Exponential Stabilization of Fully Actuated Planar Bipedal Robotic Walking With Global Position Tracking Capabilities

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 [5–8]. 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) [9–13]. 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 [12–14]. 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 [18–20].

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.

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.

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. 2.2–2.4.

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].

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 O_wX_wZ_w, respectively.

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.

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

where the derivation of $Ii:Q→ℝ3×3$ 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.

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 X_w-axis because the X_w-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.

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

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

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.

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.

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 $ϕ̃1L(qst)=ϕ̃1R(qst)=ϕ̃1(qst)$⁠.

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.

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

From Eq. (15), it can be known that P_i(q_st) is always invertible.

where $Ni=PiMi−1Bi$ is proved to be invertible and $Li=PiMi−1hi+zi$⁠.

where χ is the full state space, $03×3∈ℝ3×3$ is a zero matrix, and $I3×3∈ℝ3×3$ is an identity matrix.

where the expressions of $ΔL→R$ and $ΔR→L$ can be derived from $Δi$ and y_i and the expressions of the switching surfaces S_L→R(t, x) and S_R→L(t, x) can be obtained from S_i and y_i.

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 $x(t)=0∀t>TRk−1$ and the kth right-to-left-support impact assuming $x(t)=0∀t>TLk$⁠, 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 s_d(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 q_st.

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 $pLk(TRk−1+)=x(TRk−1+)$ for $t∈(TRk−1,+∞)$ 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 L_τ such that $|Tik(τ,w)−Tik(τ,u)|≤Lτ‖w−u‖, ∀(τ,w), (τ,u)∈U$⁠.

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 $ΔL→R(τLk−,0)=0$ and $ΔR→L(τRk−,0)=0$ hold for any k ∈ {1, 2,…}. Then, there exist positive-definite diagonal matrices K_Pi and K_Di (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 $x(0)∈Bδ(0):={x∈χ:‖x‖<δ}$⁠.

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

Let V_L(x) and V_R(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 V_L(x) and V_R(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.

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

where T_K+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 τ_K+1 is the moment of the desired (K + 1)th impact assuming x(t) = ∀t > T_K.

where $LΔR:=max(2LΔRx2,2LΔRt2Lτ2)$⁠.

hold.

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

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 $ΔL→R(τK−,0)=0$ and $ΔR→L(τK+1−,0)=0$ in Theorem 2 will always hold for any K ∈ {1, 3, 5,…} if the desired walking pattern is designed properly.

The desired walking motion q_id(t, q_st) (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 s_d(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 q_id(t, q_st) will satisfy the following conditions:

• (C1) $ΔL→R(τK−,0)=0$ and $ΔR→L(τK+1−,0)=0, ∀ K∈{1,3,5,…}$

• (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 s_d(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 s_d(t) and the low-level planning of the desired walking pattern represented by $ϕ̃ji(qst)$ are decoupled for the satisfaction of (C1).

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 $t=TK−1+$⁠, and the Kth desired switching is a left-to-right-support switching at $t=τK−$⁠. Suppose that the desired walking pattern $gL(l sin (q1),q2,q3)=0$ 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 $gi(l sin (q1),q2,q3)=0$ (i ∈ {L, R}) such that $ΔL→R(τK−,0)=0$ in condition (C1) is satisfied for any K ∈ {1, 3, 5,…}.

From Eq. (7), the desired support-leg trajectory q_1d(t) will increase monotonically within the Kth step if s_d(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 $q1d(τK−)=q1*$⁠.