Kinematics of the propulsion of a wheelchair · the limb and a high frequency of propulsion are...

Kinematics of the propulsion of awheelchair

J.G. Gruijters

D&C 2010.030

Traineeship report

Coach(es): PhD. BEng. BSc. M. Leary, RMIT, Australia

Supervisor: prof.dr. H. Nijmeijer

Eindhoven University of TechnologyDepartment of Mechanical EngineeringDynamics & Control

Eindhoven, October, 2010

Abstract

This report describes research in the field of manual wheelchair propulsion. The goal is to develop apredictive model to describe the loads in the limb of the wheelchair user. This model can eventually beused to predict performance and enable optimalisation of wheelchairs for different people and differentpurposes.During this research two models have been examined. The first model was found in the literature andis quasi-static. This model was reproduced to validate the results. An error in the original shouldermoment in the original article was found and corrected. With the knowledge of this model a new threedimensional model of an arm was build. This model also included the dynamics of the arm.Both models have been simulated. During the simulation of the first model, the seat height is changed.For the second model the seat height and a more forward / backward seat position is simulated.The results of the first model show that the seat should be at the lowest position to minimize the elbowmoment, and there exists an optimal seat position for the shoulder moment.The results of the second model are more complex. For this model the first derivative of the momentsis also taken into account, besides the peak values of the moments. These results show some contradic-tions, so more knowledge of how these factors contribute to injuries is necessary.

Contents

1 Introduction 11.1 Quantifiable factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature overview: Experimental research . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Literature Overview: Model-based research . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Richter model 42.1 Parameters of the Richter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Outputs of the Richter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 A three dimensional model 83.1 Parameters of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 The degrees of freedom at the joints . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 Lengths of the arm segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.3 Estimation of the position of the centre of mass . . . . . . . . . . . . . . . . . . . 133.1.4 Masses of the segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.5 Moments of inertia of the segments . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.6 Hand-rim forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.7 Wheel angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.8 Position of wheel hub with respect to the shoulder . . . . . . . . . . . . . . . . . 15

3.2 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Outputs of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Results of the Richter model 184.1 Changing the seat height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Comparison with SAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Comparison with ADAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Results of the three dimensional model 245.1 Varying the seat height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.1.1 The shoulder moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.1.2 The elbow moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.1.3 The wrist moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.1.4 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.2 Varying the seat position to front / rear . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2.1 The shoulder moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2.2 The elbow moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.3 The wrist moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

ii

5.2.4 Overview of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6 Conclusions and recommendations 376.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A The arm lengths 41A.1 Upper and forearm length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A.1.1 Handbook of normal physical measurements . . . . . . . . . . . . . . . . . . . . 41A.1.2 Occupational Biomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.1.3 Adultdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A.2 The used segment lengths in the 3D model . . . . . . . . . . . . . . . . . . . . . . . . . 46

B The Richter model 48B.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48B.2 The free body diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48B.3 Calculations of the angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49B.4 Calculations on the forearm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52B.5 Calculations on the upper arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53B.6 Check the calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

C Implementation of the 3D model in Matlab 56C.1 The Cartesian coordinate frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56C.2 Moments of inertia about the joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57C.3 The kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57C.4 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58C.5 The non-conservative forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Glossary

a Length between the shoulder and the wristF Magnitude of the total force acting between hand and hand-rimF (r) Magnitude of the radial force acting between hand and hand-rimF (t) Magnitude of the tangential force acting between hand and hand-rim~F Force~g Gravitational acceleration~H Angular momentumI Unity tensorCM J Moments of inertia tensor about the center of massOJ Moments of inertia tensor about the end of a beamLa Length from the shoulder to the handLfa Length of the forearmLh Length of the handLhs Length of the shoulder to the wheel hub in the Richter modelLhub Length of the shoulder to the wheel hubLua Length of the upper arm~M Moment

M Number of constraints in Grübler countm Massmi Number of constraint directions in Grübler countN Number of bodies in the system in Grübler countNDOF Number of degrees of freedom~P Linear momentumPP Push progressQnc Non-conservative forcesq Generalized coordinates~rcm Vector from the joint to the centre of massRHR Radius of the hand-rimt TimeT Kinetic energyTe Moment at the elbowTinterval Time between two discretized time stepsTs Moment at the shoulderV Potential energyβ2 Angle to describe the position of the wheel hub w.r.t. the shoulder about the ~eO2 axesβ3 Angle to describe the position of the wheel hub w.r.t. the shoulder about the ~eO3 axes

iv

θc Contact angleθe Elbow angleθf Angle of the hand-rim forceθhs Angle of the shoulder with respect to the wheel hubθr Release angleθs Shoulder angleθw Wheel angleψ1 Roll at the shoulder jointψ2 Flexion at the shoulder jointψ3 Abduction at the shoulder jointψ4 Roll at the elbow jointψ5 Flexion at the elbow jointψ6 Flexion at the wrist jointψ7 Abduction at the wrist jointψ Column containing the degrees of freedomφ Camber angle of the wheelχ Toe angle of the wheel~ω Angular velocity

Chapter 1

Introduction

The interaction between man and machine is the subject of many studies. In the past a lot of researchhas been done in the field of manual wheelchair propulsion. This research is of great importance for thedisabled who rely on their wheelchair for locomotion as well as disabled athletes. When the literatureis reviewed, it is noticeable that the majority of the research is experimental work. To save time andcosts for research projects, a model is needed which can be used to predict factors which influence theperformance or cause painful injuries.

In general two groups of users can be distinguished. The first group exists of manual wheelchairusers which use the wheelchair for locomotion and everyday tasks. Pain which is caused by using thewheelchair and the endurance required to do everyday tasks are the main aspects of consideration forthis group. The second group exists of disabled athletes, which are mostly concerned about speed,endurance and maneuverability.

The hand-propelled wheelchair is most commonly used, although crank and lever propelled wheel-chairs are becoming more popular [9]. Because of the wide use of the hand-propelled wheelchair, mostresearch is done on these kind of wheelchairs. Model based research is not commonly used in this field.Proposed models are simplified systems with calculated centers of mass [5], arm-wheel models [26] ormulti-body models of the arm [13, 21]. A model can give more insight in the propulsion kinetics andkinematics and can be used for optimalisation. However, a model is a simplification of reality, so thevalidity for each application should be considered.

The goal of this research is the development of a model to predict seat positions which have the leastchance of injuring the user. The model must predict the moments in the joints of the arm which must bedelivered by the muscles. The parameters which can be changed are the wheel position and orientation.In the model this is covered by the place of the shoulder with respect to the wheel hub. In this reportthe main focus is on everyday users, because most research is done on everyday users, so more data isknown about this group.

1.1 Quantifiable factors

The two different groups of users which are taken into consideration are the daily users and athletes.Both groups have their own parameters which are used as a measure to compare seat positions.

For the group of daily users the prevention or minimization of the causes of pain is important.The shoulder and wrist joint are the most mentioned painful joints in the literature. For example Vander Woude et al. (2001) stated that at least 50% of the wheelchair users suffer from wrist complaints.From these users 30-50% has shoulder complaints too [9]. The factors which cause the pain have beenidentified in the literature. High peak values of the forces and moments, rapidly loading of the force on

1

2 CHAPTER 1. INTRODUCTION

the limb and a high frequency of propulsion are commonly accepted to be causes of upper limb pain[1, 14, 20, 23].

For the athletes propulsion performance is the primary objective. The performance can be measuredby different quantities or ratios. The power output can be seen as a quantity to compare different set-upsof a wheelchair. Another ratio is the Mechanical Efficiency (ME), which gives the relation between thepower output at the wheels and power provided by the body. The mechanical efficiency is influencedby different factors. An increase in speed for example will decrease the mechanical efficiency [1, 9].Another ratio is the Fraction of Effective Force (FEF) which gives the percentage of the total force directedin tangent direction to the hand rim. Although a higher effective force seems to be preferable, it issuggested that this leads to higher forces and moments in the limb [8], which can cause injuries.

1.2 Literature overview: Experimental research

As stated before, most work is experimental research. The population of wheelchair users is hetero-geneous because of the large range of disabilities. To ensure a homogeneous group of participants,some work is done on non-disabled participants propelling a wheelchair. The disadvantage of havingnon-disabled participants is the possible difference in propulsion style between everyday users, athletesand the non-disabled participants. In other studies the participants are athletes [17, 22] or everydaywheelchair users [1, 2, 4, 12, 14, 18, 19, 25]. One study used a mix of non-disabled and disabled partici-pants [23].

The hand-rim forces of the participants are measured by a few studies and the results are foundin the literature. It is hard to compare the results because of the differences in set-up and researchprotocol. Similarities for everyday wheelchair users can be found between some studies if the peakheight of the total force, tangential component and radial component are compared [1, 26]. On the otherhand, another study [23] shows a different characteristic for the tangential and radial component. Thetangential force is larger than the radial force in this study, which is different to what the other studiesshow. Athletes also show a larger tangential component in comparison with the radial component, butthe peaks of the components are higher [22]. Also the angle of the wheel used for propulsion is largerand they use a different part of the wheel for propulsion compared with the push angles of everydayusers found in the literature. Seat position is also used to optimise for speed or manoeuvrability or tominimise the causes of painful injuries for everyday users. To quantify the position, the horizontal andvertical distance between the wheel hub and the shoulder joint are often used. When the hub is shiftedforward relative to the shoulder joint, the push frequency is lower [3, 20, 14] as well as the ’rate of rise’of the total force [14, 20], which is the first time derivative of the force. By shifting the hub forward thestability is decreased, which increases the chance of ”tipping over”. When the vertical displacement ofthe shoulder relative to the hub is smaller, the push angle is increased [3, 12, 14, 20] as well as the pushfrequency [3, 12]. But a too low position can cause excessive abduction at the shoulder [20]. With a largerpush angle, the range of motion at the joints is also enlarged, which is potentially a cause of injury.

With these relations, suggestions are given to optimise the seat position. The stability and the in-crease of the moments in the arm are compromised by the optimalisation. So the reaction forces andmoments should be fitted to each individual and the stability level should be adapted to the level theindividual is comfortable with. When this is compared with the seat position of athletes, a difference isseen for the horizontal position. The seats of athletes are often more shifted forward, which increasesthe horizontal distance between the shoulder and the wheel hub. A possible explanation could be thedifferent propulsion technique of the athletes and a more competitive focus.

1.3. LITERATURE OVERVIEW: MODEL-BASED RESEARCH 3

1.3 Literature Overview: Model-based research

As stated before, most work in the field of wheelchair propulsion is experimental, but some modelshave been used. Brubraker (1986) modeled the wheelchair and the user with two centres of mass, onefor the upper body and one for the lower extremities and the wheelchair. His work mainly describesthe reasons why a wheelchair should be adapted to every individual [5]. Sullivan et al. (2005) showsa model he created of the whole upper body which is patient-specific [21]. A more general model iscreated by Pennestrì et al. (2006) [13]. The model of Pennestì is based on the bones and muscles of thearm and is used to simulate an arm driving a steering wheel of a car. Like the model of Sullivan, thismodel is three dimensional. Because of a lack of good three dimensional measurements of wheelchairpropulsion in the literature at this time, the model of Richter (2001) is also examined[26]. The modelis two dimensional, which is a more simplified model, but it gives a good view on how the moments inthe limb change during propulsion. Assuming that the arm and the wheel are in one plane could be agood approximation for higher seat positions, but the lower the seat, the more abduction at the shoulder.More abduction at the shoulder causes the arm to be out of the plane of the wheel-hub.

Chapter 2

The Richter model

To create a model, the two dimensional model of Richter (2001) is examined. The reason this model isused, is because all the required inputs are defined, so no extra measurements are necessary [26]. The

Figure 2.1: The Richter model

model is a multi-body quasi-static model. The model consists of a upper arm (Lua), the forearm plushand (Lfa) and the hand-rim. The end of the forearm is constrained to the hand-rim of the wheelchair.The shoulder, at the end of the upper arm, is constraint in all translational directions. The model isquasi-static, which means that the kinetic energy is not taken into account. This leaves a static balanceof the forces and moments acting on the bodies.

A free-body diagram was made and the equations of motion are found using Newton-Euler equations.Using the Grübler count for two dimensional models the number of degree of freedom (NDOF) isdetermined to be one. The Grübler count is shown in equation (2.1). In the Grübler count the N isthe number of bodies, M the number of contraints and mi the number of constraint directions of one

4

2.1. PARAMETERS OF THE RICHTER MODEL 5

constraint.

NDOF = 3N −M∑i=1

mi (2.1)

For the degree of freedom the angle of the wheel θw is chosen. This angle is defined as the angle betweena vertical line from the wheel hub and the position of the hand on the hand-rim. All other variables areparameters to the system so these are constant during simulation or a function of θw.

2.1 Parameters of the Richter model

The parameters can be divided into the next categories:

• The lengths of the links of the upper arm (Lua) and the forearm (Lfa).

• Relative position of the shoulder with respect to the wheel hub, defined by the length betweenshoulder and wheel hub (Lhs) and the angle (θhs).

• The hand-rim forces acting on the limb as measured by Richter (2001) [26].

The hand-rim forces measured in [26] are the total force, the radial component and the tangential com-ponent at a forward velocity of the wheelchair of 1.4

[ms

]. The graph of the forces measured by Richter

is digitized and approximated with a Gaussian function using the curve fitting toolbox of Matlab [11].The angle of the total force with respect to the tangential force (θf ) can be determined using the relationtan θf = |Fradial|

|Ftangential| .The values of the inputs during the simulation are the same as used by Richter. The upper arm (Lua)

is 26.7 centimeter and the forearm is 33.3 centimeters. The angle of the position of the shoulder (θhs)is kept at zero degrees. The length between the wheel hub and the shoulder (Lhs) is equal to the lengthof the upper arm and the forearm, which is 60 centimeter.

Figure 2.2: The hand-rim forces measured by Richter[26]

The magnitude of the hand-rim forces are shown in figure 2.2. The line ”F” is the total force between thehand and the hand-rim. The line ”F(r)” is the radial component of the force and ”F(t)” is the tangentialcomponent. These forces are measured for a wheelchair propelled with a forward velocity of 1.4

[ms

]. In

the model of Richter a shoulder height of 0.6 [m] with respect to the wheel hub is used and the shoulderis straight above the wheel hub. It is likely that the hand-rim forces change if the velocity and shoulderposition are changed. If these hand-rim forces are used, the result will only give a good approximationif it is very close to this velocity and position.

6 CHAPTER 2. THE RICHTER MODEL

The forces are a function of the push progress. The push progress (PP ) is a function of the wheelangle (θw), contact angle (θc) and release angle (θr). The push progress is calculated as followed:

PP =θw − θcθr − θc

(2.2)

2.2 Model equations

To use the Newton-Euler method a free-body diagram is necessary.

Figure 2.3: The free-body diagram of the Richter model

In figure 2.3 the free-body diagram is shown. Before the Newton-Euler equations can be applied,the relation between the dependent angles and the wheel angle (θw) must be quantified. The dependentangles are the angle at the shoulder (θs) and the elbow angle (θe). These are both calculated using thetriangle created by the upper arm, the forearm and a line from the shoulder to the hand (La). The anglesare described as followed:

La =

√[Lhs sin (θhs)−RHR sin (θw)]2 + [Lhs cos (θhs)−RHR cos (θw)]2 (2.3)

θs = −tan−1(Lhs sin(θhs)−RHR sin(θw)Lhs cos(θhs)−RHR cos(θw)

)− cos−1

(L2ua + L2

a − L2fa

2 Lua La

)(2.4)

θe = π − cos−1(L2ua + L2

fa − L2a

2 Lua Lfa

)(2.5)

2.3. OUTPUTS OF THE RICHTER MODEL 7

Now the dependent angles are known as a function of the wheel angle (θw). The range of the wheelangle during the simulation is still unknown. Two assumptions are made in [26] for this purpose:

• At the contact angle, so when the hand grabs the hand-rim, the forearm is perpendicular to thehand-rim. This will lead to a contact angle (θc) which is described as:

θc = θhs − cos−1(L2

hs+(Lfa+RHR)2−L2ua

2 Lhs (Lfa+RHR)

).

• At the release angle, so when the hand releases the hand-rim, the forearm is parallel to the upperarm, which implies θe = 0. The release angle (θr) can be calculated by using:

θr = θhs + cos−1(L2

hs+R2HR−(Lfa+Lua)

2

2 Lhs RHR

).

The variable RHR is the radius of the hand-rim. The wheel angle is than simulated from the contactangle to the release angle.

Now the Newton-Euler equations can be derived. The Newton-Euler equations are as followed:∑~F = ~P =

∂

∂t(m ∗ ~rcm) = 0 (2.6)

∑~M = ~H =cm J · ~ω = 0 (2.7)

These equations hold because the model is quasi-static. This leads to the simple equations∑ ~F = 0 and∑ ~M = 0. These formulas are used to calculate the elbow and shoulder moments, by applying these

equations to the forearm and upper arm. The equations of the elbow (~Te) and shoulder (~Ts) momentsare:

~Te = −~F Lfa sin(θs + θe −

(π2− θF − θw

))(2.8)

~Ts = ~Te + ~F Lua cos(θs + θF + θw) (2.9)

In these equations the hand-rim forces are seen as a total force (~F ) (combining the radial and tangentialcomponent), which is under an angle (θF ) with respect to the tangential component.

More information about the calculations can be found in appendix B.The model was simulated using Matlab. As can be seen in equation (2.9), the outcome of the shoul-

der moment is different from what is found in [26]. To be sure that these results are correct, the out-comes of the mathematical model is verified by two different Finite Element programs, namely SAM[24] and Adams [7] 1. Another way to check the calculations, is to look at the sum of all moments aboutthe shoulder. Because the Richter-model is quasi-static, it should be equal to zero. The used equation is:∑ ~Maround shoulder = Ts − F cos(θF + θw) (Lfa cos(θs + θe) + Lua cos(θs))

+F sin(θF + θw) (Lfa sin(θs + θe) + Lua sin(θs)) = 0(2.10)

2.3 Outputs of the Richter model

The outputs of the model are the shoulder and elbow torque during the propulsion of the wheel in thequasi-static case. The calculated reaction forces in the arm are not the real reaction forces in the jointsof a human, because the muscle forces are not taken into account here. The muscle forces create themoments in the shoulder and elbow, but also create extra forces in the joints. Because these forces arenot calculated, the reaction forces are not the real joint forces during the propulsion.

The model is simulated for multiple shoulder heights. The shoulder is moved 10 centimeters up anddown, similar as in the article of Richter. This gives a view on how the moments change when the seatheight is changed.

1The model in Adams is made by Maciej Mazur, PHD student at the department of School of Aerospace, Mechanicaland Manufacturing Engineering, RMIT University, Australia

Chapter 3

A three dimensional model

To get a better understanding how the moments in the arm are influenced by the seat position, a modelis needed which includes dynamics and is capable of moving like a human arm.In this model it shouldalso be possible to use experimental data of the movements of the arm and the hand-rim forces, to givethe model a larger area of validity. The model of Pennestrì et al. (2006) shows a sophisticated model forevaluating arm muscle strength for a disabled person driving a car, which also shows all motions of ahuman arm [13]. This model is used as a source of inspiration.

In the three dimensional model the arm is modeled as three beams, namely the upper arm, the lowerarm and the hand. The three joints are the shoulder, elbow and wrist joint. For the joints the model ofPennestrì is used, but the elbow joint is modeled differently to match with the three beams structure.The degrees of freedom of the joints are modeled as followed:

• The shoulder joint is modeled as a spherical joint which leaves three angles unconstrained. Theshoulder is thus able to have abduction, flexion and roll of the arm. The three translations areconstrained.

• The elbow joint is modeled such that roll and flexion are possible and the three translations andthe abduction of the forearm are constrained.

• The wrist joint is modeled as a universal joint which leaves the abduction and flexion uncon-strained. The three translations and roll of the hand are constrained.

The Grübler count for three dimensional arm shows that the model has seven degrees of freedom.The Grübler count is shown in equation (3.1).


mi (3.1)

These degrees of freedom must be defined. To simplify the mathematics of the model, several Carthesiancoordinate frames are defined.The frames are shown in figure 3.1. The global (fixed) frame ~eO is located at the shoulder. This frame isfixed to the ground and it is drawn in gray in figure 3.1. The ~eO1 vector points straight down (directionof gravity). The ~eO2 points into the the body (the modeled arm is the right arm). The ~eO3 vector pointsforward from the shoulder.The first body fixed frame ~e1 is connected to the upper arm at the shoulder joint, such that the vector ~e11is always directed in the direction of the upper arm. If all three degrees of freedom at the shoulder arezero, the ~e1 frame coincides with the ~eO frame.

8

9

(a) View of the ~eO1 - ~eO2 plane (b) View of the ~eO1 - ~eO3 plane

Figure 3.1: The frames of the model

The second body fixed frame ~e2 is connected to the forearm at the elbow joint, such that the ~e21 vectoris always directed in the direction of the forearm. If all degrees of freedom at the elbow are zero, the ~e2

frame coincides with the ~e1 frame.The third body fixed frame ~e3 is connected to the hand at the wrist joint, such that the vector ~e31 isalways directed in the direction of the hand. If all degrees of freedom at the wrist are zero, the ~e3 framecoincides with the ~e2 frame.

All degrees of freedom are controlled by a moment at the joint. This implies seven moments in thearm, which need to be calculated. These moments are an indication of the muscle strength required forthe propulsion.

All the definitions of the frames imply a zero position where the arm is hanging down. When thepalm is pointed forward, all angles are defined as zero, as is shown in figure 3.2.

During the simulation the arm and the wheel are moving in one plane, just like the Richter model.In the literature no usable data about the positions of the joints during the propulsion was found, whichis the reason of the two dimensional simulations. A three dimensional simulation would require moreassumptions about the positions of the joints.

10 CHAPTER 3. A THREE DIMENSIONAL MODEL

Figure 3.2: The zero position of the angles

3.1 Parameters of the system

The three dimensional model has several parameters. These parameters are:

• The angles at the joints.

• The angular velocities at the joints.

• The angular accelerations at the joints.

• Lengths of the upper arm, lower arm and from the wrist to the grabbed hand-rim.

• The length from the joint to the centre of mass of the upper arm, forearm and hand.

• The masses of the upper arm, forearm and hand.

• The moments of inertia about the principle axes of the upper arm, forearm and hand.

• The hand-rim forces.

• The camber angle, toe angle and the angle from the wheel hub to the hand.

• Relative position of the wheel hub with respect to the shoulder.

These parameters are discussed in the following subsections. The parameters are known before simu-lation. The values can be obtained from the literature or these can be measured when a person is doingan experiment. The literature can help to get better insight in the general relation between seat positionand the kinematics in the arm. The experimental values can be used to minimise the kinematics in thearm for one individual. There are no experiments done yet, because of a lack of time. The values of theparameters described in the following sections will also be provided.

3.1. PARAMETERS OF THE SYSTEM 11

3.1.1 The degrees of freedom at the joints

The degrees of freedom of the system are all rotational. A clockwise rotation is always defined as positive.First the degrees of freedom (DOF) at the shoulder will be explained, than the DOF at the elbow and thewrist. Than the equations of the DOF used for the simulation are presented. Finally the calculation ofthe derivatives (velocity and acceleration) during the simulation are discussed.

(a) View of the ~eO1 - ~eO2plane

(b) View of the ~eO1 - ~eO3 plane

Figure 3.3: The angles at the shoulder

The joint at the shoulder has three degrees of freedom, namely roll, flexion and abduction of theupper arm. These are defined as ψ1, ψ2 and ψ3 respectively. In figure 3.3 the angles at the shoulder arevisible. The angle ψ3 is negative as shown in figure 3.3.

Figure 3.4: The angles at the elbow in the ~eO1 - ~eO3 plane

The elbow joint has two degrees of freedom. In figure 3.4 the angles are made visible. ψ4 and ψ5 arethe roll and the flexion of the forearm respectively.


(a) View of the ~eO1 - ~eO2plane

(b) View of the ~eO1 - ~eO3plane

Figure 3.5: The angles at the wrist

The joint at the wrist has two degrees of freedom, namely the flexion and abduction of the hand,which are ψ6 and ψ7 respectively. These are shown in figure 3.5.

During the simulation the arm and the wheel hub are in one plane. This assumption simplifies themathematics. The hand is constrained to the hand-rim. The assumption is made that the hand is alwaysperpendicular to the hand-rim, which implies that the angles are a function of the angles at the wheel(camber, toe in and the wheel angle) as well as the position of the wheel hub with respect to the shoulder,which will be explained in section 3.1.8.Because the simulation is in one plane, the abduction and roll of the upper arm are equal to zero. To getthe correct orientation of the arm, the roll of the forearm is set to π

2 radians. The flexion at the wrist isalways zero, to keep the hand in the same plane as the upper and forearm. With these assumptions theother angles can be calculated and the wheel angle becomes the driving constraint. The position of thehand-rim is dependent on the placement of the wheel hub. Because of the two dimensional simulationonly the angle β2 is of importance to describe the position of the wheel hub with respect to the shoulder.The angle β2 rotates about the ~eO2 axes. A positive value for β2 will place the wheel hub further backward.The values of the angles are as followed (in radians):

ψ1 = 0

ψ2 = −(−atan(Lhs sin(β2)−(Rhr+Lh) sin(θw)Lhs cos(β2)−(Rhr+Lh) cos(θw) )− acos(

L2ua+a

2−L2fa

2 Lua a))

ψ3 = 0ψ4 = π

2

ψ5 = −(π − acos( 12L2

ua+L2fa−((−Lh−Rhr) cos(θw)+Lhs cos(β2))

2−(−(−Lh−Rhr) sin(θw)−Lhs sin(β2))2

Lua Lfa))

ψ6 = 0ψ7 = −(−ψ5 − ψ2 + θw)

(3.2)The derivatives are approximated using finite difference approximations and the Taylor series expan-

sion. It is assumed that the functions are smooth functions in the range used during the simulation. Ifnot, this can dramatically influence the results. For the first derivative of the angles, the forward differ-ence formula (forward Euler) is used. The letter n determines which degree of freedom is calculated.The letter i is used to denote which time step is being calculated. For the last increment it is not possible


to determine an approximation of the first derivative. This value of the first derivative at the last incre-ment is set to zero. Discontinuities can be the result, which shows the reader that at the final incrementthe approximation is not giving any trustworthy results. The first derivative is calculated as followed:

ψn(i) =ψn(i+ 1)− ψn(i)

Tinterval(3.3)

where Tinterval is the time interval between the discretized time steps.The second derivative is approximated using the same methodology as used for the first derivative,

which means that the Tayler series expansion is used again. With this approximation the last two in-crements cannot be determined. At these last two increments the derivatives are set to zero. It is mostlikely that this produces discontinuities in the results, because the dynamic forces are suddenly zero.The discontinuities show that these approximations are not valid at those last two increments. Thesecond derivative is calculated as followed:

ψn(i) =ψn(i+ 2)− 2 ψn(i+ 1) + ψn(i)

T 2interval

(3.4)

The disadvantage of numerical differentiation, is that it cannot be used for ”noisy” signals. It is likelythat this method cannot be used for experimental data, because of the influence of noise on the signals.Measuring the velocity and accelerations while measuring positions of the arm can solve this problem.The measurement tools at the RMIT University are capable to measure position as well as velocity andacceleration.

3.1.2 Lengths of the arm segments

The lengths of the arm segments are an input to the model. These are the length of the upper arm(Lua), forearm (Lfa) and hand (Lh).

Table 3.1: The lengths of the upper arm, forearm and handLength of the Length of the Length of theupper arm [m] forearm [m] hand [m]

0.3308 0.2545 0.0772

For the simulation the lengths are gathered from the literature. Ratios can be derived to give thesegments the correct proportions [6]. Then the final lengths can be chosen to use for this simulation[10][16]. The used lengths are of the 50th percentile male in Great Britain. These lengths are shown intable 3.1. A more detailed discription of how the segments are calculated can be found in appendix A.

3.1.3 Estimation of the position of the centre of mass

Each segment of the arm has its own centre of mass, which can be positioned everywhere on the arm.Lacking good estimations, the assumption is made that all the centres of mass are in the middle of thebar. This implies:

• Length from shoulder to centre of mass of the upper arm: rua = 12 Lua.

• Length from elbow to centre of mass of the forearm: rfa = 12 Lfa.

• Length from wrist to centre of mass of the hand: rh = 12 Lh.


3.1.4 Masses of the segments

All segments have a mass which is adapted to fit with the segment lengths. The segment lengths arefound in the literature and an average bodyweight for these lengths was included. The masses of thesegments were also measured and presented with an average bodyweight, so these two can be matched.In this case a bodyweight of about 80 kilograms is chosen.

Table 3.2: The masses of the upper arm, forearm and handMass of the Mass of the Mass of the

upper arm [kg] forearm [kg] hand [kg]2.23 1.39 0.52

The used masses of the segments are gained from the book of Don B. Chaffin (1999)[6] and areshown in table 3.2.

3.1.5 Moments of inertia of the segments

The moments of inertia are assumed to be the principle moments of inertia, which implies that theproducts of inertia are equal to zero. This means that the inertia matrix is a diagonal matrix. Themoments of inertia are often given about the centre of mass. To get the moment of inertia about thejoints the Huygens-Steiner formula is used, which will be explained later and can be seen in equation(3.9). The moments of inertia of the segments used in the simulation are shown in table 3.3.

Table 3.3: The principle moments of inertia about the centre of mass of the upper arm, forearm andhand

Axes Name in the Moment ofmodel Inertia

[kg m2

]Upper arm ~e11 Jcm_ua_xx 26.3 10−4

~e12 Jcm_ua_yy 166 10−4

~e13 Jcm_ua_zz 157.2 10−4

Forearm ~e21 Jcm_fa_xx 11.8 10−4

~e22 Jcm_fa_yy 88.5 10−4

~e23 Jcm_fa_zz 86.1 10−4

Hand ~e31 Jcm_h_xx 2.9 10−4

~e32 Jcm_h_yy 10.4 10−4

~e33 Jcm_h_zz 8.5 10−4

3.1.6 Hand-rim forces

In the literature many examples of measurements of the hand-rim forces can be found. The used hand-rim forces are those measured in [26].

In figure 2.2 the hand-rim forces are shown. The forces used in this simulation are the radial andtangential force. These were already digitized as discussed in section 2.1. During the simulation thehand-rim forces do not change as the seat height changes. Because of this the simulation is only validin a small region about the original position.


3.1.7 Wheel angles

Because the hand is constrained to the hand-rim, the wheel angles are important for the movement ofthe arm. To describe the orientation of the wheel with respect to the shoulder, three angles are defined,namely the camber angle (χ), wheel angle (θw) and the toe angle (φ).

(a) The toe angle (b) Rotation of the wheel (c) The camber angle

Figure 3.6: The wheel angles

In figure 3.6 the wheel angles are shown. Because in the simulation the arm and wheel are in oneplane, the camber (χ) and toe (φ) angles are zero. The wheel angle (θw) is the driving constraint induring the simulation.

The assumptions as used in the Richter model will not work in this case, because of enormousaccelerations of the upper arm near the release angle. In the literature wheel angles are described fordifferent groups. For everyday users one study showed a contact angle of -16 degrees and a release angleof 50 degrees. The step size of the wheel angle between the contact and release angle is 1 degree.

3.1.8 Position of wheel hub with respect to the shoulder

The position of the wheel hub with respect to the shoulder is described by two angles and one length.The first angle rotates about the ~eO2 axis and is called β2. The other angle rotates about the ~eO3 axis andis called β3. The length is the length between the shoulder and the wheel hub and is called Lhub.

The length and the angles are shown in figure 3.7. During the simulation the angle β3 is kept zero, tokeep the arm and the wheel in one plane. The angle β2 is kept zero when the shoulder is moved up anddown. The angle β2 is changed when the shoulder is moved to the front and the rear from −5 degreestill 5 degrees, while the length Lhub is adapted to have the same shoulder height for each position.


(a) The angle β2 (b) The angle β3 (c) The length Lhub

Figure 3.7: The position of the wheel hub relative to the shoulder.

3.2 Model equations

To determine the seven moments at the degrees of freedom in the arm, the model is built as a multi-body dynamical model. The Lagrangian equations of motion are used to obtain the equations of motion.The moments at the wrist, elbow and shoulder can be calculated with the equations of motion. Forthe calculations the kinetic energy (T ), potential energy (V ) and the non-conservative forces (Qnc) areneeded.

d

dt

(∂T

∂q

)− ∂T

∂q+∂V

∂q= (Qnc)

T (3.5)

The equation (3.5) shows the Lagrangian equation of motion. The used general coordinates are theangles of the arm q = ψ = [ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ]

T . The equation (3.5) is dependent of the kineticenergy of the arm, potential energy of the arm and the non-conservative forces (Qnc) acting on the arm.The kinetic energy is dependent on the moments of inertia OJ(ψ) of the arm. The kinetic energy iscalculated using:

T =1

2ψT OJ(ψ) ψ (3.6)

The generalized coordinates are the degrees of freedom of the arm. The rotations which are not free (oneat the elbow and one at the wrist) are not represented in the equations of motion and are kept zero forthese calculations. The kinetic energy can be rewritten using [15], which will eliminate the differentiationof the the kinetic energy. The result is as follows:

d

dt

(∂T

∂ψ

)=

d

dt

(ψT OJ(ψ)

)= ψT OJ(ψ) + ψT

dOJ(ψ)dt

(3.7)

∂T

∂ψ=

1

2

∂

∂ψ

(ψT OJ(ψ) ψ

)(3.8)

The matrix OJ(ψ) represents the inertia of the arm about the joint. To determine the moments of inertiaabout the joints, the moments of inertia about the centre of mass are transformed using the Huygens-Steiner formula:

OJ = CM J+m (~rCM · ~rCM I− ~rCM ~rCM ) (3.9)

3.3. OUTPUTS OF THE SYSTEM 17

Than the potential energy needs to be calculated. The potential energy is dependent on gravity (~g) andthe position of the mass in space ( ~rcm). The potential energy of all segments of the arm are calculatedusing the following formula:

V = −m~g · ~rcm (3.10)

The gravitational acceleration is defined as ~g =[9.81 0 0

]~eO.

The first part of the non-conservative forces are the forces acting between the hand-rim and thehand. The non-conservative forces are not directly a function of the position of the arm. These forces area function of the push progress, as already seen in section 3.1.6. The second part of the non-conservativeforces are the moments at the joints in the arm. These are the outputs of the model, so these arecalculated during the simulation.

More information about how these equations are implemented in Matlab can be found in appendixC.

3.3 Outputs of the system

During the simulation the motion of the arm is described using the equations (3.2). The seven momentsin the arm are the outputs of the system during this simulation. The seven moments are designed toinfluence the seven degrees of freedom of the arm. There are three moments at the shoulder, two atthe elbow and two at the wrist joint. The axis of the moments are similar as the axis of the degrees offreedom. This leads to an estimate for the moments which must be created by the muscles to performthe task. The forces at the joints are not calculated, because the muscle forces are not calculated.

The outputs are given for different seat positions, varying about the original shoulder height of 0.6meter, similar to the simulations of the Richter model. For the first simulation the shoulder is moved 5centimeters up and down, while the shoulder remains straight above the wheel hub. During the secondsimulation the shoulder is moved 5 centimeters to the front and to the rear, while the height above thewheel hub is the same. The positions are very close to the original height, because the hand-rim forcesdo not change.

Chapter 4

Results of the Richter model

The results of the Richter model are split up into the results of the shoulder height simulation and thecomparison with Finite Element Models (FEM), using SAM [24] and ADAMS [7]. The model in ADAMSis created by Maciej Mazur, a PHD student at the department of School of Aerospace, Mechanical andManufacturing Engineering, RMIT University, Australia. The comparison with the FEM was necessarybecause the shoulder moment seen in [26] could not be reproduced.

4.1 Changing the seat height

The Richter model gives an indication of the moments produced at the elbow and shoulder. The nominalvalue of the length between the shoulder and the wheel hub (Lhs) is 0.6 meter. The shoulder is movedup and down with a maximum of 0.1 meter, similar to [26].

Figure 4.1: The elbow moment of the Richter model

18

4.1. CHANGING THE SEAT HEIGHT 19

The elbow moment is shown in figure 4.1. Because the high peak values of the moments are com-monly seen as causes of pain and injuries for everyday users, low peak values are desirable. As is seenin figure 4.1 the highest seat position has the lowest peak values.

(a) Shoulder moment of article of Richter (b) Shoulder moment as calculated

Figure 4.2: The shoulder moment of the Richter model

The formulas of the shoulder moment as seen in the article of Richter (2001) and as calculatedduring this project are not the same. In figure 4.2 both results are shown. The difference between theresults of Richter and the calculations during this project lead to different seat heights with the lowestpeak values. The results of Richter showed the lowest peak values at the lowest seat position. But whenthe calculations done during this project are reviewed, there is an optimal seat height for the absolutepeak value of the shoulder moment. This optimum is near the nominal value of the length between theshoulder and the wheel hub (Lhs).

Because of the difference in the moment at the shoulder, it is important to know that no majormistakes are made during this project. One way of testing this is looking at the sum of all momentsabout the shoulder. This should be equal to zero, because the model is quasi-static. The equation usedfor this purpose is equation (2.10). In figure 4.3 the result is shown. When the formula of the momentat the shoulder joint of the Richter article is examined, it is clear that the sum of all moments about theshoulder is not equal to zero. For the calculated moment as seen in section 2.2 the result is zero duringthe simulation.

20 CHAPTER 4. RESULTS OF THE RICHTER MODEL

Figure 4.3: The sum of all moments about the shoulder joint

4.2 Comparison with SAM

SAM [24] is a finite element method for solving mechanical problems. The simulation of the modelrequires changing the angles of the arm and changing the angle and magnitude of the force. SAM doesnot allow the forces to change over time and changing the direction of the force is also not possible. Forthis reason the model had to be modified and the adapted model was simulated in SAM as well as inMatlab. These two simulations of the same adapted model are compared in this section. During this

Figure 4.4: The model used in SAM

simulation the hand-rim forces are pointed upwards. The upward force has a constant magnitude of 70Newtons, which is in the range of the measured total force of Richter. Because the hand-rim forces arenot the same as in the original model, this simulation should only be compared with the results of thesame adapted simulation in Matlab. The angles at the shoulder and elbow are the same as for the othersimulations and the model can be seen in figure 4.4.

4.2. COMPARISON WITH SAM 21

Figure 4.5: The comparison of the shoulder torque with SAM

When the moments at the elbow of the two simulations are compared it is clear that these have thesame values. The only exception is the first simulation point. The reason of this difference is foundin the SAM program. Because the model is built when the arm is hanging down, the SAM programstarts the simulation with the arm hanging down. From the second simulation point on the shoulderand elbow angle are the real values of the model.

The results of the moment at the shoulder are more interesting. Although this simulation does notgive the solution to the original problem, it does show an agreement with the calculations done in thisreport. Figure 4.5 shows the graphs of all three the moments. It is clear that the formula as seen in [26]does not coincide with the moment obtained in SAM while the moment as calculated during this projectdoes coincide. The only exception is again the first simulation increment.

Finally the sum of all moments about the shoulder is calculated as seen in (2.10). The sum of allmoments should be equal to zero, because the model is quasi-static. In figure 4.6 it can clearly be seenthat the calculated moment and the moment obtained from SAM are equal to zero. The only exceptionis the first simulation point of the moment from SAM, but it is clear that this difference is introducedby SAM. When the sum of all moments about the shoulder is calculated with the shoulder momentof the Richter article, it is not equal to zero. This means that the simulation in SAM agrees with thecalculations done during this internship.

22 CHAPTER 4. RESULTS OF THE RICHTER MODEL

Figure 4.6: The sum of all moments about the shoulder

4.3 Comparison with ADAMS

Adams [7] is a computer program capable of simulating complex mechanical systems. The model inAdams is created by Maciej Mazur, a PhD student at the RMIT University. The results of this modelare compared with the results of the Richter model as seen in section 4.1. This is possible because theAdams program is capable of running the entire model without restrictions. So the results in Adamscan be compared with the results of section 4.1.

Figure 4.7: Comparison of the shoulder torques

4.3. COMPARISON WITH ADAMS 23

From the simulation in SAM it is expected that the analytical solution of the moment at the elbowjoint is correct. The simulation in Adams shows indeed the same values as seen in section 4.1. Whenthe moment at the shoulder joint is considered it is clear that it is the same as the calculated shouldermoment as seen in section 2.2. Because these two moments are about the same, the graphs almostcoincide.

Figure 4.8: Checking the results

Also when the sum of all moments about the shoulder joint is considered, the result is zero whenthe shoulder moment of Adams and as calculated in section 2.2 are used. The result is not equal tozero when the shoulder moment as in the Richter article is used. From these results it is clear that themoment at the shoulder joint as calculated in section 2.2 agrees with the simulation in Adams.

Chapter 5

Results of the three dimensionalmodel

The first simulation done with this model was exactly the same as the Richter model. The same armlengths are used and the angles at the wrist are kept zero. The quasi-static results were the same as seenfor the Richter model, so these are not shown in this report.

Two different simulations are made with the three dimensional model. During the first simulationthe seat is shifted up and down. In the second simulation the seat is shifted to the front and to the rear.The seat height is kept the same. The change in seat position is done by shifting the wheel hub withrespect to the shoulder.

Both the quasi-static and dynamic results will be shown in this report. The difference betweenthese two results show the importance of including the dynamics. The two values which will be lookedat are the peak values of the moments in the arm as well as the peak values of the first derivative ofthe moments. These represent the moments which the muscles must produce, so these indicate therequired muscle strength. High values of the derivative of the hand-rim forces are also considered as acause of injury in the literature. These produce high values of the derivative of the moments in the arm.So the derivative is also calculated.

5.1 Varying the seat height

This simulation involves the influence of the seat height at the moments in the arm. Because thehand-rim forces measured by Richter[26] are used, the original seat height of Richter is used and smallvariations up to 5 centimeters are applied. Because the simulation is in 2D, only the shoulder flexionmoment, elbow flexion moment and wrist abduction moment will be shown. All other moments areequal to zero.

The used velocity of the wheelchair is 1.4[ms

], which is the same as in [26]. This velocity corresponds

with the hand-rim forces used in [26], so these hand-rim forces are used during the simulation.

5.1.1 The shoulder moment

Looking at the results in figure 5.1, no change in the negative peak value can be seen for the dynamicsimulation, while this was the case for the quasi-static simulation. The negative peak value is the largestabsolute value of the shoulder flexion moment. For the positive peak value a gradual decline can be seen,but this value is not the absolute peak value. This result does not allow a clear conclusion on which seatposition will result in the lowest shoulder flexion moment.

24

5.1. VARYING THE SEAT HEIGHT 25

(a) The shoulder moment with quasi-static simulation

(b) The shoulder moment with dynamic simulation

Figure 5.1: The moment at the shoulder

In the literature the first derivative of the forces at the hand-rim is indicated as a cause of pain.Because all the simulations use the same angles at the hand-rim and the same forces, the derivative ofthe hand-rim forces are equal. There is still a difference in the derivative for the flexion moment at theshoulder. The first derivative can be estimated using numerical differentiation. A forward numericaldifferentiation method is used:

Msf (i) ≈Msr(i+ 1)−Msr(i)

Tinterval(5.1)

26 CHAPTER 5. RESULTS OF THE THREE DIMENSIONAL MODEL

Figure 5.2: The derivative of the flexion moment at the shoulder

In figure 5.2 the derivative of the flexion moment of the shoulder is shown. It is clear that the highestseat position has the highest peak values of the first derivative of flexion moment at the shoulder. Whilelowering the seat position, this peak value gradually decreases until the peak at about 20% of the pushprogress has the highest peak value. This also shows that there is a possible optimum of the seat heightfor the peak value of the derivative of the shoulder flexion moment.

5.1.2 The elbow moment

In figure 5.3 the flexion moment at the elbow is shown. The peak value of the moment for the quasi-static and dynamic simulation is the highest at the lowest seat position. An increasing seat height willdecrease the peak value. In the literature the elbow joint is not seen as a joint which is often injuredbecause of the use of a manual wheelchair. Therefore it is possible that the optimal seat position for theelbow is not the optimal position to minimize injuries.

The derivative of the elbow flexion angle can be estimated as seen in (5.1). The derivative is shownin figure 5.4. The lowest seat position has the highest peak values for the first derivative of the momentat the elbow. This result can be seen in the same way as the peak values of the moment.


(a) The elbow moment with quasi-static simulation

(b) The elbow moment with dynamic simulation

Figure 5.3: The moment at the elbow


Figure 5.4: The derivative of the flexion moment at the elbow

5.1.3 The wrist moment

In figure 5.5 the wrist abduction moment is shown. For the quasi-static case all graphs coincide, so onlyone graph is visible. This is because the hand describes the same movement independent of the changesin seat height. Because the same hand-rim forces are used, the quasi-static moments are the same. If thedynamics are also considered, a lower seat position shows the highest peak value except for the highestseat position. The highest seat position has a larger angular acceleration at the end of the push progress,which is the cause of the higher moment at the wrist. This indicates that there exists an optimum forthe peak values of the wrist moment.

The derivative of the wrist abduction moment can be estimated in the same manner as seen for theshoulder flexion moment (equation (5.1)). The derivative is shown in figure 5.6. The lowest seat positionhas the highest values for the first derivative of the moment, except for the highest seat position. Thehighest seat position has a larger angular acceleration at the end of the push progress, which causes thehigher moment at the wrist. This causes a steeper line, which has a higher first derivative.


(a) The wrist moment with quasi-static simulation

(b) The wrist moment with dynamic simulation

Figure 5.5: The moment at the wrist


Figure 5.6: The derivative of the flexion moment at the wrist

5.1.4 Overview of the results

To give a better overview of the results, the relations between a higher seat position and the absolute peakvalues and the absolute peak values of the derivatives are shown in table 5.1. A negative relation meansthe peak values decline for higher seat positions and visa versa for a positive relation. An optimummeans that in the range of the simulation there is a position which leads to the lowest values.

Table 5.1: Relation between a higher seat position and the absolute peak values of the moments andthe derivative

Moment Peak moments First derivativeShoulder About the same OptimumElbow Negative NegativeWrist About the same Optimum

5.2 Varying the seat position to front / rear

This simulation shows the effect on shifting the seat to the front and to the rear. Because the hand-rimforces as measured in [26] are used, the original seat height is used and kept constant. The angle β2is changed, with a maximum of 5 degrees, which is equivalent to a shift of the wheel hub of about 5.25centimeters to the front and to the back. A positive angle will move the axle more backwards w.r.t. theshoulder. This will change the length between the shoulder and the wheel hub.

The simulation has a small range of just over 10 centimeters in total, because the hand-rim forcesare kept the same. Also the push angle is kept the same for all simulations. Because the simulation isin 2D, only the shoulder flexion moment, elbow flexion moment and wrist abduction moment will beshown. All other moments are equal to zero.

The velocity of the wheelchair used during the simulation is the same as in [26], to correspond withthe used hand-rim forces. This velocity is 1.4

[ms

].

5.2. VARYING THE SEAT POSITION TO FRONT / REAR 31

5.2.1 The shoulder moment

In figure 5.7 the shoulder flexion moment is shown for the quasi-static and the dynamic case. The lowestpeak values are seen for the largest negative angle. So a negative angle seems to be more favourable thana positive angle, which means that the seat shifted more backwards is more favourable. In the literaturesimilar conclusions have been drawn from different experimental research [14, 20].

(a) The shoulder moment with quasi-static simulation

(b) The shoulder moment with dynamic simulation

Figure 5.7: The moment at the shoulder


The derivative of the moment can be estimated in a similar way as in equation (5.1). The derivativeis shown in figure 5.8. This graph shows that a negative angle is more favourable, which means that amore backward seat position is favourable.

Figure 5.8: The derivative of the flexion moment at the shoulder

5.2.2 The elbow moment

In figure 5.9 the flexion moment at the elbow is shown for the quasi-static and dynamic case. For thequasi-static case the peak values decline if the angle is increased, which means that a more forward seatposition is more favourable. The dynamic simulation does not show such a decline, all peak values areapproximately the same.

The derivative is estimated in a similar way as in equation (5.1). In figure 5.10 the derivative isshown. Because of the multiple local peaks the global peak value shifts from the local peak at about 5%of the push progress to the local peak at about 25% of the push progress, which means there should bean optimal position. From this graph it seems that the optimal position should be at about -2.5 degrees.


(a) Elbow moment with quasi-static simulation

(b) Elbow moment with dynamic simulation

Figure 5.9: The moment at the elbow


Figure 5.10: The derivative of the flexion moment at the elbow

5.2.3 The wrist moment

In figure 5.11 the abduction moment at the arm is shown for the quasi-static and the dynamic simulation.For the quasi-static simulation the graphs are on top of each other, because the hand makes the samemovement every simulation. This is a result of the assumption that the hand is perpendicular to thehand-rim. The peak values for the dynamic case are about the same for every angle.

The derivative of the wrist abduction moment can be estimated in a similar way as in equation (5.1).The derivative of the wrist abduction moment is shown in figure 5.12. The peak between the 5-10% ofthe push progress is the global peak value. The highest peak value is seen for an angle of -5 degrees.With an increasing angle the peak value of the derivative will decline.


(a) The wrist moment with quasi-static simulation

(b) The wrist moment with dynamic simulation

Figure 5.11: The moment at the wrist


Figure 5.12: The derivative of the flexion moment at the wrist

5.2.4 Overview of the results

An overview of the results for moving the seat to the front and the rear are shown in table 5.1. A negativerelation means the peak values decline for a more forward seat position and visa versa for a positiverelation. An optimum means that in the range of the simulation there is a position which leads to thelowest values.

Table 5.2: Relation between a more forward seat position and the absolute peak values of themoments and the derivative

Moment Peak moments First derivativeShoulder Negative OptimumElbow About the same PositiveWrist About the same Positive

Chapter 6

Conclusions and recommendations

6.1 Conclusions

The goal of the research is to develop a model of the propulsion of a manual wheelchair and use thismodel to compare seat positions. Two models have been examined.

The Richter model shows how the hand-rim forces are produced by the muscles in the arm. Thesemuscle forces are not directly included in the model, but the sum of the moments produced about thejoints are. Although the muscle forces might produce opposing moments, these resulting momentsare a measure for the kinematics in the arm. The problem which had to be solved, was the differencein shoulder moment between the calculations made during this project and found in the literature.The finite element models agreed with the calculations made during this project. When the modelis simulated for different seat heights, the results show different relations between peak values of themoments en seat position. The peak values of the elbow moment are decreasing as the seat height isdecreasing. For the shoulder moment there seems to be an optimum seat height for the seat position.Because the shoulder joint is more vulnerable for injuries, it could be wise to adapt the seat height to theoptimal seat height for the shoulder.

The three dimensional model is simulated as the arm and hand-rim are in one plane. One simula-tion shows the differences in joint moments as the seat height changes, the other one shows the jointmoments as the seat is moved to the front and the rear. This model includes the dynamics of the arm.

For the seat height there is no clear conclusion to be drawn from these simulations. The joints mostvulnerable for injuries identified in the literature are the wrist and shoulder joint. The peak values of themoments at the joints do not change much if the seat height is changed. To minimize the peak valueof the first derivative of the shoulder flexion moment, a more higher seat position seems favourable,although it increases the peak at the end of the push progress. This could lead to an optimal seatposition for the shoulder. For the first derivative of the wrist abduction moment a higher seat positionseems favourable, although the moment at the end of the push progress is increased. This increase willcause higher values for the derivative, which can become the global maximum as seen in the results.This could lead to an optimum seat height for the wrist. Because the wrist and the shoulder joint aremore vulnerable, an optimum can be found if it is made clear what the best relation is between theshoulder en wrist moment.

When the seat is moved more to the front of the wheelchair, which places the axle more backwardswith respect to the shoulder, the peak value of the first derivative of the wrist abduction moment declines.A seat which is positioned more backwards to the wheelchair decreases the peak values of the shoulderflexion moment. It also decreases the peak values of the derivative of the moment, except for the mostbackward position. This is because then the values at the end of the push progress become the global

37

38 CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS

maximum value. The peak values of the moment at the wrist do not change much. The peak value ofthe derivative of the wrist moment decreases if the seat is moved forward. Thus the shoulder and wristjoint show different relations when the seat is moved to the front of the wheelchair. To give an optimalseat position the best compromise between shoulder and wrist stresses must be made clear.

6.2 Recommendations

To obtain a better understanding about the propulsion, there are several opportunities to improve thethree dimensional model. One way is getting experimental data of the positions of the joints and thehand-rim forces. With these measurements the model can be improved by doing a three dimensionalsimulation with the corresponding hand-rim forces.

To improve the area where the model is valid, the hand-rim forces from several positions can bemeasured. This could lead to a relation between hand-rim forces and shoulder position. This relationcan then be used to find the minimum of the peak loads for a wider spectrum of seat positions and itwill increase the validity of the model.

Including the movement of the shoulder in the model will improve the validity of the model. Duringthe simulations in this report the shoulder is fixed with respect to the wheelchair. In real life wheelchairusers will use their abdominals to move the shoulder forward. This will influence the kinematics in thearm.

The optimization of the moments is another subject which needs to be solved. It is not clear how themoments in the arm should be optimized. The optimization is dependent on the mechanical strengthat the joints. Only it is not known in which proportion the stresses at the joints are least likely to injurethe user. These relations must be known before the optimal seat position can be determined.

Bibliography

[1] Boninger ML; Rory AC; Baldwin MA; Shimada SD; Koontz A. Wheelchair pushrim kinetics, bodyweight and median nerve function. Phys Med Rehabil, 80:910–915, 1999.

[2] Boninger ML; Impink BG; Cooper RA; Koontz AM. Relation between median and ulnar nervefunction and wrist kinematics during wheelchair propulsion. Arch Phys Med Rehabil, 85:1141–1145,2004.

[3] Samuelsson KAM; Tropp H; Nylander E; Gerdle B. The effect of rear-wheel position on seatingergonomics and mobility efficiency in wheelchair users with spinal cord injuries: A pilot study.Journal of rehabilitation research & development, 41:65–74, 2004.

[4] Boninger ML; Souza AL; Cooper RA; Fitzgerald SG; Koontz AM; Fay BT. Propulsion patterns andpushrim biomechanics in manual wheelchair propulsion. Arch PhysMed Rehabil, 83:718–723, 2002.

[5] Brubaker CE. Wheelchair description, an analysis of factors that affect mobility and performance.Journal of rehabilitation research and development, 23:19–26, 1986.

[6] Don B. Chaffin, Gunnar B. J. Andersson, and Bernard J. Martin. Occupational Biomechanics. JohnWiley & Sons Inc., 1999.

[7] MSC Software Cooperation. MD Adams R3. Santa Ana, USA.

[8] Deroches G; Aissaoui R; Bourbonnais D. The effect of resultant force at the pushrim on shoulderkinetics during manual wheelchair propulsion: A simulation study. IEEE Transactions on biomedicalengineering, 55:1423–1431, 2008.

[9] Van der Woude LHV; Veeger HEJ; Dallmeijer AJ; Janssen TWJ; Rozendaal LA. Biomechanics andphysiology in active manual wheelchair propulsion. Medical Engineering & Physics, 23:713–733, 2001.

[10] Judith G. Hall, Ursula G. Froster-Iskenius, and Judith E. Allanson. Handbook of normal physicalmeasurement. Oxford Medical Publications, 1989.

[11] The Math Works Inc. Matlab R2007b. Natick, Massachusetts USA, 2007.

[12] Kotajarvi BR; Sabick MB; Kai-Nan An; Zhao KD; Kaufman KR; Basford JR. The effect of seatposition on wheelchair propulsion biomechanics. Journal of Rehabilitation Research & Development,41:403–414, 2004.

[13] ] Pennestrì E; Stefanelli R; Velentini PP; Vita L. Virtual musculo-skeletal model for the biomechan-ical analysis of the upper limb. Journal of biomechanics, 40:1350–1361, 2006.

[14] Boninger ML; Baldwin M; Cooper RA; Koontz A; Chan L. Manual wheelchair pushrim biomechan-ics and axle position. Arch Phys Med Rehabil, 81:608–613, 2000.

39

40 BIBLIOGRAPHY

[15] Yoshihiko Nakamura. Advanced Robotics; Redundancy and optimalisation. Addison-Wesley LongmanPublishing Co. Inc. Boston, MA, USA, 1990.

[16] L. Peebles and B. Norris. Adultdata. Department of Trade and Industry (UK), 1998.

[17] Rodgers MM; Keyser RE; Gardner ER; Russell PJ; Gorman PH. Influence of trunk flexion onbiomechanics of wheelchair propulsion. Journal of rehabilitation research and development, 37:283–295, 2000.

[18] Richter WM; Rodriguez R; Woods KR; Axelsin PW. Stroke pattern and handrim biomechanics forlevel and uphill wheelchair propulsion at self-selected speeds. Arch Phys Med Rehabil, 88:81–87,2007.

[19] Richter WM; Rodriguez R; Woods KR; Karpinski AP; Axelson PW. Reduced finger and wrist flexoractivity during propulsion with a new flexible handrim. Arch Phys Med Rehabil, 87:1643–1647,2006.

[20] Boninger ML; Koontz AM; Sisto SA; Dyson-Hudson TA; Chang M; Price R; Cooper RA. Pushrimbiomechanics and injury prevention in spinal cord injury, recommendations based on culp-sci in-vestigations. JRRD, 42:9–20, 2005.

[21] Sullivan SR; Langrana NA; Sisto SA. Multibody computational biomechanical model of the upperbody. ASME, 2005.

[22] Goosey-Tolfrey VL; Fowler NE; Campbell IG; Iwnicki SD. A kinetic analysis of trained wheelchairracers during two speeds of propulsion. Medical Engineering & Physics, 21:259–266, 2001.

[23] Robertson RN; Boninger ML; Cooper RA; Shimada SD. Pushrim forces and joint kinetics duringwheelchair propulsion. Arch Phys Med Rehabil, 77:856–864, 1996.

[24] Artas Engineering Software. SAM6.0. Nuenen, The Netherlands.

[25] Boninger ML; Cooper RA; Robertson RN; Tudy TE. Wrist biomechanics during two speeds ofwheelchair propulsion: An analysis using a local coordinate system. Arch Phys Med Rehabil, 78:364–372, 1997.

[26] Richter WM. The effect of seat position on manual wheelchair propulsion biomechanics: A quasi-static model-based approach. Medical Engineering & Physics, 23:707–712, 2001.

Appendix A

The arm lengths

In this appendix the arm lengths for the models is discussed. At first there is an overview of the literature,then the used lengths in the three dimensional model are presented.

A.1 Upper and forearm length

In this section the publications found and used for the length of the arm are discussed. The resourceswill be discussed separately. The properties of the arms will be used in the wheelchair project.

A.1.1 Handbook of normal physical measurements

In the book by Judith Hall et al. (1989) a lot of measurements of the human body are presented. In thiscase the upper arm and forearm are the case studies.

Upper arm

The upper arm length is presented as a function of age in this book. The scale of the age is from 4 yearsold till 16 years old. In Figure A.1 the length can be seen for males and females.

In figure A.1, the lines divide the percentage of smallest people from the rest. The percentages areshown at the left of the graph. The 50th percentile is the mean of the measurement.

If the assumption is made that after the age of 16 the upper arm does not grow any longer, the lengthat the age of 16 is representative for adults. For females this could be a good assumption, because thelines at the age of 16 are approximately horizontal. If the graph of males is considered, it is very likelythe arm is still growing, because the line is not completely horizontal. Also the possible increase of themean length over time is not taken into account.

The mean line for the males and females upper arm are 32.9 and 30.2 cm respectively.

41

42 APPENDIX A. THE ARM LENGTHS

(a) Upper arm length for males (b) Upper arm length for females

Figure A.1: Upper arm lengths for males and female from the Handbook of Normal Physical Mea-surements

Forearm

Judith G. Hall et al.(1989) also provides forearm lengths classified by sex for the ages of 4 up to 16 years.The graphs can be seen in figure A.2. Just as seen for the upper arm, the lines divide a percentage ofsmallest people from the rest. The percentages are shown at the left of the graph.

If the assumption is made that the forearm does not grow anymore after people turn 16, the length ofthe forearm on the age of 16 is representative for all adults. For females this could be a good assumptionbecause the lines are all approximately horizontal. For males is is very likely that the forearm is stillgrowing, because the lines are not completely horizontal. Also the possible increase of the mean lengthover time is not taken into account.

The mean line of the males and females forearm are 25.4 and 23.1 cm respectively.

A.1. UPPER AND FOREARM LENGTH 43

(a) Forearm length for males (b) Forearm length for females

Figure A.2: Forearm lengths for males and females in Handbook of Normal Physical Measurements

A.1.2 Occupational Biomechanics

This section is based on the work by Don B. Chaffin et al. (1999). In figure A.31 the body is drawn andthe various lengths of the body parts are given as function of the body height2. For the upper arm thismeans that the length is about equal to 0, 186 ∗ H , where H is the length of the person. The forearmhas a length of 0.146 ∗ H . The ratio between the upper arm and forearm is than:

Lupper armLforearm

=0, 186 ∗ H0, 146 ∗ H

≈ 1, 27 (A.1)

If the ratio is calculated for the results in the book of Judith H. Hall et al. [10], the following results arefound:

Male32, 9

25, 4≈ 1, 29 Female

30, 2

23, 1≈ 1, 31

The largest deviation (this is the deviation of the females) is about 3% of the original ratio, given by thebook Occupational Biomechanics of Don B. Chaffin et al.. In the wheelchair the forearm is extendedwith a part of the hand. This is because the hand holds the rim, so a length has to be added to theforearm.

1from the book: , [6]2Comes originally from Roebuck, Kroemer and Thomsom, 1975


Figure A.3: Body segment lengths in proportion to body stature in Occupational Biomechanics

A.1.3 Adultdata

In the publication of Adultdata from L. Peebles et al. [16] different lengths are described of the humanbody. The upper and forearm are both measured, as well as the whole length till the grip.

From shoulder to center of gripped rod

In this subsection the whole arm, so from the shoulder till the gripped rod, see figure A.43 for moreinformation. In figure A.4 are the results of research in several countries. What is interesting for thewheelchair project is the Mean value, the 5th percentile value and the 95th percentile value. These havethe following meaning:

• The 5th percentile value is the value which divides the 5% of the smallest people of the wholepopulation from the others. This value will always be the smallest value, because of it’s meaning.

• The mean value is the value which divides the smallest half from the tallest half of the population.Because this is considered to have a normal distribution, it also is the peak value for the chance ofmeasuring in a sample.

• The 95th percentile value is the line which divides the 5% of the tallest people from the rest of thepopulation. This will always have the highest value.

To get a view of the length of the whole arm, the mean values are compared. The mean is taken asa representative for a large group, because the user of the wheelchair in the wheelchair project isn’tspecified.For the males the mean id between 621,7 [mm] and 685 [mm]. For the females it is between 560 [mm]and 625 [mm].

Upper arm

For the upper arm, the measurement in figure A.54 is chosen from the report of L. Peebles et al. [16].These measurements are taken from a arm hanging straight at the side, which represents the upper armin the model of the wheelchair project.Again the mean is taken as representative for a large group. Themean of the measurements of males are about 331 [mm], while for females it is about 306 [mm].

3Form the report: Adultdata, [16]4Form the report: Adultdata, [16]

A.1. UPPER AND FOREARM LENGTH 45

Figure A.4: Length from the shoulder to grip handle in [16]

Figure A.5: Length of the upper arm in [16]


Forearm with and without wrist

The measurements of the forearm in the report Adultdata [16] the measured length is not immediatelyusable for the wheelchair project. The reason is that the measurement starts at the back of the upper arm,which is not the place the elbow joint is modeled in the project. Therefore two separate measurementsare being used.

The first measurement is the length of the forearm without the wrist. The results are shown in figureA.6. What also can be seen in that figure is that the measurement starts at the back of the upper arm,just as stated before.

Figure A.6: The measurement of the forearm in [16]

The second measurement is the length from the back of the upper arm till a gripped rod. This lengthis used to get a comparison with the whole arm in subsection A.1.3. This measurement can be seen infigure A.7

A.2 The used segment lengths in the 3D model

Now all the information needed to get an comparison with the other literature, and the length of theforearm can be corrected. This correction is needed because this measurement does not start at theelbow joint, but at the back of the upper arm. First the data of two countries is gathered and put in onetable. The reason only these two countries are picked, is because the length of the upper arm is onlyknown for these two countries. The data, including a calculated ratio of R =

Lupper arm

Llower arm without wrist, is

shown in table A.1.The ratios in table A.1 are lower than the ratios seen with the other literature, the difference is about

10% of the original ratio found in the book of Don B. Chafin et al. [6].The reason of the difference can be explained by the measurement of the lower arm. To correct this, thelength of the back of the upper arm till the gripped rod is shortened, so that this length plus the lengthof the upper arm is equal to the length of from the shoulder to the gripped rod. The correction made tothe length from back of upper arm till gripped rod can also be used to correct the length from the back ofthe upper arm to the wrist. In table A.2 the correction for the lower arm is shown, together with the newlengths and the new ratio. The new ratios have an deviation from the original ratio found in OccupationalBiomechanics [6] of about 3%. The values from the Handbook of normal physical measurements arecompared with the values of Adultdata very alike, the deviations are all under 2%.

For the three dimensional model, the mean values of the UK males will be used for simulation. Theused segment lengths are shown in A.3.

A.2. THE USED SEGMENT LENGTHS IN THE 3D MODEL 47

Figure A.7: The measurement from the back of the upper arm till a gripped rod in [16]

Table A.1: The data gathered from AdultdataCountry and Length Length Length Length forearm Ratio

gender whole arm upper arm forearm till till gripped rod[mm] wrist[mm] wrist [mm] [mm]

UK males 662,5 330,8 288,3 365,5 1,15UK females 608,5 305,7 256,6 325,5 1,19USA males 664,5 331,8 289,2 366,6 1,15USA females 611,0 306,9 257,6 326,9 1,19

Table A.2: The corrected data for the forearmCountry and Correction New length New length Length wrist Ratio

gender [mm] forearm till forearm till from wrist tillwrist [mm] gripped gripped

rod[mm] rod [mm]UK males -33,8 254,5 331,7 77,2 1,30UK females -22,7 233,9 302,8 68,9 1,31USA males -33,9 255,3 332,7 77,4 1,30USA females -22,8 234,8 304,1 69,3 1,31

Table A.3: The lengths of the upper arm, forearm and hand used in the modelLength Length of the Length from

upper arm forearm till wrist till[mm] wrist [mm] gripped rod [mm]330,8 254,5 77,2

Appendix B

The Richter model

B.1 The model

The calculation of the wheelchair model will give some insight on how this model behaves, and wherethe formulas come from. This is also a check to see if what was done is correct.

The model is shown in figure 2.1. The angles θhs ,θc and θr are defined to the vertical axis of thewheel with clockwise is being positive. The angle θs is defined negative as it is drawn in this picture.

B.2 The free body diagram

For the free body diagram the arm is disconnected from the rim. The rim is replaced by the force therim puts on the arm. In [26] these forces are shown.The force, the tangential component and the radial component at the rim are shown in figure 2.2. Thefree body diagram is shown in figure 2.3. It shows the upper and lower arm and the forces and momentson them. This model is quasi-static, which means that the sum of all forces and moments must be equalto zero, which leads to (B.1) and (B.2)∑

~F = ~P =∂

∂t(m ~rcm) = 0 (B.1)

∑~M = ~H =cm J · ω = 0 (B.2)

For all calculations the positive x-forces are pointing to the right. The positive y-forces are pointingupward. The positive moments are counter-clockwise.The number of degrees of freedom can be calculated with the Gruebler count.


mi (B.3)

In this equation B.3 are the variables defined as followed:

Table B.1: The variables of the Gruebler count

N Number of bodiesM Number of constraintsm Number of constrained directions

48

B.3. CALCULATIONS OF THE ANGLES 49

Applying equation B.3 and the definitions in table B.1, the number of degrees of freedom are:

NDOF = 3 ∗ 2− (2 + 2 + 1) = 1

The revolute joints at the shoulder and elbow constrain two translations (vertical and horizontal). Thelast constraint is at the hand-rim contact, which constraints radial translations of the hand.

B.3 Calculations of the angles

There are six angles which must be calculated. The variable for the system is the angle of the wheel (θw).Because there is only 1 degree of freedom only one variable is needed.

Figure B.1: The diagram to determine θc and θsi

First the contact angle (θc) is calculated. The triangle used can be seen in figure B.1 and it exists ofLua, Lfa+RHR and Lhs. This triangle can be calculated using the law of cosines.

L2ua = L2

hs + (Lfa +RHR)2 − 2 (Lfa +RHR) Lhs cos(θhs − θc)

θc = θhs − cos−1(L2hs + (Lfa +RHR)

2 − L2ua

2 Lhs (Lfa +RHR)

)(B.4)

The angle θc is negative in equation B.4 because of the direction (it is defined positive at the other sideof the vertical axis).From the same figure B.1 the initial shoulder angle (θsi) can calculated.

(Lfa +RHR)2= L2

hs + L2ua − 2 Lhs Lua cos(−θsi − θhs)

θsi = −θhs − cos−1(L2hs + L2

ua − (Lfa +RHR)2

2 Lhs Lua

)(B.5)

50 APPENDIX B. THE RICHTER MODEL

Figure B.2: The diagram to determine θr

The release angle (θr) can be calculated from figure B.2. Again the law of cosines is used.

(Lfa + Lua)2= L2

hs +R2HR − 2 Lhs RHR cos (θr − θhs)

θr = θhs + cos−1

(L2hs +R2

HR − (Lfa + Lua)2

2 Lhs RHR

)(B.6)

In figure B.3 the diagram is shown to calculate the elbow angle (θe) and shoulder angle (θs).First the

Figure B.3: The diagram to determine θe and θs

length a is determined as a function of the wheel angle. This will then be the variable to shorten theequations of the elbow and shoulder angle. The calculation of a comes from the x and y position of theshoulder and the wrist. The calculation can be done with Pythagoras’ theorem and is shown in equationB.7.

a =

√[Lhs sin (θhs)−RHR sin (θw)]2 + [Lhs cos (θhs)−RHR cos (θw)]2 (B.7)

When a is known, the shoulder angle can be calculated using the law of cosines.

L2fa = L2

ua + a2 − 2 Lua a cos (−(θs − θi))

B.3. CALCULATIONS OF THE ANGLES 51

θs = θi − cos−1(L2ua + a2 − L2

fa

2 Lua a

)As used for the calculation of a, θi can expressed in the position of the shoulder and the position of thewrist.

−θi = tan−1(Lhs sin(θhs)−RHR sin(θw)Lhs cos(θhs)−RHR cos(θw)

)This will than lead to the equation of the shoulder angle (θs):

θs = −tan−1(Lhs sin(θhs)−RHR sin(θw)Lhs cos(θhs)−RHR cos(θw)

)− cos−1

(L2ua + a2 − L2

fa

2 Lua a

)(B.8)

For the calculation of the elbow angle (θe) the same laws apply.

a2 = L2ua + L2

fa − 2 Lua Lfa cos (π − θe)

θe = π − cos−1(L2ua + L2

fa − a2

2 Lua Lfa

)(B.9)

The last angle to be determined is the angle of the force (θF ). This angle is defined as the anglebetween the force and the tangential direction of the rim, as shown in figure B.4.

Figure B.4: The diagram to determine θF

To determine this angle all three trigonometric functions can be applied, because all three curves areapproximated with a Gaussian7 curve. While simulating the best function can be chosen, because of theapproximation. So the three possibilities are:

sin(θF ) =|Fr||F |

=⇒ θF = sin−1(|Fr||F |

)

cos(θF ) =|Ft||F |

=⇒ θF = cos−1(|Ft||F |

)tan(θF ) =

|Fr||Ft|

=⇒ θF = tan−1(|Fr||Ft|

)(B.10)

For the simulations the equation with the tangent (equation B.10) is used, because this one has thesmoothest curve for the used angles.


B.4 Calculations on the forearm

Figure B.5 shows the free body diagram of the forearm.

Figure B.5: The free body diagram of the forearm

∑ ~Maround elbow = ~Te − cos(θF + θw) ~F Lfa cos(θs + θe)

+sin(θF + θw) ~F Lfa sin(θs + θe) = 0(B.11)

If equation B.13 is used in equation B.11, the following is obtained:

~Te = ~F Lfa (cos(θF + θw) cos(θs + θe)− sin(θF + θw) sin(θs + θe))

~Te = ~F Lfa cos(θs + θe + θF + θw)

~Te = ~F Lfa sin(π2− θs − θe − θF − θw

)~Te = −~F Lfa sin

(θs + θe −

(π2− θF − θw

))(B.12)

Equation B.12 is the same as in the article of W.M. Richter. The reaction forces can be calculated fromequation B.13 and are as followed:

~F3 = cos(θF + θw) ~F~F4 = −sin(θF + θw) ~F

(B.13)

B.5. CALCULATIONS ON THE UPPER ARM 53

B.5 Calculations on the upper arm

Figure B.6 shows the free body diagram of the upper arm.

Figure B.6: The free body diagram of the upper arm

Forces F3 and F4 are already known from the calculations of the forearm. From the equations B.1 andB.2 can be learned: ∑

~F =

[~F1 − ~F3

~F2 − ~F3

]= 0 (B.14)

∑~Maround schoulder = ~Ts − ~Te − cos(θs) Lua ~F3 − sin(θs) ~F4 = 0 (B.15)

From equation B.14 the reaction forces can be derived:

~F1 = ~F3 = cos(θF + θw) ~F~F2 = ~F3 = −sin(θF + θw) ~F

If this is used in equation B.15, the following is obtained:

~Ts = ~Te + cos(θs) Lua cos(θF + θw) ~F − sin(θs) Lua sin(θF + θw) ~F

~Ts = ~Te + ~F Lua(cos(θs) cos(θF + θw)− sin(θs) sin(θF + θw))

~Ts = ~Te + ~F Lua cos(θs + θF + θw) (B.16)

Equation B.16 isn’t the same as in the article of W.M. Richter. The difference can be seen in figureB.7 for θhs = 0 and the original length. For the conclusions this makes a difference. In figure B.5 theshoulder torque is displayed as it is in the article with the enlongments of Lhs. The angle θhs is keptat 0 degrees. In figure B.5 the new calculated shoulder torque is displayed. What becomes clear is that

the absolute shoulder torque∥∥∥ ~Ts∥∥∥ does not become less for a lower seat position. In fact there is an

optimum somewhere between +10 cm and -10 cm of seat height difference.


Figure B.7: Difference between the torque from the article and the calculated torque

(a) The shoulder torque from the article fordifferent heights Lhs

(b) The calculated shoulder torque for differ-ent heights Lhs

B.6. CHECK THE CALCULATIONS 55

B.6 Check the calculations

To check the result, the sum of all moments about the schoulder for the whole arm (so upper andforearm) are examined. Because the calculation is quasi-static, it should be equal to 0. A figure of this isdrawn in figure B.8.

Figure B.8: The diagram to perform a check on the calculations

The sum of all moments bout the shoulder are:∑ ~Maround shoulder = Ts − ~F cos(θF + θw) [Lfa cos(θs + θe) + Lua cos(θs)]

+~F sin(θF + θw) [Lfa sin(θs + θe) + Lua sin(θs)] = 0(B.17)

This check is used to check the results of the analytical calculation as well as the numerical calculations,to see what are the differences.

Notes1Richter WM. The effect of seat position on manual wheelchair propulsion biomechanics: A quasi-static model-based approach.

Medical Engineering & Physics 2001;23;707-712.

Appendix C

Implementation of the 3D model inMatlab

In this appendix the implementation of the three dimensional model in Matlab is presented. This isdone to help anyone who would like to understand or reproduce the results. Only the general multibody dynamic model is discussed here. All inputs are already discussed in the report in section 3. Thisappendix will just show how the model is implemented in Matlab (file test.m).

The whole system which is implemented, is the Lagrange equation of motion:

d

dt

(∂T

∂q

)− ∂T

∂q+∂V

∂q= (Qnc)

T (C.1)

The general coordinates are in this case the degrees of freedom, represented with the ψ. This is repre-sented as the column ψ = [ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7]

T .

C.1 The Cartesian coordinate frames

The first step is realizing that the frames can be expressed using the angles. So every frame can beexpressed in the global~eO frame. The general way to describe these changes in orientation is the follow-ing:

~ei = Ai j ~ej (C.2)

One of the most commonly used features of these direction cosine matrices is as followed:(Ai j

)−1=(Ai j

)T= Aj i (C.3)

The arm has seven degrees of freedom. With the angles which describe the degrees of freedom, thedirection cosine matrices can be derived. The following matrices are found:

A1 0 =

C(ψ3) ∗ C(ψ2) S(ψ3) ∗ C(ψ2) −S(ψ2)C(ψ3) ∗ S(ψ2) ∗ S(ψ1)− S(ψ3) ∗ C(ψ1) S(ψ3) ∗ S(ψ2) ∗ S(ψ1) + C(ψ3) ∗ C(ψ1) C(ψ2) ∗ S(ψ1)C(ψ3) ∗ S(ψ2) ∗ C(ψ1) + S(ψ3) ∗ S(ψ1) S(ψ3) ∗ S(ψ2) ∗ C(ψ1)− C(ψ3) ∗ S(ψ1) C(ψ2) ∗ C(ψ1)

A2 1 =

C(ψ5) 0 −S(ψ5)S(ψ5) ∗ S(ψ4) C(ψ4) C(ψ5) ∗ S(ψ4)S(ψ5) ∗ C(ψ4) −S(ψ4) C(ψ5) ∗ C(ψ4)

56

C.2. MOMENTS OF INERTIA ABOUT THE JOINTS 57

A3 2 =

C(ψ7) ∗ C(ψ6) S(ψ7) ∗ C(ψ6) −S(ψ6)−S(ψ7) C(ψ7) 0

C(ψ7) ∗ S(ψ6) S(ψ7) ∗ S(ψ6) C(psi6)

AHub O =

C(χ) ∗ C(θw) S(χ) ∗ C(θw) −S(θw)C(χ) ∗ S(θw) ∗ S(φ)− S(χ) ∗ C(φ) S(χ) ∗ S(θw) ∗ S(φ) + C(χ) ∗ C(φ) C(θw) ∗ S(φ)C(χ) ∗ S(θw) ∗ C(φ) + S(χ) ∗ S(φ) S(χ) ∗ S(θw) ∗ C(φ)− C(χ) ∗ S(φ) C(θw) ∗ C(φ)

The degrees of freedom of the system, so the angles ψn, are defined in section 3.1.1. The χ, θw and φ arethe wheel angles, as seen in section 3.1.7. So these are the camber angle (χ), wheel angle (θw) and thetoe angle (φ). The S represents a sine-function, the C represents a cosine-function.

All vectors which are used are easy to express in one of the frames, and can then be described inthe ~eO frame. To every center of mass and every joint there is a vector which described their place. Forinstance the place of the center of mass with respect to the ~eO frame is described by a vector from theshoulder to the elbow and the elbow to the center of mass. Adding these vectors gives the place of thecenter of mass of the forearm.

C.2 Moments of inertia about the joints

All moments of inertia about the center of mass should be converted to the moments of inertia about thejoints. This is done with the Huijgens-Steiner formula as seen in equation (3.9). The moments of inertiaabout the shoulder are all moments of inertia converted to the shoulder. This must be done because allmasses have an angular acceleration about the shoulder if one angle has an angular acceleration. Theangles at the elbow influence the masses at the forearm and the hand, so for the elbow only these twomoments of inertia are taken in account. For the wrist only the moments of inertia of the hand is takeninto account.

C.3 The kinetic energy

In section 3.2 is seen that the kinetic energy is received by calculating the moments of inertia. So thekinetic energy is a function of the moments of inertia. The equation is split into two parts as seen in thebook Advanced robotics [15].

The first part is the following equation:

d

dt

(∂T

∂ψ

)=

d

dt

(ψT OJ(ψ)

)= ψT OJ(ψ) + ψT

dOJ(ψ)dt

(C.4)

The part of ψT OJ(ψ) is relatively easy to compute, because this is a simple matrix multiplication. Thelast part of ψT dOJ(ψ)

dt is calculated as followed (an is the n-th colummn of OJ):

ψTdOJ(ψ)dt

=[ψT ∂a1

∂ψ ψ ψT ∂a2∂ψ ψ ... ψT ∂an

∂ψ1ψ]

(C.5)

This requires a derivative of the moments of inertia with respect to the degrees of freedom. This canbe calculated using the command ”diff” in Matlab. In the test.m file, this part is stored in the ”row_ 1”variable.

The second part is as followed:

∂T

∂θ=

1

2

∂

∂θ

(θT OJ(θ) θ

)(C.6)

58 APPENDIX C. IMPLEMENTATION OF THE 3D MODEL IN MATLAB

For the derivative of the moment of inertia matrix with respect to the degrees of freedom the command”diff” is used again. In the test.m file this part is stored in the ”row_ 2” variable.

In Matlab the total kinetic energy at shoulder, elbow and wrist can be derived using ψw)_OJ+row_1−row_2. The last step is deleting the constraint angles. So for the elbow only the first two columns mustbe implemented, while for the wrist these are the second and third column. In total the T-row will haveseven columns, equal to the number of degrees of freedom.

C.4 Potential energy

Calculating the potential energy is straightforward. This is a simple multiplication and is calculated asfollowed:

V = −m~g · ~rcm (C.7)

The dot product can also be replaced by a normal matrix product:

V = −m gT rcm (C.8)

The g and rcm are now underlined to show that these are the vectors represented in a column-matrix.The Lagrangian equation of motion requires a derivation to obtain ∂V

∂q . This is again obtained usingthe command ”diff” in Matlab.

C.5 The non-conservative forces

The non-conservative forces have two parts. The first part are the hand-rim forces acting on the hand.The second part are the moments in the arm, which are the outputs of the system.

The non-conservative forces are calculated as followed:

Qnc =

nF∑i=1

(∂~ri∂ψ

)T· ~Fnci +

nM∑j=1

(∂θj∂ψ

)T· ~Mnc

j (C.9)

The hand-rim forces are the only non-conservative forces in the system. These equations can be refor-mulated to matrix multiplications. The vector ~ri is the vector pointing from the shoulder to the pointwhere the force acts on the hand. The derivative is calculated by using the ”diff” command again.

The only non-conservative moments are the moments in the arm, so the outputs of the system.These results in a [M1 M2 M3 M4 M5 M6 M7]

T column.The moments and the forces are splits in the equations of motion, to make it easier to compute the

moments in the arm.

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