Slides used in PhD Thesis Defence: Lalit Patnaik 2015, Indian Institute of Science

Dynamics and Control of a
Rimless Wheel based 2D Dynamic Walker
using Pulsed Torque Actuation
Lalit Patnaik
Research Advisor: Prof. L Umanand
Department of Electronic Systems Engineering
Indian Institute of Science
April 17, 2015

Outline
Introduction
Dynamics and Torque Regimes
Analysis of Operating Point Space
Hardware Design and Control Topology
Conclusion

Introduction
Terrestrial Locomotion Alternatives
Energetics of Locomotion
Models of Walking
Thesis Scope
Analytical Solution for Dynamics
Torque Regimes
Experimental Results
Physical Constraints
Optimal Walker
Mechanical Hardware
Electrical Hardware
System Integration
Control Scheme
Conclusion

Dynamics and Control of Rimless Wheel using Pulsed Torque
Introduction
Wheeled locomotion: efficiency
I Fast
I Energy efficient
4 / 76

Introduction
I Fast
I Energy efficient
I Continuous ground contact
I Need prepared surface: hard AND even
4 / 76

Introduction
I Fast
I Energy efficient
I Continuous ground contact
I Need prepared surface: hard AND even
Pair of surfaces Coefficient of rolling resistance
µr = (drag force)/(normal reaction)
Steel wheels on steel rails ∼0.001
Car tyre on tar/asphalt road ∼0.01
Car tyre on loose sand ∼0.1
4 / 76

Introduction
Legged locomotion: versatility
I No need of prepared surface
I Can traverse diverse terrain
5 / 76

Introduction
Legged locomotion: versatility
I No need of prepared surface
I Can traverse diverse terrain
I Intermittent ground contact
I Can be lossy: braking and ground impacts
5 / 76

Introduction
Legged locomotion gaits
Gaits
Walking Running
Static Zero Moment Point
(ZMP) based
Dynamic
t
slow fast
Never air-borne
t
Sometimes air-borne
in phase
out of phase
slow fast
t = Kinetic Energy
= Potential Energy
6 / 76

Introduction
Static walking: Jansen mechanism
−1 0 1 2 3 4 5 6 7 8
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
Link−0
Link−1
Link−2
Link−3
Link−4
Link−5
Link−6
Link−7
Link−8
Link−9
Link−10
Node−0
Node−1
Node−2
Node−3
Node−4
Node−5
Pin
CC
Link-1
L
in
k
-9
L
i
n
k
-
2
L
in
k-
0
L
i
n
k
-
3
Link-4
L
in
k
-5
L
in
k
-6
Link-7
L
in
k
-8
Link-10
Node-0
Node-1
Node-2
Node-3
Node-4
Node-5
CC Pin
−1 0 1 2 3 4 5 6 7
−6
−5
−4
−3
−2
−1
0
1
2
3
Link−1
Link−2
Link−9
Link−10
Node−1
Node−0
Link−0
Link−6
Link−5
Link−7
Link−4
Link−3
Link−8
Node−3
Node−4
Node−2
Pin
CC
Node−5
7 / 76

Introduction
Static walking: Jansen mechanism
−1 0 1 2 3 4 5 6 7 8
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
Link−0
Link−1
Link−2
Link−3
Link−4
Link−5
Link−6
Link−7
Link−8
Link−9
Link−10
Node−0
Node−1
Node−2
Node−3
Node−4
Node−5
Pin
CC
Link-1
L
in
k
-9
L
i
n
k
-
2
L
in
k-
0
L
i
n
k
-
3
Link-4
L
in
k
-5
L
in
k
-6
Link-7
L
in
k
-8
Link-10
Node-0
Node-1
Node-2
Node-3
Node-4
Node-5
CC Pin
−1 0 1 2 3 4 5 6 7
−6
−5
−4
−3
−2
−1
0
1
2
3
Link−1
Link−2
Link−9
Link−10
Node−1
Node−0
Link−0
Link−6
Link−5
Link−7
Link−4
Link−3
Link−8
Node−3
Node−4
Node−2
Pin
CC
Node−5
I Kinematics: [Jansen Mechanism Animation]
7 / 76

Introduction
Bond graph model of Jansen mechanism
Bond Graph Model of Jansen Mechanism
0
Sf : ω
35
36
1
R
:
R
1
C : C1
1
1
L
:
I
0
2
S
e
:
3
7
3
78
mTF
M1
4
5
mTF
M8
79
80
0
10
1
R : R2
9
C : C2 8
0
13
1
R : R3
12
C : C3
11
0
85
1
R : R4
84
C : C4 83
0
88
1
R : R5
87
C : C5
86
6
7
81
82
mTF
M2
14
16
15
mTF
M9
89
91
90
1
1
1
1
1
1
17
18
19
92
93
94
L
:
m
1
20
L
:
m
1
21
S
e
:
38
L
:
I
1
22
L
:
m
6
95
L
:
m
6
96
S
e
:
97
L
:
I
6
98
mTF
M3
23
24
mTF
M10
100
101
0
29
1
R : R6
28
C : C6 27
0
32
1
R : R7
31
C : C7
30
0
106
1
R : R8
105
C : C8 104
0
109
1
R : R9
108
C : C9
107
25
26
102
103
mTF
M4
mTF
M11
33
1
48
L
:
I
2
34
S
e
:
39
110
1
113
L
:
I
8
11
1
S
e
:
11
2
mTF
M5
49
50
mTF
M12
114
115
0
55
1
R : R10
54
C : C10 53
0
58
1
R : R11
57
C : C11
56
0
120
1
R : R12
119
C : C12 118
0
123
1
R : R13
122
C : C13
121
51
52
116
117
mTF
M6
59
61
60
mTF
M13
124
126
125
1
1
1
1
1
1
62
63
64
L
:
m
3
65
L
:
m
3
66
S
e
:
67
L
:
I
3
68
L
:
m
4
13
0
S
e
:
13
7
L
:
m
4
13
1
S
e
:
13
2
L
:
I
4
13
3
S
e
:
13
4
mTF
M7
70
7
1
mTF
M14
127
128
129
0
0
135
1
3
6
77
1
C
:
C
1
4
7
5
R
:
R
1
4
7
6
74
1
C
:
C
1
5
7
2
R
:
R
1
5
7
3
LINK-0
LINK-1
LINK-6
LINK-2-9-10
LINK-8
LINK-3
LINK-4-5-7
MOTOR
8 / 76

Introduction
Legged locomotion gaits
Static ZMP-based Dynamic
Stability high low intermediate
Posture sprawling upright upright
bent knees extended knees
No. of legs 4 or more 2 2 and 4
Energy high high low
Terrain uneven even intermediate
Foot type point flat any
Appearance natural unnatural natural
Natural hardly any damped utilized
dynamics
9 / 76

Introduction
Energy budget
10 / 76

Introduction
Energy budget
Once per lifetime
10 / 76

Introduction
Energy budget
Once per lifetime
Once per walk
10 / 76

Introduction
Energy budget
Once per lifetime
Once per walk Continuous
10 / 76

Introduction
Energy flow
Actuator Mechanical
Load
Mechanical
Energy
Metabolic / Electrical
Energy
Muscle / Motor
11 / 76

Introduction
Energy flow
Actuator Mechanical
Load
Mechanical
Energy
Energy
Muscle / Motor
dynamics
losses
11 / 76

Introduction
Energy flow
Actuator Mechanical
Load
Mechanical
Energy
Energy
Muscle / Motor
speed
efficiency
dynamics
losses
11 / 76

Introduction
Energy flow
Actuator Mechanical
Load
Mechanical
Energy
Energy
Muscle / Motor
speed
efficiency
dynamics
losses
I Biological systems
I minimize metabolic energy[1]
for given actuator (muscle)
[1] Minetti and Alexander, “A Theory of Metabolic Costs for Bipedal Gaits,” J. Theor. Biol. (1997)
11 / 76

Introduction
Energy flow
Actuator Mechanical
Load
Mechanical
Energy
Energy
Muscle / Motor
speed
efficiency
dynamics
losses
I Biological systems
I minimize metabolic energy[1]
for given actuator (muscle)
I Engineered systems
I minimize mechanical energy, then design actuator
[1] Minetti and Alexander, “A Theory of Metabolic Costs for Bipedal Gaits,” J. Theor. Biol. (1997)
11 / 76

Introduction
Terminology
Human walking[2]
Right
heel-
strike
Left
toe-
off
Right
mid-
stance
Left
heel-
strike
Right
toe-
off
Left
mid-
stance
Right
heel-
strike
Left
toe-
off
0 % 50 % 100 %
Time, percent of cycle
Double
support Right single support
Double
support Left single support
Double
support
Right stance phase
Left stance phase
Right swing phase
Left swing phase
Cycle (stride) duration
[2] Inman et al, Human walking (1981)
12 / 76

Introduction
Models of Walking
Models of walking
a b c
d e f g
[3] Alexander, “Mechanics of Bipedal Locomotion,” Perspectives in Experimental Biology (1976)
[4] Margaria, Biomechanics and Energetics of Muscular Exercise (1976)
[5] Alexander, “Simple models of human movement,” ASME Appl. Mech. Rev. (1995)
13 / 76

Introduction
Models of Walking
Ideal inverted pendulum walker
mid-stance velocity
14 / 76

Introduction
Models of Walking
step angle
14 / 76

Introduction
Models of Walking
impact
14 / 76

Introduction
Models of Walking
14 / 76

Introduction
Models of Walking
Practical inverted pendulum walker
15 / 76

Introduction
Models of Walking
Loss mechanisms
Braking loss
I Muscles performing negative work[6]
[6] Cavagna et al, “The role of gravity in human walking: penduluar energy exchange, external work and optimal 16 / 76

Introduction
Models of Walking
Loss mechanisms
Braking loss
I Coulomb friction in rotary joints

Introduction
Models of Walking
Loss mechanisms
Braking loss
I Coulomb friction in rotary joints
I Rolling resistance for walking

Introduction
Models of Walking
Loss mechanisms
Impact loss: instantaneous loss of kinetic energy
0 20 40 60
0
0.2
0.4
0.6
0.8
1
φm
(degree)
v
m
’/v
m
[7] Alexander, Principles of animal locomotion (2003)
[8] McGeer, “Passive dynamic walking,” Int. J. Robot. Res. (1990)
17 / 76

Introduction
Models of Walking
Achieving a sustained walk
Passive dynamic walker on downward incline
18 / 76

Introduction
Models of Walking
I Biped
I Without knees[8]
I With knees[9]
[9] McGeer, “Passive Walking with Knees,” IEEE ICRA (1990)
18 / 76

Introduction
Models of Walking
I Biped
I Without knees[8]
I With knees[9]
I Rimless wheel
I Point foot[10]
I Flat foot[11]
[9] McGeer, “Passive Walking with Knees,” IEEE ICRA (1990)
[10] Coleman et al, “Motions of a Rimless Spoked Wheel: a Simple 3D System with Impacts,”
Dynam. Stabil. Syst. (1997)
[11] Nirukawa et al, “Design and Stability Analysis of a 3D Rimless Wheel with Flat Feet and Ankle Springs,”
J. Sys. Design & Dyn. (2009)
18 / 76

Introduction
Models of Walking
Add actuator
19 / 76

Introduction
Models of Walking
Add actuator
[12] Collins et al, ”A 3D Passive-Dynamic Walking Robot with Two Legs and Knees,” Int. J. Robot. Res. (2001)
[13] Dertien, Realisation of an energy-efficient walking robot (2005)
19 / 76

Introduction
Thesis Scope
Thesis scope
Motor
Controller Load
1 2
3
20 / 76

Introduction
Thesis Scope
Thesis scope
Motor
Controller Load
1 2
3
1. Load dynamics, energetics and torque regimes
Achieving sustained forward motion with pulsed torque
20 / 76

Introduction
Thesis Scope
Thesis scope
Motor
Controller Load
1 2
3
2. Physical constraints on choice of operating points
Locating optimal operating points
20 / 76

Introduction
Thesis Scope
Thesis scope
Motor
Controller Load
1 2
3
2. Physical constraints on choice of operating points
Locating optimal operating points
3. Control topology for pulsed torque actuation of two
synchronized brushless DC (BLDC) motors
20 / 76

Closed form solution missing
I "Surprisingly, the inverted pendulum model of walking does
not appear to have been solved analytically before." [14]
I Taylor’s analytical solution uses elliptic integrals
(not closed form)
[14] Graham Taylor and Adrian Thomas, Evolutionary Biomechanics (2014)
22 / 76

Torque sources
sin
23 / 76

Closed-form analytical solution
φ(t) =
vi
vn
sinh (ωnt) + (φc + φi) cosh(ωnt) − φc
v(t) =
dφ
dt
l = vi cosh (ωnt) + (φc + φi) · vn · sinh (ωnt)
where
vn =
p
gl
ωn =
r
g
l
φc = sin φa − sin φb
24 / 76

0 0.5 1
−40
−20
0
20
40
t (s)
φ
(degree)
t (s)
v
(m/s)
0 0.5 1
0.5
1
1.5
2
numerical
analytical
numerical
analytical
Parameter values: m = 50 kg, l = 1 m, φb = 5◦, φa = 15◦
Initial conditions: φi = −25◦, vi = 1 m/s
Final condition: φf = 25◦
Error < 6% (for step angle < 25◦)
25 / 76

Regime 1 : τa > Eloss/φm
t (s)
Energy
(J)
0 0.2 0.4 0.6 0.8
0
10
20
30
40
50
0.8
1
1.2
1.4
v
(m/s)
−20
−10
0
10
20
φ
(degree)
∆φ < φm
−20 −10 0 10 20
0.4
0.6
0.8
1
1.2
1.4
φ (degree)
v
(m/s)

Regime 2 : τa = Eloss/φm
0 0.1 0.2 0.3 0.4
0
20
40
60
80
100
t (s)
Energy
(J)
1.4
1.5
1.6
1.7
1.8
1.9
v
(m/s)
−20
−10
0
10
20
φ
(degree)
a
∆φ = φm
−20 −10 0 10 20
1.4
1.6
1.8
2
φ (degree)
v
(m/s)

Regime 3 : Eloss/2φm < τa < Eloss/φm
0 0.1 0.2 0.3 0.4
0
50
100
150
200
t (s)
Energy
(J)
1.8
1.9
2
2.1
2.2
v
(m/s)
−20
−10
0
10
20
φ
(degree)
φm < ∆φ < 2φm
φ (degree)
v
(m/s)
−20 −10 0 10 20
1.6
1.8
2
2.2
2.4
2.6

Regime 4 : τa = Eloss/2φm
2.1
2.2
2.3
2.4
2.5
2.6
v
(m/s)
−20
−10
0
10
20
φ
(degree)
0 0.1 0.2 0.3 0.4
0
50
100
150
200
t (s)
Energy
(J)
∆φ = 2φm
−20 −10 0 10 20
2
2.2
2.4
2.6
2.8
φ (degree)
v
(m/s)

Regime 5 : τa < Eloss/2φm
0 0.1 0.2 0.3 0.4
0
50
100
150
200
250
t (s)
Energy
(J)
−20
−10
0
10
20
φ
(degree)
2.3
2.4
2.5
2.6
2.7
2.8
v
(m/s)
∆φ > 2φm
−20 −10 0 10 20
2
2.2
2.4
2.6
2.8
φ (degree)
v
(m/s)

Chatur: Rimless Wheel based 2D Dynamic Walker
I Sustained forward motion using pulsed torque
I All four proposed torque regimes achieved
I Oscilloscope mounted on mobile platform
31 / 76

Chatur: Rimless Wheel based 2D Dynamic Walker
I Sustained forward motion using pulsed torque
I All four proposed torque regimes achieved
I Oscilloscope mounted on mobile platform
I [Chatur Video]
31 / 76

Regime 1
1
2
3
4
1
2
3
4
Time scale 250 ms/div, dis = 3/8
(1) Accelerometer output (2 V/div)
(2) Motor-1 sector number
(3) Motor-1 phase current Ia1 (10 A/div)
(4) Torque reference (1 V/div)
32 / 76

Regime 2
1
2
3
4
1
2
3
4
33 / 76

Regime 3
1
2
3
4
1
2
3
4
34 / 76

Regime 4
1
2
3
4
1
2
3
4
Time scale 250 ms/div, dis = 1
35 / 76

Average power vs average speed
Pavg = 2 · τavg · ωavg
0 0.2 0.4 0.6 0.8
0
2
4
6
8
10
vx,avg
(m/s)
P
avg
(W)
R1
R2
R3
R4
36 / 76

Mechanical cost of transport vs normalized speed
Mechanical COT = Pavg/mgvx,avg
0 0.1 0.2 0.3 0.4
0
0.01
0.02
0.03
0.04
Normalized vx,avg
Mechanical
CoT
R1
R2
R3
R4
37 / 76

Take-off constraint
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
v2
0 = gl(3 cos φm + 2φm sin φb − 2)
[15] Usherwood, “Why not walk faster?,” Biol. Lett. (2005)
39 / 76

Take-off constraint
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
1
.
0
m
0
.
8
m
0
.
6
m
0
.
4
m
v2
0 = gl(3 cos φm + 2φm sin φb − 2)
[15] Usherwood, “Why not walk faster?,” Biol. Lett. (2005)
39 / 76

Sliding constraint
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
40 / 76

Constant speed region
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
1%
41 / 76

Constant speed region
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
0.1%
1%
41 / 76

Fall-back constraint
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
42 / 76

Steady-state constraint
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
43 / 76

Region of operation
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
take-oﬀ
sliding
fall-back
constant-speed
steady-state
region
of
operation
44 / 76

Region of operation : l = 1 m, φb = 1◦
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
45 / 76

Region of operation : l = 1 m, φb = 1◦
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
45 / 76

Optimal Walker
Average velocity: vx,avg = f2(v0, φm)
v0
(m/s)
φ
m
(degree)
1 2 3
10
15
20
25
0
.4
m
/s
1.2
m/s
0.
8
m
/s
1.6
m/s
2.
0
m
/s
Parameters: m = 65 kg, l = 1 m, φa = 10◦, φb = 1◦
46 / 76

Optimal Walker
Average power: Pavg = f1(v0, φm)
v0
(m/s)
φ
m
(degree)
1 2 3
10
15
20
25
1
0
W
2
0
W
3
0
W
4
0
W
47 / 76

Optimal Walker
Optimal operating point for a given speed
v0
(m/s)
φ
m
(degree)
1 2 3
10
15
20
25
0.4 m
/s
5
0
W
7
0
W
90
W
48 / 76

Optimal Walker
Locus of optimal operating points
v0
(m/s)
φ
m
(degree)
1 2 3
10
15
20
25
49 / 76

Optimal Walker
Optimal locii for various leg lengths
0 0.5 1
5
10
15
20
25
30
Normalized v0
φ
m
(degree)
Parameters: m = 65 kg, g = 9.81 m/s2
, φa = 10◦ and φb = 1◦
50 / 76

Optimal Walker
Power vs speed
0 1 2 3
0
100
200
300
vx,avg
(m/s)
P
avg
(degree)
, φa = 10◦ and φb = 1◦
51 / 76

Optimal Walker
Mechanical COT vs speed
vx,avg
(m/s)
Mechanical
COT
0 1 2 3
0
0.05
0.1
0.15
0.2
0.25
, φa = 10◦ and φb = 1◦
52 / 76

Mechanical Hardware
Version-1: Mild steel slotted angle frame
I Rimless wheel = hub motor + annular disc + 18 spokes
54 / 76

Mechanical Hardware
I Two wheels take care of lateral stability
54 / 76

Mechanical Hardware
I ‘Tail’ with castor wheel to resist counter torque on frame
54 / 76

Mechanical Hardware
I ‘Tail’ with castor wheel to resist counter torque on frame
I Problems: wheel alignment, assembly issues
54 / 76

Mechanical Hardware
Version-2: Welded mild steel frame
TOP VIEW
SIDE VIEW
FRONT VIEW
SOUTH-EAST
ISOMETRIC VIEW
55 / 76

Mechanical Hardware
Version-2: Welded mild steel frame
56 / 76

Mechanical Hardware
Electronics enclosure design
TOP VIEW ISOMETRIC VIEW
FRONT VIEW SIDE VIEW
Chatur
board
controller
CIPOS
boards
inverter
Heat sinks
57 / 76

Electrical Hardware
Electrical hardware
I Energy source: Lead acid batteries
58 / 76

Electrical Hardware
Electrical hardware
I Motor drive: Infineon CIPOS based 3-phase inverter
58 / 76

Electrical Hardware
Electrical hardware
I Actuator: Brushless DC (BLDC) hub motor
58 / 76

Electrical Hardware
Electrical hardware
I Controller: Cypress PSoC5 based controller card
58 / 76

Electrical Hardware
Electrical hardware
I Sensors: Motor currents, battery voltages, rotor-position,
3-axis accelerometer
58 / 76

Electrical Hardware
Electrical hardware
I Sensors: Motor currents, battery voltages, rotor-position,
3-axis accelerometer
I Communication: Radio frequency link to operator
58 / 76

System Integration
Full system after integration
Parameters for Chatur
I mass: m = 50 kg
I leg/spoke length:
l = 0.57 m
I step angle:
φm = 10◦
I Torque constant:
Kt = 0.6 Nm/A
59 / 76

System Integration
Electrical system block diagram
1
CIPOS
Inverter 1
CIPOS
Inverter 2
BLDC motor1
PWM
V sense
Fault
I sense
Rotor position
JTAG
RS232
DACs
Sensed quantities
Oscilloscope
Chatur Controller Board
RF link
BLDC motor2
Batteries
Personal Computer
60 / 76

System Integration
Load and actuator torque profiles
φ (degree)
Torque
(Nm
)
−10 −5 0 5 10
−50
0
50
−50 0 50
−200
−100
0
100
200
Torque
(Nm
)
φ (degree)
τl = Opposing load torque
τa = Actuator torque
Pattern repeats 18 times per mechanical revolution
61 / 76

Control Scheme
Inner loop: Torque control
PI Controller Limiter PWM
Logic-2
Logic-1
gate pulses
to inverter
rotor position
motor currents
d
+
-
62 / 76

Control Scheme
Non-commutating current feedback
sector
number
5 4 6 2 3 1 sector
number
5
4
6
2
3 1
Reconstructing torque feedback (Ifb) from motor phase currents
63 / 76

Control Scheme
Full control scheme
Scaling
EMA
Filter
Torque
Controller
Logic
gate pulses
(to inverter)
rotor position
motor currents
+
-
Scaling impacts
(from accelerometer)
+
Mux
+
sector change count
sector change count
of other motor
ref
(from operator)
wheel sync
term
lag
0
select
64 / 76

Control Scheme
Experimental results: Torque control
1
2
4
3
Time scale: 100 ms/div
(1) Accelerometer output: 2 V/div
(2) Motor phase current: 10 A/div
(3) Torque reference: 0.5 V/div
(4) Torque feedback: 0.5 V/div
65 / 76

Control Scheme
Experimental results: Wheel synchronization
1
2
3
4
double impact
dip in torque
reference
Time scale: 250 ms/div
(1) Accelerometer output: 2 V/div
(3) Motor-1 phase current Ia1: 10 A/div
(4) Torque reference: 1 V/div
66 / 76

Conclusion
Summary of Contributions
1. Four torque regimes in a rimless wheel based dynamic
walker using pulsed torque actuation
68 / 76

Conclusion
I Regimes are defined by ratio of energy losses (Eloss) to
available actuator torque (τa)
68 / 76

Conclusion
I For sustained forward motion: Eloss ↑⇒ pulse duty ratio ↑
68 / 76

Conclusion
I 4 types of sub-phases (unactuated rise, unactuated fall,
actuated rise, actuated fall) are concatenated in 4 different
ways to form repeating cycles yielding the 4 regimes
68 / 76

Conclusion
I 4 types of sub-phases (unactuated rise, unactuated fall,
actuated rise, actuated fall) are concatenated in 4 different
ways to form repeating cycles yielding the 4 regimes
I Proposed regimes holds for all walkers – engineered or
biological (actuation is present only for a portion of the
stance phase)
68 / 76

Conclusion
2. Closed-form analytical solution for stance phase
dynamics of 2D inverted pendulum walking using
hyperbolic functions
69 / 76

Conclusion
I Parametric expression for constant actuation and braking
torques (τa and τb)
69 / 76

Conclusion
I Error in solution within 6% (for step angle < 25◦
)
69 / 76

Conclusion
I Error in solution within 6% (for step angle < 25◦
)
I Useful tool for investigating effect of various
parameters/variables on dynamics
69 / 76

Conclusion
3. Framework of physical constraints on the choice of
operating points for a generic inverted pendulum walker
70 / 76

Conclusion
I Not all operating points (v0,φm) lead to realizable
steady-state gait
70 / 76

Conclusion
steady-state gait
I Constraint lines delimit valid region of operation of walker
(fundamental limits on walking like an inverted pendulum)
70 / 76

Conclusion
steady-state gait
I Constraint lines delimit valid region of operation of walker
(fundamental limits on walking like an inverted pendulum)
I Sub-regions that result in various regimes of walking are
identified
70 / 76

Conclusion
4. Optimal operating points in inverted pendulum walking
71 / 76

Conclusion
I Operating point with minimum Pmech for given speed is
located based on tangency of power and velocity contours
71 / 76

Conclusion
I Repeating for different speeds, optimal locus of operating
points is obtained
71 / 76

Conclusion
points is obtained
I Shape and location of optimal locus is sensitive to loss and
internal energy models
71 / 76

Conclusion
points is obtained
I Shape and location of optimal locus is sensitive to loss and
internal energy models
I Using a suitable constant step angle over a broad range of
speeds could lead to an inverted pendulum walker that is
close to optimal
71 / 76

Conclusion
5. Hardware design and control topology for pulsed torque
actuation of a synchronized dual BLDC motor driven
platform
72 / 76

Conclusion
platform
I Complete design for mechanical and electrical hardware of
hub-actuated dual rimless wheel 2D dynamic walker
(Chatur)
72 / 76

Conclusion
platform
(Chatur)
I Method for wheel synchronization: increment torque of
lagging motor
72 / 76

Conclusion
platform
(Chatur)
lagging motor
I BLDC drive with non-commutating current feedback
reduces current spikes during sector transitions.
72 / 76

Conclusion
platform
(Chatur)
lagging motor
I BLDC drive with non-commutating current feedback
reduces current spikes during sector transitions.
I Control scheme can be used in any synchronized dual
motor system with pulsed actuation
72 / 76

Conclusion
Scope for Further Work
1. Improvements in mathematical model
73 / 76

Conclusion
I Unified model for walking and rolling
73 / 76

Conclusion
I Effect of distributed mass on dynamics
73 / 76

Conclusion
2. Reduce losses in experimental prototype
73 / 76

Conclusion
I Compliance
73 / 76

Conclusion
I Compliance
I Better wheel alignment
73 / 76

Conclusion
I Compliance
I Lower detent torque
73 / 76

Conclusion
I Compliance
I Lower detent torque
I Actuators with high efficiency at low speed
73 / 76

Conclusion
3. Ability to start from standstill.
74 / 76

Conclusion
I High torque actuator
74 / 76

Conclusion
I Introduce mechanical advantage: actuated feet, variable
length spokes?
74 / 76

Conclusion
length spokes?
4. Avoid undesirable effects of castor wheel, tail piece
74 / 76

Conclusion
length spokes?
I Reduce system weight
74 / 76

Conclusion
length spokes?
I Use counterweight on stator
74 / 76

Conclusion
length spokes?
I Use counterweight on stator
I Ability to walk backward
74 / 76

Conclusion
5. Phase displaced rimless wheels
75 / 76

Conclusion
I Alternating ground impacts for spokes of left and right
wheels
75 / 76

Conclusion
wheels
I Minimize lateral dynamics: reduce distance between
wheels
75 / 76

Conclusion
wheels
wheels
6. Dynamics and energetics of turning
75 / 76

Conclusion
wheels
wheels
I Large reflected torque on the actuators
75 / 76

Conclusion
wheels
wheels
I Large reflected torque on the actuators
I How to minimize energy spent for turning?
75 / 76

Conclusion
Thank you!
Questions?
Lalit Patnaik
plalit@dese.iisc.ernet.in
76 / 76

Jansen mechanism: step length and step height
0 1 2 3 4 5 6
−6.5
−6
−5.5
−5
−4.5
Maximum foot height point
Take-off point Landing point
Step height = 1.5
Step length = 4.5
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Step length
Step height
Crank length

Vertical GRF provides actuation torque
vertical
GRF Vertical
GRF
[body
weight]
left leg right leg
Cycle time [%]
double
support
0 50 100
0
0.5
1
Actuation torque, τa ≈ (vertical GRF) × (step length)

Comparison: φ vs sin φ
0 10 20 30 40 50 60 70 80 90
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
(degrees)
(radians)
sin
φ Error
5◦ 0.13%
10◦ 0.51%
15◦ 1.15%
20◦ 2.06%
25◦ 3.25%
30◦ 4.72%

Area under velocity waveform
0 0.2 0.4 0.6 0.8
0
0.2
0.4
0.6
0.8
1
t (s)
v
(m/s)
0 0.2 0.4 0.6 0.8
−15
−10
−5
0
5
10
15
t (s)
φ
(degree)

Take-off constraint lines for various leg lengths
No actuation i.e. φa = 0
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
(a) φb = 0
0 1 2 3 4
0
20
40
60
v0
(m/s)
φ
m
(degree)
(b) φb = 5◦

Fall back constraint lines for various leg lengths
0 2 4 6
0
10
20
30
40
v0
(m/s)
φ
m
(degree)
(c) φa = 0, φb = 0
0 2 4 6
0
10
20
30
40
v0
(m/s)
φ
m
(degree)
(d) φa = 0, φb = 5◦
0 1 2 3 4
0
10
20
30
40
v0
(m/s)
φ
m
(degree)
(e) φa = 10◦
, φb = 5◦

Steady-state constraint lines for various leg lengths
0 1 2 3 4
0
10
20
30
v0
(m/s)
φ
m
(degree)
(f) φa = 5◦
, φb = 0
0 1 2 3 4
0
10
20
30
40
50
v0
(m/s)
φ
m
(degree)
(g) φa = 10◦
, φb = 0◦
0 1 2 3 4
0
10
20
30
40
v0
(m/s)
φ
m
(degree)
(h) φa = 10◦
, φb = 5◦

Specifications of Chatur
Parameter Value
Leg length, l 0.57 m
Step angle, φm 10◦
Step length, ls = 2l sin φm 0.2 m
CoM height variation, 9 mm
∆h = l(1 − cos φm)
Mass, m rimless wheels, 2 × 12 kg
frame, 7 kg
batteries, 15 kg
others, 4 kg
total, 50 kg
Motor torque constant, Kt 0.6 Nm/A
Maximum actuator torque parameter, 3◦
φa,max (at Imax = 12 A)
Braking torque parameter, φb ∼1◦

Specifications of BLDC hub motor
Parameter Value
Power 300 W
Voltage 24 V
Current 12 A
Pole pairs 25
Torque constant (Kt ) 0.6 Nm/A
Phase resistance (Rph) 0.35 Ω
Phase inductance (Lph) 0.12 mH
Mass 5.2 kg
Block rotor test

Commutation sequence of BLDC motor (CCW rotation)
Sector no. Motor current Inverter state*
direction A B C
(101)2 = 5 A+B− sw 0 x
(100)2 = 4 A+C− sw x 0
(110)2 = 6 B+C− x sw 0
(010)2 = 2 B+A− 0 sw x
(011)2 = 3 C+A− 0 x sw
(001)2 = 1 C+B− x 0 sw
*sw = switching, 0 = clamped to 0, x = both switches OFF
A
B C
B A
C A
A C
B C
C B
A B
+ -
+ -
+ -
+ -
+ -
+ -

BLDC circuit schematic & stator mmf orientations
A
B
C
A B
+ -
B A
+ - C A
+ -
A C
+ -
B C
+ - C B
+ -
North pole
5
4
6
2
3
1

PWM schemes for BLDC
A
B
C
Va
Vb
Vab
Va
Vb
Vab
Tsa=Tsb
Ts
PWM scheme with
both legs switching:
Tsa=Tsb=2Ts
PWM scheme with
one leg switching and
other leg clamped:
Tsa=Ts
Tsa
Ts

Open loop BLDC operation (No torque control)

BLDC sector identification error: click!
Solution: hardware glitch filters in PSoC

CIPOSTMinverter board specifications
Inputs (1) 24 V dc link, 12 A
(2) Power supply for gate drive (15 V)
and fault detection circuit (5 V)
(3) Six PWM control signals for the IGBTs
Outputs (1) Three pole voltages of the inverter
(2) Fault indication to controller
Dimensions 87 mm × 37 mm × 1.6 mm

Chatur controller board specifications
Inputs (1) Power
(a) 24 V, 0.6 A power supply (for flyback)
(b) 5 V aux. power supply (unused)
(2) Sensing
(a) Motor currents (Ia1, Ib1, Ia2 and Ib2)
(b) Battery voltages (Vdc1 and Vdc2)
(c) Rotor positions (3-bit digital output)
(3) Others
(a) JTAG program/debug interface
(b) Remote control signal from RF TX
(c) Fault indication from inverters
Outputs Six PWM control signals for each inverter
Dimensions 190.5 mm × 146.9 mm × 1.6 mm

Specifications of flyback power supply
Input 23–27 V, 0.6 A
Outputs (1) 5V, 0.3A (circuits on control side)
(2) 5V, 0.3 A (low voltage circuits on power side)
(3) 15V, 0.3A (CIPOSTMgate drive)
Switching 100 kHz
frequency

Electrical wiring diagram
CONN1
(JTAG)
10
Battery-1
Battery-2
Battery-3
Battery-4
Battery-5
Battery-6
Terminal Block
CIPOS
Inverter
Board-2
HS Fan-1
HS Fan-2
Motor-1
Motor-2
Chatur Controller Board
24 V
12 V
0 V
24 V
12 V
0 V
24 V
0 V
Vdc1
Vdc2
15
V
5
V
0
V
A
B
C
A
B
C
PWM
PWM
7 7
5 5
Position
Position
CONN401
(PBT)
CONN403
(PBT)
CONN7-8
(RMC)
CONN201-202
(RMC)
CONN301-302
(RMC)
CONN303-310
(FTs)
J25
(PBT)
CIPOS
Inverter
Board-1
J25
(PBT)
J24
(RMC)
J24
(RMC)
J26,30
(FT)
J26,30
(FT)
J27-29
(FTs)
J27-29
(FTs)
Note: RMC = Relimate Connector; PBT = PCB Terminal Block; FTs = Faston Tabs; HS = Heat Sink
CONN4
(RMC)
9
IDACs

Acquiring motor currents (once per PWM cycle)
ADC
Mux
channel
select
(chan)
start-of-conversion
DMA
soc
(soc)
chan

Time scales of various events/processes
Description Time
ADC conversion time 10 µs
PWM cycle time 61.2 µs
Software loop time 100 µs
Motor L/R time constant 350 µs
Sector dwell time 30–120 ms
Double impact blindfold window 300 ms
Step duration 350–900 ms

Double impact blindfold window

Slides used in PhD Thesis Defence: Lalit Patnaik 2015, Indian Institute of Science

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