PREPRINT ACCEPTED TO INTERNATIONAL JOURNAL OF CONTROL

Modelling and Control of a Knuckle Boom Crane

M. Ambrosinoa and E. Garonea

aService d’Automatique et d’Analyse des Systemes, Universite Libre de Bruxelles, Brussels,Belgium.

ARTICLE HISTORY

Compiled March 31, 2021

ABSTRACTCranes come in various sizes and designs to perform different tasks. Depending ontheir dynamic properties, they can be classified as gantry cranes and rotary cranes. Inthis paper we will focus on the so called ’knuckle boom’ cranes which are among themost common types of rotary cranes. Compared with the other kinds of cranes (e.g.boom cranes, tower cranes, overhead cranes, etc), the study of knuckle cranes is stillat an early stage and very few control strategies for this kind of crane have beenproposed in the literature. Although fairly simple mechanically, from the controlviewpoint the knuckle cranes present several challenges. A first result of this paperis to present for the first time a complete mathematical model for this kind of cranewhere it is possible to control the three rotations of the crane (known as luff, slew,and jib movement), and the cable length. The only simplifying assumption of themodel is that the cable is considered rigid. On the basis of this model, we proposea nonlinear control law based on energy considerations which is able to performposition control of the crane while actively damping the oscillations of the load.The corresponding stability and convergence analysis is carefully proved using theLaSalle’s invariance principle. The effectiveness of the proposed control approachhas been tested in simulation with realistic physical parameters and in the presenceof model mismatch.

KEYWORDSKnuckle Cranes; Robotics; Oscillation Reduction; Underactuated Systems;Nonlinear Control

1. Introduction

To handle heavy loads and materials, different types of cranes are widely used indifferent industrial fields. Although the tasks performed by the cranes are fairly simple(i.e. moving or lifting heavy materials), it is quite challenging for a human operatorto achieve accurate positioning and swing elimination simultaneously. Moreover, thereare many practical problems associated with manual operation, such as low efficiency,poor safety, etc. Therefore, the control problem of cranes has been studied for a longtime. In this paper we focused on the modelling and control of knuckle cranes, thatare among the most common types of rotary crane.

This research has been funded by The Brussels Institute for Research and Innovation (INNOVIRIS) of theBrussels Region through the Applied PHD grant: Brickiebots - Robotic Bricklayer: a multi-robot system for

sand-lime blocks masonry (ref : 19-PHD-12)

Corresponding author: Michele.Ambrosino@ulb.ac.be

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Knuckle cranes are special kinds of boom cranes. Compared with other cranes, thistype of crane has higher flexibility and lower energy consumption (Hong & Shah, 2019)and for this reason it is probably the most common kind of crane in heavy industry.Knuckle cranes are special boom cranes that have an auxiliary jib connected to theboom to enhance the maneuverability and increase the workspace of the crane. In thispaper we present for the first time a complete mathematical model for this kind ofcrane which takes into account not only the three main rotations (i.e. luff, slew, andjib movement) but also the cable dynamic and the payload oscillations.

As all cranes, knuckle cranes are nonlinear systems with underactuated dynamics.The problem of controlling underactuated systems has been a topic of great interest indifferent industrial fields (Li, Yan, & Shi, 2017; Moreno-Valenzuela & Aguilar-Avelar,2018; Scalera, Gasparetto, & Zanotto, 2019; Yan, Wang, & Xu, 2019; Yang & Xian,2020; Zhang, Zhang, Ouyang, Ma, & Cheng, 2020). The condition of underactuationrefers to a system having fewer actuators (input variables) than degrees of freedom(number of independent variables that define the system configuration). This impliesthat some of the states of the system cannot be directly commanded, which highlycomplicates the design of control algorithms. In particular for a knuckle crane (as forany crane) the non-actuated variable are the swing angles of the payload, whereas thefour actuated variables are the three main rotation (i.e. luff, slew, and jib movements)and the length of the cable. While in the past few years several solutions have beenproposed for the control of boom cranes control, the control problem of a knucklecrane is still an opening and challenging problem

Generally speaking, control schemes for boom cranes present in literature can beroughly categorised into open loop and closed loop techniques (Ramli, Mohamed,Abdullahi, Jaafar, & Lazim, 2017).

Input shaping is one of the most used open loop techniques, that can be applied inreal time, mainly for the control of the oscillations of the payload. (Maleki & Singhose,2010) proposes a input-shaping control for a in-scale boom crane to reduce the residualoscillations. (Newman & Vaughan, 2017) discusses an input-shaping strategy thatminimizes the oscillations of the payload caused by an external disturbance during theluff command. Open-loop trajectory planning methods such as S-curve trajectoriesapproaches are proposed in (Uchiyama, Ouyang, & Sano, 2012; Uchiyama, Ouyang, &Sano, 2013) to achieve anti-sway control for the payload. Open loop control schemeshave been widely used because they are easy to implement. However, open loop controlschemes are sensitive to external disturbances and to model mismatch. Therefore,several closed loop control methods have been proposed to increase robustness andachieve better performance in presence of perturbations. As closed loop technique,the Linear Quadratic Regulator (LQR) is one the most common techniques appliedto cranes (Ramli et al., 2017). In (Sun, Yang, Fang, lu, & Qian, 2017) the authorsprovided a LQR control approach for boom crane considering a fixed cable length.Another closed-loop control scheme widely used for boom crane is the ProportionalIntegral Derivative (PID) control. In (Yang, Sun, Qian, & Fang, 2017) the authorsdesigned a Proportional-Derivative (PD) controller with gravity compensation basedon the nonlinear model of the boom crane. In (Ambrosino, Berneman, et al., 2020) isshown a partial feedback linearization (PFL) based on detailed mathematical modelof boom cranes. In (Uchiyama, Takagi, & Sano, 2006) is shown a pole placementapproach for a linearized model of boom crane. In (Arnold, Sawodny, Neupert, &Schneider, 2005) a model predictive control (MPC) approach is used for a boom cranein order to reduce the swing angles as much as possible. A constraint control based

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on the theory of the Explicit Reference Governor (ERG), is discussed in (Ambrosino,Dawans, & Garone, 2020), where the authors focused on controlling a simplified modelof boom crane in the presence of constraints. A second-order sliding mode control law isproposed in (Ismail & Ha, 2013) for trajectory tracking and anti-sway control. Controlschemes based on a combination of open and closed loop techniques have also beenproposed. In (Huang, Maleki, & Singhose, 2013) a combination of input shaping andfeedback control is proposed to reduce the effect of gusts of wind.

Compared with boom cranes, the state of the art of knuckle cranes control is muchless developed. In (Krus & Palmberg, 1992) the authors focus on controlling mobileelectro-hydraulic proportional valves to move the crane to a desired position. In (Ped-ersen, Nielsen, Andersen, Hansen, & Pedersen, 2003) the authors solve the problemof controlling knuckle crane through the inverse kinematics without take into accountthe dynamic of the cable and the payload. In (Chu, Sanfilippo, Æsøy, & Zhang, 2014)an anti-sway control is shown which is performed by simplifying the dynamic modeland assuming that the tip of the crane can be controlled directly. To the best of ourknowledge, no research has been carried out to derive a detailed mathematical modelof a knuckle cranes and to develop a control strategy taking into account the nonlinearnature of this type of crane.

The main contributions of this paper are:

(1) The development of a complete mathematical model for a knuckle crane whichtakes into account all of the degrees of freedom (DoFs) of this type of cranes(e.g. the three rotations, the length of the rope and the payload swing angles).The only simplifying assumptions of the proposed model is that the cable isconsidered rigid.

(2) The design of a novel control strategy designed directly on the nonlinear modeland for which a detailed proof of asymptotic stability is provided making use ofthe LaSalle’s invariance principle.

(3) Different simulation scenarios are presented to demonstrate that the proposedcontrol method is able to perform position control of the crane while activelydamping the oscillations of the load, even in presence of model mismatch.

The rest of this paper is organized as follows. In Section 2, the dynamic model of aknuckle crane and the control objectives are provided. In Section 3, the proposed con-troller is designed, and in Section 4 the corresponding stability analysis is provided indetail. Section 5 shows the results of the simulations for the proposed control strategy.

2. Dynamic Model

A schematic view of a knuckle crane is depiced in Fig.1a. The knuckle crane consists ofa first boom of length lb and mass mb connected to the tower with one revolute joint.The auxiliary jib of length lj and mass mj is linked to the first boom by a revolutejoint. For the sake of simplicity, in this paper all the links and joints are considered tobe rigid. The cable is supposed to be massless and rigid, thus the lifting mechanismcan be described as a prismatic joint. The payload of mass m is described as a lumpedmass. The two swing angles of the payload are represented in the Fig.1b.

The configuration of the crane is conveniently described by six generalized coordi-nates. In particular: α is the slew angle of the tower, β is the luff angle of the boom, γ

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(a) Model of a knuckle crane (b) Payload swing angles.

Figure 1. Schematic view of a knuckle crane

is the luff angle of the jib, d is the length of the rope, θ1 is the tangential pendulationmainly due to the motion of the tower, and θ2 is the radial sway mainly due to themotion of the boom.

The dynamic model of the knuckle crane can be obtained using the Euler-Lagrangemethod. Firstly, we need to express the system kinematic energy T (t), which consistsof three parts: the boom kinematic energy Tb(t), the jib kinematic energy Tj(t), andthe payload kinematic energy Tp(t). Then, we define the system potential energy U(t)consisting of the boom potential energy Ub(t), jib potential energy Uj(t), and thepayload potential energy Up(t). The Lagrangian of the knuckle crane is

L(t) = T (t)− U(t) = Tb(t) + Tj(t) + Tp(t)− Ub(t)− Uj(t)− Up(t), (1)

where:

T (t) =1

8(mB((lBCβSαα+ lBCαSββ)2 + (lBCαCβα− lBSαSββ)2 + l2BC

2ββ

2))

+1

2(mJ((lBCβSαα+ lBCαSββ +

1

2(lJCγSαα) +

1

2(lJCαSγ γ))2 + (lBCαCβα

+1

2(lJCαCγα)− lBSαSββ −

1

2(lJSαSγ γ))2 + (lBCββ +

1

2(lJCγ γ))2))

+1

2(m((Cθ2SαSθ1 d− CαSθ2 d+ lBCβSαα+ lBCαSββ + lJCγSαα+ lJCαSγ γ

−CαCθ2 θ2d+ SαSθ2αd+ CαCθ2Sθ1αd+ Cθ1Cθ2Sαθ1d− SαSθ1Sθ2 θ2d)2 + (lBCββ

−Cθ1Cθ2 d+ lJCγ γ + Cθ2Sθ1 θ1d+ Cθ1Sθ2 θ2d)2 + (SαSθ2 d+ lBCαCβα+ lJCαCγα

−lBSαSββ − lJSαSγ γ + CαSθ2αd+ Cθ2Sαθ2d+ CαCθ2Sθ1 d+ CαCθ1Cθ2 θ1d

−Cθ2SαSθ1αd− CαSθ1Sθ2 θ2d)2)) +1

2Itotα

2 +1

2IBβ

2 +1

2IJ γ

2,

(2)

U(t) = gm(lBSβ + lJSγ − Cθ1Cθ2d) + gmJ(lBSβ +1

2lJSγ) +

1

2glBmBSβ, (3)

and where

4

Sα , sin(α), Sβ , sin(β), Sγ , sin(γ), Sθ1 , sin(θ1), Sθ2 , sin(θ2),

Cα , cos(α), Cβ , cos(β), Cγ , cos(γ), Cθ1 , cos(θ1), Cθ2 , cos(θ2).

The meaning of the system physical parameters in (2)-(3) are reported in Tab. 1.

Table 1. Parameters of the knuckle crane system

Parameters Physical Unitsmb Boom mass kgmj Jib mass kgm Payload mass kglb Boom length mlj jib length mItot Tower inertia moment kg ·m2

Ib Boom inertia moment kg ·m2

Ij Jib inertia moment kg ·m2

Moreover, to simplify the next analysis we introduce the auxiliary variables

• A1 = l2Bm+ (l2BmB)/4 + l2BmJ ;• A2 = l2Jm+ (l2JmJ)/4;• A3 = 2lBlJm+ lBlJmJ ;• A4 = 2lBm;• A5 = 2lJm.

The equations of the motion of the crane are derived using the Euler-Lagrangeequation

d

dt

(∂L(q, q)

∂q

)− ∂L(q, q)

∂q= ζ, (4)

where q = [α, β, γ, d, θ1, θ2]T ∈ R6 is the system state vector, and ζ =[u1, u2, u3, u4, 0, 0]T ∈ R6 is the control input vector. The dynamic model of the knucklecrane can be described by the following equations

5

Itotα+A1αC2β +A2αC

2γ + d2αm+ 2d2αθ1mCθ1C

2θ2Sθ1 + 2d2αθ2mC

2θ1Cθ2Sθ2−

−A1αβS2β −A2αγS2γ +A3αCβCγ − d2θ2mSθ1+

+2ddαm+ 2A4dαCβSθ2 + 2A5dαCγSθ2 + 2A4dαCβSθ2+

+2A5dαCγSθ2 −A3αβCγSβ −A3αγCβSγ − 2d2θ1θ2mCθ1−−d2αmC2

θ1C2θ2 +A4dCβCθ2Sθ1 +A5dCγCθ2Sθ1 − 2ddθ2mSθ1+

+A4dβCθ2SβSθ1 −A4dθ2CβSθ1Sθ2 +A5dγCθ2SγSθ1 −A5dθ2CγSθ1Sθ2−−2A4dθ2CβSθ1Sθ2 − 2A5dθ2CγSθ1Sθ2 +A4dβ

2CβCθ2Sθ1 −A4dθ21CβCθ2Sθ1−

−A4dθ22CβCθ2Sθ1 +A5dγ

2CγCθ2Sθ1 −A5dθ21CγCθ2Sθ1 −A5dθ

22CγCθ2Sθ1−

−2ddαmC2θ1C

2θ2 + 2d2θ1θ2mCθ1C

2θ2 + 2A4dαθ2CβCθ2 + 2A5dαθ2CγCθ2+

+d2θ1mCθ1Cθ2Sθ2 − 2A4dαβSβSθ2 − 2A5dαγSγSθ2 +A4dθ1CβCθ1Cθ2+

+A5dθ1CγCθ1Cθ2 + 2A4dθ1CβCθ1Cθ2 + 2A5dθ1CγCθ1Cθ2 − d2θ21mCθ2Sθ1Sθ2−

−2A4dθ1θ2CβCθ1Sθ2 − 2A5dθ1θ2CγCθ1Sθ2 + 2ddθ1mCθ1Cθ2Sθ2 = u1,

(5)

A1β + IBβ + (A1α2S2β)/2 + (A3α

2CγSβ)/2− (A3γ2CβSγ)/2 + (A3γ

2CγSβ)/2

+glBmCβ + (glBmBCβ)/2 + glBmJCβ+

(A3γCβCγ)/2−A4dSβSθ2 + (A3γSβSγ)/2−A4dθ2Cθ2Sβ−2A4dθ2Cθ2Sβ −A4dCβCθ1Cθ2 +A4dα

2SβSθ2 +A4dθ22SβSθ2

+A4dθ1CβCθ2Sθ1 +A4dθ2CβCθ1Sθ2+

2A4dθ1CβCθ2Sθ1 + 2A4dθ2CβCθ1Sθ2 +A4dαCθ2SβSθ1

+2A4dαCθ2SβSθ1 +A4dθ21CβCθ1Cθ2 +A4dθ

22CβCθ1Cθ2+

2A4dαθ1Cθ1Cθ2Sβ − 2A4dθ1θ2CβSθ1Sθ2 − 2A4dαθ2SβSθ1Sθ2 = u2,

(6)

A2γ + IJ γ + (A2α2S2γ)/2 + (A3α

2CβSγ)/2 + (A3β2CβSγ)/2−

−(A3β2CγSβ)/2 + glJmCγ + (glJmJCγ)/2 + (A3βCβCγ)/2−

−A5dSγSθ2 + (A3βSβSγ)/2−A5dθ2Cθ2Sγ − 2A5dθ2Cθ2Sγ −A5dCγCθ1Cθ2+

+A5dα2SγSθ2 +A5dθ

22SγSθ2 +A5dθ1CγCθ2Sθ1 +A5dθ2CγCθ1Sθ2+

+2A5dθ1CγCθ2Sθ1 + 2A5dθ2CγCθ1Sθ2 +A5dαCθ2SγSθ1 + 2A5dαCθ2SγSθ1+

+A5dθ21CγCθ1Cθ2 +A5dθ

22CγCθ1Cθ2 + 2A5dαθ1Cθ1Cθ2Sγ−

−2A5dθ1θ2CγSθ1Sθ2 − 2A5dαθ2SγSθ1Sθ2 = u3,

(7)

6

dm− dα2m− dθ22m−A4α

2CβSθ2 −A4β2CβSθ2 −A5α

2CγSθ2−−A5γ

2CγSθ2 − dθ21mC

2θ2 −A4βSβSθ2 −A5γSγSθ2−

−gmCθ1Cθ2 + dα2mC2θ1C

2θ2 −A4βCβCθ1Cθ2 −A5γCγCθ1Cθ2+

+A4αCβCθ2Sθ1 +A5αCγCθ2Sθ1 + 2dαθ2mSθ1 +A4β2Cθ1Cθ2Sβ+

+A5γ2Cθ1Cθ2Sγ − 2A4αβCθ2SβSθ1 − 2A5αγCθ2SγSθ1 − 2dαθ1mCθ1Cθ2Sθ2 = u4,

(8)

dCθ2(gmSθ1 −A4β2SβSθ1 −A5γ

2SγSθ1 + dθ1mCθ2 + 2dθ1mCθ2+

+A4αCβCθ1 +A5αCγCθ1 +A4βCβSθ1 +A5γCγSθ1 − 2A4αβCθ1Sβ−−2A5αγCθ1Sγ + dαmCθ1Sθ2 + 2dαmCθ1Sθ2 − 2dθ1θ2mSθ2−

−dα2mCθ1Cθ2Sθ1 + 2dαθ2mCθ1Cθ2) = 0,

(9)

−d(A4α2CβCθ2 − 2dθ2m− dθ2m+A4β

2CβCθ2 +A5α2CγCθ2+

+A5γ2CγCθ2 − (dθ2

1mS2θ2)/2 + dαmSθ1 + 2dαmSθ1 +A4βCθ2Sβ+

+A5γCθ2Sγ − gmCθ1Sθ2 +A4β2Cθ1SβSθ2 +A5γ

2Cθ1SγSθ2−−A4βCβCθ1Sθ2 −A5γCγCθ1Sθ2 +A4αCβSθ1Sθ2 +A5αCγSθ1Sθ2+

+2dαθ1mCθ1C2θ2 − 2A4αβSβSθ1Sθ2 − 2A5αγSγSθ1Sθ2 + dα2mC2

θ1Cθ2Sθ2) = 0.

(10)

Finally, the system (5)-(10) can be compactly rewritten as

M(q)q + C(q, q)q + g(q) =

[I4×4

02×2

]u. (11)

The matrices M(q) ∈ R6×6,C(q, q) ∈ R6×6, and g(q) ∈ R6 represent the inertia,centripetal-Coriolis, and gravity term, respectively. I4×4 and 02×2 are the Identitymatrix and the Null matrix, respectively.

The system matrices (see Appendix A for the detailed description) are defined as

M(q) =

m11 m12 m13 m14 m15 m16

m21 m22 m23 m24 m25 m26

m31 m32 m33 m34 m35 m36

m41 m42 m43 m44 0 0m51 m52 m53 0 m55 0m61 m62 m63 0 0 m66

, (12)

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C(q, q) =

c11 c12 c13 c14 c15 c16

c21 c22 c23 c24 c25 c26

c31 c32 c33 c34 c35 c36

c41 c42 c43 0 c45 c46

c51 c52 c53 c54 c55 c56

c61 c62 c63 c64 c65 c66

, (13)

g(q) =[0, g2, g3, g4, g5, g6

]T. (14)

Although the equation of motion (11) is quite complicated, as all mechanical sys-tems, it has several fundamental properties that can be exploited to facilitate thedesign of the control law. The two main properties that will be exploited in the nextsections are:

Property 1. The matrix 12M(q)− C(q, q) is skew symmetric which means that:

ηT[

1

2M(q)− C(q, q)

]η = 0, η ∈ R6

Property 2. The gravity vector (14) can be obtained as the gradient of the crane

potential energy (3), i.e., g(q) = ∂U(q)∂q

3. Control objective

The control objective consists of two main tasks: i) move the crane to the desiredconfiguration, ii) dampen the load swings at the same time. This control objectivecan be described compactly as

limt→∞

[α(t), β(t), γ(t), d(t), θ1(t), θ2(t)] = [αd, βd, γd, dd, 0, 0],

limt→∞

[α(t), β(t), γ(t), d(t), θ1(t), θ2(t)] = [0, 0, 0, 0, 0, 0],(15)

where αd, βd, γd, dd are the desired references for the actuated states.

To design the control law and to perform the corresponding stability and conver-gence analysis (see Section 4) we will consider the following reasonable assumptions.

Assumption 1. The payload swings are such that: |θi| < π2 , i = 1, 2.

Assumption 2. The cable length is always greater than zero to avoid singularity inthe model (11): d(t) >,∀t ≥ 0.

Assumption 3. The boom and the jib angles, according to Fig.1a, are mechanically

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constrained in the range:

−π2< β <

π

2,

−π2< γ <

π

2.

It is worth noting that in most real-world crane the Assumption 3 can be morerestrictive according to specific mechanical constraints.

4. Control design and stability analysis

The control strategy proposed in this paper consists of a nonlinear control law basedon energy consideration. To design the control law, firstly we considered a Lyapunovfunction candidate partially based on the energy of system (11). The resulting controllaw will be a proportional-derivative (PD) controller with a gravity compensation. Adetailed stability analysis of the resulting closed loop system will be carried out basedon the LaSalle’s invariance principle.

In order to develop our control law, we start from a energy function

E(t) =1

2qTM(q)q +mgd(1− Cθ1Cθ2), (16)

where the first term is the kinetic energy of the crane, whereas the second termrepresents the potential energy of the payload. If one takes the time derivative of (16),it follows that

E(t) =1

2qT M(q)q + qTM(q)q +mgd(1− Cθ1Cθ2) + θ1mgdSθ1Cθ2 + θ2mgdSθ2Cθ1 .

(17)Using (11) and Property 1, it follows that

E(t) = αu1 + β(u2 − glBCβ(m+1

2mB +mJ)) + γ(u3 − glJCγ(m+

1

2mJ)) + d(u4 +mg).

(18)

Based on (16), we can define the following Lyapunov function candidate:

V (t) =1

2qTM(q)q +mgd(1− Cθ1Cθ2) +

1

2kpαe

2α +

1

2kpβe

2β +

1

2kpγe

2γ +

1

2kpde

2d, (19)

where eα, eβ, eγ , ed are the error signals defined as:

eα = αd − α, eβ = βd − β, eγ = γd − γ, ed = dd − d. (20)

Differentiating (19) with respect to the time and using (11) and Property 1, we

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obtain

V (t) = α(u1 − kpαeα) + β(u2 − kpβeβ − glBCβ(m+1

2mB +mJ))

+γ(u3 − kpγeγ − glJCγ(m+1

2mJ)) + d(u4 − kpded +mg).

(21)

In order to cancel the gravitational terms and keep V (t) non-positive, the followingcontrol law is designed

u1 = kpαeα − kdαα, (22)

u2 = kpβeβ − kdββ + glBCβ(m+1

2mB +mJ), (23)

u3 = kpγeγ − kdγ γ + glJCγ(m+1

2mJ), (24)

u4 = kpded − kddd−mg, (25)

where kpα, kpβ, kpγ , kpd, kdα, kdβ, kdγ , kdd ∈ R are positive control gains.Substituting (22)-(25) into (21) one obtains

V (t) = −kdαα2 − kdββ2 − kdγ γ2 − kddd2 ≤ 0. (26)

The following theorem describes the stability property of the crane using the controllaw (22)-(25).

Theorem 4.1. Consider the system (5)-(10). Under Assumptions 1-3, the control law(22)-(25) makes every equilibrium point (15) asymptotically stable.

Proof. Choosing (19) as a Lyapunov function with the system (5)-(10), the controllaw (22)-(25) leads to (26). Noticing that V (0) is bounded, it is easy to infer that:

V (t) ∈ L∞ ⇒ q, eα, eβ, eγ , ed, θ1, θ2 ∈ L∞. (27)

At this point, let Φ be defined as the set where V (t) = 0, i.e.,

Φ = {q, q|V (t) = 0}. (28)

Furthermore, let Γ represent the largest invariant set in Φ where Assumptions 1-2are verified. Based on (26), Γ is the set such that:

α = 0, β = 0, γ = 0, d = 0⇒ α = 0, β = 0, γ = 0, d = 0,

eα = 0, eβ = 0, eγ = 0, ed = 0⇒ eα = φ1, eβ = φ2, eγ = φ3, ed = φ4,(29)

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where φ1,2,3,4 are constants to be determined.Plugging (29) and (22) in (5) one obtains

d2θ1Cθ1Cθ2Sθ2 − d2θ2Sθ1 − 2d2θ1θ2Cθ1 −A4θ12CβCθ2Sθ1

−A5dθ21CγCθ2Sθ1 −A4dθ

22CβCθ2Sθ1 −A5dθ

22CγCθ2Sθ1

−d2θ21Cθ2Sθ1Sθ2 +A4dθ1CβCθ1Cθ2 +A5dθ1CγCθ1Cθ2

−A4dθ2CβSθ1Sθ2 −A5dθ2CγSθ1Sθ2 + 2d2θ1θ2Cθ1C2θ2

−2A4dθ1θ2CβCθ1Sθ2 − 2A5dθ1θ2CγCθ1Sθ2 = kpαeα,

(30)

where Cβ = cos(βd − φ2) and Cγ = cos(γd − φ3).To continue the analysis, equation (30) can be rewritten as

d

dt[θ1Cθ1Cθ2(d

2Sθ2 +A4dCβ +A5dCγ)− θ2Sθ1(d2 +A4dCβSθ2 +A5dCγSθ2)] = kpαeα.

(31)Integrating equation (31) with respect to the time we obtain

[θ1Cθ1Cθ2(d2Sθ2 +A4dCβ+A5dCγ)− θ2Sθ1(d

2 +A4dCβSθ2 +A5dCγSθ2)] = kpαeαt+c1,(32)

where c1 denotes a constant to be determined.

If kpαeα 6= 0, then if t⇒∞, one would have that the left side of equation (32) tendsto infinity, i.e.

[θ1Cθ1Cθ2(d2Sθ2 +A4dCβ +A5dCγ)− θ2Sθ1(d

2 +A4dCβSθ2 +A5dCγSθ2)]⇒∞,

which conflicts with (27).As a result:

[θ1Cθ1Cθ2(d2Sθ2 +A4dCβ +A5dCγ)− θ2Sθ1(d

2 +A4dCβSθ2 +A5dCγSθ2)] = c1 (33)

kpαeα = 0. (34)

Since kpα > 0, it is clear that

eα = 0⇒ φ1 = 0⇒ α = αd. (35)

Similar to (30)-(35), by plugging (29) and (23)-(24) into (6)-(7) one achieves

A4dθ22SβSθ2 −A4dθ2Cθ2Sβ +A4dθ

21CβCθ1Cθ2 +A4dθ

22CβCθ1Cθ2

+A4dθ1CβCθ2Sθ1 +A4dθ2CβCθ1Sθ2 − 2A4dθ2θ1CβSθ1Sθ2 = kpβeβ,(36)

A5dθ22SγSθ2 −A5dθ2Cθ2Sγ +A5dθ

21CγCθ1Cθ2 +A5dθ

22CγCθ1Cθ2 +A5dθ1CγCθ2Sθ1

+A5dθ2CγCθ1Sθ2 − 2A5dθ1θ2CγSθ1Sθ2 = kpγeγ .

(37)

11

Equations (36)-(37) can be rewritten as

d

dt[−A4dθ2(Cθ2Sβ − CβCθ1Sθ2) +A4dθ1CβCθ2Sθ1 ] = kpβeβ, (38)

d

dt[−A5dθ2(Cθ2Sγ + CγCθ1Sθ2) +A5dθ1CγCθ2Sθ1 ] = kpγeγ . (39)

In the same way of (32)-(35) it is clear that:

eβ = 0⇒ φ2 = 0⇒ β = βd, (40)

eγ = 0⇒ φ3 = 0⇒ γ = γd. (41)

By using the (29) and (25) in (8)-(9)-(10), we get

− dθ22 − d(θ2

1C2θ2) + g(1− Cθ1Cθ2) = kpded, (42)

dθ1C2θ2 = −gSθ1Cθ2 + 2dθ1θ2Sθ2Cθ2 , (43)

dθ2 = −1

2(dθ2

1Cθ2Sθ2)− gSθ2Cθ1 . (44)

Substituting (35) and (42)-(44) into (30), we obtain

θ1 = 0 ∨ θ2 = ±π/2 ∨ (β = γ = ±π/2). (45)

According to Assumptions 1 and 3, the solutions β = γ = ±π/2 and θ2 = ±π/2 arenot considered. Thus, it can be concluded that

θ1 = 0⇒ θ1 = 0⇒ θ1 = 0. (46)

Using (40) and (46) into (36) one obtains

A4dθ22Cβd

Cθ2 +A4dθ22Sβd

Sθ2 +A4dθ2CβdSθ2 −A4dθ2Cθ2Sβd

= 0. (47)

Taking into account Assumption 2, equation (47) can be rewritten as:

d

dt[θ2(Cβd

Sθ2 − Cθ2Sβd)] = 0. (48)

After that, one integrates (48) in respect to the time

[θ2(CβdSθ2 − Cθ2Sβd

)] = c2, (49)

where c2 denotes constant to be determined..

12

The equation (49) can be rewritten as:

d

dt[−(Cβd

Cθ2 + Sθ2Sβd)] = c2. (50)

Integrating (50) one obtains

[−(CβdCθ2 + Sθ2Sβd

)] = c2t+ c3, (51)

where c3 is a constant. Assume that c2 6= 0, then if t ⇒ ∞, one will have that theleft side of equation (51)

[−(CβdCθ2 + Sθ2Sβd

)]⇒∞,

which conflicts with 27.Than, c2 = 0 and in (51) is easy to verify that θ2 must be a constant. Consequently,it is clear that:

θ2 = 0⇒ θ2 = 0. (52)

Including (46) and (52) into (44) and considering the Assumption 1, it can be obtainedthat:

gSθ2 = 0⇒ θ2 = 0. (53)

Including (46) and (53) into (42) we obtain

kpded = 0⇒ ed = 0⇒ d = dd. (54)

5. Simulation results

To demonstrate the effectiveness of the control law (22)-(25), five different simulationscenarios will be shown. In each of them the goal is to move the crane to a desiredposition while dampening the oscillations of the payload as much as possible. Finally,at the end of the section, the performances of the proposed control approach arecompared with a linear quadratic regulator (LQR) obtained by linearization whichrepresents the most commonly crane control technique used in the literature (Ramliet al., 2017).

To get realistic values for the simulation tests, we consider a small knuckle crane:the NK375b from NEMAASKO (Fig.2). See (NEBOMAT, 2005) for more details.

In all the simulation scenarios, without loss of generality, the desired angles areselected as αd = 60◦, βd = 30◦, γd = 22◦ and the desired cable length is selected asd = 2m according to the workspace of the knuckle crane. Matlab® and Simulink®

code is released as open-source on GitHub: https://github.com/MikAmb95/Knucklecrane simulator-.

13

Figure 2. NK375b knuckle crane

The parameters for the control law (22)-(25) are

kpα = 103, kpβ = 104, kpγ = 104, kpd = 103,

kdα = 102, kdβ = 103, kdγ = 103, kdd = 102.

Scenario 1. In this simulation we show the performance of the control law (22)-(25) considering for the parameters of the kuckle crane model (11) the nominal valuescollected in Tab.2 and setting zero initial conditions for the two swing angles of thepayload (i.e. θ1 and θ2). The simulation results are reported in Figg.3a-4b. We can seethat the three actuated angles (i.e. α, β, and γ) reach the desired angular values inaround 100 seconds. Additionally, the cable achieves the desired length. As one can seein Figg.5a-5b, the payload swing angles (i.e. θ1 and θ2) exhibit a first oscillation dueto the movement of the crane. However, when the crane reaches the desired position,the swing angles have negligible residual oscillations. As one can see from Figg.6a-6b,the inputs profile and values are reasonable and well within the typical limits of thecrane actuators.

Scenario 2. In this simulation the payload is perturbed by setting as initial swingangles: θ1(0) ≈ 11.5◦ and θ2(0) ≈ 5.7◦ as shown in Figg. 9a-9b. The proposed controlscheme is able to steer the knuckle crane to the desired configuration (see Figg.7a-8b). The behavior of the crane is similar to the previous case. However, the controlinput have an oscillating trend (see Fig.10a-10b) due to the residual oscillations of thepayload swing angles (see Figg.9a-9b). Due to the initial perturbations, the residual

Table 2. Parameters of the knuckle crane system

Parameters Physical Unitsmb 300 kgmj 250 kgm 100 kglb 2 mlj 2.3 m

14

0 20 40 60 80 100 120 140 160 180

Time[s]

0

10

20

30

40

50

60

70

[deg

]

(a) Scenario 1. Tower angle α. Blue

line: Nonlinear controller. Red line: De-sired reference.

0 20 40 60 80 100 120 140 160 180

Time[s]

5

10

15

20

25

30

35

[deg

]

(b) Scenario 1. Boom angle β. Blue

line: Nonlinear controller. Red line: De-sired reference.

Figure 3.

0 20 40 60 80 100 120 140 160 180

Time[s]

0

5

10

15

20

25

[deg

]

(a) Scenario 1. Jib angle γ. Blue line:

Nonlinear controller. Red line: Desired

reference.

0 20 40 60 80 100 120 140 160 180

Time[s]

1

1.2

1.4

1.6

1.8

2

2.2

d[m

]

(b) Scenario 1. Rope length d. Blue

line: Nonlinear controller. Red line: De-

sired reference.

Figure 4.

0 20 40 60 80 100 120 140 160 180

Time[s]

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1[d

eg

]

(a) Scenario 1. Payload swing angle θ1.

0 20 40 60 80 100 120 140 160 180

Time[s]

-2

0

2

4

6

8

10

12

2[d

eg

]

10-3

(b) Scenario 1. Payload swing angle θ2

Figure 5.

oscillations of the payload swing angles θ1 and θ2 are confined both within 1◦ in thetime window considered in the simulations.

The reader is referred to https://youtu.be/aBA6CoGARvs for a video containingextra material.

Scenario 3. In this simulation, to evaluate the robustness of our control schemew.r.t. non-perfect gravity compensation, we lift a payload with different mass than

15

0 20 40 60 80 100 120 140 160 180

Time[s]

-4

-2

0

2

4

u1[N

m]

0 20 40 60 80 100 120 140 160 180

Time[s]

8500

9000

9500

10000

u2[N

m]

(a) Scenario 1. Control input u1, u2.

0 20 40 60 80 100 120 140 160 180

Time[s]

4700

4800

4900

5000

5100

u3[N

m]

0 20 40 60 80 100 120 140 160 180

Time[s]

-981.05

-981

-980.95

u4[N

m]

(b) Scenario 1. Control input u3, u4.

Figure 6.

0 20 40 60 80 100 120 140 160 180

Time[s]

-10

0

10

20

30

40

50

60

70

[deg

]

(a) Scenario 2. Tower angle α. Blue

line: Nonlinear controller. Red line: De-

sired reference.

0 20 40 60 80 100 120 140 160 180

Time[s]

0

5

10

15

20

25

30

35

[deg

](b) Scenario 2. Boom angle β. Blue

line: Nonlinear controller. Red line: De-

sired reference.

Figure 7.

the one known by the control law. The payload mass has been set to 50 kg. The sim-ulation results are reported in Figg.11a-12b, where we can see that the crane reachesthe desired positions with a negligible error in terms of desired angles configurationsand desired cable length. Additionally, the payload swing amplitudes (Figg.13a-13b)have negligible residual swings. As one can see from Figg.14a-14b, the input values,due to the different mass of the payload, have lower values than in the previous casesbecause they strictly depend on the payload mass value, especially for the terms relat-ing to gravity compensation. As expected the input u4 in Fig.10b is roughly doubledcompared to the one in Fig.14b.

Scenario 4. In this simulation, we consider a gust of wind as external disturbanceacting on the crane to demonstrate that the proposed control scheme is robust w.r.tan external unmodeled force. As one can see in Figg. 17a-17b, when the gust of windoccurs (at around 30 seconds), initially the swing angles θ1 and θ2 increase but thenthe residual oscillations of the payload swing angles θ1 and θ2 are confined within 1◦

and 2◦, respectively, in the time window considered in the simulations. As expected,the controller is able to counteract reasonably well the external disturbance, thereforeno major change occurs during the desired trajectory (see Figg.15a-16b).

It is worth noting that the simulations shown in Scenario 3 and Scenario 4 allowus to state that even if the crane system model is not accurate, the proposed controllaw is able to effectively counteract this type of model mismatch.

Scenario 5. In this simulation scenario we will consider the presence of measurement

16

0 20 40 60 80 100 120 140 160 180

Time[s]

-5

0

5

10

15

20

25

[deg

]

(a) Scenario 2. Jib angle γ. Blue line:Nonlinear controller. Red line: Desired

reference.

0 20 40 60 80 100 120 140 160 180

Time[s]

0.8

1

1.2

1.4

1.6

1.8

2

2.2

d[m

]

(b) Scenario 2. Rope length d. Blueline: Nonlinear controller. Red line: De-

sired reference.

Figure 8.

0 20 40 60 80 100 120 140 160 180

Time[s]

-8

-6

-4

-2

0

2

4

6

8

10

12

1[d

eg

]

(a) Scenario 2. Payload swing angle θ1.

0 20 40 60 80 100 120 140 160 180

Time[s]

-6

-4

-2

0

2

4

6

2[d

eg

]

(b) Scenario 2. Payload swing angle θ2

Figure 9.

noise for the angular position measurements used by the control law. The simulationresults are given in Figg.19a-20b where we can clearly see that the presence of noisehas a very limited impact on the performance of the control law. As expected, thethree actuated angles (i.e. α, β, and γ) reach the desired angular values in around 100seconds. Additionally, the cable achieves the desired length. Moreover, at the beginningthe payload swing amplitudes (i.e. θ1 and θ2) present oscillations due to the presenceof noise in the measurements.

Comparison analysis. To show the efficiency of the proposed nonlinear controlscheme, we have performed a comparative analysis considering a linear quadratic reg-ulator (LQR). For the LQR control approach, first, the crane dynamics is linearizedaround the equilibrium point, and then, the following weight matrices have been cho-sen to stabilize the plant: Q = diag{102, 103, 103, 52, 10, 10, 10, 10, 10, 50, 102, 102} andR = diag{50, 50, 50, 10}. The weight matrices have been chosen so that in nominal con-ditions the two controllers (i.e. the proposed control scheme and the LQR controller)give similar dynamics during the crane movement (see Fig.23) and similar input val-ues for the each joint torques (see Fig.24). Although in nominal circumstances, theLQR gives good results, in case of model mismatch, this control approach tends to notbehave properly. Repeating the simulations carried out in Scenario 3, in Fig.25 it ispossible to notice that the LQR is not able to follow the desired trajectory when themass of the payload is different from the nominal one.

17

(a) Scenario 2. Control input u1, u2. (b) Scenario 2. Control input u3, u4.

Figure 10.

0 20 40 60 80 100 120 140 160 180

Time[s]

0

10

20

30

40

50

60

70

[deg

]

(a) Scenario 3. Tower angle α. Blue

line: Nonlinear controller. Red line: De-

sired reference.

0 20 40 60 80 100 120 140 160 180

Time[s]

0

5

10

15

20

25

30

35

[deg

](b) Scenario 3. Boom angle β. Blue

line: Nonlinear controller. Red line: De-

sired reference.

Figure 11.

0 20 40 60 80 100 120 140 160 180

Time[s]

0

5

10

15

20

25

[deg

]

(a) Scenario 3. Jib angle γ. Blue line:

Nonlinear controller. Red line: Desiredreference.

0 20 40 60 80 100 120 140 160 180

Time[s]

0.8

1

1.2

1.4

1.6

1.8

2

2.2

d[m

]

(b) Scenario 3. Rope length d. Blue

line: Nonlinear controller. Red line: De-sired reference.

Figure 12.

6. Conclusion

The paper proposed for the first time a detailed mathematical model of a knuckle cranewhich takes into account all of the degrees of freedom (DoFs) that characterize thistype of system (i.e. the three rotations, the length of the rope and the payload swingangles). On the basis of this model, a nonlinear control law has been developed whichis able to perform position control of the crane while actively damping the oscillations

18

0 20 40 60 80 100 120 140 160 180

Time[s]

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1[d

eg

]

(a) Scenario 3. Payload swing angle θ1.

0 20 40 60 80 100 120 140 160 180

Time[s]

-0.01

-0.005

0

0.005

0.01

0.015

2[d

eg

]

(b) Scenario 3. Payload swing angle θ2

Figure 13.

(a) Scenario 3. Control input u1, u2.

0 20 40 60 80 100 120 140 160 180

Time[s]

3500

4000

4500

5000

u3[N

m]

0 20 40 60 80 100 120 140 160 180

Time[s]

-1000

-800

-600

-400

u4[N

m]

(b) Scenario 3. Control input u3, u4.

Figure 14.

0 20 40 60 80 100 120 140 160 180

Time[s]

0

10

20

30

40

50

60

70

[deg

]

(a) Scenario 4. Tower angle α. Blueline: Nonlinear controller. Red line: De-

sired reference.

0 20 40 60 80 100 120 140 160 180

Time[s]

5

10

15

20

25

30

35

[deg

]

(b) Scenario 4. Boom angle β. Blueline: Nonlinear controller. Red line: De-

sired reference.

Figure 15.

of the load. It is worth noting that, the proposed control law is based directly onthe nonlinear model of the knuckle crane avoiding any form of linearization of themathematical model. The stability properties of the scheme have been proved usingthe LaSalle’s invariance principle. Five different simulation scenarios and a comparisonanalysis with a LQR control have been included in the paper to show the effectivenessof the proposed control approach.

19

0 20 40 60 80 100 120 140 160 180

Time[s]

0

5

10

15

20

25

[deg

]

(a) Scenario 4. Jib angle γ. Blue line:Nonlinear controller. Red line: Desired

reference.

0 20 40 60 80 100 120 140 160 180

Time[s]

1

1.2

1.4

1.6

1.8

2

2.2

d[m

]

(b) Scenario 4. Rope length d. Blueline: Nonlinear controller. Red line: De-

sired reference.

Figure 16.

0 20 40 60 80 100 120 140 160 180

Time[s]

-4

-3

-2

-1

0

1

2

3

4

5

6

1[d

eg

]

(a) Scenario 4. Payload swing angle θ1.

0 20 40 60 80 100 120 140 160 180

Time[s]

-6

-4

-2

0

2

4

6

2[d

eg

]

(b) Scenario 4. Payload swing angle θ2

Figure 17.

0 20 40 60 80 100 120 140 160 180

Time[s]

-400

-200

0

200

400

u1[N

m]

0 20 40 60 80 100 120 140 160 180

Time[s]

8500

9000

9500

10000

u2[N

m]

(a) Scenario 4. Control input u1, u2. (b) Scenario 4. Control input u3, u4.

Figure 18.

20

0 20 40 60 80 100 120 140 160 180

Time[s]

-10

0

10

20

30

40

50

60

70

[deg

]

(a) Scenario 5. Tower angle α. Blue

line: Nonlinear controller. Red line: De-sired reference.

0 20 40 60 80 100 120 140 160 180

Time[s]

0

5

10

15

20

25

30

35

[deg

]

(b) Scenario 5. Boom angle β. Blue

line: Nonlinear controller. Red line: De-sired reference.

Figure 19.

0 20 40 60 80 100 120 140 160 180

Time[s]

-5

0

5

10

15

20

25

[deg

]

(a) Scenario 5. Jib angle γ. Blue line:

Nonlinear controller. Red line: Desiredreference.

0 20 40 60 80 100 120 140 160 180

Time[s]

0.8

1

1.2

1.4

1.6

1.8

2

2.2

d[m

]

(b) Scenario 5. Rope length d. Blue

line: Nonlinear controller. Red line: De-sired reference.

Figure 20.

0 20 40 60 80 100 120 140 160 180

Time[s]

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1[d

eg

]

(a) Scenario 5. Payload swing angle θ1.

0 20 40 60 80 100 120 140 160 180

Time[s]

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

2[d

eg

]

(b) Scenario 5. Payload swing angle θ2

Figure 21.

21

(a) Scenario 5. Control input u1, u2. (b) Scenario 5. Control input u3, u4.

Figure 22.

0 50 100 150

Time[s]

0

10

20

30

40

50

60

[de

g]

0 50 100 150

Time[s]

0

5

10

15

20

25

30

35

[de

g]

0 50 100 150

Time[s]

0

5

10

15

20

25

30

[de

g]

0 50 100 150

Time[s]

0

0.5

1

1.5

2

2.5

d[m

]

0 50 100 150

Time[s]

-1

-0.5

0

0.5

1

1[d

eg

]

0 50 100 150

Time[s]

-1

-0.5

0

0.5

1

2[d

eg

]

Figure 23. Blue line: Nonlinear controller. Red line: LQR.

Figure 24. Blue line: Nonlinear controller. Red line: LQR.

22

0 100

Time[s]

-600

-400

-200

0

200

[deg

]

0 100

Time[s]

-200

-100

0

100

200[d

eg

]

0 100

Time[s]

-1000

-500

0

[deg

]

0 100

Time[s]

-150

-100

-50

0

d[m

]

0 100

Time[s]

-1500

-1000

-500

0

500

1[d

eg

]

0 100

Time[s]

-250

-200

-150

-100

-50

0

2[d

eg

]

Figure 25. Blue line: Nonlinear controller. Red line: LQR.

23

Appendix A.

The nonzero entries of the system matrices M(q), C(q, q), F (q) and g(q) are define asfollows:

m11 = Itot + d2m+A1C2β +A2C

2γ +A3CβCγ + 2A4dCβSθ2 + 2A5dCγSθ2 − d2mC2

θ1C2θ2 ,

m22 = A1 + IB, m33 = A2 + IJ ,m44 = m, m55 = dmC2θ2 , m66 = dm,

m12 = m21 = A4dCθ2SβSθ1 , m13 = m31 = A5dCθ2SγSθ1 ,

m14 = m41 = Cθ2Sθ1(A4Cβ +A5Cγ), m15 = m51 = dCθ1Cθ2(A4Cβ +A5Cγ + dmSθ2),

m16 = m61 = −dSθ1(dm+A4CβSθ2 +A5CγSθ2), m23 = m32 =1

2(A3Cβ−γ),

m24 = m42 = −A4(SβSθ2 + CβCθ1Cθ2)

m25 = m52 = A4dCβCθ2Sθ1 , m26 = m62 = −A4d(Cθ2Sβ − CβCθ1Sθ2),m34 = m43 = −A5(SγSθ2 + CγCθ1Cθ2)

m35 = m53 = A5dCγCθ2Sθ1 m36 = m63 = −A5d(Cθ2Sγ − CγCθ1Sθ2)c11 = 2ddm−A2γS2γ −A1βS2β + 2A4dCβSθ2 + 2A5dCγSθ2 −A3βCγSβ −A3γCβSγ

−2ddmC2θ1C

2θ2 + 2A4dθ2CβCθ2 + 2A5dθ2CγCθ2 − 2A4dβSβSθ2

−2A5dγSγSθ2 + 2d2θ1mCθ1C2θ2Sθ1 + 2d2θ2mC

2θ1Cθ2Sθ2

c12 = A4dCθ2SβSθ1 +AdβCβCθ2Sθ1 +A4dθ1Cθ1Cθ2Sβ −A4dθ2SβSθ1Sθ2

c13 = A5dCθ2SγSθ1 +A5dγCγCθ2Sθ1 +A5dθ1Cθ1Cθ2Sγ −A5dθ2SγSθ1Sθ2

c14 = θ1Cθ1Cθ2(A4Cβ +A5Cγ)− Cθ2Sθ1(A4βSβ +A5γSγ)− θ2Sθ1Sθ2(A4Cβ +A5Cγ)

c15 = dCθ1Cθ2(dmSθ2 −A4βSβ −A5dγSγ + dθ2mCθ2) + dCθ1Cθ2(A4Cβ +A5Cγ + dmSθ2)

−dθ1Cθ2Sθ1(A4Cβ +A5Cγ + dmSθ2)− dθ2Cθ1Sθ2(A4Cβ +A5Cγ + dmSθ2)

c16 = −dSθ1(dm+A4θ2CβCθ2 +A5θ2CγCθ2 −A4dβSβSθ2 −A5γSγSθ2)

−dSθ1(dm+A4CβSθ2 +A5CγSθ2)− dθ1Cθ1(dm+A4CβSθ2 +A5CγSθ2)

c21 =1

2(αSβ(2A1Cβ +A3Cγ + 2A4dSθ2)) +

1

2(3A4dCθ2SβSθ1) +

1

2(A4dβCβCθ2Sθ1)

+1

2(3A4dθ1Cθ1Cθ2Sβ)− 1

2(3A4dθ2SβSθ1Sθ2)

c22 =1

4(A3γSβ−γ) +

1

2(A4d(CβSθ2 − Cθ1Cθ2Sβ)) +

1

2(A4dθ2(CβCθ2

+Cθ1SβSθ2))−1

2(A4dαCβCθ2Sβ) +

1

2(A4dθ1Cθ2SβSβ)

c23 = −1

4(A3Sβ−γ)(β − 2γ))

c24 =1

2(A4βCθ1Cθ2Sβ)−A4θ2Cθ2Sβ −

1

2(A4βCβSθ2) +A4θ1CβCθ2Sβ

+A4θ2CβCθ1Sθ2 +1

2(A4αCθ2SβSβ)

24

c25 = A4dCβCθ2Sβ +1

2(A4dαCθ1Cθ2Sβ) +A4dθ1CβCθ1Cθ2 −

1

2(A4dβCθ2SβSβ)−A4dθ2CβSβSθ2

c26 = A4dθ2SβSθ2 −1

2(A4dβCβCθ2)−A4dCθ2Sβ +A4dCβCθ1Sθ2 −

1

2(A4dβCθ1SβSθ2)

−A4dθ1CβSβSθ2 −1

2(A4dαSβSβSθ2) +A4dθ2CβCθ1Cθ2

c31 =1

2(αSγ(2A2Cγ +A3Cβ + 2A5dSθ2)) +

1

2(3A5dCθ2SγSθ1) +

1

2(A5dγCγCθ2Sθ1)

+1

2(3A5dθ1Cθ1Cθ2Sγ)− 1

2(3A5dθ2SγSθ1Sθ2)

c32 = −1

4(A3Sβ−γ)(2β − γ))

c33 =1

2(A5d(CγSθ2 − Cθ1Cθ2Sγ))− 1

4(A3βSβ−γ) +

1

2(A5dθ2(CγCθ2 + Cθ1SγSθ2))

−1

2(A5dαCγCθ2Sθ1) +

1

2(A5dθ1Cθ2SγSθ1)

c34 =1

2(A5γCθ1Cθ2Sγ)−A5θ2Cθ2Sγ −

1

2(A5γCγSθ2) +A5θ1CγCθ2Sθ1

+A5θ2CγCθ1Sθ2 +1

2(A5αCθ2SγSθ1)

c35 = A5ddCγCθ2Sθ1 +1

2(A5dαCθ1Cθ2Sγ)

−1

2(A5dγCθ2SγSθ1)−A5dθ2CγSθ1Sθ2 +A5dθ1CγCθ1Cθ2

c36 = A5dθ2SγSθ2 −1

2(A5dγCγCθ2)−A5dCθ2Sγ +A5dCγCθ1Sθ2 −

1

2(A5dγCθ1SγSθ2)

−A5dθ1CγSθ1Sθ2 −1

2(A5dαSγSθ1Sθ2) +A5dθ2CγCθ1Cθ2

c41 = dθ2mSθ1 − dαm−A4αCβSθ2 −A5αCγSθ2 + dαmC2θ1C

2θ2 +

1

2(A4θ1CβCθ1Cθ2)

+1

2(A5θ1CγCθ1Cθ2)−

1

2(3A4βCθ2SβSθ1)−

1

2(A4θ2CβSθ1Sθ2)−

1

2(3A5γCθ2SγSθ1)

−1

2(A5θ2CγSθ1Sθ2)− dθ1mCθ1Cθ2Sθ2

c42 = A4βCθ1Cθ2Sβ − (A4θ2Cθ2Sβ)/2−A4βCβSθ2 +1

2(A4θ1CβCθ2Sθ1)

+1

2(A4θ2CβCθ1Sθ2)−

1

2(A4αCθ2SβSθ1)

c43 = A5γCθ1Cθ2Sγ −1

2(A5θ2Cθ2Sγ)−A5γCγSθ2 +

1

2(A5θ1CγCθ2Sθ1)

+1

2(A5θ2CγCθ1Sθ2)−

1

2(A5αCθ2SγSθ1)

c45 = dθ1m(S2θ2 − 1)− 1

2(αCθ1Cθ2(A4Cβ +A5Cγ + 2dmSθ2))

−1

2(A4βCβCθ2Sθ1)−

1

2(A5γCγCθ2Sθ1)

25

c46 =1

2(A4β(Cθ2Sβ − CβCθ1Sθ2)) +

1

2(A5γ(Cθ2Sγ − CγCθ1Sθ2))

−dθ2m+1

2(αSθ1(2dm+A4CβSθ2 +A5CγSθ2))

c51 = 2d2θ2mCθ1C2θ2 −

1

2(d2θ2mCθ1) +

1

2(A4dCβCθ1Cθ2)

+1

2(A5dCγCθ1Cθ2)−

1

2(3A4dβCθ1Cθ2Sβ)− 1

2(A4dθ1CβCθ2Sθ1)

−1

2(A4dθ2CβCθ1Sθ2)−

1

2(3A5dγCθ1Cθ2Sγ)− 1

2(A5dθ1CγCθ2Sθ1)

−1

2(A5dθ2CγCθ1Sθ2) + 2ddmCθ1Cθ2Sθ2

−1

2(d2θ1mCθ2Sθ1Sθ2)− d2αmCθ1C

2θ2Sθ1

c52 =1

2(A4dCβCθ2Sθ1)−

1

2(A4dαCθ1Cθ2Sβ)

−A4dβCθ2SβSθ1 −1

2(A4dθ2CβSθ1Sθ2) +

1

2(A4dθ1CβCθ1Cθ2)

c53 =1

2(A5dCγCθ2Sθ1)−

1

2(A5dαCθ1Cθ2Sγ)

−A5dγCθ2SγSθ1 −1

2(A5dθ2CγSθ1Sθ2) +

1

2(A5dθ1CγCθ1Cθ2)

c54 = −1

2(αCθ1Cθ2(A4Cβ +A5Cγ))− 1

2(A4βCβCθ2Sθ1)−

1

2(A5γCγCθ2Sθ1)

c55 =1

2(dαCθ2Sθ1(A4Cβ +A5Cγ + dmSθ2))−

1

2(A4dβCβCθ1Cθ2)−

1

2(A5dγCγCθ1Cθ2)

c56 =1

2(dαCθ1(dm+A4CβSθ2 +A5CγSθ2)) +

1

2(A4dβCβSθ1Sθ2) +

1

2(A5dγCγSθ1Sθ2)

c61 =1

2(3A4dβSβSθ1Sθ2)−

1

2(d2θ1mCθ1)− d2θ1mCθ1C

2θ2 −A4dαCβCθ2 −A5dαCγCθ2

−1

2(A4ddCβSθ1Sθ2)−

1

2(A5ddCγSθ1Sθ2)−

1

2(A4dθ1CβCθ1Sθ2)−

1

2(A4dθ2CβCθ2Sθ1)

−1

2(A5dθ1CγCθ1Sθ2)−

1

2(A5dθ2CγCθ2Sθ1)− 2ddmSθ1 +

1

2(3A5dγSγSθ1Sθ2)− d2αmC2

θ1Cθ2Sθ2

c62 =1

2(A4dθ2SβSθ2)−A4dβCβCθ2 −

1

2(A4ddCθ2Sβ) +

1

2(A4ddCβCθ1Sθ2)

−A4dβCθ1SβSθ2 −1

2(A4dθ1CβSθ1Sθ2) +

1

2(A4dαSβSθ1Sθ2) +

1

2(A4dθ2CβCθ1Cθ2)

c63 =1

2(A5dθ2SγSθ2)−A5dγCγCθ2 −

1

2(A5ddCθ2Sγ) +

1

2(A5ddCγCθ1Sθ2)

−A5dγCθ1SγSθ2 −1

2(A5dθ1CγSθ1Sθ2) +

1

2(A5dαSγSθ1Sθ2) +

1

2(A5dθ2CγCθ1Cθ2)

c64 =1

2(A4β(Cθ2Sβ − CβCθ1Sθ2)) +

1

2(A5γ(Cθ2Sγ − CγCθ1Sθ2)) +

1

2(αSθ1Sθ2(A4Cβ +A5Cγ))

c65 =1

2(d2αmCθ1)− d2αmCθ1C

2θ2 +

1

2(A4dαCβCθ1Sθ2) +

1

2(A5dαCγCθ1Sθ2)

+1

2(A4dβCβSθ1Sθ2) +

1

2(A5dγCγSθ1Sθ2)

c66 =1

2(dαCθ2Sθ1(A4Cβ +A5Cγ))− 1

2(A5dγ(SγSθ2 + CγCθ1Cθ2))−

1

2(A4dβ(SβSθ2 + CβCθ1Cθ2))

26

g2 =1

2glBcosβ(2m+mB + 2mJ), g3 =

1

2glJcosγ(2m+mJ),

g4 = −gmcosθ1cosθ2, g5 = gmdcosθ2sinθ1, g6 = gmdcosθ1sinθ2

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