Flexible reconfiguration in robotics pertains to a robotic system’s capacity to dynamically adjust its configuration to meet diverse tasks, environmental demands, or operational spec- ifications. This involves the deliberate design of robots with the capability to modify their physical structures, including joint arrangements, links, or components, to efficiently execute a spectrum of tasks. This study introduces the analysis and design of a modular mechanism aimed at morphological reconfiguration in robotic platforms. This adaptability enhances the versatility of robots, enabling them to adeptly handle a variety of tasks in diverse scenarios. The proposed mechanisms and technologies encompass modular designs, adjustable joints, and the capability to seamlessly switch between different end-effectors. The system com- prises three gear systems and a joint featuring a convex-toothed sphere. These gears facilitate rotation along pitch and yaw axes, while an internal gear system, utilizing a planetary mech- anism, propels a prismatic piston via a screw mechanism. The mathematical modeling, em- ploying Lagrangian mechanics, illustrates globalized dynamics. Control models, grounded in a spherical coordinate model, are scrutinized by deducing the Jacobian matrix. Demonstrating diverse robotic platform configurations, this work exemplifies the concept of flexible recon- figuration, wherein robots adeptly manage varied and dynamic tasks without necessitating manual adjustments or reprogramming. Emphasizing the modeling and numerical simula- tions of motion analysis, this paper delves into the design of a joint mechanism contributing to the dynamic adaptability and reconfigurability of robots across various tasks. Keywords: reconfigurability, robot modeling, mechanism dynamics, model-based control 1 Introduction Robotic platforms are extensively utilized across various industries, ranging from manufacturing and entertainment to home services, transportation, construction, education, scientific research, surveillance, security, manufacturing, and search and rescue operations. Over time, the adoption *Edgar A. Mart ´ınez-Garc ´ıa. (edmartin@uacj.mx) Alejandra Morales Gallegos (al173527@alumnos.uacj.mx) Rafael Torres-C ´ordoba (ratorres@uacj.mx) Evgeni Magid (magid@it.kfu.ru) 1 of robotic applications has experienced exponential growth. However, the efficacy of these plat- forms hinges on their ability to adapt to the specific tasks required in a given environment. For instance, robotic arms are meticulously designed for tasks such as assembly, selection, inspec- tion, manufacturing, and material manipulation. These tasks necessitate platform models with varying morphological designs, tailored to the specific workspace and application requirements. Traditional robotic platforms, with fixed morphologies, may excel in performing specific tasks but could be poorly suited for a broader range of applications within the same environment. On the contrary, directing research and development efforts towards physically reconfigurable robotic platforms can yield significant advantages. These platforms boast the capability to morph or adjust their physical structures to suit a broader range of tasks within a single environment. Such flexibility empowers them to capitalize on commercial opportunities more efficiently, as they can be customized to fulfill diverse tasks without requiring distinct specialized platforms. As a result, physically reconfigurable robotic platforms emerge as a more versatile solution for tackling a spectrum of tasks across different environments, ultimately enhancing their commercial viability and practicality. To enable a robot to change functionality with good performance in its programmed mission, the reconfiguration function must be practical and/or semi-automatic in the new task. Robotic reconfiguration, as a new physical capability, provides significant benefits in many daily activities of society. The applications of reconfigurable robots need to be sophisticated through engineering techniques to make them incredibly diverse. Advances in robotic engineering tend towards the modularization, reconfiguration, and self-reconfiguration of robots[3], allowing the extension of different physical and functional parts of the robot [1]. Robotics has permeated numerous sectors, with industry standing prominently among them. In industrial landscapes, the incorporation of modularized robotic platforms offers a myriad of advantages, from cost-saving measures to space optimization. By employing fewer robotic platforms, daily tasks can be exe- cuted more efficiently. Yet, the implications of robotic reconfiguration extend far beyond industrial confines. The integration of reconfigurable robots holds the promise of transformative changes in both individual lives and societal paradigms. This research is dedicated to crafting and modeling a highly adaptable module poised to con- figure and reconfigure robotic platforms with a spectrum of morphologies and functionalities. It highlights different assembly morphologies for robotic platforms, offering an array of potential reconfigurations spanning walking, rolling, and robotic arms. The primary objective is to seam- lessly integrate modularity within robotic platforms, streamlining reconfiguration processes and synchronized control to amplify their capabilities. This entails a meticulous design and model- ing approach, culminating in a modular mechanism capable of dynamically reshaping complex robotic structures when interconnected and replicated. Such a mechanism is imbued with syn- chronization functions and locomotion control algorithms to ensure suitable performance across diverse configurations. Moreover, the study embarks on designing a mechanized system featuring both rotary and prismatic articulations, augmented with redundant sensory capabilities for precise and responsive actuation. This involves an exhaustive exploration of motion dynamics and con- trol algorithms, complemented by the development of observability models tailored to the modular robotic architecture. Additionally, the research entails formulating controller schemes tailored to accommodate the intricacies of diverse robotic reconfigurations, further enriching the platform’s adaptability and versatility. Through the development of a modularized robotic mechanism and its mechanical pluralization, a realm of robotic reconfiguration unfolds, broadening the spectrum of tasks achievable by a robotic platform. In Section 2, we review research akin to our project, delving into recent works on related themes and delineating a robust conceptual framework. Transitioning to Section 3, we showcase the CAD software design of the modular mechanism, while Section 4 offers comprehensive in- sights into its instrumentation. Section 5 outlines the modeling of motion geometry for the module 2 mechanisms, alongside dynamic and kinematic models. Chapter 6 introduces the local position and velocity models, along with sophisticated global control models. Simultaneously, Section 7 explores diverse sample configurations and their functional reconfiguration. In Section 8 discusses some analysis about the results. Finally, section 9 encapsulates the conclusions derived from this comprehensive study. 2 Related Work This section discusses some essential concepts and offers a comprehensive review of contem- porary literature that supports the ongoing endeavor. Investigation into the reconfiguration and modularization of robotic structures encompasses a broad range of intricate subjects that have been consistently pursued over the years. Modularization, as a fundamental principle, streamlines construction optimization by integrating functional components designed for diverse elements. Research efforts aimed at engineering precisely tailored models to improve stability, particu- larly in navigating challenging and uneven terrains encountered by quadruped robots, have been extensively explored. In the study conducted by Kula et al. [4], a comprehensive investigation delves into the realm of reconfigurable limbs for walking robots. The research examines the me- chanical synthesis required to redefine walking cycloids, resulting in insights and showcasing the ability to optimize walking trajectories. This optimization renders the robots highly adaptable to diverse terrains, thereby significantly enhancing their overall performance. Techniques aimed at achieving robotic modularization and reconfiguration to enhance navigation across diverse multi- obstacle environments and terrains are pertinent in the field of reconfigurable robotics. The work [6] introduced the design and motion planning of a metamorphic reconfiguration walking robot and categorize walking robots into distinct types based on their structure, each offering unique movement characteristics. Furthermore, robotic reconfiguration emphasizes adaptable configura- tions to optimize performance tailored to specific tasks, such as adjusting the degrees of freedom within a structure. As demonstrated by the work [1], the proposed robot was reconfigured for machining tasks with varying numbers of axes, achieved through module reassembly and joint locking. In another context, modular configurability, which allows for multiple degrees of free- dom in movement, represents specific instances of functional self-reconfigurability. The work [2] focused on the design and locomotive control of a morphologically variable parallel sub-actuated wing, emphasizing the need for optimal performance across different flight phases. Neverthe- less, robots constructed using origami approaches leverage folding and unfolding mechanisms for self-assembly and movement. This results in lightweight, cost-effective solutions that are adapt- able to diverse applications. For instance, the work reported by [8] proposed the creation of a reconfigurable robot utilizing origami techniques, which showed significant advantages including compact size, reduced manufacturing complexity, and the utilization of easily foldable materials. Previous research has yielded modular self-assembling robotic structures. For instance, inspired by social insects like ants and swarm robotics involved simple robotic structures interconnecting to form complex structures [9]. Furthermore, mudular self-assembling approaches [10], and refined for compactness [11]. The study by [12] presented a self-assembling and self-repairing robotic structure, demonstrating concurrent self-repair and assembly during assigned tasks. Furthermore, the principles of modularization and reconfiguration in robotics are also applied to agricultural tasks, where robotic machinery is utilized for a multitude of purposes. The work by [13] pre- sented a conceptual modularized robotic platform capable of monitoring, harvesting, and fumiga- tion in agriculture. To tackle the challenge of rewiring during modular addition and removal, the study by [14] introduced a reconfigurable modular humanoid architecture. This architecture in- corporated actuators specifically engineered for mechanical and electrical compatibility between modules. Real-time communication interfaces between modules facilitating autonomous collec- 3 tive integration. Environmental interaction considerations are key application domains for robotic modularization and reconfiguration. The work [15] designed a modular robotic platform for life support applications, emphasizing flexible indoor movement within confined spaces. Similarly, [16] optimized the movement of a reconfigurable robot for pavement and sidewalk cleaning, with a turning mechanism for maneuverability in confined spaces while avoiding collisions. The chal- lenge of terrain-dependent movement for reconfigurable robots is addressed by [17], utilizing se- mantic segmentation for object perception. Besides, the study by [18] introduced a reconfigurable quadruped/biped robot for aiding the elderly and disabled, capable of adjusting its walking form for stability and user safety based on the terrain. The topology of a reconfigurable modular robot significantly influences its performance. The study [19] determined optimal geometry to enhance performance, while the work [20] suggested a modeling method for kinematic analysis based on module coordinate systems, facilitating comprehensive characterization of module interactions. The distinctive analysis of related work in Table 1 summarizes how our study findings compare with existing research. Table 1: Comparison of key aspects between this study and related research. Key features Related This study distinctions work Motion geometry optimization [4] Dynamic-based motion control of for enhancing performance. [16][19] modules multi-function robots. Metamorphic walking, adaptive [6][14] Adaptable modules to craft structures. locomotion and reconfigurable [18] adaptable modules. Adjusting structural DOFs. [1] Module’s functions reconfiguration. Modular wing configurability. [2] Crafting adaptable modules. Reconfigurable origami robotics. [8] Assembling adaptable modules. Self-assembling/repairing structure. [12] Enabling module’s functions. Agriculture modular robotics. [13] Multi-function adaptable modules. Modular robot for flexible [15] To craft adaptable modules capabilities. for morphological configurations. Robot reconfiguration for diverse [17] [3] Crafting modules for multiple terrain-dependant navigation. robotic tasks. Kinematic analysis for [20] Euler-Lagrange analysis. modular robotics. Despite substantial progress in robotic reconfiguration, the vast scope of potential applications for modularization and reconfiguration necessitates continued exploration and refinement. This work specifically focuses on enhancing the mechanical design of the actuator responsible for providing movement to the robotic platform, aiming to expand degrees of freedom and achieve a broader range of motion. 3 Modular mechanism design In this section, the proposed physical system design of the modular mechanism is presented, depicted in a CAD model. The proposed module mechanism includes three gear systems that, together, provide both rotational and prismatic movement. These gear systems offer 6 degrees of freedom (DOF) concerning the workspace, and in conjunction with its exterior design, allow for modularity. This work utilizes the rotation mechanisms yaw and pitch as illustrated in Figure 4 1, along with the spherical joint shown in Figure 2, and the prismatic movement mechanism in Figure 3. The central issue of this joint mechanisms consists of a toothed sphere or ball joint, with the orthogonal crossing of connected elements to control the degrees of freedom using a spherical coordinate approach. 3.1 Rotary motion mechanisms The rotational movements of the module, according to the inertial system in Figure 2, are provided by the pitch mechanism (Figure 1b) and the yaw mechanism (Figure 1a), which offer rotation in pitch, yaw, and combined movements that also include roll. According to the yaw mechanism shown in Figure 1a, we have, in the first place, the straight gear e20transmitting the driving motion. This motion reaches the worm gear, consisting of g21 ande22, which serves the same purpose as in the case of the pitch system: to increase power and stabilize the load applied to the module. The cylindrical gear e24essentially functions as the final transmitter of the yaw rotation. (a)Yaw mechanism. (b)Pitch mechanism. Figure 1: Mecanismos yaw y pitch In addition, the pitch mechanism (Figure 1b) tha ussually deals against gravity forces, initiates the driving motion at the straight gear e1, which is powered by an electric motor. Subsequently, the motion is transmitted to the worm gear, commonly referred to as a worm, consisting of g2and e3, which increases power and stabilizes the load posture. The helical gear e3has, on one side, a system of cylindrical planetary gears ∆4with a sun, three planets, and a straight ring gear, and on the other side, a pair of output cylindrical gears e5ande6. The planetary system called ∆4provides high torque to the system. Gears e5ande6also provide additional torque and function as the final transmitters of the pitch rotation. Further analytical discussion of this notation is provided in more detail later in Definition 1. Furthermore, the pitch and yaw mechanisms serve to transfer their respective movements to the spherical joint, as depicted in Figure 2, thereby effectuating the output motion of the module. To ensure the proper transmission of movement, both the toothed sphere and the final gears of the yaw and pitch mechanisms were designed with identical modules. This design consideration facilitates seamless movement transfer. 5 As a consequence of the resulting torque and power generated within the pitch and yaw mech- anisms, there is an amplification of torque at the spherical joint. This increased torque is essential for enabling the module to support the interconnection of other modules and accessories effec- tively. Figure 2 illustrates the arrangement of the gear systems in conjunction with the spherical joint, providing a visual representation of their integration. Figure 2: Semi-spherical joint connected to the yaw and pitch mechanisms. 3.2 Prismatic motion mechanism The spherical joint, integral to the system’s functionality, is manipulated via angular coordinates applied to its peripheral surface. Its design incorporates a hollow structure to house a third gear system, as illustrated in Figure 3b. This thoughtful arrangement serves to streamline the trans- mission of motion to a linked component featuring a screw mechanism. This screw mechanism, depicted in Figure 3a, enables the module’s prismatic movement, allowing for controlled linear motion along a defined axis. This integration of components not only enhances the versatility of the module but also ensures precise and efficient movement, contributing to the overall effective- ness of the system. The gear system in question is characterized by its intricate planetary configuration, which consists of two distinct subsystems operating in parallel. At the core of the primary system lies the sun gear denoted as es30, serving as the primary driving cylindrical gear. Working in tandem with this sun gear are four cylindrical planets, labeled as ep31,ep32,ep33, and ep34. Adjacent to this primary system lies a secondary planetary arrangement, featuring its own sun gear, es35, and an additional set of four planets: ep36,ep37,ep38, and ep39. The intricate interplay of these gears and planets is depicted in Figure 4a. Encompassing both planetary systems is a single crown gear, captured in the illustration pre- sented in Figure 4b. This crown gear acts as a pivotal component in the overall mechanism, facilitating smooth and efficient motion transmission between the various gears. The stacking of these planetary systems one atop the other, as showcased in Figure 4c, serves a dual purpose. Not only does it enable a substantial reduction in rotational speed, but it also significantly amplifies the torque output within a confined spatial footprint. 6 (a)Screw piston prismatic link. (b)Primatic screw piston section view. Figure 3: Different views of the prismatic mechanism. This thoughtful configuration underscores the attention to detail in the design process, aimed at improving both power transmission efficiency and space utilization. By integrating these plan- etary systems, the mechanism achieves improved performance and functionality, making it an indispensable component in the larger modular system. Encircling the intricate arrangement of planetary gears lies the crown gear, serving as the pivotal component in transmitting motion to the prismatic axis of the screw-type piston. When the prismatic mechanism is set into motion, the rotational movement of the crown gear initiates the extension of the axis. As the crown gear rotates, its engagement with the planetary gears facilitates the smooth trans- mission of motion along the prismatic axis. This dynamics ensures precise and controlled exten- sion of the axis, enabling the module to execute its intended function effectively. The coordinated movement between the crown gear and the planetary gears illustrates the pro- posed engineering behind the mechanism’s design. By integrating these components, the system achieves motion transmission, contributing to its overall reliability and performance. Figure 5b provides a comprehensive visual depiction showcasing both the interior and exterior views of the module. The interior image offers a detailed glimpse into the intricate mechanisms and components nestled within the module’s structure, highlighting the complexity of its design and functionality. Meanwhile, the exterior view offers a broader perspective, revealing the mod- ule’s overall form and configuration in its entirety. Through these images, the readers gain valuable insights into the module’s inner workings and external appearance, further showing its capabilities and potential applications. This compre- hensive representation serves to enhance understanding and appreciation of the module’s design and construction, underscoring its significance within the broader context of modular robotics approach of this study. 7 (a)Perpespective view. (b)top view. (c)Dual planetary gearing. Figure 4: Prismatic mechanism based on a dual cascade planetary gearing system. (a)Joint’s inner mechanisms. (b)External appearance. Figure 5: Inner and external joint’s views. 4 Module instrumentation Each mechanism, including pitch, yaw, and prismatic motion, is driven by its dedicated motor, ensuring asynchronous actuation. These motors serve as the driving force behind the module’s various movements, providing the propulsion for each mechanism’s function. To exert fine control over the speed of these motors, the Pulse Width Modulation (PWM) method is employed. Through PWM, the system achieves unparalleled control over the motors’ rotational speed by adjusting the width of the electrical pulses supplied to them. This approach allows for seamless modulation of motor speed, enabling smooth and accurate adjustments tailored to the specific requirements of each task. By leveraging PWM for speed control, the modular system attains enhanced efficiency and responsiveness across all mechanisms. This approach to motor control not only improves performance but also ensures robust execution of tasks in diverse environments, underlining the system’s versatility and reliability. Within each of the pitch, yaw, and prismatic gear systems, specialized motors are employed, each equipped with an encoder boasting a high resolution pulses per revolution. This high- resolution encoder plays a pivotal role in the system, enabling precise measurements crucial for accurately determining both position and velocity. The angular position model, derived from these measurements, serves as a cornerstone in the system’s physical sensing. This model provides in- sights into the exact orientation and movement of the mechanisms, forming the basis for informed 8 decision-making and control strategies. Through the integration of these high-resolution encoders and analytical position models, the system attains accuracy and reliability in its motion control capabilities. This issue ensures operation and precise execution of tasks across diverse scenarios, for suitable performance and functionality. The model for sensing the motor’s angle is articulated in terms of a digital pulse train, as depicted by Equation (1): φ(η) =2π Rηt (1) In this context, φsymbolizes the position angle in radians, ηtrepresents the instantaneous digital pulses (dimensionless), and Rdenotes the angular resolution coefficient measured in pulses per revolution. This model allows us to precisely determine the angle of the motor based on the pulse count received. Furthermore, leveraging this angular model and its derivative, we can calculate the angular velocity. To initiate this process, we employ the Taylor series: φ2≈φ 0!(t2−t1)0+˙φ 1!(t2−t1)1+¨φ 2!(t2−t1)2+... (2) By truncating at the second term or first derivative, the following expression captures the effect of two pulse readings at times t1andt2. ˙φ(t,η) =φ(η2)−φ(η1) t2−t1, (3) Substituting (1) into (3), we obtain: ˙φ(η) =2π Rη2−2π Rη1 t2−t1, (4) upon factoring, we obtain: ˙φ(η) =2π Rη2−η1 t2−t1 =2π R∆η ∆t . (5) Furthermore, the model for linear displacement, which plays a crucial role in our analysis, s(η) =2πr Rη. (6) Hence, by differentiating to determine linear velocity using the Taylor series, (2): ds dt(η) =v(η) =s(η2)−s(η1) t2−t1. (7) In this context, vdenotes velocity, measured in meters per second (m/s), while tsignifies time, expressed in seconds (s). Upon substituting (6) into (7), we arrive at: v(η) =2πr Rη2−2πr Rη1 t2−t1, (8) and after factorizing : v(η) =2πr Rη2−η1 t2−t1 =2πr R∆η ∆t . (9) 9 A nine-axis Inertial Measurement Unit (IMU) equipped with RS485 communication is em- ployed to precisely measure orientation, velocity, and gravitational force parameters. This IMU provides data from an accelerometer, gyroscope, and magnetometer into a cohesive output, pro- viding insights into the module’s dynamics. Positioned at the termination point of the module’s output link, as depicted in Figure 6, the IMU plays a pivotal role in determining angular and linear positions, as well as angular and linear velocities. Leveraging equations 1, 6, 4, and 8, respectively, the IMU facilitates accurate and real-time tracking of the module’s motion parameters. In combination, a planetary mechanism is utilized to model the output angle, allowing for the precise determination of linear displacement in conjunction with the encoder. Let φarepresent the angle of the planetary system’s ring gear, thereby inferring the displacement of the piston and enabling comprehensive analysis of the module’s movements and dynamics. sp=L 2 φa+ZZ tˆayawdt2 , (10) where the screw linear advance parameter is L=N n. (11) In this context, Lstands for the advance in meters (m), Nsignifies the number of parallel strings, and ndenotes the number of strings per linear metric unit. Figure 6: Strategic positioning of the geometric measurement model with the 9-axis IMU. The connections of the three DC motors with encoders and the IMU are routed to an Arduino MEGA microcontroller. However, the rotational movement of the toothed sphere, coupled with the placement of the IMU and one of the DC motors inside it, raises concerns about cable en- tanglement. To effectively address this challenge, a rotary electrical connector is employed (rec). Positioned at the base of the sphere, as depicted in Figure 7, this connector comprises a fixed side 10 and a rotating side. The fixed side is housed within the toothed sphere, while the rotating side facilitates connections to both the exterior of the sphere and the microcontroller. This approach ensures communication while accommodating the dynamic movement of the mechanism. Figure 7: Location of the rotary electrical connector to the interior motor of the spherical joint. At the module output, sensing of three dynamic variables along the pitch, yaw, and roll axes is crucial. These variables encompass position angle, velocity, and acceleration, measured respec- tively by an inclinometer, a gyroscope, and an accelerometer. The general models for the Euler angles of the link are as follows: φt= ˆφi+Z tˆ˙φgdt+ZZ tˆaa dtdt21 3, (12) as well as Eulerian angular velocities: ˙φt=d dtˆφi+˙φg+Z tˆa dtdt1 3, (13) and the comprehensive inertial model of angular accelerations: ¨φt=d2 dt2ˆφi+d dtˆ˙φg+ˆaa dt1 3. (14) In this context, φt,˙φtand ¨φtrepresent the estimator models, whereas ˆφ,ˆ˙φ, andˆ¨φdenote the direct sensor measurements. 5 Modular mechanism dynamics In this section, we derive the dynamic modeling of the mechanisms within the modular device. We analyze the kinetic and potential energies obtained from the kinematic models, employing an Euler-Lagrange approach to develop the dynamic model. Following the symbolic notation of mechanisms as introduced in the Mechanics course, for the purpose of this project, we establish the following definitions: Definition 1 (Mechanism Notation) .Let e irepresent a gear, g ia worm screw, and ∆ia planetary gear, interconnected in various ways: perpendicular, parallel, via belt, and serially. Thus, from Definition 1, we derive the representation of the pitch mechanism, outlined in the following Definition 2. 11 Definition 2 (Pitch Mechanism) .Let e 0denote the pinion of the actuator or global driving gear, meshed perpendicularly with e 1. Subsequently, with an axis collinear to g 2and this to e 3. Then e 3 connected perpendicularly to ∆4and this connected again perpendicularly to e 5. Finally, e 5is in serial connection to e 6. Similarly, for the yaw rotation mechanism, its mechanical connection is described using the following notation provided by Definition 3, Definition 3 (Yaw Mechanism) .Let e 19represent the pinion of the actuator (global driving gear) connected perpendicularly to e 20. Subsequently, e 20is connected in parallel to g 21, which trans- mits to e 22. Then, e 22is serially connected to e 23, and finally, it is connected in parallel to e 24. Essentially, considering the interconnection of mechanical elements formalizes the prismatic displacement mechanism described by the notation in Definition 4. Definition 4 (Prismatic Mechanism) .Let e 29represent the actuator pinion or global driving gear, connected in parallel to ∆30, which is also parallel to ∆35. Hence, Definitions 1, 2, 3, and 4, which delineate the physical systems, serve as the corner- stone for deriving the kinematic models in subsequent sections. 5.1 Kinematic Model In this context, all mechanisms are established as underactuated systems, typically controlled by a single variable that operates independently, often facilitated by a global driving gear e0(φ0). Kinematic model of the pitch mechanism is deduced starting from (15) and resulting in expression (20), based on Figure 1. φ1= −r0 r1 φ0, (15) and the connection of e1on a parallel face with g2, both aligned along a collinear axis, signifies a gear transmission by parallel faces and a collinear axis. This configuration indicates that: φ2=φ1. (16) The coupling of g2ande3represents a worm gear connection, wherein: φ3= −n2 n3 φ2=r0n2 r1n3 φ0. (17) Similarly, the collinear parallel axis connection between e3and the planetary gearing system ∆4(e3in parallel to ∆4, denotes a direct parallel alignment of axes: φ4=4r2 s4 (rs4+ra4)2 φ3=4r2 s4r0n2 (rs4+ra4)2r1n3 φ0, (18) and subsequently, the parallel connection between ∆4ande5is assumed, φ5=φ4. (19) Finally, building upon Definitions 1 and 4, the series connection e5-e6is elucidated by Propo- sition 1. 12 Proposition 1 (Pitch Rotation Gear) .Leveraging the deductions from earlier motion transmis- sions, we derive the rotary model of φ6. This model is subject to a mechanical advantage term intrinsic to the prior connection of mechanical elements, as depicted below: φ6=−r5 r6φ5=−4r2 s4r0r5n2 (rs4+ra4)2r1r6n3 φ0. (20) Such that riandniare defined by Definition 5. Definition 5 (Gear-Worm Screw parameters) .Let r idenote the radii [m] of the respective e i, and nirepresent the number of threads of g iconnected to the threads n i+1of e i+1. The kinematic model of the yaw mechanism begins (21) and results in Proposition (2). Firstly, there is a perpendicular connection between the driving gear e19and gear e20given by: φ20= −r19 r20 φ19, (21) and followed by a parallel, colinear shaft connection between gear e20and worm gear g21, where: φ21=φ20. (22) Next, from Definition 5, the worm gear connection between g21ande22is given by: φ22=n21 n22 φ21= −r19n21 r20n22 φ19. (23) The series connection between gears e22ande23is expressed as follows: φ23= −r22 r23 φ22=r19r22n21 r20r23n22 φ19, (24) and there’s the parallel collinear axis connection between e23ande24, φ24=φ23, (25) where the latter is given by Proposition 2. Proposition 2 (Yaw Rotation Gear) .Given the previous algebraic deductions, the kinematic ro- tary transmission model for gear φ24is obtained, expressed as follows: φ24= −r22 r23 φ22=r19r22n21 r20r23n22 φ19. (26) In this context, r isignifies the radii [m] of the corresponding e i, while n 21represents the number of teeth on gear g 21engaged with the teeth n 22of gear e 22. In the context of the kinematic model of the prismatic mechanism, the input φ29transmits motion to the planetary system ∆30. The planetary system ∆30is characterized by transmitting output motion via a carrier. The velocity at the output (carrier) of ∆30in such a planetary system is derived from the following motion relationship: vs=pc psvc. (27) 13 In this equation, vsrepresents the speed of the sun gear, pcdenotes the perimeter of the carrier, andpsstands for the perimeter of the sun gear. Consequently, the tangential velocity of the sun gear progresses at a rate ofPc Pstimes the tangential velocity along the circumferential path of the carrier. Here, PcandPssignify the perimeters of the carrier and the sun gear, respectively. Knowing that vs=rsωsandvc=rcωc, we can substitute these expressions into (27), rsωs=2πrc 2πrsrcωc, (28) where rc=ra−rs 2+rs. Hence rsωs= ra−rs 2+rs rs!ra−rs 2+rs ωc=(rs+ra)2 4rsωc, (29) and solving for ωc, where ωc=φc30andωs=φs30=φ29, we find: φc30=4r2 s (rs+ra)2φ29. (30) Hence, the output model on the carrier of the second planetary system is as follows: φc35=4r2 s (rs+ra)2φc30. (31) Substituting (30), we have the following proposition: Proposition 3 (Prismatic Rotation Gear) .Based on previous algebraic deductions, the rotation model φ35preceded by the two cascaded planetary gears is proposed, such that its kinematic model is expressed as: φc35= 4r2 s (rs+ra)2! 4r2 s (rs+ra)2φ29! =16r4 s (rs+ra)4φ29 (32) Where r arepresents the radius of the planetary system’s ring. The displacement d T35of the screw- type prismatic link is determined by: dT36=Lφc35 (33) Where L =N p, where N represents the number of strings and p denotes the pitch or advance- ment parameter,1 n. Thus, dT36=L 16r4 s (rs+ra)4φ29! (34) 5.2 Dynamic Model By formulating the Lagrangian operators consistent with the motion geometry outlined by the preceding kinematic models, our objective is to derive the dynamic model for each component of the pitch mechanism. Given that energy transmission takes place through a tooth of each gear, whose height varies periodically, we arrive at the following energy equation for each individual gear: L0=1 2I0˙φ2 0−m0gr0sin(φ0), (35) 14 L1=1 2I1˙φ2 1−m1gr1sin(φ1), (36) L2=1 2I2˙φ2 2−m2gr2sin(φ2), (37) L3=1 2I3˙φ2 3−m3gr3sin(φ3), (38) L4=1 2I4˙φ2 4−m4gr4sin(φ4), (39) L5=1 2I5˙φ2 5−m5gr5sin(φ5), (40) L6=1 2I6˙φ2 6−m6gr6sin(φ6). (41) Hereafter, we may define the Langrangian vector ⃗L= (L0,...,L6)⊤to be used by following Definition 6. Definition 6 (Energy Model of the Pitch Mechanism) .LetLidenote the Lagrangian operator for each element constituting the pitch mechanism. This operator accounts for both the rota- tional kinetic energy and the potential energy transmitted by a tooth of the preceding gear to the subsequent gear. The set of energy differentials is defined as: ⃗L=1 2 I00 0 0 0 0 0 0I10 0 0 0 0 0 0 I20 0 0 0 0 0 0 I30 0 0 0 0 0 0 I40 0 0 0 0 0 0 I50 0 0 0 0 0 0 I6  ˙φ0 ˙φ1 ˙φ2 ˙φ3 ˙φ4 ˙φ5 ˙φ6 2 −  m0gr0 0 0 0 0 0 0 0 m1gr1 0 0 0 0 0 0 0 m2gr2 0 0 0 0 0 0 0 m3gr3 0 0 0 0 0 0 0 m4gr4 0 0 0 0 0 0 0 m5gr5 0 0 0 0 0 0 0 m6gr6  sin(φ0) sin(φ1) sin(φ2) sin(φ3) sin(φ4) sin(φ5) sin(φ6) ,(42) where I iis the moment of inertia, φiand ˙φirepresent the rotation angle and velocity, respectively, miis the mass of the gear, and g denotes gravitational force. Alternatively, the abbreviated form of the preceding Lagrangian vector in this definition is established as follows: ⃗L=M˙φ2−Gg(Φ). (43) For the sake of clarity, when solving each mechanical element separately using the Lagrangian equations as outlined in Definition 6, let’s address the Euler-Lagrange differential equation accord- ing to the following expressions: ∂⃗L ∂Φ=d dt ∂⃗L ∂˙Φi! . (44) 15 In the case of all elements eiundergoing rotational motion, we calculate the torques exerted on each element individually, considering their respective rotational dynamics and interactions within the system. This comprehensive analysis allows us to discern and quantify the rotational forces acting on each element, providing valuable insights into the system’s overall mechanical behavior. For the global conductor pinion, the dynamic expression is τ0=du dt I0˙φ0 +m0gr0cos(φ0) =I0¨φ+m0gr0cos(φ0). (45) Subsequently, for gear e1, τ1=du dt I1˙φ1 +m1gr1cos(φ1) =I1¨φ+m1gr1cos(φ1), (46) when we substitute φ1from equation (15), we obtain the following: τ1=I1 −r0 r1 ¨φ0+m1gr1cos −r0 r1φ0 . (47) Likewise, τ2=du dt I2˙φ2 +m2gr2cos(φ2) =I2¨φ2+m2gr2cos(φ2), (48) when we substitute φ2from equation (16), we obtain the following: τ2=I2 −r0 r1 ¨φ0+m2gr2cos −r0 r1φ0 . (49) Similarly, τ3=du dt I3˙φ3 −(−m3gr3cos(φ3)) = I3¨φ3+m3gr3cos(φ3), (50) where upon substituting φ3from equation (17), we obtain the following: τ3=I3r0n2 r1n3 ¨φ0+m3gr3cosr0n2 r1n3φ0 . (51) Subsequently, τ4=du dt I4˙φ4 −(−m4gr4cos(φ4)) = I4¨φ4+m4gr4cos(φ4), (52) and substituting φ4into equation (18), we get the following: τ4=I44r2 s4r0n2 (rs4+ra4)2r1n3 ¨φ0+m4gr4cos4r2 s4r0n2 (rs4+ra4)2r1n3φ0 . (53) Thus, for τ5, τ5=du dt2 2I5˙φ5 −(−m5gr5cos(φ5)) = I5¨φ5+m5gr5cos(φ5), (54) and substituting φ5into equation (19), we obtain the following: τ5=I54r2 s4r0n2 (rs4+ra4)2r1n3 ¨φ0+m5gr5cos4r2 s4r0n2 (rs4+ra4)2r1n3φ0 . (55) 16 Finally, for τ6, τ6=du dt2 2I6˙φ6 −(−m6gr6cos(φ6)) = I6¨φ6+m6gr6cos(φ6), (56) when we substitute φ6into equation (20), we obtain the following: τ6=I6 −4r2 s4r0r5n2 (rs4+ra4)2r1r6n3 ¨φ0+m6gr6cos −4r2 s4r0r5n2 (rs4+ra4)2r1r6n3φ0 . (57) Theorem 1 (Pitch mechanism torque) .The differential equation governing the total system of the pitch mechanism results in a second-order nonlinear differential equation, for which finding an analytical solution is nontrivial. Hence, the following law of dynamics for the mechanical system establishes τ¨φ0,φ0 for all simultaneous elements of the pitch controlled by the same independent control variables. It is expressed as follows:  τ0 τ1 τ2 τ3 τ4 τ5 τ6 = I0 −I1r0 r1 −I2r0 r1 I3r0n2 r1n3 4I4r2 s4r0n2 (rs4+ra4)2r1n3 4I5r2 s4r0n2 (rs4+ra4)2r1n3 −4I6r2 s4r0r5n2 (rs4+ra4)2r1r6n3 ¨φ0+ m0r0cos(φ0) m1r1cos(−r0 r1φ0) m2r2cos(−r0 r1φ0) m3r3cos(−r0n2 r1n3φ0) m4r4cos(K∆4r0n2 r1n3φ0) m5r5cos(K∆4r0n2 r1n3φ0) m6r6cos(K∆4r0r5n2 r1r6n3φ0) g. (58) Similarly K∆4=r2 s4 (rs4+ra4)2(59) Where K ∆4is the mechanical advantage given by the relationship between r sand r a. Formulating the Lagrangian operators according to the motion geometry established by the kinematic models of the yaw mechanism, we derive the following Definition 7. Definition 7 (Yaw mechanism energy model) .LetLirepresent the Lagrangian operator for each element comprising the yaw mechanism. This operator considers both the rotational kinetic en- ergy and the potential energy transmitted by a tooth of the preceding gear to the subsequent gear. The set of energy differentials is defined as: L19=1 2I19˙φ2 19−m19gr19sin(φ19) (60) L20=1 2I20˙φ2 20−m20gr20sin(φ20) (61) L21=1 2I21˙φ2 21−m21gr21sin(φ21) (62) L22=1 2I22˙φ2 21−m22gr22sin(φ22) (63) L23=1 2I23˙φ2 23−m23gr23sin(φ23) (64) L24=1 2I24˙φ2 24−m24gr24sin(φ24) (65) 17 By applying the same analytical method used in the previous analysis of the pitch mechanisms, and subsequently solving the Lagrangian equations, we obtain the torques for each element. τ19=d dt I19˙φ19 +m19gr19cos(φ19) =I19¨φ19+m19gr19cos(φ19). (66) Likewise, τ20=d dt I20˙φ20 +m20gr20cos(φ20) =I20¨φ20+m20gr20cos(φ20), (67) when substituting φ20into equation (21), we obtain: τ20=I20 −r19 r20 ¨φ19+m20gr20cos −r19 r20φ19 . (68) Moreover, τ21=d dt I21˙φ21 +m21gr21cos(φ21) =I21¨φ21+m21gr21cos(φ21), (69) and substituting φ21into equation (22), τ21=I21 −r19 r20 ¨φ19+m21gr21cos −r19 r20φ19 . (70) Furthermore, τ22=d dt I22˙φ22 −(−m22gr22cos(φ22)) = I22¨φ22+m22gr22cos(φ22), (71) when substituting φ22into equation (23), τ22=I22 −r19n21 r20n22 ¨φ19+m22gr22cos −r19n21 r20n22φ19 . (72) τ23=d dt I23˙φ23 −(−m23gr23cos(φ23)) = I23¨φ23+m23gr23cos(φ23), (73) when we substitute φ23into equation (24), τ23=I23r19r22n21 r20r23n22 ¨φ19+m23gr23cosr19r22n21 r20r23n22φ19 . (74) Finally, τ24=d dt I24˙φ24 −(−m24gr24cos(φ24)) = I24¨φ24+m24gr24cos(φ24), (75) and substituting φ24into equation (26) to obtain τ24=I24r19r22n21 r20r23n22 ¨φ19+m24gr24cosr19r22n21 r20r23n22φ19 . (76) 18 Theorem 2 (Yaw mechanism torque) .The differential equation describing the total system of the yaw mechanism results in a second-order nonlinear differential equation, for which finding an analytical solution is nontrivial. Hence, the following law of dynamics for the mechanical system establishes τ¨φ0,φ0 for all simultaneous elements of the yaw controlled by the same independent control variables, and it is expressed as follows:  τ19 τ20 τ21 τ22 τ23 τ24 = I19 I20 −r19 r20 I21 −r19 r20 I22 −r19n21 r20n22 I23 r19r22n21 r20r23n22 I24 r19r22n21 r20r23n22 ¨φ19+ m19r19cos(φ19) m20r20cos −r19 r20φ19 m21r21cos −r19 r20φ19 m22r22cos −r19n21 r20n22φ19 m23r23cos r19r22n21 r20r23n22φ19 m24r24cos r19r22n21 r20r23n22φ19 g. (77) Definition 8 (Prismatic Mechanism Energy Model) .LetL29denote the Lagrangian operator for each element of the prismatic mechanism, accounting for its rotational kinetic energy, as well as the potential energy transmitted from one gear tooth to the subsequent gear. The kinetic models are defined as: L29=1 2I29˙φ2 29−m29gr29sin(φ29) (78) Lc30=1 2Ic30˙φ2 c30−mc30grc30sin(φc30) (79) Lc35=1 2Ic35˙φ2 c35−mc35grc35sin(φc35) (80) LT36=1 2mT36˙d2 T36−mT36gdT36 (81) Upon solving the previously presented equations, we derive the torque values for each indi- vidual element: τ29=du dt I29˙φ29 +m29gr29cos(φ29) =I29¨φ29+m29gr29cos(φ29), (82) as well as τc30=du dt Ic30˙φc30 −(−mc30grc30cos(φc30)) = Ic30¨φc30+mc30grc30cos(φc30), (83) upon substituting φc30into equation (30), we derive the following expression: τc30=Ic304r2 s (rs+ra)2¨φ29+mc30grc30cos 4r2 s (rs+ra)2φ29! . (84) Subsequently, τc35=du dt Ic35˙φc35 −(−mc35grc35cos(φc35)) = Ic35¨φc35+mc35grc35cos(φc35), (85) where by substituting φ35into equation (32), we derive the following: τc35=Ic3516r4 s (rs+ra)4¨φc29+mc35grc35cos 16r4 s (rs+ra)4φ29! . (86) Finally, fT36=mT36¨dT36−mT36g=mT36L 16r4 s (rs+ra)4! ¨φ29−mT36g (87) 19 Theorem 3 (Torques and force of the prismatic mechanism) .The differential equation governing the entire prismatic mechanism system leads to a second-order nonlinear differential equation, the analytical solution of which presents a nontrivial challenge. Consequently, the following law of mechanical system dynamics defines τ¨φ0,φ0 for all simultaneously pitch-controlled elements using the same independent control variables, as expressed below:  τ29 τc30 τc35 fT36 = I29 Ic30 4r2s (rs+ra)2 Ic35 16r4s (rs+ra)4 0 ¨φ29+ m29r29cos(φ29) mc30rc30cos 4r2s (rs+ra)2φ29 mc35rc35cos 16r4s (rs+ra)4φ29 mT36L 16r4s (rs+ra)4φ29 g. (88) 6 Control module model In this section, we present a kinematic control model for the module, employing variational meth- ods to regulate the motion of coordinates with respect to all independent control variables (angu- lar). From this framework, laws governing the control of joint variables are derived. The position model deals with the coordinates of the module’s link within the workspace. Its representation relies on the instantaneous variation of joint variables: yaw, pitch, and elongation. Referring to the reference system illustrated in Figure 5b, the pitch angle corresponds to rota- tion around the x-axis, while yaw variation of the module is linked to rotation around the z-axis. Additionally, the prismatic elongation of the module’s link is denoted by dt. Through analytical deduction of the system’s geometry, Definition 9 is established using spherical coordinates. Definition 9. (Position Model) Let x, y, and z represent the coordinates in the workspace, and let A be the projection of d tonto the xy plane. The following spherical model relies on the three joint variables θ0,θ1, and d t, such that x=Acos(θ0) =dT36cos(θ0)cos(θ1), (89) likewise, y=Asin(θ0) =dT36sin(θ0)cos(θ1), (90) and z=dT36sin(θ1)). (91) Here, the term A =dT36cos(θ1)denotes the projection in meters. Derived from Definition 9, Proposition 4 intricately formulates the module’s position model, leveraging the nuanced functional attributes of the gears. Proposition 4. (Intrinsic Variables Model) The coordinates in the workspace are directly dictated by the intrinsic variables of the modular mechanism, establishing a fundamental relationship between the system’s internal parameters and its spatial positioning. θ0=φ24=r19r22n21 r20r23n22 φ19, (92) θ1=φ6=−4r2 s4r0r5n2 (rs4+ra4)2r1r6n3 φ0 (93) 20 and dT36=L 16r4 s (rs+ra)4! φ29 (94) Hence, upon substituting θ0,θ1, and d T36into the previously formulated equations, we obtain the following expressions: x=Acos(θ0) =L 16r4 s (rs+ra)4φ29! cosr19r22n21 r20r23n22φ19 cos −4r2 s4r0r5n2 (rs4+ra4)2r1r6n3φ0 ,(95) likewise, y=Asin(θ0) =L 16r4 s (rs+ra)4φ29! sinr19r22n21 r20r23n22φ19 cos −4r2 s4r0r5n2 (rs4+ra4)2r1r6n3φ0 ,(96) and z=dT36sin(θ1) =L 16r4 s (rs+ra)4φ29! sin −4r2 s4r0r5n2 (rs4+ra4)2r1r6n3φ0 . (97) Letp∈R3be defined as p= (x,y,z)T, and let ˙p= (˙x,˙y,˙z)Trepresent the vector of Cartesian velocities tangent to the module’s link. Define Φ= (θ0,θ1,dt)Tsuch that the angular velocities are denoted by ˙Φ= (˙θ0,˙θ1,˙dt)T. Based on the XYZ position model, the Jacobian matrix describing the evolution of all coordinates with respect to all independent control variables simultaneously is deduced as follows: Jt= −dT36sin(θ0)cos(θ1)dT36cos(θ0)cos(θ1)cos(θ0)cos(θ1) dT36cos(θ0)cos(θ1)−dT36sin(θ0)sin(θ1)sin(θ0)cos(θ1) 0 dT36cos(θ1) sin(θ1) . (98) It is characterized by its non-singular and invertible nature, this square matrix is also non-stationary. Proposition 5. (Kinematic Law) The relationship between Cartesian velocities and joint variables is expressed as a time-varying linear system. The gain for multiple outputs relative to multiple inputs is applicable in the direct solution model.  ˙x ˙y ˙z =Jt ˙θ0 ˙θ1 ˙dT36 . (99) Likewise, the inverse model  ˙θ0 ˙θ1 ˙dT36 =J−1 t· ˙x ˙y ˙z  (100) and defining the vector of module’s joint variables as ˙Θ=˙θ0,˙θ1,˙dT36T, and the vector of Carte- sian variables in the workspace as ˙p= (˙x,˙y,˙z)T. 6.1 Kinematic Control Law Based on Proposition 5, which encapsulates the direct and inverse kinematic laws of the proposed module in a single equation, the direct solution in terms of its first-order derivative is: 21 ˙p=Jt·˙Θ. (101) Expressing it as a first-order linear differential equation, we can solve it using the method of separation of variables and derivatives with respect to dt, as follows: dp dt=Jt·dΘ dt, (102) when integrating with respect to both differentials, Zpt+1 ptdP=J·ZΘt+1 ΘtdΘ, (103) This leads to a recursive expression. Drawing from Proposition 5 as a foundational element, Theorem 4 is articulated as follows. Theorem 4. (Kinematic control law) The theorem articulates the control law, which governs both the degrees of freedom and the control joint variables, as established by pt+1−pt=J·(Θt+1−Θt). (104) Where, (I) below, denotes the initial position control law employing the joint variables as input