Towards Exact Interaction Force Control for Underactuated Quadrupedal Systems with Orthogonal Projection and Quadratic Programming

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Towards Exact Interaction Force Control for Underactuated

Quadrupedal Systems with Orthogonal Projection and Quadratic

Programming

Shengzhi Wang1, Xiangyu Chu1∗, and K. W. Samuel Au1

Abstract— Projected Inverse Dynamics Control (PIDC) is

commonly used in robots subject to contact, especially in

quadrupedal systems. Many methods based on such dynamics

have been developed for quadrupedal locomotion tasks, and

only a few works studied simple interactions between the

robot and environment, such as pressing an E-stop button. To

facilitate the interaction requiring exact force control for safety,

we propose a novel interaction force control scheme for under-

actuated quadrupedal systems relying on projection techniques

and Quadratic Programming (QP). This algorithm allows the

robot to apply a desired interaction force to the environment

without using force sensors while satisfying physical constraints

and inducing minimal base motion. Unlike previous projection-

based methods, the QP design uses two selection matrices in its

hierarchical structure, facilitating the decoupling between force

and motion control. The proposed algorithm is veriﬁed with a

quadrupedal robot in a high-ﬁdelity simulator. Compared to the

QP designs without the strategy of using two selection matrices

and the PIDC method for contact force control, our method

provided more accurate contact force tracking performance

with minimal base movement, paving the way to approach the

exact interaction force control for underactuated quadrupedal

systems.

I. INTRODUCTION

Quadrupedal systems, whose six Degree of Freedom

(DoF) ﬂoating base is considered to be passively connected

to an inertial frame, are generally underactuated. To con-

trol such systems for locomotion and manipulation in real-

world deployment, underactuation must be addressed while

designing a whole-body controller. A paradigm based on

orthogonal projection and Quadratic Programming (QP) has

been exploited. Within it, a projection matrix stems from the

contact between a foot or an additional arm and the envi-

ronment, helping eliminate contact forces and thus reducing

variables in optimization. Besides, the projection matrix

can create two spaces: motion space and constraint space,

allowing more freedom for task-oriented applications. QP

can minimize a quadratic cost that resolves underactuation

and accommodate constraints such as unilateral contact and

friction cones. This paradigm normally works for locomotion

tasks or simple interaction tasks like pressing an emergency

button [1]. However, it is still open to using such a paradigm

to achieve interaction applications that need exact force

control. For example, as shown in Fig. 1, a robot uses one leg

1The authors are with the Department of Mechanical and Au-

tomation Engineering at The Chinese University of Hong Kong,

and with the Multiscale Medical Robotics Center, Hong Kong,

China. ∗: Corresponding author. {shengzhiwang, xiangyuchu,

samuelau}@mrc-cuhk.com

Maintain

Pressing

Open Door

Enter

Elevator

Fig. 1. A scenario of a quadrupedal robot exerting desired force on

the environment. Here, exact interaction force control of the raised leg is

preferred, because, on the one hand, it can succeed to maintain pressing the

button and keep the elevator door open, and on the other hand, the robot

will keep standing without falling.

to keep pressing a button for allowing other robots to move

into an elevator, and the robot can keep standing and have

minimal base movement at the same time. In this case, exact

force control is preferred for avoiding the robot’s falling

since impedance control for inducing force may generate

external force disturbance due to an unknown environment.

Motivated by this need, in this paper, we focus on how the

underactuated quadrupedal system applies an exact force to

the environment subject to physical constraints and minimal

disturbance on the robot’s base.

Projection-based methods have a long history. Projected

Inverse Dynamics Control (PIDC) was ﬁrst proposed for

fully-actuated systems in [2], paving the way to design con-

trollers in motion and constraint space. Some researchers ex-

tended this idea to underactuated systems [3]–[6]. For exam-

ple, the work [4] used either null space motion or constraint

forces to resolve underactuation without affecting task-space

dynamics, while [6] only made use of constraint forces. The

aforementioned studies focused on motion control levels, and

their constraint forces do not need to be specially considered

in the controller. To impose more authority on the constraint

force for underactuated applications, the constraint force is

optimized within constraint space to maintain contact [7], [8].

For example, [8] designed an optimization problem to seek

the optimal contact wrenches that minimize torques while

satisfying physical constraints and compensating external

forces. Although the constraint force/wrench was specially

treated, those methods still cannot provide an exact con-

arXiv:2210.10238v1 [cs.RO] 19 Oct 2022

tact force actively since the force/wrench was implicitly

manipulated. Previously, explicitly tracking desired contact

forces in quadrupedal systems was implemented in [9] but

required planning on both desired position trajectories and

desired contact forces at the same time if imposing higher

priority on force control than motion control, because desired

accelerations affected realizable contact forces. The tracking

of planned force proﬁles may not be suitable for tasks

requiring fast reaction (e.g., tracking desired interaction force

command from users); thus, a reactive control scheme is

preferable.

To this end, to approach an exact force output, we propose

a novel interaction force control scheme for underactuated

quadrupedal systems in the sense of reaction control. It

allows the system (e.g., using a foot) to apply a force as

precisely as possible to the environment, without requiring

force planning. Our scheme uses two QP designs that resolve

the underactuation problem by splitting it into two orthogonal

spaces: motion and constraint space, and then optimizing the

cost in each space in a hierarchical order. Physical limitations

(i.e., unilateral contact, torque limits, and friction cones)

are also considered as inequality constraints in the design.

To decouple force and motion control and accomplish the

exact constraint force control as much as possible, we apply

two selection matrices, in which one of them distributes

the desired force control task to the designated joints, and

another one selects the rest joints for the underlying motion

task. For instance, as shown in Fig. 1, a quadrupedal robot

uses its front right leg for force control, while the other three

legs support its base and conduct the motion task of the base.

These two selection matrices select the front right leg and the

other three legs for the force and motion control, respectively.

In summary, our control scheme is a reactive control (e.g.,

can respond to user-deﬁned force inputs fast) that does not

rely on any motion planning techniques and force sensors

(FS), and induces minimal base motion.

The contributions of this paper are:

1) Presenting a novel interaction force control scheme for

underactuated quadrupedal systems that does not require

motion planning and FS.

2) For resolving the underactuation problem, we propose

a hierarchical QP structure that minimizes the cost

function for the motion and force control in motion and

constraint space, respectively. To reduce the coupling

effect between motion and force control, two selection

matrices are deployed, allowing us to decouple force

and motion to the greatest extent and thus approach the

exact force control.

II. BACKGROUND

A. Projected Inverse Dynamics with External Disturbance

The dynamics equation of a quadruped robot can be

expressed as:

M¨

q+h=Bτ +JT

cλ+JT

xFx,(1)

where q=qT

b,qT

jTis the generalized coordinate vec-

tor including unactuated ﬂoating base coordinates qb∈

SE(3) and actuated joint conﬁguration qj∈Rnj,M∈

R(6+nj)×(6+nj)denotes the inertia matrix, h∈R(6+nj)is

the non-linear effect consisting of Coriolis, centrifugal and

gravitational forces, B=0nj×6,Inj×njTis the selection

matrix1of actuated joints, τ∈Rnjis the actuated joint

torques, Jc∈Rk×(6+nj)represents the constraint Jacobian

with k= 3nc(ncdenotes the number of legs in contact

with the environment and the contact is assumed as a point

contact), λ∈Rkdenotes the generalized constraint force

vector used to control unactuated qb,Jx∈Rne×(6+nj)is

the Jacobian at x, and Fx∈Rneis the external disturbances

due to the interaction from human or environments.

During locomotion, the support feet should not slip, i.e.,

the constraint Jc˙

q=0must be satisﬁed. This constraint

indicates that any admissible ˙

qlies in the constraint null

space N(Jc). According to [10], the dynamics equation (1)

can be projected into two subspaces by using the orthogonal

projection matrix Pand I−Prespectively as:

P M ¨

q+P h =P Bτ

|{z }

τm

+P JT

xFx,(2)

(I−P)(M¨

q+h)=(I−P)Bτ

| {z }

τc

+JT

cλ+ (I−P)JT

xFx,

(3)

where P=I−J+

cJcimplies that the orthogonal null

space projection matrix is computed from the Moore-Penrose

pseudoinverse of Jcand projects vectors into the null space

of the constraint, such that P=P2=PT,P JT

c=0, and

P˙

q=˙

qfor all ˙

q∈ N (Jc). Then I−Prepresents the

complementary projection into N⊥(Jc).

To solve the equation of λ,¨

qmust be computed ﬁrst. How-

ever, ¨

qcannot be obtained directly through pre-multiplying

the inverse of P M in (2), as P M cannot be inverted

attributed to the rank deﬁciency of P. With the help of

additional equations (I−P)˙

q=0and its derivative

(I−P)¨

q=˙

P˙

qthat are derived from P˙

q=˙

q,¨

qcan

be solved by substituting the latter equation into (2) as:

q=M−1

c(τm−P h +˙

P˙

q+P JT

xFx),(4)

where Mc=P M +I−P. Equations (4) shows that

only the motion torques τmcontributes to the motion of

the system, therefore N(Jc)is called the motion space as

described in [6]. Similarly, because in (3) the ¨

qis ﬁxed

and constraint forces λare free to choose depending on the

constraint torques τc,N⊥(Jc)is named the constraint space.

Eventually, the constraint forces are obtained by inserting ¨

into (3) and yields:

λ= (JT

c)+h(I−P)[ ¯

M(τm−P h +˙

P˙

q) + h]−τc

+ (I−P)( ¯

M P −I)JT

xFxi,

(5)

with ¯

M=M M −1

c, and is the constraint inertia matrix and

is always invertible [10]. More details can be found in [6].

1Unlike the selection matrix in previous papers, the selection matrix here

is not square.

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TowardsExactInteractionForceControlforUnderactuatedQuadrupedalSystemswithOrthogonalProjectionandQuadraticProgrammingShengzhiWang1,XiangyuChu1,andK.W.SamuelAu1AbstractProjectedInverseDynamicsControl(PIDC)iscommonlyusedinrobotssubjecttocontact,especiallyinquadrupedalsystems.Manymethodsbasedonsuchdyn...

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Towards Exact Interaction Force Control for Underactuated Quadrupedal Systems with Orthogonal Projection and Quadratic Programming

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