This paper describes a calibration method for inertial and magnetic sensors using a batched optimization procedure. Well established sensor, motion and constraint models are applied which include sensor gains, biases, misalignments and inter-triad misalignments. For the magnetometer, hard and soft iron model parameters and local dipangle are embodied in the framework as well. The method does not require any additional equipment, is minimal restrictive with respect to the required movements, and can be performed within one minute. Our approach is applicable for both single and multi Inertial Measurement Units (IMU) and leverages from the relative pose between rigidly connected IMU's. We demonstrated that our approach resulted in improved dead reckoning estimates and showed good agreements with an optical reference system for both position and orientation estimates.
Inertial Measurement Unit (IMU's) have a profound role in the control of mobile vehicles, robotic manipulators, capturing human body motions and augmented reality applications in smartphones. Fused with a magnetometer, an IMU enables precise orientation estimates. In addition, IMU's provide velocity and position change estimates over short time periods. The quality of those estimates largely depend on correct sensor models. Hence, estimating the model parameters, known as calibration, is an important procedure that has major e ect on the eventual usability of the IMU.
The upswing of micro-electromechanical (MEMS) based IMU's resulted in many
scienti c publications describing calibration methods [
However, environmental in uences, like temperature changes and mechanical stress, cause deviations of the true parameter values over time. Hence, an easy to perform, without the need for extra equipment, calibration procedure is desired to correct for those recurred systematic errors. ? Supported by the Dutch SIA RAAK Postdoc program
While most literature focusses on the calibration of a single sensor, or only
the gyroscope and accelerometer triads, do some include the magnetometer as
well [
We propose a similar method, yet di erent approach for an easy in-use calibration of consumer grade triaxial inertial and magnetometer sensors without the need for external equipment. The novelty can be found in various aspects: { A tight coupled, batched, sensor fusion approach is used to bene t from all sensor observations in a joint manner. { Flexible in the number of rigidly connected IMU's (MIMU), their sampling rates, and the motion trajectory. { Exploiting the centripletal and tangential forces of a MIMU constellation. { Simultaneous estimation of all inertial and magnetic calibration parameters. { Simultaneous estimation of the navigation states. { Sample based handling of measurement outliers. { Robust for any magnetic disturbance and temporally or spatially inhomogenous elds.
This paper is structured as follows: we will rst outline the theoretical framework and highlight the method. Then, the method is demonstrated in three di erent situations. First, a MIMU array is calibrated, where the orientation state is compared with an optical reference system. In the second situation a MIMU array is calibrated using multiple static periods and compared with an existing popular calibration approach. Third, the method is applied on an IMU embodied in an Apple iPhone X during typical smartphone usage. 2
This section elaborates on the sensor, motion and constraint models.
Diagonal gain matrices are indicated with K, lower triangular matrices
describing the misalignments between the senstive axes with N , biases vectors with
b and Gaussian noise vectors with e [
We assume a rigid body on which one or multiple sensors (Si) are attached. Hence, the sensor's orientation with respect to the global frame can be written as:
RSiL = RSiBRBL t t (1)
Accelerometer: The output of an accelerometer at time t is modeled as the sum of the linear aSi
Si;t and gravitational acceleration gSi : yaS;it = KaSi NaSi RSiBaSBi;t + RSiLgL t + baSi + ea;t; ea;t
N (0; a) This linear acceleration at pick-up point (Si) is a summation of the body, centripetal and tangential accelerations. Latter can be expressed using the lever-arm pSBi , angular velocity and angular acceleration: aSi;t = RtBLaLB;t +
B
B LB;t pSBi + !LBB;t !LBB;t
B pSi where RGiSi is the relative orientation, or triad misalignment, between the accelerometer and gyroscope pair.
Gyroscope: The output of a gyroscope is modeled as an angular velocity measured in the gyroscope frame with respect to the inertial frame (!LGGii;t). We assume that the gyroscope and accelerometer are rigidly connected to the underlying body, hence !BSi = !SiGi = 0: ygG;ti = KgGi NgGi RGiSi RSiB!LBB;t + bgGi + eg;t; eg;t N (0; g) Magnetometer: The output of a magnetometer at time t is modeled as a local measured homogenous magnetic eld mL which is a ected by soft (D) and hard iron (o) e ects: ymM;it = DMi RSiBRBLmL + omMi + em;t; em;t
N (0; m) It should be noted that the triad misalignment between magnetometer and accelerometer is captured in matrix DMi and the magnetometer bias is captured in the o set vector oMi . The local magnetic eld is a function of the local magnetic dip angle and modeled as: mL = cos 0 sin
T (2) (3) (4) (5) (6) 2.2
Motion model: The kinematic relations of the position p, velocity v , acceleration a, orientation q, angular velocity ! and angular acceleration are given by the following proces model which is driven by a zero mean acceleration noise model:
L pt+T
L vt+T
L at+T qtL+BT = ptL + T vtL + T22 = vtL + T atL + wv = 0 + wa = qtLB !LBB;t+T = !LBB;t + T
B LB;t+T = 0 + w
atL + wp exp T 2 LB;t + w! B !LBB;t + T2
LBB;t + wq where T is the sample period, is the quaternion product operator and exp the quaternion exponential. The process noises are being described by: wXBLt
N (0;
X ); X 2 fp; v; a; q; !; ; g It should be noted that wp; wv; wa represent the same linear acceleration uncertainty, and wq; w!; w the same angular acceleration uncertainty. Zero net position: The user is asked to return the IMU to approximately the same position as where it has been picked-up. This knowledge is modeled as a zero net position change measurement: where Tp is the time di erence between pick-up and return.
Zero velocity: Whenever a zero movement period is detected using a
movingvariance detector [
Outliers are e ciently detected and suppressed by embedding a Cauchy Loss function for each observation individually. Sensor covariances are either obtained from data-sheets or derived from an Allan Variance plot. Other covariance values were chosen such that they have a realistic meaning, e.g. sub-centimeter level zero net position change, millimeter/s and millidegree/s zero velocity uncertainties. 2.3
The model parameters and navigation states are estimated simultaneously
using a large scale iterative non-linear least squares solver [
KaSi ; KgGi ; NaSi ; NgGi ; baSi ; bgGi 8i 2 S 2) Magnetometer gains, magnetometer o sets, and the magnetic dip angle: DMi ; oMi ;
RBSi ; pSBi ; RSiGi 2) Boresights, lever arms and sensor triad misalignments: 2) Body kinematics: pLB;t; vBL;t; aLB;t; qtLB; !LBB;t;
B LB;t 8i 2 S 8i 2 S 8t 2 T
Our approach has been evaluated in di erent indoor experiments. Before start of the recording sensors were running long enough to stabilize the temperature. In each experiment a di erent estimation aspect will be emphasized: 1. MIMU module, calibrated with two static poses. The orientation of the insample trial is compared with an optical reference. 2. MIMU module, calibrated with 20 static poses. The position of an out-ofsample trial is compared with a di erent calibration set and optical reference. 3. Smartphone IMU module, calibrated with two static periods, representing a typical smartphone movement. The pose of the in-sample trial is presented. 3.1
An Inertial Elements MIMU22BT multi IMU, sampled at 62:5Hz, module was
used, see Fig. 1. The MIMU was manipulated indoors in a rather arbitrary way
for about one minute while its movements were recorded by a passive optical
reference system (Vicon @100Hz). Processing the raw inertial and magnetic sensor
readings yielded the estimated calibration parameters. Uncalibrated and
calibrated sensor values are depicted in Fig. 2. In addition, the in-sample orientation
estimate was compared with the optical reference, see Fig. 3.
In our previous work we described a new method for Pedestrian Dead Reckoning
(PDR) based on MIMU sensors [
We re-used both the calibration and experimental datasets. The calibration dataset was used to calibrate the sensor using the method described in this 15 10 15 10
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paper. The experimental data set contains the intertial data of a subject who
traversed repeatedly an indoor oval path (800 m). Subsequently, the estimated
trajectory using a Kalman based PDR approach [
Figure 5a illustrates the in-sample estimates of the translational kinematics. In addition, in-sample orientation estimates are visible by the (un)calibrated magnetometer readings in Fig. 5b. 4
This paper describes a novel method for the calibration of inertial and magnetic measurement units. It allows for simultaneous estimation of calibration parameter and navigation states and exible in the number of sensors used.
The advantages, suitability and performance is demonstrated in di erent experiments. Especially experiment 2 shows the added value of proper estimated parameters when a MIMU module is used for PDR activities.
In a follow up study we would like to include the uncertainty on the parameter estimates by evaluating the Hessian matrix. In addition, the framework can be easily extended by inclusion of the sensor covariances and time-dependent bias states. 0.3 ][0.2 m p0.1 0.0 0.2 /]s 0.1 m [v 0.0 0.1 1.0 2/]sm 00..50 [ a 0.5 1.0 zyx norm zyx norm 0 2 (a) In sample position, velocity and acceleration estimates of the sensor module expressed in the global reference frame. (b) Mapping on the unit sphere (orange) of the uncalibrated (red) and calibrated (blue) magnetometer output.
Fig. 5. IMU experiment: Example of algorithm output mimicking a typical smartphone activity motion.