Global-Position Tracking Control of a Fully Actuated NAO BipedalWalking Robot

Yuan Gao and Yan Gu

Abstract— The ability to reliably track various plannedpaths with specific timing, which is termed as the global-position tracking capability in this paper, is essential to highlyversatile bipedal robotic walking. To provably guarantee global-position tracking performance for fully actuated bipedal robots,this study proposes a time-dependent state-feedback controlstrategy based on a) a full-order model of the hybrid, nonlinear,floating-based bipedal walking dynamics, b) time-dependentinput-output linearization, and c) Lyapunov-based stabilityanalysis. This study also provides general guidelines for adapt-ing the proposed control strategy to walking experiments inthe presence of a common hardware limitation of bipedalrobots, which will help to bridge the gap between theoryand experiment in bipedal walking control. Simulations andexperiments on a NAO bipedal robot were performed todemonstrate the effectiveness of the proposed walking controlstrategy.

I. INTRODUCTION

Versatility of bipedal robotic walking refers to a robot’sability to purposefully perform different motions [1]. Animportant aspect of enabling versatile walking is to real-ize reliable tracking of various planned trajectories of arobot’s global position (i.e., a robot’s based position in theworld coordinate frame). Global-position tracking controlhas been extensively investigated and successfully imple-mented on various bipedal robots through the Zero-Moment-Point (ZMP) approach [2]–[4]. Despite the high versatility,the ZMP approach can only realize relatively low agility andenergy efficiency and cannot provably guarantee the stabilityof the closed-loop walking control system [5], [6].

To produce provably stable, agile, and efficient walk-ing, the Hybrid-Zero-Dynamics (HZD) approach has beenintroduced based on full-order dynamic modeling, input-output linearizing state feedback control, and orbital sta-bilization [5], [7]. The HZD framework has realized agilebipedal gait for fully actuated [8], underactuated [9], [10],and multi-domain walking [11], as well as running [12]and robot-assisted human walking [13]. Recently, velocityregulation [4] and learning-based gait library design [14]have been incorporated into the HZD framework to enhancewalking versatility beyond periodic walking.

Inspired by the HZD approach, we have introduced time-dependent feedback control to achieve exponential global-position tracking for enhancing walking versatility of fullyactuated planar bipedal robots [15], [16]. To our best knowl-edge, it is the first time that exponential global-position

Yuan Gao and Yan Gu are with the Department of Mechanical Engi-neering, University of Massachusetts Lowell, Lowell, MA 01854, U.S.A.Emails: yuan gao@student.uml.edu, yan gu@uml.edu.

tracking has been provably produced for bipedal roboticwalking. This control strategy has been theoretically ex-tended to a 3-D bipedal robot with 9 DOFs and validatedthrough simulations [17].

In this study, our previous global-position tracking controlwill be further extended to a general three-dimensional (3-D) bipedal robot model and experimentally evaluated ona NAO bipedal humanoid robot [18]. This paper has twomain contributions. The first main contribution is a setof formally constructed sufficient conditions that provablyguarantee the exponential global-position tracking perfor-mance of the proposed control strategy. The second maincontribution is experimental guidelines that can be used toadapt the proposed global-position tracking controller to arepresentative bipedal robot platform with a common controlimplementation limitation, which will help to bridge the gapbetween theory and experiment in bipedal walking control.

This paper is organized as follows. Section II presentsthe hybrid full-order model of bipedal walking dynamics.The proposed global-position tracking control is explainedin Section III, including the derivation of trajectory trackingerrors, the construction of impact invariance conditions,and the analysis of closed-loop system stability. Section IVintroduces simulation results for control design validationand hardware implementation preparation. Details of exper-imental validation are given in Section V, including generalguidelines of controller adaptation for addressing a commonhardware limitation of bipedal robots.

II. HYBRID FLOATING-BASED DYNAMICS OFBIPEDAL ROBOTIC WALKING

This section presents a full-order model of bipedal walk-ing dynamics, which provides a faithful description of thehybrid, nonlinear dynamic behaviors of all degrees of free-dom involved in walking and will be utilized in the designof the proposed model-based feedback control in Section III.

Because bipedal robotic walking is inherently hybridinvolving both continuous dynamics (e.g., foot swinging)and discrete behaviors (e.g., foot landings), it is natural tomodel a bipedal walking process as a hybrid dynamicalsystem. The following modeling assumptions are consideredin this study: a) The walking surface is even and horizontal;b) During a single-support phase, the robot’s support footremains full, static contact with the walking surface; c) Thelanding event is modeled as an impact between rigid bodies,and a double-support phase is instantaneous [5].

The generalized coordinates of a floating-based bipedalrobot can be denoted as

2019 American Control Conference (ACC)Philadelphia, PA, USA, July 10-12, 2019

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q =[pT

b ,γγγTb ,q1, ...,qn

]T ∈Q, (1)

where Q ⊂ Rn+6 is the configuration space, pb :=[xb,yb,zb]

T ∈ R3 represents the floating-base position withrespect to (w.r.t.) the world coordinate frame, which isused to represent the robot’s global position in this study,γγγb := [φb,θb,ψb]

T represents the vector of pitch, roll, andyaw angles of the floating base w.r.t. the world coordinateframe, and q1, ...,qn represent the robot’s joint angles.

The NAO robot (Fig. 1) has 24 joints (i.e., n = 24). Dueto the mechanical coupling between the two hip joints andthe holonomic constraint that the support foot remains static,full contact with the walking surface, the robot has 23 DOFs.Since it also has 23 independent joint actuators, the robot isfully actuated.

Fig. 1. An illustration of the revolute joints of a NAO bipedal humanoidrobot. The coordinate system of the robot’s floating base is located at thecenter of the chest.

A. Continuous-Phase Dynamics

With Lagrange’s method, bipedal walking dynamics dur-ing a continuous phase can be obtained as [5]:

M(q)q+ c(q, q) = Bu+JT F, (2)

where M(q) : Q → R(n+6)×(n+6) is the inertia matrix,c(q, q) : T Q → R(n+6)×(n+6) is the sum of Coriolis, cen-trifugal, and gravitational terms, B∈R(n+6)×m (m = 23) is aconstant matrix, and u∈Rm is the joint torque vector. Here,F ∈ R7 is the vector of the ground-reaction force appliedat the support foot and the internal force between the twomechanically coupled hip joints, and J is the associatedJacobian matrix. The holonomic constraints that the NAOrobot is subject to can be described as [19]:

Jq+ Jq = 0. (3)

From Eqs. (2) and (3), the continuous-phase dynamics canbe compactly expressed as:

M(q)q+ c(q, q) = B(q)u, (4)

where the derivations of c(q, q) and B(q) are shown next.From Eq. (2), one has

q = M−1(−c+Bu+JT F). (5)

Substituting Eq. (5) into Eqs. (3) yields:

F = (JM−1JT)−1(JM−1c− Jq−JM−1Bu). (6)

Substituting Eq. (6) into Eq. (2), the expressions of c(q, q)and B(q) can be obtained as:

c(q, q) = c−JT(JM−1JT)−1(JM−1c− Jq);B(q) = B−JT(JM−1JT)−1JM−1B.

(7)

B. Switching Surface

A switching event that connects a continuous single-support phase with a subsequent instantaneous double-support phase can be described by a switching surface:

Sq(q, q) := {(q, q) ∈ T Q : zsw(q) = 0, zsw(q, q)< 0}, (8)

where zsw : Q→ R represents the swing-foot height abovethe walking surface.

C. Reset Map

When the swing foot hits the walking surface, an instanta-neous rigid-body impact occurs. The robot’s joint positionswill remain continuous upon the impact. But the jointvelocities will experience a sudden jump, which can bedescribed by the following reset map:

q+ = Rq(q−)q−, (9)

where q− and q+ denote the values of q right before andafter an impact. Here, Rq :Q→R(n+6)×(n+6) can be obtainedby solving[

M(q−) −JT (q−)JT (q−) 07×7

][q+

δF

]=

[M(q−)q−

07×1

],

where δF is a vector of the impulsive ground-reaction forceand hip coupling force, and 07×7 is a 7×7 zero matrix.

III. TIME-DEPENDENT GLOBAL-POSITION TRACKINGCONTROL

The primary control objective of this study is to achievesatisfactory tracking of the desired global-position trajectoryalong the desired global path on the walking surface for afully actuated NAO bipedal walking robot. To realize this ob-jective, a time-dependent, model-based feedback controlleris synthesized based on the dynamic model presented inSection II, input-output linearization, and Lyapunov-basedstability analysis.

A. Trajectory Tracking Errors

Let hc(q) : Q→Qc⊂Rm denote the variables of interest tobe controlled, which include: a) the robot’s global position,represented by the floating-base position, (xb,yb,zb); b) thefloating-base roll angle, φb, and pitch angle, θb; c) the swing-foot position and orientation, denoted as psw(q) ∈ R3 andγγγsw(q) ∈ R3, respectively; d) the upper-body joint angles,qupper ∈ Rnupper (nupper = 12 for the NAO robot in Fig. 1).Accordingly, hc(q) is defined as:

hc(q) :=[xb,yb,zb,φb,θb,pT

sw(q),γTsw(q),qT

upper]T

.

Let hd(t,θ(q)) :R+×R→Rm denote the desired positiontrajectories of hc(q), which are encoded by time t anda configuration-based variable θ : Q → R that increases

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monotonically within a step and represents how far a stephas progressed. The desired global path on the walkingsurface, denoted as Γd , is chosen as a straight-line path inthis study. For simplicity and without loss of generality, Γdis specified as the XW -axis of the world coordinate frameΣXW YW ZW , i.e., Γd := {(XW ,YW ) ∈ R2 : YW = 0}. The desiredposition trajectory along Γd is denoted as xd(t), which isan explicit function of time t. In previous work under theHZD framework [20], the desired trajectories are typicallydesigned as state-based alone instead of time-dependent.Here, we define the desired global-position trajectory asexplicitly time-dependent because desired global-positiontrajectories are often expressed as time functions in practicalrobotic applications.

With Γd and xd(t) defined, the desired trajectoryhd(t,θ(q)) is defined as

hd(t,θ(q)) := [xd(t),φφφ T (θ(q))]T , (10)

where φφφ(θ) :R→Rm−1 represents the desired trajectories ofyb, zb, φb, θb, psw, and γγγsw. We can use Bezier polynomialsto parameterize the desired function φφφ(θ) as [20]

φφφ(s) :=M

∑k=0

αααkM!

k!(M− k)!sk(1− s)M−k, (11)

where s := θ−θ+

θ−−θ+ is the normalized θ , αααk ∈ Rm−1 (k =0,1, ...,M) is a coefficient vector to be optimized in Sec-tion IV, θ− and θ+ denote the values of θ right before andafter an impact associated with the desired gait, and M isthe order of the Bezier polynomials.

As the control objectives of this study is to track thedesired position trajectory xd(t) while following the desiredpath Γd , the desired trajectory of yb (i.e., the first elementof φφφ(θ)) will be planned as a trajectory that varies aroundzero within a small bound. This desired trajectory design hasthree advantages as compared with our previous study [17]that focuses on driving yb strictly to zero: a) The holonomicconstraint that requires a secured foot-ground contact canbe more easily met in motion planning, thus reducing thecomputational load of planning; b) The resulting gait will bemore natural-looking; c) The robot will still be considered asfollowing a straight-line path by tracking a small-magnitudesine-wave path about the straight line.

With the trajectory tracking errors compactly expressed as

h(t,q) := hc(q)−hd(t,θ(q)), (12)

the control objective of this study becomes to drive hexponentially to zero.

B. Impact Invariance Condition

Since bipedal walking is a hybrid dynamical process,the desired trajectories hd(t,q) should respect the resetmap in Eq. (9). To meet this requirement, a set of time-dependent impact invariance conditions can be derived basedon previous state-based impact invariance construction [20].

Let TK (K ∈ Z+) denote the Kth actual impact moment,and let T0 denote the initial moment of a walking process.

Let τK denote the Kth desired impact moment assumingy = 0 and y = 0 for all t > TK−1. The time-dependent im-pact invariance conditions can be mathematically expressedas [17]:

hc(q(τ+K )) = hd(τ+K ,q(τ+K ));

hc(q(τ+K ),Rq(q(τ−K ))q(τ−K )) = hd(τ+K ,q(τ+K ),Rq(q(τ−K )q(τ−K ))),

(13)where τ

−K and τ

+K denote the moments right before and after

τK , respectively.

C. Model-based State Feedback Control

To simplify the nonlinear continuous-phase dynamics ofbipedal robotic walking, input-output linearization is utilizedto synthesize a feedback controller that realizes exponentialtracking of the desired trajectories.

The output functions are defined as the trajectory trackingerrors:

y = h(t,q). (14)

With the following input-output linearizing control law [21]

u = ( ∂h∂q M−1B)−1[( ∂h

∂q )M−1c+v− ∂ 2h

∂ t2 − ∂

∂q (∂h∂q q)q], (15)

the continuous-phase dynamics in Eq. (4) become y = v. Bychoosing

v =−Kpy−Kd y,

where Kp ∈ Rm×m and Kd ∈ Rm×m are positive definitediagonal matrices, one has y =−Kd y−Kpy.

Although the proposed control law in Eq. (15) can stabi-lize the continuous-phase dynamics with properly chosen Kpand Kd , the reset map in Eq. (9) remains uncontrolled. Thus,the closed-loop stability will be formally analyzed next.

D. Closed-Loop Stability Analysis

The stability analysis of the hybrid, nonlinear, time-varying closed-loop system will be discussed as an extensionof our previous work [17] in this subsection.

From the dynamic model presented in Section II and theproposed control law in Eq. (15), the closed-loop dynamicscan be compactly written as:x = Ax :=

[0m×m Im×m

−Kp −Kd

]x if (t,x−) /∈ S(t,x);

x+ = ∆(t,x−) if (t,x−) ∈ S(t,x),(16)

where x :=[yT , yT ]T ∈X ⊂R2m, 0m×m and Im×m are m×m

zero and identity matrices, respectively, and the expressionsof S : R+×X →R2m−1 and ∆ : R+×X →X can be readilyobtained from Eqs. (8), (9), and (14). Note that the reset mapand the switching surface are both explicitly time-dependentdue to the explicit time dependence of y.

For notational simplicity, ?(T−k−1) := ?|−k−1 and ?(T+k−1) :=

?|+k−1 will be used in the following stability analysis. Due tospace limitations, a sketch of the analysis will be presented.

According to Lyapunov-based stability analysis of hybridsystems [22], the hybrid time-varying system in Eq. (16)is locally exponentially stable if there exists a Lyapunov

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function candidate V (x) and a positive number r such thatfor any x(T0) ∈ Br(0) := {x ∈ X : ‖x‖ ≤ r}:

(A) V (x) exponentially decreases during each continuousphase;

(B) {V |+1 ,V |+2 ,V |

+3 ...} is a strictly decreasing sequence.

The condition (A) can be met by properly selecting thecontrol gains Kp and Kd in Eq. (16). Specifically, if thecontrol gains are chosen such that the matrix A in Eq. (16)is Hurwitz, then there exists a Lyapunov function candidateV (x) and positive numbers c1, c2, and c3 such that

c1||x||2 ≤V (x)≤ c2||x||2

andV (x)≤−c3V (x)

hold for all x ∈ X during the continuous phase ofthe Kth step, ∀K ∈ Z+ [21]. Then, one has V |−K ≤e−c3(TK+1−TK)V |+K−1.

Before analyzing the convergence of {V |+1 ,V |+2 ,V |

+3 ...},

we first analyze the reset map in Eq. (16). From Eq. (16),one has

‖x|+K‖=‖∆(T−

K ,x f |−K )‖≤‖∆(T−K ,x|−K )−∆(τ−K ,x|−K )‖+‖∆(τ−K ,x|−K )−∆(τ−K ,0)‖+‖∆(τ−K ,0)‖.

(17)

Suppose that the desired trajectories hd(t,θ(q)) providedby a high-level planner is continuously differentiable in t.Then, the reset map ∆ is continuously differentiable in t.It can also be proved that ∆ is continuously differentiablein x. Thus, there exists a positive number r1 and Lipschitzconstants L∆t and L∆x such that [15]

‖∆(T−K ,x|−K )−∆(τ−K ,x|−K )‖ ≤ L∆t‖Tk− τk‖ (18)

and

‖∆(τ−K ,x|−K )−∆(τ−K ,0)‖ ≤ L∆x‖x|−K‖ (19)

hold for any x(T0) ∈ Br1(0).Based on our previous study [15], it can be proved that

there exists a positive number r2 and a Lipschitz constantLTx such that

|TK− τK | ≤ LTx‖x(τK ;T+K−1,x|

+K−1)‖ (20)

for any x(T0)∈Br2(0). Here, x(t; t0,λ0) represents a solutionof ˙x = Ax with the initial condition x(t0) = λ0, ∀t > t0.

If the desired trajectories satisfy the impact invariancecondition in Eq. (13), then ‖∆(τ−K ,0)‖ = 0 always holds,which will greatly simplify the stability analysis as follows.

Equations (17) - (20) indicate that there exists a positivenumber Ld such that

‖x|+K‖ ≤ Ld(‖x(τK ;T+K−1,x|

−K−1)‖+‖x|

−K‖), (21)

where, from the above analysis, one has

‖x|−K‖ ≤√

c2

c1e−

c32c2

(TK−TK−1)‖x|+K−1‖,

and

‖x(τK ;T+K−1,x|

−K−1)‖ ≤

√c2

c1e−

c32c2

(τk−TK−1)‖x|+K−1‖.

Hence, the equations above yield

‖x|+K‖ ≤ Ld

√c2c1

e− c3

2c2(τK−TK−1)(1+ e

− c32c2

(TK−τK))‖x|+K−1‖.(22)

Equation (22) shows that the convergence rate of thesequence {x|+1 ,x|

+2 , ...} is c3

2c2, which is determined by the

control gains in Eq. (16). Hence, {V |+1 ,V |+2 ,V |

+3 ...} will be

a strictly decreasing sequence if the control gains are chosensuch that the convergence rate c3

2c2is sufficiently large.

In summary, the closed-loop hybrid system is locallyexponentially stable if the control gains in Eq. (16) arechosen such that the matrix A in Eq. (16) is Hurwitz and thatthe continuous-phase convergence rate is sufficiently high.

IV. SIMULATIONS

This section presents the MATLAB [23] and Webots [24]simulation results on a NAO bipedal humanoid robot todemonstrate the effectiveness of the proposed control strat-egy in realizing exponential global-position tracking as wellas to provide preliminary insights into hardware implemen-tation.

A. Trajectory Generation

In both simulations and experiments, the same desiredtrajectories hd(t,θ(q)) will be used for convenience ofcomparison. The desired walking path Γd and the desiredposition trajectory xd(t) along the path are assumed to beprovided by a high-level task planner because they bothdefine critical desired behaviors of a bipedal robot whenperforming high-level tasks. Without loss of generality, thedesired straight-line path is chosen as the XW -axis of theworld coordinate frame, and the desired position trajectoryalong Γd is chosen as xd(t) = 0.04t−0.01 (m).

The desired Bezier polynomials φφφ(θ) are planned throughoptimization-based motion planning. The Bezier polynomi-als are chosen as 6th-order (i.e., M = 6), and their coefficientsαααk (k = 0,1, ...,M) are optimized to satisfy the followingconstraints: a) Impact invariance conditions in Eq. (13); b)Joint position and velocity limits; c) Holonomic constraintsin Eq. (3); d) Desired gait properties (e.g., step length andmaximum swing-foot elevation).

B. MATLAB Simulations

In MATLAB simulations, the proposed time-dependentinput-output linearizing control strategy in Eq. (15) is imple-mented using the hybrid dynamic model presented in Sec-tion II. The control gains are set as Kp =KpIm×m (Kp = 484)and Kd = KdIm×m (Kd = 44). This choice of control gainswill guarantee the stability of the linearized continuous-phase dynamics in Eq. (16).

Figure 2 shows the global-position tracking results, whichindicate that the simulated NAO robot exponentially con-verges to the desired global path Γd and the desired position

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trajectory xd(t) along the path despite the large initialtracking error.

Fig. 2. Global-position tracking results in MATLAB and Webotssimulations. The initial global-position tracking error is set as xbm(0)−xd(0) = xbw(0)− xd(0) = −1 cm, which is approximately 1

3 of the steplength. (a) Tracking results of xd(t). xbm(t) and xbw(t) are the actualglobal-position trajectories along Γd obtained in MATLAB and Webotssimulations, respectively. Both of them demonstrate satisfactory tracking ofthe desired trajectory xd(t). (b) Tracking results of Γd . left foot (M) and rightfoot (M) are the actual support-foot placements in MATLAB simulations,and left foot (W) and right foot (W) are those in Webots simulations. Bothof them indicate satisfactory tracking of the desired path Γd , which is theXW -axis of the world coordinate frame.

C. Webots Simulations

Webots is a robot simulator that allows accurate simu-lation of dynamical, physical, and graphical properties ofrigid-body objects. In this study, Webots simulations areconducted to help us gain preliminary insights into hardwareimplementation on the NAO robot.

The NAO robot has a major hardware limitation, that is,the robot does not allow users to send torque commands tojoint actuators. Since the control input u in Eq. (15) is atorque signal, this limitation prevents the direct implemen-tation of the proposed control law in Eq. (15).

To enable reliable global-position tracking for the NAOrobot despite this hardware limitation, we adapt the proposedwalking control strategy for the NAO robot and use Webotssimulations to evaluate the adapted strategy on a robotmodel that emulates the NAO robot including its hardwarelimitation in actuator accessibility.

The proposed controller adaptation for the time step t isdescribed as follows:• At the time step t, the phase variable θ := xb(t)−xst is

computed based on xb(t) and the current support-footposition along the XW -axis, denoted as xst ;

• The individual joint position qd(t) is computed bysolving hc(qd(t))−hd(t,θ) = 0;

• qd(t) is then fed into NAO’s joint controller in Webots,which emulates the actual robot’s default controller anddrives the current configuration to the desired one.

To avoid drastic initial motion of the robot under large initialtracking errors, it has been found that θ can be replaced

by θd := xd(t)− xst in the above algorithm until the robotconverges sufficiently close to the desired trajectories.

Figure 2 shows the global-trajectory tracking performancein Webots simulations together with that in MATLAB sim-ulations. One can notice that the green line (xbm, MAT-LAB) exponentially converges to the desired trajectory xd(t)and that the blue line (xbw, Webots) converges to a smallneighborhood around xd(t). This discrepancy in trackingperformance is primarily due to: a) The impact models inWebots and MATLAB are different; b) The implementedcontrollers are different. However, the satisfactory trackingresults in Webots indicate the effectiveness of the proposedcontroller adaptation strategy in the presence of a NAOrobot’s hardware limitations.

V. EXPERIMENTS

In this section, we will present the experimental set-up, hardware implementation details, and the experimentalresults.

The experimental set-up for testing the global-positiontracking performance of the proposed walking strategy isshown in Fig. 3. With this experimental set-up, the robot’sgeneral coordinates q in Eq. (1) can be readily obtained forcontroller implementation.

Fig. 3. Experimental set-up for testing the global-position trackingperformance on the NAO bipedal robot. The labels refer to: (1) A Logitech4K PRO WEBCAM. (2) The world coordinate frame. (3) The referencepoints for perspective transformation. (4) AprilTag attached on the NAO’sfeet, which is utilized to determine the robot’s floating-base position, pb, inthe world coordinate frame. (5) A NAO robot. (6) The desired global path.

Due to the NAO robot’s limited actuator accessibility, itis necessary to adapt the proposed walking control strategyin Eq. (15) for hardware implementation. Since the controlstrategy adapted for Webots simulations can realize satisfac-tory global-position tracking on a robot model that emulatesa NAO robot including its hardware limitation, it will beexperimentally implemented.

Experimental results are illustrated in Fig. 4. It clearlyshows that the proposed control strategy drives the robot’sglobal position to reliably track both the desired positiontrajectory xd(t) and the desired global-path Γd .

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Figures 2 and 4 also demonstrate that steady-state track-ing accuracy is guaranteed even across rigid-body land-ing impacts in both simulations and experiments. This isbecause the impact invariance conditions are incorporatedinto motion planning such that the desired trajectory hdrespects the impact dynamics as explained in Section IV-A. Accordingly, the corresponding joint position qd , whichis used as the desired joint position in Webots simulationsand experiments, will also agree with the impact dynamics.

Fig. 4. Experimental results of global-position tracking with the proposedwalking control Strategy. The initial global-position tracking error is set asxb(0)− xd(0) = −1 cm, which is approximately 1

3 of the step length. (a)Results of tracking xd(t). As clearly shown in the plot, the proposed walkingcontrol strategy guarantees accurate tracking of the desired trajectory xd(t).(b) Results of tracking Γd . The results demonstrate that the proposedwalking control strategy guarantees accurate tracking of the desired globalpath Γd .

VI. CONCLUSIONS

In this paper, we have theoretically synthesized a time-dependent global-position tracking controller for 3-D fullyactuated bipedal robotic walking and experimentally vali-dated it on a NAO bipedal humanoid robot. The proposedcontroller was theoretically established through full-orderdynamic modeling, model-based state feedback control, andLyapunov-based stability analysis, which can achieve reli-able global-position tracking for enhancing walking versa-tility. Due to limited actuator accessibility of the NAO robot,necessary adaptation of the proposed walking controllerwas made for hardware implementation. Both simulationand experimental results validated the effectiveness of theproposed walking strategy. The results of this paper can beused to guide hardware implementation of global-positiontracking control on legged robots with the common hardwarelimitation of actuator accessibility, thus helping to bridge thegap between theory and experiment.

REFERENCES

[1] D. G. E. Hobbelen and M. Wisse, “Limit cycle walking,” in HumanoidRobots, Human-like Machines, 2007.

[2] M. Vukobratovic and B. Borovac, “Zero-Moment Point: thirty fiveyears of its life,” International Journal of Humanoid Robotics, vol. 1,no. 01, pp. 157–173, 2004.

[3] S. Kajita, M. Morisawa, K. Miura, S. Nakaoka, K. Harada, K. Kaneko,F. Kanehiro, and K. Yokoi, “Biped walking stabilization based onlinear inverted pendulum tracking,” in Proc. of IEEE/RSJ InternationalConference on Intelligent Robots and Systems, pp. 4489–4496, 2010.

[4] M. J. Powell, A. Hereid, and A. D. Ames, “Speed regulation in 3Drobotic walking through motion transitions between human-inspiredpartial hybrid zero dynamics,” in Proc. of IEEE International Confer-ence on Robotics and Automation, pp. 4803–4810, 2013.

[5] J. W. Grizzle, G. Abba, and F. Plestan, “Asymptotically stable walkingfor biped robots: Analysis via systems with impulse effects,” IEEETransactions on Automatic Control, vol. 46, no. 1, pp. 51–64, 2001.

[6] R. Tedrake, S. Kuindersma, R. Deits, and K. Miura, “A closed-formsolution for real-time ZMP gait generation and feedback stabilization,”in Proc. of IEEE-RAS International Conference on Humanoid Robots,pp. 936–940, 2015.

[7] E. R. Westervelt, J. W. Grizzle, and D. E. Koditschek, “Hybrid zerodynamics of planar biped walkers,” IEEE Transactions on AutomaticControl, vol. 48, no. 1, pp. 42–56, 2003.

[8] R. W. Sinnet, M. J. Powell, R. P. Shah, and A. D. Ames, “A human-inspired hybrid control approach to bipedal robotic walking,” Proc.of IFAC, vol. 44, no. 1, pp. 6904–6911, 2011.

[9] A. Hereid, E. A. Cousineau, C. M. Hubicki, and A. D. Ames,“3D dynamic walking with underactuated humanoid robots: A directcollocation framework for optimizing hybrid zero dynamics,” inProc. of IEEE International Conference on Robotics and Automation,pp. 1447–1454, 2016.

[10] M. J. Powell, W.-L. Ma, E. R. Ambrose, and A. D. Ames, “Mechanics-based design of underactuated robotic walking gaits: Initial experi-mental realization,” in Proc. of IEEE-RAS International Conferenceon Humanoid Robots, pp. 981–986, 2016.

[11] A. Hereid, S. Kolathaya, M. S. Jones, J. Van Why, J. W. Hurst,and A. D. Ames, “Dynamic multi-domain bipedal walking withATRIAS through SLIP based human-inspired control,” in Proc. ofACM International Conference on Hybrid Systems: Computation andControl, pp. 263–272, 2014.

[12] K. Sreenath, H.-W. Park, I. Poulakakis, and J. W. Grizzle, “Embeddingactive force control within the compliant hybrid zero dynamics toachieve stable, fast running on MABEL,” The International Journalof Robotics Research, vol. 32, no. 3, pp. 324–345, 2013.

[13] A. E. Martin and R. D. Gregg, “Stable, robust hybrid zero dynamicscontrol of powered lower-limb prostheses,” IEEE Transactions onAutomatic Control, vol. 62, no. 8, pp. 3930–3942, 2017.

[14] X. Da, R. Hartley, and J. W. Grizzle, “Supervised learning for stabiliz-ing underactuated bipedal robot locomotion, with outdoor experimentson the wave field,” in Proc. of IEEE International Conference onRobotics and Automation, pp. 3476–3483, 2017.

[15] Y. Gu, B. Yao, and C. G. Lee, “Bipedal gait recharacterizationand walking encoding generalization for stable dynamic walking,” inProc. of IEEE International Conference on Robotics and Automation,pp. 1788–1793, 2016.

[16] Y. Gu, B. Yao, and C. G. Lee, “Exponential stabilization of fullyactuated planar bipedal robotic walking with global position trackingcapabilities,” Journal of Dynamic Systems, Measurement, and Control,vol. 140, no. 5, p. 051008, 2018.

[17] Y. Gu, B. Yao, and C. G. Lee, “Straight-line contouring control offully actuated 3-D bipedal robotic walking,” in Proc. of AmericanControl Conference, pp. 2108–2113, 2018.

[18] “Softbank robotics.” https://www.softbankrobotics.com. Accessed:2018-09-20.

[19] D. Kim, J. Lee, O. Campbell, H. Hwang, and L. Sentis,“Computationally-robust and efficient prioritized whole-body con-troller with contact constraints,” arXiv preprint arXiv:1807.01222,2018.

[20] E. R. Westervelt, C. Chevallereau, J. H. Choi, B. Morris, and J. W.Grizzle, Feedback control of dynamic bipedal robot locomotion. CRCpress, 2007.

[21] H. K. Khalil, Noninear systems. No. 5, Prentice Hall, 1996.[22] M. S. Branicky, “Multiple lyapunov functions and other analysis tools

for switched and hybrid systems,” IEEE Transactions on AutomaticControl, vol. 43, no. 4, pp. 475–482, 1998.

[23] “The MathWorks, Inc.” https://www.mathworks.com/. Accessed:2018-09-20.

[24] “Cyberbotics ltd.” https://cyberbotics.com/. Accessed: 2018-09-20.

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