# Backstepping control of a rigid body, Report no. 2595 by tgv36994

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```									         Backstepping control of a rigid body

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Control & Communication
Department of Electrical Engineering
o                                 o
WWW: http://www.control.isy.liu.se
E-mail: torkel@isy.liu.se, ola@isy.liu.se

16th February 2004

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Report no.: LiTH-ISY-R-2595
Submitted to CDC’02, Las Vegas, Nevada

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Technical reports from the Control & Communication group in Link¨ping are
available at http://www.control.isy.liu.se/publications.
Abstract

A method for backstepping control of rigid body motion is proposed.
The control variables are torques and the force along the axis of motion.
The proposed control law and lyapunov function guarantee asymptotic
stability from all initial values except one singular point.

Keywords: rigid body, backstepping, lyapunov stability. aircraft
control
Backstepping Control of a Rigid Body
S. Torkel Glad and Ola H¨rkeg˚ 1
a    ard

Abstract                                  has the form
ˆ
F = m(Fa (V ) + uv V )
A method for backstepping control of rigid body mo-
tion is proposed. The control variables are torques                         ˆ     1
where V = |V | V and uv is a control variable. The ﬁrst
and the force along the axis of motion. The proposed
part, Fa , corresponds to aerodynamic or hydrodynamic
control law and lyapunov function guarantee asymp-
forces, and the second part models approximately the
totic stability from all initial values except one singular
thrust action of an engine aligned with the velocity
point.
vector. The moment M is assumed to depend on V , ω
and control variables.

1 Introduction
3 Stationary motion
An important tool for nonlinear control synthesis is
backstepping, see e.g. [4], [8]. The idea is to extend              Consider a motion with V = Vo , ω = ωo where Vo , ωo
a Lyapunov function from a simple system to systems                 are constants. The velocity equation is then
involving additional state variables and at the same
time design the feedback control to guarantee stabil-                                                       ˆ
ωo × Vo = Fa (Vo ) + uv Vo
ity. This technique has been successful in several ap-
plications, [1, 2, 3, 9]. Recently backstepping design                     ˆ       1                       ˆ
where Vo = |V | Vo . Multiplying with Vo shows that uv
ˆo
has been successfully applied to the control of aircraft,           has to satisfy
[5, 6, 7]. The aircraft dynamics is essentially described                                      ˆ
uv = −VoT Fa (Vo )
by rigid body dynamics in combination with equations
Then ωo can be calculated from
describing the aerodynamic forces. There are several
ways of designing controllers for rigid body equations                                   1 ˆ                      ˆ
ωo =          (Vo × Fa (Vo )) + γ Vo
occuring in various applications, see e. g. [10, 11]. The                               |Vo |
purpose of the present paper is to formulate a design
method for a controlled rigid body using backstepping               where γ is an arbitrary constant. If uv and M can
techniques. The design can then be specialized to air-              be chosen arbitrarily it is thus possible to achieve a
craft control problems or the control of various types              stationary motion for any value of Vo .
of vehicles.

4 Backstepping design
2 Rigid body dynamics
In this section we develop a backstepping design to
We assume that the controlled object is a rigid body                make Vo , ωo a stable equilibrium. Deﬁne
with mass m. We describe the motion in a body ﬁxed                                   uM = I −1 (M − ω × Iω)
coordinate system with the origin at the centre of mass
and obtain the equations:                                           We will regard uM as the control signal. Then the
dynamics is given by
˙             1
V = −ω × V + F
m                         (1)                   ˙                         ˆ
V = −ω × V + Fa (V ) + uv V
I ω = −ω × Iω + M
˙                                                                                                 (2)
˙
ω = uM
where V is the velocity, ω is the angular velocity, F
is the external force and M is the external torque (all             First regard the angular velocity ω (together with uv )
these quantities are vectors with three components). I              as the control variable. Let V0 be the desired velocity
is the moment of inertia. We will assume that the force             vector and introduce the Lyapunov candidate
1 Dept. of Electrical Engineering, Link¨pings universitet, SE-
o                                               1
581 83 Link¨ping, Sweden. torkel@isy.liu.se, ola@isy.liu.se
o                                                                      W1 =     (V − Vo )T (V − Vo )
2
Choose a control of the form                                                           References
1                              [1]    M. T. Alrifai, J. H. Chow, and D. A. Torrey. A back-
ω = ω des    ¯
=ω+       V × Fa (V )                stepping nonlinear control approach to switched reluctance
|V |2
motors. In Proc. of the 37th IEEE Conference on Decision
After some manipulations this gives                           and Control, pages 4652–4657, Dec. 1998.
˙                                  ˆ
W1 = −¯ T (Vo × V ) + uv (V − Vo )T V
ω               ¯                              [2]    J. J. Carroll, M. Schneider, and D. M. Dawson. In-
tegrator backstepping techniques for the tracking control
ˆ
where uv = uv − V T Fa . Choosing
¯                                                  of permanent magnet brush DC motors. In Conference
¯
ω = k1 (Vo × V ),         ¯                ˆ
uv = −(V − Vo )T V          Record of the 1993 IEEE Industry Applications Society An-
nual Meeting, pages 663–671, Oct. 1993.
then gives                                                    [3]    T. I. Fossen and ˚. Grøvlen. Nonlinear output feed-
A
1                          back control of dynamically positioned ships using vectorial
ω des = k1 (Vo × V ) +        V × Fa (V )            observer backstepping. IEEE Transactions on Control Sys-
|V |2
tems Technology, 6(1):121–128, Jan. 1998.
˙
W1 |ω=ωdes                                   ˆ
= −k1 |V0 × V |2 − ((V − Vo )T V )2 ≤ 0       [4]                                         c
R. A. Freeman and P. V. Kokotovi´. Robust Nonlin-
˙
In this expression W1 |ω=ωdes = 0 only if V = Vo (pro-        ear Control Design: State-Space and Lyapunov Techniques.
a
Birkh¨user, 1996.
vided the singularity V = 0 is avoided, which can be
done e.g. by starting so that |V − Vo | < |Vo |). The         [5]          a
O. H¨rkeg˚  ard. Flight control design using backstep-
ping. Licentiate thesis 875, Department of Electrical Engi-
lyapunov function thus guarantees convergence to the
o
desired V = Vo .
[6]         a      ard
O. H¨rkeg˚ and S. T. Glad. A backstepping design
Deﬁne                                                         for ﬂight path angle control. In Proc. of the 39th Confer-
ence on Decision and Control, pages 3570–3575, Sydney,
ξ = ω − ω des
Australia, Dec. 2000.
In the new variables the dynamics is                          [7]          a     ard
O. H¨rkeg˚ and S. T. Glad. Flight control design
˙                                          ¯ ˆ
V = −ξ × V + k1 (|V |2 V0 − (V T V0 )V ) + uv V          using backstepping. In Proc. of the IFAC NOLCOS’01, St.
Petersburg, Russia, July 2001.
˙
ξ = uM + φ(V, ξ)                                        [8]              c
M. Krsti´, I. Kanellakopoulos, and P. Kokotovi´.    c
where φ(V, ξ) =    d     des                                  Nonlinear and Adaptive Control Design. John Wiley &
dt (ω     ).   Introducing
Sons, 1995.
1                             [9]              c                     c
M. Krsti´ and P. V. Kokotovi´. Lean backstepping
W2 = k 2 W1 + ξ T ξ
2                             design for a jet engine compressor model. In Proc. of the
gives                                                         4th IEEE Conference on Control Applications, pages 1047–
1052, 1995.
˙                                      ˆ
W2 = −k1 k2 |V0 × V |2 − k2 ((V − Vo )T V )2 − k3 ξ T ξ ≤ 0
[10] C. A. Woolsey, A. M. Bloch, N. E. Leonard, and
if we select the control                                      J. E. Marsden. Dissipation and controlled euler-poincare
systems. In Proceedings of the 40th IEEE Conference on
u = k 2 V0 × V − φ − k 3 ξ                   Decision and Control, pages 3378–3383, Orlando, Florida,
˙
Since W2 = 0 only occurs for V = Vo , ξ = 0 (except for       December 2001.
the singular case V = 0, discussed above) there will be       [11] C. A. Woolsey and N. E. Leonard. Global asymptotic
convergence to V = Vo , ξ = 0.                                stabilization of an underwater vehicle using internal rotors.
In Proceedings of the 38th IEEE Conference on Decision
and Control, pages 2527–2532, Phoenix, Arizona, December
1999.
5 Conclusions

We have proposed a control law that steers the veloc-
ity and angular velocity vectors to desired values. The
control law uses external torques and a force along the
velocity vector. This conﬁguration is similar to, but
not precisely equal to the one used in aircraft control,
where control surfaces generate torques and the engine
gives a longitudinal force. However, our proposed rigid
body control could inspire new aircraft control designs.
An interesting extension would be to take the orienta-
tion into account, which would make it possible to e.g.
include the eﬀect of forces like gravity.

Acknowledgement This work was supported by the