An FPGA-Based Trigonometric Kalman Filter Approach for Improving the Measurement Quality of a Multi-Head Rotational Encoder

Abstract

This article introduces an advanced theoretical approach, named the Trigonometric Kalman Filter (TKF), to enhance measurement accuracy for multi-head rotational encoders. Leveraging the processing capabilities of a Field-Programmable Gate Array (FPGA), the proposed TKF algorithm uses trigonometric functions and sophisticated signal fusion techniques to provide highly accurate real-time angle estimation with rapid response. The inclusion of the Coordinate Rotation Digital Computer (CORDIC) algorithm enables swift and efficient computation of trigonometric values, facilitating precise tracking of angular position and rotational speed. This approach represents a notable advancement in control systems, where high accuracy and minimal latency are essential for optimal performance. The paper addresses key challenges in angle measurement, particularly the signal fusion inaccuracies that often impede precision in high-demand applications. Implementing the TKF with an FPGA-based pure fixed-point method not only enhances computational efficiency but also significantly reduces latency when compared to conventional software-based solutions. This FPGA-based implementation is particularly advantageous in real-time applications where processing speed and accuracy are critical, and it demonstrates the effective integration of hardware acceleration in improving measurement fidelity. To validate the effectiveness of this approach, the TKF was rigorously tested on a precision drive control system, configured for a direct PMSM drive in an astronomical telescope mount equipped with a standard $0.5$m telescope frequently used by astronomers. This real-world application highlights the TKF’s ability to meet the stringent positioning and measurement accuracy requirements characteristic of astronomical observation, a field where minute angular adjustments are critical. The FPGA-based design enables high-frequency updates, essential for managing the minor, precise adjustments required for telescope control. The study includes a comprehensive computational analysis and experimental testing on an Altera Stratix FPGA board, presenting a detailed comparison of the TKF’s performance with other known methods, including fusion techniques such as differential methods, $\alpha$–$\beta$ filters, and related Kalman filtering applied to one sensors. The study demonstrates that the four-head fusion configuration of the TKF outperforms traditional methods in terms of measurement accuracy and responsiveness.

1. Introduction

Over the past century, the primary focus of astronomical facility development has been the construction of increasingly large telescopes, enabling the observation of progressively fainter celestial objects. This trend reflects the pursuit of enhanced light-gathering power and higher resolution and positioning, critical for advancing our understanding of distant and dim phenomena in the universe. This approach has been a spectacular success, but as the optical precision of telescopes has increased, the requirements for precision in manufacture have increased dramatically, as well as those for the precision in setting and changing the position of the lens [1]. A very important aspect of astronomical observations is the accuracy and precision of the observation devices located on Earth [2,3]. This is particularly crucial because when observing distant objects in the sky, even the slightest deviation or inaccuracy can lead to significant errors in the data collected. High accuracy is essential to capture detailed and reliable information about celestial bodies. An additional challenge lies in tracking these objects, which involves the precise and coordinated movement of the rotational axes of astronomical instruments, such as telescopes. In this case, not only is the accuracy of the instruments paramount, but the speed and responsiveness of the system—its ability to quickly adjust and follow the movement of objects across the sky—become equally critical. The dynamics of the measurement process, including the rapid determination of position and continuous real-time adjustments, are essential for ensuring that the instruments can maintain focus on fast-moving or distant objects, enhancing both the quality and reliability of the observations. One of the recent trends is the implementation of robotic telescopes. This is often associated with a network of telescopes viewing the same phenomenon, thus increasing accuracy and resolution. Also, the takeover of the observation by the next instrument on time in the event of the field of observation being blocked or the exit of the object of interest is important [4,5]. Tracking satellites, especially those in low Earth orbits, poses a considerable challenge in the design of control drive systems. At angular velocities nearing 2 deg/s, maintaining tracking accuracy better than $1^{″}$ becomes crucial. This precision is dictated by the scale of the image projected onto the camera sensor. As a general approximation, 1 arcsecond typically corresponds to about one pixel on the camera sensor, although this value depends on the specific characteristics of the camera and the telescope’s focal length [6,7].

As mentioned, the mechanical accuracy of the measurement is very important. In many cases, it is not possible to obtain it using only one position sensor. Therefore, methods based on the synergy of measuring sensors are used. In recent years, there has been a substantial surge in the attention given to multisensor data fusion, particularly for industrial applications demanding high dynamics and precision. As part of the work carried out, the use of the Kalman filter was proposed and focused on [8]. The concept of the Kalman filter is to utilize a model of the system being observed and continuously refine it as a new set of data is collected. The filter operates by predicting the current state of the system using the previous state estimate and the system model. This prediction is then combined with the latest measurement to produce an updated estimated state. Data fusion fundamentally involves amalgamating information from diverse sources to construct a robust and comprehensive depiction of a particular environment or process [2,9,10,11]. Uncertainty forms the cornerstone of all descriptions related to sensing and data fusion processes. Introducing probabilistic models has provided a potent and consistent method for portraying uncertainty, naturally paving the way for extensive information fusion. The Kalman filter, equipped with proper model structures, can effectively compensate for each sensor deviation, be it mechanical or statistical errors.

Multisensor fusion and integration is a technology that utilizes data from multiple sensors to derive insights that are unattainable by single sensors operating independently [12]. The advancements in sensor technology, while significant, are further strengthened by the application of multisensor fusion techniques, which improve the performance, reliability, and capabilities of complex systems. Techniques for multisensor data fusion (MDF) span multiple disciplines, including measurement science, pattern recognition, statistical estimation, artificial intelligence, and others [2,13]. The diverse applications of MDF in mechatronic systems range from industrial automation and intelligent robotics to military applications, biomedical fields, and microelectromechanical (MEMS) and nanoelectromechanical (NEMS) systems [14]. In user-centric applications, object tracking stands out as an area where MDF, in conjunction with the Kalman filter, has proven effective [15]. Recent research also explores the combination of the Kalman filter with artificial neural networks, offering a promising hybrid approach that leverages the Kalman filter’s state estimation with the learning capabilities of neural networks to achieve improved tracking and measurement accuracy [16].

There is a vast body of literature on enhancing measurements and sensor fusion, with a foundation in the use of position encoders in both simple and complex mechanical systems [17,18,19,20,21]. A model of an incremental shaft encoder can be developed to facilitate sensor characterization. In [22], a model is derived by formulating a new mathematical expression for the spectral characteristics of errors introduced when a sampled signal, assumed to have a nominally constant rate, undergoes uniform quantization. This process accounts for the effects of both differential and integral nonlinearities. The study in [23] addresses the estimation of rotational velocity and acceleration from incremental encoders while accounting for sensor imperfections, with a particular focus on automotive applications such as traction and brake control. The solution is based on a notch filter. An alternative approach, presented in [24], involves the use of magnetic encoders instead of optical encoders. Magnetic encoders (ME) are sinusoidal encoders that leverage magnetic effects and are widely used in industrial control systems due to their numerous advantages, including low cost, simple design, high reliability, and the ability to operate in harsh environments. However, ME signals are often affected by noise, reducing their accuracy. To address this issue, the authors propose an advanced adaptive digital phase-locked loop as an effective solution. Another approach, as described in [25], introduces rotational geometry into the measurement process. Synergy between encoder measurements in multiple dimensions and their mutual influences is also a critical area of study. The work in [26] presents an alternative perspective by utilizing a planar inductive sensor based on the principle of electromagnetic induction. The sensor consists of a primary coil and a secondary coil, with the primary coil designed as an array of planar spiral coils arranged in a matrix configuration. The authors propose a method for mutual error cancellation, underscoring advancements in sensor fusion technology. In [9], the authors propose a two-stage filtering approach in a 3D environment. The first stage processes data from a gyroscope (GYRO), while the second stage uses magnetic compass data to correct the yaw angle. The concept was successfully validated on an FPGA platform. Another challenge in measurement systems is deviations in results. Paper [27] introduces an adaptive strong tracking Kalman filter, which mitigates tracking mismatches caused by low-resolution encoders. Additionally, the authors observe the load moment, a parameter potentially valuable for control and monitoring systems. To improve the resolution and accuracy of incremental rotary encoders, [28] proposes a phase encoding and code compensation system, achieving accuracy levels suitable for astronomical applications. Furthermore, the measurement biases of sensors during target tracking, if not properly addressed, can significantly degrade localization accuracy, as demonstrated in [29].

This solution introduces data fusion algorithms based on the Kalman filter [8], specifically employing a separated-state fusion method, which is described in detail below. The Kalman filter is widely regarded as one of the most effective and popular techniques in information processing due to its ability to estimate the state of dynamic systems from noisy measurements. Its application here is particularly relevant for combining multiple sensor inputs to improve accuracy and reliability. The study addresses a practical issue in industrial electronics, where many modern sensors produce quantized outputs (e.g., incremental encoders) or are sampled with limited resolution in digital control systems (e.g., N-bit analog-to-digital converters) [2]. This quantization or resolution limitation can introduce errors or noise in the measurement data, which the proposed Kalman filter-based data fusion method aims to mitigate, providing more accurate and stable system performance in real-time control applications [30,31,32].

Certain characteristics related to the interdependencies of trigonometric functions have been identified. A system model incorporating sinusoidal relationships can be developed based on the concept of sine–cosine angular position encoders [33]. The achievable accuracy of sine–cosine angular position encoders has been analyzed, demonstrating that a balanced design results in a constant absolute error in the encoded function value. However, when these encoders are used to transform positions into Cartesian coordinates, the resulting errors propagate to the computed Cartesian coordinates. Another widely used sensor for measuring absolute mechanical angles is the resolver, which operates based on the principle of signal decomposition [34,35]. Resolvers are extensively utilized in industrial applications due to their high accuracy, resolution, and robustness. These devices generate quadrature sinusoidal electrical signals that represent the angular position of a shaft. However, this approach necessitates the computation of nonlinear trigonometric functions, whose accuracy depends on the mapping or the computational method used. One effective method for solving trigonometric relations is the CORDIC algorithm [36,37]. The use of digital hardware such as FPGAs offers numerous advantages, particularly when employing fixed-point computation. These benefits include simplified design, rapid prototyping, reduced hardware requirements, and high-speed processing capabilities. Additionally, FPGAs provide reliability, programmability, and reconfigurability, making them ideal for applications requiring fast and efficient computations. These features make FPGAs especially suitable for implementing demanding computational tasks, such as those involving the Kalman filter and the CORDIC algorithm.

The presented solutions are used in the direct drive with a permanent-magnet synchronous motor (PMSM) of an astronomical telescope mount with very high position control requirements [4]. It is expected that the mount will ensure accurate pointing of the telescope instruments and tracking the motion of fixed stars and satellites as the Earth rotates. The primary function of telescopes lies in observing celestial objects in motion across the sky at speeds relative to the observer. This movement is a consequence of Earth’s daily rotation, translating to a rate of 15 degrees per hour, which equals $7.27\times {10}^{-5}$$\mathrm{rad}/\mathrm{s}$ or 15 $\mathrm{arcsec}/\mathrm{s}$. Objects within our solar system, such as planets, asteroids, or comets, exhibit motion concerning the backdrop of stars. The velocity of these non-star entities in relation to the stars increases as their proximity to Earth grows. The anticipated precision pertains to measurements in outer space, factoring in disturbances like wind or the flexing of the mechanical-optical system. In practical terms, this necessitates that the control error within the isolated system should not surpass several hundredths of an arcsecond.

This article is organized as follows. The first section introduces the topics of position measurement, sensor fusion, and the high demands on telescope position control. Methods in Section 2 precisely describe the issues related to our Kalman filter approach (Section 2.1) and the CORDIC (Section 2.2) algorithm. These are all the tools necessary to build up the Trigonometric Kalman Filter sensor fusion applications described in Section 3.4. Section 4 presents laboratory results, compares them with other known methods, and provides research conclusions. The overall findings of the work are summarized in Section 5, which also serves as a foundation for further exploration and investigation.

2. Methods

2.1. Kalman Filter

In Control Systems and Statistics, the Kalman filter is a recursive algorithm with infinite impulse response [8,38,39]. This means that the derivative of the known estimate ${\displaystyle \frac{d}{dt}}\underline{x}$, called the system trend, and the set of output measurements ${\underline{y}}_k$ are sufficient to estimate the current state of the system ${\underline{x}}_k$ at each time $tϵ(0,\infty )$. For the Kalman filter, it is not necessary to know the history of observations. Mathematically, the Kalman filter states can be described by two variables: ${\displaystyle \frac{d}{dt}}\underline{x}$, the trend which is the derivative of the current state estimate obtained based on knowledge of observation, and $\mathbf{P}$, the uncertainty of the error covariance matrix of the estimation process.

The Kalman filter estimation process is divided into two subsequent stages: prediction and correction. The entire process computation is performed recursively to obtain the optimum value of the corrector $\mathbf{K}$ with an assumed error that is as small as possible $\epsilon$. Prediction performs the state estimate $\widehat{\underline{x}}$ based on the trend and input signal $\underline{u}$. Correction leads to improvements in the new estimate of the exact value $\widehat{\underline{x}}$ based on the measured output $\underline{y}$ and the value of the corrector $\mathbf{K}$.

2.1.1. Prediction

This is also called time actualization and is based on knowing the derivative of the state ${\displaystyle \frac{d}{dt}}\widehat{\underline{x}}$. It is described as a relation in the presence of input $\underline{u}$:

$${\displaystyle \frac{d}{dt}}\widehat{\underline{x}}=\mathbf{F}\widehat{\underline{x}}+\mathbf{B}\underline{u},$$

(1)

and for a future discrete approach, at the kth step, it takes the form:

$${\widehat{\underline{x}}}_{k|k-1}={\mathbf{F}}_k{\widehat{\underline{x}}}_{k-1|k-1}+{\mathbf{B}}_k{\underline{u}}_k,$$

(2)

where index _k means actual step k, index _k|k−1 means computing kth step from $k-1$ previous steps, and index _k−1|k−1 is the last computation step value $k-1$. Notice that matrices $\mathbf{F}$ and $\mathbf{B}$ from the continuous time form in (1) become ${\mathbf{F}}_k$ and ${\mathbf{B}}_k$ in (2) for the kth step of computation with a desired $T_s$ step using the forward Euler method for solving ODEs of the form in (1).

Innovation and some form of filtering at this stage cause updates with a vector of state variables $\underline{x}$ of the system covariance $\mathbf{P}$, based on discrete state function (2):

$${\mathbf{P}}_{k|k-1}={\mathbf{F}}_k{\mathbf{P}}_{k-1|k-1}{\mathbf{F}}_k^T+\mathbf{Q}.$$

(3)

2.1.2. Correction

Correction, which is the actualization of the measurement, involves the introduction of the correction signal based on the measured discrete output ${\underline{z}}_k$. It is defined as the difference in response values of the observer and the measured output signal based on the state-space continuous equation:

$$\underline{y}=\mathbf{H}\underline{x},$$

(4)

which becomes like a residual when using a discrete step:

$${\tilde{\underline{y}}}_k={\underline{z}}_k-{\mathbf{H}}_k{\widehat{\underline{x}}}_{k|k-1},$$

(5)

where ${\mathbf{H}}_k$ is the discrete form of state-space matrix $\mathbf{H}$, obtained in a similar form to (2) for the kth step. Additionally, the innovation covariance system is introduced, based on knowledge of the measurement covariance $\mathbf{R}$:

$${\mathbf{S}}_k={\mathbf{H}}_k{\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^T+\mathbf{R}.$$

(6)

Based on the above, the mathematical form of the Kalman corrector ${\mathbf{K}}_k$ can be determined by:

$${\mathbf{K}}_k={\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^T{\mathbf{S}}_k^{-1},$$

(7)

which together with the calculated residual (5) corrects the state vector:

$${\widehat{\underline{x}}}_{k|k}={\widehat{\underline{x}}}_{k|k-1}+{\mathbf{K}}_k{\tilde{\underline{y}}}_k.$$

(8)

Based on a similar correction to (8) for the state vector ${\widehat{\underline{x}}}_k$, the covariance is also corrected:

$${\mathbf{P}}_{k|k}=(I-{\mathbf{K}}_k{\mathbf{H}}_k){\mathbf{P}}_{k|k-1}.$$

(9)

The provided formula for the Kalman filter, utilizing the optimal Kalman gain ${\mathbf{K}}_k$, applies exclusively to linear systems. Under the stated assumptions, both $\mathbf{P}k|k$ and ${\mathbf{K}}_k$ remain constant and can be determined once, based on the system matrices ${\mathbf{F}}_k$, ${\mathbf{B}}_k$, and ${\mathbf{H}}_k$. Typically, this computation is performed iteratively until it converges to a stable and consistent solution.

2.2. Geometric Processor

The Coordinate Rotation Digital Computer (CORDIC) algorithm is a numerical technique employed to calculate a variety of mathematical functions, including trigonometric and hyperbolic functions, square roots, and other complex operations. It works by iteratively rotating coordinates to compute these functions, relying solely on basic arithmetic operations such as addition, subtraction, and bit-shifting. This algorithm [36] is widely used in embedded signal-processing applications, particularly where trigonometric computations are required. Examples include rotary encoders, positioning systems, kinematic calculations, phase tracking, and coordinate transformations [40,41]. One notable interesting application of the CORDIC algorithm is in the computation of the arc-tangent function. This operation is crucial for phase calculation in communication systems, where the complex-envelope signals are derived from in-phase and quadrature components. It also plays a key role in angular position measurements for automotive applications, as well as in the phase computation for power system control and AC circuit analysis, as discussed in studies such as [42,43,44,45]. While the standard CORDIC algorithm is well established in the literature, its efficient implementation—especially in terms of computational accuracy and minimizing circuit complexity and power consumption—continues to be a challenge [46,47]. To address this issue, various approaches, including full-custom and semi-custom designs, such as IP macrocells synthesized on FPGA or standard-cell technologies, have been investigated, as highlighted in [37].

To effectively illustrate the functioning of the algorithm, it is beneficial to reference the relationships shown in Figure 1. This figure provides a visual representation of the key geometric dependencies involved, helping to clarify the underlying principles of the operations.

As shown on Figure 1, Cartesian position $\left[X_i,Y_i\right]$ can be represented by vector radius $R_i$ and angle ${\theta }_i$ as:

$$\begin{array}{c}\hfill Y_i=R_{i}sin{\theta }_i\\ \hfill X_i=R_{i}cos{\theta }_i\end{array}$$

(10)

where ${\theta }_i$ can be defined in a range closed by the resolution of the ith step using ${\alpha }_i$:

$${\alpha }_i={tan}^{-1}\left(2^{i-2}\right)$$

(11)

while defining a possible change in rotation relative to the angle. Two cases (negative and positive ${\alpha }_i$) of rotation are shown in Figure 1, so the expressions for the rotated components are:

$$\begin{array}{c}\hfill Y_{i+1}=\sqrt{1+2^{-2(i-2)}}{R}_{i}sin({\theta }_i\pm {\alpha }_i)\\ \hfill X_{i+1}=\sqrt{1+2^{-2(i-2)}}{R}_{i}cos({\theta }_i\mp {\alpha }_i)\end{array}$$

(12)

The computational operation of adding (or subtracting) a shifted value of $X_i$ to $Y_i$ to obtain $Y_{i+1}$, while simultaneously subtracting (or adding) a shifted value of $Y_i$ to $X_i$ to obtain $X_{i+1}$, is termed cross-addition [36].

$$\begin{array}{c}\hfill Y_{i+1}=Y_i\pm 2^{-2(i-2)}{X}_i\\ \hfill X_{i+1}=X_i\mp 2^{-2(i-2)}{Y}_i,\end{array}$$

(13)

With imperfect rotation and with increased magnitude, some corrections with a small step are necessary. Expressions ± and ∓ can be the binary operator ${\xi }_i$:

$$\begin{array}{c}\hfill Y_{i+1}=Y_i+{\xi }_i{2}^{-2(i-2)}{X}_i\\ \hfill X_{i+1}=X_i-{\xi }_i{2}^{-2(i-2)}{Y}_i,\end{array}$$

(14)

Considering (10) and (13), it is possible to obtain the $(n+1)$th iteration as:

$$\begin{array}{c}\hfill Y_{n+1}=\left[\sqrt{1+2^{-0}}\sqrt{1+2^{-2}}...\sqrt{1+2^{-2(i-2)}}\right]R_{1}sin\left({\varphi }_1+{\xi }_1{\alpha }_1+{\xi }_2{\alpha }_2+...+{\xi }_n{\alpha }_n\right)\\ \hfill X_{n+1}=\left[\sqrt{1+2^{-0}}\sqrt{1+2^{-2}}...\sqrt{1+2^{-2(i-2)}}\right]R_{1}cos\left({\varphi }_1+{\xi }_1{\alpha }_1+{\xi }_2{\alpha }_2+...+{\xi }_n{\alpha }_n\right)\end{array}$$

(15)

which can be simplified to:

$$\begin{array}{cc}\hfill Y_{n+1}& =K_{c}sin\left({\varphi }_1+\lambda \right)\hfill \\ \hfill X_{n+1}& =K_{c}cos\left({\varphi }_1+{\xi }_1{\alpha }_1+{\xi }_2{\alpha }_2+...+{\xi }_n{\alpha }_n\right)\hfill \end{array}$$

(16)

where

$$\begin{array}{cc}\hfill K_c& =\sqrt{1+2^{-0}}\sqrt{1+2^{-2}}...\sqrt{1+2^{-2(i-2)}},\hfill \\ \hfill \lambda & ={\xi }_1{\alpha }_1+{\xi }_2{\alpha }_2+...+{\xi }_n{\alpha }_n.\hfill \end{array}$$

(17)

After assuming a finite number of iterations when finding any angle:

$$-{\varphi }_1={\xi }_1{\alpha }_1+{\xi }_2{\alpha }_2+...+{\xi }_n{\alpha }_n,$$

(18)

and the ith angles are:

$$\begin{array}{cc}\hfill {\alpha }_1& ={\displaystyle \frac{\pi }{2}},\hfill \\ \hfill {\alpha }_2& ={tan}^{-1}\left(2^0\right)={\displaystyle \frac{\pi }{4}},\hfill \\ \hfill {\alpha }_3& ={tan}^{-1}\left(2^{-1}\right)=0.24497,\hfill \\ \hfill & ...\hfill \\ \hfill {\alpha }_i& ={tan}^{-1}\left(2^{-(i-2)}\right).\hfill \end{array}$$

(19)

Above set of $\alpha$ terms can be precomputed and tabulated in memory.

The simplicity and efficiency of the CORDIC algorithm have made it popular in various fields, including signal processing, graphics rendering, and digital communication systems, where rapid and accurate computation of trigonometric functions is essential.

3. Proposed Method—Trigonometric Kalman Filter Application

3.1. Four-Head Position Sensor System

The combined configuration of two four-head solutions is illustrated in Figure 2, which does not follow a standard solution typically provided by the encoder manufacturer [2,4]. The proposed measurement system, as shown in Figure 2a, involves mounting four read heads around the circumference of the encoder ring, with each read head positioned approximately every ${\displaystyle \frac{\pi }{2}}$ radians, as depicted in Figure 2b. The method and results presented indicate that the precise placement of the read heads is not critical. In our application, the primary focus is on achieving higher accuracy and independence from mechanical inaccuracies. The rotating encoder ring incorporates a high-precision scale, and each read head is capable of detecting the current position of the encoder ring at each of the specified positions (${\gamma }^{RH1}$–${\gamma }^{RH4}$), where the actual position of the ring on the shaft is denoted as $\gamma$.

The proposed mechanical solution theoretically offers accuracy four times higher and has the potential to eliminate both the fourth- and second-order signal harmonics associated with fixed mechanical displacements. It could be significant for high-precision, low-speed drives like tracking telescopes.

3.2. Measurement Modeling

The presented solution consists of four actual position measurement values, obtained by the read heads RH1 to RH4 (as shown in Figure 2), respectively:

$${\tilde{\mathbf{z}}}_k={\left[\begin{array}{cccc}{\gamma }_k^{RH1}& {\gamma }_k^{RH2}& {\gamma }_k^{RH3}& {\gamma }_k^{RH4}\end{array}\right]}^T,$$

(20)

with assumed displacements (the values are approximate and may vary within a small range):

$${\tilde{\mathbf{z}}}_0={\left[\begin{array}{cccc}0& {\displaystyle \frac{\pi }{2}}& \pi & {\displaystyle \frac{3}{2}}\pi \end{array}\right]}^T.$$

(21)

3.3. State-Space Trigonometric Model of One Head

The relationship between rotational speed ${\omega }_r$ and angular position $\gamma$ is fundamental in rotational motion, where speed is the rate at which the angular position changes over time:

$${\displaystyle \frac{d}{dt}}\gamma ={\omega }_r.$$

(22)

For uniform, constant rotational speed ${\omega }_r$, the angular position $\gamma$ described as (22) increases linearly over time according to:

$$\gamma ={\omega }_r·t+{\gamma }_0,$$

(23)

Relationship (23) is most important in control systems where precise tracking of both position and speed is necessary, as any variation in speed directly impacts the final angular position.

Based on (23), a Cartesian angle decomposition into trigonometric sin and cos components, respectively, is possible:

$$\begin{array}{cc}\hfill sin\gamma & =sin({\omega }_r·t+{\gamma }_0),\hfill \\ \hfill cos\gamma & =cos({\omega }_r·t+{\gamma }_0),\hfill \end{array}$$

(24)

Upon taking the derivative of (24), we obtain:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}sin\gamma & ={\omega }_{r}cos({\omega }_r·t+{\gamma }_0),\hfill \\ \hfill {\displaystyle \frac{d}{dt}}cos\gamma & =-{\omega }_{r}sin({\omega }_r·t+{\gamma }_0).\hfill \end{array}$$

(25)

where the tracking harmonic functions $sin\gamma$ and $cos\gamma$ can be parameterized using ${\omega }_r$ as a factor.

The traditional statistical approach in Control Systems Theory defines the state transition and measurement probability model of the system using density functions like:

$$\begin{array}{c}\hfill p\left({\underline{x}}_k|{\underline{x}}_{k-1},{\underline{u}}_k\right)\\ \hfill p\left({\underline{y}}_k|{\underline{x}}_k\right)\end{array}$$

(26)

where ${\underline{x}}_k\in {\mathbf{R}}^{n_x}$ denotes the states (hidden variables and/or parameters) of the system in time t denoted at the kth step, and ${\underline{y}}_k\in {\mathbf{R}}^{n_y}$ are known observations. The states evolve according to a first-order Markov chain process, and the observations are assumed to be conditionally independent given the states, allowing the model to be expressed as:

$${\underline{x}}_k=f\left({\underline{x}}_{k-1},{\underline{u}}_k,{\underline{v}}_{k-1}\right),$$

(27)

$${\underline{z}}_k=h\left({\underline{x}}_k,{\underline{n}}_k\right),$$

(28)

where ${\underline{u}}_t\in {\mathbf{R}}^{n_u}$ denotes the input observations, ${\underline{v}}_k\in {\mathbf{R}}^{n_x}$ and ${\underline{n}}_k\in {\mathbf{R}}^{n_y}$ are noises for the process and measurements, respectively. The state-space representation can be represented more precisely than (2) and (4) at the kth step by:

$${\underline{x}}_k={\mathbf{F}}_k\left({\underline{x}}_{k-1}\right){\underline{x}}_{k-1}+{\mathbf{B}}_k\left({\underline{x}}_{k-1}\right){\underline{u}}_k+{\underline{v}}_{k-1},$$

(29)

$${\underline{z}}_k={\mathbf{H}}_k\left({\underline{x}}_k\right){\underline{x}}_k+{\underline{n}}_k.$$

(30)

Based on (25), the continuous-time derivative of the trigonometric relation form can be expresses as:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}sin\gamma \\ cos\gamma \end{array}\right]={\omega }_r\left[\begin{array}{c}cos\gamma \\ -sin\gamma \end{array}\right]$$

(31)

where variable ${\omega }_r$ can be treated as a parameter. To find its value algebraically, we use two ways:

• From the first row of Equation (31):

  $${\omega }_r={\displaystyle \frac{\frac{d}{dt}sin\gamma }{cos\gamma }},$$

  (32)
• From the second row of Equation (31):

  $${\omega }_r=-{\displaystyle \frac{\frac{d}{dt}cos\gamma }{sin\gamma }}.$$

  (33)

Both expressions can give consistent results assuming no disturbances occur. However, in the case of the assumption of any disturbances as in formulas (29) and (30), the solution is not coherent. The appropriate one in this case is the disturbance observer, in a Kalman filter structure close to the one proposed in the work [48]. It is assumed that the disturbances $\underline{v}$ for the state and $\underline{n}$ for the measurement equations in (27) and (28), respectively, have a long-term average value near 0:

$${\displaystyle \frac{d}{dt}}\underline{v}=0,$$

(34)

$${\displaystyle \frac{d}{dt}}\underline{n}=0.$$

(35)

Focusing especially on the property of (34), we can introduce:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{\displaystyle \frac{d}{dt}}sin\gamma \\ {\displaystyle \frac{d}{dt}}cos\gamma \end{array}\right]=\mathbf{0},$$

(36)

which can be included in the vector of observed state variables, taking the form:

$$\widehat{\underline{x}}=\left[\begin{array}{c}sin\gamma \\ cos\gamma \\ {\displaystyle \frac{d}{dt}}sin\gamma \\ {\displaystyle \frac{d}{dt}}cos\gamma \end{array}\right],$$

(37)

and it involves the use of continuous-time model matrices like:

$$\mathbf{F}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$$

(38)

As noted, for this class of dynamical systems, dependency (29) has no u input, or it is null or irrelevant, so we assume:

$$\mathbf{B}=\mathbf{0}.$$

(39)

Using the forward Euler method with assumed discretization time $T_s$ to fit into (29) and (30), ${\mathbf{B}}_k$ produces zeros like its continuous form, but $\mathbf{F}$ from (38) takes a shape ${\mathbf{F}}_k$:

$${\mathbf{F}}_k=\left[\begin{array}{cccc}1& 0& T_s& 0\\ 0& 1& 0& T_s\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],$$

(40)

and similarly for $\mathbf{H}$:

$${\mathbf{H}}_k=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right],$$

(41)

which are both linear and state-independent.

3.4. Trigonometric Kalman Filter Idea

Based on the above atomic (one encoder information) data, it is possible to obtain a Kalman filter system with all data for tracking sin and cos functions for each position. A simplified diagram with all the details discussed is shown in Figure 3.

Two basic blocks can be noticed, KALMAN FILTER and CORDIC, the latter multiplied due to the functions performed.

The operation of the Kalman filter was presented in Section 2.1. Based on the above analysis and taking into account the description of the model, we can select from (37) important state-space variables so ${\widehat{\underline{x}}}_k^{RHn}$ for one head representation becomes:

$${\widehat{\underline{x}}}_k^{RHn}=\left[\begin{array}{c}\widehat{sin{\gamma }_k^{RHn}}\\ \widehat{cos{\gamma }_k^{RHn}}\\ \widehat{{\displaystyle \frac{d}{dt}}sin{\gamma }_k^{RHn}}\\ \widehat{{\displaystyle \frac{d}{dt}}cos{\gamma }_k^{RHn}}\end{array}\right]$$

(42)

which is adequate for 4 possible heads and takes the form:

$${\widehat{\underline{x}}}_k={\left[{\widehat{\underline{x}}}_k^{RHn1}{\widehat{\underline{x}}}_k^{RH2}{\widehat{\underline{x}}}_k^{RH3}{\widehat{\underline{x}}}_k^{RH4}\right]}^T$$

(43)

For the state space constructed this way and with computation time $T_s$, the ${\mathbf{F}}_{\mathbf{k}}$, ${\mathbf{H}}_{\mathbf{k}}$, ${\mathbf{B}}_{\mathbf{k}}$ matrices look like:

$${\mathbf{F}}_k=diag\left\{{\mathbf{F}}_{\mathbf{k}}^{\mathbf{RH}\mathbf{1}}{\mathbf{F}}_{\mathbf{k}}^{\mathbf{RH}\mathbf{2}}{\mathbf{F}}_{\mathbf{k}}^{\mathbf{RH}\mathbf{3}}{\mathbf{F}}_{\mathbf{k}}^{\mathbf{RH}\mathbf{3}}\right\},$$

(44)

$${\mathbf{H}}_k=diag\left\{{\mathbf{H}}_{\mathbf{k}}^{\mathbf{RH}\mathbf{1}}{\mathbf{H}}_{\mathbf{k}}^{\mathbf{RH}\mathbf{2}}{\mathbf{H}}_{\mathbf{k}}^{\mathbf{RH}\mathbf{3}}{\mathbf{H}}_{\mathbf{k}}^{\mathbf{RH}\mathbf{3}}\right\},$$

(45)

$${\mathbf{B}}_k=\mathbf{O}.$$

(46)

where each ${\mathbf{F}}_k^{RHn}$ is represented by (47) according to the number of heads:

$${\mathbf{F}}_k^{RHn}=\mathbf{1}+T_s\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$$

(47)

whereas ${\mathbf{H}}_k^{RHn}$ is one head measurement matrix for position ${\gamma }_k^{RHn}$ only, for each head from (41):

$${\mathbf{H}}_k^{RHn}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right].$$

(48)

It can be noticed, also based on previous considerations, that the signal ${\underline{x}}_k$ from (43) contains all the properties that are important to us. For such an approach, the object seems to be very simple. Therefore, the calculation of the Kalman filter and the determination of ${\mathbf{K}}_k$ can be carried out, according to the method from Section 2.1, when selecting appropriate tuning conditions $\mathbf{Q}$ and $\mathbf{R}$ from (3) and (6), respectively, considering the occurring, expected noise. It is performed once and can be subject to minor corrections during operation at most. The essence of this solution, however, lies in the appropriate processing of input and output signals in order to obtain important information from their properties.

In Figure 3, you can see several blocks with the description CORDIC. They contain the algorithm described in Section 2.2. They implement two natural calculations from this point of view: the calculation of sin and cos based on the knowledge of each measurement position ${\gamma }_k^{RHn}$ and the calculation of the output function tan from state vector ${\underline{x}}_k$.

The DIVIDE block from Figure 3 is worth discussing. It is the idea of extracting speed value ${\omega }_r$ from state variable properties. If we look closely at Equation (31), ${\omega }_r$ appears there explicitly. Based on (25), the continuous-time derivative relation form can be expressed as the following pair:

$$\begin{array}{c}\hfill {\omega }_r={\displaystyle \frac{\frac{d}{dt}sin\gamma }{cos\gamma }},\end{array}$$

(49)

$$\begin{array}{c}\hfill {\omega }_r=-{\displaystyle \frac{\frac{d}{dt}cos\gamma }{sin\gamma }}\end{array}$$

(50)

which, given the knowledge of the state variable vector for a single axis (42), takes the average form of both suspected ${\omega }_r$ for one RH:

$${\omega }_r^{RHn}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\frac{d}{dt}sin{\gamma }_k^{RHn}}{cos{\gamma }_k^{RHn}}}-{\displaystyle \frac{\frac{d}{dt}cos{\gamma }_k^{RHn}}{sin{\gamma }_k^{RHn}}}\right)$$

(51)

Taking into account the large amount of information from each of the heads, it was decided at this stage of the research to average the values as follow:

$${\widehat{\omega }}_r={\displaystyle \frac{1}{4}}\sum _{n=1}^4{\widehat{\omega }}_r^{RHn}$$

(52)

4. Laboratory Results

After numerous simulation studies, the experimental verification of the proposed control method used other existing research laboratory setups. All necessary and auxiliary tests were performed at the laboratory’s astronomical telescope drive station, the properties of which are described in detail in [4].

4.1. Laboratory Setup

4.1.1. Telescope Mount

An implementation of the system within a real astronomical telescope mount setup was explored. The mechanical design of the mount is thoroughly described in [4]. The base axis of the laboratory stand, shown in Figure 4, features a surface-mounted magnet synchronous motor, powered by a three-phase MOSFET 24V power inverter. This system is controlled by a Texas Instruments Sitara AM335x processor, alongside an Altera FPGA Cyclone IV [30]. To support this research, the system was further augmented with an additional FPGA board, the Altera Stratix IV, depicted in Figure 5, as well as interconnections between the encoder read heads (Figure 2). A detailed summary of the key parameters and specifications of the laboratory setup is provided in Appendix A.

Each drive system consists of a measurement ring and four reading heads (RH), symmetrically arranged around the measurement ring (see Figure 2). This configuration enhances positioning accuracy and reduces errors. The system is capable of measuring angles with an accuracy of 1 arcsecond. Data from the reading heads are transmitted to the control system using the BISS-C industrial protocol. The FPGA circuit is responsible for reading the data, calculated according to the proposed algorithm and sending it to the main control and acquisition system unit.

All trials were preformed on the drive’s vertical axis presented in Figure 4 after preparing special test scenarios.

4.1.2. FPGA Controller

Field Programmable Gate Arrays (FPGAs) are increasingly being used for implementing real-time embedded applications due to their high-speed processing capabilities, flexibility, and parallelism. Unlike traditional processors, FPGAs allow developers to design custom hardware architectures tailored specifically to application requirements, optimizing both performance and resource efficiency. This makes them particularly well suited for real-time applications where low latency and precise timing are essential, such as control systems, digital signal processing, and data acquisition. In addition, FPGAs support a high level of parallelism, enabling multiple processes to be executed simultaneously. This parallelism is a major advantage in embedded applications, with tasks like sensor data processing.

The DE4 presented in Figure 5 is powered by an Altera Stratix IV GX series 40 nm FPGA chip directly coupled to abundant peripherals and interfaces. This board is commonly used for complex and intensive computing tasks, like digital signal processing (DSP), image processing, network processing, and scientific computations due to its high processing power and large memory capacity. The DE4 board is widely used in research, academia, and industry for data-intensive processing in machine learning and deep learning, real-time processing in control systems, DSP, and video processing, networking and communication protocols testing and simulation, and hardware acceleration for scientific computations.

The central component in the functional diagram presented in Figure 6 is the CORDIC algorithm, which plays a critical role in parallel calculations. In the system design and testing phases, the pre-built IP module, ALTERA_CORDIC, was using to parallelize the implementation. It was chosen because it is more hardware-efficient than direct multiplication and division. The IP is optimized for FPGA architectures, allowing for lower resource usage and faster processing. The LPM_DIVIDE IP used is part of the LPM (Library of Parameterized Modules), which includes various configurable IP cores for common arithmetic, logic, and memory functions. LPM_DIVIDE can handle both integer and fixed-point division, making it versatile for applications requiring accurate division operations. It typically operates synchronously with the clock, allowing seamless integration with other clocked operations in a design. The auxiliary code includes procedures for reading data from the BISS-C interface heads and formatting them into a valid 32-bit form, as well as communication procedures compatible with the supported software. The Serial Tab Logic Analyzer tool was used to facilitate communication and data collection. In the device’s functional part, memory-sharing-based communication is implemented, and it operates in a dedicated RAM area. The entire code for the Kalman filter system was written in VHDL. Most matrices were initialized as arrays of arrays of std_logic_vector(31 downto 0) flattened into the appropriate size std_locic_vector, with calculations performed using custom libraries synthesized with the Intel HLS Compiler.

4.2. Experiment During Small Speed Changes’ Analysis

Before beginning the experimental tests on the actual test stand with the FPGA part, an extensive series of simulation tests were conducted to validate the proposed method and to compare the accuracy and quality of the estimation results against the initial assumptions as well as other established solutions. In the experimental phase, the performance of the algorithm was also tested on the ARM A8 (Texas Instruments AM3358) floating-point unit (referred to hereafter as the CPU results) to evaluate applicability and efficiency.

Given the demanding requirements for controlling an astronomical mount and its operation, a test was designed to capture the worst-case scenarios of mechanical and power electronics design. The test region was set within a negligible angular displacement of $\pm 10$ arcseconds, intended as a distinguishing feature for assessing the quality of angular position mapping and reproduction. Additionally, as mentioned, the low torque value was expected to induce a significant disturbance of $\pm 0.25$ A in the measurement and regulation of the working current in the range $0\to 1$ A. An example of this can be seen in Figure 7. This test profile served as a reference for further studies, especially those with results presented in the following figures.

Due to the ease of measurement and the predictability of current $i_q$’s behavior, it was decided to maintain a constant value of $i_q=1$ A, and examine the response of the telescope’s axis position. Observing the electric drive’s behavior during the assumed scenario, it can be seen that such operation was static, involving the overcoming of frictional forces and stresses. The behavior itself was not the subject of this work but served as a representative example of the system’s response.

To provide an initial demonstration of the algorithm’s operation, Figure 8a,b present the results of position and velocity observations for the specified scenario. These figures illustrate the performance of the algorithm when calculated on both CPU (values indexed as ^C) and FPGA (values indexed as ^F) platforms, respectively. The comparison aims to highlight any differences in accuracy, response speed, and quality between the two implementations, thereby validating the effectiveness of the algorithm across different computation hardware.

At first glance, the results shown in Figure 8 appear to be qualitatively similar, with no immediate, noticeable discrepancies. However, a deeper analysis in the time domain is recommended to detect any subtle variations in performance, such as differences in signal noise, response time, or transient behavior. This extended analysis could provide further insight into how the algorithm’s performance on each platform handles dynamic changes and reacts, ultimately confirming the accuracy and robustness of the method under more rigorous conditions.

A qualitative comparison of the results regarding position $\gamma$ for the same set of input data and tuning over the entire experimental time frame is shown in Figure 9a. There were no significant quantitative differences; they were always within the noise range, which can be seen in detail in the magnification in Figure 9b. Clear differences and a reduction in noise level from the measurement in the estimated position value ${\widehat{\gamma }}^C$ (for the CPU result), ${\widehat{\gamma }}^F$ (for the FPGA result) could only be observed at that level of analysis. One can notice the trend of changes being maintained with a much lower noise value in relation to the measured values ${\gamma }^{RHx}$.

The accurate determination of the rotational speed ${\omega }_r$ is a crucial aspect of a control system, as it directly impacts precision and stability, especially during tracking operation responses [3]. This speed value is essential not only for ensuring efficient performance but also for making adjustments to maintain the desired operating conditions of telescopes. The estimated speed response for the assumed test scenario from Figure 7 is shown in Figure 10 and Figure 11.

This test is fully represented in Figure 10a; small qualitative differences were noticeable between CPU and FPGA implementation results, which prompted enlarging certain test steps. Figure 10b clearly illustrates the static properties, especially the response to mechanical disturbances (observed oscillations) and measurement disturbances (noticeable noise). For a deeper analysis, portions of the test from Figure 10a are enlarged in Figure 11a,b, showing both positive and negative position changes, which correspond to positive and negative estimated speed responses.

The differences in response based on the implementation method were noticeable, with the FPGA solution achieving a more precise replication of the actual speed. A clear distinction existed between the fixed-point and floating-point implementations. The fixed-point approach on the FPGA demonstrated smoother transitions and reduced latency compared to the floating-point implementation, highlighting its suitability for tasks requiring high responsiveness and real-time accuracy.

4.3. Discussion of Results

From the studies presented above, assessing the quality of mechanical quantity observations remains challenging. To gain a more comprehensive understanding, it would be valuable to compare these results with other established methods, particularly those that are widely recognized and applied in similar fields. Additionally, examining the performance against well-documented techniques could help validate the method’s effectiveness, especially in terms of precision, accuracy, and adaptability across different scenarios. However, the assumed scenario from Figure 7 seemed to be exemplary and exhibited all the extreme noticeable properties.

For comparison, well-established methods were utilized, starting with the M-derivative method [49], followed by the $\alpha$–$\beta$ and $\alpha$–$\beta$–$\gamma$ filters. The results of the cross-sectional analysis are presented in Figure 12a. These findings are based on the proposed scenario depicted in Figure 7, with particular attention given to the rapid positive changes emphasized in Figure 12b. The M-derivative method estimates velocity using Euler’s method, which employs a standard derivative calculation. This approximation functions as a backward finite difference, calculating the difference between the current position and the position from the previous sampling interval, and dividing this result by the sampling time [49]. For comparison, tracking filter structures such as $\alpha$–$\beta$ and $\alpha$–$\beta$–$\gamma$, as described in detail in [50], were also applied. These filters are rooted in optimal filtering theory and are commonly used for tracking applications. A key parameter, the tracking index $T^2{\displaystyle \frac{{\sigma }_{\gamma }}{{\sigma }_{\omega }}}$, represents the ratio of position uncertainty due to target maneuverability to sensor measurement accuracy. This parameter plays a fundamental role in initiating and guiding the tracking process, and it was further selected through simulation tests for the best result.

Table 1. Comparison of different methods with TKF.

Observer Type	σγ [10−6]	σω [10−6]	f0 [HZ]	Comments
Direct RH1	$5.85$	364,000	$10\times {10}^3$
Direct RH2	$5.77$	686,000	$10\times {10}^3$
Direct RH3	$5.89$	583,000	$10\times {10}^3$
Direct RH4	$5.67$	852,000	$10\times {10}^3$
$\alpha$–$\beta$ RH1	$0.169$	$8.33$	470	${\sigma }_{\gamma }={10}^{-4}$, ${\sigma }_{\omega }={10}^{-2}$
$\alpha -\beta -\gamma$ RH1	$0.0943$	$2.74$	363	${\sigma }_{\gamma }={10}^{-4}$, ${\sigma }_{\omega }={10}^{-2}$
TKF CPU	$1.82$	$1.76$	$\sim 900$	$q=1$, $r=1$, unstable
TKF CPU	$4.54$	$4.49$	890	$q=10$, $r=1$
TKF CPU	$5.45$	$5.48$	860	$q=100$, $r=1$
TKF CPU	$5.68$	$5.63$	830	$q=1000$, $r=1$
TKF FPGA	$4.42$	$4.36$	890	$q=10$, $r=1$

presents the indices for the individual types of observers, and additionally for TKF, the results for various selections of the q and r coefficients. In the analysis, the variance of random variable data x made up of N observations was defined as:

$${\sigma }_x={\displaystyle \frac{1}{N}}\sum _{i=1}^N{|x_i-\mu |}^2,$$

(53)

where

$$\mu ={\displaystyle \frac{1}{N}}\sum _{i=1}^N{x}_i.$$

(54)

and N is the number of samples during stable position $\gamma$ or speed ${\omega }_r$.

The TKF exhibited a fast response and a broad estimation bandwidth, both of which are essential in dynamic systems requiring high precision in real-time tracking. Notably, the TKF filter offered superior accuracy in both position $\widehat{\gamma }$ and speed ${\widehat{\omega }}_r$ estimations when compared to traditional approaches, with the proposed TKF solution achieving the highest fidelity in reproducing these values.

An analysis of the results presented in Table 1 reveals that the proposed solution consistently achieved superior quality, as indicated by lower position variance $\sigma \gamma$, compared to other known and proposed above methods. Notably, the comparison between the variance of the measured values $\gamma$ from $RH1-RH4$ and the estimated values $\widehat{\gamma }$, particularly when using the fixed-point FPGA implementation of TKF, consistently favored the fused estimate, indicating enhanced accuracy and reliability in the estimation process.

In addition, the FPGA implementation of the TKF filter benefited from its increased bit precision, which contributed to more robust and reliable data processing. This heightened bit accuracy enabled the FPGA to maintain a broader bandwidth for velocity estimation, which is particularly beneficial in applications that demand high-speed adjustments and accurate tracking over varying conditions. Furthermore, the integral function’s additive property enhanced the consistency and stability of the velocity estimation over extended periods, thereby reducing error accumulation and even improving the overall performance of the control system.

5. Conclusions

This paper made a significant contribution by introducing an original estimation scheme for sensor fusion involving four encoders. This scheme achieved exceptionally high accuracy in position estimation and angular speed measurement, meeting the demands for both precision and response frequency.

A noteworthy aspect of this approach was the definition of harmonic variables described as trigonometric functions. Their properties were utilized to establish internal relationships within the Kalman filter framework, leading to the development of the Trigonometric Kalman Filter.

The demonstrated benefits of the TKF algorithm in high-precision systems underscore its applicability in scenarios involving minute positional variations that demand exceptional accuracy. This study not only detailed the successful FPGA implementation of the TKF algorithm for control systems but also highlighted ongoing efforts to further minimize control errors and optimize dynamic performance. The findings from this research were validated through comparative testing against established position and velocity reproduction systems, as shown by experiments conducted in a laboratory environment. Furthermore, there are plans to integrate the proposed algorithm into an industrial drive system with constrained computational resources, emphasizing the approach’s practicality and scalability.

Although the topics covered have garnered significant attention, there is still considerable potential for further exploration and advancement in this area. There are still aspects of the Kalman filter that can be refined to achieve even greater accuracy. Features such as the incorporation of an acceleration signal or the use of variable nonlinear structures, similar to Unscented Kalman Filters (UKFs) and Extended Kalman Filters (EKFs), could enhance accuracy and provide faster responses while increasing the frequency response.

In conclusion, this paper provides a basis for further research and development in the field of advanced control techniques, specifically in sensor fusion for high-performance direct-drive systems with high dynamics permanent-magnet synchronous motors.