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Synchronised trajectory-tracking control of multiple 3 ... - IEEE Xplore

Synchronised trajectory-tracking control of multiple 3 ... - IEEE Xplore

Fig. 1Photograph

Fig. 1Photograph of 3-DOF helicopter with ADS. Pitch axis: The pitch axis is controlled by the differenceof 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 pitchtracking, 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 controller should guarantee the stabilityof the attitude trajectory tracking errors of all involvedsystems. Secondly, the controller should also guaranteethe stability of the synchronisation errors. Thirdly, thecontroller 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 moretracking 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 performanceof synchronisation controller, i.e. how the trajectory ofeach 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 informationof 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 controller 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

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