Control of Robotic Systems Project

7/30/2019 Control of Robotic Systems Project

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EE 585

Control of Robotic Systems

FINAL EXAM

By

smail KAYAHAN

18/05/2011


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1. IntroductionIn this project first of all a robot which has two degree of freedom will be implemented into

the SIMULINK program. After this implementation to study on it, computed torque control

will be designed with a PD controller in the outer loop. This computed torque is used for

linearizing the dynamics of the robot structure and by the coefficients of the PD controller the

error of the actual and desired angles will be tried to make zero. After that part admittance

filter will be implemented in to the simulation for a force control. In all analysis in the

admittance control section the torques will be constant for both joints. In the first part of the

admittance control section, the parameters of inertia (Md), viscosity(Bd) and stiffness(Kd)

will be constant values for observing the desired and actual angular positions of the joints

together. Lastly by changing inertia, viscosity and stiffness parameters step by step the actual

angular positions of the joints will be observed and tried to find which parameter or

parameters has an impact on the fixing the position of robot.

2. General Block Diagram of the System

Position Controller

DYNAMICS OF

ROBOT

PLANT

ADMITTANCE

FILTER

-

+

qd

TORQUE

EXTERNAL

ErrorSENSOR

qa , qa, qa

ext

ext

Figure 1-General Block Diagram of the System

In my study because there is no sensor and it is only a simulation on the SIMULINK the

external torque information cannot be feed back to the admittance filter. To realize this

situation a external torque block implemented to the model.


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Results

2.1Results of Part cIn this part a ramp reference trajectory like and is used for desired angles.

The robot is controlled through a computed torque control theory and the desired and actual

angular positions are displayed in the Figure 2. In this graph it can be seen that the control

algorithm works and actual angular positions follow the desired angular positions for both

joints.

Figure 2 - Angular Positions for Actual and Desired Ramp Inputs with Coputed Torque

At this point another important parameter for the control is absolute mean errors of the actual

angular positions and error graphs of the joints. The error graph shows that error cannot

reaches to a steady zero value and it oscillates around zero. However the absolute mean error

is very small with respect to the input.

Joint Absolute Mean Error

q1 1.1079e-004

q2 3.9703e-004


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Figure 3 - Error Plot for Both Joint with Ramp Input

Lastly for this part of the study, the controller gains can be explained.

Control Paramater Gain

Kp 20000

Kv 500


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2.2Results of Part dAt this part external torques are used for the input of the plant but with an admittance

filter. The torques are constant like the filter parameters. The Figure 4 below shows the

desired and actual positions together.

Figure 4- Desired and Actual Angular Positions of the Robot with Admittance Parameters

This figure shows that with the constant torques robots joint angles change differentlybecause of the different torques. Moreover it shows that our robot follows the desired

angle which is determined by the admittance filter precisely.

The figure below indicates the error change of the joints with respect to time. While at the

beginning of the action it has variable error, it becomes steady in a short time with a very

small value with respect to the desired inputs.


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Figure 5 - Error of Joint Positions


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2.3Results of Part e

2.3.1 Different Inertia Parameter

While the viscosity and stiffness parameters are same for two cases, if the inertia

parameter of the admittance filter increases, the position for the angles give a under-

damped response because of the change of the characteristic equation of the filter. In

this case it is reasonable because with the increment of the inertia, the ability of giving

a over-damped response becomes harder.


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2.3.2 Different Viscosity Parameters

While the inertia and stiffness parameters are same for two cases, if the viscosity

parameter of the admittance filter decreases, the position for the angles give a under-

damped response because of the change of the characteristic equation of the filter. In this

case it is also reasonable because with the increment of the viscosity, the ability of giving

a over-damped response becomes easier.


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2.3.3 Different Stiffness Parameters

3. Discussion and ConclusionTwo degree of freedom robot is implemented into the SIMULINK program for simulating its

angular positions and errors with desired inputs. Then a PD Controller with computed torque

control algorithm is designed and the actual positions and errors are observed with the PD

controller parameters..The errors for both joints cannot be zero any time but its mean has very

close value to the 0. After that part admittance filter is implemented in to the simulation for a

force control. In all analysis in the admittance control section the torques is constant for both

joints. In this first part of the admittance control section, the parameters of inertia (Md),

viscosity(Bd) and stiffness(Kd) are constant and with this constant parameters the actual

positions of both joints are observed.

Lastly by changing inertia, viscosity and stiffness parameters step by step the actual angular

positions of the joints is observed and it is seen that the stiffness of the admittance parameter


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has a linear effect on the fixed position value of the angular position. The fixed angular

positions of the joints with different stiffness values are showed in the table below.

Stiffness

Parameter

Fixed Angular

Position of joint 1

Fixed Angular

Position of joint 2

2 0.75 0.5

1 1.5 1

4 0.38 0.25

This table shows that if we increase the stiffness parameter which can be called as spring

coefficient, the fixed angular position decreases linearly. This can be proved by theoretically

as follows.

If we take the laplace of the torque equation of the admittance function with initial conditions

of the angles as zero, it will be

Because torque is constant we can rewrite it as;

Moreover if we divide the equation with M and write it as a function of Q(s);

And finally if we apply final value theorem which determine the fixed point of the angular

position;

So because the torque is constant, if we change the K parameter the fixed point will be

effected inverse proportionally.

Control of Robotic Systems Project

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