Method and apparatus for powered descent guidance
A method and apparatus for landing a spacecraft having thrusters with non-convex constraints is described. The method first computes a solution to a minimum error landing problem for a convexified constraints, then applies that solution to a minimum fuel landing problem for convexified constraints. The result is a solution that is a minimum error and minimum fuel solution that is also a feasible solution to the analogous system with non-convex thruster constraints.
1. A method of computing a thrust profile to land a spacecraft at or near a surface target, wherein the spacecraft is subject to a non-convex thrust constraint, comprising the steps of:
computing a minimum landing error descent solution for the spacecraft subject to a convexified non-convex constraint, wherein the descent solution includes a landing error range; and
determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum landing error descent solution for the spacecraft subject to the convexified non-convex constraint.
2. The method of claim 1 , wherein the step of determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum landing error descent solution subject to the convexified non-convex thrust constraint comprises the steps of:
computing a minimum fuel descent solution for the spacecraft subject to the convexified non-convex constraint and the landing error range; and
determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum fuel descent solution for the spacecraft subject to the convexified non-convex constraint.
3. The method of claim 2 , wherein the step of computing a minimum landing error descent solution for the spacecraft subject to the convexified non-convex thrust constraint comprises the steps of computing
min
t
f
,
T
c
(
·
)
,
Γ
(
·
)
r
(
t
f
)
2
subject to the first set of constraints comprising the constraints of:
{umlaut over ( r )}( t )= g+T c ( t )/ m ( t ), {dot over ( m )}( t )=αΓ( t )
∥ T c ( t )∥≦Γ( t ), 0<ρ 1 ≦Γ( t )≦ρ 2
r ( t )∈ X∀t∈└ 0, t f ┘
m (0)= m wet , m ( t f )≧ m dry
r (0)= r 0 , {dot over (r)} (0)= {dot over (r)} 0
r ( t f ) T e 1 =0 , r ( t f )=0
wherein
r(t) is a position vector of the spacecraft as a function of flight time;
{dot over (r)}(t) is a velocity vector of the spacecraft as a function of flight time;
{umlaut over (r)}(t) is an acceleration vector of the spacecraft as a function of flight time;
t is the time from beginning of powered landing;
t(0) is the time at the beginning of powered landing
t(f)=t f is the time at the ending of powered landing;
T c (t) is a net thrust force vector acting on the spacecraft as a function of flight time;
m(t) is a mass of the spacecraft as a function of flight time;
{dot over (m)}(t) is a mass change of the spacecraft as a function of flight time;
X is the set of feasible positions of the spacecraft;
m(0) is the spacecraft mass at the beginning of the thrust profile;
m(t f ) is the spacecraft mass at the end of the thrust profile;
m dry is the initial mass of the spacecraft without fuel;
m wet is the initial mass of the spacecraft with fuel;
α
=
1
I
sp
g
e
cos
ϕ
,
wherein I sp is the specific impulse of the thruster, g e is the earth's gravitational constant, T 1 and T 2 are the lower and upper limits of the thrust force that can be provided by each thruster;
ρ i =nT 1 cos φ is the minimum thrust value available from the thruster(s);
ρ 2 =nT 2 cos φ is the maximum thrust value available from the thrusters(s);
e i is a unit vector of all zeros except the i th row, which is unity; and
Γ(t)=slack variable that bounds thrust magnitude.
4. The method of claim 3 , wherein
min
t
f
,
T
c
(
·
)
,
Γ
(
·
)
r
(
t
f
)
2
subject to the first set of constraints is computed as
min
N
,
η
Ey
N
2
,
subject to a second set of constraints comprising:
E
u
Y
k
η
≤
e
σ
T
Y
k
η
for
k
=
0
,
…
,
N
μ
1
(
t
)
[
1
-
(
Fy
k
-
z
0
(
t
k
)
)
+
(
Fy
k
-
z
0
(
t
k
)
)
2
2
]
≤
e
4
T
Y
k
η
≤
μ
2
(
t
)
[
1
-
(
Fy
k
-
z
0
(
t
k
)
)
]
Ey
k
∈
X
k
=
1
,
…
,
N
Fy
N
≥
ln
m
dry
y
N
T
e
1
=
0
,
E
v
y
N
T
=
0
y
k
=
Φ
k
[
r
0
r
.
0
ln
m
wet
]
+
Λ
k
[
g
0
]
+
Ψ
k
η
k
=
1
,
…
,
N
wherein
t
k
=
k
Δ
t
,
k
=
0
,
…
,
N
σ
(
t
)
=
Γ
(
t
)
m
(
t
)
u
(
t
)
=
T
c
(
t
)
m
(
t
)
z
(
t
)
=
ln
m
(
t
)
η
=
[
p
0
⋮
p
m
]
,
wherein p j is a vector of parameters and p j ∈R 4
E=[I 3×3 0 3×1 ], F=[ 0 1×6 1], E u =[I 3×3 0 3×1 ], E v =[0 3×3 I 3×3 0 3×1 ]
Y
k
η
=
[
u
(
t
k
)
σ
(
t
k
)
]
,
wherein u(t k )=is the control input, and
σ
(
t
k
)
=
Γ
(
t
k
)
m
(
t
k
)
μ 1 (t)=ρ 1 e −z 0 , wherein z 0 =a lower bound on ln m(t)
μ 2 (t)=ρe −z 0 , wherein z 0 =a lower bound on ln m(t)
y
k
=
[
r
(
t
k
)
r
.
(
t
k
)
z
(
t
k
)
]
r(t k ) is r(t)t k
{dot over (r)}(t k ) is {dot over (r)}(t) at t k
z(t k ) is z(t) at t=t k
e 4 T =A transpose of a vector of all zeros except the fourth element, which is unity
y N T =A transpose of y k at the final time step N
Φ k , Λ k , Ψ k , Y k are discrete time state transition matrices describing the solution to {umlaut over (r)}(t)=u(t)+g and
z
.
=
m
.
(
t
)
m
(
t
)
=
-
ασ
(
t
)
wherein g is the acceleration of gravity of the planet or object the spacecraft is landing on.
5. An apparatus for computing a thrust profile to land a spacecraft at or near a surface target, wherein the spacecraft is subject to a non-convex thrust constraint, comprising the steps of:
a spacecraft processor;
a memory, communicatively coupled to the processor, the memory for storing instructions comprising
instructions for computing a minimum landing error descent solution for the spacecraft subject to a convexified non-convex constraint, wherein the descent solution includes a landing error range; and
instructions for determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum landing error descent solution for the spacecraft subject to the convexified non-convex constraint;
a thruster, communicatively coupled to the spacecraft processor, for maneuvering the spacecraft according to the determined thrust profile.
6. The apparatus of claim 5 , wherein the instructions for determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum landing error descent solution subject to the convexified non-convex thrust constraint comprise:
instructions for computing a minimum fuel descent solution for the spacecraft subject to the convexified non-convex constraint and the landing error range; and
instructions for determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum fuel descent solution for the spacecraft subject to the convexified non-convex constraint.
7. The apparatus of claim 6 , wherein the instructions computing a minimum landing error descent solution for the spacecraft subject to the convexified non-convex thrust constraint comprises the instructions for computing
min
t
f
,
T
c
(
·
)
,
Γ
(
·
)
r
(
t
f
)
2
subject to the first set of constraints comprising the constraints of:
{umlaut over ( r )}( t )= g+T c ( t )/ m ( t ), {dot over ( m )}( t )=αΓ( t )
∥ T c ( t )≦Γ( t ), 0<ρ 1 ≦Γ( t )<ρ 2
r ( t )∈ X∀t∈└ 0, t f ┘
m (0)= m wet , m ( t f )≧ m dry
r (0)= r 0 , {dot over (r)} (0)= {dot over (r)} 0
r (t f ) T e 1 =0 , r ( t f )=0
wherein
r(t) is a position vector of the spacecraft as a function of flight time;
{dot over (r)}(t) is a velocity vector of the spacecraft as a function of flight time;
{umlaut over (r)}(t) is an acceleration vector of the spacecraft as a function of flight time;
t is the time from beginning of powered landing;
t(0) is the time at the beginning of powered landing
t(t)=t f is the time at the ending of powered landing;
T c (t) is a net thrust force vector acting on the spacecraft as a function of flight time;
m(t) is a mass of the spacecraft as a function of flight time;
{dot over (m)}(t) is a mass change of the spacecraft as a function of flight time;
X is the set of feasible positions of the spacecraft;
m(0) is the spacecraft mass at the beginning of the thrust profile;
m(t f ) is the spacecraft mass at the end of the thrust profile;
m dry is the initial mass of the spacecraft without fuel;
m wet is the initial mass of the spacecraft with fuel;
α
=
1
I
sp
g
e
cos
ϕ
,
wherein I sp is the specific impulse of the thruster, g e is the earth's gravitational constant, T 1 and T 2 are the lower and upper limits of the thrust force that can be provided by each thruster;
ρ i =nT 1 cos φ is the minimum thrust value available from the thruster(s);
ρ 2 =nT 2 cos φ is the maximum thrust value available from the thrusters(s);
e i is a unit vector of all zeros except the i th row, which is unity; and
Γ(t)=slack variable that bounds thrust magnitude.
8. The apparatus of claim 7 , wherein
min
t
f
,
T
c
(
·
)
,
Γ
(
·
)
r
(
t
f
)
2
subject to the first set of constraints is computed as
min
N
,
η
Ey
N
2
,
subject to a second set of constraints comprising:
E
u
Y
k
η
≤
e
σ
T
Y
k
η
for
k
=
0
,
…
,
N
μ
1
(
t
)
[
1
-
(
Fy
k
-
z
0
(
t
k
)
)
+
(
Fy
k
-
z
0
(
t
k
)
)
2
2
]
≤
e
4
T
Y
k
η
≤
μ
2
(
t
)
[
1
-
(
Fy
k
-
z
0
(
t
k
)
)
]
Ey
k
∈
X
k
=
1
,
…
,
N
Fy
N
≥
ln
m
dry
y
N
T
e
1
=
0
,
E
v
y
N
T
=
0
y
k
=
Φ
k
[
r
0
r
.
0
ln
m
wet
]
+
Λ
k
[
g
0
]
+
Ψ
k
η
k
=
1
,
…
,
N
wherein
t
k
=
k
Δ
t
,
k
=
0
,
…
,
N
σ
(
t
)
=
Γ
(
t
)
m
(
t
)
u
(
t
)
=
T
c
(
t
)
m
(
t
)
z
(
t
)
=
ln
m
(
t
)
η
=
[
p
0
⋮
p
m
]
,
wherein p j is a vector of parameters and p j ∈R 4
E=[I 3×3 0 3×1 ], F=[ 0 1×6 1], E u =[I 3×3 0 3×1 ], E v =[0 3×3 I 3×3 0 3×1 ]
Y
k
η
=
[
u
(
t
k
)
σ
(
t
k
)
]
,
wherein u(t k )=is the control input, and
σ
(
t
k
)
=
Γ
(
t
k
)
m
(
t
k
)
μ 1 (t)=ρ 1 e −z 0 , wherein z 0 =constant
μ 2 (t)=ρe −z 0 , wherein z 0 =constant
y
k
=
[
r
(
t
k
)
r
.
(
t
k
)
z
(
t
k
)
]
r(t k ) is r(t)t k
{dot over (r)}(t k ) is (t) at t k
z(t k ) is z(t) at t=t k
e 4 T =A transpose of a vector of all zeros except the fourth element, which is unity;
y N T =A transpose of y k at the final time step N
φ k , Λ k , Ψ k , Y k are discrete time state transition matrices describing the solution to {umlaut over (r)}(t)=u(t)+g and
z
.
=
m
.
(
t
)
m
(
t
)
=
-
ασ
(
t
)
wherein g is the acceleration of gravity of the planet or object the spacecraft is landing on.
9. An apparatus for computing a thrust profile to land a spacecraft at or near a surface target, wherein the spacecraft is subject to a non-convex thrust constraint, comprising:
means for computing a minimum landing error descent solution for the spacecraft subject to a convexified non-convex constraint, wherein the descent solution includes a landing error range; and
means for determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum landing error descent solution for the spacecraft subject to the convexified non-convex constraint.
10. The apparatus of claim 9 , wherein the means for determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum landing error descent solution subject to the convexified non-convex thrust constraint comprises:
means for computing a minimum fuel descent solution for the spacecraft subject to the convexified non-convex constraint and the landing error range; and
means for determining the thrust profile for the spacecraft subject to the non-convex constraint from the computed minimum fuel descent solution for the spacecraft subject to the convexified non-convex constraint.
11. The apparatus of claim 10 , wherein the means for computing a minimum landing error descent solution for the spacecraft subject to the convexified non-convex thrust constraint comprises means for computing
min
t
f
,
T
c
(
·
)
,
Γ
(
·
)
r
(
t
f
)
2
subject to the first set of constraints comprising the constraints of:
{umlaut over ( r )}( t )= g+T c ( t )/ m ( t ), {dot over (m)} ( t )=−αΓ( t )
∥ T c ( t )∥≦Γ( t ), 0<ρ 1 ≦Γ( t )≦ρ 2
r ( t )∈ X∀t∈└ 0, t f ┘
m (0)= m wet , m ( t f )≦ m dry
r (0)= r 0 , {dot over (r)} (0)= {dot over (r)} 0
r ( t f ) T e 1 =0 , r ( t f )=0
wherein
r(t) is a position vector of the spacecraft as a function of flight time;
{dot over (r)}(t) is a velocity vector of the spacecraft as a function of flight time;
{umlaut over (r)}(t) is an acceleration vector of the spacecraft as a function of flight time;
t is the flight time of the spacecraft
t(0) is the time at the beginning of powered landing
t(f)=t f is the time at the ending of powered landing;
T c (t) is a net thrust force vector acting on the spacecraft as a function of flight time;
m(t) is a mass of the spacecraft as a function of flight time;
{dot over (m)}(t) is a mass change of the spacecraft as a function of flight time;
X is the set of feasible positions of the spacecraft;
m(0) is the spacecraft mass at the beginning of the thrust profile;
m(t f ) is the spacecraft mass at the end of the thrust profile;
m dry is the initial mass of the spacecraft without fuel;
m wet is the initial mass of the spacecraft with fuel;
α
=
1
I
sp
g
e
cos
ϕ
,
wherein I sp is the specific impulse of the thruster, g e is the earth's gravitational constant, T 1 and T 2 are the lower and upper limits of the thrust force that can be provided by each thruster;
ρ 1 =nT 1 cos φ is the minimum thrust value available from the thruster(s);
ρ 2 =nT 2 cos φ is the maximum thrust value available from the thrusters(s);
e i is a unit vector of all zeros except the i th row, which is unity; and
Γ(t)=slack variable that bounds thrust magnitude.
12. The method of claim 11 , wherein
min
t
f
,
T
c
(
·
)
,
Γ
(
·
)
r
(
t
f
)
2
subject to the first set of constraints is computed as
min
N
,
η
Ey
N
2
,
subject to a second set of constraints comprising:
E
u
Y
k
η
≤
e
σ
T
Y
k
η
for
k
=
0
,
…
,
N
μ
1
(
t
)
[
1
-
(
Fy
k
-
z
0
(
t
k
)
)
+
(
Fy
k
-
z
0
(
t
k
)
)
2
2
]
≤
e
4
T
Y
k
η
≤
μ
2
(
t
)
[
1
-
(
Fy
k
-
z
0
(
t
k
)
)
]
Ey
k
∈
X
k
=
1
,
…
,
N
Fy
N
≥
ln
m
dry
y
N
T
e
1
=
0
,
E
v
y
N
T
=
0
y
k
=
Φ
k
[
r
0
r
.
0
ln
m
wet
]
+
Λ
k
[
g
0
]
+
Ψ
k
η
k
=
1
,
…
,
N
wherein
t
k
=
k
Δ
t
,
k
=
0
,
…
,
N
σ
(
t
)
=
Γ
(
t
)
m
(
t
)
u
(
t
)
=
T
c
(
t
)
m
(
t
)
z
(
t
)
=
ln
m
(
t
)
η
=
[
p
0
⋮
p
m
]
,
wherein p j is a vector of parameters and p j ∈R 4
E=[I 3×3 0 3×1 ], F=[ 0 1×6 1], E u =[I 3×3 0 3×1 ], E v =[0 3×3 I 3×3 0 3×1 ]
Y
k
η
=
[
u
(
t
k
)
σ
(
t
k
)
]
,
wherein u(t k )=is the control input, and
σ
(
t
k
)
=
Γ
(
t
k
)
m
(
t
k
)
μ 1 (t)=ρ 1 e −z 0 , wherein z 0 =constant
μ 2 (t)=ρe −z 0 , wherein z 0 =constant
y
k
=
[
r
(
t
k
)
r
.
(
t
k
)
z
(
t
k
)
]
r(t k ) is r(t)t k
{dot over (r)}(t k ) is (t) at t k
z(t k ) is z(t) at t=t k
e 4 T =A transpose of a vector of all zeros except the fourth element, which is unity
y N T =A transpose of y k at the final time step N
Φ k , Λ k , Ψ k , Y k are discrete time state transition matrices describing the solution to {umlaut over (r)}(t)=u(t)+g and
z
.
=
m
.
(
t
)
m
(
t
)
=
-
ασ
(
t
)
wherein g is the acceleration of gravity of the planet or object the spacecraft is landing on.