### Inhalt des Dokuments

# Observability of Motion States in Magnetometer-Free Inertial Motion Tracking of Kinematic Chains

### From Fachgebiet Regelungssysteme TU Berlin

### Abstract

Inertial measurement units are commonly used in a growing number of application fields to track or capture motions of kinematic chains, such as human limbs, exoskeletons or robotic manipulators. A major challenge is the presence of magnetic disturbances that results in unreliable magnetometer readings. Recent research revealed that this problem can be overcome by methods that exploit the kinematic constraints of the joints. These methods rely on the assumption that the motion of the kinematic chain is sufficiently rich to assure/yield observability of the relative pose. While it has been demonstrated that the methods converge in a number of specific applications and scenarios, a formal investigation of the crucial requirement has not yet been presented and no specific conditions for observability have so far been given. It therefore remains unclear when and why convergence to the true motion states can be assured. In this project, we perform observability analyses to determine precise conditions under which the relative pose of the body segments is indeed observable. We aim to support these theoretical results by simulation studies and validation in experimental data.

### People involved

- Karsten Eckhoff
- Ive Weygers
- Manon Kok
- Thomas Seel

## Description

The objective of this project is to answer the question under which conditions the motion states of a kinematic chain can be uniquely determined from the sensor readings of 6D-IMUs which are attached to the segments of the kinematic chain. To this end a kinematic model, in form of a state-space representation, is used to describe the motion of the kinematic chain. The kinematic model consists of the motion dynamics of the segments, a measurement model for the IMU readings and kinematic constraints which result from the connection of the segments via joints. Subsequently, it can be investigated whether the system, described by the state-space representation, is observable. When the system is observable the internal states (motion states) of the system can be uniquely reconstructed from the inputs (IMU readings) and the outputs (kinematic constraints) of the system. Hence, the observability of the motion states depends on the IMU readings and consequently on the movement of the kinematic chain. For different types of kinematic chains an observability analysis has been performed which led to precise conditions on the movement of the kinematic chain.

### Considered kinematic chains

In the context of this project, four different types of kinematic chains have been considered; Fig. 1 shows the different types. The kinematic chains can be distinguished by the number of segments, the degrees of freedom of the joints connecting the segments and whether the kinematic chain possesses a sparse sensor setup (not every segment of the kinematic chain is equipped with a sensor) or not.

The kinematic chains (b) and (d) are special cases of the kinematic chain (a). Furthermore kinematic chain (d) can be seen as a special case of the kinematic chain (c) if the middle segment has no spatial extent and the two joint axes are non-parallel to each other.

Note that the shown kinematic chains could also be a part of a larger kinematic chain.

### Kinematic Constraints

Depending on the type of kinematic chain different constraints can be utilized. The most general constraint is the position constraint, which can be exploited whenever two segments are connected via a joint. The constraint does not depend on the degrees of freedom of the joint. The position constraint utilizes that both segments share at least one common point, the joint center. In the case of a joint with only one degree of freedom the two segments share all point which lay on the joint axis. Furthermore, there exist kinematic chains which exploit that the relative orientation between the connected segments is constraint due to a restricted number of degrees of freedom of the joint. For a joint with only one degree of freedom it can be exploited that the coordinates of the joint axis (expressed in the coordinate systems of the two segments) have to be the same when they are transformed into a common coordinate system. Moreover, for a joint with two degrees of freedom it can be exploited that the connection via 2D-joint prohibits a rotation around the axis which is perpendicular to the two joint axes. In order to utilize the constraints parameters which are describing the structure of the kinematic chain need to be known. For the position constraint the position of the joint center expressed in the coordinate systems of the segments need to be known and for the 1D-constraint/2D-constraint the coordinates of the joint axis/axes expressed in the coordinate systems of the segments need to be known. There exists methods to identify these parameters from magnetometer-free IMU readings in case they are not known.

### Observability Statements

The observability analysis for kinematic chain (a), which utilizes the position constraint, showed that the relative orientation between the two segments can be uniquely determined whenever the specific force (the sum of the linear acceleration and the acceleration due to gravity) of the joint center and its derivative are non-parallel to each other. Fig. 2 shows the results of a simulation study to validate the obtained observability statement.

For kinematic chain (b), when only the 1D-constraint is exploited, the following observability condition has been obtained: The relative orientation between the two segments can be uniquely determined whenever the angular velocities of the segments are non-parallel to the joint axis.

More detailed results of the observability analysis and a simulation study for kinematic chain (c) can be found in the publications that are and will be provided below.

### Publications

- K. Eckhoff, M. Kok, S. Lucia, T. Seel.
**Sparse Magnetometer-free Inertial Motion Tracking–A Condition for Observability in Double Hinge Joint Systems**. In*21st IFAC World Congress*, pages 1–8, Berlin, Germany, 2020. - Bibtex| PDF | Link
**Author :**K. Eckhoff, M. Kok, S. Lucia, T. Seel**Title :**Sparse Magnetometer-free Inertial Motion Tracking–A Condition for Observability in Double Hinge Joint Systems**In :**In*21st IFAC World Congress*,**Date :**2020