Magnetometer-free Realtime Inertial Motion Tracking by Exploitationof Kinematic Constraints in 2-DoF Joints

Daniel Laidig∗1, Dustin Lehmann∗1, Marc-Andre Begin2, and Thomas Seel1

Abstract— Inertial Measurement Units (IMUs) are used totrack the motion of kinematic chains in a wide variety ofrobotic and biomedical applications. However, inertial motiontracking is severely limited by the fact that magnetic fieldsare inhomogeneous in indoor environments and near electronicdevices. Methods that use only accelerations and angular ratesfor orientation estimation yield no absolute heading informationand suffer from heading drift. To overcome this limitation,we propose a novel method that exploits an orientation-basedkinematic constraint in joints with two degrees of freedom(DoF), such as cardan joints, saddle joints, the human wrists,elbow or ankles. The method determines the relative heading ofthe joint segments in real time by minimization of a nonlinearcost function. A filter for singularity treatment ensures accuratetracking during motion phases for which the cost functionminimum is ambiguous. We experimentally validate the methodin metacarpophalangeal (MCP) joints between the palm andthe fingers. Accurate relative orientation tracking is achievedcontinuously despite several singular motion phases and eventhough the heading components of the 6D orientations driftby more than 360 degrees within ten minutes. The proposedmethod overcomes a major limitation of inertial motion track-ing and thereby facilitates the use of this technology in roboticand biomechanical applications.

I. INTRODUCTION

In various robotic and biomechanical applications, such asrehabilitation robotics, feedback-controlled neuroprostheticsor human motion assessment, motion tracking plays a centralrole and is nowadays increasingly performed using inertialsensors [1], [2], [3], [4]. In contrast to marker-based opticalmotion tracking systems, inertial sensors are completelywearable and neither require an elaborate camera setup nora clear line of sight.

Inertial and magnetic sensors, also known as 9D inertialmeasurement units (IMUs), comprise 3D accelerometers, 3Dgyroscopes and 3D magnetometers. Sensor fusion of thesemeasurements yields the complete orientation of the sensorwith respect to a fixed inertial frame, which is crucial fordetermining the velocity and position of the sensor as wellas the joint angles between neighboring segments in roboticor biomechanical limbs. However, 9D sensor fusion onlyyields accurate orientation estimates if the magnetic fieldis homogeneous. Magnetometer readings are known to behighly unreliable in indoor environments and near ferro-magnetic material and electronic devices. Abundant research

*Both authors contributed equally to this work.1Daniel Laidig, Dustin Lehmann and Thomas Seel are with Control

Systems Group, Technische Universitat Berlin{laidig,seel}@control.tu-berlin.de

2Marc-Andre Begin is with Department of Aeronautics and Astronautics,Massachusetts Institute of Technologybegin@mit.edu

shows that inside buildings the local magnetic field vectormay easily vary by more than a factor two in less than 20centimeters [5], [6], [7], [8]. Therefore, 9D sensor fusionfails in numerous environments that are relevant for mostrobotic and biomechanical applications [9].

In 6D inertial sensor fusion, the magnetometer readingsare omitted, and the orientations of the sensors are deter-mined from the measured accelerations and angular rates.The heading of such an orientation exhibits an unknown andarbitrary offset, i.e. there is no absolute heading information.

This can be overcome by a calibration pose ensuringprecise initial alignment of the sensors at the beginning ofa measurement. As shown in [9], this approach leads toimproved results compared to conventional 9D sensor fusion,especially in realistic indoor environments. However, theheading still slowly drifts due to integration of gyroscopebias around the vertical axis. Even with very precise biascalibration, this drift easily reaches 90 degrees and morewithin a few minutes. This renders it impossible to determinelong-time stable joint angles from 6D sensor orientations.

In recent literature it has been shown that the missingrelative heading information can be obtained by exploitingkinematic models and constraints of different joint types.

(1) For joints with one major degree of freedom (DoF)such as the knee joint, the joint angle can be calculatedwithout the use of magnetometers [10], [11]. Moreover,the relative heading information can be determined in realtime by using an orientation-based constraint [12], whichfacilitates 3D visualization of the body segments.

(2) For rotational joints with two degrees of freedomsuch as the elbow or wrist, an angular-rates-based constraint

B2

S2

B1

S1

j2

j1

β0

Fig. 1. Kinematic model of a 2-DoF joint with segments B1, B2, jointaxes j1, j2 and inertial sensors S1, S2. Right: View in the j1-j2-plane. Theangle β0 describes the deviation of the angle between j1 and j2 from 90◦.

was used to perform sensor-to-segment calibration and si-multaneously identify the relative heading of both segments[13]. The limited degrees of freedom in the relative segmentorientation of the elbow and the hand have been exploitedin a magnetometer-free method that requires initial sensor-to-segment calibration protocols [14], [15].

(3) In joints with up to three rotational degrees of freedom,position-based constraints can be formulated in models thatuse the orientation, velocity and position of each segment asstates and assume the sensor-to-joint positions to be known.This approach was realized in an optimization-based methodfor offline motion analysis [16], which also allows for ad-ditional hinge joint constraints. A similar but online-capablemethod, which uses a sliding-window approach, is describedin [17]. However, both [16] and [17] still require the use ofmagnetometers for the initial orientation. Another approachbased on an extended Kalman filter (EKF) is proposed in[18], but it has not yet been validated in biological joints.

Experimental validation in previous studies has almostexclusively been carried out under idealistic assumptionsof either (1) rigid mechanical setups with perfect jointkinematics, (2) short time durations without sufficient focuson long-time stability, or (3) motions that persistently excitethe degrees of freedom of the joints and thus yield sufficientrelative-heading information at all times. Motions duringwhich the joint segments remain close to singularities of theconstraints (i.e. poses with unobservable relative heading)for more than a few seconds have barely been considered ordiscussed in previous studies.

In the present paper, we propose a novel magnetometer-free method for relative realtime motion tracking in 2-DoFjoints. A new, orientation-based constraint is proposed andexploited to determine the relative heading by window-basedonline optimization. In contrast to previous approaches, theproposed method requires neither an initial-pose calibrationnor sensor-to-joint position parameters, its long-time stabilityis validated experimentally in the metacarpophalangeal joints(MCP) of the hand, and its advantage over a conventionalmagnetometer-free method is demonstrated.

II. METHOD

Two segments B1 and B2 are connected by a joint with twodegrees of freedom, which can be modeled as a kinematicchain of two hinge joints as shown in Fig. 1. Denote the jointaxes as j1, j2 ∈ R3, with Euclidean norm ‖j1‖ = ‖j2‖ = 1.The coordinates of j1 are fixed in the frame B1 and thecoordinates of j2 are fixed in the frame B2. Without lossof generality, define the coordinate systems B1 and B2 suchthat j1 = [ 0 0 1 ]T and j2 = [ 0 1 0 ]T.

The angle between j1 and j2 is fixed, but not necessarily90◦. As shown in Fig. 1, we use β0 to denote the deviationof this angle from 90◦.1 This kinematic model is generalenough to describe many different joints such as saddlejoints, Cardan joints, the human wrist, elbow, or the ankle.

1For the elbow joint, this angle β0 is commonly called carrying angleand approximately 5–15◦, see for example [19].

Two inertial sensors S1 and S2 are placed on the segmentsin known orientation, i.e. the orientations of the sensors w.r.t.the segments S1B1q and S2B2q are known. This can be achievedby precise attachment or by using automatic sensor-to-segment calibration methods [13], [20]. The sensors measurethe angular rates ω1(t) and ω2(t) as well as the accelerationsa1(t) and a2(t) in local coordinates, perform 6D sensorfusion, potentially on-chip and for example according to thealgorithm described in [21], and report their orientations,described by the quaternions S1E1q and S2E2q. We can easilycalculate the body segment orientations BiEiq = SiEiq⊗BiSiq, i =1, 2, with ⊗ denoting quaternion multiplication.

Without the use of magnetometers, the absolute headingof each sensor is unknown, which can be described bythe orientations being estimated in two different referenceframes, E1 and E2. At each moment in time, the differencebetween the reference frames E1 and E2 is only a rotationaround the vertical axis, which has the coordinates [ 0 0 1 ]T

in both E1 and E2. Let δ(t) be the angle of this rotation, thenthe vector representation of the corresponding quaternion is

E2E1q(δ) =

[cos(δ2

)0 0 sin

(δ2

)]T. (1)

In 6D sensor fusion, i.e. without a common reference forthe heading, δ(t = 0) is unknown. Even if δ(t = 0) wouldbe known, for example from a known initial calibrationpose, this knowledge soon loses its value, because δ(t) driftsslowly due to non-zero gyroscope bias in both IMUs.

B2E1q

B2E2qB2E2q

β0

β0 6=β0 B1E1q

estimatedorientation

actualorientation

δ

Fig. 2. Orientations of segments B1 and B2. Calculating the relativeorientation between B1E1q and B2E2q without accounting for the difference inheading δ leads to a wrong joint orientation and wrong joint angles.

A. Orientation-based kinematic constraint for 2-DoF joints

Calculating relative segment orientations and joint anglesdirectly from B1

E1q and B2E2q leads to false results, cf. Fig.2. Instead, we must first determine an estimate δ(t) of theheading difference δ(t) and compensate it by bringing bothbody orientations into a common reference frame, i.e.

B2B1q = B1E1q

−1 ⊗ B2E1q = B1E1q−1 ⊗ E2E1q(δ(t))⊗ B2E2q. (2)

Note that any estimate δ(t) yields a relative orientationB2B1q =: [ q0 q1 q2 q3 ]

T. The intrinsic z-x′-y′′ Euler angle

decomposition of this relative orientation is

α = atan2(2q0q3 − 2q1q2, q20 − q21 + q22 − q23), (3)

β0 = arcsin(2q0q1 + 2q2q3), (4)

γ = atan2(2q0q2 − 2q1q3, q20 − q21 − q22 + q23). (5)

According to the definitions of B1 and B2 above, α and γare estimates of the joint angles, corresponding to rotationsaround j1 and j2, respectively, and the angle β0 correspondsto the fixed angle between between j1 and j2. In general, afalse δ(t) 6= δ(t) leads to a false β0 6= β0, cf. Fig. 2. Thisdeliberation leads to the orientation-based constraint

arcsin(2q0q1 + 2q2q3)− β0 = 0, (6)

which can be exploited to find the true heading differenceδ(t) without using any magnetometer readings.

B. Singularity detection

The kinematic constraint (6) becomes singular, i. e. ityields no relative-heading information, if the segment orien-tations are such that one of the joint axes is perfectly vertical.In such a case, a change of relative heading and a rotationaround the vertical joint axis cannot be distinguished, and theconstraint is perfectly fulfilled for arbitrary values δ ∈ R.

Due to inaccuracies in the estimated sensor orientations,this singularity will already deteriorate the results if any ofthe joint axes is close to vertical. In order to obtain a measurefor how close the axes are to vertical, we transform the jointaxes to the global coordinate system2

jE11 (t) = B1E1q(t)⊗ jB11 ⊗ B1E1q(t)−1 =: [ j1x j1y j1z ]T, (7)

jE22 (t) = B2E2q(t)⊗ jB22 ⊗ B2E2q(t)−1 =: [ j2x j2y j2z ]T, (8)

and define the sample rating r(t) as the product of thelenghts of their horizontal-plane projections (cf. Fig. 3):

r(t) :=‖jE11,proj‖ ‖jE22,proj‖, (9)

‖jE11,proj‖ =‖[ j1x j1y ]T‖, ‖jE22,proj‖ = ‖[ j2x j2y ]

T‖,

which goes to zero if any of the axes gets close to vertical.

2For quaternion multiplication, we implicitly regard three-dimensionalvectors as quaternions with zero real part.

j2j1

j1,proj

j2,proj

Fig. 3. Projections of the joint axes j1 and j2 in the horizontal plane.

Ts Twin Test

w=1

w=2

w=3

t1 t2 t3 tt2,Nt2,1 · · ·

Fig. 4. Graphical representation of the estimation windows, the estimationtime instants tw , the sample time instants tw,k and the durations Ts, Test,Twin.

C. Optimization-based estimation of the heading error

Instead of using the constraint sample-by-sample to es-timate δ(t), we propose a window-based estimation overmultiple samples. This is done to account for measurementinaccuracies and for the fact that biological joints onlyapproximately fulfill the 2-DoF joint constraint.

At regular time intervals Test, we perform an estimation ofδ(t) based on data in a time window immediately precedingthe estimation time. We denote the corresponding quantitieswith an index w ∈ N+, i.e. the estimation time instants are

tw = wTest. (10)

Each window consists of N samples with a sample indexk = 1, . . . , N , taken at a regular sampling interval Ts (whichmight be larger than the sampling rate of the sensor at whichsensor fusion is performed) at the sampling times

tw,k = wTest + (k −N)Ts, (11)

leading to a window duration of Twin := TsN .This implies that only previous data is used to estimate

δ(t), making this approach suitable for use in realtimeapplications. See Fig. 4 for a visual representation of theestimation time windows.

Let B2B1q(tw,k) =: [ q0 q1 q2 q3 ]T be the relative orien-

tation that (2) yields for a given δ. For each sample of thewindow, we define the weighted constraint error

ew,k(δ) = (arcsin(2q0q1 + 2q2q3)− β0) rw,k (12)

with k = 1, . . . , N and rw,k := r(tw,k).As the heading difference δ(t) only changes slowly over

time, we assume that it is constant during each window andfind the angle δw that minimizes the sum of squares of theerrors given in (12) over the window w:

δw = argminδ

N∑k=1

ew,k(δ)2. (13)

We determine analytical expressions for the Jacobian

[Jw]k =∂ew,k∂δ

∈ RN×1 (14)

and use a Gauss-Newton algorithm to minimize the error foreach window w, while using the estimate of the previouswindow as the initial value for each new window.

D. Singularity treatment and heading filter

The sample rating rw,k allows us to focus on the partsof a window that contain most relative-heading information.However, if the segments stay close to the singularity fordurations that exceed the window duration Twin, the estimateof the corresponding windows might be wrong. To mitigatethis, we first introduce the window rating

rw :=

√√√√ 1

N

N∑k=1

r2w,k. (15)

Then we design a filter that• smoothes the δw trajectory, taking the rating into ac-

count• extrapolates linearly if the rating is below a certain

threshold• does not introduce a delay if δw changes linearly.To achieve this, we determine a filtered estimate δf,w by

applying the following nested adaptive filter that estimatesthe heading bias bw and uses it to extrapolate the headingdifference whenever the rating is below a certain threshold:

bw = bw−1 + swkb,w(δw − δw−1 − bw−1) (16)δf,w = δf,w−1 + bw + swkδ,w(δw − δf,w−1 − bw) (17)

sw =

{rw rw ≥ rmin

0 else(18)

with a rating threshold rmin ∈ [0, 1] and filter gains

kb,w := max

(1− exp

(− ln 2

Tsτb

),1

w

), (19)

kδ,w := max

(1− exp

(− ln 2

Tsτδ

),1

w

). (20)

τb and τδ are tunable half-error time constants for the biasand heading filter, respectively. The filter gains are increasedfor small w to ensure fast initial convergence.

E. Heading correction

We finally use (1), (2) and the filtered estimate δf(t) todetermine the correction quaternion E2E1q(t), the orientationof the second segment B2 in the reference frame of thefirst segment E1, and the relative orientation between bothsegments. The resulting quaternions can be used for jointangle calculation and for 3D visualization.

III. EXPERIMENTAL VALIDATION

The proposed method is validated experimentally in themetacarpophalangeal joints (MCP) of the index and middlefingers of the human hand. We design an experiment thatallow us to obtain an approximate ground-truth reference forthe heading difference δ and to investigate long-time stabilityof the obtained estimates during time motions with andwithout sufficient excitation, i.e. during motions that yieldsufficient relative-heading information and during motionsfor which the segments remain close to the singularity.

j1

j2

MCPPIPDIP S2

S1

Fig. 5. Experimental setup and example scenes from the conductedexperiments.

A. Setup

To measure the 6D orientation of the finger segments, weuse a recently developed modular finger and hand motioncapturing system [9] as shown in Fig. 5. One inertial sensoris attached to the back of the hand and tracks the movementsof the metacarpal bones of both fingers. The second andthird sensor are attached to the proximal phalanges of theindex and middle finger. The MCP joints, which connect theproximal finger segments to the metacarpals, are approximate2-DoF joints. The segment-to-sensor attachment is known byprecise attachment. The IMUs measure the angular velocityand the acceleration at a rate of 100Hz, and 6D sensorfusion is performed using the algorithm presented in [21].The abduction axis j1 of the MCP is fixed in the coordinatesystem of the back of the hand. The flexion axis j2 is fixedin the coordinate system of the proximal phalange. Since theaxes are approximately perpendicular we assume that β0=0.

B. Obtaining an approximate reference

For validation, we define a reference pose that is takenseveral times during the experiment: The hand is lyingflat on a horizontal surface with all fingers straight. Forthat pose the coordinate systems of all three sensors areapproximately aligned. Therefore, whenever that pose istaken, the reference value δref is determined as the headingdifference that minimizes the rotation angle (maximizes thereal part) of the relative quaternion B2B1q in (2). Assumingthat the heading drift is slow and approximately linear, welinearly interpolate δref between the reference time instants.

Since this method relies on precise attachment of thesensors as well as exact positioning of the segments duringresting phases, it yields only an approximate ground-truthreference, which we will only use for qualitative comparisonand to examine whether the relative heading estimates δf(t)of the proposed method are long-time stable.

C. Conducted experiments

Two long-time experiments were conducted. The firstexperiment with a total duration of 8 min 41 s was designedwith distinct movement phases to show phases with naturalmovement and longer phases with rest near the singularities.See Table I for an overview of the different movement phasesand their duration.

TABLE IOVERVIEW OF MOVEMENT PHASES FOR THE FIRST EXPERIMENT

Phase Description Start End Duration

M1 normal movement 0 s 90 s 90 sM2 slow movement 90 s 153 s 63 sM3 normal movement 153 s 247 s 94 sM4 no movement, j2 vertical 247 s 290 s 43 sM5 normal movement 290 s 382 s 92 sM6 no movement, j1 vertical 382 s 424 s 42 sM7 normal movement 424 s 521 s 97 s

The motion phases were chosen to include phases withand without excitation. Phases with movement are expectedto yield good estimation results (M1, M3, M5, M7). Duringphases with approximately vertical axes (M4, M6), the datacontains almost no heading information, but the singularitytreatment as described in Section II-D should mitigate this.

The second experiment consists of functional movementslike counting, typing, moving objects in and out of a boxto test the method in real world applications. The secondexperiment has a total duration of 3 min 26 s. Videos of theperformed motions can be found at [22].

IV. RESULTS AND DISCUSSION

For brevity, we restrict the detailed analysis in this paperto the first experiment and the index finger. More detailedresults for both fingers and both experiments can be foundat [22].

The settings for the estimation are chosen as follows:Every Test = 1 s we analyze a window with a duration ofTwin = 20 s, consisting of N = 100 samples taken everyTs = 0.2 s. We choose the filter time constants as τb = 15 sand τδ = 30 s, with a rating threshold of rmin = 0.4.

A. Cost function analysis

To analyze the properties of the optimization problem forthe different motion phases, we evaluate the cost function in(13) for each time instant tw and for many evenly spacedangles δ ∈ [0◦, 360◦]. Fig. 6 shows the cost function for theindex finger over the entire experiment. It can be seen thata distinct global minimum exists for phases with movement,while the cost function becomes flat and exhibits a distinctsecond local minimum for phases without movement and avertical joint axis (M4 and M6). During those phases norelative-heading information is available, and the minimado not reflect the true value of δ. From the overall trend,it can be seen that the minimum drifts slowly with anapproximately constant slope of 0.6

◦

s .

δ[◦

]

t [s]

e(δ,t)

M1 M2 M3 M4 M5 M6 M7

Fig. 6. Cost function for the index finger during the experiment. Theminimum for each time instant tw , i.e. the unfiltered heading estimate, ismarked with a red dot.

δ[◦

]

r w[-

]

δ (estimated by constraint)δf (filtered estimate)δref (determined from resting poses)

M1 M2 M3 M4 M5 M6 M7

t [s]

e relori

[◦]

6D+KinCon6D+BiasEst+InitReset (5 s)6D+BiasEst+InitReset (20 s)

accuracy of reference

Fig. 7. Top: Estimated values for δ(t) and δf(t) as well as the referenceδref(t) (left axis) and window rating rw (right axis). Bottom: Relativeorientation error for the proposed method and the method described in [9].

B. Realtime heading error estimation

For realtime estimation of δ(t), the proposed algorithm isapplied to the recorded data, i.e. for the estimation at a giventime instant tw only data from the past is used according to(13). The time series of the estimated δ(t) and δf(t) as wellas the approximate ground-truth δref(t) are shown in Fig. 7.The reference is depicted with a band of ±10◦ to indicate theaforementioned approximate accuracy of the reference value.It can be seen that during the phases with low window ratingrw, the unfiltered δ(t) diverges from the reference. However,the filtered estimate δf(t) remains close to the true value,even for long periods of time near the singularities.

C. Alternative 6D method

We additionally evaluate the 6D method proposed in [9].During a time window without movement at the start ofthe measurement, the gyroscope readings are averaged toobtain an estimate of the gyroscope bias. After removingthis bias, 6D sensor fusion is performed. At the beginningof the measurement, an initial-pose reset is performed, which

assumes a known pose to determine and correct a fixedheading offset. However, it is an inevitable fact that theresulting orientations are only valid for a limited time untilthey are deteriorated by integration drift resulting fromresidual and time-varying gyroscope bias. To obtain long-time stable motion tracking, the subject would have to repeatthe initial reset procedure at least once per minute.

D. Error of the relative orientation

As explained above, accurate relative heading estimation iscrucial for obtaining accurate relative segment orientations,which are required to determine joint angles and forwardkinematics. We use δref(t) to determine the approximate truerelative orientation for each MCP joint and compare it to therelative segment orientation that is estimated by the proposedmethod. Denote this error by erelori(t).

Fig. 7 shows erelori(t) for the method presented in [9](6D+BiasEst+InitReset) with bias estimation windows of5 s and 20 s and for the new method based on kinematicconstraints (6D+KinCon). As expected, when using initialreset and bias estimation, erelori slowly drifts, while theproposed constraint-based method yields accurate resultsthroughout the entire measurement.

V. CONCLUSION

We proposed and evaluated a novel method formagnetometer-free inertial motion tracking for joints withtwo degrees of freedom. The method uses an orientation-based constraint to determine the relative heading be-tween the sensor orientations. A window-based optimizationscheme was introduced, which renders the method realtime-capable. Furthermore, we examined a method for rating thereliability of the optimization result and proposed an adaptivefilter that takes this rating into account and extrapolates theestimate when no relative-heading information is available.

With the proposed method it is possible to track the rela-tive orientations and joint angles of joints with two degreesof freedom in real-world applications with realistic magneticenvironments. While previous research is often limited toshort-time analysis of rich motions, we demonstrated thatthe proposed method yields long-time stable results andis capable of handling phases in which no movement isperformed or in which the movement does not containheading information. Furthermore, in contrast to previouslyproposed methods, the current method works on arbitrarymotions without the need to perform an initial reset with apredefined pose and without the need to determine sensor-to-joint position parameters.

Future work will focus on using the introduced constraintto simultaneously perform joint axis estimation in additionto relative heading estimation to facilitate plug-and-playmagnetometer-free motion tracking.

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[22] Supporting material. [Online]. Available: http://www.control.tu-berlin.de/2DoF Joint Heading Tracking