Nonlinear Tracking Control for Control Moment
Gyroscope with Nonholonomic Constraint
M2014SC013 Chinatsu MURAI
Supervisor: Isao TAKAMI

Abstract
This research describes the design of tracking controller with integrator to compensate friction for Control
Moment Gyroscope which is a first-order nonholonomicsystem. Backstepping method is used to design the controller for the chained system. Stability of the system
is guaranteed theoretically based on Lyapunov function.
The eﬀectiveness of the controller is shown by simulations and experiments.

1Introduction

In this paper, a control strategy for a class of nonholonomic systems is proposed. It is proven than nonholonomic systems cannot be stabilized by using linear
time-invariant state feedback even if the controllability
is ensured in the sense of nonlinear system [1]. The
chained form is applied as the canonical one to the firstorder nonholonomic system [2].
Control Moment Gyro (CMG), which is treated inthis research, is also a first-order nonholonomic system. CMGs are applied to attitude control of large
scale spacecraft. They provide huge torque by tilting
a gimbal, which contains spinning rotor. CMGs are
generally used in collaboration. In this research, control for single CMG is discussed as a fundamental research for CMGs. Recently eﬃcient methods for single
CMG are proposed, for example tracking controller using backstepping approach [3], optimal controller basedon center-stable manifold computation [4], and tracking
controller with an integrator [5].
There exists friction in CMG which cause the diﬃculty
to control. This paper describes the design of a tracking
controller which makes the rotating bodies track the reference trajectory without error despite the friction of thesystem. The contribution of this research is the proposalthat the rotor can provide torques to move the gimbals
while keeping stability despite the friction of the system.
The tracking controller is designed for the chained form
system based on backstepping approach. The influenceof friction is dissolved by the controller with integrator. Stability of the system is guaranteed theoretically
based on Lyapunov function. The eﬀectiveness of thecontroller is illustrated by simulations and experiments.

2

Mathematical Model

The schematic model of CMG is shown in Figure 1.
CMG consists of four rigid bodies, rotor1, gimbal2, gimbal3 and gimbal4, giving four angular degree of freedom.
Note that gimbal3 and gimbal4 do not have any drive
sources. Gimbal3 is locked in this research. The angle
q1 and the angular velocity ω1 are defied as the rotation
about a2 of rotor1 relative to gimbal2. The angle q2
and the angular velocity ω2 are defined as the rotation
about b1 of gimbal2 relative to gimabl3. Gimbal2 has
hardware restriction on its motion range and the singularity. Thus the trajectory of gimbal2 is discussed under0 < q2 < (π/2) in this research. The angle q4 and theangular velocity ω4 are defined as the rotation about c3

Figure 1 Schematic Model of CMGof gimbal4. An input of motor1 attached in rotor1 is T1 .
Another input of motor2 attached in gimbal2 is T2 .
2.1

Nonlinear Dynamics

The equation of motions of each bodies, rotor1, gimbal2 and gimbal4, are obtained as eq.(1)-(3) where Ii isthe moment of inertia. The friction is Fi .
IR1y ω̇1 + IR1y ω̇4 sin q2 + IR1y ω2 ω4 cos q2 = T1 + F1 (1)(IG2x + IR1x )ω̇2 − IR1y ω1 ω4 cos q2− I1 ω42 sin q2 cos q2 = T2 + F2(2)IR1y ω̇1 sin q2 + (I2 + I1 sin2 q2 )ω̇4+ I1 ω2 ω4 sin 2q2 + IR1y ω1 ω2 cos q2 = F4

(3)

The constraint equation is derived as eq.(4) by the integral of eq.(3) when the initial angular momentums ofeach bodies are zero.
IR1y ω1 sin q2 + (I2 + I1 sin2 q2 )ω4 = 0
2.2

(4)

Chained Form

According to the ordinary converting algorithm [6],the constraint equation (4) is converted to the chained
form system applying the following conversions.
{{x1 = q1
u1 = ω1x2 = α(q2 ) ,(5)u2 = β(q2 )ω2x3 = q4
d−IR1y sin q2α(q2 ), β(q2 ) =2dq2
I2 + I1 sin q2
Then the chained form system is obtained as eq.(6).
[] []]ẋ1
1 0 [u1ẋ2 =0 1(6)u2ẋ3x2 0α(q2 ) =This system ensures global controllability because achained form system globally satisfies Lie algebra rankcondition.

3Controller Synthesis

3.3We design the controller that makes gimbal2 track thetrajectory under the restriction and the angle of gimbal4
track the reference trajectory without error. Hence theerror dynamics is constructed for gimbal2 and gimbal4.
In addition, the controller with an integrator is considered for gimbal4 because of compensating the friction.
3.1

Error Dynamics

Let the reference trajectories of gimbal2 and gimbal4
be q2ref and q4ref . According to the coordinate and input conversions (5), the reference trajectories in chained
form are defined as eq.(7). The tracking error variables
are defined as eq.(8).
refrefrefref
ref refxref
2 = α(q2 ), x3 = q4 , u2 = β(q2 )q̇2(7)ref
x2e = x2 − xref
2 , x3e = x3 − x3(8)Then the state variable of the error dynamics is defined
as xe = [x1 x2e x3e s3 ], where s3 is the integral of theerror x3e . The error dynamics is defined as eq.(9).
ẋ = u1 1ẋ2e = u2 − uref2(9)ẋ
=xu−ẋref3e213ṡ3 = x3e3.2

Control Design of Kinematic System

The error dynamics (9) is divided into the following
two subsystems (10) and (11).
∆ : ẋ = u2 − uref2
 1 2e ẋ1 = u1
∆2 :ẋ = x u − ẋref3
 ṡ 3e= x 2 133e(10)(11)

The two subsystems are stabilized using backstepping
approach. Firstly the feedback controller is applied aseq.(12) to stabilize the subsystem ∆1 .
u2 = uref
2 − k1 x2e ,k1 > 0(12)Secondly the subsystem ∆2 is stabilized using backstepping controller. The feedback controller is applied aseq.(13) to stabilize s3 by regarding x3e as a virtual input.
x3e = −k2 s3 , k2 > 0(13)Then the error σ between the left side and right side ofeq.(13) is defined as eq.(14).
σ = x3e + k2 s3(14)Consider to stabilize the dynamics of σ because it isnecessary for x3e to satisfy eq.(13).
Theorem 1 The dynamics of s3 , σ becomes asymptotically stable if the following equation is satisfied.
{( 2)}
u1 = ẋref
3 + k2 − 1 s3 − (k2 + k3 ) σ /x2 ,k3 > 0, x2 ̸= 0 (15)Proof If the Lyapunov function candidate is chosen
as eq.(16), the time derivative is calculated as negativefunction (17) by using eq.(15).
1 2 1 2s + σ >02 3 2V̇1 = −k2 s23 − k3 σ 2 < 0
V1 =(16)(17)

Control Design of Dynamic System

The chained form system inputs u1 and u2 are designed as eq.(12) and (15). From the input conversions(5), the error ξ1 between β(q2 )ω2 and the right handside of eq.(12), and the error ξ2 between ω1 and and theright hand side of eq.(15) are defined.
ξ1 = β(q2 )ω2 − (uref2 − k1 x2e )(18){ ref ( 2)}ξ2 = ω1 − ẋ3 + k2 − 1 s3 − (k2 + k3 ) σ /x2(19)Consider to stabilize the dynamics of ξ1 and ξ2 because
it is necessary for ω1 and β(q2 )ω2 to satisfy eq.(15) and(12), respectively.
Theorem 2 The dynamics of x2e , ξ1 becomes asymptotically stable if the following equation is satisfied.
{( 2)ω̇2 = u̇ref2 + k1 − 1 x2e − (k1 + H1 ) ξ1}−γ(q2 )ω22 /β(q2 )(20)
dβ(q2 )
H1 > 0, γ(q2 ) =dq2
Proof If the Lyapunov function candidate is chosen
as eq.(21), the time derivative is calculated as negativefunction (22) by using eq.(20).
1 21x2e + ξ12 > 022V̇2 = −k1 x22e − H1 ξ12 < 0V2 =(21)(22)

Theorem 3 The dynamics of s3 , σ and ξ2 becomes
asymptotically stable if the following equation is satisfied.
{ω̇1 = G1 s3 + (G2 − x32 )σ + (G3 − H2 x22 )ξ2} 2
ref+G4 ẋref/x2 , H2 > 0 (23)3 + x2 ẍ3Where Gi (i = 1, 2, 3, 4) are the functions of x2 and ξ1
as follows.
G1 = 2k2 x2 − k23 x2 + k3 x2 − k22 ξ1 + k1 k22 x2e
ref−k22 ẋref2 + ξ1 − k1 x2e + ẋ2 ,G2 = k2 k3 x2 + k22 x2 − x2 + k32 x2 + k2 ξ1ref−k1 k2 x2e + k2 ẋref
2 + k3 ξ1 − k1 k3 x2e + k3 ẋ2 ,G3 = −k2 x22 − k3 x22 ,G4 = −ξ1 + k1 x2e − ẋref2 , x2 ̸= 0
Proof If the Lyapunov function candidate is chosen
as eq.(24), the time derivative is calculated as negative
function (25) by using eq.(23).
1 2 1 2 1 2s + σ + ξ2 > 02 3 22V̇3 = −k2 s23 − k3 σ 2 − H2 ξ22 < 0V3 =(24)(25)

Then eq.(20) and (23) are substituted for eq.(1)-(3)to derive the dynamic system inputs T1 and T2 . The
inputs are calculated as follows.
{
T1 = G1 s3 + (G2 − x32 )σ + (G3 − H2 x22 )ξ2}
ref22+G4 ẋref
3 + x2 ẍ3 − f1 x2 /f2 x2 (26)

{( 2)
T2 = (IG2x + IR1x ) u̇ref2 + k1 − 1 x2e}
− (k1 + H1 ) ξ1 − γ(q2 )ω22 − f3 β(q2 ) /β(q2 ) (27)
f1af1 =
2I2 + I1 sin q2 − IR1y sin2 q2✓ ✻✵✠✝
✒✑✏ ✹✵✎✍✌☞☛ ✷✵f1a = I1 ω2 ω4 sin2 q2 cos q2 − I2 ω2 ω4 cos q2+IR1y ω1 ω2 sin q2 cos q2✡✠✟ ✵✞✝✈

I2 + I1 sin2 q2
f2 =IR1y (I2 + I1 sin2 q2 − IR1y sin2 q2 )f3 =❛☞FisFic sgnωi + Fiv ωi✡
✠✟✞❛ ✵ ✺✵✵ 0q4ref [rad] =− 1 sin( π4 t − π2 ) + 12
 12✷✵✵✺✵✶✵✵t✁✂✄ ☎✆✝

♣✎✏♣✏✆✄✑✇✒✏ ✁✓t✄✔✎✕t✏✎✎✄r✄✎✄✓✖✄✶✺✵✶❛✎ ✵ ✄✑✏✍✌✵ ✂☞☛ ✵ ✁✡❛
✵ ✷✵
✵

(28)

π
Initial conditions are given as [ q1 q2 q4 ] = [ 0 180][rad]. References are given as eq.(29) and (30).

 π

18


7− 36π sin( π4 t − π2 ) + 12q2ref [rad] = 7π32572 π sin( 2 t − 2 π) + 72 π✶✺✵Figure 3 Simulation of q2(ωi = 0)(ωi ̸= 0)(i = 1, 2, 4)✶✵✵t ✁✂ ✄☎✆✶☛The eﬀectiveness of this research is illustrated by simulations. Simulations including friction are executed.
The friction are measured by some experiments. Theequation of friction is defined as eq.(28), where Fis , Ficand Fiv (i = 1, 2, 4) are the coeﬃcients of static friction,coulomb friction and viscous friction respectively.
Fi =✺✵✶ ✺✍✌Simulation{✵Figure 2 Simulation of ω1

IR1y ω1 ω4 cos q2 + I1 ω42 sin q2 cos q2
IG2x + IR1xx2 ̸= 0, β(q2 ) ̸= 0

The dynamical system of CMG becomes asymptoticallystable by inputs T1 and T2 . Here note that q2 = 0 is an
singularity because the torque becomes infinite if q2 = 0in eq.(27).

4♣✔✕♣✕☎✂✖
✇✗✕ ✘t✂✙✔✚t✕✔

✺✵✶✵✵t☎✆✝ ✞✟✠♣✒✓♣✓✟✝✔
✇✕✓ ☎✖t✝✗✒✘t✓✒✒✝r✝✒✝✖✙✝
✶✺✵✷✵✵(t < 4)
Figure 4 Simulation of q4(4 ≤ t ≤ 8)
✵ ✂(t > 8)✓✒ ✵ ✁(29)(t < 4)(4 ≤ t ≤ 8)
(t > 8)✑✏✎ ✵ ✷✍

(30)✌☞

k1 = 5, k2 = 0.01, k3 = 0.04, H1 = 10, H2 = 1.8✵☛✡

✠ ✲✵ ✷✲✵ ✁✵Gain parameters are chosen by trial and error as follows.

♣✔✕♣✕✞✆✖
✇✗✕ ✄✘t✆✙✔✚t✕✔

✺✵(31)

Simulations are shown in Figure 2-6. The solid line
shows the proposed method (proposed) and the dashedline shows the controller without integrator (w/o integrator). The dotted line shows the reference.
It canbe seen that the response of ω1 is stabilized in Figure 2.
The angle of gimbal2 has constraint 0 < q2 < (π/2). As
can be seen that the response of q2 tracks the reference
under the restriction in Figure 3. The angle of gimbal4,
which does not have any drive sources, is shown in Figure 4. The response of q4 tracks the reference without
error by applying proposed method in case that the friction exist in the system. Control torques are shown in
Figure 5, 6. Constraints of the torques are |T1 | < 0.6[Nm] and |T2 | < 2.4 [Nm]. In the proposed method, thesystem can be controlled within the constraints.

✶✵✵t✄☎✆ ✝✞✟✶✺✵✷✵✵

Figure 5 Simulation of torque T1✵ ✶♣✒✓♣✓✆✄✔
✇✕✓ ✁✖t✄✗✒✘t✓✒✑✏✍✌✎ ✵ ✵✺✵☛✡

☞✠✟ ✲✵ ✵✺✞✲✵ ✶✵✺✵✶✵✵t✁✂✄ ☎✆✝

✶✺✵Figure 6 Simulation of torque T2✷✵✵

5✵ ✶ExperimentThe eﬀectiveness of the proposed controller is confirmed by experiments. The experimental results areshown in Figure 7-11. The solid line shows the experiment and the dashed line shows the simulation. As canbe seen that the experimental results of rotor1 and gimbal2 are similar to the simulation results, qualitatively
in Figure 7, 8. The experimental result of gimbal4 isoscillated a little but it becomes stabilized in Figure 9.
The control torques are shown in Figure 10, 11. Thesystem can be controller within the limits.
The pro✔ ✻✵✡
✞✓✒✑ ✹✵✏✎✍✌☞☛✡
✠✟✞✄❡✕✄✖✁✂✄✗t✆✁✂s✘✙t✁✚✗✺✵✶✵✵t✁✂✄ ☎✆✝

✶✺✵✶✞❛ ✵ ✺✵✵✺✵✶✵✵t✁✂✄ ☎✆✝

✄❡✎✄✏✁✂✄✑t✆✁✂s✒✓t✁✔✑✶✺✵

Figure 8 Experiment of q2✶❛✏ ✵ ☎✒✑✎✍✵ ✄✌☞ ✵ ✂☛❛✵ ✁✵✵✺✵✶✵✵t✆✝✞ ✟✠✡

✞❡✓✞✔✆✝✞✕t✠✆✝s✖✗t✆✘✕✶✺✵

Figure 9 Experiment of q4✵ ✄✔✓ ✵ ✁✒✑✏ ✵ ✂✝❡✕✝✖☎✆✝✗t✟☎✆s✘✙t☎✚✗✵☞☛✠✟ ✲✵ ✵✺
✞

✲✵ ✶✵

✺✵✶✵✵t✁✂✄ ☎✆✝

✶✺✵Figure 11 Experiment of torque T2

✵

✠✟

✍✌

☛✡

In this research, a nonlinear tracking control of CMG
to compensate the friction for nonholonomic system isproposed. Firstly chained form system from the equation of motion is derived. The state equation is converted into the chained system using ordinary algorithm.
Secondly the tracking controller with integrator basedon the backstepping method is designed. The integral of
the error between the angle of gimbal4 and the reference
trajectory is included to the controller. The integrator
makes states track the reference without error despite
the friction in the system. The stability of the systemwith integrator is guaranteed theoretically by consisting Lyapunov function. Finally the eﬀectiveness of theproposed method is illustrated by simulations and experiments.

✡

✎✵☞

6✶ ✺☛

✍✌

posed controller has usefulness for CMG control fromthe experiments.

Figure 7 Experiment of ω1❛☞✎ ✵ ✵✺✵✈ ✲ ✵✵✍✌✄❡✒✄✓✁✂✄✔t✆✁✂s✕✖t✁✗✔

✑✏✡ ✲✵ ✂✲✵ ✁✵✺✵✶✵✵t☎✆✝ ✞✟✠✶✺✵Figure 10 Experiment of torque T1

Conclusion

References[1] R. W. Brockett, ”Asymptotic stability and feedback
stabilization,” Diﬀerential Geometric Control Theory, R. S. Millman and H. J. Sussmann, Eds. Boston,
MA: Birkhauser, pp. 181-191, 1983.
[2] R. M. Murray and S. S. Sasty, ”Nonholonomic Motion Planning: Steering Using Sinusoids,” IEEETransactions on Automatic Control, Vol. 38, No. 5,pp. 700-716, 1993.
[3] M. Reyhanoglu and J. van de Loo, ”State feedback
tracking of a nonholonomic control moment gyroscope,” Proc. the 45th IEEE CDC, pp. 6156-6161,
2006.
[4] K. Ishikawa and N. Sakamoto, ”Optimal control forcontrol moment gyros - Center-stable manifold approach,” Proc. the 53rd IEEE CDC, pp. 5874-5879,
2014.
[5] S. Washizu, C. Murai, I. Takami and G. Chen, ”Nonlinear Control for First-Order Nonholonomic Systemwith Hardware Restriction and Disturbance,” Proc.
the 10th Asian Control Conference 2015, pp. 20812086, 2015.
[6] F. Matsuno and J. Tsurusaki, ”Chained form transformation algorithm for a class of 3-states and 2inputs nonholonomic systems and attitude controlof a space robot,” Proc. the 38th IEEE CDC, pp.
2126-2131, 1999.