Human Movement Analysis with Inertial Sensors

Abstract

Present-day systems for human movement analysis are not portable, have a limited capture volume and require a trained technician to analyze the data. To extend the use and benefits to non-laboratory settings the acquisition should be robust, reliable and easy to perform. Ideally, data collection and analysis would be automated to the point where no trained technicians are required. Over the last decade several inertial sensor approaches have been put forward that address most of the aforementioned limitations. Advancements in micro-electro-mechanical sensors (MEMS) and orientation estimation algorithms are boosting the use of inertial sensors in motion capture applications. These sensors currently are the most promising opportunity for non-restricted human motion analysis. In this chapter we will describe the types of sensors used, followed by an overview of their use in the biomechanics community (Sect. 16.1); provide the necessary background of basic mathematics for those that want to refresh the basics of kinematics (case studies and appendix). The limitations of traditional systems can be dealt with due to the redundant information available to obtain orientation estimates. There are several different methods to derive orientation from sensor information; we will highlight the main groups of algorithms and the various ways in which they use the available data (Sect. 16.3). The chapter furthermore contains two hands-on examples to derive orientation (case study 1) and extract joint angles (case study 2, Sect. 16.4).

Appendices

Appendix: Theoretical Basis for Human Movement Analysis with Inertial Sensors

Kinematics is the branch of mechanics that describes the motion of points, bodies (objects) and systems of bodies (groups of objects) without consideration of the causes of motion. Therefore, kinematics is not concerned with the forces, either external or internal, that cause the movement. It includes the description of linear and angular displacements and its time-derivatives: velocities and accelerations. A complete and accurate quantitative description of the simplest movement requires a huge volume of data and a large number of calculations, resulting in an enormous number of graphic plots. Therefore, it should be kept in mind that any given analysis may use only a small fraction of the available kinematic variables.

1.1 Cartesian Reference Systems

A reference system is an adequate and arbitrary system where the position of any point (or solid) is referenced. A Cartesian reference system is formed by three perpendicular axes, which origin is located at the common intersection of the axes, determining the 3 dimensions of the space. Any point in the space is therefore located with respect to this reference system by three coordinates, one by each axis: $\left( {x,\,y,\,z} \right)$ (Fig. 16.6).

Fig. 16.6

Cartesian reference system

Two types of reference systems are commonly defined for human movement analysis:

• Fixed reference system, also called absolute or inertial, which is a Cartesian reference frame fixed to the world, coincident with the view of the external observer. In this reference system magnitudes related with global body movements are defined, as the movement of the body center of mass or trunk bending and rotations.

• Relative reference system, also called segment reference system, is a Cartesian reference frame fixed to the moving segment. A common variable measured on this reference system is joint movement.

The relative system commonly defined for human movement analysis has its origin coincident with the body center of mass, whose directions axis $(X - Y - Z)$ are coincident with the main body axis as follows: $X$ is the anterior axis (also called direction), pointing forward, $Y$ is the vertical axis (also called direction), pointing upwards, and $Z$ is the medial–lateral axis (also called direction), pointing right.

This body-centered reference system also contains the main body planes:

• Sagittal plane: divides any part of the body into right and left portions. It is perpendicular to z (medial–lateral) axis. Flexion and extension takes place in the sagittal plane.

• Frontal plane: divides any part of the body into front and back portions. It is perpendicular to x (anterior) axis. Abduction and adduction take place in the frontal plane.

• Transverse plane divides any part of the body into upper and lower portions. It is perpendicular to y (vertical) axis. Internal and external rotation takes place in the transverse plane. Also called medial and lateral rotation.

Three-Dimensional Kinematics

2.1 Matrix Notation for Reference Systems

The human musculoskeletal system is composed of a series of jointed links, which are commonly approximated as rigid bodies. Six independent parameters, the degrees of freedom, DOF, are needed to describe the location ($(x,y,z)$ coordinates with respect to reference system axes) and orientation ($(\alpha ,\beta ,\gamma )$ angles with respect to reference system planes) of a segment in space. Those six coordinates $(x,y,z,\alpha ,\beta ,\gamma )$constitute the degrees of freedom of a segment, and therefore uniquely define its spatial location and orientation at any time instant.

Most of the mechanical quantities one has to deal with in motion analysis, such as linear and angular position, velocity and acceleration of the markers and segments, are vectors. Because a vector has both magnitude and direction, one can describe the same vector in several different perspectives, depending on the intention or objective of the analysis. Describing a vector in a particular perspective is in essence equivalent to computing its components based on the coordinate system of the particular perspective.

Matrices are a form of mathematical notation suitable for operations among coordinate systems and vectors. A reference system can be defined using three vectors that represent each system’s axes, whose length is the unity. Therefore, the unit vector of the axes reference system, hence unit coordinate vectors (Fig. 16.7 right), can be expressed follows, where $\varvec{i},\varvec{j},\varvec{k}$ are the unit vectors of the $X - Y - Z$ respectively: $\varvec{i} = [1,0,0] ,\, \varvec{j} = [0,1,0] ,\, \varvec{k} = [0,0,1]$. Using this notation, the global coordinate system can be expressed by sorting $\varvec{i},\varvec{j},\varvec{k}$ vectors into a matrix as follows:

Fig. 16.7

Vector coordinates (left). Unitary vectors that define a reference system (right)

$$ \left[ {\varvec{i},\varvec{j},\varvec{k}} \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

2.2 Rotation Matrix

As shown in Fig. 16.7 left, a vector can be expressed as the sum of the component vectors projected over the $i,j,k$ vectors:

$$ \varvec{V}_{x} = v_{x} \cdot \varvec{i };\quad \varvec{ V}_{\varvec{y}} = \varvec{v}_{\varvec{y}} \cdot\varvec{j };\varvec{ }\quad \varvec{V}_{\varvec{z}} = \varvec{v}_{\varvec{z}} \cdot\varvec{k }; $$

$$ \varvec{V} = \left[ {\varvec{v}_{\varvec{x}} ,\varvec{v}_{\varvec{y}} ,\varvec{v}_{\varvec{z}} } \right] = \varvec{i} \cdot v_{x} + \varvec{j} \cdot v_{y} + \varvec{k} \cdot v_{z} $$

In other words, one can not only describe the same vector in several different perspectives, but also change the perspective from one to another depending on the situation and needs. This changing perspective of describing a vector is called vector transformation or axis transformation, and it is done through rotation matrices. A rotation matrix is the mathematical form for expressing in a compact way the orientation of a reference system, usually a mobile one, with respect to another, usually fixed reference system.

To transform a vector from one reference frame to another is equivalent to changing the perspective of describing the vector from one to another. A transformation alters not the vector, but the components as follows:

$$ v\varvec{'}_{x} = \varvec{v}\cdot\varvec{i'} = \left( {v_{x} \cdot\varvec{i} + \varvec{v}_{\varvec{y}} \cdot\varvec{j} + \varvec{v}_{\varvec{z}} \cdot\varvec{k}} \right)\cdot\varvec{i'} = \varvec{v}_{\varvec{x}} \cdot\varvec{i}\cdot\varvec{i'} + \varvec{v}_{\varvec{y}} \cdot\varvec{j}\cdot\varvec{i'} + \varvec{v}_{\varvec{z}} \cdot\varvec{k}\cdot\varvec{i'} $$

$$ \varvec{v'}_{\varvec{y}} = \varvec{v}\cdot\varvec{j'} = \left( {\varvec{v}_{\varvec{x}} \cdot\varvec{i} + \varvec{v}_{\varvec{y}} \cdot\varvec{j} + \varvec{v}_{\varvec{z}} \cdot\varvec{k}} \right)\cdot\varvec{j} = \varvec{v}_{\varvec{x}} \cdot\varvec{i}\cdot\varvec{j'} + \varvec{v}_{\varvec{y}} \cdot\varvec{j}\cdot\varvec{j} + \varvec{v}_{\varvec{z}} \cdot\varvec{k}\cdot\varvec{j'} $$

$$ \varvec{v'}_{\varvec{z}} = \varvec{v}\cdot\varvec{k'} = \left( {\varvec{v}_{\varvec{x}} \cdot\varvec{i} + \varvec{v}_{\varvec{y}} \cdot\varvec{j} + \varvec{v}_{\varvec{z}} \cdot\varvec{k}} \right)\cdot\varvec{k'} = \varvec{v}_{\varvec{x}} \cdot\varvec{i}\cdot\varvec{k'} + \varvec{v}_{\varvec{y}} \cdot\varvec{j}\cdot\varvec{k'} + \varvec{v}_{\varvec{z}} \cdot\varvec{k}\cdot\varvec{k'} $$

$$ \left[ {\begin{array}{*{20}c} {v'_{x} } \\ {v'_{y} } \\ {v'_{z} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varvec{i}\cdot\varvec{i'}} & {\varvec{j}\cdot\varvec{i'}} & {\varvec{k}\cdot\varvec{i'}} \\ {\varvec{i}\cdot\varvec{j'}} & {\varvec{j}\cdot\varvec{j'}} & {\varvec{k}\cdot\varvec{j'}} \\ {\varvec{i}\cdot\varvec{k'}} & {\varvec{j}\cdot\varvec{k'}} & {\varvec{k}\cdot\varvec{k'}} \\ \end{array} } \right]\cdot\left[ {\begin{array}{*{20}c} {v_{x} } \\ {v_{y} } \\ {v_{z} } \\ \end{array} } \right] $$

In those transformations, $\varvec{i},\varvec{j},\varvec{k}$ are the unit vectors of the $X - Y - Z$ system, and $\varvec{i'},\varvec{j'},\varvec{k'}$ are the unit vectors of the $X - Y - Z$ system. Therefore, the transformation matrix from the global reference frame (frame G) to a particular local reference frame (frame L) can be written as:

$$ \left[ {\begin{array}{*{20}c} {v'_{x} } \\ {v'_{y} } \\ {v'_{z} } \\ \end{array} } \right] = {}_{{}}^{L} \varvec{T}_{G} \cdot\left[ {\begin{array}{*{20}c} {v_{x} } \\ {v_{y} } \\ {v_{z} } \\ \end{array} } \right] $$

$$ {}_{{}}^{L} \varvec{T}_{G} = \left[ {\begin{array}{*{20}c} {\varvec{i}\cdot\varvec{i'}} & {\varvec{j}\cdot\varvec{i'}} & {\varvec{k}\cdot\varvec{i'}} \\ {\varvec{i}\cdot\varvec{j'}} & {\varvec{j}\cdot\varvec{j'}} & {\varvec{k}\cdot\varvec{j'}} \\ {\varvec{i}\cdot\varvec{k'}} & {\varvec{j}\cdot\varvec{k'}} & {\varvec{k}\cdot\varvec{k'}} \\ \end{array} } \right] $$

Obviously, in human movement analysis the local reference frame is typically fixed to a segment or a body part, whereas the global reference system is reference system fixed to the laboratory, the global reference system.

Similarly, ${}_{{}}^{G} \varvec{T}_{L}$ is the inverse rotation matrix of ${}_{{}}^{L} \varvec{T}_{G}$ which can also be derived as:

$$ {}_{{}}^{\varvec{G}} \varvec{T}_{\varvec{L}} = \left[ {\begin{array}{*{20}c} {\varvec{i'}\cdot\varvec{i}} & {\varvec{j'}\cdot\varvec{i}} & {\varvec{k'}\cdot\varvec{i}} \\ {\varvec{i'}\cdot\varvec{j}} & {\varvec{j'}\cdot\varvec{j}} & {\varvec{k'}\cdot\varvec{j}} \\ {\varvec{i'}\cdot\varvec{k}} & {\varvec{j'}\cdot\varvec{k}} & {\varvec{k'}\cdot\varvec{k}} \\ \end{array} } \right] $$

However, a special feature of the rotation matrices is that they are orthonormal, that is, the vectors $\varvec{i},\varvec{j},\varvec{k}$ are orthogonal and unitary. Therefore, the inverse of those matrices are in fact their transpose, so the change is straightforward:

$$ {}_{{}}^{G} \varvec{T}_{L} = \left( {{}_{{}}^{L} \varvec{T}_{G} } \right)^{ - 1} = \left( {{}_{{}}^{L} \varvec{T}_{G} } \right)^{T} $$

A series of transformations can be performed through successive multiplication of the transformation matrices from the right to the left. Once the transformation matrices from the global reference frame to the local reference frames are known, computation of the transformation matrices among the local reference frames is simply a matter of transposition and multiplication of the transformation matrices. Hence the transformation matrix from one local reference frame (A) to another (B) can be easily obtained through cascading of the transformation matrices:

$$ {}_{{}}^{B} \varvec{T}_{A} = {}_{{}}^{B} \varvec{T}_{G} \cdot{}_{{}}^{G} \varvec{T}_{A} = {}_{{}}^{B} \varvec{T}_{G} \cdot\left( {{}_{{}}^{A} \varvec{T}_{G} } \right)^{ - 1} = {}_{{}}^{B} \varvec{T}_{G} \cdot\left( {{}_{{}}^{A} \varvec{T}_{G} } \right)^{T} $$

2.3 How to Extract Rotation Angles Using Euler Convention

As shown above, the components of a free vector change as the reference frame changes. Figure 16.8 shows two different reference frames: the $X - Y$ system and the $X^{'} - Y'$ system. Vector $\varvec{v}$ can be expressed as $\varvec{v}(x, y)$ in the $X - Y$ system, $\varvec{ v}(x', y')$ in the $X' - Y'$ system. The relationships between $X - Y$ and $X' - Y'$ can be obtained from the geometric relationships:

$$ x^{\prime} = x \cdot cos\left( \Upphi \right) + y \cdot sin\left( \Upphi \right) $$

$$ y^{\prime} = y \cdot cos\left( \Upphi \right) - x \cdot sin\left( \Upphi \right) $$

In matrix form:

$$ \left[ {\begin{array}{*{20}c} {x'} \\ {y'} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {cos\left( \Upphi \right)} & {sin\left( \Upphi \right)} \\ { - sin\left( \Upphi \right)} & {cos\left( \Upphi \right)} \\ \end{array} } \right]\cdot\left[ {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right] $$

It is straightforward to expand these manipulations between axes to the three-dimensional space:

$$ x^{\prime} = x \cdot cos\left( \Upphi \right) + y \cdot sin\left( \Upphi \right) $$

$$ y^{\prime} = y \cdot cos\left( \Upphi \right) - x \cdot sin\left( \Upphi \right) $$

$$ z' = z $$

In matrix form:

$$ \left[ {\begin{array}{*{20}c} {x'} \\ {y'} \\ {z'} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {cos\left( \Upphi \right)} & {sin\left( \Upphi \right)} & 0 \\ { - sin\left( \Upphi \right)} & {cos\left( \Upphi \right)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\cdot\left[ {\begin{array}{*{20}c} x \\ y \\ z \\ \end{array} } \right] $$

Therefore, the rotation around the z axis from $X - Y - Z$ reference system to $X^{'} - Y^{'} - Z'$ is expressed as follows:

Fig. 16.8

Two-dimensional vector transformation between two coincident reference frames

$$ \varvec{R}_{z} \left( \Upphi \right) = \left[ {\begin{array}{*{20}c} {cos\left( \Upphi \right)} & {sin\left( \Upphi \right)} & 0 \\ { - sin\left( \Upphi \right)} & {cos\left( \Upphi \right)} & 0 \\ 0 & 0 & 1 \\ \end{array} } \right] $$

Rotation around the $X$ and $Y$ axis can be obtained in a similar procedure:

$$ \varvec{R}_{x} \left( \Upphi \right) = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & {cos\left( \Upphi \right)} & {sin\left( \Upphi \right)} \\ 0 & { - sin\left( \Upphi \right)} & {cos\left( \Upphi \right)} \\ \end{array} } \right] $$

$$ \varvec{R}_{y} \left( \Upphi \right) = \left[ {\begin{array}{*{20}c} {cos\left( \Upphi \right)} & 0 & { - sin\left( \Upphi \right)} \\ 0 & 1 & 0 \\ { - sin\left( \Upphi \right)} & 0 & {cos\left( \Upphi \right)} \\ \end{array} } \right] $$

Once rotation around each of the three reference system axis is defined, an arbitrary rotation, which is usually composed by rotations around all three axes, can be seen as a composition of three sequential rotations, which is actually what the Euler theorem states.

Using Euler theorem, any arbitrary rotation can be de-composed into three sequential rotations. For example, consider the sequence $X - Y' - Z''$. This means that we rotate about $X$ axis first, $\theta_{1}$ degrees. As result we get the new orientation, given by $X' - Y' - Z'$. Then a rotation around the new $Y'$ axis $\theta_{2}$ degrees is made, resulting in the new orientation $X'' - Y'' - Z''$. A final rotation around the new $Z''$ axis $\theta_{3}$ degrees is made, resulting in the resulting final orientation $X''' - Y''' - Z'''$.

Taking each rotation matrix, any point $\varvec{p}\left( {x,y,z} \right)$expressed in the $X - Y - Z$ reference system (local) can be transformed (expressed) in the global reference system $X''^{'} - Y''^{'} - Z'''$ as follows:

$$ \left[ {\begin{array}{*{20}c} {p_{x'''} } \\ {p_{y'''} } \\ {p_{z'''} } \\ \end{array} } \right] = \left[ {\varvec{R}_{z} \left( {\theta_{3} } \right)} \right]\cdot\left[ {\varvec{R}_{y} \left( {\theta_{2} } \right)} \right]\cdot\left[ {\varvec{R}_{x} \left( {\theta_{1} } \right)} \right]\cdot\left[ {\begin{array}{*{20}c} {p_{x} } \\ {p_{y} } \\ {p_{z} } \\ \end{array} } \right] = {}_{{}}^{L} \varvec{R}_{G} $$

Expanding the former, and using shorthand notation where $c1 = cos\left( {\theta_{1} } \right)$ and $s2 = sin\left( {\theta_{2} } \right)$

$$ \left[ {\begin{array}{*{20}c} {p_{x'''} } \\ {p_{y'''} } \\ {p_{z'''} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {c2 \cdot c3} & {s3 \cdot c1 + s1 \cdot s2 \cdot c3} & {s1 \cdot s2 - c1 \cdot s2 \cdot c3} \\ { - c2 \cdot c3} & {c1 \cdot c3 - s1 \cdot s2 \cdot s3} & {s1{\text{c3 + c1}}s2 \cdot s3} \\ {s2} & { - s2 \cdot c2} & {c1 \cdot c2} \\ \end{array} } \right] \cdot \left[ {\begin{array}{*{20}c} {p_{x} } \\ {p_{y} } \\ {p_{z} } \\ \end{array} } \right] $$

From the above matrix, $\theta_{1}$, $\theta_{2}$ and $\theta_{3}$ angles can be obtained as follows:

$$ \theta_{2} = arcsin\left( {{}_{{}}^{L} \varvec{R}_{G} \left( {3,1} \right)} \right) $$

$$ \theta_{1} = arcos\left( {\frac{{{}_{{}}^{L} \varvec{R}_{G} \left( {3,3} \right)}}{{cos\left( {\theta_{2} } \right)}}} \right) $$

$$ \theta_{3} = arcos\left( {\frac{{{}_{{}}^{L} \varvec{R}_{G} \left( {1,1} \right)}}{{cos\left( {\theta_{2} } \right)}}} \right) $$

This example corresponds to a rotation sequence around $X - Y' - Z'$ axes. However, any other rotating sequence can be used. In theory, there are 12 possible correct rotation sequences, by the combination of the $X - Y - Z$ rotations.

2.4 International Society of Biomechanics Standards

One of the characteristics of Euler theorem, is that the value of $\theta_{1}$, $\theta_{2}$ and $\theta_{3}$ angles depend on the rotation sequence assumed, this makes the comparison of data among various studies difficult, if not impossible. On the other hand, some rotation sequences are closer to representing joint rotations in clinically relevant terms, which makes the application and interpretation of biomechanical findings easier and more welcoming to clinicians.

The international society of biomechanics has made recommendations for the definitions of segment coordinate systems as well as for rotation sequences for reporting joint movement (Grood and Suntay 1983; Wu et al. 2002). The purpose was to present these definitions to the biomechanics community so as to encourage the use of these recommendations, to provide first hand feedback, and to facilitate the revisions. It was hoped that this process will help the biomechanics community to move towards the development and use of a set of widely acceptable standards for better communication among various research groups, and among biomechanists, physicians, physical therapists, and other related interest groups. Those recommendations include the definitions for major human joints as ankle, hip, spine, shoulder, elbow, wrist and hand.