Contact Optimization for Non-Prehensile Loco-Manipulation via Hierarchical Model Predictive Control Alberto Rigo Yiyu Chen Satyandra K. Gupta and Quan Nguyen

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Contact Optimization for Non-Prehensile Loco-Manipulation via

Hierarchical Model Predictive Control

Alberto Rigo, Yiyu Chen, Satyandra K. Gupta, and Quan Nguyen

Abstract— Recent studies on quadruped robots have focused

on either locomotion or mobile manipulation using a robotic

arm. Legged robots can manipulate heavier and larger objects

using non-prehensile manipulation primitives, such as planar

pushing, to drive the object to the desired location. In this

paper, we present a novel hierarchical model predictive control

(MPC) for contact optimization of the manipulation task. Using

two cascading MPCs, we split the loco-manipulation problem

into two parts: the ﬁrst to optimize both contact force and

contact location between the robot and the object, and the

second to regulate the desired interaction force through the

robot locomotion. Our method is successfully validated in

both simulation and hardware experiments. While the baseline

locomotion MPC fails to follow the desired trajectory of the

object, our proposed approach can effectively control both

object’s position and orientation with a minimal tracking error.

This capability also allows us to perform obstacle avoidance for

both the robot and the object during the loco-manipulation task.

I. INTRODUCTION

Legged robots have great potential to interact with the

environment and have demonstrated signiﬁcant performance

for locomotion, such as high-speed running and robust walk-

ing on challenging terrains [1], [2], [3], [4], [5], [6], [7],

[8]. With the existing control and planning algorithms, most

applications for quadruped robots focus on navigation and

inspection which always try to avoid objects/obstacles even

if they are movable [9], [10], [11]. In this paper instead,

we are interested in realizing the capability of legged robots

leveraging their body during locomotion to manipulate a

heavy object.

In mobile manipulation, robots can exhibit different modes

of interaction with the object. For example, mobile robots

equipped with a robotic arm [12], [13], [14], [15], [16]

can enable basic manipulation tasks such as door opening,

pick-and-place, and load carrying. However, such setups

are limited to small payload and object dimension due to

the payload limit of the portable robot arm. For legged

robots, manipulation with their feet is also an intriguing

idea as quadrupedal animals can use their legs or limbs for

manipulation[17], [18]. However, this setup is unsuitable for

loco-manipulation tasks, which require the robot to move

and manipulate the object simultaneously because it requires

both locomotion and manipulation. If one of the legs is

used for manipulation, the locomotion task will become

1Alberto Rigo, Yiyu Chen, Satyandra K. Gupta, and Quan Nguyen, are

with the Department of Aerospace and Mechanical Engineering, University

of Southern California, Los Angeles, CA, 90089 rigo@usc.edu,

yiyuc@usc.edu,quann@usc.edu,guptask@usc.edu

Fig. 1: Motion snapshots of Unitree A1 robot manipulating a 5kg object

to follow a circular trajectory.

challenging for quadruped robots. Therefore, this paper tack-

les the problem of loco-manipulation for quadruped robots

using a planar pushing motion. For large and heavy objects,

non-prehensile manipulation such as planar pushing offers

excellent advantages. When the object is too large or too

heavy to be grasped, pushing becomes one of the options

to drive it to the desired state. In addition, this method

also allows quadruped robots to manipulate objects without

adding an additional robotic arm.

Pushing is a widely used motion primitive and has been

thoroughly studied by the manipulation community. The

mechanics of planar pushing is well-studied in [19], [20],

[21]. Some motion planning algorithms [22], [23] are in-

troduced to ﬁnd open-loop trajectories to drive the object

to the target pose, assuming that the manipulator always

sticks with the object for the entirety of the push. To handle

the complexity associated with frictional contact interac-

tions, motion planning algorithms developed by the robotic

manipulation community manage to handle different mode

sequences [24], [25], [26]. Nevertheless, these approaches are

computationally heavy due to the nonlinear and non-convex

optimization programs. A recent work in [27] proposes a

real-time controller to reason across different contact modes,

including sticking and sliding, using an online approximation

for the ofﬂine mix-integer program.

The recent developments on model predictive control

for legged robot locomotion [28], [1], [16] suggest that

optimal control action can be computed online given a

proper contact schedule. However, these works mainly focus

on locomotion. To simultaneously achieve locomotion and

manipulation tasks, we propose a novel hierarchical MPC

framework including (1) high-level manipulation MPC to

arXiv:2210.03442v1 [cs.RO] 7 Oct 2022

Desired

Object-robot

Trajectory

Contact

Optimizer MPC Loco-Manipulation

MPC

Contact Force

[𝒇𝒄𝟏. . . . . 𝒇𝒄𝑵]

Contact location

[𝒅𝟏.....𝒅𝑵]Forces to

Torques

Mapping

Swing Leg

Controller

Current

Object and

Robot States

Gait

Generator

Robot/Simulation

Fig. 2: Control Architecture

optimize for both contact force and contact location of

the manipulation task; and (2) low-level loco-manipulation

MPC to regulate the interaction force between the robot

and the object while maintaining the desired locomotion

performance. Both MPC problems are solved effectively in

real-time. Numerical and experimental validation have shown

that our approach outperform locomotion MPC or heuristic

approach for loco-manipulation. Thanks to the capability of

optimizing contact location, our approach can allow legged

robots to manipulate heavy objects effectively with a highly

accurate position and orientation tracking. This also enables

the execution of collision-free trajectory for both the robot

and the object.

The rest of the paper is organized as follows. Section II

introduces the object-robot system for non-prehensile body

loco-manipulation. Section III presents the proposed control

architecture and the two MPC in detail. Then, Section IV

shows simulation and hardware experiments results. Finally,

Section V draws conclusion remarks.

II. SYSTEM OVERVIEW

In this paper, we are interested in pushing an arbitrary

object, following a planned trajectory in terms of xand

yworld frame position and heading angle ψ. We assume

we know all the geometric and inertial characteristics of

the object, and we have the feedback on its heading angle

and center of mass position. Due to the limitations of the

pushing primitive, to move the object to the desired location,

we have to align its heading angle toward that location.

Leveraging the position tracking of the quadruped robot,

we can optimize the contact point between the robot head

and object to push forward and, at the same time, rotate

the object to align the heading angle to the desired one.

Without changing the contact location, we would not be able

to control the heading angle of the object. The nonlinearity

of the loco-manipulation problem is solved by splitting it into

two separate linear parts, the ﬁrst responsible for determining

the required manipulation action to be exerted on the object;

the second responsible for the locomotion under the effect

of the contact interaction.

III. PROPOSED FRAMEWORK

The high-level control comprises the swing leg controller

and two Model Predictive Controllers (MPC) in a hierar-

chical structure, as depicted in Fig. 2. First, the contact

optimizer MPC is used to compute the required control input,

i.e., contact force and contact point on the object surface,

to drive the manipulated object to the desired states. Then,

the loco-manipulation MPC is responsible for tracking the

planned trajectory for the object-robot system based on the

output of the contact optimizer MPC. The two MPCs use

the same prediction horizon, so the predicted values for the

contact interaction by the contact optimizer MPC are used

as inputs for the loco-manipulation MPC.

A. Contact Optimizer MPC

This ﬁrst controller uses a simpliﬁed model of manipulated

object dynamics. In this paper, we are interested in con-

trolling the object’s position and heading angle. Therefore

we can use the following simpliﬁed rigid body dynamics

equations:

m¨

pobj =fµ+fc(1)

Iz˙ωz=fc×d(2)

where prepresents the position of the object in the world

frame, fµis the frictional force between object and ground,

and fcis the contact force applied to the object by the robot

in world frame, ωzis the angular velocity of the object in

the vertical direction with respect to its center of gravity, and

dis the vector between the contact point and object center

of mass in world frame. If we consider the contact force

and the contact point as control variables for the problem,

the previous set of equations is nonlinear. We can make

some assumptions and simpliﬁcations to use them as model

dynamics in a linear MPC. First, with a small enough MPC

frequency update, we can assume that the contact force will

change only by a small amount; hence the contact force used

in the eq. (2) is the known value of force computed at the

previous controller update, fc0. Then, we can further assume

the contact force fcis always in the x-direction of the object

body frame, simplifying the deﬁnition of the contact point

d. It becomes the distance from the center of mass of the

object in the y direction of the object body frame, as seen in

Fig 3. With these assumptions, equations (2) are now linear

and can be used to represent the object dynamics in the state

space form:

˙x=Ax +Bu (3)

where x=ψ x y ωz˙x˙y g,u=fcd, and

the matrices are







03×3I3×303×1

03×303×3



−µ





01×301×30







, B =







03×2





fc00

0 cos ψ

0 sin ψ





01×2







(4)

where we assumed that the frictional force fµis expressed as

−µmg for both xand ydirections, the body frame contact

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ContactOptimizationforNon-PrehensileLoco-ManipulationviaHierarchicalModelPredictiveControlAlbertoRigo,YiyuChen,SatyandraK.Gupta,andQuanNguyenAbstractRecentstudiesonquadrupedrobotshavefocusedoneitherlocomotionormobilemanipulationusingaroboticarm.Leggedrobotscanmanipulateheavierandlargerobjectsusingn...

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Contact Optimization for Non-Prehensile Loco-Manipulation via Hierarchical Model Predictive Control Alberto Rigo Yiyu Chen Satyandra K. Gupta and Quan Nguyen

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