Adaptive control for delta parallel robot

Adaptive control for delta parallel robot Điều khiển thích ứng cho robot song song tam giác Adaptive control for delta parallel robot

Reason for selection the topic

Industrial robots have revolutionized various sectors by enhancing productivity and reducing costs while improving product quality and competitiveness In the mechanical engineering industry, their applications are extensive, particularly in automotive engineering, welding, casting, metal spraying, disassembly, transportation of workpieces, and product assembly.

A parallel robot is a multi-body system characterized by a closed kinematic loop structure, where links are interconnected by movable joints Despite their complex design and control challenges, parallel robots offer significant advantages over serial robots, including the ability to handle large loads and exhibit high rigidity due to their geometric configuration They are capable of executing intricate operations with high precision Consequently, a thorough investigation into the dynamics and control of parallel robots is essential to fully leverage their benefits, making it a scientifically relevant and practically significant topic.

Research Motivation

Since their introduction, industrial robots have been utilized across various sectors to enhance productivity and reduce costs by replacing human labor Their primary objectives include improving product quality and competitiveness while boosting labor efficiency In the mechanical engineering field, these robots play a crucial role in automotive engineering, welding, casting, metal spraying, disassembly, workpiece transport, and product assembly.

A parallel robot is a multi-body system characterized by a closed kinematic loop structure, where links are interconnected by movable joints Despite their intricate kinematic design and challenges in control and design, parallel robots offer significant advantages over serial robots These include the ability to support heavy loads, enhanced rigidity due to their geometric configuration, and the capability to execute complex tasks with high precision.

Delta parallel robots are renowned for their speed and precision, making them ideal for applications like pick-and-place tasks However, to fully leverage their capabilities, advanced control strategies are essential due to the complex dynamics and control challenges they present.

This research focuses on developing adaptive control methods for Delta parallel robots, which are essential for managing uncertainties and variations in their dynamics and operating environments By thoroughly investigating adaptive control, we aim to maximize the advantages of Delta parallel robots while addressing practical deployment challenges This study is significant for the mechanical engineering industry, as it enhances the performance and applicability of Delta parallel robots, contributing to technological advancements.

Research Aims and Objectives

Robotic manipulators are mechanical structures used in various industries for repetitive tasks, such as manufacturing, flight simulation, medical procedures, micromotion, and pick-and-place operations They ensure high output while maintaining exceptional quality Industrial robots are categorized into serial and parallel manipulators based on their mechanical structure In serial manipulators, links are connected in a sequence, while in parallel systems, links connect a mobile platform to a fixed one, allowing for greater speed and acceleration However, designing control systems for parallel robotic manipulators presents challenges due to their multivariable nature, non-linear elements, and significant interaction among robot links Generally, the closed-loop kinematic chain mechanism is more complex for forward kinematics than the traditional open-loop mechanism, leading to extensive research on the kinematic and dynamic analyses of these systems.

One ofithe most encounterediproblems iniengineeringiapplications isi the challengesito be addressed The nonlinearitiesiof the parallel robots, structurediand unstructuredidynamic uncertainties and various operatingiconditions are amongithe typicalichallengesito beifaced [2]

To effectively regulate parallel robotic manipulators, various traditional control strategies have been utilized, including PD control, PD control with gravity compensation, and a combination of PD control with gravity pre-compensation, proportional control, and velocity feedback These methods partially rely on the dynamic model of the robotic system A significant factor contributing to the suboptimal performance of PD controllers is the arbitrary calculation of controller gains, often determined through trial and error Additionally, computed torque control, a widely adopted technique for parallel robots, depends on inverse dynamics and necessitates knowledge of the matrices that characterize the robot's dynamics.

Research contents

Adaptive control is an advanced method that has gained considerable interest recently for effectively managing the complexities of parallel manipulators, such as Delta robots Unlike traditional control strategies, adaptive control techniques can modify and refine control parameters in real-time, making them ideal for systems operating in dynamic and uncertain environments.

Adaptive control offers a significant advantage by continuously updating control parameters in response to the system's behavior, effectively addressing uncertainties and ensuring robust performance This adaptability allows the controller to achieve precise tracking, even as the system dynamics change over time.

Adaptive control strategies have been extensively researched to improve the performance of Delta robots These approaches typically integrate online parameter estimation algorithms with adaptive feedback control laws By continuously assessing the system's response and adjusting control inputs in real-time, adaptive controllers effectively reduce the impact of uncertainties and disturbances, leading to enhanced tracking accuracy and increased system robustness.

Adaptive control techniques provide the flexibility to integrate additional sensors or feedback signals, allowing controllers to adjust to varying operating conditions and enhance performance in real-time This adaptability is crucial in industrial applications, where accurate control is vital for processes like assembly, manipulation, and machining.

Research methods

Adaptive control is an effective strategy for enhancing the agility and robustness of Delta robots, particularly under uncertain and dynamic conditions By continuously adjusting to system and environmental changes, adaptive controllers significantly improve the performance and versatility of parallel manipulators, facilitating their broader implementation in industrial applications.

Recent advancements in computational intelligence control methods, particularly fuzzy logic systems (FLS) and artificial neural networks (ANN), have been applied to trajectory tracking control problems A study demonstrated the use of a fuzzy controller to manage the trajectory tracking of a Delta parallel robot, where the controller's parameters were optimized using a particle swarm optimization algorithm While the fuzzy logic controller showed good performance in a closed loop, challenges such as unclear defuzzification operations and high computational demands were noted On the other hand, ANN, particularly adaptive B-Spline neural networks (BSNN), have been utilized for recognizing parameters in nonlinear systems and designing adaptive control systems The BSNN, characterized by its two-layer structure, uses B-Spline functions in the input layer and a linear neuron in the output layer, effectively reducing trajectory tracking error by adjusting the gains of three PD controllers to regulate position and orientation.

Thesis layout

The thesisiincludes thei following imain parts:

Chapter 1 provides a comprehensive literature review on the dynamics and control of spatial parallel robots, examining both domestic and international research This analysis informs the selection of a thesis direction that is both scientifically significant and highly applicable in practical scenarios.

Chapter 2: Theirequired mathematicaliframework formithe inverseikinematicsihas beeniprovided

Chapter 3: Hasibeen devotedito scrutinizingithe controllers’idesigniprocess; subsequently, theiefficiency andirobustness of controllersihave been analyzedi with the aim ofievaluating theitracking performancesiin complicatedipaths andisurmounting unmodeledidynamics

Simulationiof Delta robotihas beenicarried outiiniChapteri4,iwhere the aforementionediapproaches wereiimplementedion the simulated manipulator.

LITERATURE REVIEW OF THE PROBLEM OF

Robots have a parallel structure

Robots have evolved significantly due to their diverse applications, with the concept of parallel-structured robots first introduced by Gough and Whitehall in 1962 The application of these robots gained traction when Stewart developed a flight simulator using a parallel mechanism in 1965 Today, parallel robots have advanced considerably and are now capable of achieving six degrees of freedom (6 dof).

A parallel-structured robot features a movable platform connected to a fixed base, driven by multiple parallel branches, or legs The number of legs typically matches the number of degrees of freedom, with actuators positioned on either the fixed base or the legs themselves As a result, these robots are often referred to as platform robots.

Figure 1 1: Aischematic viewi of Delta iparallel robot includingi1 : Fixed-base,i 2:

Actuator, 3 : Rear-arm,i4 : Forearm,i5 :itraveling-plate.

Compare serial robots and parallel robots

Traditional robot design typically features a series of rigid links connected by rotating or sliding joints, with one end fixed as the base and the other equipped with an operating end, resembling a human arm While this serial robot configuration offers a large working space, it suffers from limited load-bearing capacity, making it large and costly due to the need for link stiffness and effective force transmission These robots can be categorized into spatial and planar operations, with the Puma robot exemplifying spatial manipulation The load capacity of such manipulators is constrained by the driving torque of the motors, resulting in a cumulative load effect where each joint's load increases the burden on the previous joints Consequently, these robots exhibit low resistance to inertia, leading to reduced operational speeds and accelerations A review of technical specifications reveals that many of these robots are heavy and large yet only capable of handling lightweight objects, with transmission errors compounding at each operational stage.

Parallel robots offer exceptional rigidity and load-carrying capabilities due to their actuators working in tandem to support the load Their design ensures high positional accuracy, as errors from each leg are averaged out rather than compounded, unlike in serial robots Additionally, while kinematic chains impose spatial constraints, conventional designs of parallel robots exhibit low inertia properties, enhancing their performance.

Parallel robots are utilized in various fields, such as CNC machining, high-precision manufacturing, automation in the semiconductor industry, and high-speed electronics assembly A comparison between serial and parallel robots highlights their distinct advantages and applications, underscoring the importance of selecting the appropriate robot type for specific tasks.

Table 1: Compare serial robots and parallel robots

No Feature Serial robot Parallel robot

4 Load/mass ratio Lower Higher

5 Inertial load More Less than

7 Design/control complexity Simple Complex

The Stewart platform, a highly studied type of parallel robot, features a stable octagonal base with a triangular top and a pedestal below, all connected by six legs, providing six degrees of freedom This design is particularly beneficial for applications such as flight simulators and recreational seating, where controlled acceleration is essential despite large loads and limited space In contrast, achieving similar functionality with a six-degree-of-freedom chain robot is challenging due to the substantial size required for the base actuator to support other components and the load The original concept of the platform was introduced by Gough.

[20] in 1949 to test tires and later applied by Stewart [22] in flight simulators Since then, many variations have been proposed by different authors and they are called “Stewart Platforms”.

Applications of parallel robots

Parallel robots have been widely applied in life Some specific applications include:

In 1949,i Eric Gough iintroduced the basiciprinciple and developedi a device i called:

The Universal Tire-Testing Machine, also known as the Universal Rig, was utilized for tire testing by Dunlop and officially began operation in 1955 This innovative device features a hexagonal moving plate, with each corner linked to translational drive stages through ball joints, while the impact links connect to the base via Cardan joints, allowing for variable link lengths due to its linear drive mechanism The machine remained in use until 2000 and is currently displayed at the British Science Museum.

Figure 1 4: Delta robot applied in food technology [23]

The Delta robot, developed by Reymond Clavel in the early 1980s, utilizes parallelograms to achieve three degrees of translational motion and one degree of rotational motion This innovative design has led to the granting of 36 patents, including significant ones from WIPO, the US, and Europe Delta robots are primarily employed in food packaging lines and various lifting applications A recent survey by Corves provided a comprehensive analysis of the different variations, application areas, and market assessments for Delta robots.

Elekta Company (Sweden), a company specializing in medical equipment, used the Delta robot to make a device to hold a 20 kg microscope used in surgery (Figure 2.4)

Figure 1 5: SuriScope product in action, Humboldt University (Berlin, Germany)

A European project to create the CRIGOS (CompactiRobot foriImageiGuided Orthopedic)irobot uses theiGough-Stewart mechanismito provideidoctors with a high- performanceibone surgeryitool (Figure 2.5)

Figure 1 6: CRIGOS robot is used for bone regeneration surgery

Delta robots are widely utilized in high-speed pick and place operations, including sorting, packaging, and assembly of small components Their ability to swiftly and accurately transfer items between locations is crucial for industries such as electronics and food processing.

Delta robots play a crucial role in the packaging industry by efficiently performing tasks like filling, sealing, and boxing Their swift and precise handling of objects enhances productivity on packaging lines, leading to increased throughput and reduced labor costs.

Delta robots play a crucial role in product assembly, especially within the electronics and automotive sectors Their ability to handle delicate and small components with exceptional precision ensures that each part is accurately positioned and assembled, enhancing overall efficiency and quality in manufacturing processes.

Delta robots, integrated with advanced vision systems, excel in inspection and quality control by swiftly detecting defects and inconsistencies in products This capability ensures that only items meeting stringent quality standards proceed through the production process.

In 3D printing technology, Delta robots are leveraged for their unique structural advantages, which include high speed, precision, and a large build volume relative to their footprint

Figure 1 8: Print head of 3D printer

Delta robots offer significantly faster print speeds than traditional Cartesian 3D printers, thanks to their lightweight arms and efficient movement patterns This speed advantage makes them ideal for rapid prototyping and the production of parts that require quick turnaround times.

Delta 3D printers are designed with a large cylindrical build volume, enabling the creation of taller objects without needing additional space This unique feature is particularly beneficial for producing items like vases, statues, and other vertical structures.

Delta robots offer smooth and precise motion, ensuring high-quality prints with enhanced detail and minimal defects Their exceptional precision is especially advantageous for handling complex geometries and intricate designs.

Delta robots are engineered to significantly reduce vibrations and oscillations during the printing process, enhancing overall print quality Their stable platform ensures precision and accuracy, making them particularly effective for high-resolution prints.

Delta 3D printers offer a flexible and adaptable design, enabling easy customization for diverse printing requirements This versatility supports the integration of various print heads and materials, along with enhanced features like multi-material printing and dual extrusion capabilities.

Delta robots feature a compact and vertical design that optimizes space utilization in workshops and production settings, making them ideal for small businesses and hobbyists looking to maximize their workspace.

Delta robots can be seamlessly integrated into automated production lines, facilitating continuous 3D printing By collaborating with other robotic systems, they enhance efficiency in tasks like part removal, quality inspection, and post-processing, creating a fully automated workflow for 3D printing operations.

Some related studies

A parallel robot is a complex mechanical system characterized by its loop structure, necessitating dynamic calculations for effective design and control quality enhancement The existing literature extensively covers the theoretical frameworks and calculation methods for the dynamics of string robots, providing a wealth of information on establishing dynamic equations for multibody systems with loop configurations.

[31, 32] After that, the problem of kinematics and dynamics was mentioned more specifically in the documents on parallel robots [21, 31]

Wittenburg [32] was one of the first to propose the use of structure separation concepts to solve the dynamics of many-body systems with a loop structure Nakamura

Schiehlen and others have utilized active joints to parameterize the configuration space in addressing the dynamics of parallel robots Various approaches have been analyzed to tackle the dynamic challenges associated with these robots, employing different motion equations Notably, some researchers have applied the Lagrange-Euler equation in conjunction with the virtual work principle to formulate solutions Additionally, Staicu and colleagues introduced a recursive matrix method to further advance the understanding of these dynamic problems.

Recent research by Corves's group explored the inverse dynamics of Delta parallel robots through three methods: Lagrange multiplier, virtual work principle, and Newton-Euler equations, finding that numerical solutions yielded similar results, with the factorized Lagrange method demonstrating the shortest computational time S B Park utilized a model of two particles at the ends of the parallelogram of the Delta robot to derive the equations of motion and calculate inverse dynamics using the multiplicative Lagrange method Additionally, Q Zhang employed ANSYS software to assess the robot's durability, deformation, and natural oscillation frequency, while R Kelaiaia focused on static deformation analysis for deformation compensation to enhance the robot's accuracy.

The 3-PRS parallel robot, developed by M.S Tsai's research team, addresses the forward kinematics problem using numerical methods and the inverse dynamics problem through the multiplier Lagrange method Additionally, model-based control methods have been experimentally applied to real robots, paving the way for further advancements by authors Q Xu and Y Li.

52] further developed a model-based control method using neural networks Recently, a variant of this robot was also developed by H Sun [53] and Q Li [58]

Mueller's group has built a model of a residual parallel robot with PKM drive [54-

57] In which, the inverse dynamics problem is solved using numerical methods and numerical simulation of model-based control methods

In the literature, researchers often derive equations of motion by employing simplified assumptions, such as decoupling specific joints and modeling the system as a tree structure with constraint equations This approach, known as subsystem decomposition, applies dynamic principles to tree systems while neglecting the torques at the decoupled joints.

Recent studies on spatial Delta parallel robots have employed various methods to derive the equations of motion, including the Lagrange multipliers method, the principle of virtual work, Newton-Euler equations, and subsystem decomposition In these studies, the connection between the driving links and the moving platform is typically represented as either a homogeneous link or as two concentrated masses at the ends However, there has yet to be a comparative evaluation of these two modeling approaches.

1.4.2 Trajectory Tracking Control of Parallel Robots

The literature on robot control is extensive, with various approaches discussed by Spong and Vidyasagar, as well as Sciavicco and Siciliano However, these studies often overlook the unique challenges associated with parallel robots Research by Ghorbel and Murray indicates that the dynamic equations governing parallel robots closely resemble those of serial robots, allowing many control algorithms for serial robots to be applicable to parallel systems Despite this, there remains a gap in developing methods that leverage the distinct advantages of parallel robots Early efforts by Nakamura introduced driving surplus coordinates to optimize objective functions, while Kock advanced the field with concepts like dynamic stiffness control and preloading Additionally, F Aghili proposed a model-based control method utilizing dependent coordinates, and Harada implemented an impedance control strategy in the task space for coordinating two cooperating parallel robots through the pseudo-inverse of the Jacobian matrix to manage control forces Currently, trajectory tracking control of the end-effector in parallel robots is a significant area of focus.

Recent studies have focused on enhancing Delta robot control quality by developing control laws derived from simplified dynamic models, where each parallelogram link is represented by two point masses at its ends Linearization techniques have been employed to create straightforward control laws B Hemici and colleagues designed PID and H∞ controllers based on this linear model for robust trajectory tracking Similarly, A Mohsen combined PD and PID control laws with fuzzy logic for effective end-effector trajectory tracking Kenmochi utilized Lagrange equations to derive motion differential equations and applied exact linearization for robust position control Canahure developed PD and Linear Quadratic Regulator (LQR) control laws based on inverse kinematics, although significant deviations from the desired trajectory were noted This simplified dynamic model was also explored by J Du and others.

Y Lou [73] for real-time robot control using the inverse dynamics method PID control adaptive control using neural networks were also applied for trajectory tracking control of Delta robots [76-79]

Various studies employ different controllers to direct the end-effector along a predetermined trajectory Although these controllers typically meet the established criteria, there is a lack of comparative research that evaluates their performance and provides recommendations for optimal usage in specific scenarios.

In summary, there are currently not many studies on adaptive control for Delta parallel robots.

Research in Vietnam

Author Nguyen Van Khang explores the complexities of kinematics and dynamics in multi-body systems with closed-loop structures, detailing methodologies for formulating dynamic equations using the Lagrange equations of the second kind For parallel robots, he employs the subsystem decomposition and Lagrange multipliers methods to derive equations of motion Additionally, he presents two numerical methods for solving the inverse dynamics problem of parallel robots, which are based on Lagrange multipliers and reduced differential equations in minimal coordinates, the latter being particularly useful for robot control Nguyen Van Khang has also developed three algorithms to convert differential-algebraic equations into ordinary differential equations, focusing on the forward dynamics problem His research group has further investigated inverse kinematics and sliding mode control using neural networks for redundantly actuated robots, with findings published in relevant studies.

Nguyen Quang Hoang's research group has developed a stabilization solution for the constraint equations of multi-body systems with closed-loop structures, utilizing the sliding mode principle Additionally, they have introduced numerical methods to effectively address the inverse kinematics problem in both parallel and serial robots.

The research group of Chu Anh My [85-87] analyzed the design and kinematics of a hybrid parallel-serial robot for applications in welding

The research group led by Pham Van Bach Ngoc has developed a parallel Hexapod robot designed for mechanical processing applications, enabling traditional machine tools to achieve complex surface details previously deemed unattainable This innovative device has been successfully manufactured and is currently showcased at the Institute of Mechanics Additionally, the group has made significant strides in creating a control method for a three degrees-of-freedom translational Delta parallel robot.

Le Hoai Quoc and his research team employed screw theory and Plücker coordinates to investigate the singular configurations and load capacity of parallel manipulators featuring externally distributed actuation legs within their workspace.

The research group led by Nguyen Xuan Vinh and colleagues [15] identified a set of special configurations of parallel manipulators using screw theory

The authors Tuong Phuoc Tho [16, 88] designed mechanical structures for a specific model of Delta robot, based on which they calculated inverse kinematics to facilitate robot control

Nguyen Minh Thanh's research group [17, 18, 89] modeled and analyzed factors influencing workspace during the modeling process Additionally, the group proposed an optimization method for parallel manipulator design using genetic algorithms

Authors Trinh Hoang Kien and Pham Huy Hoang [19] simulated the workspace and inverse dynamics of a planar three-degree-of-freedom parallel robot, specifically the 3RRR mechanism

Current research primarily emphasizes addressing kinematic challenges, formulating motion equations, and developing methods for their resolution However, there is a noticeable lack of focus on adaptive control issues related to parallel delta robots in the existing literature.

Identifying the research problem of the thesis

This thesis presents an adaptive neural network controller designed to effectively track the trajectories of a delta parallel robot (DPR) with three degrees of freedom Notably, this approach does not rely on a detailed mathematical model or precise parameter values of the DPR The performance of the proposed adaptive controller was evaluated against a traditional PD+GI controller, with both analytical and numerical results demonstrating the robust and efficient capabilities of the adaptive neural network controller.

Conclusion of chapter 1

The dynamics and control of parallel spatial robots present complex challenges that necessitate further research These robots possess distinct advantages over serial robots, which has piqued the interest of scientists This study synthesizes findings from various domestic and international sources to address a key research problem: enhancing control quality for parallel robots The focus is on improving adaptability and operational efficiency in diverse and uncertain working conditions By implementing adaptive control, robots can automatically adjust their control parameters based on real-time sensor data and system feedback, thereby reducing errors and enhancing stability and accuracy during operation.

THEORETICAL MODELING OF DEVELOPING

Model analysis of delta parallel robots

Figure 2 1: Scheme of theiDelta robot The structureiof the robotiincludes the followingicomponents (Figure 3.1 and

- Three active links A1B₁, A2B₂, A3B₃, connected toithe fixed base by rotaryijoints and driveniby three motors, which are mounted on the fixed base (1)

- Three passive links B1D₁, B2D₂, B3D₃, each forming a parallelogram structure

Due to the properties of the parallelogram links, theimoving platform isia rigid bodyithat translates inispace Thus, theirobot has threeidegrees of freedomidetermined by theithree coordinates (θ₁, θ₂, θ₃) inithe jointispace

The robot model depicted in Figure 3.1 features parallelogram links, which complicate direct kinematic calculations To streamline this process, we create a kinematic model by substituting the parallelogram links with single rods of equal length These rods are attached to the robot's driving links via Cardan joints at points B₁, B₂, and B₃, while their other ends connect to the moving platform using spherical joints at points D₁, D₂, and D₃, as illustrated in Figure 3.2.

Based on the kinematic model of the robot, we define the coordinate systems by establishing a fixed coordinate system Ox y z 0 0 0 attached to the stationary base, with the origin O located at the center of this base In this configuration, the Oz 0 axis is oriented vertically upward, the Ox 0 axis extends through A1, which is the center of the driving motor for leg 1, and the Oy 0 axis is arranged to ensure that the coordinate system is right-handed.

Three additional fixed coordinate systems Ox y z i i i i(i=1, 2, 3) are chosen to analyze the legs A1B1D1, A2B2D2, A3B3D3 of the robot In each system, the Oz i axis coincides with

The Ox, Oy, and Oz axes form a right-handed coordinate system with the Oz axis intersecting at point Ai Alongside the original fixed coordinate system (R0), there exist three additional fixed coordinate systems (i, i, i).

Ox y z (i=1, 2, 3) denoted as ( ) ( ) ( ) R 0 ' , R 0 '' , R 0 ''' These systems ( ) ( ) ( ) R 0 ' , R 0 '' , R 0 ''' are created by rotating the system ( ) R 0 around the Oz 0 axis by angles  i (where  i are 0, 120, and

Analysis of Link Movements

From Figure 3.2, we observe that the robot has a total of seven moving links, including:

The three driving links A1B₁, A2B₂, and A3B₃, each with a length of L1, rotate around an axis perpendicular to the Oxz plane at point Ai To analyze their movements, we utilize the angular coordinates θ₁, θ₂, and θ₃ The traveling platform (B) acts as a rigid body that translates through space, necessitating the examination of point P, the center of the moving platform The coordinates of point P in the reference system (R₀) are denoted as Xp, Yp, and Zp.

Reverse kinetics

This paper presents a reverse kinetics approach to achieve the desired angular position of an actuator based on a specified endpoint in Cartesian space Utilizing a geometrical method, this model does not take into account factors such as weight and moment of inertia.

Forward kinetics

A forward motion is used to reach the end position using the upper leg angles The representative geometry of a robot leg is shown in Fig 3.3:

1 1 1 cos cos cos sin sin i i i

To determine the endpoint position P(x P, y P, z P) of a delta robot, we first input the angles (θ i and α i) to calculate the position B 1 (x B1, y B1, z B1) using equation (3.4) Subsequently, we substitute this result into the equation to find the coordinates of point D 1 (x D1, y D1, z D1) By constructing the forward kinematics model of the delta robot in Simulink with the specified angles (θ1, θ2, θ3), we can accurately derive the endpoint position P(x P, y P, z P) of the delta robot.

Dynamicimodel of Delta Parallel Robot

The dynamic analysis of parallel robots is more complex than that of serial robots due to the influence of closed-loop chains within the system This dynamic model connects the robot's movements to the torques generated by the actuators Due to the manipulator's intricate kinematics, the motion equations for the actuating torques are derived using the Lagrangian method, which reformulates classical mechanics by integrating the principles of momentum and energy conservation.

   (2.5) where Q j is the function of generalized pair, i is the index of the number of constraints,

L is Lagrangian function, j index ofigeneralized coordinate andi n is the numberiof generalized coordinates q j iare the generalizedicoordinates j q, j are theigeneralized coordinates derivatives,i i denotes theiLagrange multipliers,  i is theikinematic constraint equation

In the dynamic analysis of the Delta parallel robot, three generalized coordinates are employed, reflecting its three degrees of freedom However, due to the complexity of its kinematics, three additional redundant coordinates are incorporated, resulting in a total of six generalized coordinates (n = 6).

Figure 3.3 presents the schematic of aikinematic chain, which aids in formulating the dynamic equations of the Delta parallel robot

Figure 2 3: Schematic diagram of Delta Parallel Robot

To establish the differential equations governing the dynamics of the Delta parallel robot, we combine two systems of equations derived from (3.1) The first system incorporates the Lagrange multipliers as unknown variables.

 (2.7) where ˆQ j represents a generalized external force applied to the system The second system of equations pertains to the actuation forces and can be expressed as:

0 cos cos cos cos sin sin sin sin sin 1,2,3 i bi x i i i i ai i i y i i i i ai i i z ai i bi

(2.9) where L ai ,L bi are the lengthsiof each armiand lower armirespectively; a b i , i are theiradii of the fixed andimobile platforms, respectively;i i is the separationiangle between each kinematicichain of theiDelta Parallel Robot ( 1 =0 ,   2 0 ,   3 $0  )

2.5.1 Kinetic energy of Delta Parallel Robot

Theitotal kinetic energyiof the DeltaiParallel Robot isicalculated usingithe following expression:

The kinetic energy of each arm in a Delta Robot is represented by K_ai, while K_p denotes the kinetic energy of the mobile platform, and K_bi refers to the kinetic energy of each forearm These energies can be calculated using specific equations that define their relationships and contributions to the overall system dynamics.

The massiof the mobileiplatform is m p

Where I i isithe inertia momentiof each upperiarm, I mi is theiinertia moment ofieach motor, and whoseimathematical expression canibe expression:

The massiof each upperiarm is m ai Thereforei(3.8) can presented as:

Theikinetic energy ofithe lower armiis represented below:

The Delta Parallel Robot features angular velocity for each actuator and incorporates lower arm mass, enabling it to execute both translational and rotational movements within a specified workspace This capability is enhanced by the inclusion of passive spherical joints in the forearms, contributing to its versatility and efficiency in various applications.

2.5.2 Potential energy of Delta Parallel Robot

The delta robot operates in three-dimensional space, requiring a formulation of the system's potential energy (U) that accounts for vertical movement and gravitational acceleration Unlike planar robots, which have a potential energy of U = 0 due to their restricted horizontal motion, the potential energy of the delta robot is defined by a specific equation that reflects its unique operational dynamics.

The potential energy of the forearm, denoted as U_bi, along with the potential energy of the arm, U_ai, and the potential energy of the mobile platform, U_p, can be represented by the equation (2.16) To determine the necessary energy magnitudes, one can refer to the derivation outlined in expression (3.12).

U = −m gp (2.17) ai ai ai sin i

2.5.3 Lagrange function of Delta Parallel Robot

By substituting equation (3.8) and (3.12) intoiLagrangian (L = K – U ) of ai parallel robot ofi n degree of freedom gives the following equation:

We obtain the set of equations to be solved to calculate the Lagrange multipliers

By substituting (3.16) into (3.3), defined in (3.1)

2 i x i cos i i cos i ai cos cos i i p 3 bi x px i p b a L m m p f

2 i y i sin i i sin i ai sin i cos i p 3 bi y py i p b a L m m p f

2 i z ai sin i p 3 bi z p 3 bi c pz i p L m m p m m g f

Where f px , f py , f pz iare the componentsi x y z, , of aniexternal force onithe mobile platform.iSolving (3.4) givesithe set ofidifferential equations describingithe dynamic behavioriof the DeltaiParallel Robot, asifollows:

3 3 3 1 cos cos cos a b a a b a a b a m m gL m m gL m m gL

Conclusion of chapter 2

In this chapter, we created a dynamic model for Delta parallel robots, emphasizing the complexities of their multi-body systems with closed-loop structures We tackled the challenges of formulating motion equations by employing multi-body dynamics software tools, which facilitated the development of differential-algebraic equations for the robots that would be difficult to derive explicitly.

The chapter provides a structural analysis of Delta parallel robots, focusing on their components and degrees of freedom By simplifying the actual model into a kinematic model, we enhance the analysis of the robot's movement This kinematic model substitutes the parallelogram links with single rods of equivalent length, enabling a clearer examination of the robot's spatial translation.

We developed a dynamic model of the Delta parallel robot using the Lagrangian approach, integrating classical mechanics with energy conservation principles This formulation included three generalized coordinates and extra redundant coordinates to address the robot's kinematic complexity The outcome was a set of coupled differential equations that accurately described the robot's dynamics, encompassing actuator torques and external forces influencing the system.

The kinetic and potential energy equations for the Delta parallel robot were established, considering the mobile platform, upper arms, and lower arms These energy equations were incorporated into the Lagrangian function, creating a detailed framework for examining the robot's dynamic behavior.

This chapter lays a strong groundwork for comprehending and modeling the dynamics of Delta parallel robots, paving the way for deeper exploration and optimization in later sections The methods and equations introduced are essential for enhancing the design and control of these intricate robotic systems.

ADAPTIVE CONTROLLER FOR DELTA

Adaptive Speed Control for DC Motors

An innovative adaptive control technique utilizing iB-Spline Neural Networks is introduced for effectively tracking angular speed reference trajectories in direct current motors This method generates control signals without the need for a detailed mathematical model or precise parameters of the nonlinear dynamic system The robust adaptive tracking control scheme relies solely on velocity output signal measurements, removing the requirement for real-time acceleration, current, or disturbance signal estimates Experimental results confirm the efficiency and robustness of this approach under challenging motor operation conditions, such as variable-speed reference trajectories and unknown load torque Laboratory tests on a direct current motor validate the feasibility and effectiveness of the proposed adaptive output feedback trajectory tracking control method.

DC motors come in various configurations, with one specific type providing notable operational benefits by maintaining a relatively constant rotor speed, even when load torque fluctuates up to its nominal value The connection diagram of the motor under investigation is illustrated in Figure 3.1.

Figure 3 1: Direct current motor equivalent circuit

In the field winding, key elements include field winding inductance (Lf), field resistance (rƒ), external variable resistance (rfx), field current (if), and field voltage (uf) For the armature winding, the primary components are armature winding inductance (Lɑ), armature resistance (Rɑ), armature current (iɑ), and armature voltage (uɑ) Additionally, mutual winding inductance (Lɑf) and rotor speed (ω) are also critical parameters in the system.

The voltage source provides power to both the field winding and the armature winding, resulting in a total current that is the sum of the two circulating currents By analyzing the voltage-current relationship for resistive and inductive elements, and grouping the field resistive elements, we derive the DC motor model.

The relationship betweenielectrical andimechanical systems isidetermined by af f a L

The equation J d b L i i dtω = − ω + − τ describes the dynamics of a rotor, where J represents the moment of inertia, b is the viscous damping coefficient, and τ L i is the load torque The electric torque is expressed as τ e = L i i a f f a This mathematical model illustrates a coupled nonlinear dynamic system, indicating that traditional linear controllers can only ensure satisfactory performance near specific equilibrium points By analyzing the mathematical model, the equilibrium points of the system can be identified.

2 2 2 f f af L in a af in a f bR R L u i L u bR R

= − + (3.6) wherei u in = u a = u f Theiequilibrium points areiinfluenced by theiapplied voltage atithe terminals, theiload torque, andispecific motor parameters.iAs a result,ithe motor demonstratesiseveral equilibrium statesicorresponding to differentidesired operational scenarios

Inithis subsection, aicontemporary controlistrategy utilizingisliding mode is examined Slidingimode controliis renownedifor its robustnessiagainstiparametric uncertaintiesiand externalidisturbances [96]

To start, theispeed trackingierror variableiis definedias e=  d − , wherei d denotesithe referenceishaft speed Foricontroller design, weiintroduce the stateivariables z 1 =e andi z 2 = e A controlistrategy for theiDC motor employingisecond-order sliding modesican be outlinedias follows:

 = − (3.8) with u 1 > 0, u 2 > 0 iand s = kz 1 +ż 1 , whereik > 0 is theicontroller gain Noteithat the sliding surface i s necessitatesithe availabilityiof the accelerationisignal asiwell

Upon differentiatingi s with irespect to timeiand incorporating Equationsi (3.7) and

(4.8), theievolution of theisliding variableidynamics is described by

1 1 1 2 a f dz b z b z t cu u dt = − − + − (3.10) wherei 1 , af f a b L b c

 = + + + − − − −  (3.11) wherei b i , i = 2, ‚5 ,iare constantsidetermined byiuncertain motoriparameters The system'sibehavior in slidingimode remains independentiof theseiparameters b i , c,iand disturbancesiin  The iLyapunov function employedito ascertain theistability of the proposedisystem is detailedi in [96]

The proof of stability requires calculating the time derivative of the Lyapunov function and the second derivative of the sliding variable to assess the impact of uncertainty This analysis reveals that the trajectory of the uncertain state in the phase plane spirals towards the origin, ultimately converging to it within a finite time frame.

To implement the super-twisting sliding mode controller effectively, it is essential to solve Equations (3.7)-(3.11) simultaneously at each sampling instance, relying on real-time rotor speed data To reduce voltage fluctuations, integrating a low-pass filter into the control signal is beneficial Additionally, utilizing the concept of equivalent control requires a thorough understanding of all motor parameters.

3.1.3 Adaptive B-Spline Controller Design for DC motors

The complexities of DC motors and the limitations of traditional linear controllers make it challenging to effectively regulate key variables To overcome this, robust control laws are essential for accurately tracking speed trajectories over time This work proposes the use of neural network-based architectures to enhance system robustness, alongside B-spline methods, which are known for their efficiency in addressing both theoretical and practical problems with speed and precision.

The adaptiveicontroller designibased on B-splineineuralinetworks comprisesitwo maini stages:

1 Initial stage involvesidefining the structureiand characteristics ofiinputs and theitraining rules

Adaptive algorithms have increasingly replaced or improved traditional methods in various applications A notable example is the detailed training of neural networks for power electronics, which is presented in [100] The architecture of a B-spline neural network (BSNN) is depicted in Figure 4.2, showcasing a single layer with two basis functions that emphasize the essential elements of an adaptive scheme This design comprises an input space, a layer of basis functions with output weights, and the final BSNN output Key parameters such as inputs, the number and shape of basis functions, and the learning rate are established prior to configuration Once set up, the BSNN exhibits strong performance under different operational conditions, as confirmed by offline validation using data collected from a DC motor.

The adaptiveicontroller adopts a similariconfiguration as depicted iniFigure 3.2:

1 Aniinput (error betweeniactual and reference speed)

2 Twoithird-order basis functionsidepicted in Figure 3.2,iassuming normalized inputi signals

3 Twoiweights that updateiat each samplingiinterval

4 Anioutput derived fromithe sum ofitwo components: theiproduct of basis functionioutputs and theiricorrespondingiweights

Figure 3 2: The structure of daptive controller [101]

The off-line training procedure for the B-spline neural network (BSNN) is illustrated in Figure 3.3 This stage involves defining the key elements of the BSNN structure using input-output data (voltage-speed) from a DC motor Initially, a layer with two basis functions is established, as shown in Figure 3.2, due to its effectiveness in controlling a single error signal and favorable performance criteria observed in previous studies The shape of these basis functions ensures a smooth output response from the neural network While various higher-order shapes were evaluated, similar performance was achieved, albeit with increased computational requirements.

The number of weights corresponds to the number of basis functions, with a consistent learning rule applied across all scenarios Initially, a low learning rate, typically set at 1x10^-3, is used for stability during experimental trials, and it can be adjusted iteratively for improved neural network response without sacrificing stability Off-line training experiments are conducted under specific operating conditions, as detailed in Table 2, ensuring the input voltage for the motor ranges from 0 to 110 V The BSNN's successful performance is evaluated based on whether its output aligns with expected values under these conditions During this stage, parameters such as the number of inputs, learning rate, number of basis functions, and their shape can be adjusted as needed.

The second loop of the offline training phase focuses on optimizing the motor's closed-loop response by subjecting it to various operational conditions with the adaptive controller Key adjustments aim to refine the learning rate while ensuring stable learning conditions, ultimately achieving satisfactory performance in settling time and overshoot Continuous adaptation during online operation is essential to accommodate new operational or parametric variations of the motor, as demonstrated in this study.

In this research, the rotor speed deviation is the sole input for the BSNN, facilitating efficient control signal generation Three configurations were examined to define basis functions: (a) two multivariable functions of three and four orders, (b) multivariable functions of two and three orders, and (c) single-variable functions of three orders Each configuration shows different performance levels, with computational demands varying based on the type and order of the basis functions This ANN structure has proven effective in other dynamic systems, as supported by prior studies.

[102–104] Theidataset utilized inithis study, detailed iniTable 2, was chosenibecause represents a typical input-output relationship observed in DC motor operations

Figure 3 3: Flowchart depicting the offline training process of the neural network controller

Table 2: Operationiconditions for off-lineitraining

Load i Torque  L (Nm) Rotor i Speed  (rad/s)

The proposed controller focuses on achieving robustness, design simplicity, and ease of experimental implementation, all of which are supported by laboratory results Figure 3.2 illustrates the structure of the proposed neural controller, with the output defined according to reference [98].

In the neural network structure, the output is calculated using the equation \( a = \sum w_i \cdot f_i \), where \( w_i \) represents the weighting factors and \( f_i \) denotes the outputs of the basis functions With only two weights in this case, the number of weights \( p \) is set to 2 The output of the basis functions exhibits a nonlinear relationship with the input values, which is influenced by the specific shape of the functions employed The proposed controller incorporates two third-order monovariable functions, and the weight vector is updated according to an instantaneous learning rule as defined in the referenced literature.

Delta Parallel Robot Apdaptive Controller

Section 4.1 discusses the application of controllers based on artificial neural networks (ANNs) for the Delta Parallel Robot, which is powered by three DC motors These ANNs effectively manage nonlinear systems, tackling issues like parametric uncertainty, unknown dynamics, and significant coupling among state variables in the dynamic system's mathematical model.

Instantaneous learning neural networks, such as B-spline neural networks (BSNNs), offer significant advantages over traditional models like back-propagation (BP) and radial basis function (RBF) networks A primary benefit of BSNNs is their ability to continuously adapt control laws in real-time, eliminating the need for offline training that BP and RBF networks require before deployment This reliance on offline training poses challenges, as any significant changes in operating conditions necessitate a new training session In contrast, BSNNs efficiently map input patterns to corresponding weights with minimal computational overhead, making them a more flexible and efficient choice for dynamic environments.

BP and RBF ANNs often have longer execution times and higher computational requirements, which can lead to difficulties in converging to optimal solutions In contrast, BSNNs offer a reliable alternative for applications like adaptive filters, real-time control, nonlinear systems modeling, and pattern recognition The online training strategy of BSNNs allows for continuous adjustment of ANN weights in the Delta Parallel Robot, ensuring effective performance across different operating conditions.

Control law of the neuro-controller

The components comprising theistructure of theiB-spline neural network are comprises three key elements: anil-dimensional space forinormalized inputs, a collection ofiP basis functions, andithe output function ofi the BSNN

Figure 3 5: The diagram illustrates how the proposed BSNN is utilized for updating the gains in the PD controller

Figure 3.5 illustrates the essential elements of the B-spline neural network (BSNN) employed for trajectory control of the Delta Parallel Robot Central to the BSNN framework are the basis functions, which are determined by a collection of control point vectors These basis functions are formulated using a periodic expression, as detailed in several studies [106].

[107], renowned for their numerical stability, computational efficiency, and capability to accommodate diverseidistributions of controlipoints The outputiof the ANNi is determinediby a linearicombination of theseibasis function outputs

A key characteristic of the BSNN is its instantaneous training, which simplifies the process to a linear optimization problem This simplification arises because the adjustable weights function as linear coefficients, making the output linearly dependent on the weight set The output of the BSNN can be expressed in a specific mathematical formulation.

In the neuro-controller for the Delta Parallel Robot, the output vector \( a \) is represented as a \( P \)-dimensional vector, with the weights vector \( w \) consisting of only four elements to enhance computational efficiency, as depicted in Figure 3.5 The adaptive control scheme adjusts the gains of three PD controllers, one for each actuator, with the mathematical equations for each PD controller illustrated in Figure 3.6.

The equation \( \tau = K_P \theta + K_D \frac{d\theta}{dt} \) describes the relationship between proportional gain \( K_P \) and derivative gain \( K_D \), where \( \theta_d \) represents the desired angular displacement and \( \theta \) indicates the actual angular displacement The block diagram in Figure 3.6 illustrates the adaptive neuro-controller designed for trajectory tracking of the Delta Parallel Robot As shown in Figure 3.5, each controller in the system features two inputs and one output, establishing the structure of the Artificial Neural Network (ANN).

The equation K = NN w (3.17) defines the relationship where m = P, v, with NN m representing the BSNN used to compute K m, and w as the associated weights vector This formulation guarantees the convergence of the desired trajectory for the Delta Parallel Robot to a constrained solution space, even in the presence of bounded external or internal perturbations Figure 3.6 depicts the process of tuning the gains of the PD controller During the offline phase, control vector points are limited based on input magnitudes, such as error and its derivative, leading to the generation of monovariable base functions In the online phase, data from the offline phase is utilized to compute the BSNN output, which in turn adjusts the controller gain values, with each gain being managed by its respective BSNN.

Learning Rule

The performance function is essential for establishing the learning rule, computational complexity, and overall model quality This thesis prioritizes a simple learning rule for practical applications, leading to the selection of the Medium Quadratic Error (MQE) performance function, recognized for its reliable and superior performance across various scenarios.

To enhance output performance in real-time, learning rules that function instantaneously are developed by minimizing the Mean Quadratic Error (MQE) This is achieved through gradient descent techniques, which can be implemented in two primary ways: Batch learning, where network weights are updated once per training cycle after evaluating all examples, and Online learning, which updates weights after each individual example is processed, allowing for continuous adaptation to incoming data.

In computational engineering and neural networks, the gradient descent method is preferred for its simplicity and efficiency in model optimization In neuro-controller design, downward gradient rules are applied, and weight updates, denoted as Δw(t - 1), are executed using an instantaneous learning rule, as outlined in equation (3.14).

(3.18) wherei is the ilearning relationship, a iis the vectorithat contains theioutput of thei base functions, w iare the weightsivector, and e x = y(t) −ŷ(t), wherei y(t) and ŷ(t) iare the actual outputiand the desiredioutput of theiBSNN respectively

The input values for the network range from -4.55% to 4.55%, with a learning rate of 0.412 and a control point vector of [-1.7, -0.8, 0.7, 1.4] By defining these parameters for the BSNN, we achieve outstanding performance from the neurocontroller, as demonstrated by the following results.

This chapter introduces an advanced control strategy for DC motors applied to a Delta Parallel Robot (DPR) The proposed iB-spline neural networks-based adaptive control technique shows significant promise in tackling challenges related to nonlinear dynamic systems and varying operating conditions Key contributions and findings of this work highlight its effectiveness in enhancing the performance and adaptability of robotic systems.

- Aniadaptive control algorithmifor DC motors usingiB-spline neural networksi was developed, which does not require detailed mathematical modeling or exact parameter values of the motor

- Theiproposed control schemeionly requires velocityimeasurements, avoiding the needifor real-time measurementsior estimations ofiacceleration, current, andidisturbance signals

- Experimentaliresults confirmed theiefficient and robustiperformance of theicontrol approach underihighly demanding conditions,isuch as variable-speedi reference trajectoriesiand unknowniload torque

- A comprehensive DC motor model was presented, including both field and armature windings, and their respective elements such as inductance, resistance, and voltage

- Theimathematical model highlights the coupledinonlinear dynamic systemi of the motor, emphasizing the limitations of conventional linear controllers

3 Control Design Using Super-Twisting Sliding Mode

- Aisecond-order sliding modeicontrol methodology was described, which is robustito parametric uncertaintiesiand exogenousidisturbances

- The stability of the proposed system was established using Lyapunov functions,

- The design of an adaptive controller based on B-spline neural networks was detailed, focusing on a two-stage process involving offline training and online learning

- The controller structure, input parameters, and learning rate were optimized toiachieve satisfactory performanceiover aiwide range ofioperatingiconditions

5 Application to Delta Parallel Robot

- The adaptive control strategy was extended to a Delta Parallel Robot equipped with three DC motors

- The BSNN-based controller proved advantageous over traditional ANN methods, offering real-time adaptability without the need for extensive offline training

- The neuro-controller successfully adjusted the gainsiof three PDicontrollers to regulateithe trajectory ofi the iDPR, demonstrating effective performance in trajectory tracking tasks

- The learning rule for the BSNN was based on minimizing the mean quadratic error (MQE) using downward gradient methods

- Both batch and online learning approaches were discussed, with online learning being preferred for its efficiency and simplicity in updating the network weights

In summary, this chapter presents an effective solution for controlling DC motors and Delta Parallel Robots, demonstrating adaptability to changing conditions while ensuring high performance This approach is particularly well-suited for real-time applications in robotics and dynamic systems Future research may focus on optimizing neural network parameters and expanding this control strategy to accommodate more complex, higher-dimensional systems.

CHAPTER 4 SIMULATION RESULTS AND DISCUSSIONS CONCLUSIONS, LIMITATIONS AND FURTHER WORK

Simulation results

This chapter presents simulation results to assess the effectiveness of the proposed adaptive neural trajectory tracking controller based on a Bounded-Saturation Neural Network (BSNN) The performance of the BSNN-based neural controller is compared with a well-tuned Proportional-Integral-Derivative (PID) controller with gravity compensation Both controllers serve as straightforward extensions of the PD-with-gravity-compensation (PDgc) control law for bounded inputs Notably, they demonstrate superior capability in approximating a PDgc control signal within the limited range of control variables compared to previously proposed algorithms Consequently, implementing these controllers results in significant closed-loop performance improvements The desired reference input for the Delta Parallel Robot, modeled as an Archimedes spiral, is defined by the equations P_xd = cos(t)r_s, P_yd = sin(t)r_s, and P_zd.

= 0.06 i + 0.006t iwith r s = i 0.0006t and t i = [0 9π] Theinominal system parametersiof theiDPR for thisi case are idescribed in Tablei 3

L bi Loweriarm length 524imm m ai Upperiarm mass 0.300iKg m bi Loweri arm mass 0.270 i Kg m p Mobileiplatform mass 0.20iKg

I mi Motorimoment of inertia 3.7e -6 i Kgm 2

The initialiconditions of angularipositions and angularivelocities of theirobot used

In this study, reference signal values are stored in an n-dimensional vector, with each coordinate updated every 0.001 seconds Figures 5.1 and 5.2 illustrate that both controllers closely track the reference signal; however, the PD BSNN controller demonstrates lower error compared to the PD+G controller during transient states Consequently, with proper conditioning of the algorithm, satisfactory performance can be achieved in various trajectory tracking tasks involving reference signals.

TSE (Transient State Error): Calculated as the average integral of errors during the transient phase

SSE (Steady State Error): Calculated as the final error when the system is stable

The system meets requirements when the error for the measured angles theta is TSE < 0.6 and SSE < 0.1

Controller TSE 1 SSE 1 TSE  2 SSE 2 TSE  3 SSE 3

Controller TSE Px SSE Px TSE Py SSE Py TSE Pz SSE Pz

Theiaverage errors ofithe PD+G controlleriand PD BSNNicontroller are compared iniTables 4 andi5 Table 4ipresents the transientistate error (TSE) andisteady state error

(SSE)ifor angularidisplacement, while Tablei5 comparesiTSE and SSEifor Cartesian displacement,imeasured in millimetersi(mm) and degrees,irespectively

The simulation results clearly show that the proposed control method is highly effective for regulating the Delta Parallel Robot It demonstrates outstanding performance features, including a rapid transient response, minimal steady-state error, and the ability to adapt to new system conditions These qualities highlight its efficiency in real-world control applications.

The adaptive PD+G control method demonstrated effective trajectory tracking for the Delta parallel robot in simulation, as illustrated in Figure 4.1 The robot's actual path closely aligns with the desired trajectory, showcasing the method's capability to minimize tracking errors and ensure stability.

In this article, we compare the PD+G method with the B-Spline Neural Network (BSNN) method for controlling Delta parallel robots The results indicate that the BSNN method significantly surpasses the PD+G method in trajectory tracking accuracy and overall control performance, as illustrated in the accompanying image.

This comparison highlights the superior performance of the BSNN method, which offers enhanced precision and stability in controlling Delta parallel robots

Figure 4 2: Tracking trajectory of Delta Parallel Robot in PD BSNN

Besides, Motor Velocity, Input torque and Power of delta robot are also calculated and simulated in Figures 4.3 to 4.5 to prove that the BSNN algorithm works effectively

The BSNN controller demonstrates effective management of the Delta parallel robot's motor velocity, enabling precise, stable, and rapid tracking of the desired trajectory.

Figure 4 4: Input torque of Delta Parallel Robot in BSNN controller

The input torque graph of the BSNN-controlled Delta parallel robot offers essential insights into the robot's dynamic behavior and operational efficiency, enabling an assessment of the BSNN controller's effectiveness in regulating torque levels for optimal performance and stability.

The power graph of the Delta parallel robot controlled by the BSNN provides crucial insights into energy management and operational efficiency This data is instrumental in assessing how effectively the BSNN controller optimizes power consumption while improving the robot's overall performance.

Limitations

The study's findings remain untested on actual hardware, highlighting a significant gap in real-world applicability Although simulations offer important insights, it is crucial to implement the adaptive control method on physical Delta robots to ensure its robustness and performance in genuine industrial settings.

The exploration of adaptive control solutions for Delta robots in practical applications is currently insufficient, highlighting a significant gap between theoretical concepts and real-world implementation To bridge this divide, it is essential to conduct more in-depth investigations and experimental validations that evaluate the effectiveness of adaptive control methods in overcoming the diverse challenges faced by Delta parallel robots across various industrial environments.

These limitations emphasize the importance of conducting rigorous real-world

Further worrks

To address the limitations identified in the study on adaptive control for Delta parallel robots, the following steps are recommended for future research:

Real-world Testing and Validation: Conduct extensive testing and validation of the adaptive control method on physical Delta robots in real industrial environments This involves:

- Implementing the adaptive control algorithm on actual hardware to assess its performance and robustness

- Collecting data from real-world scenarios to validate theoretical predictions and simulations

- Analyzing the impact of practical factors such as varying loads, environmental conditions, and operational constraints on the control system's effectiveness

Integration of Practical Solutions: Explore and develop specific adaptive control solutions tailored to practical applications of Delta robots This includes:

- Investigating industry-specific challenges and requirements to refine the adaptive control algorithm

- Collaborating with industrial partners to implement and evaluate the adaptive control method in relevant settings

- Iteratively refining the control strategy based on feedback from industry stakeholders and performance metrics gathered during practical deployment

Comparison with Existing Methods: Compare the performance of the adaptive control method with other established control strategies for Delta parallel robots This involves:

- Benchmarking against traditional control methods (e.g., PID control) and advanced techniques (e.g., neural network-based control)

- Evaluating metrics such as trajectory tracking accuracy, energy efficiency, and response time to identify strengths and areas for improvement

Optimization and Tuning: Optimize the adaptive control parameters and tuning procedures to enhance control performance under varying operational conditions This includes:

- Employing systematic optimization techniques (e.g., evolutionary algorithms, gradient descent) to fine-tune control parameters

- Investigating adaptive tuning algorithms that can autonomously adjust control parameters based on real-time feedback and system dynamics

Long-term Stability and Reliability Studies: Conduct long-term studies to assess the stability and reliability of the adaptive control method over extended operational periods This involves:

- Monitoring the performance degradation over time and developing strategies to maintain control effectiveness

- Addressing issues such as sensor drift, component wear, and environmental changes that can impact control system performance

Future research should concentrate on enhancing the applicability, robustness, and practicality of adaptive control methods for Delta parallel robots This focus will help bridge the gap between theoretical advancements and real-world industrial applications.

This chapter presents the simulation results for the Delta Parallel Robot (DPR) utilizing an adaptive neural trajectory tracking controller based on a B-spline neural network (BSNN) The performance of the BSNN-based neuro-controller was compared to a well-tuned PID controller with a gravity compensator (PD+G) Key findings from the simulations indicate the effectiveness of the BSNN approach in enhancing trajectory tracking performance.

- Theidesired reference inputifor the DPRiwas an Archimedesispiral trajectory, specified byi P xd = cos(t)r s , P yd i = sin(t)r s and P zd = i -0.6 + 0.006t with r s i = 0.005t and t i =

- Theinominal dynamic parameters ofithe DPR were provided,iincluding the lengths

- The initial conditions for angular positions and velocities were set to ( q i = [0.3212 0.3212 0.3212 0 0 0] )

- Both theiPD+G and the PDiBSNN controllers were able to follow the reference trajectory closely However, theiPD BSNN controller exhibited a smaller error in the transient state compared to theiPD+G controller

The comparison of average errors for angular and Cartesian displacements revealed that the PD BSNN controller outperformed the PD+G controller, exhibiting significantly lower transient state errors (TSE) and steady state errors (SSE) across all displacement measurements.

- Table 4ihighlighted that the PDiBSNN controller had an angular displacement TSE of 0.0015 degrees and SSE of 3.67e -5 degrees, which were much lower than those of the PD+G controller

- Table 5 showed that the PD BSNN controller had a Cartesian displacement TSE of 0.0052 mm and SSE of 3.07e -4 mm, again outperforming the PD+G controller

- The PD BSNN controller demonstrated superior performance in terms of faster transientiresponse, minimal steadyistate error,iand adaptive capability to varying system conditions

- Theisimulation results confirmedithat the proposediBSNN-based adaptive control methodiis highly effective for controlling the DPR, providing robust and precise trajectory tracking under diverse operating conditions

In conclusion, the BSNN-based adaptive neuro-controller significantly outperforms traditional PID controllers with gravity compensators by adapting in real-time to changes in system parameters and external disturbances This capability makes it an ideal solution for high-performance control of the Delta Parallel Robot The findings highlight the potential for applying this advanced control strategy to other complex and nonlinear robotic systems.

The results of my implementation show that the adaptive controller is highly effective in tracking the complex trajectories of Delta Parallel Robots (DPR) Extensive testing reveals that this adaptive control strategy significantly improves the precision of the DPR's mobile platform displacement In comparison to the traditional Proportional-Derivative plus Gravity (PD+G) controller, the adaptive controller demonstrates superior performance in terms of accuracy and responsiveness.

A comparative analysis of the PD+G control scheme and the iB-Spline Neural Network (BSNN) control scheme reveals that BSNN-based adaptive control excels in meeting the dynamic requirements needed for the DPR to navigate complex trajectories This neural control approach continuously adjusts to the robot's real-time motion demands, guaranteeing consistent and reliable performance.

In contrast, the traditional PD+G controller exhibits a significant drop in performance under comparable conditions, as it fails to dynamically adjust to changes in trajectory and load This limitation leads to less precise control, rendering it unsuitable for applications that demand high accuracy and adaptability.

In summary, the adaptive neural control strategy surpasses traditional control methods in performance metrics Its ability to adapt to a wide range of complex trajectories positions it as an optimal solution for improving the operational efficiency of Delta Parallel Robots across diverse industrial and research applications.

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Phạm Văn Bạch Ngọc và Trần Minh Tuyến (2014) đã thực hiện nghiên cứu về thiết kế mẫu robot song song 3 bậc tự do tịnh tiến Nghiên cứu này được công bố trong tuyển tập công trình của Hội nghị cơ học kỹ thuật toàn quốc, nhân dịp kỷ niệm 35 năm thành lập Viện Cơ học, tại Hà Nội, trang 173.