# Stability Analysis of a Helicopter with an External Slung Load System

Authors: Kary Thanapalan
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

This paper describes the stability analysis of a helicopter with an underslung external load system. The Lyapunov second method is considered for the stability analysis. The system is considered as a cascade connection of uncertain nonlinear system. The stability analysis is conducted to ensure the stabilisation of the helicopter system and the positioning of the underslung load at hover condition. Stability analysis and numerical results proved that if desired condition for the stability is met, then it is possible to locate the load at the specified position or its neighbourhood.


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## Body

## 1. Introduction

Recently, research on helicopter carrying external underslung loads has gained great attention in the aerospace research community for the past few decades due to the reevaluation and extension of the ADS-33 and the inherent stability problems associated with this system [1–3]. Helicopters have the ability to carry large and bulky loads externally on a sling. This capability is important in many applications, ranging from lifting heavy loads to saving life. Importantly, when lives are under risk and rapid rescue operations are needed, this operation is vital. The stability of the helicopter will be disturbed by the underslung load, which is a huge obstacle for an accurate pickup or placement of the loads [4]. Thus, it is necessary to resolve the stability problems associated with the system to ensure the stabilisation of the helicopter system and the positioning of the underslung load under various complicated situations.From the review of popular helicopter control methods, it is clear that, in the past years, considerable attention has been paid to the design of controller to obtain a satisfactory helicopter handling quality [5]. The control problem has been tackled using different approaches ranging from linear quadratic control [6], eigenstructure assignment [7], classical SISO techniques [8], to sliding mode control [9]. Apart from the methods emphasised above there are many other techniques which are reported for complex modern control system design ranging from quantitative feedback theory to singular perturbation method [10].The extensive studies of the reported controller design methods evidenced that the helicopter control and the control of a helicopter with an external underslung load are very active research areas. The research in this area is mainly motivated by the factor that the current control methods cannot provide full satisfaction to the desired design requirements on flight handling quality, stability, robustness, and so forth.In this paper, stability analysis for the helicopter with an underslung external load system is discussed. The key advantage of the proposed method is that the analysis takes the system uncertainty into account. The proposed method can give a guaranteed stability region for the systems considered. The paper begins by presenting a mathematical model of the system and then describing the stability analysis with a numerical example to illustrate the applicability, accuracy, and effectiveness of the proposed method.


## 2. System Model

Considering the control of a helicopter with an underslung load, the dynamical models of both the helicopter and load have some terms which are uncertain. The uncertainties may arise from the helicopter to carry an unknown load or the immeasurable parameters in the dynamical models. The uncertainties may also arise from computational errors of the dynamical effects such as aerodynamics. Therefore for a realistic model uncertainties must be taken into account during the controller design.A mathematical model of the helicopter described in [11] and an underslung load model presented in [4] are adopted in this work. Considering the two models, a mathematical model for a helicopter carrying an underslung load can be obtained.Firstly, the underslung load is considered to be suspended from a single suspension point that is subject to motion and therefore modelled as a driven spherical pendulum. The equations that describe the load dynamics are obtained by first considering motion with reference to the longitudinal suspension angleθL in the X-Z plane (Figure 1). This is then repeated for the lateral case involving φL and the Y-Z plane. These are then combined to obtain the model for the motion of the load. The underslung load system has six inputs, longitudinal, lateral, and vertical velocities together with the corresponding accelerations of the helicopter, whilst the outputs are the longitudinal and lateral directional suspension angles. The load is subject to an isotropic aerodynamic force (proportional to the square of its airspeed) such as what would be experienced by a spherical shaped load. Aerodynamic interaction with the helicopter that may occur, for example, due to rotor downwash, has been ignored. Finally, the sling itself is assumed to be rigid and contribute zero aerodynamic force of its own. With these assumptions, the equations governing the load motion can be derived as follows.Figure 1
Coordinate system for the longitudinal motion in theX-Z plane.For the case of the longitudinal motion in theX-Z plane, the mathematical model is described below:(1) θ ¨ L = - g l x sin ⁡ θ L + cos ⁡ θ L l x X ¨ 0 + sin ⁡ θ L l x Z ¨ 0 + k D sign ⁡ X ˙ L cos ⁡ θ L M L l x X ˙ 0 2 + k D sign ⁡ Z ˙ L sin ⁡ θ L M L l x Z ˙ 0 2 - 2 k D M L sign ⁡ X ˙ L cos 2 ⁡ θ L X ˙ 0 + sign ⁡ Z ˙ L sin 2 ⁡ θ L Z ˙ 0 θ ˙ L - k θ θ ˙ L  + k D l x M L sign ⁡ X ˙ L cos 3 ⁡ θ L + sign ⁡ Z ˙ L sin 3 ⁡ θ L θ ˙ L 2 .Define θ~L=[θLθ˙L]T=[θL1θL2]T; then the load model can be rewritten as follows:(2a)θ˙L1=θL2,(2b)θ˙L2=-glxsin⁡θL1+kDlxMLsign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL22-kθθL2+kDsign⁡X˙Lcos⁡θL1MLlxX˙02+kDsign⁡Z˙Lsin⁡θL1MLlxZ˙02+cos⁡θL1lxX¨0+sin⁡θL1lxZ¨0-2kDMLsign⁡X˙Lcos2⁡θL1X˙0+sign⁡Z˙Lsin2⁡θL1Z˙0θL2.The helicopter model is considered as the second subsystem. To simplify the analysis, the linear helicopter model [4] is considered, which is expressed in the state space form x-˙Ht=Ax-Ht+Bu-(t): (3)ϕ˙θ˙ψ˙u˙v˙w˙p˙q˙r˙=0000001000000000100000000010-gcos⁡θe0XuXvXwXpXq-weXr-veg00YuYvYwYp+weYqYr-ue0-gsin⁡θe0ZuZvZwZp-veZq+ueZr000LuLvLwLpLqLr000MuMvMwMpMqMr000NuNvNwNpNqNrϕθψuvwpqr+000000000000Xθ1cXθ1sXθ0Xθ0TYθ1cYθ1sYθ0Yθ0TZθ1cZθ1sZθ0Zθ0TLθ1cLθ1sLθ0Lθ0TMθ1cMθ1sMθ0Mθ0TNθ1cNθ1sNθ0Nθ0Tθ1cθ1sθ0θ0T.For linearization, it is assumed that the external forces X, Y, and Z and moments L, M, and N can be represented as analytic functions of the disturbed motion variables and their derivatives [4]. Thus the forces and moments can be written in Taylor’s expansion form. Then, the linearized equations of motion about a general trim condition can be written as in the state space form x-˙=Ax-+Bu- and the system matrix A and control matrix B are derived from the partial derivatives of the nonlinear function F-, that is, (4)x-˙=F-x-,u-,t,A=∂F-∂x-x-=x-e,B=∂F-∂u-x-=x-e.Now, the longitudinal rotational motion is described by the pitch angle θ and pitch rate q together with the translation motion components u,w so the equation of longitudinal motion can be written as follows:(5)θ˙q˙u˙w˙=01000MqMuMw-gcos⁡θeXq-weXuXw-gsin⁡θeZq-ueZuZwθquw+00Mθ1sMθ0Xθ1sXθ0Zθ1sZθ0θ1sθ0.By applying a linear transformation T- such that xH(t)=T-z(t) and T- is defined by(6)T-=100001a11a1200100001,where (7)a11=-Zθ1sMθ0-Zθ0Mθ1sXθ1sZθ0-Zθ1sXθ0,a12=-Xθ0Mθ1s-Xθ1sMθ0Xθ1sZθ0-Zθ1sXθ0,and letting q-=(q-a11u-a12w), then the system equation (5) is transformed into the following form:(8)θ˙q-˙u˙w˙=X11X12X13X14X21X22X23X24X31X32X33X34X41X42X43X44θq-uw+00B21B22Xθ1sXθ0Zθ1sZθ0θ1sθ0,where(9)X11=0,X12=1,X13=a11,X14=a12X21=a11gcos⁡θe+a12gsin⁡θe,X22=Mq-a11Xq-we-a12Zq-ueX23=Mq-a11Xq-we-a12Zq+uea11+Mu-a11Xu-a12ZuX24=Mq-a11Xq-we-a12Zq+uea12+Mw-a11Xw-a12Zw,X31=-gcos⁡θe,X32=Xq-we,X33=Xq-wea11+Xu,X34=Xq-wea11+Xw,X41=-gsin⁡θe,X42=Zq+ue,X43=Zq+uea11+Zu,X44=Zq+uea12+Zw,B21=Mθ1s-a11Xθ1s-a12Zθ1s,B22=Mθ0-a11Xθ0-a11Zθ0.Using (8) the system model can be rearranged to include the variables θ and q- into the load model. For the stability analysis purpose, an extra term is introduced into the system model, which is zero with the expression(10)κ1kDsign⁡X˙Lcos⁡θL1uMLlx+κ2kDsign⁡X˙Lcos⁡θL1wMLlx-κ1kDsign⁡X˙Lcos⁡θL1uMLlx-κ2kDsign⁡X˙Lcos⁡θL1wMLlx,where κi>0 (i=1,2) are small positive constants.With this arrangement for the longitudinal motion of the helicopter with an underslung load combined system model can be written as follows:(11a)θ-˙Lt=f1θ-Lt+G1θ-LtpxHt+qθ-Lt,xHt+Ht,θ-L,xHt(11b)x˙Ht=f2θ-Lt,xHt+G2u~t,where (12) θ - L t = θ L 1 θ L 2 θ q - T , x H = u w T , u ~ t = θ 1 s θ 0 T , p x H = u 2 + κ 1 u w 2 + κ 2 w T , f 1 θ - L t = θ L 2 - g l x sin ⁡ θ L 1 + k D l x M L sign ⁡ X ˙ L cos 3 ⁡ θ L 1 + sign ⁡ Z ˙ L sin 3 ⁡ θ L 1 θ L 2 2 - k θ θ L 2 q - X 21 θ + X 22 q - G 1 θ - L t = 0 0 k D sign ⁡ X ˙ L cos ⁡ θ L 1 M L l x k D sign ⁡ Z ˙ L sin ⁡ θ L 1 M L l x  0 0 0 0 q θ - L t , x H t = 0 cos ⁡ θ L 1 l x u ˙ + sin ⁡ θ L 1 l x w ˙ - 2 k D M L sign ⁡ X ˙ L cos 2 ⁡ θ L 1 u + sign ⁡ Z ˙ L sin 2 ⁡ θ L 1 w θ L 2 - κ 1 k D sign ⁡ X ˙ L cos ⁡ θ L 1 u M L l x - κ 2 k D sign ⁡ X ˙ L cos ⁡ θ L 1 w M L l x H t , θ - L t , x H t = 0 0 a 11 u + a 12 w X 23 u + X 24 w , f 2 θ - L t , x H t = X 31 θ + X 32 q - + X 33 u + X 34 w X 41 θ + X 42 q - + X 43 u + X 44 w , G 2 = X θ 1 s X θ 0 Z θ 1 s Z θ 0 .It is assumed that the longitudinal motion is primarily controlled by longitudinal cyclic commands θ1s and main rotor collective θ0.For the case of the lateral motion in theY-Z plane, with the coordinate system described in Figure 2, the load model is(13)φ¨Lyz=-glysin⁡φLyz+cos⁡φLyzlyY¨0+sin⁡φLyzlyZ¨0+kDsign⁡Y˙Lcos⁡φLyzMLlyY˙02+kDsign⁡Z˙Lsin⁡φLyzMLlyZ˙02-2kDMLsign⁡Y˙Lcos2⁡φLyzY˙0+sign⁡Z˙Lsin2⁡φLyzZ˙0φ˙L-kφφ˙L+kDlyMLsign⁡Y˙Lcos3⁡φLyz+sign⁡Z˙Lsin3⁡φLyzφ˙Lyz2.Define φ~L=φLφ˙LT=φL1φL2T; then the model can be written as follows:(14a)φ˙L1=φL2(14b)φ˙L2=-glysin⁡φL1+kDlyMLsign⁡Y˙Lcos3⁡φL1+sign⁡Z˙Lsin3⁡φL1φL22-kφφL2+kDsign⁡Y˙Lcos⁡φL1MLlyY˙02+kDsign⁡Z˙Lsin⁡φL1MLlyZ˙02+cos⁡φL1lyY¨0+sin⁡φL1lyZ¨0-2kDMLsign⁡Y˙Lcos2⁡φL1Y˙0+sign⁡Z˙Lsin2⁡φL1Z˙0φL2.Now, for the helicopter model the lateral rotational motion is described by the roll angle ϕ and roll rate p together with the translation motion components v,w so the equation of lateral motion can be written as follows:(15)ϕ˙p˙v˙w˙=01000LpLvLwgYp-weYvYw0Zp-veZvZwϕpvw+00Lθ1cLθ0TYθ1cYθ0TZθ1cZθ0Tθ1cθ0T,now using a linear transformation T-1 such that xH(t)=T-1z(t) and T-1 is defined by(16)T-1=100001b11b1200100001,where (17)b11=-Zθ1cLθ0T-Zθ0TLθ1cYθ1cZθ0T-Zθ1cYθ0T,b12=-Yθ0TLθ1c-Yθ1cLθ0TYθ1cZθ0T-Zθ1cYθ0T.Let p-=(p-b11v-b12w); then (15) can be written in the following form:(18)ϕ˙p-˙v˙w˙=Y11Y12Y13Y14Y21Y22Y23Y24Y31Y32Y33Y34Y41Y42Y43Y44ϕp-vw+00N21N22Yθ1cYθ0TZθ1cZθ0Tθ1cθ0T,where(19)Y11=0,Y12=1,Y13=b11,Y14=b12,Y21=-b11g,Y22=Lp-b11Yp+we-b12Zp-ve,Y23=Lp-b11Yp+we-b12Zp-veb11+Lp-b11Yv-b12Zv,Y24=Lp-b11Yp+we-b12Zp-veb12+Lp-b11Yw-b12Zw,Y31=g,Y32=Yp+we,Y33=Yp+web11+Yv,Y34=Yp+web12+Yw,Y41=0,Y42=Zp-ve,Y43=Zp-veb11+Zv,Y44=Zp-veb12+Zw,N21=Lθ1c-b11Yθ1c-b12Zθ1c,N22=Lθ0T-b11Yθ0T-b12Zθ0T.Using (18) the helicopter with underslung load for lateral motion in the Y-Z plane can be described by(20a)φ-˙Lt=f1φ-Lt+G1φ-LtpxHt+qφ-Lt,xHt+Ht,φ-L,xHt(20b)x˙Ht=f2φ-Lt,xHt+G2u^t,where (21)φ-˙Lt=φ˙L1φ˙L2ϕ˙p-˙T,pxHt=v2w2T,f1φ-Lt=φL2-glysin⁡φL1+kDlyMLsign⁡Y˙Lcos3⁡φL1+sin⁡Z˙Lsin3⁡φL1φL22-kφφL2p--b11gϕ+Lp-b11Yp+we-b12Zp-vep-,G1φ-Lt=00kDsign⁡Z˙Lcos⁡φL1MLlykDsign⁡Z˙Lsin⁡φL1MLly0000,qφ-Lt,xHt=0cos⁡φL1lyv˙+sin⁡φL1lyw˙-2kDMLsign⁡Y˙Lcos2⁡φL1v+sign⁡Z˙Lsin2⁡φL1wφL2,Ht,φ-Lt,xHt=00ϕ˙-p-Y23v+Y24w,u^t=θ1cθ0TT,f2φ-Lt,xHt=Y31ϕ+Y32p-+Y33v+Y34wY42p-+Y43v+Y44w,G2=Yθ1cYθ0TZθ1cZθ0T.It is assumed that the lateral motion is primarily controlled by lateral cyclic commands θ1c and the tail rotor collective θ0T.Figure 2
Coordinate system for the lateral motion in theY-Z plane.


## 3. Stability Analysis

The goal is to analyse the stability of the combined system to ensure that it is possible to stabilise the helicopter with underslung load system modelled by (11a), (11b), (20a), and (20b), in a real environment with uncertainties. In this paper, for the stability analysis the Lyapunov second method is applied for the helicopter with an underslung external load system. The analysis for the longitudinal motion is discussed first. The system equations can be considered to have two main parts, that is, known and unknown (or partly known). The known terms formed the nominal part of the system model. The unknown or partly known part can be considered as the uncertainty to the system. The whole system is then modelled by a nominal part with the addition of uncertainty. In fact, the known elements in the subsystem (11a) are characterised by the prescribed triple (f1,G1,p) and it is desired that the nominal part of the system is stable.The basic notations and concepts required for the analysis are described first. The state space is denoted byX≔Rn and the control space by u-≔Rm, where 1≤m≤n. The Euclidean inner product (on X or u- as appropriate) and induced norm are denoted by ·,· and ·, respectively. Let C(Rp;Rq) and C1(Rp;Rq) denote the space of all continuous functions and the space of continuous functions with continuous first-order partial derivatives, respectively, and let C∞(Rp;Rq) denote the space of functions whose partial derivatives of any order exist and are continuous, mapping Rp→Rq. For a real-valued continuous scalar function x→v(x), defined on Rn,∇:v~→∇v~∈Rn denotes the gradient map. The Lie derivative of v along a vector field f:Rn→Rn is denoted by Lfv~:Rn→R which is defined by(22)Lfv~x=∇v~x,fx.The Lie bracket of vector fields f,g∈C∞(Rn,Rn) is the vector field f,g∈C∞(Rn,Rn) defined by f,g=(Dg)f-(Df)g, where (Df) denotes the Jacobian matrix of f and (Dg) denotes the Jacobian matrix of g.In this paper, nonlinear systems with the following format are considered:(23)x˙t=fxt+Gxtu~t,where xt∈Rn, u~∈Rm. In general mathematical models of dynamical systems are usually imprecise due to modelling errors and exogenous disturbances [12]. Equation (23) can be considered as the nominal part of the system model and the uncertainty can be modelled as an additive perturbation to the nominal system model; more specifically, the structure of the system has the form(24)x˙t=fxt+Gxtu~t+ϑxt,ut,where ϑx(t),ut models the uncertainty in the system.System (24) is globally asymptotically stable to the zero state if the system exhibits the following properties.(i) Existence and Continuation of Solutions. For each x∈Rn, there exists a local solution x:0,t1→Rn (i.e., an absolutely continuous function satisfying (24) almost everywhere (a.e.) and x0=x0) and every such solution can be extended into a solution on 0,∞.(ii) Boundedness of Solutions. For each ħ>0, there exists rħ>0 such that xt∈rħBn, for all t≥0 on every solution x:0,∞→Rn with x0∈ħBn, where Bn denote the open unit ball centred at the origin in Rn.(iii) Stability of the State Origin. For each δ>0, there exists dδ>0 such that xt∈δBn for all t≥0 on every solution x:0,∞→Rn with x0∈dδBn.(iv) Global Attractivity of the State Origin. For each ħ>0 and ε>0, there exists Tħ,ε≥0 such that xt∈Bn for all t≥Tħ,ε on every solution x:0,∞→Rn with x0∈ħBn.Consider a nonlinear system described by the ordinary differential equation as follows:(25)x˙t=ft,xt,xt0=x0,where f:R×X→X and ft,0=0 for all t. To analyse the stability of (25), Lyapunov’s second stability analysis method is applicable. The Lyapunov approach is to show that a candidate “Lyapunov function” is nonincreasing along all solution to (25) by means that do not require explicit knowledge of solutions to (25). From this, appropriate conclusion can be drawn regarding stability concepts relating to solutions of the differential equation (25). An essential part of Lyapunov’s method is the determination of the time derivative of the candidate “Lyapunov function” along all solution of the dynamical system.Consider a Lyapunov candidatet,x→v~t,x:R×X→R which satisfies the condition v~∈C1R×X, in which case its time derivative along solutions to (25) is given by (26)v˙t,xt=∂v~t,xt∂t+∇v~t,xt,ft,xtfor almost all t∈R.LetWxt denote a positive definite function. If v~t,xt satisfies(i)
v~t,0=0 for all t≥0,(ii)
Wxt≤v~t,xt for all xt∈Φ, 0⊂Φ⊂X and all t≥0,(iii)
v˙(t,x(t))<0 in Φ,then v~t,xt is said to be a Lyapunov function in Φ. If v~˙t,xt in Φ, then v~t,xt is said to be a weak Lyapunov function.A setΕ is said to be an invariant set with respect to the dynamical system x˙=fx if (27)x0∈Ε⟼xt∈Ε∀t∈R+.In other words Ε is the set of points such that if a solution of x˙=fx belongs to Ε at some instant initialized points at t=0, then it belongs to Ε for all future time.Now, a setΕ⊂x is said to be a local invariant manifold for (25) if, for any x0∈Ε, x(t) with x(0)=x0 is in Ε for t<T where T>0. If T=∞, then Ε is said to be an invariant manifold.Now considering the longitudinal motion (Figure1) for the helicopter with an underslung external load system, choose a Lyapunov function candidate for the first subsystem v1 as(28)v1θ-L=12ς1θL12+ς2θL22+ς3θ-ς4q-2+ς5q-2,where ςi (i=1,2,3,4,5) are design parameters to be determined. Then(29)v˙1θ-L=∇v1θ-L,f1θ-L+G1θ-LtpxHt+qθ-Lt,xHt+Ht,θ-L,xHt=Lf1v1θ-L+∇v1θ-L,G1θ-LtpxHt+qθ-Lt,xHt+Ht,θ-L,xHt.From (28), ∇v1(θ-L) can be obtained as (30)∇v1θ-L=ς1θL1ς2θL2ς3ς3θ-ς4q--ς4ς3θ-ς4q-+ς5q-T.Substituting ∇v1(θ-L) and f1θ-L into the derivative of Lf1v1θ-L, we have(31)Lf1v1θ-L=ς1θL1θL2-kθς2θL22-ς2glxsin⁡θL1θL2+kDlxMLς2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL23+ς3ς3θ-ς4q-q-+-ς3ς4θ+ς42+ς5q-X21θ+X22q-.Checking the term of -sin⁡(θL1)θL2, it can be seen that -sin⁡θL1θL2≤-(2/π)θL1θL2 when θL1 and θL2 are both positive or negative. The situation of θL1 and θL2 having different signs will help with the system stability. Then, (32)Lf1v1θ-L≤ς1θL1θL2-kθς2θL22-ς2glx2πθL1θL2+kDlxMLς2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL23+ς3ς3θ-ς4q-q-+-ς3ς4θ+ς42+ς5q-X21θ+X22q-.If the design parameter ς1 is chosen as ς1=(2g/lxπ)ς2, Lf1v1θ-L satisfies the following:(33)Lf1v1θ-L≤-kθς2θL22+kDlxMLς2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL23+ς3ς3θ-ς4q-q-+-ς3ς4θ+ς42+ς5q-X21θ+X22q-.Further analysis on (33) will start from examining the first two terms. Rewrite these two terms. So (34)-kθς2θL22+kDlxMLς2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL23=-kθς2θL22+kDlxMLς2θL2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL22≤-kθς2θL22+kDlxMLς2max⁡θL2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL22=-ς2θL22kθ-kDlxMLmax⁡θL2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1.If the hinge friction is big enough to satisfy the following: (35)kθ>kDlxMLmax⁡θL2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1,then(36)λL=kθ-kDlxMLmax⁡θL2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1≥0,-kθς2θL22+kDlxMLς2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL23≤-ς2θL22λL.Now, the following analysis has been conducted for (33) and the load movement for the longitudinal motion is described by(37)X˙L=X˙0-lxθL2cos⁡θL1Z˙L=Z˙0-lxθL2sin⁡θL1.Around the hover condition X0=0 and Z0=0, the signs for X˙L and Z˙L are opposite to the one of θL2 provided that θL1 is positive.Considering the term(sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1)θL23 for all possible combinations of the signs of θL2 and θL1, the analysis follows below:(i)
IfθL1>0 and θL2>0, then sign⁡X˙L=-1, sign⁡Z˙L=-1, cos3⁡θL1>0, and sin3⁡θL1>0. Therefore, (sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1)θL23<0.(ii)
IfθL1<0 and θL2>0, then sign⁡X˙L=-1, sign⁡Z˙L=+1, cos3⁡θL1>0, and sin3⁡θL1<0. So (sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1)θL23<0.(iii)
IfθL1>0 and θL2<0, then sign⁡X˙L=+1 and sign⁡Z˙L=+1, cos3⁡θL1>0, and sin3⁡θL1>0. So the following is true: (sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1)θL23<0.(iv)
IfθL1<0 and θL2<0, then sign⁡X˙L=+1 and sign⁡Z˙L=-1, cos3⁡θL1>0, and sin3⁡θL1<0. So (sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1)θL23<0.Therefore(38)-kθς2θL22+kDlxMLς2sign⁡X˙Lcos3⁡θL1+sign⁡Z˙Lsin3⁡θL1θL23≤-kθς2θL22.Now examining the rest of the terms of Lf1v1θ-L, then (39)ς3ς3θ-ς4q-q-+-ς3ς4θ+ς42+ς5q-X21θ+X22q-=ς32θq--ς3ς4q-2-ς3ς4X21θ2-ς3ς4X22θq-+ς42+ς5X21θq-+ς42+ς5X22q-2=-ς3ς4-ς42+ς5X21q-2-ς3ς4X21θ2+ς32-ς3ς4X22+ς42+ς5X21θq-.For the system, θ and θL1 always have different signs, θ and q- always have different signs, and q- and θL1 have the same signs. So (40)ς32-ς3ς4X22+ς42+ς5X21θq-≤0ifς32-ς3ς4X22+ς42+ς5X21≥0.Therefore(41)ς3ς3θ-ς4q-q-+-ς3ς4θ+ς42+ς5q-X21θ+X22q-≤-ς3ς4-ς42+ς5X21q-2-ς3ς4X21θ2.Choose the design parameters to satisfy(42)ς3ς4-ς42+ς5X21>0.If X21>0, then (43)-ς3ς4-ς42+ς5X21q-2-ς3ς4X21θ2<0.If X21<0, then (44)-ς3ς4-ς42+ς5X21q-2-ς3ς4X21θ2<0in the region of -[ς3ς4-(ς42+ς5)X21]q-2<ς3ς4X21θ2. Since θ is caused by the load motion here the value of θ should be much smaller than q-. The inequality (44) is easy to be satisfied.In summary of the above analysis and by defining(45)Ψ1θ-L=kθς2θL22+ς3ς4-ς42+ς5X21q-2+ς3ς4X21θ2,Ψ2θ-L=ς3ς4-ς42+ς5X21q-2+ς3ς4X21θ2,the following lemma can be derived.Lemma 1.
Defining a Lyapunov function (28) and choosing the design parameters to satisfy ς1=(2g/lxπ)ς2,[ς3ς4-(ς42+ς5)X21]>0, and [ς32-ς3ς4X22+(ς42+ς5)X21]≥0, then within the region specified by -[ς3ς4-(ς42+ς5)X21]q-2<ς3ς4X21θ2(i)
v1(0)=0 and v1(θ-L)>0, ∀θ-L≠0,(ii)
v1θ-L→∞ as θ-L→∞,(iii)
Lf1v1θ-L≤-Ψ1θ-L, ∀θ-L, around the hover condition or Lf1v1θ-L≤-Ψ2(θ-L), ∀θ-L, if (35) holds.Both functionsΨ1 and Ψ2 are nonnegative.Recall the control termp(xH(t)) in the first subsystem of (11a) and (11b); it can be seen that DpxH-1 exists for all xH(t). The unknown vector fields, qθ-L(t),xH and Ht,θ-L(t),xH, model the uncertainties imposed onto the system. Since qθ-L(t),xH is directly mapped into the “control” space of xH(t) it can be considered as a matched uncertainty. Ht,θ-L(t),xH is unknown and it does belong to the control space of xH(t); so it represents the mismatched uncertainty in the system [12].Generally, the range of the (longitudinal) load suspension angle is within-π/2<θL<π/2. The helicopter velocities and load suspension angle have maximum operational values; therefore the uncertainties in the system are bounded. With the Lyapunov function defined in (28), (46)Lg1v1θ-L=∇v1θ-L,g1=kDMLlxς2θL2.In G1θ-L the columns g1 and g2 are the same; therefore (47)Lg2v1θ-L=kDMLlxς2θL2.By considering the maximum values of cos⁡(·) and sin⁡(·) functions, the bounding values relating to the uncertainty qθ-L(t),xH can be estimated as follows:(48)qθ-L,xH≤1lxu˙+w˙+1ML-kD2+κlxu+w+kθlxθ-L2≤kθlxθL2+kDML2+κlxu+w+1lxu˙+w˙≤kθlxθL2+kDML2+κlxpxH+α2t≤μ1Lg2v1θ-L+α1pxH+α2t,where μ1=MLkθ/kD, α1=(kD/ML)(2+κ/lx) (κ=max⁡κ1κ2), and α2t is defined by α2t=max⁡(1/lx)u˙+w˙>0.For the mismatched uncertaintyHt,θ-L(t),xH the following analysis is conducted to obtain its bounding function. The mismatched uncertainty can be rewritten as (49)Ht,θ-Lt,xH=00a11u+a12wX23u+X24w=0000a11a12X23X24uw.Let A-=a11a12X23X24; then (50)Ht,θ-L,xH≤A-xHt.Define a positive function as (51)θ⌢θ-L=θL12+θL22+θ2+q-2+ε-,where ε- is a very small positive constant. Choosing β1=(MLlx/kDς2)A- then the mismatched uncertainty Ht,θ-L(t),xH is bounded by the following inequality:(52)Ht,θ-L,xH≤θ⌢-1θ-Lβ1∑i=12Lgiv1θ-LpixH,where gi denotes the ith column of the matrix function G1θ-L and pi is the ith component of pxH, respectively. As β1 has a design parameter ς2 involved it is easy to have β1 to lead (52) to be true. For the matched uncertainty, we have (53)qθ-L,xH≤kθlxθL2+α1pxH+α2t.Following the above analysis, the following lemma can be derived.Lemma 2.
The uncertainties (52) and (53) are bounded and satisfy, if α1, β1≥0, and α1+β1<1, for α1=(kD/ML)(2+κ/lx) and β1=(MLlx/kDς2)A-.


## 4. Numerical Results

Using a simulation software the linearized model for a UH-60 helicopter is obtained with the following state and input matrices:(54)A=-0.00000.0003-0.0000-0.00000.0000-0.00001.0000-0.00230.0503-0.0003-0.0000-0.0000-0.0000-0.0000-0.0000-0.00000.99890.0463-0.00010.0000-0.0000-0.00000.00000.0000-0.0000-0.04641.0002-1.9386-28.593-0.0030-0.0195-0.01260.0168-1.20005.70760.268332.8694-0.36400.00270.0126-0.0311-0.0013-4.1348-2.23500.37130.3138-1.40340.00040.01500.0034-0.33400.28660.63072.3410-0.9144-0.47830.00170.0226-0.02460.0007-6.5893-2.1301-0.05980.0631-0.67440.00080.00290.00460.00250.2059-1.4255-0.1636-0.7549-0.0119-0.00020.00090.00280.0006-0.0749-0.1032-0.2214B=-0.00000.0000-0.00000.00000.00000.0000-0.0000-0.0000-0.00000.0000-0.0000-0.0000-2.21900.57340.3170-0.00193.5381-0.0453-0.09630.2007-1.56290.0237-6.0740-0.10141.9943-0.0714-0.05410.07310.0446-0.10970.0553-0.0281-0.8197-0.00820.1573-0.0635.The desired condition α1+β1<1 can be verified by considering a typical case that the UH-60 helicopter carries a load weight 1000 lb by a 15 ft slung length, and let kD=75 lb/ft.For the UH-60 helicopter model, with reference to the general mathematical model presented in Section2, we have Xθ1s=0.5734, Xθ0=0.3170, Zθ1s=0.0237, Zθ0=-6.0740, Mθ1s=-0.1097, and Mθ0=0.0553. So the following parameters can be calculated to have the values a11=-0.2 and a12=-0.02. For Xu=-0.0195, Zu=0.0150, Mu=0.0029, Mq=-1.4255, Xq-we=5.7076, and Zq+ue=0.6307, we have X23=0.06. And also, for Xw=0.0168, Zw=-0.3340, and Mw=0.0025, we have X24=0.01. Based on all the above parameters, σmaxA-=0.0041. In this case, let κ=0.01, α1=(kD/ML)(2+κ/lx)≈0.15, and β1=(MLlx/kD)σmaxA-=0.82. Therefore, α1+β1=0.15+0.82=0.97 which satisfies α1+β1<1. Another example considers that the UH-60 helicopter carries a load weight 500 lb by a 15 ft slung length and let kD=50 lb/ft; then α1=0.2 and β1=0.62; therefore α1+β1=0.82. Therefore, this assumption α1+β1<1 is realistic for the system discussed in here. The numerical values vary with respect to the slung load configuration. It may represent the system dynamic variations. To investigate the stability characteristics of dynamic variations two series of simulation studies were carried out.Firstly, the slung length was kept constant at 15 ft and the load varied between the limits 500–2000 lb withkD=75 lb/ft, producing the system poles listed in Table 1. The corresponding root loci are sketched in Figures 3 and 4. Figure 4 clearly shows that the positions of the system poles crossed the imaginary axis as the load weight increased.Table 1
System poles locations for a fixed slung length (l=15 ft) but with different load weights.
Helicopter 500 lb 1000 lb 1500 lb 2000 lb −1.0919 −1.1159 −0.7848 −0.6221 −0.9369  −6.3938 −6.4301 −9.2203 −7.9848 −7.5613  −0.0032 −0.0048 −0.0034 −0.0052 −0.0052  −0.0977 −0.0818 −0.0888 −0.0972 −0.0694  −0.3045 −0.3036 −0.3394 −0.3318 −0.3333  −0.3159 ± 0.4363i −0.3199 ± 0.4628i −0.2295 ± 0.5050i −0.2366 ± 0.5974i −0.1834 ± 0.4968i  −0.0489 ± 0.3898i −0.0211 ± 0.3868i 0.0074 ± 0.3445i 0.0119 ± 0.4406i 0.0264 ± 0.3199i  — −0.0316 ± 1.5043i −0.1020 ± 1.7380i −0.0368 ± 1.8753i −0.1160 ± 2.0003i  — −0.0736 ± 1.5019i −0.7662 ± 1.1680i −0.7349 ± 1.9064i −1.2367 ± 0.8995iFigure 3
System root-locus diagram showing pole locus for constant slung length (15 ft) as load weight is increased.Figure 4
Enlarged view of Figure3 showing poles locus near to Re(poles) = 0 axis.Secondly, load weight was kept constant and the slung length varied between the limit of 10–20 ft. Table2 summarises the results of variation of the system poles and the corresponding root loci are illustrated in Figures 5 and 6.Table 2
System poles for a fixed load (1000 lb) but with different slung lengths.
Helicopter 10 ft 15 ft 20 ft −1.0919 −1.0560 −0.7848 −0.8687  −6.3938 −9.9252 −9.2203 −8.5565  −0.0032 −0.0061 −0.0034 −0.0114  −0.0977 −0.2012 −0.0888 −0.0953  −0.3045 −0.2606 −0.3394 −0.3479  −0.3159 ± 0.4363i −0.1809 ± 0.6219i −0.2295 ± 0.5050i −0.1636 ± 0.6092i  −0.0489 ± 0.3898i 0.0643 ± 0.4577i 0.0074 ± 0.3445i 0.0799 ± 0.4594i  — −0.1055 ± 1.9883i −0.1020 ± 1.7380i −0.1247 ± 1.4470i  — −0.8847 ± 1.2464i −0.7662 ± 1.1680i −0.6352 ± 1.0384iFigure 5
System root-locus showing pole locus for constant load weight (1000 lb) as slung length is increased.Figure 6
Enlarged view of Figure5 showing poles locus near to Re(poles) = 0 axis.The pole locus indicates that the system poles have the variation trends of moving further towards right direction on thes-plane with increase of load weight and slung lengths. From the simulation results it is clearly shown that the system becomes less stable when the load and/or the slung length increases. The simulation results are compared with the stability analysis presented above.Now, consider the case of fixed slung length of 15 ft with different load weights. In the cases of a UH-60 helicopter that carries a load weight 500 lb and 1000 lb, thenα1=0.3 and 0.41 with β1=0.41 and 0.82, respectively; therefore α1+β1<1; thus the system is stable. But when the load weight is 1500 lb and 2000 lb, then α1=0.1 and 0.075 and also β1=1.23 and 1.64, respectively; therefore α1+β1>1; thus the system is unstable. For the case of the constant load weight (1000 lb) as slung length is increased, then for a 10 ft and 15 ft slung length α1=0.15 and β1=0.55 and 0.82, respectively; therefore α1+β1<1; thus the system is stable. But when the slung length increased to 20 ft then α1=0.15 and β1=1.09; therefore α1+β1=1.24; that is, α1+β1>1; thus the system is unstable. However, the system can be marginally stable; if the UH-60 helicopter carries a load weight 725 lb by a 20 ft slung length, then α1=0.2 and β1=0.8; therefore α1+β1=1. For this set of analysis kD is assumed to be kD=75 lb/ft. However, kD can vary depending on the weather condition and an example of kD=50 lb/ft is shown above for illustrative purpose. This analysis suggests that the maximum value of 20 ft slung length may be used for slung load operation. However, the load weight limit can vary with respect to the capacity of the load carrying hooks, helicopter weight, environmental conditions, and so forth.The examples presented here show that the simulation results are correlated with the stability analysis. Thus, desired condition for the stability is met and it can locate the load at the specified position or its neighbourhood.


## 5. Conclusions

In this paper stability analysis of a helicopter with an underslung load system is investigated. The investigation identified the conditions for the stabilisation of the system and the positioning of the underslung load at hover condition. Stability analysis and numerical results proved that the desired condition for the stability is met; then it is possible to locate the load at the specified position or its neighbourhood. An illustration example is also given in the paper. The method results in a guaranteed load positioning accuracy which depends on the design parameters and the method can be applicable to any individual helicopter with external load system.



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*Source: 4195491-2016-06-05.xml*