1Experimental Validation on Flatness based Control
of Flexible Robot Arm
Elisha D, Markus
Department of Electrical,Electronic and Computer Engineering, Central University of Technology, Bloemfontein,
South Africa
Email:emarkus@cut.ac.za
Abstract—This paper discusses the practical implementation
of a flatness based control for a flexible joint robot arm. Using
differential flatness theory, reference trajectories are generated
for a flexible joint robot and then a tracking controller is
implemented. The vibrations experienced by the robot arm
are sufficiently damped and nonminimum phase behaviour is
eliminated. The control shows fast transcient response as desired
for flexible robots. Experimental results proves the effectiveness
of the flatness based control approach.
keywords: Flexible robot arm, Differential flatness, nonlin-
ear control, trajectory tracking.
I. INTRODUCTION
This study is part of an ongoing research by the authors
on designing state feedback controllers for the flexible joint
robot arm. The experimental implementation of a flatness
based controller for the flexible robot arm is hereby presented
in this paper. The controller eliminates undesired vibrations
arising from the flexible arm while maintaining a fast re-
sponse to reference tip angle trajectories. Flexible joint robot
arms have some advantages when compared to their rigid
counterparts owing to their low inertia, lighter weight, lower
energy consumption, faster movements, compliance, low cost
and wider reach. However, trajectory generation and tracking
for these types of robots is quite tasking. The convention used
in the design of controllers for these classes of systems is
first to linearize the nonlinear dynamics of the manipulator by
feedback linearization [1]–[6]. This enables the use of linear
techniques for controller design. Such approach to controller
design for the flexible manipulator leaves tracking errors since
the nonlinear system dynamics are not fully captured in the
design.
In this paper, the controller design for trajectory tracking
of the flexible joint robot is carried out using differential
flatness. The theory of differential flatness first introduced by
Fliess et al. [7] has been successfully used in motion planning
and control for nonlinear systems [8]–[11]. A major benefit
of differential flatness based control is its ability to simplify
trajectory planning and improve stabilization in task space
[12]–[14]. A system is said to be differentially flat when a
set of variables (called flat output) equal in dimension to the
number of inputs is found for a system such that all the
states and inputs of the system are expressible in terms of
these outputs and their higher derivatives. The flatness property
trivializes exact linearization of nonlinear dynamics as is the
case with robotic dynamics and can significantly reduce the
burden of a robot control problem, as well as the computational
overhead involved [8], [11], [15], [16].
Flatness based control takes advantage of the nonlinear
structure of the system by computing flat output(s) and their
derivatives [7], [17]. The robot arm is made to follow the
trajectory of these outputs which are functions of its states and
inputs [12], [18]. The diffeomorphic property of flat systems,
usually classified by endogenous feedback [18] enables system
trajectories to be generated thereby replacing the tedious
dynamical computations of such systems. The flexible joint
robot arm is modeled and controlled using its flatness property.
Trajectories are then be generated and a linear controller is
designed to track these trajectories as closely as possible. A
similar work to this study found in literature is [19] where the
authors considered vibration control for a flexible link robot
using differential flatness.
II. MATHEMATICAL MODEL
The model used for the study is the standard Quanser
flexible joint manipulator platform [20] shown in Fig. 1. The
robot is oriented horizontally which eliminates gravity, hence
the potential energy due to the springs is zero. The robot arm
is attached to the motor by two linear springs in a tendon-like
fashion. This results in flexibility at the joint. We define θ as
the motor angular displacement and α as the joint twist or link
deflection. The position of the the arm end effector is given
as the sum of the two angles (θ+α) which is our generalized
coordinates. The nonlinear dynamic model of the flexible joint
robot is formulated using Lagrange equations [21].
From the Lagrangian, the energy equation for the flexible
manipulator is formulated as:
L = K − V (1)
where
K = Kh +Kl
V = Vg + Vs (2)
The kinetic and potential energy of the hub and link are
defined as follows:
Kh =
1
2Jhθ˙
2 is the Kinetic energy of the hub
Kl =
1
2Jl(θ˙ + α˙)
2 is the Kinetic energy of the load
Vg = 0 is the potential energy due to gravity
978-1-5090-3334-5/16/$31.00 c©2016 IEEE
2Fig. 1. Experimental Set up
Vs =
1
2Ksα
2 is the potential energy due to the springs
Jh and Jl are the motor and link inertia respectively. m is the
link mass, h is the height of the center of mass of the link.
Ks and g represents the spring stiffness and gravity constant
respectively.
L is now defined as:
L =
1
2
Jhθ˙
2 +
1
2
Jl(θ˙ + α˙)
2 − 1
2
Ksα
2 (3)
The equations of motion according to the Lagrangian will
be:
d
dt
(
∂L
∂
.
α
)
− ∂L
∂α
= −Bα˙ (4)
d
dt
(
∂L
∂
.
θ
)
− ∂L
∂θ
= τ (5)
B and τ represents the generalised forces comprising of
damping due to the springs and torque due to the motor.
Solving equation (4) and (5), we obtain the following
equations:
JL(α¨+ θ¨) +Ksα = −Bα˙ (6)
JLα¨+ (JL + Jh)θ¨ = τ (7)
From Fig. 2, the mesh equation for the armature circuit is:
U = Ui +RI + LI (8)

 

.
θ 
Fig. 2. Motor circuit
Fig. 3. Tip of flexible joint robot arm (Top view)
U is the supply voltage of the motor, Ui is the induced
voltage and I the current through the armature circuit. Rm is
the ohmic resistance. L is the motor windings. For mechanical
systems, the current dynamics is much faster hence may be
neglected. The equation becomes:
U = Ui +RI (9)
Defining a motor constant Ku which includes the gear ratio,
the relationship between Torque and the applied voltage is:
τ =
Ku
Rm
(U −Kuθ˙) (10)
Where
.
θ = w, i = τKu and Ui = Kuw
Fig. 3 illustrates the model of the flexible arm showing the
motor and link deflection angles.
Defining the state variables as:
x1 = α
x2 = α˙
x3 = θ
x4 = θ˙ (11)
Equations 6 and 7, can be represented in the form:
x˙ = f(x) + g(x)u (12)
where [22]
f(x) =
⎡
⎢⎢⎢⎢⎢⎣
x2
−JL+JhJLJh Ksx1 − JL+JhJLJh Bx2 +
K2u
JhRm
x4
x4
Ks
Jh
x1 +
B
Jh
x2 − K
2
u
JhRm
x4
⎤
⎥⎥⎥⎥⎥⎦
978-1-5090-3334-5/16/$31.00 c©2016 IEEE
3g(x) =
[
0
Ku
JhRm
0 − Ku
JhRm
]T
(13)
III. DIFFERENTIAL FLATNESS ANALYSIS OF ARM
A. Differential Flatness Overview
Given a nonlinear system of the form:
x˙ = f(x, u) (14)
where: x ∈ n is the state vector and u ∈ m is the input
vector.
The system in (12) is said to be differentially flat if there
exists a variable or set of variables y ∈ m called the flat
output of the form:
y = h(x, u, u˙, u¨, ......, u(p)) (15)
such that:
x = γ1(y,
.
y,
.
y, ......, y(q)),
and
u = γ2(y,
.
y,
.
y, ......, y(q+1)) (16)
p and q being finite integers, and the system of equations
d
dt
γ1(y,
.
y,
.
y, ......, y(q+1)) =
f(γ1(y,
.
y,
.
y, ......, y(q)), γ2(y,
.
y,
.
y, ......, y(q+1))) (17)
are identically satisfied [17].
B. Determination of the Flat output
Choosing the tip position of the manipulator as [21]
y = θ + α (18)
And using the state representation of equation 12, the
expression for α and θ may be given as [22]:
α¨ = −β1Ksα− β1Bα˙+ β2θ˙ − β3U (19)
θ¨ = β4α+ β5Bα˙− β2θ˙ + β3U (20)
where β1 = JL+JhJLJh Ks, β2 =
K2u
JhRm
, β3 = KuJhRm , β4 =
Ks
Jh
,
β5 =
B
Jh
Adding 19 and 20, we obtain:
y¨ = α¨+ θ¨ = α(β4 − β1Ks) +Bα˙(β5 − β1) (21)
y
-KT
.
(x) (x)y f g u= +
(x)y z=
x
T
v K y= −
Desired 
Trajectories
+
-
y*
1 2
(x) (x)u A A v= +
Fig. 4. Block diagram of flatness based controller design
IV. CONTROLLER DESIGN
Having obtained the flat output and dynamics of the manipu-
lator in terms of the flat output, the controller for for trajectory
tracking will be designed. The control law is chosen to satisfy
the controller using the error dynamics:
e(4) +K3e
(3) +K2e¨+K1e˙ +K0e = 0 (22)
where e = y − y∗,
y∗ is the desired reference trajectory to be tracked.
Ki, i = 0, 1, 2, 3 the controller gains are chosen to have
hurwitz coefficients so that the polynomialP 4+K3P 3+.....K0
has all its root strictly in the left complex plane leading to the
trajectory tracking error dynamics:
s4 +K3s
3 +K2s
2 +K1s+K0 = 0 (23)
Figure 4 shows the block diagram of the flatness based non-
linear feedback controller design for the flexible manipulator.
The resulting controller design is a multi-loop system. The
inner loop linearizes the manipulator dynamics while the outer
loop stabilizes and tracks the trajectories. z(x) represents the
transformation of the robot states to the flat output.
Using the expression for the flat output in equation 18, after
some manipulations, the motor voltage required to drive the
arm in terms of the flat output will be substituted as:
U(t) = Kuy˙ +
[
RmJL+RmJh
Ku
]
y¨
+
[
RmJLB
KuKs
− KuJLKs
]
y(3) − RmJhJlKuKs y(4)
(24)
A. Trajectories of Motion
One key benefit of flatness based control is the simplifica-
tion of trajectory planning and tracking of these trajectories.
Using the flatness property of the manipulator, the desired
trajectories of motion and the input required to track them
from rest to rest could be solved as an interpolation problem
without integrating the system equations. Newton interpolation
method is employed to generate the coefficients of the flat
output polynomial. For the 4th order dynamics of the flexible
manipulator already expressed, the flat output and its deriva-
tives are parameterised at an instant in time t = t1 to another
instant t = t2. The problem is to generate a desired trajectory
of motion for between these two points. The interpolation
polynomial for the fourth order flexible manipulator system
is given by [17]:
y∗(τ) = α0+α1τ+α2τ2+α3τ3+ .......+α2n+1τ2n+1 (25)
978-1-5090-3334-5/16/$31.00 c©2016 IEEE
4where n = 4 and
τ =
t− t1
t2 − t1 (26)
Equation (25)gives the desired trajectory for the flat output as
y∗. Differentiating equation this equation, we obtain:
y˙ ∗ (τ) = α1+2α2τ +3α3τ2+ ....+(2n+1)α2n+1τ2n (27)
and so on.
B. Trajectory Planning
For the fourth order dynamics of the robot arm, the flat
output y is used to derive the reference trajectories for y∗,
y˙∗, y¨∗, and y(3)∗ using the boundary conditions t1 = 0 and
t2 = 14s; y, y˙, y¨, y(3) at t1 = [0, 0, 0, 0] rads respectively and
y, y˙, y¨, y(3) at t2 = [2, 0, 0, 0] rads respectively.
A ninth degree polynomial with ten coefficients was used
for the trajectories. The coefficients of the polynomial were
determined as:
⎡
⎢⎢⎢⎢⎢⎢⎣
a5
a6
a7
a8
a9
⎤
⎥⎥⎥⎥⎥⎥⎦
=
⎡
⎢⎢⎢⎢⎢⎢⎣
1 1 1 1 1
5 6 7 8 9
20 30 42 56 72
60 120 210 336 504
120 360 840 1680 3024
⎤
⎥⎥⎥⎥⎥⎥⎦
−1 ⎡
⎢⎢⎢⎢⎢⎢⎣
a0
a1
a2
a3
a4
⎤
⎥⎥⎥⎥⎥⎥⎦
(28)
The trajectories are used in the controller and the results are
presented in the next section.
V. SIMULATION AND RESULTS
The result in Fig. 5 shows model validation for the flexible
robot arm setup. The result indicates that the parameter
estimation of the variables in simulation is close to the real
system.
The robotic controller set up was tested for disturbance
rejection using arbitrary robot movements and flat output
trajectories. The results show a fast response to disturbances
as shown in Fig. 6 and 7 respectively. This is important to
ensure that vibrations are well dampened and nonminimum
phase behaviour is well compensated in the controller. As can
be seen in the figures, link deflections quickly dampen to zero
with the designed controller. The motor angle is also seen
to reject disturbances in the flat output trajectories which is
desirable. Fig. 8 presents the results of tracking an arbitrary
sine wave. Using the designed controller, the robot arm is seen
to track closely the sine wave movements. The flat output
trajectory tracking is shown in Fig. 9. The results shows θ
movements and α deflections as tracked by the controller. As
shown, a fast response to arm movements is clearly seen. The
robot arm position y is able to move from 0 to 2 radians in
less than a second. The controller successfully tracked these
movements without overshoots or delays. This is despite high
link deflections seen in α. This shows the effectiveness of the
proposed flatness based control.
0 2 4 6 8 10 12 14
u i
n V
-5
0
5
0 2 4 6 8 10 12 14
α
 in
 D
EG
-20
0
20
t in SECS
0 2 4 6 8 10 12 14
θ
 in
 D
EG
-200
-100
0
100
measurement
simulation
Fig. 5. Model validation
0 5 10 15 20 25 30 35 40 45 50
y i
n d
eg
-50
0
50
0 5 10 15 20 25 30 35 40 45 50
α
 in
 D
EG
-40
-20
0
20
t in SECS
0 5 10 15 20 25 30 35 40 45 50
θ
 in
 D
EG
0
20
40
60
measurement
Fig. 6. Disturbance Rejection
VI. CONCLUSION
In this paper, an experimental validation was conducted for
the flatness based control of a flexible joint robot arm. The
nominal control for the robot was designed and stabilized
using differential flatness. The proposed controller is used to
track the reference trajectories that were interpolated using the
flat output. The validated results show fast robot response to
arbitrary movements and disturbance rejection. The tracking
results of the flat output reference trajectories also shows close
tracking performance. These results attest to the effectiveness
of the flatness based control for flexible robots. Further work
will involve control of flexible robots with higher degrees of
freedom.
ACKNOWLEDGMENT
The experiment in this study was carried out at the Systems
Theory and Control Engineering Laboratory at Saarland Uni-
versity Saarbruken, Germany. The authors would like to thank
Prof Rudolph and his team for their invaluable contribution of
978-1-5090-3334-5/16/$31.00 c©2016 IEEE
50 2 4 6 8 10 12
y i
n D
EG
-100
0
100
200
Measured y
Reference yr
0 2 4 6 8 10 12
α
 in
 D
EG
-20
-10
0
10
t in SECS
0 2 4 6 8 10 12
θ
 in
 D
EG
-100
0
100
200
Fig. 7. Disturbance Rejection
t in SECS
98 100 102 104 106 108 110 112
θ
 + 
α
 in
 D
EG
-40
-30
-20
-10
0
10
20
30
40
Measurement
Simulation
Fig. 8. Tracking a sine wave
0 2 4 6 8 10 12 14 16 18
y i
n R
ad
ian
s
-1
0
1
2
Measured y
Reference yr
0 2 4 6 8 10 12 14 16 18
α
 in
 R
ad
ian
s
-2
0
2
t in Secs
0 2 4 6 8 10 12 14 16 18
θ
 in
 R
ad
ian
s
0
100
200
Fig. 9. Tracking the flat output in radians
allowing us to use their robotic laboratory for the experiment.
This work was carried out under the DHET grant supported by
the Central University of Technology Free State South Africa
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978-1-5090-3334-5/16/$31.00 c©2016 IEEE
Dear Mr Elisha Markus, 
Congratulations, your submission, "Experimental Validation on Flatness based 
Control of Flexible Robot Arm", has been accepted for presentation at PRASA 
which is being held 2016-11-30 at Stellenbosch. The reviews that led to this 
decision are attached. 
Please address any reviewer revisions and upload the IEEE formatted, camera 
ready version of your paper by the 26 October 2016. Note that registration 
is required for a paper to be presented and included in the proceedings. See 
http://blogs.sun.ac.za/prasarobmech2016/registrations/ for registration 
details. 
Thank you and looking forward to your participation in this event. 
Best regards, 
Japie Engelbrecht 
Stellenbosch University 
jengelbr@sun.ac.za 
Reviewer 1: 
-----------------------------
Relevance: Broad interest 
Technical and methodological soundness: Average 
Clarity of presentation: Poor 
Originality: An extension of earlier work 
Comments: 
The derivation of the model and controller needs to be made clearer. There 
are some steps in the derivation (e.g. equation 12, 18 and 24) that is very 
hard to follow. More intermediate steps need to be added. Furthermore 
equation 18.1 seems to contradict equation 6. 
The experimental setup needs to be described in more detail. It is very 
confusing if the reader is not familiar with 'the standard Quanser flexible 
joint manipulator'. I suggest moving figure 3 to the introduction. 
The axis labels on figure 9 are incorrect. 
Should 'x3' be present in the last vector in equation 12? 
I assume the alpha in equations 14-16 is not the same alpha used elsewhere. 
This is very confusing. 
The text on figure 4 does not match the symbols used elsewhere in the paper. 
Subject: [2016prasarobmech] Editorial Decision on Paper
Date: 2016/10/08 8:21 PM
From: "Japie Engelbrecht" <jengelbr@sun.ac.za>
To: "Mr Elisha Markus" <emarkus@cut.ac.za>
Page 1 of 2
2016/11/26about:blank
Functions f and g are not defined. 
I suggest a significant rewrite of the modelling and controller design 
sections. 
-----------------------------
Reviewer 2: 
-----------------------------
Relevance: Average interest 
Technical and methodological soundness: Very good 
Clarity of presentation: Average 
Originality: Some original ideas 
Comments: 
* Article is well written with a few clearly visible typing errors 
(unnecessary repetition of words) 
* The dynamic model is derived well, although very basic clearly does 
capture the dominant dynamics. This is proved by means of experiments on a 
practical setup which is good 
* Flatness theory linearizes the non-linear plant successfully. 
* See a glitch or vibration in data of last graph without any discussion. 
-----------------------------
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