Control Heuristics for Soft Landing Problem'
Togay Tanyolag and Hakan Yasarcan
Industrial Engineering Department
Bogazici University
Bebek — Istanbul 34342 — Turkey
caloynat@gmail.com; hakan.yasarcan@boun.edu.tr
Abstract
In this paper, we developed two different control heuristics for the soft landing problem.
The first heuristic is adapted from the mass spring damper model using the similarity of
the equations of the soft landing model given in this paper to the equations of the mass
spring damper model; both models can be reduced to a second order linear differential
equation. The second one is a bang-bang heuristic that first allows the spacecraft to fall
freely, but after a critical point is reached, it uses the reverse force thruster at its
maximum power until the touchdown. Bang-bang heuristic minimizes the time needed to
land. However, it may crash the spacecraft in the presence of an error in the parameter
estimates, or an error in the velocity or height readings, or an overlooked factor such as
a delay in changing the level of the force created by the reverse force thruster, which is
known as actuator delay. The mass spring damper based control heuristic requires a
longer landing time, but it is more robust compared to the bang-bang control heuristic in
the sense that it is less sensitive to the errors in parameter values, errors in readings, and
presence of an actuator delay.
Keywords: soft landing; spacecraft; control heuristic; mass spring damper; bang-bang;
error in parameter estimates; error in readings; actuator delay.
Introduction
In some cases, landing on the surface of a celestial body is a part of a space
exploration program. In such cases, soft landing becomes a problem to be addressed. A
reasonable landing process requires a control heuristic that will ensure the safety of the
spacecraft, which practically means a soft touchdown of the spacecraft to the surface of
the celestial body at the end of the landing process. Note that the crash force (the force
created at the time of touchdown) is a complex result of the crash velocity (the velocity
with which the spacecraft touches the surface), the landing gear parameters of the
This research is supported by a Marie Curie International Reintegration Grant within the 7th European
Community Framework Programme (grant agreement number: PIRGO7-GA-2010-268272) and also by
Bogazici University Research Fund (grant no: 5025-10A03P9).
spacecraft, the mass of the spacecraft, and the gravitational force. Out of these four
important components that determine the crash force, a control heuristic can only have an
effect on the crash velocity. Moreover, this effect is indirect. Control heuristic determines
the control force, control force results in the net force, net force determines the
acceleration, acceleration gradually adjusts the velocity, and the value of velocity at the
time of touchdown becomes the crash velocity. Therefore, it’s not an easy task to manage
the crash velocity at around a desired level. Moreover, the heuristic that will be employed
should also manage the length of the time needed to land at a reasonably low value
because a long landing duration requires extensive fuel usage. The two criteria,
minimizing the crash velocity and minimizing the length of the time needed to land, are
contradictory, which makes the soft-landing problem a challenging task. A control
heuristic aiming to satisfy the two criteria, should allow the vehicle descend to the
surface rather quickly, but make it decelerate safely to low velocity values before the
instant of landing (Liu, Duan, and Teo, 2008; Zhou et al., 2009). The landing dynamics
of Apollo 15 is an example of this strategy (Figure 1).
5000
4000 —
3000
é
2000
1000
0
12 31 56 72 79 88 101 114 123 132 140 148 158 169
Second
Figure 1: The landing dynamics of Apollo 15
In plotting the dynamics observed in Figure 1, we connected to the Apollo 15 entry of
the Wikipedia website (http://en.wikipedia.org/wiki/Apollo_15; accessed on 16
September 2011) and time coded the landing video on the page
(http://en.wikipedia.org/wiki/File:Apollo_15_landing_on_the_Moon.ogg; accessed on 16
September 2011). Note that Apollo 15 was the fourth to land on the Moon (30 July 1971).
In this paper, we first presented the stock-flow diagram and the equations of the soft
landing model. Later, we developed two control heuristics; a mass spring damper based
control heuristic and a bang-bang control heuristic. The mass spring damper based
control heuristic is adapted from the mass spring damper model using the similarity of the
equations of the soft landing model given in this paper to the equations of the mass spring
damper model; both models can be reduced to a second order linear differential equation.
The bang-bang heuristic dynamically calculates a critical point. It first allows the
spacecraft to fall freely, but after the critical point is reached, it uses the reverse force
thruster at its maximum power until the touchdown.
The behaviors obtained from the two control heuristics are also presented and
discussed in the paper. Bang-bang heuristic minimizes the time needed to land under the
assumed conditions. However, this aggressive management of the time needed to land
may make it crash the spacecraft under problematic conditions. We tested the
performances of the two heuristics in the presence of an error in the parameter estimates;
in the presence of an error in the velocity or height readings; and in the presence of an
overlooked factor such as a delay in changing the level of the force created by the reverse
force thruster, which is known as actuator delay. The mass spring damper based control
heuristic requires a longer landing time, but it is more robust compared to the bang-bang
control heuristic in the sense that it is less sensitive to the errors in parameter values,
errors in readings, and presence of an actuator delay.
The Model Structure and Equations
In this study, we first constructed a stock-flow model of the soft-landing problem,
which is given in Figure 2. This diagram represents only the physical structure of the
problem described in the previous section; it does not represent the controller (e.g. a
human decision maker, a computer). Height and Velocity are the two stock variables in
the model. Velocity, which is a stock variable, is at the same time the one and only flow
of Height. Velocity has a single flow too; Acceleration. Height is controlled via Velocity,
Velocity via Acceleration, Acceleration via Net Force, and Net Force via Control Force
(equations 1-7). The control feedback loop also includes the controller (Figure 3), which
determines Control Force of the reverse force thruster via Desired Control Force. Note
that the natural inputs to the controller are Height and Velocity.
Height, =1000 [m] (dd)
? One Newton amounts to the force needed to increase the velocity of a one kilogram body of mass by one
meter per second in one second (N = kg: m/s”).
Height,, », = Height, +Velocity,-DT [ml] (2)
Velocity, =-10 [m/s] GB)
Velocity, , y» = Velocity, + Acceleration: DT [m/s] (4)
Acceleration = Net Force / Mass [m /s° (5)
Mass = 1000 [kg] (6)
Net Force =—Gravitational Force + Damping Force + Control Force [Nv] (7)
Gravitational
Acceleration
Desired
Control Force
Max Force
Gravitational
Mass Force Control Force
Net Force
Acceleration oN,
Suspension
Damping Spring Coefficient
A Force
Height
ity’ Suspension
Seah Damper Coefficient
Spring
Compression
Figure 2: Simplified stock-flow diagram of the soft landing model
+
Net TOES gy
Acceleration
Control Force +
Velocity
CONTROLLER
Desired Control
Force a,
Figure 3: Causal-loop diagram of the control feedback loop structure
The simplifying model assumptions are given below:
e The movement of the spacecraft in the horizontal axes is not modeled. Spacecraft is
assumed to move only vertically.
Positive Height, Velocity, Acceleration, and force directions are upward from the
surface.
e There is no atmosphere in the landing area, thus no air friction exists that would
cause a drag force on the vehicle.
Gravitational Acceleration is assumed to be constant during landing, it does not
change with the distance to the surface.
e Mass is a constant, the change in the mass due to fuel consumption is ignored.
There are no delays caused by actuators; Desired Control Force generated by the
controller affects Control Force without a time lag.
Information flow from the system to the controller is perfect and instantaneous;
There are no errors or delays caused by measurement processes.
e Upon touching the ground, the thruster is off and is not switched on again. The
simplified model diagram in Figure 2 and Equation 8 do not reflect this assumption.
The aim of this paper is not to discuss the modeling process. Moreover, by giving the
simplified version of the model in Figure 2, we aim to improve the readability of the
manuscript and prevent digression. If more details are needed, see Yasarcan and
Tanyolag (2012).
The rest of the model equations follow:
Control Force =
0, Height <0
: Height > 0, (8)
Desired Control Force, [N]
Desired Control Force < Max Force
Max Force, otherwise
Max Force = 30,000 [N] (9)
Gravitational Force = Mass -Gravitational Acceleration [NV ] (10)
Gravitatio nal Acceleration = 8.87 |m/s? | (11)
Damping Force =
0, Spring Compression = 0
Suspension Suspension (12)
; Spring ; eal
Spring . __|-| Damper -Velocity, otherwise
_ Compression .
Coefficient Coefficient
0, Height >0
Spring Com pression = ee , [m] (13)
— Height, otherwise
Suspension Spring Co efficient = 17,740 [N /m] (14)
- Bd N-s
Suspension Damper Coefficient = 2,803 | —— (15)
m
3. A Mass Spring Damper Based Control Heuristic
The stock-flow model given in Figure 2 represents only the physical structure of the
soft landing problem. However, the simulated behavior discussed in the previous section
is generated by the model including the suggested mass spring damper based control
heuristic, which is assumed to be used by the controller (Figure 3) in producing the
values for Desired Control Force. The aim of this section is to present the formulations
of this heuristic.
Yasarcan and Barlas (2005) uses a procedure in developing control heuristics for
control problems involving information delay or indirect control via a secondary-stock.
This procedure adapts a well known successful heuristic for control problems involving
material supply line delay, using the similarity of the differential equations of control
problems involving different types of delay structures. The model presented in this paper
can be reduced to a second order linear differential equation because it contains two stock
variables, which are defined by approximate integral equations (Equation 2 and Equation
4). The mass spring damper model is well studied and it is known how to obtain a certain
behavior by adjusting the model parameter values. Furthermore, it can also be
represented by a second order linear differential equation. Utilizing an approach similar
to the approach of Yasarcan and Barlas (2005), we developed a heuristic based on the
similarity of the differential equations of the mass spring damper model and the model
presented in this paper’.
|i}
@:
Figure 4: Mass spring damper schematic
The schematic given in Figure 4 is a well known one. The differential equation of a
non-driven (i.e. Fexernai = 0) mass spring damper model with mass m, spring constant k,
and damper coefficient c is given below:
m-X+ce-x+k-x=0 (16)
In Equation 16, x represents displacement, x represents velocity, and ¥ represents
acceleration. This equation can be described by using stock-flow concepts, x and x
being the stocks and their associated flows being x and ¥ respectively. Note that x is a
flow and a stock at the same time. As a further clarification, —k-x is the spring force
(F
spring
,) and —c-x is the damper force (F,
damper
). The net force applied on the body of
mass is the sum of these two forces (F,,, = F, + F ing =—¢€:X—k-+x). According to
‘net damper spring
Newton’s second law of motion mass times acceleration is equal the net force acting on
the body (F,
net
=m-x). Therefore, mass times acceleration is equal to the sum of the
spring force and damper force. Hence, Equation 16 is obtained.
° The authors of this paper acknowledge that it is Dr. I. Emre Kése who suggested us to use the mass spring
damper model for this purpose.
The damping ratio ¢ of the mass spring damper model defined by Equation 16 is:
(17)
The dynamics of the mass spring damper model can be underdamped, overdamped, or
critically damped depending on the value of the damping ratio ¢. For € values under 1, the
dynamic behavior is underdamped and for € values over 1, it is overdamped. The case
where the damping ratio ¢ is exactly | is called critically damped. When the dynamic
behavior is underdamped, the spring dominates the movement and the body oscillates. In
the critically damped case, the body asymptotically approaches the rest condition without
an overshoot. In the overdamped case, the damper dominates the dynamics and the body
approaches the rest condition slower compared to the critically damped case (Astrém and
Murray, 2008). As a summary, the importance of € is that determining its value
determines the dynamics of the mass spring damper model.
The suggested control heuristic is adapted from the mass spring damper model that is
defined by Equation 16. Height, Velocity, Acceleration, and Mass in our model
corresponds to x, «, ¥, and m in Equation 16, respectively. In the heuristic, we named
k as Height Coefficient and c as Velocity Coefficient. Thus, Equation 16 becomes:
. Velocity . Height .
Mass « Acceleration + . Velocity + . - Height =0 (18)
Coefficient Coefficient
Utilizing Newton’s second law of motion, the following can be written:
. . Velocity . Height .
Desired Net Force =-| . Velocity — + Height [N] (19)
Coefficient Coefficient
The reverse force thruster should also counteract Gravitational Force. Hence,
Desired Control Force, which is the output of the heuristic and an input to Control Force
(see Equation 8 and Figure 2), can be given as:
Desired Control Force = Desired Net Force + Gravitational Force [N] (20)
The parameters of the adapted heuristic, Height Coefficient and Velocity Coefficient
values are set to 10 [N/m] and 200 [NV : s/m], respectively. Consequently, the damping
ratio ¢ for our model becomes:
_ Velocity Coefficient _ 200 _ (21)
2- {Mass Height Coefficient 2- 1000-10
g
The value of the damping ratio means that the suggested control heuristic produces a
critically damped behavior for the height of the spacecraft.
3.1. Selection of the Controller Parameters
Decreasing Control Force Damping Factor (CFDF) shortens the landing duration and
increases the final velocity (See Figure 5). Long landing durations and also great final
velocity values should be avoided. Therefore, a CFDF value with a reasonable landing
duration and final (crash) velocity should be selected.
140 60
120 50
100
>
2
5
2 80 8
2
8 30 5
8 a
5 60 3
3
20 £
40
20 10
0 0
0 05 1 #15 2 25 3 35 4 45 5
Control Force Damping Factor
— — Crash Time Crash Velocity (absolute)
Figure 5. Crash Time (the time of touchdown) and
the absolute value of Crash Velocity (the velocity at the time of touchdown)
variation with different Control Force Damping Factor values
The final velocity should be less than -10 m/s to be able to obtain a safe landing, so
CFDF should minimally be 1.6. The CFDF value 2 has a special mathematical
significance; it is the minimal value that makes the vehicle asymptotically’ seek the
ground level and is not affected by the initial conditions. Due to this mathematical
property, CFDF is taken as 2.
3.2. Adjustment to the Mass Spring Damper Based Heuristic
Two adjustments to the heuristic is necessary:
e As mentioned in the previous sub-section, there is a problem with the asymptotical
approach; the vehicle continues to hover on the ground with a very small distance
away from the surface. We corrected this problem by adding Desired Final Velocity
to the heuristic. Note that Equation 19 implicitly assumes that the heuristic seeks
Velocity = 0. Therefore, we replaced Equation 19 with Equation 22. The existence
of a negative Desired Final Velocity makes the vehicle approach the ground level
with an acceptable velocity. Hence, the problem of the infinite landing duration due
to the asymptotical seek is avoided.
The heuristic should stop engines at the time of first touchdown. We replaced
Equation 20 with Equation 24 so that upon touching the ground, the thruster is off
and is not switched on again. The variable Landing State is given in equations 25-
26. Note that Landing State equations are valid for the bang-bang control heuristic
too.
Desired Net Force =
Velocity . Desired Final Height . (22)
- a -| Velocity — - . - Height [N ]
Coefficient Velocity Coefficien t
Desired Final Velocity = —1.2 [m /s] (23)
Desired Desired Gravitatio nal
eure [ * [ ane } Landing State =0
fe
Control |= 4\ Net Force Force [N. ] (24)
Force 0, Landing State =1
Landing State, =0 [dimension! ess] (25)
* To be mathematically correct, asymptotical seek of the goal takes indefinite time. This issue will be
addressed in the next sub-section.
Landing State, ,y, =
VDT, Landing State, = 0, Height, <0
26)
Landing State, + 0
. [dimension! ess] ‘
i otherwise
3.3.Dynamic Behavior of Landing Obtained by Using the Mass Spring
Damper Based Heuristic
As described in the previous section, Height is controlled via Velocity (Equation 2),
Velocity via Acceleration (Equation 4), Acceleration via Net Force (Equation 5), and Net
Force via Control Force (Equation 7). The control feedback loop also includes the
controller, which determines Control Force applied by the reverse force thruster via
Desired Control Force. In order to obtain a reasonable value for Desired Control Force,
the controller should consider the system state variables (i.e. Height and Velocity). Only
by doing so is it possible to reach the aim of landing the spacecraft as gently and as fast
as possible. Even under the simplifying assumptions listed in the previous section, the
control task remains a challenging one because it is quite difficult to appropriately
consider the system state information in the decisions. The main reason for the difficulty
is that the control task requires simultaneous control of Height and Velocity, which —due
to the physical structure of the problem— can only be indirectly affected by the reverse
force thruster; Height and Velocity have inertia; their values do not change
instantaneously (see Figure 2 and equations 1-7).
The stock-flow model given in Figure 2 and defined by equations 1-9 describes the
structure of the soft landing problem excluding the controller. The formulations of the
heuristic suggested for the controller is explained in the next section. The dynamic
behavior presented in figures 6-4 is generated by simulating the model including the
controller with the proposed heuristic for 60 seconds (equations 1-13 and equations 19-
20).
The dynamic behavior of Height is given in Figure 6. Initially, the change in Height
(ie. Velocity) is relatively fast and, as the spacecraft approaches to the surface, the
change in Height slows down. Hence, the behavior obtained by the control heuristic is a
reasonable one; by a fast initial decline, the heuristic tries to decrease the time to land; by
a slow final approach, it keeps the impact force well below harmful values. At the instant
of touchdown, the value of Velocity is -2.04 meters per second (-7.35 km/h) creating a
maximum impact force of circa 14,782 Newton, approximately 1.67 times the weight of
the spacecraft on the target celestial body (8,870 Newton). The weight corresponds to the
adTs
model variable Gravitational Force, which is the force that the landing gear must bear
when the spacecraft is standing still on the ground.
1: Height
1 10009 5
s} 500: 1
a
HL
1 0
0.00 15.00 30.00 45.00 60.00
IPage 1 Second
Figure 6: Dynamic behavior of Height
The dynamic behavior of Velocity and Net Force acting on the vehicle during landing
are given in figures 7 and 8, which further explain the dynamic behavior obtained by the
control heuristic. At first, the heuristic allows the spacecraft to accelerate in the negative
direction towards the landing surface (see Figure 7, approximately within the time range
of 0-10 seconds) by keeping Net Force negative (ie. Control Force less than
Gravitational Force, see figures 8 and 9). Aiming to decrease the duration of landing,
Velocity continues to increase during this initial period. After this initial phase, Velocity
decreases until the vehicle touches the surface (see Figure 7, approximately within the
time range of 10-55 seconds). In this later phase, the heuristic produces more Control
Force than Gravitational Force (Figure 9) resulting in a positive Net Force (Figure 8). At
the moment of landing, Control Force is turned off and Damping Force, which is zero
throughout the simulation up to this point, takes over and stops the vehicle (see figures 8
and 9, approximately around 55 seconds).
4: Velocity
1 10:
1
a
1 -15:
1
4
1 -40:
0.00 15.00 30.00 45.00 60.00
Page 1 Second
Figure 7: Dynamic behavior of Velocity
1: Net Force
1 9000:
Ry
F é i A
1
1 -9000'
0.00 15.00 30.00 45.00 60.00
Page 1 Second
Figure 8: Net force acting on the vehicle during landing
-13-
4: Damping Force 2: Gravitational Force 3: Control Force
15000:
5000:
1
2
3 -5000:
0.00 15.00 30.00 45.00 60.00
Page 1 Second
Figure 9: Absolute values of the forces acting on the vehicle during landing?
4. A Bang-Bang Control Heuristic
The bang-bang principle relies on the fact that a system can be controlled in minimal
time using properly all available power throughout the whole control (LaSalle, 1960).
Based on this principle, we developed a bang-bang control heuristic for our model. The
purpose is to let the vehicle descend with only the effect of Gravitational Force in an
accelerating fashion up to a point in time, and then apply the maximum possible force
until touchdown. Note that; in our model, Control Force is in the positive Height
direction and Gravitational Force is the only force in the negative direction that can pull
the vehicle towards the ground. The maximum force generated by the reverse force
thruster creates Maximum Acceleration (Equation 27). Actually, during the time the
maximum force is applied, the spacecraft is moving towards the celestial body (in the
negative direction). Therefore, Maximum Acceleration decelerates the negative speed of
the vehicle to a desired level (Equation 28).
Maximum . 3
= (Max Force — Gravitatio nal Force )/ Mass [m /s | (27)
Acceleration
Desired Final Velocity = —2 [m /s| (28)
5 In order to ease the comparison of the different forces acting on the vehicle, the directions of the forces
are ignored on this diagram.
We call the time that Max Force is first applied as Deceleration Start Time.
Acceleration in the positive direction (or deceleration in the negative direction) is a
constant and equal to Maximum Acceleration (Equation 27) between Deceleration Start
Time and Crash Time (the time of touchdown). The bang-bang heuristic dynamically
decides when to use the reverse force thruster at its maximum power, by looking at
current Velocity, current Height, Maximum Acceleration, and Desired Final Velocity
values. In order to be able to bring the current Velocity to Desired Final Velocity at the
time of touchdown, there should be a sufficient remaining distance between the vehicle
and the surface of the planet (i.e. Height) for the given Maximum Acceleration and
current Velocity values (Equation 29).
Desired Net Force =
Desired Maximum
Max Force, Velocity? —| Final 2 2+ Height -| Possible Iv] (29)
Velocity Acceleration
—Gravitional Force, otherwise
4.1. Derivation of Desired Net Force Equation
Equation 29 is equivalent to Equation 30, where current Velocity corresponds to vo,
Desired Final Velocity to vi, current Height to Ax, and Maximum Acceleration to a.
Equation 31 is a natural result of conservation of mechanical energy (Serway and
Faughn, 1989) and Equation 30 is obtained by dividing Equation 31° by mass.
v, -y =2-a-Ax [m/s?| (30)
1 2 1
=m-v, -=m-y,? =m-a-Ax [J]
2 2 G1)
4.2.Dynamic Behavior of Landing Obtained by Using the Bang-Bang
Heuristic
The dynamic behavior of Height is given in Figure 10. Initially, the change in Height
(ie. Velocity) is relatively fast and, as the spacecraft approaches to the surface, the
® One Joule amounts to the work done by applying a force of one Newton through a distance of one meter.
(J=N-m=kg-m’/s’).
change in Height slows down. At the instant of touchdown, the value of Velocity is -3.28
meters per second (-11.81 km/h) creating a maximum impact force of circa 17,869
Newton, approximately 2.01 times the weight of the spacecraft on the target celestial
body (8,870 Newton).
1: Height
1 1000, 4-1:
mM
iad)
0.00 5.00 10.00 15.00 20.00
IPage 1 Second
1 0
Figure 10. Dynamic behavior of Height in the Bang-Bang Heuristic
The dynamic behavior of Velocity and Net Force acting on the vehicle during landing
are given in figures 11 and 12, which further explain the dynamic behavior obtained by
the control heuristic. At first, the heuristic allows the spacecraft to accelerate with the
effect of Gravitational Acceleration in the negative direction towards the landing surface
(see Figure 12, approximately within the time range of 0-11 seconds) by keeping Net
Force equal to Gravitational Force (see figures 12 and 13). Aiming to decrease the
duration of landing, Velocity continues to increase during this initial period. After this
initial phase, Velocity decreases until the vehicle touches the surface (see Figure 11,
approximately within the time range of 11-17 seconds). In this later phase, the heuristic
produces Control Force equal to Max Force (Figure 12) resulting in a positive Net Force
(Figure 13). At the moment of landing, Control Force is turned off and Damping Force,
which is zero throughout the simulation up to this point, takes over and stops the vehicle
(see figures 12 and 13, approximately around 17 seconds).
-16-
4: Velocity
1 50:
"1. va
4
1 -50" a
yi:
1 -150°
0.00 5.00 10.00 15.00 20.00
Page 1 Second
Figure 11. Dynamic behavior of Velocity
1: Net Force
1 30000:
—_
1 1000059
a 1 bees
1 -10000:
0.00 5.00 10.00 15.00 20.00
Page 1 Second
Figure 12. Net force acting on the vehicle during landing
aif
1: Damping Force 2: Gravitational Force 3: Control Force
1 30000:
2
3
1
2 15000
, \
1
2
3 opt 1 1 i
0.00 5.00 10.00 15.00 20.00
Page 1 Second
Figure 13. Absolute values of the forces acting on the vehicle during landing
5. The Comparison of the Two Control Heuristics
The mass spring damper heuristic (MSD) and the bang-bang heuristic presented in the
previous sections have different characteristics. The differences between the two
heuristics and the difference in the resulting behavior is explained in this section and a
summary is provided in Table 1.
Table 1. Comparison of Mass Spring Damper and Bang Bang Heuristics
MSD Bang-Bang
Changes in Control Force smooth catastrophic
-3.28
(Numerical error is big.
Crash Velocity [m/s] -2.04 | Theorethically, it should be -2.)
17869
(Numerical error is big.
Theorethically, it should be close
to the result generated by the MSD
Max Landing Force [N] 14782 based heuristic.)
Crash Time [s] 55.46 16.66
Sensitivity to errors in parameters low high
Sensitivity to variable readings low high
Sensitivity to Actuator Delay low high
The qualitative comparison of the velocity figures 7 and 11 gives a preliminary
insight to the difference in the smoothness of the control. Furthermore, the comparison of
the net force figures 8 and 12 reveals that the bang-bang control heuristic makes a sudden
jump in the force, whereas the mass spring damper heuristic changes force gradually.
Max Instantaneous Change in Force (Equations 42-43)’ quantify the momentary
difference in force as 30,000 Newton in the bang-bang heuristic. In the mass spring-
damper heuristic, however, Max Instantaneous Change in Force is very small and only
exists due to the discrete nature of the simulation; it approaches zero as DT (simulation
time step) goes to zero’. Therefore, changes in Control Force is smooth with the mass
spring damper based control heuristic and catastrophic with the bang-bang heuristic.
Crash Velocity is another important criterion like Crash Time, and equations 34 and
35 are necessary for monitoring it. According to our simulation runs, the bang-bang
heuristic and the mass spring damper heuristic landed with velocities of -3.28 m/s and
-2.04 m/s, respectively. It is worth noting that Crash Velocity of the bang-bang heuristic
should theoretically be equal to -2 m/s, which is the value of Desired Final Velocity (see
Equation 28). Extreme forces used by the bang-bang heuristic results in bigger simulation
errors compared to the mass spring damper based heuristic. As a summary, the two
heuristics land the vehicle at the same speed.
The variable Crash Time (i.e. the duration of the landing process) captured by the
performance measure equations 32 and 33 are about 55 seconds for the MSD heuristic
and about 17 seconds for the bang-bang heuristic. This result was not a surprising one as
the bang-bang heuristic is the minimum-time solution for our problem. Considering that
minimizing the time to land is one of the main criteria, bang-bang heuristic seems very
successful. However, the aggressive management of the time needed to land may make
the bang-bang heuristic crash the spacecraft under problematic conditions.
5.1. An Error in one of the Parameter Values
To be able to compare the deterioration in the results, we assumed that the estimate of
Mass used in the heuristics is 950 kg instead of 1000 kg. The dynamic behavior generated
by the two heuristics in the presence of this error is given in figures 14 and 15. The Crash
Velocity values for the MSD and bang-bang heuristics deteriorate to -2.21 m/s and -25.59
m/s, respectively. These values suggest that, in the case of a parameter estimation error, a
great deterioration in the bang-bang heuristic occurs, whereas MSD succeeds in making a
reasonable landing.
7 The performance measure equations are given in the appendix.
*In our simulations, we set DT (simulation time step) equal to 2° (1/512) seconds.
Height
1 10005~
1 500 :
-~—__|
1 0 =
0.00 16.00 30.00 45.00 60.00
Page 1 Second
Figure 14. Landing behavior generated by the MSD heuristic in the presence of an error
in the Mass estimate
4: Height
1 1000, 41
i
1 500, Xi
1
1 o ri
0.00 5.00 10.00 16.00 20.00
IPage 1 Second
Figure 15. Landing behavior generated by the bang-bang heuristic in the presence of an
error in the Mass estimate
5.2. An Error in Height Readings
We assumed that there is an error in Height readings; it is read as 10 meters more
than it actually is at all times during the simulation. The dynamic behavior generated by
the two heuristics in the presence of this error is given in figures 16 and 17. The Crash
Velocity values for the MSD and bang-bang heuristics deteriorate to -2.86 m/s and -20.76
m/s, respectively. Similar to the case with an error in the parameter estimates, the
Boies
behavior generated by the bang-bang heuristic deteriorates in a qualitatively significant
manner.
4: Height
1 1000: ry
1 500 i
~~ __|
1 0 =
0.00 16.00 30.00 48.00 60.00
lPage 4 Second
Figure 16. Landing behavior generated by the MSD heuristic in the presence of an error
in Height readings
1: Height
1 1000, 9= 4:
1
‘: 500,
1,
1 0 ="
0.00 5.00 10.00 15.00 20.00
Page 1 Second
Figure 17. Landing behavior generated by the bang-bang heuristic in the presence of an
error in Height readings
5.3. The Presence of an Actuator Delay
In this sub-section, we assumed that there is an overlooked factor present in the
model, an actuator delay (i.e. a delay in changing the level of the force created by the
reverse force thruster) of 2 seconds. The presence of this delay creates no significant
“91 =
change in the behavior generated by the MSD heuristic (Figure 18) and the new Crash
Velocity value generated by this heuristic is -1.75 m/s. However, a huge deterioration in
the behavior generated by the bang-bang heuristic is observed (Figure 19). The new
Crash Velocity value generated by this heuristic is -120.48 m/s. Similar to the cases with
an error in the parameter estimates and with an error in readings, MSD heuristic manages
a safe landing, proving its robustness. However, bang-bang heuristic is quite unreliable in
the presence of aforementioned issues.
1: Height
1 10009 4
1 500:
1
rt
a)
1 0: :
0.00 15.00 30.00 45.00 60.00
IPage 1 Second
Figure 18. Landing behavior generated by the MSD heuristic in the presence of actuator
delay
4: Height
1 1000, ma
ft 500,”
i
\ |
1 0
0.00 5.00 10.00 15.00 20.00
Page 1 Second
Figure 19. Landing behavior generated by the bang-bang heuristic in the presence of
actuator delay
=22«
6. Conclusions and Future Research
In this study, we first developed a soft landing model using System Dynamics
methodology. The soft landing challenge can simply be summarized as trying to land a
spacecraft on the surface of a celestial body as gently and as fast as possible. The main
reason for the challenge is that the control task requires simultaneous control of the
height and velocity of the spacecraft, which have inertia and can only be indirectly
affected by the reverse force thruster. We also presented two control heuristic. The first
one is adapted from the mass spring damper model and the second one is a bang-bang
heuristic. According to the initial simulation runs that we obtained, the bang-bang
heuristic quickly lands the spacecraft at the end of a very brief landing period. However,
it is not robust in the sense that it is over sensitive to the presence of errors in the
parameter estimates and errors in the velocity or height readings. It is also very sensitive
to the presence of an actuator delay (a delay in changing the level of the force created by
the reverse force thruster). On the other hand, the mass spring damper based control
heuristic requires a longer landing time, but it is more robust compared to the bang-bang
control heuristic in the sense that it is much less sensitive to the aforementioned
problems.
Note that, a longer actuator delay may make the mass spring damper control heuristic
create problematic behaviors too. In the continuation of this study, we plan to focus on
addressing this issue by further improving the mass spring damper based control
heuristic. In order to overcome the possible problematic behaviors, we plan to adapt and
use the heuristics developed by Yasarcan and Barlas (2005) and Yasarcan (2011), which
are specifically suitable for this kind of control problems. It is also possible to develop a
soft landing game based on the model as a platform for learning and dynamic decision
making experimentation.
Acknowledgement
We presented an earlier draft of this work in the paper titled “4 Soft Landing Model
and a Mass Spring Damper Based Control Heuristic” at the 29th International
Conference of the System Dynamics Society, 24-28 July 2011, Washington DC — USA.
We improved that previous work and divided it into two parts. One part is this paper and
the other part is the paper titled “4 Soft Landing Model and an Experimental Platform as
an Introductory Control Design Too!’, which is also presented at this conference.
-23-
References
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Appendix: Performance Measure Equations
Performance measure equations are used to evaluate the landing. Crash Time gives
the duration of the landing beginning with the initial conditions until the moment of
touchdown (equations 32-33). Crash Velocity is the velocity value at the moment of
touchdown (equations 34-35). Max Landing Force reports the maximum force that is
generated by the landing gear after touchdown (equations 36-37). Force Ratio gives a
scale of Maximum Landing Force compared to Gravitational Force (Equation 38). Note
that at static equilibrium the landing gear withstands Gravitational Force. Thus, Force
Ratio = | is the theoretical minimum. Max Acceleration gives the maximum acceleration
of the vehicle during landing (equations 39-40). Instantaneous Change in Net Force is the
absolute value of the change in Net Force between two consecutive time steps. It is
necessary for the calculation of Maximum Instantaneous Change in Net Force, which
reports the maximum of the changes in Net Force between two consecutive time steps.
The value of the Maximum Instantaneous Change in Net Force is a measure for the
smoothness of the control (equations 41-43).
Crash Time, = 0 {s] (32)
; ; t/DT, CrashTime, =0, Height, <0
Crash Time,» = Crash Time, +
| ble»
, otherwise
Crash Velocity, =0 [m /s] (34)
om } 7
Velocity 7
Is an
Crash Velocity, /DT , CrashVelocity, =0, Height, <0 f / |
+ m/s
Velocity ), 0, otherwise
Max Landing Force, =0 [N] (36)
Max Landing Force, y, =
Damping Force,
Max
" — Max Landing Force, ) ( Damping Max Landing (7)
Landing | +4 ~—————__——_—_~,, > [N]
DT Force Force
Force ‘
0, otherwise
Force Ratio = Maximum Landing Force: [dimension! ess] (38)
Gravitational Force
-25-
Max Acceleration, = 0 |m / s°|
Max Acceleration, , y- =
Max Acceleration,
Acceleration
Acceleration,
— Max Acceleration . Max
, Acceleration, >
DT
0, otherwise
Instantaneous
. = ABS(Net Force,,y, — Net Force,) [N]
Change in Net Force ),,_,,
Max Ins tan tan eous Change in Net Force, =0 [NV ]
Max Instantaneous Change in Net Force,, yr =
Max Instantaneous Change in Net Force,
Max
Instantaneous
Instantaneous
Change in al .
Changein
Net Force Instantaneous
+ ‘ \ Net Force
1 Changein
DT
Net Force
0, otherwise
Max
Instantaneous
Change in
Net Force
) lo]
G9)
(40)
(41)
(42)
i]
(43)
-26-
