orthogonal system. Assessing spatial orientation, or How not to be afraid of Mahoney and Madjwick filters

If any two mutually perpendicular vectors of unit length are chosen on the plane (Fig. 7), then an arbitrary vector in the same plane can be expanded in the directions of these two vectors, i.e., represent it in the form

where are numbers equal to the projections of the vector onto the directions of the axes. Since the projection onto the axis is equal to the product of the length and the cosine of the angle with the axis, then, recalling the definition of the scalar product, we can write

Similarly, if in three-dimensional space choose any three mutually perpendicular vectors of unit length, then an arbitrary vector in this space can be represented as

In a Hilbert space, one can also consider systems of pairwise orthogonal vectors of this space, i.e., functions

Such systems of functions are called orthogonal systems of functions and play an important role in the analysis. They are encountered in various problems of mathematical physics, integral equations, approximate calculations, the theory of functions of a real variable, etc. to create general concept Hilbert space.

Let us give exact definitions. Function system

is called orthogonal if any two functions of this system are orthogonal to each other, that is, if

In three-dimensional space, we required that the lengths of the vectors of the system be equal to one. Recalling the definition of the length of a vector, we see that in the case of a Hilbert space, this requirement is written as follows:

A system of functions that satisfies the requirements (13) and (14) is called orthogonal and normalized.

Let us give examples of such systems of functions.

1. On the interval, consider the sequence of functions

Every two functions from this sequence are orthogonal to each other. This is verified by simple calculation of the corresponding integrals. The square of the length of a vector in Hilbert space is the integral of the square of the function. Thus, the squares of the lengths of the sequence vectors

the essence of integrals

i.e. our vector sequence is orthogonal but not normalized. The length of the first vector of the sequence is and all

the rest have length. By dividing each vector by its length, we get an orthogonal and normalized system trigonometric functions

This system is historically one of the first and most important examples of orthogonal systems. It arose in the works of Euler, D. Bernoulli, D'Alembert in connection with the problem of string vibrations. Its study played an essential role in the development of the whole analysis.

The appearance of an orthogonal system of trigonometric functions in connection with the problem of string vibrations is not accidental. Each problem of small oscillations of a medium leads to a certain system of orthogonal functions describing the so-called natural vibrations given system (see § 4). For example, in connection with the problem of vibrations of a sphere, so-called spherical functions appear; in connection with the problem of vibrations of a circular membrane or cylinder, so-called cylindrical functions appear, etc.

2. We can give an example of an orthogonal system of functions, each function of which is a polynomial. Such an example is the sequence of Legendre polynomials

i.e., there is (up to a constant factor) the order derivative of . We write the first few polynomials of this sequence:

Obviously, there is a degree polynomial in general. We leave it to the reader to verify for himself that these polynomials are an orthogonal sequence on the interval

The general theory of orthogonal polynomials (the so-called orthogonal polynomials with weight) was developed by the remarkable Russian mathematician P. L. Chebyshev in the second half of the 19th century.

Expansion in orthogonal systems of functions. Just as in three-dimensional space, each vector can be represented

as a linear combination of three pairwise orthogonal vectors of unit length

in the function space, the problem arises of expanding an arbitrary function into a series in terms of an orthogonal and normalized system of functions, i.e., of representing a function in the form

In this case, the convergence of series (15) to a function is understood in the sense of the distance between elements in the Hilbert space. This means that the root-mean-square deviation of the partial sum of the series from the function tends to zero at , i.e.

This convergence is usually called "average convergence".

Expansions in various systems of orthogonal functions are often encountered in analysis and are an important method for solving problems of mathematical physics. So, for example, if an orthogonal system is a system of trigonometric functions on the interval

then such an expansion is the classical expansion of a function into a trigonometric series

Let us assume that expansion (15) is possible for any function from the Hilbert space, and find the coefficients of such an expansion. To do this, we multiply both sides of the equality scalarly by the same function of our system. We get equality

from which, due to the fact that at is determined by the value of the coefficient

We see that, as in ordinary three-dimensional space (see the beginning of this paragraph), the coefficients are equal to the projections of the vector onto the directions of the vectors .

Recalling the definition of the scalar product, we obtain that the coefficients of the expansion of a function in terms of the orthogonal and normalized system of functions

are determined by the formulas

As an example, consider the orthogonal normalized trigonometric system of functions given above:

We have obtained a formula for calculating the coefficients of the expansion of a function into a trigonometric series, on the assumption, of course, that this expansion is possible.

We have established the form of the expansion coefficients (18) of a function in terms of an orthogonal system of functions under the assumption that such an expansion takes place. However, an infinite orthogonal system of functions may turn out to be insufficient for expanding any function from the Hilbert space in terms of it. For such a decomposition to be possible, the system of orthogonal functions must satisfy an additional condition, the so-called completeness condition.

An orthogonal system of functions is called complete if it is impossible to add to it a single function that is not identically zero and orthogonal to all functions of the system.

It is easy to give an example of an incomplete orthogonal system. To do this, we take some orthogonal system, for example, the same

system of trigonometric functions, and exclude one of the functions of this system, for example, the Remaining infinite system of functions

will still be orthogonal, of course, will not be complete, since the function : excluded by us is orthogonal to all functions of the system.

If the system of functions is not complete, then not every function from the Hilbert space can be expanded in terms of it. Indeed, if we try to expand a zero function orthogonal to all functions of the system in such a system, then, by virtue of formulas (18), all coefficients will be equal to zero, while the function is not equal to zero.

The following theorem holds: if a complete orthogonal and normalized system of functions in a Hilbert space is given, then any function can be expanded into a series in terms of the functions of this system

In this case, the expansion coefficients are equal to the projections of the vectors onto the elements of the orthogonal normalized system

The Pythagorean theorem in § 2 in the Hilbert space allows us to find an interesting relation between the coefficients and the function. Denote by the difference between and the sum of the first terms of its series, i.e.

The design of the PLA is a LSI, made in the form of a system of orthogonal tires, in the nodes of which there are basic semiconductor elements - transistors or diodes. Setting up the PLA for the required logical transformation (programming the PLA) consists in the appropriate organization of connections between the basic logical elements. Programming of the PLM is performed either during its manufacture, or by the user using a programmer device. Due to such properties of PLM as simplicity structural organization And high speed performing logical transformations, as well as a relatively low cost, determined by manufacturability and mass production, PLM are widely used as an element base in the design of computer systems and industrial automation systems.

There are no good "mechanical systems" to follow even at this level. In my opinion, there has never been a successful "mechanical" system at all, which would be described by a linear model. It does not exist now, and in all likelihood will never exist, even with the use of artificial intelligence, analog processors, genetic algorithms, orthogonal regressions, and neural networks.

Let us clarify the meaning of the norm - G. In an (n+1)-dimensional space, an oblique coordinate system is introduced, one axis of which is the line Xe, and the second axis is an n-dimensional hyperplane G, orthogonal to g. Any vector x can be represented as

Parabolic regression and the system of orthogonal

For definiteness, we confine ourselves to the case m = 2 (the transition to the general case m > 2 is carried out in an obvious way without any difficulties) and represent the regression function in the system of basis functions if > 0 (x), (x), ip2 to) that are orthogonal (on set of observed

The mutual orthogonality of the polynomials (7-(JK) (on the observation system xlt k..., xn) means that

With such planning, called orthogonal, the X X matrix will become diagonal, i.e. the system of normal equations splits into k+l independent equations

The system of points with the fulfillment of the condition of orthogonality (plan of the 1st order)

Obviously, the strain tensor in a rigid motion vanishes. It can be shown that the converse is also true: if at all points of the medium the strain tensor is equal to zero, then the law of motion in some rectangular coordinate system of the observer has the form (3.31) with an orthogonal matrix a a. Thus, a rigid motion can be defined as the motion of a continuous medium, in which the distance between any two points of the medium does not change during the motion.

Two vectors are said to be orthogonal if their dot product is zero. A system of vectors is called orthogonal if the vectors of this system are pairwise orthogonal.

About Example. System of vectors = (, 0, . . ., 0), e% = (0, 1, . . ., 0), . .., e = (0, 0, . . . , 1) is orthogonal.

The Fredholm operator with kernel k (to - TI, 4 - 12) has in the Hilbert space (according to Hilbert's theorem) a complete orthogonal system of eigenvectors . This means that φ(τ) form a complete basis in Lz(to, T). Therefore, I am.

The orthogonal system of n-zero vectors is linearly independent.

The above method of constructing an orthogonal system of vectors t/i, Yb,. ..> ym+t for a given linearly independent

For a biotechnical well drilling system, where the amount of physical work remains significant, studies of the biomechanical and motor-power areas of activity are of particular interest. The composition and structure of labor movements, the number, dynamic and static loads and the forces developed were studied by us on the Uralmash-ZD drilling rigs using stereoscopic filming (two synchronously operating cameras using a special technique at a frequency of 24 frames per 1 s) and the ganiographic method using a three-channel medical oscilloscope. The rigid fixation of the optical axes, parallel to each other and perpendicular to the line of the base (of the filming object), made it possible to quantitatively study (based on perspective-orthogonal conjugate projections over film frames, as shown in Fig. 48) working postures, trajectories of movement of the centers of gravity of workers when performing individual operations, techniques, actions and determine efforts, energy costs, etc.

A promising approach to identifying independent alternatives is to identify independent synthetic factor indicators. The original system of factor indicators Xi is transformed into a system of new synthetic independent factor indicators FJ, which are orthogonal components of the system of indicators Xr. Transformation is performed using the methods of component analysis 1. Mathematical

One of constituent parts ADAD is a module for the three-dimensional design of complex piping systems. The graphical database of the module contains three-dimensional elements of pipelines (connections, taps, flanges, pipes). The element selected from the library is automatically brought into line with the characteristics of the pipeline system of the designed model. The module processes drawings and creates two- and three-dimensional images, including the construction of isometric models and orthogonal projections of objects. There is a choice of parts for pipelines, types of coatings and types of insulation according to a given specification.

Relations (2.49) show how the solution of equations (2.47) should be constructed. First, the polar expansion of the tensor of is constructed and the tensors p "b ncc are determined. Since the tensors a "b and p I are equal, the matrix s has the form (2.44), (2.45) in the main coordinate system of the tensor p. We fix a matrix Su. Then aad = lp labsd. From aad, au is calculated from the equation aad = biljd x ad. The "orthogonal part" of the distortion is found from (2.49) id = nib sd.

The remaining branches do not satisfy condition (2.5 1). Let's prove this statement. The matrix x \u003d A 5, f \u003d X Mfs is orthogonal. Denote by Xj the matrix corresponding to the first matrix s" (2.44), and by Xj the matrix corresponding to any other choice of the matrix sa (2.44).

Equal to zero:

An orthogonal system, if it is complete, can be used as a basis for space. In this case, the decomposition of any element can be calculated by the formulas: , where .

The case when the norm of all elements is called an orthonormal system.

Orthogonalization

Any complete linearly independent system in a finite-dimensional space is a basis. From a simple basis, therefore, one can pass to an orthonormal basis.

Orthogonal decomposition

When decomposing the vectors of a vector space in an orthonormal basis, the calculation of the scalar product is simplified: , where and .

What are we talking about

The appearance on Habré of a post about the Madgwick filter was in its own way a symbolic event. Apparently, the general interest in drones has revived interest in the problem of estimating body orientation from inertial measurements. At the same time, traditional methods based on the Kalman filter have ceased to satisfy the public - either because of the high requirements for computing resources that are unacceptable for drones, or because of the complex and non-intuitive parameter settings.

The post was accompanied by a very compact and efficient implementation of the filter in C. However, judging by the comments, the physical meaning of this code, as well as the entire article, remained vague for someone. Well, let's be honest: the Madgwick filter is the most intricate of the group of filters based on generally very simple and elegant principles. These principles will be discussed in my post. There will be no code here. My post is not a story about any specific implementation of the orientation estimation algorithm, but rather an invitation to invent your own variations on given topic, which can be very many.

Orientation view

Let's remember the basics. In order to estimate the orientation of a body in space, it is first necessary to choose some parameters that together uniquely determine this orientation, i.e. essentially an orientation connected system coordinates relative to a conditionally fixed system - for example, the NED (North, East, Down) geographic system. Then you need to make kinematic equations, i.e. express the rate of change of these parameters in terms of the angular velocity from the gyroscopes. Finally, vector measurements from accelerometers, magnetometers, etc. must be included in the calculation. Here are the most common ways to represent orientation:

Euler angles- roll (roll, ), pitch (pitch, ), heading (heading, ). This is the clearest and most concise set of orientation parameters: the number of parameters is exactly equal to the number of rotational degrees of freedom. For these angles, we can write kinematic Euler equations. They are very popular in theoretical mechanics, but they are of little use in navigation problems. First, knowing the angles does not allow you to directly convert the components of any vector from a bound to a geographic coordinate system or vice versa. Secondly, at a pitch of ±90 degrees, the kinematic equations degenerate, roll and heading become uncertain.

Rotation matrix is a 3x3 matrix by which to multiply any vector in the associated coordinate system to get the same vector in the geographic system: . The matrix is always orthogonal, i.e. . The kinematic equation for it has the form .
Here is a matrix of angular velocity components measured by gyroscopes in a coupled coordinate system:

The rotation matrix is slightly less visual than the Euler angles, but unlike them, it allows you to directly transform vectors and does not lose its meaning for any angular position. From a computational point of view, its main drawback is redundancy: for the sake of three degrees of freedom, nine parameters are introduced at once, and all of them must be updated according to the kinematic equation. The problem can be slightly simplified by using the orthogonality of the matrix.

rotation quaternion- a radical, but very unintuitive remedy against redundancy and degeneration. This is a four-component object - not a number, not a vector, not a matrix. The quaternion can be viewed from two angles. First, as a formal sum of a scalar and a vector , where are the unit vectors of the axes (which, of course, sounds absurd). Secondly, as a generalization complex numbers, which now uses not one, but three different imaginary units (which sounds no less absurd). How is quaternion related to rotation? Through Euler's theorem: a body can always be transferred from one given orientation to another by one finite rotation through some angle around some axis with a direction vector . These angle and axis can be combined into a quaternion: . Like a matrix, a quaternion can be used to directly transform any vector from one coordinate system to another: . As you can see, the quaternion representation of the orientation also suffers from redundancy, but much less than the matrix one: there is only one extra parameter. A detailed review of quaternions has already been on Habré. It was about geometry and 3D graphics. We are also interested in kinematics, since the rate of change of the quaternion must be related to the measured angular velocity. The corresponding kinematic equation has the form , where the vector is also considered a quaternion with a zero scalar part.

Filter schemes

The most naive approach to calculating orientation is to arm ourselves with a kinematic equation and update any set of parameters we like according to it. For example, if we have chosen a rotation matrix, we can write a loop with something like C += C * Omega * dt . The result will be disappointing. Gyroscopes, especially MEMS, have large and unstable zero offsets - as a result, even at complete rest, the calculated orientation will have an infinitely accumulating error (drift). All the tricks invented by Mahoney, Madgwick and many others, including myself, were aimed at compensating for this drift by involving measurements from accelerometers, magnetometers, GNSS receivers, lags, etc. Thus was born a whole family of orientation filters based on a simple basic principle.

Basic principle. To compensate for the orientation drift, it is necessary to add to the angular velocity measured by the gyroscopes an additional control angular velocity constructed on the basis of vector measurements of other sensors. The control angular velocity vector should tend to match the directions of the measured vectors with their known true directions.

Here lies a completely different approach than in the construction of the corrective term of the Kalman filter. The main difference is that the control angular velocity - not a term, but a factor with the estimated value (matrix or quaternion). This results in important benefits:

An estimation filter can be built for the orientation itself, and not for small deviations of the orientation from that given by gyroscopes. In this case, the estimated values will automatically satisfy all the requirements imposed by the problem: the matrix will be orthogonal, the quaternion will be normalized.
The physical meaning of the control angular velocity is much clearer than the corrective term in the Kalman filter. All manipulations are done with vectors and matrices in the usual three-dimensional physical space, and not in the abstract multidimensional space states. This greatly simplifies the refinement and tuning of the filter, and as a bonus, it allows you to get rid of large matrices and heavy matrix libraries.

Now let's see how this idea is implemented in specific filter options.

Mahoney filter. All the mind-bending mathematics of Mahoney's original article was written to justify simple equations (32). Let's rewrite them in our notation. If we ignore the estimation of zero offsets of gyroscopes, then there will remain two key equations - the actual kinematic equation for the rotation matrix (with the control angular velocity in the form of a matrix ) and the law of formation of this very velocity in the form of a vector . Assume for simplicity that there are no accelerations or magnetic pickups, and thanks to this, measurements of the acceleration of free fall from accelerometers and tension are available to us magnetic field Earth from magnetometers. Both vectors are measured by sensors in a linked coordinate system, and in the geographic system their position is known for sure: it is directed upwards, to magnetic north. Then the Mahoney filter equations will look like this:

Let's look closely at the second equation. The first term on the right side is the vector product. The first factor in it is the measured gravitational acceleration, the second is the true one. Since the factors must be in the same coordinate system, the second factor is converted to the associated system by multiplying by . The angular velocity, constructed as a vector product, is perpendicular to the plane of multiplier vectors. It allows you to rotate the calculated position of the associated coordinate system until the multiplier vectors coincide in direction - then the vector product will be zeroed and the rotation will stop. The coefficient sets the rigidity of such feedback. The second term performs a similar operation with the magnetic vector. In fact, the Mahoney filter embodies the well-known thesis: the knowledge of two non-collinear vectors in two different coordinate systems makes it possible to uniquely restore the mutual orientation of these systems. If there are more than two vectors, then this will give a useful measurement redundancy. If there is only one vector, then one rotational degree of freedom (movement around this vector) cannot be fixed. For example, if only the vector is given, then roll and pitch drift can be corrected, but not yaw.

Of course, in the Mahoney filter, it is not necessary to use a rotation matrix. There are also non-canonical quaternion variants.

Virtual gyro platform. In the Mahoney filter, we applied the steering angular velocity to a coupled coordinate system. But you can apply it to the calculated position of the geographic coordinate system. The kinematic equation then takes the form

It turns out that such an approach opens the way to very fruitful physical analogies. Suffice it to recall what gyroscopic technology began with - headings and inertial navigation systems based on a gyro-stabilized platform in a gimbal suspension.

www.theairlinepilots.com

The task of the platform there was to materialize the geographic coordinate system. The orientation of the carrier was measured relative to this platform by angle sensors on the suspension frames. If the gyroscopes drifted, then the platform drifted after them, and errors accumulated in the readings of the angle sensors. To eliminate these errors, we introduced Feedback from accelerometers installed on the platform. For example, the deviation of the platform from the horizon around the northern axis was perceived by the eastern axis accelerometer. This signal made it possible to set the control angular velocity, which returns the platform to the horizon.

We can use the same visual concepts in our problem. The written kinematic equation should then be read as follows: the rate of change of orientation is the difference between two rotational movements - the absolute motion of the carrier (the first term) and the absolute motion of the virtual gyroplatform (the second term). The analogy can be extended to the law of formation of the control angular velocity. The vector embodies the readings of the accelerometers supposedly standing on the gyro platform. Then, from physical considerations, we can write:

Exactly the same result could have been reached formally by doing vector multiplication in the spirit of the Mahoney filter, but now not in a connected, but in a geographical coordinate system. Is it just necessary?

The first hint of a useful analogy between platform and strapdown inertial navigation appears in an ancient Boeing patent. Then this idea was actively developed by Salychev, and recently by me too. Obvious advantages of this approach:

The control angular velocity can be formed on the basis of understandable physical principles.
Naturally, the horizontal and course channels are separated, which are very different in their properties and methods of correction. In the Mahoney filter they are mixed.
It is convenient to compensate for the influence of accelerations by using GNSS data, which are issued in geographic, rather than related, axes.
It is easy to generalize the algorithm to the case of high-precision inertial navigation, where the shape and rotation of the Earth must be taken into account. I have no idea how to do this in the Mahoney scheme.

Madgwick filter. Madgwick chose the hard way. If Mahoney, apparently, intuitively came to his decision, and then justified it mathematically, then Madgwick from the very beginning showed himself to be a formalist. He undertook to solve the optimization problem. He reasoned thus. Set the orientation to the rotation quaternion. In the ideal case, the calculated direction of some measured vector (let us have it) coincides with the true one. Then it will be. In reality, this is not always achievable (especially if there are more than two vectors), but you can try to minimize the deviation from exact equality. To do this, we introduce a minimization criterion

Minimization requires gradient descent - moving in small steps in the direction opposite to the gradient, i.e. opposite to the fastest increase of the function . By the way, Madgwick makes a mistake: in all his works he does not enter at all and insistently writes instead of , although in fact he calculates exactly .

Gradient descent eventually leads to the following condition: to compensate for the orientation drift, you need to add to the rate of change of the quaternion from the kinematic equation a new negative term proportional to:

Here Madgwick deviates a little from our "basic principle": he adds a correction term not to the angular velocity, but to the rate of change of the quaternion, and this is not exactly the same thing. As a result, it may turn out that the updated quaternion will no longer be a unit and, accordingly, will lose the ability to represent orientation. Therefore, for the Madgwick filter, artificial normalization of the quaternion is a vital operation, while for other filters it is desirable, not optional.

Influence of accelerations

Until now, it has been assumed that there are no true accelerations and that accelerometers measure only free-fall acceleration. This made it possible to obtain a vertical standard and, with its help, to compensate for the drift of roll and pitch. However, in the general case, accelerometers, regardless of their operating principle, measure apparent acceleration- vector difference of true acceleration and free fall acceleration . The direction of the apparent acceleration does not coincide with the vertical, and errors due to accelerations appear in the roll and pitch estimates.

This can be easily illustrated using the analogy of a virtual gyro platform. Its correction system is designed in such a way that the platform stops in the angular position in which the signals of the accelerometers allegedly installed on it are nulled, i.e. when the measured vector becomes perpendicular to the sensitivity axes of the accelerometers. If there are no accelerations, this position coincides with the horizon. When horizontal accelerations occur, the gyroplatform deviates. We can say that the gyro platform is similar to a heavily damped pendulum or plumb line.

In the comments to the post about the Majwick filter, the question flashed of whether it is possible to hope that this filter is less susceptible to accelerations than, for example, the Mahoney filter. Alas, all the filters described here operate on the same physical principles and therefore suffer from the same problems. You can't fool physics with math. What then to do?

The simplest and crudest method was invented back in the middle of the last century for aircraft vertical gyro: reduce or completely reset the control angular velocity in the presence of accelerations or course angular velocity (which indicates entering a turn). The same method can be transferred to current strapdown systems. In this case, the accelerations should be judged by the values , and not , which in the turn are themselves zero. However, in magnitude it is not always possible to distinguish true accelerations from the projections of the free fall acceleration due to the very tilt of the gyro platform that needs to be eliminated. Therefore, the method works unreliably - but it does not require any additional sensors.

A more accurate method is based on the use of external speed measurements from a GNSS receiver. If the speed is known, then it can be numerically differentiated and the true acceleration can be obtained. Then the difference will be exactly equal regardless of the movement of the carrier. It can be used as a vertical standard. For example, one can set the control angular velocities of the gyroplatform in the form

Sensor zero offsets

A sad feature of consumer-grade gyroscopes and accelerometers is the large instabilities of zero offsets in time and temperature. To eliminate them, only one factory or laboratory calibration is not enough - it is necessary to re-evaluate during operation.

Gyroscopes. Let's deal with zero offsets of gyroscopes. The calculated position of the associated coordinate system moves away from its true position with an angular velocity determined by two counteracting factors - zero offsets of the gyroscopes and the control angular velocity: . If the correction system (for example, in the Mahoney filter) managed to stop the drift, then it will be in the steady state. In other words, the control angular velocity contains information about an unknown active perturbation . Therefore, you can apply compensatory assessment: we do not know the magnitude of the disturbance directly, but we know what corrective action is needed to balance it. This is the basis for estimating the zero shifts of gyroscopes. For example, Mahoney's score is updated according to the law

However, his result is strange: estimates reach 0.04 rad / s. Such instability of zero offsets does not happen even with the worst gyroscopes. I suspect the problem is that Mahoney does not use GNSS or other external sensors - and suffers from the effects of accelerations to the full extent. Only on the vertical axis, where accelerations do not harm, the estimate looks more or less sensible:

Mahony et al., 2008

accelerometers. Estimating the zero offsets of accelerometers is much more difficult. Information about them has to be extracted from the same control angular velocity. However, in rectilinear motion the effect of zero offsets of accelerometers is indistinguishable from the inclination of the carrier or the misalignment of the installation of the sensor unit on it. No additives to accelerometers are created. The additive appears only during a turn, which makes it possible to separate and independently evaluate the errors of gyroscopes and accelerometers. An example of how this can be done is in my article. Here are pictures from there:

Instead of a conclusion: what about the Kalman filter?

I have no doubt that the filters described here will almost always have an advantage over the traditional Kalman filter in terms of speed, code compactness and ease of customization - this is what they were created for. As for the accuracy of estimation, everything is not so clear-cut here. I have seen unsuccessfully designed Kalman filters, which, in terms of accuracy, noticeably lost to a filter with a virtual gyro platform. Madgwick also argued the benefits of his filter with respect to some Kalman estimates. However, for the same orientation estimation problem, at least a dozen different Kalman filter circuits can be built, and each will have an infinite number of tuning options. I have no reason to think that the Mahoney or Madgwick filter will be more accurate the best possible Kalman filters. And of course, the Kalman approach will always have the advantage of universality: it does not impose any strict restrictions on the specific dynamic properties of the system being evaluated.

1) O. such that (x a , x ab)=0 at . If, in addition, the norm of each vector is equal to one, then the system (x a ) is called. orthonormal. Complete O. s. (x a ) called. orthogonal (orthonormal) basis. M. I. Voitsekhovsky.

2) O. s. coordinates - a coordinate system, and which coordinate lines (or surfaces) intersect at right angles. O. s. coordinates exist in any Euclidean space, but, generally speaking, do not exist in an arbitrary space. In a two-dimensional smooth affine space O. s. can always be introduced at least in a sufficiently small neighborhood of each point. O.'s introduction is sometimes possible with. coordinates in the case. In O. with. metric tensor g ij diagonals; diagonal components gii accepted as Lame coefficients. Lame coefficient O. s. in space are expressed by the formulas

Where x, y And z- Cartesian rectangular coordinates. The element of length is expressed through the Lame coefficients:

surface area element:

volume element:

vector differential operations:

The most commonly used O. s. coordinates: on the plane - Cartesian, polar, elliptical, parabolic; in space - spherical, cylindrical, paraboloidal, bicylindrical, bipolar. D. D. Sokolov.

3) O. s. functions - a finite or counting system (j i(x)) of functions belonging to the space

L2(X, S, m) and satisfying the conditions

If l i=1 for all i, then the system is called orthonormal. It is assumed that the measure m(x) defined on the s-algebra S of subsets of the set X is countably additive, complete, and has a countable base. This is O.'s definition of s. includes all considered in the modern analysis of O. of page; they are obtained for various concrete realizations of the measure space ( X, S, m).

Most Interest represent complete orthonormal systems (j n(x)), which have the property that for any function there is a unique series converging to f(x) in the space metric L2(X, S, m) , while the coefficients with p are determined by the Fourier formulas

Such systems exist due to the separability of space L2(X, S, m). A universal method for constructing complete orthonormal systems is provided by the Schmidt orthogonalization method. To do this, it suffices to apply it to some complete L2(S, X, m) a system of linearly independent functions.

In theory orthogonal rows in are generally considered by O. of page. space L2[a, b](that special case when X=[a, b], S- a system of Lebesgue measurable sets, and m is the Lebesgue measure). Many theorems on the convergence or summability of the series , , with respect to general o. s. (j n(x)) spaces L2[a, b] are also true for series in orthonormal systems of the space L2(X, S, m). At the same time, in this particular case, interesting concrete orthogonal systems have been constructed that have good properties of one kind or another. Such, for example, are the systems of Haar, Rademacher, Walsh-Paley, Franklin.

1) Haar system

where m=2 n+k, , m=2, 3, ... . Haar series represent a typical example martingales and true for them general theorems from martingale theory. Moreover, the system is a basis in Lp, , and the Fourier series in the Haar system of any integrable function converges almost everywhere.

2) Rademacher system

represents an important example of O. of page. independent functions and has applications both in probability theory and in the theory of orthogonal and general functional series.

3) Walsh-Paley system is defined through the Rademacher functions:

where are the numbers q k are determined from the binary expansion of the number n:

4) The Franklin system is obtained by orthogonalization by the Schmidt method of the sequence of functions

It is an example of an orthogonal basis of the space C continuous functions.

In the theory of multiple orthogonal series, systems of functions of the form

where is the orthonormal system in L2[a, b]. Such systems are orthonormal on the m-dimensional cube J m =[a, b]x . . .x[ a, b] and are complete if the system (j n(x))

Lit.:[l] Kaczmarz S., Steinhaus G., Theory of orthogonal series, trans. from German, M., 1958; The results of science. Mathematical analysis, 1970, M., 1971, p. 109-46; there, p. 147-202; Dub J., Probabilistic processes, trans. from English, M., 1956; Loev M., Theory of Probability, trans. from English, M., 1962; Sigmund A., Trigonometric series, trans. from English, vol. 1-2, M., 1965. A. A. Talalyan.

- a finite or countable system of functions belonging to the Hilbert space L2 and satisfying the conditions of the function gnas. weighing O. s. f.,* means complex conjugation...
Physical Encyclopedia
is the group of all linear transformations of an n-dimensional vector space V over a field k that preserve a fixed non-degenerate quadratic form Q on V)=Q for any)...
Mathematical Encyclopedia
is a matrix over a commutative ring R with identity 1, for which the transposed matrix coincides with the inverse. The determinant of O. m. is equal to +1 ...
Mathematical Encyclopedia
- a network for which the tangents at some point to the lines of different families are orthogonal. Examples of O. s.: asymptotic network on a minimal surface, line curvature network. A. V. Ivanov...
Mathematical Encyclopedia
is an orthogonal array, OA is a matrix of size kx N, the elements of which are the numbers 1, 2, .....
Mathematical Encyclopedia
- see Isogonal trajectory...
Mathematical Encyclopedia
- English: System "generator - motor" Regulated electric drive, the converting device of which is an electric machine converting unit Source: Terms and definitions in the electric power industry ...
Construction dictionary
- see Projection...
Big encyclopedic polytechnic dictionary
- the procedure for determining the results of elections, in which mandates are distributed among the parties that nominated their candidates for a representative body in accordance with the number of votes they received ...
Glossary of legal terms
- a kind of proportional electoral system. By end results resembles a proportional system with panache and preferential voting ...
Glossary of legal terms
- organs of the human body involved in the process of reproduction of offspring ...
medical terms
- a series of four types of genes that encode polymorphic proteins found on the surface of most nucleated cells ...
medical terms
- order n Matrix...
- a special case of parallel projection, when the axis or projection plane is perpendicular to the direction of projection ...
Great Soviet Encyclopedia
- system of functions (), n = 1, 2,..., orthogonal with weight ρ on the interval, i.e. such that Examples. Trigonometric system 1, cos nx, sin nx; n = 1, 2,..., - O. s. f. with weight 1 on the segment...
Great Soviet Encyclopedia
- ORTHOGONAL system of FUNCTIONS - system of functions??n?, n=1, 2,.....
Big encyclopedic dictionary

"ORTHOGONAL SYSTEM" in books

Section XXIV The old system of trench warfare and the modern system of marches

From the book Strategy and Tactics in the Art of War author Jomini Genrikh Veniaminovich

Section XXIV The old system of trench warfare and modern system marches System of positions refers to the old way of conducting methodical warfare with armies sleeping in tents, having supplies at hand, engaged in observing each other; one army

19. The concept of "tax system of the Russian Federation". Correlation between the concepts of "tax system" and "tax system"

From the book Tax Law the author Mikidze S G

19. The concept of "tax system of the Russian Federation". Correlation between the concepts of "tax system" and "tax system" The system of taxes is a set of federal taxes established in the Russian Federation, regional and local taxes. Its structure is enshrined in Art. 13–15 of the Tax Code of the Russian Federation. In accordance with

From the book How it really was. Reconstruction true history author Nosovsky Gleb Vladimirovich

23. Geocentric system of Ptolemy and heliocentric system of Tycho Brahe (and Copernicus) Tycho Brahe's system of the world is shown in fig. 90. In the center of the world is the Earth, around which the Sun revolves. However, all the other planets are already revolving around the Sun. Exactly

23. Geocentric system of Ptolemy and heliocentric system of Tycho Brahe (and Copernicus)

From the author's book

23. Geocentric system of Ptolemy and heliocentric system of Tycho Brahe (and Copernicus) Tycho Brahe's system of the world is shown in fig. 90. In the center of the world is the Earth, around which the Sun revolves. However, all other planets are already revolving around the Sun. Exactly

orthogonal matrix

TSB

orthogonal projection

From the book Big Soviet Encyclopedia(OR) author TSB

Orthogonal system of functions

From the book Great Soviet Encyclopedia (OR) of the author TSB

49. The judicial system and the system of law enforcement agencies according to the "Fundamentals of the Legislation of the USSR and the Union Republics" 1958

From the book History of State and Law of Russia author Pashkevich Dmitry

49. The judicial system and the system of law enforcement agencies according to the "Fundamentals of the Legislation of the USSR and the Union Republics" 1958. Fundamentals of the legislation on the judiciary established the principles of construction judicial system USSR, principles of peer review

The system of objective (positive) law and the system of legislation: correlation of concepts

From the book Jurisprudence author Mardaliev R. T.

The system of objective (positive) law and the system of legislation: correlation of concepts The system of objective (positive) law is internal structure law, dividing it into branches, sub-sectors and institutions in accordance with the subject and method of legal

29. Mandatory system of government and the system of local self-government in the period of a class-representative monarchy

author

29. order system management and the system of local self-government during the period of the estate-representative monarchy

86. The judicial system and the system of law enforcement agencies according to the "Fundamentals of Legislation of the USSR and the Union Republics" 1958

From the book Cheat Sheet on the History of the State and Law of Russia author Dudkina Ludmila Vladimirovna

86. The judicial system and the system of law enforcement agencies according to the "Fundamentals of Legislation of the USSR and Union Republics" 1958 Since 1948, the procedural legislation of the USSR and the republics has undergone significant changes: 1) people's courts have become

31. French government system, suffrage and electoral system

From the book Constitutional Law foreign countries the author Imasheva E G

31. System government agencies France, suffrage and electoral system France has a mixed (or semi-presidential) republican government. The system of government in France is built on the principle of separation of powers.Modern France

44. System of public authorities of France, suffrage and electoral system

From the book Constitutional Law of Foreign Countries. Crib author Belousov Mikhail Sergeevich

44. French government system, suffrage and electoral system France is a mixed (semi-presidential) republic, the system of government is based on the principle of separation of powers.

Chapter IV. Dual Head Compliance System. Insect system. Minisystem

From the book Su Jok for everyone by Woo Pak Jae

Chapter IV. Dual Head Compliance System. Insect system. Mini-system Double Head Correspondence System There are two head correspondence systems on the fingers and toes: the "human type" system and the "animal type" system. The "human type" system. Border

First emotional center - skeletal system, joints, blood circulation, immune system, skin

From the book Everything will be fine! by Hay Louise

The first emotional center - the skeletal system, joints, blood circulation, immune system, skin Healthy state of the organs associated with the first emotional center, depends on the feeling of security in this world. If you are deprived of the support of family and friends that you