FIELD OF THE INVENTION The present invention relates to a method and apparatus for calibrating sensor systems and, particularly to a method and apparatus for calibrating motion sensor systems.

BACKGROUND OF THE INVENTION The great majority daily lives of people living in the information era are closely related to the development, acquisition, transmission and processing of information resources. As windows of information sensing, acquisition and detecting, sensors of different functions play a very important role in the signal detection and information processing systems. Sensors are devices or means capable of sensing ( or responsive to ) specified measurement values and converting them into usable output signals according to certain rules. Sensors ordinarily comprise sensitive element which directly responds to the measurement values, converting element which generates output signals, and corresponding electronic circuits. Motion sensors, as sensors having their own coordinate systems, are capable of converting the motion signals of an object into detectable electrical signals, such as the acceleration sensor and gyrocompass sensor. The acceleration sensor and gyrocompass sensor are ordinary measurement instruments for shock and vibration measurements and motion tracking in many fields such as industry and national defense, which are especially applicable in vibration measurement and motion tracking in the fields of seismology, architecture, military, transportation, machinery, navigation and the like. The acceleration sensor is a measurement instrument for converting the physical signal of acceleration into an electrical signal which is easy to be measured. The output value of the measurement of the acceleration sensor is a voltage value reflecting the acceleration. For example, the acceleration sensor in the form of an IC chip manufactured by the Hitachi Metals Ltd, Tokyo, Japan is a three-dimentional piezoresistance acceleration sensor capable of detecting the accelerations in the three axial directions ( X, Y and Z ). This sensor is a very small and thin three-dimensional acceleration sensor of the semiconductor type which is highly sensitive, shock-proof and pressing proof. More information relating to this acceleration sensor can be obtained from the web site http://www.hitachimetals. co.jp/e/p rod/prod 06/p06 10.htm. Those informations are incorporated herein by reference. The gyrocompass sensor is a measurement equipment for converting the physical signal of angular velocity into electrical signal which is an easy to be measured. Similarly, the output value of measurement of the gyrocompass sensor is an electrical signal reflecting the angular velocity. In order to obtain an accurate motion locus, two or more motion sensors are ordinarly needed in a sensor system, for example, two three-dimensional motion sensors are needed in systems, such as the three-dimensional hand-writing recognition system, inertial measurement unit, robotic arm motion measurement system, aerial guidance system, appliance remote controller and the like, to sense the motion of the sensor system in the three-dimensional space. If the two or more motion sensors in a sensor system are motion sensors of the same type, for example, both of them are acceleration sensors or gyrocompass sensors, then it is necessary to calibrate the positional relationship between the coordinate systems of those two or more motion sensors, such that the output values of those two or more motion sensors in the system are transformed into the same coordinate system, so as the motion locus of the sensor system can be tracked by measuring the output values of those motion sensors. The positional relationship between two motion sensors in a sensor system comprises a translational relationship and a rotational relationship therebetween. The translational relationship between two motion sensors is determined by the design and construction of the system electronic circuitry board, which is relatively easy to determine by measurement or other approaches. The rotational relationship between two motion sensors refers to rotating the coordinate system of one of the sensors by a specific angle required by the coordinate system of the other sensor, such that the rotational angle of the coordinate system of one of the sensors can achieve the requirements of design with respect to the coordinate system of the other sensor, for example parallel to each other. There are two existing approaches for determining the rotational relationship between motion sensors: the first one is the approximation method, which is realized by an approximation process during the design and production of the electronic circuitry board. For example, two or more sensors are parallelly placed during the design of the electronic circuitry board, and the parallelism and freedom from rotation of the coordinate systems of the two or more sensors are realized by an as-strict-as-possible manufacturing process in their production. This approach imposes severe requirements upon the manufacturing process. The second approach is the measurement approach, in which the angles between the two or more acceleration sensors are measured by angle measurement instrument. The requirement on the measurement accuracy of the angle measurement instrument is very high in order to obtain accurate measurements. Therefore, there is a need for a method and apparatus for rapidly, conveniently and speedily calibrating the rotational relationship between two or more motion sensors in a sensor system, for lowering the manufacturing technological requirements of the sensor systems, and for reducing the dependency on the precise angle measurement instruments.

OBJECT AND SUMMARY OT THE INVENTION It is an object of the present invention to provide a method of calibrating the rotational relationship between a first motion sensor and a second motion sensor in a sensor system, comprising the steps of: determining a minimal number of measurements based on the number of dimensions of the coordinate system of said first motion sensor and the number of dimensions of the coordinate system of said second motion sensor; measuring said sensor system for a specific number of measurements to obtain the output values of said first motion sensor and said second motion sensor during each of the measurements, said specific number of measurements is not less than said minimal number of measurements; and obtaining a rotational relationship between said first motion sensor and said second motion sensor based on said measured output values. Since there are systematic errors and observational errors, therefore, a specific number of measurements are carried out on said sensor system, such that the influeuce on the calibration caused by the systematic errors and the observational errors can be reduced by increasing the number of measurements to a number greater than the minimal number of measurements, and a more accurate calibration result can be achieved. According to an embodiment of the present invention, where the number of measurements is greater than the minimal number of measurements, the rotational relationship between the first motion sensor and the second motion sensor is obtained as follows: obtaining the residual error in a reference coordinate system of the output values of the first motion sensor and the second motion sensor, and obtaining the rotational relationship by processing the residual error according to an optimized strategy. It is another object of the present invention to provide a calibration apparatus for calibrating the rotational relationship between a first motion sensor and a second motion sensor in a sensor system, the apparatus comprises: a determining means for determining a minimal number of measurements based on the number of dimensions of the coordinat system of said first motion sensor and the number of dimensions of the coordinate system of said second motion sensor; a measuring means for measuring said sensor system for a specific number of measurements to obtain the output values of said first motion sensor and said second motion sensor during each of the measurements, said specific number is not less than said minimal number of measurements; and an acquisition means for obtaining the rotational relationship between said first motion sensor and said second motion sensor based on said measured output values. It is yet another object of the present invention to provide a computer program product for calibrating the rotational relationship between a first motion sensor and a second motion sensor in a sensor system, said computer program comprises: code for determining a minimal number of measurements based on the number of dimensions of the coordinate system of said first motion sensor and the number of dimensions of the coordinate system of the second motion sensor; code for performing a specific number of measurements on said sensor system to obtain the output values of said first motion sensor and said second motion sensor during each of the measurements, said specific number is not less than said minimal number of measurements; and code for obtaining the rotational relationship between said first motion sensor and said second motion sensor based on said measured output ralues. It is still another object of the present invention to provide a motion tracking system for obtaining the motion locus of the system, said system comprises at least a first motion sensor and a second motion sensor, and further comprises a calibration apparatus according to the present invention for calibrating the rotational relationship between the coordinate system of said first motion sensor and the coordinate system of said second motion sensor, and a motion tracking means for obtaining the motion locus of said first motion sensor and said second motion sensor according to the rotational relationship between the coordinate system of said first motion sensor and the coordinate system of said second motion sensor. Other objects and effects of the present invention will be more obvious and understandable with reference to the description of the accompanying figures and the contents of the claims, and by following a more comprehensive understanding of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS The present invention and the advantages thereof will be further elucidated with reference to the exemplary embodiments and the attached drawings, wherein: Fig. 1 is a flow chart showing the method of calibrating the rotational relationship of two motion sensors in a calibration sensor system according to the present invention; Fig. 2 is a schematic diagram showing an apparatus for calibrating the rotational relationship of two motion sensors in a sensor system according to the present invention; and Fig. 3 is a schematic diagram showing a motion tracking system according to the present invention. Like features are denoted by like reference signs throughout the drawings. DETAILED DESCRIPTION OF THE INVENTION Fig. 1 shows a flow chart of the method of calibrating the rotational relationship of two motion sensors in a sensor system according to the present invention. In this embodiment, the two motion sensors in the sensor system are both acceleration sensors having three dimensional coordinate system. Let one of the two acceleration sensor be the first sensor and the another acceleration sensor the second sensor. Firstly, a specific number of measurements of the output values of the two motion sensors in the sensor system is determined (SI lO). According to the basic principles of geometry, if both the coordinate system of the first sensor and the coordinate system of the second sensor are unidimensional coordinate systems, the rotational relationship between those two sensors can be obtained by measuring at least one set of output values of the sensors. If both the coordinate system of the first sensor and the coordinate system of the second sensor are two-dimensional coordinate systems, the rotational relationship between those two sensors can be obtained by measuring at least one sets of output values of the sensors. If both the coordinate system of the first sensor and the coordinate system of the second sensor are three-dimensional coordinate systems, the rotational relationship between those two sensors can be obtained by measuring at least three sets of output values of the sensors. In this embodiment, both the coordinate system of the first sensor and the coordinate system of the second sensor are three-dimensional coordinate systems, thus the minimal number of measurements is 3. In order to reduce the systematic errors and observational errors, the specific number of measurements is greater than 3. Secondly, the acceleration of the first sensor and the acceleration of the second sensor are measured (S 120). The acceleration of the sensor system is rendered unchanged during measurement, that is, the accelerations of the first sensor and the second sensor are measured, respectively, during static or parallelly translation period of the sensor system. It is relatively difficult to keep the sensor system in the parallelly translation state and perform measurement, and by comparision, it is relatively easy to realize the static measurement condition. Further, the residual error in one reference coordinate system of the accelerations of the first sensor and the second sensor during this measurement is obtained (S 130). Under the condition of the static or parallelly translation state of the sensor system, the difference in the same coordinate system between the acceleration of the first sensor and the acceleration of the second sensor equals to zero, the acquirement of the rotational relationship between said coordinate systems can be realized by calculating the residual error in one reference coordinate system between the acceleration of the first sensor and the acceleration of the second sensor. The reference coordinate system can be the coordinate system of the first sensor or the coordinate system of the second sensor, or it can be the world coordinate system as well. In this embodiment, the coordinate system of the first sensor is taken as the reference coordinate system to acquire the residual error between the accelerations of the two sensors. Let ao,, and ai,, be the accelerations of the first sensor and the second sensor, respectively, in the ith measurement, then the residual error in the coordinate system of the first sensor between the acceleration of the first sensor and the acceleration of the second sensor in the ith measurement is ao,, - Rai,, . Then, it is determined whether the number of measurements has reached the specific number (S 140). If the number of measurements is less than the specific number of measurements, the measurements of the accelerations of the first sensor and the second sensor continue. Let the sensor system be in different attitude during each of the measurements. If the measurements are made in the static state of the sensor system, the attitude of the sensor system is changed at the beginning of each new measurement; if the measurements are made in the parallel translation of the sensor system, the attitudes of the sensor system in the measurements are different from one another with respect to the world coordinate system. If the number of measurements has reached the determined number of measurements, then the rotational relationship between the coordinate systems of the first sensor and the second sensor is obtained under the rule of sum of squares of the residual errors being minimum (S 150). Let R be the parameter of the rotational relationship between the first sensor and the second sensor, and this parameter can be expressed by a orthogonal matrix of parameters. While R is a 3 x 3 matrix of 9 variables, however, R is a matrix of only three degrees of freedom since R is a orthogonal matrix of parameters. According to Euler theorem, any rotation can be expressed by three rotational angles called Euler's angles. The rotational relationship between the first sensor and the second sensor can also be expressed by an Euler's angle. According to the X axis rule, a rotation can be expressed by the three Euler's angles (φ, θ, ψ): the first angle of rotation, φ, is the angle of rotation around the Z axis; the second angle of rotation, θ e [0, π], is the angle of rotation around the X axis; and the third angle of rotation, ψ, is the angle of rotation further around the Z axis. R can be expressed by R=BCD, where B, C and D are each a matrix of rotation. Thereby, B, C and D can be expressed as follows by the use of the Euler's angles:

co&φ wnψ 0 — mn ώ oos cii 0 0 0 1

1 0 0 0 cosø sin θ

cos φ sin^'; 0 B — gint^ cas-φ (J 0 0 1

Appropriate Euler's angles (φ, θ, ψ) can be found with the method of searching,

such that the sum of squares of the residual errors of the accelerations of two sensors n in the coordinate system of the first sensor is minimum, i.e. min /J| fl0, ~ ^a\ ι Il •

The acquisition of this Euler's angle makes it possible to transform the coordinate

system of the second sensor to the coordinate system of the first sensor, it can also

transform the coordinate system of the first sensor to the coordinate system of the second sensor, and it can further transform the coordinate system of the first sensor and the coordinate system of the second sensor to the world coordinate system. In summary, the acquisition of the Euler's angle can transform the coordinate system of the first sensor and the coordinate system of the second sensor to the same coordinate system. In the above embodiment, if both the coordinate system of the first sensor and the coordinate system of the second sensor are two dimensional coordinate system, the rotational relationship between the two sensors can also be expressed by the

orthogonal matrix of parameters R. Although R is a 2x2 matrix of four variables,

however, R is a orthogonal matrix of parameters, thus R is a matrix of only one

cosθ sinθ degree of freedom. R can be expressed as R= . If the number of - sin θ cos θ

measurements is greater than one, when the sum of squares of the residual errors of the accelerations of the two sensors in the coordinate system of the first sensor is minimum, a more optimized value of R can be obtained.

In the above embodiment, if both the coordinate system of the first sensor and that of the second sensor are uni-dimensional coordinate systems, the measurement values of the sensors are the projections of their accelerations on the direction of their coordinate systems, and for the same acceleration, the measurement values of the two sensors are in a certain (fixed) proportion with each other. Let R be the coefficient of proportion between the measurement values of the first sensor and the measurement values of the second sensor, where R is a real number. Let A0 be the measured acceleration of the first sensor, and Ai be the measured acceleration of the second sensor. They are expressed in the form of matrix as follows:

A =, where ao,i and ai;1 represent the output values of the two

sensors in the ith measurement. The rotational relationship between the two sensors is obtained under the rule

of the sum of the squares of the residual errors of the accelerations of the two sensors

in the coordinate system of the first sensor being minimum, i.e.

m}n Z^ \\ aoj - Raii ι=l

Let E=A0-RA1 (1), the problem becomes min E E (2),

it can be obtained from (2) that dETE = 0 (3), δR

In the above embodiment, if the number of dimensions of the coordinate system of the first sensor is different from that of the second sensor, then the minimal number of measurements is determined by the small number of dimensions of the coordinate system in those two coordinate systems. Because the manufacturing cost of the motion detecting system can be greatly increased by the inconsistency of the numbers of dimensions of the coordinate systems of two sensors, there fore, in practical applications, the numbers of dimensions of the coordinate systems of the first and the second sensor are equal in most situations. In the above embodiment, if a plurality of motion sensors exist (more than three motion sensors), the coordinate system of one of those motion sensors can be taken as the reference one, and the rotational relationships between each of the other motion sensors and this reference motion sensor can be calibrated, respectively, by the use of the above mentioned method. Thus, the rotational relationships among the coordinate systems of a plurality of motion sensors can be determined. Fig. 2 is a schematic diagram showing an apparatus for calibrating the rotational relationship of two motion sensors in a sensor system according to the present invention. The two motion sensors in this embodiment are both three-dimensional acceleration sensor. The calibration apparatus 200 comprises: a determining means 210 for determining the minimal number of measurements based on the numbers of dimensions of the coordinate system of said first sensor and the coordinate system of said second sensor; a measuring means 220 for measuring said sensor system for a specific number of measurements to obtain the accelerations of said first motion sensor and said second motion sensor during each measurement, said specific number being not less than said minimal number of measurements; and an acquisition means 230 for obtaining the rotational relationship between said first motion sensor and said second motion sensor based on said measured accelerations. In this embodiment, the coordinate system of the first sensor and that of the second sensor are both three-dimensional, and the determining that means determines at least three measurements are needed. The measuring means 220 is used for measuring said sensor system for the specific number of measurements to obtain the accelerations of said first motion sensor and said second motion sensor during each of the measurements, the accelerations are three-dimensional, and said specific number is not less than said minimal number of measurements (3), i.e., the specific number is equal to or greater than 3. When the specific number of measurements is equal to 3, the acquisition means 230 acquires directly the rotational relationship between the first motion sensor and the second motion sensor based on the three sets of accelerations of the sensors measured by the measuring means 220. As stated previously, the rotational relationship between the first sensor and the second sensor can be expressed by a orthogonal matrix of three degrees of freedom, that is, it can be expressed by the Euler's angles (φ,θ,ψ). Those Euler's angles can be calculated based on the measured three sets of accelerations of the first sensor and the second sensor, and the rotational relationship between the first motion sensor and the second motion sensor can thus be obtained, while the coordinate system of the first motion sensor and that of the second motion sensor are transformed into the same coordinate system. When the specific number of measurements is greater than 3, the acquisition means 230 comprises a residual error acquisition means 232 and an optimized processing means 234. The residual error acquisition means 232 is used for acquiring the residual errors of the accelerations of the first sensor and that of the second sensor in one reference coordinate system. The optimized processing means 234 is used to obtain the rotational relationship by processing said residual errors based on an optimized strategy. Said strategy can be: the sum of the squares of the residual errors of the output values of the first motion sensor and the second motion sensor is minimum; the sum of the absolute values of the residual errors of the output values of the first motion sensor and the second motion sensor is minimum; or the weighted sum of the residual errors of the output values of the first motion sensor and the second motion sensor is minimum. The present invention can also be implemented by an appropriately programmed computer, with a computer program installed on the computer, that can be a computer program product for calibrating the rotational relationship between a first motion sensor and a second motion sensor in a sensor system, the computer program comprises: code for determining a minimal number of measurements based on the number of dimensions of the coordinate system of said first motion sensor and the number of dimensions of the coordinate system of said second motion sensor; code for measuring said sensor system for a specific number of measurements to obtain the output values of said first motion sensor and said second motion sensor during each of the measurements, said specific number being not less than said minimal number of measurements; and code for acquiring the rotational relationship between said first motion sensor and said second motion sensor based on said measured output values. This computer program product can be stored on a storage carrier. This portion of program code can be provided to a processor to form a machine, such that the code executed on the processor becomes an apparatus for implementing the above mentioned functionality. Fig. 3 is a schematic diagram showing a motion tracking system according to the present invention. The motion tracking system 300 comprises two motion sensors, a first motion sensor 310 and a second motion sensor 311. The system further comprises: a calibrating means 200, and a motion tracking means 400 for acquiring the motion locus of the motion tracking system 300 based on the rotational relationship between the coordinate system of said first motion sensor and the coordinate system of said second motion sensor. The calibrating means 200 is used to calibrate the rotational relationship between said first motion sensor and said second motion sensor. The calibrating means 200 receives the output values of the first motion sensor 310 and the second motion sensor 320, obtains the rotational relationship of those two motion sensors, and then transmits the obtained rotational relationship to the motion tracking means 400. The motion tracking means 400 is used to obtain the motion locus of said first motion sensor and said second motion sensor based on the rotational relationship between the coordinate system of said first motion sensor and that of said second motion sensor. When the calibrating means 200 has completed calibration, the motion tracking means 400 receives the output values of the first motion sensor 310 and the second motion sensor 320, and transforms the output values of the first motion sensor and the second motion sensor to the same coordinate system for performing motion tracking based on the rotational relationship between the coordinate system of said first motion sensor and the coordinate system of said second motion sensor. It should be noted that the above mentioned embodiment is explanatory only rather than to limit the present invention. Those skilled in the art are capable of designing many alternative embodiments without departing from the scope of the appended claims. In the claims the reference signs in the parentheses shall not be understood as to limit the claims. The word "comprise" does not exclude other means or steps not listed in the claims. The article "a" preceding a component does not exclude the existence of a plurality of such components. The present invention can be implemented by hardware in-cluding several specific components, as well as by an appropriately programmed computer.