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Efficient monocular vision method used for measuring the angular rate and acceleration in rotation motion

Ming Yang, Hanglei An, Sifan Mo, Huinan Gong, Bingbing Dong, and Guanggui Chen

Open Access Open Access

Abstract

The rotation motion is one of the most basic dynamic units, whose accurate measure-ment is essential to the aspects of inertial navigation, mechanical manufacturing, robot control. Currently, the laser interferometry (LI), the sensor-based and autocollimator-based methods are commonly applied to the rotation motion measurement. However, they are always difficult to meet the requirements of high-accuracy measurement and wide frequency range. In this study, a novel monocular vision-based measurement method is investigated, which determines the angular rate and angular acceleration by using the improved line segmentation detector (LSD) method with sub-pixel accuracy and simultaneously improves the measurement accuracy and frequency range. Additionally, its hardware implementation only requires a simple and flexible vision measurement system. Several comparison experiments with the LI and grating sensor-based method demonstrate that the investigated method can obtain the measurement resolution of 0.0005° and accuracy of 0.480% in the range from 0.001 to 10 Hz.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The rotation motion has two forms: uniform velocity and non-uniform velocity, it is the basic constituent unit of plane and space motions. Currently, the rotation motion is widely existed in the applications of inertial navigation, mechanical manufacturing, precision angular vibration calibration, etc. [1–3]. The accurate measurement of angular rate and acceleration is critical to ensure the performance of these applications. Therefore, it is urgently important to develop an appropriate measurement method for the rotation motion.

The commonly used rotation motion measurement methods include: laser interferometry (LI), sensor-based method (SM) as well as autocollimator-based method (ACM) [4–6]. The LI usually uses a laser interferometer that combines a prism to measure the angular displacement, which can get the accuracy of 1% in a relatively wide frequency range [7,8]. However, this method suffers from the influence of high-cost, low-flexibility, and complex operation [9–11]. The ACM with an autocollimator to get the rotation motion, whose accuracy can highly reach to sub arc-second, while it is only suitable for the static and small angle [12,13]. The SM usually utilizes a circular grating ruler or an angular vibration sensor to achieve the angular displacement measurement, which can obtain the accuracy of 0.1° [14,15], while its accuracy is strongly depended on the characteristic of the used sensor and its frequency range is limited [16,17]. In short, these methods are difficult to meet the increasing demands for the high-accuracy angular vibration measurement and wide frequency range in practical engineering.

Recently, the machine vision method has been increasingly used in the precision angle and angular vibration measurement because of its prominent advantages in flexibility, efficiency, simplify, low-cost, etc. [18–20]. Yang M et al. [21] presented a Zernike moment-based method for measuring the rotation angle in plane motion orbit, which can get the accuracy of 0.098° in the range 0.01-10 Hz. J. J. Lee et al. [22] investigated a dynamic rotational angle measurement method for the large civil structures, whose accuracy is approximately to 1%. W. Li et al. [23] and J. Jin et al. [24] proposed the calibration pattern spot array-based measurement methods to determine the rotational angle, whose accuracy can highly reach to several arc-seconds. H. Dong et al. [25] described the chessboard-based rotational angle measurement method, whose accuracy is about 0.1°. F. Zhang et al. [26] accomplished the rotation angle measurement by applying the spot images, which also can get the accuracy of several arc-seconds. In order to measure the rotation angle with the strong anti-interference, the visual encoder-based method is also popularly investigated. Such as: H. Kim et al. [27] used a RGB camera combines the visual encoder to measure the angular rate, whose upper frequency can get 1000 Hz; X. Jia et al. [28] applied a 24-bit encoder to measure small rotation angle, its accuracy is approximately to 20 arc-second; H. Cheng et al. [29] investigated the visual encoder-based rotation angle and angular displacement measurement method, it is suitable for the small and large angle and its accuracy can highly reach to 0.1%. Additionally, J. Zhong et al. [30,31] developed the linear camera-based measurement method with the composite fringe pattern, which can determine the rotation speed of rotary machinery. Although these vision methods can get certain accuracy and meet the demands at some specific conditions, they are always failed to accomplish the high-accuracy dynamic rotation angle measurement in a wide frequency range with an economical, flexible, and efficient system.

A novel monocular vision-based (MV) measurement method used for the angular rate and acceleration in rotation motion is investigated, which improves their measurement accuracies in a relatively wide frequency range. This MV method obtains the angular rate and acceleration information by measuring the dynamic rotation angle, which ensures the accuracy by adopting the improved line segmentation detector (LSD) method to accurately extract the rotation motion feature edges. The implantation of this investigated method on hardware only requires a low-cost camera and an economic feature mark. Compared with the current measurement methods, the investigated MV method can improve the measurement performance in a considerable frequency range, reduce the cost as well as simplify the operation.

The remainder of this article is organized as follows: Section 2 describes the monocular vision-based rotation motion measurement system. Section 3 presents the proposed measure-ment approach for the angular rate and angular acceleration in the rotation motion. Several comparison experiments and results analysis are provided in Section 4. Section 5 conducts the measurement uncertainty of the investigated monocular vision method for the rotation motion. Ultimately, Section 6 ends with the conclusion.

2. Monocular vision-based rotation motion measurement system

Figure 1 displays a schematic diagram of the MV-based rotation motion measurement system used for the angular rate and angular acceleration. A turntable can provide the uniform rotation motion and the non-uniform angular vibration with a specific displacement amplitude in a wide frequency range. A high-contrast feature mark that consists of four same circles and a rectangle surrounded by these circles is firmly mounted on the working surface of this turntable, which has the consistent rotation motion characteristic with the working surface. A CMOS camera whose optical axis perpendicular to the working surface is used to collect the motion sequence images of this mark.

 figure: Fig. 1.

Fig. 1. Schematic drawing of the monocular vision-based rotation motion measurement system.

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The designed high-contrast feature mark and its corresponding rotation motion edge are shown in Fig. 2. The l in the ROI (Region of Interest) that comprised of the centers of the four circles is the equivalent edge of the two long rectangular edges lL and lR, which has the consistent rotation motion with the working surface of the turntable. The MV method with the improved LSD [32,33] is applied to accomplish the high-accuracy extraction of the edges {lLi} and {lRi} in the mark motion sequence images. Whereafter, the angular rate and angular acceler-ation can be determined by calculating the rotation angles {ri} between the extracted {li} in the collected sequence images.

 figure: Fig. 2.

Fig. 2. (a) Sketch of the designed high-contrast feature mark; (b) the rotation motion edge of this mark.

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When the working surface rotates with the constant linear velocity, its angular rate ωr can be calculated by

$${\omega _r} = {\phi / t}$$
of which, ϕ is the rotation angle, its value ranged from 0° to 360°; t is the corresponding motion time, which is the rotation period when ϕ equals to 360°. Since the non-uniform rotation motion is usually excited by the sinusoidal signal, the angular displacement rs(t) can be expressed as
$${r_s}(t )= {r_{sp}}\cos ({2\mathrm{\pi }{f_v} + {\varphi_i}} )$$
where, rsp, fv, and φi are the amplitude, frequency, and initial phase of rs(t), respectively.

3. Monocular vision-based rotation motion measurement method

The flowchart of MV-based measurement rotation motion method used for the angular rate and angular acceleration is illustrated in Fig. 3. The captured mark motion sequence images are denoted as {F j (x, y)}, where sub-script j = 1, 2, …, N, and N is the number of collected images. Firstly, the template matching method [34] with a series of circles with different sizes is used to reliably determine the ROIs in the mark images captured at arbitrary shooting distance and rotation positions. Secondly, the improved LSD method is utilized to accurately extract the rotation motion edge l j of the jth frame F j (x, y) which is the equivalent rectangular long edges in the mark image. Finally, the angular rate and angular acceleration are calculated by the fitted edges {l j} based on the least square method (LSM).

 figure: Fig. 3.

Fig. 3. Flowchart of the monocular vision-based measurement method used for the angular rate and angular acceleration in the rotation motion.

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3.1 LSD-based rotatory edge extraction with sub-pixel accuracy

In order to eliminate the influences of background similar edges and noises, the improved LSD method extracts the rotation edges only in the determined ROIs of {F j (x, y)}. For each frame F j (x, y), it is first scaled to 80% in order that coping with the aliasing and quantization artifacts. Whereafter, the gradient magnitude G (x, y) at the pixel (x, y) is calculated by using a 2 × 2 mask, and is given by the following form

$$G({x,y} )= \sqrt {g_x^2({x,y} )+ g_y^2({x,y} )}$$
where,
$$\left\{ {\begin{array}{ {c}} {{g_x}({x,y} )= \frac{{{F_j}({x + 1,y} )+ {F_j}({x + 1,y + 1} )- {F_j}({x,y} )- {F_j}({x,y + 1} )}}{2}}\\ {{g_y}({x,y} )= \frac{{{F_j}({x,y + 1} )+ {F_j}({x + 1,y + 1} )- {F_j}({x,y} )- {F_j}({x + 1,y} )}}{2}} \end{array}} \right.$$
and the level-line angle is computed as
$${\theta _{level}} = \arctan [{ - {{{g_x}({x,y} )} / {{g_y}({x,y} )}}} ]$$

According to the set gradient threshold T, the classification of initial line support regions can be obtained by eliminating the pixels whose G (x, y) smaller than T. Then the region growing is achieved by adding the unused pixel in its neighborhood whose level-line angle is equal to the initial region angle θRegion.

A line support region can be approximated as a geometrical rectangle rec whose center (cx, cy) is given by

$$\left\{ {\begin{array}{{c}} {{c_x} = \frac{{\sum\nolimits_{k \in R \textrm{egion}} {G(k )\cdot x(k )} }}{{\sum\nolimits_{k \in R \textrm{egion}} {G(k )} }}}\\ {{c_y} = \frac{{\sum\nolimits_{k \in R \textrm{egion}} {G(k )\cdot y(k )} }}{{\sum\nolimits_{k \in R \textrm{egion}} {G(k )} }}} \end{array}} \right.$$
where, G (k) is the gradient magnitude of pixel (x (k), y (k)) in the region; x (k) and y (k) are the corresponding horizontal and vertical coordinates, respectively. The rectangle's main orienta-tion angle is set to the angle of the eigenvector corresponded to the smallest eigenvalue of the following matrix
$${\boldsymbol M} = \left[ {\begin{array}{{cc}} {{m^{xx}}}&{{m^{xy}}}\\ {{m^{xy}}}&{{m^{yy}}} \end{array}} \right]$$
of which,
$$\left\{ {\begin{array}{{c}} {{m^{xx}} = \frac{{\sum\nolimits_{k \in R \textrm{egion}} {G(k )\cdot {{({x(k )- {c_x}} )}^2}} }}{{\sum\nolimits_{k \in R \textrm{egion}} {G(k )} }}}\\ {\frac{{{m^{xx}} = \sum\nolimits_{k \in R \textrm{egion}} {G(k )\cdot {{({y(k )- {c_y}} )}^2}} }}{{\sum\nolimits_{k \in R \textrm{egion}} {G(k )} }}}\\ {\frac{{{m^{xy}} = \sum\nolimits_{k \in R \textrm{egion}} {G(k )\cdot ({x(k )- {c_x}} )({({y(k )- {c_y}} )} )} }}{{\sum\nolimits_{k \in R \textrm{egion}} {G(k )} }}} \end{array}} \right.$$

The validation of a rectangle is depended on the pixels in this rectangle whose θlevel are equal to the its main orientation angle. Therefore, the number of false alarms (NFA) is adopted to validate the effectiveness of the approximated rectangles. In order to avoid the phenomenon that two straight lines constituted a rectangle occurred, the aligned points density of arbitrary rec is calculated by the formula

$$d = \frac{{ka}}{{\textrm{length}(rec) \cdot \textrm{width}(rec)}}$$
where, ka is the number of aligned points. When the d of the rec is greater than or equals to 0.7, it can be taken as a reasonable rectangle. Otherwise, this rec will be divided into two smaller rectangles. Finally, the endpoint coordinates of these rectangle center lines were obtained, i.e., the sub-pixel edge endpoints of the rotation edges.

3.2 Angular rate and angular acceleration measurement

The extracted sub-pixel rotation edge endpoints of the {F j (x, y)} by the LSD method described in Section 3.1 can be denoted as {l j (x, y)}. Whereafter, the LSM is used to fit the edge endpoint coordinates of l j (x, y) and obtain its fitting edge line l j. In order to simplify calculating, the first frame F 1 (x, y) is selected as the reference frame, whose rotation edge corresponds to angle is taken as zero. Ultimately, the angles {θj} corresponding to the {F j (x, y)} is calculated by using the slope of the fitting edge lines {lj}.

The angular displacements rs (tj) of the turntable at the corresponding sampling time tj can be obtained by the calculated θj via considering its quadrantal distribution. In each rotation motion period, the angular rate ωr can be solved by

$${\omega _r} = {{\sum\limits_{j = 1}^N {{r_s}({{t_j}} )} } / {\sum\limits_{j = 1}^N {{t_j}} }}$$
it is worth noting that when the time is greater than one period, the corresponding rotation angle must be compensated for the n·360°, and n is the period number. Since the excitation signal of the turntable is sine, the angular acceleration ra (tj) at the sampling time tj can be calculated by the following formula
$${r_a}({{t_j}} )= \omega _v^2{r_s}({{t_j}} )$$

In order to get the angular acceleration amplitude and initial phase, the sine approximation method (SAM) described in [35,36] shown in formula (12) is adopted to fit the N acceleration {ra (tj)} and their corresponding sampling time {tj}.

$${r_a}({{t_j}} )= {A_a}\cos ({{\omega_v}{t_j}} )- {B_a}\cos ({{\omega_v}{t_j}} )+ {C_a}{t_j} + {D_a}$$
where, Aa and Ba are the corresponding sinusoidal acceleration components; Ca is an ultra-low-frequency linear disturbance; and Da is an offset. The SAM applying Ca and Da to eliminate the influences of the disturbance and offset, thus to improve the accuracy of acceleration amplitude measurement. The Aa, Ba, Ca, and Da are obtained by solving the overdetermined equations that constituted by the N acceleration ra (tj) and their sampling time tj. Finally, the fitted angular acceleration amplitude rap can be got by
$${r_{ap}} = \sqrt {A_a^2 + B_a^2}$$
and the corresponding initial phase φai can be calculated by arctan (Ba/Aa).

4. Experiment verifications and results analysis

4.1 Experimental setup

Figure 4 displayed the MV-based rotation motion measurement experimental setup used for the vertical turntable. The turntable (NIM-RT010201) can provide the uniform and non-uniform rotation motions in the range from 0.001 to 10 Hz. The high-contrast mark that comprised of four same circles with diameter of 15 mm and a rectangle with the size of 60 mm x 15 mm was mounted on the working surface of this turntable. The CMOS camera (MER2-160-227U3M) with the maximum resolution 1.6 megapixel and maximum frame rate 227 fps to collect this mark’s motion sequence images. The processing units (LENOVO Y9000X) with the RAM 16 GB and CPU i7-10875H was used to process the collected sequence images.

 figure: Fig. 4.

Fig. 4. Setup for the monocular vision-based rotation motion measurement: (I) turntable, (II) high-contrast mark, (III) isolation platform, (IV) CMOS camera, (V) processing units.

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Additionally, the SM with a circular grating ruler (REXM20) and the LI with a heterodyne interferometer (Polytec OFV 5000), were adopted to verify the performance of the investigated MV method for angular rate and acceleration measurements. The CMOS camera with the frame rate 1-200 fps to collect the mark’s motion sequence images in the range of 0.001-10 Hz. In order to guarantee the measurement accuracy of the rotation motion, the number of collected images and sampled periods was set at least 20 and 5, respectively.

4.2 Angular rate measurement results

Since the LI is difficult to measure the full angle when the turntable produces uniform motion, only the SM was also selected to validate the measurement accuracy of the investigated MV method for angular rate. Figure 5 shown the measured angular rate results by the SM and MV methods at the frequencies between 0.001 and 10 Hz, ten times repeat measurement were accomplished at each frequency, as we shown in Dataset 1 (Ref. [37]). In the whole range, the angular rate of the MV method was highly agree with that of the SM, and their maximum absolute relative deviation is less than 0.3%. The maximum relative standard deviation (RStd) of the MV method was 0.235%, which is slightly less than the corresponding 0.297% of the SM.

 figure: Fig. 5.

Fig. 5. The measured angular rate by the SM and investigated MV method in the range of 0.001-10 Hz: (a) the average value, (b) the relative standard deviation.

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4.3 Angular acceleration measurement results

In order to fully validate the measurement accuracy of the investigated MV method for angular acceleration, the LI and SM were simultaneously applied to the angular acceleration. The tested frequencies were selected according to the vibration calibration regulation in the range from 0.001 to 10 Hz, and the angular acceleration at each frequency was measured by these methods ten times, respectively. Figure 6 shown the measured angular acceleration results by the SM, LI, and investigated method, as we shown in Dataset 2 (Ref. [38]). In the entire range, the measured acceleration amplitude average values by the SM, LI, and MV method were almost overlapped, because the results of the MV method were highly similar to those of the SM and LI, and their corresponding relative deviations were less than 0.200% and 0.149%, respectively. The maximum RStd of the MV method in the range of 0.001-10 Hz was 0.225%, which is less than the corresponding 0.257% of the SM and 0.314% of the LI. Although the RStd of the MV method slightly increased when the frequency is greater than 2 Hz, it is also can meet the measurement accuracy demand of the actual rotation motion.

 figure: Fig. 6.

Fig. 6. The measured angular acceleration amplitudes by the LI, SM, and investigated MV method in the 0.001-10 Hz: (a) the average value, (b) the relative standard deviation.

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4.4 Discussion

As shown in Figs. 5(b) and 6(b), the RStd of the investigated MV method was slightly less than that of the SM at a few higher frequencies. The reason for this phenomenon is the angular displacement amplitude of turntable’s working surface rapidly decreased when the frequency is greater than 2 Hz, the relative resolution of the circular grating ruler used in the SM is slightly insufficient compared with that of the MV method by reducing the distance between the camera and turntable’s working surface. Additionally, since the measurement repeatability of the LI is affected by the slight angular velocity at lower frequencies, the RStd of the MV method was less than that of the LI in the range from 0.001 to 0.01 Hz. The RStd of these three methods were highly resembled and less than 0.1% at the frequencies between 0.01 and 2 Hz, this is due to they have the consistent measurement resolution in this range.

The test frequency of the investigated MV method in this experiment was limited to the range of 0.001-10 Hz, because the repeatability of the LI will be significantly decreased when the frequency is too low and that of the SM is worse when the frequency is greater than 10 Hz. The frequency of this MV method even can reach to the quasi-static, and obtained the similar accuracy with that in the range of 0.001-10 Hz. Actually, its upper frequency can also easily higher than 20 Hz by shortening the shooting distance between the CMOS camera and the turntable’s working surface, using the telecentric lenses or the CMOS camera with higher frame rate. In this experiment, the circle mismatch of the mark image will appear when the rotational speed exceeds 9000 r/min. The reliable measurement of higher speed has to simultaneously reduce the mark size and shooting distance.

5. Angular acceleration measurement uncertainty

In order to further illustrate the measurement accuracy of the investigated MV method, its corresponding uncertainty evaluation for the angular acceleration was achieved. The turntable distortion, the motion feature extraction, the environmental noises and disturbances, and the repeatability in the whole measurement procedure were taken as main uncertainty sources, as provided in Table 1. The uncertainty components of these sources were assessed according to the recommended methods in the GUM, then their relative contributions were obtained by their statistical distributions. According to the propagation criterion, the expanded measurement un-certainties of the MV method for angular acceleration was 0.480%. Compared with the reported accuracy in the relevant literatures, this uncertainty is able to meet the high-accuracy measure-ment demand of rotation motion.

Tables Icon

Table 1. The evaluated measurement uncertainty of the investigated MV method for the angular acceleration in rotation motion

View Table

6. Conclusion

This study investigated a novel monocular vision-based measurement method, which is able to determine the angular rate and angular acceleration in the rotation motion simultaneously. This investigated method utilizes the improved LSD method to accurately extract the rotary feature edges of motion sequence images, which can improve the measurement accuracy and frequency range. The implementation of the investigated method on hardware only requires an economic feature mark and a CMOS camera. Comparison experiments with the commonly used LI and SM confirmed that the investigated method has the ability to obtain the satisfactory accuracy in the range from 0.001 to 10 Hz. The uncertainty of the investigated method for the angular acceleration measurement is assessed, which can reach to 0.480%. However, the proposed MV method is slightly sensitive to the illumination variation and its upper frequency is usually limited to dozens of Hz, and the shooting distance has to be adjusted to ensure the measurement accuracy at the smaller angular displacements. In the future, we concentrate on applying the investigated method to achieve the higher frequency rotation motion measurement as well as high-accuracy angular vibration calibration and traceability.

Funding

National Key Research and Development Program of China (2017YFF0205003); National Natural Science Foundation of China (52265066, 62203132).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. Dataset 1 can be found in [37]. Dataset 2 can be found in [38].

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37. M. Yang, “The measured angular rates by the SM and MV method,” figshare (2022) [retrieved 08 October 2022], https://doi.org/10.6084/m9.figshare.21299820

38. M. Yang, “The measured angular accelerations by the SM, LI, and MV method,” figshare (2022) [retrieved 08 October 2022], https://doi.org/10.6084/m9.figshare.21299823.

Supplementary Material (2)

NameDescription
Dataset 1       The measured angular rates by the SM and MV method.
Dataset 2       The measured angular accelerations by the SM, LI, and MV method.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. Dataset 1 can be found in [37]. Dataset 2 can be found in [38].

37. M. Yang, “The measured angular rates by the SM and MV method,” figshare (2022) [retrieved 08 October 2022], https://doi.org/10.6084/m9.figshare.21299820

38. M. Yang, “The measured angular accelerations by the SM, LI, and MV method,” figshare (2022) [retrieved 08 October 2022], https://doi.org/10.6084/m9.figshare.21299823.

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Figures (6)
Fig. 1.
Fig. 1. Schematic drawing of the monocular vision-based rotation motion measurement system.

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Fig. 2.
Fig. 2. (a) Sketch of the designed high-contrast feature mark; (b) the rotation motion edge of this mark.

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Fig. 3.
Fig. 3. Flowchart of the monocular vision-based measurement method used for the angular rate and angular acceleration in the rotation motion.

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Fig. 4.
Fig. 4. Setup for the monocular vision-based rotation motion measurement: (I) turntable, (II) high-contrast mark, (III) isolation platform, (IV) CMOS camera, (V) processing units.

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Fig. 5.
Fig. 5. The measured angular rate by the SM and investigated MV method in the range of 0.001-10 Hz: (a) the average value, (b) the relative standard deviation.

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Fig. 6.
Fig. 6. The measured angular acceleration amplitudes by the LI, SM, and investigated MV method in the 0.001-10 Hz: (a) the average value, (b) the relative standard deviation.

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Tables (1)

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Table 1. The evaluated measurement uncertainty of the investigated MV method for the angular acceleration in rotation motion

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Equations (13)

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ω r = ϕ / t
r s ( t ) = r s p cos ⁡ ( 2 π f v + φ i )
G ( x , y ) = g x 2 ( x , y ) + g y 2 ( x , y )
{ g x ( x , y ) = F j ( x + 1 , y ) + F j ( x + 1 , y + 1 ) − F j ( x , y ) − F j ( x , y + 1 ) 2 g y ( x , y ) = F j ( x , y + 1 ) + F j ( x + 1 , y + 1 ) − F j ( x , y ) − F j ( x + 1 , y ) 2
θ l e v e l = arctan ⁡ [ − g x ( x , y ) / g y ( x , y ) ]
{ c x = ∑ k ∈ R egion G ( k ) ⋅ x ( k ) ∑ k ∈ R egion G ( k ) c y = ∑ k ∈ R egion G ( k ) ⋅ y ( k ) ∑ k ∈ R egion G ( k )
M = [ m x x m x y m x y m y y ]
{ m x x = ∑ k ∈ R egion G ( k ) ⋅ ( x ( k ) − c x ) 2 ∑ k ∈ R egion G ( k ) m x x = ∑ k ∈ R egion G ( k ) ⋅ ( y ( k ) − c y ) 2 ∑ k ∈ R egion G ( k ) m x y = ∑ k ∈ R egion G ( k ) ⋅ ( x ( k ) − c x ) ( ( y ( k ) − c y ) ) ∑ k ∈ R egion G ( k )
d = k a length ( r e c ) ⋅ width ( r e c )
ω r = ∑ j = 1 N r s ( t j ) / ∑ j = 1 N t j
r a ( t j ) = ω v 2 r s ( t j )
r a ( t j ) = A a cos ⁡ ( ω v t j ) − B a cos ⁡ ( ω v t j ) + C a t j + D a
r a p = A a 2 + B a 2
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