ACCURACY ANALYSIS OF A METHOD FOR DETERMINING THE RELATIVE POSITION OF TWO OBJECTS

V. I. Lukovnikov

(Presented by the scientific-technical seminar of the Department of Electrical Equipment, AEMF)

A known effective method for determining the angle of relative rotation \(\Psi\) of two objects about a common axis uses correlation of the optical image of one of them with its reference standard (mask), located on the other, when torsional oscillations are imparted to the standard [1]. The relative error in determining \(\Psi\) in this case will be determined by the equation

\[ \delta U_\psi=\delta U_{\text{oe}}+\delta t_{\text{emch}}, \tag{1} \]

where \(\delta U_{\text{oe}}\) is the relative error introduced by the optoelectronic part of the device implementing the indicated method of measuring \(\Psi\), and \(\delta t_{\text{emch}}\) is the relative error of the electromechanical part of the device, which ensures a specified law of oscillation (scanning) of the standard.

The quantity \(\delta U_{\text{oe}}\) is determined by noise, aging of elements, drift of operating points, phase shifts, etc., and is usually considerably smaller than the quantity \(\delta t_{\text{emch}}\), which depends on the inaccuracy of the scanning law.

For a distorted sinusoidal scanning law

\[ \varphi(t)=\varphi_m(1+\delta\varphi_m)\sin[2\pi f_{\text{col}}(1+\delta f_{\text{col}})t]+\varphi_m\cdot\delta\varphi \]

it is not difficult to obtain, in the case of determining \(\Psi\) through a reference pulse (the first method according to [1]),

\[ \delta t'_{1\text{emch}}= 1-\{\arcsin[(\Psi^*-\delta\varphi)(1+\delta\varphi_m)^{-1}] + \arcsin[\delta\varphi(1+\delta\varphi_m)^{-1}]\} [(1+\delta f_{\text{col}})\arcsin\Psi^*]^{-1}, \tag{2} \]

and in the case of using clock pulses (the second method according to [1])

\[ \delta t'_{2\text{emch}}= 1-\arcsin[(\Psi^*-\delta\varphi)(1+\delta\varphi_m)^{-1}] \cdot(1+\delta f_{\text{col}}) \cdot\arcsin\Psi^*{}^{-1}, \tag{3} \]

where \(\varphi_m,\ \delta\varphi_m,\ f_{\text{col}},\ \delta f_{\text{col}}\) are the value and relative error due to changes in the amplitude and frequency of oscillations; \(\delta\varphi\) is the relative error due to a change in the position of the reference axis of the oscillation angle \(\varphi\).

Fig. 1

a) \( \delta t'_{\text{emch}} \) versus \( \Psi^* \).
Curves:
1. \( \delta\varphi = 0.2 \)
2. \( \delta\varphi = 0.4 \)
3. \( \delta\varphi = 0.5 \)
4. \( \delta\varphi = 0.6 \)
5. \( \delta\varphi = 0.8 \)
6. \( \delta\varphi = 1.0 \)

\( \delta\varphi_m = 0 \), \( \delta f_{\text{col}} = 0 \).

b) \( \delta t'_{\text{emch}} \) versus \( \Psi^* \).
Curves:
1. \( \delta\varphi = -0.2 \)
2. \( \delta\varphi = -0.4 \)
3. \( \delta\varphi = -0.6 \)
4. \( \delta\varphi = -0.8 \)

\( \delta\varphi_m = 0 \), \( \delta f_{\text{col}} = 0 \).

For a distorted triangular scanning law

\[ \varphi(t)=4\varphi_m f_{\text{col}}(1+\delta f_{\text{col}})(1+x\varphi_m)t-\varphi_m\cdot\delta\varphi \]

(during a half-period), we obtain, respectively,

\[ \delta t''_{1\text{emch}}=\delta f_{\text{col}}+\delta\varphi_m, \tag{4} \]

\[ \delta t''_{2\text{emch}}=\delta f_{\text{col}}+\delta\varphi_m+\delta\varphi\cdot{\Psi^*}^{-1}. \tag{5} \]

A comparison of the scanning laws shows, in terms of accuracy, in

(a)

(b)

Fig. 2

in favor of the triangular one, since when it is used the error \(\delta\varphi\) does not contribute to the total error \(\delta t'_{1\mathrm{emch}}\), if the first method of determining \(\Psi\) is applied. However, the sinusoidal scanning law is optimal from the point of view of dynamics.

A comparison of the methods for determining \(\Psi\) confirms the higher accuracy of the first for both scanning laws. This is especially pronounced for \(\Psi\) close to zero, which is very important. Indeed,

\[ \lim_{\psi \to 0}\delta t'_{1\mathrm{emch}} = 1- \left[ (1+\delta f_{\mathrm{kol}}) \sqrt{(1+\delta\varphi_m)^2-\delta\varphi^2} \right]^{-1}, \]

whereas

\[ \lim_{\psi \to 0}\delta t'_{2\mathrm{emch}} \to \infty . \]

a)

b)

Fig. 3

When \(\delta\varphi=0\), both methods have the same accuracy; that is, expressions (2), (3) and (4), (5) coincide.

Figures 1, 2, and 3 present particular dependences of the errors \(\delta f'_{1\mathrm{emch}}\) and \(\delta t'_{2\mathrm{emch}}\) on \(\Psi^*\), the analysis of which makes it possible to obtain a number of conclusions.

It turns out that, when \(\Psi^*\) and the elementary error \((\delta\varphi,\ \delta\varphi_m,\ \delta f_{\mathrm{col}})\) have the same signs, the magnitude of the total error does not exceed unity for \(\Psi^* > \delta\varphi,\ \delta\varphi_m,\ \delta f_{\mathrm{col}}\). In the opposite case the error may even be infinite.

When \(\Psi\) and the elementary errors have different signs, regions appear, shown in Figs. 1b, 2b, and 3b by dashed lines, in which a number of values \(\Psi^* \le 1\) cannot be measured. The size of these regions depends substantially on the magnitude of the corresponding elementary errors. The minimum of the error \((\delta t'_{1\mathrm{emch}})_{\min}=0\) at the points

a)

Vertical axis: \(z'\)

Horizontal axis: \(\beta_1\)

Axis markings: \(0,\ 20^\circ,\ 40^\circ,\ 60^\circ\); \(0.2,\ 0.4,\ 0.6,\ 0.8\)

Curve labels:

1. \(\alpha_1 = 45^\circ\)
2. \(\alpha_1 = 60^\circ\)
3. \(\alpha_1 = 70^\circ\)
4. \(\alpha_1 = 80^\circ\)
5. \(\alpha_1 = 85^\circ\)
6. \(\alpha_1 = 88^\circ\)

Additional curve label: 7

b)

Vertical axis: \(\delta t'_{\mathrm{EMU}}\)

Horizontal axis: \(\psi^{*}(S)\)

Axis markings: \(0,\ 0.2,\ 0.4,\ 0.6,\ 0.8\); \(0.4,\ -0.8,\ -1.6,\ -2.4\)

Curve labels:

1. \(x_0(\psi^{*})\)
2. \([\delta t'_{\mathrm{EMU}}(+\delta \psi_m)]_0\)
3. \([\delta t'_{1\mathrm{EMU}}(\delta \psi)]_0\)
4. \([\delta t'_{\mathrm{EMU}}(-\delta \psi_m)]_0\)
5. \([\delta t'_{2\mathrm{EMU}}(\psi^{*})]_{\min}\)

Fig. 4

$\dot{\Psi}^{*}=\delta\varphi$ for $\delta\varphi_m=\delta f_{\mathrm{col}}=0$. At the same time, $(\delta t'_{2\mathrm{emc}})_{\min}=0$ is equal to zero only when $\delta\varphi=0$, which is of no practical interest; while for $\delta\varphi\ne 0$ the extremal points are determined by the equality

\[ \varkappa_0=\frac{\arcsin \dot{\Psi}^{*}}{\arcsin(\dot{\Psi}^{*}-\delta\varphi)} =\frac{\sqrt{1-(\dot{\Psi}^{*}-\delta\varphi)^2}}{\sqrt{1-\dot{\Psi}^{*2}}}. \tag{6} \]

The graphical solution of equation (6) is shown in Fig. 4,a, where the notations $\alpha_1=\arcsin \dot{\Psi}^{*}$, $\beta_1=\arcsin(\dot{\Psi}^{*}-\delta\varphi)$ have been introduced. Only the intersection points of the curves $Z'_1=\alpha_1\cdot\beta_1^{-1}\cdot\cos\alpha_1$, where $\alpha_1$ was varied, and $Z'_2=\cos\beta_1$, marked with asterisks, have practical meaning. With their aid the auxiliary curve $\varkappa_0=\varkappa_0(\dot{\Psi}^{*})$ shown in Fig. 4,b (curve 1) was obtained.

Using the dependence $\varkappa_0(\dot{\Psi}^{*})$, the points at which $\delta t'_{2\mathrm{emc}}$ is minimal for $\varphi_m\delta=\delta f_k\delta=0$ can be determined from the expression

\[ \delta\varphi=\dot{\Psi}^{*}\left[1-\sqrt{(1-\varkappa_0^2)\dot{\Psi}^{*-2}+\varkappa_0^2}\right]. \tag{7} \]

Thus, for example, the dependence of $(\delta t'_{2\mathrm{emc}})_{\min}$ on $\dot{\Psi}^{*}$ was obtained, represented in Fig. 4,b by curve 5. Its deviation from the line connecting the extrema of the curves $\delta t'_{2\mathrm{emc}}(\dot{\Psi}^{*})$ in Fig. 2 does not exceed 8%.

For the cases $\delta\varphi=\delta f_{\mathrm{col}}=0$, and also $\delta\varphi=\delta\varphi_m=0$, the minimum of the errors $\delta t'_{1\mathrm{emc}}$ and $\delta t'_{2\mathrm{emc}}$ is observed at $\dot{\Psi}^{*}=0$.

If the comparison is made only for the first method of determining $\dot{\Psi}^{*}$, then it can be seen that the greatest contribution (in comparison with $\delta\varphi$) to the total error is made by the errors $\delta f_{\mathrm{col}}$ and $\delta\varphi_m$ when $\dot{\Psi}^{*}\le 0.707$, when $\dot{\Psi}^{*}$ and the errors have the same signs (curves 2 and 3, Fig. 4,b), and when $0\le \dot{\Psi}^{*}\le 1$, when $\dot{\Psi}^{*}$ and the errors have different signs (curves 4 and 3, Fig. 4,b).

REFERENCES

1. P. I. Gos’kov, V. I. Lukovnikov. Reports of the 6th Scientific-Technical Conference on Problems of Production Automation, vol. 3, Tomsk, 1969.