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Abstract

The use of sensors is a primary need in robotic systems. There are sensors that generate a signal whose frequency depends on input stimulus. The application of such sensors is desirable due to their short response time, accuracy, and resolution. For proper use of these sensors, adequate frequency measurement is required. The principle of rational approximations is a method for frequency estimation that has advantages over other measurement methods. Some of them include not a fixed sampling time, insensitivity to jitter, and accuracy limited by the reference stability. Nevertheless, there are some measurement parameters with a not well-researched effect in measurement process. The objective of this work is to elucidate how the phase of input signals (measurand and reference) affects the frequency measurement process.

Keywords

Frequency measurement phase sensors

Introduction

During operation, mobile robotic systems accomplish plenty of signal processing tasks that depend on the robot application. Measurement of physical parameters is the way in which a robot receives or perceives environmental conditions. This awareness of the environment allows the robot to fulfill its purpose.

Most of the novel sensing technologies require the use of piezoelectric material-based sensors; some examples have been reported elsewhere.^1
–3 These kind of sensors work under the piezoelectric effect; as a consequence, when these sensors are stimulated, there is a variation of their frequency.⁴ For this reason, these sensors are known as frequency domain sensors (FDS). Accelerometers are other examples of FDS used in robots.^5,6 Due to the characteristics of FDS devices, measurement of variations in their natural frequency is the best approach for their applications.

Usually in the systems where FDS are used, implementations of frequency measurement methods or specialized equipment are used. Some frequency measurement methods include conventional counters, reciprocal counting, interpolating reciprocal counting, and time-stamping counter⁷; other techniques include the methods of dependent count: method for measurement of absolute values, method for measurement of relative values, and universal method of dependent count⁸; additionally, some of the mentioned techniques have been adapted or implemented for measuring picosecond time intervals.⁹

The direction of robotic development aims to develop ubiquitous applications, while the robots can increase their autonomy and intelligence. Improving measurement capabilities is one step to achieve these tasks. For this reason, in this work, one particular aspect of the “principle of rational approximations” is analyzed. This is a method for estimating the desired frequency from a signal, with plenty of advantages over other frequency estimation methods.^7
–9 These improvements include not a fixed sampling time, insensitivity to jitter, and accuracy limited by the reference stability.^10

–15 The application of such a technique has been proposed for several tasks, like sensors for chemical compounds^16,17 or automotive industry.¹²

The aim of this work is to provide an analysis of how the phase of signals, during measurement, affects the approximation of an unknown frequency. This value can be generated by a sensor embedded in a robotic system.

Theoretical background

The principle of rational approximations is a method for frequency measurement. This technique has its theoretical fundamentals on number theory, particularly, the mediant fractions. During measurement, a digital multiplication of two input signals is required. As stated by Hernandez Balbuena et al.,¹⁰ a signal to measure (S_x) and a reference (S₀) go into an AND gate. This digital multiplication is also known as a signal comparison process. The last generates a third signal ( $S_{x} & S_{0}$ ), where a pulse train is produced. Prior to signal comparison, both signals are discretized; later the pulse width (τ) is adjusted $t_{d} < τ < T_{0}$ , where t_d corresponds to the minimum detectable time allowed by the circuits implementing the principle of rational approximations, and T₀ is the period of reference signal. As reported by previous works,^14,15 the amount of pulses in $S_{x} & S_{0}$ and their duration are defined by the pulse width and the reference (f₀) and unknown (f_x) frequency values.

Considering n as the number of coincidence and P_n and Q_n as the quantity of pulses until the nth coincidence for S_x and S₀, respectively, an approximation to f_x can be obtained in each n coincidence

f_{x} = f_{0} \frac{P_{n}}{Q_{n}}

Considering T₀ as the period of the reference signal, the measurement time (M_t) to until any nth coincidence is calculated as

M_{t} = T_{0} Q_{n}

To evaluate the accuracy of measurements, the relative error β can be calculated as

β = \frac{f_{e} - f_{0} \frac{P_{n}}{Q_{n}}}{f_{e}}

where f_e is the nominal or expected frequency value. An analysis of the principle of rational approximations is shown in Figure 1. In general terms, the usual theoretical considerations were used, which will be discussed below.

Figure 1.

The principle of rational approximations: (a) signal coincidence process during measurand approximation and (b) relative error.

Under the knowledge that a periodic signal is represented as

v (t) = A sin (2 π f t + ϕ)

where A is the signal amplitude or the peak deviation of the function from zero. f is the signal frequency or the number of oscillations (cycles) that occurs each second of time, and ϕ is the phase which specifies (usually in radians) where in its cycle the oscillation is at $t = 0$ .

As stated before, for application of the principle of rational approximations, it is required to “discretize” or to “digitize” both input signals; in this process, it is generated as a signal that is on high level during the time corresponding to pulse width τ. Considering equation (4), its discretized form is shown in equation (5), where T is the period corresponding to f and $ϕ_{t}$ is the phase expressed in terms of time

v (t) = {\begin{matrix} 1 & 0 \leq t < τ \\ 0 & τ < t < T \end{matrix}

Equation (5) generates a periodic square wave that keeps the same frequency as equation (4); for this reason, the phase ϕ can be used to express where the square wave starts using $ϕ_{t}$ , and the phase time can be referred as starting time. Another interesting observation is that the phase can only take values such as $0 \leq ϕ < T$ .

Considering the models provided by equations (4) and (5), the measurand S_x and reference S₀ signals can be modeled in terms of measurement time M_t as follows

S_{x} (f_{x}) = {\begin{matrix} 1 & ϕ_{t x} \leq M_{t} < τ \\ 0 & τ < t < T_{x} \end{matrix}

S_{0} (f_{0}) = {\begin{matrix} 1 & ϕ_{t 0} \leq M_{t} < τ \\ 0 & τ < t < T_{0} \end{matrix}

The parameters $ϕ_{t x}$ and $ϕ_{t 0}$ are the phase of S_x and S₀, respectively. The signal coincidence process of Figure 1(a) illustrates a signal coincidence process when $f_{x} = 5$ MHz, $f_{0} = 8$ MHz, $τ = 62.5$ ns, and $ϕ_{t x} = ϕ_{t 0} = 0$ . The signal coincidence process shows how the overlapping in time generates pulses in $S_{x} & S_{0}$ that have a variable duration. As shown in the literature,^14,15 the coincidence time $t_{0 x}$ is the time during both input signals overlap, and it is the duration of the pulses of coincidence in the signal $S_{x} & S_{0}$ . From Figure 1(a), the variations in duration of $t_{0 x}$ are variable, but for all pulses of coincidence, the values of $t_{0 x}$ are within interval $0 < t_{0 x} \leq τ$ .

For each coincidence, an approximation to f_x is obtained using equation (1). The last allows us to evaluate the accuracy of measurement process through equation (3). Relative error during measurement time is presented in Figure 1(b). As it is expected, β decreases in a time as short as 50 µs. The analysis, as shown in Figure 1, shows the functioning of the principle of rational approximations, which has been reported and analyzed in previous publications.^{10,13

–16,18} An important characteristic is that the mentioned reports lack realism. For this reason, the aim of this work is to evaluate the effect of phase from input signals, which is closer to reality than previous reports.

Phase analysis

In order to evaluate how the phase of input signals affects the measurement process, three different cases can be analyzed: a random phase in S_x or S₀ and a random phase in both signals. Each case was analyzed using the algorithms proposed in a previous publication,¹⁹ after their implementation in MATLAB. But for the herein proposed computational experiments, the phase of input signals was randomized. This was done after the multiplication of signal period with a random number between 0 and 1. Additionally, the phase value in all cases is lower than one-third of the pulse width. The resulting value was used as the starting time of generated signal. Using the corresponding period, each previously exposed case can be analyzed. For the case where random phase is required in S_x or S₀, five signals with random phase were generated. For the third case, five signals with random phase in S_x and S₀ were obtained.

Effect of measurand random phase

For the case where random phase is on S_x, results of experiment are presented in Figure 2. Particularly, the relative error is shown in Figure 2(a). The first evident observation is that β has a value greater than zero at the beginning of measurement time, and it decreases over time. According to the general case, illustrated in Figure 1, the worst approximations are obtained at the beginning of measurement process, when $P_{n} / Q_{n} = 1$ , considering that n has different values ranging from $1$ until a coincidence number when $P_{n} \neq Q_{n}$ . The existence of $P_{n} = Q_{n}$ is due to an apparent phase condition of both input signals, which is lost when M_t increases.

Figure 2.

Effect of measurand (f_x) random phase $ϕ_{t x}$ in frequency measurement process: (a) relative error β and (b) coincidence time $t_{0 x}$ .

In the same way as discussed in the calculations presented in Figure 1, in the herein presented cases of the phase analysis $τ = 62.5 ns$ , the phase value generated during the experiments is presented in Table 1. All the $ϕ_{t x}$ values are shorter than τ, which guarantees that there is only one overlapping for one pulse in S_x and S₀.

Table 1.

Considered measurand phase in each experiment.

Number of experiments	$ϕ_{t x}$ (s)
1	$4.37 \times 10^{- 8}$
2	$3.54 \times 10^{- 8}$
3	$4.21 \times 10^{- 8}$
4	$8.95 \times 10^{- 9}$
5	$9.47 \times 10^{- 9}$

A detailed observation of Figure 2(b) indicates that the maximum peaks corresponding to $ϕ_{t x} = 0$ , $ϕ_{t x} = 9.47 \times 10^{- 9}$ , and $ϕ_{t x} = 8.95 \times 10^{- 9}$ occur at the same M_t. These plots are overlapped in the light blue line that is the closest to zero value of β. Additionally, most of the peaks corresponding to $ϕ_{t x} = 4.21 \times 10^{- 8}$ , $ϕ_{t x} = 3.24 \times 10^{- 8}$ , and $ϕ_{t x} = 4.37 \times 10^{- 8}$ occur at the same measurement time.

In general, the results from analysis of relative error variations during measurement time show that when the phase is in the order of $1 \times 10^{- 9}$ , there is no observable effect in the approximated value of f_x; but in the case when the order is $1 \times 10^{- 8}$ , there is a significant effect on the measured frequency value. In all cases, small variations of the phase, in the vicinities of the order of magnitude, generate the same behavior of β during the same M_t.

Additionally, variations on the phase magnitude order generate different packets of coincidences which lead to uncertainty in the measurement process. Another important observation is that the best approximations generated at the kernel of the packet of coincidences do not fulfill the condition of $t_{0 x} = τ$ , which is stated by Murrieta-Rico et al.¹⁵ But in the case of the best coincidences ( $β = 0$ ), there are the greatest values of $t_{0 x}$ . These phenomena indicate that a variation in the phase of S_x generates a permanent “time” displacement of the best coincidences, but an approximation to zero from the positive values of β.

Effect of reference random phase

In difference to previously studied case, now the effect of random phase $ϕ_{t 0}$ in reference S₀ is explored. Table 2 presents the considered $ϕ_{t 0}$ values, and Figure 3 shows behavior of measurement process when there is random phase in S₀.

Table 2.

Considered reference phase in each experiment.

Number of experiments	$ϕ_{t 0}$ (s)
1	$7.01 \times 10^{- 9}$
2	$1.32 \times 10^{- 8}$
3	$9.07 \times 10^{- 9}$
4	$3.72 \times 10^{- 8}$
5	$2.32 \times 10^{- 8}$

Figure 3.

Effect of reference (S₀) random phase $ϕ_{t 0}$ in frequency measurement process: (a) relative error β and (b) coincidence time $t_{0 x}$ .

In a similar way to $ϕ_{t x} \neq 0$ , two different behaviors when $ϕ_{t 0} \neq 0$ are observed. On the one hand, when $ϕ_{t 0}$ takes values of $7.01 \times 10^{- 9}$ and $9.07 \times 10^{- 9}$ , β values overlap along the same line; as shown before, this is expected of small pulse width variations or a phase condition in the magnitude order of $1 \times 10^{- 9}$ . On the other hand, when $ϕ_{t 0} \neq 0$ and it takes values of $1.32 \times 10^{- 8}$ , $3.72 \times 10^{- 8}$ , and $2.32 \times 10^{- 8}$ , there is an overlapping of the relative error generated with these values (Figure 3(a)). The last phenomenon is characteristic of $ϕ_{t 0}$ corresponding to magnitude order of $1 \times 10^{- 8}$ .

Another important observation is that in the case when phase is in the order of $1 \times 10^{- 8}$ , and it is a deviation in the reference signal, there is an approach from a negative value of β to zero. After observing the coincidence time variations in Figure 3(b), the maximum values of $t_{0 x}$ correspond to $0 \times 10^{- 9}$ , $7.01 \times 10^{- 9}$ , and $9.07 \times 10^{- 9}$ , which explain the “apparent” alignment of the relative error plots, and the non-observable dark blue line generated with $t_{0 x} = 0$ .

Since the reference signal starts with a phase condition difference of zero, the best coincidences have a permanent displacement, but such time overlappings remain the kernel in packets of coincident pulses, which can be observed in the variations of coincidence time $t_{0 x}$ , as shown in Figure 3(b).

Combined effect of measurand and reference random phase

In the last case, random phase values of $ϕ_{t x}$ were considered $ϕ_{t 0}$ . The phase deviations evaluated in each experiment are presented in Table 3. The signal relative error calculated for these experiments is shown in Figure 4(a).

Table 3.

Considered reference phase in each experiment.

Number of experiments	$ϕ_{t x}$ (s)	$ϕ_{t 0}$ (s)
1	$5.16 \times 10^{- 9}$	$8.83 \times 10^{- 9}$
2	$4.71 \times 10^{- 8}$	$3.81 \times 10^{- 8}$
3	$2.09 \times 10^{- 8}$	$2.32 \times 10^{- 8}$
4	$4.15 \times 10^{- 8}$	$6.93 \times 10^{- 9}$
5	$1.14 \times 10^{- 8}$	$4.12 \times 10^{- 8}$

Figure 4.

Combined effect of measurand (S_x) and reference (S₀) random phase ( $ϕ_{t x}$ , $ϕ_{t 0}$ ) in frequency measurement process: (a) relative error β and (b) coincidence time $t_{0 x}$ .

In cases when $ϕ_{t x} > ϕ_{t 0}$ , there is an approximation of β from positive values to zero; on contrary when $ϕ_{t 0} > ϕ_{t x}$ , β approximates from negative values to zero. These observations, as discussed before, are result of the “most delayed” beginning of the signal comparison process, which means that the phase of S_x or S₀ is the “dominant” one.

How the relative error behaves is how the desired frequency is approximated. In other words, if β approaches to zero from positive values, the f_x value approximates from greater to smaller values. This means that the phase condition can be how the relative error approaches to zero, or how direct frequency measurements are generated.

In these experiments, the phase of signals is considered to be a constant value; for this reason, the maximum coincidence time is given as follows

t_{0 x} = τ - (ϕ_{t x} - ϕ_{t o})

when the relative error approaches from above zero to zero; in the case of an approximation from below zero to zero is given by

t_{0 x} = τ - (ϕ_{t 0} - ϕ_{t x})

The validity of equations (8) and (9) can be observed in Figure 4(b), where the variations of $t_{0 x}$ during M_t are presented.

Conclusions

For robotic systems, frequency measurement is a task required for different processes; one case is measurement of frequency generated by sensors. The principle of rational approximations is a tool suitable for quantifying the variations in the frequency generated by a sensor under stimulus.

In this work, the effect of phase in the estimation of a desired signal was presented. The principle of rational approximations’ functioning was evaluated under variations in the phase of measurand and reference frequency. As a result, a relationship in the relative error and phase of input signals was discussed. The presented analysis allows to determine which kind of phase condition is presented.

Footnotes

Acknowledgment

The authors thank the Universidad Autónoma de Baja California y Universidad Nacional Autónoma de Mexico for providing facilities for research and experimentation.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Universidad Nacional Autónoma de México under grant DGAPA-UNAM IN107817, PRODEP, and the Consejo Nacional de Ciencia y Tecnología basic-science proposal A1-S-3349.

ORCID iD

Fabian N Murrieta-Rico

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Effect of phase in fast frequency measurements for sensors embedded in robotic systems

Abstract

Keywords

Introduction

Theoretical background

Phase analysis

Effect of measurand random phase

Effect of reference random phase

Combined effect of measurand and reference random phase

Conclusions

Footnotes

Acknowledgment

Declaration of conflicting interests

Funding

ORCID iD

References