Robotic Mobility
• Free Space
• Relative Orbital Motion
• Airless Major Bodies (moons)
• Gaseous Environments (Mars, Venus, Titan)
– Lighter-than-”air” (balloons, dirigibles)
– Heavier-than-”air” (aircraft, helicopters)
• Aquatic Environments (Europa)
© 2010 David L. Akin - All rights reserved
http://spacecraft.ssl.umd.edu
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1
Course Overview
ENAE 788X - Planetary Surface Robotics
Propulsive Motion in Free Space
• Basic motion governed by Newton’s Law
→
−
→
−
F = ma (actually, F = m ẍ)
• Over a distance d and time t,
d
∆V = 2
t
(required to accelerate and decelerate)
• The rocket equation (relates propellant to ΔV)
mf inal
− V ∆V
= e exhaust
mo
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Course Overview
ENAE 788X - Planetary Surface Robotics
Cost of Propulsive Maneuvering
• Assuming ∆V � Vexhaust
• Use the Taylor’s Series expansion of e
∆V
mf inal
≈1−
mo
Vexhaust
• Since mo=minitial=mprop+mfinal,
Vtravel
mprop
2
mprop
d
or
≈2
≈
mo
Vexhaust
mo
Vexhaust t
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3
Course Overview
ENAE 788X - Planetary Surface Robotics
Hill’s Equations (Proximity Operations)
Linearized equations of motion relative to a target in circular orbit in
a rotating Cartesian reference frame
ẍ = 3n2 x + 2nẏ + adx
ÿ = −2nẋ + ady
Ref: J. E. Prussing and B. A. Conway, Orbital Mechanics
Oxford University Press, 1993
z̈ = −n2 z + adz
�
µ
n=
a3
adx, ady, adz are disturbing accelerations (e.g., thrust, solar pressure)
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Course Overview
ENAE 788X - Planetary Surface Robotics
Clohessy-Wiltshire (“CW”) Equations
Force-free solutions to Hill’s Equations
sin (nt)
2
x˙o + [1 − cos (nt)]y˙o
x(t) = [4 − 3 cos (nt)]xo +
n
n
2
4 sin (nt) − 3nt
y˙o
y(t) = 6[sin (nt) − nt]xo + yo − [1 − cos (nt)]x˙o +
n
n
z˙o
sin (nt)
z(t) = zo cos (nt) +
n
ż(t) = −zo n sin (nt) + żo cos (nt)
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5
Course Overview
ENAE 788X - Planetary Surface Robotics
“V-Bar” Approach
Ref: Collins, Meissinger, and Bell, Small Orbit Transfer Vehicle (OTV) for On-Orbit Satellite
Servicing and Resupply, 15th USU Small Satellite Conference, 2001
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6
Course Overview
ENAE 788X - Planetary Surface Robotics
“R-Bar” Approach
• Approach from along
the radius vector (“Rbar”)
• Gravity gradients
decelerate spacecraft
approach velocity - low
contamination approach
• Used for Mir, ISS
docking approaches
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Ref: Collins, Meissinger, and Bell, Small Orbit Transfer
Vehicle (OTV) for On-Orbit Satellite Servicing and
Resupply, 15th USU Small Satellite Conference, 2001
7
Course Overview
ENAE 788X - Planetary Surface Robotics
Hopping (Airless Flat Planet)
Vv , h
Use F=ma for vertical motion
V
γ
Vh , d
1
2
˙
gt
h
=
V
t
−
Vv = −g
v
2
tf lt = 2Vv /g
Constant velocity in horizontal direction produces
d = Vh tf lt
V h Vv
=2
g
Vh = V cos γ; Vv = V sin γ
V2
V 2 sin γ cos γ
=
sin (2γ)
d=2
g
g
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Course Overview
ENAE 788X - Planetary Surface Robotics
Hopping (Airless Flat Planet)
Vv , h
Horizontal distance is maximized when sin (2γ) = 1
V
γ
Vh , d
γopt
�
V = gd
hmax
hmax
π
= = 45o
2
∆Vtotal
�
�2
1
Vv
Vv
− g
= Vv
g
2
g
√ 2
2
gd
d
V
=
=
=
4g
4g
4
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9
V2
dmax =
g
�
= 2V = 2 gd
V
Vv = √
2
Course Overview
ENAE 788X - Planetary Surface Robotics
An Example of Propulsive Gliding
Video shown in class
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10
Course Overview
ENAE 788X - Planetary Surface Robotics
Propulsive Gliding (Airless Flat Planet)
T = mg
g
Assume horizontal velocity is V
V
∆Vh = 2V
(includes acceleration and deceleration)
tf lt = d/V
∆Vv = gtf lt
gd
=
V
Total ΔV becomes
∆Vtotal
gd
= ∆Vv + ∆Vh = 2V +
V
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MARYLAND
11
Course Overview
ENAE 788X - Planetary Surface Robotics
Propulsive Gliding (Airless Flat Planet)
Want to choose V to minimize
∂
∂V
�
gd
2V +
V
�
=0
Vopt =
∆Vtotal = 2
�
�
�
gd
+ gd
2
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gd
2− 2 =0
V
12
gd
2
√ �
2
= 2 2 gd
gd
Course Overview
ENAE 788X - Planetary Surface Robotics
Delta-V for Hopping and Gliding
2000
1800
Delta-V (m/sec)
1600
1400
1200
1000
800
600
400
200
0
0
200000
400000
600000
800000 1000000 1200000
Distance (m)
Ballistic Hop
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Propulsive Glide
Course Overview
ENAE 788X - Planetary Surface Robotics
Hopping (Spherical Planet)
� �
�
2 1
−
v= µ
r
a
∆v = 2vo
a(1 − e2 )
p
=
r=
1 + e cos ν
1 − e cos θ
� �
�
�
�
2
1 − e cos θ
1−e
2
a=r
−
v= µ
2
1−e
r
r(1 − e cos θ
−r(1 − e cos θ)(−2e) + (1 − e2 )r(− cos θ)
∂v
=0⇒
=0
2
2
∂e
r (1 − e cos θ)
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14
Course Overview
ENAE 788X - Planetary Surface Robotics
Hopping (Spherical Planet)
2er − 2e2 r cos θ − r cos θ + re2 cos θ = 0
eopt
cos θe2 − 2e + cos θ = 0
√
2 ± 22 − 4 cos2 θ
1 ± sin θ
=
=
2 cos θ
cos θ
+ produces e > 1 (hyperbolic orbit); − gives elliptical orbit
eopt
1 − sin θ
=
cos θ
aopt = r
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�
1 − eopt cos θ
1 − e2opt
�
Course Overview
ENAE 788X - Planetary Surface Robotics
Propulsive Gliding (Airless Round Planet)
2
V
ω2 r =
r
g
Assume horizontal velocity is V
V
∆Vh = 2V
(includes acceleration and deceleration)
tf lt = d/V
∆Vv =
Total ΔV becomes
∆Vtotal
�
2
V
g−
r
�
tf lt
gd dV
−
=
V
r
gd dV
−
= ∆Vv + ∆Vh = 2V +
V
r
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Course Overview
ENAE 788X - Planetary Surface Robotics
Propulsive Gliding (Airless Round Planet)
∂
∂V
Want to choose V to minimize
�
gd dV
−
2V +
V
r
�
=0
Vopt
�
∆Vtotal = 2
�
gd
=
d
2
−
r
�
gd
+ gd
d
2− r
�
∆Vtotal
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gd
d
2− 2 − =0
V
r
2 − dr
d
−
gd
r
d�
=2 2−
gd
r
17
�
gd
2 − dr
Course Overview
ENAE 788X - Planetary Surface Robotics
Hopping on Flat and Round Bodies
9000
Delta-V (m/sec)
8000
7000
6000
5000
4000
3000
2000
1000
0
0
1000000 2000000 3000000 4000000 5000000 6000000
Distance (m)
Ballistic Hop
Propulsive Glide
Hop on Sphere
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Glide on Sphere
Course Overview
ENAE 788X - Planetary Surface Robotics
Nondimensional Forms
V
Define ν ≡ √
dg
νf lat
glide
d
ρ≡
r
hmax
η≡
d
√
=2 2
1
νf lat hop = 2
η=
� 4
νspherical glide = 2 2 − ρ
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(0 ≤ ρ ≤ 1)
Course Overview
ENAE 788X - Planetary Surface Robotics
Multiple Hops
• Assume n hops between origin and destination
• At each intermediate “touchdown”, vv has to be
reversed
∆Vtotal = 2V + 2(n − 1)Vv
Vv
Vv
tpeak =
ttotal = 2ntpeak = 2n
g
g
�
�
2n
Vv = 2ghmax
2η
V h Vv
d = Vh ttotal =
νv =
g
n
�
hmax
V
dg
1
1
η≡
ν≡√
d/n Vh = 2nV νh = 2 2nη
dg
v
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Course Overview
ENAE 788X - Planetary Surface Robotics
Multiple Hop Analysis
�
∆ν = 2ν + 2(n − 1)νv
∆ν = 2 νv2 + νh2 + 2(n − 1)νv
�
�
1
2η
2η
+
+ 2(n − 1)
∆ν = 2
n
8nη
n


�
�
�
1
1
2
∂∆ν
2


= �
−
=0
+ (n − 1)
2
2η
∂η
n 8nη
nη
1
+
n
8nη
1
Analytically messy, but note that for n = 1 ⇒ ηopt =
4
(In general, solve numerically)
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Course Overview
ENAE 788X - Planetary Surface Robotics
Optimal Solutions for Multiple Hops
ηopt
∆ν
0.3
3
0.25
2.5
0.2
2
0.15
1.5
0.1
1
0.05
0.5
0
0
1
10
100
1
1000 10000
10000
Number of Hops (n)
Number of Hops (n)
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100
22
Course Overview
ENAE 788X - Planetary Surface Robotics
Hopping Between Different Altitudes
Relative to starting point, landing elevation ≡ h2
vv1 �= vv2
v1 = (vh , vv1 )
v2 = (vh , vv2 )
2
vv1
1 vv1
1 2 t
peak =
hpeak =
h = vv1 t − gt
g
2 g
2
�
vv1 = 2ghpeak
1 2
From peak, vv = −gtf all ; h = hpeak − gtf all
2
�
2
1 vv2
2
h2 = hpeak −
(hpeak − h2 )
tf all =
2 g�
g
vv2 =
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2g(hpeak − h2 )
23
Course Overview
ENAE 788X - Planetary Surface Robotics
Optimal Hop with Altitude Change
�
vv1
+
g
�
2
(hpeak − h2 )
g
�
d = vh (tpeak + tf all ) = vh
��
�
2hpeak
2
+
(hpeak − h2 )
d = vh
g
g
�
�
�
�
√
d g = vh
2hpeak + 2(hpeak − h2 )
√
d g
�
vh = �
2hpeak + 2(hpeak − h2 )
��
�
�
2 +
2
∆v =
vh2 + vv1
vh2 + vv2
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�
Course Overview
ENAE 788X - Planetary Surface Robotics
Nondimensional Form of Equations
h2
v
hpeak
;λ≡
Remember that ν ≡ √ ; η ≡
d
d
dg
��
�
�
2 +
2
∆ν =
νh2 + νv1
νh2 + νv2
�
�
νv1 = 2η
νv2 = 2(η − λ)
1
�
νh = √
2η + 2(η − λ)
��
�
�
∆ν = � √
1
�
2η + 2(η − λ)
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�2
��
�
�
+ 2η + �
25
1
�
√
2η + 2(η − λ)
�2
+ 2(η − λ)
Course Overview
ENAE 788X - Planetary Surface Robotics
Optimization of Height-Changing Hop
• This is not going to be one where you can take the
derivative and set equal to zero, so use the
equation to find a numerical optimization
• Set λ = 0 to check for plain hop solution
�
∆ν = 2
1
1
+ 2η ⇒ ηopt =
8η
4
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Course Overview
ENAE 788X - Planetary Surface Robotics
Trajectory Design for Height Change
3
∆ν
2.5
ηopt
2
1.5
1
0.5
Height Change λ
0
-3
-2
-1
0
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1
2
3
Course Overview
ENAE 788X - Planetary Surface Robotics
Apollo Concept of Lunar Flying Vehicle
from “Study of One-Man Lunar Flying Vehicle - Final Report Volume 1: Summary” North American Rockwell,
NASA CR-101922, August 1969
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Course Overview
ENAE 788X - Planetary Surface Robotics
Apollo 15 Revisited: LFV Sortie
Basic assumptions
• Vehicle inert mass=300 kg
• Crew mass=150 kg
• Science package=100 kg
• Total propellant=130 kg
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Course Overview
ENAE 788X - Planetary Surface Robotics
Apollo 15 Revisited: Leg 1
Base camp to bottom of rille
• Distance 3 km
• Altitude change -150 m
• ΔV=139 m/sec
• Propellant used=22 kg
• Collect 20 kg of samples
at landing site
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30
Course Overview
ENAE 788X - Planetary Surface Robotics
Apollo 15 Revisited: Leg 2
Propulsive glide along
bottom of rille
• Distance 2 km
• No net altitude change
• ΔV=160 m/sec
• Propellant used=25 kg
• Collect 20 kg of samples
at landing site; leave 25
kg science package
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31
Course Overview
ENAE 788X - Planetary Surface Robotics
Apollo 15 Revisited: Leg 3
Hop to top of mountain
• Distance 15 km
• Altitude change 1600 m
• ΔV=310 m/sec
• Propellant used=46 kg
• Collect 30 kg of samples
at landing site; leave 50
kg science package
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32
Course Overview
ENAE 788X - Planetary Surface Robotics
Apollo 15 Revisited: Leg 4
Return to base
• Distance 12 km
• Altitude change -1450 m
• ΔV=278 m/sec
• Propellant used=37 kg
• Return with 25 kg of science
equipment and 70 kg of
samples
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33
Course Overview
ENAE 788X - Planetary Surface Robotics
Apollo 15 Revisited: Discussion
• Current minimum estimates are for 400 kg of
residual propellants in Altair at landing - would
support three equivalent sorties
• Presence of water ice or ISRU propellant
production at outpost would easily support
moderate flier mission requirements
• Challenges in routine refueling of cryogenic
propellants on the lunar surface, reliable flight and
landing control system
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34
Course Overview
ENAE 788X - Planetary Surface Robotics
Landing Impact Attenuation
• Cannot rely on achieving perfect zero velocity at
touchdown
• Specifications for landing conditions
– Vertical velocity ≤ 3 m/sec
– Horizontal velocity ≤ 1 m/sec
1
1
2
Kinetic Energy = mv = m(vh2 + vv2 )
2
2
Max case 500 kg vehicle =⇒ E = 2500N m
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Course Overview
ENAE 788X - Planetary Surface Robotics
Mars Phoenix Lander
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Course Overview
ENAE 788X - Planetary Surface Robotics
Apollo Lunar Module
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Course Overview
ENAE 788X - Planetary Surface Robotics
Landing Deceleration
• Look at 3 m/sec vertical velocity
• Constant force deceleration
F
1 2
v = d = adesired d
2
m
1
mv 2 = F d
2
tdecel =
v
adesired
adesired
d�cm�
1/6 g
281
1/2 g
92
0.61
1g
0.31
2g
46
23
0.15
3g
15
0.10
• Spring deceleration
F = kx
mv
k= 2
d
2
�
1
F dx = mv 2
2
kd
apeak =
m
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1 v2
d=
2 adesired
38
td �sec�
1.88
Course Overview
ENAE 788X - Planetary Surface Robotics
Effect of Lateral Velocity at Touchdown
• Resolve torques around landing gear footpad
τtot
θ̈ =
Itot
w
h
mg
Fv
�
Fh
Fh h − Fv w − mgw
θ̈ =
Icg + m�2
• Worst cases - hit obstacle (high force), landing
downhill
• Issue: rotational velocity induced is counteracted
by vehicle weight
• Will vehicle rotation stop before overturn limit?
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39
Course Overview
ENAE 788X - Planetary Surface Robotics
Atmospheric Neutral Buoyancy
• Given an enclosed volume V of gas with density ρ
• Lift force is V(ρatm-ρ) - must be ≥mg
– on Earth ~1 kg lift/cubic meter of He
– on Mars ~10 gms lift/cubic meter of He
• Horizontal velocity at equilibrium is identical to
wind speed
• Interior pressure generally identical to ambient
(except for superpressure balloons)
• Can generate low density through choice of gas,
heating
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Course Overview
ENAE 788X - Planetary Surface Robotics
Dynamic Atmospheric Lift
Lift
Drag
Thrust
Weight
For steady, level flight:
1 2
D = ρv ScD
2
1 2
L = ρv ScL
2
L=W
T =D
L
L
W =L=D =T
D
D
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41
1 2
L
L = ρv ScD
2
D
Course Overview
ENAE 788X - Planetary Surface Robotics
Looking for Equation for Range
T ve
propulsive power
=
Efficiency =
fuel power
ṁf h
ηoverall
−W
dW
= −ṁf g = L T
dt
D ṁ g
T ve
=
ṁf h
f
−W ve
dW
= h L T ve =
dt
g D ṁ h
f
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42
−W ve
h L
g D ηoverall
Course Overview
ENAE 788X - Planetary Surface Robotics
More Aerial Range
Rewrite and integrate
dW
=
W
−ve dt
=⇒ ln W = C −
h L
g D ηoverall
Initial conditions -
−ve t
h L
g D ηoverall
at t = 0 W = Winit → C = ln Winit
hL
Winit
ηoverall ln
Range =
gD
Wf inal
L
VD
Winit
ln
Range =
g SF C Wf inal
Breguet Range Equation
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43
Course Overview
ENAE 788X - Planetary Surface Robotics
Liquid Mobility - Subsurface
• Generally same governing equations as neutral
buoyancy (e.g., balloons) in atmosphere
• Volume is set by rigid shell - overall density must
equal fluid density
• Thrust provided by propeller(s), tracks/wheels,
towing
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44
Course Overview
ENAE 788X - Planetary Surface Robotics
Liquid Mobility - Surface
• Displacement Vρ>mg
• Have to overcome drag of fluid
• Also have to consider interface drag with surface larger on smaller vehicles
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Course Overview
ENAE 788X - Planetary Surface Robotics