KR102478451B1 - Computing apparatus and state estimation method for system using the same - Google Patents

Abstract

A method for estimating the state of a system is disclosed. The state estimation method includes identifying a manifold in which a state of a system is defined; determining a stable embedding function suitable for the manifold; adding a noise model to the system; generating a stochastic discrete-time system by discretizing the system to which the noise model is added; and estimating a state of the system by applying an unscented Kalman filter defined in Euclidean space to the stochastic discrete system.

Description

Computing device and state estimation method of a system using the same {COMPUTING APPARATUS AND STATE ESTIMATION METHOD FOR SYSTEM USING THE SAME}

An embodiment of the present specification relates to a technique for estimating the state of a system defined on a manifold in Euclidean space based on an unscented Kalman filter using stable embedding.

State estimation using filtering can be applied to various fields such as robot control and drone control. Here, the state may be a variable representing the state of the system, such as position, speed, rotation, and temperature of the system. Among the Kalman filters, an unscented Kalman filter can relatively accurately estimate a state estimation value of a nonlinear system, and is thus used in the related art field.

Recently, when the target system is defined on a manifold in Euclidean space, 1) a method of projecting the final value obtained by state estimation using an unscented Kalman filter onto a manifold 2) Lee algebra when the manifold is a Lee group 3) In the case of a Riemannian manifold, define a sigma point on the tangent plane and transform the coordinates between the Riemannian manifold and the tangent plane A method using , etc. could be used for state estimation. However, such a technique has a problem in that the error of state estimation is relatively large or a relatively long calculation time is required.

Accordingly, there is a need for a technique capable of improving the calculation time of the state estimation of the system defined in the manifold and the error of the estimated value.

Embodiments of the present specification are proposed to solve the above problems, and relate to a technique for estimating a state of a system defined on a manifold in Euclidean space based on an unscented Kalman filter using stable embedding. The technical problem to be achieved by the present embodiment is not limited to the technical problems described above, and other technical problems may be inferred from the following embodiments.

In order to achieve the above object, a method for estimating a state of a system according to an embodiment of the present specification includes identifying a manifold in which a state of a system is defined; determining a stable embedding function suitable for the manifold; adding a noise model to the system; generating a stochastic discrete-time system by discretizing the system to which the noise model is added; and estimating a state of the system by applying an unscented Kalman filter defined in Euclidean space to the stochastic discrete system.

According to an embodiment, the step of identifying the manifold may include extending the system into Euclidean space.

According to an embodiment, the stable embedding function may have a minimum value in the manifold through a Euclidean dot product with the system extended to the Euclidean space.

According to an embodiment, the determining of the stable embedding function may include adding the determined stable embedding function to the expanded system.

According to an embodiment, the adding of the noise model may include generating a stochastic continuous-time system by adding the noise model to the expanded system to which the stable embedding function is added. can

According to an embodiment, generating the stochastic discrete system may include generating the stochastic discrete system by discretizing the stochastic continuous system based on Euler discretization.

According to an embodiment, the unscented Kalman filter defined in the Euclidean space generates a sigma point using a preset method and repeatedly performs time update and measurement update. It can be characterized as doing.

According to an embodiment, the step of estimating the state of the system may include iteratively estimating the state of the system by repeatedly applying the unscented Kalman filter.

In order to achieve the above object, a non-transitory computer-readable storage medium according to an embodiment of the present specification includes a medium configured to store computer-readable instructions, and the computer-readable instructions are executed by a processor. In this case, the processor: identifying a manifold in which the state of the system is defined; determining a stable embedding function suitable for the manifold; adding a noise model to the system; generating a stochastic discrete-time system by discretizing the system to which the noise model is added; and estimating the state of the system by applying an unscented Kalman filter defined in Euclidean space to the stochastic discrete system.

In order to achieve the above object, an arithmetic device for estimating a state of a system according to an embodiment of the present specification includes a memory for storing at least one instruction; and identifying a manifold in which the state of the system is defined, determining a stable embedding function suitable for the manifold, adding a noise model to the system, and discretizing the system to which the noise model is added to obtain a stochastic It may include a processor that generates a stochastic discrete-time system and estimates a state of the system by applying an unscented Kalman filter defined in Euclidean space to the stochastic discrete-time system. there is.

Details of other embodiments are included in the detailed description and drawings.

According to an embodiment of the present specification, one or more of the following effects are provided.

First, a manifold defined by the state of the extended system in Euclidean space is identified, an appropriate stable embedding function is determined, and a stochastic continuous system generated by adding a noise model is discretized, and an unscented Kalman filter defined in Euclidean space is applied. In this case, the state of the system can be estimated relatively accurately compared to a general unscented Kalman filter.

Second, even when the state of the system defined in the manifold is estimated by applying the unscented Kalman filter using the stable embedding according to the present specification, the calculation time is improved.

The effects of the embodiments are not limited to the effects mentioned above, and other effects not mentioned will be clearly understood by those skilled in the art from the description of the claims.

1 is a diagram illustrating a method of estimating a state of a system according to an exemplary embodiment.
2 is a diagram for explaining the asymptotic stability of a manifold M in a system according to an exemplary embodiment.
3 is a diagram illustrating an arithmetic device according to an exemplary embodiment.
4 is a diagram showing a graph of the average of estimation error over time by comparing an unscented Kalman filter (UKF-SE) using a stable embedding function according to an embodiment with an unscented Kalman filter developed exclusively for other manifolds.

The terms used in the embodiments have been selected as general terms that are currently widely used as much as possible while considering the functions in the present disclosure, but they may vary depending on the intention or precedent of a person skilled in the art, the emergence of new technologies, and the like. In addition, in a specific case, there are also terms arbitrarily selected by the applicant, and in this case, the meaning will be described in detail in the corresponding description. Therefore, terms used in the present disclosure should be defined based on the meaning of the term and the general content of the present disclosure, not simply the name of the term.

In the entire specification, when a part is said to "include" a certain component, it means that it may further include other components, not excluding other components unless otherwise stated. In addition, terms such as “~unit” and “~module” described in the specification mean a unit that processes at least one function or operation, which may be implemented as hardware or software, or a combination of hardware and software.

The expression of “at least one of a, b, and c” described throughout the specification means 'a alone', 'b alone', 'c alone', 'a and b', 'a and c', 'b and c' ', or 'all of a, b, and c'.

A “terminal” referred to below may be implemented as a computer or portable terminal capable of accessing a learning device or other terminals through a network. Here, the computer includes, for example, a laptop, desktop, laptop, etc. equipped with a web browser, and the portable terminal is, for example, a wireless communication device that ensures portability and mobility. , IMT (International Mobile Telecommunication), CDMA (Code Division Multiple Access), W-CDMA (W-Code Division Multiple Access), LTE (Long Term Evolution), etc. It may include a handheld-based wireless communication device.

Hereinafter, with reference to the accompanying drawings, embodiments of the present disclosure will be described in detail so that those skilled in the art can easily carry out the present disclosure. However, the present disclosure may be implemented in many different forms and is not limited to the embodiments described herein.

Hereinafter, embodiments of the present disclosure will be described in detail with reference to the drawings.

1 is a diagram illustrating a method of estimating a state of a system according to an exemplary embodiment.

Referring to FIG. 1 , in step S110, the computing device may check a manifold in which a system state is defined. Equation 1 below represents a general equation of state of a nonlinear continuous-time system defined in a manifold.

[Equation 1]

에서 정의된

차원의 다양체로서, 다양체에 시스템의 상태가 정의되어 있으므로 다양체의 종류가 확인될 필요가 있다. 또한,

은 상태 미분 방정식,

은 다양체 M의 접선 번들(tangent bundle), x는 시스템의 상태,

는 시스템의 입력,

는 센서 측정 방정식,

는 센서 측정값을 나타낼 수 있다. 여기서, 다양체 관련하여 관련 기술 분야에서 통상의 기술자에게 자명한 내용이 적용될 수 있다. Here, M is a Euclidean space

defined in

As a dimensional manifold, since the state of the system is defined in the manifold, the type of the manifold needs to be identified. Also,

silver state differential equation,

is the tangent bundle of manifold M, x is the state of the system,

is the input of the system,

is the sensor measurement equation,

may represent a sensor measurement value. Here, the contents obvious to those skilled in the art in relation to the manifold can be applied.

로 확장하면 다음의 수학식 2와 같을 수 있다. 이때, 다음의 수학식 3과 같은 조건이 성립할 수 있다. In step S120, the computing device may extend the system into Euclidean space. Specifically, the system of Equation 1 in the Euclidean space in the manifold M

Expanding to Equation 2 may be as follows. At this time, a condition such as Equation 3 below may be satisfied.

[Equation 2]

[Equation 3]

이고,

일 수 있다. 수학식 3의 조건은 수학식 2의 확장 시스템이 수학식 1의 원래 시스템과

일 때 같다는 것을 나타낼 수 있다. 따라서, 다양체 M은 수학식 2의 확장 시스템의 불변 집합(invariant set)일 수 있다. At this time, in Equation 2 and Equation 3

ego,

can be The condition of Equation 3 is that the extended system of Equation 2 is equal to the original system of Equation 1.

can be shown to be the same when Thus, the manifold M may be an invariant set of the extension system of equation (2).

In step S130, the computing device may determine a stable embedding function suitable for the manifold. Here, the condition for finding the stable embedding function V is as shown in Equation 4 below.

[Equation 4]

는 유클리드 내적을 나타내는 기호이고, 수학식 4를 만족하는 함수가 안정 임베딩 함수로 결정될 수 있다. here,

is a symbol representing the Euclidean dot product, and a function satisfying Equation 4 can be determined as a stable embedding function.

를 부가하여 다음의 수학식 5와 같은 변형된 시스템을 생성할 수 있다. In step S140, the computing device may add a stable embedding function to the expansion system. Specifically, in the expansion system of Equation 3, the gradient of the stable embedding function V multiplied by a constant

It is possible to create a modified system such as Equation 5 below by adding.

[Equation 5]

은 상수로서 안정 임베딩 상수를 나타낼 수 있다. 안정 임베딩 함수 V가 수학식 5에서 정의된 바와 같이, 다양체 M에서 최소값 0을 가지므로

이 성립할 수 있다. 따라서, 수힉식 5의 변형된 시스템은 수학식 1의 원래 시스템과

일 때 같으며 다양체 M은 수학식 5의 불변 집합일 수 있다.

는 V가 감소하는 방향을 가리키며, 다양체 M은 수학식 5의 시스템에서 점근적 안정성(asymptotic stability)을 가질 수 있다. 점근적 안정성은 다양체 M의 근방(neighborhood)의 점에서 출발한 시스템의 모든 흐름이 다양체 M으로 수렴한다는 것을 나타낼 수 있다. 시스템에서 다양체 M의 점근적 안정성에 대한 이해를 위해 도 2를 참조한다. here,

As a constant, may represent a stable embedding constant. Since the stable embedding function V has a minimum value of 0 in the manifold M, as defined in Equation 5,

this can be achieved Therefore, the modified system of Equation 5 is different from the original system of Equation 1.

, and the manifold M may be an invariant set of Equation 5.

indicates the direction in which V decreases, and the manifold M can have asymptotic stability in the system of Equation 5. Asymptotic stability can indicate that all flows in a system starting from a point in the neighborhood of a manifold M converge to a manifold M. See Figure 2 for an understanding of the asymptotic stability of manifold M in the system.

In step S150, the computing device may add a noise model to the system. Since a noise model is required to apply the Kalman filter, a stochastic continuous time system as shown in Equation 6 below can be generated by adding noise to the modified system of Equation 5.

[Equation 6]

는 상태 x의 초기 조건을 나타내고,

는 프로세스 잡음(process noise)을 나타내고,

는 측정 잡음을 나타낼 수 있다.

는 평균이

이고 공분산 행렬이

인 가우시안 확률 분포를 나타낼 수 있다. 프로세스 잡음 공분산 행렬

는 적절한 값으로 결정될 수 있고, 측정 잡음 공분산 행렬

는 센서 명세서로부터 적절한 값으로 결정될 수 있다. here,

denotes the initial condition of state x,

represents the process noise,

may represent measurement noise.

is the average

and the covariance matrix is

can represent a Gaussian probability distribution. Process noise covariance matrix

can be determined as an appropriate value, and the measurement noise covariance matrix

may be determined as an appropriate value from the sensor specification.

을 가지는 오일러 이산화(Euler discretization)로 이산화하여 다음의 수학식 7의 확률적 이산 시스템을 생성할 수 있다. In step S160, the computing device may generate a stochastic discrete-time system. Specifically, in order to apply the unscented Kalman filter, the system of Equation 6 is time-stepped.

A stochastic discrete system of Equation 7 can be generated by discretizing with Euler discretization having .

[Equation 7]

이고,

일 수 있다. here,

ego,

can be

이며,

이고,

이며,

이다. 오일러 이산회 이외에도 1차, 2차, 3차를 포함하여 다양한 이산화 방법을 적용하여 이산화할 수 있다. Also, the stochastic properties are:

is,

ego,

is,

to be. In addition to Euler discretization, it can be discretized by applying various discretization methods including first order, second order, and third order.

In step S170, the computing device may estimate the state of the system by applying an unscented Kalman filter. Also, in step S180, the arithmetic device may iteratively estimate the state of the system using an unscented Kalman filter.

Specifically, the state estimation value may be repeatedly obtained by applying an unscented Kalman filter defined in Euclidean space to the stochastic discrete system of Equation 7. At this time, a more accurate state estimation value can be obtained due to the effect of stable embedding. Any unscented Kalman filter defined in Euclidean space can be applied to Equation 7, and the core process of a general unscented Kalman filter defined in Euclidean space is as follows. A general stochastic discrete system is shown in Equation 8 below.

[Equation 8]

이고,

는 상태,

는 입력,

는 센서 측정 값,

는 프로세스 잡음,

는 측정 잡음,

이다. 상태 추정 값을

으로 초기화하고 추정값 오차 공분산 행렬을

으로 초기화할 수 있다. 그리고 증강 상태(augmented state)는 다음의 수학식 9와 같을 수 있다. here,

ego,

state,

is the input,

is the sensor measurement value,

is the process noise,

is the measurement noise,

to be. state estimate

and initialize the estimate error covariance matrix to

can be initialized with And the augmented state may be as shown in Equation 9 below.

[Equation 9]

이다. 증강 상태 추정 값은

로 주어질 수 있고, 증강 추정 값 오차 공분산은

일 수 있다. 무향 칼만 필터는 시그마 포인트(sigma point)를 생성하고, 시간 업데이트(time update)를 하고, 측정 업데이트(measurement update)를 하는 것을 각

마다 반복 수행할 수 있다. 시그마 포인트

와 가중치

를 생성하는 과정은 다음의 수학식 10과 같다. here,

to be. The augmented state estimate is

, and the augmented estimate error covariance is

can be Unscented Kalman filters generate sigma points, do time updates, and do measurement updates.

It can be repeated every time. sigma point

and weights

The process of generating is as shown in Equation 10 below.

[Equation 10]

는 행렬의 i번째 열이고,

이다. 자주 사용되는 상수는

이다. 수학식 10에서 시그마 포인트와 가중치 생성법은 여러 방법 중 하나이며, 다양한 생성법이 적용될 수 있다. here,

is the ith column of the matrix,

to be. Frequently used constants are

to be. In Equation 10, the sigma point and weight generation method is one of several methods, and various generation methods may be applied.

The key equation of time update is shown in Equation 11 below, and the key equation of measurement update is shown in Equation 12 below.

[Equation 11]

[Equation 12]

는 센서 측정값을 나타내고,

는 시간 계단 k에서의 상태 추정값을 나타낸다. 안정 임베딩을 이용한 무향 칼만 필터는 일반적인 무향 칼만 필터에 비해

이 정확하며 계산 시간도 비슷할 수 있다. here,

represents the sensor measurement value,

denotes the state estimate at time step k. An unscented Kalman filter using stable embedding is compared to a general unscented Kalman filter.

is accurate and the computation time may be similar.

는 V가 감소하는 방향을 가리키므로 다양체 M의 특정 근방에서 시작한 수학식 5의 모든 흐름은 다양체 M에 수렴할 수 있으며, 이를 점근적 안정성이라 할 수 있다. 도 2는 일 실시 예에 따다 다양체를 2D 상의 원으로 표현하였으며, 유클리드 공간

에서 정의된

차원의 다양체 일 수 있다. 2 is a diagram for explaining the asymptotic stability of a manifold M in a system according to an exemplary embodiment.

indicates a direction in which V decreases, so all flows in Equation 5 that started in a specific vicinity of the manifold M can converge to the manifold M, which can be referred to as asymptotic stability. Figure 2 is a manifold expressed as a circle on 2D according to an embodiment, Euclidean space

defined in

It can be a manifold of dimensions.

3 is a diagram illustrating an arithmetic device according to an exemplary embodiment. The computing device of FIG. 3 is a device capable of executing the method for estimating the state of the system described above, and FIG. 3 shows components related to the present embodiment, but is not limited thereto, and other general-purpose components other than those shown in FIG. 3 are shown. Additional components may be included.

Referring to FIG. 3 , an arithmetic device 300 may include a memory 310 and a processor 320 . The memory 310 and the processor 320 may communicate with each other through a bus (not shown), and the memory 310 and the processor 320 may refer to units that process at least one function or operation. and may be implemented in hardware, software, or a combination of hardware and software.

In an embodiment, the computing device 300 may include a memory 310 that stores various data. For example, at least one instruction for operating the computing device 300 may be stored in the memory 310 . In this case, the memory 310 and the processor 320 may perform various operations based on these instructions. As the instructions stored in the memory 310 are executed by the processor 320, the processor 320 may perform the aforementioned process of estimating the state of the system. Memory 310 may be volatile memory or non-volatile memory. For example, the memory 320 may store data processed by the processor 320 and data to be processed. The memory 320 may include a random access memory (RAM) such as dynamic random access memory (DRAM) and static random access memory (SRAM), a read-only memory (ROM), an electrically erasable programmable read-only memory (EEPROM), and a CD-ROM. It may include ROM, Blu-ray or other optical disk storage, Hard Disk Drive (HDD), Solid State Drive (SSD) or flash memory.

에 대한 그래프를 나타내는 도면이다. 4 is an average of estimation error over time by comparing an unscented Kalman filter (UKF-SE) using a stable embedding function according to an embodiment with an unscented Kalman filter developed exclusively for other manifolds.

It is a diagram showing a graph for .

An unscented Kalman filter using a stable embedding function can estimate a state estimation value more accurately, but may take a similar calculation time compared to a general unscented Kalman filter. Therefore, there is an effect of obtaining a more accurate state estimation value while taking a similar calculation time. In addition, since it does not require a discretization method or coordinate transformation to maintain the structure of the manifold, only a stable embedding function and a general unscented Kalman filter defined in Euclidean space can be used, so anyone can use it easily.

, 위성 궤도 좌표계는

, 위성의 바디(body) 좌표계는

로 나타낼 수 있다. 위성의 확률 연속시간 시스템은 다음의 수학식 13과 같다.As a result of comparing the unscented Kalman filter (UKF-SE) using the stable embedding function with the unscented Kalman filter developed exclusively for other manifolds through simulation, it can be confirmed that the improvement is shown in FIG. 4 . As an embodiment, a specific satellite system may be used as a system defined in the manifold to be used for simulation. The Earth-centered inertial coordinate system is

, the satellite orbit coordinate system is

, the body coordinate system of the satellite is

can be expressed as The probability continuous-time system of the satellite is as shown in Equation 13 below.

[Equation 13]

는

다양체에 정의된 자세(attitude) 쿼터니언(quaternion),

는 관성 좌표계에 따른 위성 각속도,

는

와 함께 다음의 수학식 14과 같을 수 있다. here,

Is

the attitude quaternion defined on the manifold;

is the satellite angular velocity according to the inertial coordinate system,

Is

It may be the same as Equation 14 below.

[Equation 14]

는 궤도 좌표계에 따른 위성 각속도이고 수학식 15와 같을 수 있다. At this time,

Is the satellite angular velocity according to the orbital coordinate system and may be expressed as Equation 15.

[Equation 15]

는 중력 기울기 토크(gravity gradient torque)로 수학식 16과 같을 수 있다. At this time,

Is the gravity gradient torque and may be equal to Equation 16.

[Equation 16]

는 위성의 관성 모멘트 행렬이고,

는

의 확률 분포를 따르는 프로세스 잡음이고,

은

단위 행렬이고,

이고,

은

의 회전 행렬로 다음의 수학식 17과 같을 수 있다.

is the moment of inertia matrix of the satellite,

Is

is the process noise following the probability distribution of

silver

is the identity matrix,

ego,

silver

As a rotation matrix of , it may be as shown in Equation 17 below.

[Equation 17]

은 상수로 관성 좌표계에 따른 궤도의 각속도이고,

는 지구의 중력 상수이고,

는 위성의 질량 중심과 지구 중심과의 거리이고,

는

의 3번째 열일 수 있다. 위성 시스템의 초기 조건은

이고,

,

,

일 수 있다.

is the angular velocity of the trajectory according to the inertial coordinate system as a constant,

is the Earth's gravitational constant,

is the distance between the center of mass of the satellite and the center of the Earth,

Is

It may be the third column of The initial condition of the satellite system is

ego,

,

,

can be

The measurement sensor of the satellite consists of a magnetometer and an acceleration gyro, and the measurement model of the magnetometer is as shown in Equation 18 below.

[Equation 18]

는 바디 좌표계에서 측정된 지구의 자기 벡터량이고,

은 궤도 좌표계에서 측정된 지구의 자기 벡터량이며 다음의 수학식 19과 같다. here,

is the Earth's magnetic vector quantity measured in the body coordinate system,

is the amount of the Earth's magnetic vector measured in the orbital coordinate system and is expressed in Equation 19 below.

[Equation 19]

는 지구의 자기 쌍극자 모멘트이고,

는 자기 쌍극자가 기울어진 각도,

는 궤도가 기울어진 각도,

는 지구의 회전 속도이다. 수학식 18에서

는 자력계 측정 잡음으로

의 확률 분포를 따르고

이다.here,

is the magnetic dipole moment of the earth,

is the angle at which the magnetic dipole is tilted,

is the angle of inclination of the orbit,

is the rotational speed of the earth. in Equation 18

is the magnetometer measurement noise

follows the probability distribution of

to be.

The acceleration gyro model is as shown in Equation 20 below.

[Equation 20]

는 위성의 측정된 각속도,

는 가속도 자이로 측정 잡음으로

의 확률 분포를 따르고,

이다. 이때 측정은 10Hz의 샘플링 주파수를 이용하여 측정되었다. here,

is the measured angular velocity of the satellite,

is the acceleration gyro measurement noise

follows the probability distribution of

to be. At this time, the measurement was performed using a sampling frequency of 10 Hz.

가 6차 다양체인

에 정의되어 있으며,

는

와 동형인 쿼터니언 공간일 수 있다.

에 적합한 안정 임베딩 함수는 다음의 수학식 21과 같다.According to an embodiment, a method of applying an unscented Kalman filter using a stable embedding function to a satellite system is as follows. The satellite system of Equation 13 is

is a 6th degree manifold

is defined in

Is

It can be a quaternion space that is isomorphic to .

A stable embedding function suitable for is shown in Equation 21 below.

[Equation 21]

를 부가하면 다음의 수학식 22와 같다. The function of Equation 21 may be a stable embedding function that satisfies all conditions of Equation 4. Extend the system of Equation 13 to Euclidean space and

When is added, the following Equation 22 is obtained.

[Equation 22]

이다. 수학식 22를 이산화하고 전술한 유클리드 공간에서 정의된 무향 칼만 필터를 적용할 수 있다. here,

to be. Equation 22 can be discretized and an unscented Kalman filter defined in the aforementioned Euclidean space can be applied.

를 다음 수학식 23과 같이 구할 수 있다. The mean squared error to check the performance of the Kalman filter,

can be obtained as in Equation 23 below.

[Equation 23]

는 시간 t=kh에서의 위성의 연속시간 경로에서의 위성의 실제 자세이고,

는 무향 칼만 필터로 계산한 시간 t=kh에서의 자세 추정값이고,

는 N=1000개의 시뮬레이션에 대한 평균이고, 위첨자 (i)는 N개의 시뮬레이션에서 i번째 시뮬레이션을 나타낸다.

는 얼마나 위성의 자세 추정 값의 실제 추정 값과의 평균 오차를 나타내며, 값이 작을수록 정확한 추정을 나타낼 수 있다. here,

is the actual attitude of the satellite on its continuous-time path at time t = kh,

is the attitude estimate at time t = kh calculated by an unscented Kalman filter,

is the average over N=1000 simulations, and the superscript (i) denotes the ith simulation in N simulations.

represents the average error of the satellite attitude estimation value and the actual estimation value, and the smaller the value, the more accurate the estimation.

를 사용한 경우에 대응하는 안정 임베딩 함수를 이용한 무향 칼만 필터(UKF-SE)에 대한 시뮬레이션 비교 대상으로, 다음과 같은 무향 칼만 필터들을 사용하였다. 1) Standard UKF는 유클리드 공간에서 정의된 일반적인 무향 칼만 필터이고, 2) Projection은 유클리드 공간에서 정의된 일반적인 무향 칼만 필터의 최종 추정 값을 최적화 알고리즘으로 다양체에 사영시키는 방법을 나타내고, 3) Normalization은 유클리드 공간에서 정의된 일반적인 무향 칼만 필터의 최종 추정값을

를 이용하여 다양체

에 사영시키는 방법을 나타내고, 4) Lie algebra는 다양체가 리 군일 때 사용하는 무향 칼만 필터로 리 군 제한적인 이산화 방법, 리 군과 리 대수 사이의 좌표변환을 사용하는 방법을 나타내고, 5) Tangent space는 다양체가 리만 다양체일 때 사용하는 무향 칼만 필터로 리만 다양체 제한적인 이산화 방법, 다양체와 접평면 사이의 좌표변환을 사용하는 방법을 나타낸다.

As a simulation comparison target for the unscented Kalman filter (UKF-SE) using a stable embedding function corresponding to the case of using , the following unscented Kalman filters were used. 1) Standard UKF is a general unscented Kalman filter defined in Euclidean space, 2) Projection represents a method of projecting the final estimated value of a general unscented Kalman filter defined in Euclidean space onto a manifold with an optimization algorithm, 3) Normalization is Euclidean The final estimate of a typical unscented Kalman filter defined in space

manifold using

4) Lie algebra is an unscented Kalman filter used when the manifold is a Li group, and represents a method using a discretization method limited to the Li group and a coordinate transformation between the Li group and Li algebra, 5) Tangent space is an unscented Kalman filter used when the manifold is a Riemann manifold, a Riemann manifold restricted discretization method, and a method using coordinate transformation between the manifold and the tangent plane.

[Table 1]

와 1000번 시뮬레이션했을 때의 계산 시간을 나타내며, 표 1에서도 확인되는 바와 같이 안정 임베딩을 이용한 무향 칼만 필터인 UKF-SE의 경우 다른 방법 대비 상대적으로 빠른 계산 시간을 나타내며, 가장 낮은 오차를 갖는 것을 확인할 수 있다. 즉, 도 4와 표 1을 통해 안정 임베딩을 이용한 무향 칼만 필터인 UKF-SE의 경우 다른 방법 대비 상대적으로 적은 계산 시간이 필요하며, 추정 값의 정확도가 상대적으로 개선된 방법임을 확인할 수 있다. Table 1 above shows the error of the unscented Kalman filter estimation value.

and the calculation time when simulated 1000 times. As can be seen in Table 1, UKF-SE, an unscented Kalman filter using stable embedding, shows a relatively fast calculation time compared to other methods and has the lowest error. can That is, in the case of UKF-SE, which is an unscented Kalman filter using stable embedding, through FIG. 4 and Table 1, it can be confirmed that a relatively less calculation time is required compared to other methods, and the accuracy of the estimated value is relatively improved.

The foregoing embodiments are merely examples and other embodiments may be implemented within the scope of the claims described below.

Claims (10)

identifying a manifold that is related to the state of the system and is included in some dimension of the Euclidean space;
creating a system in which the system extends from the manifold into the Euclidean space;
determining a stable embedding function that satisfies a predetermined condition in relation to the manifold;
applying the stable embedding function to the system expanded into the Euclidean space;
generating a stochastic continuous time system by applying process noise to apply an unscented Kalman filter to the system to which the stable embedding function is applied;
generating a stochastic discrete-time system by discretizing the stochastic continuous-time system by applying a preset discretization method; and
Estimating the state by applying the unscented Kalman filter to the stochastic discrete-time system,
A method for estimating the state of a system. delete According to claim 1,
A stable embedding function that satisfies the certain conditions is
Characterized in that, among the at least one embedding function applied to the state, an embedding function that satisfies the condition that the system extended to the Euclidean space and the operation result is 0,
A method for estimating the state of a system. delete delete delete delete delete As a non-transitory computer-readable storage medium,
a medium configured to store computer readable instructions;
The computer readable instructions, when executed by a processor, cause the processor to:
identifying a manifold that is related to the state of the system and is included in some dimension of the Euclidean space;
creating a system in which the system extends from the manifold into the Euclidean space;
determining a stable embedding function that satisfies a predetermined condition in relation to the manifold;
applying the stable embedding function to the system expanded into the Euclidean space;
generating a stochastic continuous time system by applying process noise to apply an unscented Kalman filter to the system to which the stable embedding function is applied;
generating a stochastic discrete-time system by discretizing the stochastic continuous-time system by applying a preset discretization method; and
and estimating the state by applying the unscented Kalman filter to the stochastic discrete-time system. As an arithmetic device for estimating the state of the system,
A memory that stores instructions readable by a computer; and
contains a processor;
The processor connected to the memory,
It is related to the state of the system and identifies a manifold included in some dimension of the Euclidean space, creates a system in which the system extends from the manifold to the Euclidean space, and satisfies a certain condition in relation to the manifold. Determining a stable embedding function, applying the stable embedding function to the system expanded into the Euclidean space, and applying process noise to apply an unscented Kalman filter to the system to which the stable embedding function is applied A stochastic continuous-time system is generated, a stochastic discrete-time system is generated by discretizing the stochastic continuous-time system by applying a predetermined discretization method, and the stochastic discrete-time system is generated. Estimating the state by applying the unscented Kalman filter to the temporal system,
computing device.

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