Method and measuring device for determining the position of an object

Abstract

The invention relates to a method for determining the position of an object by ascertaining the rotational angle of the object. Relative rotational angle increments Δθ k are measured at successive measurement times t k , where k = 1, and are combined in order to form cumulated rotational angle increments θ k for each measurement time t k . The cumulated rotational angle increments θ k for the measurement times t k are calculated iteratively from a number n of relative rotational angle increments Δθ k , which were measured at subsequent measurement times t k , where k = 1,..., n, the respective cumulated rotational angle increment θ k-1 determined for the previous time interval, and the transcendent function (I) with ϛ as the function of the cumulated rotational angle increment θ k-1 at the respective previous measurement time t k -1, an approximation of the transcendent function (I) being carried out using Padé approximations. By means of the Padé approximations of the transcendent function, an automatic computer-supported calculation of trigonometric functions of angles θ k or rotations from given angles θ k can also be carried out in a very resource-efficient and precise manner.

Description

[0001] German Aerospace Center eV Attorney file: Königswinterer Straße 522-524 V / DLR-0799-WO 53227 Bonn-Oberkassel 2022 / 224 Germany Date: 8 January 2024 Method and measuring device for determining the position of an object The project that led to this application has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 101004205. The invention relates to a method for determining the position of an object from the rotation rates of the object, wherein relative rotation angle increments ^^ to ^^ consecutive measurement points ^^ ^^ with k = 1, …, n and for the respective measurement times ^^ ^ to cumulative rotation angle increments ^^ together ^ ^^The invention further relates to a measuring device for determining the position of an object from the rotation rates of the object, wherein the measuring device comprises a sensor unit for measuring relative rotation angle increments ^^ ^^ for one another other subsequent measurement points ^^ ^ with k = 1, …, n and a data processing unit ^ which is set up to measure at consecutive times ^^ ^^ measured relative rotation angle increments ^^ ^^ for the respective measurement times ^^ ^^ to cumulative rotation angle increments ^^ ^^to summarize. Gyroscopes are used to measure the rate of rotation (angular velocity) around a predefined axis. By using at least three gyroscopes that are oriented so that the direction vectors along the axes of their measurement are linearly independent, the angular velocity vector ω of a rotation around any axis of rotation oriented in space can be determined. If the gyroscopes are firmly attached to an object, such as a vehicle (e.g. a rocket), the (relative) change in position of the vehicle (orientation in space) can be calculated from the rate of rotation by integrating the signal. If the initial orientation JG / JG is known, the absolute orientation of the vehicle in space is then known. Knowledge of the orientation is the input variable for attitude control and for determining the position when orientation-dependent forces act on the vehicle, e.g. caused by drive or braking systems fixed to the vehicle.Gyroscopes are usually bundled with acceleration sensors for measuring linear acceleration in a sensor package (an inertial measurement unit (IMU)), which is available as a module and is the central building block for determining the orientation and position of the vehicle. In addition to the sensors, the IMU also contains signal processing stages with the aim of generating a signal at the user interface for further use in the on-board computer. After digitization and possible filtering of the signals, the frequency is reduced so that a sufficiently low-frequency signal that meets the specifications of the interface protocol is available at the interface for further processing. In the case of gyroscopes, it is often not the angular rates ω that are measured directly, but rather their integrals, i.e. those over a sampling time step. Δ ^^ ^^ integrated (ie cumulative) rotation rate, which, for example, in the time step of ti−1 night i , i ∈ N by ^ ^^ ^^ In the case of a time-constant axis of rotation, this integral corresponds to the (relative) angle of rotation (angle increment) ^^ ^^ ^^ by which the IMU moves during this sampling time step Δ ^^ ^^ If a cumulative angle of rotation is to be displayed on the user interface Δ ^^ ^ ^ ^ ^ ^^ ^^ output that covers a longer time range ∆ ^^^^, ^^ ^^ ^^ = ∑ ^ ^^ ^ =1 Δ ^^ ^^ , n ∈ N from to ^ (i.e. a frequency difference compared to the input frequency ^^ lower frequency < refers, the n individual angles of rotation ∆ ^^ ^^―1+ ^^ , k = 1,..., n, which within the time interval [t I−1 , t I] are measured. To understand the notation, it is pointed out that the time interval [ ^^ ^^―1, ^^ ^^], which is also referred to as time domain I in the following, is divided into ^^ time intervals [ ^^meas, ^^―1, ^^meas, ^^] by the sequence of ^^ measurement times ^^meas, ^^, ^^ = ^^ where ^^ ^^ = ^^ ^^, with the individual lengths Δ ^^ ^^ = ^^meas, ^^ ― ^^ can be different. The ^^ -th time range can be preceding, ( ^^ -1)-th time range [ ^^ ^^―2 , ^^ ^^―1 ]). For example, if the sampling frequency of the gyroscopes is f = 2 kHz, the measured angle increments are set to a constant sampling rate of Δt = f -1 = 5 * 10 −4 s and the frequency at the user interface f out = 100 Hz, then ^^ = ^^ ^^ ^^ ^^ ^^ = 20der with time intervals Δt = 5 * 10−4 s measured rotation angle increments∆ ^^ ^^ to sum the angle increment related to a time step Δtout = 1 / fout = 0.01 s at the output, i.e. the relative angle of rotation by which the IMU rotates during a time step Δt out has rotated. However, these considerations are only correct if the rotation occurs around a temporally constant axis of rotation. This assumption is usually not met. For example, if a rigid body rotates without the influence of external torques around an axis that is not one of the three principal axes of inertia, the angular velocity vector in the body-fixed reference system performs a precessional motion and its direction is not constant over time. In the context of navigation, this phenomenon is referred to as "coning". In the case of a temporally variable axis of rotation, the following changes occur: 1. The integral Δφ measured by the sensors k the rotation rate ω does not directly result in the angle increment ∆ ^^^^ , which directly indicates the rotation axis and the angle of rotation that occurred during the sampling interval. Instead, the relative angle of rotation increment ∆ ^^ ^^ from the integrals of the current and past angular rates. 2. The direct summation of temporally successive angular increments associated with the time intervals ∆ ^^ ^^ , ∆ ^^+1 does not result which leads to rotation in the time interval [t i−2+k ,t i+k ] corresponding angular increment. Instead of simply summing the angular increments, a more complicated mathematical operation must be performed. John E. Bortz, "A New Mathematical Formulation for Strapdown Inertial Navigation," In: IEEE Transactions on Aerospace and Electronic Systems AES-7.1 (1971), pp. 61–66. DOI: 10.1109 / TAES.1971.310252, proposes a (nonlinear) differential equation for the angle of rotation θ(t) as a function of time, which depends on the rate of rotation ω(t), and is as follows: (2.1) In the above equation, the multicomponent rotation angle θ(t) is shown in bold to distinguish it from its norm θ = ‖ ^^‖2. This differential equation can be simplified for small rotation angles θ by expanding it into a Taylor series around rotation angles θ = 0, retaining only the leading order(s). It can then be expressed, for example, in a time interval [t I-1 , t I ], i.e. for the angle increment ^^ ^^ ^ ^ ^ ^ ^^ ^^ To do this, ω(t) is approximated in this interval by a polynomial of (n – 1)-th degree by ^ ^ ^^ ^ Its n coefficients can be calculated from n measured integrated rotation rates Δφ k, k = 1, ... , n must be determined. To do this, the coefficients must be inserted into the polynomial approximation, the integral calculated, and then the equations must be solved for the coefficients. This method already requires n measured angle increments Δφ k of time intervals that together represent the time range of the cumulative angle of rotation ∆ ^^ ^^ and provides the cumulative angle of rotation ^^ ^^ ^ ^ ^ ^ ^^ ^^ of the interval [t I−1 , t I MB Ignagni. “Efficient class of optimized coning compensation algorithms,” In: Journal of Guidance, Control, and Dynamics 19.2 (1996), pp. 424–429. DOI: 10.2514 / 3.21635. eprint: https: / / doi.org / 10.2514 / 3.21635 describes a higher-order algorithm. However, for higher orders ^^ the computational effort increases significantly and the approximation of the measured data by high-degree polynomials ^^ ― 1leads to strong oscillations at the edges of the interpolation interval, known as Runge's phenomenon. R.A. McKern. "A study of transformation algorithms for use in a digital computer," MA thesis. Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1968. URL: http: / / hdl.handle.net / 1721.1 / 14164 discloses an algorithm for combining the relative rotations from consecutive time intervals. This algorithm calculates the unit quaternion of the relative position change during the entire time interval via the detour of unit quaternions from the angular increments of two consecutive time intervals. The previously described algorithm for calculating the cumulative angle of rotation is used. ^^ ^^ ^ for the interval tI] used, However, only for n = 2 consecutive angle increments. Then the obtained angle of rotation ^^ ^^ ^ ^^ ^ ^^ ^^ the unit quaternion of the relative position change in the time interval t I ] calculated as: ^ The angle vector in the argument of the exponential function and on the right side of the equation is to be interpreted as a so-called pure quaternion, which only has one with the vector components of the angle increment ∆ ^^ ^ ^ ^ ^ ^^ ^^ formed imaginary part. In addition, the trigonometric functions are approximated by their Taylor polynomials up to the 3rd order. The result is then used to calculate ^^0 and the quaternion 0 0 ^^ = 1 to calculate the quaternion of the relative change in position recursively according to: ^ ^ ^^ ^ Here, the quaternions are to be multiplied using the quaternion product. Analogous to formula (2.3), the angle increments ^^ ^^ ^^―1 ,  ^^ ^^ ^^ belonging quaternions ^ ^^ ^^ calculated and then in analogy to the formula ^^―1 (2.4) to obtain the ^ ^ ^ ^―2 ^^ = ^ ^ ^ ^ ― ― 1 2^^ ^ ^ ^ ^―1 ^^ . (2.5) The angular increment of the combined rotations is then recovered by inversion of function (2.3). The corresponding relation for the angle increment obtained from the vector components of the angular increment is ^^ ^^ ^ ^ ^ ^ ^^ ^^ formed quaternion as follows: This solution for the special case of n=2 temporally consecutive angle increments ^^ ^^ ^^―1 ,   ^^ ^^ ^^ can be applied to any number n of angle increments for a time range and a corresponding angle of rotation ^^ ^^ ^ ^ ^^ ^^ ^^ be generalized.The iterative calculation of the angle of rotation ^^ ^^ ^ ^^^ = ^^ of the^ ^ ^^ ^ ^^ ^^ ^^ ^^ ^^ wrote through the procedure Set ^^ Für ^^ = Calculate ^ ^^ out of ^^ ^^ ^^ with equation (2.3) Be 0 ^^ ^^ and ^^ with equation (2.5) Calculate ^^ out of ^ 0 ^ ^^ with equation (2.6) Calculating the angle of rotation increment ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^ is done indirectly via quaternions and is therefore based on the three equations (2.3), (2.4), and (2.5). This requires the calculation of various transcendental functions (cos, sin, arcsin), e.g., after approximation by their Taylor polynomials, which increases the computational effort. After each ^^-th recursive update of the quaternion, equation (2.6) is evaluated to determine the iterative rotation angle increment from the quaternion ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^ This also involves increased computing effort, which ^^ -th recursion step. When approximating the trigonometric functions in equation (2.3), the quaternion is no longer a unit quaternion. However, the relation of equation (2.6) only applies to unit quaternions. Its application to non-normalized quaternions leads to an additional (numerical) error. The calculation of equation (2.6) for sufficiently large angular increments ^^ ^^ ^ ^ ^ ^ ^^ ^^ , such as those that occur when a large number n of increments ^^ ^^ ^^When approximated by its Taylor polynomial, it requires consideration of relatively high orders to achieve the desired numerical accuracy. This increases the computational effort. The known methods for determining the position from a rotation angle ^^ ^^ ^ ^ ^ ^ ^^ ^^ a time range, e.g. the interval , t I ] by summarizing a Number of n angle increments Δφ k at points in time ^^ ^^ , e.g. ^^ = the together form the time domain, are very computationally intensive and therefore require a relatively large amount of computing time and computing capacity, as well as relatively powerful and thus complex electronics with hardware computing power. DE 102022121662 B3 discloses a method for determining the position of an object by determining the object's rotation angle, wherein rotation angle increments are measured for successive measurement times and combined to form a cumulative rotation angle for a cumulative time domain containing the time points.The cumulative angle of rotation is calculated for a cumulative time domain as a function of exponents of at least a second-order approximation of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3). Iteratively cumulative angle of rotation increments for each time point are determined from the exponent function, each with a measured angle of rotation increment for the respective time interval and the cumulative angle of rotation increment determined for the previous time point or a predefined initial angle of rotation increment as input variables. The cumulative angle of rotation for the time domain is determined by the last cumulative angle of rotation increment determined at the last measurement point. The iterative angle of rotation increments are calculated directly without using quaternions.US 2018 / 0231385 A1 discloses a method for determining the pose parameters of an inertial measurement unit (IMU) sensor, comprising the steps of collecting measurement data generated by IMU sensors, using a processor to temporally integrate the measurement data, including any errors, generating a temporally continuous error propagation model, and integrating the data. The error propagation model can be used to generate compensation gradients for the pose parameters. EP 1637840 A1 discloses a method for compensating conicity in a strapdown inertial navigation system that uses groups of five consecutive incremental rotation angles of a body-fixed coordinate system measured by orthogonally mounted gyros at regular measurement intervals. Each group of five measurements is obtained during a group interval corresponding to five measurement intervals.The cone-compensated angular displacement of the body-fixed coordinate system around a fixed axis in space during a p-th group interval is obtained by summing the five measured incremental angles and a cone compensation term. The cone compensation term consists of the sum of half the cross product of a first and a second vector sum and the weighted sum of three vector cross products. The second vector sum is the sum of the five incremental rotation angles in a group. The first vector sum is the sum of the second vector sum over p groups. The multiplier and multiplicand of each vector cross product is a weighted sum of five measured incremental angles. The cone-compensated angular displacement can be summed over p groups to obtain an accurate estimate of the vector rotation angle over several group intervals. Based on this, it is the object of the present invention to provide an improved method.This object is achieved by the method having the features of claim 1 and the measuring device having the features of claims 11 and 12. Advantageous embodiments are described in the subclaims. It is proposed that cumulative rotation angle increments be used to determine the position of an object. ^^ ^^ for the measurement times ^^ ^^ iteratively from a number n of relative rotation angle increments ^^ ^^ which are measured at consecutive times ^^ ^^ with k = 1, …, n were measured, the cumulative rotation angle increment determined for the previous time interval ^^ ^^―1 and the transcendent Funktion ^^ (Ϛ) = Ϛ 1 2(1 ― Ϛ ^^ ^^ ^^ Ϛ) mit = ^^―1 ^^―1 as a function of ^^ / 2 ⋅ ^^ / 2cumulative rotation angle increment ^^ ^^―1 at the previous measurement time ^^be calculated. This is done by approximating the transcendental function by Padé approximations. The rotation angle increments and angles contain angle information for the required spatial directions and can have vectors with the vector components of the three units i, j, k of the unit quaternion described above or the three spatial directions x, y, z. Even if the rotation angle increments and angles are not explicitly referred to as vectors in the present application and the claims for the sake of simplicity, the embodiment as a vector or representations by skew-symmetric matrices thereof is encompassed. The method has the advantage that the calculation of the iterative rotation angle increments is carried out directly without detours via quaternions. Only a transcendental function with as a function of the cumulative Rotation angle increment ^^ to the previous measurement time ^^ ^^―1This is done using Padé approximations of the transcendental function. In this way, the only transcendental function to be calculated can be approximated with high precision by a rational function and thus calculated efficiently and with a predetermined number of operations. Padé approximations themselves are well-known tools for numerical approximations. The approximations of the transcendental function 1 ^^(Ϛ) =Ϛ2 (1 ― ^^ ^^ ^^ Ϛ) with Ϛ as a function of the cumulative rotation angle increment ^^―1 ^^ by approximations exploits the fact that the function ^^(Ϛ) has special properties that make its Padé approximations so accurate and thus efficient. This also allows for estimating the maximum approximation error. In addition, only even powers of the argument are used. required, so that the calculation of the square root of the transcendental function is only for the range 0 ≤necessary because the function for the double angle from it This also allows us to directly avoid the singularity of the function at = ^^, where the approximation collapses. In addition to relative cumulative angle increments ^^ ^^ , whose norm must be limited to ensure a given precision, absolute angular increments and thus, in particular, the angle of rotation parameterizing the position of the object (body) can also be calculated with a given precision. The only transcendental function to be calculated ^^(Ϛ) is nevertheless approximated with high precision by a rational function and can thus be calculated efficiently and with a predetermined number of calculations. Furthermore, it is possible to calculate the trigonometric functions (sine and cosine) of the cumulative rotation angle increments. ^^ ^^and therefore also the corresponding rotation matrix without further approximation or calculation of transcendental functions from the result for the transcendental function ^^(Ϛ) This allows for very precise and fast calculation of trigonometric functions and rotations from angles, even with reduced processor power. ^^ ^^ from the result of the transcendental function ^^(Ϛ)This solves the following technical problems: 1) Absolute angle increments can be transmitted to an on-board computer instead of relative angle increments. This ensures that in the event of a temporary failure of communication between the IMU and the on-board computer, resulting in the loss of even individual angle increments, this failure / error can be compensated for without other information sources (sensors). This makes it possible to determine the position from the gyroscope measurement data directly within the IMU, independent of the on-board computer. 2) The need to use lookup tables and / or time-delayed iteration algorithms (with possibly an accuracy check as a termination condition and thus an unspecified number of arithmetic operations) for calculating the transcendental functions is eliminated.Instead, a direct calculation of the required transcendental function can be performed with a predetermined number of calculation operations and guaranteed accuracy. 3) Using the rotation matrix, the translational velocity and position can be calculated from the accelerations / relative velocity increments measured by the IMU's acceleration sensors. In combination with the attitude and a model for gravity, complete inertial navigation within the IMU appears possible, independent of the onboard computer. 4) Trigonometric functions and rotation matrices can also be calculated efficiently and precisely for given rotation angles in other application areas, such as computer graphics / computer vision and CAD. The calculation of the cumulative rotation angle increments. ^^ ^^ for a particular measurement point ^^ ^^can be performed as a function of the exponent of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3), which is approximated to at least second order in the second argument. Iteratively cumulated rotation angle increments ^^ ^^ for the measurement times ^^ ^^ with k = 1,…, n each from the function of the approximate exponent of the Baker formula with one at the time of measurement ^^ measured relative rotation angle increment ^^ ^^ and the one for the ^^ previous point in time ^^ ^^―1 certain cumulative angle increment ^^ ^^―1 or a specified starting angle increment ^^ 0 as input variables. This allows a precise summation / cumulation of the angle increments, even in the case of a temporally non-constant rotation axis, in a very efficient manner with reduced computational effort. ^^ ^^a new (measured) relative angle increment, which is abbreviated to ^^ ^^ is called, then this by ^ ^^ added to the existing cumulative angle increment and then results in the new cumulative angle increment ^^ ^^ This process is abbreviated below as Algorithm 1, where the intermediate result is ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^ which is designated from the time ^^ ^^―1 starting from, an integer number of k relative angle increments ^^ ^^ taken into account. The stored angle increment ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^is reset to zero after a number of n updates. This creates a cumulative, but still relative (i.e., based on the respective start time ^^ the accumulation-related) angle increment ^^ ^ ^ ^ ^ , ^ ^^ generated, ^ ^^ which is the number of n relative rotations with angles ∆ ^^ ^^ , k = 1, ...,n resulting total rotation is parameterized. Algorithm 1 for the iterative calculation of ^ of the interval[ ^^ ^^―1 = ^^ ^^ ^^ ^ ∆ ^^ ] ^^ 1: procedure ^^ ^ ^ ^ ^ ^^ ^^ (∆ ^^ 1 , …, ∆ ^^ ^^ ) 2: Set ^^ ^ ^ ^ ^ , ^ 0 ^ ^^ = 0 3: for k = 1,…,n do 4: Calculate ^^ ^ ^ ^^ , ^ ^ ^ ^ ^^ out of ^^ ^ ^^ ^, ^ ^ ^^ ^― ^ 1 und ^^ ^ ^^ according to equation (1.2)The calculation of the function ^^ in line 4 of Algorithm 1 requires in particular the approximate calculation of only a single transcendental function ( 1.3) which is the value of the last cumulative angle increment, i.e. ^^ ^ to be evaluated (cf. the explicit (3.5a) below). Here, the function ^^(Ϛ) straight, so that only the squared norm ( θ ^^―1 ) 2  is required and thus no calculation of the square root is necessary. In addition, the algorithm is based on a simpler iteration and should simplify the analysis of the propagation of measurement uncertainties and measurement noise compared to the quaternion-based algorithm described in the introduction to the state of the art. These properties allow a direct implementation on the hardware (for example, a Field Programmable Gate Array (FPGA)) of the IMU. The algorithm can thus be implemented as part of a sensor package and the signal processing of the sensor data for output to the user interface. It calculates angular increments with a reduced frequency, which can then be used in the on-board computer to calculate the position. The Padé approximation allows a particularly efficient calculation of the transcendental function in conjunction with the iterative algorithm 1. The rotation angle increments ^^ ^^on consecutive Measurement times ^^ ^ with k = 1, …, n measured (or from measured ^^ ^ ^^ ^^ calculated) and to a cumulative angle of rotation ^^ ^ for one the Measurement times ^^ ^^ comprehensive cumulation time range I by calculating a cumulative angle of rotation ^^ ^ for a cumulative Time range ^^ as a function of the exponent of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3). Iteratively cumulated rotation angle increments ∆ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^ for the times ^^ ^^ with k = 1,…, n each from the function of the exponents with one rotation angle increment each ∆ ^^ ^^ the measurement time ^^ ^^and the cumulative rotation angle increment determined for the previous time interval[ ^^ ^^―1, ^^ ^^―1+ ^^] ∆ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^ ― ^ 1 or a specified output angle increment ∆ ^^ ^ ^ ^ ^ , ^ 0 ^ ^^ as input variables. The cumulative angle of rotation ∆ ^^ ^ ^ ^ ^ ^^ ^^ for the time domain ^^ results as a function of the last cumulative rotation angle increment ∆ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^ , which for the last time ^^ ^ ^ of the interval of the time range I. It can be assigned to this cumulative rotation angle increment ∆ ^^^ ^ ^ ^ , ^ ^ ^ ^ ^^ For the iteration, the initial rotation angle increment ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^ = ^ 0 for k = 0 with an initialization with the value zero. To determine the cumulative angle of rotation ^^ ^^ ^ ^ ^ ^ ^^ ^^ for the time range [ ^^ I-1 , ^^ I ] with the measured rotation angle increments ^^ ^^ ^^ for the respective times with k = 1,…, n as ^^ ^^ The input variables are calculated before the first iteration step. The cumulative angle increments are then ^^ ^^ ^ ^ ^ ^ , ^ ^ ^^ ^^ as intermediate results iteratively with k = 1 to n from the cumulative rotation angle increment determined or specified for the previous iteration step k-1 ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^ ― ^ 1 and the one for the time ^^ ^^ , which belongs to the respective iteration step k, measured rotation angle increment ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^as a function of the exponent of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3). This exponent only needs to be approximated by its Taylor polynomial in its second argument, since the first exponent is exact. In each iteration step, the evaluation of only one equation is required, which will be explained in more detail later as equation (3.5). This has the following consequences: – There is only one transcendental function, ^^(Ϛ) = ^ to calculate, which is efficiently approximated by the Padé approximation. – The iteration is simplified because (apart from the initialization required in each case) no additional computational step is required at every nth time step. This is an important advantage, especially for implementation on very limited hardware (such as a Field Programmable Gate Array (FPGA)). – The analysis of the propagation of measurement uncertainties and measurement noise is facilitated. – The Padé approximation of the only transcendental function, ^^ θ ^^―1 2 ist more accurate than the approximation of equation (1.3) by its Taylor polynomial of the same order. For large cumulative rotation angles ^^ ^^ ^ ^ ^ ^ ^^ ^^and thus, especially for a larger number n, greater accuracy is achieved. – By selecting the two orders for the approximation separately, the method can be easily adapted to the given requirements (e.g., the maximum rotation rate, sensor frequency, frequency divider, number of angular increments to be summed) for the user. In the case of a time-varying rotation axis, the following changes arise (considered in the proposed method): 1. The integral Δφ measured by the sensors k the rotation rate ω does not directly result in the angle increment ^ the direct axis of rotation and the angle of rotation of the ^ rotation that occurred during the sampling interval. Instead, ^^ from the ^^ Integrals of the current and past rotation rates.2. The direct summation of temporally successive angle increments associated with the time intervals ^^ ^ ,∆ ^^ does not yield the angular increment corresponding to the rotation in the time interval [ ^^meas, ^^―2, ^^meas, ^^]. Instead of the simple summation of the angular increments, a more complicated mathematical operation must be performed. The method is based on the complex mathematical operations required for time-varying axes of rotation, rather than a simple summation of the angular increments. The reason for this is that rotations about non-parallel axes of rotation do not commute. Rather, the rotations in (three-dimensional) space form a group, the rotation group SO(3), which is a so-called Lie group. The mathematical field of Lie groups forms the theoretical framework from which the representations of the group elements in the form of rotation matrices or in the form of unit quaternions arise. The rotation matrices are special (their determinant is 1) orthogonal (their transpose is their own inverse) matrices in three dimensions.The name of the group, SO(3), is derived from the terms Special Orthogonal Rotation Matrices in 3 Dimensions. Unit quaternions are the extension of complex numbers to three independent imaginary units, whose norm, calculated from the real and imaginary parts, is 1 and which are multiplied using a noncommutative, norm-preserving product. A rotation can be represented by a unit quaternion, although the representation is not unique; both q and -q can be used to represent a specific rotation. Depending on the chosen representation (in the form of vectors, unit quaternions, etc.), the operations described above for calculating the angular increments and their summation follow. The cumulative angle of rotation. ^^ ^^ ^ ^ ^ ^ ^^ ^^ for a time range I, which includes a number of n consecutive points in time^^meas, ^^with k = 1, …, n, this can be done by ^^^ ^^―1 , ^^ ^^ ^^ recursively (ie iteratively) from a ^ 2 2 Number n of rotation angle increments ∆ ^^ ^^ , which were measured at consecutive times tmeas,k with k = 1, …, n were measured, and the cumulative angle of rotation increment determined for the previous time interval ^^ ^^―1 The zeroth order exponent can be γ(0) ^^ ^^―1 ^^2 ,  =   ^ ^―1 ^^ 2. The first-order exponent can be ^ ^^ =   ^^ 2 ^^   +    ^ ^^ ^ ^^ ^^ w obei N 0a given integer value a m dependent on the value N0 given by the Padé approximation, and that for the time ^^ ^^ cumulative angle increment ^ 2 ^ ^^ = ^^ ^^ ^^―1 ^^2, each from the sum of the Development coefficient γ (mα) ^^ 2, the exponent of the Baker-Campbell Hausdorff formula in the zeroth, first and at least second order m α = 0, 1 and 2. ^ ^ For the measurement time ^^meas, ^^ a cumulative rotation angle increment ^ ^―1 ^^ each from the sum of the expansion coefficients (mα) of the Baker-Campbell-Hausdorff formula in the zeroth, first and at least second order m α = 0, 1, and 2. The calculation is based on only a single transcendental function, which reduces the time and computational effort. ^ ^ ^^ ^^―1 ^^ ^^ ^^can the sum of the arguments of the Approximations in the zeroth, first, second and third order m α= 0, 1, 2 and 3. By also considering an expansion coefficient of the higher third order, the accuracy can be further increased. Here, too, only the single transcendental function is included, which can be calculated efficiently and accurately using the Padé approximation. Furthermore, the (automated, computer-aided) calculation of a rotation matrix of the active rotation between a first coordinate system F of the object and a coordinate system determined by the cumulative angle of rotation ^^ ^^The second coordinate system G obtained by the rotation described above can be approximated with the results of the Padé approximations containing trigonometric functions, and the translational velocity and / or position of the object can be calculated from the also measured accelerations or relative velocity increments using this rotation matrix. Beyond the processing of sensor data from an IMU and the efficient calculation of the attitude and position of objects, an efficient calculation of trigonometric functions of given angles can be realized automatically and with high accuracy with reduced computing power. This can be used in a wide variety of applications in various fields, e.g., in computer vision. From a given angle ^^ ^^ can be calculated using the Padé approximation of the function ^^(Ϛ)efficiently and in particular without approximating any other transcendental functions trigonometric functions, such as the functions ^^ ^^ 2 ^^ Ϛ 2Ϛ and ^^ ^^ ^^2Ϛ.The invention is explained in more detail below with reference to the accompanying drawing with an embodiment. It shows: Figure 1 - Time course of the ^^ = measured or the value derived from the measurements ^^ the measurement times^^meas, ^^determined inputs ^^ ^^ ^^, the k-th iteration step for determining the k-th cumulative angle increment Figure 2 - for the position of an object; Figure 3 - flowchart of the method for determining a cumulative rotation angle increment from measured rotation angle increments (for simplicity, for constant sampling time steps Δ ^^ ^^ = Δ ^^); Figure 4 - diagrams of the relative errors when using the Padé approximation to calculate the transcendental function ^^ (Ϛ); Figure 5 - Diagrams of the relative errors when using the Padé approximation directly from ^^(Ϛ) and a regularized expression to calculate the transcendental function ^^ Figure 6 - Relative error of the approximation of ^^ ^^ ^^ and^^ ^^ ^^ ^^0( ^^), which on a Padé approximation of ^^(Ϛ) based; Figure 7 - Relative error of the approximation by the Taylor polynomials with degrees 2n+1, 2n of^^ ^^ ^^( ^^)and^^ ^^ ^^( ^^); Figure 8 - Relative error of the approximations of^^ ^^ ^^ θand^^ ^^ ^^ θby the Taylor polynomials with degrees 2n+1, 2n of ^^ ^^ ^^ and use in the expressions for^^ ^^ ^^ θand^^ ^^ ^^ θof half the angle; Figure 9 - Further flowchart of the process. Figure 1 shows a time course of the ^^ = ^^ measured or from the Measurements at the measurement times ^^ measured values ​​for the rotation ^angle increments, ie the inputs ^^ ^^ the k-th iteration step to determine the k-th cumulative angle increment ^^ ^^ ^ o ^, u ^^ t and the cumulative angle of rotation, ie the measurement result as output ^^ ^^ ^ o ^, u ^^ t at the time ^^ ^ This will make the ^ Time sequence with the measurement times of the summation for different time intervals k = 1, …., n of the summary to the low-frequency measurement result. To simplify the representation, constant sampling time steps Δ ^^ ^^ = Δ ^^ were assumed. The bold typeface of the quantities denotes vectors and the normal typeface of the quantities denotes scalars, such as the norm of the rotation angle vector. Figure 2 shows a sketch of a measuring device 1 for determining the position of an object 2. The measuring device has at least one inertial measuring unit IMU, which is used to measure the rotation angle increments ^^ ^^ ^^ consecutive points in time ^^ ^^ is configured to record rotating (circular) movements of the inertial measurement unit (IMU) and the associated object 2 in the three mutually orthogonal spatial axes (X-, Y-, and Z-axis) of the Cartesian coordinate system. These measured rotation angle increments ^^ ^^ ^^are fed as input values ​​to a data processing unit, which is used to calculate a cumulative angle of rotation ^^ ^^ ^ o ^ ut = ∆ ^^ ^ o ^, u ^^ t for a cumulative time range I, which includes consecutive points in time ^^ with i = 1,…, n. In addition, ^^It is optionally conceivable to measure the accelerations or relative speed increments as components along the three spatial axes for the speed and, if applicable, position and to feed them to the data processing unit. The data processing unit 3 can, for example, be a microprocessor or microcontroller that executes a computer program with program instructions that cause the microprocessor or microcontroller to execute the method steps of the claimed method. However, the data processing unit 3 can also be implemented as hard-wired hardware logic, such as a field-programmable gate array or similar. The method enables a direct calculation of the cumulative angle of rotation ^^ ^^ iteratively from the angle of rotation increment saved in the last step and the currently measured ^^ ^^ ^^The rotation matrix obtained in this process can be used to calculate the velocity and, if applicable, position from the measured accelerations or velocity increments. The basis for the summation of the measured rotation angle increments is the Baker-Campbell-Hausdorff (BCH) formula, which originates from the theory of Lie groups and specifies a commutation law for certain linear operators. It provides the following formula for the multiplication of two elements obtained by the exponential mapping: e α , e β of a Lie group the exponent γ(α, β) of the resulting element e γ(α, β) , which therefore has the relation e γ(α, β) = e α e β (3.1) is satisfied. The result γ(α, β) is expressed by the Baker-Campbell-Hausdorff formula as a formal series expansion in the total order m of ^^ and ^^ and therefore has the following schematic form: ^^ ^^ ^^ ^^^^ ^^ ^^ ^^^^ ^^ ^^ ^^ ^^ with the coefficient^^( ^^ ^^, ^^ ^^), whereby the actual form is considerably more complicated due to the general non-commutability of the group elements^^ ^^ ≠ ^^ ^^. The general Baker-Campbell-Hausdorff (BCH) formula is not yet available as a formal series in a form suitable for the summation of temporally consecutive ones with the time intervals [t i−3+k , t i-2+k ], [t i-2+k , t i-1+k ] associated angle increments ∆ ^^ ^^―1 , ∆ ^^ ^^ is suitable to determine the actual rotation angle increment belonging to the rotation in the time interval[ti−3+k, ti-1+k], ie the rotation angle ^^ ^^ ^ o ^ utfor the time domain I (in this case consisting of two subintervals). The BCH formula still needs to be specialized for the special case of the Lie group SO(3). Furthermore, a suitable evaluation / calculation of the series expansion must be performed. After specializing to SO(3), a suitable evaluation of the expression, which is still present as a series, is performed. The specialization to the Lie group SO(3) initially only provides the required relation for the summation of two angle elements, for the previously described case ^ ^^ where γ is still given as a series expansion in the total order m as in equation (3.2) and ^^ ^^ is the newly accumulated rotation angle increment resulting from the newly measured relative rotation angle increment ∆ ^^ ^^ and the previously calculated cumulative rotation angle increment. ^^The evaluation of the series expansion according to equation (3.2) of the function γ in equation (3.3) is performed under the assumption that only its second argument is small and therefore only in this argument can an approximation be made by the Taylor polynomial. This is necessary because when the relation (3.3) is applied iteratively to a total of n elements according to the rule: ^ ^ ^^ the first argument of γ contains the (iterative) cumulative rotation angle increment ^ from the previous k–1 steps. This is therefore for large k ≤ n way small. Therefore, an approximation of the Taylor series in the first argument by its Taylor polynomial can lead to an unacceptably large numerical error. The BCH formula can now be approximated for the Lie group SO(3) as follows, in order to find the exponents ^^ ^^ the orders m β = 0, 1, 2: ^ ^^ ^ ^^ with the transcendental function The only function contained in the system of equations and to be calculated by approximation is the transcendental function ^^(Ϛ) . The derivation of the approximation of at least second order m β The Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3) will be described in detail later. It is sufficient to use only the second argument ^^2 = ^^ ^^ an approximation at least in the orders m β = 0, 1, 2. The expressions are in the first argument ^^ ^ exakt. There is only one transcendent function B θ 2 1   = B θ ^^―1 2, which can be efficiently and accurately approximated using the Padé approximation. ^ and the of given maximum expected values ​​for the angle increments and their number n, which in a single angle increment ^^ ^^ ^ ^^ ^ ^^ ^^ should be summarized. When using equations (3.5), the required order can be chosen to be significantly lower with the same numerical accuracy than when using equations (2.3), (2.5), and (2.6). This reduces complexity and computational effort, i.e., calculation time and hardware expenditure. The iterative calculation of the cumulative rotation angle ^^ ^^ ^ ^^^ ^^ ^^ of the interval [ ^^I-1, ^^I] using at least a second-order approximation of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3) can be performed on the basis of equations (3.4) and (3.5a), (3.5) using procedure 1 as follows: Prozedur ^^ ^^ ^^ Set For ^^ iterative: Calculate ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^from the (iterative) cumulative rotation angle increment ^^ ^^ ^ ^^ ^, ^ ^ ^^ ^― ^ 1 and the measured rotation angle increment ^^ ^^ ^^ according to equation (3.4) with the explicit form (3.5a), (3.5) Set the resulting cumulative angle of rotation ^^ ^^ ^ ^ ^ ^ ^^ ^^ equal to the last (iterative) cumulative rotation angle increment ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^Figure 2 shows a corresponding flow diagram of this method or the above procedure for determining a cumulative rotation angle increment from rotation angle increments measured with a rotation angle sensor for successive time intervals. Calculating the Explicit Expression for the BCH Formula for the Lie Group SO(3). In the following, an expression for the BCH formula for the Lie group SO(3) is derived, which describes rotations in three-dimensional Euclidean space. The strategy here is to expand the formula in orders of only its second argument ^^2 = ^^ ^^ ^ ^^^ ^^ ^^ and to ignore the dependence on the first argument ^^ 21 to summarize so that the formula with respect to its first argument ^ 2 ^1 is exact. The result is therefore an expansion of the formula in orders of its second argument with simultaneous resumption all dependencies on its first argument. M. Müger, "Note on the theorem of Baker-Campbell-Hausdorff-Dynkin," https: / / www.math.ru.nl / ~mueger / PDF / BCHD.pdf, provides an overview of some fundamentals, facts, and proofs of the BCH formula. Section 8, and especially Remark 8.2 on page 24, points out that, based on a special gradation of the algebra based on the word length in the second argument instead of the total word length in both arguments, an expansion of the BCH formula in the second argument can be carried out. The corresponding expression is given up to first order in the second argument, but without specializing to the Lie group SO(3) and explicitly resuming the series containing the Bernoulli numbers. 1.Calculation of the zeroth and first order,^^ ^^ = 0,1 The first order simple steps of specialization and resumption are carried out using the generating function of the Bernoulli numbers, which is given by ^. ^ Given this, the expression of the BCH formula given as a series in M. Müger's "Note on the theorem of Baker-Campbell-Hausdorff-Dynkin" yields the following explicit representation in resumptive form: contains only even orders of so that the calculation of equation (A.2) no calculation of the standard ^^ the square root contained in the vector This is advantageous for numerical calculations. 2. Basics of the formalism and the relationship between the differential equation of angle increments and the BCH formula. The basics and steps required for the derivation of the second order are presented below. The starting point is the exponential map, which is also defined for arguments α that are elements of a Lie algebra, i.e., the tangent space of a Lie group at the element of the identity, and which can be represented by matrices. The exponential map is defined via its Taylor series ^ ^ where α is an element of a representation of a Lie algebra. This element α can be given, for example, by the pure quaternions in the case of rotations. Powers in the above equation are to be understood as being carried out with the corresponding product, which is defined for the representation, for example, the matrix product in the case of a matrix representation or the quaternion product in the case of a representation by quaternions. Since different elements α, β of a Lie algebra generally do not commute with each other, i.e. ^^ ^^ ≠ ^^ ^^ holds, the derivative of the exponential mapping is not given by the known expression. Rather, applying the derivative to the series representation according to equation (A.4) yields the following expression: where the n-fold commutator is defined as follows: and represents the adjoint action of the element α on another element β. From the derivation according to equation (A.5) follows the relation (A.7) This can be inverted. For this purpose, the generating function of the Bernoulli numbers is used, which is given in equation (A.1). The inverse of the relation (A.7) is then given by A property of practical importance for numerical calculations now becomes apparent: the series in the inverse according to (A.8) contains only even powers of α. Therefore, the expression, which is then specialized for the Lie group SO(3), depends only on even powers of the norm of α and thus does not contain square roots. First, the relation for the generating function is projected onto its even and odd powers, yielding the equations: This result is used to put the inverse according to equation (A.8) into the following form: (A.11) This expression (A.11) is now specialized to the Lie group SO(3), or its double covering, the Lie group SU(2), represented by the unit quaternions. For two elements θ, ω from their Lie algebra ^^ ^^(2) the multiple commutators are reduced as follows: By repeated application of this relation, the expression for equation (A.11) is After identifying the simple commutator with the cross product and reducing the double commutator / cross product the result (A.13) is in fact the differential equation for the angular increment θ as a function of the angular velocity ω, which is given in equation (2.1). The result of equation (A.8) and, in the special case of the Lie group SO(3), its explicit expression by (A.13) are closely related to the Baker-Campbell-Hausdorff formula, as will be shown below. The Baker-Campbell-Hausdorff formula provides the Lie-algebra-valued element γ as a function of two Lie-algebra-valued elements α, β, as a solution of the following relation ^^ ^^( ^^, ^^) = ^^ ^^ ^^ ^^ . (A.15) To show this connection, the exponent γ is expanded in orders of β, as is presented, for example, in M. Müger's “Note on the theorem of Baker-Campbell-Hausdorff-Dynkin”. For this purpose, a parameter t ∈ R is introduced and the expansion is written as where clearly ^^ (0) = ^^( ^^, 0) = ^^is. The result for^^ in the development of equation (A.16) can be directly derived from the Equation (A.11) can be obtained by replacing α by γ(α, t β). This then leads to the following differential equation for γ: By inserting t = 0 into the above equation, a result for ^^ A comparison of this result with equation (A.11) shows that the differential equation for the Lie algebra element α for a given Lie algebra element β = e −α ∂ t e α is identical to the equation for the linear contribution in β in the BCH series according to equation (A.16). In addition, by inserting t = Δt, the approximation of γ by^^ ≅ ^^ + ^^ (1) ( ^^, ∆ ^^ ^^) + ^^(∆ ^^ ^^) 2 to a solution of the differential equation (A.11) to the linear order in Δt β. 3 Calculation of the second order, ^^ ^^ = 2 In the following, the term ^^ (2)of the BCH series (A.16). This contains the second-order terms in β. First, the derivative of relation (A.11) must be calculated. This leads to: An equation for γ (2) is analogous to the procedure in the case for γ (1) To do this, α can be replaced by γ(α, t β) and ^^ = 0, ^^ = 0 can be inserted. This gives To obtain an explicit result for the special case of the Lie group SO(3), the series in (A.20) is summed to a closed expression. The procedure is analogous to the derivation of (A.13), using equation (A.12) to reduce the multiple commutators. Furthermore, ^^ = ^ 2 ^1 , ^^ = ^ 2 ^2to obtain the result in the variables used in the main text. To use the reduction formula (A.12), it can first be established that a general sequence with argument f k can be divided into sequences of only even and only odd parts as follows: (A.21) Using this identity, the inner sum in (A.20) can be divided as follows: where ϑ is the heavyside step function. Inserting this result into the series in (A.20) yields the Taylor series: A closed-form expression for the first of the two series is already known from (A.9). A closed-form expression for the second series can be obtained by applying a suitable derivative operator to the known result: Analytical continuation of the expressions by identification z = i θ1 then leads for the term of the BCH series, which is quadratic in θ2, to the result which depends only on a single transcendental function, B(ζ), which is already contained in the expression for γ (1) appears and is given in (A.3). This circumstance is advantageous for numerical calculations, since only a single transcendental function needs to be approximated and calculated. 4. Explicit expression for the BCH formula up to second order, m β = 0, 1, 2 The explicit result of the BCH formula for SO(3) in the expansion (A.16), ie up to and including the quadratic order terms in the second argument ^ 2 ^2 wird im The following is summarized and simplified. The term is simply represented by the first argument ^^1 of the function. The term is given by (A.13) 2nd, if it ^^ →  ^^1 and ^^ →  ^^2 or alternatively, the expression (A.2) is used directly. The term γ (2) is shown in (A.25). This gives the formula for the BCH Now the following identities are used: to first calculate and simplify the multiple commutators appearing in equation (A.26) Thus, the formula (A.26) can be gradually reduced to the following form: 5. Calculation of higher orders, m β ≥ 3 The described procedure can also be directly applied to the calculation of higher orders. In particular, the order mβ = 3 in θ2 , i.e. the term γ (3) in the BCH series (A.16) can be used to achieve even greater accuracy. The calculation of the exponents γ (mα) in the zeroth, first and at least second order mα = 0, 1 and 2 requires an approximate calculation of only a single transcendental function ^^(Ϛ) = Ϛ 1 2(1 ― ^^ ^^ ^^ Ϛ). This is with the value half the squared norm of the last cumulative angle increment, i.e. =^ ^^ mit θ ^^―1 =   ^^ ^^―1 ⋅ ^^ ^^―1 Here, the function ^^(Ϛ) is even, so that only the squared norm ( ^^ ^^―1 ) 2 and therefore no calculation of the square root is necessary. For sufficiently small 0 ≪ 1 , i.e. small norm (θ ^^ ) 2 ≪ 1 of the cumulative Angle increment ^^ ^^ can the function ^^(Ϛ) in equation (1.3) can be approximated by its low-order Taylor polynomial. This case of relatively small cumulative angles occurs, for example, when only a finite number n of relative angle increments in ∆ ^^ ^^ whose norm is also small, i.e. ^^ ^^ ^^ ≪ 1Since the cumulative angle is output after n time steps and then its internally stored value is set back to 0, the cumulative angle increments are ^^ ^^ also by relative angle increments that are related to their respective starting point of the accumulation. The accuracy of the approximation of the function ^^(Ϛ) by its Taylor polynomial is thus determined by how large n and the relative angle increments to be summed ^^ with k = 1, ...,n. For example, for n relative Angle increments ^^ ^^ constant norm ∆ ^^ 1 = • • • = ^^ = the size of the ^^ ∆ ^^ Product n ∆ ^^ crucial for the accuracy of the data used to evaluate ^^ (Ϛ)from equation (1.3). The renormalization of the angle increments is explained below. In order to obtain the absolute angle of rotation and thus the position of the body in space, all relative angle increments from a point in time at which the position was known ^ ^^ This represents a cumulation of any ^ many relative angle increments ^^ ^^ without periodic resetting of the The number n and therefore also the argument Ϛ , with which the function ^^(Ϛ) in (1.3) are then a priori not bounded upwards. The approximation of (1.3) by the Taylor polynomial then leads to an ever increasing and ultimately unbounded and thus unacceptable error. This problem of the upper unbounded norm of the angle increments and thus of an unbounded argument Ϛcan be solved as shown below. This is one of the prerequisites for approximating the periodic transcendental function (1.3) with an upper bound on the loss of accuracy by, for example, its Taylor polynomial. It is pointed out that a rotation in two dimensions can be achieved by a single angle of rotation ϕ which can be restricted to the interval [-π, π] or, if the sign ± of the direction of rotation is specified, to the interval [0, π]. This means that the angle of rotation ϕ and all angles of rotation ϕ ^^ = ϕ ± 2π ^^, ^^ ∈ ^^ describe the same rotation. Therefore, if a rotation angle ϕ > π be given, it is always possible to reduce it to its representative by subtracting suitable integer multiples of 2π ϕ ′= ϕ ―2πκ0 ,which lies within the interval [-π, π]. For a given ϕ > 0, the quantityκ0 = ^^ ^^ ^^(ϕ) | 2 ϕ π | chosen, whereby [ ]the floor function, then ϕ ′ = ϕ ―2πκ0 in fact, the representative who fulfills the condition will receive it. A similar procedure is possible for a rotation in three dimensions. Such a rotation is defined by a vector-valued angle of rotation ^^ parameterized, whose norm (length) ^^ = ^^ ∙ ^^ the total (scalar) angle of rotation and its direction (unit vector) ^^ ^^ = ^ ^ ^ ^ set the rotation axis. ^^ > 0 is, lays ^^ ^^also determines the direction of rotation. The right-fist or corkscrew rule can be applied here, where the thumb of the right hand points in the direction of the vector and the fingertips of the clenched right hand indicate the direction of rotation for rotations with a positive angle, i.e., the norm of the vector-valued angle of rotation. Analogous to rotations in two dimensions, in three dimensions every rotation can be parameterized by a vector-valued angle of rotation ^^′ = ^^′ ^^ ^^′, whose norm satisfies 0 ≤ ^^′ ≤ ^^. If a general angle of rotation ^^ is given, whose norm ^^ not in the interval [0,π], it can be replaced by a representative ^^′ with minimal norm0 ≤ ^^′ ≤ ^^, which consists of ^^ can be calculated as follows ^ ^ ′ = (θ ― 2πκ0) ^^ ^^ (3.1) Here, ^^0 ∈ Norm and direction vector of the representative with minimum Norm is then through ^^ ′= |θ ― 2πκ0|, ^^ θ′ = sgn(θ ― 2πκ0) ^^ ^^ (3.2) given, ie the condition 0 ≤ ^^′ ≤ ^^is met. In the next step, the value is determined which is κ0 to be chosen to obtain the representative with minimal norm. A general angle of rotation ^^ has a standard ^^ which are either in the interval2 ^^ ^^ ≤ ^^ < (2 ^^ + ^^or in the interval (2 ^^ + 1)^^ ≤ ^^ < 2( ^^ + 1)^^ ^^, ^^ ∈ ℕ, In the first and the latter case, the choice of the constant κ0 = ^^ orκ0 = ^^ + 1to the following expressions: These can also be represented as follows In fact, the standard ^^′ now the condition0 ≤ ^^′ ≤ ^^. The two cases can be solved using the floor function [ ] compact to A comparison of the expressions from equations (3.5) and (3.1) shows that for a rotation angle ^^ , whose norm lies in the interval κπ ≤ θ < (κ + 1)π, is the constantκ0 = κ + 2 1 must be chosen in order to then use equation (3.1) to determine the representative ^^′ with minimal norm. This can be verified directly if the norm of the representative ^^ ' using the relation ^^ ― 2 ^ 2 ^ = 1 ― ( 2 ―1) ^^ , which is valid for any ^^ ∈ ℕ, is calculated. This satisfies the condition 0 ≤ ^^′ ≤ ^^. In the algorithm for the accumulation of relative angle increments described in detail above, at each sampling time ^^ ^^ the square of the norm of the stored cumulative angle or angle increment ^^ ^^―1 which is abbreviated here again to ( ^^ ^^―1 ) 2 Even if the norm of the angle increment ^^ ^^―1 is minimal, so the condition 0 ≤ is fulfilled, is ^^ ≤ ^^this is not necessarily the case for the cumulative rotation angle increment ^^ ^^ the case that with the algorithm based on equation (1.2) and the current relative angle increment ^ ^^ was calculated. ^ Assuming that ^^ ^^―1 already replaced by its representative with minimal norm, so the condition 0 ^^ is fulfilled, and ≤ ^^ that the current relative angle increment ^^ ^^ the normκ ^^π ≤ Δθ ^^ < (κ ^^ + 1)π, so for example for 0 ≤ ^^ ^^ < ^^ the constant κ ^^ = 0, for ^^ ≤ ∆ ^^ ^^< 2 ^^ then κ ^^ = 1 etc., the squared norm up to and including ^^ where an upper limit can be determined using the triangle inequality for vectors. A comparison of this estimate with equation (3.4) shows that the norm of the current relative angle increment determines the value of the constant κ which is required for the calculation of the representative of minimal norm. In the case that all relative angle increments have a sufficiently small norm, i.e. ^^ ^^ < for all k, all are and thus result in the ^^ κ ^^ = 0Equation (3.5) the cases κ = 0 , in which the cumulative angle increment already has a minimum norm or κ = 1 In practice, the restriction ^^ ^^ < ^^ be the rule, because usually the sampling time of the gyroscopes is adjusted to the expected system dynamics so that only a small rotational movement occurs during a single sampling time step, i.e. ^^ is. ^^ ≪ ^^ In the generic case, when individual κ ^^ are to be chosen such that κ ^^π ≤ Δθ ^^^holds, there are, for example, the following two possibilities for the determination of ^ κ in the (k + 1)th time step: — First, a determination of (3.7) is carried out. Then a calculation of κ as the natural number less than or equal to the square root of κs is, so With the resulting κ This gives the relative, i.e., the remainder that remains after subtracting the maximum possible integer multiples of ^^2 from the square of the norm ( ^^ ^^)2. Due to the construction, 0 ≤ ^^ always holds. With this result, the representative of the minimal norm can then be constructed by transforming equation (3.5) as follows: — Alternatively, to calculate (albeit only approximately) the square root of the constant κs To avoid this, integer values κ' over the interval1 ≤ κ ′ can be iterated, either starting forward from the lower or ^^ from the upper interval limit. Then the current value can be κ ' respectively calculated and the change of sign depends on the size εκ′ The largest value of κ ', for the εκ′ ≥ 0 is valid, then the value for κ and ε κ ≥ 0. With these values, the representative with minimal norm can then be calculated using equation (3.10). The expression (3.10) for calculating the representative with minimal norm requires the calculation of the square root ^^ 1 + ^^. In general, κπ ≤ θ ^^+1< (κ + 1)π, from which it follows that ^^ then satisfies the condition 0 ≤ ε < 2 1κ. Thus is ^^not necessarily small, so that the square root cannot simply be approximated by its Taylor polynomial of a fixed order. However, as already mentioned above, in practice it will predominantly be the case that the norm of the relative angle increments ^^ ^^ the condition ∆ ^^ ^^ This is already ≪ ^^ a condition that must be met for the applicability of the algorithm based on the Baker-Campbell-Hausdorff formula and therefore does not represent an additional restriction. Then, from equation (3.6), it follows that the norm of the cumulative angle ^^ ^^ on ^^ ^^ ≤ ^^ ^^―1 ^^ ^^ ≤ ^^ + ∆ ^^ is limited. ^^ A calculation of ^^ e.g., using (3.11) with κ = κ ′ = 1 then results in ^^ is limited as follows If all points in time ^^ ^^ can be derived from the specifications of the specific application, so ∆ ^^ so that ^^ ≤ ∆ for all k, then results as a limitation for ^^ the condition0 ≤ ^^ ≤ 2 ∆ ^ ^ ^^ By increasing the sampling rate, the ^^ ^^ and thus also ∆ be reduced. A ^^ Approximation of 1 + ^^ by its Taylor polynomial becomes possible. The maximum order of the Taylor expansion, i.e. the degree of the Taylor polynomial to achieve a desired accuracy, can be determined. From equation (3.5) we get where the remainder term can be approximated using Stirling's formula for the asymptotic behavior of the factorials as follows The error in the norm caused by using the Taylor polynomial of degree n0 can now be estimated as follows The calculation of the representative of minimal norm can be carried out using the following procedures: Algorithm 2: Determination of the representative of minimal norm to ^^ ^^ It is believed that ( ^^ ^^ ) 2 exists and the condition 0 ≤ ^^ ^^ ≲ ^^ + ∆ ^^is satisfied. 1: procedure ^^ 2: Calculate ^^ = ( ^ ^ ^ ^ ―1 3: if ^^ > 0 then 4: Calculate ^^′ ^^ using equation (3.13) (for K = 1 and n0 determined from equation (3.15)) to 5: else 6: set ^^′ ^^ = ^^ ^^ From this it is easy to calculate n0for a given accuracy and given ∆ ^^ When checking the norm of the cumulative angular increment and, if necessary, replacing it with the representative of the minimum norm in each time step, in the case of relative angular increments that satisfy the condition 0 ≤ ∆ ^^ ^^ < ^^fulfill, the case κ = 1 the only one in which the replacement of the cumulative angle increment by its representative of minimal norm is to be carried out. Therefore, the simplified Algorithm 2 can be formulated. In the case that the norm ^^ is not restricted, the more general algorithm ^^ 3. Algorithm 3: Determination of the representative of minimal norm to ^^ ^^ It is believed that ( ^^ ^^ ) 2 is present. 1: procedure ^^′ ^^ ( ^^ ^^ ) 2 , ^^ ^^ 2: Calculate ^^ = ( ^^ ^^)2 ^ ^2 3: if ^^ > 0 then 4: Calculate κ ^^ = | ^^| 5: Calculate κ = [ κ ^^] 6: Calculate ^^′ ^^ to 7: else 8: Set ^^′ ^^ = ^^ ^^An alternative can be, for example, the variant given as Algorithm 4. Modifications to the calculation of ^^′ ^^ , e.g. in the form of a replacement of ^^ by^^ = ^^ ― 1and subsequent evaluation of the Taylor series from equation (3.13) up to the desired order n0, can also be easily performed if required. Algorithm 4: Determination of the representative of minimal norm to ^^ ^^ It is believed that ( ^^ ^^ ) 2 is present. 1: procedure ^^′ ^^ ( ^^ ^^ ) 2 , ^^ ^^ 2: Calculate ^^ = ( ^ ^ ^ ^^ ^2 )2 3: if ^^ > 1 then 4: Set κ ′ = | ^^| 5: while ^^ > κ ′ 2 do 6: Set κ = κ ′ 7: Calculate  κ ′ = κ +1 8: Calculate ^^′ ^^ to 9: else 10: Set ^^′ ^^ = ^^ ^^The approximation of the transcendental function ^^(Ϛ) from equation (1.3). The evaluation of the transcendental function defined in equation (1.3) ^^(Ϛ) is at any time t k , to which a new relative angle increment ^^ exists for ^^ the calculation of the cumulative angle increment ^^ ^^ required. In the k-th time step, half the norm of the result for the cumulative Angle increment from the previous time step, i.e. by = ^^ ^^―1 2 . The application of the algorithm described above for renormalizing the angle increments guarantees that the norm ^^ ^^―1 the condition 0 ≤ ^^ ≤ ^^ In the following section, numerical approximations for ^^ (Ϛ) which allow ^^(Ϛ)efficiently and with the desired precision over the entire required range of values. The function defined in equation (1.3) has the following Taylor series expansion: evaluated ∈ ℕ The Riemannian is defined over the series (3.21) Inserting equation (3.20) into equation (3.19) then gives for the Taylor series von ^^ (Ϛ) where ^^ ^^0+1 the remainder of the approximation of ^^ by the Taylor polynomial of degree 2 ^^0in Ϛ It is important that Ϛ (2n), ie the Riemann not with the argument ϚThe Padé approximation proposed by the invention is considerably more accurate than the use of the Taylor polynomial for the approximation, with approximately the same computational effort. The suitability of the Padé approximation is based on the fact that the definition in equation (3.21) of the Riemann With the first ones of the Taylor expansion and taking into account the fact that according to equation (3.23) quotients of two with large argument in lowest order asymptotically approach 1 (one), the expression for the remainder term can be written to all orders in the argument Ϛ can be summed approximately as a geometric series. This gives the remainder R If the remainder term in equation (3.22) is replaced by its approximation defined in equation (3.24)^^ ^^0+1(Ϛ), then the following approximation ^^ ^^0 (Ϛ) für ^^(Ϛ) erhalten: The above approximation has only rational coefficients, although it apparently contains transcendental numbers such as π and Ϛ(2 ^^) contains, because it holds Ϛ(2 ^^) = ^^( ^^) ^^ 2 ^^ , where^^( ^^) ∈ ℚ is a rational number that depends on n. The first coefficients appearing in equation (3.25) are explicitly (3.26) Figure 4 shows the relative errors when using the approximation by equation (3.25) for different values ​​of ^^0 and when using the Taylor polynomial with different degree n in Ϛ 2 across the entire relevant value range 0 ≤ Ϛ ≤ ^ 2 ^ It is important to note that for^^ = ^^0 +2, the two approximations have a comparable computational effort. For all combinations of^^ = ^^0 +2, the approximation by equation (3.25) is more accurate than the approximation by the corresponding Taylor polynomial. Thus, the approximation by (3.25) already reaches^^0 = 10 The precision preset in the Mathematica software “$MachinePrecision” = 15.9546 (number of decimal places for the representation of machine-precision numbers) is used over almost the entire value range. This is only possible with the approximation by Taylor polynomials for ^^ = 24 and thus with approximately twice the computational effort. Using the asymptotic behavior of the Riemannian given in equation (3.23), the error of the approximation can be reduced by equation (3.25) and explain why this is a much more accurate and thus more efficient calculation of the function than the approximation by the corresponding Taylor polynomial. ^^(Ϛ) The approximation performed in equation (3.24), on which equation (3.25) is based, causes the following approximation errors when also taking into account the next leading term of the asymptotic expansion (3.23) of the Riemann equation This error is monotonically increasing as a function of and thus maximal at the upper interval limit, i.e. at There we have the relative error of the approximation of the transcendental function ^^(Ϛ) by the equation (3.25) (3.28) This error tends to increase for increasing ^^0 with exponential speed to zero. In contrast, the maximum error of the approximation by the Taylor polynomial of degree^^ = ^^0 +2in Ϛ 2 by the remainder term from equation (3.22) after substitution^^0→ ^^ = ^^0 +2. For this, for large n, after replacing the Riemannian by its constant asymptotics, it can be determined from the Equation (3.23) a lower one as follows Also, the maximum is assumed to be at the upper limit of the interval of , so = ^ 2 ^ at . The relative error of the approximation of ^^ (Ϛ) by the Taylor polynomial of degree ^^ is therefore n against it for the degree^^ = ^^0 +2with comparable computational effort an error of This is . If one wants to achieve a comparable error with the Taylor approximation, one finds by equating the relative errors the relation from which it follows that^^ > 2 ^^0like that of the approximation (3.25) using a Taylor series, the number of terms to be calculated and thus the computational effort must be about twice as large. The approximation of the transcendental function ^^(Ϛ)is now further improved. This exploits the fact that the approximation by equation (3.25) is a special case of the Padé approximation, which approximates a function by a rational function of two polynomials in the function argument, whose function value and first derivatives at the expansion point coincide with those of the function to be approximated. In the case of the approximation by equation (3.25), the function and the numerator polynomial and the denominator polynomial have the degrees ^^0 +1 or1 in Ϛ 2 In general, the Padé approximation of ^^ (Ϛ) with a numerator polynomial of degree ^^ and a denominator polynomial of degree ^^ in Ϛ 2 through given and agrees a m b n Function value B(0) and the first^^ = 1,..., ^^ + ^^derivatives ^^ ^^ ^^. With the number^^0 = ^^0 +2 = ^^ + ^^it turns out that for given 0 ≤ ^^0≤ 10 the Padé approximations with^^ = ^^0 ― ^^and ^^ = ^^ and various ^^ = 1,…, ^^0 different exactly the function ^^(Ϛ) approximate. For the choice ^^ = ^^0― = ^^ results in the interval 0 ≤ Ϛ ≤ ^^ the Padé approximation with the 2greatest overall accuracy. These special cases for each ^^0 can therefore be denoted by the following equation (3.34) for the Padé approximation The coefficients are as follows: The integer numerators and denominators given above are linear combinations of products of Ϛ-function values, where each term has the same even transcendentality in the sum. Each of these combinations of fixed transcendentality is then removed by dividing by a corresponding power of π. The resulting integer numerators and denominators should therefore also appear in the relationships between, such as They follow, for example, from the shuffle and stuffle relations. Figure 5 shows the relative errors | ^^ ^ when using the Approximation of ^^ by equation (3.25) for different values ​​of ^^0(diagram a)) and using the Padé approximation of^^^^ ^^ ^^é, ^^0(Ϛ) with equation (3.34) for different values ​​of ^^0 (Diagram b)) over the entire relevant value range 0 ≤ ^^ shown. For^^0 = ^^ +2have 2 0the two approximations have the same number of terms. The computational effort for calculating^^ ^^ ^^ ^^ ^^, ^^0+2(Ϛ) is even somewhat lower compared to that for^^ ^^ ^^ ^^, ^^0(Ϛ), because of the more uniform distribution of the total degree ^^0 on numerator and denominator polynomials of fewer powers of Ϛ 2 must be calculated. For all combinations of^^0 = ^^0 +2, the approximation of the function ^^(Ϛ) is even more accurate than the approximation by equation (3.25) and can be calculated with slightly less computational effort than the approximation by equation (3.25). Thus, the approximation of the function ^^(Ϛ) already at ^^0 = 9 already over almost the entire value range, the precision preset in the Mathematica software “$MachinePrecision” = 15.9546 (number of decimal places for the representation of machine-precision numbers), while the approximation (3.25) only at ^^0 = 10It should be noted that, unlike the exact expression in equation (1.3), none of the three approximations by equations (3.22), (3.25) and (3.2) has an apparent pole at Ϛ = 0. The approximations can therefore be applied over the entire range of values 0 ≤ be used. The approximation of the transcendental function ^^ (2) (Ϛ) explained. In the calculation of the cumulative angle increment according to equation (1.2) in the explicit form given in (A.30), in addition to the transcendental function ^^(Ϛ), the combination This has, as already ^^ in equation (1.3) itself apparently has a pole at = 0. However, the function is regular and at Ϛ = 0 results in Analogous to the procedure for the construction of approximations of ^^(Ϛ)the various approximations presented there for ^^ (2) (Ϛ) For example, the Taylor series of ^^ (2) (Ϛ) through If the Padé approximation of from equation (3.34) is used directly in equation (3.36), this gives the best result for these approximations. This is not surprising, because the Taylor series in equation (3.37) of ^^ (2) (Ϛ) does not have the property of the Taylor series of ^^(Ϛ)in equation (3.22), the coefficients converge exponentially to a constant for large n (or exhibit exponential behavior themselves). However, when using equation (3.34) directly in the approximation according to equation (3.36), the influence of the apparently present pole is still visible. A particular disadvantage, however, is that using this strategy requires separate treatment of the apparently present pole, i.e., the value = 0. The aforementioned disadvantage can be avoided and at the same time the best approximation result of all investigated approximations can be constructed. This is achieved if, based on the Padé approximation by equation (3.34), an expression for ^^ (2) (Ϛ) which is regularized by removing the apparently existing pole. For this purpose, the formulation of the Padé approximation (3.34) can be extended by introducing two polynomials How The use of this representation of the Padé approximation of ^^(Ϛ) in the definition of ^^ (2) (Ϛ) in equation (3.36) allows us to determine from the obtained approximation the apparently existing pole at Ϛ = 0 as follows Here designated ^^ ( ^^ 2 ^^ ) ^^, ^^0 (Ϛ) the resulting approximation, which is based on the Padé approximation of ^^(Ϛ) but no longer contains any numerically problematic apparent poles and is therefore regular. Figure 5 shows the relative errors when directly using the Padé approximation (3.25) to calculate ^^ (2) (Ϛ) and when using the regularized approximation ^^ ( ^^ 2 ^^ ) ^^, ^^0(Ϛ)from equation (3.40) for different values ​​of ^^0 across the entire relevant value range 0 ≤ ^^ shown. 2Diagram a) shows the relative errors where the Inserting the approximation of^^ ^^ ^^ ^^ ^^, ^^0 from equation (3.34) into equation (3.36). Diagram b) shows the relative error when given by the regularized approximation from equation (3.40) Especially for ^^0 > 8 and small is the regularized expression of of equation (3.40) is considerably more accurate than the approximation based on the use of the Padé approximation of ^^(Ϛ) into the definition of equation (3.36) of ^^ (2) (Ϛ) The results of the approximations can be used to calculate the cumulative angle from relative angle increments to determine the position of the object. For this purpose, the operation given in line 4 of Algorithm 1 can be refined by 4: Calculate ^^ ^ ^ ^^ , ^ ^ ^ ^ ^^ out of ^^ ^ ^^ ^, ^ ^ ^^ ^― ^ 1 und ^^ ^ ^ ^ ^ ^^ ^^ according to equation (3.4), i.e., in equation (1.2) with the explicit form according to equations (3.5a), (3.5). This operation evaluates the following equation based on equation (3.5) adjusted using the Padé approximations explained above: (3.41) This now includes the following steps, assuming that ^^ ^^― ^^ either directly before calling the following Algorithm 5 in this time step or at the end of the procedure in the last time step using, for example, Algorithm 2, the representative with the minimum norm. Algorithm 5 Calculation of ^^ ^^ aus ^^ ^^―1 und ∆ ^^ ^^ 1: procedure ^^ ^^ ( ^^ ^^―1 , ∆ ^^ ^^ ) 2: Calculate (θ^^―1 ) 2 = ^^ ^^―1 ∙ ^^ ^^―1 3: Calculate the polynomials P and a us ^^― ^^ using equation (3.38) with given N0 4: Calculate^^ ^^ ^^ ^^ ^^, ^^0(Ϛ)from P, Q, using equation (3.39) 5: Calculate equation (3.39) 6: Calculate ^^ out of ^^ , ∆ ^^ according to equation 1.2 using ^^^^ ^^ ^^ ^^, ^^0, ^^( ^^ 2 ^^ ) ^^, ^^0 in equation (3.41)It is noted that in the above abbreviated notation ^^ ^^ = ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^ and ∆ ^^ ^^ = ^^ ^^ ^ ^ ^ ^ ― ^^ 1 ^^ + ^^The accuracy is increased by terminating the expansion of equation (3.41) after the second order in ^^ 2 ^^ ^^ ^^ limited, which is the most relevant case in practice ∆ ^^ ≪ 1 However, it covers the entire spectrum. If necessary, an extension of equation (3.41) can be carried out using the methods described above without any conceptual obstacles. The calculation of trigonometric functions and the rotation matrix is ​​explained below. Depending on the measuring principle, the acceleration sensors in the IMU's sensor package either measure the acceleration directly or they measure relative velocity increments, which, in analogy to relation (1.1), are given by Here, the acceleration of the acceleration sensor with respect to an inertial system I, expressed in the reference system S of the sensor at the same time t at which the value of the acceleration a(t) is present. Therefore, if the sensor measures velocity increments, these are the result of an integration of the acceleration in the co-moving reference system S. In particular, equation (1.1) therefore also includes rotational movements that the sensor performs during the integration time. The velocity increments according to equation (1.1) cannot therefore be used directly to determine the velocity and position of the vehicle with respect to an inertial system or other reference system. Instead, velocity increments are required that have been calculated by integrating the acceleration in a coordinate system that is fixed at least in the integration interval. The corresponding integral is of the form where the rotation matrix ( ^) ^^ the acceleration measured in the sensor system S at time t into a value measured at the starting time t i-1 the integration of the sensor system corresponding but not co-rotating coordinate system is transformed, ie it is ^ ^ ^ ^ ( ( ^ ^ ^ ^ ^ ) ^― , ^^ 1) ^^( ^^) = ^ ^ ^ ^ ( ( ^ ^ ^ ^ ^ ) ^―1) ^^ ^ ^ ^ ^ ( ( ^ ^ ^ ^ ^ ) ^) , ^^ ^^( ^^′) and thus ^ ^ ^ ^ ^ , ^― ^^ 1 ∆ ^^ ^^ which is based on the sensor reference system Si-1 = S(t i-1 ) and obtained by integration. In order to add up a number n of these relative velocity increments, the individual velocity increments are again integrated into a common reference system, e.g. S I-1 at the start time ^^ the summation, to transform. The summation is (without gravity) Herein is ^ ^ ^ ^ ^ ^ ^ ^ ― ― 1 1+ ^^ ^^ which is from the start time ^^ ^^―1 from cumulative relative angle increment ^^ ^^ ^ ^ ^ ^ , ^ 0 ^ ^^ associated rotation matrix. This motivates why the respective cumulative angle increment ^^ ^^ ^ ^ ^ ^ ,^ 0 ^ ^^ associated rotation matrix ^^ ^ wird. The algorithm is not only relevant for the application described above, the processing of sensor data from an IMU and the efficient calculation of a vehicle's position and attitude. Furthermore, the efficient calculation of trigonometric functions presented here should have a wide range of applications in various fields, e.g., computer vision. It will be shown below that the algorithm used for updating the cumulative angle increment ^^ ^^ already calculated result for the function value of the function ^^(Ϛ) can be used to efficiently and especially without approximating any further transcendental functions the (trigonometric) functions ^^ ^^ 2 ^^ Ϛ 2Ϛand ^^ ^^ ^^2Ϛ. With these results, the associated rotation matrix ^ ^^^ ^^ ^^( ^^) of the active rotation between the coordinate systems F and G, which are mapped to each other by means of the rotation, can then be directly calculated. The active rotation transforms the basis vectors ^^ ^^^^ of the coordinate system F into the basis vectors ^^ ^ ^^^ = ^ ^^^ ^^ ^^(2Ϛ) of the coordinate system G obtained after applying the rotation described by the angle ^^. Knowledge of this matrix is ​​an essential building block for calculating the translational velocity and ultimately the position from the accelerations or relative velocity increments measured by the acceleration sensors. Ϛ = identified, whereby θ is and the first Argument in update of the cumulative angle increment underlying equation (3.41) and θ ^^―1 = ^^ ^^―1∙ ^^ ^^―1 ist und der Representative of the minimum norm of the cumulative angle increments from the last time step. However, the following algorithm is not specifically designed for application to the cumulative angle increments ^^ ^^―1 Rather, ^^1any angle and Ϛ = θ 2 1 be half the amount. First, expressions for (trigonometric) functions are derived that depend on half the norm of the cumulative angle increment, i.e. where the dependency by means of ^^(Ϛ) From the definition of ^^(Ϛ) in equation (1.3) follows which after insertion in the arguments 0 ≤ ≤ are valid, the following expressions are derived 2can The argument Ϛ = θ 2 1 = θ ^^―1 is only half the norm of the cumulative 2angle increment. However, the trigonometric functions of the norm itself are required. These can be obtained from the known relations between trigonometric functions with single and double angles. This leads to An advantage of the fact that the function ^^(Ϛ) only depends on half the norm of the angle of rotation, is that by doubling the argument of the trigonometric functions, the square roots are eliminated and the expressions in equation (3.45) result, which are rational functions in Ϛ 2 and ^^(Ϛ) These contain in the form of combinations ^^ ^^ 2 ^^ Ϛ 2Ϛ and^^ ^^ ^^2Ϛ, the even power series in Ϛare, then there is no longer even a dependence on the square root from the calculation of the norm itself. By using the approximations according to equation (3.25) or (3.34), the trigonometric functions can be very efficiently approximated by purely rational expressions. When using equation (3.34), Figure 6 shows the relative errors of the approximations of sin ^^ (Diagram a)) and cos θ (Diagram b)) with the Padé approximation according to equation (3.34) in the approximations according to equation (3.46) in the relevant range of values 0 ≤ ^^ ≤ ^^. Figure 7 shows the relative errors of the approximations of sin ^^ (diagram a)) and cos θ (diagram b)) for the direct approximation by Taylor polynomials of comparable degree. Figure 8 shows the relative errors of the approximations of 2 sin ^ 2 ^ cos ^ 2 ^ (Diagram a)) and 1 ― 2 sin 2 ^ 2^ (Diagram b)) for the direct approximation by Taylor polynomials of comparable degree. It can be seen that the relative error of the approximations of sin ^^ because of the zero at ^^ = ^^The use of the Padé approximation by equation (3.34) in the approximations derived in this section according to equation (3.46) provides a more accurate approximation over the entire value range than a direct approximation by a Taylor polynomial of comparable degree. The approximation of the trigonometric functions with half an angle by Taylor polynomials and the subsequent calculation of the functions with the whole angle as argument is a significant improvement over the direct approximation by the corresponding Taylor polynomials. However, the method developed here is considerably more accurate, especially for smaller angles, with comparable computational effort. A further advantage of the approximation by equation (3.46) is that the condition sin 2 ^^ + cos 2 ^^ = 1 regardless of the chosen order N0. The one with a vector-valued rotation angle ^^associated rotation matrix of the active rotation is where L is a vector L = (L1,L2, L3)T constructed from the matrix representation of the three Lie algebra generators of the rotation group SO(3). Thus, for the construction of the rotation matrix according to equation (3.47), the following combination is required in particular: This gives the following expression for the rotation matrix according to equation (3.47) This expression is also a rational function in and ^^(Ϛ) , ∙ ^^ and its evaluation does not require evaluation of transcendental and it contains only even powers of so that the square root ∙ ^^ does not need to be calculated. When applying rotation to represent the vector v in a different coordinate system, equation (3.50) can be modified as follows Algorithm 6 calculates the rotation matrix itself for given ^^, ^^2 and ^^(Ϛ), e.g., approximated via equation (3.34). It should be noted here that the calculation of the combination ^^ = is not required if the Algorithm 5 previously executed also requires the combination C, so the result can simply be stored within Algorithm 5 and then used. Algorithm 6 Calculation of ^ ^ ^ ^^ ^ ^^ out of ^^ , ^^ 2 and B ^ 2 ^ ^ 1: procedure ^ ^ ^ ^ ^^ ^ ^ ^, ^^ 2 , B ^ 2 ^ 2: Calculate ^^ 3: Calculate ^ ^ 2 ^ from2 ^ ∙ ^^ 4: Calculate Calculate the rotation matrix from ^ 2 ^ ∙ ^^, 2 2 5: ^ 2 ^ ∙ ^^ , ^^^ 2 ^ using equation (3.50) The following Algorithm 7 is a slight modification of Algorithm 6, which shows how the coordinates ^^ ^^ a vector ^^ based on equation (3.51) directly in the reference system G from the coordinates ^^ ^^ in the reference system F instead of calculating the rotation matrix. Algorithm 7 Calculation of ^^ ^^ out of ^^ , ^^ 2 and B ^ 2 ^ , ^^ ^^ 4: Calculate calculate^^ ^^ ^^ ^^ ∥ ^ 2 5: Be ^^ ^^ ^^ , ^^ , ^^ 2 ^ , ^^ ^^using equation (3.51) Both 7, e.g., for the efficient and precise calculation of trigonometric functions and rotations in real-time applications, e.g., for calculating rotations in the field of computer graphics / computer vision and CAD. Figure 9 shows a flow chart of the method, in which at least one gyroscope in the x-direction, in the y-direction and in the z-direction measures the angular increments. ∆ ^^ ^ 1 ^ ,2,3 These can be calculated from the integral of the angular velocity ω 1,2,3 (t) over a time from a previous measurement time t k-1 up to a current measurement time t kFor the sake of simplicity, only the equations for the angle increments have been presented. To include the velocity increments, each measurement result of a gyroscope is simply to be considered as being extended by the measurement result of the velocity increment in the corresponding axial direction, and then corresponding steps are to be carried out using the algorithms for the velocity and / or position update in a manner analogous to the steps shown for the angle increments. The description of Figure 9 below is also limited to the angle increment updates for the sake of simplicity. The three measured values ​​for the angle increments ∆ ^^ ^ 1 ^ ,2,3 can be vectorized to create a rotation angle increment vector ∆ ^^ ^^ This allows the individual angle increments ∆ ^^ ^^ recursively from the exponent ^^ ^^ ^^ ^^ ^^with the delay z -1 for inclusion of the previous rotation angle increment vector ∆ ^^ The calculation of the cumulative rotation angle increments ^^ occurs with the exponent ∆ ^^ ^^ , ^ ^ ^ ^ ^^ = 2 ^^ ^ ^^ 2 ^, ^ ^^^ ^^ ∆ ^^ ^―^ 1 , ∆ 2 ^^ ^^ recursively with a delay z -1 to include the previously calculated cumulative rotation angle increment ∆ ^ from the previously determined individual rotation angle increments ^^ The last cumulative angle of rotation increment calculated in this way then forms the cumulative angle of rotation ∆ ^^ ^ ^ ^ ^ ^^ ^^ as a measurement result.

Claims

German Aerospace Center eV Attorney file: Königswinterer Straße 522-524 V / DLR-0799-WO 53227 Bonn-Oberkassel 2022 / 224 Germany Date: January 8, 2024 Patent claims 1. Method for determining the position of an object by determining the angle of rotation of the object, wherein relative angle of rotation increments ∆ ^^ ^^ at consecutive measurement points ^^ ^^ with k = 1, …, n and for the respective measurement times ^^ ^^ to cumulative rotation angle increments ^^ ^^ together- characterized by calculating the cumulative rotation angle increments ^^ ^^ for the measurement times ^^ ^^iteratively from a number n of relative rotation angle increments ∆ ^^ ^^, which at successive measurement times ^ with k = 1, …, n were measured, ^ ^^the cumulative rotation angle increment ^^―1 determined for the previous one and d 1^^ the transcendental function ^^(Ϛ) = (1 ― ^^ ^^ ^^ Ϛ) with as a function of the cumulative ^ ^ ^^―1 at the previous measurement time ^^ ^^―1 , where an approximation of the transcendental function ^^ (Ϛ) =Ϛ 1 2(1 ― ^^ ^^ ^^ Ϛ) by Padé approximations.

2. Method according to claim 1, characterized by calculating the cumulative rotation angle increments ^^ ^^ for a particular measurement point ^^ ^^ as a function of the exponent of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3), where iteratively cumulated rotation angle increments ^^ ^^ for the measurement times ^^ ^^with k = 1,…, n each from the function of the approximate exponent of the Baker-Campbell-Hausdorff formula with one at the time of measurement ^^ ^^ measured relative rotation angle increment ∆ ^^ ^^ and the one for the previous point in time ^^ ^^―1 certain cumulative JG / JG - 2- Rotation angle increment ^^ ^^―1 or a specified starting angle increment ^^ 0 as input variables.

3. Method according to claim 2, characterized by determining a cumulative angle of rotation ^^ ^^ ^ ^ ^ ^ , ^ ^ ^ ^ ^^ for the measurement time ^^ ^^ by which the last time ^^ ^^ of the time range I, which includes the measuring times k = 1, …, n, determined last cumulative rotation angle increment ^^ ^^4. Method according to one of claims 1 to 3, characterized in that the cumulative rotation angle increment ^^ ^^ for the measurement time ^^ ^ with the ^ Function of the approximate exponent ^ ^^ the Baker-Hausdorff Formula iteratively from a number n of relative rotation angle increments ∆ ^^ ^^ , which are measured at consecutive times ^^ ^^ with k = 1, …, n were measured, and the cumulative angle of rotation increment determined for the previous time interval ^^ ^^―1 where the contribution to the zeroth order exponent ^ ^^ ^^ ^ the contribution to First-order exponents^^ ^^ ^ ^^ ^^ +  B θ ^^―1 2 ^^ ^^― ^^ 2 ∙ ^^ 2^^ ^^ ^^ ^^― ^^2 ― θ ^^―1 2 ^^ 2 ^^ ^^ and the contribution to the exponent in 2 2 ∙ 2 2 × 2, with the transcendental function ^^(Ϛ) = Ϛ2w obei ein [ ] and the parameters a m and b m of the value ^^0 dependent given coefficients of the Padé approximation, and that for the time ^^ ^^ cumulative - 3-D rehwinkelinkrement ^^ ^^ each from the sum of the contributions^ ^^ ^ ^ 2 ^^ to the exponent in the zeroth, first and at least second order m α = 0, 1 and 2.

5. Method according to claim 1 or 2, characterized in that the function ^ ^^ the sum of the arguments of the Approximations in the zeroth, first, second and third order mα = 0, 1, 2 and 3.

6. Method according to one of the preceding claims, characterized in that for determining the cumulative angle of rotation Δθ Iout for the time range^^ ^ ^^ ^^ ^^ the measured angle of rotation increments ∆ ^^ until ^^ for the respective times ^^ ^^ with k = 1,…, n as input variables the starting angle increment ^^ ^^ ^ ^ ^ ^ , ^ 0 ^ ^^ for k = 0 with the value zero and the cumulative rotation angle increments ^^ ^^ iteratively with k = 1 to n from the cumulative rotation angle increment determined or specified for the previous iteration step k-1 ^^ ^^― ^^ and the one at the time ^^ ^^ , which belongs to the respective iteration step k, measured rotation angle increment ∆ ^^ ^^ as the transcendent function ^^(Ϛ) = 1 (1 ― ^^ ^^ ^^ Ϛ)comprising function of the exponent of the Baker-Campbell Formula for the Lie rotation group SO(3) using a Padé approximation of the transcendental function ^^(Ϛ) 7. Method according to one of the preceding claims, characterized by calculating a rotation matrix of the rotation between a first coordinate system F of the object and a coordinate system F determined after application of a rotation angle ^^ ^^ described rotation obtained second coordinate system G with the expressions of the function ^^(Ϛ) and in particular their Padé approximations calculated trigonometric functions and calculation of the translational velocity and / or - 4- Position of the object from the cumulative rotation angles transformed into the second coordinate system ^^ ^^ 8. Method for the automatic computer-aided calculation of trigonometric functions of angles ^^ ^^ or rotations from given angles ^^ ^^, characterized by calculating the transcendental function ^^(Ϛ) = Ϛ 1 2(1 ― ^^ ^^ ^^ Ϛ)with as Function of the given angle ^^ ^^ , where an approximation of the transcendental function ^^(Ϛ) = Ϛ 1 2(1 ― ^^ ^^ ^^ Ϛ)by Padé- Approximations are carried out, and the trigonometric function or rotation is obtained as a result of the calculation of the transcendental function.

9. Method according to claim 8, characterized in that the approximation of the transcendental function is carried out by evaluating the equation ^^ 1 + ∑ ^20 ^^= ^^ ^^Ϛ2 ^^where N0is a given integer value and the away mdependent on the value N0 given by the Padé approximation.

10. Method according to claim 9, characterized by obtaining the trigonometric functions as a result of the Padé approximation of the transcendental function by 11. Measuring device for determining the position of an object by determining the rotation rates of the object, wherein the measuring device comprises a sensor unit for - 5- Measuring relative rotation angle increments ∆ ^^ ^^ for consecutive points in time ^ and has a data processing unit which is set up to ^ ^^ at consecutive points in time ^^ ^^ measured relative rotation angle increments ∆ ^^ ^^ to a cumulative rotation angle increment ^^ ^^ for the respective measurement times ^^ ^^ to cumulative rotation angle increments ^^ ^^to summarize, characterized in that the data processing unit for calculating the cumulative rotation angle increments ^^ ^^ for the measurement times ^ iteratively from a number n of ^ ^^ relative rotation angle increments ∆ ^^ ^^ , which are measured at consecutive times ^^ ^^ with k = 1, …, n were measured, the cumulative angle of rotation increment determined for the previous time interval ^^ ^^―1 and the transcendental function^^ ^^ ^^ Ϛ)with as function of the cumulative rotation angle increment at the respective previous measurement time ^^ ^^―1 and for approximating the transcendental function ^^(Ϛ)1= ― ^^ ^^ ^^ Ϛ) by Padé approximations.

12. Measuring device for determining the position of an object and / or for the automatic computer-aided calculation of trigonometric functions of angles^^ ^^ or rotations from given angles ^^ ^^ with a data processing unit, characterized in that the data processing unit is configured to carry out the steps of the method according to one of claims 1 to 10.

13. A computer program comprising instructions which, when the computer program is executed by a data processing unit, cause the data processing unit to carry out the steps of the method according to one of claims 1 to 10.

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