Ballscrew Efficiency Modeling in a Crank-Slider...

Ballscrew Efficiency Modeling in a

Crank-Slider Prosthetic Knee Actuator Holly Warner*

Department of Mechanical Engineering

Cleveland State University

Cleveland, Ohio 44115

December 2, 2014

Abstract

Results from a previously developed model of an actuator for use in the simulation and

optimization of a prosthetic knee joint design indicated the potential for successful energy

regeneration. This model took into account losses due to the resistance of the motor.

However, no mechanical losses were included. This project addresses that modeling

deficiency, providing further insight into the actuator’s energy regeneration capacity.

This is approached by two methods, the development of a friction torque model for the

ballscrew for use in a simulation in which the design parameters are static, having been

previously optimized, and the implementation of a more basic method of accounting for

friction based on efficiency for use in future optimization simulations where the

parameter details are less defined. Results for each approach are presented. In both cases

the capacity for the system to perform energy regeneration is indicated.

1. Introduction

A proof-of-concept simulation for a prosthetic knee crank-slider actuator design

including energy regeneration using ultracapacitors has been formerly established. It can

be described by the bond graph shown in Figure 1.

The input on the left is a knee torque profile. From left to right, the elements represent a

torsion spring, crank-slider geometry, nut mass, screw lead, motor inertia, motor

constant, armature resistance, a DC-DC power converter, and a capacitor. Detail on the

bond graph modeling method can be found in [1]. Within this model eight parameters

were optimized by using an evolutionary algorithm with a fitness function maximizing

* Email address: [email protected]

𝑀𝐾(𝑡) . .

𝑆𝐸 1

𝐶 : 1 𝐾

1/𝐺 . .

𝑀𝑇𝐹 1

𝐼 : 𝑚

1/𝑙 . . 𝑇𝐹 1

𝐼 : 𝐽𝑚

𝑎 . . 𝐺𝑌 1

𝑢 . . 𝑇𝐹

𝑅 : 𝑅𝑎

𝐶 : 𝐶

Figure 1 – Bond graph for the original, frictionless simulation


Crank-Slider Prosthetic Knee Actuator H. Warner

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the stored electrical charge. These parameters include the capacitance of an

ultracapacitor, the spring constant of a torsion spring, two link lengths, two constant

angles, the ballscrew lead, and the initial charge of the capacitor. For the set of

parameters that reached the maximum energy storage condition, the ultracapacitor energy

increased by 12.7 Joules during one stride, which was almost half of the 26.5 Joules that

were available. While the motor’s resistance was accounted for, this model neglected

frictional losses in the transmission components. Inclusion of such losses would provide

useful insight as to the actuator’s ability to function in an energy regenerating mode. The

primary element for which this can be modeled is the ballscrew.

Friction within a ballscrew has been a topic of much study ranging from the development

of highly complex models to the simplest efficiency accounting as well as some

intermediate approaches. Complex models of a ballscrew system includes variables such

as rolling contact, lubrication, sliding, ball-to-ball contact, and the return system, among

others, such as in [2]. Simplifications modeling only a portion of these effects have been

established. For example, in [3] the model was based primarily on bearing-related effects.

Considering the problem from a general perspective, it has been modeled with

modification as a basic screw as well [4]. Experimental modeling has also been applied to

this problem [5]. Frequently, manufacturer’s recommend that a ballscrew be modeled by

use of a 90% efficiency term [6]. Selecting from among these options is really dependent

on the accuracy required for the application and the available information.

Because the model defined within the proof-of-concept system optimization is general

with the focus being placed on regeneration capacity, only the lead of the ballscrew is

being included in the simulation as this models the kinematic and kinetic transformations

that this element implies. This is only one parameter that defines a ballscrew; the

remainder of the parameters, diameter, length, and preload among others, have been left

to be determined during future mechanical design work. Accordingly, the ability to model

friction is limited because few details of the screw are known. Two separate approaches

to adding a friction model to the simulation that are feasible within these constraints were

identified.

An optimal set of parameters has been previously defined for the frictionless model.

Within this set a ballscrew lead value is specified. A specific ballscrew with this lead

value could be selected based on guidelines given by ballscrew manufacturers and a

complex friction model developed from the screw’s now known parameters. This model

can be simulated for a given parameter set, providing insight into friction’s effects on the

system’s power flow and energy usage.

Additionally, and perhaps most commonly, the frictional losses of a ballscrew can be

modeled by a simplified method, accounting for the efficiency of the screw which is

stated by manufacturers to be about 90% [6]. The ballscrew friction model developed by

Olaru, et al. has shown close agreement with this value for a variety of speeds and a

range of contact loads, indicating the sufficiency of this method for optimization

purposes [2]. The results of the complex friction model should also provide confirmation

of this approach. Accounting for this efficiency figure and adding it to the current



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optimization simulation will provide a second means for modeling the ballscrew’s

friction. This approach will be feasible for optimization as it is not dependent on screw

geometry.

In Section 2 the two approaches, complex friction modeling for a given parameter set and

general efficiency friction modeling for use in optimization, are developed in detail. A

method for tracking energy throughout the system is also described. The results of

simulation of both approaches and optimization of the second approach are presented in

Section 3. Lastly, Section 4 provides several concluding observations and possibilities for

future work.

2. Methods

2.1. Complex friction model

According to [4], the friction of a ballscrew is greatly dependent on the preload of the

ballnut and primarily of the coulomb type. It is therefore stated that the friction torque

could be represented by the equation typically describing the torque to raise or lower a

load as shown in (1) with the force 𝐹𝑃 set equal to the preload rather than axial load force.

𝜏𝑓𝑟𝑖𝑐 = 𝐹𝑃𝑅 (2𝜋𝑅𝜇±𝑙 cos(𝛼)

2𝜋𝑅 cos(𝛼)∓𝜇𝑙) (1)

In addition to the preload the pitch radius 𝑅, friction coefficient 𝜇, screw lead 𝑙, and

thread angle 𝛼 must be known. For a high-precision ballscrew with a light preload in the

ballnut a value of 𝜇 = 0.005 may be used for the coefficient of friction. Additionally, the

thread angle 𝛼 for a ballscrew is 45°. [4] The remaining parameters are dependent on the

geometry of the specific ballscrew being considered. The first set of signs given in (1) is

for extension of the screw, while the second set is for compression.

2.2. Ballscrew selection

The selection of a ballscrew is primarily dependent on the axial load. To estimate the

maximum dynamic axial load the original simulation was run with the best parameter set.

This set of parameters for the original system as shown in Figure 1 is given in Table 1.

The parameters which were optimized are shown in the lower portion of the table.

Parameter Symbol Value Units

Motor Constant 𝛼 .054 Nm/A

Armature Resistance 𝑅𝑎 .0821 Ohm

Estimated Nut Mass 𝑚 2.1 kg

Motor Inertia 𝐽𝑚 1.29 kg m2

Capacitance 𝐶 221.54 F

Spring Constant 𝐾 47.64 Nm/rad



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Link Length 𝑎 .055 m

Link Length 𝑑 .25 m

Angle 𝛾 1.32 rad

Angle 𝜙𝐿 1.17 rad

Screw Lead 𝑙 .000809 m/rad

Initial Capacitor Charge 𝑞𝑐0 6726 C

Table 1 – Original system parameter set. Optimized

parameters are given in the lower section of the table.

The geometry constants are illustrated in Figure 2, a concept drawing of the crank-slider

driven prosthesis. Within this figure the long link on the left represents the ballscrew. At

the base of this link is the motor which is attached to the shank. At the top a triangular

link is shown at which point the user’s socket would be connected.

Figure 2 – Concept drawing of

crank-slider driven prosthetic leg

An equation expressing the axial load of the ballscrew was derived (2) and added to the

original system simulation.

𝐹𝑎𝑥𝑖𝑎𝑙 =1

𝑙((𝑚𝑙2 + 𝐽𝑚)�̈�𝑚 +

𝛼2

𝑅𝑎�̇�𝑚 −

𝛼𝑢

𝑅𝑎𝐶𝑞𝑐) (2)

The simulation was run, and the peak force was extracted. The value determined was

𝐹𝑎𝑥𝑖𝑎𝑙 = 1421 N ≈ 320 lbf. Coupled with the value of the screw lead, 0.2 in/rev or

5.08 mm/rev, this information was sufficient to select a ballscrew as it also defined the

required preload value. The optimal preload value is 10% of the maximum force

according to [4]. This is a value of 𝐹𝑃 = 142.1 N ≈ 32 lbf. The selected screw must be

capable of having this preload applied.

γ

a

d

𝜙𝐿



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A search was conducted among several ballscrew manufacturers. The final selection was

a PowerTrac 0631-0200 SRT RA screw and SEL 10408 nut assembly from Nook

Industries. The datasheet for this product can be found in Appendix 1. It is able to handle

a dynamic load of up to 815 lbf and preloads up to 233 lbf, and it possesses the required

lead. While a ballscrew does not have a pitch radius as defined in the typical sense for

power screws or gears, the ball circle radius is a reasonable approximation. For the Nook

Industries screw the ball circle radius was 𝑅 = 0.00801 m.

2.3. System equations with complex friction model

The augmented bond graph representing the complete system is shown in Figure 3. It was

able to be developed with the same causality for each of the energy storage elements as

the original model. No algebraic loops were introduced through the addition of the R

element.

Figure 3 – Bond graph for including complex ballscrew friction model

To form the complete function Φ𝑓𝑟𝑖𝑐 , the sign conventions of the bond graph and (1) were

handled as shown in (3), an expression in the form of coulomb friction.

Φ𝑓𝑟𝑖𝑐(𝑓) = |𝜏𝑓𝑟𝑖𝑐| 𝑠𝑖𝑔𝑛(𝑓) (3)

The completed model includes a total of three states. Accordingly, three equations of

motion may be systematically derived from the bond graph. This development is

illustrated in Appendix 2. Equations (4), (5), and (6) show the final result.

�̇�𝑘 = 𝐺𝑙�̇�𝑚 (4)

�̈�𝑚 =1

𝐽𝑚+𝑚𝑙2[𝑙𝐺𝑀𝑘(𝑡) − 𝑙𝐺𝐾𝜙𝑘 −

𝛼2

𝑅�̇�𝑚 +

𝛼𝑢

𝑅𝐶𝑞𝑐 −Φ(�̇�𝑚)] (5)

𝑖𝑐 =𝛼𝑢

𝑅�̇�𝑚 −

𝑢2

𝑅𝐶𝑞𝑐 (6)

2.4. Simulation with complex friction model

Having established the system’s dynamic equations, the control 𝑢 must be addressed to

construct a complete simulation. An open-loop approach was selected, remaining

𝑀𝐾(𝑡) . .

𝑆𝐸 1

𝐶 : 1 𝐾

1/𝐺 . .

𝑀𝑇𝐹 1

𝐼 : 𝑚

1/𝑙 . . 𝑇𝐹 1

𝐼 : 𝐽𝑚

𝑎 . . 𝐺𝑌 1

𝑢 . . 𝑇𝐹

𝑅 : 𝑅𝑎

𝐶 : 𝐶

𝑅 : Φ𝑓𝑟𝑖𝑐(𝑓)



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consistent with the original simulation. To determine 𝑢, the equations of motion were

manipulated such that a direct solution was possible based on reference data. This is

termed 𝑢-inversion and shown in Appendix 3.

Simulink® was used to create the system simulation. This was implemented by

constructing the system equations in block diagram form. While no optimization was

intended for this simulation, it was developed within the same framework as the original

simulation such that optimization could be possible if required in the future given further

development. Additionally, a specialized function was used to implement the friction

model. Within a MATLAB embedded function block logic was assembled to provide

switching between the screw extension and compression variations of the friction torque

equation and to apply the sign function.

Lastly, auxiliary MATLAB code was developed and additional blocks were added to the

Simulink® diagram to track the power flow and energy usage of the system. Alongside a

general energy accounting an efficiency term for the ballscrew was calculated. A sum of

the power entering and a sum of the power exiting the 1 junction connecting the R

element to the bond graph, excepting the power entering the R element, were computed.

These were integrated to determine the total energy entering and exiting the junction.

Division of the exiting energy value by the entering energy value produced an efficiency

term for the ballscrew. The MATLAB code for this calculation as well as the system

Simulink® and the other MATLAB code can be found in Appendix 4.

2.5. Generalized friction model

As previously stated, for the sake of optimization definition of the complete parameter set

required for the complex friction model is not preferable. As an alternative a simulation

could account for the losses based on a general efficiency term unrelated to the ballscrew

parameters. For ballscrews this efficiency is typically cited to be at least 90% [6].

Applying this concept to the basic transformer model shown for the ballscrew in

Figure 1, requires a loss of the power conservation property of the bond graph as

illustrated in general terms in (7) for the case where the screw is converting power in the

mechanical rotation domain to power in the mechanical translation domain and 𝜂 < 1.

𝐹𝑠𝑐𝑟𝑒𝑤 �̇� = 𝜂 𝜏𝑠𝑐𝑟𝑒𝑤 �̇� (7)

The equations describing a ballscrew within a bond graph are separated into the

kinematic relationship and kinetic relationship as shown in (8) and (9).

�̇� =1

𝑙�̇� (8)

𝜏𝑠𝑐𝑟𝑒𝑤 = 𝑙𝐹𝑠𝑐𝑟𝑒𝑤 (9)

Using these equations to substitute back into the right hand side of the power equality

given in (7), one can see that the efficiency coefficient must only be applied to either (8)

or (9). Since the friction torque is a kinetic variable, it follows that the efficiency



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coefficient should be applied to (9). It cannot simply be said, however, that (10) always

holds true as it is possible for the screw to be driven by the force, backdriving, as shown

in (11).

𝜏𝑠𝑐𝑟𝑒𝑤 = 𝜂𝑙𝐹𝑠𝑐𝑟𝑒𝑤 (10)

𝐹𝑠𝑐𝑟𝑒𝑤 = 𝜂1

𝑙𝜏𝑠𝑐𝑟𝑒𝑤 (11)

For the power equality to hold for both of these cases and the equation to be of the form

(9) as implemented in the bondgraph, the coefficient cannot simply be set to 0.9, though

the efficiency is always 90%. The solution to this is to use the equation of the form (10)

where two values of 𝜂 are switched between. The first is obvious; 𝜂𝐹 = 0.9 for the case

where the force is driving the screw. The second is found by solving (11) for 𝜏𝑠𝑐𝑟𝑒𝑤. This

requires 𝜂𝜏 =1

0.9 and is for the case where the torque is driving the screw.

Replacing (9) in the bond graph system equation derivation process with the form (10)

leads to a slight alteration to the second equation of motion (13) of the original set of

equations. The complete set is shown below for the original frictionless system with the

addition of the efficiency term.

�̇�𝑘 = 𝐺𝑙�̇�𝑚 (12)

�̈�𝑚 =1

𝐽𝑚+𝑚𝑙2𝜂[𝜂𝑙𝐺𝑀𝑘(𝑡) − 𝜂𝑙𝐺𝐾𝜙𝑘 −

𝛼2

𝑅�̇�𝑚 +

𝛼𝑢

𝑅𝐶𝑞𝑐] (13)

𝑖𝑐 =𝛼𝑢

𝑅�̇�𝑚 −

𝑢2

𝑅𝐶𝑞𝑐 (14)

The derivation of (12), (13), and (14) is not given as it follows the derivation of the

previously discussed friction model set of equations nearly identically.

2.6. Simulation with generalized friction model

Control within this simulation was once again defined in an open loop manner by

𝑢-inversion by the same method as was described before. Similar to the previously

developed simulation the system equations were implemented in block diagram form. An

embedded MATLAB function was prepared for the selection of 𝜂. Based on the power

convention of the bond graph the switching was defined such that 𝜂 = 0.9 would be

selected for positive power flow through the TF element, corresponding to backdriving,

and 𝜂 =1

0.9 would be selected for negative power flow in which case the screw is being

driven by the torque.

Code for an energy balance was completed for this model as well as calculation of the

efficiency across the ballscrew TF element. While the system energy would no longer

balance because of the non-power conserving change made to the system equations, the



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difference will represent the losses due to the screw’s friction torque. In addition,

dividing the time integral of the input power of the TF by the time integral of the output

power of the TF yielded an efficiency term to be compared with the expected value of

90%.

2.7. Optimization with generalized friction model

Upon completion of the simulation, optimization of the generalized friction model could

be accomplished. This was approached by use of an evolutionary optimization algorithm,

namely biogeography-based optimization. Biogeography-based optimization is an

algorithm based upon the migration and emigration of species to and from various

isolated habitats where the habitats represent problem solutions and species characterize

solution features [7]. In addition to the basic algorithm, mutation and elitism were

utilized. Each run was begun with a random seed. A total of five optimization runs were

completed with the algorithm parameters given in Table 2.

Parameter Value

Population Size 200

Number of Generations 100

Number of Elite Individuals 2

Probability of Mutation 0.02

Table 2 – Biogeography-based optimization parameters

The optimization algorithm selected minimizes a given cost. As the goal for this

optimization is the maximization of capacitor charge for one gait cycle, the cost function

was defined as (15).

𝐶𝑜𝑠𝑡 = −(𝑞𝑐,𝑓𝑖𝑛𝑎𝑙 − 𝑞𝑐,𝑖𝑛𝑖𝑡𝑖𝑎𝑙) (15)

While maximization of energy gain is more applicable, this cost function based on

maximizing the capacitor charge was selected to remain consistent with the original

simulation such that a direct comparison may be made.

The optimized parameters were selected as shown in Table 3, which also includes the

search space. All of the parameters were allowed to vary throughout a continuous search

space except for the ballscrew lead 𝑙 for which a discrete set was defined.

Parameter Minimum Value Maximum Value Units

𝐶 0 500 F

𝐾 0 100 Nm/rad

𝑎 0 .15 m

𝑑 0 .3 m

𝛾 0 𝜋 rad

𝜙𝐿 0 𝜋 rad

𝑙 1 6.350 mm/rev



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𝑞𝑐0 0 8000 C

Table 3 – Optimization parameter ranges

Lastly, several constraints were placed on the acceptable solutions to help ensure

feasibility. This was implemented through penalizing the cost function of any solution

not meeting the constraints by setting it to infinity. Specifically, the geometry variables

were required to result in real values for the variable length link (ballscrew) and for the

transformer ratio G. Additionally, solutions for the 𝑢-inversion were required to be real

and between zero and one. The MATLAB code and Simulink® code for the optimization

and simulation are given in Appendix 5.

3. Results

3.1. Complex friction model results

The parameters within the simulation were set to the values given in Table 1. The

simulation was run with an input reference knee moment with a length of one stride.

Mathematically perfect tracking as predicted by the 𝑢-inversion technique was attained,

as shown in Figure 4.

Figure 4 – Tracking results for complex friction model

The attributes of the electrical system throughout the simulation time are depicted in

Figure 5. With the inclusion of the friction losses the capacitor still charged for the given

parameter set.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

70

80

Time (s)

Knee A

ngle

(d

eg)

rms error = 7.9578e-05

Reference Data

Simulated



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Figure 5 – Capacitor charge, voltage, and current for one stride.

The plot of power flow in Figure 6 shows an alternating pattern between the power

provided and required by the human and the power delivered to and extracted from the

capacitor. This illustrates the power exchange expected for regeneration.

Figure 6 – Power flow for one stride

The change in energy for each component was also evaluated and is shown in Figure 7.

The sum of the final values of each component’s change in energy was on the order of

10-6 proving that the system model is truly energy conserving. An overall gain of 7.95 J

was observed in the capacitor. The net available energy entering the system from the

human was 26.58 J. The capacitor stored energy represents approximately 30% of this

available energy.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.96724

6726

6728

Cap

acit

or

Charg

e, q c

Charge gained = 0.2620 C

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.930.35

30.36

30.37

Cap

acit

or

Vo

ltage, v C

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-20

0

20

Time (s)

Cap

acit

or

Curr

ent,

i c

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-600

-400

-200

0

200

400

600

Time (s)

Pow

er

(W)

Positive corresponds to power entering bond graph

Mechanical Power from Human

Spring Power

Linear Kinetic Power

Rotational Kinetic Power

Joule Power Losses

Capacitor Power

Friction Power Losses



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Figure 7 – Energy distribution for one stride

It was also noted that the losses related to the motor resistance were two to three times

greater than the friction losses, suggesting that this may be an important area to

investigate with respect to the overall actuator’s efficiency.

Finally, the efficiency of the ballscrew actuator was plotted in Figure 8.

Figure 8 – Ballscrew efficiency

At its lowest point the efficiency of the screw was 97%. This suggests that the

manufacturer’s value of 90% is more than sufficient in accounting for the friction losses.

However, it should be noted that the efficiency is highly dependent on the preload of the

ballscrew. While large variations in the required value of the preload are not expected for

different screws, further trials for different parameter sets would be required to determine

the range of this variation.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-40

-30

-20

-10

0

10

20

30

Time (s)

E

nerg

y (

J)

Negative corresponds to energy gained by a component

Mechanical Energy from Human

Spring Energy

Linear Kinetic Energy

Rotational Kinetic Energy

Joule Losses

Capacitor Energy

Friction Losses

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Time (s)

Bal

lscre

w e

ffic

iency,



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3.2. Generalized Friction Model and Optimization Results

For each of the five optimization runs convergence was achieved. An example plot is

shown in Figure 9.

Figure 9 – Example convergence plot from Trial 4

Associated with each trial a best parameter set was determined. The results of the five

trials are shown in Table 4.

Optimized System Parameters Capacitor

Charge

Gained

(C)

Capacitor

Energy

Gained

(J)

Friction

Losses

(J) 𝑪

(F)

𝑲

(Nm/rad)

𝒂

(m)

𝒅

(m)

𝜸

(rad)

𝝓𝑳

(rad)

𝒍 (mm/rev)

𝒒𝒄𝟎

(C)

Trial 1 124.43 31.44 0.049 0.28 1.55 1.34 4 4140 0.057 23.98 9.84

Trial 2 117.99 32.32 0.060 0.30 1.77 1.29 5 3850 0.056 0.26 9.66

Trial 3 137.90 31.07 0.036 0.078 2.74 1.51 3 4613 0.056 12.30 9.68

Trial 4 234.83 30.46 0.035 0.29 0.55 1.28 3 7423 0.060 1.32 9.50

Trial 5 123.23 31.57 0.061 0.24 2.11 1.36 5 4094 0.055 -13.90 9.43

Table 4 – Optimization results for five runs

While it cannot be concluded from the results of the individual trials that a global

optimum was found by the optimization algorithm, it is clear that particular ranges of the

parameters tend to cause the maximum capacitor charge. The consistency across the

capacitor charge gained and friction losses columns between trials is striking in that this

suggests that even with the variations among the system parameters, the algorithm may

have been consistently approaching the same limit.

It is also notable that the energy gained by the capacitor was highly variable. This merits

further investigation especially as it seems intuitively that an increase in capacitor charge

should indicate an increase in the capacitor’s stored energy. One potential cause of this to

be considered is the initial kinetic energy in that the energy within a capacitor is related to

the square of the capacitor voltage. The voltage across the capacitor is related to the

angular velocity of the motor. For example, in the case where the energy gain is large, a

large initial kinetic energy is observed. This issue poses an opportunity for future work.

0 10 20 30 40 50 60 70 80 90 100-0.1

0

0.1

Generation

Min

imu

m C

ost

(C

)-

(qc,f

inal -

qc,i

nit

ial )



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Because the timing of the efficiency switching could not be exactly determined during

the 𝑢-inversion process for this case, the tracking, as shown in Figure 10, is no longer

mathematically perfect; however, it is still sufficiently accurate for gait.

Figure 10 – Representative tracking plot, Trial 4

A set of plots demonstrating the electrical features of system is given in Figure 11.

Within the first plot one can observe the overall charge of the capacitor. The voltage

remains fairly constant. There is significant variation in the current.

Figure 11 – Typical capacitor charge, voltage, and current curves, Trial 4

Once again an alternating pattern between the capacitor and human power flows is

observed in Figure 12, representing the power exchange between the human and

prosthesis. Additionally, one can see indications of the switching of the efficiency value

within this plot as it causes discontinuities in the power flow.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

70

80

Time (s)

Knee A

ngle

(d

eg)

rms error = 0.0011812

Reference Data

Simulated

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.97420

7422

7424

Cap

acit

or

Charg

e, q c

Charge gained = 0.05952 C

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.931.73

31.74

31.75

Cap

acit

or

Vo

ltage, v C

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-20

0

20

Time (s)Cap

acit

or

Curr

ent,

i c



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Figure 12 – Example power flow plot from Trial 4

A plot exhibiting the change in energy within the elements of the system was also created

and is shown in Figure 13. While an exact energy balance could not be computed for this

case as it was forced into a non-energy conserving state, the model could be checked by

comparing the energy balance and the frictional losses value. These values were

Figure 13 – Plot of individual components’ change in energy, Trial 4

equivalent to the thousandths of Joules, confirming that the system was balanced, though

not energy conserving.

Finally, the results of the efficiency model were plotted in Figure 14 to confirm its

accurate performance. A brief transient is observed. This is likely due to the fact that the

𝜂 is set to one in the case that the ballscrew output power, which is used for switching, is

zero.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-600

-400

-200

0

200

400

600

800

Time (s)

Pow

er

(W)

Positive corresponds to power entering bond graph

Mechanical Power from Human

Spring Power

Linear Kinetic Power

Rotational Kinetic Power

Joule Power Losses

Capacitor Power

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-30

-20

-10

0

10

20

30

Time (s)

E

nerg

y (

J)

Negative corresponds to energy gained by a component

Mechanical Energy from Human

Spring Energy

Linear Kinetic Energy

Rotational Kinetic Energy

Joule Losses

Capacitor Energy



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Figure 14 – Characteristic efficiency plot, Trial 4

4. Conclusion

Completion of this project represents the development and validation of two methods of

adding mechanical losses to the crank-slider actuator model for a prosthesis. The first

method explored an approach to modeling the frictional losses of the ballscrew based

upon its geometry and preload. The alternative method was the implementation of

generalized friction losses based upon an efficiency percentage, decoupling the ballscrew

geometry from the addition of friction to the simulation. This second method was pursued

because it allows greater freedom in optimization; the ballscrew may simply be modeled

by its lead value.

The results of simulation of the first method indicated that the manufacturer’s value of

90% efficiency is beyond sufficient. Additionally, for the specific parameter set used in

this simulation, regeneration capacity was maintained. The predicted regeneration

capacity of the complex friction model simulation is equivalent to 26% of the energy

needed for driving an ankle motor according to Winter’s data for a fast walking pace [8].

Brief evaluation indicated that this method was greatly dependent on the preload of the

ballscrew. A more detailed assessment of the required preload and its variation in relation

to other parameter sets should be completed.

Simulation and optimization of the second method was completed for a total of five trials.

For many parameters a clear range arose among the best solutions. Even at 90%

efficiency the system was capable of charging the capacitor. No conclusion with respect

to the system’s energy regeneration potential may be made at this point. Further work

must be done in this area as a large amount of variability was predicted within this result.

One potential cause of this apparent inconsistency has been proposed, and future work

will include its investigation.

Several directions within the broader scope of this model’s implementation will be

addressed in the future. First, more extensive optimization may be completed, including

multi-objective optimization. Within a closed-loop control system the ability to not only

charge the capacitor but also to track reference data becomes imperative. Other potential

elements of the cost function include minimizing user effort, tracking ground reaction

forces, and minimizing control signals. Secondly, dependent on the conclusions formed

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Time (s)

Bal

lscre

w e

ffic

iency,



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with respect to the variablility of the capacitor energy gain, the optimization cost may be

better executed as energy gain rather than change in capacitor charge. Third, the results

will be applied to a crank-slider actuator model within a larger control system

development simulation.

References

[1] D. Karnopp, D. Margolis and R. Rosenberg, System Dynamics: Modeling,

Simulation, and Control of Mechantronic Systems, 5th ed., Hoboken: Wiley, 2012.

[2] D. Olaru, G. C. Puiu, L. C. Balan and V. Puiu, "A New Model to Estimate Friction

Torque in a Ball Screw System," in Product Engineering: Eco-Design, Technologies

and Green Energy, Dordrecht, Springer, 2004, pp. 333-346.

[3] S. Jindrich, "A Study on the Ball Screw Friction Torque," in Student's Conference,

2011.

[4] A. H. Slocum, Precision Machine Design, Englewood Cliffs: Prentice-Hall, 1992.

[5] A. Kamalzadeh, "Precision Control of High Speed Ball Screw Drives," 2008.

[6] T. McNier and J. G. Johnson, "Specifying, Selecting, and Applying Linear Ball

Screw Drives," Thomson Industries, Inc, Wood Dale.

[7] D. Simon, "Biogeography-Based Optimization," in IEEE Transactions on

Evolutionary Computation, 2008.

[8] D. Winter, "Energy Generation and Absorption at the Ankle and Knee during Fast,

Natural, and Slow Cadences," Clinical Orthopaedics and Related Research, pp. 147-

154, 1983.



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Appendix 1: Ballscrew datasheet



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Appendix 2: Complex friction model equation derivation

Figure A2.1 – Reference bond graph for including complex ballscrew friction model

Derive first state space equation:

�̇�2 = 𝑓2 = 𝑓3 = 𝐺𝑓4 = 𝐺𝑓6 = 𝐺𝑙𝑓7 = 𝐺𝑙𝑓8 =𝐺𝑙

𝐽𝑚𝑝8 (A2.1)

Derive second state space equation:

�̇�8 = 𝑒8 = 𝑒7 − 𝑒9 − 𝑒10 = 𝑙𝑒6 − 𝛼𝑓11 −Φ(𝑓10) = 𝑙(𝑒4 − 𝑒5) − 𝛼𝑓12 −Φ(𝑝8

𝐽𝑚)

= 𝑙(𝐺𝑒3 − 𝑒5) −𝛼

𝑅𝑎𝑒12 −Φ(

𝑝8

𝐽𝑚) = 𝑙(𝐺(𝑒1 − 𝑒2) − 𝑒5) −

𝛼

𝑅𝑎(𝑒11−𝑒13)−Φ(

𝑝8

𝐽𝑚)

= 𝑙(𝐺(𝑀𝑘(𝑡) − 𝐾𝑞2) − 𝑒5) −𝛼

𝑅𝑎(𝛼𝑓9 − 𝑢𝑒14) − Φ(

𝑝8

𝐽𝑚)

= 𝑙(𝐺(𝑀𝑘(𝑡) − 𝐾𝑞2) − 𝑒5) −𝛼

𝑅𝑎(𝛼𝑓8 −

𝑢

𝐶𝑞14) − Φ(

𝑝8

𝐽𝑚)

= 𝑙(𝐺(𝑀𝑘(𝑡) − 𝐾𝑞2) − 𝑒5) −𝛼

𝑅𝑎(𝛼

𝐽𝑚𝑝8 −

𝑢

𝐶𝑞14) − Φ(

𝑝8

𝐽𝑚) (A2.2)

Require 𝑒5 from derivative causality to complete (A2.2):

𝑝5 = 𝑚𝑓5 = 𝑚𝑓6 = 𝑚𝑙𝑓7 = 𝑚𝑙𝑓8 =𝑚𝑙

𝐽𝑚𝑝8 (A2.3)

Taking time derivative of (A2.3):

𝑒5 = �̇�5 =𝑚𝑙

𝐽𝑚�̇�8 (A2.4)

Substituting (A2.4) into (A2.2)

�̇�8 = 𝑙 (𝐺(𝑀𝑘(𝑡) − 𝐾𝑞2) −𝑚𝑙

𝐽𝑚�̇�8) −

𝛼

𝑅𝑎(𝛼

𝐽𝑚𝑝8 −

𝑢

𝐶𝑞14) − Φ(

𝑝8

𝐽𝑚) (A2.5)

𝑀𝐾(𝑡) . .

𝑆𝐸 1

𝐶 : 1 𝐾

1/𝐺 . .

𝑀𝑇𝐹 1

𝐼 : 𝑚

1/𝑙 . . 𝑇𝐹 1

𝐼 : 𝐽𝑚

𝑎 . . 𝐺𝑌 1

𝑢 . . 𝑇𝐹

𝑅 : 𝑅𝑎

𝐶 : 𝐶

𝑅 : Φ𝑓𝑟𝑖𝑐(𝑓)

1

2

3 4

5

6 7

8

9

10

11 12

13 14



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Solving (A2.5) for �̇�8:

�̇�8 =𝐽𝑚

𝐽𝑚+𝑚𝑙2[𝑙𝐺𝑀𝑘(𝑡) − 𝑙𝐺𝐾𝑞2 −

𝛼2

𝑅𝑎𝐽𝑚𝑝8 +

𝛼𝑢

𝑅𝑎𝐶𝑞14 −Φ(

𝑝8

𝐽𝑚)] (A2.6)

Derive third state space equation:

�̇�14 = 𝑓14 = 𝑢𝑓13 = 𝑢𝑓12 =𝑢

𝑅𝑎𝑒12 =

𝑢

𝑅𝑎(𝑒11 − 𝑒13) =

𝑢

𝑅𝑎(𝛼𝑓9 − 𝑢𝑒14)

=𝑢

𝑅𝑎(𝛼𝑓8 −

𝑢

𝐶𝑞14) =

𝑢

𝑅𝑎(𝛼

𝐽𝑚𝑝8−

𝑢

𝐶𝑞14)

(A2.7)

Change of variables:

𝑞2 = 𝜙𝑘, knee angle

𝑝8 = 𝐽𝑚𝜃𝑚, motor momentum

𝑞14 = 𝑞𝑐, capacitor charge

Taking time derivatives:

�̇�2 = �̇�𝑘, knee angular velocity

�̇�8 = 𝐽𝑚�̈�𝑚, knee inertial force

�̇�14 = 𝑖𝑐, capacitor current

Substitute change of variables to form final set of system equations:

�̇�𝑘 = 𝐺𝑙�̇�𝑚 (A2.8)

�̈�𝑚 =1


𝛼2

𝑅�̇�𝑚 +

𝛼𝑢

𝑅𝐶𝑞𝑐 −Φ(�̇�𝑚)] (A2.9)

𝑖𝑐 =𝛼𝑢

𝑅�̇�𝑚 −

𝑢2

𝑅𝐶𝑞𝑐 (A2.10)



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Appendix 3: 𝑢-inversion

Starting from the system equations:

�̇�𝑘 = 𝐺𝑙�̇�𝑚 (A3.1)

�̈�𝑚 =1


𝛼2

𝑅𝑎�̇�𝑚 +

𝛼𝑢

𝑅𝑎𝐶𝑞𝑐 −Φ(�̇�𝑚)] (A3.2)

𝑖𝑐 =𝛼𝑢

𝑅𝑎�̇�𝑚 −

𝑢2

𝑅𝑎𝐶𝑞𝑐 (A3.3)

Multiply (A3.3) by 𝑞𝑐:

𝑞𝑐𝑖𝑐 =𝛼𝑢

𝑅𝑎𝑞𝑐�̇�𝑚 −

𝑢2

𝑅𝑎𝐶𝑞𝑐2 (A3.4)

Integrate (A3.4) with respect to time:

∫ 𝑞𝑐𝑖𝑐 𝑑𝜏𝑡

0= ∫

𝛼𝑢

𝑅𝑎𝑞𝑐�̇�𝑚 −

𝑢2

𝑅𝑎𝐶𝑞𝑐2 𝑑𝜏

𝑡

0 (A3.5)

Perform change of variables 𝑞𝑐 = 𝑥3, 𝑖𝑐 = �̇�3:

∫ 𝑥3�̇�3 𝑑𝜏𝑡

0= −

1

𝑅𝑎𝐶∫ (𝑢𝑥3)

2 𝑑𝜏𝑡

0+

𝛼

𝑅𝑎∫ (𝑢𝑥3)�̇�𝑚 𝑑𝜏𝑡

0 (A3.5)

Continue integration:

1

2(𝑥3(𝑡)

2 − 𝑥3(0)2) = −

1


2 𝑑𝜏𝑡

0+

𝛼


0 (A3.6)

𝑥3(𝑡)2 = 2 [−

1


2 𝑑𝜏𝑡

0+

𝛼


0] + 𝑥3(0)

2 (A3.7)

Solve (A3.2) for 𝑢𝑥3:

𝑢𝑥3 =𝑅𝑎𝐶

𝛼[(𝐽𝑚 +𝑚𝑙2)�̈�𝑚 − 𝑙𝐺𝑀𝑘(𝑡) + 𝑙𝐺𝐾𝜙𝑘 +

𝛼2

𝑅𝑎�̇�𝑚 +Φ(�̇�𝑚)] (A3.8)

Substitution of (A3.8) into (A3.7) gives an expression for 𝑥3(𝑡)2 in terms of values that

are either known or could be taken from reference data. Taking the square root of this

result provides an expression for 𝑥3(𝑡). Finally, dividing (A3.8) by this expression for

𝑥3(𝑡) yields a direct solution for 𝑢. The implementation of these final steps can be

completed numerically within MATLAB and are, therefore, not shown here.



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Appendix 4: Complex friction model MATLAB code and Simulink®



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Appendix 5: Generalized friction model MATLAB code and Simulink®



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