The design and simulation of an active force control architecture for an electric drive for an active knee prosthesis

1

Faculty of Engineering Technology

The design and simulation

of an active force control architecture for an electric drive

Cedric J. Rodriguez M.Sc. Thesis October 2020 Report number: BE-760

Supervisors:

dr. R. Pawlik

dr. M. Seyr

Research Hub

Otto Bock Healthcare Products GmbH

1110 Wien

Austria

ir. E. E. G. Hekman

dr. ir. M. Abayazid

Department of Biomechanical Engineering

Faculty of Engineering Technology

University of Twente

P.O. Box 217

7500 AE Enschede

The Netherlands

“An algorithm is a methodical set of steps that can be used to make calculations, resolve problems and reach decisions. The algorithms controlling

vending machines work through mechanical gears and electric circuits.

The algorithms controlling humans work through sensations, emotions and thoughts.”

Y.N. Harari, Organisms are algorithms, Homo Deus, 2015

Preface

This work is the final assignment in the pursuit of a double degree in the programs Industrial Design Engineering and Biomedical Engineering at the University of Twente. The Industrial Design Engi- neering program focuses on Emerging Technology Design with specialisation in Biomedical Product Development. The Biomedical Engineering program focuses on the development of (bio)robotic solu- tions such as the control of prostheses and orthoses. This work combines these two types of expertise into one master’s thesis by contributing to the development of an active knee prosthesis for one of the leading companies in the field namely Ottobock Healthcare. More specifically, this assignment contributes to the development and simulation of a novel architecture to control an active drive for an active knee prosthesis.

I firstly would like to thank Roland Pawlik for providing the opportunity for me to work at Ottobock Healthcare in Vienna. I am also thankful for his guidance throughout the whole project. I would also like to thank Martin Seyr for his dedication during the development of the simulation models and for his critical feedback on the control. I would also like to thank Dirk Seifert for all his organisational and technical support. I also would like to show my great gratitude to Wolfgang Sauberer that helped me day in and day out on the design, implementation and debugging of the control architecture. I would like to thank Christoph Zahalka for his help on acquiring the measurements and Johannes Zajc for his help with the verification and documentation for patient testing. My gratitude also goes to Gabriela Hörig that helped me with debugging the prototype at any time that this was needed. I would like to thank the rest of the students of Ottobock Healthcare for the pleasant lunch breaks and fun evening celebrations. Next to my colleagues at Ottobock, I would also like to show my appreciation to my supervisors at the University of Twente. I would like to thank Edsko Hekman for his knowledge of biomechanics and his support throughout the assignment. I would also like to thank Momen Abayazid and Bart Koopman for their feedback during the project. All in all, I am very thankful for the experiences and knowledge that I gained throughout this work. I am also happy for student goodbye drinks and the many after-work events such as the Christmas celebration and cross-country skiing trip. Vienna has been a very inspiring and motivating environment to work, laugh and learn.

I finally would like to show my gratitude to my family and friends for all their support. I would like to thank my parents and my siblings which have always been there for me. I would like thank Katya for supporting me through all the ups and downs. I am very gratefully for these opportunities and that I was able to share them with the great people around me.

v

vi Preface

Summary

There is much potential for active knee designs but the benefits strongly depend on the appropriate use and integration of actuators, sensory systems, and the control scheme. Therefore, it is interesting for Ottobock Healthcare to explore new active control strategies.

The goal of this work is to design and simulate an active force control architecture for an electric drive for an active knee prosthesis

A control architecture describes the structure of and connections between different controllers to acquire a high-level system goal. The required performance of the architecture has been defined based on the desired quantitative performance of the electric drive. The architecture should be able to support four control modes: torque, impedance, speed and position control. These control modes can be divided into three levels of control: (1) the high-level control where the user state and intent is detected, (2) the mid-level control where the state of the user is translated to the state i.e. current, of the electric drive and (3) the low-level control where the physical current in the actuator is regulated.

The structure of the low-level and mid-level controllers of the architecture have been designed and validated using simulation models of the motor, transmission and sensors. The low-level (current) control uses Field-Oriented Control which uses a feedforward method to decouple the currents in the rotor coordinate system and two closed-loop PI controllers to acquire the zero steady state error. The controllers have been tuned to have a rise time of 0.425 ms and an overshoot of 20% using the Internal Model Control method. The mid-level speed controller is a cascade of the current controller which also uses a closed-loop PI controller. The position controller is then a cascade of the speed controller which uses a simple P controller. The performance of the speed controller has been simulated to acquire a settling time of 0.2 s without surpassing the nominal current of the actuator of 3.6 A. For the position controller, both objectives could not be satisfied. To reach the set point within 2 s, a short peak current of 11 A is required using this structure. The mid-level torque control is a cascade of the current controller which uses a model of the transmission for feedforward control. This model could potentially compensate for efficiency losses and inertia effects. The measured torque step response on the test bench showed large overshoots of 62.5% and long settling times of 70 ms due to a compliant transmission. In simulation, friction compensation increases the settling times and the inertia com- pensation increases the oscillation frequency. Virtual damping could be added to the controller. The simulations suggest that virtual damping can cause a reduction of the overshoots up to 0% and reduce the settling times down to 25 ms. A cascade of the force controller is the mid-level impedance con-

vii

viii Summary

troller where the high-level controller determines the emulated stiffness and damping of the knee joint.

The performance of the control architecture using the torque and impedance controllers has been validated with the DCD prototype. The maximum peak and continuous torque that have been acquired by the prototype, is 6.0 N m in a quasi-static test case. This torque is higher than the minimum peak torque of 3.5 N m but is not up to the desired continuous torque of 10 N m. One reason for the limited output torque is the conservative current limit in the current controller of 3.3 A. The emulated stiffness of the impedance controller could be set up to 24 N m/rad which is higher than the minimum stiffness of 3.5 N m/rad but it is still much lower than 244 rad/s. The difference between the set point stiffness and the measured stiffness during quasi-static situations is not expected to be distinguishable by the user. When evaluating the impedance behaviour dynamically, the kinematic response has a spring like behaviour. The oscillation frequencies have been measured and evaluated for stiffness set points up to 16 N m/rad.

To conclude, the goal of this work is to design and simulate an active force control architecture for an electric drive. The architecture is composed out of high-level, mid-level and low-level control.

The mid-level control consists of a torque, impedance, speed and position controller that determine the desired actuator current. The low-level current controller will physically regulate the current. In simulation, the speed controller showed good performance. The controller could reach its set points within 0.2 s. The position control can fully flex the knee joint within 2 seconds yet surpasses the nominal current in simulation. The measured step response of the torque controller in the prototype shows large overshoots and settling times due to the compliant characteristics of the transmission. The impedance controller on the other hand shows proper impedance behaviour both in quasi-static as in dynamic test cases.

The future steps of this work should be focused on measuring the speed and position performance on a test bench and in the prototype to validate the simulated behaviour. Additionally, the control structure of the positions controller could be improved and evaluated through simulation. Regarding the torque controller, the feedforward model could be improved by incorporating virtual damping.

Finally, the performance of the impedance, torque, speed and position control should be tested with

users to evaluate the subjective performance of the control architecture.

Contents

Preface v

Summary vii

1 Introduction 1

1.1 Motivation . . . . 1

1.2 Goal of the assignment . . . . 2

1.3 Report organization . . . . 3

2 Technical Background 5 2.1 Objective . . . . 5

2.2 Physiological Gait . . . . 5

2.3 Control strategies for active knee designs . . . . 5

2.4 Direct Coupling Design . . . . 6

2.4.1 Electric drive . . . . 6

2.5 Operating system . . . . 8

3 Control Architecture Design 11 3.1 Objective . . . . 11

3.2 Torque & impedance control . . . . 11

3.3 Speed & position control . . . . 13

3.4 List of Requirements . . . . 13

3.5 Control Architecture . . . . 16

4 Plant & Sensor Modelling 17 4.1 Objective . . . . 17

4.2 Motor Model . . . . 17

4.3 Transmission Model . . . . 20

4.4 Sensor models . . . . 22

5 Controller Designs 25 5.1 Objective . . . . 25

5.2 Low-Level Controller . . . . 25

5.3 Mid-Level Controllers . . . . 29

ix

x Contents

5.3.1 Speed controller . . . . 29

5.3.2 Position controller . . . . 32

5.3.3 Torque controller . . . . 33

5.3.4 Impedance controller . . . . 37

5.4 Discussion . . . . 37

6 Performance Evaluation 41 6.1 Objective . . . . 41

6.2 Methods . . . . 41

6.3 Materials . . . . 42

6.4 Results . . . . 42

6.5 Discussion . . . . 44

7 Conclusions and Future Work 47 7.1 Conclusions . . . . 47

7.2 Discussion . . . . 49

7.3 Future Work . . . . 50

References 51

Appendices

List of Figures

2.1 Physiological gait cycle of human during walking illustrating the two phases (Stance Phase/Swing Phase) and eight intervals during one stride. [1] . . . . 6 2.2 A schematic of the Permanent Magnet Synchronous Machine to illustrate the key

components using a exploded view (left) and from a top view (right) . . . . 8 2.3 A schematic of the planetary gearbox to illustrate the key components using a exploded

view of the three stages . . . . 9 3.1 (a) Ideal knee joint external torques of the active electric drive during stair ascent

based on walking data of healthy subjects. Positive torques correspond to torques in flexion direction. (b) Expected angles and zero-crossings of the angular velocity of the knee joint during stair ascent. Measured by Riener et al. and Spyropoulos et al.

and adapted by the author. [2, 3] . . . . 12 3.2 (a) Ideal knee joint external torques of the active electric drive during stair ascent

(SA), level walking (LW) and sit-to-stand movements (STS) based on walking data of healthy subjects. Positive torques correspond to torques in flexion direction. (b) Expected angles of the knee joint during stair ascent (SA), level walking (LW) and sit-to-stand movements (STS). Measured by Riener et al. and Spyropoulos et al. and adapted by the author. [2, 3] . . . . 13 4.1 Spatial representation of the stator coordinate system (A, B, C-axis) and rotor coordi-

nate system (d, q-axis) of a PMSM with one pole pair . . . . 18 4.2 (a) Simulated average currents in the d- and q-direction (sim. ¯ i

and ¯ i

) including the

uncertainty of the simulation (σ) compared to the measured average output currents in the d- and q-direction (meas. ¯ i

and ¯ i

) including the uncertainty of the measurements.

(b) Simulated average motor output torques (sim. ¯ τ

) of the actuator including the uncertainty of the simulation (σ) compared to the measured average output torques (meas. ¯ τ

) including the uncertainty of the measurements . . . . 20 4.3 Measured output torques of the transmission at different current set points (left) and

output angular velocities and model output torque of the transmission at different current set points and output angular velocities (right) . . . . 22 5.1 Closed-loop control scheme for the current controller where R

(z) is a discrete two

dimensional input reference current and I

(s) is the actual current in the d- and q-direction. The plant H

(s) represents the electrical dynamics in actuator. . . . 26

xi

xii LIST OF FIGURES

5.2 Block scheme representation of the discrete regulator R

and the decoupling W where the input is the discrete current error E

(z) and the sampled current I

(z). The output is the discrete voltage signal U

(z). . . . 27 5.3 (a) Thirty current step responses at zero angular velocity (b) Comparison between the

simulated current response and the measured current response at zero angular velocity 29 5.4 (a) Simulation results on the effect of the model, offset and scaling errors on the

performance of the current control in dynamic test cases and (b) Simulation and measurement results on the performance of the current control with a constant rotor angular velocity of 105 rad/s . . . . 30 5.5 Closed-loop control scheme for the speed controller cascade where R

(z) is a

discrete one dimensional input reference speed and ω

(s) is the continuous output speed. The plant H

(s) represents the mechanical dynamics of the drives. . . . 30 5.6 Block scheme representation of the discrete regulator R

where the input is the discrete

angular velocity error E

(z). The discrete current output R

(z) is determined by a PI controller with a limitor and an antiwindup scheme. . . . 31 5.7 Simulation results of the speed controller describing (a) the knee angular velocity

response and (b) the current draw of the motor during a step change from 0 to 1.13 rad/s for different proportional and integral controller gains, respectively. . . . 32 5.8 Simulation results of the position controller describing (a) the knee angle response

and (b) the current draw of the motor during a step change from 0 to 2.3 rad for different proportional controller gains, . . . . 33 5.9 Various friction compensation methods during the zero-crossing of the angular veloc-

ity with a driving and breaking efficiency of 0.88 . . . . 35 5.10 (a) Measured and simulated torque overshoots and low frequency oscillations during

a torque step change due to the compliant plastic coupling and transmission (b) Simulated angular velocity of the rotor due to the compliant behaviour of the plastic coupling and the gearbox . . . . 36 5.11 (a) Simulated torque compensation using different friction compensation profiles in

the torque controller during torque step responses (b) Simulated compensated input torque overshoot and low frequency oscillations during torque step responses due to the compliant gearbox . . . . 37 5.12 (a) Simulated torque step responses when compensating different percentages of the

rotor inertia in the torque controller and (b) simulated torque step responses when using different amounts of virtual damping in the torque controller to reduce overshoot and settling times . . . . 38 6.1 Performance of the torque control during a torque step change including crossing the

play using (a) the torque controller and (b) the impedance controller . . . . 43 6.2 Quasi static test case results evaluating the impedance performance for stiffness set

points of 4, 8, 12, 16, 20 and 24 N m/rad and an equilibrium angle of 0 rad. . . . . 43 6.3 Normalised knee joint kinematic response for stiffness set points of (a) 4 N m/rad,

(b) 8 N m/rad, (c) 12 N m/rad and (d) 16 N m/rad . . . . 45

List of Tables

2.1 Magnitude of the parameters of the Permanent Magnet Synchronous Machine used in the DCD including the expected standard deviation [4] . . . . 7 3.1 Active regions during the gait cycle during level walking, stair ascent and sit-to-stand

movements . . . . 12 3.2 List of Requirements for the control of the electric drive . . . . 15 6.1 Set and measured stiffness in N m/rad during the quasi static test case of the

impedance controller where θ

and θ

are two knee angles on the impedance line in rad and τ

and τ

are the torques at those angles in N m . . . . 44

xiii

xiv LIST OF TABLES

List of Symbols

U

Input voltage vector in the stator-coordinate system U

Input voltage vector in the rotor-coordinate system

U ¯

Augmented input voltage vector in the rotor-coordinate system u

Input voltage scalar (d-component)

u

Input voltage scalar (q-component)

I

Phase current vector in the stator-coordinate system I

Rotor current vector in the rotor-coordinate system

i

Current scalar (d-component)

i

Current voltage scalar (q-component) R

Phase-phase resistance

R

Phase resistance of a motor L

Phase-phase inductance

L

Phase inductance

L

Phase inductance scalar (d-component) L

Phase inductance scalar (q-component)

Ψ

Stator flux vector in the stator-coordinate system Ψ

Stator flux vector in the rotor-coordinate system Ψ

Stator flux scalar (d-component)

Ψ

Stator flux scalar (q-component) Ψ

Permanent magnet flux scalar

p Number of pole pairs of a motor ϑ

Rotor electrical angle

ϑ

Rotor mechanical angle ϑ

Knee angle

ϑ

Equilibrium angle

xv

xvi LIST OF TABLES

ω

Rotor electrical angular velocity ω

Rotor mechanical angular velocity

ω

Knee angular velocity

µ Angular velocity dependent friction coefficient for motor loss µ

Load dependent friction coefficient when driving

µ

Load dependent friction coefficient when yielding µ

Load dependent friction coefficient when yielding τ

Motor electrical torque

τ

Motor output torque τ

Knee torque

τ

Knee reference torque

i

Transmission ratio between the rotor and stage 1

i

Transmission ratio between the gearbox stage 1 and stage 2 i

Transmission ratio between the gearbox stage 2 and stage 3 J

Inertia of rotor with respect to ω

J

Inertia of one stage of the gearbox with respect to ω

J

Inertia of the knee head with respect to ω

J

Inertia of knee shank with respect to ω

J

Inertia on the left side of the gearbox with respect to ω

J

Inertia on the right side of the gearbox with respect to ω

A Linear state matrix of the current in rotor coordinate system B Linear input matrix of the current in rotor coordinate system W

Coupling matrix of currents in rotor coordinate system

W ˆ

Estimated coupling matrix of currents in rotor coordinate system W

Back EMF matrix in rotor coordinate system

W ˆ

Estimated back EMF matrix in rotor coordinate system

K

Proportional gain matrix

K

Integral gain matrix

K

Anti-windup gain matrix

List of Acronyms

DCD Direct Coupling Design

PMSM Permanent Magnet Synchronous Machines FOC Field-Oriented Control

SVPWM Space Vector Pulse Width Modulation IMC Internal Model Control

ADC Analog-to-digital converter PWM Pulse-width modulation

xvii

xviii LIST OF TABLES

Chapter 1

Introduction

This chapter contains the motivation for and relevance of this work. It also describes the context of the project and finally the research goals.

1.1 Motivation

Passively supporting prosthetic devices for locomotion have been around for centuries. [5] Two decades ago, the first microprocessor-controlled knee prosthesis (MPK) introduced a new generation of pas- sive assistance. [6] Some studies showed that the users of MPKs producing variable damping i.e.

variable breaking torques, experience biomechanical improvements during their locomotion such as an enhanced smoothness of gait and a decrease in hip work production. [7–9] Despite the suggested benefits, the research has been inconclusive about the general advantages of MPKs with respect to non-microprocessor-controlled knees (NMPKs). [10]

For both NMPKs and MPKs, there are still common deficiencies when comparing the passive gait of prosthetic users with the gait of healthy subjects. Both users perform compensatory movements during day-to-day activities that result in asymmetric loading between both limbs. On the long term, this can lead to secondary complications such as back pain. [11] One reason for these compensatory movements, is the lack of active assistance during their locomotion. This establishes the opportunity to move from passive prosthetic devices to more capable active designs that also produce driving torques.

Active behaviour of the knee prosthesis could impel equal loading which in turn means a reduction of these compensatory movements. [12] It is also suggested that an active knee could furthermore result in a lower energy consumption of the users during activities such as stair ascent. [13] Although there is much potential, general claims on the benefits of active knees still require more research. [14]

One reason for this is that the benefits strongly depend on the appropriate use and integration of actuators, sensory systems, and the control scheme. [15] Therefore, it is interesting to explore new design concepts and develop different active control strategies.

1

2 Chapter 1. Introduction

1.2 Goal of the assignment

The Direct Coupling Design (DCD) includes a newly developed electric drive which consists of a Permanent Magnet Synchronous Machines (PMSM) connected to a three stage planetary gearbox.

This drive will control the driving torque, speed and position of the knee to acquire the active be- haviour of the knee joint.

The goal of this work is to design and simulate an active force control architecture for an electric drive.

To accomplish this goal, the architecture will be decomposed into separate controllers. The structure of these underlying controllers of the force control architecture will be determined based on the required quantitative performance. Secondly, a model will be created of the drives to be able to design and simulate different test cases using the underlying controllers. Subsequently, the parameters of the underlying controllers of the architecture will be determined and the response will be simulated.

Fourthly, the performance of the control architecture will be validated within a prototype. These goals can be split up into sub-goals as follows:

1. Design an active force control architecture for the electric drive (a) Determine the desired driving torques during human gait

(b) Determine the desired positions and angular velocities in sitting situations (c) Construct the list of control requirements for the electrical drive

(d) Determine the underlying controllers, environments and interfaces to create the control architecture

2. Develop a simulation model of the electric to test the low-level and mid-level controllers (a) Develop a submodel of the electric motor and validate the predictive performance of the

submodel

(b) Develop a submodel of the transmission and validate the predictive performance of the submodel

(c) Develop a submodel of the sensors

3. Design the low-level and mid-level controllers of the architecture and test the controllers on the simulation models

(a) Design the current controller and validate performance of the control (b) Design the speed controller and validate performance of the control (c) Design the position controller and validate performance of the control (d) Design the torque controller and validate performance of the control (e) Design the impedance controller and validate performance of the control

4. Evaluate the performance of the speed, position, torque and impedance using the DCD prototype

1.3. Report organization 3

1.3 Report organization

The remainder of this report is organized as follows. Chapter 2 will present the background knowledge that is required for reading and understanding the topics of this work. In Chapter 3, the required per- formance of the DCD will be specified and the modules of the control architecture will be constructed.

Subsequently, Chapter 4 will discuss the mathematical equations that form the models of the drives.

Chapter 5 will discuss the design of the low-level and mid-level controller designs using the plant

model. Chapter 6 will evaluate the control architecture within the prototype of the DCD. Finally, in

Chapter 7, conclusions and recommended future work will be given.

4 Chapter 1. Introduction

Chapter 2

Technical Background

2.1 Objective

Before jumping into the design of the control architecture, this chapter will present the background knowledge that is required for reading and understanding the topics of this work. It shines light on the physiological gait of healthy humans and a method to describe it. It also summarizes the previous work that has been conducted on control strategies to support active control. Finally, the electromechanical components of the Direct Coupling Design are listed that have been determined prior to this work.

2.2 Physiological Gait

During each stride of locomotion, the alignment between the body and the supporting legs are chang- ing in a repetitive and recognisable pattern of motion. These patterns of motion depend on the activity (also called modes) that are being executed such as level walking or stair descent. There are several modes that could benefit from additional active support for example: level walking, stair ascent, slope ascent and sit-to-stand. [2] All these modes require an application of positive power due to the increase in potential energy of the body.

Level walking, stair and slope ascent can be divided into two phases and eight intervals, see Figure 2.1. There are a wide variety of ways to define the phases of the sit-to-stand mode. For this work, the sit-to-stand mode will be divided into four phases. In Chapter 3, the active and passive section of the gait cycle will be highlighted.

2.3 Control strategies for active knee designs

To support the active forces during the gait cycle, various control strategies have been developed. Fluit et all have provided an overview of the current available micro-controlled prosthetic knee designs both in the commercial and research context. [16] This paper identified one commercial active knee prosthesis and five active prosthetic designs within the research context. The mechatronic design of the active prostheses vary from using Series Elastic Actuators (e.g. Ossur Power Knee, CSEA, CYBERLEGs-BETA) to using ball-screws (e.g. Myoelectric prosthesis) or belt-chain transmissions

5

6 Chapter 2. Technical Background

Figure 2.1: Physiological gait cycle of human during walking illustrating the two phases (Stance Phase/Swing Phase) and eight intervals during one stride. [1]

(SCSA). [16] A recent publication presents an additional design (Utah Knee) that is developed in the research context. [17] This fully powered active knee utilises an active variable transmission to increase the operational torque and speed range of a DC Motor. Additionally, engineers at Reboocon are developing a new active knee that is expected to be released on the market soon. [18]

In many of the powered knee prosthesis control strategies, a torque control strategy is utilised.

The torque control strategy ensures the active assistance during the gait cycle without constraining the knee to a pre-defined kinematic trajectory. Impedance control is a common implementation of the torque control because it ensures that the knee joint produces torque that is suitable for each gait phase. [15] Impedance control is either achieved using a physical spring [19, 20] or by emulating a spring using controllers. [21–24]

To acquire impedance control, the control architecture of active low limb prostheses and orthoses can generally be divided hierarchically intro three layers: high-level, mid-level and low-level con- trol. [25] The high-level is focused on determining the user state and intent. The user state could be the mode of locomotion and the phase of this mode, see Figure 2.1.The intention is sent to the mid-level controller to convert the user intention into a device state. This could be the velocity of the knee joint or the current of the actuator. Finally, the low-level control is the execution layer where the controller focuses on minimizing the error between the desired state of the device and the actual state.

This is important for controller tracking and rejecting disturbances. This layered approach results in an interpretable cascade of controllers where software modules can flexibly be reused or adjusted.

2.4 Direct Coupling Design

2.4.1 Electric drive

The electric motor have been chosen to be a PMSM due to the high power-to-weight ratio. Addi-

tionally, this PMSM has a relatively low rotor inertia. A low rotor inertia reduces the back driving

torques in passive sections of the gait cycle. Additionally, an absolute encoder is added to the

motor to measure the angle of the rotor. This motor is then connected to a three stage planetary gear-

2.4. Direct Coupling Design 7

Parameter Symbol Value σ Unit

Phase resistance R

0.300 0.015 W

Phase inductance in d L

198 · 10

19.8 · 10

H Phase inductance in q L

198 · 10

19.8 · 10

H Permanent magnet flux Ψ

2.57 · 10

6.43 · 10

V s/rad

Pole pairs p 7 - −

Rotor inertia w.r.t. ω

J

69 · 10

- kgm

Table 2.1: Magnitude of the parameters of the Permanent Magnet Synchronous Machine used in the DCD including the expected standard deviation [4]

box. A planetary gearbox is chosen due to the high efficiency, large gear ratio and high backdrivability.

Permanent Magnet Synchronous Machine

Synchronous Machines are a family of electric motors that require an alternating current to rotate the motor shaft. Alternating currents create a magnetic field due to the electromagnets in the stator. The rotor then aligns to the magnetic field because of the reluctance of the rotor, hence called a Reluctance Motor. Another possibility and considered in this thesis are rotors that align with the magnetic field of the stator due to a permanent magnet integrated onto the rotor. As the name suggests, the PMSM is of this class and is illustrated in Figure 2.2.

The PMSM of interest is rated at 100 W with a maximum efficiency of 88 % and is controlled with a nominal voltage of 18 V . It furthermore contains twelve stator slots and seven rotor poles pairs which are surface mounted onto the rotor. Based on the manufacture’s datasheet, it has a phase-phase resistance of 0.46 Ω. When measuring the resistance using a LCR meter, the phase-phase resistance was 0.600 W. The resistance is assumed to have a normal distribution with a standard deviation of 5%, see Table 2.1. The phase inductance is assumed to have a normal distribution with a standard deviation of 10%. The motor has an effective torque constant of 29.8 mN m/A and is scaled to the magnetic flux using

. The standard deviation is expected to be 2.5% based on the measured magnitude of the back EMF voltage. The rotor inertia of the motor is 69 gcm

. [4]

Angular hall encoder

The encoder consists of four hall sensors that use one pole magnet to determine the absolute position of the rotor. The encoder has a resolution of 12 bit and can be used as an absolute and incremental encoder. The sampling rate of this encoder is determined by the angular velocity of the rotor. [26]

Current sensors

Additionally to the angular hall encoder, three currents sensors are available to measure the currents

at each phase of the motor. These sensors consist of a linear Hall sensor circuit that measure the

strength of the magnetic field when current is flowing through a copper conduction path. The sensor is

combined with an Analog-to-digital converter (ADC) that has a resolution of 10 bit and a measurement

range of ±20 A. [27]

8 Chapter 2. Technical Background

Hall sensor Housing

Hall sensor magnetRotor with shaft

Permanent magnet Air gap

Stator

Copper windingsFront flange Ball bearing

Motor phases

Phase C

Phase A

Phase B

Figure 2.2: A schematic of the Permanent Magnet Synchronous Machine to illustrate the key com- ponents using a exploded view (left) and from a top view (right)

Planetary gearbox

Each planetary stage consists of an internal gear, several planetary gears, a carrier and a sun gear.

For this prototype, a high gear ratio is required which could not be acquired with one stage. Hence, multiple stages are coupled together to form the gearbox. More specifically, the chosen gearbox consists of three stages depicted in Figure 2.3. The gear ratio of the first, second and third stage is 4, 4 and 5, respectively. The final gear ratio of 80 is a multiplication of ratios of all three stages.

The satellite gears are connected using needle bearings while the sun gear is rigidly connected to the carrier. The maximum continuous output velocity of the gearbox is 337 °/s and can output a maximum continuous torque of 20 N m. Furthermore, the maximum intermittent output torque at a duty cycle of 20% is 32 N m. The maximum efficiency is documented as 90%. Finally, the backlash (play) is equal or smaller than 0.5 °. [28]

2.5 Operating system

Motor Control Platform

Finally, the Motor Control Platform is the newly introduced environment. This environment consist

of a fast processor that can update up to 20 kHz. This platform is able to control the bridges for the

electric motor. An additional battery for the electric drive is directly connected to this platform.

2.5. Operating system 9

Stage 1 (4:1) Stage 2 (4:1) Stage 3 (5:1)

Housing Internal gear

Output shaft Sun gear Satellite gear shaft Satellite gear

Sintered bearing Needle bearing Planet carrier

Figure 2.3: A schematic of the planetary gearbox to illustrate the key components using a exploded

view of the three stages

10 Chapter 2. Technical Background

Chapter 3

Control Architecture Design

3.1 Objective

A control architecture describes the structure of and connections between different controllers to acquire a high-level system goal. This chapter will present the main design considerations for this control architecture. In the first section, the driving torques and impedance behaviour of human gait will be analysed. Thereafter, the desired speeds and positions during sitting situations using a knee prosthesis will be discussed. These analyses will be used to define the quantitative requirements for the control of the electric drive. Using these requirements, the controllers and their interfaces in the control architecture will be determined.

3.2 Torque & impedance control

The desired injection or dissipation of energy in the knee joint is determined by the concentric or eccentric power, respectively. Using typical knee angle data of healthy subjects, the expected zero- crossings of the knee angular velocity could be determined during for example stair ascent, see Figure 3.1(b). Using these zero-crossings, the active and passive torque regions can be determined, see Fig- ure 3.1(a). It is beneficial for the energy consumption that the electric drive solely generates torques during these active regions. Let us take for example the active region 3% to 42% of the cycle time during stair ascent. During this interval, the active electric drive should be tracking a bell shaped torque signal in the extension direction. At the beginning of the second active region of stair ascent (51%), of the electric drive can be seen. This requires the active drive to apply a step change of 0.20 N m/kg. The end of the active region will turn off at zero torque. The expected duration of the both active regions have been calculated using the average duration of the stair ascent cycle time of 1.41 s which results in a time of 550 and 338 ms. [2] The results of this analysis can be seen in Table 3.1.

11

12 Chapter 3. Control Architecture Design

0 10 20 30 40 50 60 70 80 90 100 Cycle time [%]

-1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.4 0.2 0

-1.6

Normalised knee torque [Nm/kg]

(a)

0 10 20 30 40 50 60 70 80 90 100 Cycle time [%]

20 0 40 60 80 100 120

-20

Knee angle [degrees]

(b)

Knee angle Zero-crossing Active torque Passive torque Active region

Figure 3.1: (a) Ideal knee joint external torques of the active electric drive during stair ascent based on walking data of healthy subjects. Positive torques correspond to torques in flexion direction. (b) Expected angles and zero-crossings of the angular velocity of the knee joint during stair ascent. Measured by Riener et al. and Spyropoulos et al. and adapted by the author. [2, 3]

Mode Active regions of the gait cycle Start End Duration Peak torque

Level walking Mid Stance 15% 28% 144 ms −0.48 N m/kg

Terminal Stance 41% 49% 89 ms 0.22 N m/kg

Initial Swing and Mid Swing 73% 76% 33 ms −0.04 N m/kg Terminal Swing and Initial Contact 97% 6% 100 ms 0.38 N m/kg Stair ascent Mid Stance and Terminal Stance 3% 42% 550 ms −1.08 N m/kg

Pre-Swing and Mid Swing 51% 75% 338 ms 0.22 N m/kg Sit-to-Stand Moment Transfer to Extension 23% 55% 810 ms −1.14 N m/kg Table 3.1: Active regions during the gait cycle during level walking, stair ascent and sit-to-stand

movements

This stair ascent analysis has also been conducted for level walking and sit-to-stand movements.

The resulting active and passive torques can be seen in Figure 3.2 and are summarised in Table 3.1.

The duration of the active regions are estimated using the average cycle time of 1.11 and 2.53 s for level walking and sit-to-stand movements. [2, 3]

These analyses show the interesting regions where the electric drive can potentially assist the user with active torque assistance during stair ascent, level walking and sit-to-stand movements. It shows that the shortest assistance has a duration of 33 ms and the longest has a duration of 810 ms. It also determines the peak torques of each region with a maximum of 0.48 N m/kg during level walking, 1.08 N m/kg during stair ascent and 1.14 N m/kg during sit-to-stand.

Impedance control

The knee torques are not independent of the knee angles but show a spring like behaviour in different

3.3. Speed & position control 13

0 10 20 30 40 50 60 70 80 90 100 Cycle time [%]

-1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.4 0.2 0

-1.6

Normalised knee torque [Nm/kg]

(a)

0 10 20 30 40 50 60 70 80 90 100 Cycle time [%]

20 0 40 60 80 100 120

-20

Knee angle [degrees]

(b)

Active torque^STS Passive torque^STS Passive torque^LW Active torque^LW Passive torque^SA Active torque^SA

Stair ascent Level walking Sit-to-stand

Figure 3.2: (a) Ideal knee joint external torques of the active electric drive during stair ascent (SA), level walking (LW) and sit-to-stand movements (STS) based on walking data of healthy subjects. Positive torques correspond to torques in flexion direction. (b) Expected angles of the knee joint during stair ascent (SA), level walking (LW) and sit-to-stand movements (STS). Measured by Riener et al. and Spyropoulos et al. and adapted by the author. [2, 3]

phases of the gait cycle. During Mid Stance and Terminal Stance of level walking for example, a constant stiffness of 244 N m/rad and 3.33 N m/rad can be seen. [19] Therefore, impedance control is a common method for deploying force control as described by Section 2.3.

3.3 Speed & position control

It would be reasonable to extend or flex within several seconds up to desired position. It is therefore desired that the knee should be able to go from a fully flexed position of 130° to a fully extended position of 0° in 2 s and vice versa. This means that the speed controller should be able to control the speed of the knee joint up to 65 °/s which is 1.13 rad/s.

3.4 List of Requirements

A List of Requirements is a list of quantitative criteria which characterises the desired control per- formance of the electric drive, see Table 3.2. The list is intended to help design and evaluate the quantitative performance of the architecture and underlying controllers in Chapter 5 and 6. This does not mean that the subjective performance is less important than the quantitative performance. In contrary, the subjective feel of the prosthesis is central to the design of the control architecture of the electric drive. Yet, to assess the subjective performance, a baseline architecture should be designed that thereafter can be tested with users.

The control architecture should provide active assistance such as torque and impedance control

(Requirement 1.3 and Requirement 1.4) and should have speed/position control capabilities (Require-

14 Chapter 3. Control Architecture Design

ment 1.5 and Requirement 1.6), see Section 1.2.

The lowest peak torque of 0.04 N m/kg is during Initial Swing when level walking. This means that the minimum peak torque during torque control is 3.2 N m (Requirement 2.1) when assuming a body mass of 80 kg. Based on Section 3.2, it is desired that the torques tracking of the electric drive reaches up to 1.14 N m/kg body mass. Yet, the maximum peak output torque of the gearbox is 32 N m. Therefore, the maximum required peak torque output of the control architecture is limited to the plant design and is 32 N m (Requirement 2.2). This peak torque is enough for full active assistance of users up to 80 kg body mass during level walking and partial active support during sit-to-stand and stair ascent. Also, the maximum continuous torque is limited by the gearbox with a torque of 10 N m (Requirement 2.3). The performance of the torque controller will be evaluated by assessing the torque step response curve. This is due to the various step changes seen in Section 3.2. The overshoot of 40%

of the torque step response curve is expected to be safe and comfortable for the user (Requirement 2.4). The accuracy of the steady state torque is desired to be 10% of the set point (Requirement 2.5).

The maximum 2% settling time is determine based on the shortest duration of the activation in Section 3.2 of 33 ms (Requirement 2.6). The minimum and maximum desired knee stiffness is 3.33 N m/rad and 244 N m/rad based on the stance phase stiffness during level walking (Requirement 3.1 and 3.2).

Finally, the speed and position requirements (Requirements 4.1 to 4.6) are set based on Section 3.3.

The maximum current draw of the motor is set to the nominal current (Requirement 4.7). During

speed and position control, the motor is operated for multiple seconds. This could result in the motor

heating up if it is operated above the nominal current of 3.6 A.

3.4. List of Requirements 15

Nr Requirements Limit Value Unit

1 General

1.3 The electric drive should have torque control capabilities set - - 1.4 The electric drive should have impedance control capabilities set - - 1.5 The electric drive should have speed control capabilities set - - 1.6 The electric drive should have position control capabilities set - -

2 Torque control

2.1 Output peak torque on knee joint rotating at 0 rad/s min 3.2 N m 2.2 Output peak torque on knee joint rotating at 0 rad/s max 32 N m 2.3 Output continuous torque on knee joint rotating up to 2π rad/s max 10 N m

2.4 Overshoot of knee torque step response max 40 %

2.5 Steady state error of knee torque step response max 10 %

2.6 Settling time of knee torque step response max 33 ms

3 Impedance control

3.1 Virtual stiffness of the knee joint min 3.33 N m/rad

3.2 Virtual stiffness of the knee joint max 244 N m/rad

4 Speed/Position control

4.1 Output continuous angular velocity in both directions min 1.13 rad/s

4.2 Steady state error of the knee angular velocity max 10 %

4.3 Overshoot of the knee angular velocity max 0 %

4.4 Output knee joint angle (extension) min 0 °

4.5 Output knee joint angle (flexion) max 130 °

4.6 Time to fully extend knee joint from fully flexed position max 2 s

4.7 Motor current draw max 3.6 A

Table 3.2: List of Requirements for the control of the electric drive

16 Chapter 3. Control Architecture Design

3.5 Control Architecture

A control architecture describes the structure of and connections between different controllers to acquire a high-level system goal. These controllers should be mapped onto the digital environments described in Section 2.5.

To define the controllers, the goals of the control of the electrical drive should be specified. The control of the electric drive can be split up into three different goals: (1) Determine the desired behaviour of the prosthesis such as the torque, impedance, speed or position control based on the user state and intent, (2) transform the desired prosthesis state to the desired state of the electric drive and (3) physically achieve the desired magnitude of the electric drive and suppress disturbances. The three goals can be mapped onto the environments of the operating system of the prosthesis depending on how fast the dynamics are expected to be.

The second goal is the mechanical relation between desired knee state and desired motor state. The mechanical dynamics within the motor are faster and should therefore be placed on the Motor Control Platform running around 1 kHz. The components responsible for this relation is referred to as the Mid-Level Controller. This will be for example the relation between the knee joint angular velocity and the motor current when using the speed controller or the relation between the knee stiffness and the motor current when using the impedance controller.

The final controller will regulate the current in the motor to get the desired motor torque, position

or velocities. This will require a faster update frequency and will also be placed in the Motor Control

Platform. This is the Low-Level Controller and it will run at frequencies of around 10 kHz or higher.

Chapter 4

Plant & Sensor Modelling

4.1 Objective

The objective of this chapter is to develop simulation models of the electrical drive i.e. the plant, that could be used to design and also validate the low-level and mid-level controllers of the control archi- tecture through simulation. Developing and tuning the controllers through models and simulations provide several benefits compared to designing it in a prototype namely: signal accessibility, flexibility and scalability. The signal accessibility describing the dynamics in simulation can be saved without limitations of sensors, update rates and data transfers. This is especially important when designing the low-level controller which updates tens of thousands of times per second. Secondly, using models and simulations provide the flexibility to quickly test different structures, controllers and components.

Adjustments to the controllers or architecture can be done easily when new information is acquired.

Finally, the architecture is expected to be utilised in future variants of the DCD and also in different projects. Therefore, the models can and have to be applicable to other prototypes.

There are two ways to model the dynamics of the plant namely numerical and analytical. [29] A numerical approach uses FEM models to estimate the internal dynamics of the system of interest based on a detailed spatial model. This limits us to components where this detailed information is available and reduces the scalability. It is therefore more interesting to use the analytic approach which requires far less parameters. This makes it more practical to evaluate different system characteristics.

The plant model can be divided into two submodels: the motor model (Section 4.2) and the transmission model (Section 4.3). Additionally, a model of the current sensors, hall sensor and angular velocity sensors will also made (Section 4.4).

4.2 Motor Model

The motor model should describe the dynamic relation between the input voltages on the three phases and the output motor current and motor torque. These motor dynamics can be separated into three sections namely: voltage-current dynamics, current-torque relation and the torque losses. The analytical method for modelling the voltage-current dynamics of an electric three phase motor is based

17

18 Chapter 4. Plant & Sensor Modelling

A B

C

d q β

α Rotor θ

Permanent magnet Stator Copper windings

Motor phase

Figure 4.1: Spatial representation of the stator coordinate system (A, B, C-axis) and rotor coordinate system (d, q-axis) of a PMSM with one pole pair

on the ideal voltage equation

U

= R

I

+ d

dt Ψ

(4.1)

where U

is the input voltage vector in the stator coordinate system denoted by superscript S. [30]

Furthermore, R

represent the phase resistance, I

the phase current and Ψ

the stator flux. The voltages, currents and flux can be depicted spatially in the stator coordinate system along the A, B and C-axis which represent the three phases on the PMSM, see Figure 4.1.

The voltage equation can be transformed into a local coordinate system that is fixed to the rotor using the Clark-Park transformation. [31] The rotor coordinate system is beneficial because the torque forming and non-torque forming current components are directly mapped to the q and d-axis, respectively. The d-component is always parallel with the magnetic flux of the permanent magnet and the q-component always perpendicular to it. Additionally, a commonly used flux model is added to Equation 4.1 where the permanent magnet flux is mapped to the d-component and includes the current dependent flux. [32, 33] Rewriting this all to a current model results in

d dt I

=





−

d

0

0 −

q



 I

+





0 ω

Ld

−ω

q

0



 I

+





1 Ld

0

0

q



 U

+



 0

−ω

q

Ψ



 (4.2)

where the first term describes an uncoupled second-order system, the second term denotes the angular velocity dependent coupling between the two current components, the third term includes the voltage inputs and the final term describes the current generated by the velocity dependent back EMF voltage. The model consist of three input signals

h

u

u

ω

ⁱ

. Using Equation 4.2, the current to torque relation can be determined which results in

τ

= 3

2 p(i

Ψ

+ (L

− L

)i

i

)) (4.3)

4.2. Motor Model 19

where τ

is the electrical torque produced by the motor. [32] The output torque of the motor can now be calculated by incorporating the mechanical losses inside the motor. This could be the friction in the bearings or the air resistance around the rotor. These losses are angular velocity dependent [34]

and will be thus modelled as a viscous loss

τ

= τ

− µω

(4.4)

where µ is the friction coefficient, τ

is the final output torque of the motor. The static friction, Stribeck effect and Coulomb effect have been ignored due to the relative low amount of static cases of the motor. [35]

The voltage-current dynamic model in Equation 4.2 and the current-torque relation in Equation 4.3 and 4.4 together form the model of the motor. These equations have been implemented a simulation application called Simulink. [36, 37]

Model validation

To ensure that this motor model predicts realistic motor currents and torques, the predictions of the model have been compared to measurements on a test bench. The steady state voltage-current relation and the current-torque relation have been evaluated separately. For the voltage-current relations, −0.20 and 3.33 V is applied in the d- and q-direction on a test bench. By adding a braking load, a constant rotor speed of 167.55 rad/s was acquired. Subsequently, the currents in the d- and q-directions have been measured.

To get an estimation of the uncertainty of the model, thirty simulation runs have been executed in which the motor resistance, inductance, and torque constant were altered based on the expected value, the uncertainty and the uncertainty distribution of the motor parameters listed in Table 2.1. The standard deviation of the currents across these thirty runs will then give the range where the measured steady state current is expected to be based on the uncertainty of the parameters of the model. The average difference in the simulated and measured current is 10 mA in the q-direction and 20 mA in the d-direction, see Figure 4.2(a). This suggests that there is a good match between the voltage-current relation of the motor in simulation and on the test bench. Also, the standard deviation of both the currents in the d-direction and the q-direction is smaller than 100 mA.

Next to the voltage-current relation, also the current-torque relation in the model has been validated.

This is done by controlling the current inside the motor to a fixed value and measuring the motor

output torque. This is again compared with the predicted output torques of the model. The model

uncertainty is generated by using different parameters based on the uncertainty distribution of the

torque constant and the friction coefficient. Figure 4.2(b) depicts the simulated and measured output

torques for rotor angular velocities between 100 and 325 rad/s. The results show that the largest

average difference is 5 mN m. This is at slower angular velocities. This could be due to the Stribeck or

Coulomb friction effects that have been neglected in the model. The larger variance of the measured

20 Chapter 4. Plant & Sensor Modelling

0 0.01 0.02 0.03 0.04 0.05 0.06

Time [s]

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

Current [A]

Sim.

Meas.

Sim.

Meas.

i_d± σ i_d± σ i_q± σ

i_q± σ

(a) (b)

100 150 200 250 300 350

Angular velocity [rad/s]

0.01 0.02 0.03 0.04 0.05 0.06 0.07

Torque [Nm]

Sim. τ_m ± σ Meas. τ_m ± σ

Figure 4.2: (a) Simulated average currents in the d- and q-direction (sim. ¯i

and ¯ i

) including the uncertainty of the simulation (σ) compared to the measured average output currents in the d- and q-direction (meas. ¯ i

and ¯ i

) including the uncertainty of the measurements.

(b) Simulated average motor output torques (sim. τ ¯

) of the actuator including the uncertainty of the simulation (σ) compared to the measured average output torques (meas. ¯ τ

) including the uncertainty of the measurements

torque losses comparison to the model uncertainty is expected to come from the high noise levels in the measured torque signal.

4.3 Transmission Model

The transmission model is a dynamic model of two rigid bodies that uses the Lagrange equations to calculate the kinematics of the bodies based on the motor torque on the fast moving side and the knee joint load on the slow moving side. A transmission with play has been chosen because play causes fast changing angular velocities and back EMF voltages. This is expected to have a strong influence on the performance of the current controller. Also, crossing the play will result in impact forces that can corrupt the torque control. Thus, the model will have a discontinuous structure consisting of one inertia on each side of the play of the transmission. When the two inertias are out of contact, the models function as a system with two degrees of freedom. When colliding, an impulse response is calculated depending on the state of both interias and the angular velocities are set to zero. When in contact and no loss of contact, the system is constrained to a one degree of freedom model with a combined inertia.

Division of inertias

The planetary gearbox consists of three stages that reduce the angular velocity of the motor. In this

gearbox, there are many sources of play while the transmission model consists of two inertias and one

source of play. This thus requires a simplification of the number of rigid bodies. The inertias of the

gearbox are divided in the middle where the first stage and half of the second stage will be on the left

4.3. Transmission Model 21

(fast moving) side. Also, the rotor interia is added to the left side

J

= (i

i

i

)

J

+ (i

i

)

J

+ 1

2 i

J

(4.5)

where J

represents the combined inertia on the fast moving side. On the right (slow moving) side, the inertia of the other half of the second stage is added with the third stage and the knee head inertia

J

= 1

2 i

J

+ J

+ J

(4.6)

where J

represents the combined inertia on the slow moving side. Additionally, the motor torque on the fast moving side is scaled by the total gear ratio of all the stages

τ

= (i

i

i

)τ

(4.7)

where τ

is the output torque of the motor on the rotor at the fast moving side and the τ

is the scaled up torque acting on the left side of the inertia.

Dynamic equations

When there is no contact, each inertia will accelerate depending on the torque on each side of the gearbox

"

ω ˙

ω ˙

#

=

"

J

0 0 J

#

"

τ

τ

#

= M

τ (4.8)

where ω

and ω

are the accelerations of individual inertias. During the collision between the two inertias, an impulse equation is used to calculate the impact force. Subsequently, the angular velocity is set to zero thus dissipating all energy. During contact, the model is constrained to a one degree- of-freedom system. Furthermore, the forward and backward efficiency compensation is considered due to the sliding effect of the gears. The angular velocity dependent friction is not included into the model. If the actuator is driving, which is expected to be most of the time, the acceleration of the knee is calculated using

ω ˙

= ˙ ω

= τ

µ

− τ

J

µ

+ J

(4.9)

where µ

is the (forward) efficiency. The acceleration of the knee during yielding can also be determined by

ω ˙

= ˙ ω

= τ

− τ

µ

J

+ J

µ

(4.10)

where µ

is the (backward) efficiency.

22 Chapter 4. Plant & Sensor Modelling

Figure 4.3: Measured output torques of the transmission at different current set points (left) and output angular velocities and model output torque of the transmission at different current set points and output angular velocities (right)

Model validation

The transmission model consist of several important parameters that determine the dynamics namely:

the magnitude of the play, the inertias of the rotor and the stages and finally the efficiencies. The play is specified to be a maximum of 0.5°. [28] When measuring the play inside the prototype, a play of 1.0° is measured. This is because there is some play between the transmission and the knee head. The inertia of the rotor is based on the manufacturers datasheet of 69 · 10

kgm

. The inertia of a stage is estimated by measuring the shapes of the stages in combination with the mass. This results in an inertia of 2.89 · 10

kgm

.

The efficiency based on the manufacturer datasheet is 0.9. To validate this assumption, a current controlled motor is used to generate a constant motor output torque. Using a variable brake, the output torque of the gearbox is evaluated for angular velocities of the knee joint of −6 to 6 rad/s. The transmission will mostly be used in driving situations, therefore only output torques in the driving quadrants have been measured. The results of the expected model torques and the transmission output torques are depicted in Figure 4.3. The linear relationship between the current and the output torque is clearly visible. Additionally, angular velocities dependent losses do not influence the output torque.

This matches properly with the assumptions of the previous section. Finally, the model driving ef- ficiency has been reduced to minimize the mean square error between the model and the measured output torque predictions. This resulted in a driving efficiency of 0.88.

4.4 Sensor models

The control architecture for the electric drive is based on two types of sensors namely: the current

sensors and the hall sensor. The current sensors and the hall sensor are used to estimate the two

components of the currents in the rotor coordinate system specified in Equation 4.2. The hall sensor