Tutorial for the application of Kane's Method to model a small-size helicopter

Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 Tutorial for the application of Kane’s Method to model a small-size helicopter Luis A. Sandino1 , Manuel Bejar2 and Anibal Ollero1,3 1 University of Seville, 2 University Pablo de Olavide, 3 Center for Advanced Aerospace Technologies Seville, SPAIN lsandino@us.es, mbejdom@upo.es, aollero@catec.aero Abstract This paper presents a tutorial for applying Kane’s method to derive a mechanical model of a small-size helicopter. Throughout this development main concepts underlying Kane’s methodology are explained in detail. Furthermore, references to other approaches in the field of classical mechanics are included to analyze the advantages provided by Kane’s method when compared to other alternatives. Since this work considers the most general case of two rigid bodies, fuselage and main rotor, the resulting model will account for most significant modelling issues in the mechanics behaviour of a small-size helicopter, such as gyroscopic effects. Finally it is emphasized that the adopted pedagogical approach should contribute to illustrate the method to beginners. Keywords: UAS, Helicopter, Mechanical modelling, Kane’s method 1 INTRODUCTION The various approaches to obtain equations of motion for mechanical systems can be classified as either vector approaches based on the direct use of D’Alembert’s or Jourdain’s principles such as Newton-Euler equations or Kane’s equations, or scalar approaches based on Hamilton’s or Gauss’ variational principles such as Lagrange’s equations or the Gibbs-Appell equations [3]. Although the equations obtained using the various approaches are equivalent in the sense that they generate the same numerical results in simulation, Kane’s method holds some unique advantages. Both Lagrange’s equations and Kane’s equations use generalized coordinates and thus embed configuration constraints. Therefore, multibody mechanisms yet possess few degrees of freedom, are better served by either one of these methods than by standard application of Newton-Euler equations. Kane’s method, because it uses generalized coordinates yet is based on a vector approach, is sometimes referred to as Lagrange’s form of D’Alembert’s Principle. But Kane’s method also uses generalized speeds, which allows motion constraints to be embedded. So Kane’s method has been compared to the Jourdain’s Principle [10] and Kane’s equations have been likened to the Gibbs-Appell equations [2, 9] and Maggi’s equations [4]. The use of generalized speeds in Kane’s method enables the selection 1 ISBN-978-84-615-6269-5 162 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 of an independent set of motion variables for non-holonomic constrained systems which allows to obtain a model in first order differential equations form. In Lagrangian approach, where only generalized velocities are available, the method of Undetermined Multipliers should be used to handle motion constraints. As mentioned above, the use of generalized speeds can lead to more compact equations of motion. Indeed, a convenient choice of generalized speeds can result in equations that are diagonal in the generalized speed derivatives; that is, uncoupled dynamical equations. A further advantage of Kane’s method over Lagrange’s method derives from its membership in the vector approach classification. The projection of constraint forces which are known to be non-working forces, allows such forces to be disregarded from the outset of the analysis, which leads to savings in the operations required during formulation. The paper is organized as follows. Section 2 introduces variables used in Kane’s equations for deriving a kinematic model, and compares them to those used in Lagrange’s equations. After this, an application example given by the kinematic model of a small-size helicopter is presented according to the pedagogical nature of this work. In section 3, dynamics corresponding to Kane’s equations for a generic mechanical system are presented, highlighting the relationship with the projection of Newton-Euler’s equations into a space tangent to the configuration space. Again, these general concepts are better illustrated by presenting their application to a small-size helicopter. Finally, section 4 is devoted to the conclusions. 2 DERIVATION OF KINEMATICS AND REFERENCES TO LAGRANGE’S APPROACH Let S be a mechanical system with n degrees of freedom moving in an inertial reference frame N subjected to m motion constraints. For the sake of generality, the system consists of Pl (l = 1, · · · , λ ) particles (points possessing mass) and Bk (k = 1, · · · , ν) rigid bodies which centres of mass are denoted by BO k . Points of external force application on rigid bodies Bk are denoted by Pf ( f = 1, · · · , φ ). Thus, the union of the former reference points is Pj ( j = 1, · · · , µ) = Pl ∪ BO k ∪ Pf where µ = λ + ν + φ . The first step in order to build a kinematic model for S is the definition of a set of configuration variables that describe location of reference points Pj∗ = Pl ∪ BO k and orientation of reference frames fixed within each body Bk . In Kane’s method as well as in Lagrange’s method, generalized coordinates qi (i = 1, · · · , n) are used as configuration variables. Generalized coordinates specify point locations and reference frame orientations as relative to each other rather than all relative to inertial reference frame N. In this way, they describe only the allowed configurations for S and therefore encapsulate some configuration constraints. For example, if a revolute joint connects two bodies, only the joint angle is needed to describe the orientation of the second body if orientation of the first is known. In general, instead of the set of 3λ + 6ν variables that specifies the location and orientation of all components of S when defining them relative to a common origin in N, a set of generalized coordinates only consists of n = 3λ + 6ν − c independent variables, where c is the number of configuration constraints encapsulated. In order to characterize motion of S, an additional set of variables called motion variables is also required. In Lagrange’s equations these are given by the generalized coordinate derivatives, also called generalized velocities q̇i (i = 1, · · · , n). However, motion variables in Kane’s method are defined as functions generally linear 2 ISBN-978-84-615-6269-5 163 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 in the generalized coordinate derivatives and non-linear in the generalized coordinates. The use of such functions, called generalized speeds ur (r = 1, · · · , n), provides significantly more compact and thus computationally efficient equations: h u1 ··· iT un = Yn×n · h q̇1 ··· q̇n iT + Zn×1 (1) where matrices Y and Z are functions of generalized coordinates qi (i = 1, · · · , n) and possibly of time t. The reciprocal expressions of the generalized coordinate derivatives in terms of the generalized speeds are called kinematic differential equations: h q̇1 ··· q̇n iT = Wn×n · h u1 ··· un iT + Xn×1 (2) where W = Y−1 and X = Y−1 Z. In Kane’s method, in the same way that configuration constraints are encapsulated within the definition of a set of generalized coordinates, any motion constraint of the system that may arise for physical reasons (e.g., rolling constraints) can be fully embedded by a wise choice of generalized speeds. This will lead to manageable first order dynamical differential equations when describing the entire motion of the system. Additionally, it is often convenient the consideration of differentiated configuration constraints also as motion constraints1 . In contrast to Kane’s method, if motion constraints are present when applying Lagrangian approach, they must be carried along as part of the system description and handled using Lagrange’s Method of Undetermined Multipliers [2]. Concerning the steps required to embed motion constraints, firstly they should be expressed as linear relationships among the generalized speeds [5]: A· h u1 ··· un iT +B = 0 (3) where matrices A and B are functions of generalized coordinates qi (i = 1, · · · , n) and possibly of time t. Given m motion constraints, there will be m dependent generalized speeds in the set ur (r = 1, · · · , n) that can be expressed in terms of the remaining p = n − m independent generalized speeds. After selecting these p independent motion variables, equation (3) can be transformed into the following expression between dependent and independent generalized speeds: h u p+1 ··· un iT = C· h u1 ··· up iT +D (4) Thus, the embedding process is achieved substituting (4) into every kinematic expression containing dependent generalized speeds before formulating dynamical equations. As pointed out before, the definition of the generalized speeds, that is up to the analyst, can have a considerable impact on the compactness of motion equations. However these effects will not become evident until dynamical equations are formulated in a subsequent section of this paper. This section concludes with Table 1 summarizing the differences between Kane’s and Lagrange’s methods when deriving a kinematic model for the general system previously analyzed. 1 The terms configuration constraint and motion constraint have been used in favour of the roughly equivalent terms holonomic constraint and non-holonomic constraint so that a differentiated holonomic constraint, which is still properly referred to as a holonomic constraint, can be grouped with non-holonomic constraints. In simple words, configuration constraints are expressed as equations involving only configuration variables and motion constraints are expressed as equations involving both motion and configuration variables. 3 ISBN-978-84-615-6269-5 164 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 Lagrange’s method Kane’s method generalized coordinates generalized coordinates qi (i = 1, · · · , n) qi (i = 1, · · · , n) generalized velocities generalized speeds q̇i (i = 1, · · · , n) ur (r = 1, · · · , p) Constraints encapsulated c c+m Constraint equations m - Configuration variables Motion variables Table 1: Summary of differences between Kane’s and Lagrange’s method when defining a kinematic model Application example for tutorial purposes: small-size helicopter As advanced in the beginning, this paper intends to improve the understanding of Kane’s methodology by presenting an application example: a small-size helicopter. The system is depicted in Fig. 1 and consists of 3 components: fuselage, main rotor and tail rotor. As pointed out in [7, 8], for most commercially available smallsize helicopters the inertial effects of the main rotor (gyroscopic effects) become the main component influencing the rotational dynamics of the whole mechanical system whereas the tail rotor inertial influence is negligible. This means that main rotor and fuselage will be considered as rigid solids (ν = 2) and tail rotor will only act as an application point of force on the fuselage (φ = 1). Since there are no particles in the system, λ can be considered nil. Figure 1: Helicopter description: reference frames, centres of mass and dimensions of interest for modelling purposes Helicopter motion will be described in an inertial reference frame N where a dextral set of orthogonal unit vectors ni (i = 1, 2, 3) is fixed. Fuselage is denoted by F, whereas its mass and centre of mass are given by mF and F O respectively. A dextral set of orthogonal unit vectors fi (i = 1, 2, 3) is fixed in F and then the central inertia O dyadic of fuselage can be expressed as IF/F = IF11 f1 f1 + IF22 f2 f2 + IF33 f3 f3 . Main rotor is denoted by MR and is modelled as a thin solid disk with constant angular speed ωMR . The same nomenclature criterion for fuselage stands for this case, that is mass and centre of mass are denoted respectively by mMR and MRO . A dextral set of orthogonal unit vectors mri (i = 1, 2, 3) is also fixed in MR, which allows the definition of the central inertia 4 ISBN-978-84-615-6269-5 165 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 O dyadic of main rotor as IMR/MR = IMR11 mr1 mr1 + IMR11 mr2 mr2 + 2 · IMR11 mr3 mr3 with mr3 = f3 being the rotation axis. Scalar constants IF11 , IF22 , IF33 and IMR11 are the principal moments of inertia of the corresponding bodies. An additional reference point O fixed in F is used to describe the location of F O , MRO and tail rotor T RO by means of the corresponding dimensions defined in Fig. 1: O O pO→F = dO−F O ,3 f3 O pO→MR = dO−MRO ,3 f3 pO→T R = dO−T RO ,1 f1 (5) Assumption that F O is located on the rotation axis of the main rotor will be achieved in the real system by an appropriate placement of the equipment on the fuselage. Under all these definitions and assumptions, the centre of mass H O of the whole system relative to O is given by: O pO→H = O O mF · dO−F O ,3 + mMR · dO−MRO ,3 mF pO→F + mMR pO→MR = f3 = dO−H O ,3 f3 mF + mMR mF + mMR (6) Once system geometry and mass distribution have been specified, next step is the definition of the generalized coordinates that have been introduced previously. Since rotation axis and centre of mass position for MR are O totally fixed in F for simplicity (mr3 = f3 and pO→MR = dO−MRO ,3 f3 ), orientation of MR relative to F can be described by only one generalized coordinate. Therefore the two rigid solid compound that models the small-size helicopter possesses c = 5 configuration constraints and consequently the number of degrees of freedom is given by n = 6ν − c = 7. The position of centre of mass H O in the inertial reference frame N is described by generalized coordinates qi (i = 1, 2, 3): pN O →H O = q1 n1 + q2 n2 + q3 n3 (7) Generalized coordinates qi (i = 4, 5, 6) are the Euler-angles (roll, pitch and yaw) corresponding to successive rotations (body123 order, see [6]) that describe the orientation of F in the inertial reference frame N. Furthermore, generalized coordinate q7 is the angle corresponding to the rotation of MR relative to F. Thus, unit vectors ni , fi and mri are geometrically related by the direction cosine matrices shown in Table 2. f1 f2 f3 mr1 mr2 mr3 n1 c5 c6 −c5 s6 s5 f1 c7 −s7 0 n2 c4 s6 + s4 s5 c6 c4 c6 − s4 s5 s6 −s4 c5 f2 s7 c7 0 n3 s4 s6 − c4 s5 c6 s4 c6 + c4 s5 s6 c4 c5 f3 0 0 1 where si = sin(qi ) and ci = cos(qi ). Table 2: Direction cosine matrices Once generalized coordinates are defined, kinematic equations can be formulated as follows: N HO v N , N dpO→H O ω F , f1 dt N df dt 2 = q̇1 n1 + q̇2 n2 + q̇3 n3 · f3 + f2 N df dt 3 · f1 + f3 N df 1 dt (8) · f2 = (s6 q̇5 + c5 c6 q̇4 )f1 + (c6 q̇5 − s6 c5 q̇4 )f2 + (q̇6 + s5 q̇4 )f3 F ω MR = q̇7 f3 (9) (10) Generalized speeds ui (i = 1, · · · , 7) are defined in such a way that (8), (9) and (10) can be written in a more 5 ISBN-978-84-615-6269-5 166 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 compact way: ui , q̇i (i = 1, 2, 3) ⇒    u4 , s6 q̇5 + c5 c6 q̇4    u5 , c6 q̇5 − s6 c5 q̇4 ⇒      u6 , q̇6 + s5 q̇4 u7 , q̇7 N vH O = u1 n1 + u2 n2 + u3 n3 (11) = u4 f1 + u5 f2 + u6 f3 (12) NωF F ω MR ⇒ = u7 f3 (13) The former leads to the following kinematic differential equations: q̇i = ui (i = 1, 2, 3) q̇4 = − (s6 u5 − c6 u4 ) /c5 q̇5 = s6 u4 + c6 u5 q̇6 = u6 + s5 (s6 u5 − c6 u4 ) /c5 q̇7 = u7 (14) As was explained before, configuration constraints are embedded in the definition of generalized coordinates itself. In contrast, in order to embed motion constrains, their mathematical formulation and subsequent replacement into equations of motion is required. In this case, the only motion constraint is given by the fact that the main rotor is assumed to be rotating with constant angular speed ωMR : u7 = ωMR (15) Therefore, there is m = 1 dependent generalized speeds and p = n − m = 6 independent generalized speeds. Hence, last equation from (14) can be removed from the analysis and the remaining six equations constitute the kinematical differential equations and form the first half of the equations of motion for the system under study. 3 DERIVATION OF DYNAMICS AND REFERENCES TO NEWTON-EULER’S APPROACH After deriving a kinematic model that describes the configuration and motion of the target system, next step in Kane’s method is the definition of some auxiliary kinetic expressions: partial velocities, partial angular velocities, generalized active forces, and generalized inertia forces. Subsequently these variables will be related to each other by applying mechanics principles to obtain the final dynamic model. 3.1 Partial velocities and partial angular velocities Position of reference points Pj ( j = 1, · · · , µ) with respect to N are given by vectors pPj = p(q1 , · · · , qn ,t). An expression for the velocity vector N vPj in terms of q̇i (i = 1, · · · , n) is obtained by applying the chain rule to these vector functions: N Pj v = N dpPj dt = h N ∂ pP j ∂ q1 ··· N ∂ pP j ∂ qn i h · q̇1 ··· q̇n iT + N ∂ pPj ∂t ( j = 1, · · · , µ) (16) where the left superscript indicates that differentiation is performed in frame N. Substituting (2) into (16) yields: N Pj v = h N ∂ pP j ∂ q1 ··· N ∂ pP j ∂ qn i ·W· h u1 ··· un iT + h N ∂ pPj ∂ q1 ··· N ∂ pPj ∂ qn i ·X+ N ∂ pPj ∂t (17) Finally, (4) must be used to remove dependent generalized speeds from (17). Once this last replacement is N Pj P P done, the r-th partial velocity vr j of Pj is then defined as the coefficient for ur (r = 1, · · · , p), i.e., vr j = ∂ ∂ uvr , 6 ISBN-978-84-615-6269-5 167 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 P in the resultant expression. The portion of N vPj not containing any ur is denoted by vt j and called the velocity remainder of Pj in N. Rewriting (17) according to this nomenclature yields: N Pj v = h P v1 j ··· P v pj i h · u1 ··· up iT P + vt j (18) By a similar reasoning, the angular velocity of rigid bodies Bk (k = 1, · · · ν) can always be expressed in terms of the independent generalized speeds [5]: N ω Bk = h B ω1k ··· B ω pk i h · u1 ··· up iT B + ωt k (19) B where the r-th partial angular velocity ω r k of Bk is again defined as the coefficient for ur (r = 1, · · · , p), i.e., N B B B ω r k = ∂ ∂ωur k . The portion of N ω Bk not containing any ur is denoted ω t k and called the angular velocity remainder of Bk in N. Partial velocities will be defined only for those points Pj subjected to applied forces and/or possessing mass, whereas partial angular velocities will be defined only for those rigid bodies Bk subjected to applied torques and/or possessing inertia. Application example for tutorial purposes: small-size helicopter (continuation 1) Considering (11), (12), (13) and using the expressions: N N Fi v = N HO ω MR = N v + N ω F × pH O →F i ω F + F ω MR (20) where Fi is any point fixed in F, Table 3 with partial velocities and partial angular velocities for points Pj ( j = F O , MRO , T RO ) and rigid bodies Bk (k = F, MR) of the system under study can be constructed by inspection. r 1 2 3 4 5 6 FO n1 n2 n3 (dO−H O ,3 − dO−F O ,3 )f2 (dO−F O ,3 − dO−H O ,3 )f1 0 O vMR r n1 n2 n3 (dO−H O ,3 − dO−MRO ,3 )f2 (dO−MRO ,3 − dO−H O ,3 )f1 0 O vTr R n1 n2 n3 dO−H O ,3 f2 −dO−H O ,3 f1 − dO−T RO ,1 f3 dO−T RO ,1 f2 ω Fr 0 0 0 f1 f2 f3 ω MR r 0 0 0 f1 f2 f3 vr Table 3: partial velocities and partial angular velocities 3.2 Generalized active forces and generalized inertia forces According to [5], generalized active forces are defined as dot product of partial velocities and active (i.e. applied) forces and dot product of partial angular velocities and active torques. Then, for each reference point Pj subjected to applied forces and for each rigid body Bk subjected to applied torques, generalized active forces are given by: P (Fr )Pj , vr j · RPj B (Fr )Bk , ω r k · TBk (r = 1, · · · , p) P vr j (21) B Pj , ω r k where is the r-th partial velocity of Pj , RPj is the resultant of all active forces acting on is the r-th partial angular velocity of Bk and TBk is the resultant of all active torques acting on Bk . The r-th generalized active force Fr can then be determined by summing the results over all reference points Pj and all rigid bodies Bk : µ Fr = ν ∑ (Fr )Pj + ∑ (Fr )Bk j=1 k=1 7 ISBN-978-84-615-6269-5 168 (r = 1, · · · , p) (22) Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 Inertia force R∗Pj and inertia torque T∗Bk are defined by [5] as: R∗Pj , −mPj N aPj O O T∗Bk , −IBk /Bk · N α Bk − N ω Bk × IBk /Bk · N ω Bk where mPj is the mass of Pj , N aPj , N d N vP j (23) Bk /BO k is the acceleration of Pj in N, I dt is the inertia dyadic of Bk about N N B N Bk is the angular velocity of B in N and N α Bk , d ω k its centre of mass BO k k (i.e. the central inertia dyadic), ω dt ∗ is the angular acceleration of Bk in N. Note that RPj is equal to the time derivative in N of linear momentum O LPj = mPj N vPj of Pj and T∗Bk is equal the time derivative in N of angular momentum HBk = IBk /Bk · N ω Bk of Bk , both with negative sign. Therefore, generalized inertia forces are defined as dot product of partial velocities and inertia forces and dot product of partial angular velocities and inertia torques. Then, for each reference point Pj possessing mass and for each rigid body Bk possessing inertia, generalized inertia forces are given by: P (Fr∗ )Pj , vr j · R∗Pj B (Fr∗ )Bk , ω r k · T∗Bk (r = 1, · · · , p) (24) P B where vr j is the r-th partial velocity of Pj , R∗Pj is the inertia force for Pj , ω r k is the r-th partial angular velocity of Bk and T∗Bk is the inertia torque of Bk . The r-th generalized inertia force Fr∗ can then be determined by summing the results over all points Pj and all rigid bodies Bk : µ Fr∗ = ν ∑ (Fr∗ )Pj + ∑ (Fr∗ )Bk j=1 (r = 1, · · · , p) (25) k=1 Application example for tutorial purposes: small-size helicopter (continuation 2) Concerning forces and torques applied to the system (See Fig. 2), main rotor generates a force FMR = fMR,3 f3 applied at point MRO and torques MMR,i = tMR,i fi (i = 1, 2, 3) applied to MR, whereas tail rotor generates a force FT R = fT R,2 f2 applied at point T RO and a torque MF = tT R,2 f2 applied to F. Force of gravity W j = −m j gn3 ( j = F, MR) applied at centres of mass F O and MRO is also considered, where g is the acceleration of gravity. Figure 2: Forces and torques applied to the helicopter Now, using equations (21) and (22) the generalized active forces for the first and fourth partial velocities can be obtained as follows: O O O · RMRO + vT1 R · RT RO + ω F1 · TF + ω MR F1 = vF1 · RF O + vMR 1 · TMR 1 3 = n1 · WF + n1 · (FMR + WMR ) + n1 · FT R + 0 · MF + 0 · ∑ MMR,i i=1 = fMR,3 s5 − fT R,2 c5 s6 O O O · RMRO + vT4 R · RT RO + ω F4 · TF + ω MR F4 = vF4 · RF O + vMR 4 4 · TMR = (dO−H O ,3 − dO−F O ,3 )f2 · WF + (dO−H O ,3 − dO−MRO ,3 )f2 · (FMR + WMR )+ 3 + dO−H O ,3 f2 · FT R + f1 · MF + f1 · ∑ MMR,i = tMR,1 + dO−H O ,3 fT R,2 i=1 8 ISBN-978-84-615-6269-5 169 (26) Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 Following the same procedure, the remaining generalized active forces are given by: F2 = fT R,2 (c4 c6 − s4 s5 s6 ) − fMR,3 s4 c5 F3 = fMR,3 c4 c5 + fT R,2 (s4 c6 + c4 s5 s6 ) − (mF + mMR )g F5 = tMR,2 + tT R,2 F6 = tMR,3 + dO−T RO ,1 fT R,2 (27) Regarding generalized inertia forces for the first and fourth partial velocities, using equations (23) to (25) yields: O O ∗ · R∗MRO + ω F1 · T∗F + ω MR F1∗ = vF1 · R∗F O + vMR 1 1 · TMR O O O O = n1 · (−mF N aF ) + n1 · (−mMR N aMR ) + 0 · (−IF/F · N α F − N ω F × IF/F · N ω F )+ + 0 · (−I F4∗ O = vF4 MR/MRO · R∗F O MR/MRO · N α MR − N ω MR × I O + vMR 4 · R∗MRO + ω F4 · T∗F · N ω MR ) = −(mF + mMR )u̇1 ∗ + ω MR 4 · TMR O O = (dO−H O ,3 − dO−F O ,3 )f2 · (−mF N aF ) + (dO−H O ,3 − dO−MRO ,3 )f2 · (−mMR N aMR )+ O O O O + f1 · (−IF/F · N α F − N ω F × IF/F · N ω F ) + f1 · (−IMR/MR · N α MR − N ω MR × IMR/MR · N ω MR ) = K456 u5 u6 − K4p4 u̇4 + K45 u5 (28) Following the same procedure, the remaining generalized inertia forces are given by: F2∗ = −(mF + mMR )u̇2 F3∗ = −(mF + mMR )u̇3 F5∗ = K546 u4 u6 − K5p5 u̇5 + K54 u4 F6∗ = K645 u4 u5 − K6p6 u̇6 (29) where parameters Kxxx have the following values: K456 = IF22 − IF33 − IMR11 + K4p4 = IF11 + IMR11 + mF · mMR · (dO−F O ,3 − dO−MRO ,3 )2 mF + mMR mF · mMR · (dO−F O ,3 − dO−MRO ,3 )2 mF + mMR K45 = −2IMR11 ωMR K546 = IF33 − IF11 + IMR11 − K5p5 = IF22 + IMR11 + mF · mMR · (dO−F O ,3 − dO−MRO ,3 )2 mF + mMR mF · mMR · (dO−F O ,3 − dO−MRO ,3 )2 mF + mMR K54 = 2IMR11 ωMR K645 = IF11 − IF22 K6p6 = IF33 + 2 · IMR11 (30) Note that all operations in (26) and (28) are performed after expressing all vectors in the reference frame N using the corresponding relationships of Table 2. 3.3 Equations of motion Classical Newton-Euler’s equations for each reference point Pj and each rigid body Bk of system S can be expressed as: RPj − L̇Pj = 0; TBk − ḢBk = 0 (31) where RPj is the resultant of forces acting on Pj , LPj is the linear momentum of Pj , TBk is the resultant of all torques acting on Bk and HBk is the angular momentum of Bk . Projecting the first and the second equations in 9 ISBN-978-84-615-6269-5 170 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 P B (31) into the vector space formed by vr j and ω r k respectively (which is, in fact, equivalent to apply Jourdain’s principle) and summing both results over all bodies of system S yields: µ ν P B µ ν P B ∑ vr j · RPj + ∑ ω r k · TBk + ∑ vr j · (−L̇Pj ) + ∑ ω r k · (−ḢBk ) = 0 j=1 k=1 j=1 (r = 1, · · · p) (32) k=1 Thus, comparing (32) with equations (21) to (25) leads to: Fr + Fr∗ = 0 (r = 1, · · · , p) (33) which are known as Kane’s equations. This set of first order differential equations in the generalized speeds is called dynamical differential equations and constitutes the second half of the equations of motion of system S. These equations together with the kinematic differential equations in (2) will govern the behaviour of system S. Application example for tutorial purposes: small-size helicopter (continuation 3) Finally, equation (33) is applied to obtain the dynamical differential equations, resulting in: (mF + mMR )u̇1 = fMR,3 s5 − fT R,2 c5 s6 (mF + mMR )u̇2 = fT R,2 (c4 c6 − s4 s5 s6 ) − fMR,3 s4 c5 (mF + mMR )u̇3 = fMR,3 c4 c5 + fT R,2 (s4 c6 + c4 s5 s6 ) − (mF + mMR )g K4p4 u̇4 = tMR,1 + dO−H O ,3 fT R,2 + (K456 u6 + K45 ) u5 K5p5 u̇5 = tMR,2 + tT R,2 + (K546 u6 + K54 ) u4 K6p6 u̇6 = tMR,3 + dO−T RO ,1 fT R,2 + K645 u4 u5 (34) where parameters Kxxx are those listed in (30). Equations (14) together with (34) form the equations of motion for the helicopter. It is remarkable that these equations are uncoupled in the generalized speed derivatives thank to the choice of generalized speeds made in (11), (12) and (13). This fact makes the model suitable for numerical integration. Note also that constraint forces and torques (i.e. interaction forces and torques between main rotor and fuselage) were not present during the formulation. The reason is that the vector space used to project NewtonEuler’s equations to derive Kane’s equations is orthogonal to those forces and torques, which allows to disregard them from the outset of the analysis. 3.4 Comparison of computational efficiency between Kane’s and Newton-Euler’s methods The procedure presented in this paper to obtain the equations of motion of a small-size helicopter can be easily performed with the help of MotionGenesis software [1], a symbolic manipulator for the analysis of mechanical systems that implements Kane’s method. In order to compare computational efficiency between Kane’s and Newton-Euler’s methods, equations (14)-(34) and the model corresponding to direct application of NewtonEuler’s equations (31) were generated using Autolev2 scripts. Each script was executed several times in a laptop with an Intel Core i5 M480 processor at 2.67 GHz. The average CPU time was 1.58 seconds for the script corresponding to Kane’s method and 2.35 seconds for the one corresponding to Newton-Euler’s method. The size of MATLAB code generated by Autolev was 16 kBytes in the first case and 24 kBytes in the second one. This code 2 version 4 of MotionGenesis software. 10 ISBN-978-84-615-6269-5 171 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 was used to implement the helicopter model together with the controller proposed by [8] in MATLAB-Simulink with a sample time of 0.01 seconds. The resulting system was executed in a PC/104 Single Board Computer with MATLAB-Simulink’s xPC real-time operating system. The simulation consisted of the application of a sequence of reference step signals to the variables q1 , q2 , q3 and q6 during 60 seconds. The average Task Execution Time (TET)3 was 2.24 × 10−4 seconds for the model obtained with Kane’s method and 4.85 × 10−4 seconds for the model obtained with Newton-Euler’s method. These results are summarized in Table 4a. Regarding accuracy, Table 4b shows the maximum absolute difference between variables qr (r = 1, · · · 6) of both models during real-time simulation. As can be seen the difference between the two models is very small and the computational efficiency of the Kane’s method is better. Newton-Euler’s method r
| − qN−E max |qKane r r 1 3.3398 × 10−4 2 3.3527 × 10−4 3 4.2196 × 10−5 4 3.9635 × 10−5 5 4.0048 × 10−5 6 2.4435 × 10−6 Kane’s method Average CPU time 2.35 s 1.58 s (Autolev scripts) Generated MATLAB code 24 kB 16 kB 4.85 × 10−4 s 2.24 × 10−4 s size Average TET (Model real-time execution) (a) Computational efficiency (b) Accuracy Table 4: Comparison between Kane’s and Newton-Euler’s methods 4 CONCLUSIONS This paper presents a pedagogical development of Kane’s method through its application to model a small-size helicopter. To this end, the formulation of the equations of motion emphasizes the interpretation of the concepts underlying in Kane’s methodology in order to ease its understanding for beginners. Concerning the model itself, this work considers the general case of two rigid bodies, fuselage and main rotor. Hence, the resulting modelling structure accounts for most significant modelling issues in the mechanics behaviour of small-size helicopters. The method has proved to hold some unique advantages when compared to other traditional approaches. On the one hand, the use of generalized coordinates allows to embed configuration constraints. On the other hand, the adoption of generalized speeds enables the derivation of a compact model in first order differential equations form. Indeed, an adequate choice for generalized speeds can result in equations that are uncoupled in the generalized speed derivatives. Another remarkable advantage is that constraint forces are disregarded from the outset of the analysis. Finally it should be also noted that equations of motion generated by Kane’s method are characterized by their easy implementation and computational efficiency. This efficiency is two-fold: on the one hand, equations are 3 This value is the measured CPU time to run the model equations and post outputs during each sample interval. 11 ISBN-978-84-615-6269-5 172 Proc. of the 1 st Workshop on Research, Development and Education on Unmanned Aerial Systems (RED-UAS 2011) Seville, Spain, Nov 30th - Dec 1 st , 2011 obtained with the fewest symbolic operations, on the other hand, obtained equations require the fewest numerical calculations for their solution. ACKNOWLEDGEMENTS This work have been partially supported by Junta de Andalucia excellence project P09-TEP-5120 and European Commission projects PLANET (FP7-ICT-2009-5) and CONET (FP7-ICT-2007-7). References [1] Motiongenesis Kane 5.x. http://www.motiongenesis.com/, 2011. [2] H. Baruh. Analytical Dynamics. McGraw Hill, 1998. [3] H. Baruh. Another look at the describing equations of dynamics. Journal of the Chinese Society of Mechanical Engineers, Transactions of the Chinese Institute of Engineers, Series C/Chung-Kuo Chi Hsueh Kung Ch’eng Hsuebo Pao, 21(1):15–23, 2000. [4] M. Borri, C. Bottasso, and P. Mantegazza. Equivalence of kane’s and maggi’s equations. Meccanica, 25(4):272–274, 1990. [5] T. R. Kane and D. A. Levinson. Dynamics, Theory and Applications. McGraw Hill, 1985. [6] T. R. Kane, P. W. Likins, and D. A. Levinson. Spacecraft Dynamics. McGraw Hill, 1983. [7] K. Kondak, M. Bernard, N. Losse, and G. Hommel. Elaborated modeling and control for autonomous small size helicopters. VDI Berichte, (1956):207, 2006. [8] K. Kondak, M. Bernard, N. Meyer, and G. Hommel. Autonomously flying vtol-robots: Modeling and control. In IEEE International Conference on Robotics and Automation, pages 736–741, 2007. [9] J.G. Papastavridis. A panoramic overview of the principles and equations of motion of advanced engineering dynamics. Applied Mechanics Reviews, 51(4):239–265, 1998. [10] J. Piedboeuf. Kane’s equations or jourdain’s principle? Systems, volume 2, pages 1471–1474, 1993. 12 ISBN-978-84-615-6269-5 173 In IEEE Midwest Symposium on Circuits and