design optimization of electromechanical exoskeleton

DESIGN OPTIMIZATION OF

ELECTRO-MECHANICAL

EXOSKELETON Group 3

Ayushi Srivastava, USC ID: 9664835920 Joel Agrawal, USC ID: 4108114994

Mahadevan Ramachandran, USC ID: 1378622539 VVenkata Vinaya Krishna Leeladhar Piratla, USC ID: 2831670672

ABSTRACT

This project aims to optimize the design of an electro-mechanical exoskeleton to support a

disabled person's legs and allow him/her to walk. The optimization focuses on the

optimization of the material that members of the frame is made from and the optimum angle

through which the motors at the hip and knee joints rotate as well as the rotational velocity

in order to minimize power consumption from a standard battery pack.

NOMENCLATURE

Mpthigh Mass of average human thigh Mpcalf Mass of average human calf Mpankle Mass of average human ankle Methigh Mass of the thigh member of the exoskeleton Mecalf Mass of the calf member of the exoskeleton Meankle Mass of the ankle member of the exoskeleton Mthigh Total mass of thigh Mcalf Total mass of calf Mankle Total mass of ankle Θ1 Angle of rotation of motor 1 Θ2 Angle of rotation of motor 2 Θ3 Angle of rotation of motor 3 Θ4 Angle of rotation of motor 4 t1 Time taken by motor 1 to rotate by θ1

t2 Time taken by motor 2 to rotate by θ2

x(1) Θ1 x(2) Θ2 x(3) Θ3 x(4) Θ4 x(5) t1 x(6) t2

Table of Contents 1. Introduction .......................................................................................................................................... 4

2.1 Design project definition phase ........................................................................................................ 5

2.1(b). Societal need ................................................................................................................................. 5

2.1(c). Functional Requirements ............................................................................................................... 5

2.1(d). Design Constraints ......................................................................................................................... 6

2.1(e). Sketch History ............................................................................................................................ 7

2.1(f) Design Matrix ............................................................................................................................ 11

2.1(g). Geometry Management Strategy ................................................................................................ 12

2.1(h). Discipline Analysis Models........................................................................................................... 12

2.2 NUMERICAL IMPLEMENTATION PHASE .......................................................................................... 13

2.2(a). Discipline Analysis Models and N2 diagram: ............................................................................... 13

2.2(b). Block diagram of analysis modules.............................................................................................. 14

2.2(c). Starting point values for design parameters ............................................................................... 15

2.2(d). Design of Experiment .................................................................................................................. 16

2.3 Optimization phase ......................................................................................................................... 17

2.3(a). Gradient-based algorithm ........................................................................................................... 17

2.3(b). Formal Optimization problem statement ................................................................................... 18

Minimization of mass of exoskeleton frame to using CES EDUPACK: .............................................. 19

2.3(c). Optimizer convergence and better results: ................................................................................. 28

2.3(d). Sensitivity Analysis....................................................................................................................... 29

2.3(e). Active Constraints ........................................................................................................................ 30

2.3(f). Scaling in design problem ............................................................................................................. 31

2.3(g). Heuristic technique ...................................................................................................................... 32

2.3(h). Comparison of results with gradient-based approach ................................................................ 34

2.3(i). Multiple objective Optimization formulation ............................................................................... 34

2.3(j). Pareto Front .................................................................................................................................. 35

2.3(k). 3 View drawing ............................................................................................................................ 35

Calculation of exoskeleton and total mass ................................................................................................. 36

Motor Power calculation ............................................................................................................................. 37

Power of Motor 1 (Hip joint)................................................................................................................... 37

Power of Motor 2 (Knee joint) ................................................................................................................ 37

Power of Motor 3 (Hip joint)................................................................................................................... 37

Power of Motor 4 (Knee joint) ................................................................................................................ 37

1. Introduction

The marriage of man and machine has long been a touchstone of sci-fi and now real life. Video

games like, Mech Warrior, BattleTech etc. have had relationships maxing out on mechanical

muscle.

Innovations are being made in the field of exoskeleton technology for advancing robotic-

assisted prospects not in service of soldiers, but in service of the disabled. Earlier robotic-

assistance was focused on military applications, now there is a new wave in the medical field to

help those in the need of support to walk, stand, and carry heavy objects and to treat ailments

such as paralysis, general fatigue, etc.

Exoskeleton can be described as a device that is worn by a user to perform specific tasks:

assisting walking, supporting heavy loads, etc. The mechanical exoskeleton is gaining much

attention as a method of providing walking assistance for paraplegic patients, since these

devices provides the disabled a similar experience like human walking and gives full mobility to

their body. Many paraplegic patients who are unable to walk have a dream of walking again

and exoskeleton-type walking assistive devices helps them to walk, whereas wheelchairs can

only make them move.

Notable studies have been made recently on exoskeleton for disabled. For example,

researchers at Harvard recently developed an exoskeleton to increase the strength of the

wearer. Stanford researchers are working on technology that mimics the processes of the

human brain, this could drastically change the way humans interact with prosthetic limbs. An

Israeli company, Argo Medical Technology, has recently developed an exoskeleton to support

the lower limb called ReWalk. There are many more examples of innovation being made in this

field.

Our aim in this project is to optimize the power consumption and reduce the total mass of the

exoskeleton by selecting an appropriate material.

2.1 Design project definition phase

2.1(b). Societal need As mentioned above, there exists a need for an alternate mode of enabling disabled people to

move that doesn’t confine them to a wheelchair. The exoskeleton that was initially developed

to aid in military applications can be modified for this purpose and many prototypes are

currently available. As the device is to offer maximum mobility for the user, it is imperative that

it is not limited to a power source that is immobile like a power outlet. This implies the use of a

battery on board the device (if electrical actuators are considered or pressurized fluid tanks if

hydraulic/pneumatic actuators are considered). In order to maximize the efficacy of the device,

the battery must be able to power it for the longest duration possible, which can be

accomplished either by increasing the battery capacity or by reducing the power consumed by

the actuators. Increasing battery capacity might not always be feasible since the energy density

for a specific battery is almost a constant and to increase the capacity, the battery needs to be

made larger which might hinder the design. The alternate i.e. decreasing the power

consumption by decreasing overall mass and optimizing motor torques might be the best way

to improve performance. Another added benefit is the reduced cost of the device owing to the

lighter mass and the need of a smaller battery.

2.1(c). Functional Requirements

Figure 1: Functional requirements for the Mechanical Exoskeleton

The above figure represents all the functional requirements that the mechanical exoskeleton

must possess in order to be a feasible concept that can be prototyped. However, the functional

requirements of ‘supporting body weight’ and the power required for ‘movement of the

exoskeleton’ are selected to be optimized using different optimization techniques since they

can be optimized numerically. Functional requirements such as ‘attach to the body’ can be

done using a simple harness, ‘shock absorption’ needs a simple spring/damper system and the

Mechanical Exoskeleton

Support for Leg

Support Body Weight

Attach to the body

Power Unit

Movement of Exoskeleton

Auxillary Power

Movement Control Unit

Control the Actuators

Motion of joints

Balance Control

Shock Absorption

‘movement control unit’ will be a computer program which cannot be optimized numerically.

While design parameters are generated for all functional requirements, the motors used at the

joints and the frame/load bearing members are the primary focus of this project.

2.1(d). Design Constraints The exoskeleton design is subject to a number of constraints based on feasibility, cost, necessity

etc. The major constraints that impact the design are:

Dimensions of the exoskeleton members: For an average human being, the lengths of the

thighs, calves and feet and the width of the waist are constant and the frame that attaches to

them must also have similar lengths in order to work efficiently. These dimensions form the

basic constraints that must be met for any design. Based on an average human leg, the

following dimensions are obtained for the exoskeleton frame:

Table 1: Exoskeleton member lengths

Dimension (mm) Dimension (in)

Lwaist1 450 17.7165

Lwaist2 250 9.8425

Lhip 100 3.937

Lthigh 420 16.5354

Lcalf 480 18.8976

Lfoot 60 2.3622

Support load: The exoskeleton must be able to withstand the weight of a full grown human

being in addition to its own mass and the mass of the battery. It must also be able to bear

impact loads during walking without any deformation or failure of any of the members. This

load can be minimized by reducing the mass of the exoskeleton which is done by optimizing the

material. The minimum load to be borne by the exoskeleton is:

Mass of exoskeleton + mass of battery + mass of an average human being

Where the mass of an average human being is taken as 80kg (176.84lbs). The mass of the

exoskeleton is calculated at a later stage when the material is chosen.

Battery capacity: The chosen battery must have sufficient capacity to power the exoskeleton for

a fixed duration. The duration of walking isn’t discussed in this project and hence the battery

capacity isn’t quantified. However, the size, mass and type of battery will play a major role in

the complete design of exoskeletons where trade-offs between the mass and capacity will need

to be performed.

Range of motion: Each joint in the exoskeleton (2 hip and 2 knee joints) possess one D.O.F

(rotation about an axis perpendicular to the plane of motion of the members). Hence the hip

joints and rotate between 240o and 315o while the knee joints can rotate between 230o and

300o. These angles have been approximated by observation of an actual human gait and

indicate the final positions of the thigh and calf over half a cycle.

Figure 2: Range of motion - Constraints

2.1(e). Sketch History

For the initial design we started with the following kinematic model. Using this design, we

found out the dimensions for the different parts of the exoskeleton depending upon the

average height of the person. Then using Solidworks we made the final design and did the final

calculations based on that model. A circular pipe shape has been selected for the links due its

advantage of lightness, since the shape has high area inertia of moment compared with the

other cross sectional shapes in all directions, meaning better strength in shear and bending

moment for the unit mass.

Initially we were considering many design parameters to satisfy the functional requirements

discussed earlier in section 2.1 (c), but due to the different optimization goals we decided on

dropping some of the design parameters. Parameters like, springs for shock absorption,

gyroscope for balance control, and different sensors were not considered in the further analysis

as they have negligible effect on the total power which is being consumed by the motors.

Design parameters which were being initially considered:

Frame

Harnesses

Primary battery

Secondary battery

Processor

Sensor

Motors

Springs

Design parameters which we decided on dropping as they have no effect on the total power of to run the motor, which we are planning to optimize:

Secondary Battery

Processor

Sensor

Springs

Design Parameters that were not considered at all:

Hydraulics

Pneumatics (These were not considered as the components are very heavy and will increase the overall mass of the exoskeleton.)

Design Parameters that can be considered in the future:

Neural controls

Climbing and jumping applications

Figure 3: Initial Kinematic Model

Table 2: Dimensions of different parts of the exoskeleton (for initial model)

Figure 4: Final design of the frame

2.1(f) Design Matrix

The following design matrix is made by considering all the Functional Requirements and Design

Parameters.

Frame Harnesses Primary battery

Secondary battery Processor Sensor Motors Springs

Support for body weight 1 1 0 0 0 0 0 1

Attachment to the body 1 1 0 0 0 0 0 0

Movement of exoskeleton 1 0 1 0 1 1 1 1

Auxiliary power 0 0 0 1 0 0 0 0

Control of actuators 0 0 1 0 1 1 1 0

Balance controls 0 0 1 0 1 1 1 0

Motion of joints 1 0 1 0 0 0 1 0

Shock absorption 0 0 0 0 0 0 0 1

The design matrix we got was highly coupled. Now as we proceeded with the analysis we

removed some of the design parameter as they were not affecting the total power which is

being consumed by the motors. Thus the final matrix we got is the following:

Frame Primary battery Motors

Support for body weight 1 0 0

Movement of exoskeleton 1 1 1

Motion of joints 1 1 1

Only these parameters are being used for the analysis as only they have an effect on power.

2.1(g). Geometry Management Strategy

Geometry management for our system is based on a “Real Physics” and ‘’Low fidelity’’ model.

This is due to fact that our functional requirements and parameterization are done on the basis

of observations as per the physical behavior of the model and not by actual experimentation.

Also, we do not have a large number of variables in our design which accounts for the low

fidelity. The dimensions of our design as well as material property indices of stiffness and

fracture toughness were used for calculating the mass of the exoskeleton frame. The maximum

and minimum values of the angular rotations were assumed as per general observations and

then a physical model was developed to minimize power.

For our design motion, actual walking patterns were observed and the range of values were

obtained for different angular rotations were assumed to create a design motion cycle.

2.1(h). Discipline Analysis Models

MATLAB: To optimize the power consumption required to operate the exoskeleton for various

input values of angular velocities and angular rotations, MATLAB is used. The motor torque

values will be used to generate the range of angles using an optimization code. The Sequential

Quadratic Programming algorithm was used to generate the optimum values for the angular

rotations for which the power would be minimized. Genetic Algorithm was also used to

optimize the power consumption so as to compare with the SQP algorithm and choose the best

result.

CES EDUPACK: Since minimization of mass is an important objective for our design, material

selection of the support structure was done using this software. Material indices for stiffness

and fracture toughness were used in penalty functions to determine the best candidate

material.

SOLIDWORKS: We used this analysis tool to parameterize the design and finalize the design

model.

The selected analysis tools are sufficient to model our design efficiently. The team had prior

experience in these tools due to past projects and research which helped in the design process.

Our assumptions are sufficient and satisfactory to optimize the power requirement and

minimize the mass of the entire system.

2.2 NUMERICAL IMPLEMENTATION PHASE

2.2(a). Discipline Analysis Models and N2 diagram:

In our case we have mainly two analysis modules, to optimize mass of the Exoskeleton through

CES Edupack and to optimize the power consumption by the unit through Multidisciplinary

Optimization.

Using CES Edupack we are selecting a material for the exoskeleton using the following material

indices. The constraints were put on the values of Density, Young’s Modulus and Fracture

toughness so that we can get the material which has minimum cost, maximum stiffness and

maximum fracture toughness.

For Optimization of Mass Using CES Edupack

Material Indices:

M1=

Minimize cost

M2=

Maximize Stiffness per unit volume cost

M3=

Maximize Fracture toughness for displacement limited design

Material constraints used:

Property Min/Max Value Units

Density Max 0.0965 lb/in^3

Young’s Modulus Min 10.3*10^6 psi

Fracture toughness Min 30 ksi in^0.5

For Optimization of Power

Objective Function Equation:

The objective function for optimizing power

f=

+ …

+ …

+ …

N2 Diagram:

Figure 5: N2 diagram

2.2(b). Block diagram of analysis modules Module 1:

CES EduPack is used to screen the best candidate material.

Solidworks is used to generate the shape of the exoskeleton.

Cost is used as a constraint.

Shape, l, b, w, Weight it has to support

Support for leg

Maximize Stiffness(E)

Maximize Fracture Toughness (K1c)

Minimize mass (m)

Module 2:

Found the optimum values of theta (1-4), t1 and t2 according to the different parameter values of

the frame.

Module 3:

2.2(c). Starting point values for design parameters

Starting point values for our design were based upon human observations and taking prior

designs into considerations. The average height and mass of a human being were assumed and

accordingly estimated. Similarly, the dimensions of the exoskeleton structure were based upon

the average dimensions of a human’s limb.

These starting point values were sufficient to determine the mass of the system and also

formulate the physical equations required to calculate the power requirement for each of the

four motors. These values helped to keep the design calculations in a feasible range as per the

functional requirements mentioned.

The starting set taken for to calculate the objective function is as follows:

x0 = [ 5.1 ; 5 ; 4.3 ; 4.1 ; 0.7 ; 1.3 ];

Power = 138.14W

l, b, w Movement

Control unit

Angles of motion(Theta 1-4 )

Time : t1, t2

Angles of motion(Theta 1-4 )

Time : t1, t2 Power Unit

Minimize total power

consumption

2.2(d). Design of Experiment

For the design of experiments, we chose to go with the full factorial approach. For our project,

we decided to take all the possible combinations of the design parameters of Θ1, Θ2, Θ3, Θ4, t1, t2

since we did not have a very large number of parameters to solve and through this method we

could also cover a bigger design space. The individual powers required for each motor were

calculated and the final total power was plotted in the form of a scatter plot.

The following code was used in MATLAB to model the full factorial experiment:

Figure 6: Design of Experiment – full factorial code

Figure 7: Design of Experiment – scatter plot

2.3 Optimization phase

2.3(a). Gradient-based algorithm

The gradient-based algorithm chosen for our optimizer was the Sequential Quadratic

Programming (SQP) algorithm. We used the FMINCON in-built function from MATLAB to find

the minimum value of the power required after formulating our objective function. This

function was used since we had a combination of inequality and equality constraints in out

design and the hessian would have taken a long time to compute if other gradient based

methods had been selected.

The code used to implement SQP in MATLAB is as follows:

Figure 8: Gradient based algorithm – main function using Fmincon

Figure 9: Gradient based algorithm – objective function

Figure 10: Gradient based algorithm – constraints function

2.3(b). Formal Optimization problem statement

Design variables ] = [ ]

The objective function ->

min [ ] =

+ …

+ …

+ …

subject to:

The six parameters which have been considered here are directly linked with the functional

requirements. The angular rotation values are responsible for the motion of the joints and the

four motors provide the torque to lift the knee and leg to provide a motion cycle.

Additional Objective:

We considered an additional objective to minimize the mass of the frame of the exoskeleton by

performing a material selection analysis to get the best candidate material for our design

model.

The following procedure was used:

Minimization of mass of exoskeleton frame to using CES EDUPACK:

Selection Criteria

Aluminum alloys are generally used when properties like density, stiffness and fracture

toughness are of importance for material selection. Aluminum 6022 T43 alloy is the most

commonly used alloy used for these property related applications. The values of these three

properties below of Aluminum 6022T43 are taken as constraint values and are given as

selection criteria in the software to filter materials to select the best candidate material.

Material Density: Less than Aluminum 6022 T43

Young’s Modulus: More than Aluminum 6022 T43

Fracture Toughness: More than or equal to Aluminum 6022 T43

Function Exoskeleton frame

Constraints Material Density Young’s Modulus Fracture toughness All dimensions are specified Corrosion resistant

Objectives Minimize mass Maximize Stiffness Minimize cost

Free variable Choice of material

Translation

Property Min/Max Value Units

Density Max 0.0965 lb/in^3

Young’s Modulus Min 10.3*10^6 psi

Fracture toughness Min 30 ksi in^0.5

Material indices used

Cost can be minimized by minimizing the overall mass of the material used.

M1=

Minimize cost

M2=

Maximize Stiffness per unit volume cost

M3=

Maximize Fracture toughness for displacement limited design

We begin screening by graphing Young’s modulus vs. Density for all materials in level 3 of the Ces Edupack database.

Density (lb/in^3)0.001 0.01 0.1

Yo

un

g's

mo

du

lus (

10

^6

psi)

1e-6

1e-5

1e-4

0.001

0.01

0.1

1

10

100

Screening out the materials that do not satisfy initial requirements for Density and Young’s Modulus of Aluminum 6022 T43 and eliminating failed part records

This graph still contains many possible materials, and hence needs to be refined further

Eliminating the materials that do not meet the thermal expansion and Fracture toughness criteria.

Density (lb/in^3)0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115

Yo

un

g's

mo

du

lus (

10

^6

psi)

2

5

10

20

50

100

Zooming in and adding guidelines corresponding to

We are left with mainly with carbon fiber composites, two ceramics and a few Aluminum alloys.

Graphing Fracture toughness vs. Young’s Modulus

Density (lb/in^3)0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.08 0.082 0.084 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1 0.102 0.1040.1060.108 0.11 0.112

Yo

un

g's

mo

du

lus (

10

^6

psi)

10

20

50

Cyanate ester/HM carbon fiber, UD composite, quasi-isotropic laminate

PEEK/IM carbon fiber, UD composite, 0° lamina

Epoxy/HS carbon fiber, UD composite, 0° lamina

BMI/HS carbon fiber, UD composite, 0° lamina

Epoxy/aramid fiber, UD composite, 0° lamina

Aluminum, A356.0, sand cast, T6

Aluminum, 443.0, sand cast, F

Aluminum, 2297, wrought, T87



Alumino silicate/Nextel 720. 45Vf - quasi-isotropic laminate

Alumino silicate/Nextel 720, 45Vf - woven fabric

Epoxy SMC (carbon fiber)

Cyanate ester/HM carbon fiber, UD composite, 0° lamina

It is found that most carbon fiber composites except Epoxy SMC and Cyanate ester/HM carbon fiber, quasi-isotropic lamina have high values of K1C/E.

Plotting cost vs. Density,

Young's modulus (10^6 psi)10 20 50

Fra

ctu

re t

ou

gh

ne

ss (

ksi.

in^

0.5

)

25

30

35

40

45

50

55

60

65

70

75





Aluminum, 8090, wrought, T851Aluminum, A356.0, sand cast, T6



Alumino silicate/Nextel 720, 45Vf - woven fabric


Cyanate ester/HM carbon fiber, UD composite, 0° lamina



Restricting cost of the material to $30/lb, we can eliminate BMI/HS carbon fiber, PEEK/IM carbon fiber, Cyanate ester/HM carbon fiber and Alumino

Silicate/Nextel 720.

Density (lb/in^3)0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14

Pri

ce

(U

SD

/lb

)

1

10

100

1000

Aluminum, 5754, wrought, O











As can be seen from the graphs, the aluminum alloys are heavier for the same mechanical

properties. However, some of them are cheaper and offer only slightly reduced property values.

Hence to determine the choice of material we consider a penalty function and consider the

following five materials:

Table 3: Material Properties

Material Price (USD/lb) Density (lb/in^3) Young’s modulus (10^6 psi)

Fracture toughness (ksi in^0.5)

Epoxy SMC (Carbon fiber)

11 0.056 16 30

Aluminum 8090 T851

6.45 0.092 11.9 31


18 0.0565 20.55 60.85


28.7 0.0499 10.15 56.9


6.5 0.0934 11.45 31.4

The table represents the properties of five of the possible materials that look the best.

Table 4: Material Index Values


Aluminum 8090 T851




M1 1.623 1.6852 0.9833 0.6983 1.6472

M2 4.091 3.847 2.693 1.512 3.713

M3 1.875 2.605 2.961 5.606 2.742

Another consideration is the resistance to corrosion due to exposure to the environment.

Epoxy/aramid fiber, UD composite, 0° lamina is fairly durable against UV radiation while the

others are much better. Since the exoskeleton could be exposed to outdoor environment,

better durability against UV radiation is preferred and hence Epoxy/aramid fiber, UD

composite, 0° lamina is not a very good choice.

The choice among these is evaluated using a penalty function which serves as a mean to

aggregate the various objectives into a single objective function, formulated such that its

minimum defines the most preferable solution.

Z= C1M1 + C2M2 + C3M3

The exchange constants are defined as follows:

C1=10: Mass and cost are the primary parameters that are being considered for the design as

the lightest material will reduce the load on the person to use the exoskeleton

C2=5: Stiffness is the next important parameter as the frame needs to maintain its shape to

ensure the person does not fall down while walking.

C3=1: The fracture toughness although important is not as important as the other parameters

as the exoskeleton not be subjected to very high stresses.

Table 5: Penalty Function values


Aluminum 8090 T851




Z values 38.56 38.692 26.259 20.149 37.779

Aluminum 8090 T851 has the highest Z values and can be regarded as a good material for the

application. However, Aluminum 8090 and Aluminum 8091 are no longer being produced and

they have been replaced by newer alloys.

The material with the next highest value is considered.

Hence, Epoxy SMC (Carbon Fiber) is the chosen material for the exoskeleton. Frame made from

this material will be the lightest and stiffest material with respect to cost considerations.

After finding the best candidate the functional requirement for the frame of the exoskeleton is

solved by minimization of the mass of the exoskeleton frame by choosing the appropriate

density of the material.

Total mass of the exoskeleton frame = 3.992 lb ~ 4lb

2.3(c). Optimizer convergence and better results:

The following results were obtained from the SQP optimizer program:

>> main_sqp

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in feasible

directions, to within the default value of the optimality tolerance and constraints are satisfied

to within the default value of the constraint tolerance.

<stopping criteria details>

x =

5.4950

5.2300

4.0200

4.1800

0.5000

0.5000

fval =

-364.4690

theta =

315.0122

300.0007

240.0006

230.0082

t =

0.5000

0.5000

Hence the minimum optimized power found is -364.469 W.

The value obtained is much better than the starting power value obtained. The power obtained from the

initial assumption of variables was lower but the constraints were violated which could not be seen as a

feasible design.

The values obtained seems to be the global minimum since we didn’t get better results for multiple

iterations with different starting points. The obtained value also satisfies all constraints and hence can

be viewed as global minimum.

2.3(d). Sensitivity Analysis

The objective function is defined as:

f=

+ …

+ …

+ …

Calculating the differential values with respect to the individual parameters:

f *= -364.4691

Normalizing,

Θ1 has the largest impact since most constraints are built on it and its normalized value is greatest which

implies that it is most sensitive to a cause a change in the power consumption of the motors.

2.3(e). Active Constraints

All the following constraints were in-active for our design:

When we relaxed any of the constraints, the objective function value converged to a very small

amount close to 0. This case is possible when both the legs of the exoskeleton are straight and

Since this condition is not practical and power consumption here is 0, we take

into consideration the unrelaxed constraint equations at which minimum power consumption

comes out to be -364.469 W.

2.3(f). Scaling in design problem

We started the scaling process by finding the hessian matrix for our objective function. The

expressions used were:

H= [dfx11, dfx12, dfx13, dfx14, dfx15, dfx16;

dfx21, dfx22, dfx23, dfx24, dfx25, dfx26;





]

Where,

x1=Θ1

x2=Θ2

x3=Θ3

x4=Θ4

x5=t1

x6=t2

dfx11, dfx22 and so on represent the double differentiation of objective function f with respect

to x1, x2 and so on.

dfx12 and similar variables represent the differentiation of objective function f with respect to

the variable x1 and then again differentiating it with respect to x2.

The following results were obtained for the Hessian matrix:

H= 1.0e+03 * [

0.1543 0 0 0 0.5482 0

0 -0.0107 0 0 0 0.1972

0 0 0.2290 0 -0.1699 0

0 0 0 0.0599 0 -0.0165

0.5482 0 -0.1699 0 -2.3461 0

0 0.1972 0 -0.0165 0 -0.5696]

On finding the Eigen values for the above matrix we got the following results:

λ1 =-247.1288598848391

λ2 = 291.5159004076417

λ3 = 216.07269844075182

λ4 = 62.58174522541338

λ5 = -632.5308318048131

λ6 = 49.549086579399635

From these values we found that the Hessian condition number = λmax/λmin = -0.461

Since this value is not very high hence our design variables were not scaled and the new

optimum function value was not required to be found.

2.3(g). Heuristic technique For our design problem, we chose to use the genetic algorithm optimizer. This was due to the

fact that Particle Swarm Optimization was difficult to implement with 6 variables. Also, from

the genetic algorithm, the iteration history along with max fitness values can be plotted easily.

Genetic Algorithm provides various options like mutation, crossover rate to be tuned for better

efficiency of result.

The following results were obtained from the Genetic Algorithm code:

Figure 11: Genetic Algorithm – Generation history plot

Figure 12: Genetic Algorithm - Average fitness plot

The optimizer produced the following results:

The minimum power required: -317.995 W

2.3(h). Comparison of results with gradient-based approach

The gradient based approach produced the following result for power consumption:

364.469 W

The value obtained from the heuristic technique is:

317.995 W

We can see the values obtained are close to each other. In this case, the heuristic approach

produced better results than the gradient based approach.

Genetic Algorithm tuned parameter set:

Table 6: Parameter Tune for GA code

Probability of crossing over

Probability of mutation Fitness

0.5 0.06 672.715 0.7 0.1 317.995

0.75 0.15 377.122 0.8 0.2 543.85

0.6 0.3 405.64

0.8 0.4 344.456

We can see that the fitness function has minimum value for the combination:

Probability of crossing over = 0.7; Probability of mutation = 0.1

2.3(i). Multiple objective Optimization formulation

For our design problem, there was only one problem to optimize i.e the power consumption.

Our mass was minimized by choosing the best density material, thereby reducing the mass as

much as possible. Optimizing the mass using a numerical technique wasn’t feasible since the list

of densities, moduli, etc. could not be imported into MATLAB.

Hence, our design model is the lightest and consumes the least power.

2.3(j). Pareto Front

Since our design model is based on a single objective function, hence the Pareto front

estimation was not found.

2.3(k). 3 View drawing

Figure 14: Side view of Exoskeleton

Figure 134: Front View of Exoskeleton

Figure 15: Top View of Exoskeleton

Appendix Calculation of exoskeleton and total mass

Density of material (Epoxy SMC/Carbon fiber) based on material optimization using CESEdupack:

ρ = 0.056 lb/in3

Radius of the members r = 10mm ~ 0.4in

Lthigh = 420mm = 16.65in

Lcalf = 480mm = 18.89in

Mass of thigh member Methigh = πr2Lthighρ

= π*0.42*16.5*0.056

= 0.464 lb

Mass of calf member Mecalf = πr2Lcalfρ

= π*0.42*18.89*0.056

= 0.532 lb

The ankle support and the base of the exoskeleton is considered to be made of steel instead of Epoxy

SMC/Carbon fiber since forming the composite into the complex shape of the foot will be difficult. The

mass of the ankle and its attachments is assumed to be 1lb per foot.

Meankle = 1lb

The total mass that the motors will need to move is calculated by adding the mass of the human leg to

the exoskeleton mass. To find the mass of the leg, we consider the following mass percentages of the

thigh, calf and foot compared to the total mass of the body:

Segment Average percentage Mass for an 80kg person

Thigh 14.47 28.65

Calf 4.57 9.05

Foot 1.33 2.65

Adding the masses of the members to the above mass,

Mthigh = 29.114 lb

Mcalf = 9.582 lb

Mankle = 3.65 lb

Motor Power calculation

Power of Motor 1 (Hip joint) P1 = Power consumed by motor 1

T1 = Torque for motor 1

ω1 = Angular velocity of motor 1

P1 = T1 * ω1

P1 =

Power of Motor 2 (Knee joint) P2 = Power consumed by motor 2



P2 = T2 * ω2

P2 =

Power of Motor 3 (Hip joint) P3 = Power consumed by motor 3



P3 = T3 * ω3

P3 =

Power of Motor 4 (Knee joint) P4 = Power consumed by motor 4



P4 = T4 * ω4

P4 =

design optimization of electromechanical exoskeleton

Documents