Real-time Trajectory Optimization and Control for Ball Bumping with Quadruped Robots Qiayuan Liao Zhefeng Cao Hua Chen and Wei Zhang

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Real-time Trajectory Optimization and Control for Ball Bumping with

Quadruped Robots

Qiayuan Liao, Zhefeng Cao, Hua Chen, and Wei Zhang

Abstract— This paper studies real-time motion planning and

control for ball bumping motion with quadruped robots. To

enable the quadruped to bump the ﬂying ball with different

initializations, we develop a nonlinear trajectory optimization-

based planning scheme that jointly identiﬁes the take-off time

and state to achieve accurate ball hitting during the ﬂight

phase. Such a planning scheme employs a two-dimensional

single rigid body model that achieves a satisfactory balance

between accuracy and efﬁciency for the highly time-sensitive

task. To precisely execute the planned motion, the tracking

controller needs to incorporate the strict time-state constraint

imposed on the take-off and ball-hitting events. To this end, we

develop an improved model predictive controller that respects

the critical time-state constraints. The proposed planning and

control framework is validated with a real Aliengo robot.

Experiments show that the problem planning approach can

be computed in approximately 60 ms on average, enabling

successful accomplishment of the ball bumping motion with

various initializations in real-time.

I. INTRODUCTION

Jumping and interacting with objects in the air is one

of the most amazing behaviors that animals can perform.

Quadrupedal animals like leopards can leap and catch birds;

dogs can jump in the air and hit the ball using their heads.

Usually, this type of acrobatic behavior consists of multiple

phases, including jumping, interacting with the target object,

and landing phases. Ball bumping motion, which requires the

quadruped to jump into the air and hit a falling ball toward

a goal location, is one of the most representative acrobatic

motions for quadrupeds. Different from standard quadrupedal

locomotion on the ground, such a ball bumping motion is

highly time-sensitive, i.e., if the juggler does not arrive at

the expected position with the speciﬁc velocity at a proper

time, then the ball can not reach the desired region. In this

paper, we aim to tackle the speciﬁc ball bumping problem

with quadruped robots, which serves as a starting point for

exploring more highly dynamic motion planning and control

frameworks for future robotic applications.

A. Related Works

Dynamic locomotion for quadruped robots has attracted

a considerable amount of research attention during the past

decade. Thanks to the rapid advancements of both hardware

All authors are with the Department of Mechanical and

Energy Engineering, Southern University of Science and

Technology, Shenzhen, China.

liaoqiayuan@gmail.com,

12150041@mail.sustech.edu.cn, chenh6@sustech.edu.cn,

zhangw3@sustech.edu.cn

. Qiayuan Liao is also with the School

of Electromechanical Engineering, Guangdong University of Technology,

Guangzhou, China.

Laptop Aliengo

Tennis Ball

Releaser

Fig. 1:

Left:

A quadruped robot bumps the ball into a trash can.

Right: experiment setup.

and algorithms, quadrupedal robots have demonstrated im-

pressive locomotion skills on ﬂat and uneven terrains [1]–[4].

Fundamentally speaking, achieving jumping motion for

quadrupeds can be formulated as trajectory optimization

problems in general [5]. MIT Cheetah 3 is capable of

jumping onto a desk with a height of 30 inches by an ofﬂine

trajectory optimization that considers full-body kino-dynamics

in a 2D vertical plane [6]. Chigonoli et al. [7] proposed a

hierarchical planning framework with centroidal dynamics and

joint-level kinematics to achieve Omnidirectional jumping,

which takes on average 0.55 s to ﬁnd a reference trajectory

plan. Nguyen et al. [8] synthesized a full-body trajectory

optimization to achieve 3D jump with quadrupeds, which

takes several minutes to solve. Among the pioneering works

trying to plan and control jumping motions for quadrupeds,

Park et al. [9], [10] adopted a 2D single rigid body model

for quadrupedal dynamics and developed an event-triggered

jumping controller that accomplishes jumping over obstacles

on MIT Cheetah 2 quadruped. Li et al. [11] studied the

jumping motion of quadrupeds with only two rear legs and

developed a hierarchical planning and control framework

that can be implemented in real-time based on a spring-

loaded inverted pendulum model [12], which is veriﬁed with

simulations. As the reversed process of jumping, landing

control with quadrupeds has also been considered. Jeon et

al. [13] developed a supervised learning-based warm start

interface for nonlinear landing trajectory planning to improve

the performance of quadrupedal landing. How to exploit

the conservation of angular momentum to help modulate

the robot’s conﬁguration during the ﬂight phase has also

attracted recent research attentions [14], [15], which further

improves the landing performance of quadrupeds. Despite

these amazing demonstrations of jumping motion control

for quadrupeds, problems studying quadruped jumping with

physical interactions with other objects during the ﬂight phase

have not been studied adequately in the literature.

More recently, the problem of operating quadrupeds to

arXiv:2210.05195v1 [cs.RO] 11 Oct 2022

physically interact with other objects to achieve more compli-

cated tasks has attracted more research attention. Ji et al. [16]

considered the soccer shooting problem with a quadruped

robot and tackled the problem via a hierarchical reinforcement

learning strategy. Shi et al. [17] adopted a learning-based

approach to enable a quadruped robot to manipulate balls

with its legs. All these early investigations focus on scenarios

where the target object is static or quasi-static. How to design

effective planning and control strategies for interaction with

moving objects remains lacking.

In addition to quadrupeds, acrobatic motions for bipedal

robots have also been studied extensively in the literature.

In particular, investigations on jumping control with bipeds

date back to Raibert’s seminal works [18], [19]. Hierarchical

planning and control frameworks have been long employed

in the synthesis of acrobatic bipedal motions. Wensing et

al. [20] proposed a conic optimization-based task-space

control to achieve motions like ball kicking and standing broad

jump with a humanoid robot. Xiong et al. [21] developed

an optimization-based controller for biped hopping with

biped Cassie. Chigonoli et al. [22] employed the trajec-

tory optimization-based approach for synthesizing dynamic

acrobatic motions with a humanoid robot. Reinforcement

learning-based approaches have also demonstrated impressive

performance in achieving agile biped behaviors [23], [24].

Poggensee et al. [25] implemented ball-juggling on biped

Cassie, the motion is a periodic orbit, and there is no ﬂying

phase while bounding.

B. Contributions

The main contributions of this work are summarized

as follows. First, to incorporate the challenging time-state

constraint in the ball hitting problem, we develop a real-

time optimization scheme to ﬁnd the critical time-state

pair based on a two-dimensional single rigid body model.

Such an online planning scheme allows us to ﬁnd the

appropriate take-off state and time to achieve precise ball

hitting, which addresses the main challenge in accomplishing

the ball bumping problem. Second, to enable the quadruped

to accurately execute the planned motion, we develop an

improved model predictive controller that actively adjusts the

ground reaction forces based on the time remaining for the

jumping motion. Such a modiﬁcation effectively guarantees

the execution of the time-restricted motion, making the ball

bumping motion practically achievable. Third, we demonstrate

the effectiveness and performance of the proposed framework

on a real quadruped platform, successfully accomplishing the

ball bumping motion with various initializations.

II. PROBLEM STATEMENT

Our goal is to control the quadruped to bump the falling

ball to the desired landing point. Throughout this paper,

we assume the initial state of the ball is given, including

the released position, velocity, and the released time

. As

depicted in Fig. 2, the whole process can be split into three

phases. The ﬁrst phase is takeoff, in which the quadruped

starts to jump at

according to the released time

of the

tjttth

Jump start

Takeoff

Collision

Landing point te

Initial position

and velocity

Fig. 2: The process of ball bumping with the quadruped robot.

ball, then executes the online planned jumping motion and

ﬁnally takes off at

. The second is a ﬂying phase, and

the quadruped will ﬂy off the ground based on the takeoff

velocity, then collide with the ball at

in order to let the ball

reach the desired landing region. The last one is the landing

phase, where the quadruped will keep balance after contact

with the ground.

The ball bumping motion studied in this paper requires an

active real-time plan of a feasible trajectory for the quadruped

in response to the falling ball’s state and accurate execution

of the planned trajectory. Such a problem calls for a proper

balance between the efﬁciency and accuracy of the proposed

approach. In this section, we provide our modeling of the

quadruped-ball system used for planning the ball bumping

task and then give an overview of the proposed framework.

A. Modeling of Jumping Quadruped and the Falling Ball

1) quadruped model: In order to simplify the planning

of jumping motion while reserving acceptable accuracy, we

consider a 2D planar single rigid body model(2D-SRBM) for

quadruped in the sagittal plane. The state is the position of the

center of mass(COM) and the pitch angle of the quadruped,

denoted as

xq∈SE(2)

. The dynamics are given as follows

dtxq

xq=0313

0303xq

xq

+



03×203×2

12/m 12/m

I−1[rf]×I−1[rr]×



ff

fr+



04×1

0



(1)

where

[a]×

is deﬁned as an operator such that

[a]×b=

a1b2−a2b1

for all

a,b∈R2

and

0m×n

are the n by

n and m by n zeros matrices respectively.

I∈R2×2

is the

inertial matrix in x-z plane. As shown in Fig. 3,

are the

vectors from the COM to the front and rear feet respectively.

ff,frare the corresponding ground reaction forces.

frff

xb(th−)˙

xb(th+)

xq(th)

lhhbody

Fig. 3: Left:2D single rigid body model; Right: collision model.

2) ﬂying ball/quadruped model: Because of the lightweight

legs and the short duration of the ﬂying phase, it is hard to

adjust the angular momentum of the torso in the air. Therefore,

we can consider the position trajectory of the quadruped torso

between time

and

as a ballistic trajectory. As for the

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Real-timeTrajectoryOptimizationandControlforBallBumpingwithQuadrupedRobotsQiayuanLiao,ZhefengCao,HuaChen,andWeiZhangAbstractThispaperstudiesreal-timemotionplanningandcontrolforballbumpingmotionwithquadrupedrobots.Toenablethequadrupedtobumptheyingballwithdifferentinitializations,wedevelopanonlinear...

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Real-time Trajectory Optimization and Control for Ball Bumping with Quadruped Robots Qiayuan Liao Zhefeng Cao Hua Chen and Wei Zhang

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