Transfer function of a position sensor.

Andrés Marrugo, PhD
Universidad Tecnológica de Bolívar.

The transfer function of a small position sensor is evaluated experimentally. The sensor is made of a very small magnet and the position with respect to the centerline (see Figure 2.24) is sensed by the horizontal, restoring force on the magnet. The magnet is held at a fixed distance, $h$, from the iron plate. The measurements are given in the table below.


Displacement, d [mm]	0	0.08	0.16	0.24	0.32	0.4	0.48	0.52
Force [mN]	0	0.576	1.147	1.677	2.187	2.648	3.089	3.295

Fig. 2.24 A simple position sensor.

Find the linear transfer function that best fits these data.
Find a transfer function in the form of a second-order polynomial ($y = a+bf+cf^2$), where $y$ is the displacement and $f$ is the restoring force by evaluating the constants $a$, $b$, and $c$.
Plot the original data together with the transfer functions in (1) and (2) and discuss the errors in the choice of approximation.

Solution

Let's begin by plotting the data.



In [1]:

    
import matplotlib.pyplot as plt
import numpy as np
%matplotlib inline

d = np.array([0,0.08,0.16,0.24,0.32,0.4,0.48,0.52])
f = np.array([0,0.576,1.147,1.677,2.187,2.648,3.089,3.295])

plt.plot(f,d,'*')
plt.ylabel('Displacement [mm]')
plt.xlabel('Force [mN]')
plt.show()

We can see that the data is approximately linear, but not quite.

The linear transfer fuction that best fits the data is found by performing a linear fit in the least squares sense. If we go back to the book and review how to carry out linear approximation of nonlinear transfer functions.



In [4]:

    
from IPython.display import IFrame
IFrame('../pdfs/linear-approximation.pdf',
       width='100%', height=400)









    Out[4]:

We see that we need to fit the data to a line with equation $d=af+b$, and we need to compute the coefficients $a$ and $b$ that provides a best fit in the least squares sense.

To do this in python we use the polyfit function.



In [3]:

    
# polyfit computes the coefficients a and b of degree=1
a,b = np.polyfit(f,d,1)

print 'The coefficients are a =',a,'b =',b

d1 = a*f+b
plt.plot(d1,f,':b',label='Fitted line')
plt.plot(d,f,'*')
plt.ylabel('Displacement [mm]')
plt.xlabel('Force [mN]')
plt.axis([0,0.6,0,3])
plt.show()









    



The coefficients are a = 0.157661058706 b = -0.0131058771534

We have obtained the linear fit to the data. Several points are not exactly on the line, therefore there's always an error with respect to the ideal transfer function. Probably a second order fit might be better.

For the transfer function in (2), $y = a+bf+cf^2$, we have to find $a$, $b$, and $c$.



In [5]:

    
# polyfit computes the coefficients a and b of degree=1
c2,b2,a2 = np.polyfit(f,d,2)

print 'The coefficients are a =',a2,'b =',b2,'c =',c2









    



The coefficients are a = 0.00187368040221 b = 0.126233269703 c = 0.00926531112283

Now we plot both transfer functions.



In [9]:

    
d2 = a2+b2*f+c2*f**2
tf2=plt.plot(f,d2,'--k',label='2nd order fit')
tf1=plt.plot(f,d1,':b',label='Linear fit')
tf0=plt.plot(f,d,'*',label='Exact output')
plt.ylabel('Displacement [mm]')
plt.xlabel('Force [mN]')
plt.legend(loc='upper left')
plt.show()

Now let's compute the errors.



In [15]:

    
# Linear fit error
error1 = np.sum(np.abs(d - d1))/len(d)
error1_max = np.max(np.abs(d - d1))
print 'Mean error linear fit: ', error1
print 'Max error linear fit: ', error1_max

# Error fitting to a second order degree polynomial
error2 = np.sum(np.abs(d - d2))/len(d)
error2_max = np.max(np.abs(d - d2))
print 'Mean error 2nd order fit: ',error2
print 'Max error 2nd order fit: ',error2_max









    



 Mean error linear fit:  0.00877563500446
Max error linear fit:  0.0136126887162
Mean error 2nd order fit:  0.00136479637705
Max error 2nd order fit:  0.00234194838585

Owing to the nonlinearity of the data, the average and maximum errors are lower for a second degree polynomial fit (error 2). However, the error for a linear fit can be considered acceptable, while maintaining a simple and linear transfer function.

This page was written in the IPython Jupyter Notebook. To download the notebook click on this option at the top menu or get it from the github repo.