HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY MASTER THESIS Design an adaptive controller and a state observer based on neural network for the 4DOF parallel robot NGUYEN MANH CUONG Contro
OVERVIEW
The four degrees of freedom parallel robot (4DOFPR) model
Robotic systems are rapidly developing and increasingly adopted across diverse economic and social fields because they tackle complex and dangerous tasks, perform repetitive work with high accuracy, and operate with remarkable flexibility Beyond precision and consistency, robots can work in hazardous environments, handle heavy loads and toxic substances, and adapt to varying environmental conditions As a result, these capabilities boost productivity and quality, prevent accidents, and reduce labor costs.
Parallel robots are increasingly prevalent across industry, defense, medicine, and entertainment, reflecting state-of-the-art technology A wide range of parallel‑architecture configurations has been studied, including six‑DOF robots used in medical surgery and rehabilitation, as well as structures applied to flight and automotive simulation These models exploit the core advantages of parallel design—low inertia, high payload, and smooth transmission—to deliver reliable, high‑precision performance Whether in realistic car simulations or training environments for trainees and drivers, constructing driving‑simulation models based on recent parallel architectures and motion platforms provides a practical approach to mastering automobile dynamics.
Car driving simulation models are developed to mitigate the impact of unexpected forces on drivers in real-world and virtual reality environments, with applications in healthcare and rehabilitation.
Robot modeling is essential to describe system motion Geometric analyses of six-DOF constrained parallel robots and forward/inverse kinematics for Quanser’s Hexapod illustrate the high-precision potential of six-DOF designs, but coordinating six actuators increases complexity in trajectory-tracking controllers, especially under substantial uncertainties Therefore, configurations with fewer joints and DOFs can mitigate actuator hysteresis and redundancy, improving practicality in application and controller design under uncertainty To further cut computational load and redundant constraints, researchers have developed four-DOF platforms that combine vertical translation along OZ with rotations about the OX and OY axes.
Figure 1.1 Parallel robot applied in the car motion simulator
Figure 1.2 Parallel robot applied in rehabilitation system [4]
Based on the references and analyses of the above scientific works, this thesis aims to reduce computational complexity and eliminate redundant constraints while preserving the necessary motion It focuses on a four-degree-of-freedom parallel robot platform featuring translational movement along the Z axis and rotational movements about the X and Y axes.
Trajectory tracking controllers and state observers
In robot control, especially orbital tracking control problems, modern methods focus on designing control algorithms that can handle uncertainties, perturbations, and unknown structural components in the system model while still ensuring stability and high-quality tracking The 4-DOFPR parallel robot model is commonly affected by nonlinear uncertain elements in practical applications, particularly external forces acting in different directions on the system By integrating robust and adaptive strategies, these approaches aim to maintain performance under disturbances and model inaccuracies, delivering reliable tracking and stable operation in challenging environments.
Parallel structures are treated as nonlinear models in the control design field, a characterization that has drawn substantial attention from researchers seeking effective solutions for nonlinear control systems To address the challenges posed by these models, a variety of control methodologies for nonlinear dynamics have been developed, focusing on ensuring stability, robustness, and improved performance Among these approaches, innovative strategies tailor nonlinear control techniques to the specific features of parallel-structure systems, enabling more reliable and adaptive control in practical applications.
3 that have been interested in is the Backstepping technique as in [12], [13], [14],
[15], and [16] in order to ensure the quality of trajectory tracking control However, when uncertainties or unmodeled components exist in the system model, the
The “explosion of terms” phenomenon adversely affects control quality Sliding mode control (SMC) is widely used for its robust performance in the presence of unknown elements, as demonstrated in [17], [18], and [19] However, SMC can cause chattering that may damage the system [20], and it imposes a heavy computational burden for high-order systems A hybrid approach that combines these two controllers exploits their complementary strengths to enhance control performance, improve robustness, and reduce computational cost, as indicated in [20], [21], [22], and [23] Nevertheless, the combined controller cannot cope with the chattering and remains a challenge that calls for effective mitigation strategies.
To address the explosion of terms in dynamic surface control (DSC) based on a multiple sliding-surface controller and the Backstepping approach, this work adopts a low-pass filter at each computation step While these filters mitigate the term explosion inherent in DSC, their accuracy depends on the chosen filter time constant and is governed by intricate mathematical conditions that can vary with the frequency characteristics of real-world experimental devices In response to these challenges, this study proposes an alternative that replaces purely analytic filters with a neural network to approximate the virtual signals, thereby reducing chattering and easing the mathematical burden The neural network–based approximation provides a robust mechanism to handle dynamic uncertainties without the stringent filter tuning required by traditional DSC implementations.
Within control theory, noise components are commonly treated as an inevitable part of any dynamic system, and rigorous noise analysis is essential for improving the stability and accuracy of a 4DOFPR system In particular, stochastic disturbances present real challenges by degrading performance and reliability, highlighting the need for robust disturbance rejection and control strategies tailored to the 4DOFPR mechanism.
Regarding non-Gaussian noises, the modified extended Masreliez–Martin filter described in [26] provides an efficient approach for nonlinear systems when environmental disturbances influence the entire system Additionally, [27] addresses stochastic parameters by estimating stochastic nonlinear systems Building on the cautious considerations published in [28], these approaches together offer a robust framework for managing uncertainty in nonlinear dynamics.
Stochastic disturbances from unknown varying input forces are assumed to act on the actuators of the 4DOFPR system along the vertical direction, driven by body weight, moment disturbances, and other unknown components A broad range of external and internal stochastic noises arise from factors such as friction, vibrations, sudden force changes, and shifts in environmental conditions, all of which constitute uncertainties This thesis treats the 4DOFPR as a model susceptible to stochastic uncertainty elements.
As noted above, many conventional nonlinear control methods, including sliding mode control (SMC) and backstepping, struggle to deliver improved performance when obtaining an accurate model is difficult due to model uncertainty, unmodeled dynamics, and time-varying parameters; these identification challenges limit control accuracy and robustness, especially in the presence of disturbances, prompting the use of more resilient approaches.
Uncertainties in control systems fall into matched and unmatched categories, and achieving high-quality control with conventional methods typically requires upfront identification of mathematical models that contain these uncertainty elements Adaptive controllers, whose updating laws are designed from the system’s mathematical properties to compensate for unknown components, have been proposed, yet their convergence heavily depends on the particular system model and chosen control parameters, making them unable to handle the full range of uncertain elements or variations in the 4DOFPR model As a result, such controllers may not adapt well to changing uncertainties and diverse model specifications, motivating the investigation of an alternative approach that uses neural networks to approximate the uncertainties.
A more modern and efficient approach approximating uncertainties by neural networks was appreciated as a promising approach [34], [35], [36], [37], and [38]
In addition, this methodology has increasingly become a novel method for the control design of the complex system affected by unknown uncertainties and disturbances [34] In [36], a radius basis function neural network (RBFNN) has been constructed to approximate the value of torques in a web transport system According to [37], wheel slip phenomena and external disturbances have been compensated by using the adaptive neural network to guarantee the tracking quality for mobile robots Meanwhile, an adaptive controller based on RBFNN as in [38] has solved the problem of the dual-arm robot containing the noises
Moreover, the idea to use the notable technique in approximating the nonlinear function of the Radial Basis Function neural network was conducted in
[39] and [40] By online tuning network parameters, the updated law of the RBFNN shown in [40], [41], and [42] was derived based on the Lyapunov theory, guaranteeing the minimization of the stochastic disturbance in terms of the overall system Besides using RBFNN, the fuzzy logic in which the weight value is based on the adaptive law has been designed [43] to effectively deal with the uncertain dynamics problem by estimating the appropriate values for the controller
By analyzing the above studies, solutions, and approaches, this thesis proposes an adaptive controller based on Backstepping-SMC and an RBFNN to guarantee the tracking performance of a four-degree-of-freedom parallel robot, contributing to improved tracking accuracy, robustness against uncertainties and external disturbances, and enhanced adaptability for complex robotic dynamics.
This work addresses controller design for the uncertain model of the 4DOF parallel robot (4DOFPR) Unlike the methods in [31], [32], and [33], the proposed approach based on a radial basis function neural network (RBFNN) does not heavily rely on the mathematical characteristics of parallel robots Consequently, the RBFNN-based controller can cope with a broader range of uncertainties and bounded disturbances, thanks to its strong nonlinear approximation capabilities and its ability to compensate for external disturbances.
The proposed adaptive controller integrates Backstepping with sliding mode control (SMC), leveraging Backstepping’s nonlinear cancellation and SMC’s robustness to enhance overall system performance Unknown nonlinear parts are attenuated, improving robustness and adaptivity The typical drawbacks of these controllers—explosion of terms in Backstepping and chattering in SMC—are markedly alleviated because a neural network approximates and compensates the uncertain dynamics that drive these phenomena Consequently, the hybrid controller not only mitigates the limitations of the individual techniques but also enhances the system’s robustness and adaptive behavior.
The Radial Basis Function Neural Network (RBFNN) is designed to estimate all unknown components of the dynamic system and to compensate for disturbances from input torque and external forces, enabling the controller to maintain high control quality even when only imperfect information about the system is available By capturing model uncertainties and external disturbances, the RBFNN-based controller delivers robust performance under information gaps and varying operating conditions Unlike the approach in [37], this strategy leverages the universal approximation capability of RBFNNs to produce accurate estimates of unmeasured dynamics and to sustain stable, precise control throughout the operating envelope.
Conclusion
Chapter 1 provides an overview of the four-degree-of-freedom (4-DoF) parallel robot and its wide range of applications Building on recent studies of nonlinear controllers for trajectory tracking, this work proposes an adaptive controller based on a radial neural network to address robot model uncertainty and the limitations of model-based controllers A review of prominent state-observer sets is also included From the tracking control problem for a 4-DoF parallel robot and the analysis of related literature, the scope and research objectives of the thesis are defined and presented.
DESIGN AN ADAPTIVE CONTROLLER AND A STATE
Mathematical model of the 4DOFPR
Figure 2.1 (a) Robot coordinate; (b) Vector diagram of 4DOFPR
This section presents the robot’s kinematic and dynamic models and frames the global control design problem Figure 2.1(a) shows the overall system and its coordinates, while Figure 2.1(b) defines the global coordinate frame chosen for analysis The system state comprises rotational angles about the Ox, Oy, and Oz axes and the vertical translational position of the upper platform panel Robot motion is produced by coordinating vertical actuation of three pistons with the rotation of a shaft mounted beneath the base platform The kinematic model establishes the mapping between piston displacements and the platform position P, a transformation that is detailed in the following subsection.
Model-based robot control relies on a computationally efficient formulation of the motion equations by describing the dynamic model with a set of independent joint coordinates, known as minimal coordinates These minimal-coordinate formulations are widely used for nonlinear motion control of robot manipulators because they eliminate redundancy and streamline computation In this approach, the robot’s joint coordinates serve as the independent variables that fully capture the system dynamics, enabling efficient real-time control even for complex, constrained mechanisms.
9 manipulator that is subjected to kinematic loop constraints are split into a set of dependent and independent coordinates
Let the system state vector be p = [ p_z, α, β, γ ]^T, where p_z denotes the vertical position and α, β, and γ are the rotation angles about the OX, OY, and OZ axes, respectively In addition, the piston lengths and OZ rotation are described by the vector [1, 2, 3]^T together with l_i (i = 1, 2, 3) and the term γ^T q, with l_i representing the length of piston i The distance between origin O and the base platform OA is c, while a and b are the radii of the upper and base platforms, respectively According to the vector diagram, a vector equation of the 4DOFPR is computed as:
A B = OP + PB - O A - OO i i i A i A , (2.1) where i=1, 2, 3 The coordinates of A 1 , A 2 , and A 3 in the local coordinate PX Y Z 1 1 1 associated with the base panel’s center are 1 sin( ) cos( ) 0 ,
A base =a −a Assuming that a b= , the coordinates of B 1 , B 2 , and B 3 in the local coordinate
O XYZ A associated with the upper panel’ center are
B is defined as B = [aπ; −aπ]^T The centers of the base and the upper panel are OA = [0; 0; c]^T and p_de = [0; 0; p_z]^T in the global coordinate frame The fixed coordinates of the i-th base point A_i and the i-th top point B_i are obtained by A_i = T_A A_i^base + O_A and B_i = T_A B_i^base + O_A, where T_A denotes the transformation from the base frame to the global frame and O_A is the global position of the base center Applying this transformation to all base and top points yields their coordinates in the global axis.
B =T B +p with cos sin 0 sin cos 0
Cosine and sine are the foundational trigonometric functions that connect angles to side lengths in triangles and describe circular motion and oscillations When these functions appear in succession or in combination, they model repetitive, wave‑like behavior across physics, engineering, signal processing, and computer graphics These functions are periodic with a 2π cycle, and their core relationships—such as sin^2(x) + cos^2(x) = 1, plus the angle‑addition and double‑angle formulas—enable compact expressions for complex angles and signals By understanding sine and cosine together, you can analyze rotations, vibrations, and harmonic motion, design precise animations, filters, and simulations, and apply Fourier analysis to decompose signals In short, the repetition of cos and sin in mathematical expressions captures the essence of cyclical patterns that underlie much of science and technology.
Finally, the length of the robot’s legs is computed as
2, l i = A B i i (2.2) where A B i i = −B i A i From (2), q can be computed from p
To achieve more precise movement, controllers commonly use a dynamic model that incorporates the system's kinetic energy, so forces, accelerations, mass, inertia, and other parameters can be identified more accurately With a controller designed from a dynamic model, system reliability improves and control quality increases However, obtaining exact parameter values is challenging, leading to a mismatch between the actual system and its mathematical representation Therefore, it is practical to treat the dynamic robot model as a nominal model with bounded uncertainties and measurement noises.
According to [25], the dynamic model of the robot based on the Euler-Lagrange form is described by
12 p py p p dc p p px p dc p p p dc m I m m m m m m I m m m m m m a m m
In this piston-cylinder-mobile-panel system, the masses and inertia are defined as m_cover for the piston covering, m_piston for the pistons themselves, m_motor for the motors mounted on the cylinders, m_panel for the mobile panels, and I_panel for the inertia of the mobile panels about the OxOy axes The configuration parameters 1, 2, dc, p, px, and py describe the geometric and positional aspects of the mechanism, while γ2 (gamma_2) represents a dynamic coefficient that influences the system’s response These quantities—cover mass, piston mass, motor mass, panel mass, and panel inertia—together form the essential inputs for the dynamic model of the piston-cylinder-mobile-panel assembly, guiding analysis, simulation, and optimization of the mechanism’s performance.
9.8( / 2) g m s is the gravity acceleration F= F 1 F 2 F 3 T is defined as a control signal vector.
Controller design for 4DOFPR
2.2.1 Backstepping aggregated with SMC (BASMC)
The controller designed by combining the two controllers can ensure that the robot tends to track the desired trajectory with the identified mathematical model
By combining the Backstepping technique and SMC method, the controller takes
11 advantage of the two control algorithms, which means it can eliminate the system’s nonlinear elements and increase robust behavior to external noises
To facilitate the control design procedure, initially, assuming that M C, , and
All robot system parameters are assumed to be precisely known, so the controller can be designed under ideal conditions with complete information Consequently, a reference trajectory for the dynamic model q_d(t) = [l1^d(t), l2^d(t), l3^d(t), γ^d(t)]^T is computed from the robot’s reference path and is defined as p_d(t) = [p_zd(t), α^d(t), β^d(t), γ^d(t)]^T using forward kinematics To describe the tracking process clearly, the initial tracking error in each control period is defined as the difference between the actual joint configuration q and the reference q_d, i.e., e(t) = q(t) − q_d(t).
With the tracking error vector ξ1, the control objective is to drive ξ1 to the vicinity of zero A virtual control is defined as the derivative of q (q̇), chosen to satisfy the Lyapunov candidate function V(ξ1), ensuring stable convergence of the tracking error toward zero.
V = ξ ξ 1 T 1 (2.5) From a first-order derivative of V 1 , it is straightforward to show that
V =ξ ξ 1 T 1 =ξ q q 1 T − d (2.6) the equation of q has a form of α q= d −k ξ 1 1 (2.7) Next, the vector depicting the error between q and α is defined as
Alternatively, we can write q ξ= + 2 α, and V 1 turns into
k 1 is a diagonal positive definite matrix In addition, ξ 2 is also considered as a sliding manifold to compute the system control input
Figure 2.2 Structure of BASMC controller
Figure 2.2 shows the structure diagram of the BASCM controller The tracking error is used as the input to the Backstepping controller, which computes the virtual control signal corresponding to the ideal velocity of the robot’s link lengths and the rotation about the Oz axis Once the desired control signal is derived, the virtual signal error is fed into the Sliding Mode Control to calculate the actual control signal for the system.
Taking derivative of ξ 2 results in
Applying the Backstepping design steps, we derive a second Lyapunov candidate function to compute the system input, as captured by equation (2.10) This Lyapunov function is constructed to guarantee global stability of the closed-loop system under the proposed control law In this way, the input calculation explicitly accounts for the system’s global stability through the Lyapunov framework.
V = +V 2ξ ξ T 2 2 (2.11) Taking derivative of V 2 leads to
The control signal F comprises two parts: an equivalent control signal and a switching control signal having the form of the SMC controller The equivalent control F eq is chosen as follows
F Cq D M ξ α (2.13) which holds the system state on the defined sliding manifold Besides, the switch control F sw drives the system state to the sliding surface
Let the switching control signal be given by equation (2.14), with k² = diag(k21, k22, k23, k24) and k³ = diag(k31, k32, k33, k34) as diagonal positive definite matrices One of the main factors driving chattering in this setup is the signum function embedded in the control law To mitigate this effect, the satlins function is commonly chosen, and the switching control signal in (2.14) is transformed into a smoothed version using the satlins function.
Eventually, the control signal is obtained by
There is a trade-off between the robustness of the control system and the smoothness of the control signal, and the proposed method can reduce chattering only within a certain operating range In theory, the chattering phenomenon is primarily caused by unmodeled dynamic elements in the system A radial basis function neural network (RBFNN) can estimate the effects of these unmodeled dynamics, thereby mitigating chattering and enhancing controller performance within the identifiable range.
V = −ξ k ξ T 1 1 1 −ξ k T 2 2 sat ξ 2 −ξ k ξ T 2 3 2 (2.17) where k , 1 k , and 2 k 3 are chosen as above, V 2 0 which satisfied the Lyapunov stability standard
Remark 1 BASMC controller is constructed based on the Backstepping technique; therefore, the phenomenon called “explosion of terms” is caused by iteratively and incrementally taking derivative of the virtual control signal (2.7) Especially when the system contains uncertain elements such as discontinuous functions, this calculation can have an adverse effect on system performance Additionally, even though the satlins function in (2.15) has the ability to reduce the chattering’s impact, there will be a trade-off in the system’s robust characteristic since this model-based controller cannot capture all the dynamics in the 4DOFPR model, which is often subjected to uncertainties, noises, and external forces
2.2.2 RBFNN-based (RBFNNB) adaptive controller
Accurately identifying the system matrices M, C, and D and the external disturbance τd is challenging in practice As a result, a model-based controller like BASMC relies on nominal values of these matrices, which can reduce tracking quality due to the explosion of terms and the associated chattering This section introduces a controller designed to overcome issues caused by uncertain elements and external disturbances, providing robust performance even when model information is imperfect.
Consider M, C, and D as the actual system-parameter matrices, with their nominal counterparts denoted by M̂, Ĉ, and D̂ The deviations ΔM, ΔC, and ΔD represent the differences between the actual and nominal values, i.e., ΔM = M − M̂, ΔC = C − Ĉ, and ΔD = D − D̂, capturing the model uncertainties Without loss of generality, these deviations allow the true matrices to be written as M = M̂ + ΔM, C = Ĉ + ΔC, and D = D̂ + ΔD, expressing the impact of uncertainty on the system parameters.
M M ΔM C C ΔC and D= +D ΔD, the robot system (2.3) is transformed into
To facilitate the control design steps, (2.18) is rewritten as
Finally, system dynamic equations turn into
Conducting the same procedure as in the previous section, with the defined uncertainties, ξ 2 becomes
Iterative calculation of α causes an explosion of terms, so we adopt a radial basis function neural network (RBFNN) to estimate the uncertainties and supply a suitable derivative estimate for α The RBFNN’s online learning capability enables real-time approximation of highly nonlinear dynamics, allowing the controller to operate without requiring full or precise prior knowledge of the system By treating α as unknown and defining the uncertainty vector σ = [σ1, σ2, σ3, γ]^T, the proposed adaptive scheme uses the RBFNN to significantly reduce chattering and enhance robust performance.
= − = -1 − -1 + + ξ 2 q α M F M Cq D σ (2.22) The control signal F=F eq +F sw with F eq and F sw are rewritten as
Equation (2.24) defines F_M k ξ_k ξ, and (2.23) shows that σ̂ ∈ R^{4×1} is the neural network’s output vector Here σ is treated as the target value for σ̂, and the Radial Basis Function Neural Network (RBFNN) is designed to guarantee that σ̂ converges to σ with arbitrarily small error, or to identify an appropriate value of σ to ensure the system’s stability.
Figure 2.3 shows the RBFNN’s structure with three layers: an input layer, a hidden layer, and an output layer The neural structure is constructed with
Let ε denote the approximation error, with ε ≤ ε0, where ε0 is an arbitrarily small positive constant W and Ŵ are the ideal weight matrix and the updated weight matrix, respectively, and the difference W − Ŵ is treated as the error weight matrix The Radial Basis Function (RBF) is employed to calculate the output of the hidden layer as follows.
Let the system state q be treated as the input vector, δ denote a center point, and η represent the width of the radial-basis function h Substituting the control signals from (2.23) and (2.24) into (2.22) yields the error dynamics for ξ2 as shown in equation (2.28) The adaptive law for the RBF neural network weights is defined by Ŵ′ = Γ h ξ = (T2 − μ ξ Ŵ2), where Γ ∈ R^{n×n} is a positive definite matrix and n is the number of hidden-layer nodes, and the learning rate is chosen in the interval (0,1).
Figure 2.4 Structure of the adaptive controller
Figure 2.4 shows the proposed controller architecture based on a radial basis function neural network (RBFNN) In contrast to BASMC’s structure, the RBFNN uses the robot state as input to estimate the appropriate σ̂ values, employing an online weight-update rule within the adaptive controller.
The second Lyapunov candidate function is reselected as
V sign tr sign tr sign tr
With the updated law (2.29), V 2 is rendered as
V = −ξ k ξ T 1 1 1 −ξ k T 2 2 sign ξ 2 −ξ k ξ T 2 3 2 +ξ ε T 2 + ξ 2 tr W W W T − (2.33) Applying the Cauchy-Schwarz inequality results in
It can be shown that
(2.35) where k 2min and k 3min are the minimum values of k 2 and k 3 If the bounded condition in the following inequality is satisfied
Using a Lyapunov-based design, the network’s update law is crafted and proven to guarantee global asymptotic stability The Lyapunov function is constructed so that its derivative along the network trajectories is nonpositive, ensuring convergence of the states to the equilibrium and validating the method’s convergence Consequently, the proposed update rule provides a stable network behavior with guaranteed convergence under the stated assumptions.
Conclusion
This chapter proposes a neural network-based adaptive controller and a high-gain state observer for the trajectory tracking control problem of a four-degree-of-freedom parallel robot with an uncertain model The adaptive controller ensures system stability even when the model contains uncertainties, while the high-gain observer provides accurate state estimation to support robust trajectory tracking.
The approach addresses parameter variation, uncertainties, and noise components, while the proposed controller minimizes the effects of explosion of terms and chattering on the control quality of the system In addition, the state observer can estimate the system states even when model errors are present To solve the control problem, this framework delivers robust performance under imperfect modeling and disturbances by combining accurate state estimation with smooth, reliable control actions.
This study designs a backstepping controller integrated with sliding mode control to solve the tracking control problem for a four-degree-of-freedom parallel robot, assuming an ideal robot model A set of simulations verifies the method, evaluates its performance, and analyzes its advantages and limitations Based on this analysis, an adaptive controller is proposed to further improve the control quality.
Propose a neural network-based adaptive control law for a four-degree-of-freedom parallel robot model, built on the BASMC controller and a radial basis function neural network (RBFNN) The RBFNN is capable of online approximation of uncertain dynamics and can compensate for the adverse effects of conventional nonlinear controllers and sensor noise, improving control precision and robustness By learning the uncertain components in real time, the adaptive scheme enhances tracking performance and resilience to disturbances A high-gain state observer is also employed to estimate the system states, reducing the complexity and burden of data from the sensor system while maintaining accurate state information.
SIMULATION RESULTS
Results of the RBFNN based adaptive controller (RBFNNB)
This section presents the simulation results of 4DOFPR using the RBFNNB controller Assuming that, the robot experiences the model’s uncertain elements and external noises
The following ternary function describes reference trajectories
( )t = + t+ t + t , d o 1 2 3 p a a a a (3.1) in which, the coefficients a a a o , , 1 2 ,and a 3 are computed from
(3.2) where p o , v o , and t o are the initial position, velocity and time, while p f , v f , and t f those of the end The desired trajectory p d ( )t is constructed using the parameters in Table 3.1
Initial position parameters End position parameters
6 o m s rad s t rad s rad s m rad rad rad
3 f m s rad s t rad s rad s m rad rad rad
1 8; 2 10; 3 10. k = k = k RBFNN parameters Neural number: nP;
Besides, system specifications include: a = b = 0.25 (m); c = 0.05 (m); a = 0.050 mz ( )
, and nominal values of robot’s dynamic coefficients are m (kg) p ; m =0.5(kg) 1
; m =3(kg) ; 2 m =3(kg) Additionally, controller gains, as well as RBFNN dc parameters, are given in Table 3.2
The simulation scenario is conducted with nominal values of matrices M C, , and
D; therefore, the system parameters are also nominal values Moreover, the system is influenced by an external force in the vertical direction, whose value is described in Figure 3.1
This study compares three controllers—BASMC, the Dynamic Surface Control Neural Network (DSCNN) from [25], and the proposed RBFNNB controller—to illustrate their effectiveness When the system faces external disturbances and parameter uncertainties, model-based controllers like BASMC fail to deliver acceptable control quality despite their robustness, whereas the DSCNN and the proposed RBFNNB offer improved robustness and accuracy.
Figure 3.2 shows that the system inherits dynamics from sliding mode control (SMC) Under external disturbances, the tracking process exhibits steady-state errors, reducing precision Specifically, when the CDS is subjected to an unexpected external force, the system is unlikely to fully compensate for the resulting deviations because the mathematical model lacks sufficient information to capture the disturbance and its effects, hindering real-time disturbance rejection.
Self-learning neural-network controllers do not require any system-specific information, enabling control performance even when the mathematical model is inaccurate or external disturbances are bounded Consequently, DSCNN and RBFNNB outperform BASMC Figure 3.3 shows the state-p tracking errors, with BASMC exhibiting the largest errors and RBFNNB the smallest In the study cited as [25], DSCNN is evaluated with neural-network error ε while low-pass-filter errors are neglected, effectively removing a learning-rate term from the adaptive law; as a result, DSCNN’s convergence is slower than that of RBFNNB, yielding a larger tracking error for DSCNN.
(a) Motion trajectory of p z (b) Motion trajectory of
(c) Motion trajectory of (d) Motion trajectory of
(a) Tracking error of p z (b) Tracking error of
(c) Tracking error of (d) Tracking error of
Initially, the Radial Basis Function Neural Network (RBFNN) requires time to estimate unknown system information, causing fluctuations in the early responses of both DSCNN and the RBFNN when these estimates substantially affect control quality, as shown in Figures 3.2 and 3.3 With proper design, the RBFNNB architecture leverages the RBFNN to simultaneously estimate suitable values for uncertain parts and to compensate for drawbacks of model-based controllers such as Backstepping and Sliding Mode Control (SMC) As a result, the fluctuations of the RBFNNB are nontrivial yet effectively mitigated Furthermore, when the system experiences abrupt external disturbances at 1 s, 2 s, and 4 s, the RBFNNB-controlled system promptly returns to the stable state because the RBFNN provides near-immediate estimates (approximately 0.08 s), as shown in Figure 3.4.
Figure 3.4 shows that the solid lines represent the ideal values that guarantee the system’s stability and are unknown to the controllers In the proposed controller, a radial basis function neural network (RBFNN) is designed to estimate these values, with the estimates shown as a dashed line At the initial step, the neural network begins estimating the parameter σ, and its convergence speed depends on the network structure and the chosen learning rate Through its self-learning ability, the RBFNN continually refines the estimate, enabling improved stability and enhanced controller performance.
26 generate unknown values that track the idea values even when sudden system information changes
(a) Approximated value of 1 (b) Approximated value of 2
(c) Approximated value of 3 (d) Approximated value of
(a) Motion trajectory of l 1 (b) Motion trajectory of l 2
(c) Motion trajectory of l 3 Figure 3.5 Motion trajectory of q
(a) Tracking error of l 1 (b) Tracking error of l 2
(c) Tracking error of l 3 Figure 3.6 Tracking error of q
The tracking performance of q is shown in Figures 3.5 and 3.6, with the trends for q closely following those of p Overall, the proposed RBFNNB controller delivers the best tracking performance, outperforming BASMC and DSCNN.
Simulation results of the adaptive controller using the high-gain state
The high-gain state observer is to generate the input state signal for the adaptive controller designed in the previous chapter Both the controller and the
Neural network-based control relies on accurate position and velocity information provided by the state observer to compute the control signal and to estimate values required for guaranteeing system stability Consequently, the control performance is largely determined by the observer’s output quality To assess this relationship, a simulation scenario is built to analyze observer accuracy and its influence on the control system, especially under uncertainties, model errors, and noise.
For this scenario, the reference trajectory’s parameters are given in Table 3.3 The initial position for 4DOFPR is reselected as l 01 =0.55( ),m l 02 =0.45(m),
12 o m s rad s t rad s rad s m rad rad rad
3 f m s rad s t rad s rad s m rad rad rad
Figure 3.7 Uncertain parts in the robot model
To verify the system’s insensitivity to model errors and uncertainties, we assume ideal model parameters mp: mp1 = 0.5 kg, mp2 = 3 kg, mp3 = 3 kg, while the nominal model parameters are md = 25 kg, m1 = 1 kg, m2 = 2 kg The system also contains uncertain elements as indicated in Figure 3.7 These uncertain parts, together with model variations, degrade the quality of both the controller and the observer Nevertheless, as noted previously, the RBFNNB adaptive controller can estimate the appropriate controller values to counteract these effects. -**Support Pollinations.AI:**🌸 **Ad** 🌸Creating content with precision? Discover Pollinations.AI for smarter writing; [Support our mission](https://pollinations.ai/redirect/kofi) to keep AI accessible.
The control parameters are chosen as shown in Table 3.2, and the observer parameters are set to ε_ob = 0.01 and k_i1 = 1.0, k_i2 = 2.0 for i = 1, 2, 3, 4 Since the goal is to estimate the robot’s velocity values, the robot’s position remains necessary for the observer, so the observer’s initial position is assumed to coincide with the robot’s initial position.
Figure 3.8 and Figure 3.9 illustrate the observer outputs, showing that the estimated position tracks the actual position with minimal errors—about 1×10^-4 m for the leg position and 1×10^-4 rad for the angle γ—under the assumption that there is no initial discrepancy These small errors indicate that the measured quantities can be used to compute the observer outputs, and the position-value comparison demonstrates the high-gain performance of the high-gain observer in accurately tracking the true state.
Figure 3.9 demonstrates that the velocity observer can track the actual velocity, with speeds rising rapidly at the initial moment because the robot’s starting position differs from the desired trajectory The controller then generates the appropriate input to quickly steer the system toward the reference trajectory Even in the presence of uncertain components and model variation, the velocities of the robot’s legs and the angular velocity about the OZ axis stay aligned with their true values, showing only small deviations within an acceptable range The velocity error behavior is depicted in Figure 3.10 Initially, the observer requires time to compute the correct value, which produces an internal oscillation, as shown in Figure 3.10.
Figure 3.10 shows that the time interval remains minimal (about 0.03 s) and the convergence speed is fast, underscoring a primary advantage of the high-gain observer At steady state, the maximum velocity error of the legs is only about 6×10^-3 m/s, while the angular velocity error is approximately 5×10^-4 rad/s The results also reveal a small steady error in the observed velocity deviations, since equation (2.46) indicates that with a sufficiently small observer parameter epsilon_ob, the adverse effect of model error can be mitigated However, because 0 < epsilon_ob < 1, this feature cannot eliminate errors entirely, and the observer gain must be chosen to keep the deviation within an acceptable range From this analysis, the high-gain observer achieves fast convergence while simultaneously minimizing the impact of uncertainty components and model noise.
RBFNN-based control can approximate wide ranges of uncertainty while delivering high-gain outputs that generate values closely tracking the ideal references for unknown parts, thereby ensuring system stability (see Figure 3.11) This translates into high control quality and improved tracking performance (Figure 3.12), enabling the robot states to follow their references with small errors—approximately 1.2×10^-3 meters (Figure 3.13) A steady-state error appears in the observer, as noted in the preceding evaluations By contrast, using the 4DOFPR configuration keeps the errors at an acceptable level.
Figure 3.11 Estimated values from RBFNN
To thoroughly demonstrate how varying ε_ob mitigates the impact of model errors, we evaluate multiple ε_ob values The results indicate that smaller ε_ob leads to better system performance, but if ε_ob becomes excessively small, peaking phenomena [44] can arise and destabilize the system.
Figure 3.14 Observed position with different values of ob
Figure 3.15 Observed velocity with different values of ob
Figures 3.14 and 3.15 show that when ε_ob is 0.05 or 0.1, the observed quality is not as good as at ε_ob = 0.01 The value ε_ob = 0.01 yields superior observed quality compared with the other tested values By contrast, with ε_ob = 0.1 the observed quality deteriorates, indicating that larger ε_ob values reduce performance.
35 error is the most significant deviation Specifically, the steady error exists in both position and velocity values Thus it can have a significant deteriorative impact on the system performance.
Conclusion
This chapter presents simulation results for the proposed adaptive controller based on an RBF neural network, comparing its performance with established methods reported in the literature It also analyzes the controller’s performance with the support of a high-gain observer, demonstrating improved estimation accuracy and enhanced overall control performance.
Compared with conventional controllers, the proposed approach shows superior performance by coping with uncertain parts and external noises that challenge tracking Specifically, the tracking errors of RBFNNB are significantly smaller than those of the other methods, even in the presence of assumed external forces and unknown model parameters for both simulation scenarios Moreover, a standard high-gain observer can ensure an acceptable quality of the observed states, and the control performance remains guaranteed when the input is generated from the observer However, both the controller and observer parameters need to be moderately tuned, and an appropriate tuning method can markedly improve system performance The future work section will discuss this problem in more detail.
CONCLUSION AND FUTURE WORK
Results of the thesis
This work proposes a novel adaptive control algorithm based on a radius basis function (RBF) neural network to tackle the trajectory tracking problem of a four-DOF parallel robot The model explicitly accounts for uncertainties, exogenous disturbances, and system variability, and the controller achieves robust system stability while minimizing the degradation in tracking performance typical of conventional nonlinear controllers, even when paired with a high-gain state observer The main contributions include the development of an RBF-based adaptive control law, a formal stability analysis under uncertain dynamics, and comprehensive validation showing improved tracking accuracy and disturbance rejection in a real-time robotic setting.
This thesis tackles controller design for the uncertain model of a 4-DOF parallel robot (4DOFPR) The proposed approach uses a radial basis function neural network (RBFNN) whose effectiveness does not depend on the exact mathematical form of the parallel robot, enabling robust operation across a broad range of uncertainties and bounded disturbances The controller leverages the RBFNN's nonlinear approximation and disturbance compensation to enhance stability and performance despite model errors and external noises Neural network outputs are compared to nominal uncertain values to verify the validity of uncertainty estimation and to quantify the influence of the neural network on overall system performance (Publication 1).
An adaptive controller that combines Backstepping with Sliding Mode Control (SMC) is proposed to capitalize on the strengths of both methods for nonlinear dynamics cancellation and enhanced system robustness The neural network compensates the unknown elements, significantly reducing the explosion of terms and chattering typically associated with these controllers As a result, the proposed controller surpasses the limitations of the individual techniques and delivers improved robustness and adaptive behavior for the overall system.
An adaptive controller is designed using a high-gain state observer to estimate robot states and reduce the complexity of sensor systems The high-gain observer ensures observation quality, thereby guaranteeing control quality and maintaining system stability even under model errors Simulation results verify the theoretical analysis and demonstrate the efficiency of the proposed controller.
During the thesis work, research on control algorithms and adaptive control for nonlinear systems and parallel robots was published at leading scientific conferences and in prestigious national and international journals (Publications 1, 2, 3, 9, 12, 14).
Future work
Parallel robots contend with uncertainties, model variations, and exogenous disturbances, along with numerous constraints on both robot state and actuator motion Incorporating these constraints into the control law design improves control quality and yields more reliable control signals for the system.
The observation quality of a high-gain state observer depends largely on the careful selection of its parameters, especially when a wide range of model errors is present Since each stage of operation requires different parameter values, targeted tuning can significantly improve overall system performance Fuzzy logic rules can be designed to derive suitable parameter values from system knowledge and the robot model’s input and output data, and this approach can also be used to tune control parameters and the neural network to further enhance control performance.
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