International audienceThis paper deals with a method used to design the control law of the μDrone MAV. This vehicle uses six propellers to fly and the dynamic model approximation for the motion is a MIMO linear time-invariant system. As we want to design a linear regulator, it is necessary to build a robust feedback control law. The LQ state feedback regulator design is applied to a standard model, tacking into account some perturbations. This is why the model is augmented with a perturbation vector and an observable subsystem is extracted in order to build a state estimator whose gain is the solution of a LQ problem. The subsystem is then decomposed into a controllable set and an uncontrollable one. The use of an asymptotic rejection strategy of the influence of uncontrollable modes gives the possibility to find a state feedback applied only to the controllable ones. Here again feedback matrix is chosen as the solution of a LQ problem. To compute the weighting matrices of quadratic criterions we use a “partial observability gramian”. The great advantage of this method is due to the use of only three scalars to synthesize the control law

Friedt, Jean-Michel

Lang, Bernard

Paquier, Pascal

HAL - Université de Franche-Comté

One way to design the control law of a mini-UAV.Bernard Lang, Pascal Paquier, Jean-Michel FriedtTo cite this version:Bernard Lang, Pascal Paquier, Jean-Michel Friedt. One way to design the control law of a mini-UAV.. 3rd US-European Competition and Workshop on Micro Air Vehicle Systems (MAV’07)& European Micro Air Vehicle Conference and Flight Competition (EMAV’07)., Sep 2007,Toulouse, France. 1, pp.1-7, 2007. <hal-00179460>HAL Id: hal-00179460https://hal.archives-ouvertes.fr/hal-00179460Submitted on 16 Oct 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.   1 One way to design the control law of a mini-UAV B. Lang* , P. Paquier† National Engineering Institute in Mechanics and Microtechnologies (ENSMM), Besançon, France and J.-M. Friedt‡ Business or Academic Affiliation 2, Besançon, France This paper deals with a method used to design the control law of the μDrone MAV. This vehicle uses six propellers to fly and the dynamic model approximation for the motion is a MIMO linear time-invariant system. As we want to design a linear regulator, it is necessary to build a robust feedback control law. The LQ state feedback regulator design is applied to a standard model, tacking into account some perturbations. This is why the model is augmented with a perturbation vector and an observable subsystem is extracted in order to build a state estimator whose gain is the solution of a LQ problem. The subsystem is then decomposed into a controllable set and an uncontrollable one. The use of an asymptotic rejection strategy of the influence of uncontrollable modes gives the possibility to find a state feedback applied only to the controllable ones. Here again feedback matrix is chosen as the solution of a LQ problem. To compute the weighting matrices of quadratic criterions we use a “partial observability gramian”. The great advantage of this method is due to the use of only three scalars to synthesize the control law. Nomenclature zyx ,,  = centre of gravity coordinates of the MAV   = roll angle   = pitch angle   = yaw angle iu  = actuator control input for solid iS  iF  = thrust produced by the ith actuator iM  = moment produced by the ith actuator i  = time constant of the ith actuator ik  = DC gain of the ith actuator u  =  Tuuuuuu654321 input vector y  =  Tzyx   output vector ry  = reference vector x  = state vector of initial linear MAV model 18R  CBA ,,    = matrices of initial state-space model                                                  * Associate Professor, ENSMM, blang@ens2m.fr. † Coordinator of the UAV project, ENSMM, Address/Mail Stop for second author. ‡ Research and Development Engineer, Department Name, Address/Mail Stop for fourth author (etc).    2 px  = perturbation vector  I. Introduction HIS paper deals with a method used to design the control law of the μDrone. This MAV uses six propellers to fly and it is not obvious to control the six brushless motors simultaneously. In order to build a dynamical model of the MAV, we worked on the assumption that it can be described with seven solids in interaction. After a linearization of the equations in the vicinity of a horizontal trim, the model is a multi-input, multi-output (MIMO) linear time-invariant dynamic system. This approximation can be used only if it is possible to build a robust feedback control law.  The interest of the present paper is to provide a simple method to reach the needed robustness, using the LQ state-feedback regulator design applied to a “standard model”, tacking into account some perturbations. This method is detailed in Ref. 1 and very recently in Ref. 2.  The paper is organized as follows. Section II gives some details on the MAV model and the control design is explained in section III. Section IV gives complements about the choice of adjustment parameters.  II. Model of the μDrone The μDrone is a “gyro dyne” with six propellers. We worked on the assumption that it can be described with seven solids in interaction. Solid 0S  is the body. Solids 1S  and 2S  are made up of a fix pitch propeller, the motor shaft and the blades holder. The two propellers are counter-rotating and provide the lift of the MAV. Solids 3S  and 4S  include a variable pitch propeller, the motor shaft and the variable pitch system. These propellers provide the trim stabilization. Solids 5S  and 6S  are like the two precedents and provide MAV propulsion. The detailed description of this mechanical system modeling can be found in Ref. 3. Let iF  denotes the thrust produced by the ith actuator and iM  the associated moment. If iu  denotes the control input of the actuator, we assume that  (1) )(.)()(.)(tFktMtuktFdtdFiiiiiiii With this assumption and after linearization of the mechanical model, the MIMO linear time-invariant system, denoted DRO, is an approximation of the MAV behavior4,5. DRO is a system with 6 inputs, 6 outputs, and state-vector x  is of size 18.  (2) xCyuBxAdtxd... The eigenvalues of the A  matrix are so defined: 10 are zero, 6 are real negative and 2 are purely imaginary. The rank of controllability matrix is 2 and that of observability matrix is 4.  III. Control design of the MAV  T    3 In order to build a “standard model” we introduce a perturbation vector px  and we work on the assumption that each perturbation is constant. It is then possible to give a simple model for these perturbations  (3) pppxAdtxd.  where pA  is a square matrix of zeros. In our application 2Rxp and the perturbations are moments acting on roll and pitch rates. The initial system DRO is augmented and becomes  (4) eeeeeexCyuBxAdtxd...       with         OCCOBBAOAAAeepe ,,12 As we want to elaborate the control law with a state-feedback, it is necessary to build a state estimator. The first step is to compute the observability staircase form of  eeeCBA ,,  and extract an observable subsystem  (5) oeoeooeooxCyuBxAdtxd... It is then possible to build a state estimator  (6) )ˆ..(.ˆ.ˆoeooeooeooxCyKuBxAdtxd  The gain matrix oK is chosen as the solution of the LQ problem find ooxKKv .  that minimize   dtvRvxQxJoTooToo.....0   for    vCxAdtxd TeooTeoo..   and define a positive scalar ok such as  (7)  ToooKKkK .  At this point, the problem is to make a good choice for the weighting matrices oQ and oR of the quadratic criterionoJ . In order to adjust easilyoQ , we use a “partial observability gramian” defined as   (8) dttACCtATGeoeoTeoTTeoooooo)..exp(..)..exp()(0  where ooooTkT .  with oT  positive, called the filtering horizon and ok the shape factor.  As the pair  eoeoCA ,  is observable, )(oooTG is asymmetric positive matrix, and it is possible to choose    (9)  IRTGTQooooooo1)(.     with I  the identity matrix.  The state estimator is then  (10)     yuKBxCKAdtxdoeooeooeoo.ˆ..ˆ Now let us compute the controllability staircase form of  eoeoeoCBA ,,  with the similarity transformation matrix ocT       4  (11)   cnceoceonceoccnceoceoeonccncxxCCyuBOxxAAOAxxdtd...21       with    ooccncxTxx.  As 21eoA   is not a matrix of zeros, uncontrollable modes ncx  perturb controllable modescx . In order to reject the influence of uncontrollable modes it is possible to use a linear transformation  (12) )(.)()(~)(.)()(txTtxtxtxGtvtuncaccnca    where aG  and aT  are constant matrices, leading to   (13)   ncaeoceoncceoceocncaeoceoncaaeoceoceoccxTCCxCyvBxGBATTAAxAdtxd..~......~.~21  Now, it is often possible to find aG  and aT  such as   (14)       OTCCOGBATTAAaeoceoncaeoceoncaaeoceo....21  With these conditions equations (11) become   (15) ceocnceoncnceocceoccxCyxAdtxdvBxAdtxd~...~.~  where the pair  eoceocBA ,  is controllable  Once more, it is possible to compute the solution of the LQ problem  find  ccxKv~.  which minimize  dtvRvxQxJcTccTcc...~..~0  for  vBxAdtxdeocceocc.~.~   In order to adjust weighting matrices cQ  and cR  of the quadratic criterioncJ  , define a positive scalarck ,   a positive control horizon cT  such as   (16) cockTT /  and the “partial observability gramian”    (17) dttACCtATGeoceocTeocTTeoccocc)..exp(..)..exp()(0   It is then possible to choose  (18) eocTeocceoccocTeocccCCQBTGBTR.).(.. Now, it is possible to elaborate the feedback control      5  (20) rrccyKxKv .~.     where rK  is a constant matrix and ry  the reference vector. In order to determine rK  matrix we use equations (15) in which vector v  is replaced by expression (20)  (21)  ceocrreoccceoceoccxCyyKBxKBAdtxd~...~..~ In permanent rate we wantryy  . As   ceoceocKBA .  is a stability matrix, it is invertible and   (22)  rreocceoceoceocyKBKBACy .....1  Then, the choice   (23)   11 .)..( eocceoceoceocrBKBACK   leads to the solution. In our application, this last matrix is not invertible; it is the reason why we use the expression of the pseudo inverse and it appears that it is not possible to choose reference r  except for the zero value. Now the use of expression (12) leads to the control vector    ncarrccncaxGyKxKxGvu ..~..   which gives  (24)  rrcnccaacyKxxKGTKu ...   but with the transformation matrix ocT  in (11) it is possible to express u  as a feedback control  (25)  rrooccaacyKxTKGTKu ....   According to separation principle and using state estimator expression (10), it follows easily the regulator equations  (26) yyDxCuyyBxAdtxdrregoregrregorego.... with       OKDTKGTKCKKBBTKGTKBCKAArregoccaacregoreoregoccaaceoeooeoreg.......  IV. Choice of the adjustment parameters   Among the main objectives, the stability of state matrix estimator regA  in (26) is essential. It is clear that parameters ookT ,  and ck  will influence stability, acting on oK  and cK  matrices. The first step is to assign a fixed value to ok  and ck  parameters; often 5 is a good value to start. Now the filtering horizon oT  is adjusted    6 so that regA  is a stability matrix; it is often effective to start with the slowest time constant value of the initial model.  In order to quantify the robustness stability of the feedback, the regulator, without reference, can be converted to a transfer function matrix. Starting with    (27) oregooregoxCuyKxAdtxd... we obtain   (28) )().()( sysKsu   where   (29)   occaaccoeooceoeocTKGTKKKKCKKKBAsIKKsK......)(1 (It is interesting to notice the symmetry of expression (29) in relation to matrix gains cKK  andoK ). Similarly, the initial model of the MAV is    (30) )().()( susDROsy   Now define   (31) )().()( sDROsKsLu    the input loop transfer   (32) )().()( sKsDROsLy    the output loop transfer   (33) )(max1,)(max1min jLjLMyust      the static margin   (34) )(.max1,)(.max1min jLjLMyudyn   the dynamic margin Then, when regA  is a stability matrix, the stability of the global feedback is robust in relation to any relative error such as   (35) stMjDROjDRO)()(    it is why stM  is expressed in %  Similarly, the stability of the feedback is robust in relation to any relative error such as    (36) dynMjDROjDRO.)()(  Notice that:  TddynMM   where TdM  is the time delay margin.  Now it is not difficult to compute static and dynamic margin for different values of ookT ,  andck . It then may be possible to find a triplet for which the stability margins and the dynamic performances are fulfilled.  The reason why this strategy is often efficient is due to the natural robustness of a LQ problem solution and to the notion of loop transfer recovery (LTR). In our application the following results are obtained stM  42.2 %, dynM  32.7 ms.      7 References Books           1De Larminat P., Automatique, commande des systèmes linéaires, deuxième édition, Hermès, 2000, Chaps.16, 19-20. 2De Larminat P., Automatique appliquée, Hermès, 2007.  Reports, Theses, and Individual Papers 3Verdot, T., and Passave, J., “Lois de comportement et système de navigation inertielle,” PFE, ENSMM, Besançon, France 2005. 4Paquier, P., Brocard, G., Lang, B., and Friedt, J.-M., “Projet μDrone – ENSMM (Besançon) Dossier de synthèse,” Concours international universitaire de drones miniatures, ENSMM, Besançon, Sept. 2005. 5Bonneuil, A., Mairiniac, L., Poudevigne, E., and Poupon, E., “Vers une plateforme avionique autonome,” PFE, ENSMM, Besançon, France, 2007.  μDrone project : <URL : http://www.ens2m.fr/drone>     

One way to design the control law of a mini-UAV.

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One way to design the control law of a mini-UAV.

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