Modeling and characterization of a 2-DOF variable stiffness joint

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Modeling and characterization of a

2-DOF variable stiffness joint

J.D. (Jelle) Zult

BSc Report

C e

Dr.R. Carloni E. Barrett, MSc Dr. H.K. Hemmes

October 2016 043RAM2016 Robotics and Mechatronics

EE-Math-CS University of Twente

P.O. Box 217

7500 AE Enschede

The Netherlands

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2

Abstract

A newly developed robotic wrist is analyzed, in order to obtain a model that can accurately predict its

dynamic behavior. The wrist can move in two degrees of freedom, in which the coupled output stiffness

can be varied. Such a device is called a variable stiffness joint (VSJ) and it applies the technology of

the variable stiffness actuator (VSA). An overview of the current design is made to get an understanding

of its operation. Then the identified subsystems are individually explored, leading to the construction

of a sub-model for each of them. Friction plays an important role in the behavior of the system, so

a fair amount of attention is paid to modeling these effects. The two most important subsystems are

then tested in order to check the corresponding models for inaccuracies and acquire their missing

parameter values. To be able to perform the necessary measurements, two experimental setups have

been designed and built. The experimental results are evaluated on quality and validity, and interpreted

accordingly. Earlier data is compared to simulations of the complete model, in an attempt to draw

conclusions on the modeling and characterization process and the final result that this yields.

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Introduction

Interaction between robots and humans will become more and more common as technology progresses, and therefore it is relevant to design systems that allow for this interaction to be safe and intuitive. One obstacle that conventional robotics poses is that the typical stiff and sturdy robot-arm motion can be harmful to people that engage in physical interaction with them. Furthermore, the solution of simply adding compliance to these robotic arms is not satisfactory either, as this would come at the expense of the precision and strength needed to perform certain tasks. To overcome this dilemma, an alternative solution has been found in variable stiffness actuators (VSA’s). Implementing this kind of technology means that robots will be able to vary between compliant and precise motion, depending on the situa- tion. The SHERPA project [1] wishes to employ VSA’s for the robotic arm that is in development. This arm will be used in rescue operations in the mountains, where a team of ground robots and aerial robots search for people in danger. The robot arm will be mounted on a ground robot, and should be able to safely interact with the aerial robots.

This bachelor assignment is embedded in the topic of variable stiffness actuators. One particular device will be treated, which is a 2-DOF variable stiffness joint (VSJ) that has been designed and built by a master student as a part of his master thesis [2]. It is planned to be used as the wrist joint for the robotic arm and contains a 2-DOF VSM. The problem is that there is no accurate dynamic model yet that describes the relation between its three DC motors as an input and the 2-dimensional end-effector motion as an output. This kind of model is necessary to enable the design of controllers for a wide range of applications. It can also be used to make improvements to the design and test how this would effect its dynamic behavior.

The main focus of this bachelor assignment is characterizing the complete 2-DOF VSJ system.

Several subsystems will be modeled with a theoretical approach. Then unknown parameter values can be found by doing experiments on different subsystems. At the same time, data will be collected and analyzed in order to determine whether the modeled subsystems and the entire model are sufficiently close to reality, which could lead to changes in the models that are used.

1.1 Variable stiffness actuators

As mentioned above, variable stiffness actuators allow a robotic manipulator to change its compliance according to the situation. This means that when the actuators are set to be compliant, the manipulator is able to comply with any external impacts and absorb energy. This in contrast with the case that the actuators are set to be stiff, so that external impact is resisted.

There are two known ways of varying joint stiffness in a robotic arm. A controller could be used to emulate compliant motion, which is called ’active stiffness’. One great benefit is that a relatively simple actuator design could be used, which results in more compactness and a system that is easier to model [3]. On the other hand there is the possibility to use a variable stiffness mechanism (VSM), which uses actual mechanical stiffness. This provides a higher energy efficiency [4], but also a safer and more reliable system. This is because there is no additional response time in a mechanical system. An active stiffness system has to read sensor data, process it and give an output, which takes time and makes for a less reliable system.

When employing a mechanical system, one has to think of a mechanism where stiffness can be varied quickly and efficiently. Several VSM’s have shown to be reasonable options for achieving this goal. These include varying the pre-load of non-linear springs [5], changing the properties of an elastic material [6], adjusting the transmission ratio between output and spring [7, 8] and using artificial mus- cles that work together, in the way that the human body achieves variable stiffness [9]. The variable transmission in the vsaUT-II [7, 10] has shown to provide independent stiffness adjustment that requires a low amount of work. This is the concept that has been implemented in each VSJ of the robotic arm.

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6 CHAPTER 1. INTRODUCTION

1.2 Content of this report

Next to performing the necessary tasks to achieve the goal of this bachelor assignment, there should also be a critical look on the results that are produced. The research question that is to be answered in this report is therefore formulated as follows:

• How well does the acquired characterized model describe the behavior of the 2-DOF variable stiffness joint?

All of the content of the following chapters is, directly or indirectly, related to an attempt to answer this question. First an overview will be given in chapter 2 of the prototype that has previously been built.

This should support the construction of the model of the system and its subsystems in chapter 3. This

modeling process will be explained step by step, such that important choices and assumptions made

can be evaluated on validity and effectiveness. Chapter 4 gives a description of the experiments that

are performed to characterize certain parts of the system. It also includes a discussion on the quality of

the experimental results and an analysis of the final characterized model that has been acquired. The

final conclusion will be given in chapter 5, where an attempt is made to summarize what has been found

and translate this to an answer to the research question.

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Chapter 2

Current design

The 2-DOF VSJ designed by [2] makes use of the variable transmission principle. However, to realize two degrees of freedom, the choice was made not to simply stack two 1-DOF VSJ’s on top of each other, as this would make for a bulky design with high output inertia [11]. The output inertia is ideally very low for swift and agile motion. The system should additionally be compact enough to be able to employ it as a wrist joint on top of the rest of the robotic arm. These major requirements, amongst others, lead to important design choices. One of these is using a coupled VSM, where only one VSM governs the stiffness of both output DOF’s. For low output inertia, the amount of components that can be base-mounted has been maximized. These components include each of the three actuators and the pivot varying mechanism inside the VSM. Other choices are a vertical stacking orientation for the mechanics of each DOF, and using planetary gears in a particular way to connect actuator, VSM and output together, but the details will be explained later.

2.1 Topological design

Figure 2.1: Topological design, constructed based on figures and description by [2]

Figure 2.1 shows the conceptual topological design of the 2-DOF VSJ. There are three base-mounted actuators. A

1

and A

2

are the primary actuators that govern the equilibrium of the two angles θ

x

and θ

y

of the end-effector. The stiffness actuator, A

^s

, controls the pivot position in the VSM, q. This is the pivot position in the variable transmission of both internal elastic elements. For each DOF there is a planetary gear set, which acts as a three-port transmission. These ports are for each DOF connected to one of the primary actuators, one of the elastic elements in the VSM and one of the two inputs of the 2-DOF differential mechanism. This differential mechanism is a set of gears that transforms two input angles φ

1

and φ

2

into end-effector angles θ

x

and θ

y

.

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8 CHAPTER 2. CURRENT DESIGN

2.2 First prototype

Figure 2.2 illustrates the spatial design using a schematic and conceptual sketch. It shows how the variable pivot makes for a varying transmission ratio between the ring gear (outer gear) of the planetary gearbox and the springs within the elastic element. The other connections to the planetary gears are also displayed: the planet carrier (attached to the three middle gears) is connected indirectly to the end-effector and the primary actuator directly to the sun gear. The precise working of each component will later be described in the modeling phase. Several belt and gear transmissions have been used as well. Because of this high number of transmissions used, a large effect of friction is expected, as it adds up for each transmission. Because of this notion - which is also evident from the measurements in [2] - the modeling of friction will get a fair amount of attention.

Figure 2.2: Schematic illustration of the spatial design, constructed based on figures and description by

[2]

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Chapter 3

Dynamic Model

The working principles of the prototype have been described in chapter 2. Now that this has been established, the next step is to construct a model that describes the dynamic behavior of this system sufficiently accurate, while making sure it does not become too complicated. The model should facilitate the design of control algorithms, but the possibility of employing it for estimator-based control should also be considered [12], which requires fast computation. Moreover, a more complicated model generally requires more parameter values to be found through experimentation, leading to a more extensive characterization process.

Next to this, there is the desire for the model to be modular, meaning that each subsystem in the prototype gets its own sub-model, which can then be connected for acquiring the full model. This modular approach is very intuitive, but also quite convenient in this case. The robotic wrist is still under development, meaning that when subsystems are redesigned, the corresponding sub-model can easily be re-characterized without having to analyze the entire system. It also comes in handy for the redesign process itself, when one would like to see what effect the modification of a single subsystem has on the behavior of the whole system. Lastly, certain sub-models might in future be re-used for other parts of the robotic arm, for instance the pivot mechanism.

For these reasons the general aim will be obtaining a modular model that is competent, yet compu- tationally efficient and as simple as possible.

The model will be constructed through a method called ’port-based modeling’ [13]. This yields a so called ’bond graph’, which describes the energy flow between different ’ideal elements’ of the system in a compact diagram. From such a bond graph the differential equations of the described system could be derived. However, this will be done by software called ’20-sim’ instead. This program can be used to directly generate simulations from the constructed bond-graph. Altogether this strategy allows for a convenient and efficient modeling process. However, a good understanding of bond-graphs is needed to follow all modeling steps in this chapter.

3.1 System Overview

The need for a modular model can be fulfilled quite intuitively when using bond graphs. The output and input ports of different sub-models can simply be connected, although of course one has to pay attention to what type of connection is made. In this case, every port connection between two subsystems has been made between two 1-junctions, so that there is a common flow. This is logical when considering the physical connections to be rigid.

The complete overview of the subsystems and their interconnections is shown in figure 3.1. Three motors act as an input to the system, as well as the two torques that act on either direction of the end-effector (in the figure on both sides of the differential mechanism). Now each subsystem will have to be characterized sufficiently accurate, such that the behavior of the complete model resembles the behavior of the real system for a given input.

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10 CHAPTER 3. DYNAMIC MODEL

Figure 3.1: Overview of the bond graph structure of the entire robotic wrist

There are however some convenient shortcuts for the entire characterization process. First of all, the three motors are identical, so only one motor characterization will be enough. This also holds for the two planetary gear modules (PG1 and PG2) and for the two elastic elements as well. Then there are the two transmission modules, which contain the cumulative behavior of all transmissions between the PG module and the differential mechanism. These can probably be described with the same model, although differences are expected in the parameter values.

The VSM consists of two parts: the pivot mechanism and the elastic element. These are, as can be seen in figure 3.1, only connected through the pivot position q as a variable. It is hereby assumed that the elastic element has no influence on q, there is only the influence of q on the elastic element.

In other words, the connection of the pivot to the lever in the elastic element is considered to be non- backdrivable.

3.2 VSM - elastic element

The most basic way to model any elastic element is considering it to act as a linear spring. The current prototype has been designed to show this linear behavior for any given pivot position q, which it does for small displacements. Earlier measurements however have shown that this is not necessarily the case (figure 3.7). At some point of displacement, the curve becomes non-linear and even non-continuous.

This raises the question if the current design acts linearly enough and why the discontinuity occurs. A previous explanation has been that a stiction threshold could cause structural stiffness to be measured, in the region before the threshold value. [2] This, however, seems unlikely because the discontinuity would have to occur at the same torque threshold in every measurement, which is not the case. An alternative explanation is that one of the springs might lose its pretension at some point. And because the spring only pulls, a discontinuity is reached when it cannot pull anymore. This explanation seems logical, but it is not trivial what exact effect this would have.

For these reasons there is need for an analysis of the specific elastic element that is used. A minimal

amount of approximations or assumptions will be used, because those would be in the way of trying to

answer the question of how linear the design essentially is.

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3.2. VSM - ELASTIC ELEMENT 11

Figure 3.2: Approach to the analysis of the elastic element

Figure 3.2 displays the approach to this analysis. It should result in the computation of a reaction torque, depending on q and the angular displacement φ. This reaction torque will be in opposite direction to φ, while q will not be affected.

3.2.1 Geometric Relations

A geometric overview and a simplified schematic of the design are shown in figure 3.3. It can be seen that two variables will completely describe the position of the lever. These are the pivot position (q), and the angle of the ring to which one side of the lever is connected (φ). Note that q is zero at the center of the ring, and φ is zero when the lever is in equilibrium. The reaction torque will be derived from the extensions of the two springs and what torques these create.

Figure 3.3: Simplified schematic of the elastic element and a geometric overview with arbitrary scaling The angle of the lever around the pivot (α) is calculated as follows, where R is the radius of the ring.

α = arctan

R · sin(φ) q + R · cos(φ)

(3.1) The position of the point where the lever attaches to the ring can be expressed in a vector, ~ X

R

= R, taking the center of the ring as the origin.

X ~

R

= R ·

sin(φ)

− cos(φ)

(3.2) Likewise, the unit vectors of the length (~ u

L

) and width (~ u

W

) of the lever are constructed.

~ u

L

= − sin(α) cos(α)

~ u

W

= cos(α) sin(α)

(3.3)

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12 CHAPTER 3. DYNAMIC MODEL These vectors are used in combination with the length (L) and width (W ) of the lever, to end up with the position of point where the right hand spring attaches to the lever, ~ X

a

. The other end of the spring is attached to the fixed world. The position of this attachment (x

0

, y

₀

) is subtracted from ~ X

_a

to get the vector of the spring, ~ X

_s,a

.

X ~

a

= ~ X

R

+ L · ~ u

L

+ W · ~ u

W

X ~

s,a

= ~ X

a

− x

0

y

0

(3.4) The unit normal vector of the spring is determined using the spring vector and its length, x

s,a

.

~ n

_a

= 1 x

s,a

X ~

_s,a

(2)

− ~ X

_s,a

(1)

where x

_s,a

= | ~ X

_s,a

| (3.5)

Taking the dot product between the unit normal vector and the vector from the pivot to the right hand corner leads to the arm (r

a

) that the spring force makes around the pivot.

r

a

= ~ n

a^•

X ~

a

− 0 q

(3.6) The spring force is calculated from the spring length, its rest length (s

0

) and linear spring constant k.

The torque on phi resulting from the right-hand spring, T

a

, is computed by multiplying the spring force by r

a

. Assuming mirror symmetry and equal springs, the torque contribution by the left-hand spring (T

b

) is equal to the negative of T

a

if angle φ were in the opposite direction.

T

a

(φ, q) = r

a

· (x

s,a

− s

0

) · k T

b

= −T

a

(−φ, q) (3.7) The sum of these two torques must be divided by r

2

(see figure 3.3) and multiplied by ring radius R.

This results in the following expression:

T

φ

= (T

a

+ T

b

) · sin(α)

sin(φ) (3.8)

3.2.2 Model and Simulations

The mathematical steps in the previous section have been processed into an equation sub-model called geom computation (see figure 3.4). Its output is T

φ

, the reaction torque, which is negated before being used as an input to the modulated effort-source. This source applies this effort on ω(= ˙ φ), to which the module port is connected.

Furthermore, the inertia of the lever around the point of rotation is in reality not constant, but com- pared to the inertia of the rest of the system these variations can be neglected. The contribution of the lever inertia itself might also be neglected instead. Lastly, there will be some friction resulting from the springs and from the pivot-lever connection, which is expected to be quite small compared to the total system as well.

Figure 3.4: Bond-graph model of the elastic element in the VSM

One of the questions that this model has to answer is whether small angle approximations leading to

a linear spring are justified or not. The maximum angle that is reached depends on the stiffness setting

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3.2. VSM - ELASTIC ELEMENT 13

q and on the maximum expected load in real operation. There is however a physical limit built into the prototype at |φ| = 1.40 rad. Figure 3.5 shows a simulation of what the equilibrium angle is for different stiffness settings and a load of 1 N m. Values of q higher than 0 have not been displayed, because at this point the compliance will be high enough such that the physical limit is reached.

Figure 3.5: Simulation result of deflection φ for different stiffness settings (q) and a load of 1 N m

The plot in figure 3.6 has been constructed by investigating how linear the simulated torque-deflection curve is for each stiffness setting, given their respective maximum deflection. This plot shows the RMS between the simulated torque-deflection curve and the curve if it were perfectly linear, at different stiff- ness settings (q).

Figure 3.6: Simulation result: non-linearity of the elastic element with pulling springs, expressed as the RMS between the actual curve and an equivalent linear curve

When fitting the model to earlier measurements, one can see that about the same discontinuity

arises (figure 3.7). It also shows how different stiffness settings result in roughly the same torque-

deflection curves. This clearly shows how not properly pre-tensioning the springs will yield this type of

result when measuring the torque-deflection curve.

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14 CHAPTER 3. DYNAMIC MODEL

Figure 3.7: Comparison between the simulated result (left) and earlier measurements from [2] (right)

3.3 Planetary Gear Module

The planetary gear set can be considered the heart of the prototype, as it is the module where three different other modules come together. These are the primary actuator, the elastic element of the VSM and the output (which leads to one of the DOFs in the end-effector). Inaccuracies and mistakes in the characterization of this module could have a large effect on the overall behavior of the system.

Therefore, a lot of focus has been put on the gear set in order to obtain a sufficiently competent model.

For the sake of a modular model, a stand-alone sub-model will be constructed for the gear set. Its three ports are referred to as the sun, the ring and the carrier, which can be connected to any three other sub-models. In the complete model, the sun is connected to a primary actuator, the ring to the VSM and the carrier to the output.

The gear set is regarded as a 3-port transformer with several imperfections as a result of friction.

One could also consider some play between the gears as an imperfection, but the current prototype is well enough engineered so that this effect is not noticeable.

At first, a simple version of the system is modeled, without any of these imperfections. It is based on the equation that relates the sun velocity (ω

S

), the ring velocity (ω

R

) and the carrier velocity (ω

C

) to each other.

(N

R

+ N

S

)ω

C

= N

R

ω

R

+ N

S

ω

S

(3.9)

Where N

R

and N

S

are the number of gear teeth in the ring and sun gear respectively. Note that the number of teeth of a planet carrier gear is not a separate required parameter as it is geometrically determined by N

R

and N

S

. Eq. 3.9 can be rewritten to express ω

C

in ω

R

and ω

S

:

ω

C

= 1 1 +

^N_N^S

R

ω

R

+ 1 1 +

^N_N^R

S

ω

S

(3.10)

This relation can be translated into a bond-graph model, as displayed in figure 3.8. A zero-junction is used to sum two flows, which come from the ring velocity and the sun velocity. Both flows go through a transformer first, of which the parameters are:

r

R,C

= 1 1 +

^N_N^S

R

r

S,C

= 1 1 +

^N_N^R

S

(3.11)

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3.3. PLANETARY GEAR MODULE 15

Figure 3.8: Basic bond-graph structure of the planetary gear set

This structure is the backbone of the model, so now one can consider how friction affects this system.

The first step is to analyze where the frictional effects take place. In other words, where there is motion.

Now for a 2-port transformer this would be very simple, because if there is a certain velocity at 1 port, there is proportional motion at the other port (assuming a rigid connection). This means that there is only one resistive element necessary - this could be at either side of the transformer - to characterize the friction in the system. In this case, however, there is a 3-port transformer. The complication is that a velocity at one port, does not always result in the same velocity at the other two ports. At one of the ports there could be arbitrary motion, on which the motion of the third port will depend. On top of that, one has to realize that the relative velocities of the gears with respect to each other are also not necessarily proportional to any of the absolute velocities at the three ports.

To get more insight into this problem, the spatial configuration of the planetary gear is considered in figure 3.9. On the left side all the angular velocities are indicated. These are the sun velocity (ω

S

), ring velocity (ω

R

), planet velocity (ω

P L

) and the carrier velocity (ω

C

). Of course these velocities are all related, as just two of them define the whole system.

Figure 3.9: Schematic drawing of the gear set configuration and geometry

On the right-hand sketch, two velocities are shown that are tangential to the gears. These represent gear motion on which a certain friction works, where v

1

is between the sun and planet gear and v

2

is between the ring and planet gear. These tangential velocities can be expressed in angular velocities, using the sun gear radius R

S

and ring gear radius R

R

:

v

1

= (ω

S

− ω

C

) · R

S

v

2

= (ω

R

− ω

C

) · R

R

(3.12) Note that v

1

and v

2

are related through the motion of the planet gear:

v

1

= −v

2

= ω

P L

· R

P L

(3.13)

The force that results from motion between two gears will be some function of the tangential velocity:

F = f (v). Such a force results in a torque that works in opposite direction on the angular velocity that

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16 CHAPTER 3. DYNAMIC MODEL causes it. The reaction torques on the sun (T

S

), carrier (T

C

) and ring (T

R

) are shown below:

T

_S

= − f (v

1

)

R

_S

T

_R

= − f (v

2

)

R

_R

T

_C

= f (v

2

)

R

_R

+ f (v

1

)

R

_S

(3.14)

These relations can be processed into the bond-graph structure (figure 3.10) by adding two velocity differences in the form of zero-junctions: ω

S

− ω

C

and ω

R

− ω

C

. As can be seen in the model, the efforts that these velocity differences generate have a negative contribution to the effort on the sun and ring gear and have a positive contribution to the effort on the carrier. This is consistent with equation 3.14 if one realizes that the tangential velocities v

1

and v

2

are proportional to the velocity differences. Then the following holds, where f

SC

and f

RC

represent the functions used in their respective R-element:

f (v

1

)

R

_S

= f ((ω

S

− ω

C

) · R

S

)

R

_S

= f

SC

(ω

S

− ω

C

) (3.15)

f (v

2

) R

R

= f ((ω

R

− ω

C

) · R

R

) R

R

= f

RC

(ω

R

− ω

C

) (3.16)

Figure 3.10: Bond-graph model with friction

The next question is what function should be used to characterize the friction that occurs at each resistive element. The most simple friction models will be considered first, which are a combination of viscous friction and simple stiction. These seem competent enough for the current system. However, if this does not turn out to be sufficient, a more complex model can be considered later.

For the friction between the gears, it is expected that stiction is going to dominate the behavior and viscous friction will be negligible. This will first be described with an arctan function, as it is continuous and simple, yet seemingly competent enough under the current conditions. The arctan-type friction model has two parameter values:

f

gear

(∆ω) = k

a

· arctan(k

b

· ∆ω) (3.17)

The fact that this function is continuous allows for faster computation than a sign function would. If however emulating true stiction turns out to be more important, the function can be easily transformed to a sign function, where the second parameter value disappears:

f

gear

(∆ω) = k

a

· π

2 · sign(∆ω) (3.18)

This function can also be modified in certain ways to increase computational speed around the region v = 0, which is discussed in section 4.1.3.

For the bearings primarily viscous friction is expected, while stiction is assumed negligible compared to the stiction between the gears. This results in the following equation:

f

_bearing

(ω) = b · ω (3.19)

This means that there are now two friction elements that are described by an arctan function, and

three that are described by viscous friction. The parameter values used in these will be determined

experimentally. By fixing one of the three ports, which is setting its velocity to zero, measurements can

be done concerning the other two ports. Fixing either the sun gear or the ring gear results in the two

simplified bond graph models in figure 3.11.

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3.3. PLANETARY GEAR MODULE 17

Figure 3.11: Simplified bond-graph models where the sun (L) or the ring (R) is fixed

The velocity is zero within the red circle, so the contributions to the two zero-junctions are zero as well. The situation where the sun is fixed results in the following frictional torque on the carrier, which is deduced from the corresponding simplified bond-graph model:

T

C1

= −b

C

·ω

C

−k

a,1

·arctan(k

b,1

·ω

C

)− b

R

r

RC

·ω

R

+k

a,2

·arctan(k

b,2

(ω

R

−ω

C

))− k

a,2

r

RC

·arctan(k

b,2

(ω

R

−ω

C

)) (3.20) Here, subscript 1 represents constants used in the gear friction between sun and carrier, while subscript 2 represents constants in the gear friction between ring and carrier. Furthermore, b

C

is the constant in the carrier, b

R

is the constant in the ring and b

S

the constant in the sun. Substituting ω

R

= ω

_C

/r

_RC

and rearranging yields the following equation:

− T

C1

= (b

C

+ b

_R

r

_RC²

) · ω

C

+ k

a,1

· arctan(k

b,1

· ω

C

) + k

a,2

· ( 1 r

RC

− 1) · arctan(k

b,2

· ( 1 r

RC

− 1) · ω

C

) (3.21) This equation can be further simplified if the terms inside both arctan functions are equal, which will be investigated.

k

b,1

· ω

C

= k

? b,2

· ( 1 r

RC

− 1) · ω

C

(3.22)

From equation 3.11 it is clear that 1/r

RC

− 1 = N

S

/N

R

. This ratio of gear teeth, N

S

/N

R

, is equal to the ratio of gear radii, R

S

/R

R

. From the notion in equation 3.13 that v

1

= −v

2

, further conclusion can be made. Because the connection between the gears is similar, it is assumed that the friction resulting from these velocities is equal as well. Taking this assumption in mind, a relation between k

b,1

and k

b,2

can be deduced. The following steps follow from this assumption and combining equations 3.13:

f (v1) = f (−v2) → f

_SC

(ω

_S

− ω

C

) · R

_S

= f

_RC

(−(ω

_R

− ω

C

)) · R

_R

(3.23) R

S

R

· k

a,1

· arctan(k

b,1

· (ω

S

− ω

C

)) = k

a,2

· arctan(−k

b,2

· (ω

R

− ω

C

)) (3.24) eq. 3.9 → ω

S

− ω

C

= − R

R

S

(ω

R

− ω

C

) (3.25)

R

S

R

· k

a,1

· arctan(−k

b,2

· R

R

S

(ω

R

− ω

C

)) = k

a,2

· arctan(−k

b,2

· (ω

R

− ω

C

)) (3.26) As it turns out, an additional relation can be made from this analysis (between k

a,1

and k

a,2

), which could have been expected. The following relations are found:

k

_a,2

= R

S

R

_R

· k

_a,1

, k

_b,2

= R

R

_S

· k

_b,1

(3.27)

This also means that equation 3.22 is indeed correct. Equation 3.21 reduces to:

− T

C1

= (b

C

+ b

_R

r

²_RC

) · ω

C

+ k

a,1

(1 + N

_S²

N

_R²

) · arctan(k

b,1

· ω

C

) (3.28) In the same way, but now by letting ω

R

= 0, the fixed ring equation is obtained:

− T

C2

= (b

C

+ b

_S

r

_SC²

) · ω

C

+ k

a,1

( N

_S

N

R

+ N

_R

N

S

) · arctan(k

b,1

N

_R

N

S

· ω

C

) (3.29)

The two equations that are obtained together contain five unknown parameter values:

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18 CHAPTER 3. DYNAMIC MODEL

• Viscous friction coefficient in the carrier bearing, b

C

in the ring bearing, b

R

in the sun bearing, b

S

• Parameters of the arctan model assumed between the gears k

a,1

and k

b,1

(on which k

a,2

and k

b,2

are dependent)

These can be determined experimentally, by fitting measurement data to these equations. Additionally the used models can be checked for consistency. The detailed approach is explained in section 4.1.

3.4 VSM - pivot mechanism

The pivot mechanism makes use of a planetary gear set to convert rotational motion into translation.

Unlike the planetary gear module discussed in section 3.3, this gear set has its ring gear fixed, which greatly simplifies the associated bond-graph model (figure 3.12). Basically what remains is a two port transformer, with a certain inertia and friction resulting from the gears. An extra complication will be in the conversion of rotation of the gears into the translation of the pivot. This is done by taking the sine of the resulting carrier angle φ

C

and multiplying this by the ring gear radius with a gain. The input is the velocity of the motor that drives this mechanism. The belt transmission and the transmission of the planetary gear are combined into a single transformer.

Figure 3.12: Bond-graph of the pivot mechanism

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3.5. DIFFERENTIAL 19

3.5 Differential

The outputs of both planetary gear sets together will provide the 2D motion of the end-effector. This happens through the differential mechanism, which transforms the velocity of the planetary outputs, ˙ φ

₁

and ˙ φ

2

, into two velocities of the end-effector (figure 3.13). One of them is around the x-axis ( ˙ θ

x

) and the other around the y-axis ( ˙ θ

y

).

Figure 3.13: The differential mechanism, extracted from [2]

The transformation that this mechanism gives has been described by [2]:

θ ˙

_x

θ ˙

y

= 1 2

1 1

d1

d₂

−

^d_d¹

2

φ ˙

₁

φ ˙

2

where d

₁

d

2

= 1 in this case (3.30)

In order to translate it into a single-bond bond-graph model, this transformation is decomposed as follows:

θ ˙

x

= 1

2 ( ˙ φ

1

+ ˙ φ

2

) θ ˙

y

= 1

2 ( ˙ φ

1

− ˙ φ

2

) (3.31)

One zero-junction will be used to get the velocity sum, while another one is used to get the velocity difference between ˙ φ

₁

and ˙ φ

₂

. A transformer is used to add the factor of

¹₂

. The result can be seen in figure 3.14

Figure 3.14: Bond-graph model of the differential mechanism

Both resulting velocities have an inertial element attached, based on the inertia of the end-effector

around either axis. Note that the differential mechanism itself only experiences friction when there is

a difference between the velocities at the inputs. As this velocity difference is proportional to ˙ θ

y

, the

resistive element has been positioned at the corresponding one-junction.

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20 CHAPTER 3. DYNAMIC MODEL

3.6 Conclusion

The dynamic models of each of the subsystems have been made, based on theoretical reasoning and a few earlier measurements and observations. In this process several assumptions have been made in order to maintain an effective model that is not over-complicated. The different sub-models have been linked together to obtain the model for the whole prototype, in the way that this was planned in section 3.1.

The remaining step for completing the characterization is to find the values to all of the unknown parameters that are used in the sub-models. This is rather straightforward for parameters like inertia values, which are easily determined through ’Solidworks’ in which the prototype has been designed.

For other parameters like friction coefficients this requires experimental data, which is acquired in the next chapter.

Additionally, the experiments that follow will be used to evaluate the validity of some of the choices

made in this chapter. Not all assumptions will hold under the results that are found and thus a few

adaptations are going to be made. As characterization is a back and forth process, these adaptations

have not been presented as a part of the constructed model, but rather as a result of the performed

experiments. Therefore, the sub-models that have been found should at this point be regarded provi-

sional.

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Chapter 4

Experimental

4.1 Planetary Gear Module

4.1.1 Measurement model

In section 3.3 two equations had been derived which will be used in this experiment. The fixed sun equation is:

− T

C1

= (b

C

+ b

_R

r

²_RC

) · ω

C

+ k

a,1

(1 + N

_S²

N

_R²

) · arctan(k

b,1

· ω

C

) (4.1)

If this holds, a series of measurements should result in a data curve of the following the form:

− T = c

1

· ω

C

+ c

2

· arctan(c

3

· ω

C

) (4.2)

There are two quantities that must be measured here. The angular velocity of the carrier ω

C

can be measured through the incremental encoder of the motor that would set the carrier into motion. Then a reaction torque will arise from the friction in the gear set (−T ). This torque can be measured by correctly placing a force/torque sensor in the setup. This sensor can measure forces or torques in 6 degrees of freedom, but only the torque about one particular axis will be of interest. The next step will be to mechanically fix the sun gear while letting the carrier and ring gear free to rotate.

For the fixed ring equation the same applies.

− T

C2

= (b

_C

+ b

_S

r

_SC²

) · ω

_C

+ k

_a,1

( N

_S

N

R

+ N

_R

N

S

) · arctan(k

_b,1

N

_R

N

S

· ω

C

) (4.3)

The fixed ring gear experiment should result in a torque-velocity curve of the same form:

− T = c

₄

· ω

_C

+ c

₅

· arctan(c

₆

· ω

_C

) (4.4)

4.1.2 Experimental setup

The experiment required a custom testing setup to be designed and built. The two most important parts in this setup are the driving motor and the torque sensor. The motor could be controlled to move to a certain angle with a set velocity. For each velocity, the torque sensor could then measure the reaction torque caused by friction in the gear set.

Page 21

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22 CHAPTER 4. EXPERIMENTAL

Figure 4.1: Setup design for the planetary gear set measurements

In figure 4.1 the setup design is shown. The top part is the driving motor, which is impaired from moving around its own axis by the motor plate that is attached to the base at the right-hand side. Below the motor, there is a connection to the torque sensor through an adapter. The lower side of the sensor then connects to the planet carrier of the gear set through another adapter. In this way the sensor measures the reaction torque that is felt by the planet carrier. The gear set is clamped inside the base structure. Additionally, two ’fixers’ are attached to the base. These are used to either restrict the sun gear motion or the ring gear motion, as necessary for doing the experiments.

The setup has been designed to be compatible with the current gear set used in the prototype. This turned out to be quite a challenge, as some parts of the prototype were unexpectedly glued together.

For this reason the design presented in figure 4.1 may seem unnecessarily complicated or bulky.

4.1.3 Results and Discussion

The planned experiments have been performed on two devices: the first (or old) prototype and the second prototype. The data for each experiment consists of a series of measurements at different constant velocities, ranging from about 0.33rad/s to 6rad/s in both directions. These each yield a data point and combined this results in a curve. At lower velocities the measurement has been done with a gradually increasing velocity, by plotting the measured torque over time against this velocity over time.

Old Prototype

Figure 4.2 displays the fixed sun result for the old prototype, with the fitted arctan model on the right-

hand side. One can notice that the mid-section of the curve is far from smooth, even after filtering out the

noise. This can be seen in later measurements as well and is mainly the result of an angle dependence

that has been observed in the raw measurements. This is very inconvenient, as it blurs out any possible

unexpected behavior at these very low velocities. The Stribeck effect [14] is an example of what might

be overlooked now.

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4.1. PLANETARY GEAR MODULE 23

-6 -4 -2 0 2 4 6

velocity (Rad/s) -0.15

-0.1 -0.05 0 0.05 0.1 0.15 0.2

torque (Nm)

-6 -4 -2 0 2 4 6

-0.1 -0.05 0 0.05 0.1 0.15 0.2

torque (Nm)

Figure 4.2: Old prototype - Sun fixed - Fit: T = c

1

∗ v + c

₂

∗ arctan(c

₃

∗ v) where c

₁

= 0.011; c

₂

= 0.052; c

₃

= 4;

For the fixed ring result (figure 4.3) the same problem has arisen, where it is not possible to say what exactly happens around v = 0 rad/s. Sadly it was very hard to reduce this angle dependence and unclear whether it was due to building errors in the prototype. It is more likely that the problem lay in the setup, more specifically in an observed misalignment. This could also be concluded from the fact that the effect was still observed in the second prototype, although this might also be because this prototype contains a few re-used parts from the old one. Nonetheless, the curve fits have been made, assuming that the averaged values would yield reasonably valid results.

-6 -4 -2 0 2 4 6

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

torque (Nm)

-6 -4 -2 0 2 4 6

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1

torque (Nm)

Figure 4.3: Old prototype - Ring fixed - Fit: T = c

4

∗ v + c

5

∗ arctan(c

6

∗ v) where c4 = 0.012; c5 = 0.012; c6 = 6;

Second Prototype

The fixed sun and ring result for the second prototype can be seen in figure 4.4 and 4.5 respectively,

including the fitted models. By comparing this to the old prototype result, one can see that the redesign

of this module has led to a decrease of friction by roughly 10 times. As mentioned before the same

angle dependence can be seen as in the old prototype, although the amplitude of this effect has scaled

with the highly reduced friction.

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24 CHAPTER 4. EXPERIMENTAL

-6 -4 -2 0 2 4 6

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

torque (Nm)

-6 -4 -2 0 2 4 6

-0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02

torque (Nm)

Figure 4.4: Second prototype - Sun fixed - Fit: T = c

1

∗ v + c

₂

∗ arctan(c

₃

∗ v) where c

₁

= 0.0015; c

₂

= 0.005; c

₃

= 10;

Note that, by looking at the constants that were found with the fitting curves, c

2

is found to be the same as c

5

. These two constants basically quantify the amount of stiction in the fixed sun and fixed ring situation. On the other hand, the viscous damping coefficient in the fixed ring case (c

4

) is much smaller than that in the fixed sun case (c

1

). This is to be expected as the large ring bearing adds a lot of this viscous damping.

-6 -4 -2 0 2 4 6

-0.01 -0.005 0 0.005 0.01 0.015

torque (Nm)

-6 -4 -2 0 2 4 6

-0.01 -0.005 0 0.005 0.01 0.015

torque (Nm)

Figure 4.5: Second prototype - Ring fixed - Fit: T = c

4

∗ v + c

5

∗ arctan(c

6

∗ v) where c4 = 0.0001; c5 = 0.005; c6 = 12;

Angle Dependence

To further investigate whether the irregularities in the measurements were caused by an angle depen- dence, a torque measurement at a low and constant velocity was performed. Normally this should result in a constant torque after the motor reaches its equilibrium velocity. However, the result in the right-hand plot in figure 4.6 shows that there is quite a significant deviation in this curve. Moreover, the mirror image of this curve reoccurs when letting the motor rotate in the opposite direction at the same low and constant velocity. By comparing this result qualitatively to one of the raw torque measurements for a varying velocity (left-hand plot in figure 4.6) the shape of the deviations in the curves seem to be quite similar.

It is also important to note that the size of the angle dependent deviations, which were measured

with the old prototype, is similar to the size of the deviations found in the corresponding measured

torque-velocity curve.

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4.1. PLANETARY GEAR MODULE 25

0 1000 2000 3000 4000 5000 6000

steps -150

-100 -50 0 50 100 150

velocity(RPM) scaled torque

0 500 1000 1500 2000 2500 3000

-0.018 -0.017 -0.016 -0.015 -0.014 -0.013 -0.012 -0.011 -0.01 -0.009

Torque (Nm)

Figure 4.6: Qualitative comparison between a raw torque measurement (scaled) for a varying velocity (left) and the angle dependence curve (right)

Simulation

Fitting the data to the model turned out to be impossible for the values found in the old prototype experiment. In other words, there is no combination of parameters for which the model reproduces the results found in both experiments. This could mean that some of the assumptions made in the model were false regarding the old prototype.

For the second prototype the assumption that there was a negligible amount of stiction in the bear- ings had to be revised. This resulted in the conclusion that there must in fact be a pretty significant amount of stiction in these bearings. After the model had been updated with this extra effect, there were no problems in determining the necessary parameters from the constants (c

1

− c

6

) that were found. This resulted in a single model that could predict the torque-velocity curve for both the fixed ring and fixed sun case accurately (see figure 4.7).

Figure 4.7: Fitted model simulations with ’true stiction’ - the fixed sun result in solid red and the fixed ring result in transparent red

In addition the arctan friction model has been replaced by a friction model that can better describe the sticking behavior around zero velocity. A sign function as proposed before would become easily unstable, because of the big leap in reaction torque around ω = 0. This can be improved by introducing the following logical operation (4.8) in the resistive element in 20-sim:

Figure 4.8: Proposed stiction model in a resistive element

Here, k

1

represents the first parameter that an equivalent arctan model would have. Then around

ω = 0 a thin slope is used as a transition region, which has a small width defined as w. One has to

be aware that instability is not ruled out when using this operation. For instance when choosing w too

small, unwanted oscillations can occur more easily. On the other hand, choosing w too large results in

too much of a loss of sticking behavior.

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26 CHAPTER 4. EXPERIMENTAL

4.2 VSM - elastic element

The elastic element will yield a reaction torque for a certain displacement, based on the stiffness setting.

In this experiment this torque-displacement curve is measured for several stiffness settings. The overall result will be shown later in this section, as well as a number of interesting measurements.

4.2.1 Experimental setup

To measure the torque that the elastic element generates, a torque sensor is used. This sensor is placed between a hand grip and an adapter that connects to the lever (figure 4.9). This lever is forced to follow the same trajectory as it would in the wrist, by a bearing clamped inside the body. Like the hand grip suggests, the setup is driven manually. The encoder at the top will then track the degree of deflection. The stiffness setting is changed by manually placing the pivot in one of the holes in the ground plate.

Figure 4.9: Setup of the VSM experiment

4.2.2 Results and Discussion

The three most compliant pivot positions tested (q = 2.0 − 0.8 cm) yield a torque-deflection curve that is qualitatively similar to the curve in figure 4.10. It is mostly linear, although some curving can be noticed at the highest deflections. There is a small amount of hysteresis to be seen, which is due to friction. It is important to note that differences in velocities do not result in differences in hysteresis. Hence, the major part of the friction must be independent of velocity as well. The velocities that have been applied range from nearly 0 to how fast a human can move the gripper. The velocities that can be expected in real application almost certainly lie within this range, so that this observation will always hold.

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

displacement (rad) -0.06

-0.04 -0.02 0 0.02 0.04 0.06

torque (Nm)

low velocity high velocity fit: k=0.063

Figure 4.10: Torque-deflection curve for q = 1.4 cm

For the next three pivot positions (q = 0.2to − 1.0 cm) an additional feature to the curve can be

seen (figure 4.11). At a certain point of deflection, a very sudden deviation from the linear segment is

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4.2. VSM - ELASTIC ELEMENT 27 observed. In an analysis in section 3.2 this was already presumed to be a consequence of a spring losing its pretension. During the experiment this was also noticed, which almost certainly confirms that hypothesis. Moreover, the simulation result in figure 3.6 predicted the highest non-linearity to occur between q = 0.3 and q = −0.6 cm. This roughly coincides with the occurrence of this effect in the experiments, although q = −1.0 cm falls outside of this range.

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

displacement (rad) -0.25

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

torque (Nm)

Figure 4.11: Torque-deflection curve for q = −0.4 cm

Figure 4.12 shows yet another type of behavior, corresponding to the stiffest settings (q = −1.6 to

−2.2 cm). The loss of tension of one of the springs is not seen anymore, as higher torques allow for less deflection. There is however quite some play now, where the torque seems to be constant for low deflections. The reason for this has been observed to be due to play in the pivot with respect to the ground plate. This has become visible because of the large forces accompanied to these stiffness settings. It can be considered a flaw in the robustness of the measurement setup, and should not occur in the prototype.

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

displacement (rad) -1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

torque (Nm)

Figure 4.12: Torque-deflection curve for q = −2.2 cm

The observation of equal hysteresis for different velocities is made for the other values of q as well.

The conclusion of the friction being mostly velocity-independent seems to hold in all use-cases.

If one would chose to take a small angle approximation, leading to a linear torque-deflection curve, the corresponding stiffness can be determined from the linear regime of the measured curves. In figures 4.10, 4.11 and 4.12 it is shown how this stiffness information is extracted from the measured curves.

The dependence of the linear stiffness on q has been plotted in figure 4.13. This figure also displays

the comparison of this result with two different models.

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28 CHAPTER 4. EXPERIMENTAL

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

pivot position q (cm) 0

1 2 3 4 5 6 7 8

stiffness K (Nm/rad)

measured K simulation K fitted simplified model K

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

pivot position q (cm) 10^-2

10^-1 10⁰ 10¹

stiffness K (Nm/rad)

measured K simulation K fitted simplified model K

Figure 4.13: Comparison between the experimental result of the linear stiffness (k) for different stiffness settings (q) and two models

The complicated model that followed from theoretical analysis results in a stiffness curve that per- forms well around q = 0, but is more off at very high or low values of q. This can be seen more clearly in the logarithmic plot on the right hand side.

The plot also leads to the observation that the curve of the measured data is almost a straight line when the y-axis is logarithmic. Therefore an alternative model is introduced that is much simpler and even turns out to be more accurate, based on the measured data. It consists of an e-power (which on itself results in a straight line), with a constant added to it, giving it a slight curvature:

K = a

1

∗ exp(a

2

∗ q) + a

3

(4.5)

The fitted result is displayed in the figure, where a

1

= 0.2, a

2

= −1.45 and a

3

= 0.02. Having this method lead to a better result is on one side a bit disappointing, as the first method required a lot more effort.

On the other hand it is very convenient, because this simple model will require a lot less computation time when running simulations of the whole system. It could also have been expected that a theoretical model would fall short on imperfect situations. In this case, a parasitic structural compliance had not been taken into account, which becomes visible when K is high.

4.3 Complete system simulations

The following table (4.1) contains all the parameters that are measured or estimated for use in the full model simulations. Note that some values turned out to be negligible (like the sun bearing friction coeffi- cient) so these are not mentioned in this table. Other parameters have been combined (the moments of inertia) or found redundant. This is the case for the motor parameters and pivot mechanism parameters, because for the simulations these have been simplified to velocity sources.

Parameter Symbol Value

Inertia of (2x)

-the ring gear and what is attached to it I

R

6.47e-005 kg mˆ2 -the carrier and what is attached to it I

C

1.90e-004 kg mˆ2 Planetary gear module friction coefficients(2x)

-carrier bearing (viscous) b

C

0.00010 N m s / rad

-carrier bearing (stiction) k

C

0.00026 N m

-ring bearing (viscous) b

R

0.00080 N m s / rad

-ring bearing (stiction) k

R

0.00205 N m

-gear friction (from sun to carrier) k

1

0.00125 N m

Transmissions friction coefficients (2x, estimated)

-transmissions (viscous) b

T

0.003 N m s / rad

-transmissions (stiction) k

T

0.010 N m

VSM coefficients ( as in K = a

1

∗ exp(a

2

∗ q) + a

3

) (2x)

-a

1

a

1

0.2 -a

2

a

2

-1.45 (with q in cm)

-a

3

a

3

0.02 N m / rad

Differential mechanism (estimated)

- gear friction coefficient (stiction) k

D

0.002 N m Table 4.1: Parameters used for the final simulations

For the first simulation in figure 4.14 the primary actuators have been fixed at zero velocity. Then by

giving the end-effector a range of angles, the 2-dimensional stiffness characteristic of the whole system

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4.3. COMPLETE SYSTEM SIMULATIONS 29 has been simulated. At first this resulted in a circular shape (the most transparent curve). Now the measurement that have been done before in [2] resulted in a clear ellipse instead. The explanation for this could be found in extra parasitic compliance in one of the DOFs, which in combination with stiction present in the system could yield a similar elliptical curve. This has been shown on the left hand side of figure 4.14, by gradually increasing an extra compliance in one of the DOFs. In this case the parasitic compliance is probably caused by the extra long belt transmission used in one of the DOFs, possibly in combination with a small difference between the two VSM modules used.

Figure 4.14: Two dimensional stiffness characteristic at a 1.1 rad displacement: simulated for several values of parasitic compliance (left) and a comparison to (slightly scaled) older data from [2] (right)

In the second simulation in figure 4.15 one can see what happens when letting the primary actuators move the end-effector around. When using higher stiffness settings (the transparent curves) the wrist will follow the actuators faster. When on the other hand using lower stiffness settings, there is going to be more lag for the wrist in following the actuators.

Figure 4.15: End effector (θ

x

, θ

_y

) and elastic elements (φ

1

, φ

₂

) deflection over time as a result of circular

motion generated by the two primary actuators. Stiffness settings: q=-1.5 (transparent lines) to q=0.5

(solid lines)

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30 CHAPTER 4. EXPERIMENTAL

(31)

Chapter 5

Conclusion

The goal of this bachelor assignment has been to find and characterize a model for the prototype of a 2 DOF variable stiffness joint. This has been done with a modular approach. The models for each identified subsystem have been created by port-based modeling. These include the planetary gear module, the elastic element of the VSM, the pivot mechanism, the differential mechanism, the motor and a subsystem representing several transmissions cumulatively. This resulted in a collection of sub- models, that would then be combined into the complete model through rigid connections. This model should according to the posed requirements be competent, yet computationally efficient and as simple as possible. The research question can be answered by evaluating how well this has been done.

In the experimental phase, two sub-models have been fully characterized that are also considered the most important: the planetary gear module and the elastic element.

For the planetary gear module a satisfactory characterization has eventually been realized, although some significant problems have arisen. First of all an observed misalignment in the setup is suspected to have caused the unwanted angle dependence, polluting the raw measurements. This could have been one of the reasons why the interpreted measurements of the old prototype did not fit in the model.

The reason could also have been that some significant effects present in the old prototype were left unaccounted for in the model. A more robust experimental setup could have given more clarity on this question mark. The results of the second prototype measurements, on the other hand, fitted well with the model, after acknowledging an inaccurate assumption concerning friction in the bearings. Another choice that has been found invalid was the use of the arctan model to describe stiction. By introduc- ing an alternative model, the sticking behavior could be better described. There might however be a better computational model that can ensure true sticking behavior, while maintaining stability and low simulation times.

For the elastic element only small problems were encountered. The model that had been con- structed based on geometric relations proved competent in predicting earlier found non-linearities. The analysis showed how the springs losing their pretension were primarily responsible for the significant non-linearity found at several stiffness settings. By removing this effect, a linear spring type model would become a good alternative. By introducing a new relation between stiffness setting q and the apparent stiffness K, a much simpler alternative model could be made, which could be fitted even better to the measured stiffness curve. It also greatly reduced simulation times and the possibility of unstable behavior. During the experiments a flaw was found in the measurement setup in the fact that some play arose in the pivot at high stiffness settings, but that could be accounted for in the interpretation of the results. Both models characterized for this subsystem performed well, although the simple version performed even better for the reasons mentioned above.

The other sub-models are left to be provisional, as no experiments have been performed yet to find the necessary parameters and be able to validate them. However, it is still important to obtain at least some results from the complete model. Therefore, some of these parameter values have been esti- mated while others have been made redundant by simplifying the corresponding sub-model. Previous data of the 2-dimensional stiffness characteristic of the whole wrist could be compared to that of the complete model. The measurements showed an elliptical stiffness curve, unlike the circular curve that was simulated from the model. However, the addition of a reasonable amount of compliance in one of the DOFs resulted in a good match. The question remains whether this is justified or not and in which sub-model the compliance should be applied. On the other hand it does mean that the assumption that all connections and transmissions are rigid will not hold anymore. It will be interesting to see if this effect is still going to be present in the second prototype.

All in all the modeling and characterization of the 2-DOF VSJ has been successful, apart from a few problems encountered and the lack of important data necessary for validation of the integrated model.

Modeling and characterization of a 2-DOF variable stiffness joint