An optimal open-loop strategy for handling a ﬂexible beam with a robot manipulator Shamil Mamedov12 Alejandro Astudillo12 Daniele Ronzani12

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An optimal open-loop strategy for handling a ﬂexible beam

with a robot manipulator

Shamil Mamedov1,2, Alejandro Astudillo1,2, Daniele Ronzani1,2,

Wilm Decr´

e1,2, Jean-Philippe No¨

el1, Jan Swevers1,2

Abstract— Fast and safe manipulation of ﬂexible objects

with a robot manipulator necessitates measures to cope with

vibrations. Existing approaches either increase the task ex-

ecution time or require complex models and/or additional

instrumentation to measure vibrations. This paper develops a

model-based method that overcomes these limitations. It relies

on a simple pendulum-like model for modeling the beam, open-

loop optimal control for suppressing vibrations, and does not

require any exteroceptive sensors. We experimentally show that

the proposed method drastically reduces residual vibrations

– at least 90% – and outperforms the commonly used input

shaping (IS) for the same execution time. Besides, our method

can also execute the task faster than IS with a minor reduction

in vibration suppression performance. The proposed method

facilitates the development of new solutions to a wide range of

tasks that involve dynamic manipulation of ﬂexible objects.

I. INTRODUCTION

Many industries extensively use ﬂexible materials [1].

Naive handling of ﬂexible objects with a robot arm may in-

troduce large vibrations. Existing feedback solutions [2], [3]

require accurate sensing of the vibrations using an additional

sensor and complex analytical or data-driven models. On the

other hand, existing feedforward solutions increase the task

execution time [4]. Therefore, the industry can substantially

beneﬁt from new methods for fast handling of ﬂexible objects

that are strong in performance and simple in implementation.

This paper investigates the general problem of manipulat-

ing a ﬂexible beam with a rigid robot arm [2]. The prob-

lem involves modeling, parameter estimation, control, and

perception. We assume a structured industrial environment

and do not address the perception. For sensing vibrations

of the beam, we do not use exteroceptive sensors – such

as external force-torque sensors at the end-effector or a

camera – only a joint torque sensor/estimator. This constraint

on instrumentation increases the problem’s complexity and

the practical value of the developed solutions for economic

reasons.

Any model-based control method requires a model of the

system. Beams are inﬁnite dimensional systems; they are

accurately modeled by partial differential equations that are

computationally demanding to solve and are seldom used

in control and trajectory planning. In robotics, for accurate

modeling of ﬂexible beams, researchers make simplifying

assumptions, e.g., separability of spatial and time modes as

This research was supported by the FWO-Vlaanderen through SBO

project ELYSA for cobot applications (S001821N).

1The MECO Research Team, KU Leuven, 3000 Leuven, Belgium.

2The DMMS Lab, Flanders Make, 3001 Leuven, Belgium.

Fig. 1. A stroboscopic photo of the Franka Panda handling a ﬂexible beam

in the assumed mode method [5], or apply discretization

methods such as the ﬁnite element method [6], [3]. Another

related approach is ﬂexible multibody dynamics in relative

(to the rigid body mode) [7] or absolute nodal coordinates

[8]. The model parameters in the above-mentioned methods

are often obtained from CAD models because, in prac-

tice, it is difﬁcult to estimate them. Data-driven methods

(system identiﬁcation) approach modeling beam dynamics

differently; they infer the model structure from data [2].

In this paper, we use a simple lumped parameter model

for the beam modeling that only considers the ﬁrst natural

frequency. The model is computationally fast compared with

more accurate models and physically interpretable, unlike

purely data-driven methods [2]. Parameters of the model

can be estimated from data or analytically from material

properties.

The most crucial aspect of ﬂexible object handling lies

in vibrations suppression. Input shaping (IS) is one of the

most well-established techniques for suppressing vibrations

of linear systems [4]. Despite its simplicity, input shapers

modify the original trajectory and extend the motion time. In

robotics, modiﬁcation of the joint trajectories yields changes

in the end-effector trajectory. To avoid end-effector trajectory

changes, IS can be applied to the normalized arc length

[9] or the operational space trajectory [10]. To counteract

increased motion time, [9] proposes accelerating the original

motion by the amount of delay. Zhout et al. [6] developed

a nonlinear IS for suppressing ﬂexible payload vibrations.

Instead of shaping joint accelerations, the authors shaped

modal excitation forces. However, retrieving joint velocities

or accelerations from shaped modal forces is not trivial and

sometimes not unique.

Neither linear nor nonlinear input shapers can handle

arXiv:2210.00578v1 [cs.RO] 2 Oct 2022

input constraints; for example, a feasible trajectory in the

joint space after being shaped in the operational space

might become infeasible. In contrast, optimal control can

explicitly consider state and input constraints. In [11], the

authors used direct time-optimal control for the sloshing-free

transport of liquids with a robot arm. They formulated the

optimal control problem (OCP) in the operational space, with

the decision variables being the end-effector translational

accelerations. Indirect optimal control [12], [7] and model

predictive control (MPC) [13], [14], [15] have also been

used for vibration suppression. MPC solves an underlying

OCP at every sampling instant and is naturally more robust

to model errors. However, it comes at a high computation

cost and complexity since MPC requires online/real-time

measurements and computation. To reduce the computational

burden, researchers often drastically reduce the horizon of the

MPC or linearize the nonlinear dynamics [15], [13] rendering

MPC conservative.

We approach vibration suppression from a direct optimal

control point of view similar to [11]. However, instead of

formulating an OCP in the operational space, we do it in the

joint space where constraints can be handled more naturally,

and arm dynamics can be fully exploited. In terms of control

strategy, we opt for an open-loop formulation instead of

feedback control, as feedback control requires real-time esti-

mation of the beam vibrations, which are challenging to ob-

tain without exteroceptive sensors. Because of the underlying

optimization problem, our approach can be computationally

slow compared with solutions that leverage IS. To address

it, we propose an efﬁcient numerical implementation that

substantially reduces computation time compared with a

naive implementation.

To brieﬂy summarize, our contributions are:

•a model that is simple yet effectively captures the

complex dynamics of the system;

•an OCP for handling the beam that drastically reduces

residual vibrations and allows to trade-off vibrations

suppression and task execution time;

•experimental validation of the proposed method and

comparison with IS.

This paper is organized as follows: Section II addresses

the modeling of the robot arm and the beam, Section III

introduces two methods for estimating the parameters of the

beam model. In Section IV, we discuss the proposed control

method; in Section V, we present experimental results,

followed by a discussion. Section VI concludes the paper

and provides directions for future research.

II. MODELING

In this section, we develop a model of the setup and

discuss underlying assumptions.

A. Arm dynamics

For a robot arm with ndof degrees of freedom (dof), let

q∈Rndof be the vector of joint positions and assume that:

Assumption 1: The internal joint controller can accurately

track the joint reference trajectories.

Fig. 2. Schematic representation of approximating a beam attached to

the end-effector of a robot arm with a simple pendulum of length land a

lumped mass mattached to the end-effector by means of a passive revolute

joint with stiffness kand damping c

Then, a double integrator model accurately describes the

arm’s dynamics:

q=u,(1)

where ¨

q∈Rndof is the vector of joint accelerations, and u∈

Rndof is the vector of inputs (reference joint accelerations).

B. Beam dynamics

For modeling the beam dynamics, we make another critical

and simplifying assumption:

Assumption 2: The beam can be approximated by a sim-

ple pendulum connected to the end-effector of an arm

through passive revolute joint with stiffness kand damping

c, as shown in Fig. 2.

By making such assumption, we consider only the ﬁrst

natural frequency of a beam and only the lateral vibrations.

To derive the pendulum dynamics using the Lagrange

formulation [16, Ch. 7], let p0

m∈R3denote the position

of the pendulum mass min the robot’s base frame

m=p0

b+lR0

bRz(θ)i,(2)

where p0

bis the position of the origin of the frame {b}

in the base frame {0},R0

b∈SO(3) is the transformation

from frame {b}to the base frame {0},Rz(θ)∈SO(3) is a

rotation matrix around Zbaxis, θis the angular position of

the pendulum and i= [1 0 0]>is a unit vector. From now

on, we drop superscript 0for convenience. Differentiating

(2) yields the velocity of the pendulum mass

pm=˙

pb+l˙

θRb

dRz(θ)

dθ i+l˙

RbRz(θ)i.(3)

Given the expressions for the position and velocity of the

pendulum mass we formulate the kinetic K, potential P,

and dissipation Dfunctions necessary to derive the dynamic

equation:

K=1

2˙

m˙

pm, P =1

2kθ2−mg>pm, D =1

2c˙

θ2,(4)

where g= [0 0 −9.81]>m/s2is the gravity acceleration

vector. Using the Lagrange equations

dt ∂L

∂˙

θ−∂L

∂θ =−∂D

∂˙

θ,(5)

with L=K−Pbeing the Lagrangian, and properties of

the rotation matrices, we obtain the ﬁnal expression for the

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摘要：

Anoptimalopen-loopstrategyforhandlingaexiblebeamwitharobotmanipulatorShamilMamedov1;2,AlejandroAstudillo1;2,DanieleRonzani1;2,WilmDecr´e1;2,Jean-PhilippeNo¨el1,JanSwevers1;2AbstractFastandsafemanipulationofexibleobjectswitharobotmanipulatornecessitatesmeasurestocopewithvibrations.Existingapproach...

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An optimal open-loop strategy for handling a ﬂexible beam with a robot manipulator Shamil Mamedov12 Alejandro Astudillo12 Daniele Ronzani12

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