Coordinate Systems#

Coordinate systems (or frames) are an important topic when working with IMUs and are getting even more complicated when these IMUs are attached to a human body. This library makes a couple of deliberate choices when it comes to the definition of coordinate systems and related naming conventions. Therefore, please read this document carefully before using the pipelines available in gaitlab.

TL;DR(A)#

This TL;DR(Again) is intended as quick reference, after you already read the full documentation below.

Accelerometer measures a positive value, if movement occurs in the positive (+) direction of the axis
Gyroscope measures a positive value, if rotation occurs in the direction of the arrow in the image below
Gravity results in a measure of +1g, if an axis is pointing upwards, and -1g, if the axis is pointing downwards

Foot Sensor Frame (FSF)

forms a right-handed coordinate system with axes called X, Y, Z
uses right-hand-rule around each axis for definition of positive direction of the Gyroscope
defines axes’ directions as up (Z), to the tip of the shoe (X), and to the left (Y)

Foot Body Frame (FBF)

consists of the 3 axes ML (medial to lateral), PA (posterior to anterior), and SI (superior to inferior)
is not right handed and should not be used for any physical calculations
produces the same sensor signal independent of the foot (right/left) for the same anatomical movement (e.g. lateral acceleration = positive acceleration)
follows convention of directions from [1]

World/Global Frame (GF)

uses the same axes definitions as the FSF during the stance phase, but
is stationary, i.e. in contrast to FSF and FBF not moving
is used to describe all positions/velocity and orientations after the Trajectory Reconstruction
is fixed during an observation period
The observation period depends on the algorithm used in the Trajectory Reconstruction and is usually either a stride or a gait sequence level

Foot Frame Overview#

gaitmap foot frame — The positive directions for the accelerometer (straight arrows) and the gyroscope (circular arrows) for the foot sensor frame (left) and foot body frame (right). Click here for fullscreen version of image#

Table showing the expected signal (positive or negative and in which axis) when a certain movement (displacement or rotation) of a foot happens for the sensor (FSF) and the body frame (FBF).#
	FSF		FBF
	Left Foot	Right Foot	Left Foot	Right Foot
Displacements
anterior	+acc_x	+acc_x	+acc_pa	+acc_pa
posterior	-acc_x	-acc_x	-acc_pa	-acc_pa
lateral	+acc_y	-acc_y	+acc_ml	+acc_ml
medial	-acc_y	+acc_y	-acc_ml	-acc_ml
inferior	-acc_z	-acc_z	+acc_si	+acc_si
superior	+acc_z	+acc_z	-acc_si	-acc_si
Rotations
eversion	+gyr_x	-gyr_x	-gyr_pa	-gyr_pa
inversion	-gyr_x	+gyr_x	+gyr_pa	+gyr_pa
dorsifelxion	-gyr_y	-gyr_y	+gyr_ml	+gyr_ml
plantarflexion	+gyr_y	+gyr_y	-gyr_ml	-gyr_ml
external rotation	+gyr_z	-gyr_z	-gyr_si	-gyr_si
internal rotation	-gyr_z	+gyr_z	+gyr_si	+gyr_si

gaitmap foot frame signal templates — Example signals of 3D-accelerometer and 3D-gyroscope sensor signals for sensor-frame and body-frame coordinate system definitions. The signal section corresponds to one left-/right stride, corresponding to the DTW template definition. Click for here for the fullscreen version of image.#

Sensor vs Body Frame#

When working with IMUs you always need to differentiate between the local sensor coordinate system and the global world coordinate system. The former describes everything from the view of the sensor (i.e. when the sensor is rotated or moved the coordinate system is adjusted as well). The latter describes the sensors orientation and movement relative to a global reference frame set by global objects relevant for the performed measurement. If measuring human movement we usually use two separate global frames: (1) A global frame that is relative to the space the human is moving in. This frame is important to describe global displacements and rotations of the human during movement. This “real global frame” is explained in more detail below. (2) A frame that uses the human body as the reference. This frame makes it easy to map the motion to anatomical movement directions. For example, the forward direction should point towards the anterior site of our human and upwards should be defined the superior direction.

Body Frame#

If our sensor is rigidly attached to the trunk (or head) of a human a transformation from the sensor frame (SF) to the body frame (BF) can be achieved by a single rotation matrix describing the relative orientation from sensor to body.

However, if we attach sensors to two opposite-sided extremities of a human, this is not as easy anymore. As the movements of our extremities are described as mirror images of each other, it makes sense to choose a BF that results in the same sensor signal for the same movement, independent of which extremity we look at. For example, a lateral raise of the leg should produce a positive acceleration signal in both sensors. This requires that the BF of one sensor is the mirror-image (mirrored at the sagittal plane) of the other’s. As mirroring is a non-cartesian transformation, it is not possible to find rotation matrices that can transform the local SFs into their respective BFs. Hence, performing the transformation will require the mirroring of individual axis, which breaks the right-handedness of the coordinate system. If we further want to match the positive directions of the Gyroscope with same anatomical rotational directions on both sides, we also need to break the right-hand-rule for the movement direction around an axis.

Warning

Therefore, we must not use such a BF to perform any sort of physical calculations that involve the signal for more than one axis! In particular rotations cannot be performed in this coordinate system.

This inability to perform certain mathematical options potentially requires us to switch from the SF to the BF multiple times during an analysis pipeline. For example, initial sensor alignment, filtering, and calibration needs to be performed in the SF. The identification of certain movements and events is usually done best in the BF, so that the same algorithm can be used for each foot. Orientation and position estimations of the sensors need to be performed in the SF again and the results need to be converted back to the BF to get final biomechanical measures and joint angles.

Foot Coordinate System#

Depending on where the sensors are attached at the extremities it might make sense to choose different BFs that align better with the directions of movement the specific body segment can perform. The following section describes the chosen SF and BFs for sensors attached anywhere at the feet. These frames are called foot sensor frame (FSF) and (left/right) foot body frame ((L/R)FBF).

To make the transformation from the FSF to the FBFs easy, this library uses an FSF that already aligns with all mature axes of movements (see Foot Frame Overview). Specifically, sensor axes independent of the foot are expected to point up (Z), to the tip of the shoe (X), and to the left (Y). The forward direction for rotations are determined by the right-hand-rule around these axes.

Note

This means to use the provided functions for coordinate conversion in this library, you are expected to rotate your IMU data to fit this coordinate system. This cannot always be done precisely. The required precision of alignment will depend on the exact algorithms used and the final biomechanical parameters of interest. This is discussed further in the section about Alignment with the Foot Sensor Frame.

To transform FSF into FBFs, only renaming and axis flips are required (see table Required transformation for accelerometer and gyroscope from FSF to FBF for both feet). FBFs’ axes FBFs are denoted by ML (medial to lateral), PA (posterior to anterior), and SI (superior to inferior). The order of naming directly indicates the positive direction of the respective axis. All rotations are named by the axes they occur around. Note, that positive direction of rotation is not determined by the right-hand-rule. Rather, forward directions for axes and directions of rotation are directly taken from the recommendations given in [1] (see Foot Frame Overview).

Required transformation for accelerometer and gyroscope from FSF to FBF for both feet#
Left Foot		Right Foot
LFBF	FSF	RFBF	FSF
acc_pa	acc_x	acc_pa	acc_x
acc_ml	acc_y	acc_ml	-acc_y
acc_si	-acc_z	acc_si	-acc_z
gyr_pa	-gyr_x	gyr_pa	gyr_x
gyr_ml	-gyr_y	gyr_ml	-gyr_y
gyr_si	-gyr_z	gyr_si	gyr_z

World/Global Frame#

At this point, it is important to emphasize again that the SF/BF moves with the sensor/subject. Therefore, the alignment information for the FSF/FBF provided above, of course only hold true while the test subject is in its default pose (e.g. flat foot phase for foot mounted IMUs). During movement the orientation of the sensor can not be easily described by terms like “forward” or “upwards”. Hence, the SF is not suitable to calculate and express measures that require a fixed global frame. Additionally, the BF is unsuitable to talk about movements of the entire human, as this frame moves with the subject. For example, to estimate the stride length, we need a clear definition of a forward direction over the entire duration of a stride and a reference point that does not move with the subject. Therefore, we need a “global” fixed reference frame (GF) for the duration of our observation. The GF does not move or rotate if the sensor or the subject move. For an example, motion capture systems (e.g. camera based systems) always report their information in such a global reference frame, because the measurement devices themself are fixed in a global frame.

For inside-out systems, like body-attached IMUs, it is difficult to establish such a fixed coordinate system. Rather complicated algorithms must be used to project the local measurements of an IMU into a fixed global frame. During the gait pipeline, this step is performed by the Trajectory Reconstruction. The orientation, velocity, and position provided by these algorithm is always expressed in the GF. However, for the sake of simplicity, no additional prefix is used to indicate that in the column names. One just has to remember that all positions and orientations provided after the Trajectory Reconstruction are expressed in the global frame.

Note that the orientation of this fixed global frame is usually determined by the orientation of the FSF/FBF at the beginning of the observation period. Many algorithms consider a single stride as an observation period. This means that when comparing two strides the GFs do not necessarily line-up, as the GF for each stride is defined based on the orientation of the sensor at the beginning of the stride.

Algorithmic Implementation#

Alignment with the Foot Sensor Frame#

Aligning the coordinate system of a sensor with the gaitmap coordinate system can be a complicated, but usually requires two steps:

Rough Alignment of sensor axis using prior knowledge of the sensor orientation (or other methods)
Alignment of the z-axis using gravity during a static moment
(:py:func`~gaitmap.preprocessing.sensor_alignment.align_dataset_to_gravity`)

The alignment should be performed in this order, because the orientation of the sensor after the gravtiy alignment can not be easily predicted, if the alignment error is large. Therefore, prior knowledge should be used to already rotate the sensor before performing a gravity alignment.

For a full example on how to perform these alignment steps see our Preprocessing Example.

Note

Perfect alignment will be impossible as every sensor attached to a foot or a shoe can not be attached perfectly flat. You should aim to make the alignment as good as possible, but small misalignment in the heading of the sensor will not negatively effect the performance of any algorithm.

Transformation into the Foot Body Frame#

Once the data is properly aligned to the gaitmap-FSF, it is very easy to transform it into the respective BF. For this you can use the function convert_to_fbf if you have a multi-sensor dataset or convert_left_foot_to_fbf and convert_right_foot_to_fbf if you have a single-sensor dataset. Note that in both cases you need to specify which foot each sensor was attached to, as the transform from the FSF to FBF is different for each foot (see table Required transformation for accelerometer and gyroscope from FSF to FBF for both feet). For examples please refer directly to the documentation of these functions.

Note

All coordinate transform functions create a full copy of your dataset.