WO2009031064A2

WO2009031064A2 - Extracting inertial and gravitational vector components from acceleration measurements

Info

Publication number: WO2009031064A2
Application number: PCT/IB2008/053399
Authority: WO
Inventors: Bin Yin; Victor M. G. Van Acht; Nicolaas Lambert
Original assignee: Koninklijke Philips Electronics N.V.
Priority date: 2007-09-03
Filing date: 2008-08-25
Publication date: 2009-03-12
Also published as: WO2009031064A3; TW200921103A

Abstract

This invention discloses method of extracting inertial Vi and gravitational Vg acceleration vector components from acceleration vector Va = Vi + Vg measured by an accelerometer when the absolute orientation of the accelerometer coordinate system relative to the earth coordinate system varies over time. A sequence of acceleration readout vectors Va [k] from the accelerometer is received. For a given sampling moment k a temporal estimate (I)g [k] is determined for the gravitational acceleration vector Vg, the temporal estimate being determined in the accelerometer coordinate system using the acceleration readout vectors from the accelerometer. A first estimate for the inertial acceleration vector (II)i [k] is set to zero. A measure of the difference between the readout vector Va [k] and the sum of (I)g [k] + (II)i [k] is determined and a measure of the difference in length between the estimate (II)g [k] and the actual gravitational acceleration vector Vg is determined. If the determined measures do not equal substantially to zero the estimates (II)g [k] and (II)i [k] are replaced by a next improved estimate and above mentioned steps are repeated.

Description

Method of extracting inertial and graviational vector components from acceleration vectors measured by an accelerometer

FIELD OF THE INVENTION

The present invention relates to a method and a device for extracting inertial and gravitational acceleration vector components from acceleration vector measured by an accelerometer when the absolute orientation of the accelerometer coordinate system relative to the earth coordinate system varies over time.

BACKGROUND OF THE INVENTION

An activity monitor (AM) based on the measurement of acceleration of a human body is getting more widely used. The applications range from those in medical and healthcare domain where the monitor aids to analyze and evaluate patients' vital body signs such as ECG and EMG signals to improve the quality of diagnosis, to those in consumer lifestyle domain where it enables the estimation of the amount as well as the way of energy expenditure associated with physical activities so that people can know how healthy their lifestyles are and make plans for improvement if necessary. In literature, it has been shown that human's daily physical activities (PAs), including postures, motions and their transitions, can be recognized by pattern classification techniques using acceleration data collected by a body-worn AM device. An AM device consists of, nowadays typically, a single or multiple tri-axial (3D) accelerometers. A DC accelerometer measures both gravitational and inertial accelerations. With a 3D accelerometer, the gravitational components in three axes of the sensor coordinate system will give the orientation (or tilt), relative to the earth- fixed reference coordinate system, of the body part where the sensor is attached. This is depicted graphically in Figure 1 showing on the left side a sensor coordinate system and on the right side an earth- fixed reference coordinate system. If the orientation information is available, inertial accelerations in both the sensor and earth-fixed coordinate systems can be readily obtained. The information about orientation and inertial accelerations will significantly improve the performance of activity recognition algorithms, in terms of the number of postures and motions that can be recognized and the accuracy of the recognition results, especially combined with a prior knowledge of the sensor orientation relative to the relevant anatomical axis of the wearing subject.

For a 3D accelerometer the orientation of a sensor coordinate system relative to the earth-fixed reference coordinate system may be defined as the absolute orientation and that relative to the relevant anatomical axis of its wearing subject as the relative orientation. Once the gravitational components from the axes of a 3D accelerometer have been derived, the absolute orientation of the sensor can be readily expressed in the form of rotation matrix, Euler angles or Quaternion.

Separating gravitational components from inertial ones when a subject stays relatively static can be realized by low-pass filtering or simply averaging the readout from each sensor axis over a certain time interval {David Mizell, Using Gravity to Estimate Accelerometer Orientation, Proceedings of the 7th IEEE ISWC, 2003). In this case, the inertial components are negligible, so a simple averaging usually gives good results. However, as a subject moves, the readout of an accelerometer from its three axes results from both gravitational and inertial accelerations, plus that the absolute orientation of the sensor changes in time because of the subject's movement, meaning both gravitational and inertial components in all three axes' readout are time-varying. Then, the low-pass filtering or averaging method will have difficulties in reliably separating the gravitational and inertial accelerations. Extra inertial sensors can help. The use of a 3D gyroscope and a 3D magnetometer on top of a 3D accelerometer has been disclosed (MTi and MTx User Manual and Technical Documentation, Xsens Technologies B. V., 2006). The magneto and accelerometers are responsible for measuring inertial accelerations as well as slow, usually under 1 Hz, rotational movements, while the gyroscope follows absolute orientation changes during fast rotational movements. The problem with this reference is that the gyroscopes are generally expensive and, more importantly, power-hunger, so they are not ideal for portable and wireless applications, like in a body-worn AM device.

BRIEF DESCRIPTION OF THE INVENTION The object of the present invention is to overcome the above mentioned drawbacks by providing a method and a device where the acceleration readout of an accelerometer alone suffices to estimate the absolute orientation and inertial accelerations and that is also able to follow the variation of the absolute orientation if the subject wearing the device is moving. According to one aspect the present invention relates to a method of determining estimates for inertial V₁ and gravitational V_g acceleration vector components from acceleration vector V_a = V₁ + V_g measured by an accelerometer when the absolute orientation of the accelerometer coordinate system relative to the earth coordinate system varies over time, the method comprising: receiving acceleration readout vectors V_a [k] from the accelerometer, determining, for each respective sampling moment k, a temporal estimate V [k] for the gravitational acceleration vector V , the temporal estimate being determined in the accelerometer coordinate system using the acceleration readout vectors from the accelerometer, defining a target function J=J; +J₂, where Ji is proportional to the difference between the acceleration readout vectors V_a [k] and the sum of a temporal estimate V_g [k] for the gravitational acceleration vector V and an estimate V₁ [k] for the inertial acceleration vector, J₁ oc jψ_a [k] - V₁ [k] - V_g [k])\ , and where J ₂ is proportional to the difference between the actual length of the gravitational vector V_g and the length of the difference of the acceleration readout vector V_a [k] and the estimate V₁ [k] for the inertial acceleration vector,

wherein the estimate for the inertial acceleration V₁ [k] is determined for the inertial acceleration by means of minimizing the target function J Thus, it is possible to separate the inertial and the gravitational acceleration vectors from the acceleration readout data measured by an accelerometer when the subject wearing the accelerometer is moving. Therefore, since only acceleration data are needed a more efficient and cost effective way is provided to extract both orientational and inertial acceleration components from the accelerometer readout data. In one embodiment, the minimizing of J; and J2 is performed by determining an improved estimate V\k + 1] for the subsequent sampling moment k+1, and an improved estimate V_g [k + 1] for the gravitational acceleration vector V_g being determined by V_g [k + l] = V_a[k + I]- V₁ Ik ₊ I]. For each subsequent step k+1 a new temporal estimate V [k + 1] for the gravitational acceleration vector V is determined. Since the estimate V\k + 1] has been improved the inertial and the gravitational components will converge towards the actual values of the inertial and the gravitational vector components and provide highly accurate results.

In one embodiment, the target function J = J₁ + J₂ is defined as:

where £{} represents the mathematical expectation and wi and W2 are nonnegative weighing factors, and wherein the improved estimate V₁ [k + 1] for the inertial acceleration vectors are determined by:

where

with μ as a positive constant that controls the update rate and the stability of the estimation, and where E{ } has been replaced by its instantaneous value.

Accordingly, Ji is a cost function for measuring the distance between

V_g = V_a - V₁ and V_g , and J 2 is a cost function for measuring how much the length of V_g deviates from the actual gravitational acceleration vector V_g . Therefore, these two cost functions define a two step process, namely that in the first step (in Ji) a "learned" V₁ value is provided (namely V₁ ) by using temporarily estimated V_g and in the second step (in Ji) the estimate of the acceleration vector V is corrected to an improved value V if its length does not equal the true length of the gravitational acceleration vector V , namely

⁼ v ^xl ⁺y_g ^{2 + z}l • Accordingly, this method achieves the goal effectively in two steps, each of which has a reduced number of unknowns. The equations for Ji and J 2 are both nonnegative and will equal zero (up to a certain noise level) when the estimates are correct. E{} may be considered to give an averaged value, or mean, of a variable. For example, a variable S[k] takes values S[O], S[I], ..., S[N-I], S[N] at sampling moments 0, 1,...,N-I and N, an approximation of E{S[k]} will be (S[O]+S[l]+...+S[N-l]+S[N])/(N+l). This may be reforming by saying that target function J is minimized in least mean square (LMS) sense.

Minimizing J; forces the estimate of V₁ to approach the vector V₁ (which equals V_a - V ), which should be at the vicinity of V₁ when V is a reasonable temporary estimate of V_g, while minimizing J 2 makes sure that in the meanwhile the length of V_g has a length as close as possible to the length of the gravitational acceleration vector V_g .

In one embodiment, the temporary estimate V_g is given by:

with V_a [k] being a sequence of the acceleration readout data from the 3D-accelerometer in a discrete time domain, k being the sampling moment, and 2K+1 the number of samples around V_a [k] .

The advantage of introducing such a first estimate is that the number of unknown components in V₁ = [x_t ,y, ,z_t ] and V_g = [x_g , y_g ,z _g \ becomes reduced from six unknown components (X₁^z₁ and x_g,y_g,z_g) to only three unknown components, namely x_uyι,Zι.. By defining the estimate for the gravitational V_g acceleration vector in that way, a very reasonable first estimate for the acceleration vector is typically provided because the inertial acceleration tends to average to something close to zero. As an example, if a subject runs with certain acceleration it will after some time slow down (otherwise the subject would approach an infinite speed). This is especially the case during a periodic movement, such as walking. Therefore, by selecting a reasonable time domain, i.e. that covers this time period from where the speed is increasing until the speed decreases, the temporal estimate V_g [k] will be close to the gravitational acceleration vector V . The sampling rate is preferably selected such that both the increase and the decrease around the inertial acceleration V_a [k] are covered (it will sum up to something close to zero).

In one embodiment, the temporary estimate V_g is determined using low-pass filtering method. In one embodiment, the acceleration readout vectors V_a [k] are received in discrete time or in a continuous time domain.

By the term discrete time domain is meant that a sampling domain, where the readouts from the accelerometer are discrete (e.g. every 1/10 of second data are collected), but this may just as well be done in a continuous time domain meaning that the data are continuously collected. In this case, the sum in the equation for the estimate for the gravitational vector V_g [k] would be replaced by an integral, and therefore the complete method including the definition of J and update of unknowns is to be implemented in a continuous time domain.

According to another aspect, the present invention relates to a computer program product for instructing a processing unit to execute the above mentioned method steps when the product is run on a computer or computerized device.

According to another aspect, the present invention relates to a device adapted to estimate inertial V₁ and gravitational V acceleration vector components from acceleration vector V_a = V_t + V measured by the device when the absolute orientation of the device coordinate system relative to the earth coordinate system varies over time, comprising: a receiver for receiving acceleration readout vectors V_a [k] from the device, a processor for determining, for each respective sampling moment k, a temporal estimate V_g [k] for the gravitational acceleration vector V_g , the temporal estimate being determined in the accelerometer coordinate system using the acceleration readout vectors from the accelerometer, wherein the processor is further adapted to determine an estimate V₁ [k] for the inertial acceleration by means of minimizing a target function J=J; +J₂, Ji being proportional to the difference between the acceleration readout vectors V_a [k] and the sum of a temporal estimate

V [k] for the gravitational acceleration vector V and an estimate V₁ [k] for the inertial acceleration vector, J₁ oc ^ [k] - V₁ [k] - V [k])\ , and J₂ being proportional to the difference between the actual length of the gravitational vector V_g and the length of the difference of the acceleration readout vector V_a [k] and the estimate V₁ [k] for the inertial acceleration vector,

In one embodiment, the device is selected from: - a three axial accelerometer, an activity monitor (AM), a rehabilitation monitoring system adapted to record limb trajectories during exercising,

The aspects of the present invention may each be combined with any of the other aspects. These and other aspects of the invention will be apparent from and elucidated with reference to the embodiments described hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the invention will be described, by way of example only, with reference to the drawings, in which

Fig. 1 depicts graphically when a three axial (3D) accelerometer coordinate system in Fig. l(a) is moving relative to an earth fixed coordinate system shown in Fig. l(b), Fig. 2 depicts the relationship of the accelerometer readout vector V_a , the gravitational acceleration vector V_g and the inertial acceleration vector V₁ in the accelerometer coordinate system,

Fig. 3 shows that in principle, any point on the sphere satisfies equation (6), Fig. 4 is a flowchart of a method according to the present invention, and Figs. 5-6 show experimental results depicting orientation estimation with a slowly moving Xsens unit, Figs. 7-8 show experimental results depicting orientation estimation for relatively fast movement.

DESCRIPTION OF EMBODIMENTS

Figure 1 depicts graphically when a three axial (3D) accelerometer coordinate system 101 in Fig. l(a) is moving relative to an earth fixed coordinate system 102 shown in Fig. l(b), where the y-axis is considered to be west and the x-axis is considered to be local magnetic north. The latter is shown as a left-handed system, but in a right handed system the z axis points upwards. As illustrated here, as the subject 100 moves, e.g. the subject is running and the 3D-accelerometer is placed in the sole of the shoe, the accelerometer coordinate system moves with respect to the fixed earth fixed coordinate system. This results in the measured accelerometer vector values, which may be formulized as V_a = [x_a ,y_a ,z_a ], where x_a , y _a , z_a are the acceleration components of the measured accelerometer vector in the accelerometer coordinate system as shown in Fig. l(a). The measured acceleration consists of an inertial acceleration V₁ and gravitational acceleration V_g , i.e.

V_a = V + V_g . (1)

Equation (1) may also be expressed in the following form: x_a = _Xl + x_g ; (2) y_a = y, + y_g ; O) z_β = z, + V (4)

Equations (2)-(4) may be expressed both in discrete (or sampling) domain signal where where x, y, and z may be expressed as a function of sampling moment k, i.e. x_a(k), ya(k) and z_a(k). However, the present invention may just as well be implemented in continuous domain signals. In the following, the discrete domain signal will be used.

Accordingly, the aim of the present invention is to extract the gravitational and the inertial acceleration components from the acceleration readout V_a of the accelerometer such that the inertial acceleration and the absolute orientation of the subject wearing the accelerometer may be followed over time.

In addition to equations (l)-(4) it is known that the gravitational acceleration vector is of a fixed length, namely g. This may also be expressed in the following way:

IK IL = H^{+ y}\^{+ z}l =^l ^ after normalizing the gravitational acceleration vector V with g. The problem of extracting the gravitational and the inertial acceleration components from the acceleration readout V_a of the accelerometer is not one step solvable because the number of unknown components in the vector V_g and V₁ is six, while the number of equations is only four, namely equations (2)-(5).

Figure 2 depicts the relationship of the accelerometer readout vector V_a , the gravitational acceleration vector V_g and the inertial acceleration vector V₁ in the accelerometer coordinate system. The radius of the sphere is one, i.e. after the normalization of the acceleration vector V_g . This means that the acceleration vector V_g will always lie on the sphere, but have various directions as the accelerometer coordinate system moves relative to an earth fixed coordinate system. Figure 3 shows that in principle, any point on the sphere satisfies equation (5), and this leads to arbitrary solutions V^' _g and V ^', , which illustrates the abovementioned problem of having unknowns more than equations. According to the present invention, a tracking algorithm or a method is provided which learns V₁ using temporarily estimated gravitational acceleration vector V_g , and meanwhile corrects the estimate of V_g if its length does not equal one. In this manner, the algorithm achieves the goal effectively in two steps each of which has a reduced number of unknowns.

Figure 4 is a flowchart of a method according to the present invention for extracting inertial V₁ and gravitational V acceleration vector components from acceleration vector V_a = V_t + V measured by a accelerometer when the absolute orientation of the accelerometer coordinate system relative to the earth coordinate system varies over time. In a first step (Sl) 401 a sequence of acceleration readout vectors V_a [k] are generated by the accelerometer, where k is the sampling moment. This means that e.g. every second the acceleration is measured. The sampling moment k may be considered as time, i.e. F_a [lsec] , V_a [2 sec] , V_a [3 sec], or V_a [nr.\], V_a [nr.2], V_a [nr.3] etc. As described earlier, each element of the V_a [k] = [x_a ,y_a ,z_a ] readout vector contains inertial and gravitational components (see equations (2)-(4)), and since it is assumed that the accelerometer coordinate system relative to the earth coordinate system varies over time the inertial and gravitational components are time dependent.

In order to be able to extract the gravitational x_g[k], y_g[k], z_g{k] and the inertial components Xi[k], j^[k], z,[k] shown in equations (2)-(4) from the acceleration vector V_a [k] for a given moment, i.e. given time for V_a [l sec] , V_a [2 sec] , V_a [3 sec] , respectively, a temporal estimate V_g [k] for the gravitational acceleration vector V_g is determined (S2) 403 for a given moment k. In one embodiment, this temporal estimate is determined by the following equation: V_g[k] = -L- ∑V_a[k-m] (6)

ZK + 1 _m=__κ where V_a [k] being a sequence of the acceleration readout data from the 3D-accelerometer in a discrete time domain, k being the sampling moment, and 2K+1 the number of samples around V_a [k] for averaging. As an example, if e.g. the sampling moment &=3sec, i.e. the gravitational and the inertial components are to be extracted for V_a [3 sec] and K=I, then equation (6) becomes 1/(2* 1+1 )( V_a [2] + V_a [3] + V_a [4])= V_g [3] . This is clearly just the average value of the gravitational components over time interval 2-4 seconds. When this time interval is selected reasonable this estimate typically is very good guess because it is unlikely that the inertial forces last very long, but tend to average to something close to zero. As an example, if a person wearing an accelerometer jumps up inertial acceleration vector is initially largest, becomes zero at maximum height (at the instant where the person stops) and becomes negative when the person touches the ground. If the sampling frequency is high enough, the sum of these inertial acceleration vectors over time would be zero, and thus the sum (or the average) of the accelerometer readouts would be equal to the gravitational acceleration. Here it is of course important to select a reasonable time interval to which a reasonable V_g[k] is provided. This depends on the application or the implementation of the accelerometer, e.g. the type of sport. If the application is jump, a reasonable time interval could e.g. be 1-2 seconds.

For said given moment k, a first estimate for the inertial acceleration vector

V₁ [k] is chosen. This value can in principle be arbitrarily chosen, but usually it is initialized to be zero.

Now, a two step process is performed. In the first step (S3) 405 a measure of the difference between the readout vector V_a [k] and the sum of V_g [k] + V₁ [k] is determined. In one embodiment, this is done by defining a target function J = J₁ + J₂ (S4) 407, where:

where Ji gives the measure of the distance between the measured vector V_a [k] and the estimate one V_a [k] , and J2 measures the difference in length between the actual gravitational vector V [k] (of length 1 after the normalization) and the estimated one. E{] represents the mathematical expectation, wi and W2 are nonnegative weighing factors. Both J; and J2 are nonnegative and will equal substantially to zero, i.e. up to a certain noise level, when the estimates are correct. Accordingly, V_g and V₁ can be learned by minimizing the target function J.

The second step in the two step process (S5) 409 is to improve the estimates for the gravitational and the inertial vectors, if J; and J₂ are not substantially equal to zero. The term substantial means up to a certain noise level, or when a pre-defined threshold value (e.g. the difference is equal or lower than 0.001) is reached. In one embodiment, the method solved this in an iterative way. Assuming that neither J; nor J ₂ equal to zero, the following steps are, according to one embodiment, performed:

V_g [k + l] = V_a[k + l] - V, [k +

where

where μ as a positive constant that controls the update rate and the stability of the estimation and El } has been replaced by its instantaneous value. As equation 9 shows an improved estimate is determined for both the inertial and the gravitational acceleration vectors, namely V₁ [k + 1] and V_g [k + 1] for the subsequent moment k+1. Parallel to this a new estimate is determined for the gravitational vector, namely \ - m] in accordance to equation 6 and step (S4) 407 is

repeated. Accordingly, the target function J is minimized in a least mean square sense

(LMS). Minimizing J₁ forces the estimate of V₁ to approach the vector V -V_g , which should be at the vicinity of V₁ when V_g is a reasonable temporary estimate of V_g , while minimizing

J 2 makes sure that in the meanwhile the estimate of V₁ leads to differential vector V₀ - V₁ , which actually is the estimate of V , having a length as close as possible to one. Accordingly, the introduction of the temporary estimate V in J; effectively reduces the number of unknowns to 3, namely only the inertial acceleration vector components. With w;=0, the searching would be drifty and might end up at a random solution that is incorrect. Since the equation 7 does not guarantee a unit length for V_g , the existence of J2 provides an opportunity to correct V₁ if it would lead to V_g , the estimate of V_g , that has a length deviating from 1. A solution compromising J; and J 2 will be iteratively reached, and, depending on the magnitude of the factors wi and W2, Ji and J 2 will contribute differently to the estimation process. For instance, wi can be set more heavily if V_g is considered to be a good estimate. A special case is that W2 is set to zero, implying that the temporary estimate V_g is good enough and no further correction is needed.

Usually, one can get a reasonably good V_g simply by low-pass filtering or averaging V_a , in particular, in the case of cyclic movements, like walking or cycling where V_g varies around its nominal value, or more generally, when the gravitational component V_g varies slowly compared to the inertial component V₁ . Although there are exceptions, normal body movements in daily activities mostly fall into those categories.

Since only acceleration data are needed, the proposed method, therefore, provides a useful and more efficient and cost-effective way to extract both orientational and inertial acceleration information compared to the prior art solutions.

Experimental results:

An Xsens inertial measurement unit, consisting of 3D gyroscopes, 3D accelerometers and 3D magnetometers, is used to obtain 3-dimensional acceleration data as well as reference gravitational components, see MTi and MTx User Manual and Technical Documentation, Xsens Technologies B.V., 2006, hereby incorporated by reference. The sampling frequency is 100Hz. The proposed algorithm, or the method discussed previously, runs on the acceleration data and the extracted gravitational components \x_g ,y_g ,z_g ) are compared with the reference ones.

The results are shown in figure 5 - 8. These figures show the orientation estimation with an Xsens unit moving slowly and relatively fast, respectively (with W₁ = 0 or W₁ > 0), where the lengths of the reference V_g and the estimated one are shown in the bottom right figure.

A slow and smooth movement is made with the Xsens unit in the first experiment. The reference ^g ' ^g ' ^g components calculated by the Xsens, and the estimated ones with ^{Wl ~} and with ^{Wl >} , are shown in Figure 5 and 6, respectively. A relatively faster movement is made in the second experiment shown in Figure 7 and 8, respectively. The reference and estimated components with ^{Wl ~} and with ^Wl are shown in Figure 7 and 8, respectively. In both experiments, without the constraint set by ^l (i.e., ^w! ), a random and incorrect solution is obtained although the estimated ^g satisfies

H ^gH , while with ^w! the estimated gravitational components resemble the reference ones very well. The results also demonstrate that the corrective effect due to the presence of

² does bring improvement, especially in the case of fast movements.

Figure 9 shows a device 900 according to the present invention comprising a receiver (R) 901 and a processor (P) 902. The receiver for receiving a sequence of acceleration readout vectors V_a [k] from the device, and the processor adapted, for a given sampling moment k perform the method steps as disclosed in Fig. 4. The device may be a three axial accelerometer, an activity monitor (AM), a rehabilitation monitoring system adapted to record limb trajectories during exercising, a vital body parameter measuring device, a joystick, and the like. The device may further comprise a memory having stored therein an algorithm to perform the method steps as disclosed in Fig. 4. The device may also be a computer, e.g. LAP, laptop, PC computer, mobile telephone and the like, which is capable of receiving the acceleration readout vectors V_a [k] measured by an accelerometer and thus process the data external, such that the gravitational and the inertial acceleration components are extracted from the acceleration readout vectors V_a [k] .

Certain specific details of the disclosed embodiment are set forth for purposes of explanation rather than limitation, so as to provide a clear and thorough understanding of the present invention. However, it should be understood by those skilled in this art, that the present invention might be practiced in other embodiments that do not conform exactly to the details set forth herein, without departing significantly from the spirit and scope of this disclosure. Further, in this context, and for the purposes of brevity and clarity, detailed descriptions of well-known apparatuses, circuits and methodologies have been omitted so as to avoid unnecessary detail and possible confusion.

Reference signs are included in the claims, however the inclusion of the reference signs is only for clarity reasons and should not be construed as limiting the scope of the claims.

Claims

CLAIMS:

1. A method of determining estimates for inertial V₁ and gravitational V_g acceleration vector components from acceleration vector V_a = V₁ + V_g measured by an accelerometer when the absolute orientation of the accelerometer coordinate system (101) relative to the earth coordinate system (102) varies over time, the method comprising: - receiving acceleration readout vectors V_a [k] from the accelerometer (401), determining, for each respective sampling moment k, a temporal estimate V [k] for the gravitational acceleration vector V (403), the temporal estimate being determined in the accelerometer coordinate system using the acceleration readout vectors from the accelerometer, - defining a target function J=J; +J₂ (407), where Ji is proportional to the difference between the acceleration readout vectors V_a [k] and the sum of a temporal estimate

V_g [k] for the gravitational acceleration vector V_g and an estimate V₁ [k] for the inertial acceleration vector, J₁ oc jψ_a [k] - V₁ [k] - V_g [k]ϊ , and where J ₂ is proportional to the difference between the actual length of the gravitational vector V_g and the length of the difference of the acceleration readout vector V_a [k] and the estimate V₁ [k] for the inertial acceleration vector, J₂ oc l(v_a [k] - V₁ [A - ψ_g

, wherein the estimate for the inertial acceleration V₁ [k] is determined for the inertial acceleration by means of minimizing the target function J

2. A method according to claim 1, wherein the minimizing of J; and J 2 is performed by determining an improved estimate V₁ [k + 1] for the subsequent sampling moment k+1, and an improved estimate V [k + 1] for the gravitational acceleration vector V (409) being determined by V [k + l] = V_a [k + l]- V₁ [k + l].

3. A method according to claim 1, wherein the target function J = J₁ + J₂ is defined as:

V, [k ₊ l] = V_t [k]₊ μ[-£r[k]

where

^ dJ

BV₁ ^[k] = -*r fr.-^fl-VA .-^fte with μ as a positive constant that controls the update rate and the stability of the estimation.

4. A method according to claim 1 , wherein the temporary estimate V is given by:

with V_a [k] being a sequence of the acceleration readout data from the accelerometer in a discrete time domain, k being the sampling moment, and 2K+1 the number of samples around V_a [k] .

5. A method according to claim 1, wherein the temporary estimate V_g is determined using low-pass filtering method.

6. A method according to claim 1, wherein the acceleration readout vectors V_a [k] are received in discrete time or in a continuous time domain.

7. A computer program product for instructing a processing unit to execute the method steps of claim 1 when the product is run on a computer.

8. A device (900) adapted to estimate inertial V₁ and gravitational V_g acceleration vector components from acceleration vector V_a = V_t + V_g measured by the device when the absolute orientation of the device coordinate system relative to the earth coordinate system varies over time, comprising: a receiver (902) for receiving acceleration readout vectors V_a [k] from the device, a processor (903) for determining, for each respective sampling moment k, a temporal estimate V_g [k] for the gravitational acceleration vector V_g , the temporal estimate being determined in the accelerometer coordinate system using the acceleration readout vectors from the accelerometer, wherein the processor (903) is further adapted to determine an estimate V₁ [k] for the inertial acceleration by means of minimizing a target function J=J; +J₂, Ji being proportional to the difference between the acceleration readout vectors V_a [k] and the sum of a temporal estimate V [k] for the gravitational acceleration vector V and an estimate V₁ [k] for the inertial acceleration vector, J₁ oc jψ_a [k] - V₁ [k] - V_g [k])\ , and J ₂ being proportional to the difference between the actual length of the gravitational vector V and the length of the difference of the acceleration readout vector V_a [k] and the estimate V₁ [k] for the inertial acceleration vector,

J₂ ~ \k [*]- 1 [k}]\ -\_Vg [k\\ .

9. A device according to claim 8, wherein the device is selected from: a three axial accelerometer, an activity monitor (AM), a rehabilitation monitoring system adapted to record limb trajectories during exercising, a vital body parameter measuring device, a joystick.