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, wherein rotational angle increments Δθk for successive measurement points in time meas,k , in which k = 1,..., n, are measured and combined in order to form a cumulated rotational angle ΔθI out for a cumulation period I which contains the points in time tmeas,k. The invention is characterized by the process of calculating the cumulated rotational angle ΔθI out for a cumulation period I as a function of exponents of an approximation of at least the second order of the Baker-Campbell-Hausdorff formula for the Lie rotational group SO(3), wherein iteratively cumulated rotational angle increments ΔθI,k out for the points in time tmeas,k, in which k = 1,..., n, are determined in each case from the function of the exponents using a respective measured rotational angle increment Δθk for the respective time interval tmeas,k and the cumulated rotational angle increment ΔθI,k-1 out determined for the previous point in time tmeas,k-1 or a specified starting rotational angle increment ΔθI,0 out as input variables, and the cumulated rotational angle ΔθI out for the period I is determined using the last cumulated rotational angle increment ΔθI,n out determined at the last measurement point in time tn .

Description

[0001] Method and measuring device for determining the position of an object

[0002] The project leading to this application has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 101004205.

[0003] The invention relates to a method for determining the position of an object by determining the rotation rates of the object, wherein rotation angle increments A0 k for consecutive times t meas>k measured and to a cumulative angle of rotation AO^ut for a time t meas>k comprehensive cumulative time period I. Reference is made to Figure 1.

[0004] The invention further relates to a 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 measuring rotation angle increments A0 kfor consecutive times t meas>k and a data processing unit which is arranged to carry out at successive times t meas>k measured rotation angle increments A0 k to a cumulative rotation angle A0' ou t for a time range I of the interval + £ comprising the times t-meas.fc with the integers k = 1 , ... , n k=1 At k ], where At k the length of each

[0005] Time interval t mpa . k — t mp:K k i is.

[0006] To understand the notation, it is pointed out that the time domain I (the time interval by the sequence of n measurement times t meas>k , k = where t n = t, is, in n time intervals [t m eas,fc-i>t m eas,fc], where the individual lengths At k = t meas>k — t meas,fc-i can be different. The / -th time range can have a preceding, ( / -1 )-th time range connect.

[0007] Gyroscopes are used to measure the rotation rate (angular velocity) around a predefined axis. Using at least three gyroscopes oriented so that the direction vectors along the axes of their measurement are linearly independent, the angular velocity vector w of a rotation around an arbitrarily oriented axis in space can be determined.

[0008] If the gyroscopes are permanently attached to an object, such as a vehicle (e.g., a rocket), the (relative) change in position of the vehicle (spatial orientation) can be calculated from the angular rate by integrating the signal. If the initial orientation is known, the absolute orientation of the vehicle in space is then known. Knowledge of the orientation is the input for attitude control and for determining the position when orientation-dependent forces act on the vehicle, e.g., caused by vehicle-mounted propulsion or braking systems.

[0009] Typically, the gyroscopes are bundled with acceleration sensors for measuring linear acceleration in a sensor package (an inertial measurement unit (IMU)), which is available as a module and serves as the central component for determining the vehicle's orientation and position. In addition to the sensors, the IMU also contains signal processing stages with the goal 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 for further processing at the interface.

[0010] In the case of gyroscopes, it is often not the rotation rates w that are measured directly, but their integrals, ie the integrated over a sampling time step (ie cumulative) rotation rate, which, for example, in the journal of In the case of a time-constant axis of rotation, this integral corresponds to the (relative) angle of rotation (angle increment) A0 k by which the IMU moves during this sampling time step If a cumulative angle of rotation is to be output to the user interface, which is based on a longer Time range (i.e. a lower frequency / out < f n ), the n individual angles of rotation L6 i ~' [ +k , k = \ n, which are measured within the time interval t / ], are added together.

[0011] For example, if the sampling frequency of the gyroscopes is f = 2 kHz, so that the measured angle increments are set to a constant sampling rate of s and the frequency at the user interface / out = 100 Hz -4, then n of the rotation angle increments measured at time intervals Af = 5 * 10 s to calculate the angle increment related to a journal s at the output, i.e. the relative angle of rotation by which the IMU rotates during a time step Af ou t has turned.

[0012] However, these considerations are only correct if the rotation occurs around a temporally constant axis of rotation. This assumption is generally 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 frame undergoes 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:

[0013] 1 . The integral Lcp measured by the sensors kthe rotation rate w does not directly result in the angle increment A0 k , which directly indicates the axis of rotation and the angle of rotation during the sampling interval. Instead, A0 k from the integrals of the current and past rotation rates.

[0014] 2. The direct summation of temporally successive angle increments A0 associated with the time intervals [C-2+k, ti-i+k], [ti-i+k, ti+k] k , A0 k+1 does not yield the angular increment corresponding to the rotation in the time interval [f / -2+kJ / +k]. Instead of simply summing the angular increments, a more complex mathematical operation must be performed.

[0015] 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(f) as a function of time, which depends on the rate of rotation w(f) and is as follows: with0 - 0 = 0 - 0) = 00, 0 = ||0||2. (2.1 )

[0016] In the above equation, the multicomponent rotation angle θ(f) is shown in bold (for clarification purposes) to distinguish it from its norm ||0||2.

[0017] This differential equation can be simplified for small angles of rotation θ by expanding it into a Taylor series around angle θ = 0, retaining only the leading order(s). It can then be expressed, for example, in a time interval [f M , t / ], i.e. for the angle increment A0p MtTo do this, w(f) in this interval is approximated by a polynomial of (n - 1 )-th degree by

[0018] Its n coefficients can be calculated from n measured integrated angular rates Lcp 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 k = 1,..., n equations solved for the coefficients. This procedure already requires n measured angle increments Lcp k of time intervals which together form the time range of the cumulative angle of rotation A9 k and provides the cumulative angle of rotation A0p Mt of the interval fj.

[0019] 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 (https: / / doi.org / 10.2514 / 3.21635) describes a higher-order algorithm. However, for higher orders n, the computational effort increases significantly, and the approximation of the measured data by high-degree polynomials n — 1 leads to strong oscillations at the edges of the interpolation interval, known as Runge's phenomenon. RA McKern: “A study of transformation algorithms for use in a digital computer”, MA thesis, Massachusetts Institute of Technology, Dept, of Aeronautics and Astronautics, 1968 (http: / / hdl.handle.net / 1721.1Z14164) discloses an algorithm for combining the relative rotations from successive time intervals.This algorithm calculates the unit quaternion of the relative position change during the entire time interval from the angular increments of two consecutive time intervals via the unit quaternions. The previously described algorithm for calculating the cumulative angle of rotation A0p is used. Mt for the interval t / ] is used, but only for two orders n = 2. Then, with the obtained angle of rotation A0p Mt the unit quaternion of the relative position change in the time interval [ / M , t / ] is calculated as:

[0020] The rotation 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 a value related to the vector components of the angle increment A0p Mt formed imaginary part. In addition, the trigonometric functions are approximated by their Taylor polynomials up to the 3rd order.

[0021] The result is then used to calculate, starting from a starting time t0 and the quaternion = 1 the quaternion of the relative change in position is calculated recursively according to:

[0022] The quaternions must be multiplied using the quaternion product.

[0023] In analogy to formula (2.3), the angle increments A0 k-1 , A0 k belonging quaternions and then in analogy to the formula (2.4) to obtain the quaternion Then the angle increment of the combined

[0024] Rotations by inversion of function (2.3). The corresponding relation for the angle increment A0(„ zt formed quaternion as follows:

[0025] This solution for the special case of n=2 temporally consecutive angle increments A0 k-1 , A0 k can be set to any number n of angle increments for a time range and an associated angle of rotation A0p Mt be generalized.

[0026] The iterative calculation of the angle of rotation can then be described by the procedure:

[0027] procedure

[0028] Set oQ = 1

[0029] For k = l,...,n

[0030] Calculate from A0 k with equation (2.3)

[0031] Calculate from equation (2.5)

[0032] Calculate ^ using equation (2.6)

[0033] The calculation of the rotation angle increment A0p” tis carried out 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 every n-th recursive update of the quaternion, equation (2.6) is evaluated to derive the iterative rotation angle increment A0^ from the quaternion. t This also involves increased computational effort, which occurs every nth recursion step. When approximating the trigonometric functions in equation (2.3), the quaternion is no longer a unit quaternion. However, the relation in equation (2.6) only applies to unit quaternions. Applying it to non-normalized quaternions leads to an additional (numerical) error.

[0034] The calculation of equation (2.6) for sufficiently large angular increments A0o Mt, such as those that arise when a large number n of increments A0 k When approximated by its Taylor polynomial, it requires consideration of relatively high orders to achieve the desired numerical accuracy. This increases the computational effort.

[0035] The known methods for determining the position from a rotation angle A0p Mt a time range, e.g. the interval [f / _ 1; tj\ by summarizing a number of n angle increments Acp k at times t meas>k , e.g. which together form the time domain, are very computationally intensive and therefore require a relatively large amount of computing time, computing capacity and require relatively powerful and therefore complex electronics with hardware computing power.

[0036] Based on this, it is the object of the present invention to create an improved method.

[0037] The object is achieved by the method having the features of claim 1 and the measuring device having the features of claim 7. Advantageous embodiments are described in the subclaims.

[0038] It is proposed that to determine the position of an object, rotation angle increments A9 k at consecutive measurement times t meas>k with k = 1 , ... , n measured (or from measured 4cp k calculated) and to a cumulative angle of rotation A9 I 0U t for one of the measurement times t meas k comprehensive cumulation time range I by calculating a cumulative angle of rotation A9 I 0U t is calculated for a cumulative time range I as a function of exponents of an approximation of at least second order of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3). Iteratively cumulative rotation angle increments A9 I k 0Ut for the times t meas>k with k = 1 ,... , n each from the function of the exponents with one rotation angle increment A9 k of the measurement time t k and the cumulative angle increment A9 determined for the previous time interval I k 0U t or a given output angle increment A9 I ' 0 out as input variables. The cumulative angle of rotation A9 I 0U t for the time range I is a function of the last cumulative rotation angle increment A9 I n 0U t, which for the last time interval t n of the interval of the time range I. It can be assigned to this cumulative rotation angle increment A9 I n 0Ut are equivalent to.

[0039] The rotation angle increments and angles contain angular 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, respectively. Although the rotation angle increments and angles are not explicitly referred to as vectors in the present application and claims for the sake of simplicity, the embodiment is encompassed as vectors or representations by skew-symmetric matrices thereof.

[0040] The method has the advantage that the iterative rotation angle increments are calculated directly without recourse to quaternions. In each iteration step, only one equation, explained in more detail later as equation (3.5), is required. This has the following consequences:

[0041] - It is only a single transcendental function, possibly approximated by its Taylor polynomial.

[0042] - The iteration is simplified because (apart from the initialization required in each case) no additional computational step is required for every nth journal. This is a significant advantage, especially for implementation on highly limited hardware (such as a Field Programmable Gate Array (FPGA)).

[0043] - The analysis of the propagation of measurement uncertainties and measurement noise is facilitated.

[0044] - A polynomial approximation of the only transcendental function, e.g. by its Taylor polynomial should be more accurate than the approximation of equation (2.6) by its Taylor polynomial of the same order. For large cumulative rotation angles Ad o ' ut ur|d thus, greater accuracy is achieved, especially for a larger number n. - By separately selecting the two orders for the approximation in a Taylor series in two variables, the method can be easily adapted to the given requirements of the user (e.g., the maximum rotation rate, sensor frequency, frequency divisor, number of angular increments to be summed).

[0045] In the case of a time-varying axis of rotation, the following changes occur:

[0046] 1 . The integral Lcp measured by the sensors k the rotation rate w does not directly result in the angle increment A0 k , which directly specifies the rotation axis and the angle of rotation during the sampling interval. Instead, l\Q k from the integrals of the current and past rotation rates.

[0047] 2. The direct summation of consecutive time intervals rt niPa s fr-2Tn1Pa s fr-il ftmeas fr-i Trn Pa s frl associated angle increments A0 k ' 7 , A0 k does not result in the rotation in the time interval [t me as,fc-2Tmeas,fc]. Instead of simply summing the angular increments, a more complex mathematical operation must be performed.

[0048] The method relies on the complex mathematical operations required for time-varying rotation axes, rather than a simple summation of angular increments. The reason for this is that rotations around non-parallel rotation axes 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 arise in the form of rotation matrices or unit quaternions.

[0049] The rotation matrices are special (their determinant is 1) orthogonal (their transpose is their inverse) matrices in three dimensions. The name of the group, SO(3), is derived from the terms Special Orthogonal Rotation Matrices in 3 Dimensions.

[0050] 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 are multiplied by means of a non-commutative, norm-preserving product.

[0051] Two unit quaternions each represent one rotation.

[0052] Depending on the selected representation, the operations described above for calculating the angle increments and their summation then follow.

[0053] The cumulative angle of rotation A0^ ut for a time range I, which includes a number of n consecutive times t meaSikwith k = 1 , ... , n, can be used with the function of the exponent y^ > OMt , yJ recursively (ie iteratively) from a number n of rotation angle increments A9 k , which were measured at consecutive measurement times tmeas.k with k = 1 , ... , n, and the respective for the previous

[0054] Time interval determined cumulative angle increment be calculated.

[0055] The zeroth order exponent can be included.

[0056] First order exponent can be Y (i:) ( > Ae °y \ + > Ae 'p 1 x y - + B the transcendental function B(Q +

[0057] 7 > i QÖ? r'io

[0058] + d? + ife + + 63SS1287S +°« ) mit dem Bernoulli numbers ß 2k and the

[0059] Operator 0(^ 12 ) taken into account.

[0060] For the measurement time tmeas k a cumulative rotation angle increment each from the sum of the exponents zeroth, first and at least second order m a = 0, 1 and 2.

[0061] The calculation is based on a single transcendental function, which reduces the time and computational effort. The exponent function can be the sum of the arguments of the

[0062] Approximations in the zeroth, first, second and third order m a = 0, 1, 2, and 3. By including a higher third-order exponent, the accuracy can be further increased. This also includes only a single transcendental function.

[0063] An approximation of the transcendental function for the l-th time range. This is based on the Bernoulli numbers B2k as coefficients and the argument 0 introduced for simplified representation. X, which serves as a placeholder for the previous cumulative rotation angle Aö^r 1 The approximation can be from k equal to 1 up to a maximum iteration level k = k chosen as a function of the accuracy. max (here k is the k-th order term in the Taylor expansion, not to be confused with the k-th journal).

[0064] The output angle increment A9 I 0 O ut for k = 0 can be specified with the value zero when initializing. To determine the cumulative angle of rotation Aö^ut for the time range [tpiTi] with the measured angle of rotation increments A0 1 up to A0 n for the respective times t meas>k with k = 1 , ... , n as input variables, the initialization is carried out before the first iteration step. Subsequently, the cumulative rotation angle increments A0 I k out as intermediate results iteratively with k = 1 to n from the cumulative rotation angle increment AO^-^ut determined or specified for the previous iteration step k-1 and the time t meas fc , which belongs to the respective iteration step k, measured rotation angle increment A0 I k calculated as a function of exponents of the approximation of at least second order of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3).

[0065] The determination of the rotation rate of the object can be carried out as a function of the cumulative rotation angle AO^ut determined for the time range [ti-iTi] and the time At / ;OUt = Sfc=i At fc The rotation rate w can be calculated from the quotient of the cumulative rotation angle Aö^ut and the time At / out be calculated.

[0066] The invention is explained in more detail below with reference to an exemplary embodiment of the invention, which is shown in the accompanying drawing.

[0067] Figure 1 - Time course of fc = measured or from the measurements at the measuring times t meas k determined inputs Aß k , the k-th iteration step to determine the k-th cumulative angle increment Output 40^ at time t /

[0068] Figure 2 - Sketch of a measuring device for determining the position of an object;

[0069] Figure 3 - Flowchart of the method for determining a cumulative rotation angle increment from measured rotation angle increments;

[0070] Figure 4 - Further flowchart of the process.

[0071] Figure 1 shows a time course of k = measured or from the measurements at the measuring times t meas>k determined measured values ​​for the angle of rotation increments, ie the inputs Aß k , the k-th iteration step for determining the k-th cumulative angle increment Aß a ^ tand the cumulative angle of rotation, ie the measurement result as output A6 I ^ Jt at time t ; . This means that the time sequence with the measurement times t meas k , the summation for different time intervals k = 1 , ...., n and the summary to the low frequency measurement result.

[0072] 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 A0 k consecutive times t meas>k is configured to record rotating (circular) movements of the inertial measurement unit IMU and the connected object 2 in the three mutually orthogonal spatial axes (X-, Y-, and Z-axis) of the Cartesian coordinate system. These measured rotation angle increments A0 kare fed as input values ​​to a data processing unit, which calculates a cumulative angle of rotation Aö^ut for a cumulative time range I, which includes successive times t meas k with i = 1 , ... , n.

[0073] The data processing unit 3 can be, for example, a microprocessor or

[0074] The data processing unit 3 can be a 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.

[0075] The method allows a direct calculation of the cumulative angle of rotation A9' ou t iteratively from the angle of rotation increment A0 stored in the last step and the currently measured kThe basis is the Baker-Campbell-Hausdorff (BCH) formula, which originates from the theory of Lie groups and gives a commutation law for certain linear operators. It provides the following equation for the multiplication of two elements obtained by the exponential mapping e a , eß of a Lie group the exponent y(a, ß) of the resulting element e^- ß), which thus satisfies the relation e / (a. ß) = e a eß (3.1 ) is satisfied. The result y(a, ß) is given by the Baker-Campbell-Hausdorff formula as a formal series expansion in the total order m of f and g and therefore has the following schematic form: with the coefficient c(m ai mp), where the actual form is due to the general non-commutability of the group elements a ß ß a is much more complicated.

[0076] 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 series with the time intervals [ti- 3+k , tj.2+k], [ti-2+k, tj.i +k ] associated angle increments A9 k ~ 1 , L6 k is suitable to determine the actual angle of rotation increment corresponding to the rotation in the time interval [f / -3+k, ti-i+k], ie the angle of rotation A9' out for 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). In addition, 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 S0(3) initially only provides the required relation for the summation of two angle elements, for the previously described case. where y is still given as a series expansion in the total order m as in equation (3.2) and A0o Mt here the interval [f z-3+k , f / . 7+k ] is related.

[0077] The evaluation of the series expansion according to equation (3.2) of the function y 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 y contains the (iterative) cumulative rotation angle increment the previous k-1 steps. This is therefore not necessarily small for large k < n. Therefore, an approximation of the Taylor series in the first argument by its Taylor polynomial can lead to an unacceptably large numerical error.

[0078] The BCH formula can now be approximated for the Lie group SO(3) as follows to determine the exponents the orders m ß = 0, 1 , 2 to obtain:

[0079]

[0080] The only function contained in the system of equations that can be calculated by approximation is the transcendental function B(Q). This contains the Bernoulli numbers ß 2fc and the operator O(^ 12 ).

[0081] The derivation of the approximation of at least second order m ß The Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3) will be described in detail later.

[0082] It is sufficient if only in the second argument 02= ~ e i ne Approximation by the Taylor polynomial at least in orders m ß = 0, 1 , 2 is performed.

[0083] The expressions are in the first argument exactly.

[0084] 01 A0 / ,fc — 1

[0085] Only a single transcendental function B(y) = B( — ^-) occurs, which, depending on the application, can be approximated by its Taylor polynomial. A comparison of the coefficients of the Taylor polynomial containing the Bernoulli numbers B2k with those of the Taylor polynomial in equation (2.6) shows that the former decrease by approximately a factor of 1 / 10 as the order increases beyond the third order, while the latter decrease by only a factor of between 5 / 10 and 1.

[0086] Therefore, for numerical calculations where the transcendental function B( and equation (2.6) must be approximated by their Taylor polynomials, it is advantageous to calculate the expression according to equation (3.5) instead of using the conventional algorithm based on equation (2.6). AClI'k — 1

[0087] The maximum required orders of the Taylor polynomials for B( °2 ut) and the expression (2.6) depend on the desired numerical accuracy for given maximum expected values ​​for the angular increments and their number n, which can be included in a single angular increment A0o Mt should be summarized.

[0088] When using equations (3.5), the required order can be chosen significantly lower than when using equations (2.3), (2.5), and (2.6) while maintaining the same numerical accuracy. This reduces complexity and computational effort, i.e., computation time and hardware requirements.

[0089] The iterative calculation of the cumulative rotation angle A0p Mt of the interval [tj.^ tj] using the approximation of at least second order of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3) can be carried out on the basis of equations (3.4) and (3.5) by the procedure as follows:

[0090] procedure

[0091] Set

[0092] For iterative:

[0093] Calculate A0{'^ from the (iterative) cumulative rotation angle increment A0^7 1 and the measured rotation angle increment A0 k according to equation (3.4) with the explicit form (3.5)

[0094] Set the resulting cumulative rotation angle A0o Mt equal to the last (iterative) cumulative rotation angle increment

[0095] Figure 2 shows an associated flow chart 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.

[0096] Calculation of 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.

[0097] The strategy here is to expand the formula in orders of only its second argument 02= — and the dependence on the first argument

[0098] = — y— in all orders, so that the formula is exact with respect to its first argument. The result thus represents an expansion of the formula in orders of its second argument 02, while simultaneously summing all dependencies on its first argument.

[0099] 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 performed. The corresponding expression is given up to first order in the second argument, but without specializing to the Lie group SO(3) or explicitly resuming the series containing the Bernoulli numbers.

[0100] 1 . Calculation of the zeroth and first order, = 0.1

[0101] The first-order simple steps of specialization and resumption are carried out using the generating function of the Bernoulli numbers, which is given by

[0102] _ - _ y°° B n "no z — 1 ^n=0 n[ z (A.1 ) is given, the following explicit representation in resumptive form results for the expression of the BCH formula given as a series in M. Müger: “Note on the theorem of Baker-Campbell-Hausdorff-Dynkin”: The Taylor series of the function (2fc) contains only even orders of $ 2 , so that the calculation of equation (A.2) does not require the calculation of the of the vector This is an advantage in numerical calculations.

[0103] 2. Basics of the formalism and connection between the differential equation of angle increments and the BCH formula

[0104] The principles and steps required to derive the second order are presented below.

[0105] The starting point is the exponential map, which is also defined for arguments a 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 a is an element of a representation of a Lie algebra. This element a 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.

[0106] Since different elements a, ß of a Lie algebra do not usually commute, so aß ß a, 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 element a on another element ß.

[0107] From the derivation according to equation (A.5) follows the relation

[0108] 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 relation (A.7) is then given by

[0109] 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 a. Therefore, the expression, which is then specialized for the Lie group SO(3), depends only on even powers of the norm of a 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:

[0110] 1

[0111] It follows immediately that B1= — ~ B = 0 for n > 1. So:

[0112] This result is used to put the inverse according to equation (A.8) into the following form:

[0113] (A.11 )

[0114] 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 6, w from their Lie algebra su(2), the multiple commutators reduce as follows:

[0115] By repeated application of this relation, the expression for equation (A.11 ) is then 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 0 as a function of the angular velocity w, which is given in equation (2.1 ).

[0116] 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 yields the Lie-algebra-valued element / as a function of two Lie-algebra-valued elements a, ß, as a solution of the following relation eY (a,ß) = e a e ß (A.15)

[0117] To demonstrate this relationship, the exponent y is expanded in orders of ß, as is shown, for example, in M. Müger: “Note on the theorem of Baker-Campbell-Hausdorff-Dynkin.” For this purpose, a parameter t ER is introduced and the expansion is written as where clearly y(°) = y(cr, 0) = a.

[0118] The result for yW in the expansion of equation (A.16) can be obtained directly from equation (A.11) if a is replaced by y(a, t ß). This then leads to the following differential equation for y. (A.17)

[0119] By inserting t = 0 into the above equation, a result for yCD is obtained due to equation (A.16), which is as follows:

[0120] A comparison of this result with equation (A.11 ) shows that the differential equation for the Lie algebra element a for a given Lie algebra element ß = e~ a d t e a is identical to the equation for the linear contribution in ß in the BCH series according to equation (A.16). In addition, by inserting t = At ​​the approximation of y by y = a + becomes a solution of the differential equation (A.11 ) to the linear order in Af ß.

[0121] 3 Calculation of the second order, m p = 2

[0122] In the following, the term of the BCH series (A.16). This contains the second-order terms in ß.

[0123] First, the derivative of relation (A.11) must be calculated. This leads to:

[0124] An equation for is analogous to the procedure in the case of To do this, a can be replaced by y(a, t ß) and ß = 0,t = 0 can be inserted. This gives

[0125] To obtain an explicit result for the special case of the Lie group S0(3), the series in (A.20) is summed to a closed expression. The procedure for deriving (A.13) can be analogous, using equation (A.12) to reduce the multiple commutators. Furthermore, 0 0 a = y,ß = y is directly set to 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:

[0126] (A.21 )

[0127] Using this identity, the inner sum in (A.20) can be divided as follows: where is the heavyside step function. Inserting this result into the series in (A.20) yields the Taylor series:

[0128] 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:

[0129] Analytical continuation of the expressions by identification z = i 0 X then leads for the term of the BCH series, which is quadratic in 02, to the result which depends only on a single transcendental function, B(£), which is already included in the expression for 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.

[0130] 4. Explicit expression for the BCH formula up to second order, = 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 -^-, is summarized and simplified below. The term y<°) is simply given by the first argument y of the function. The term is given by (A.13) if 0 0 in it X and co 02 or alternatively, the expression (A.2) is used directly. The term is shown in (A.25). This gives the formula for the BCH

[0131] Now the following identities are used: to first calculate and simplify the multiple commutators appearing in equation (A.26)

[0132] Thus, the formula (A.26) can be gradually reduced to the following form:

[0133] 5. Calculation of higher orders, > 3 The described procedure can also be directly applied to the calculation of higher orders. In particular, the order m ß = 3 in 02, so the term in the BCH series (A.16) can be used to obtain even greater accuracy. Figure 4 shows a flowchart 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 A <Pi,2,3 vornimmt. Diese können aus dem Integral der Winkelgeschwindigkeit Wi, 2.3(f) über eine Zeit von einem vorhergehenden Messzeitpunkt t meas ,ki up to a current measurement time t meas ,k. The three measured values ​​for the angle increments A <p k 2,3 can be vectorized to a rotation angle increment vector Acp k to obtain.

[0134] This allows the individual rotation angle increments A0 k recursively from the exponent A0 k (A <p k-1,A <p k ) with the delay z' 1 to include the previous rotation angle increment vector A <p k-1 be calculated.

[0135] The calculation of the cumulative rotation angle increments is done recursively with the exponent with a delay z' 1 to

[0136] Inclusion of the previously calculated cumulative rotation angle increment Aö^r 1 from the previously determined individual rotation angle increments A0 k .

[0137] The last cumulative rotation angle increment calculated in this way then forms the cumulative rotation angle A0o Mt as a measurement result.

Claims

Patent claims 1 . Method for determining the position of an object by determining the angle of rotation of the object, wherein angle of rotation increments A9 k at consecutive measurement times t meas>k with k = 1 , n measured and to a cumulative angle of rotation A9 I 0U t for one the times t meas>k comprehensive cumulation time period I, characterized by Calculating the cumulative rotation angle A9 I 0U t for a cumulative time domain I as a function of exponents of an approximation of at least second order of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3), where iteratively cumulated rotation angle increments A9 I k 0U t for the times t meas>k with k = 1 , ... , n each from the function of the exponents with one at time t k measured rotation angle increment A9 kand the one for the previous time t mea s,fc-i determined cumulative rotation angle increment A9 I k ' 1 0ut or a given output angle increment A9 I '° 0U t are determined as input variables and the cumulative angle of rotation A9 I 0U t for the time range I by the last time t n determined last cumulative rotation angle increment A9 I n 0Ut is intended.

2. Method according to claim 1, characterized in that the cumulative angle of rotation A0o Mt for a time range I, which includes a number of n consecutive times t meas>k with k = 1 , ... , n contains, with the function of the exponent y^ ^"' , iteratively from a number n of angles of rotation increments A9 k , which are measured at consecutive times t meas>kwith k = 1 , ... , n were measured, and the cumulative rotation angle increment Aö^r determined for the previous time interval 1 where the zeroth order exponent y(°) is the first order exponent Order a and the second-order exponent transcendental function + + sÜttSs +°( , ; 1 4 den Bernoulli numbers ß2“ and the Remainder of order o(^ 12 ) and that for the time t meas>k cumulative rotation angle increment ^-J from the sum the exponents in the zeroth, first and at least second order m a = 0, 1 and 2. Method according to claim 1 or 2, characterized in that the function of the exponent ^— ) is the sum of the arguments of the Approximations in the zeroth, first, second and third order m a = 0, 1, 2 and 3. Method according to one of the preceding claims, characterized by Approximation of the transcendental function for the k-th time domain with the Bernoulli numbers B2I as coefficients and 0 X equal to the previous cumulative angle of rotation Aö^ 1 up to a maximum iteration level l = l chosen as a function of accuracy max Method according to one of the preceding claims, characterized in that for determining the cumulative angle of rotation Aö^ut for the time range At fc ] with the measured rotation angle increments A9 1 up to A0 n for the respective times t meas>k with k = 1 , n as input variables the output rotation angle increment A9 I 0 out for k = 0 is given the value zero and the cumulative rotation angle increments A0 I k O U t iteratively with k = 1 to n from the cumulative rotation angle increment AO^-^ut determined or specified for the previous iteration step k-1 and the rotation angle increment A0 measured at time t-meas.fc belonging to the respective iteration step k k as a function of exponents of the approximation of at least the second order of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3). Method according to one of the preceding claims, characterized by determining the rotation rate of the object as a function of the time domain + Zk=i At fc ] determined cumulative angle of rotation Aö^ut and the time Sfc=i Atfc. 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 measuring angle of rotation increments A0k for consecutive times tmeas.fc and a data processing unit which is arranged to process the data at consecutive times t meas>k measured rotation angle increments A0 k to a cumulative rotation angle A0' ou t for one the times t meas>k comprising a time range I, characterized in that the data processing unit for calculating the cumulative angle of rotation AO^ut for a cumulative time range I as a function of exponents of an approximation of at least second order of the Baker-Campbell-Hausdorff formula for the Lie rotation group SO(3), wherein iteratively cumulative angle of rotation increments A0 I k out for the times t meas . / c with k= 1 , n each from the function of the exponents with a rotation angle increment A0 for the respective time t meas>kand the cumulative angle of rotation increment AO^-^ut determined for the previous time tmeas.ki or a given initial angle of rotation increment A0 I '° ou t are determined as input variables and the cumulative angle of rotation AO^ut for the time t meas n by the last time t meas “determined last cumulative rotation angle increment A0 I n out Measuring device according to claim 7, characterized in that the A data processing unit configured to carry out the steps of the method according to any one of claims 2 to 6. A computer program comprising instructions which, when executed by a data processing unit, cause the computer program to carry out the steps of the method according to any one of claims 1 to 6.

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