Nonclassical Parametric Variational Technique to Manipulability Control of a Serial-Link Robot That Is Used in Treatment of Femoral Shaft Fractures

. Robot-assisted intramedullary nailing is a minimally invasive surgical procedure commonly used to treat femur fractures. Despite its bene ﬁ ts, there are several disadvantages associated with this technique, such as frequent malalignment, physical fatigue, and excessive radiation exposure for medical personnel. Therefore, it is crucial to ensure that robotic surgery for fracture reduction is precise and safe. Precise calculation and regulation of the robot ’ s reduction force are of utmost importance. In this study, we propose a manipulator that utilises robot assistance and indirect contact with the femur to e ﬀ ectively reduce fractures in the shaft. The dynamics of the reduction robot are analysed using the implicit function theorem, which allows us to address the reduced problem. A parametric approach is presented to tackle the initial algebraic constraints, enabling the approximation of the state-space solution while simultaneously controlling the class of constraints in a multiway manner. This approach simpli ﬁ es the problem from an in ﬁ nite-dimensional one to a ﬁ nite-dimensional one, leading to an approximate solution obtained by solving a set of control linear algebraic equations. The proposed robotic-assisted system enhances fracture repositioning while reducing radiation exposure for both the patient and the medical sta ﬀ . Through numerical results and their practical application, we have developed an e ﬃ cient method that yields positive outcomes.


Introduction
In recent years, there has been a growing trend of analysing and synthesising linear time-invariant control systems in the form of differential-algebraic equations [1][2][3][4].One significant class of these systems, known as DAC-SYS (differential-algebraic control systems), comprises systems controlled by mechanical differential algebra [5][6][7].These systems find wide application in various fields, including robot dynamics, machine dynamics, and vehicle dynamics [8][9][10], with the principles of multibody system theory being employed in each case.When subsystems are coupled by constraints or when kinematic linkages such as joints are utilised, constraint equations can be explicitly incorporated, resulting in a mathematical model with differential-algebraic equations [11,12].The physical relevance of DAC-SYS is clearly manifested in this natural system description.The concept of DAC-SYS has been introduced [13][14][15], and in recent years, tools for simulation, identification, analysis, and design of these systems have been developed.The algorithms for numerical simulation are addressed [16,17], while for mechanical DAC-SYS, a recent study focuses on the design and manipulability analysis of a serial-link robot (S-LR) [18][19][20].This paper is aimed at investigating the kinematics of the robotic DAC-SYS, specifically under the influence of algebraic kinematic constraints at the joints.This information allows critical points of the variational formulation to be determined, and all efficiency functions with consistent initial conditions can be consolidated into a performance index.This motivated our research to identify the optimal control method for a robotic manipulation system subject to S-LR constraints.

Manipulability Analysis and Inverse Kinematics
Finding a workspace representation appropriate for regulating end effector motion is the main goal of this section.
Dexterity and maneuverability must be quantified in order to plan tasks that include positioning and orienting the robot's end tip as well as for the design and control of the robot.It is additionally helpful for planning trajectories and computing inverse kinematics.For assessing a robot's capacity to move around a workplace, manipulability measurement is very crucial.Figure 1 illustrates the manipulator's coordinate systems.The robot's velocity can be determined by Jacobian from the joint velocities using the following notation for the manipulator tip's kinematic equation expressed by V = F W , where V ∈ ℝ 6 is the robot tip's operational spatial vector, W ∈ ℝ 6 is the joint's spatial vector, and F stands for transfer function.
The robot's differential kinematic equation can be expressed as follows: The vector product method yields the robot's Jacobean matrix as follows: where z i is a joint axis-corresponding unit vector; the vector from the {i}-origin coordinate's to the {6}-coordinate's origin is represented by s 0,i 6 .It is possible to formulate equation (3) as We refer to [15,[18][19][20][21] for more details on the analysis of mechanical design of this model.
The manipulability index of the current robotic manipulator was found to range between 0.7 and 1 using the Monte Carlo approach [14,20].This indicates that there is an inverse kinematic solution and that the manipulator is operable throughout the full work area.

Robotic frame
Foot position where T and T 6 represent the joint 5 and 6's respective constant velocity twists.One can see [19][20][21][22] for further details.
For a given position and orientation of the tool end tip, the deformation of the prismatic joints and the rotational orientations of the rotary joints of the robot manipulators must be determined using the inverse kinematic solution.The manipulator's position control and trajectory planning can both benefit from using the inverse kinematic model.
The robot velocity of end effector V p a0 of the moving point p a with respect to inertial point p 0 is given by where dA is the adjoint map given by where R ba is the rotation matrix that determines how p a is oriented in relation to p 0 and p ba is the vector that connects p a to p b .Choosing of variable is extremely advantageous for controlling the links' velocity since it leads to the separation of the kinematic and dynamic components.Thus, the findings in this section can be used to the analysis and development of control rules for mechanically limited manipulators, such as the surgical robots of Raven and da Vinci [22,23].

Reduced Representation with Insertion Consistent Initial Algebraic Constraints
Recalling from Section 2, the developed robotic system can be modeled as a time-invariant DAC-SYS.
14 Rank E = n 0 < n that ensures the existence of nonsingular matrices G and H, where which transform ( 12) and ( 13) into its dynamic decomposition equivalent form with Consider The dynamic composite system can now be written as Equations ( 18)-( 20) form the Hessenberg DAC-SYS of size two with W 1 2 W 2 1 = 0 and W 1 2 B 1 = 0, and this S-LR reduction counterpart is guaranteed to have the same finite poles, input and output transformation zeros, and dynamic order as the character equation.

Reduction Robot System
The Hessenberg DAC-SYS is a particularly significant category of general differential-algebraic equations having a distinctive structure.With the manipulability control of index-two Hessenberg DAC-SYS, a variational formulation approach was devised.
We are now looking for a suitable function, such that the core features lead to a solution of DAC-SYS ( 18)- (20).
Assume that the invertible W 1 2 W 1 1 W 2 1 represent the Jacobean condition.As a result, by repeatedly differentiating the algebraic equation, the implicit function theorem [24,25] can be used to estimate the unknown state x 2 until it can be estimated explicitly as ; then, the reduced robot system can be a dynamic equation defined on consistent initial algebraic constraints.
It is clear that the null space of W 1 2 and W 1 2 W 1 1 is nonempty; then, the class of consistent initial condition is dropped to become

28
with x 1 , u convex set and K 1 , K 2 > 0 (symmetric), and finally, we look for min (F 1Ω ) on the class of admissible pairs Ω.

Variational Formulation of DAC-SYS.
In this section, we will briefly examine how the reduction robot system ( 22)-( 26) transforms into an identical variable (functional) whose crucial points are the answer to ( 22)-( 26).
(i) Define the linear operator , 29 with where

31
where the separable space Dom L = x 1 , u ϵc 1 ℝ n × c ℝ n has such a supremum norm that fills all the normed space in the stated domain (ii) Definition of variational formulation in this case is challenging because of x 1 , x 2 = t f t 0 x 1 x 2 dt, and with regard to x 1 , x 2 and d/dt, the operator L is not symmetric.
Then, redefine x 1 , x 2 = x 1 , Lx 2 to guarantee the variational formulation existence The critical points of F 2Ω are the solution of ( 19)-( 23) over Ω only in the event that , does not degenerate on either the domain or the range of L x 1 , u (iii) The critical points of F 2Ω can be concluded from , u linear part of the variation = 0 33 (iv) F 1Ω and F 2Ω are quadratic in x 1 and u (v) Since x 1 , u is certain and positive in relation to , , then minimum of F 2Ω is the solution of ( 22)-(26) (vi) Since the Banach space is separable in the solvable topological space C ℝ n , suppose that ϕ i , ψ i are linearly independent basis functions.Afterwards, the variable x 1 , u can be defined as a finite linear combination of these basis functions: i and set the outcome to zero: An approximation of a solution to DAC-SYS is then formed by solving the acquired system of linear algebraic equations.

Numerical Illustration
Using the findings of the inverse kinematic analysis in equations ( 12) and ( 13) and also some design parameters, including the link lengths and lower limb anatomic parameters, the objective of our study would be to design an S-LR manipulator which satisfied the specifications for comprehensive fracture fragment alignment with in target operation.The design parameters can be defined as follows: Set G = V and H = U −1 .Under the transformation G, H , DAC-SYS ( 12) and ( 13) can be expressed as follows: x 11 x 12 x 13 x 14 , According to W 1 2 W 2 1 = 0, W 1 2 B 1 = 0, the performance index is obtained:

39
F 1Ω and F 2Ω are given in Section 3.
All the states are measurable.Set Derive F α j i , β k i with respect to its unknown parameters α j i , β k i and equalize the result to zero.Overall, the system behaved smoothly, allowing the user to fully manage the endoscope through the selection of parameters.The system also stayed steady during all selections (see Figure 2).In comparison to the intended trajectory, the system's inaccuracy was rather tiny (see Figure 3).
Using the aforementioned parameters, a MATLAB simulation was employed to invade the manipulator workspace.The simulation results are shown in Figures 2 and 3.

Conclusions
This contribution provides the necessary tools for addressing linear mechanical differential-algebraic systems.The focus was primarily on analysing the forward and inverse kinematics of the developed robotic manipulator to support femoral shaft fracture reduction surgery, thereby enhancing the safety of the S-LR system.The manipulator's design incorporates three rotational joints and three prismatic joints.
The results of the workspace study and kinematic analyses demonstrate that the mechanism can access any location within the specified workspace using the given design parameters.Modelling the mechanical system naturally leads to differential-algebraic systems, which can be explored using variational methods.It is assumed that the solvability space is a separate Banach space to ensure the existence of a countable linearly independent set of functions.The method is based on utilising the implicit function theorem and differentiation index.The differentiation index is repeatedly used to solve the algebraic equations for the unknown state-space parameters and generate reduced-order dynamic equations defined on a class of coherent initial algebraic constraints.The average operating time was approximately 7 minutes, while the time required for the S-LR system to stabilise was around 6 seconds, demonstrating the efficiency of the proposed method.
In this particular case, the system is expressed using an operator from the variant formulation, and the solution is presented as a linear combination of basis components from the setting space.The simulation of the S-LR reduction system is performed using MATLAB software to ensure the validity and accuracy of this technique.

Planned future work includes the following:
(i) There is a plan to investigate the dynamics of the joints in the robotic system to further enhance the safety of S-LR

ϑ 5 ϑ 6 −sin ϑ 5 ϑ 6
depicts the relationship between the robot's first and last links' velocities.In Figure1, the reference states are depicted.From equation (5), we can write

36
Rank E = < n; this lead us to use singular value decomposition to compute U which is orthonormal eigenvectors of EE T , and V is orthonormal eigenvectors of E T E.

Figure 3 :
Figure 3: System performance errors under numerical control.