Journal of Rehabilitation Robotics, 2013, 1, 93-98 93

E-ISSN: 2308-8354/13 © 2013 Synergy Publishers

Estimation of Forces and Moments of Lower Limb Joints from Kinematics Data and Inertial Properties of the Body by Using Inverse Dynamics Technique

Subhra Chowdhury and Neelesh Kumar*

Biomedical Instrumentation Unit, CSIR-CSIO, Sector 30-C, Chandigarh-160030, India

Abstract: In this paper, the forces and its moments acting on hip, knee & ankle joints of the body have been estimated with the help of kinetic models for better biomechanics understanding of human gait. This helps in accurate measurement of segmental masses, acceleration, joint centers and moment of inertia acting at various joints. Free Body

Diagram (FBD) and Link Segment Model (LSM) are used for computing forces & moments using Inverse Dynamics (ID) technique. Available lower limb walking model is limited in terms of number of joint forces and moments are analyzed; so, the improved biomechanical model for kinetic analysis of human walk involving lower limb joints & muscles is

proposed which estimate the forces acting on the hip joint, knee joint & ankle joints. This was also performed to understand the cause of deviation in any movement by estimating the patterns of forces acting on lower limb joints. Result analysis provides input parameter for the development of prosthetic foot design. by informing the force and

moment values of lower limb joints. This analysis will also help for quantification of lower limb prosthetics.

Keywords: Lower limb prosthesis, Moment, Inverse dynamics, Free body diagram, Link segment model.

1. INTRODUCTION

Biomechanical modeling uses inverse dynamics

technique to compute kinetic and kinematic variables

such as moment, ground reaction forces, accelerations

etc. Dynamic measurement of these variables is a

complex task which inducts inaccuracies. Estimation of

ground reaction forces and moment of forces of lower

limb joints is possible when full kinematic description

and inertial properties are available. Kinetic models are

used to describe forces and moment of forces

associated with linear and angular acceleration acting

on the lower limb joints and muscles which includes

both external forces and internal forces.

Human Gait modeling involves dynamic locomotive

model to predict the forces and moments at lower limb

joints. These ID equations were derived using the free-

body diagram and using inverse dynamics [1]. A

mathematical model for the dynamics of human

locomotion was proposed. Using moment histories, a

typical gait pattern can be demonstrated [2]. The total

moment of force values of ankle, knee, and hip joint

can support the body during stance when combined

together. The resultant moment was defined as the

addition of moment of knee joint and the subtraction of

moment of hip and ankle joint [3]. To estimate the force

and moment of lower limb joint muscles, a computer

model was used to simulate the effects of joint

*Address correspondence to this author at the Biomedical Instrumentation Unit, CSIR-CSIO, Sector 30-C, Chandigarh-160030, India; Tel: 91-172-2637165; Fax: 91-172-2637165; E-mail: neel5278@gmail.com

replacement [4]. The kinematic data were obtained in

X, Y, and Z direction. A model was used to control the

whole body balance during human walking. It assessed

the effects of forces, acceleration & joint moments

which are acting on foot & hip but the balance of trunk

& swing leg was maintained by active hip abduction

moment [5, 6]. Human gait is a dynamic process of

transferring body weight without loss of equilibrium

from one place to another and lower limb prosthesis

can mimic the natural load bearing process which is an

integral part to share load bearing for locomotion [7].

The dynamic equations of motion were used to

calculate the proximal end forces and moments from

the distal end forces and moments for each body

segment. The method started at the foot and continued

up the limb [8]. Motion is caused by the imbalances of

the internal and external forces and moments at joints

[9]. Once the forces and concentrated joint moments

are determined, the contribution of each individual

muscle can be approximated. Modeling also permits

the study of motions in response to applied forces [10].

Direct dynamic modeling is a difficult task and the

process involves imposing forces on the system and

differential equations [11]. The method of determination

of forces and moments at each lower limb joint can be

performed by simple repetition of the free body model

of each segment [12]. Kinematic and kinetic patterns

were examined to understand variations and their

cause and effect relationships. Kinetic force pattern

variability was high for the hip and knee joints [13].

Force variability between subjects was considerable

with variations in cadence [14]. Walking is the most

convenient way to travel short distances and the heel is

94 Journal of Rehabilitation Robotics, 2013, Vol. 1, No. 2 Chowdhury and Kumar

the first part of the foot to touch the ground in walking;

therefore they play an important role in between the

body and ground during walking [15]. The study of

human gait is broadly classified into kinematics (i.e.,

the study of the motion of bodies with respect to time,

displacement and velocity either in a straight line or in a

rotary direction) and kinetics. The study of the forces

associated with motion and forces resulting from

motion are known as kinetics. The study of kinetics of

human movement plays an important role because it

allows us to gain the basic mechanics of movement. It

is useful for finding the cause of deviation of any

movement by estimating various patterns of the forces

and explains how to calculate force and moment of

forces using kinematic and inertial properties. Here the

reaction forces and muscle moments were calculated

using Link-segment model. An inverse dynamics

technique was used to compute forces and moment of

forces based on the kinematics and inertial properties

of the body. In free body diagram, segments are

broken which is used for presenting and calculating

unknown forces, moment of forces. Existing lower limb

models used inverse dynamics technique to estimate

lower limb joints but the numbers of joints analyzed are

limited. In this project the force and moment estimation

of each lower limb joint performed to give an idea for

limitation of force bearing values of lower limb

prosthetics. The aim of this paper is to provide design

inputs for the prosthetic feet development by giving the

approximated force and moment values of lower limb

joints. This experiment was performed to predict

quantification of lower limb prosthetics.

2. MATERIAL AND METHODS

To study the cause of deviation during different

phases of human gait, these experiments are useful

which uses Newtonian mechanics. It started with initial

conditions and then applying input values (i.e.

segmental mass, acceleration, moment of inertia etc.),

moment and force values were calculated with the help

of the properties such as segmental mass, center of

mass and radius of gyration.

For computing the internal kinetics of human

movement, it is necessary to have kinematics and

anthropometric parameters. The physical

characteristics like subject’s body weight can be

calculated by using Kistler force platform. As multiple

segment analysis of lower limb kinetic model assumes

frictionless joints and body segment treated as rigid bar

so the effect of friction was not assumed here. The

inertial parameters such as segmental mass, location

of center of mass, acceleration and segmental mass

moment of inertia are difficult to determine for a living

person, so it can be calculated with the help of values

from Dempster’s body segment parameter table. The

table has been created showing segmental mass as

proportion of total body mass, location of center of

gravity and radius of gyration (for angular acceleration)

as proportion of segmental length. With the help of the

approximate proportion values given in Dempster's

table, inertial values for a subject assuming weight

75Kg was calculated (Table 1) as here the force and

moment values for a 75kg subject have been

estimated. These inertial values of a subject play an

important role for estimating lower limb force and

moment values.

2.1. Mechanical Model of Lower Limb Joints

Based on the anatomical model (Figure 1A), a link

segment model (Figure 1B) is developed to estimate

internal forces and moment of forces of lower limb

joints. In the link segment model each segment is

replaced by product of mass and moment of inertia

located at each segment’s center of mass. A free body

diagram is a simple sketch including all of the forces

and moment of forces which helps to visualize the

direction of reaction forces acting on the body. Free

body diagram of lower limb segments which is mapped

to the anatomical model is shown in Figure 1C. The link

segment model is used to calculate the ground reaction

forces & muscle moments but a FBD of each segment

is also required which is broken at the joints & the

forces that act across each joint is estimated. In case of

FBD of lower limb, the segments represents the lower

limb joints were drawn in minimalist form (single line),

Table 1: Anthropometric Values for a 75 Kg Male

Acceleration (m/s2) Segment Mass (Kg) Moment of Inertia (Kg.m

2) Length (m)

X Y

Angular acceleration (rad/s

2)

Foot 1.2 0.011 0.195 -4.39 6.77 5.12

Leg 2.4 0.064 0.335 -4.01 2.75 -3.08

Thigh 6.0 0.130 0.30 6.58 -1.21 8.62

Estimation of Forces and Moments of Lower Limb Joints Journal of Rehabilitation Robotics, 2013, Vol. 1, No. 2 95

which are free from the other bodies. Segment’s center

of mass was indicated and from that point

accelerations were drawn. After that external reaction

forces and moment of forces along with unknown

forces were drawn with their directions. At the mass

center, the force of gravity was drawn and at the

proximal end of each segment reaction forces were

present.

2.2. Inverse Dynamics Technique

Net forces and moments cannot be measured

directly as they represent the net affect which produce

forces and moment of forces of each joint. So,

estimation of forces and moment of forces of hip joint,

knee joint and ankle joint were derived from the

kinematics and inertial properties of the body by

applying the process of inverse dynamics in the

developed FBD for motion analysis which explores

kinetics acting on human lower limb muscle and bone-

joints and finally presented the method for computing

the kinetics of human movements. For solving

unknown forces at ankle joint, knee joint and hip joint of

lower extremity, lower extremity was sectioned into

three segments (thigh, leg and foot) to determine the

interaction between them.

2.3. Estimation of Forces and Moments Using Newton-Euler Equation

The effects of force and moments on lower limb of

human body can be assessed using Newton-Euler

equation. The forces and moment acting at lower limb

joints were estimated by using parameters from

Dempster’s model (Table 1). In human body, a set of

forces act at a given point of time and it can be

possible to combine the forces into single resultant

force vector. Newton’s 2nd

law can then be considered

for the single resultant force where the resultant forces

were divided into unknown and known forces and the

unknown forces formed a single net force which can be

solved. The effects of forces & moment of forces acting

at the joints were determined where acceleration, mass

of each segment is known. The calculated net forces

and moment of forces represent the summation of net

effects of forces of respective lower limb joint structures

in producing movement. Each of the lower limbs joint is

responsible for the individual steps of human

movement. The effects of forces & moment of forces

acting at the joints were determined by using equation

1.1 and 1.2. In equation 1.1, ‘F’ represents force, ‘m’

represents mass, ‘a’ represents acceleration, In case of

equation 1.2, ‘M’ represents moment of force ‘I’

moment of inertia and ‘ ’ angular acceleration.

Newton Equation (Linear):

Fx = ma (1.1)

Euler Equation (Angular):

M = I (1.2)

2.4. Multi-Segment Analysis of Lower Limb

Complete analysis of human lower limb joint

segments was started at the most distal segment

(Foot) and continues proximally. For lower limb, only

the foot has three unknown parameters which can be

solved by using three equations. In the case of

proximal joints of lower limb (i.e. thigh or leg) there are

six unknown parameters at each joint. So the solution

Figure 1: (A) Anatomical model; (B) Link segment model; (C) Free body diagram of lower limb.

96 Journal of Rehabilitation Robotics, 2013, Vol. 1, No. 2 Chowdhury and Kumar

started with the most distal joint (ankle joint) that has

only one joint and after finding those three unknown

values one can precede to the adjacent segment. By

using Newton’s third law, it can be stated that the

forces on the distal end of one segment must be equal

and opposite to those of proximal end of the adjacent

segment. So, the unknown forces at ankle joint must be

in opposite direction to the proximal end of knee joint

and same for hip joint also. This is the method for

solving kinetics of a stance limb but when solving the

kinetics of swing limb, the process is same but the GRF

values are zero.

3. RESULTS

Matlab programs were developed for calculating

forces & moment of forces acting on the ankle joint,

knee joint and hip joint of lower limb joints, when the

ground reaction forces and the body inertial properties

are constant. By using the equations developed in the

MATLAB program the vertical, horizontal force and

moment values have been estimated. The other

program estimates the forces and moment for the

stance phase in the gait cycle through which force and

moments estimation of a human body is possible when

each segment’s mass, location of center of gravity and

segmental moment of inertia are known. Results

obtained are reported in Table 2. It can be concluded

that in case of lower limb joints, force values of the

horizontal direction are lesser than the force values of

vertical direction. The moment values decreases from

the ankle to hip joint and vertical force values increases

from the distal to proximal joint.

4. DISCUSSION

The project has been performed for evaluating

forces and moment values of lower limb joints during

walking so that one can predict the quality for

prosthetic feet during gait. The project also provides a

reference database for further research in clinical gait

analysis.

Inverse dynamics technique and Euler equation was

used to find out the force values and to estimate the

moment values by using free body diagram of the joints

of lower limb.

4.1. Interpretation

Estimating the forces and the moment of forces by

using MATLAB from the developed lower limb model

shows that the absolute moment value increases from

the ankle to hip joint and force value increases from the

distal to proximal joint. The knee and ankle forces

produced as a result of this work would be very

essential while designing the prosthetic foot. By using

these outcomes from this experiment, quality prediction

of a developed prosthetic foot is possible.

MATLAB Program Equations

a) To find the values of forces and moments of

ankle joint

A_x=m_f*a_x;

A_y=((m_f*g)+(m_f*a_y));

M_A=((I*alpha)+(A_x*d_y)+(A_y*d_x)):

b) To find the values of forces and moments of

knee joint

K_x=((m_l*a_x)-A_x);

K_y=((m_l*a_y)+(m_l*g)+(A_y));

M_K=((I*alpha)+(A_x*d_y_a)+(A_y*d_x_a)+(K_x*d_y_

k)+(K_y*d_x_k)+(M_A));

c) To find the values of forces and moments of Hip

joint

H_x=((m_t*a_x)-K_x);

H_y=((m_t*a_y)+(K_y)+(m_t*g));

M_H=((I*alpha)+(H_x*d_y_h)+(H_y*d_x_h)+(K_x*d_y_

k)+( K_y*d_x_k)+(M_K));

Table 2: Estimated Force and Moment Values for Lower Limb Joints

Lower limb Joint Horizontal force (AX) (N) Vertical force (Ay) (N) Moment (M) (Nm)

Ankle joint -5.27 19.9 -1.1

Knee joint -14.9 50 -9.1

Hip Joint 24.6 101.6 -11.7

Estimation of Forces and Moments of Lower Limb Joints Journal of Rehabilitation Robotics, 2013, Vol. 1, No. 2 97

Earlier model has analyzed the limited number of

joint forces and moments thus the improved model is

proposed. The force and moment values of hip joint are

greater than the values of knee and ankle joint because

hip joint can bear the maximum amount of forces

during walking and the pressure generated by the

ankle and knee joint are lesser than the hip joint. So,

the moment and force values of hip joint have

maximum value whereas ankle joint has minimum

value.

4.2. Implication

From the results of the force and the moment

estimation of lower limb joints, lower limb prosthetic

design inputs can be obtained which can mimic the

natural foot function. With the help of the force and

moment values, the force bearing limit of a prosthetic

foot can be obtained.

4.3. Limitations

The described model has some limitations which

are needed to be overcome and are as follows: it

assumes generalized and frictionless joints along with

fixed segmental length and fixed center of mass so the

effect of friction and joint structures were not

considered. The present model is sensitive to its input

data; measurement errors (i.e. skin motion artifacts and

captured motion) which can affect the moment data.

This model can estimate forces during stance phase of

gait cycle and has limitation for swing phase as it is

critical to estimate anthropometric parameters during

this phase. For development of prosthetic knee it is

needed to get the information for full gait cycle,

therefore besides the stance phase data, swing phase

data needs to be evaluated.

5. CONCLUSION

The paper has objective to evaluate forces and

moment of lower limb joints during walking. This

estimation predicts the shock absorption values and

quality for prosthetic feet during gait. The findings from

the modeling of lower limb walking can be used as a

design input for prosthetic foot development; a

prosthetic foot can be developed which can mimic the

natural foot function. It is useful for finding the cause of

deviation of any movement by estimating various

pattern of the forces and also explains how to calculate

force and moment of forces using kinematic and inertial

properties. The estimation of lower limb joint forces and

moment of forces can predict the force and moment

values i.e. shock absorption limits for lower limb

prosthetic design which can give the information about

performance of prosthetic knee and significance in

developing the idea about prosthetic feet design. It

provides a reference database for further research in

clinical gait analysis. The information obtained from this

experiment suggests any rectification required for the

lower limb prosthetic. The knee and ankle forces

produced as a result of this work would be very

essential while designing the prosthetic foot. By using

these outcomes from this experiment, quality prediction

of a developed prosthetic foot is possible.

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Received on 24-09-2013 Accepted on 26-11-2013 Published on 31-01-2014

DOI: http://dx.doi.org/10.12970/2308-8354.2013.01.02.3

© 2013 Chowdhury and Kumar; Licensee Synergy Publishers.

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