Fast and Accurate Approximation Methods for Trigonometric and Arctangent Calculations for Low-Performance Computers

Kusaka, Takashi; Tanaka, Takayuki

doi:10.3390/electronics11152285

Cited by 7 publications

(11 citation statements)

References 38 publications

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“…Therefore, this nonlinear function is processed by approximation to achieve high speeds. The residual correction method proposed in our previous study [ 7 , 8 ] allowed calculations using only square root operations, as shown in the following equation. When

, the maximum approximation error was less than 0.5 deg as showin in Figure 6 .…”

Section: Methodsmentioning

confidence: 99%

“…In human body motion measurement, and the posture measurement of robots and UAVs, real-time performance is required; thus, a high processing speed is vital. For this reason, we have been developing a fast approximate computation method for nonlinear functions as a fast attitude computation algorithm that can be used in embedded systems [ 7 , 8 ], and have used it in the above-mentioned wearable sensors and powered suits. This has enabled posture measurement on MCUs with small internal memory, where conventional mathematical libraries cannot be incorporated.…”

Section: Introductionmentioning

confidence: 99%

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Stateful Rotor for Continuity of Quaternion and Fast Sensor Fusion Algorithm Using 9-Axis Sensors

Kusaka¹,

Tanaka

2022

Sensors

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Advances in micro-electro-mechanical systems technology have led to the emergence of compact attitude measurement sensor products that integrate acceleration, magnetometer, and gyroscope sensors on a single chip, making them important devices in the field of three-dimensional (3D) attitude measurement for unmanned aerial vehicles, smartphones, and other devices. Sensor fusion algorithms for posture measurement have become an indispensable technology in cutting-edge research, such as human posture measurement using wearable sensors, and stabilization problems in robot position and posture measurement. We have also developed wearable sensors and powered suits in our previous research. We needed a technology for the real-time measurement of a 3D human body motion. It is known that quaternions can be used to algebraically handle 3D rotations; however, sensor fusion algorithms for three sensors are presently complex. This is because these algorithms deal with the post-rotation attitude (pure quaternions) rather than rotation information (the rotor) to avoid a double covering problem involving the rotor. If we are dealing with rotation, it may be possible to make the algorithm simpler and faster by dealing directly with the rotor. In this study, to solve the double covering problem involving the rotor, we propose a stateful rotor and develop a technique for uniquely determining the time-varying states of the rotor. The proposed stateful rotor guarantees the continuity of the rotor parameters with respect to angular changes, and this paper confirms its effectiveness by simulating two rotations around an arbitrary axis. In addition, we verify experimentally that a fast sensor fusion method using stateful rotor can be used for attitude calculation. Experiments also confirm that the calculated results converge to the desired rotation angle for two spatial rotations around an arbitrary axis. Since the proposed stateful rotor extends and stabilizes the definition of the rotor, it is applicable to any algorithm that deals with time-varying quaternionic rotors. In this research, an algorithm based on a multiply–add operation is designed to reduce computational complexity as a high-speed calculation for embedded systems. This method is theoretically equivalent to other methods, while contributing to power saving and the cost reduction of products.

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, the maximum approximation error was less than 0.5 deg as showin in Figure 6 .…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stateful Rotor for Continuity of Quaternion and Fast Sensor Fusion Algorithm Using 9-Axis Sensors

Kusaka¹,

Tanaka

2022

Sensors

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“…1(b). The thickness at different locations of the lens can then be obtained by relating the phase map of each single vortex beam ϕ 1,2 (x, y) as follows: 41,42 h = hmax[sin (ϕ 1 (x, y)) + sin (ϕ 2 (x, y))],…”

Section: Numerical Analysis a Acoustic Lens Designmentioning

confidence: 99%

Underwater double vortex generation using 3D printed acoustic lens and field multiplexing

Ellouzi,

Zabihi,

Aghdasi

et al. 2024

APL Materials

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The generation of acoustic vortex beams has attracted an increasing amount of research attention in recent years, offering a range of functions, including acoustic communication, particle manipulation, and biomedical ultrasound. However, incorporating more vortices and broadening the capacity of these beams and associated devices in three dimensions pose challenges. Traditional methods often necessitate complex transducer arrays and are constrained by conditions such as system complexity and the medium in which they operate. In this paper, a 3D printed acoustic lens capable of generating a double vortex pattern with an optional focusing profile in water was demonstrated. The performance of the proposed lens was evaluated through computational simulations using finite element analysis and experimental tests based on underwater measurements. The results indicate that by altering the positioning of the vortices’ axes, it is possible to control both the intensity and the location of the pressurized zone. The proposed approach shows promise for enhancing the effectiveness and versatility of various applications by generating a larger number of vortices and freely tailoring the focal profile with a single lens, thereby expanding the practical uses of acoustic vortex technology.

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“…Azim [7] presents a more modern approach wherein one can also do the approximation without technology. Kusaka [8] seeks to increase computation speeds for low performing computers, while Kumar [9] does the same thing for faster computers.…”

Section: Other Approaches Approximating the Trigonometric Functionsmentioning

confidence: 99%

Rotation without Imaginary Numbers, Transcendental Functions, or Infinite Sums

Bakhos¹

2023

JAMCS

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Quaterns are introduced as a new measure of rotation. Rotation in quaterns has an advantage in that only simple algebra is required to convert back and forth between rectangular and polar coordinates that use quaterns as the angle measure. All analogue trigonometric functions also become algebraic when angles are expressed in quaterns. This paper will show how quatern measure can be easily used to approximate trigonometric functions in the first quadrant without recourse to technology, infinite sums, imaginarynumbers, or transcendental functions. Using technology, these approximations can be applied to all four quadrants to any degree of accuracy. This will also be shown by approximating \(\pi\) to any degree of accuracy desired without reference to any traditional angle measure at all.

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Fast and Accurate Approximation Methods for Trigonometric and Arctangent Calculations for Low-Performance Computers

Cited by 7 publications

References 38 publications

Stateful Rotor for Continuity of Quaternion and Fast Sensor Fusion Algorithm Using 9-Axis Sensors

Stateful Rotor for Continuity of Quaternion and Fast Sensor Fusion Algorithm Using 9-Axis Sensors

Underwater double vortex generation using 3D printed acoustic lens and field multiplexing

Rotation without Imaginary Numbers, Transcendental Functions, or Infinite Sums

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