Fig. 1Photograph **of** 3-DOF helicopter with ADS. Pitch axis: The pitch axis is **control**led by the difference**of** the forces generated by the propellersJ p€ b ¼ Kf l h ðV f V b Þ¼K f l h V d ð2Þwhere J p is the moment **of** inertia **of** the system about thepitch axis, l h is the distance from the pitch axis to eithermotor, V d is the difference between the voltage applied tothe front and back motors. If the force generated by the frontmotor is higher than the force generated by the back motor,the helicopter body will pitch up (positive). The pitch angleis limited to within ð p=2; p=2Þ mechanically duringexperiment.. Travel axis: The only way to apply a force in the traveldirection is to pitch body **of** the helicopter. The correspondingdynamic equation **of** travel axis is:J t €g ¼ K f l a sin b sinða þ a 0 ÞðV f þ V b Þþ K f l h cosða þ a 0 ÞðV f V b Þ¼ K f l a sin b sinða þ a 0 ÞV s þ K f l h cosða þ a 0 ÞV d ð3Þwhere g is the travel angle, J t is the moment **of** inertia aboutthe travel axis. Moreover, if ða þ a 0 Þ¼p=2; i.e. the arm isin horizontal position, the travel motion becomesJ t €g ¼ K f l a sin b sinða þ a 0 ÞV s ð4Þwhich can be verified easily from Fig. 2.From this modelling we know that the elevation accelerationis a function **of** the sum **of** the voltages applied to the twoF fbl al hF bpitch axisa+a 0travel axiselevationaxismotors, and the pitch acceleration is a function **of** differencebetween them. If the pitch angle b and elevation angle a areconstants and b is a small value, the travel motion becomeJ t €g ¼ Kbð5Þwhere K ¼ K f l a V s sinða þ a 0 Þ and this equation means thatthe travel acceleration is governed by the pitch angle.Considering these modelling characteristics and assumingthe travel motion can be achieved by high-precise pitch**tracking**, we can simplify the 3-DOF attitude dynamics to a 2-DOF one, which includes elevation and pitch motion, asgiven in (6)"J e#K f l a cos b0 " # mg sinðaþa€a0 Þþ K f cos b ¼0€bV sð6Þ0 V dJ pK f l hand in matrix formatJ Y € þ NðY; m; K f Þ¼vð7Þwhere J 2 R 22 ¼ diag J e =K f l a cos bJ p =K f l h is themoment **of** inertia, Y 2 R 2 ¼½a bŠ T is the attitude(elevation and pitch) vector, NðY; m; K f Þ2R 2 ¼mg sinða þ a 0 Þ=K f cos b 0 Tis the nonlinear term, andv 2 R 2 ¼½V s V d Š T is **control** voltage vector. For p=2

n 3-DOF helicopters. JðtÞ 2R 2n and _ JðtÞ are defined asthe synchronisation error and the error derivative vectors,respectively. They then have the following expressions:EðtÞ ¼ D ½ e T 1 ðtÞ e T 2 ðtÞ e Ti ðtÞ e T n ðtÞ Š T ð10Þ_EðtÞ ¼ D ½ _e T 1 ðtÞ _e T 2 ðtÞ _e Ti ðtÞ _e T n ðtÞ Š T ð11Þ_JðtÞ ¼ D ½ e T 1 ðtÞ e T 2 ðtÞ e Ti ðtÞ e T n ðtÞ Š T ð12Þ_JðtÞ ¼ D ½ _e T 1 ðtÞ _e T 2 ðtÞ _e Ti ðtÞ _e T n ðtÞ Š T ð13Þe i ðtÞ ¼ D Y d ðtÞ Y i ðtÞ ð14Þ_e i ðtÞ ¼ D _Y d ðtÞ _Y i ðtÞ ð15Þwhere Y d 2 R 2 and _Y d 2 R 2 are the desired trajectories forattitude angles and angular velocities **of** all 3-DOFhelicopters. The definition **of** the synchronisation error isgiven in the subsequent Section.To track the **trajectory** synchronously for **multiple** 3-DOFhelicopters, one must satisfy the following three criteria.First, the designed **control**ler should guarantee the stability**of** the attitude **trajectory** **tracking** errors **of** all involvedsystems. Secondly, the **control**ler should also guaranteethe stability **of** the synchronisation errors. Thirdly, the**control**ler should regulate the attitude motion to trackthe desired **trajectory** at the same rate so that thesynchronisation errors go to zero simultaneously.In short, the **control** objective becomes EðtÞ!0; JðtÞ!0as t!1:e 1 ðtÞ ¼e 1 ðtÞe 2 ðtÞ ¼e 2 ðtÞe 3 ðtÞ ¼e 3 ðtÞ..e n ðtÞ ¼e n ðtÞe 2 ðtÞe 3 ðtÞe 4 ðtÞe 1 ðtÞð18ÞThe synchronisation error in (18) has been used in [14] forthe synchronisation **control** **of** **multiple** roboticmanipulators.Another more complicated synchronisation error formulain (19) can be obtained by applying the synchronisationtransformation matrix T given in (20)e 1 ðtÞ ¼2e 1 ðtÞ e 2 ðtÞ e n ðtÞe 2 ðtÞ ¼2e 2 ðtÞ e 3 ðtÞ e 1 ðtÞe 3 ðtÞ ¼2e 3 ðtÞ e 4 ðtÞ e 2 ðtÞ.e n ðtÞ ¼2e n ðtÞ e n ðtÞ e n 1 ðtÞ232I I II 2I I.T ¼ . . . . .. . .674I 2I I5I I 2Ið19Þð20ÞIn (18, 19) each individual helicopter’s synchronisationerror is a linear combination **of** its **tracking** error and one ortwo adjoining helicopters’ **tracking** errors. With more**tracking** errors involved one may expect to achieve betterperformance. However, it is compromised by the computationalchallenge. In this paper the synchronisation errors in(18, 19) are applied for our investigation.3.2 Generalised synchronisation errorSynchronisation error is introduced to identify the performance**of** synchronisation **control**ler, i.e. how the **trajectory** **of**each 3-DOF helicopter converges with respect to each other.There are various ways to choose the synchronisation error.For example, in [2] the authors include the error information**of** all systems into the synchronisation error **of** each system.However, when there is a large number **of** involved systems,this synchronisation strategy will lead to intensive onlinecomputational work. In this paper we propose a morefeasible and efficient synchronisation error JðtÞ; which is alinear combination **of** attitude **tracking** error EðtÞ.JðtÞ ¼TEðtÞð16Þwhere T 2 R 2n2n is a generalised synchronisation transformationmatrix. By choosing a different matrix T wecan form different synchronisation errors. For example, ifwe choose the following synchronisation transformationmatrix T2IT ¼64IIII. . . . . .I37I5Iwe will get the following synchronisation error formula:ð17Þ3.3 Coupled attitude errorFor **control**ler design a coupled attitude error E ðtÞ 2R 2nthat contains both the attitude **trajectory** **tracking** error E(t)and the synchronisation error JðtÞ is further introducedZ tE ðtÞ ¼EðtÞþBT T J dt ð21Þwhere E ¼ D ½e T1 e T2 e Tn ŠT ; B2R 2n2n ¼ D diag½B B BŠis a positive-definite coupling gain matrix and B 2 R 22 isalso diagonal matrix.Correspondingly the coupled angular velocity error canbe expressed as_E ðtÞ ¼ _EðtÞþBT T JðtÞð22ÞFor the synchronisation transformation matrix T in (17), thecoupled attitude errors becomee 1ðtÞ ¼e 1 ðtÞþBe 2ðtÞ ¼e 2 ðtÞþBe 3ðtÞ ¼e 3 ðtÞþB.e nðtÞ ¼e n ðtÞþBZ tZ0tZ0t0Z t0ðe 1 ðtÞðe 2 ðtÞðe 3 ðtÞ0e n ðtÞÞdte 1 ðtÞÞdte 2 ðtÞÞdtðe n ðtÞ e n 1 ðtÞÞdtð23ÞIEE Proc.-Control Theory Appl., Vol. 152, No. 6, November 2005 685