Hybrid MPC for Attitude Control of Spacecraft withImpulsive Thrusters

P. Sopasakisa, D. Bernardinia,c, H. Strauchb,S. Bennanid and A. Bemporada,c

a Institute for Advanced Studies Lucca,b Airbus DS, c ODYS Srld European Space Agency

July 14, 2015

Control objectives

I Operate in barbecue mode with ωx = 5 deg · s−1,

I Achieve high pointing accuracy using impulsive thrusters

I Low actuation count

I Reasonable computational complexity

1 / 15

I. Attitude Modelling

Reference frame

Body-Fixed Frame:

2 / 15

Nonlinear dynamical model

Notation: Θ̃x, Ψ̃x: pitch and yaw error angles, ωi: angular velocity abouti-axis, ωn = (1− Jxx/Jyy)ωx.1

d

dt

Θ̃x

Ψ̃x

ωxωyωz

=

0 ωx 0 1 0−ωx 0 0 0 1

0 0 0 0 00 0 0 0 ωn0 0 0 −ωn 0

Θ̃x

Ψ̃x

ωxωyωz

+

+

0 0 00 0 0J−1xx 0 0

0 J−1yy 0

0 0 J−1zz

τxτyτz

1A. Kater, “Attitude Control of Upper Stage Launcher During Long Coasting Period,” Master’s thesis, Lehrstuhk furAutomatisierungs und Regelungstechnik, Christian-Albrechts-University Kiel, Germany, 2013.

3 / 15

Linear Dynamical Model

Assumption: The spin rate equilibrates very fast, i.e., ωx(t) ' ωrx.

Model is now written in two parts - the spin model2:

ω̇x = J−1xx τx

and the nutation/precession model:

d

dt

Θ̃x

Ψ̃x

ωyωz

=

ωrx 1

−ωrx 1ωrn

−ωrn

Θ̃x

Ψ̃x

ωyωz

+

J−1yyJ−1zz

[τyτz]

2The spin rate ωx is controlled with a simple but efficient P-controller.

4 / 15

Linear Dynamical Model

We introduce the state variable

x(t) =[Θ̃x(t) Ψ̃x(t) ωy(t) ωz(t)

]′,

and the input vectoru(t) =

[τy(t) τz(t)

]′,

the nutation/precession model, after discretisation, is written concisely as

x(k + 1) = Ax(k) +Bu(k).

5 / 15

II. Impulsive Thrusters

Minimum impulse effect

Minimum impulse effect: thrusters cannot produce arbitrarily smalltorques, thus uk is constrained in

U = [−umax,−umin] ∪ {0} ∪ [umin, umax]

We introduce the binary vectors δ− and δ+ so that3

δ−(k) = [u(k) ≤ −umin],

δ+(k) = [u(k) ≥ umin],

and the auxiliary variable ηk defined as4

ηi(k) = [δ−i (k) ∨ δ+i (k)] · ui(k)

3Symbols ≤ and ≥ are element-wise comparison operators.4Here ηi(k) stands for the i-th component of vector η(k) at time k.

6 / 15

Minimum impulse effect

To detect thruster actuation we use the variable

v(k) = [δ−(k) ∨ δ+(k)]

and we recast the system dynamics as

z(k + 1) = Az(k) +Bη(k) + f,

γ(k + 1) = γ(k) + [ 1 1 ] v(k),

where γ(k) is the total actuation count up to time k on which we impose:

γ(k) ≤ γmax.

7 / 15

III. Hybrid MPC

Penalty function

For a sequence of control actions πN = {u(k), v(k), η(k), δ±(k)}N−1k=0 wedefine the total penalty function with horizon N as

V (πN ;x0, γ0) , VN (x(N), γ(N)) +

N−1∑k=0

`(x(k), z(k)),

where ` is the stage cost

`(x, z) , ‖Qx‖p︸ ︷︷ ︸Penalises the

pointing inaccuracy

+‖Rz‖p

and VN is the terminal cost,

VN (x(N), γ(N)) , ‖QNx(N)‖p + ρ(γ(N)− γ0)︸ ︷︷ ︸Penalises the tot.actuation count

along the horizon

.

8 / 15

Hybrid MPC - problem formulation

MPC problem formulation:

minπN

V (πN ;x0, γ0)

s.t. x(0) = x0, γ(0) = γ0,

Hybrid dynamics, for k ∈ N[0,Nu−1],

δ±(k) = 0, for k ∈ N[Nu,N−1],

9 / 15

IV. Simulations

Barbecue mode

2 4 6 8 10 12 14 16 18 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

time [s]

Ro

ll ra

te e

rro

r [d

eg

]

Absolute value of tracking error on ωx

Figure : ωx converges fast to its set-point.

10 / 15

Pointing Accuracy

−4 −3 −2 −1 0 1 2−6

−5

−4

−3

−2

−1

0

1

2

pitch error [deg]

ya

w e

rro

r [d

eg

]

Tracking error on pitch and yaw

PD

LQ

HMPC

Figure : HMPC achieves higher pointing accuracy compared to optimally tunedPD and LQR controllers (which do not account for the MI effect).

11 / 15

Actuation Count

50 100 150 200 250 300

−200

−100

0

100

200

LQ

R T

orq

ue

[N

m]

50 100 150 200 250 300−200

−100

0

100

200

PD

To

rqu

e [

Nm

]

50 100 150 200 250 300

−4000

−2000

0

2000

HM

PC

To

rqu

e [

Nm

]

Time [s]

50 100 150 200 250 3000

200

400

600

LQ

R T

orq

ue

[N

m]

50 100 150 200 250 300−200

−100

0

100

200

PD

To

rqu

e [

Nm

]

50 100 150 200 250 300−5000

0

5000

HM

PC

To

rqu

e [

Nm

]Time [s]

Figure : HMPC achieves a significantly lower actuation count. Left: ωy, Right:ωz.

12 / 15

...with additive disturbances

−12 −10 −8 −6 −4 −2 0 2 4 6−4

−2

0

2

4

6

8

10

12

14

pitch error [deg]

yaw

err

or

[deg]

Tracking error on pitch and yaw

Tsim

=500

PD

LQ

HMPC

Figure : Simulations in presence of a constant additive disturbance on uk of6Nm.

13 / 15

Performance assessment

Performance indicators: (i) actuation count on axes y and z, (ii) totalactuation count, (iii) total squared deviation

Jr =

Tsim∑k=0

(Θ̃2x(k) + Ψ̃2

x(k)).

Table : Simulation results over an interval Tsim = 300s (nominal conditions).

Thruster activationsJrx-axis y-axis z-axis total

PD controller 9 7 34 50 0.4741LQ controller 9 21 36 66 0.3858Hybrid MPC 9 4 5 18 0.0811

14 / 15

Performance assessment

Table : Simulation results over an interval Tsim = 300s (with additivedisturbances of 6Nm).

Thruster activationsJrx-axis y-axis z-axis total

PD controller 9 11 30 50 0.4454LQ controller 9 26 40 75 0.3878Hybrid MPC 9 16 5 30 0.0880

15 / 15

Thank you for your attention