Adaptive Robust Tracking Control for Hybrid Models of Three-Dimensional Bipedal Robotic Walking Under Uncertainties

Bipedal robots can potentially be used to perform locomotion tasks for a wide range of real-world operations such as home assistance, emergency response, and search and rescue. One common control approach toward accomplishing these locomotion tasks is trajectory tracking control. However, a robot's trajectory tracking performance is typically affected by uncertainties such as parametric uncertainties and unmodeled uncertainties [1–3]. Parametric uncertainties exist when the estimated values of the system's model parameters such as a robot's link masses do not match their true values, whereas unmodeled uncertainties include nonlinear functions such as joint frictions. Without proper treatment, these uncertainties can deteriorate tracking performance and even cause instability of bipedal robots, which makes it necessary to design locomotion controllers to mitigate the negative effects of uncertainties.

To mitigate uncertainties for ensuring the stability and tracking performance of continuous systems, adaptive and robust control has been extensively studied. Adaptive control [4–6] could achieve accurate steady-state tracking under parametric uncertainties by exploiting online adaptation laws to update the estimation of model parameters, but it may not be effective under unmodeled uncertainties such as disturbances [7]. On the contrary, robust control [8,9] could guarantee satisfactory transient tracking performance under unmodeled uncertainties but cannot guarantee accurate final tracking under parametric uncertainties without relying on high-gain feedback or discontinuous control action.

To combine the complementary advantages of adaptive control (AC) and robust control, the inherent conflict of adaptive and robust control design methodologies needs to be resolved. The conflict is that the estimated model parameters provided by the adaptive control may grow unbounded under unmodeled uncertainties, which will then induce aggressive robust control action for estimation error mitigation. To resolve the conflict, an adaptive robust control approach [10,11] adds a known bound to the parameter estimation to avoid unbounded parameter drifting. The approach has been used to significantly improve the transient and steady-state trajectory tracking performance for various continuous systems [12–16] in the presence of parametric and unmodeled uncertainties.

By modeling a walking robot as a continuous system, the previously reviewed adaptive control and robust control strategies have been exploited to stabilize and improve the trajectory tracking performance for walking robots [17–22].

However, treating a walking robot as a continuous system in controller design may not be appropriate because bipedal walking robots are inherently hybrid dynamical systems, involving both continuous motions (e.g., foot swinging motions) and state-triggered jumps (e.g., sudden jumps in a robot's joint velocities upon a foot-landing impact) [3,23–27]. These jumps are particularly difficult to handle in controller design because they are nonlinear and triggered when the system's state satisfies certain conditions (e.g., the swing foot striking the walking surface). Furthermore, they cannot be directly controlled due to their infinitesimal periods of duration [28].

To provably stabilize bipedal walking robots by explicitly addressing their hybrid dynamical behaviors, the hybrid zero dynamics framework has been proposed based on orbital stabilization [29] and recently extended to mitigate uncertainties that are prevalent in practical applications [1,2]. Yet, as orbitally stabilizing controllers are not suitable for accurate trajectory tracking [30], these approaches and the underlying stability analysis methods cannot be used to solve the problem of trajectory tracking control.

Beyond the scope of walking controller design, robust trajectory tracking control for general hybrid systems with state-triggered jumps has been investigated based on Lyapunov stability analysis [31]. Still, the previous work does not explicitly analyze the evolution of Lyapunov function across the uncontrolled state-triggered jumps and thus may not be used to directly inform controller design.

To realize reliable trajectory tracking for hybrid systems with state-triggered jumps (including the fully actuated walking robots), we have introduced a Lyapunov-based trajectory tracking controller design based on an explicit analysis of the evolution of the Lyapunov function across state-triggered jumps [32–35]. This controller design has recently been extended and experimentally implemented to achieve provably stable quadrupedal robotic walking on a dynamic rigid surface that rotates in the inertial frame [36]. However, these previous control approaches utilize input–output linearization and thus may not be effective in guaranteeing a good tracking performance under modeling errors or disturbances. To this end, this study extends our previous control approach to address parametric and unmodeled uncertainties for a class of hybrid systems that include fully actuated bipedal walking robots.

This study addresses the expansion of the adaptive robust controller design methodology from continuous systems to fully actuated hybrid systems with state-triggered jumps for provably guaranteeing system stability and tracking performance under parametric and unmodeled uncertainties. The specific contributions of this study are:

• To synthesize adaptive robust control laws for fully actuated hybrid systems with state-triggered jumps by incorporating the construction of multiple Lyapunov functions into control Lyapunov function.

• To analyze the stability, tracking performance, and parameter estimation convergence of the closed-loop system by explicitly examining the effects of state-triggered jumps on the evolution of Lyapunov function under uncertainties.

• To provide sufficient conditions based on these analyses under which the proposed control law guarantees the trajectory tracking performance in the presence of parametric and unmodeled uncertainties.

• To demonstrate the validity of the theoretical results through simulations of a fully actuated three-dimensional (3D) bipedal robot.

Some of the results from this paper have been reported in Ref. [37]. The new, substantial contributions of this paper include: (a) important details and discussions of the proposed controller design and stability analysis are added, including the full proof of Theorem 1 along with all the remarks in Secs. 3 and 4; (b) the convergence analysis of parameter estimation under no unmodeled uncertainties is added as a new section (Sec. 5), including Theorem 2 and its proof, which was missing in Ref. [37]; (c) comparative simulation results between four controllers are presented in Sec. 6 to illustrate the advantages of the proposed controller, whereas only the simulation results of the proposed controller were presented in Ref. [37]; and (d) simulation results of the proposed controller under parametric uncertainties are also added in Sec. 6, which were missing in Ref. [37].

The paper is structured as follows: Section 2 presents the full-order hybrid dynamic model of fully actuated bipedal walking robots. Section 3 introduces the continuous-phase adaptive robust control law. Section 4 derives the proposed sufficient closed-loop stability conditions, which can be used to guide the design of adaptive robust controllers for fully actuated hybrid systems. Section 5 explains the analysis of parameter estimation convergence in the absence of uncertain nonlinearities. Section 6 reports the simulation results on 3D bipedal robotic walking. Section 7 concludes the paper.

This section presents the hybrid model of bipedal walking dynamics, which serves as a basis for controller derivation.

Bipedal walking dynamics are inherently hybrid, involving continuous motion (e.g., foot swinging) as well as state-triggered jumps (e.g., sudden jumps in joint velocities when the swing foot strikes the walking surface), as illustrated in Fig. 1.

The robot is fully actuated when its number of degrees-of-freedom (DOFs), denoted as n, matches the number of independent actuators, denoted as m. As this study addresses the controller design for fully actuated walking, the equality m = n holds in the remaining derivation and analysis of the paper.

where $q∈Q$ and $u∈U$ are the vectors of joint positions and torques, respectively, $β∈Ωβ$ is the vector of model parameters with uncertain values, $M:Q×Ωβ→ℝn×n$ is the inertia matrix, $C:TQ×Ωβ→ℝn×n$ is the Coriolis matrix, $G:Q×Ωβ→ℝn$ is the gravitational term, $f̃:ℝ+×TQ→Ωf̃$ is the vector of unmodeled uncertainties (e.g., disturbances and joint friction), and $Bu∈ℝn×n$ is an input matrix. $Q⊂ℝn$ is the robot's configuration space when the support foot remains in full, static contact with the ground and the joint position limits are met. TQ is the tangent space of Q. $U⊂ℝm$ is the set of the robot's admissible joint torques. $Ωβ⊂ℝnp$ and $Ωf̃⊂ℝn$ are known bounded sets.

The continuous-phase dynamics in Eq. (1) has the following properties [38,39]:

for any$q∈Q$and$β∈Ωβ$, where$In×n∈ℝn×n$is an identity matrix.

Property 2. The Coriolis matrix$C$can be selected such that the matrix$Ṁ−2C$is skew-symmetric.

where$qr∈ℝn$is any reference vector.

State-Triggered Jumps. When a continuous phase ends, the robot's swing leg strikes the ground causing a discontinuity in the joint positions and velocities. The joint positions of the two legs experience a sudden jump because the swing and the support legs switch roles. The joint velocities may also experience a sudden jump due to the coordinate swap and the rigid-body impact associated with a swing-foot landing. These jumps cannot be directly controlled due to their infinitesimal duration.

where $⋆+$ and $⋆−$ represent the values of $⋆$ just before and after an impact, respectively. The derivation of $Δ$ is given in Ref. [40].

where z_sw is the height of the swing foot above the ground. Note that the switching surface S_q is not a function of the unknown parameters $β$ if $β$ are link masses.

Given that uncertainties, such as unmodeled joint frictions and unknown link masses, are usually bounded during real-world applications [41], it is assumed in this study that uncertainties are bounded:

This section presents an adaptive robust control approach that stabilizes the continuous phase of the uncertain hybrid system in Eq. (6). To guarantee the stability and tracking performance of the overall hybrid system, sufficient conditions are derived in Sec. 4 for guiding the gain tuning of the control law.

Without loss of generality, we choose the joint positions $q$ as the variables to be controlled. Note that a fully actuated system has full control authority over all of its n joint positions and thus can directly command all joints to track n desired trajectories $qd(t)$⁠. The tracking error is $e=q−qd$⁠.

The control objective is to derive a control law that stabilizes the uncertain hybrid system in Eq. (6) and guarantees a bounded final tracking error for achieving provably stable bipedal robotic walking.

We propose to achieve this control objective by extending the continuous adaptive robust controller (ARC) design methodology [42] from continuous systems to fully actuated hybrid systems with state-triggered jumps, because (a) only the continuous phase of the hybrid system in Eq. (6) can be directly driven but not the jumps and (b) ARC is effective in addressing uncertainties for continuous systems as reviewed in Sec. 1.

A continuous ARC consists of a nonlinear robust feedback term, a parameter adaptation law, a feedforward term, and a linear stabilizing feedback term (see Fig. 2). The robust feedback term mitigates the effects of unmodeled nonlinear uncertainties. The parameter adaptation law estimates the actual values of the model parameters. The feedforward term compensates for the nonlinear dynamics using estimated parameter values. The linear stabilizing feedback term stabilizes the continuous-phase system.

The robust feedback term of an ARC is crucial to guarantee the stability and transient tracking performance of a continuous system under unmodeled uncertainties. We choose sliding mode control (SMC) [42] as the robust feedback term because previous work [19] has validated its enhanced performance in rejecting uncertainties for legged locomotion.

where x_c and y_c are the state and the output of the dynamic compensator, respectively. $Ac⊂ℝnc×nc, Bc⊂ℝnc×nc, Cc⊂ℝn×nc$⁠, and $Dc⊂ℝn×n$ are matrices that define the dynamic compensator (n_c = n). These matrices are chosen such that (A_c, B_c, C_c, D_c) is observable and controllable.

where $q̇r:=q̇d−yc$⁠. Then, the goal of the SMC is to guarantee that the system state remains within the sliding mode, that is, $ξ$ remains zero.

where the definition of $hβ$ is given in Sec. 3.2. The ideal SMC is then $−hsξ||ξ||$⁠.

where $ℏ(−hsξ||ξ||)$ is a continuous approximation of the ideal SMC with an approximation error $ε(t)$ (⁠$ε(t)>0$ for any t > 0).

A valid choice of the continuous approximation $ℏ$ should satisfy the following two conditions [42]:

1. $ξTℏ(−hsξ||ξ||)≤0$

2. $hs||ξ||+ξTℏ(−hsξ||ξ||)≤ε(t)$

where $Ks∈ℝn×n$ is any symmetric positive definite matrix, and ε₁ and ε₂ are any positive scalars. $ϕh:=ϕhs(t,q,q̇,q̇r,q̈r)+ε1$⁠, where $ϕ(t)$ is any positive scalar function on t > 0. $c1=||ξ||−ϕhε2ϕh$⁠. Then, the approximation error ε is $ε(t)=(1+ε2)ϕh(t).$

which helps reduce the unpleasant transient response of $ξ$⁠.

Remark 2 (Response within sliding mode). The transfer function matrix from $ξ$ to $e$ is $(sIn×n+Gc(s))−1$⁠, where $Gc(s):=Cc(sInc×nc−Ac)−1Bc+Dc$⁠. By carefully assigning the poles of the matrix, the desired response within the sliding mode can be achieved.

The parameter adaption law estimates the actual values of the uncertain model parameters $β$⁠. By reducing the modeling error, it enhances the final tracking accuracy without relying to high-gain feedback control. With the estimated parameters, a feedforward term can be formed to compensate for the nonlinear continuous-phase dynamics, as shown in Fig. 2.

This parameter estimation law can cause an unbounded estimation error under unmodeled uncertainty $f̃$⁠. However, the robust feedback term is given in Sec. 3.1 demands the knowledge of the estimation error bound. The key to resolve this conflict is to impose a bound on the parameter estimation $β̂$⁠. To this end, different modifications of $β̂$ have been proposed [42]. We choose to utilize a smooth modification so as to allow the explicit analysis of the convergence of $β̂$ based on Lyapunov theory in Sec. 5.

Let $π(β̂)$ be a smooth projection of $β̂$ with sufficiently smooth, bounded derivatives and satisfying the conditions below

1. $π(β̂)=β̂$ if $β̂∈Ωβ$⁠.

2. $π(β̂)∈Ωβ̂:=[βmin−εβ βmax+εβ]$ if $β̂∈ℝnp$⁠, where $εβ∈ℝnp$ is a known vector of small positive numbers.

where the scalars π_i, $β̂i$⁠, β_imin, β_imax, and $εiβ$ are the ith elements of the vectors $π, β̂, βmin, βmax$⁠, and $εβ$⁠, respectively.

Remark 3 (Boundedness of $||Y||$⁠). For a general legged robot with all joints revolute, the boundedness of $||Y||$ can be guaranteed as follows. The norms $||M(q,β)||, ||C(q,q̇,β)||$⁠, and $||G(q,β)||$ are all bounded within the robot's joint limit (i.e., $(q,q̇)∈TQ$⁠) [39]. Suppose the reference trajectory q_d is planned as physically feasible (i.e., q_d, $q̇d$⁠, and $q̈d$ are all bounded). Then, the trajectory q_r, $q̇r$⁠, and $q̈r$ are also bounded within the robot's joint and actuation limits. Thus, $||Y(q,q̇,q̇r,q̈r)||$ is bounded.

where $Kξ∈ℝn×n$ is any symmetric positive definite matrix. The robust feedback term u_r and the adaptation law for $β̂$ are given in Eqs. (12) and (16), respectively.

With properly selected control gains of the ARC, the stability and tracking performance of the continuous phases of the uncertain hybrid systems in Eq. (6) can be ensured. Yet, the state-triggered jumps remain uncontrolled, and thus the ARC may not be effective in stabilizing the overall hybrid system. Therefore, in Sec. 4, we derive sufficient conditions under which the continuous ARC provably guarantees the stability and tracking performance for the overall hybrid system under uncertainties.

This section analyzes the stability of the uncertain hybrid systems with state-triggered jumps in Eq. (6) under the proposed continuous ARC. The outcome of the analysis is a set of sufficient conditions that can be used to guide the selection and tuning of control gains for guaranteeing the stability and tracking the performance of the closed-loop hybrid system.

The key of the proposed stability analysis is to explicitly analyze the effects of state-triggered jumps on system stability and tracking performance. Such an explicit analysis is necessary as the jumps are uncontrolled because of their infinitesimal duration. However, the analysis is complex mainly because the occurrence timing of these jumps is an implicit function of the state.

where $Az:=[AcBc−Cc−Dc]$⁠, $Bz:=[0In×n]$⁠, and $Cz:=[0In×n]$⁠.

with $||x||:=||z||2+||ξ||2.$

The expression of $Δz(t,x−,β)$ can be obtained from Eqs. (4), (9), and (14). The expression of S_x can be obtained from Eqs. (5) and (10). Note that $Δz$ is explicitly time-dependent because the reference trajectory q_d is explicitly time-dependent.

Theorem 1. The continuous-phase tracking control law in Eq. (20) locally stabilizes the equilibrium point of the hybrid system in Eq. (22) if its control gains are selected such that

(C5) The matrixA_z in Eq. (22) is Hurwitz.

(C6) The state$x$converges sufficiently fast during continuous phases.

for any $z(0)∈ℝnc+n$⁠.

holds for all $ξ(0)∈{ξ:||ξ||≤rξ}$⁠.

Because $ξ$ is reset to zero at the initial timing of each walking step, $Vξ$ is reset to zero. Hence, only the evolution of V_z across the jump $Δz$ needs to be analyzed.

To simplify notations, $⋆(Tk−)$ and $⋆(Tk+)$ are denoted as $⋆|k−$ and $⋆|k+$⁠, respectively, in the following analysis.

where $τk+1$ (⁠$k∈{1,2,…}$⁠) is the planned initial timing of the $(k+1) th$ walking step.

Suppose that the reference trajectories are generated to be compatible with the estimated reset map [29], we have $Δz(τk+1,0,β̂π)=0$⁠.

where the positive numbers L_T, L_x, and $Lβ$ are Lipschitz constants.

for any $x(0)∈Br2(0)$ [45]. Note that the Zeno behavior associated with hybrid dynamical systems is excluded from the system in this study because of our focus on the local stability of the desired trajectories.

where $LΔx:=2(LTkT+Lx)2$⁠.

where $ΔTk:=Tk+1−Tk$ is the duration of the kth walking step.

where $δk:=kt2LΔxkt1e−kt3ΔTk$ and $bk:=2kt2Lβ2βM2+kt2LΔxεmaxkt1kt3(1−e−kt3ΔTk).$

This inequality indicates the exponential convergence of $||x||$ to a bounded number during the overall hybrid process.▪

Remark 4 (Satisfaction of the proposed stability condition (C5)). The condition (C5) can be met by properly choosing the matrices A_c, B_c, C_c, and D_c of the dynamic compensator in Eq. (21) such that $Az=[AcBc−Cc−Dc]$ is Hurtwiz.

Remark 5 (Satisfaction of the proposed stability condition (C6)). By the proof of Theorem 1, the condition (C6) is met if the continuous-phase convergence rate $kt3$ satisfies the inequality in Eq. (51). The rate can be assigned by properly choosing the control gains as explained below. From Eqs. (26), (33), and (36), we have $kt3=min(λmin(Qz)λmax(Pz),2λmin(Kξ)kM)$⁠, where P_z can be obtained by solving the Lyapunov equation in Eq. (23) with Q_z chosen as a symmetric positive definite matrix (e.g., an identity matrix), $Kξ$ is a control gain, and k_M can be estimated based on joint limits and the expression of the inertia matrix $M$⁠.

Remark 6 (Boundedness of parameter estimation error). Although the convergence of the parameter estimation error $β̃π$ is not explicitly considered in the stability analysis, it is indeed bounded at the steady-state because the parameter estimation by definition is bounded, as indicated by the conditions (C3) and (C4). In Sec. 5, we will show that the parameter estimation error converges to zero in the absence of unmodeled uncertainties, i.e., $f̃=0$⁠, when certain conditions on the reference trajectory are met.

This section derives the sufficient conditions under which the continuous ARC exponentially eliminates the parameter estimation error in the absence of the unmodeled uncertainties (i.e., $f̃=0$⁠).

Theorem 2. Let the conditions (C5) and (C6) hold along with the following condition:

for any$t≥t0$, where T, t₀, and ε_d are some positive numbers.

Then, in the absence of the unmodeled nonlinearity, i.e., $f̃=0$, the proposed continuous-phase control law in Eq. (20) locally asymptotically stabilizes the equilibrium point of the hybrid system in Eq. (22) and drives the parameter estimation to the true value.

where π_i and β_i (⁠$i∈{1,2,…,np}$⁠) are the ith element of the function $π$ and the vector $β$⁠, respectively.

The norm of $||x̃||$ is defined as $||x̃||:=||x||2+||β̃||2.$

Recall T_k (⁠$k∈{1,2,…}$⁠) is the initial moment of the kth walking step. Let $(x¯,β̃¯)$ be the solution to the continuous-phase subsystem of Eq. (60) on $t>Tk$ with initial condition $[x¯T(Tk+),β̃¯T(Tk+)]T=x̃(Tk+)$⁠. Then, $[x¯T,β̃¯T]T=x̃$ on $t∈(Tk,Tk+1]$⁠.

The inequality in Eq. (62) implies that $x¯∈L2∩L∞$ and $β̃¯∈L∞$⁠. From the continuous-phase subsystem of Eq. (60), $x¯̇∈L∞$⁠. Thus, $x¯$ is uniformly continuous. Also, for $x¯$ and $β̃¯$⁠, all terms of the continuous-phase subsystem are uniformly continuous, and thus $x¯̇$ is uniformly continuous. Then, by Barbalat's lemma [4], $x¯→0$ locally exponentially on $t>Tk$⁠. Accordingly, $x$ locally exponentially diminishes on $t∈(Tk,Tk+1]$⁠. Thus, from the continuous-phase subsystem of Eq. (60), $Yβ¯π$ locally exponentially approaches zero on $t∈(Tk,Tk+1]$⁠. If the reference trajectories satisfy the persistent excitation condition (i.e., the condition (C7)), then the parameter estimation error $β̃π$ exponentially diminishes on $t∈(Tk,Tk+1]$ [4].

for any $x̃(0)∈Br̃t1(0)$ and $t∈(Tk,Tk+1], k∈{1,2,…}$⁠; that is, the augmented state $x̃$⁠, including the parameter estimation error $β̃π$⁠, exponentially diminishes during continuous phases.

By the definition of the Lyapunov function $Vβ$ in Eq. (56), the discrete event of state-triggered jumps does not cause a sudden change in $Vβ$⁠. Therefore, $β̃+=β̃−$⁠, and accordingly $β̃π+=β̃π−$⁠.