Suborbital Mobility and Landing Dynamics
• Ballistic hops on flat airless bodies
• Propulsive glides on flat airless bodies
• Hops and glides on spherical bodies
• Nondimensional forms
• Multiple hops
• Altitude changes
• Landing stability
• Landing dynamics
• Discussion of midterm project goals
UNIVERSITY OF
MARYLAND
© 2008 David L. Akin - All rights reserved
http://spacecraft.ssl.umd.edu
1
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Hopping (Airless Flat Planet)
Vv , h
Use F=ma for vertical motion
V
γ
Vh , d
1
2
˙
h
=
V
t
−
gt
Vv = −g
v
2
tf lt = 2Vv /g
Constant velocity in horizontal direction produces
d = Vh tf lt
Vh Vv
=2
g
Vh = V cos γ; Vv = V sin γ
V 2 sin γ cos γ
V2
d=2
=
sin (2γ)
g
g
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MARYLAND
2
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Hopping (Airless Flat Planet)
Vv , h
Horizontal distance is maximized when sin (2γ) = 1
V
γ
γopt
Vh , d
V =
!
hmax
hmax
π
= = 45o
2
∆Vtotal
gd
!
"2
Vv
1
Vv
= Vv
− g
g
2
g
√ 2
2
V
gd
d
=
=
=
4g
4g
4
UNIVERSITY OF
MARYLAND
3
V2
dmax =
g
!
= 2V = 2 gd
V
Vv = √
2
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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
UNIVERSITY OF
MARYLAND
4
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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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MARYLAND
5
gd
2− 2 =0
V
!
!
gd
2
√ "
2
= 2 2 gd
gd
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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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MARYLAND
6
Propulsive Glide
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Hopping (Spherical Planet)
∆v = 2vo
! "
#
2 1
v= µ
−
r
a
p
a(1 − e2 )
r=
=
1 + e cos ν
1 − e cos θ
! "
!
"
#
2
1 − e cos θ
2
1−e
a=r
v= µ
−
2
1−e
r
r(1 − e cos θ
∂v
−r(1 − e cos θ)(−2e) + (1 − e2 )r(− cos θ)
=0⇒
=0
2
2
∂e
r (1 − e cos θ)
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MARYLAND
7
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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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8
!
1 − eopt cos θ
1 − e2opt
"
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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
UNIVERSITY OF
MARYLAND
9
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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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MARYLAND
10
gd
d
2− 2 − =0
V
r
2 − dr
d
−
gd
r
d"
=2 2−
gd
r
!
gd
2 − dr
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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
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MARYLAND
11
Hop on Sphere
Glide on Sphere
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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
UNIVERSITY OF
MARYLAND
12
(0 ≤ ρ ≤ 1)
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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
d = Vh ttotal =
Vh Vv
νv =
g
n
!
hmax
V
dg
1
1
η≡
ν≡√
d/n Vh = 2nV νh =
dg
2 2nη
v
UNIVERSITY OF
MARYLAND
13
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Multiple Hop Analysis
!
∆ν = 2ν + 2(n − 1)νv
∆ν = 2 νv2 + νh2 + 2(n − 1)νv
!
!
2η
1
2η
∆ν = 2
+
+ 2(n − 1)
n
8nη
n


(
$
%
1
∂∆ν
2
1
2


+ (n − 1)
= #
−
=0
2
2η
∂η
n 8nη
nη
1
+
n
8nη
1
Analytically messy, but note that for n = 1 ⇒ ηopt =
4
(In general, solve numerically)
UNIVERSITY OF
MARYLAND
14
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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
1000 10000
100
10000
Number of Hops (n)
Number of Hops (n)
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MARYLAND
1
15
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Hopping Between Different Altitudes
Relative to starting point, landing elevation ≡ h2
vv1 != vv2
v1 = (vh , vv1 )
v2 = (vh , vv2 )
2
vv1
1
v
1 2 t
v1
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 −
tf all =
(hpeak − h2 )
2 g!
g
vv2 =
2g(hpeak − h2 )
UNIVERSITY OF
MARYLAND
16
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Optimal Hop Between Different Altitudes
!
vv1
+
g
"
#
2
(hpeak − h2 )
g
$
d = vh (tpeak + tf all ) = vh
!"
#
2hpeak
2
d = vh
+
(hpeak − h2 )
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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MARYLAND
17
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Nondimensional Form of Equations
v
hpeak
h2
Remember that ν ≡ √ ; η ≡
;λ≡
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)
&2
UNIVERSITY OF
MARYLAND
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!$
"
"
+ 2η + #
1
%
√
2η + 2(η − λ)
&2
+ 2(η − λ)
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Optimization of Height-Changing Hop
• This is not going to be one where 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
!
1
1
∆ν = 2
+ 2η ⇒ ηopt =
8η
4
UNIVERSITY OF
MARYLAND
19
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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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MARYLAND
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1
2
3
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
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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MARYLAND
21
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Mars Phoenix Lander
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MARYLAND
22
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Apollo Lunar Module
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MARYLAND
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Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Landing Deceleration
• Look at 3 m/sec vertical velocity
• Constant force deceleration
1 2
F
v = d = adesired d
2
m
1
mv 2 = F d
2
tdecel =
v
adesired
• 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
adesired
d!cm"
1/6 g
281
1/2 g
92
0.61
1g
46
0.31
2g
23
0.15
3g
15
0.10
td !sec"
1.88
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Effect of Lateral Velocity at Touchdown
• Resolve torques around landing gear footpad
τtot
θ̈ =
Itot
w
h
mg
Fv
!
Fh h − Fv w − mgw
θ̈ =
Icg + m"2
Fh
• Worst cases - hit obstacle (high force), landing
downhill
• Issue: rotational velocity induced is
counteracted by vehicle weight
• Will vehicle rotation stop before overturn limit?
UNIVERSITY OF
MARYLAND
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Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Landing Vehicle Model
from R. E. Lavender, “Touchdown Dynamics Analysis of Spacecraft for Soft Lunar Landing”
NASA TN D-2001, Jan. 1964
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MARYLAND
26
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Planning for October 9
• ENAE 483 Initial Concept Review
– Each of six teams has 10 minutes to present their
work
– Discussion of options considered, reference missions,
candidate requirements, analyses performed
– Sketch/drawing of “final” candidate design(s) with
critical parameters (e.g., mass, performance)
• ENAE 788D Initial Concept Review
– Individuals turn in document (presentation slides or
report) on Mars GAS concept
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MARYLAND
27
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
Discussion of 788D Initial Project
• Perform research on NASA Mars program,
other countries’ plans for Mars exploration
• Consider possible experiment payloads
– Technology development for Mars sample return
– Surface transport concepts (e.g., innovative rovers,
balloons, aircraft, kites?, ballistic hoppers?
– Subsurface access (drills, excavation)
– Precursor information for human exploration (Cr(VI))
– Rough terrain access (wall climbers)
– Outside-the-box thinking?
UNIVERSITY OF
MARYLAND
28
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design
More Discussion of 788D Project
• Assume total discretionary payload of 100 kg,
allocated in 10 kg units
– Can use multiple 10 kg units if candidate payload
needs to be bigger
– Utility of payload must be commensurate with size!
• Assume the lander structure will provide DTE
communications relay and limited local power
– Can use UHF for relay through orbital assets
– Have to supply power if independent of lander
UNIVERSITY OF
MARYLAND
29
Suborbital Mobility and Landing Dynamics
ENAE 483/788D - Principles of Space Systems Design