Non-Linear and Linearized System Analysis of Robotic Arm

PROJECT PROBLEM
This projects seeks to investigate non-linear system is both linearized version and nonlinear
version. Robotic arm has been used to study the aforementioned. In the robotic arm, the output is
the desired controlled angle of levers displacement. The state space model of the robotic arms
has to be formulated and the simulation performed in the Matlab.
BRIEF REVIEW OF THE METHODOLOGY
All physical systems that does not obey the principle of superposition are referred to as non-
linear systems.
In practice, most of the existing systems in control engineering are non-linear systems whose
composite of differential equations are made of non-linear variables (Poola , 2019).
The continuous system described as a linear system is defined by linear differential
equations that consist of constant coefficient as shown in the equation below.
dn y ( t )
dt n + an−1
dn−1 y ( t )
dt n−1 +...+ a1
dy ( t )
dt +ao y ( t ) =u( t)
In the equation above,
n−i s the syste m' s order
y−is the output of the system
And u−i s theinput ¿ the sytem , which is the external influence .
Besides the system being influenced by the external conditions, some system have
internal inbuilt conditions that might influence the behavior of the system. This
internal conditions are called initial condition, represented by the derivative
expressions below.
y ( to ) , dy ( to )
dt and dn−1 y ( to )
d tn−1
In state space model, linear system is represented by the equation below;
̇x (t )= Ax (t)
The general expression of non-linear system is as shown in the equation below.
̇x ( t ) =f (x , t)
Dynamically, f ( x , t) is the component of a nonlinear function for the state x ( t )
unlike in the linear system where;

f ( x , t ) =A
Non-linear systems are transformed into linear system within linear operating
ranges for easier analysis, modelling and implementation of the actual system,
which in real sense are non-linear. Linearization of nonlinear system is procedurally
computed using three basic concepts namely; the point of equilibrium, Taylor series
and the Jacobian.
The concept of equilibrium point.
Equilibrium point is the specific operating point of the system determined by
Jacobian linearization of the systems that are non-linear. Given a non-linear system
f ( x), the equilibrium point xo is chosen for the nonlinear system if and only if the
dynamics of the chosen point is equal to zero. Mathematically;
f ( xo ) =0
Alternatively;
x ( to ) =xo
In a nutshell, the system dynamic would remain stuck at the equilibrium point
forever in the future time. The dynamics behavior of the system at the equilibrium
point is similar to the behavior of the system at the point in the neighborhood of the
equilibrium point.
Concept of Taylor series.
Linearization is literally approximation of a non-linear system which is mainly
computed by Taylor series expansion of the related polynomial equations. For
instance, the function f ( x , u ) is approximated with Taylor series and neglecting
higher order terms becomes;
f ( x , u ) =f ( x0 , u0 ) + ∂ f ( x , u )
∂ x |( x0 ,u0 )
( x−x0 ) + ¿ ∂ f ( x , u )
∂ u |( x0 ,u0 )
( u−u0 )
Dynamic equations of the system are evaluated using this Taylor’s series
MATHEMATICAL LINEARIZATION OF THE MODELLING
The mathematical model if the single link robotic manipulator with a flexible join is given by the
expression below.

I ̈θ1+ mgl sinθ1+ k ( θ1−θ2 ) =0
J ̈θ2−k ( θ1−θ2 ) =u
Where θ1 and θ2are angular positions, I , J are moments of inertia, m∧l are the link’s mass and
length respectively, k is the link’s spring constant. Introducing the change of variables as
x1=θ1 , x2= ̇θ1 , x3=θ2 and x4 = ̇θ2
The manipulator’s state space nonlinear model equivalent model is given by
̇x1=x2
̇x2=−mgl
I sin x1− k
I ( x1−x3 )
̇x3=x4
̇x4 = k
J ( x1− x3 ) + 1
J u
Taking the nominal points as ( x1 n , x2 n , x3 n , x4 n , un ), then the matrices A and B become.
A=
[ 0 1 0 0
−k +mgl sin x1 n
I
0
k
J
0
0
0
k
I
0
−k
J
0
1
0 ] , B=
[ 0
0
0
1
J ]
Assuming that the output variable is equal to the link’s angular position that is
y=x1
The matrices C and D are given as
C= [ 1 0 0 0 ] D=0
Mathematical modelling of non-linear equations
The following numerical values are used for system parameter
mgl=5, I =J=1, k =0.08
Substituting the values.
The first dynamic
̇x1=x2

The second dynamic
̇x2=−mgl
I sin x1− k
I ( x1−x3 )
̇x2=−5
1 sin x1− 0.08
1 ( x1−x3 )
̇x2=−5 sin x1−0.08 x1 +0.08 x3
The third dynamic
̇x3=x4
The forth dynamic
̇x4 = k
J ( x1− x3 ) + 1
J u
̇x4 =0.08 x1−0.08 x3 +u
Non linearized dynamic equations of the model are summarized as shown below;
̇x1=x2
̇x2=−5 sin x1−0.08 x1 +0.08 x3
̇x3=x4
̇x4 =0.08 x1−0.08 x3 +u
Linearizing the non-linear system
Linearization sequence was executed by finding the equilibrium point then applying partial
derivative using Taylor’s series to solve for the dynamic values of the state polynomial equation.
Finding the equilibrium points in terms of ( x10 , x20 , x30 , x40∧u10).
From the polynomial equations of the non-linear system, all state variables are modified as
shown below.
x (t )=xo ( t )+ ∆ x
y ( t ) = yo ( t ) + ∆ y
u ( t )=uo ( t )+ ∆ u
The state equations therefore become;
̇x1=x20 ( t ) +∆ x2