Hybrid spiral-dynamic bacteria-chemotaxis algorithm with application to control two-wheeled machines
Goher et al. Robot. Biomim.
Hybrid spiral-dynamic bacteria-chemotaxis algorithm with application to control two-wheeled machines
K. M. Goher 0 3
A. M. Almeshal 1
S. A. Agouri 2
A. N. K. Nasir 2
M. O. Tokhi 2
M. R. Alenezi 1
T. Al Zanki 1
S. O. Fadlallah 4
0 Department of Informatics and Enabling Technologies, Lincoln University , Lincoln , New Zealand
1 Electronics Engineering Technol- ogy Department, College of Technological Studies, Public Authority for Applied Education and Training (PAAET) , Adailiya , Kuwait
2 Department of Automatic Control and Systems Engineer- ing, The University of Sheffield , Sheffield , UK
3 Department of Informatics and Enabling Technologies, Lincoln University , Lin- coln , New Zealand
4 Mechanical Engineering Department, Auckland University of Technology , Auckland , New Zealand
This paper presents the implementation of the hybrid spiral-dynamic bacteria-chemotaxis (HSDBC) approach to control two different configurations of a two-wheeled vehicle. The HSDBC is a combination of bacterial chemotaxis used in bacterial forging algorithm (BFA) and the spiral-dynamic algorithm (SDA). BFA provides a good exploration strategy due to the chemotaxis approach. However, it endures an oscillation problem near the end of the search process when using a large step size. Conversely; for a small step size, it affords better exploitation and accuracy with slower convergence. SDA provides better stability when approaching an optimum point and has faster convergence speed. This may cause the search agents to get trapped into local optima which results in low accurate solution. HSDBC exploits the chemotactic strategy of BFA and fitness accuracy and convergence speed of SDA so as to overcome the problems associated with both the SDA and BFA algorithms alone. The HSDBC thus developed is evaluated in optimizing the performance and energy consumption of two highly nonlinear platforms, namely single and double inverted pendulum-like vehicles with an extended rod. Comparative results with BFA and SDA show that the proposed algorithm is able to result in better performance of the highly nonlinear systems.
Spiral dynamics; Bacteria chemotaxis; Two-wheeled inverted pendulum with new configuration; PD-like fuzzy logic control; Hybrid fuzzy logic control
Optimization algorithms play a dominant role in
solving real problems [38, 58]. Bacterial foraging algorithm
(BFA)  and spiral-dynamics algorithm (SDA) [50, 51]
are well-known optimization techniques in solving
realworld problems. Evolutionary algorithms (EA) have been
used extensively in literature: soft computing techniques
, particle swarm optimization [53, 55], incremental
encoding , neural stochastic multi-scale optimization
, multi-objective optimization [12, 23], multi-criteria
optimization  and fuzzy logic and genetic
Nasir et al. [33, 34, 36] proposed linear and nonlinear
adaptive BFA where the bacteria step size is varied based
on the combination of bacteria and iteration index. Chen
and Lin , Farhat and El-Hawary  and Huang and
Lin  utilized index and total number of chemotaxis
to vary bacteria step size within a specified range. Niu
et al. , Yan et al.  and Xu et al.  varied the
step size within a user-defined range using
combination of index and total number of iterations. Supriyono
and Tokhi  developed various versions of BFA based
on linear and nonlinear mathematical formulations to
establish relationship between bacteria step size and
their current fitness value. This relationship enables
bacteria to have different step sizes in similar iteration
as well as through the whole operation. There are other
adaptive approaches considered the variation of the
step size based on fitness value [16, 28, 29, 44, 45, 54].
Nasir et al. [30–32] proposed adaptive spiral-dynamic
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algorithm (ASDA) to establish relationship between
spiral radius (r) and fitness value of each search point.
They introduced schemes to make variation in the spiral
radius within a specific range, enabling each search point
to have different spiral radius in moving from one
location to another location. Moreover, the movement step
of each search agent was made with respect to its fitness
value at the current location. As a result of the variation,
there was improvement to the performance mainly on
the accuracy of the final solution.
Hybrid optimization techniques
Hybrid approach is the combination of two or more
algorithms aimed to retain the advantages and eliminate
the weaknesses of the original algorithms. This includes
the synergization between different groups such as
bioinspired, nature-inspired, etc. Biswas et al. [10, 11]
proposed hybrid BFA-PSO where a chemotactic strategy of
bacteria was designed to represent exploitation part of
the algorithm, while the exploration of optimum
location was accomplished by PSO. The same approach using
a constant step size was implemented by Korani ,
where the PSO operator was used to determine new
direction of bacteria motion. Ghaffar et al.  adopted
a modified PSO operator to determine new direction of
bacteria to avoid local optima solution. Biswas et al. 
proposed chemotactic differential evolution algorithm
where adaptive chemotactic strategy of bacteria has been
used to improve fitness accuracy of classical differential
evolution (DE). Sinha et al.  implemented the same
approach on an electric power system. Kim et al.  and
Kim  used GA and BFA to tune a PID controller for
automatic voltage regulation. Panigrahi and Ravikumar
 and Hooshmand et al.  incorporated Nelder–
Mead method into bacteria chemotaxis phase to enhance
the search strategy and improve bacteria location. Other
hybrid approaches involving BFA [41, 59] used bee
colony algorithm and Tabu search.
Limitations of BFA and SDA
BFA is a well-known bio-inspired algorithm. It has a
comparable or better performance compared to other types
of optimization algorithm . Therefore, it has been
adopted by many researchers worldwide to solve
realworld problems in many areas . However, BFA has a
slow convergence speed and longer computation time.
Due to this issue, the application of original BFA in online
and offline tuning for solving a complex real-world
problem is unsatisfactory . On the other hand, SDA is a
relatively new and a simple algorithm developed inspired
from natural spiral phenomena on earth. It has a
relatively fast convergence speed which can complement the
drawback of BFA performance. Previous study showed
that SDA has a similar or better performance compared
to other differential evolutionary (DE) and particle swarm
optimization (PSO) algorithms [50, 51]. However, SDA
has a premature convergence issue where it hardly
provides an optimal solution for complex problems.
Hybrid spiral‑dynamic bacteria‑chemotaxis
A hybrid bacteria-chemotaxis spiral-dynamic algorithm
(HSDBC) has been proposed by Nasir et al. [30–32] to
synergize the chemotactic strategy of bacteria and ASDA. The
chemotaxis phase in BFA was designed such that it
represents exploration stage and placed at the first phase of the
algorithm, while the ASDA as the exploitation stage and
was placed at the second phase of the algorithm. The
combination simplified the BFA algorithm and greatly reduced
the total computation time of BFA. Moreover,
comparison with original algorithms concluded that it improved
the accuracy of the final solution and had the capability to
avoid the local optima problem. HSDBC is a new variant
of hybrid-type BFA-SDA algorithm developed to solve the
issues aforementioned above. Our previous study showed
that the algorithm outperformed both BFA and SDA
algorithms in terms of accuracy in finding a global optima
solution. Compared to BFA, the total computation time
has been significantly reduced and its convergence speed
has been considerably increased [31, 37].
Full description of the HSDBC algorithm for
n-dimensional optimization is shown in Fig. 1. The description of
the associated parameters used in the algorithm is shown
in Table 1, and the corresponding flow chart is given in
Fig. 2. The HSDBC algorithm has been tested to model
and control nonlinear systems including flexible robot
manipulator and a twin rotor system using a PD-like FLC
Contribution overview and paper organization
Establishing the optimal control strategy for nonlinear
dynamic systems, specifically inverted pendulum-based
systems, has been and still remains a field of interest for
a countless number of research studies due to their
various promising real-life applications including personal
transport systems and wheelchairs. This paper presents
an extended study of the proposed algorithm in solving
complex problem of two-wheeled inverted pendulum
systems. We will implement HSDBC algorithm to control
two different configurations of two-wheeled machines.
A detailed simulation study of the HSDBC algorithm
using several unimodal and multimodal benchmark
functions can be found in the work of Nasir and Tokhi .
A hybrid fuzzy-like PD and I controller is designed and
implemented on the two systems.
Step 0: Preparation
Select the number of search points (bacteria)
m ≥ 2 , parameters
0 ≤θ tumble,θ swim < 2π , 0 < rtumble, rswim < 1 of
Sn (r,θ ), maximum iteration number, kmax and
maximum number of swim, Ns for bacteria
Set k = 0, s = 0 .
Step 1: Initialization
Set initial points xi (0) ∈ Rn, i = 1, 2,...m in the
feasible region at random and center x*as
x* = xig (0) , ig = arg mini f (xi (0)), i = 1, 2,..., m .
Step 2: Applying bacteria chemotaxis
xi (k +1) = Sn (rtumble,θ tumble )xi (k) − (Sn (rtumble,θ swim ) − In )x*
i = 1, 2,..., m .
(a) Check number swim for bacteria i.
If s < Ns , then check fitness,
Otherwise set i = i +1, and
return to step (i).
(b) Check fitness
If f (xi (k +1)) < f (xi (k), then update xi ,
Otherwise set s = Ns , and return to step (i).
(c) Update xi
i = 1, 2,..., m .
Step 3: Updating x*
Step 4: Checking termination criterion
If k = kmax then terminate. Otherwise set
k = k +1, and return to step 2
Fig. 1 HSDBC algorithm for n-dimensional optimization
Table 1 HSDBC algorithm parameters
This paper is organized as follows: “Background”
section introduces both ASDA and ABFA optimization
algorithms, along with an explanation of the principle
of HSDBC algorithm. In order to test and validate the
proposed HSDBC algorithm on real dynamic systems,
two case studies are considered in the study and are
introduced in “Methods” section. “Case study I: single
IP with an extended rod” section describes in details
the first case study that involves a single inverted
pendulum (IP) system. A double IP system with an
extended rod is considered as the second case study
and is presented in “Case study II: double IP with an
extended rod” section. The results of the investigation
are presented at the end of each of the previously
mentioned sections, sections “Case study I: single IP with
an extended rod” and “Case study II: double IP with an
extended rod”. At last, the paper is concluded in
An inverted pendulum as a typical multi-input
multi-output system has the characteristics of nonlinear,
multivariable and close coupling Luo et al. . The uniqueness
and wide application of technology derived from this
unstable system has drawn interest of many researchers
including Akesson et al. , Askari et al.  and Balan
et al. [6, 7]. There are various applications of IP
configuration including design of walking gaits, wheelchairs, and
personal transport systems.
The system considered in this paper is a two-wheeled
machine (TWM) with an extendable rod as described
by Goher et al.  and verified by Almeshal et al. [3,
4]. This system stabilizes it extendable intermediate
body (IB) by controlling the wheel movements in a
desired manner. A TWM is designed such that either
i = i + 1
s = s + 1
Randomly place xi(k) in search space
Compute f (xi (k))
Set x* = xig (0) as center of spiral
ig = arg mini f (xi (0)),i = 1,2,..., m
xi(k +1) = Sn(rtumble,θtumble )xi(k) − (Sn(rtumble,θtumble ) − In)x*
xi(k +1) = Sn(rswim ,θswim )xi(k) − (Sn(rswim ,θswim ) − In)x*
Compute f (xi (k))
Set x* = xig (k +1) as center of spiral
ig = arg mini f (xi (k +1)),i = 1,2,..., m
Case study I: single IP with an extended rod
The system comprises a rod on an axle incorporating two
wheels as shown in Figs. 3 and 4. The numerical parameters
of the system are described in “Appendix 1”. Full details on
the system description are available in Almeshal et al. [3, 4].
Mathematical modeling of the single IP with an extended
Lagrange-Euler formulation is used to derive the
system dynamic model using the following n-coordinates
= Qi, i = 1, 2, . . . , n
where Qi is generalized force vector and qi is generalized
coordinate vector. The coordinate vector is selected as:
and the force vector as:
Fig. 3 Two-wheeled vehicle with an extendable intermediate body
k = k + 1
Fig. 2 Flowchart of the HSDBC algorithm
the center of mass of the robot is above or below the
axle joining two wheels. Statically unstable TWM have
evoked a lot of interest in present decade . Two case
studies are used to test and validate the developed
algorithm; single IP and double IP with an extended
rod. For consistency, the two systems are considered
to move along an inclined surface. The results of the
simulation are shown in a comparative manner with
three different optimization algorithms; BFA, SDA and
Fig. 4 Schematic diagram of a single IP vehicle on an inclined plane
Fig. 5 System block diagram
Fig. 6 Fuzzy PD + I controller
IB at the vertical upright position, keep the cart wheels
within a specified linear position from a predefined
reference while moving on an inclined surface, and to control
the linear displacement of the payload along the IB.
The inputs to the three control loops are the error
signal, change of error and the sum of previous errors. The
system inputs are the driving force Fc, the linear
actuator force Fa and the disturbance force Fd. FLC controllers
are developed based on Mamdani-type fuzzy inference
engine with (25) fuzzy rules shown in Table 2.
The optimization process is constrained within the
stability region of the system. Each parameter has a feasible
interval that guarantees the stability of the system within
the defined gain limits. Table 3 presents the limits of each
parameter which represent the search space of each of
Table 3 Boundary limits of the controller gain parameters
Table 2 Fuzzy rule base
Qi = [Fc Fd Fa]T
The system equations of motion of the model can be
Driving further the above equations yields the
following nonlinear equations of motion of the system:
+ C16Q˙ θ˙ cos(θ + α) + C11 sin α = Fc
− θ˙ sin θ (C10g + Mug (C5 + Q) + Mmg (C6 + Q)) = Fd
Detailed explanations of the constant parameters
appearing in Eqs. (4)–(9) are formulated in “Appendix 2”.
Three independent control loops, shown in Fig. 5, are
implemented on the system. Fuzzy PD-like combined
with conventional integrator is designed as shown in
Fig. 6. The three control loops are working to: stabilize the
the three addressed algorithms. Those parameters were
obtained through a manual tuning exercise of the system.
The performance index of the system is chosen as the
minimum mean squared error (MSE) of each control
loop. The MSE is calculated for each control loop of the
vehicle system using the following equations:
Objective_Function1 = min
Objective_Function2 = min
Objective_Function3 = min
(Yd − Ym)2
(Qd − Qm)2
The objective function of the system is calculated based
on the total MSE which can be expressed as:
The parameters used to implement the three
optimization algorithms are shown in Tables 4, 5 and 6 and the
calculated optimized parameters are shown in Table 7.
The data shown in Table 8 gives the minimum cost
functions due to the implementation of the three
optimization algorithms where the HSDBC algorithm was able
to give the minimum cost function compared to the BFA
and SDA optimizations.
Four consecutive simulation runs of the system model
yielded the performance of the system as shown in Fig. 7.
Table 4 BFA parameters
Table 5 SDA parameters
Table 6 HSDBC parameters
Table 7 Optimized gain values
Table 8 Cost functions
Minimum cost function value
As noted from Fig. 6; the three optimization algorithms;
BFA, SDA and HSDBC, resulted generally in a
satisfactory performance of the system. However, HSDBC
algorithm showed a superior performance in
minimizing the percentage overshoot in the payload
displacement as appeared in Fig. 7c. As per the tilt angle shown
in Fig. 7b, all the three algorithms behaved the same in
terms of minimizing the level and period of oscillations.
As per convergence graph shown in Fig. 8 show that the
three algorithms resulted in similar convergence of the
cost function within around 25 iterations. However, the
HSDBC was faster in convergence of the cost function
too early if compared to the BFA and SDA algorithms.
Attention has been focused on energy consumption
in this investigation. The control effort components as a
measure of energy consumption are shown in Fig. 9. It is
noted that the control effort required in the transient range;
the three algorithms yielded nearly close results. However,
the HSDC was more robust as it resulted less oscillation
of the control effort components in the magnified areas
of the plots. Significant amount of energy saving has been
achieved specifically in the cart and tilt angle control efforts
as appeared in Fig. 9a, b. Furthermore, the HSDBC resulted
in a great improvement in the control effort for the payload;
this can be demonstrated by the significant improvement
shown in Fig. 9c in terms of less oscillations and the short
time taken by the control signal to stabilize.
Case study II: double IP with an extended rod
In this case study, an additional link is added and hence
increasing the degrees of the freedom (DOF) and the
complexity of the structure. The double IP with such
configuration shown in Fig. 10 is mimicking the scenario of
Fig. 7 Performance of the single IP. a Linear displacement of the cart,
b tilt angle of the intermediate body, c linear displacement of the
Fig. 8 Cost function convergence
a wheelchair on only two wheels which has been studied
significantly by Ahmad and Tokhi .
The design of the two-wheeled robotic vehicle is based
on double inverted pendulum system with a movable
payload moving on an inclined surface with five DOF.
The increased DOFs will enable the vehicle to
maneuver freely in all directions and in different environments.
Moreover, the second link provides an extended height
to lift up the payload to a demanded height. The system
equations of motion are presented with five highly
coupled differential equations as follows:
2 C27δ¨L + 2C1δ¨R + C6θ¨1 cos(θ1 + α)
+ 0.5(C25 + C26Q)
θ¨2 cos(θ2 + α) − θ˙22 sin(θ2 + α)
2 C27δ¨R + 2C1δ¨L + C6θ¨1 cos(θ1 + α)
+ 0.5(C25 + C26Q)
θ¨2 cos(θ2 + α) − θ˙22 sin(θ2 + α)
Fig.9 Control effort components in a single IP. a Cart control effort,
b tilt angle control effort, c linear actuator driving force
Fig.10 Axonometric diagram of a double IP vehicle
2C2θ¨1 + (C5 + M2uL1(C8 + Q) + ML1(C9 + Q))
θ¨2 cos(θ1 − θ2) − θ˙2(θ˙1 − θ˙2)sin(θ1 − θ2)
+ θ˙2 cos(θ1 − θ2) Q˙ (M2uL1 + ML1)
+ C6(δ¨L + δ¨R)cos(θ1 + α)
+ C6(δ˙L + δ˙R)sin(θ1 + α)(θ˙12 − θ˙1)
+ θ˙12θ˙2 sin(θ1 − θ2)
(C5 + M2uL1(C8 + Q) + ML1(C9 + Q))
− g C14θ˙1 sin θ1 = 0.5(TR + TL)
C19Q¨ − 0.5θ˙22(2C19Q + C22)
− C23θ˙1θ˙2 cos(θ1 − θ2)
− 0.5C25θ˙2(δ˙L + δ˙R)cos(θ2 + α)
+ g C18 cosθ2 = Fa − Ffa
θ¨2(C19Q2 + C20Q + C21) + θ˙2(2C19Q2 + C22)
+ θ¨1 cos(θ1 − θ2)(C23Q + C24)
− θ˙1(θ˙1 − θ˙2)sin(θ1 − θ2)(C23Q + C24)
+ C23θ˙1 cos(θ1 − θ2)
+ 0.5(δ¨L + δ¨R)cos(θ2 + α)(C25Q + C26)
− 0.5(δ˙L + δ˙R)θ˙2 sin(θ2 + α)(C25Q + C26)
+ 0.5C25(δ˙L + δ˙R)cos(θ2 + α)
− θ˙1 θ˙2 sin(θ1 − θ2)(C23Q + C24)
+ 0.5 θ˙22(δ˙L + δ˙R)sin(θ2 + α)(C25Q + C26)
− g θ˙2 sin θ2(C17 + C18Q) = TM − TfM − LdFd (16)
A robust hybrid fuzzy logic control strategy (FLC) with
five control loops is developed. The control strategy block
diagram is presented in Fig. 11, to control the vehicle and
to counteract the disturbances occurring due to different
The control system of the vehicle consists of five hybrid
FLC controllers with a total of 15 gain parameters. The gain
parameters were first tuned heuristically in order to test the
controller as well as to find the boundaries of the search
space of those gain parameters. The same optimization
algorithms, SDA, BFA and HSDBC, are implemented in order to
optimize the vehicle control system parameters. The
performance index of the system is chosen as the minimum mean
squared error (MSE) for each control loop and defined as:
MSE 1 = min
MSE 2 = min
MSE 3 = min
MSE 4 = min
MSE 5 = min
(Qd − Qm)2
The objective function is chosen as the summation of
the MSE of the system expressed as:
Minimization of the objective function J is used to find the
optimal controller gain parameters that result in the
minimum control loop errors in the stability region of the system.
Fig. 11 Block diagram of the vehicle control system
With the complexity of the model, slight changes in the
control gain parameters will result in oscillations in the
system response and may lead to instability of the vehicle.
Constrained optimization techniques are used to avoid
this problem occurring while optimizing the control
system parameters. The optimization process is constrained
within the stability region of the system. This is achieved
by defining a feasible interval for each control
parameter shown in Table 9, which assures the stability of the
Results and discussion
This simulation scenario allows comparing the
performance of the HSDBC with other similar optimization
algorithms. Tables 10, 11 and 12 provide the simulation
Table 9 Boundary limits of the controller gain parameters
Table 10 BFA parameters
Table 11 SDA parameters
Table 12 HSDBC parameters
parameters used for BFA, SDA and HSDBC algorithms,
respectively. The optimized control gain parameters
reported by each optimization algorithm are presented in
Table 13, whereas Table 14 provides the minimum cost
function calculated by each of the optimization
algorithms. Clearly, the HSDBC algorithm has found the
minimum cost function value of 0.3682.
Figure 13 shows the system response based on the
optimized control parameters obtained by the
implementation of the BFA, SDA and HSDBC algorithms in
comparison to the manual-tuned gain parameters. It
can be noted that BFA, SDA and HSDBC are of much
similar effect on the system response by finding
stable solutions, lowering the overshoots and improved
steady-state error. However, HSDBC algorithm has a
superior performance in minimizing the percentage
overshoot and the settling time for the linear
displacement of the left and right wheel as shown in Fig. 13a,
Table 13 Optimized gain values
Table 14 Cost functions
Minimum cost function
Fig. 12 Cost function convergence plot for BFA, SDA and HSDBC
b and the tilt angles of the two pendula as shown in
Fig. 13c, d. Furthermore, HSDBC-optimized gain
parameters clearly improved the settling time of the
payload actuator displacement as depicted in Fig. 13e.
As can be noticed from the cost function convergence
plots shown in Fig. 12, the HSDBC algorithm cost
function has converged into the minimum value within
approximately 25 iterations. However, the BFA and SDA
algorithms seem to need more iterations to settle into
their best-found minimum values presented in Table 14.
HSDBC has successfully found the minimum cost
function and proved its speed in convergence. In terms of
the control output components shown in Fig. 14, the
control efforts was minimized by the implementation of
HSDBC algorithm for the left wheel, first link and the
payload linear actuator. However, the heuristic tuning
yields better results in case of the right wheel and the
second link. This seems to be accompanied with a poor
response of the system, in terms of increased
disturbance period and higher gain values, if compared to the
results obtained by the HSDBC algorithm.
A novel hybrid spiral-dynamics bacteria-chemotaxis
(HSDBC) optimization algorithm has been proposed.
Chemotactic strategy of bacteria through spiral
tumble and swim actions of bacteria is adopted to improve
exploration strategy of SDA. Moreover, spiral radius
Fig. 13 Performance of the double IP. a The linear displacement of the left wheel, b the linear displacement of the right wheel, c the tilt angle of
the first link, d the tilt angle of the second link, e the payload linear actuator displacement
and angular displacement of spiral model is made
adaptive to enhance the movement of bacteria within
feasible region. Incorporating these two schemes have
successfully saved the SDA from getting trapped into
local optima point and provides faster convergence.
The proposed algorithm has been utilized to optimize
the performance of two different IP platforms; single
and double IP with a new configuration of an extended
intermediate body. Simulation results have shown that
the proposed hybrid algorithm outperformed its
predecessor algorithms (BFA and SDA) in terms of increased
convergence speed and better fitness accuracy.
Furthermore, implementation of the HSDBC yielded significant
Fig. 14 Control effort components in double IP case. a Left wheel torque, b right wheel torque, c first link torque, d second link torque, e linear
actuator driving force
saving in the energy consumption of the two tested
Future work will consider investigating standard PID
tuning methods, such as Ziegler–Nichols method, and
evaluating and comparing their performance with the
KG initiated the concept and developed the system of two-wheeled machine
with extended rods. AN developed the HSDBC algorithm. AA and SA
contributed to the modeling and simulation of the system. OT was overseeing the
entire research and he gave technical insights in the control part. MA and TA
helped with the editing of the final draft. Final editing of the manuscript is
managed by KG. All authors read and approved the final manuscript.
The authors of this paper would like to thank Lincoln University in New Zealand
by offering the funding support for this publication.
The authors declare that they have no competing interests.
This research is originally funded by a governmental PhD scholarships and
research grants from various countries including New Zealand, Malaysia,
Kuwait and the UK.
Appendix 1: Single IP parameters
Appendix 2: Single IP constants
C1 = 4Ll + Ll2 + 4LlLu
C2 = 4Ll + 2Lu
C3 = 4Ll2 + 4L2u + 8LlLu
C4 = 4Ll + 4Lu
C5 = 2Ll + Lu
C6 = 2Ll + 2Lu
C7 = Mc + Ml + Ma + Mu + Mm
C8 = Mc + Ml + Ma + Mu + Mm
C9 = 12 MlLl2 + 21 Jl + 21 MaLa2 + 21 Ja
C10 = MlLl + MaLa
C11 = (Mc + Ml + Ma + Mu + Mm)g
1 1 1 1
C12 = 2 Mu + 2 Mm + 2 Mu + 2 Mm
1 1 1
C13 = 2 (C2 + C4)Mu + 2 Mu(C2 − 2Lg) + 2 Mm(C4 − 2Lg)
C14 = C9 + 2 C1Mu + C3Mm + 2 (Mu + Mm)Lg2
− C5MuLg + 2 C1Mu + 2 MmLg2 − C6MmLg
C15 = C10 + C5Mu + C6Mm
C16 = Mu + Mm
C17 = C5Mu + C6Mm
C18 = C10 + C17
C19 = C10 + Mu(C5 + Q) + Mm(C6 + Q)
C20 = C10g + Mug (C5 + Q) + Mm(C6 + Q)
Lu = C5 + Q
Lm = C6 + Q
Jm = Mm(Q2 + (C4 − 2Lg)Q + Lg2 − 2LgC6 + C3)
Appendix 3: Double IP constants
C1 = 0.125Rw(Mm + M1 + M2l + Ma + M2u + M)
C3 = M2lLc22 + MaLa
C4 = 0.5Mm(cos α + sin α) + 0.5M1(cos α + sin α)
C10 = 4Lc22 + L22u + 4L2uLc2
C11 = 4Lc22 + 4L22u + 8L2uLc2
C15 = M2lLc2 + MaLa
C16 = M1 + Mm + M2l + Ma + M2u + M
C17 = C15 + M2uC8 + MC9
C18 = C8 + C9
C19 = M2u + M
C20 = 2C8M2u + 2C9M
C21 = C3 + M2uC10 + MC11
C22 = 2M2uC8 + 2MC9
C23 = M2uL1 + ML1
C24 = C5 + M2uL1C8 + ML1C9
C25 = M2uRw + MRw
C26 = C7 + M2uRw C8 + MRw C9
C27 = C1 + C12
C28 = Rw (M2u + M)
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