Paper Kinematic Consistency Revised.pdf

Kinematic Data Consistency in the Inverse Dynamic Analysis of

Biomechanical Systems

M.P.T. SILVA and J.A.C. AMBRÓSIO

IDMEC-Instituto de Engenharia Mecânica, Instituto Superior Técnico

Avenida Rovisco Pais, 1

1049-001 Lisboa, Portugal.

Abstract: Inverse dynamic analysis is used in the study of human gait to evaluate the reaction forces

transmitted between adjacent anatomical segments, and to calculate the net moments-of-force that result

from the muscle activity about each biomechanical joint. The quality of the results, in terms of reaction

and muscle forces, is greatly affected not only by the choice of biomechanical model but also by the

kinematic data provided as input. This three-dimensional data is obtained through the reconstruction of

the measured human motion. A biomechanical model is developed representing human body components

with a collection of rigid bodies interconnected by kinematic joints. The data processing, leading to the

spatial reconstruction of the anatomical point coordinates, uses filtering techniques to eliminate the high

frequency components arising from the digitization process. The trajectory curves, describing the

positions of the anatomical points are obtained using a form of polynomial interpolation, generally cubic

splines. The velocities and accelerations are then the polynomial derivatives. This procedure alone does

not ensure that the kinematic data is consistent with the biomechanical model adopted, because the

underlying kinematic constraint equations are not necessarily satisfied. In the present work, the

reconstructed spatial positions of the anatomical points are corrected by ensuring that the kinematic

constraints of the biomechanical model are not violated. The velocity and acceleration equations of the

biomechanical model are then calculated as the first and second time derivatives of the constraint

equations. The solution to these equations provides the model with kinematically consistent velocities

and accelerations. The procedures are demonstrated through the application to a normal cadence stride

period and the results discussed with respect to the underlying principles of the techniques used.

Key words: biomechanical model, kinematically consistent data, inverse dynamics, data filtering, motion

reconstruction, natural coordinates.

1. Introduction

The evaluation of the muscular actions and internal forces at the human body articular

joints is of major importance in different areas of medicine, sports, physical rehabilitation

or biomedical engineering. However, there are no experimental methodologies that can

measure these forces directly. Therefore, the human motion studies rely on mathematical

and computational models in general, and multibody biomechanical models in

particular, to evaluate the intersegmentar reaction forces as well as the muscle forces

and their net moments-of-force about the anatomical joints without external interference

on the motion of the subject of the analysis.

The construction of a supporting biomechanical model requires the knowledge of its

important functions and the identification of the experimental or numerical data

required as input and the system dynamics response that is to be reported [1].

Furthermore, their application in the context of inverse dynamics analysis requires that

the kinematics of the human motion, i.e., the position, velocities and accelerations of the

anatomical points, is known in advance. This data set is generally obtained by standard

reconstruction methods based on the Direct Linear Transformation technique (DLT) [2].

The biomechanical multibody models used in the inverse dynamics procedure are

developed using any of the multibody formulations available [3]. The most common

procedures use classical dynamic approaches to obtain the reaction forces and the

moments-of-force at the joints by formulating and solving the dynamic equilibrium

equations of each anatomical segment of the human body, starting from the segments

further away from the torso and moving inwards along the kinematic chain under

analysis. The unknown forces and moments at each segment are used, after being

calculated, as external applied forces to the preceding segment in the kinematic chain

[4]. Alternatively, the general multibody dynamic formulations assemble and solve, in

a systematic way, the complete set of the equations of motion of the system, for each

time step. These models are general and allow for their use in different motion

scenarios, even those that include kinematical loop closures, for instance when the

hands grip a golf club or a steering wheel. The biomechanical model used in this work

for gait analysis was proposed by Celigueta [5] and by Silva et al. [6]. The model has

33 rigid bodies connected by 32 kinematic joints representing 16 anatomical segments.

The muscle action may be obtained by having each particular group of muscles, defined

as those with similar functions and common anatomical insertions, modeled

independently and included in the biomechanical model [7]. This leads to an

indeterminate problem, in terms of the unknown forces, that can be solved using the

optimization theory [8]. Alternatively the actions of the different muscle groups can be

lumped as moments about anatomical joints leading to a determinate inverse dynamics

problem [9]. It can be shown that the problem of force sharing by the muscles can be

viewed as an optimal problem where some of the constraints simply impose that the

muscle net moments-of-force about any given joint must be equal to those obtained in

the determinate inverse dynamic analysis. This procedure is generally referred to as

static optimisation [10]. Therefore, the correct evaluation of the net moments-of-force

obtained in the inverse dynamic analysis procedure in fundamental, regardless of the

methodology used to evaluate the muscle and anatomical joints reaction forces.

The kinematic data required for the inverse dynamic analysis of the biomechanical

model needs the trajectories of 23 anatomical points, located at the anatomical joints

and segments extremities, which are obtained by using the DLT procedure [2]. The

ground reaction forces must be acquired with force platforms synchronized with the

cameras [11]. Both of these acquisition processes are prone to errors due to signal noise,

operator imprecision or finite precision of the equipment. Therefore, the experimentally

acquired data must be filtered before it can be used in the inverse dynamic analysis.

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Different filtering methods can be used in the experimental data filtering, for noise

reduction. Among these, the Butterworth 2 nd order filters [12] and the Fourier series

with optimal regularization [13] are the most commonly used. The selection of the best

filter for each type of application is a question not completely settled [14]. Taking into

account that the applications foreseen in the present work aim to gait analysis, which is

characterized by a certain noise level in the points trajectories and force platform data, it

is used the 2 nd order Butterworth filter with the zero phase-shift technique and cut-off

frequencies on the range of 2-8 Hz [12].

Either because the acquisition of the kinematic data is generally done independently of

the biomechanical model effectively used or due to the filtering procedure, the

processed kinematic data does not ensure that the kinematic constraints associated to the

biomechanical model are fulfilled. The inverse dynamic analysis also requires that the

system velocities and accelerations are known. A common process to obtain those

involves the use a polynomial interpolation of the coordinates and its time derivatives.

This procedure does not ensure that the constraint velocity and acceleration equations

are fulfilled, even if the position data is kinematically consistent. Consequently,

spurious joints reaction forces and net moments-of-force, associated to the constraint

violations, are generated in the solution of the inverse dynamic problem [15].

To ensure the consistency of the kinematic data with the constraints of the

biomechanical model it is proposed here that the kinematic positions are modified in

order to fulfil the constraint equations. Furthermore, the velocity and acceleration of the

system are obtained by using the velocity and acceleration equations respectively. The

proposed methodology is applied to the analysis of the human gait with a normal

cadence in order to obtain the intersegmentar forces and the muscle net moments-of-

force. The need and implications of the use of the filtering procedures and the

enforcement of the kinematic data consistency are discussed in the process.

2. Input Data Conditioning

Inverse dynamic analysis of a mechanical system requires, the knowledge of its motion

and all external applied forces. The motion of the system is described by the kinematic

information necessary to define the position and orientation of each anatomical

component during the analysis period. The external applied forces, obtained using force

plates, provide all information necessary for the construction of the system force vector.

The motion of the system consists of the trajectory of a set of anatomical points located

at the joints and extremities of the subject under analysis, as depicted in Figure 1. In the

present work, these curves are obtained through a digitization process in which the

images collected by four video cameras are used to reconstruct the three-dimensional

coordinates of the anatomical points. This reconstruction process uses DLT [2,16] to

convert the two-dimensional coordinates of the video images into three-dimensional

Cartesian coordinates.

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15 ≡ 16

14 ≡ 17

5 ≡ 6

11 12

13 ≡ 18

12 ≡ 19

11 ≡ 20

4 ≡ 7

3 ≡ 8

2 ≡ 9

Figure 1. Set of anatomical points.

1 ≡ 10

The motion reconstruction process, by nature, introduces high frequency noise into the

trajectory curves of the reconstructed anatomical points. This noise occurs due to

several factors such as digitalization errors and the finite resolution of the two

dimensional images. In order to make these trajectories suitable for use in the inverse

dynamic analysis, a filtering procedure is applied with the objective of reducing the

noise levels. A Butterworth 2 nd order, low-pass filter [12,17], with the zero phase shift

technique is applied with properly chosen cut-off frequencies, reducing the noise levels

and smoothing the trajectory curves.

A similar filtering procedure is applied to the external applied forces, in order to reduce

the noise levels introduced during the acquisition process. The external forces are

measured using force plates that convert an analog electric signal into a digital signal.

The noise introduced in the external force measurement is associated with factors such

as the cross talk between force plates and the conversion from an analog to a digital

signal. In the present work three force plates are used in the measurement of the ground

reaction forces. In Figure 2, the apparatus of the gait lab, with the three force plates and

the four video cameras, is presented. The use of four video cameras not only improves

the chances for the anatomical points to be visible all time in at least one of the cameras,

but also provides an extra set of equations for the DLT method.

Cam #3

Top View

Cam #2

Plate #1

Plate #2

Plate #3

Subject

Forward Direction

Cam #4

Cam #1

Figure 2. Overall apparatus of the gait lab.

-4-

15 ≡ 16

14 ≡ 17

5 ≡ 6

13 ≡ 18

11 12

12 ≡ 19

11 ≡ 20

4 ≡ 7

3 ≡ 8

2 ≡ 9

1 ≡ 10

Top View

Cam #3

Cam #2

Plate #1

Plate #2

Plate #3

Subject

Forward Direction

Cam #4

Cam #1

The ground reaction forces are obtained independently for each foot during the trial. In

order to reduce the noise levels in the external force curves and center-of-pressure

curves, the Butterworth 2 nd order low pass filter is once again applied with cut-off

frequencies ranging from 10-20 Hz for the forces and 3-5 Hz for the center-of-pressure

curves. The choice for the correct cut-off frequency is based on a residual analysis [12].

3. Biomechanical Model Description

A whole body response biomechanical model of the human body is used in the inverse

dynamic analysis of the stride period. This model is defined using 33 rigid bodies. The

rigid bodies are connected by revolute and universal joints in such a way that 16 major

anatomical segments can be identified. A description of the 16 anatomical segments

and their corresponding rigid bodies is presented in Table I. Figure 3 illustrates the 16

anatomical segments and the underlying kinematic structure of rigid bodies and

kinematic joints. Considering this kinematic structure, an open loop topology can be

identified, with a base body described by rigid body number 7, and 5 kinematic

branches defined by the 4 limbs and the head/neck. The model has 44 degrees-of-

freedom that correspond to 38 rotations about 26 revolute joints and 6 universal joints,

plus 6 degrees-of-freedom that are associated with the free body rotations and

translations of the base body.

Revolute joints in this model are defined in two different ways, as depicted in Figure 4,

for the example case of the elbow joint. The revolute joints between two adjacent

anatomical segments are defined using an anatomical point and a joint direction unit

vector that are shared by the adjacent rigid bodies. When describing the axial rotation

within an anatomical segment, revolute joints are defined using two anatomical points

that are shared by two different rigid bodies of the same anatomical segment.

T able I. Anatomical segments description with the topology of rigid bodie s.

Segment Nr.

Description

Rigid Body Index

Lower torso

6,7,8

Upper torso

19,20,21,22,23,24,25

Head

Right upper arm

17,18

Right lower arm

15,16

Left upper arm

26,27

Left lower arm

28,29

Right upper leg

4,5

Right lower leg

2,3

Left upper leg

9,10

Left lower leg

11,12

Neck

31,32

Right hand

Left hand

Right Foot

Left foot

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