## General description

This program system is generated in MATLAB. It is oriented to be an user one so that it can be used even for the future researches in the areas of Robotics, Theory of elasticity and Theory of oscillations, as well as in the area of control of robotic systems with rigid and elastic elements.

The developed program package is used for synthesis and analysis of
manually modeled robot system with 6 degrees of freedom as presented in
Figures 1 and 2. The author of the robot mathematical model, used as a
base for the Program system **EBTLOM** formation, is also Mirjana Filipovic.

The analyzed robotic system is formed with simultaneously elastic motor
gears and elastic links. Movement of a link peak and final movement of
a robot peak can be generated through it. There is a possibility of
synthesis and analysis of the influence of dynamic environmental force
on the movement dynamics of the observed robot. In papers [1] and [2]
the example of robot configuration movement was simulated when in the
first phase a robot moved free from the influence of the environmental
force when in one moment the phenomenon of collision with the
environment and possible stuck occurred and in the next phase a robot
moved under the influence of environmental force dynamics. A collision
moment with the environment was also modeled, as well as possible stuck
during collision or during the contact with the environment (due to
his present friction force). Program system **EBTLOM** was
tested under different conditions in papers [1] and [2]. It was used to
obtain and analyze simulation results which were confirmed by the
established and published scientific theoretical theses related to the
new interpretation of Euler-Bernoulli equation, its solution as well as
a new form of motor mathematical model.

**Fig. 1. **Mechanism of a robot with “two” kinematic pairs moving along the trajectory.

### Area to which technical solution refers

Robotics, Theory of elasticity and Theory of oscillations.

### Problem being solved by the technical solution

This technical solution solves a problem of the efficient implementation of a robot model with rigid and elastic elements in its construction, as well as testing its behavior under the real designed conditions of the robot task realization.

### State of the problem solution in the world

More than 40 years ago, or precisely in 1967, Meiroitch standardized “assumed modes technique”. At that moment it was a very great contribution which should have encouraged scientists to new thinking, new researches and new ideas. Then it was a result of a visionary who interpreted Euler-Bernoulli equation in a completely new way. It should have been a flywheel in the development of the Theory of elasticity and the Theory of oscillations and also Robotics which was in expansion at that time.

The author processed a certain implementation of Euler-Bernoulli equations by which he presumed the elastic deformation to be a value that should be defined in advance by both amplitude and frequency and that way formed elastic deformation was to be entered in the dynamic model. Regarding state of technology and data process possibility at that time, as well as technical needs of the time, this solution was a very great contribution to the science and technic. Even today in some areas of the theory of construction, the regulations which define allowed elastic deformation amplitudes and allowed construction oscillations are applied. These allowed values are the most often the result confirmed by many experiences, measurements and scientific solutions also based on Meiroitch‘s “assumed modes technique”. It should be emphasized that in any case solutions such as Meiroitch’s must not be denied but they should just be further applied in areas where it is not possible to form the system model, e.g. because of its complexity.

Many researchers, not finding any other solutions, applied
Meiroitch’s one in description of the real dynamics of robotic systems
elastic deformations, overlooking the fact that Meiroitch’s solution was
derived under the conditions of predefined elastic deformation by both
amplitude and frequency, or they used many ways to modify Meiroitch’s
solution. Until now the authors implemented elastic deformations as
values based on Meiroitch’s principles and they did not get real values
as a result of robotic systems movement. Not finding any other
soultions, it is obvious that researches were reduced in last 15 years.
It is believed that “assumed modes technique” was and it still is and
it can be useful in some other research areas and that it was misused
in *Robotics, Theory of elasticity and Theory of oscillations.*

### The essence of technical solution

Nowadays development of knowledge of robot system dynamics modeling provides that new models which will treat the elastic deformation as a dynamic value can be set and analyzed. This research area is directed in that way in order to describe this theory in the real environment, without presumptions, i.e. restrictions of elastic deformation on which the present researches have been based.

With a new knowledge collected by many generations ,the intensive development of new technology areas such as robotics, particularly strengthened by the development of technical means of arithmetic data processing, requested and provided that elastic deformation could be observed as a real dynamic value depending on system parameters.

The elastic deformation is a dynamic value by both amplitude and frequency and it is a result of total system movement i.e. external and internal, dynamic and static forces.

Such elastic deformation should exist in the dynamics of robotic system movement. That means that synthesis of robotic system dynamics should be processed based on completely new, different principles comparing to “assumed modes technique” with models that are based on familiar, classic dynamics, theory of elasticity and theory of oscillations, where elastic deformations are described as dynamic values of internal and external load that influence the total dynamics of robotic systems movement.

The Robotics, area we deal with, is very important because modeling of
dynamics of robotic systems movement with rigid and elastic elements
comes directly from it. The Robotics itself is the area that can
provide solution and presents the foundation of further researches in
many other areas. The reason is simple: Robotics has significantly
progressed in last 40 years. The importance of further researches but
now based on new principles defined in papers [1] and [2] as well
as in program system **EBTLOM** should be especially emphasized.

Elastic deformation cannot be defined in advance (by both amplitude and frequency) and thus entered into the system, but completely opposite. Elastic deformation is a dynamic value that depends on the total dynamics of robotic system movement. That means that elastic deformation amplitude and its frequency change depending on forces (inertial, Coriolis, centrifugal, gravitational, as well as coupling forces between present modes, environmental force action) and it also depends on mechanism configuration, mass, segment lengths of reference trajectory selection, dynamic characteristics of motor movement etc. Elastic deformation also exists in a state of inaction and then it depends on gravitational force i.e. mechanism configuration. That means that elastic deformation depends on all characteristics of robotic system and it can be calculated in any selection moment. When we conduct mechanism through reference trajectory, the elastic deformation also exists but now on reference level without the influence of disruption.

Euler-Bernoulli equation was written in 1750. It was derived by Bernoulli, physicist and Euler, mathematician, his lifelong friend and associate. At that time they did not even dream about robotics and knowledge that we now have now. However, although it was created more than 250 years ago, Euler-Bernoulli equation is present even today and it can be very logically connected with today’s robotics knowledge. I especially emphasize robotics that, as mentioned before, is developing very intensively, and in this case it imposes solutions to some other areas such as the Theory of oscillations and the Theory of elasticity. It could be opposite. But that is less important. The most important is that we have solution. It is important to set continuity in researches through years, decades, centuries. Besides, if we analyze the past properly, there is a hope to build the future. One day our descendants will make a critical review of our equations, our assumptions and our interpretation and our program systems and we should be aware of that because we are responsible for what we leave behind ourselves. But we are not responsible only for equations we write but also for the equations we refer to. In this sense choice is very different.

Here we have one of many innovations regarding the known assumptions. In robotics reference trajectory is defined in a clearly kinematics i.e. geometrical way and now in the presence of elasticity elements in defining reference trajectory we can also include values of elastic deformations at the reference level i.e. at the level of knowing elasticity characteristics. The same can be said for defining Jacobian matrix and matrix of transformation.

### Detailed description of characteristics

The procedure for defining inverse and direct kinematics in case when conductor elasticity and segment elasticity are built in the robot construction is implemented within this technical solution. In this research phase there is one mode segment. The selection of Denavit-Hartenberg parameters is made, Jacobian and transformation matrices are formed and on these bases direct and inverse robot kinematics is designed. That is a precondition for design of dynamic robot model. All coupling elements are present through inertial matrix, Coriolis and centrifugal forces, as well as through matrix of stiffness and damping. A new form of motor mathematical model is implemented as well as dynamics of environmental forces. The collision effect is modeled. Robot moves freely in the first phase approaching the environment without restrictions.

Shortly before a second a “collision” happens which acts as a system disturbance. The assumption that the speed of a robot peak can be reduced to zero and that the robot peak gets into the contact with the environment under such conditions is an absolutely idealized case because then the “collision” effect would not even occur. In the real environment, when a robot peak is approaching the environment, the desired peak speed should be as lesser as possible because of the real restrictions (choice of control law, gaps…) and the robot peak should get into contact with the environment at that moment in order to reduce “collision” effect to minimum. The case of “collision” of elastic robotic system with the dynamic environment is designed. The conditions of appearance of “jam” during the “collision” is defined as well as possible appearances of “jam” at the contact of the elastic mechanism with the dynamic environment due to the presence of strong friction.

The phenomenon of “collision” and “jam” is included in the simulation results. It is considered that from the moment of “collision” the robot peak is constantly in the contact with the dynamic environment.

**Fig. 2. **Planned elastic robotic system in contact with the environmental force dynamics.

### References

**[1] **Mirjana Filipović,** "Euler-Bernoulli Equation based on the knowledge of the classical dynamics”, ***Engineering & Automation Problems, International Journal,* No 1 (2009) 18-34.

**[2] **Mirjana Filipović,** " Elastic Robotic System with Analysis of Collision and Jamming ", **7th
International Symposium on Intelligent Systems and Informatics - SISY
2009, Subotica, Serbia (25-26 September 2009), pp. 33-38.