A taichi task arm length estimation method, system and product considering satellite relative motion compensation

Abstract

This invention discloses a method, system, and product for estimating the arm length of a Taiji mission considering satellite relative motion compensation. The method includes: reading ephemeris data of key gravitational bodies during the simulation period, establishing the orbital dynamics equations of the satellite formation, solving the equations using a high-precision numerical integration algorithm, and outputting satellite orbit simulation data; generating basic measurements including noise and satellite-to-solar measurements based on the satellite orbit simulation data, and correcting the inter-satellite distance measurements in the basic measurements by optical signal propagation time to achieve satellite relative motion compensation and outputting high-precision observation data; receiving the high-precision observation data, combining it with the orbital dynamics equations of the satellite formation, recursively updating and optimizing the state variables of the satellite formation based on a Kalman filter algorithm, and outputting a high-precision estimate of the inter-satellite arm length; and visualizing the estimated satellite arm length and its error. This invention significantly improves the autonomy and reliability of space gravitational wave detection missions.

Description

Technical Field

[0001] This invention belongs to the field of aerospace technology, specifically relating to a method, system, and product for estimating the arm length of the Taiji mission considering satellite relative motion compensation. Background Technology

[0002] Space gravitational wave detection missions (such as the Taiji Program) measure gravitational wave signals by performing laser interferometry with sub-picometer precision between satellite formations with arms spanning millions of kilometers. However, the actual orbits of satellites deviate from the theoretical designs due to factors such as multi-body gravitational perturbations during their operation in orbit. To ensure the optical phase-locking of the interferometer and the validity of scientific data, it is necessary to continuously and accurately estimate the inter-satellite arm lengths in real time.

[0003] Several arm length estimation techniques exist in the industry, but they all have certain drawbacks: Currently, there are arm length estimation techniques based on spaceborne measurements (inter-satellite ranging, velocity measurement, and clock difference measurement), but these do not consider the relative motion correction for light flight time in the measurement model, resulting in insufficient model accuracy in the large-scale space environment of heliocentric orbits, limiting further improvement in estimation accuracy. There are also orbit determination schemes based on ground-based tracking stations (such as DSN and VLBI), which can also be used for arm length estimation and perform well with ground support. However, in actual missions, the ground station support coverage is limited (only about 8 hours every 7 days), making continuous autonomous operation impossible and unsuitable as the sole reliance for arm length estimation. Therefore, existing technologies cannot achieve arm length estimation that simultaneously satisfies high accuracy, high stability, and fully autonomous operation in the absence of ground support. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art by systematically introducing the aforementioned correction into a high-precision measurement model, optimizing the autonomous state determination process, and constructing a method, system, and product for estimating inter-satellite arm length with high precision and high stability under conditions lacking ground support.

[0005] In view of this, the present invention proposes a method for estimating the arm length of the Taiji mission considering satellite relative motion compensation, comprising: Step 1: Read the ephemeris data of key gravitational bodies during the simulation period, establish the orbital dynamics equations of the satellite formation, solve the equations using a high-precision numerical integration algorithm, and output the satellite orbit simulation data; Step 2: Based on the satellite orbit simulation data, generate basic measurements including noise and satellite-to-sun measurements, and correct the optical signal propagation time of the inter-satellite distance in the basic measurements to achieve satellite relative motion compensation and output high-precision observation data. Step 3: Receive high-precision observation data, combine it with the orbital dynamics equations of the satellite formation, and recursively update and optimize the state variables of the satellite formation based on the Kalman filter algorithm to output a high-precision estimate of the inter-satellite arm length. Step 4: Visualize the estimated satellite arm length and its error.

[0006] As an improvement to the above method, step 1 includes: Step 1-1: Obtain the position information of the Sun, Earth, Moon, Mercury, Venus, Mars, Jupiter, and Saturn during the simulation period; Step 1-2: Based on the position information of multiple celestial bodies, construct dynamic equations describing the evolution of the satellite formation state over time:

[0007] in, Indicates satellite acceleration. This represents the sequence number of the gravitational source in the solar system. The gravitational constant, For the first The mass of a gravitational source This is the satellite's position vector relative to the gravitational source. The magnitude of the position vector; Steps 1-3: Solve the dynamic equations using the 7th-8th order Runge-Kutta method of numerical integration; Steps 1-4: Output simulation results.

[0008] As an improvement to the above method, the basic measurements in step 2 include: inter-satellite ranging, inter-satellite Doppler effect measurements, and clock drift; the satellite-to-sun measurements include the satellite's azimuth angle, elevation angle, and radial velocity relative to the sun.

[0009] As an improvement to the above method, in step 2, the correction amount for the optical signal propagation time correction... for:

[0010] in, The velocity vector of the receiving star at the moment of signal reception. and are the position vectors of the receiving star and the transmitting star at the moment of reception, respectively, and c is the speed of light.

[0011] As an improvement to the above method, in step 3, the state variables of the satellite formation include: the position, velocity and clock parameters of the satellites.

[0012] Secondly, the present invention provides a Taiji mission arm length estimation system that considers satellite relative motion compensation, comprising: The satellite orbit dynamics simulation module is used to read the ephemeris data of key gravitational bodies during the simulation period, establish the orbit dynamics equations of the satellite formation, solve the equations using a high-precision numerical integration algorithm, and output satellite orbit simulation data. The satellite measurement generation module is used to generate basic measurements and satellite-to-sun measurements containing noise based on satellite orbit simulation data, and to correct the optical signal propagation time of the inter-satellite distance in the basic measurements to achieve satellite relative motion compensation and output high-precision observation data. The filter module receives high-precision observation data, combines it with the orbital dynamics equations of the satellite formation, and recursively updates and optimizes the state variables of the satellite formation based on the Kalman filter algorithm, outputting high-precision estimates of inter-satellite arm lengths; and The display module is used to visualize the estimated satellite arm length and its error.

[0013] Thirdly, the present invention provides a computer program product, including a computer program that, when executed by a processor, implements the above-described method steps.

[0014] Compared with the prior art, the advantages of the present invention are: 1. A high-precision arm length estimation method integrating satellite relative motion compensation is proposed. By systematically introducing the correction of light flight time into the measurement model, the method improves the problem of incomplete model in existing schemes at the scale of millions of kilometers, and effectively improves the model accuracy of arm length estimation.

[0015] 2. A fully autonomous arm length estimation scheme based entirely on onboard measurement equipment is proposed. All observation data comes from the high-precision measurement instruments carried by the satellite formation itself, without relying on the support of ground-based telemetry and control systems (such as Deep Space Network (DSN)). This achieves continuous and high-precision arm length estimation without ground intervention, significantly improving the autonomy and reliability of space gravitational wave detection missions.

[0016] 3. By fully utilizing the existing onboard measurement equipment of the satellite formation, the system innovatively integrates measurements of the solar relative angle and solar radial velocity. These measurements provide the system with an absolute external inertial space reference, effectively constraining the overall rotation and radial motion errors of the satellite formation, thereby significantly improving the shortcomings of pure inter-satellite measurement geometry without the need for additional hardware. Attached Figure Description

[0017] Figure 1 This is a flowchart for arm length estimation; Figure 2 This is a graph showing the arm length estimation error (0-160000 seconds). Figure 3 This is a graph showing the arm length estimation error (1500-160000 seconds). Figure 4 This is a histogram of the arm length estimation error. Detailed Implementation

[0018] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0019] Example 1 Embodiment 1 of the present invention proposes a method for estimating the arm length of the Taiji mission considering satellite relative motion compensation, comprising: Step 1: Read the ephemeris data of key gravitational bodies during the simulation period, establish the orbital dynamics equations of the satellite formation, solve the equations using a high-precision numerical integration algorithm, and output the satellite orbit simulation data; specifically: Step 1-1: Obtain the position information of the Sun, Earth, Moon, Mercury, Venus, Mars, Jupiter, and Saturn during the simulation period; Step 1-2: Based on the position information of multiple celestial bodies, construct dynamic equations describing the evolution of the satellite formation state over time:

[0020] in, Indicates satellite acceleration. This represents the sequence number of the gravitational source in the solar system. The gravitational constant, For the first The mass of a gravitational source This is the satellite's position vector relative to the gravitational source. The magnitude of the position vector; Steps 1-3: Solve the dynamic equations using the 7th-8th order Runge-Kutta method of numerical integration; Steps 1-4: Output simulation results.

[0021] Step 2: Based on the satellite orbit simulation data, generate basic measurements including noise and satellite-to-sun measurements, and perform optical signal propagation time correction on the inter-satellite ranging in the basic measurements to achieve satellite relative motion compensation and output high-precision observation data; among them, the basic measurements include: inter-satellite ranging, inter-satellite Doppler effect measurements, and clock drift; the satellite-to-sun measurements include the satellite's azimuth angle, elevation angle, and radial velocity relative to the sun.

[0022] Correction amount for optical signal propagation time correction for:

[0023] in, The velocity vector of the receiving star at the moment of signal reception. and are the position vectors of the receiving star and the transmitting star at the moment of reception, respectively, and c is the speed of light.

[0024] Step 3: Receive high-precision observation data, combine it with the orbital dynamics equations of the satellite formation, and recursively update and optimize the state variables of the satellite formation based on the Kalman filter algorithm to output a high-precision estimate of the inter-satellite arm length; among which, the state variables of the satellite formation include: satellite position, velocity and clock parameters.

[0025] Step 4: Visualize the estimated satellite arm length and its error.

[0026] Example 2 Embodiment 2 of the present invention provides a Taiji mission arm length estimation system considering satellite relative motion compensation, comprising a satellite orbital dynamics simulation module, a satellite measurement generation module, a filter module, and a display module. The flowchart of the entire arm length estimation process is shown below. Figure 1 As shown: 1. Satellite orbit dynamics simulation module: Satellite orbital dynamics simulation involves the following steps: Step S1: Read ephemeris data Acquiring high-precision position data of major celestial bodies within the solar system is essential for calculating the multi-body gravitational perturbations experienced by satellites. This invention utilizes the ephemeris database provided by the JPL Horizons system to obtain the precise position coordinates of key gravitational bodies such as the Sun, Earth, and Jupiter during the simulation period. This step handles interface communication with the ephemeris data, data parsing, and timestamp alignment. The processed celestial body position information is then output to the subsequent dynamic equation establishment and solution unit, laying the foundation for building a high-fidelity satellite orbital dynamics model and ultimately serving high-precision inter-satellite arm length estimation.

[0027] (1) in, Represents the gravitational force acting on the satellite. The numbers representing the gravitational sources in the solar system (including the Sun, Earth, Moon, Mercury, Venus, Mars, Jupiter, and Saturn). The gravitational constant, For the first The mass of a gravitational source This is the satellite's position vector relative to the gravitational source. Let be the magnitude of the vector at that position.

[0028] Step S2: Establish satellite orbital dynamics equations Based on multi-body position data provided by the ephemeris reading module, a dynamic model describing the evolution of the satellite formation state over time is constructed. The satellite orbital state is described by an 18-dimensional vector, including the position and velocity information of the three satellites. The orbital dynamics are dominated by multi-body gravity. According to Newton's second law and the law of universal gravitation, the satellite acceleration is the vector sum of the gravitational forces from all gravitational sources (including the Sun, planets, and the Moon), and its equation is: (2) The position vector of the satellite relative to the gravitational source is provided by ephemeris data. The dynamics of the clock component are described by the differential relationship between clock offset and frequency offset. This module integrates the above physical laws into a complete set of differential equations, providing a mathematical model for subsequent numerical solutions.

[0029] Step S3: Solve the satellite orbital dynamics equations This step is executed by the dynamic equation solving module, which is responsible for numerically integrating the continuous differential equations established in step 2 to obtain the system state at discrete time points. Since the dynamic equations are complex and cannot be solved analytically, this module employs a high-precision 7th-8th order Runge-Kutta method for numerical integration. This solver receives the initial state vector and the dynamic equations, and progresses step-by-step with a fixed simulation step size to calculate the satellite's precise position, velocity, and clock parameters at future times. This module ensures the accuracy and numerical stability of orbit integration during long-term simulations, and its output serves as the true orbital reference for subsequent state estimation and arm length estimation.

[0030] Step S4: Output simulation data The simulation results of this module are output to the satellite measurement generation module as input to generate satellite measurement data.

[0031] 2. Satellite Measurement Generation Module: This module is used to simulate various high-precision inter-satellite and satellite-to-solar observation data actually acquired during satellite operation in orbit. The module receives satellite orbit data from the satellite orbit dynamics simulation module and generates accurate measurements including noise, providing observational input for subsequent Kalman filter state estimation.

[0032] Step S1: Read satellite orbit simulation data This step is executed by the data interface unit and marks the starting point of the measurement simulation process. This unit specifically reads the satellite orbit data generated by the satellite orbit dynamics simulation module from its final output; specifically, the position vector sequence of the three satellites in the J2000 ecliptic coordinate system. With velocity vector sequence These orbital data form the basis for all subsequent inter-satellite geometric measurements, providing the necessary input for building accurate measurement models.

[0033] Step S2: Generate satellite clock simulation data This step is performed by the clock simulation unit, independent of the orbital data processing flow. This unit generates time-related clock parameters specifically based on the physical characteristics of the satellite's onboard clock. Among these parameters is the frequency offset sequence. It is directly generated from the clock noise model, while the clock offset sequence The integral relationship is obtained by integrating the frequency shift over time, and is as follows: (3) (4) This step simulates clock measurement data with errors that conform to actual physical characteristics by selecting a specific noise model required by the user. This simulated clock data will be used in subsequent clock error measurement models.

[0034] Step S3: Select the measurement model and the correction model This step, based on the task stage and accuracy requirements, independently configures the combination of measurement types and the depth of correction of the physical model, and is the core decision-making step in determining the system's working mode and measurement accuracy.

[0035] Step S4: Construct a measurement model This step, based on the configuration decision in step 3, uses the satellite orbit data read in step 1 and the clock data generated in step 2 to perform corresponding physical model calculations, generating a complete set of raw observation data, but without high-order physical corrections. The output of this step is directly used as the input for satellite relative motion compensation in the subsequent step 5, where the correction module refines it. Finally, the combined output generated in step 4 and corrected in step 5 constitutes the final result of the satellite measurement generation module, serving as a high-precision observation output to the subsequent square root sequential Kalman filter module for arm length estimation.

[0036] Step S4.1: Basic Measurement Model This step is responsible for generating the core observation data required. The basic measurement model, based on the geometric motion relationships and clock characteristics between satellites, constructs a complete observation system including inter-satellite relative position measurements (range measurement), Doppler effect measurements (velocity measurement), and clock drift measurements. The subscript ij represents the satellite number, and the mathematical models for each measurement are defined as follows: (5) (6) (7) in The geometric distance between satellites For ranging noise, and These are the laser carrier frequencies for receiving and transmitting, respectively. This represents the projection of the interstellar relative velocity along the line of sight. For Doppler noise measurement, This is for measuring noise in clock error measurements.

[0037] The basic measurement model contains 18 measurements (6 distance measurements, 6 velocity measurements, and 6 clock errors), which form the basis of the observation dataset for subsequent higher-order corrections and Kalman filtering.

[0038] Step S4.2: Measurement of solar relative angle and radial velocity This step is responsible for generating satellite observation data of the sun. This unit is based on the relative position and motion of the satellite and the sun, specifically including measurements of the satellite's radial velocity relative to the sun and the angle between the sun and the satellite.

[0039] The observed value of a star's solar azimuth angle is defined as: (8) The observed value of a star's solar elevation angle is defined as: (9) The observed radial velocity of a star's satellites relative to the Sun is defined as: (10) in This is the unit direction vector pointing from the satellite to the sun.

[0040] The three satellites generated a total of nine measurements, including six angular measurements and three radial velocity measurements. These observations provide the system with an absolute inertial spatial pointing reference and radial motion constraints, effectively improving the observability of the state estimation system in terms of absolute attitude and velocity. All measurements were calculated based on the satellite orbit data provided in step 1.

[0041] Step S5: Satellite relative motion compensation This step is used for satellite relative motion compensation and is a key step in improving the accuracy of the measurement model. Based on the post-Newtonian orbital mechanics theory, the inter-satellite distance (i.e., the time of flight of light) generated in step 4 is compensated for the relative motion effect between satellites to eliminate the light signal propagation error caused by the relative motion of satellites and the gravitational potential of the sun.

[0042] Calculate and incorporate the optical signal propagation time correction caused by the relative motion of the satellite. (11) in The velocity vector of the receiving star at the moment of signal reception. and are the position vectors of the receiving star and the transmitting star at the moment of reception, respectively, and c is the speed of light.

[0043] 3. Filter module: This module uses the Kalman filter algorithm to fuse the dynamic model and observation data, achieving high-precision estimation of the satellite arm length.

[0044] This module receives various observation data from the satellite measurement generation module and, in conjunction with the satellite orbital dynamics model, recursively updates and optimizes the satellite's position, velocity, clock parameters, and other state variables using advanced state estimation algorithms.

[0045] This module employs a recursive estimation algorithm enhanced with numerical stability, effectively fusing prior dynamic information with multi-source observation data. Its workflow comprises two basic steps: state prediction and measurement update. First, the system state is updated over time based on the dynamic model to obtain predicted values. Then, the predicted values ​​are corrected using measurements taken at the current moment to obtain the optimal state estimate. The algorithm is specifically designed to suppress numerical divergence issues that may arise from model nonlinearity or long-term recursion, ensuring estimation stability throughout the entire mission cycle.

[0046] Finally, the module outputs an optimized high-precision state estimation sequence for the satellite formation, and directly calculates the inter-satellite arm lengths based on this sequence. This result is the final output of the entire system, and its accuracy directly verifies the comprehensive performance of the autonomous state estimation and arm length estimation system.

[0047] 4. Display Module: The visualization module outputs the arm length estimation results and visually represents these results. Currently, the visualization module primarily generates the following results: 1. Arm length estimation results; 2. Error diagram of arm length estimation results; 3. Error distribution histogram; 4. Standard deviation of arm length estimation error.

[0048] It is worth noting that in the embodiments of the above system, the modules included are divided according to functional logic, but are not limited to the above division, as long as the corresponding functions can be achieved; in addition, the specific names of each functional module are only for easy distinction between each other and are not used to limit the scope of protection of the present invention.

[0049] Example 3 Embodiments of the present invention may also provide a computer program product, including a computer program. When the computer program is executed by a processor, it can implement the various steps in the above method embodiments.

[0050] Simulation Example 1: This embodiment constructs a simulation of arm length estimation under Taiji mission conditions, with a simulation time of 0-160,000 seconds. The initial satellite position is determined by the Earth's position at the initial moment. After reading the Earth's position at the initial moment, the satellite formation position is generated. The formation is set to lead the Earth by 20°, and the initial distance between satellites is 3×10⁻⁶. 9 m is an equilateral triangle with an included angle of 60° between the satellites.

[0051] The initial simulation conditions are: arm length 3×10 9 The simulation time step is 1 / 3 second, and the initial position error is 2 × 10⁻⁶ meters. 4 meters, with an initial velocity error of 1×10 -3 Meters per second, arm length ranging noise is 1 meter, Doppler measurement noise is 1×10⁻⁶. 3 Hertz, clock error measurement noise is 1 Hz, solar velocity noise is 1 × 10⁻⁶ -2 meters per second, with a solar angle measurement noise of 1×10⁻⁶. -5 Radius. The estimation error of the arm length of the experimental design over time is as follows: Figure 2 As shown. The initial position error set in the simulation initial conditions is 2 × 10⁻⁶. 4 The initial arm length estimate was in meters, therefore it contained a large error. However, using an arm length estimation method, the error was rapidly reduced within a very short time.

[0052] Simulation Example 2: To observe the change in arm length estimation error over time after the arm length error stabilizes, this embodiment constructs an arm length estimation simulation under Tai Chi mission conditions with a simulation time of 1500-160000 seconds. Figure 3 As shown, after the arm length estimation error decreases rapidly, it can be maintained at a relatively stable level for a long period of time.

[0053] Simultaneously, an error distribution histogram is plotted and the standard deviation is calculated as a standard to measure the accuracy of the arm length estimation, such as... Figure 4 As shown. The arm length estimation error is approximately 3 × 10⁻⁶. -3 Meters, or millimeters, represent an order of magnitude improvement in estimation accuracy compared to existing centimeter-level estimates.

[0054] To address the need for high-precision autonomous estimation of inter-satellite arm length in space gravitational wave detection missions, a method and system integrating optical time-of-flight correction is proposed. Considering the model errors caused by the lack of relative motion effects in traditional methods, the accuracy issues resulting from the lack of diverse measurement methods such as solar relative angle and radial velocity measurements, and the over-reliance on ground-based telemetry and control support, a satellite orbital dynamics simulation module, a satellite measurement generation module, a filter module, and a display module are constructed. This enables millimeter-level inter-satellite arm length estimation accuracy based entirely on on-board observation data, without ground support, and provides high-precision arm length estimation services and visualization for satellite formations.

[0055] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for estimating the arm length of a Taiji mission considering satellite relative motion compensation, comprising: Step 1: Read the ephemeris data of key gravitational bodies during the simulation period, establish the orbital dynamics equations of the satellite formation, solve the equations using a high-precision numerical integration algorithm, and output the satellite orbit simulation data; Step 2: Based on the satellite orbit simulation data, generate basic measurements including noise and satellite-to-sun measurements, and correct the optical signal propagation time of the inter-satellite distance in the basic measurements to achieve satellite relative motion compensation and output high-precision observation data. Step 3: Receive high-precision observation data, combine it with the orbital dynamics equations of the satellite formation, and recursively update and optimize the state variables of the satellite formation based on the Kalman filter algorithm to output a high-precision estimate of the inter-satellite arm length. Step 4: Visualize the estimated satellite arm length and its error.

2. The method for estimating the arm length of the Taiji mission considering satellite relative motion compensation according to claim 1, characterized in that, Step 1 includes: Step 1-1: Obtain the position information of the Sun, Earth, Moon, Mercury, Venus, Mars, Jupiter, and Saturn during the simulation period; Step 1-2: Based on the position information of multiple celestial bodies, construct dynamic equations describing the evolution of the satellite formation state over time: ； in, Indicates satellite acceleration. This represents the sequence number of the gravitational source in the solar system. The gravitational constant, For the first The mass of a gravitational source This is the satellite's position vector relative to the gravitational source. The magnitude of the position vector; Steps 1-3: Solve the dynamic equations using the 7th-8th order Runge-Kutta method of numerical integration; Steps 1-4: Output simulation results.

3. The method for estimating the arm length of the Taiji mission considering satellite relative motion compensation according to claim 1, characterized in that, The basic measurements in step 2 include: inter-satellite ranging, inter-satellite Doppler effect measurements, and clock drift; the satellite-to-sun measurements include the satellite's azimuth, elevation, and radial velocity relative to the sun.

4. The method for estimating the arm length of the Taiji mission considering satellite relative motion compensation according to claim 1, characterized in that, In step 2, the correction amount for the optical signal propagation time correction. for: ； in, The velocity vector of the receiving star at the moment of signal reception. and are the position vectors of the receiving star and the transmitting star at the moment of reception, respectively, and c is the speed of light.

5. The method for estimating the arm length of the Taiji mission considering satellite relative motion compensation according to claim 1, characterized in that, In step 3, the state variables of the satellite formation include: the satellite's position, velocity, and clock parameters.

6. A Taiji mission arm length estimation system considering satellite relative motion compensation, characterized in that, include: The satellite orbit dynamics simulation module is used to read the ephemeris data of key gravitational bodies during the simulation period, establish the orbit dynamics equations of the satellite formation, solve the equations using a high-precision numerical integration algorithm, and output satellite orbit simulation data. The satellite measurement generation module is used to generate basic measurements and satellite-to-sun measurements containing noise based on satellite orbit simulation data, and to correct the optical signal propagation time of the inter-satellite distance in the basic measurements to achieve satellite relative motion compensation and output high-precision observation data. The filter module is used to receive high-precision observation data, combine it with the orbital dynamics equations of the satellite formation, and recursively update and optimize the state variables of the satellite formation based on the Kalman filter algorithm to output high-precision estimates of inter-satellite arm lengths. and The display module is used to visualize the estimated satellite arm length and its error.

7. A computer program product, comprising a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the method described in any one of claims 1-6.

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