Design, geometry evaluation, and calibration of a gyroscope-free inertial measurement unit

Sensors and Actuators A 162 (2010) 379–387

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Design, geometry evaluation, and calibration of a gyroscope-free inertial measurement unit Patrick Schopp a,∗ , Lasse Klingbeil b , Christian Peters a , Yiannos Manoli a,b a Department of Microsystems Engineering - IMTEK, University of Freiburg, Georges-Koehler-Allee 102, 79110 Freiburg, Germany b HSG-IMIT – Institute of Micromachining and Information Technology, Wilhelm-Schickard-Str. 10, 78052 Villingen-Schwenningen, Germany

a r t i c l e

i n f o

Article history: Received 9 October 2009 Received in revised form 22 December 2009 Accepted 4 January 2010 Available online 25 January 2010 Keywords: Gyroscope-free inertial measurement unit Geometry evaluation Calibration Unscented Kalman filter

a b s t r a c t This work reports on a gyroscope-free inertial measurement unit (GF-IMU) that only comprises linear accelerometers in order to directly measure the transversal acceleration as well as the angular acceleration and velocity. The accuracy of the calculated body motion depends on the geometrical setup and the variance of the employed accelerometers. Therefore, an analytical evaluation of the sensor placement for the propagation of the sensor variance is provided here. The positions and orientations of the sensors within the body frame have to be known precisely in order to calculate the exact inertial movement of the body. With the calibration scheme presented here it is possible to identify these parameters completely even without any previous knowledge. Furthermore, the variance of the parameters can be determined, which can be used to evaluate the performance of the calibration. Using only acceleration sensors it is not possible to determine the direction of a rotation. To overcome this drawback, an Unscented Kalman filter (UKF) is applied to merge the information of the angular acceleration and the angular rate and thus robustly estimate the sign of the angular velocity. Measurements on a 3D-rotation table were carried out to exemplarily demonstrate the accuracy improvements after the calibration. Thereby, the RMS error of the angular rate was reduced by a factor of 2.8. © 2010 Elsevier B.V. All rights reserved.

1. Introduction An inertial measurement unit (IMU) is a sensor unit that measures the relative movement of body. A conventional IMU comprises three accelerometers, measuring the transversal acceleration, and three gyroscopes, measuring the angular velocity. Such sensor systems are widely used in localization and navigation applications but also for control purposes as for example the vehicle dynamic control (VDC) in passenger vehicles [1,2]. However, gyroscopes are often afflicted with certain drawbacks in their characteristics, i.e. high bias drift, low shock resistance, bad durability [3,4], whereas accelerometers show a better performance. Especially physical reliability is a major demand for automotive applications. A gyroscope-free IMU (GF-IMU) senses the complete relative body movement only by accelerometers. Thus, the mentioned disadvantages of gyroscopes can be eliminated. Furthermore, since only accelerometers are utilized the angular acceleration of the vehicle can be measured directly

∗ Corresponding author. Tel.: +49 761 203 7458; fax: +49 761 203 7592. E-mail address: [email protected] (P. Schopp). 0924-4247/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2010.01.019

without differentiation. This is particularly valuable for rollover detection in crash situations [5]. In modern passenger vehicles many safety and control applications utilize inertial sensors. Examples of these systems are: airbag control system, antitheft system, anti-lock braking system (ABS), or vehicle navigation systems. These subsystems mostly work independently. Thus, each subsystem incorporates its own sensors. Luxury class vehicles contain up to 15 axes of inertial sensors [2]. As an example, Fig. 1 illustrates the distributed sensor network of accelerometers used by the airbag control system [1,2]. By reusing these already available sensors one common GFIMU can provide the desired information about the relative body movement for various subsystems, which leads to a large cost reduction. This paper is organized as follows. Firstly, the kinematic fundamentals are presented in Section 2 in order to provide a better understanding of the previous research carried out in the GF-IMU approach, which is then discussed in Section 3. In this context we also state how the contents of this work account for the state of the art. Next, the placement of the GF-IMU accelerometers is analyzed in Section 5. A calibration scheme to determine all of the parameters of the GF-IMU is presented in Section 6. The Unscented Kalman filter that is applied as data processing algorithm will be explained

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Fig. 1. Positions of the accelerometers (circles) used by an airbag system.

Fig. 2. Point P in the body frame OBody relative to the navigation frame ONavigation .

in Section 7. Accuracy measurements of a prototype are discussed in Section 8 before the paper is concluded in Section 9. 2. Problem formulation Considering a point P at a certain position r fixed in a rigid body OBody as shown in Fig. 2, its acceleration aP can be expressed by: aP = aB + ω˙ B × r + ωB × (ωB × r),

(1)

where the relative movement of the body OBody with respect to the navigation frame ONavigation is described by the transversal acceleration of the body aB , the angular velocity ωB , and the angular acceleration ω˙ B . The terms of the equation are:

aB ω˙ B × r ωB × (ωB × r)

body acceleration, as described above tangential acceleration centripetal acceleration

It can be seen from (1) that the acceleration occurring at distinct positions r of a rigid body are different if the body undergoes a rotation or an angular acceleration. Therefore, the main concept of a GF-IMU is to calculate the relative body movement, i.e. the transversal and angular acceleration as well as the angular velocity, by measurements of the acceleration at different points of the body. Since (1) cannot be directly solved for those terms all GFIMU approaches incorporate several acceleration measurements and thus several acceleration sensors into a set of equations, which then can be solved for the variables of the inertial body motion. 3. Previous work The minimum number of accelerometer axes to determine the relative body movement is six [6–14]. However, by using six accelerometers only the transversal and the angular acceleration of the body can be determined. The angular velocity cannot be determined directly and has to be calculated via an integration step. Furthermore, the applied differential equations involve the angular velocity in order to compute the angular acceleration. Thus, errors accumulate. Padgaonkar et al. [13] faced this problem by incorporating nine accelerometers in total. By the redundancy of nine

sensors, the angular acceleration can be determined directly without the knowledge of the angular velocity since those terms can be eliminated from the equations. Another way to directly determine the angular acceleration with only six sensors is the arrangement of the accelerometers in a special cube configuration first presented by J.-H. Chen et al. [7]. With this special arrangement the terms depending upon the angular velocity are always zero and thus the angular acceleration can be obtained independently of the angular velocity. Further investigations of the cube configuration were made by Tan and Park [14]. Mostov et al. investigated the suitability of the cube configuration for automotive applications [12]. T. Chen et al. proposed a coplanar setup of a GF-IMU detecting the angular and transversal acceleration in [8–10]. For such a coplanar configuration the minimum number of sensors is nine [10]. A medical application of a six sensor GF-IMU was developed by Ang et al. They applied three dual-axes accelerometers to sense tremor in hand-held microsurgical instruments [6]. However, all GF-IMU concepts that only measure the angular and transversal acceleration have the drawback that the angular velocity has to calculated via an integration step which naturally leads to drift errors over time. The minimum number of accelerometers necessary to directly calculate the angular velocity for a 6D motion of a body is twelve [15]. In [16] we presented a simple matrix equation to calculate the entire relative body movement using twelve sensors and also an implementation of such a GF-IMU. However, it is not possible to determine the direction of the angular rotation using only accelerometers since the angular velocity is present in its quadratic form within the system equations that derive from (1). Therefore, the sign of the angular velocity is lost and the determined angular velocity is not unique. Parsa et al. [17,18] solve this problem by choosing the correct solution, and therefore the sign of the angular velocity, by the integration of the angular acceleration. In contrast to this, we apply an Unscented Kalman filter to determine the correct angular velocity [19,20]. This non-linear, recursive filtering technique provides great robustness and noise cancellation. At the same time its predictor–corrector structure allows to integrate more useful sensor information as for example the position and velocity measurements of a global positioning system (GPS) receiver. In [19] we utilized the GF-IMU in conjunction with a GPS receiver to determine the position and orientation of a moving vehicle. Using twelve accelerometers enables a nearly arbitrary placement of the accelerometers. However, the positions and orientations of the accelerometers have to be known in order to calculate the relative body movement correctly. In [20] we proposed a calibration scheme to determine all of these parameters entirely without any previous knowledge, which makes an arbitrary placement of the accelerometers feasible. Here, a modified version of the calibration scheme is developed. With this modified version it is possible to determine how well the calibration was performed without further measurements. Furthermore, an analytical geometry evaluation of an exemplary sensor configuration is provided. General conclusions about the placement of the accelerometers are derived from this analysis. It is also shown how to determine the noise of any sensor configuration without the use of simulations or measurements.

4. System equations In order to calculate the acceleration aS measured by a sensor that is attached at position r within a body the sensitivity axis s and the sensor’s metrological signal offset a0 must be incorporated into equation (1): aS = sT (aB + ω˙ B × r + ωB × (ωB × r)) + a0 .

(2)

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381

There, s is a three-dimensional vector that represents the accelerometers sensitivity within the body frame. By extracting the terms that contain the information about the body movement (2) can be rewritten into a scalar product of two vectors:

⎤T ⎡ aB,x ⎤ aB,y ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ aB,z ⎥ sz ⎢ ⎥ ⎢ ⎢ ⎢ s r − s r ⎥ ⎢ ω˙ B,x ⎥ ⎥ ⎢ z y yz ⎥ ⎢ ⎢ s r − s r ⎥ ⎢ ω˙ B,y ⎥ ⎥ ⎢ xz z x ⎥ ⎢ ⎢ s r − s r ⎥ ⎢ ω˙ B,z ⎥ ⎥ ⎢ yx xy ⎥ ⎢ ⎥ + a0 =⎢ ⎥ 2 ⎥ ωB,x ⎢ −(sy ry + sz rz ) ⎥ ⎢ ⎢ ⎥ ⎢ −(s r + s r ) ⎥ ⎢ ω2 ⎥ ⎢ xx z z ⎥ ⎢ B,y ⎥ ⎢ −(s r + s r ) ⎥ ⎢ 2 ⎥ ⎢ x x y y ⎥ ⎢ ωB,z ⎥ ⎢ s r +s r ⎥ ⎢ ⎢ x y y x ⎥ ⎢ ωB,x ωB,y ⎥ ⎥ ⎣ s r +s r ⎦ ⎣ x z z x ωB,x ωB,z ⎦ ⎡

aS

sx sy

sy rz + sz ry

(3)

ωB,y ωB,z

= c z + a0 . Row vector c contains the time-invariant parameters, i.e. position r and sensitive axis s of the sensor, whereas the column vector z holds the information about the body motion. Remembering the GF-IMU concept, (2) must be solved for the variables that represent the inertial body movement using a set of equations. The body motion consists of three 3D vector quantities (aB , ωB , ω˙ B ). Nevertheless, with a set of only nine equations it is not possible to find a unique solution for ωB since (2) is linear in aB and ω˙ B but ωB is present in a quadratic form [15]. However, by incorporating twelve sensors it is possible to directly compute the quadratic terms of ω as well as aB and ω˙ B . In doing so, the system becomes linear:

⎡

y

⎤

⎡

⎤

⎡

⎤

aS1 c S1 a0,S1 . . . = ⎣ .. ⎦ = ⎣ .. ⎦ z + ⎣ .. ⎦ aS12 c S12 a0,S12 = A z + a0 .

(4)

The 12 × 12 coefficient matrix A is composed of the row vectors for each of the 12 sensor equations. Thus, A only contains the timeinvariant parameters about the positions and the sensitive axes of the sensors. By inverting A it is possible to calculate the relative body movement, held by vector z, for a given measurement vector y applying: z = A−1 (y − a0 ) .

(5)

This leads to the only requirement for the placement of the sensors: matrix A must be invertible. This is the case, if the row vectors c S1 , . . . , c S12 of A are linearly independent. Therefore, a valid sensor configuration can be found by applying a simple rule of thumb: If the sensors are clustered into four sensor nodes of three sensors each, then the sensitive axes within one sensor node have to be linearly independent and the positions of the sensor nodes may not be coplanar. For 3D accelerometers the first requirement is usually fulfilled since these are measuring the acceleration in x, y, and zdirection. Fig. 3 shows two valid sensor configurations. Both have a symmetrical ground plane. Configuration A features three coplanar sensor nodes, whereas configuration B has two sensor nodes above the ground plane. 5. Geometry evaluation If the parameters of a valid GF-IMU are precisely known, the relative body movement can be calculated without any error. However, the noise of the accelerometers induces noise into the calculation of the motion variables. The influence of the sensor noise

Fig. 3. Two possible sensor configurations (A and B) with (a) one sensor node and (b) two sensor nodes above the ground plane at height h.

on the variance of these results depends on the geometrical setup of the GF-IMU. For a general linear system f = Mx the covariance f of the results can be calculated by applying the law of error propagation: f = M x M T , where x is the covariance matrix of the random variable vector x. Thus, in order to evaluate the variance of the inertial body movement for any arbitrary GF-IMU sensor configuration this results in: T

z = A−1 y A−1 .

(6)

The variance of the body motion lies on the main diagonal of the covariance z . Since the errors of the accelerometers are assumed to be uncorrelated only the main diagonal of the covariance y is filled with the variances of the single sensors. By evaluating (6), the noise of the determined relative body movement can be calculated for every valid GF-IMU configuration without the use of simulations or measurements from an actual measurement setup.

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Table 1 The analytical variances of relative body movement for both configurations A and B.



z2

A diag(Conf. ) z

B diag Conf. z

 2 (aB,x )

1 2  2 S 1 2  2 S 1 2  2 S

1 2  2 S 1 2  2 S 1 2  2 S

 2 (aB,y )  2 (aB,z )  2 (ω˙ B,x )  2 (ω˙ B,y )  2 (ω˙ B,z ) 2  2 (ωB,x ) 2  2 (ωB,y ) 2  2 (ωB,z )

 2 (ωB,x ωB,y )  2 (ωB,x ωB,z )  2 (ωB,y ωB,z )

3 2  8d1 8d2 1 2d2

1 h2

+

h2

S

S2

1 2  2d1 2  2d1 2d2 1 2d2

2  S2 1

+

2  S + 12 S2 h 2 1 +

1 h2

+

h2

S2

1 2  8d3 8d2

S

2  S2 1

+

1 h2

+

h2

S

1 2  8d1 8d2 1 4d2

1 4h2

+

1 4h2

2  S2 S

S2

1 2  4d1 2  4d1 4d2 1 4d2

+



2  S + 12 S2 4h 2 1 +

1 4h2

+

4h2

S2

1 2  8d1 8d2

+

1 4h2

+

1 4h2

S

2  S2 S

However, this is not an analytical investigation of the relationship between noise propagation and GF-IMU geometry and is only valid for a certain completely defined configuration. A general relation cannot be derived. For this purpose matrix A must be inverted with the parameters r S and sS remaining arbitrary. However, not all combinations of the sensor positions and sensitive axes result in an invertible matrix A. Furthermore, the large number of 72 arbitrary geometry parameters makes it difficult to derive an expression from which the relationship of output variance and geometry can easily be understood. Therefore, constraints to the sensor setup are introduced for this analysis in order to reduce the parameters: • The sensors are grouped into four sensor nodes that include three sensitive axes each as stated in the given rule of thumb. Within one sensor node the sensitive axes are aligned along the x, y, and z axis of the body frame, respectively. • The sensor nodes are placed symmetrically around the body’s origin within the x, y plane. They form a quadratic ground plane of the body. • Since not all sensor nodes may be coplanar exactly one or two sensor nodes must be located out of the ground plane. Following these restrictions to the sensor setup two possible sensor configurations remain, which are parameterized by the distance d and the height h (Fig. 3). Either one (configuration A) or two (configuration B) senor nodes are placed out of the ground plane at height h. The distance d parameterizes the dimension of the GF-IMU in the x–y plane. For both configurations matrix A is constructed, analytically inverted, and applied to (6) assuming that all sensors have the same variance S2 . The resulting analytical variances of both configurations are summarized in Table 1. The variance of the body acceleration aB is the same for both configurations and is independent of the geometry. However, the variance of the rotational movements of configuration B is always smaller than those of configuration A. Thus, a configuration with two sensor nodes located out of the ground plane always has lower variance in the rotational results. For arbitrary parameters d and h the variance of the angular movements follows a reciprocal quadratic function. For a fixed length d a certain value of h has to be chosen in order to receive a sufficiently low value for the variance. Any further improvements on the variance requires a large increase of h. The same is true for a fixed h and an arbitrary d.

6. Calibration The GF-IMU algorithm calculates the relative body movement based on the parameters held by matrix A and the offset vector a0 . Hence, these parameters have to be known precisely in order to compute exact results of the inertial body motion. Inaccuracies as sensitivity errors, offset, and cross coupling effects occur within the determined body motion if the parameters of the GF-IMU are incorrect. Fig. 4 shows a typical cross coupling effect within the determined angular acceleration for a rotation around the body’s z-axis (Fig. 4a). For this evaluation, the GF-IMU parameters are extracted from the construction plan of the prototype and a simple offset adjustment of the sensor signals as standard calibration procedure is applied. Fig. 4b illustrates that the imposed angular velocity ωB,z is misinterpreted as an angular acceleration ω˙ B,x due to incorrect orientation parameters although ωB,x remains constant. Using the calibration scheme presented in this paper, the entire parameters of the setup can be identified and therefore errors within the relative body movements are minimized (Fig. 4c). Furthermore, no initial values have to be known beforehand, which makes an arbitrary placement of the sensors feasible. Measurements from the GF-IMU and reference motion data form the basis of the parameter estimation. In order to store all parameters in one common matrix (4) can be reformulated to:

⎡

⎤

a0,S1  .. ⎦ z . . 1 a0,S12

c S1 .. ⎣ y= . c S12

(7)

This is a linear set of equations, where the parameters of the ith sensor equation is stored in the ith row vector of the common parameter matrix. By incorporating n measurements y and their corresponding reference motion data system z an overdetermined

is constructed. For one row vector c Si a0,Si this yields to:



yi,1

...

yi,n



= c Si



= c Si

Yi

a0,Si



z1 . . . zn

a0,Si Z.

1

...

1

, (8)



This equation can be solved for the parameter vector c Si a0,Si by calculating the Moore–Penrose pseudoinverse of the matrix that holds the reference motion data:



c Si

a0,Si

= Yi Z + .

(9)

By evaluating (9) for every ith row vector the entire parameters of the GF-IMU are determined. The Moore–Penrose pseudoinverse can be calculated accurately using a singular value decomposition. This has to be done only once, since it does not change for the different parameter vectors. The Gauss–Markov theorem states that for a linear model a best linear unbiased estimator of its parameters is given by the Least Squares estimator if the errors of the observations, i.e. the measurements, are uncorrelated, their expectation of the errors is zero, and all errors have the same variance [21]. Since these requirements are all true for the GF-IMU system, the method of Least Squares is a optimal estimator for the parameters of the applied model. It is also possible to estimate the variance of the determined parameters. For one row vector [c i a0,i ] an estimator for its covariance is given by [22]: Cov([ c Si



2 ZZ T a0,Si ]) = Si

−1

,

(10)

2 is the variance of the ith accelerometer. The variances where Si of the parameters may be used for hypothesis test and confidence

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Fig. 5. GF-IMU prototype on the 3D rotation table.

Fig. 4. Cross coupling effect due to inaccurate parameters. The imposed reference rotation ωB,z is shown in (a). The measured angular acceleration ω˙ B,x is shown for a simple offset adjustment of the sensors and parameters extracted from the construction plan (b) as well as for parameters determined by calibration (c).

intervals. They also indicate the performance of the carried out calibration. In our experiments the values of the variances estimated for c S are in the order of 10−8 whereas the variances estimated for a0,S is in the order of 10−6 . Thus, the metrological signal offset of the sensors is estimated with a lower accuracy than the placement of the sensors. The reference motion data and the measurements are collected during a calibration run on a 3D rotation table (Fig. 5). The measurement time was 120 s with a sampling rate of 250 Hz. In order to construct a well-defined overdetermined set of reference states and GF-IMU measurements an overlay of all three rotational degrees of freedom of the table was used. The resulting motion imposed on the GF-IMU achieves a very good coverage of the orientation state space, which can be seen in Fig. 6a. To sample many different combinations of angular velocities distinct angular rates were set for the three rotational degrees of the table. The resulting angular velocities of the GF-IMU are shown in Fig. 6b for 20 s of the complete calibration run. After determining the parameters of the GF-IMU prototype the measured standard deviations of the relative body movement are compared to the standard deviations numerically evaluated using (6). Fig. 7 shows the resulting standard deviations z for all entries of vector z. The small differences between the numerically and experimentally evaluated standard deviations indicate that the calibration performed well for the GF-IMU. The remaining differences derive from the imperfection of the applied sensor model (2). Composite non-linearity errors, as non-linear sensor output cross-axis sensitivity or hysteresis are not considered in this model. Therefore,

Fig. 6. Reference motion used for calibration. An overlay of all rotational degrees of freedom of the rotation table was used. (a) The trajectory of the normal vector of the GF-IMU around the center of rotation (star). The angular velocities imposed on the GF-IMU (b) are displayed for 20 s.

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following equation set: aB,k = aB,k−1 + vaB , ωB,k = ωB,k−1 + (ω˙ B,k−1 + vω˙ B )T,

(11)

ω˙ B,k = (ω˙ B,k−1 + vω˙ B ).

Fig. 7. Comparison of the standard deviations of the relative body motion evaluated numerically with the parameters determined by calibration and experimentally by measurements of the GF-IMU prototype.

the calibration scheme is not able to compensate for these errors. The incorporation of sensor imperfections into the sensor model is one topic of further investigations. 7. Unscented Kalman filter Vector z in (5) incorporates the angular velocity in a quadratic form. Thus, the sign of ωB , which represents the direction of the rotation, is lost. To overcome this drawback, an Unscented Kalman filter (UKF) is applied to merge the information of the angular acceleration and the angular rate and thus robustly estimate the sign of the body’s rotation. Furthermore, the UKF provides an effective noise reduction on the GF-IMU results. As it will be explained later, a non-linear filtering structure such as the UKF is necessary for the GF-IMU. The UKF does not approximate the non-linearity of the system. Instead, the probability distribution, that needs to be propagated through a non-linear model, is approximated. Thus, the model can be utilized without any further simplification. This accounts for a good convergence at highly non-linear systems. In contrast to this, other kinds of non-linear Kalman filters such as the Extended Kalman filter (EKF) linearize the applied models. A comprehensive explanation of the UKF can be found in [23,24]. The UKF algorithm has a fixed structure. Thus, the focus of this section lies on the models employed for the Kalman filter, which are different for every application. The recursive predictor–corrector structure of the UKF makes it possible to incorporate prior system knowledge into the filtering process. Therefore, a prediction x− of the system state x is computed based on the previous system state applying a process model. Subsequently, the prediction is corrected by the actual observation y to an estimate xˆ of the system state. The same is done in parallel with the covariance P of the system state. Thus, using the UKF and any other Kalman filter it is also possible to estimate the uncertainty of the system state. A schematic of the UKF algorithm is provided in Fig. 8. The process model predicts the state ahead from the previous time-step k − 1 to time-step k and is of the form: xk = f (xk−1 , vk−1 ). The uncertainty of the process stated by f is expressed by the process noise v. For the GF-IMU process model it was assumed that between two time-steps the angular and transversal acceleration of the body remains constant. The process model is defined by the

Here, T is the time between two time-steps and is the sampling rate of the GF-IMU. The assumption of constant acceleration is driven by time invariant Gaussian noise. This accounts for the fact that there actually can be a change of acceleration during time T . Hence, the process noise vector v must hold both the uncertainty vaB for a change in the transversal acceleration and the uncertainty vω˙ B for a change of the angular acceleration. Since the values of v can be chosen freely, it can be considered as a tuning parameter of the UKF. If large uncertainties are assumed than the filter algorithm will put more trust in the measurements. For small uncertainties the algorithm will rely more on the process model. The entries of vaB were set to 0.1 m/s2 and those of vω˙ B to 0.5◦ /s2 for each axis in the evaluations presented here, since these acceleration values are assumed to be the standard deviation of the body acceleration during one time step. The prediction x− of the state vector is corrected by the observation y at time-step k by comparing both with respect to their variances. However, in order to have comparable information the predicted state has to be transformed to a predicted measurement. This is done by applying the observation model h. It is of the form: y k = h(xk , nk ), where nk is the observation noise. For the GF-IMU (5) can be applied to calculate the measurements of the accelerometers for a given relative body movement. Therefore, vector z has to be constructed beforehand. Since z incorporates the angular velocity in a quadratic form a non-linear operation has to be performed to calculate these entries. Hence, a non-linear filtering is necessary for the GF-IMU. To incorporate the noise of the accelerometers the observation noise nk is added to the result of (5) which is computed as: y k = A z + a0 + n,

(12)

where n is the standard deviation of the 12 accelerometers held by a 12 × 1 vector and is assumed to be time invariant. The interaction of process and observation model provides that the angular velocity is updated both by the quadratic terms of ωB within z, resulting from the centrifugal acceleration, and the same at time by the integration of ω˙ B , which derives from the tangential acceleration. Thereby, drift errors of ωB are prevented since the centrifugal acceleration directly determines the amplitude of the rotation. Concurrently, the tangential acceleration provides a robust estimation of the sign of ωB by the integration of ω˙ B .

Fig. 8. Unscented Kalman filter framework.

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Table 2 Sensor configuration of the GF-IMU determined by calibration. The sensitive axis and the position are relative to the body frame. Sensitive axis x

Fig. 9. Photography of the GF-IMU prototype. The twelve accelerometers are located on four satellite sensor boards. Each sensor board incorporates a sensor sensitive in the x, y, and z-axis.

8. Experimental results In order to demonstrate the feasibility of the GF-IMU concept a prototype was developed using single axis accelerometers (Fig. 9). The sensors are placed on satellite sensor boards that are connected to a microcontroller mainboard for data readout. The accelerom√ eters feature a noise density of 350 ␮g/ Hz (x and y-axis) and √ 500 ␮g/ Hz (z-axis) [25] and are limited to a bandwidth of 50 Hz. Both types have a non-linearity of 1% and a cross sensitivity of 5%. Before evaluating the accuracy the GF-IMU prototype is calibrated. Then the accuracy of the filter results is evaluated for the rotation around the body’s z-axis. The results are compared to a simulation based on ideal parameters and to a non-calibrated GFIMU. The measurements as well as the calibration run are carried out on the 3D rotation table. Reference motion data is collected using the internal measurement instrumentation of the table. The parameters determined by calibration differ from the parameters extracted from the construction plan. Fig. 10 shows a

Fig. 10. The x–y positions of the 12 sensors (S1,. . .,S12) of the GF-IMU prototype as given in the construction plan and as determined by calibration.

Offset (m/s2 )

Position (cm)

y

z

x

y

z

S1 S2 S3

0.02 0.87 0.01

0.88 −0.01 0.01

0.00 0.00 0.90

−1.97 −0.43 1.30

−8.16 −8.45 −8.47

6.41 6.42 6.37

−0.32 −0.29 −5.73

S4 S5 S6

0.89 −0.06 0.02

−0.06 −0.90 0.02

0.01 0.01 0.88

−8.43 −8.77 −8.94

1.48 0.04 −1.85

6.14 6.16 6.11

−0.30 −0.77 −5.77

S7 S8 S9

0.01 −0.89 −0.01

−0.88 −0.02 −0.02

0.00 0.00 0.89

1.29 −0.30 −2.12

8.23 0.84 8.38

6.41 6.41 6.41

−0.41 −0.34 −5.75

S10 S11 S12

−0.89 0.02 0.00

0.02 0.88 −0.01

0.00 0.03 0.89

8.49 8.79 8.73

−1.58 −0.02 1.73

0.16 0.19 0.15

−0.45 −0.39 −5.65

top-view on the x–y positions of the accelerometers both for the positions extracted from the construction plan and obtained by calibration. Table 2 summarizes the GF-IMU parameters determined by calibration. The sensor positions and orientations are extracted from the construction plan for the evaluation of the non-calibrated GF-IMU. However, the calibration shows that the used sensors have a large offset in their signals (Table 2). With this large parameter error the GF-IMU results are afflicted with unstable estimations of the sign of the angular velocity. Hence, for the evaluation without calibration, the accelerometer signals are offset adjusted, since this can easily be applied to the GF-IMU by measuring the acceleration in the steady state where no relative body movement is present. Please note, that this is not the true metrological offset of the accelerometers, since their misalignment adds a part of the g-force to the measured signal. In order to synthesize measurements for the simulation the reference motion data provided by the rotation table is applied to (4). The artificial accelerometer measurements are calculated using the parameters determined by calibration, in order to have a comparable sensor configuration. A normal distributed noise which has the same variance as the acceleration sensors is added to the synthesized measurements. Fig. 11 shows the angular velocity ωB,z determined by the calibrated GF-IMU. The UKF correctly estimates the sign of the angular velocity. However, a negative offset within the results can be noticed when no angular rate is present. At high angular velocities this offset vanishes. This can be explained by the non-linearity of the observation model and the non-ideality of the accelerometers. The quadrature of the angular velocity within the observation model distorts the distribution of the predicted measurement. This distribution is approximated by the UKF by a Gaussian distribution. Thus, the mean predicted measurement corresponds to a biased predicted angular velocity. This effect has a large impact at small angular velocities and decreases at large values. The predicted state is subsequently corrected by the accelerometer measurements. However, the efficiency of the correction decreases with increasing noise level of the sensors. Furthermore, combined non-linear errors of the accelerometer characteristics as non-linear sensor outputs, cross-sensitivity, or hysteresis as well as quantization noise or outliers within the signals distort the distribution of the measurements. This again has a negative effect on the correction of the predicted state. The RMS error for the rotation around the body’s z-axis for all three evaluations is summarized in Table 3. All entries of the relative body movement for the calibrated GF-IMU show a smaller RMS error than for the non-calibrated GF-IMU, e.g. the RMS error

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9. Conclusion

Fig. 11. Measured angular rate ωB,z and its absolute error |Err| with parameters determined by calibration.

of ωB,z of the calibrated GF-IMU is by a factor 2.8 smaller compared to the non-calibrated configuration. For all evaluations, the angular acceleration ω˙ B,z is afflicted with larger error than ω˙ B,x and ω˙ B,y . This large error can be explained by the noise of the reference measurement from the rotation table, which is larger than the GF-IMU noise for ω˙ B,z determined by the UKF. The sensitive axis of the accelerometers is accurately identified by the presented calibration scheme. Therefore, faulty reconstructions of the angular acceleration are compensated, which can be seen in the RMS error results (Table 3). Consequently, this reduces the errors of the angular velocity, since the integrated angular acceleration is merged with angular velocity by the UKF. The lower RMS error of the simulation results account for the imperfect characteristics of the utilized accelerometers.

Table 3 RMS error of the GF-IMU evaluated with offset adjusted sensor signals and parameters based on the construction plan, as well as after calibration. The simulation is based on ideal sensor position and orientation parameters. RMS error

Construction plan based

After calibration

Ideal simulation

Body acceleration, aB (m/s2 ) x 0.14 y 0.05 z 0.07

0.03 0.02 0.05

0.02 0.02 0.02

Angular velocity, ωB (◦ /s) x 17.75 y 17.08 z 18.17

8.53 9.94 6.54

1.18 1.10 0.92

4.03 4.18 13.20

2.37 2.12 12.13

Angular acceleration, ω˙ B (◦ /s2 ) x 67.80 y 30.07 z 29.67

A gyroscope-free inertial measurement unit (GF-IMU) is presented that only incorporates linear acceleration sensors in order to completely determine the inertial body movement. The transversal acceleration as well as the angular acceleration and velocity can be directly determined by a linear set of equations. The positions and sensitive axes of the accelerometers account for 72 possible parameters for the placement of the sensors. In this work the sensor placement is simplified such that two arbitrary parameters, the width of the ground plane and the height, remain. Thus, the GF-IMU parameter matrix can be analytically inverted and an analytical calculation of the sensor noise propagation is possible. The presented GF-IMU approach features a nearly arbitrary placement of the sensors. However, either one or two sensor nodes have to be placed out of plane. The analytical evaluation of these possible configurations yields that a setup with one sensor node out of plane features a lower accuracy. For both analyzed configurations, the variances of the angular body movement follow a reciprocal quadratic function of the distances and are linearly dependent on the sensor variances. The presented calibration scheme is able to determine all parameters of the GF-IMU entirely by measurements of the GF-IMU and reference measurements. Hence, an arbitrary placement of the sensors is made feasible. It can also be proved that the calibration algorithm is optimal for the given linear set of equations. In addition, the variance of the determined parameters can be calculated. Thus, the performance of the parameter estimation can be determined without further evaluations. However, since the applied sensor model does not incorporate non-ideal sensor characteristics, the calibration is not able to compensate them. It is not possible to determine the sign of the angular velocity using a GF-IMU. Therefore, an Unscented Kalman filter (UKF) is applied to the GF-IMU in order to estimate the sign of the angular rotation and perform a noise cancellation.

Acknowledgment This work is supported by the German Research Foundation (Deutsche Forschungsgemeinschaft - DFG) under Grant Number GR1103.

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Biographies Patrick Schopp received the Dipl.-Ing. degree in Microsystems Engineering from the University of Freiburg, Germany, in 2008. He is now a student within the Ph.D. program “Embedded Microsystems” at the same University. His research interests include inertial navigation, sensor fusion, and localization algorithms.

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Lasse Klingbeil received his Dr. rer. nat. in experimental physics from the University of Bonn, Germany. He then spent two years as a postdoctoral fellow at the Autonomous Systems Laboratory of the “Commonwealth Scientific and Industrial Research Organization” (CSIRO) in Brisbane, Australia. He is now a research scientist at the “Institute of Micromachining and Information Technology” of the “Hahn-Schickard-Gesellschaft” (HSG-IMIT) in Villingen-Schwenningen, Germany. His research interests are inertial sensors and systems, sensor networks, sensor fusion, state estimation, and localization algorithms. Christian Peters received the Dipl.-Ing. in Microsystems Engineering in 2003 and the Dr.-Ing. degree from the Albert-Ludwigs-University, Freiburg, Germany, in 2009, both with highest honors. His Diploma thesis entitled “CMOS Based Stress Sensors - Novel Experimental Setup and Design” was awarded with the VDI award for excellent diploma thesis. His Ph.D. thesis “Components for Advanced Telemetric Powering of an Osteosynthesis Sensor System” was awarded with the Wolfgang Gentner award from the University of Freiburg for outstanding scientific work. Christian Peters is author of several international papers and journals. He is member of IEEE and VDI. Yiannos Manoli received the B.A. degree (summacum laude) in physics and mathematics from Lawrence University in Appleton, Wisconsin, in 1978, the M.S. degree in electrical engineering and computer science from the University of California, Berkeley, in 1980, and the Dr.-Ing. Degree in electrical engineering from the Gerhard Mercator University in Duisburg, Germany, in 1987. In 1985, he joined the newly founded Fraunhofer Institute of Microelectronic Circuits and Systems, Duisburg, Germany. From 1996 to 2001, he held the Chair of Microelectronics as full professor with the Department of Electrical Engineering, University of Saarland, Saarbrücken, Germany. In July 2001, he was appointed Chair of Microelectronics, Department of Microsystems Engineering (IMTEK), University of Freiburg, Germany. Since May 2005, he also serves as one of the three directors at the “Institute of Micro and Information Technology” of the “Hahn-Schickard Gesellschaft” (HSG-IMIT), Villingen-Schwenningen, Germany. His current research interests are the design of low-voltage/low-power mixed-signal CMOS circuits for embedded microsystems, energy harvesting, sensor read-out as well as A/D-converters. Prof. Manoli received Best Paper Awards from ESSCIRC 1988, PowerMEMS 2006, MWSCAS 2007 and MSE 2007. The last award was dedicated to Spicy VOLTsim (www.imtek.de/svs) a webbased application for the animation and visualization of analog circuits which also received the Multi-Media-Award of the University of Freiburg in 2005. When the Technical Faculty introduced the Best Teaching Award in 2008 Professor Manoli was the first to receive this honor. Professor Manoli has served on the committees of such conferences as ISSCC, ESSCIRC, DATE and ICCD, and was Program Chair (2001) and General Chair (2002) of the IEEE International Conference on Computer Design (ICCD). He is a member of Mortar Board, Phi Beta Kappa, IEEE, VDE and of the Editorial Board of the “Journal of Low Power Electronics”.