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## Theoretical Approach

**Résumé**

Nous étudions au sein de ce chapitre les principaux modèles théoriques d’analyse de la marche. Après avoir présenté les études analysant le genou humain, nous indiquons dans quelle mesure leurs conclusions aﬀectent nos décisions de conception pour notre méthode. Le suivi de mouvement du genou humain rend nécessaire la création d’un modèle cinéma-tique biomécanique. Ce modèle l’articulation du genou ainsi que les deux segments du corps connectés par l’articulation.

Dans la section 2.2, nous présentons les diﬀérents modèles ainsi que le modèle ciné-matique biomécanique choisi pour notre analyse. Nous y explicitons les deux référentiels principaux ainsi que les systèmes de coordonnées utilisés pour les analyses de l’articulation – le premier correspondant aux recommandations des conventions de l’ISB (International So-ciety of Biomechanics) et le deuxième étant défini à partir des données des mouvements des articulations et s’appuyant sur la théorie des torseurs.

Une fois les modèles définis, nous développons dans la section 2.3 les notions d’attache-ment des câbles du robot parallèle aux segments du corps, permettant de considérer ces segments comme les organes eﬀecteurs des robots parallèles. Nous expliquons la géométrie et expliquons la construction d’un collier flexible permettant l’attachement des câbles et l’installation de capteurs additionnels.

A partir de l’utilisation des modèles décrits et des colliers flexibles utilisés (les organes eﬀecteurs du robot parallèle), nous pouvons finalement considérer les méthodes de suivi des articulations. Dans la section 2.4, nous expliquons les paramètres à identifier ainsi que les diﬀérentes étapes de fusion de données eﬀectuées à partir d’un filtre de Kalman.

**Overview**

In this chapter, we discuss the specifics of the standard studies that focus on knee analysis and outline how their conclusions have aﬀected our decisions in the design and conception of our method. Tracking a human joint necessitates that a biomechanical kinematic model is created. This model represents the joint and the body segments connected by the joint. Section 2.2 deals with the choice of joint model for the knee.

To observe the motion of this model, our parallel robot system is designed to be attached to the human body and treat the body as its end-eﬀector. However, cables cannot be directly attached to the human body and this entails that some type of collar must be strapped. We also intend to add additional sensors to facilitate analysis and also allow easy comparisons with existing solutions. Thus, the next stage of our analysis details the robot end eﬀector – the flexible collar. The construction and geometry are described in Section 2.3.

Finally, once we have the model of the joint, and the robot end-eﬀector, we focus on steps taken to track the joint. Data from the sensors allow us to estimate the pose of the collar, and thus the body segments. This is described in Section 2.4. This chapter gives a top-down view of the approach we use.

### Biomechanical modeling

**Anatomical & Technical Frames**

Observing the motion of human joints involves observing the body segments linked by the joints. To track the knee joint, we must observe the femur and tibia and deduce the joint parameters from the motion data. We do this by attaching coordinate systems to the reference frames fixed to the tibia and femur and observe the relative motion between the two frames. These frames, based on anatomical features and landmarks, can be called Anatomical Frames.

An image-based computational method [Chao et al. 2009] allows for creating three dimensional models to which we can accurately assign coordinate systems using landmarks. However, such meth-ods are diﬃcult to implement in a clinical setting for regular diagnostic purposes. Thus we are tasked with using sensor data and their common local frame, called as the Technical Frame [Cappozzo 2009]. A rigid transformation between the Anatomical and Technical Frames can be obtained from some anatomical calibration methods.

We first describe the classical joint coordinate system that is used for the knee joint, followed by a description of the functional axes based system actually used here.The classical joint coordi-nate system is a type of Anatomical Frame defined using anatomical features, while the functional coordinate system uses Technical Frames to define an anatomical frame.

**Classical Joint Coordinate System**

A mathematical model of the body segments can be created by setting up the joint coordinate system using the recommendations of the International Society of Biomechanics [Wu and Cavanagh 1995], based on the work by E. S. Grood [Grood and Suntay 1983]. This system possesses the advantage of defining rotations that are sequence independent. They are also easy to interpret clinically.

Referring to Fig. 2.1, the sagittal plane for a body-segment is defined as the plane normal to the axis Z in the figure. In case of normal walk, this is the plane in which the body stays approximately, and which includes the vector defining the direction of motion. In this example figure, YX defines the plane, while X is the direction of motion. The sagittal planes for femur and tibia will thus be the planes parallel to this, passing through the center of the respective body segments. The front direction in this plane is referred to as the anterior direction, and the back as the posterior.

The body segment features are further identified by their respective directions from the center. A median plane is the plane parallel to the sagittal plane but passes through the center of the body, and divides it into the right and left halves. Features on body segments which are closer to the median plane are identified as “medial”, while those farther away are identified as “lateral”. Thus, on the right half of the body, the lateral direction is given by the vector normal to the median plane, pointing to the right, as given by axis Z (or axes Z1, Z2, Z3 etc) in Fig. 2.1. For body segments on the left half, this is the medial direction.

Finally, the transverse plane is normal to the sagittal and median plane, and divides the body into the upper and lower halves. The plane allows us to define the proximal direction as that vector pointing towards the transverse plane, while the distal direction as that pointing away. In Fig 2.1, the axes Y2 and Y3 (i.e pointing upward) refer to the proximal directions, while Y7 (pointing upwards, but axis fixed to upper body) defines the distal direction.

We describe the coordinate system for each bone first, and follow it by defining the joint reference axes. The figure 2.2 shows the coordinate system and we provide the definitions here.

#### Femoral coordinate system

The origin is located at the center of inter-condylar eminences of the femur.

Yp The body-fixed axis given by the proximal-distal direction. Proximal direction is considered posi-tive.

Xp Axis in femoral sagittal plane (along anterior-posterior direction), positive direction directed anteriorly.

Zp As defined by right-hand rule.

Tibial coordinate system

Origin located at the center of tibial condyles.

Yd The body-fixed axis given by the proximal-distal direction. Proximal direction is considered posi-tive.

Xd Axis, along anterior-posterior direction, directed anteriorly.

Zd As defined by right-hand rule.

Joint coordinate system

Yj Along Yd, i.e the tibial axis. Rotations about this axis are the internal/external rotations

Zj Along Zp. Rotations about this axis are the flexion/extension rotations.

Xj This is a floating axis, the common perpendicular to Yj and Zj. The positive direction is as defined by the right hand rule. Rotations about this axis are the ab/adduction rotations.

**Functional Coordinate System (FCS) – Independence from Anatomical Land-marks**

A cursory glance at the ISB recommendations in the section 2.2.2 tells us that the system is highly dependent on the location of anatomical landmarks. Studies show [Marin et al. 2003] that joint angles are not reproducible if they are calculated using coordinate systems set up with anatomical landmarks. In fact, locations of landmarks themselves are subject to errors, as reported in [Croce et al. 1999] where inter-examiner landmark position errors were up to 25 mm. While flexion/extension angle calculations are not dramatically aﬀected, this leads to crosstalk eﬀects and subsequent overestimation of ab/adduction and internal/external angles.

A functional approach defines a coordinate system using active motion data and is free from measurements of anatomical landmarks.

**Instantaneous Helical Axes**

One of the fundamental results used in Screw Theory is that any pose change of a rigid body can be eﬀected by a single rotation about an axis, followed by a displacement along the axis [Ball 1900]. This instantaneous axis (or helical axis – HA) for the motion of the femur with respect to the tibia can be obtained without the need for anatomical landmarks. Thus, knee joint motion can be interpreted in terms of the HA [Wolf and Degani 2006], by observing the motion of the HA and then determining the rotation of the femur with respect to the tibia, about the HA.

Let B denote the 3×3 orthonormal matrix that describes the orientation of the femur with respect to a global fixed frame. Let M denote the corresponding matrix for the tibia. Then the relative motion between the two, at time step i in a sequence of motion, is given by the rotation matrix J* _{i}* as,J

*= B*

_{i}

_{i}*M*

^{T}*The instantaneous axis is the axis of rotation*

_{i }*η*, corresponding to the rotation matrix J

*, which also provides us the angle of rotation about this axis.*

_{i}**Functional Alignment**

However, the above described approach is very diﬃcult to interpret clinically. For example, the helical axis in a motion involving flexion will not correspond to that in internal/external rotations. These axes also may not correspond to anatomical features and make it diﬃcult for a clinician to interpret the results. Thus we develop a Functional Coordinate System – an alignment that uses helical axes to define a clinically interpretable and anatomical landmark independent coordinate system.

The methods are described by [Gamage and Lasenby 2002; Mannel et al. 2004; Marin et al. 2003; Rivest 2005], but assume that data are represented in a coordinate system fixed to one of the body segments. These and other methods, surveyed in [Ehrig et al. 2007], provide a framework to define an anatomically interpretable functional coordinate system. The axis identification is done using data obtained from a reproducible motion.

A full squat is one such reproducible motion. It loads the knees equally and the entire range of flexion can be covered. The motion is broken down into finite sequences of motion, and the axes obtained from each step can be used to define an average axis with respect to the individual body segments [Mannel et al. 2004].

The Functional Alignment approach that we intend to use bases its ideas on the above mentioned methods and refines it. The method is defined in [Ball and Greiner 2012] and we provide a brief overview here. The method uses Eq. (2.1) as a starting point to derive a spatial average of the set of rotation matrices J* _{i}* in order to get the single best-fit Axis of Rotation (AoR). The goal is to derive joint-specific reference frames M

*and B*

_{A}*that re-interpret J around a movement-determined AoR. We calculate ‘functionally aligned’ rotation matrices F*

_{A}*: F*

_{m}*[*

_{m}*i*] = B

_{A}*J[*

^{T}*i*]M

*and decompose these matrices to Cardan angles.*

_{A }The Cardan angle representation expresses the orientation using rotations about three indepen-dent axes. This is particularly convenient as it allows us to align the axes with the anatomical features [Chao 1980]. The orientation could be expressed as flexion/extension, ab/adduction and internal/external rotation of the knee (or any joint). These angles are clinically understood and provide an easy reference to the clinician over the rotation matrices obtained from the Functional Alignment approach. A Cardan angle representation would thus facilitate analysis and diagnosis for a physiotherapist.

Thus, our task now is to obtain the body-segment pose matrices B and M for a sequence of motion to be analyzed, using the measurements provided by a set of sensors.

** Biomedical Robot Attachment**

**Collars**

Our aim is to obtain a set of transformation matrices to describe the pose of the thigh (femur) and the shank (tibia). For this purpose, we use a number of skin-based markers placed on the thigh and the shank. We also attach cable to the body segments. However, as briefly noted earlier, the cables cannot be attached directly to the body. Thus, we must use some collars that strap on the body segments and allow the cables of the parallel robot to be attached.

To ensure that markers on the same body segment are fixed rigidly with respect to each other, we attach them too via these collars. These collars thus act as the end eﬀector for the parallel robot, and hold the skin based markers and sensors. As each patient’s anatomical features vary, a rigid collar will be highly impractical and cost-ineﬀective. Thus, the collars must be adjustable.

In our setup, we propose to use two collars attached to the tibial shank (figure 2.3a), and one on the thigh (figure 2.3b). These adjustable collars are a series of aluminium plates connected by one-degree-of-freedom hinges, with each plate fitted with a pressure sensor. One link in this serial kinematic chain is a flexible, elastic strap that accounts for the variations in the patient’s characteristics. The hinges allow the collar shape to be changed to fit the patient limb as closely as possible, while the elastic strap permits the collar to be held firmly against the skin (figure 2.3c).

One advantage of this collar is that it reduces overall eﬀect of STA. As sensors are mounted on this collar, local variations in the skin surface position do not aﬀect sensor position. While STA will not be completely eliminated, it will aﬀect the entire collar and thus aﬀect all sensors. Thus, any individual sensor noise can be filtered out. As the plates comprising the collars are fitted with pressure sensors, they allow us to estimate muscle contractions and motions. Although not used in this thesis, the muscle contraction measurements from the pressure sensors would provide us information regarding local STA and may possibly aid in correcting their eﬀects.

The collars are used to hold accelerometer systems (MARG sensors), optical markers, force sensors and include attachment points to connect the passive and active wire systems. The collars are attached to the thigh and tibia, close to anatomical landmarks. We note that variations in patient anatomy results in changes in collar location and reinforces the need for the functional coordinate system described earlier.

The construction of the collar also allows for attaching variable length resistive wires between the collars on the tibia and the thigh.

**Construction & Geometry**

The collar used to fix sensors consists of styrene plates connected by hinge joints. These hinge joints can be fixed at a constant angle to function as a rigid joint. The plates can be unscrewed from the hinges and quickly replaced. The modular design of the collar ensures that the shape and size can be changed quickly. As a result, these collars can be adapted for use not just on diﬀerent patients, with diﬀering anatomical dimensions, but also for other experiments to measure motion of other joints.

Of the collars currently developed for the tibia, the lower collar, to be attached just above the ankle (see figure 2.3a) uses 5 plates, while the upper collar, attached just below the knee, uses 6 plates. All plates in the tibial collar except the front plates are rectangular planar plates with 4 screws fixed in a rectangular array. The two front plates have been molded in order to resemble the front crest of the tibial bone. This ensures that there is minimum possible slipping between the collar and the tibia. The femoral collar (figure 2.3b) has 6 rectangular planar plates with 4 screws each. Due to the higher amount of muscle around the femur, a shape-fit molded plate will not prevent slip and is thus not used in this collar.

The hinge angles and the length of the flexible strap will be the only parameters of the collar that will vary with each experiment. The collars are adjusted for each patient by adjusting the hinge angles and tightening the elastic strap. The hinge angles when fixed ensure that the collar shape does not change and prevent random collar movements.

In our experiment, the collars house the sensors. Since the shape of each collar changes with each experiment, the position of reference points on each plate, with respect to the first plate, change too. For each experiment with a new user, we need to determine the hinge angles in order to know the location of the reference and attachment points in the global reference frame.

**Table of contents :**

**1 Introduction **

1.1 Introduction

1.2 Gait Analysis

1.2.1 Overview

1.2.2 The Knee Joint

1.2.3 Knee joint tracking

1.3 Robots

1.3.1 Serial Robots

1.3.2 Parallel Robots

1.3.3 Robotized Rehabilitation

1.4 MARIONET-REHAB Gait Analysis system

1.5 Contents

**2 Theoretical Approach **

2.1 Overview

2.2 Biomechanical modeling

2.2.1 Anatomical & Technical Frames

2.2.2 Classical Joint Coordinate System

2.2.3 Functional Coordinate System

2.3 Biomedical Robot Attachment

2.3.1 Collars

2.3.2 Construction & Geometry

2.3.3 Labeling

2.4 Joint Pose Estimation

2.4.1 Unidentified Parameters

2.4.2 Sensor Data

2.4.3 Homogeneity & Decoupling

**3 MARIONET-REHAB **

3.1 Motivations

3.2 Wire driven Parallel Robots

3.2.1 Active Wire Measurement System

3.2.2 Passive Wire Measurement System

3.2.3 Combined system

3.3 Inertial Sensors

3.3.1 Synchronization

3.3.2 Drift free orientation

3.3.3 Output data

3.3.4 Reference Frame correction

3.3.5 Additional Sensors

3.4 Optical Motion Capture

3.4.1 ARENA software

3.4.2 Output data & timestamps

3.4.3 Reference frame correction

3.5 In-shoe Pressure Sensors

3.6 Other sensors

3.6.1 Collar Force Pads

3.6.2 IR Reflective sensors

**4 Calibration & Pose Estimation **

4.1 Sensor Collar Calibration

4.1.1 Kinematic Model

4.1.2 Parameter Identification

4.2 Pose Estimation

4.2.1 Notations

4.2.2 Data Track Cleaning

4.2.3 Collar Run-time Recalibration

4.2.4 Collar Pose Estimation

**5 Experiments & Analysis **

5.1 Experimental Trial

5.1.1 Sensor Configuration

5.1.2 Trials

5.2 Post-processing

5.2.1 Filtering

5.2.2 Data synchronization

5.2.3 Collar recalibration

5.3 Identifying flaws

5.3.1 Optical Motion Capture System

5.3.2 Mathematical model

5.4 Proposed improvements

**6 Concluding Remarks**

6.1 Contributions

6.2 Perspectives & Improvements

6.2.1 Sensor Placement

6.2.2 Collar Modifications

6.2.3 Setup Times

6.2.4 Mathematical Model

6.3 Future Scope

**Bibliography **

**Appendices **

A Mathematical Notes

A.1 Pose Estimation

A.1.1 Notations

A.1.2 Unweighted Least Squares Solution

B MARIONET-REHAB Software

B.1 Data Structures

B.2 Configuration File Format

B.2.1 Tibia & Femur definitions

B.2.2 Frame definition