L(τK+2)2)
(34)
where LΔL is a constant that can be obtained similarly to LΔR, and x̃L(t) is defined as 
x̃L(t):=eAL(tTK+1)x|K+1+,t>TK+1
(35)
According to Eq. (23), the following inequalities hold: 
VR|K+1c1Rx|K+12,VR(x̃R(τK+1))c1Rx̃R(τK+1)2VL|K+2c1Lx|K+22,VL(x̃L(τK+2))c1Lx̃L(τK+2)2VL|K+1+c2Lx|K+1+2,VR|K+2+c2Rx|K+2+2
(36)
Furthermore, from Eqs. (26), (27), (31), and (35) 
VR(x̃R(τk+1))ec3R(τk+1TK)VR|K+
(37)
and 
VL(x̃L(τk+2))ec3L(τk+2TK+1)VL|K+1+
(38)

hold.

Combining Eqs. (32), (34), and (36)(38), one obtains 
VR|K+2+c2Lc2Rc1Rc1LLΔLLΔRe(c3LΔτK+1+c3RΔτK)(1+ec3L(TK+2τK+2))(1+ec3R(TK+1τK+1))VR|K+
(39)

where ΔτK:=τK+1TK and ΔτK+1:=τK+2TK+1. Note that ΔτK is the desired duration of the (K + 1)th step, which is known right after the Kth actual impact occurs.

From Eqs. (23) and (31), 
x̃R(τk+1)c2Rc1Rec3R2c2RΔτkx|K+
(40)
holds, and thus from Eqs. (32) and (40), one has 
|TK+1τK+1|Lτc2Rc1Rec3R2c2RΔτKx|K+
(41)
Similarly 
|TK+2τK+2|Lτc2Lc1Lec3L2c2LΔτK+1x|K+1+
(42)
Hence, for any ε > 0, there exist sufficiently large c3L and c3R and a positive number l* such that 
ec3L(TK+2τK+2)1+εandec3R(TK+1τK+1)1+ε
(43)
hold for all x(0)Bl*(0).
Then, it can be obtained from Eqs. (39) and (43) that 
VR|K+2+c2Lc2Rc1Lc1RLΔLLΔR(1+ε)2e(c3LΔτK+1+c3RΔτK)VR|K+
(44)
holds for any x(0)BδR(0) where δR:=min(rR*,hR*,l*).
Similarly 
VL|K+3+c2Lc2Rc1Lc1RLΔLLΔR(1+ε)2e(c3LΔτK+3+c3RΔτK+2)VL|K+1+
(45)
holds for any x(0)BδL(0) where δL can be obtained similarly to the above analysis.
Note that c3i (i ∈ {L, R}) is determined by KPi and KDi. Hence, if the PD gains are sufficiently large such that Ai is Hurwitz and that there exists a positive number δ=min(δL,δR) such that 
c3LΔτK+1+c3RΔτK>2ln(c2Lc2Rc1Lc1RLΔLLΔR(1+ε))
(46)
and 
c3LΔτK+3+c3RΔτK+2>2ln(c2Lc2Rc1Lc1RLΔLLΔR(1+ε))
(47)
hold for any x(0)Bδ(0) and any K ∈ {1, 3, 5,…}, then VR|K+2+<VR|K+ and VL|K+3+<VL|K+1+ hold for any x(0)Bδ(0) and any K ∈ {1, 3, 5,…}; i.e., the closed-loop system in Eq. (24) is locally exponentially stable.

The stability conditions in Eqs. (46) and (47) indicate that the output function should converge to zero sufficiently fast so as to diminish the possible divergence caused by reset maps. In previous studies, rapidly exponential convergence of output function has been proposed to deal with the possible expansiveness of a landing impact [21], which can also be applied here to further increase the convergence rate.

In Sec. 5, it will be shown that the assumption of ΔLR(τK,0)=0 and ΔRL(τK+1,0)=0 in Theorem 2 will always hold for any K ∈ {1, 3, 5,…} if the desired walking pattern is designed properly.

Desired Walking Pattern Design

The desired walking motion qid(t, qst) (i ∈ {L, R}) in Eq. (13) is completely defined by the desired global position trajectory and the desired walking pattern. Suppose that the desired global position trajectory sd(t) is determined by the high-level task planner, which is continuously differentiable and monotonically increasing for t > 0. Then, the remaining task of motion planning is walking pattern design, which should guarantee that the desired motion qid(t, qst) will satisfy the following conditions:

  • (C1) ΔLR(τK,0)=0 and ΔRL(τK+1,0)=0,K{1,3,5,}

  • (C2) forward walking direction;

  • (C3) ground-contact constraints including the friction cone and the unilateral constraint;

  • (C4) joint position and velocity limits; and

  • (C5) joint torque limits.

Since the last four conditions (C2)–(C5), can be easily met through trajectory optimization, they are not further discussed in this paper. The first condition (C1) essentially states that the desired gait should respect the reset map; i.e., if x(τK)=0, then x(τK+)=0 should always hold. As presented in Sec. 4, the first condition (C1) is important because it can greatly simplify the stability analysis of the hybrid time-varying closed-loop control system. However, the satisfaction of (C1) is not straightforward as it involves both the desired global position trajectory and the desired walking pattern.

In this section, a new method of walking pattern design is proposed, which guarantees that (C1) is always satisfied for any feasible sd(t) that is continuously differentiable and monotonically increasing for t > 0. This is advantageous because the high-level planning of the desired global position trajectory sd(t) and the low-level planning of the desired walking pattern represented by ϕ̃ji(qst) are decoupled for the satisfaction of (C1).

Hybrid Invariance of Desired Motion.

The condition (C1) can be decomposed into two parts. One part requires that the desired walking pattern should respect the reset map, which can be satisfied based on the same method of walking pattern design for constructing HZD [13]. The other part is tricky to meet, which requires that the desired position trajectory of the support leg should respect the reset map. As indicated in Eq. (7), the desired support-leg trajectory is updated at the beginning of each actual step, and thus it depends on when and where the last actual swing-foot touchdown occurs.

When the walking process begins with the left-in-support continuous phase, the Kth step (K ∈ {1, 3, 5…}) is a left-in-support step starting at t=TK1+, and the Kth desired switching is a left-to-right-support switching at t=τK. Suppose that the desired walking pattern gL(lsin(q1),q2,q3)=0 has at least one intersection with the switching surface SL(q,q˙). Also, suppose that one of these intersections has left leg position at q1*. In the following, we will develop the conditions on q1* and the desired walking pattern gi(lsin(q1),q2,q3)=0 (i ∈ {L, R}) such that ΔLR(τK,0)=0 in condition (C1) is satisfied for any K ∈ {1, 3, 5,…}.

From Eq. (7), the desired support-leg trajectory q1d(t) will increase monotonically within the Kth step if sd(t) is planned as feasible and monotonically increasing. Then, the Kth desired landing moment t=τK, if exists, can be uniquely determined from the equation q1d(τK)=q1*.

Assuming x(τK)=0, one has 
q(τK)=HqL(q1*)andq˙(τK)=Hq˙L(q1*)q˙1d(τK)
(48)
where 
HqL(q1):=[q1ϕ̃1L(q1)ϕ̃2L(q1)]andHq˙L(q1):=[1dϕ̃1L(q1)dq1dϕ̃2L(q1)dq1]
(49)
Then, at t=τK+, one can obtain the following equations from Eq. (5) 
q(τK+)=q(τK)=HqL(q1*),q˙(τK+)=IL(q(τK))q˙(τK)=ĨL(q1*)Hq˙L(q1*)q˙1d(τK)
(50)

where ĨL(q1*):=IL(HqL(q1*))=IL(q(τK)).

Assuming x(τK+)=0, one has 
q(τK+)=HqR(q2d(τK+))andq˙(τK+)=Hq˙R(q2d(τK+))q˙2d(τK+)
(51)
where 
HqR(q2):=[ϕ̃1R(q2)q2ϕ̃2R(q2)]andHq˙R(q2):=[dϕ̃1R(q2)dq21dϕ̃2R(q2)dq2]
(52)
Because sd(t) is continuously differentiable for t > 0, one has 
sd(τK+)=sd(τK)ands˙d(τK+)=s˙d(τK)
(53)
Also, s(t) is continuous for t > 0. Then, one obtains 
q2d(τK+)=ϕ̃1L(q1*)
(54)
and 
q˙2d(τK+)=cos(q1*)cos(ϕ̃1L(q1*))q˙1d(τK):=γL(q1*)q˙1d(τK)
(55)
Therefore, if ϕ̃ji(i{L,R},j{1,2}) can be designed to satisfy 
HqL(q1*)=HqR(ϕ̃1L(q1*))
(56)
and 
ĨL(q1*)Hq˙L(q1*)=Hq˙R(ϕ̃1L(q1*))γL(q1*)
(57)
then x(τK+)=ΔLR(τK,0)=0 holds for any K ∈ {1, 3, 5,…}.
We will now derive conditions to guarantee that ΔRL(τK+1,0)=0 in the condition (C1) holds for any K ∈ {1, 3, 5,…}. Similarly, suppose that the desired walking pattern gR(lsin(q2),q1,q3)=0 has at least one intersection with the switching surface SR(q,q˙) and that one of these intersections has right leg position at q2*. If ϕ̃ji(i{L,R},j{1,2}) is designed to satisfy 
HqR(q2*)=HqL(ϕ̃1R(q2*))
(58)
and 
ĨR(q2*)Hq˙R(q2*)=Hq˙L(ϕ̃1R(q2*))γR(q2*)
(59)
with ĨR(q2*):=IR(HqR(q2*)) and γR(q2*):=(cos(q2*)/cos(ϕ̃1R(q2*))), then ΔRL(τK+1,0)=0 holds for any K ∈ {1, 3, 5,…}.

Walking Pattern Parameterization With Beziér Curves.

Similar to the previous study [12], the functions ϕ̃ji(i{L,R},j{1,2}), which define the desired walking pattern, can be parameterized by the Mth-order Beziér curves as 
ϕ̃1L(q1):=ϕ¯1L(λL)=p=0MαLpM!p!(Mp)!λLp(1λL)Mpϕ̃2L(q1):=ϕ¯2L(λL)=p=0MβLpM!p!(Mp)!λLp(1λL)Mpϕ̃1R(q2):=ϕ¯1R(λR)=p=0MαRpM!p!(Mp)!λRp(1λR)Mpϕ̃2R(q2):=ϕ¯2R(λR)=p=0MβRpM!p!(Mp)!λRp(1λR)Mp
(60)
where αip and βip (i ∈ {L, R}, p ∈ {0, 1,…, M}) are unknown parameters to be determined, and the variables λL(q1) and λR(q2) are defined as 
λL(q1)=q1q10q1*q10andλR(q2)=q2q20q2*q20
(61)

where q10 and q20 are the initial support-leg angles of the left-in-support and the right-in-support phases, respectively, determined by the desired walking pattern and the switching surfaces.

The functions ϕ¯ji(λi)(i{L,R},j{1,2}) have the following properties [12]:

  1. (1)

    ϕ¯1i(0)=αi0,ϕ¯2i(0)=βi0;

  2. (2)

    ϕ¯1i(1)=αiM,ϕ¯2i(1)=βiM;

  3. (3)

    dϕ¯1i/dλi(0)=Mαi1αi0,dϕ¯2i/dλi(0)=Mβi1βi0; and

  4. (4)

    dϕ¯1i/dλi(1)=MαiMαi(M1),dϕ¯2i/dλi(1)=MβiMβi(M1).

These properties can be utilized to construct hybrid invariance of the desired motion.

An Example of Walking Pattern Design With Third-Order Beziér Curves.

In this subsection, the procedure of obtaining the unknown parameters αip and βip (i ∈ {L, R}, p ∈ {0, 1,…, M}) of ϕ̃ji(i{L,R},j{1,2}) in Eq. (60) is explained. Suppose M = 3. From Eqs. (56)(61) and the properties of ϕ̃ji(i{L,R},j{1,2}), one has 
q1*=αR0,q2*=αL0,q10=αR3,q20=αL3
(62)
 
λL(q1)=q1αR3αR0αR3,λR(q2)=q2αL3αL0αL3
(63)
 
βL3=βR0,βL0=βR3
(64)
 
γL(αR0)[3αR1αR0αL0αL313βR1βR0αL0αL3]=ĨL(αR0)[13αL3αL2αR0αR33βL3βL2αR0αR3]
(65)
 
γR(αL0)[13αL1αL0αR0αR33βL1βL0αR0αR3]=ĨR(αL0)[3αR3αR2αL0αL313βR3βR2αL0αL3]
(66)
The function ϕ̃ji(i{L,R},j{1,2}) should also satisfy the following switching conditions obtained from Eq. (3) (forward walking direction is assumed): 
lcos(αR0)lcos(αL3)=0lsin(αR0)+lsin(αL3)3αL3αL2αR0αR3<0lcos(αL0)lcos(αR3)=0lsin(αL0)+lsin(αR3)3αR3αR2αL0αL3<0
(67)

There are 16 unknown parameters of the four third-order Beziér curves, and there are ten equations and two inequality constraints in Eqs. (64)(67). Therefore, the function ϕ̃ji(i{L,R},j{1,2}) can be determined through numerical search in order to satisfy the first condition (C1).

Simulation Results

In this section, a fully actuated planar biped with three revolute joints is simulated to show the validity of the proposed walking strategy. We first compare our proposed walking strategy with previous studies to show that we can achieve exponential position tracking in Cartesian space but the previous study can only achieve exponential velocity tracking. Then, a symmetric walking pattern is simulated with two different desired global position trajectories sd(t)–one with a constant walking speed and the other with a time-varying walking speed—to show the versatility of the proposed walking strategy. The simulation results also show that the desired gait respects the reset map regardless of the choice of sd(t) when the desired walking pattern is designed as introduced in Sec. 5. Furthermore, exponential tracking of an asymmetric walking pattern is illustrated through simulations. Finally, effects of the continuous-phase convergence rate on the closed-loop stability are analyzed through simulations, which validates the stability conditions in Sec. 4.

The simulated biped model is shown in Fig. 3. The definitions of l and r are illustrated in Fig. 3, and the masses m, MH, and MT are lumped at the center of each link.

With the walking pattern design proposed in Sec. 5, a symmetric walking pattern and an asymmetric walking pattern are generated, which will be utilized in the following simulations. The obtained swing-leg patterns are shown in Fig. 4 (the trunk patterns are omitted).

Comparison With Previous Work on Orbital Stabilization.

In the previous work on orbital stabilization [11], the bipedal gait is characterized by the support and the swing legs. Thus, only a symmetric walking pattern can be exponentially tracked. Besides walking pattern tracking, another control objective of the previous work is velocity tracking in Cartesian space.

In order to compare our proposed walking strategy with the previous orbital stabilization [11], the desired walking pattern is chosen as the symmetric walking pattern in Fig. 4(a). The desired global position trajectory sd(t) is defined as monotonically increasing with a constant speed. Because the previous walking strategy focuses on velocity tracking in Cartesian space, its desired global motion is defined as s˙d(t).

The simulation results corresponding to the previous work are shown in Fig. 5. From Fig. 5, it is clear to see that exponential tracking of the desired symmetric walking pattern is achieved. However, there is always a nonzero steady-state tracking error of the desired global position trajectory sd(t), although the desired global velocity trajectory s˙d(t) can be exponentially tracked.

In contrast, with our proposed walking strategy, we can realize exponential walking pattern tracking and exponential global position tracking, as shown in Fig. 6.

This comparison clearly illustrates that our proposed walking strategy can greatly improve walking versatility as compared with the previous work on orbital stabilization.

Exponential Tracking of Different Global Position Trajectories.

In this subsection, we will show two sets of simulated bipedal walking with the same desired symmetric walking pattern (see Fig. 4(a)) but different desired hip trajectories sd(t). Without loss of generality, the control gains are chosen the same for both cases: KPL = KPR = diag[28, 28, 28] and KDL = KDR = diag[11, 11, 11]. These control gains are chosen such that the matrices AL and AR in Eq. (20) are both Hurwitz and that the conditions in Eqs. (46) and (47) are met under relatively large initial tracking errors. The two sets of bipedal walking share the same initial conditions 
s(0)sd(0)=0.1(m),s˙(0)s˙d(0)=0.1(m/s)q2(0)ϕ̃1L(q1(0))=0.1(rad)q3(0)ϕ̃2L(q1(0))=0.1(rad)q˙2(0)dϕ̃1Ldq1(q1(0))q˙1(0)=0.1(rad/s)q˙3(0)dϕ̃2Ldq1(q1(0))q˙1(0)=0.1(rad/s)
(68)
Due to the identical control gains, one has A = AL = AR. Then, without loss of generality, the Lyapunov functions during the left-in-support and the right-in-support phases are chosen the same 
VL(x)=VR(x)=xTWx
(69)

where W is the solution of the Lyapunov equation AW + WAT + Q = 0 with Q = I6 × 6 [30].

Figure 7 shows the results with sd(t) = 0.6t − 0.1(m), and Fig. 8 corresponds to sd(t)=2.3e0.3(t+0.5)+0.6t2.1(m). From the plots, we can see that exponential tracking of the desired hip trajectory sd(t) with either a constant or a time-varying walking speed is achieved under the same desired walking pattern. It shows that a walking pattern generated through the proposed gait design method in Sec. 5 can be automatically incorporated with an arbitrary hip trajectory sd(t) that is differentiable and monotonically increasing. Note that the Lyapunov function plot in Fig. 7 shows a relatively large jump at the first impact at t = 0.5 s while the one in Fig. 8 shows no significant jump at the first landing impact at t = 2 s. Because the desired global position trajectory in Fig. 8 has a much lower velocity in the first few seconds than that in Fig. 7, the duration of the first step is much longer in Fig. 8, which results in the much smaller trajectory tracking error right before the first impact and the much smaller jump of the Lyapunov function right after the first impact in Fig. 8.

Stable Asymmetric Walking.

As stated earlier, the left-right gait characterization enables planning and tracking of an asymmetric walking pattern, which is illustrated with simulation results in this subsection. Figure 9 shows simulated bipedal walking with the initial conditions in Eq. (68), the desired asymmetric walking pattern in Fig. 4(b), and the desired global position trajectory sd(t) = 0.6t − 0.1(m). The control parameters are chosen as KPL = KPR = diag[12, 12, 12] and KDL = KDR = diag[7, 7, 7], and the Lyapunov functions are defined as Eq. (69).

As shown in Fig. 9, the desired asymmetric walking pattern as well as the desired global position trajectory is exponentially tracked, which validates the high versatility of our proposed walking strategy.

Effects of Continuous-Phase Convergence Rate on Closed-Loop Stability.

Theorem 2 introduced in Sec. 4 indicates that the continuous-phase convergence rate determined by the PD gains should be sufficiently fast to guarantee the stability of the closed-loop hybrid dynamical system in Eq. (24). In this subsection, two sets of simulated bipedal walking (see Figs. 10 and 11) are presented under different PD gains. They share the same initial conditions, desired walking pattern, desired hip trajectory, and Lyapunov function definitions as in Fig. 9. However, the PD gains corresponding to Fig. 10 are larger than those in Fig. 9, and the PD gains corresponding to Fig. 11 are less than those in Fig. 9. Accordingly, it can be seen from Figs. 911 that the actual motion converges to the desired motion faster in Fig. 10 but slower in Fig. 11 as compared with Fig. 9. Therefore, this comparison validates Theorem 2 and shows that a higher convergence rate during the continuous phases results in faster closed-loop convergence.

Conclusion

In this paper, provably stable, fully actuated, planar bipedal robotic walking has been achieved with improved versatility as compared with previous studies. In order to define both symmetric and asymmetric walking patterns, the left and the right legs were used to characterize a bipedal gait. Under the left–right gait characterization, the full-order hybrid walking dynamics were modeled. With the output function designed as the global position tracking error and the walking pattern tracking error, an input–output linearizing controller was then synthesized to exponentially drive the output function to zero. Closed-loop stability conditions were analyzed based on the construction of multiple Lyapunov functions, which essentially requires that the continuous-phase convergence rate of the output function should be sufficiently fast in order to overcome the possible divergence caused by landing impacts. A new method of walking pattern design was proposed, which guarantees that the low-level planning of the desired walking pattern can be decoupled from the high-level planning of the desired global motion. Provable exponential stabilization and high versatility of the proposed walking strategy were validated with simulated bipedal walking.

Acknowledgment

Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Funding Data

  • Division of Information and Intelligent Systems, National Science Foundation (Grant No. IIS-0916807).

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