The Butterfly-robot: theoretical behaviour of small perturbations

and their effect on stability

Joris Knol (TU/e: 0863192, NTNU: 996324)DC 2016.041

July 22, 2016

Abstract

The butterfly robot balances a ball on a particularly shaped, ever rotating disc. Previous researchprojects have focused on the control of this robot.

This report examines the small dynamical behaviours, which are usually neglected in control theorem. Toestimate the crucial factors of small scale perturbations, energy losses and fluctuations are explored. Thedynamical behaviour of the most crucial perturbation is examined in more detail. Additionally, bouncingeffects on small bumps are more closely examined and related to a loss of contact and slip. Also, asimulation model is proposed to further examine the dynamical effects of small scale perturbations.

Conclusions are drawn on the influence of small perturbations, their practical effect on the robot.

1

Contents

1 Introduction 3

2 Hypothesis on causes of instability based on energy dissipation 52.1 Energy losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Friction energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Change in mass and inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.5 Energy fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Interpretation of the resulting energy fluctuations . . . . . . . . . . . . . . . . . . . . . . . 82.3 Conclusions and summary energy fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Impact dynamics and loss of contact due to bumps 123.1 Impact dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Loss of contact and resulting slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Conclusions and summary impact dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Forces Simulation Model 174.1 Contact forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.2 Different contact models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.3 Hunt and Crossley model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.4 Constrained forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Ramp representing the contact area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 Conclusions and summary simulation model . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Conclusions and recommendations 255.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Appendix 27

7 References 28

2

Chapter 1

Introduction

This report considers the dynamical effects of small pertubations on the butterfly robot (BR for short).

The butterfly robot consists of a rotating disk with a butterfly-like shape and a ball on top, see Figure1.1. The disk is actuated with a motor. The ball can freely roll along the disk and its position is measuredusing an optical camera. The disk is constantly rotating mainly in one direction. While rotating the ballis balanced using algorithms with the camera’s input and controlling the motor’s output. The result isthat the ball is balanced in an unstable equilibrium point, actuated by the motor in 1 degree of freedom(DOF) and controlled with an algorithm [1].

Figure 1.1: Butterfly robot balancing a ball while rotating

Although empirical results show that the algorithm balances the ball adequately, the computer simula-tions show differences in comparison with those empirical results. These differences are mainly visible inthe speed of the ball.

Standard control theorems may not be sufficient to predict this speed difference. This might be dueto the simplifications in the control theorems. Therefore, instead of using control theorems, this reportfocuses on the dynamical behaviour of the BR system. Like the small dynamics due to friction and/orsurface imperfections, that are normally not considered when working with the control theorems.

Previous research has been conducted on the BR. For example, [2] discusses and experiments on themotion of the BR, and [3] derives an energy efficient controller. The works of H. Francke [4] and S. vanden Eijnden [5] focused on the robustness of the controller in a simulation. To our knowledge, little orno research has been done in the particular field of small scale dynamics.

This report is organized as follows:

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First, hypothesises are made about different perturbations that influence the BR. This is done by usingenergy and energy fluctuations. Conclusions will be made about the most important factors and theirinfluence on the BR’s dynamics.Second, the influence of small bumps influencing the ball is explored.Lastly the new simulation is derived, which can be used to examine different perturbations in moredetail.

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Chapter 2

Hypothesis on causes of instabilitybased on energy dissipation

Hypothesises are drawn on the perturbation(s) that influence the BR the most. Multiple perturbationsare researched. These perturbations are small in nature and cannot be compensated by the controller,simply because the sensor and processor are not fast and/or accurate enough. The goal is to find themost crucial perturbations. This chapter only focuses on the amount of influence of the perturbationson the dynamics, but does not explore the resulting dynamical behaviour.

To compare the influence of different perturbations the influence must be given a value. The amountof influence is roughly equal to the losses in energy. Energy losses over a small timespan will havemore influence on the BR than energy losses over a longer timespan. Thus by comparing the energyfluctuations over time, the amount of influence can be estimated.

First an ideal system is assumed, without any perturbations. The energy in this ideal system is computedusing basic energy formulas. The energy of this ideal model can be compared to the energy fluctuationsdue to the perturbations. The different perturbations considered are:- energy loss due to height differences;- elastic deformation;- difference in parameters;- sway and friction.

The energy losses are derived using the results taken from the simulation created by H. Francke [4] andS. van den Eijnden [5]. Their model uses the degrees of freedom (DOF) as given in Figure 2.1. TheseDOF are described as the path that the centr of the ball travels along the BR’s track: s, the distance ofthe center of the ball to the track: w, the rotation of the ball: ψ and the rotation of the BR: θ. In theirsimulation the ball is considered not to leave the track, w = 0, and the rotation of the ball is a functionof the travelled distance, ψ ≡ f(s). In other words, the simulation considers an ideal system.

The simulation provides the different DOF values at certain points in time, roughly every 0.001 secondsover a timespan of 10 seconds.

2.1 Energy losses

First the ideal system is assumed, the energy in the system without the perturbations is computed. Theenergy in the system under ideal circumstances is the summation of kinetic and potential energy. All theparameters of the BR can be found in the Appendix. This includes the inertia and mass of the ball andBR: mb, Jb and Jf , and the effective radius by which the ball rolls over the BR: R. If the total energyin the system is known, the input energy is also known. And since all energy is provided by the motor,

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Figure 2.1: DOF used in [4,5]

the input energy of the motor can be derived.

Esys =1

2(mbs

2 + Jbφ2 + (Jbr + Jb +mbr

2b )θ

2) + sin(φ− θ)rbmbg | rb = ‖~rb‖ (2.1)

~rb = ~ρ+R~τ = (0.1095− 0.04 cos(2φ))

sin(ϕ− θ)cos(ϕ− θ)

0

+R~τ (2.2)

The function of ~ρ describes the contour of the BR and is taken from [4,5]. The energy the BR’s controllerput into the system is the same function initialised as Ein = 0 at t = 0.

Ein =1

2(mbs

2 + Jbφ2 + (Jbr + Jb +mbr

2b )θ

2) + sin(φ− θ)rbmbg − Esys(t = 0) (2.3)

The time derivative of this function tells us the change in energy. The time derivatives are computednear the end of this chapter.

2.1.1 Friction energy

Friction energy losses can be split in two categories:- rolling friction of the ball- friction in the joint of the BR.The rolling friction is a function of the normal force and the travelled distance s,

Erol = µrol

∫ s

0

||Fn(s)||ds. (2.4)

6

The friction in the joint is similar, only using angle θ and torque Tfric,

Ejoint = µjoint

∫ θ

0

||Tfric||(θ)dθ. (2.5)

Herein is the Tfric an estimation, this function can be found in the Appendix. The estimation is madeassuming that the BR is placed on a roll-bearing.

2.1.2 Kinetic energy

The kinetic energy is split in rotational energy and translation energy.

The change in rotational energy considered. If the ball has a slightly smaller radius, then the energyneeded for the rotational speed of the ball increases.

Erot =s2

2

(Jb

(R∗)2 −

JbR2

), (2.6)

with R∗ being the decreased radius.

The translation energy change of the ball can be due to a sideways movement. The ball may sway dueto the misalignment of the BR’s joint, meaning the axis of rotation is not perfectly horizontal which itideally would be. The resulting sway in z-direction will be:

Etr =mb

2v2z | vz = f(φ, φ, θ, θ); (2.7)

vz =d

dt

(rBR(φ) sin(εjoint) sin(θ)

)= sin(εjoint)

(rBR(φ, φ) sin(θ) + rBR(φ) cos(θ)θ

)(2.8)

rBR(φ, φ) = 0.080 sin(2φ)φ (2.9)

with εjoint being the error angle between joint and horizontal axis of the motor.

2.1.3 Potential energy

Small bumps and shape imperfections of the track and ball result in a change in potential energy. Forinstance, the ball can have an ellipse shape resulting in a sinusoidal movement along the track. Themaximum height difference is given as δh.

There are 3 situations.The first is when δh is a constant, in other words, there is a dimension error. For instance, when theradius of the ball is slightly bigger. Consequently it has a low fluctuation in energy.The second situation occurs when δh is a function of s. For instance when the ball is an ellipsoid. Inwhich case the ball will move up and down in a sinusoidal movement: δh ∝ sin(ψ). In this case theenergy fluctuations are larger than the first situation.The last situation is considered a bump over which the ball must roll, resulting in a change in height.This will cause harsh energy spikes. Situation 3 will cause the biggest energy fluctuations and is thereforethe most crucial of the three.

In order to calculate the energy fluctuations of situation 3, we use situation 1 for the potential energydifference between a track with and without a bump. This gives the maximum change in potential energyat a certain point in time. This change must be divided by the amount of time it takes for the ball toreach the top of the bump to compute the energy fluctuation of situation 3.

First situation 1 is computed. This will be used to compute situation 3 later in this chapter.

Epot = mbδh~n · ~g (2.10)

7

Apart from height differences, there can be some storage of energy in elastic deformation of both theball and the track. It is assumed to be pure elastic deformation. Stated differently, there are no plasticdeformations that result in a loss of energy. On a side note, the difference in height as a result ofelastic deformation is insignificant for the potential energy losses, because the ball and BR are relativelystiff.

Eelas =1

2kbt

(||Fn||kbt

)2

=||Fn||2

2kbt(2.11)

where kbt is a replacing constant for the stiffness combining the ball and track. Again, this kbt can befound in the Appendix.

2.1.4 Change in mass and inertia

Consider the case when the BR and ball are heavier. The energy in the system then varies according tothe following equation.

Emass =1

2(m∗b s

2 + J∗b φ2 + (J∗br + J∗b +m∗br

2b )θ

2) + sin(θ + φ)rbm∗bg − Esys (2.12)

with J∗b being the increased inertia and m∗b the increased mass.

2.1.5 Energy fluctuations

The energy fluctuations can be given as the time derivative of the energy functions. As mentioned, thecomputations are done with the use of data from previous work [4,5]. This data includes the value of thedifferent DOF over a time of 10 seconds with timesteps of approximately 0.001 seconds. Because of thisthe conventional derivatives can not so easily be computed. The easiest method to process this data isby dividing the change in energy of one step over the time of that step. Therefore, the time derivativeof any of the above energy functions, herein given as function E(t), is the change in these functions E(t)over the time step T .

E(t) ≡ E(t+ T )− E(t)

T(2.13)

2.2 Interpretation of the resulting energy fluctuations

This section interprets the energy fluctuations. The energy fluctuations are derived with the ’Matlab’program. They are calculated in extreme conditions to make the results more explicit. In order toachieve these extreme conditions, some assumptions and approximations on variables are made.

Not all variables of the BR are provided, as a result some variables are approximated. The followingapproximations are made:- the mass and inertia increase is 1% for Emass.- the radius is decreased 1% for Erot.- the height difference for Epot is equal to δh = 0.0005[m].- the rolling friction of the ball is considered µrol = 0.001.- the friction torque in the joint is considered Tjoint = 10−5Fload, an average for ball-bearings with

diameter 0.01[m], herein is Fload = ‖~Fn + ~Ft + mBR~g‖ with mBR (the mass of the BR) estimated atmBR = 0.1[kg].- the angle of misalignment in Etr is chosen as εjoint = π

80 .- the combined stiffness of ball and BR is estimated at kbt = 106[N/m2]. This stiffness is chosen lowerthan the actual value, to increase the effect.

With these assumptions and estimations plots are made. See figures 2.2, 2.3 and 2.4. Many of theseassumptions are fairly rough, but the goal is to define the most influential perturbations. Their exactinfluences are not relevant to this hypothesis.

8

Plots of the energy and energy fluctuations are found in figures 2.2, 2.3 and 2.4.In the first figure, the energy and change of energy in the ideal system is visualised. This figure can becompared with the other figures to understand the order of magnitude.The second figure, Figure 2.3, shows the energy losses, both plots use the same type of data: en-ergy(fluctuation) vs. time, but note that the scale is different. It is clear that the sources of energy lossare, in order of large to small: Ejoint, Erol, Emass, Epot, Erot, Etr and Eela.In the third figure, Figure 2.4, the energy fluctuations are shown. Again both plots have an equal func-tion, and in this case also an equal scale. The order of large fluctuations to small is now: Emass, Ejoint,

Epot, Erot, Erol, Etr and lastly Eela.

Remarkable, when comparing energy loss and its fluctuations, is that Emass is now largest, and Erol isless important. Overall it is clear that the energy fluctuations due to the heavier mass, the potentialenergy and joint friction are most important. Another noteworthy feature is that most of the fluctuationspikes are around the same moment in time.

Even though the energy loss due to mass and joint friction seems to have more influence, the influenceof potential energy loss might actually be far greater. As stated previously, the potential energy can bea result of 3 situations: a constant increase due to, for instance, an oversized ball, sinusoidal increase,or a sudden increase due to a bump. The potential energy loss plotted is the first situation, meaningthat the ball is constantly at a greater distance δh from the track. Now consider the last situation, theball hits a bump and in a short timespan, the ball reaches the top. If this is considered for the energyfluctuation then it will maximumly become:

Epot,bump =Epot,maxTbump

=1.45 · 10−5

Tbump(2.14)

Tbump =

√R2 − (R− δh)2

saverage(2.15)

Using the data, the energy fluctuation becomes Epot,bump ≈ 9·10−4, which is at least 20 times higher thanthe other fluctuations. Therefore the bumps on the BR or ball are considered the most crucial factor.The fluctuation in energy is roughly 10% of the average change in energy provided by the controller. Thesize of these bumps is in the scale of surface roughness, meaning the ball will almost continuously hit anew bump. This will lead to a significant ’noise’-like disturbance on the net change in energy.

When the investigated perturbations are combined, they show steep increases in energy at overlappingmoments in time. In other words, the summation of all factors rather than one is also worth considering.The energy fluctuations of the bumps in the track are still most important, because these fluctuationsare larger than the fluctuations of all the other perturbations combined.

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(a) Total energy within the ideal system (b) Total energy fluctuations of the ideal system

Figure 2.2: Energy (fluctuations) of the ideal system

(a) (b)

Figure 2.3: Energy of different perturbations

2.3 Conclusions and summary energy fluctuations

Small scale perturbations effect the dynamical behaviour of the BR. This chapter hypothesised whichperturbations are the most influential. These can be found by computing their energy and energyfluctuations.

Bumps in the track appear to have the largest energy fluctuations. Bumps with a height of 0.0005[m]reach roughly 20 times the energy fluctuation of the other perturbations. This is mainly due to the abruptheight change in a short time span. The dynamics of these bumps are therefore worth investigatingfurther, see Chapter 3.

Other considered perturbations were:- energy loss due to height differences Epot;- elastic deformation Eela;- an increase in mass and inertia Emass;- sway due to misalignments Etr;

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(a) (b)

Figure 2.4: Energy fluctuations of different perturbations

- increase in rotational speed of the ball, due to a decreased radius Erot;- friction in the joint of the BR Ejoint;

- rolling friction of the ball Erol.These perturbations were put in order of their estimated influence according to the energy fluctuations.The order of large to small fluctuations is: Emass, Ejoint, Epot, Erot, Erol, Etr and lastly Eela.

When these perturbations are combined, they show steep increases in energy at overlapping momentsin time. In other words, the summation of all factors rather than one is also worth considering. Incomparison: the energy fluctuations of the bumps in the track are still significantly higher than all theseother perturbations combined.

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Chapter 3

Impact dynamics and loss of contactdue to bumps

The previous chapter hypothesised about the influence of different perturbations. The most crucialperturbation appears to be the influence of bumps. This chapter focuses on the resulting dynamicalbehaviour due to these bumps in the track or the surface of the ball.

Previous work [4] revealed that under ideal circumstances, the normal forces are sufficient to keep theball from leaving the track. To investigate whether the ball leaves the track or not, the focus of thischapter is on the loss of contact between the ball and BR. A loss of contact results in ’slip’. Whenslipping, the ball will move at its own (angular) speed regardless of the speed changes of the rotatingdisk.

The loss of contact due to a single bump is short and will not influence the dynamics much. However,numerous bumps are located along the BR. The summation of these small moments of slip can eventuallylead to a significant influence.

Assuming the ball and track are both stiff, the time of impact is insignificantly small. As a result of this,a discrete time model is sufficient by taking the time just before and after impact. This discrete timemodel of the ball hitting a bump is made in section 3.1.

3.1 Impact dynamics

Consider the case when a horizontal moving, rolling ball comes into contact with a small bump, seeFigure 3.1. The velocities just after collision, ~v+ and ψ+, derived from the initial velocities, ~v and ψ, arecalculated using energy equations. Assuming there is no energy dissipation due to the impact (the trackand ball are considered stiff), the energy before and after impact are equal. The impact is considered tohave a small timespan.

The only energy of importance is the kinematic energy. The energy before impact is fully known. Themovement after impact is restricted. The ball can only move as if it were hinged at the point of impact,point ’O’ in Figure 3.1, with angular velocity ω.

Ekin = E+kin ⇒ mb‖~v‖2+Jbψ

2 = (Jb +mbR2)ω2 (3.1)√

mb‖~v‖2+Jbψ2

Jb +mbR2= ω. (3.2)

Note that generally ω can theoretically either be positive or negative according to these equations.However the situation clearly restricts that it can only be positive. With this ω the velocities ~v+ and

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Figure 3.1: Impact model of the ball colliding with a bump

ψ+ are determined asψ+ = −ω, (3.3)

~v+ =

ycωxcω0

=

(R− h)ω

−√R2 − (R− h)2ω

0

. (3.4)

3.2 Loss of contact and resulting slip

Figure 3.2: Different paths of the ball after collision

Using the computed velocities ~v+ and ψ+ after impact, the path of the ball can be computed. Threedifferent paths are considered.Path ’A’ is when the ball bounces higher than the bump.Path ’B’ is when the ball only looses contact before it reaches the top of the bump.Path ’C’ is when the ball would not loose contact. All paths will start in the direction of the initialvector ~v+.

In order to determine which path is taken, the trajectory of the ball is computed as if the ball does nothave any contact with the track at all.

Path ’A’ is taken when the top of the trajectory is higher than the bump.Path ’B’ is taken when the top of the trajectory is lower than the bump and the downward acceleration

13

of the trajectory is lower than the acceleration would have been if the ball had not lost contact.Path ’C’ is taken when neither path ’A’ or ’B’ occurs.

For the occurrence of path ’A’, the top of the trajectory is calculated.For the trajectory the ball is considered to have no contact, in which case the only force acting on theball is gravity.

~g =

−9.81~τ(2)

−9.81~n(2)

0

(3.5)

Let ~X(2) indicate the second component in the vector.

The path of the ball, as a function of time, can then be used to calculate the maximum height of theball. The time at which the ball reaches the maximum height is:

tmax = −~v(2)

~g(2)(3.6)

~r+b (t = tmax) = ~v+tmax +

1

2~gt2max (3.7)

Path ’B’ occurs when the ball looses contact before it reaches the top of the bump. This is sketched inFigure 3.2 as path ’B’. This path ’B’ occurs when the path of the ball is less curved then the circularpath ’C’. Determining if this occurs can be done by comparing the second-order displacement derivativeof trajectory ’B’ and path ’C’. The second derivative of path ’C’ is:

d2yCdx2

= y′′C =d2

dx2

(√R2 − (x+ xc)2 − yc

)= −

(R2

R2 − (x+ xc)2

) 32

(3.8)

To do the same with path ’B’, the position y must first be written as a function of x.

~r+b (t) = ~v+t+

1

2~gt2 (3.9)

yB = ~v+(2)t+

1

2~g(2)t

2 (3.10)

x = ~v+(1)t+

1

2~g(1)t

2 ⇒ t =−~v+

(1) −√~v+2

(1) + 2~g(1)x

~g(1)(3.11)

however, if ~g(1) = 0, then

t =x

~v+(1)

(3.12)

This will eventually lead to:

d2yBdx2

= y′′B = −

(~g(1)~v

+(2) − ~g(2)~v

+(1)

~v+2(1) − 2~g(1)x

) 32

| ~g(1) 6= 0, (3.13)

y′′B =~g(2)

~v+2(1)

| ~g(1) = 0. (3.14)

The ball will loose contact when y′′B(x = 0) > y′′C(x = 0).

The results of above equations are plotted using the data from [4,5], similar to the plots made in Chapter2. The bounce heights and their second derivatives are plotted against the travel distance of the ballalong the BR: s in Figure 2.1. These plots show when paths ’A’, ’B’ and ’C’ occur. The first set of plotscan be seen in figures 3.3 and 3.4.

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(a) Travelled distance s, vs. height y (b) Travelled distance s, vs. double derivative y′′

Figure 3.3: Bounce prediction over the travelled distance

The irregularity at the start of the plots 3.3 (a) and (b) is caused by the start up of the BR fromstand-still. This part of the plots will not be considered. The part of interest is after s > 1.

In Figure 3.3a one can observe what the maximum bounce height would be if the ball were to encounter abump at that location. The bump height h in these graphs is h = 0.0005[m]. While the BR is operating,the speed of the ball relative to the track and angle of the track vary, resulting in different bounceheights. The height of the bump h would be equal to the maximum height the ball can achieve, if it isin constant contact with the track h = yC,max; situation ’C’ as stated earlier in Figure 3.2. Figure 3.3bshows the double position derivative y′′ of both situation ’B’ and ’C’. When y′′B(x = 0) > y′′C(x = 0) theball loses contact with the track. In these graphs is clearly observed that situation ’A’ does not occursince yA < yC ∀ s > 1, however, situation ’B’ does occur since at some points in time y′′B > y′′C .

Visualising when situation ’A’ or ’B’ occurs compared to the bump height h, is done by plotting thedifferences in height and the double derivative. The difference in height yA − yC is visualised in Figure3.4a, the difference in the double derivative y′′B − y′′C in Figure 3.4b.

(a) increasing h vs. maximal bounce height y (b) increasing h vs. maximal double derivative y′′

Figure 3.4: Influence bump height

In Figure 3.4a the difference in height is constantly decreasing, it is safe to assume that situation ’A’ does

15

not occur. On a side note: situation ’A’ does not occur with the current speed of the ball. The speed ofthe ball may differ if the controlling algorithm is changed. Figure 3.4b shows us that if the bump heightis larger than h > 1.6 · 10−4[m] situation ’B’ occurs. So in order to prevent a loss of contact the bumpson the BR should be kept lower than 1.6 · 10−4[m], according to this discrete-time estimation.

The compute the magnitude of slip, the angular velocity of the slipping ball and the velocity of theBR are compared. This can simply be derived from ∆ψ = ψtair −∆ψ. Herein is ∆ψ the difference inrotation it would have had if it were constantly in contact. And tair is the time the ball does not touchthe BR.

This tair is hard to compute, but can roughly be estimated using the worst-case-scenario. The ball canmaximally cross path ’C’ at its top. The time it takes to reach the top is computed using (3.9),(3.10),and (3.11). This leads to the function yB(x). Then taking the derivative and equalising this to zeroprovides the x-coordinate of the top.

yB(x) =~g(2)

(~v(1) +

√~v2

(1) + 2~g(1)x)2

2~g2(1)

−~v(2)

(~v(1) +

√~v2

(1) + 2~g(1)x)

~g(1)(3.15)

x =~v(2)

(~g(1)~v(2) − 2~g(2)~v(1)

)2~g2

(2)

(3.16)

By substituting this x into equation (3.11) tair is obtained. note that t in that equation (3.11) is nowtair. The ∆ψ derived using this method is equal to ∆ψ ≈ 0.004[rad] when the criteria y′′B > y′′C holds.This seems to be very small, but the track is filled with these bumps, since the bump height is only0.0005[m], which is the same as a rough surface. The region of the BR where slip may occur is roughly0.03[m]. This value is deducted by reading the Figure 3.3b (the region where y′′B > y′′C). Assuming therecan be up to 30 bumps of this size in this part of the contour, then the total slip per rotation of the BRwill be equal to 0.12[rad].

3.3 Conclusions and summary impact dynamics

The goal of this chapter was to inspect the dynamical behaviour of the BR as a result of bumps in thetrack or ball. These bumps can result in slip when the ball bounces. The slow velocities of the ball usuallymake it hard to determine slip. In this chapter a possible source of slip (loss of contact with the track)was investigated. Naturally, when the ball bounces higher than the bump there will be a loss of contact.Yet, discrete time simulations show that this does not occur, for the speed of the ball is insufficientregardless of the height of the bump. Another possibility is for the bounce trajectory’s acceleration tobe smaller than the acceleration if it were in continuous contact with the bump. Consequently the ballinitially looses contact and then regains contact before the top of the bump is reached. The discretetime model predicts that this occurs if the bump height is sufficiently large. The loss of contact can beprevented if the bumps are smaller than h < 1.6 · 10−4[m] according to a discrete time approximation.Additionally, using a worst-case-scenario estimation and a bump height of h = 5 · 10−4[m], the ball slips≈ 0.004[rad] per bump. Do note that slip only occurs on curtain parts of the track, where the velocity ofthe ball is high enough. Though the slip as a result of a single bump is insignificant, the summation ofnumerous bumps along the BR could result in a more significant amount of slip, ≈ 0.12[rad] per rotationof the BR.

16

Chapter 4

Forces Simulation Model

The dynamical effects of the most crucial perturbation, the bumps, can be further researched by exam-ining the dynamical effects not only on the ball, but on the entire BR system. A simulation model isproposed for further investigation of the entire BR system with perturbations. This simulation focuseson the contact area between the ball and BR. Consequently, this leads to a simple model of a ball rollingon a ramp. The simulation does not limit the movement of the ball, meaning it is possible for it to leavethe track. The ball’s position is derived from the forces acting on it. This can easily be done, sincethe simulation considers a ball rolling on a ramp. Naturally the forces may become more complicatedif more effects are considered in this model. For now an ideal system is used, without bumps or otherimpurities.

(a) Schematic representation of BR (b) Schematic representation of BR: ramp

Figure 4.1: Schematic representation

The model chosen for this simulation is presented in Figure 4.1a.All vectors are given in Cartesian coordinates. The position of the ball is also given in Cartesiancoordinates. These are the same type of coordinates that the camera will use in a experimental set up.The input torque of the controller is u.The distance between the center of the ball and the track is equal to w in normal direction ~n.Perpendicular to ~n is the tangential vector ~τ .The angles ϕb and ϕt represent the angle between their respective position vectors and the x-axis.

17

4.1 Contact forces

There are 3 main forces and 1 torque considered in this system. These are:- the gravitational forces;- the contact forces in normal direction;- the constraint forces in tangential direction;- the input torque delivered on the BR.

(a) Forces on the ball (b) Forces on the BR

Figure 4.2: Different frames

Only the gravitational force on the ball is of interest, because the BR is fixed. The gravitational forceon the ball is equal to:

~Fg =

0−9.81mb

0

. (4.1)

The contact force ~FHC is more difficult to determine. Therefore, an approximation of this force is used.The constraint force ~Fcon prevents the ball from slipping and is restricted by ~FHC .The torque u is determined by the controlling algorithms, these algorithms are briefly discussed at theend of this chapter for this is not part of this research.

4.1.1 Equations of motion

Using the 3 main forces and input torque u, the motion of both the ball and BR can easily be derivedwith Newton’s Law. The main forces produce a torque on both the ball and the BR.

~Fcm = ~Fcon + ~FHC (4.2)

Tb = ~Fcm × ~nR (4.3)

TBR = ~Fcm × ~rt + u (4.4)

The equations of motion then become:

~rb =~Fcm + ~Fgmb

(4.5)

18

ψ =TbJb

(4.6)

θ =TBRJf

(4.7)

~rb =

∫ ~Fcm + ~Fgmb

dt+ ~rb,0 (4.8)

ψ =

∫TbJbdt+ ψ0 (4.9)

θ =

∫TBRJf

dt+ θ0 (4.10)

~rb =

∫∫ ~Fcm + ~Fgmb

dt2 + ~rb,0t+ ~rb,0 (4.11)

ψ =

∫∫TbJbdt2 + ψ0t+ ψ0 (4.12)

θ =

∫∫TBRJf

dt2 + θ0t+ θ0 (4.13)

The forces and torque need to be computed. This model can be extended for bumps and other pertur-bations, if the forces of those perturbations are computed outside this model. For now only standardforces (without perturbations of any kind) are considered, these are the forces seen in Figure 4.2.

4.1.2 Different contact models

As mentioned, ~FHC needs to be approximated. There are multiple methods to approximate this contactforce. It can be considered that the ball is in collision with the track. Since the BR is constantly moving,this collision is continuous. Multiple methods for making impact force estimations exist; Gilardi andSharf [6] give a systematic overview of these methods. According to Gilardi and Sharf the dynamicmodelling of impact in mechanical systems can be split in two parts: discrete and continuous time.Discrete time only describes the situation before and after the impact, using coefficients to approximatethe behaviour during the actual impact. In the case of the BR, the behaviour during the impact is mostimportant and therefore a model that uses continuous time is needed.

The impact models that use continuous time are mainly based on spring-damper systems. The Hertzimpact model seems to be the most commonly used model and uses a non-linear spring (no damper)approximation.

FHz = kwn | w < 0, (4.14)

where k and n represent an elasticity factor based on material and shape, w represents the compression.A non-linear spring is used to represent the increase in contact area between the two bodies. In otherwords, this model represents a contact area in 3D space. However, the Hertz model does not use damp-ing. This damping is needed because our model must be able to estimate the influence of multiple kindsof perturbations. If the resulting perturbation is not damped, the perturbations will keep stacking witha resulting bounce becoming larger and larger as time progresses.

4.1.3 Hunt and Crossley model

Gilardi and Sharf also introduce an expansion of the Hertz model. This model, by Hunt and Crossley[7] (HC model), expands the Hertz model with a non-linear damper. The difference between the originalHertz model and the HC model is clearly visualised in [8]:

FHC = kwn + dwpwq | w < 0. (4.15)

19

Gilardi and Sharf state that the constants p and q are typically set to n and 1 respectively and refer to[7,9,10]. This model provides a smooth transaction to before, during and after the impact, because whenu = 0 the FHC = 0 regardless of speed. The only problem is that the damping factor d is experimentallydetermined [11,12]. For the BR case an experimental damping factor is not available at this moment.The damping factor can be chosen fairly small to compensate for the stacking, without influencing withthe dynamics too much.

The HC model has been used in multiple papers [13,14]. The paper by Daniel Jacobs and KennethWaldron [14] provides experimental data to verify the HC model. Other projects that use the samemodel and are similar to the problem at hand are, a ball rolling over a bump. The paper by AlirezaMoazenahmadi, Dick Petersen and Carl Howard [13] use the HC model to determine the influence ofa damaged ball-bearing. Their model consist of a ball rolling through a slit and using the HC modelto model the dynamics. This is fairly similar to this experiment except instead of a slit, a bump isconsidered.

Since the radius of the sphere is much smaller than the radius of the track, the constants k and n arederived using the following equations [14]:

k =4

3

( 1

Eb+

1

EBR

)−1√R ; n =

3

2. (4.16)

The w and w are calculated as:w = ‖~rb · ~n‖−‖~rt · ~n‖−R, (4.17)

w = (~rb − ~rt) · ~n. (4.18)

The position vector of the track (~rt) and its time-derivative are derived from the polar coordinates givenpreviously in [4,5].

~rt =

cos(ϕt − θ)sin(ϕt − θ)

0

rt(ϕt, θ) (4.19)

with the vector being the direction and rt(ϕt, θ) the magnitude:

rt = 0.1095 + 0.04 cos(2ϕt − 2θ). (4.20)

Using above equations the derivative is equal to

~rt =

cos(ϕt − θ)sin(ϕt − θ)

0

rt(ϕt, θ, ϕt, θ) +

− sin(ϕt − θ)(2ϕt − 2θ)

cos(ϕt − θ)(2ϕt − 2θ)0

rt(ϕt, θ) (4.21)

with rt(ϕt, θ, ϕt, θ) beingrt(ϕt, θ, ϕt, θ) = −2b sin(ϕt − θ)(2ϕt − 2θ). (4.22)

4.1.4 Constrained forces

The constrained force ~Fcon is the force that keeps the ball from slipping, bound by the friction forces.To prevent slipping, the ball’s angular velocity is described by the velocity difference between the balland track.

ψ+ =(~rt − ~rb) · ~τ

R(4.23)

The force needed can be computed from the angular acceleration. Using the fact that the simulationuses time steps, the angular acceleration ψDT is

ψDT = ψ+ − ψ =(~rt − ~rb) · ~τ

R− ψ. (4.24)

20

This acceleration is bound by the maximum friction force. But this force is given in continuous time.The discrete time equivalent is derived using the discrete-time step T .

ψmax = −ψmin =FfricJb

=Fnµ

Jb⇒ ψDT,max = −ψDT,min =

FHCµT

Jb. (4.25)

Let µ be the appropriate friction coefficient, which can be either a static (µst) or dynamic (µdy) coeffi-cient.

µ = µst | ψDT,min ≤ ψDT ≤ ψDT,max, (4.26)

µ = µdy | ψDT,min > ψDT ∨ ψDT > ψDT,max. (4.27)

Finally, the constrained force can be derived using Newton’s law.

~Fcon =

(Jb sat

ψDT,max

ψDT,minψDT

TR

)~τ (4.28)

4.2 Ramp representing the contact area

As mentioned, the ball is considered to be on a ramp rather than a curve. This is done to simplify themodel and because the focus is on the contact area. This simplification is possible if the contact pointbetween ball and BR and tangent of that point are known. This point and tangent must be computedconstantly. Finding the point on the BR is fairy hard, because the contour of the BR is given in polarcoordinates and is a function of θ. The ball can be located in any arbitrary point in 2D space, meaningthere may not always be contact. By computing the point on the BR closest to the ball’s center thecontact point can be found. For this, two mathematical tricks are used: first the coordinate systemis changed and then a Taylor expansion is used to approximate the point. With the closest point thetangent can be computed.

The direction of the ramp is equal to the tangent of the closest point of the BR to the center of the ball.In order to find this closest point, the coordinate system is changed, where the ~rb is the vertical axis, seeFigure 4.3.

Figure 4.3: Schematic representation of distance

21

The distance D between an arbitrary point on the BR and the ball equals:

D =√

(yt − yb)2 + (xt − xb)2. (4.29)

The actual distance is not of interest for this computation but merely a representation, we simplify thisequation to a non-linear distance Dnl. Also because the ball is on the y-axis, the xb equals zero. Theequation becomes:

D2 = Dnl = (yt − rb)2 + (xt)2. (4.30)

The contour of the BR is given in polar coordinates, with radius rt(ϕt, θ) from equation 4.20. Rewritingthis to Cartesian coordinates gives

~r∗t =

[xtyt

]= rt(ϕt, θ)

[sin(ϕ∗t )cos(ϕ∗t )

]; ~r∗b =

[0rb

]. (4.31)

Putting this into equation 4.30 results in the non-linear distance needed to compute the closest pointbetween ball and track.

Dnl =(

(0.1095 + 0.040 cos(2ϕt−2θ)) cos(ϕ∗t )− rb)2

+(

(0.1095 + 0.040 cos(2ϕt−2θ)) sin(ϕ∗t ))2

. (4.32)

The closest point on the track (as a function of ϕ∗t ) can be found by taking the derivative of Dnl.When dDnl

dϕ∗t

= 0 the minima and maxima can be found. We are only interested in the minimum. The

differentiation of Dnl and then deriving ϕ∗t = f(θ, ϕt, ϕb) is not possible using a direct approach andwill result in loops. Therefore, a 2nd order Taylor expansion (around ϕ∗t = 0) is used. Thereby reducingequation 4.32 to a polynomial.

DT = αϕ∗t2 + βϕ∗t + γ +O3, (4.33)

where O3 covers the higher order terms of the Taylor expansion. For simplicity, these higher terms willbe neglected from here on.

0 =dDT

dϕ∗t∼= 2αϕ∗t + β (4.34)

Taking the 2nd order Taylor expansion will result in only the minimum.

ϕ∗t∼=

β

2α(4.35)

The Taylor expansion described above is made with the use of the computer program ’Matlab’, in whichcase equation 4.35 will lead to:

ϕ∗t∼= −

b(2a sin(2θ)− 2rb sin(2θ) + 2b cos(2θ) sin(2θ))

arb − 4b2(2 cos(2θ)2 − 1)− 4ab cos(2θ) + 5brb cos(2θ)(4.36)

with a = 0.1095 and b = 0.04. Knowing ϕt = ϕb + ϕ∗t ,

ϕt ∼= ϕb −b(2a sin(2θ)− 2rb sin(2θ) + 2b cos(2θ) sin(2θ))

arb − 4b2(2 cos(2θ)2 − 1)− 4ab cos(2θ) + 5brb cos(2θ)(4.37)

Now that the closest point to the ball is known, the slope of the BR can be computed. This can bedone trivially by computing the normal ~n and tangent ~τ . We know the point on the track closest tothe ball, which can only mean that the normal is in the same direction as the vector between ball andtrack.

~n =rb − rt‖rb − rt‖

⇒ ~τ =

0 1 0−1 0 00 0 1

~n | rb > rt ∀t. (4.38)

Another method to compute the tangent ~τ is by deriving it directly from the BR-contour equation. First,the derivative in polar-coordinates must be translated to the derivative in a Cartesian coordinate system.In this case, the normal and tangential direction are used as axis.

The derivative ∆rt(ϕt)∆ϕt

is trivially determined:

∆rt(ϕt)

∆ϕt= −0.08 sin(2ϕt + 2θ) ⇒ ∆rt(ϕt) = −∆ϕt0.08 sin(2ϕt + 2θ) (4.39)

22

Let ϕt increase slightly, then the tangential increase ∆τ is equal to

∆τ = tan(∆ϕt)rt(ϕt). (4.40)

Linearising this around ∆ϕt = 0 and rotating the result from normal and tangential direction to x andy provides the tangent.

~τ = −[∆y∆x

]= −

[cos(ϕt) sin(ϕt)− sin(ϕt) cos(ϕt)

] [∆rt∆τ

]= −

[∆rt cos(ϕt) + ∆τ sin(ϕt)∆τ cos(ϕt)−∆rt sin(ϕt)

](4.41)

~τ = −[rt(ϕt) sin(ϕt)− 0.08 sin(2ϕt + 2θ) cos(ϕt)rt(ϕt) cos(ϕt) + 0.08 sin(2ϕt + 2θ) sin(ϕt)

](4.42)

~τ = −[(0.1095 + 0.04 cos(2ϕt + 2θ)) sin(ϕt)− 0.08 sin(2ϕt + 2θ) cos(ϕt)(0.1095 + 0.04 cos(2ϕt + 2θ)) cos(ϕt) + 0.08 sin(2ϕt + 2θ) sin(ϕt)

](4.43)

This equation can be used to verify the computed closest point on the BR. This implies that this tangentand the previous tangent should be equal within a small margin. Comparing the tangent shows thatthere is a slight error because of the Taylor expansion. The main cause for this error is that the ballis a distance ’R’ away from the track. To solve this problem the ideal path of the ball should be usedrather than the contour of the BR. Then taking the same steps: computing distance and using the Taylorexpansion will result in an accurate estimation. This has not been done yet due to a lack of time andit is recommended that this is further investigated in a subsequent project. However, this method hasbeen tested and found accurate when the non-linear distance (Dnl) is small.

4.3 Controller

The controlling algorithm is not extensively discussed in this report, because it is already described in[1,15] and is not part of this project. In this chapter only the controller itself is given. For more detailedinformation the references to papers are given when additional information is available.

The controller of the BR is the same as in [4,5]. The concept of the controller used was originally providedby [1,15]. They used the periodic solution the BR should turn to balance the ball developed by [16].The periodic solution prescribes the ideal angle the BR should be positioned in over time, with the onlyinput ϕ. The controller uses transverse coordinates. These transverse coordinates represent both theerror between the BR’s ideal angle and actual angle, and the error between the ball’s ideal and actualposition. The transverse coordinates are y, its derivative y and z. The y and y coordinates representthe error between the BR’s actual and ideal angle, θ and Θ(ϕ) respectively. The coordinate z representsthe error of the ball versus the periodic solution in the ϕ-ϕ plane. Note that all the equations, and thusalso the DOF, in this entire section are taken from [4,5].

~x⊥ =

yyz

=

θ −Θ(ϕ)

θ −Θ(ϕ)′ϕϕ− ϕ∗(ϕ)

(4.44)

Herein are Θ(ϕ) and ϕ∗ the ideal paths the ball should follow. The controller gains are a function ofϕb.

c = ky(ϕb)y + ky(ϕb)y + kz(ϕb)z (4.45)

Eventually leading to a function u dependent only on the DOF ϕb and y.

u =

c+[L−1(N +M−1CL

[yϕb

]+M−1G)

]1

[L−1M−1]1,1, (4.46)

with

L =

[1 Θ′

0 1

]| Θ′ =

dΘ

dϕb(4.47)

23

N =

[Θ′′ϕ2

b

0

]| Θ′′ =

d2Θ

dϕ2b

. (4.48)

The M , C and G matrices are taken from the BR dynamics when written as

M(~x⊥)~x⊥ + C(~x⊥, ~x⊥)~x⊥ +G(~x⊥) =

[u0

]. (4.49)

Herein are L, N , M , C and G to compensate for the internal forces and torque. The specifics arenot given in this report, for it is not part of the research, however one can find the derivation of thecontroller gains: ky(ϕ), ky and kz(ϕ) and paths Θ(ϕ) in [1]; the derivation of A(~x⊥, ~x⊥) and B(~x⊥, ~x⊥)can be found in [4] and [5]. The explicit periodic solution Θ(ϕ) can be found in [16] as well as in [4] and[5].

4.4 Conclusions and summary simulation model

In order to examine the perturbations in continuous time, a basic simulation model is proposed to whichperturbations can easily be added.

The model simplifies the BR system to a ball rolling over a ramp. This ramp is computed by first findingthe closest point on the BR to the ball’s center. The BR shape is given in polar coordinates and needsto be rewritten in Cartesian coordinates. A third order Taylor expansion is used to find the closest pointto the ball. After the closest point is found, the slope of the BR at that point is computed.

With the point and slope known, a simplified model of the BR-system can be used, because you cancalculate the contact forces. With these forces the position and movement of the ball and BR can becomputed.

The simplified model is adjustable to suit different situations, such as a bump. As long as the forces onthe ball are calculated, practically any perturbation can be implemented.

The equations of motion (4.2-4.13), combined with the computed forces and torques, are used in thesimulation. It is possible to expand it for different perturbations and BR shapes. The model computesthe interaction forces between ball and BR as the ~FHC and ~Fcon forces, which are relatively simplecalculations, the other equations are only needed to make the simplified ramp-like model. All thatneeds to be expanded are the interaction forces (~Fcm) in equation (4.2) by adding forces as a result ofperturbations.

24

Chapter 5

Conclusions and recommendations

5.1 Conclusions

The small scale dynamics of the Butterfly robot are, to our knowledge, yet unexplored. Most researchis done on the control of the system. This report focuses on the physics of dynamics on a fairly smallscale.

The research succeeded in finding the most crucial perturbation, a source of slip and proposes a simulationmodel to further investigate the influence of small perturbations. From the research done multipleconclusions are drawn. These conclusions are listed and numbered below.

1.Different perturbations were considered in order to find the most influential one. Though it is debatable ifthese perturbations are able to cause instability, small bumps on the track seem to have some interestingeffects. Leading to both high energy fluctuations and a possible source of slip. The fluctuations as aresult of bumps are roughly 20 times greater than the other considered perturbations, and roughly 10%of the average nominal change of energy in the system.

Other considered perturbations were: energy loss due to height differences, elastic deformation, differencein parameters, sway and friction. A noteworthy feature is that most of these energies have a steep increaseat the same time, where the velocity of the ball is greatest. In other words, the summation of all factorsrather than one is worth considering. However, the nominal energy fluctuations of the ideal BR systemare still significantly higher than all the examined perturbations combined.

2.Bumps are a possible source of slip. Slip occurs due to a loss of contact between the ball and thetrack. The most logical possibility is when the ball bounces higher than the bump, resulting in a lossof contact. Yet, discrete time simulations show that this possibility does not occur, for the speed of theball is insufficient regardless of the height of the bump.

Another possibility is for the bounce trajectory’s acceleration to be smaller than the acceleration if itwere in continuous contact with the bump. Consequently, the ball initially looses contact and thenregains contact before the top of the bump is reached. The discrete time model predicts that this occurswhen the bump height is more than h < 1.6 · 10−4[m].

Using a worst-case-scenario estimation and a bump height of h = 5·10−4[m], the ball will slip ≈ 0.004[rad]per bump. Do note that slip only occurs on curtain parts of the track. Although the slip caused by asingle bump is insignificant, the summation of numerous bumps along the BR could result in a significantamount of slip, ≈ 0.12[rad] per rotation of the BR.

3.In order to examine the effects of perturbations in continuous time, a simulation model is proposed.A model that computes the interaction forces between ball and BR. The model works as follows: the

25

model finds the closest point on the BR to the ball’s center of mass and the slope of the ramp. Themodel is adjustable to suit different perturbations, such as a bump. As long as the forces on the ball arecalculated, practically any perturbation is possible to implement.

5.2 Recommendations

This research is a starting point for exploring the effects of small-scale perturbations. The recommenda-tions for further research are listed and numbered below.

1.For further research it is recommended to expand the continuous time simulation model for bumps,preferably multiple bumps in a row. Additionally, the amount of slip as a result of the bumps isestimated at this point in time. Instead of proceeding with the currently used discrete time model, itsrecommended to work towards a continuous time model, which will provide more accurate data. Theproposed simulation model can be used as a basis for this.

2.Another recommendation is to gather experimental data, the energy input can fairly easily be derivedfrom the energy consumption of the motor. The energy input can then be compared with the energyfluctuations as described in Chapter 2 and with the ideal energy input. The discrepancy can be used tofurther investigate the impact of the small scale perturbations on the BR.

3.Lastly, up to this point, the BR is mainly considered a 2D system. Expanding the (existing) model(s)to 3D might lead to interesting and unexpected results. For instance, the ball is supported at 2 sides, ifboth have bumps then the effect can either be nullified or amplified.

26

Chapter 6

Appendix

Parameters and estimations

The parameters used are measured parameters, as provided by [4,5]. For experimental purposes, someof these parameters have been varied. The variations are chosen relatively large to see the effects inworst-case-scenarios. All the measured and varied parameters are listed below in Table 6.1.

Measured parametersParameter Value/Function Descriptionmb 3.00 · 10−3 [kg] Mass of the ballJb 5.48 · 10−7 [Nm2] Inertia of the ballJf 1.58 · 10−3 [Nm2] Inertia of the butterfly robotdf 2.55 · 10−2 [m] distance between the two plates the ball rolls uponRb 1.66 · 10−2 [m] The actual radius of the ball

R 1.08 · 10−2 [m] Effective radius of the ball: R =

√Rb −

d2f4

Estimations for simulationsmf 0.1 [kg] Estimated mass of the butterfly robotδh 5 · 10−4 [m] Height of the bumpsm∗b 3.03 · 10−3 [kg] Increased mass of the ball: m∗b = 1.01mb

J∗b 5.53 · 10−7 [Nm2] Increased inertia of the ball: J∗b = 1.01JbJ∗f 1.60 · 10−3 [Nm2] Increased inertia of the butterfly robot: J∗f = 1.01JfR∗ 1.07 · 10−2 [m] Decreased effective radius: R∗ = 0.99Rµrol 0.001 [−] Rolling friction of the ballεjoint

π80 [rad] Horizontal joint misalignment

kbt 106 [N/m2] The combined stiffness of the ball and BRTjoint(Fload) 10−5Fload [Nm] The friction torque of the joint

Fload ‖~Fn + ~Ft +mBR~g‖ [n] Force applied onto the joint

Table 6.1: Parameters

27

Chapter 7

References

[1]

Anton S. Shiriaev, L. B. Freidovich, I. R. Manchester ’Can we make a robot ballerina perform apirouette? Orbital stabilization of periodic motions of underactuated mechanical systems’ in AnnualReviews in Control, vol: 32, pp 200-211, Dec. 2008

[2]

Kevin M. Lynch, Naoji Shiroma, Hirohiko Arai Kazuo Tanie ’The Roles of shape and motion indynamic Manipulation: The Butterfly Example’ in in IEEE International Conference on Roboticsand Automation, pp 1958-1963, May 1998

[3]Massimo Cefalo, Leonardo Lanari, Giuseppe Oriolo ’Energy-based control of the butterfly robot’Universita di Roma ”La Sapienza”, Rome, Italy, 2006

[4]H. M. Francke ’Motion planning and analysis of the underactuated ”Butterfly” robot’ Universityof Technology, Eindhoven, Netherlands, DC 2015.092, Nov. 2015

[5]

S. J. A. M. van den Eijnden ’Orbital stabilization and robustness analysis of the Butterfly robot withpre-defined periodic trajectories’ University of Technology, Eindhoven, Netherlands, DC 2015.093,Nov. 2015

[6]G. Gilardi, I. Sharf ’Literature survey of contact dynamics modelling’ in Mechanism and MachineTheory, vol: 37, pp 1213-1239, 2002

[7]K. H. Hunt, F. R. E. Crossley ’Coefficient of restitution interpreted as damping in vibroimpact’ inJournal of Aplied Mechanics, vol: 42, pp 440-445, 1975

[8]

Margarida Machado, Pedro Moreira, Paulo Flores, Hamid M. Lankarani ’Compliant contact forcemodels in multibody dynamics: Evolution of the Hertz contact theory’ in Mechanism and MachineTheory, vol: 53, pp 99-121, 2012

[9]H. M. Lankarani, P. E. Nikravesh ’A contact force model with hysteresis damping for impact analysisof multi-body systems’ in Journal of Mechanical Design, vol: 112, pp 369-376, 1990

[10]

D. W. Marhefka, D. E. Orin ’A compliant contact model with nonlinear damping for simulationof robotic systems’ in IEEE Transactions on Systems, man, and Cybernetics, vol: 29, pp 566-572,1999

[11]Amir Haddadi, Keyvan Hashtrudi-Zaad ’A New Method for Online Parameter Estimation of Hunt-Crossley Environment Dynamic Models’ Acropolis Convention Center, Nice, France, Sept. 2008

[12]W. R. Chang, I. Etsion, D. B. Bogy ’An Elastic-plastic Model for Contact of Rough Surfaces’ inASME Journal of Tribology, Vol. 109, pp. 257-263,1987.

28

[13]

Alireza Moazenahmadi, Dick Petersen and Carl Howard ’A nonlinear dynamic model of the vi-bration response of defective rolling element bearings’ The University of Adelaide, Australia, Nov.2013

[14]

Daniel A. Jacobs, Kenneth J. Waldron ’Modeling Inelastic Collisions With the Hunt-Crossley ModelUsing the Energetic Coefficient of Restitution’ Stanford University, Stanford, USA, in Journal ofComputational and Nonlinear Dynamics, vol: 10, Mar 2015

[15]

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