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Index

1. Introduction
2. Theory
3. System
4. Simulation Approach
5. Observations
6. Analysis & Calculations
7. Result
8. Discussion & Achievements
9. Conclusion
10. References

Summary

In this report we will fulfill the objective of the simulation of the Simulation of a two-link planar anthropomorphic simulator. The purpose of the system, here we will perform the simulation of the two link simulator in which we will use the provided theoretical equations provided in the assignment. With the help of a calculation & simulation software we will design the system based upon the mathematical equations. Following which, we will redesign them in the state space equations where we can deploy them in the Simulink software and thus, complete the task of mathematical modeling and check the result by verifying the graph output values of the system in the scopes via the plot of the scopes. In between the testing of the Simulink model we will variate the value of the system in the friction coefficients but the other parameters will be kept constant as per the shared assignment document.in the conclusion part we will discuss the future aspects of the system along with artificial intelligence in designing the robot controller.

1. Introduction

In this assignment, we will perform the simulation of a two-link planar anthropomorphic simulator arm. The objective of this exercise is to develop the understanding about the modeling of the system and their simulation in the computing software such as MATLAB, Simulink and learn the skill of simulating the systems of robotics to predict the conditions which might affect the system before developing them in actual. The tasks/thought-flow are defined in each of the steps which are sorted out sequentially as follows:

• Theory: Description of the relevant background theory behind the simulation.
• System: Description and diagram of the system
• Simulation Approach: Explanation of procedures and how the simulation was performed.
• Observations: Presentation of observations, including the recording of observations.
• Analysis and Calculations: Analysis of observations, calculations and comparison with theory.
• Results: Presentation of the analysed results, including appropriate graphs (including graphs)
• Discussion and achievement: Logical deductions from analysis and results, achievement and independent contribution.
• Conclusions The major conclusions reached at end of the analysis of the simulation results.
• References: A complete and self-contained reference list

With the above points keeping in mind we will proceed with the exercise and share the results and inferences.

2. Theory

In the exercise of designing the system, we have taken into account the general conditions provided the in the theoretical section. As per the shared assignment document, the two link manipulator (see Ref.1) anthromorphic arm is shown in the vector form .This has helped us in the interpretation of the system and in designing the mathematical model which happens to be exactly as per the equations of the system provided in the assignment. Following is the diagram figure:

As per the shown diagram we have formulated the equations for mathematical modeling of the above system, here the general equations formed as follows:

Where the I, ω, T and Ɵ are the inertia, angular speed, external torque and angular displacement.

From the above equations we have derived that for a unitary mass body will unit motion will be, decided from the above motion equations, here the torques applied will be the response of the servo motor which have to counter the inertia, the mass the length and all the other counter forces including the tip mass of the body located at the top position of the manipulator arm M2. In the above system, following conditions are assumed for the simulation of the system.

Also the gyration(see Ref.2 ) constant with the reference to the center of gravity reference of the 2-link .As the system is designed as per the specifications in the diagram of the assignment , the point needs to be kept in mind that, till now the system is designed in keeping it in an isolated conditions which means that we are ignoring the external disturbances such as the atmospheric pressure ,the height of the object ,the wind direction etc.so ,we will take into account about those conditions in the later sections. But, since the system has included the most core part of the system, we can conclude that the system has quite close similarity with the system to be developed in the real world.

Such type of the approaches is even though have many exceptions, but the ideal model testing predicts the behavior of the system in the case of the developing them. Also the approach of the system is quite optimal because the system has provided the facility of having the testing at various input levels which will provide the systems responses accordingly. Say, for the variable input of the value of the coefficient of friction at various input current can provide us the possibility of the system stability at various current levels. This also provide us the information as to which material of the bearings is best suited for the making of two links of the manipulator.

As the system developed in the isolation, the acceleration due to gravity is kept constant (g=0), it was made so because the inertia of the mass is taken into account along with the center of gravity of the reference is considered in the calculation, along with the calculation Kcg which his nothing but the center of gravity constant with respect to the manipulator base.

So in my opinion the strategy is quite well defined which provide us the flexibility to test the system in most possible circumstances and its response accordingly. Now, moving back to the equations of the system, we converted them in the state space form so that we can make the mathematical model of the system which are as follows:

Where the inertia matrix is defined as below:

And its inverse is:

So from the above equations we can observe that the coefficient of friction and the angular displacement is responsible for countering the inertia of the robot which brings us to the conclusion of finalizing the equations in state space as:

3. System

Now, since the motion and other mathematical equations are finalized, the next approach is to defined the control system modeling, as the Simulink software has the functionality of determining the system parameters via the mathematical modeling, we decided first to design the system based upon the assignment information and following system arises.

For this system we decided that the system must be designed so that the controlled torque can be generated if we deploy the feedback (see Ref.3) strategy, means we will feed back the system with a signal which can cancel out all the disturbances such as the inertia, gravity interferences, friction, manipulator inertia torques and the Coriolis and the other centrifugal forces. This is computed is based upon the Euler –Lagrange method. Also after the countering force of the system, the working forces are provided by the dynamic errors and ensured that they must operate in the prescribed manner.

As shown in the diagram that the qdes signal is fed in to the system, with its derivative is added as the disturbances, and fed into the system, which accordingly generated the system output. This output is fed back to the system along with the desired output which can differentiate the error and can be corrected by the ‘h’ system which is nothing but the Euler-Lagrange gyration system for the correction of the error system and on finals, provide the desired output.

So, as per the above figure the computed control torque is,

Whereas the counter controlled torque is equation is:

So from the above equation we conclude that the system stable and the error signals were countered by the counter signals qdes and qact which thus cancels out the error by feeding back the signal to the system and calibrating it to the desired value.

4. Simulation Approach

In this section, we will now observe the procedure of the simulation as the designing the system is to be done in the Simulink and a systematic approach is required on the same. So for this task we decided to move ahead with the procedure guidelines provided in the assignment.so the guidelines are as follows:

1. First we decided to develop the basic robot model which is the core of the system by employing the state space equations in the previous calculated section.

2. In the next section we moved forward with designing the inverse robotic system in the form of a block of “subsystem”. Here we have used the I-1 robot system state space model. Point to be kept in the mind that g=0 and µ2= µ(Due to similar surfaces in both the link).then this system is connected to the basic robot system as shown in the diagram below.

Fig. (The I-1 Robot Subsystem)

 

Fig. (Robot System)

3. Next, the two link manipulator section is designed, based upon the general case of mathematical equations base line torque on the error and the external torque joints in general, as,

This satisfies the following equation:

Where T_1c & T_2c are the countering torques for the system, in which, the ΔT is the difference between the two torques. e₁ &I₁ are the error signals & the inertia for the two links.

4. For performing the two link manipulator simulation, the modeling requires the inertia equations in which the system values are defined as per the uniform two link system where L is kept as 1m and weight is kept as 1kg each with the tip mass friction coefficient as µ.

For employing the above values, the matrix used for the controlled torques is:

Here the g as earlier told is kept zero. For the inertia values the calculation is done by taking some assumptions. That is:

L₁=L₂=L_1cg=L_2cg=L       m₁=m₂=m    M1=µ₁m    M₂=µ₂m

K²_1cg=k² _2cg=1/12

So on substituting above value we got the equations of inertia as:

₁₁= 5/4(mL²) + mL²(µ₁+µ₂)+1/144

I₂₁= (mL²)/2+µ₂ (mL²)

I₂₂₌ mL²(1/4+ µ₂ )+(1/144)

These values will provide the inertia which then will be included in the torque equation.

5. Now for the simulation part, the system is thus decided to have a movement from (0,0) to (0.6,0.8) which is executed by the movement of the first link up to 6.8º and the outer link up to 240 º. In the above value the system is thus, have variating value of µ1 =0,0.1, µ2= µ=0.1,0.2,0.5,1.0 2.0 and 10. The maximum angular limit of the system is taken to be 30 º.

6. Perform the simulation by providing the simulation values in the system and the signal for a different values and the different simulation results are to be recorded.

Fig. (Complete System)

5. Observations

On performing the above simulation of the system we observed that the dynamics responses are exponential in nature for the input theta of the system. Which means that there is a steady approach for the system is achieved in the system. As seen from the graphs below:

Fig.(Thetad1)

Fig.(Thetad2)

Fig.(V₁)

Fig.(V₂)

Fig.(Theta1)

Fig.(Theta2)

Form the above figures of the system it is clear that for the particular type of the input to the system the system has a linear approach where the system responds to the particular of the transient applied to the system.

From the above graphs we observed that, for a particular kind of observational input, we get particular kind of the output from the system. The behavior of the system is also observed as per the theoretical approach which will be shown in the next section.

6. Analysis & Calculations

From the above responses we have determined some very crucial observations. The observations are based upon the formulation done prior to the modeling of the system and its simulation part.

Say, for the system we have developed,

As per the theoretical calculation assumptions of the system, the inertia we calculated for it are: –

I₁₁= 5/4(mL²) + mL²(µ₁+µ₂) +1/144 = (5/4) +(0.1+0.1) +1/144=1.4569

I₂₁= (mL²)/2+µ₂ (mL²) = 0.5+0.1 = 0.6

I₂₂₌ mL²(1/4+ µ₂ )+(1/144) = (0.25+0.1)+1/144= 0.3569

which on eventually on practical approach achieved as per the following graphs as:

Fig. (I11 Practical Value)

Similarly, for the other values we have obtained the values of the inertia and consequently the torque as the similar values of the system which shows us that the system is behaving the similar way as per the theoretical interpretation of the system.

7. Result

From the observations and the other graph values of the system, we have observed that the system is stable and will perform as per the calculation of the system for the different values of the coefficient of friction of the system among which we have countered various counter forces and we have also determined the additional forces which are considered as disturbances such the error in the destined value and the actual value of the system.

Also the transients observed in the system, which are due to the response of the system against the step input has revealed that the inertia forces, the Coriolis forces and the centrifugal forces (see Ref.5) are present in the system which also affects the systems performances in which the inertia practical graph depicts that the difference in the value of 1.459 to the graph value of 1.5 depicts that there is some additional disturbance which needs to be countered.

Also, the gyration factor ( ) of the system which  feds back the error signal difference in the angular position is calibrated via the formula also shows some variations .Thus, this modeling error needs more calibration via the addition of the integrator which on an average ,will provider us the value of the error so that we can correct the error and can design a better system.

In the working of the system the kinetic and the potential energy of the system is remained conserved as observed by the systems response in the clock wise and anti-clockwise direction and the magnitude will be determined by the graph values, and when substituted in the values:

We got that the system is conserved for the deduced energy of the system thus, proving that the system is stable for the variations.

8. Discussion & Achievements

So from the above results and analyses we have observed that the system is behaving as per the theoretical predications and the formulation of the torque equations and the corresponding angular speed and angular displacement relations. Even though there are some modifications needed for the system to be working perfectly and performing close to the real system but since some physical constraints limits them, but with the additional development of the feedback system which as countered the physical forces has enabled the system to perform the function of the manipulation of the two link system by countering the forces which is determined by the difference of the error signal which is calculated from the matrix system of the error signal.

Similarly the state space modeling of the system has provided us quietly impressive methodology of the system mapping and the analyzing is made easy, but for the disturbances calculation and the consideration of the other interferences, the method needs to modified along with this method.

Also we have observed that the calculation of the, velocity, torques and the system angular displacements shows some variations with respect to the systems determined output.

In the I^-1 robot system the gain of the system has produced singularity errors on the lower values of the coefficient of the friction which concluded that the system has more friction values which needs to be countered and the corresponding system values of the further implementation will accordingly respond in the desired manner.

The input to the system provided by the exponential factor which effects the response time till the step response is reached needs to be observed keenly. This has been, on employing the derivative to the system, has produced the output in the system which deteriorates with the longer span of time.

Along with that, the auxiliary controlling the output of V1 and V2 has the values of ωn which is nothing but the angular speed of the nth order of the system is determined from the differential rate of the rotational movement. The error here arises due to the centrifugal force of the system which is a factor cannot be avoided. As the center of gravity of the system gets shifted by the application of the force on the links, then accordingly to balance the system, this system must be also considered, because the cores disturbance is identified is in the motion of the center of gravity of the system.

9. Conclusion

In my opinion, since the disturbances and the error are of operational nature and they arises on the operation itself, we need to look for a method which can counter them with the development itself .So I would like to recommend the neurological method and the implementation of the artificial intelligence of the system which can evaluate the systems transient behavior  and then accordingly manipulate the systems input ,which ultimately provides the desired output of the system in the form of the motion of the system.

Thus, the use of the manipulator along with the behavior evaluator system can solve the problem of dynamic errors and the provide us the desired output. As the similar system cases are observed in many places and many different sectors, the error is solved by the application of artificial intelligence.

Let’s us have a brief information of the system.

Artificial intelligence

Artificial Intelligence (see Ref.4) or machine intelligence is the intelligence demonstrated by the machines in which the system evaluates and understands the environment in which the machine is present and then accordingly responds. For example,

Let’s say, a machine makes a whole in a wall, if it’s a normal machine, then it will make the whole till it is button is pressed, but a machine with the artificial intelligence first understands the input, then evaluates/processes the task, executes it and finally switches off automatically when completed.

Among the two above applications  the former one will damage itself if not controlled ,whereas the latter one will alert he human about the task and the hazards, and prevents it, this provides us a new domain of technology where we will develop the machine with the brain that can respond as per the circumstances and situations and thus ,in this manner we can have much better systems of development ,such as the flood hit areas can be easily countered by the AI based machines which many times are not been able to provide assistance due to weather conditions, the areas where the humans do not have access. Due to scarcity of the doctors and medical practitioners in the globe can be solved by the help of self-operating AI based systems where the doctor can connect with the machine without actually being present at the location. The modern industrial sections where the pollution and human danger are at par can be replaced by the Ai based systems which not only completes the task but also saves the human lives.

On the final note, the problem of the system of errors can be solved by the application of the AI based system and this can help the manipulator to perform without the error in the systems operational values and the other parameters and can transform from theoretical approach to real world reality.

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