Time-Varying ALIP Model and Robust Foot-Placement Control for Underactuated Bipedal Robot Walking on a Swaying Rigid Surface Yuan Gao1 Yukai Gong2 Victor Paredes3 Ayonga Hereid3 Yan Gu4

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Time-Varying ALIP Model and Robust Foot-Placement Control for

Underactuated Bipedal Robot Walking on a Swaying Rigid Surface

Yuan Gao1, Yukai Gong2, Victor Paredes3, Ayonga Hereid3, Yan Gu4

Abstract— Controller design for bipedal walking on dynamic

rigid surfaces (DRSes), which are rigid surfaces moving in

the inertial frame (e.g., ships and airplanes), remains largely

uninvestigated. This paper introduces a hierarchical control

approach that achieves stable underactuated bipedal robot

walking on a horizontally oscillating DRS. The highest layer

of our approach is a real-time motion planner that generates

desired global behaviors (i.e., the center of mass trajectories

and footstep locations) by stabilizing a reduced-order robot

model. One key novelty of this layer is the derivation of the

reduced-order model by analytically extending the angular

momentum based linear inverted pendulum (ALIP) model from

stationary to horizontally moving surfaces. The other novelty

is the development of a discrete-time foot-placement controller

that exponentially stabilizes the hybrid, linear, time-varying

ALIP model. The middle layer of the proposed approach is

a walking pattern generator that translates the desired global

behaviors into the robot’s full-body reference trajectories for

all directly actuated degrees of freedom. The lowest layer

is an input-output linearizing controller that exponentially

tracks those full-body reference trajectories based on the full-

order, hybrid, nonlinear robot dynamics. Simulations of planar

underactuated bipedal walking on a swaying DRS conﬁrm that

the proposed framework ensures the walking stability under

difference DRS motions and gait types.

I. INTRODUCTION

Bipedal robots can aid in various critical real-world ap-

plications such as search and rescue, emergency response,

and warehouse management. Those applications may de-

mand robots to navigate on nonstationary walking platforms,

such as shipboard ﬁreﬁghting, inspection, and maintenance.

Enabling stable legged locomotion on a nonstationary rigid

platform, which we call a dynamic rigid surface (DRS) [1],

is a fundamentally challenging control problem due to the

high complexity of the robot dynamics that is nonlinear,

hybrid, and time varying [2]. To that end, the objective of

this study is to derive and validate a hierarchical control

approach that enables stable bipedal underactuated walking

on a rigid swaying surface (e.g., a vessel’s deck).

A. Related Work

Various control approaches have been created to realize

provably stable bipedal robot walking on stationary rigid

1Y. Gao is with the College of Engineering, University of Massachusetts

Lowell, Lowell, MA 01854, USA. yuan gao@student.uml.edu.

2Y. Gong is with the Robotics Department, University of Michigan, Ann

Arbor, MI 48105, USA. ykgong@umiche.edu

3V. Paredes and A. Hereid are with the Department of Mechanical and

Aerospace Engineering, the Ohio State University, Columbus, OH 43210,

USA. paredescauna.1@buckeyemail.osu.edu, hereid.1@osu.edu.

4Y. Gu is with the School of Mechanical Engineering, Purdue University,

West Lafayette, IN 47907, USA. yangu@purdue.edu.

Fig. 1. The default controller of the Digit humanoid robot seems to fail

to guarantee stable walking on a DRS that sways at a frequency of 0.5 Hz

and a magnitude of 5 cm.

surfaces, among which the most widely studied one is the

hybrid zero dynamics (HZD) method [3]. The HZD approach

stabilizes bipedal walking by explicitly treating the full-

order, hybrid, nonlinear robot dynamics. For underactuated

robots (e.g., bipeds with point feet), the HZD method

exploits input-output linearization to transform the nonlinear

robot dynamics associated with the directly actuated degrees

of freedom (DOFs) into a linear time-invariant system, which

is then stabilized based on the well-studied linear system

theory. Due to the use of input-output linearization, internal

dynamics exist, and its solutions (e.g., periodic orbits) are

typically unstable for walking robots. The HZD method

constructs a reduced-order zero dynamics manifold that

agrees with the overall hybrid dynamics and searches for

stable periodic orbits on that manifold.

Due to the high dimensionality and strong nonlinearity of

a full-order robot model, real-time generation of stable de-

sired trajectories based on the full-order model can be com-

putationally prohibitive for achieving robust bipedal walking

on stationary uneven terrains. To that end, researchers have

integrated reduced-order model based planning with full-

order model based control. X. Xiong et al. developed a

hybrid linear inverted pendulum (LIP) model to approximate

the hybrid walking dynamics of an underactuated bipedal

robot [4]–[6]. Y. Gong et al. proposed a new variant of

the LIP model that uses the angular momentum about the

contact point, instead of the linear velocity of CoM, as

a state variable [7], [8], which is called the “ALIP”. V.

Paredes et al. introduced a LIP template model to generate

a stepping controller with an adaptive learning regulator to

ensure stable walking on a bipedal humanoid robot [9].

Yet, due to the time-varying movement of the surface-

foot contact point/region, the dynamic model of bipedal

walking on a DRS is explicitly time-varying [2], [10], which

is fundamentally different the typical time-invariant robot

dynamics during static-surface locomotion.

Recently, the control problem of stabilizing legged loco-

motion on a DRS has been initially studied. To provably

stabilizes quadrupedal walking on a vertically moving DRS,

arXiv:2210.13371v2 [cs.RO] 29 Nov 2022

A. Iqbal et al. introduced a nonlinear control approach that

explicitly handles the time-varying DRS accelerations and

the hybrid, nonlinear robot dynamics during quadrupedal

walking [2]. To enable physically feasible and computation-

ally efﬁcient planning for legged locomotion on a vertically

moving surface, the classical continuous-time LIP model

has been analytically extended, resulting in a time-varying,

homogeneous LIP model [11], [12]. Still, the walking robot

considered is fully actuated, and thus the inherent instability

associated with underactuated walking does not exist. Also,

the surface is assumed to move only in the vertical direction

in these studies. In fact, our recent experiment validation

of the proprietary controller of the Digit humanoid robot

(developed by Agility Robotics) seems to indicate that

horizontally moving surfaces might be substantially more

challenging for bipedal robots to handle (see Fig. 1).

B. Contribution

This study introduces a hierarchical control approach that

achieves stable underactuated bipedal walking on a swaying

rigid surface by explicitly treating the robot’s hybrid, time-

varying dynamics and simultaneously exploiting the comple-

mentary advantages of full-order and reduced-order models

for walking stabilization. The speciﬁc contributions are:

a) Analytically extending the ALIP model from stationary

surfaces to a horizontally swaying DRS, resulting in a

hybrid time-varying ALIP model.

b) Synthesizing a discrete-time footstep controller that

exponentially stabilizes the hybrid, time-varying ALIP

model, and formulating an optimization problem to ﬁnd

the desired CoM trajectories and the stabilizing footstep

locations.

c) Developing a three-layer control approach that ensures

the stability for the hybrid, time-varying, nonlinear

unactuated robot dynamics by stabilizing the proposed

ALIP model and by mitigating the model inaccuracy of

the ALIP model with proper full-order trajectory design

and tracking control.

d) Demonstrating the proposed approach enables a full-

order robot to stably walk on a DRS under different

surface motions and gait types.

This paper is structured as follows. Section II introduces

the derivation of a hybrid time-varying ALIP model for

bipedal walking on a swaying DRS. Section III proposes a

discrete-time footstep controller that exponentially stabilizes

the hybrid ALIP model for DRS walking, and presents

the formulation of the higher-layer planner of the pro-

posed approach that produces the desired global trajectories.

Section IV explains the translation of the desired global

trajectories into the full-body references trajectories for the

directly controlled variables. Section V presents the lower-

layer controller that exponentially tracks the desired full-

body trajectories. Section VI reports simulation validation

results. Section VII provides the concluding remarks.

II. TIME-VARYING ANGULAR MOMENTUM BASED

LINEAR INVERTED PENDULUM (ALIP) MODEL

This section introduces the derivation of the proposed

ALIP model that captures the essential robot dynamics

associated with bipedal walking on a horizontally swaying

DRS. The ALIP model serves as the basis of the higher-layer

planner introduced in Sec. III.

Different from the classical LIP model [13] that takes a

robot’s center of mass (CoM) position and velocity as its

state, we choose to use the CoM position and the angular

momentum about the surface-foot contact point as the state,

resulting in an ALIP model [7]. Compared with the classical

LIP model, such a state choice allows the ALIP model to

be more accurate in representing the true robot dynamics in

the presence of velocities jumps at foot landings, large peak

motor torques, or aggressive swing leg motions [7].

A. Angular Momentum about Foot-Surface Contact Point

This subsection introduces the mathematical expression of

a bipedal robot’s angular momentum about the foot-surface

contact point. For simplicity, this study only considers bipeds

with point feet.

1) Continuous swing phase: During a swing phase, one

foot of the biped contacts the walking surface, and the other

moves in the air. Let Sdenote the contact point attached to

the walking surface. For a DRS, the point Smoves in the

world frame. Let Adenote a stationary point in the world

frame that instantaneously coincides with Sat the given time.

During a single-support phase, the three-dimensional (3-

D) vector of angular momentum about the point S, denoted

as LS, has the following relation with the angular momentum

about the point A, denoted as LA:

LS=LA+pSA ×(mvCoM).(1)

Here the vector pSA is the position of point Arelative to

the point S, expressed in the world frame. Note that pSA =0

because the point A instantaneously coincides with the point

S at the given time. The scalar constant mis the total mass

of the robot. The vector vCoM is the absolute CoM velocity

with respect to (w.r.t.) the world frame.

We will use the relation in Eq. (1) to derive the dynamics

of LSfor legged locomotion on a DRS (Sec. II-B).

2) Discrete foot-switching event: At the end of the swing

phase, the swing foot touches the walking surface. Without

loss of generality, we assume that the support foot begins to

swing just after the swing foot touchdown.

Across a foot landing event, the position of the contact

point jumps. Let k(k∈ {1,2,...}) indicate the kth foot

landing event. Let Lkbe the angular momentum about the kth

contact point on the walking surface. Let p(k+1)→kdenote the

position vector pointing from the (k+1)th to the kth contact

point. Let (·)−and (·)+respectively represent the values of

(·)just before and after the kth foot-landing instant.

Just before the kth foot-landing instant, the angular mo-

mentum about the new contact point, L−

(k+1), can be related

to the previous one, L−

k, as follows:

L−

k+1=L−

k+p(k+1)→k×(mv−

CoM ).(2)

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Time-VaryingALIPModelandRobustFoot-PlacementControlforUnderactuatedBipedalRobotWalkingonaSwayingRigidSurfaceYuanGao1,YukaiGong2,VictorParedes3,AyongaHereid3,YanGu4AbstractControllerdesignforbipedalwalkingondynamicrigidsurfaces(DRSes),whicharerigidsurfacesmovingintheinertialframe(e.g.,shipsandairpla...

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Time-Varying ALIP Model and Robust Foot-Placement Control for Underactuated Bipedal Robot Walking on a Swaying Rigid Surface Yuan Gao1 Yukai Gong2 Victor Paredes3 Ayonga Hereid3 Yan Gu4

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