**2.2 New Scientific Achievements**

**4.1.1 Modelling Robot Manipulator Dynamics**

In my research the modelling of robot manipulators (RMs) dynamics, mapping the position, velocity and acceleration of joints to forces and torques exerted to the structure is based on the Lagrange formulation, which ensures the appropriate structure of the dynamic model that is commonly used in control algorithms.

RMs are known to be highly nonlinear multi-input multi-output systems. To preserve the known structure of the Lagrange formulation, common to all system equations of RMs and other dynamic systems such as navigation dynamics of missiles and aeroplanes, I have chosen the grey-box modelling approach.

Forces exerted to joints of the RM are the sum of four components modelling
consequently the torque resulting from the inertia (H), the Coriolis effects and
centrifugal forces (C), the gravity forces (g) and the viscose friction (f). Individual
knowledge of all these components is important for precise, model based robot control
algorithms, yet it is impossible to directly, explicitly measure these components in any
general real life system. When designing a grey-box model advantage can be taken of
other commonly known facts of robotics like *H and g are nonlinear functions of joint *
positions and the driving torque 𝜏 is linear in the joint accelerations. The centrifugal and
Coriolis effects are quadratic in the joint velocities and nonlinear in the joint positions
and f is linear in joint velocity [80].

The dynamic model identification method uses the measured resultant torque and joint
variables along suitably chosen paths for every joint. The application of Lagrange
dynamic equations for a robot manipulator in the joint space formulates the resultant
torque 𝜏_{𝑖} acting on the i^{th} joint for all the *p joints of the RM as a function of following *
vectors:

∑^{𝑝}_{𝑗=1}(𝑫_{𝑖𝑗}(𝒒) ∙ 𝒒̈_{𝑗})+ ∑^{𝑝}_{𝑗=1}∑^{𝑝}_{𝑘=1}(𝒒̇_{𝒋}∙ 𝑫_{𝑖𝑗𝑘}(𝒒) ∙ 𝒒̇_{𝑘})+ 𝑫_{𝑖}(𝒒) + 𝑓_{𝑖} = 𝜏_{𝑖} (39)
where: q is the vector of joint positions; 𝒒̇ are joint velocities; 𝒒̈ are joint accelerations;

∑^{𝑝}_{𝑗=1}(𝑫_{𝑖𝑗}(𝒒) ∙ 𝒒̈_{𝑗}) is commonly referred to as 𝑯 ∙ 𝒒̈ or 𝕁 ∙ 𝒒̈ the inertia matrix
component of the torque; ∑^{𝑝}_{𝑗=1}∑^{𝑝}_{𝑘=1}(𝒒̇_{𝒋}∙ 𝑫_{𝑖𝑗𝑘}(𝒒) ∙ 𝒒̇_{𝑘}) is commonly referred to as 𝑪 ∙ 𝒒̇

or ℂ ∙ 𝒒̇, describing the centrifugal forces and the Coriolis effects; 𝑫_{𝑖}(𝒒) is commonly
referred to as g the gravitational force component; 𝑓_{𝑖} stands for the viscous friction; and
𝜏_{𝑖} is the resultant torque acting on the i^{th} joint – identities in the common robotics
notation are:

𝐻_{𝑖𝑗} = 𝐷_{𝑖𝑗}(𝒒), 𝐶_{𝑖𝑘} = ∑^{𝑝}_{𝑗=1}𝑞̇_{𝑗} ∙ 𝐷_{𝑖𝑗𝑘}(𝒒), 𝑔_{𝑖} = 𝐷_{𝑖}(𝒒), 𝑓_{𝑖} = 𝑐𝑜𝑛𝑠𝑡_{𝑖}, (40)

where: 𝐷_{𝑖𝑗}, 𝐷_{𝑖𝑗𝑘}, 𝐷_{𝑖} are in general, highly nonlinear scalar functions of * q, the joint *
position vector. They may contain sin(*) and cos(*) functions of joint positions and/or of
their products and sums defined by the geometry of the RM.

There are well known general relations that can be used to reduce the number of equation (40) is not possible. The only information on the output of the system joint is the resultant torque (39). The identification of all nonlinear functions under these terms is a considerable problem.

By my Hypothesis III.a this paper will propose and present the validity of a new method
that will identify the RM dynamics through finding the 𝐷_{𝑖𝑗} nonlinear functions of
equation (39) as TSK FLSs, while calculating 𝐷_{𝑖𝑗𝑘} nonlinear functions as in equation
(41). All linear parameters of the system will be determined by SVD based robust LS
method. Nonlinear parameters will be evolved by multi-objective GA and fine-tuned by
a gradient descent method.

**4.1.2 ** **Modelling Multi-rotor Flight Dynamics **

The complete dynamics of an aircraft, taking into account aero-elastic effects, flexibility of wings, internal dynamics of engines, and the whole set of changing environmental variables is quite complex and somewhat unmanageable for the purpose of autonomous control engineering [10]. This paper deals with the flight dynamics of multi-rotors.

Multi-rotor UAV manoeuvres are controlled by varying angular speeds of its propellers.

Each rotor blade produces a thrust and a torque, whose combination generates the main trust, the yaw torque, the pitch torque, and the roll torque acting on the multi-rotor.

Motors produce a force proportional to the square of the angular speed and the angular acceleration of the rotor; the acceleration term is commonly neglected as the speed transients are short thus exerting no significant effects. Motors of a multi-rotor can only turn in a fixed direction, so the produced force can be always presumed positive.

Motors are set up so that opposites form pairs rotating in the same direction (clockwise/counter-clockwise), while their neighbouring motors are rotating in the opposite direction (counter-clockwise/clockwise). This arrangement is chosen so that gyroscopic effects and aerodynamic torques are cancelled in trimmed flight. The main trust is the sum of individual trusts of each motor. The pitch torque is a function of difference in forces produced on one pair of motors, while the roll torque is a function of difference in forces produced on other pair of motors. The yaw torque is sum off all motor reaction torques due to shaft acceleration and blades drag. The motor torque is opposed by a general aerodynamic drag.

For a full navigation dynamic model of a multi-rotor system both (i) the centre of mass
position vector of 𝝃 = (𝑥, 𝑦, 𝑧) in fixed frame coordinates and (ii) the orientation Euler
angles: roll, pitch, yaw angles *(ϕ,θ,ψ) around body axes X, Y, Z are considered. Using *
the Euler-Lagrange approach it can be shown how the translational forces 𝑭_{𝜉}, applied to
the rotorcraft due to main trust, can be fully decoupled from the yaw, pitch and roll

𝑚 ∙ (𝝃̈ + 𝑔 ∙ [0 0 1]^{𝑇}) = 𝑭_{𝜉} (42)
where: 𝝃̈ is the second time derivative (acceleration) of the E-frame coordinates of the
rotorcraft centre of mass 𝝃 = (𝑥, 𝑦, 𝑧), 𝑚 is the multi-rotor mass; 𝑔 is the gravitational
constant, which is acting only along the third axis 𝑧⃗; 𝑭_{𝜉} is the vector of translational
forces.

𝕁(𝒒) ∙ 𝒒̈ + ℂ(𝒒, 𝒒̇) ∙ 𝒒̇ = 𝝉 (43)

where: 𝕁 is a 3x3 matrix, called the inertia matrix, ℂ is also a 3x3 matrix that refers to Coriolis, gyroscopic and centrifugal terms, 𝒒 = [𝜙, 𝜃, 𝜓] is the state vector of Euler angles, its time derivatives are 𝑑𝒒/𝑑𝑡 = 𝒒̇ = [𝜙̇, 𝜃̇, 𝜓̇] and 𝑑𝒒̇/𝑑𝑡 = 𝒒̈ = [𝜙̈, 𝜃̈, 𝜓̈].

For the scope of this research we shall address only equation (43) as that is the complex nonlinear challenging part of the multi-rotor flight dynamics model to be identified.

Equation (43) can be analysed as three resultant torques τ*i* acting along the [ϕ, θ, ψ] axes
for i,j,k∈(ϕ, θ, ψ) as:

∑ (𝐷_{𝑗} _{𝑖𝑗}(𝒒) ∙ 𝑞̈_{𝑗}) + ∑ ∑ (𝑞̇_{𝑗} _{𝑘} _{𝑗}∙ 𝐷_{𝑖𝑗𝑘}(𝒒) ∙ 𝑞̇_{𝑘}) = 𝜏_{𝑖} (44)
The similarity with RM dynamics equation (39) is obvious, the first component of
equation (44) is the inertia matrix part expansion, the second is the Coriolis matrix term
expansion, whose components are highly nonlinear functions containing 𝑠𝑖𝑛(𝒒) and
𝑐𝑜𝑠(𝒒) components, and also their products and sums defined by the rigid body system
geometry as described in [67].

There are general relations that can be used for reducing the number of unknown inertia and Coriolis components: 𝕁 is symmetric and ℂ is defined by Christoffel symbols of 𝕁:

𝐷_{𝑖𝑗𝑘} = (𝜕𝐷_{𝑖𝑗}/𝜕𝑞_{𝑘}+ 𝜕𝐷_{𝑖𝑘}/𝜕𝑞_{𝑗}− 𝜕𝐷_{𝑗𝑘}/𝜕𝑞_{𝑖})/2 (45)
These properties result in further inherent relations as:

𝐷_{𝑖𝑗} = 𝐷_{𝑗𝑖}, 𝐷_{𝑖𝑗𝑘} = 𝐷_{𝑖𝑘𝑗}, 𝐷_{𝑘𝑖𝑗} = −𝐷_{𝑗𝑖𝑘}, 𝐷_{𝑘𝑗𝑘} = 0, ∀𝑖, 𝑘 ≥ 𝑗 (46)
It should be noted that direct measurement of any single 𝐷_{𝑖𝑗𝑘} or 𝐷_{𝑖𝑗} component of
equation (44) is not possible. Measurable data vector pairs are (𝒒,̈ 𝝉) angular
accelerations as system input and resultant torques proportional to rotation speed of
motors as system output.

Determining all 𝐷_{𝑖𝑗𝑘} and 𝐷_{𝑖𝑗} nonlinear functions is a considerable grey-box
identification problem, but when achieved the model is usable in efficient robust and
precise model based control implementations, as this model preserves all 𝒒̈, 𝒒̇ state
variables in an explicit form.

By my *Hypothesis III.b this paper will propose and present the validity of a new *
method that will identify the multi-rotor flight dynamics equation (44) 𝐷_{𝑖𝑗} components
by specially constructed continuous and periodic TSK FLSs, while calculating 𝐷_{𝑖𝑗𝑘}
nonlinear functions as in equation (45). For modelling multi-rotor flight dynamics I
propose to extend the definition of TSK FLSs in a way that they become periodic and of
continuous output, even for the attitude Euler angle system inputs 0-2π transitions.

**4.1.3 ** **Validating Quality of Complex Nonlinear Dynamic System Genetic Fuzzy **