Optimal Energy Shaping Control for a Backdrivable Hip Exoskeleton Jiefu Zhang1 Jianping Lin12 Vamsi Peddinti3 and Robert D. Gregg3 Abstract Task-dependent controllers widely used in ex-

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Optimal Energy Shaping Control for a Backdrivable Hip Exoskeleton

Jiefu Zhang1, Jianping Lin1,2, Vamsi Peddinti3, and Robert D. Gregg3

Abstract— Task-dependent controllers widely used in ex-

oskeletons track predeﬁned trajectories, which overly constrain

the volitional motion of individuals with remnant voluntary

mobility. Energy shaping, on the other hand, provides task-

invariant assistance by altering the human body’s dynamic

characteristics in the closed loop. While human-exoskeleton

systems are often modeled using Euler-Lagrange equations, in

our previous work we modeled the system as a port-controlled-

Hamiltonian system, and a task-invariant controller was de-

signed for a knee-ankle exoskeleton using interconnection-

damping assignment passivity-based control. In this paper, we

extend this framework to design a controller for a backdrivable

hip exoskeleton to assist multiple tasks. A set of basis functions

that contains information of kinematics is selected and corre-

sponding coefﬁcients are optimized, which allows the controller

to provide torque that ﬁts normative human torque for different

activities of daily life. Human-subject experiments with two

able-bodied subjects demonstrated the controller’s capability

to reduce muscle effort across different tasks.

I. INTRODUCTION

Lower-limb exoskeletons have proved to be powerful in

rehabilitation and restoring mobility, while their controller

design remains a challenge. Most commercial exoskeletons

like ReWalk and Ekso Bionics [1] fall into task-dependent

controllers tracking predeﬁned trajectories, which are not

appropriate for people with remnant voluntary mobility.

Besides, the need of detecting users’ intention for the transi-

tion between task-dependent controllers makes it difﬁcult to

perform a continuum of tasks and may cause injury when

detection goes wrong. Moreover, the controller parameter

tuning is a laborious, technical challenge, which hinders

applying exoskeletons to a larger population.

To overcome these limitations, task-independent control

frameworks have been introduced. In [2], an integral admit-

tance shaping controller for single degree-of-freedom (DoF)

exoskeletons was proposed, which provided assistance by

modifying the dynamic response of the coupled system.

Experiments showed that larger motion can be achieved

with the same muscle effort. Based on delayed output

feedback control, a uniﬁed controller was designed in [3],

which is capable to provide assistance under various walking

*This work was supported by the National Science Foundation under

Award Number 1949869 and by the National Institute of Biomedical Imag-

ing and Bioengineering of the NIH under Award Number R01EB031166.

The content is solely the responsibility of the authors and does not

necessarily represent the ofﬁcial views of the NSF or NIH.

1Electrical Engineering and Computer Science, University of Michigan,

Ann Arbor, MI 48109, USA. 2State Key Laboratory of Mechanical System

and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong

University, Shanghai 200240, China. 3Robotics, University of Michigan,

Ann Arbor, MI 48109, USA.

Corresponding author: Robert D. Gregg. Contact: {zjiefu, jplin,

rdgregg}@umich.edu

speeds and environments. Learning-based methods have also

been investigated in [4], which allowed subject-independent

hip joint moment estimation over different tasks based on

wearable sensor data. However, the “black box” nature of

these learning-based algorithms make it difﬁcult to guarantee

safety outside the training dataset.

As a trajectory-free control method, energy shaping pro-

vides task-invariant assistance by altering the human body’s

dynamics in the closed-loop system and has been extensively

investigated for exoskeleton control [5]. In [6], a potential

energy shaping-based control method was proposed to pro-

vide body-weight support (BWS) for exoskeletons using a

controlled Lagrangian. However, the control law depends on

contact conditions, which change with different gait phases.

A uniﬁed controller was proposed in [7], which provides

task-invariant assistance with respect to different human

input and contact conditions. While these potential energy

shaping methods only provided BWS, total energy shaping

can further regulate velocity by modifying the mass/inertia

matrix, which was investigated in [8]. The ability to provide

greater assistance compared with potential energy shaping

alone was shown by simulation. However, the control law

requires computationally-intensive inversion of the shaped

mass/inertia matrix, which is also susceptible to singularities

due to underactuation.

While the above energy shaping strategies used the con-

trolled Lagrangian method, one can also model the human-

exoskeleton system as a port-controlled Hamiltonian system.

Then the interconnection and damping assignment passivity-

based control (IDA-PBC) method can provide extra shaping

freedom compared with the controlled Lagrangian counter-

part [9]. By altering the interconnection structure of the

port-controlled Hamiltonian equations, additional velocity-

dependent assistance can be provided without modifying the

mass/inertia matrix. The IDA-PBC method has been applied

to control a knee-ankle exoskeleton in [10] and [11], which

proved its ability to achieve task-invariant control for primary

activities of daily life (ADL). In this paper, we extend the

method in [11] and design a task-invariant controller for a

commercial hip exoskeleton using IDA-PBC. Hip exoskele-

tons do not have access to kinematic information of knee and

ankle joints, which makes the previous kinematic models no

longer suitable. Instead of modeling the legs separately as

in [10] and [11], we adopt a complete point-footed biped

model including a trunk, stance leg, and swing leg. Since

the controller is based on the same model for both stance

and swing leg, this obviates the need for a foot force sensor

to switch between control laws for different legs.

The contributions of this paper are summarized as follows.

arXiv:2210.03777v2 [cs.RO] 25 Mar 2023

First, the IDA-PBC method is extended to control a hip

exoskeleton using the uniﬁed system model for both stance

and swing phase. Second, the proposed controller only needs

data from onboard sensors (hip joint encoders and thigh

IMU), which makes it more practical in daily life settings.

We begin in Section II by modeling the human-exoskeleton

system and reviewing the corresponding matching condition.

In Section III, a passivity-based data-driven method is used

to optimize the controller to ﬁt normative human torques.

The hardware implementation and human subject experiment

are presented in Section IV, showing that the controller

is capable of assisting multiple tasks. Finally, Section V

concludes the paper.

II. ENERGY SHAPING OF HUMAN-EXOSKELETON

SYSTEM

In this section, we model the human-exoskeleton system

as a port-controlled Hamiltonian system and give the energy

shaping-based control law. We also review the matching

condition for feasible control laws derived in [11].

Fig. 1. Left: Movex hip exoskeleton produced by Enhanced Robotics. The

Raspberry Pi and IMU are added for research proposes. Right: Kinematic

model of human-exoskeleton system.

A. System Modeling

Consider the 5-degree of freedom (DoF) human-

exoskeleton system shown in Fig. 1. The Cartesian coor-

dinate of stance feet (px,py)is deﬁned with respect to

the inertial reference frame (IRF). The angle between left

thigh and trunk is deﬁned as θl, and the angle between

right thigh and trunk is deﬁned as θr. The global thigh

angle φis deﬁned as the angle between right thigh and

the vertical axis. The generalized coordinate of the model is

q= [px,py,φ,θl,θr]∈R5in the conﬁguration space Q=R5.

Deﬁne the conjugate momenta as p=M(q)˙q∈R5, where

M(q)∈R5×5is the positive deﬁnite mass/inertia matrix and

˙q∈R5is the generalized velocity vector. Then we obtain the

port-controlled Hamiltonian (PCH) system characterized by

the Hamiltonian: H(q,p) = 1

2pM−1(q)p+V(q), where V(q)

is the potential energy. The state-space form of the PCH

dynamics can be given as

˙q

˙p=05×5I5×5

−I5×505×5∇H+0

τ+ATλ,(1)

where ∇H= [∇qH,∇pH]∈R10 is the gradient of the Hamil-

tonian. The torque τ=τexo +τhum ∈R5is the sum of

exoskeleton input τexo =Bu and human input τhum =Bv,

where u,v∈R2are the exoskeleton and human input torque

applied to hip joints and B= [02×3,I2×2]T∈R5×2is the

mapping matrix. Since the number of actuated coordinates is

less than the number of generalized coordinates, the system

is underactuated. For sake of simplicity, we omit qand p

from now on.

The holonomic contact constraints in the system (1) can

be expressed as a(q) = 0c×1, where cis the number of

constraints. This can be written in a matrix form a(q) =

A(q)q. Since a(q) = 0 is independent of time, A(q)can

be obtained by solving ˙a(q) = ∇a(q)˙q=A(q)˙q=0. In our

case, a(q) = [px,py]T=01×2,A= [Ac,02×2] = [I2×2,02×3].

The Lagrangian multiplier λ∈R2represents the ground

reaction forces, which is mapped to the system through A.

By differentiating A˙q=0 along time and plug (1) into the

equation, we obtain λas

λ=Wn−∇qhA(∇pH)Ti(∇pH)T+A∇2

p2H

×h(∇qH)T−τio ,

where W=hA∇2

p2HATi−1∈R2×2.

B. Review of Matching Conditions for Port-Controlled

Hamiltonian System

In this part, we brieﬂy review the matching condition of

a 5-DoF system derived in [11]. Consider the closed-loop

system with ubeing controlled while vremains open-looped.

The desired closed-loop Hamiltonian ˜

H=1

2pT˜

M−1p+˜

where ˜

V=V+ˆ

Vrepresents the desired potential energy

with shaping term ˆ

V. Therefore, ˜

N=∇q˜

HT= (∇qH)T+

∇qˆ

HT=N+ˆ

N, where Nand ˆ

Nrepresents the correspond-

ing gravitational vectors. We let ˜

M=Mremain unchanged,

which simpliﬁes the matching process. However, with cer-

tain structure of the interconnection matrix being satisﬁed,

velocity-dependent shaping can still be achieved [12].

Consider the desired closed-loop system

˙q

˙p=0I

−I J2∇˜

H+0

Bv +AT˜

λ+Text ,(2)

where Text denotes the external input which helps preserve

the passivity of (2). J2=−JT

2∈R5×5is skew-symmetric

deﬁned as J2=h(∇qp)T−(∇qp)i+h(∇qQ)T−∇qQi=

(∇qQ)T−∇qQ, since ∇qp=0. Q(q)∈R5is any smooth

vector-valued function with respect to qwhich allows it to

provide extra shaping DoF [13]. The closed-loop GRF can

then be expressed as

λ=Wn−∇qhA(∇pH)Ti(∇pH)T+A∇2

p2H

·h∇q˜

HT−J2(∇pH)T−Bv −Text io .

By the standard form of matching condition in [14],

system (1) and (2) match if

−∇qH+B(u+v) + ATλ

=−∇q˜

H+J2∇p˜

H+Bv +AT˜

λ+Text .

(3)

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OptimalEnergyShapingControlforaBackdrivableHipExoskeletonJiefuZhang1,JianpingLin1;2,VamsiPeddinti3,andRobertD.Gregg3AbstractTask-dependentcontrollerswidelyusedinex-oskeletonstrackpredenedtrajectories,whichoverlyconstrainthevolitionalmotionofindividualswithremnantvoluntarymobility.Energyshaping,ont...

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Optimal Energy Shaping Control for a Backdrivable Hip Exoskeleton Jiefu Zhang1 Jianping Lin12 Vamsi Peddinti3 and Robert D. Gregg3 Abstract Task-dependent controllers widely used in ex-

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