Design Backstepping Controller for 3-DOF Delta Parallel Robot .... Design Dynamic Surface Sliding-Mode Controller for 3-DOF Delta Parallel Robot 41 4.. The response of the joint angle mo
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Advisor(s)
Ph.D Le Xuan Hai, Faculty of Applied Sciences, Director at Intelligent Control Lab,
International School, Vietnam National University, Hanoi.
Abstract
Delta parallel robots are currently gaining popularity in high-precision industrial applications However, achieving precise trajectory tracking of these robots in complex environments remains a challenge due to factors like imprecise dynamic models, unidentified parameters, nonlinearities, and external disturbances In this research, we propose the non-linear control method designed for 3-DOF (Degree-Of-Freedom) parallel robots The two purposes of this research are 1) to develop a backstepping control (BC) algorithm for rapid and accurate trajectory tracking, and 2) to investigate the design of sliding mode dynamic surface control (SMDSC) to ensure robust performance even under external disturbances This approach demonstrates good asymptotic stability, making the overall system highly resistant to parameter variations Simulations conducted in MATLAB/Simulink validate the effectiveness of the proposed method These simulations show successful control of the robot's motion along the desired trajectory, with minimal errors and fast dynamical response Additionally, the proposed method offers a more flexible and adaptable control approach compared to traditional methods, making it well-suited for handling input uncertainties and external disturbances.
Keywords
Delta Robot, backstepping control, sliding mode control, dynamic surface control, tracking control.
SUMMARY REPORT IN STUDENT RESEARCH
Literature Review
Background
Parallel Delta robots, also known as parallel robots, are highly specialized robots that have found extensive applications in various industries, including manufacturing These robots are designed to tackle complex challenges and enhance efficiency in industrial settings [1] The unique design of 3-DOF Parallel Delta robots, which consists of three (or four) degrees-of-freedom (DOF) parallel arms, allows for precise and accurate movements, making them ideal for tasks that require high levels of precision [2] In addition to manufacturing, Parallel Delta robots have also been utilized in other fields such as healthcare and agriculture In healthcare, robotic technologies have been used during the COVID-19 pandemic to collect samples in a contactless manner and enable doctors to remotely provide care [3] This demonstrates the versatility of Parallel Delta robots in different applications beyond manufacturing Furthermore, the use of Parallel Delta robots in intelligent farms has been explored to improve asset efficiency, product quality, and environmental sustainability [4] This highlights the potential of these robots to revolutionize various industries and contribute to advancements in automation and decision-making processes Overall, Parallel Delta robots play a crucial role in enhancing efficiency and precision in diverse fields, including manufacturing, healthcare, and agriculture Their unique design and capabilities make them valuable assets in tackling complex challenges and driving innovation in various industries [5] As research and development in robotics continue to advance, Parallel Delta robots are expected to play an increasingly important role in shaping the future of automation and intelligent systems The literatures offer a diverse range of Delta robot designs, encompassing both translational/rotational configurations and variations in the robot’s topology For example, a modification of the Delta robot with only three rotational degrees-of-freedom (DOFs) was proposed in [6] to eliminate some spherical joints The Delta robot’s dynamical model is described by a system of three complex equations, coupled with three restriction equations This model is nonlinear and the presence of constraints prevents overshooting within the robot’s joints
Traditional control methods like PD/PID (Proportional-Derivative/ Proportional-Integral-Derivative) have been successfully implemented for controlling Delta robots [7,8] Nevertheless, several limitations have been identified in some research Ardestani et al (2013) [9] studied the
10 feasibility of applying fuzzy PD and PID supervisory control for a parallel mechanism, indicating the limitations of traditional PID controllers in industrial applications Lu et al (2015) [10] and Lu et al (2016) [11] both presented alternative control methods for delta robots, including fuzzy logic controllers tuned with PSO and PID-type interval type-2 fuzzy logic controllers, respectively These studies suggest that there are more advanced and optimized control strategies available for delta robots beyond traditional PID methods
While traditional control methods can struggle with complex nonlinear systems, several advanced techniques offer promising alternatives BC stands out for its systematic and recursive design approach By breaking down the system into manageable subsystems, BC avoids the loss of valuable nonlinear information and effectively achieves stabilization and tracking goals This makes BC a popular choice for researchers tackling challenging control problems in robotics Additionally, SMDSC presents another attractive alternative By combining the robustness of sliding mode control with the design flexibility of dynamic surface control, SMSDC offers strong performance guarantees even in the presence of uncertainties and external disturbances.
Backstepping Control (BC)
In the early 1990s, backstepping emerged as a novel approach for designing adaptive controllers This recursive, Lyapunov-based scheme tackles the challenge of controlling "strict-feedback" systems [12-17] For systems transformable into the parametric-strict feedback form, backstepping guarantees global regulation and tracking properties A key advantage of backstepping lies in its systematic procedure for designing stabilizing controllers through a step-by-step algorithm This methodology ensures a structured approach to constructing both feedback control laws and associated Lyapunov functions
Backstepping's flexibility in avoiding linearity cancellation enables stabilization and tracking objectives [12-17] However, its limitations include restricted parameter update laws and a high dynamic order in adaptive backstepping for systems with unknown parameters While tuning function controllers minimize order, their nonlinearity becomes complex for high-order systems Thus, recursive forms arise, balancing order and complexity, but face challenges in deriving recursive control laws and ensuring stability for non-affine systems.
11 design inherent to backstepping can lead to an "explosion of terms," making it potentially more intricate compared to traditional methods
Despite these limitations, backstepping control offers a powerful alternative to traditional methods
By linking the selection of control Lyapunov functions (CLFs) with the design of strict-feedback controllers, it ensures global asymptotic stability [12-17] This method also boasts different variations to address specific control challenges Integrator backstepping [13] tackles limitations of feedback linearization, while block backstepping [14] offers wider applicability For systems with unknown parameters, adaptive backstepping [18-20] utilizes parameter estimates Robust backstepping [21-23] proves effective for handling uncertainties
Backstepping technique is a recursive design method for constructing both feedback control laws and Lyapunov control functions in a systematic manner It involves dividing a strict-feedback nonlinear system of degree n into n subsystems, designing feedback control laws and Lyapunov control functions for these subsystems [14]
Consider an n-degree strict-feedback SISO nonlinear system:
In the given, x n =[ , , , ]x x 1 2 x n T ∈R n are system state vectors, u R∈ is the system control input, y R∈ is the system output, f i (.) and (.)g i with i=1,2, ,nare known nonlinear parametric functions of the system To ensure strict feedback interconnection of the system, g i (.) 0≠
The objective of this technique is to design a control law uusing the backstepping technique are as follows:
Step 1: Considering the first subsystem:
The time derivative of z 1 is given by:
Choosing a Lyapunov function for the first subsystem:
The time derivative of V 1 is given by:
Where α 1 is the virtual control signal for the first subsystem,x 2 =z 2 +α 1
Choosing the virtual control signal α 1 for the first subsystem:
With the chosen virtual control signal α 1 , the Lyapunov derivative V1becomes:
In which the term −c z 1 1 2 makes the system stable, and the term g x x 1 ( ) 1 2 will be eliminated in the next step
Step 2: Taking the derivative of z 2 to t
Choosing a Lyapunov function for the second subsystem:
The time derivative of V 2 is given by:
In which α 2 is the virtual control signal for the second subsystem, x 3 = −z 3 α 2
Choosing the virtual control signal for the second subsystem:
In which the terms −c z 1 1 2 −c z 2 2 2 make the system stable, and the term g x x z z 2 ( , ) 1 2 2 3 will be eliminated in the next step
Taking the time derivative, we have
Where x i =[ , , , ] ,x x 1 2 x i T i=1,2, ,n−1,α i − 1 is the virtual control signal for the i-1subsystem Choosing a Lyapunov function for the i subsystem:
1 1 i i i z + =x + −α (1.21) Where α i is the virtual control signal for i subsystem, x i + 1 =z i + 1 +α i
Choosing the virtual control signal for the i subsystem:
−∑ make the system stable, and the term g x z z i ( ) i i i + 1 will be eliminated in the next step
Taking the time derivative, we have
Where x n =[ , , , ] ,x x 1 2 x n T α i − 1is the viral control signal for the n-1 subsystem
Choosing the Lyapunov function for n subsystem:
Taking the time derivative of V n :
Choosing the virtual control signal for n subsystem:
Equation (1.31) shows that with the backstepping control design technique, we can find a control u that makes the strict feedback nonlinear system (1) stable, and z=[ , , , ]z z 1 2 z n T converages to a neighborhood of zero z 0 (1.32)
Thus, we can see that z 1 = −x 1 x d converages to a neighborhood of zero, or x 1 converages to a neighborhood of x d , which satisfies the signal tracking requirement
Equation (1.30) shows that the control signal u is only determined when the nonlinear parametric functions f i (.)and g i (.)with i=1,2, ,nare known functions of the system When these functions are uncertain or unknown, the control signal u cannot be determined.
Sliding Mode Control (SMC)
Sliding mode control (SMC) stands out as a powerful technique in nonlinear control systems due to its exceptional properties of accuracy, robustness, and ease of implementation and tuning [1] This method strategically guides the system's state variables onto a designated region in the state space, referred to as the sliding surface Once the system reaches this surface, SMC maneuvers the states to remain within a close vicinity Consequently, the design of an SMC system can be viewed as a two-stage process The first stage involves meticulously crafting a sliding surface that ensures the desired system behavior is achieved during the sliding motion The second stage focuses on selecting a suitable control law that effectively attracts the system's state towards the designated sliding surface [26]
Two key advantages make SMC an attractive control strategy Firstly, the dynamic behavior of the controlled system can be meticulously shaped by strategically choosing the sliding function Secondly, the closed-loop response exhibits remarkable insensitivity to specific uncertainties within the system This principle extends to bounded model parameter uncertainties, external disturbances, and nonlinearities [26] From a practical standpoint, SMC offers a viable solution for controlling nonlinear processes that are susceptible to external disturbances and significant model uncertainties
Consider the nonlinear SISO system:
( , ) ( , ) x= f x t +g x t u (1.33) ( , ) y h x t= (1.34) Where yand udenote the scalar output and input variable, and x R∈ n denotes the state vector The control aim is to make the output variable y to track a desired profile y DES that is, it is required that the output error variable e y y= − DES tends to some small vicinity of zero after a transient of acceptable duration
As mentioned, SMC synthesis entails two phases:
The first phase is the definition of a certain scalar function of the system state, says σ( ) :x R n →R
Often, the sliding surface depends on the tracking error e y together with a certain number of its derivatives: σ σ= ( , , ,e e e ( ) k ) (1.35)
The function σshould be selected in such a way that its vanishing, σ =0, gives rise to a “stable” differential equation any solution e t y ( )of which will tend to zero eventually
The most typical choice for the sliding manifold is a linear combination of the following type e c e 0 σ = + (1.36)
The number of derivatives to be included (the “k” coefficient in (1.38)) should be k r= −1, where r is the input-output relative degree of (1.33)-(1.34)
With properly selected c i coefficients, if one steers to zero the σvariable, the exponential vanishing of the error and its derivatives is obtained
If such property holds, then the control task is to provide for the finite time zeroing of , “forgetting” any aspects
From a geometrical point of view, the equation σ efines a surface in the error space, that is called “sliding surface” The trajectories of the controlled system are forced onto the sliding surface, along which the system behavior meets the design specifications
A typical form for the sliding surface is the following, which depends on just a single scalar parameter, p d k dt p e σ = + (1.39) 1; k= σ = +e pe (1.40) 2; 2 2 k= σ = +e pe p e+ (1.41)
The choice of the positive parameter p is almost arbitrary, and define the unique pole of the resulting “reduced dynamics” of the system when in sliding
The integer parameter k ison the contrary rather critical, it must equal to r-1, with r being the relative degree between y and u
The successive phase (PHASE 2) is finding a control action that steers the system trajectories onto the sliding manifold, that is, in other words, the control is able to steer the σ variable to zero in finite time
There are several approaches based on the sliding mode control method:
- Standard (or first-order) sliding mode control
- High-order sliding mode control
First-order sliding mode control
The control is discontinuous across the manifold σ =0 sgn( ) u= −U σ (1.42) That is:
U is a sufficiently large positive constant
Figure 1 Typical evolution of the σvariable starting from differential initial conditions
In steady state control variable, u will commute at a very high (theoretically infinite) frequency between the values u=U and u=-U (Fig 2)
Figure 2 Typical evolution of the control signal u (the dashed line representsσ )
In contrast to "electrical" applications that typically employ PWM control signals, discontinuous high-frequency switching control (Fig 2) is not suitable for mechanical systems This is due to the oscillations and various issues it introduces when used in these applications.
In order to solve the above problem (referred to as “chattering phenomenon”) approximate (smoothed) implementations of sliding mode control techniques have been suggested where the discontinuous “sign” term is replaced by a continuous smooth approximation Two examples follow:
Unfortunately, this approach is effective only in specific case, the one is when hard uncertainties are not present and the control action that counteract them can be set to zero in the sliding mode
Figure 3 Smooth approximations of sliding mode control
Second-order Sliding Mode Control
Dynamic Surface Control (DSC)
Using the above described smooth approximations, some problems are attenuated, at the price of a loss of robustness
Second-order sliding mode control algorithms are a powerful alternative that completely solves the chattering issue without compromising the robustness properties as well.
Dynamic surface control (DSC) is a powerful technique for controlling complex nonlinear systems
It offers robustness to uncertainties and has found applications in various fields, including automated vehicles [27], ship control [28], and robotics [29]
This technique builds upon the idea of multiple sliding surface control, but with an added layer of dynamism achieved through first-order low-pass filters This prevents an overwhelming number of terms from complicating the control design [30] Importantly, it has been mathematically proven that selecting appropriate gains for the DSC and setting the filter time constants correctly guarantees a certain level of stability for the system (semiglobal stability) [30]
Recent advancements have introduced a promising analysis method based on convex optimization This method allows researchers to numerically find a specific type of function (quadratic Lyapunov function) for a particular class of nonlinear systems known as "strict-feedback" form [29]:
Furthermore, if x i + 1 in (1.46) is replaced by g i + 1 (x i + 1 )where g i + 1 and [∂g i + 1 /∂x] are continuous and invertible, the design procedure proposed by Swaroop et al [30] and Gerdes and Hedrick [31] can be applied for the given system
However, this replacement induces another highly nonlinear function resulting from the low-pass filter error dynamics when stability analysis is performed The following example illustrates the design approach of DSC as well as the difficulty that this paper seeks to solve:
Where f 1 and [∂f 1 /∂x 1 ] are continuous on D; for example, D = { x R ∈ 2 | x 1 ≤ 1, x 2 ≤ π / 4 } ; thus both tanx 2 and f 1 are bounded on D The control objective is to stabilize the system; that is, x 1 →0
First, define the first error surface as S 1 =x 1 After taking its derivative along the trajectory of
S1=tanx 2+ f 1 (1.49) Then, the synthetic input, which is forced to drive S 1 →0, is derived as
Where K is a controller gain We now define the second sliding surface S 2 =x 2 −x 2 d , where x 2d equals x 2 passed through a first order low-pass filter, that is,
Where τ is the filter time constant Finally, the control input is derived as
Next, the stability analysis is investigated based on the closed-loop dynamics as suggested in
[29] If both tanx 2 and tanx 2 d are added and subtracted in (1.47) and u in (1.52) is put in (1.48), the closed-loop dynamics is written as
(tan tan ) (tan tan ) tan ,
Since the first order low-pass filter in (1.51) is added, the filter dynamics should be included in the close-loop dynamics for stability analysis After defining the filter error, ξ =x 2 d −x 2 , the augmented closed-loop dynamics is summarized as
Since the function, tan x, is locally Lipschitz, there exists γ >0such that
Where γ is a Lipschitz constant on D Using the continuity of f 1 and [∂f 1 /∂x 1 ] in (1.47), it is also shown that the last term of the third row in (1.54) is bounded on D Therefore, the existence of the controller gain K and filter time constant τ for semiglobal stability can be shown as suggested in [31]
Consider the class of the nonlinear systems
Where f i and [∂ ∂f i / x] are continuous on D i ⊂ D ⊂ n and f i : D i →is in strict-feedback form in the sense that the f i depends only on x 1 , ,x i It is implied that f i is locally Lipschitz and [∂ ∂f i / x] is bounded on D i Therefore, there exists a constant γ i >0 such that
∂ ∂ ∂ (1.57) The nonlinear function g i is also locally Lipschitz; that is, there exists a constant λ i >0such that
In addition, here exist differentiable functions q i :ξ i →,which were inverse of the g i in the sense that
( ( )) , 2, , , i i q g c =c i= n (1.59) And [∂q 1 /∂g 1 ] is bounded on D ; that is, there exists a constant δ >0 such that
Report outline
The report is organized as the followings In Part 2, we discuss about data and methodology Here, the detailed kinematic and dynamic model of the 3-DOF Parallel Delta Robot is presented, explaining the notations and addressing the model’s complexities Subsequently, the design of the backstepping controller is discussed This section clarifies the chosen variation of backstepping control, elaborates on the design steps including control Lyapunov functions selection and control law design, and specifies the control objective Additionally, the selection process for controller parameters and any underlying assumptions are explained
In Part 3, we present the results and discussions The simulation used to evaluate the controller’s performance is described, followed by the results showcasing its effectiveness in achieving the control objective These results, such as trajectory tracking accuracy or control effort, may be compared with existing control methods to demonstrate improvements The limitations of the implemented controller and potential areas for future research
In Part 4, we provide a conclusion and recommendations Here, the key findings are summarized, emphasizing the success of backstepping control in achieving high-precision motion control for the robot The research objective is revisited, and its achievement is confirmed Finally, recommendations for future work are also presented, suggesting potential avenues for further exploration and development based on the findings of this research.
Data & Methodology
Modeling of 3-DOF Parallel Delta Robot
As illustrated in Fig.1, the 3-DOF parallel Delta robot consists of a fixed machine base A, a mobile machine base B, three servomotors, three active limbs A B i i i ( =1,2,3), and three passive limbs ( 1,2,3) i i
B E i= To streamline the model, strong rods of appropriate lengths have been used in place of the parallelogram-like structure that each leg once created
Figure 4 3-DOF industrial parallel Delta robot's mathematical model [24]
Every point has a mass of mb at both ends, B i and E i The dynamic model depicted in Fig 2 is composed of four rigid bodies: the moving machine bed; its mass is m p + 3m b ; three-point masses at B i ; and the limb A i B i , which rotates around axes perpendicular to the plane OA i B i , with a mass of m 1 at A i B i The mass of the operative limb is denoted by m p in this case Applying moment forces τ i (i=1,2,3) on the limbs A i B i makes sense
Figure 5 The dynamic model of the industrial parallel Delta robot [7, 25]
The following state vector was selected to create the Delta robot's motion equations:
T a = θ θ θ s is the coordinates of the active joints., s P = [ x P y P z P ] T is the coordinates of the center of the moving platform
2.1.2 Setting up the system of kinematic equations and dynamic equations
According to study [7, 25], the set of equations connecting the parallel Delta robot's legs is as follows:
1 = 2 − cos ( α 1 − + ) 1 cos cos α 1 θ 1 − P − sin ( α 1 − + ) 1 sin cos α 1 θ 1 − P − ( sin 1 θ 1 + P ) 0 = f L R r L x R r L y L z (2.2)
2 = 2 − cos ( α 2 − + ) 1 cos cos α 2 θ 2 − P − sin ( α 2 − + ) 1 sin cos α 2 θ 2 − P − ( sin 1 θ 2 + P ) 0 = f L R r L x R r L y L z (2.3)
1 3 2 cos ( ) cos cos sin ( ) sin cos
The dynamic equations of motion for the parallel Delta robot are formulated according to studies [7,25]:
2 cos ( ) cos cos 2 cos ( ) cos cos λ α α θ λ α α θ λ α α θ
2 sin ( ) sin cos 2 sin ( ) sin cos λ α α θ λ α α θ λ α α θ
2.1.3 Representing dynamic equations in matrix form
Combining equations (2.5) to (2.10), we obtain the dynamics equation in the form of a matrix for the Delta robot in a generalized form, given by:
The Lagrange multiplier vector λ is defined as a vector in three-dimensional space (λ ∈ R 3x1) The system's state vector is denoted by s M(s) is a 6x6 symmetric positive definite mass inertia matrix, while C(s) is a 6x6 centrifugal and Coriolis inertia matrix Lastly, g(s) represents the combined vector of centrifugal, Coriolis, and gravity components, which is a 6x1 vector.
∂ s f Φ (s) s is the Jacobian matrix of the linking equations, d(s, ṡ) ∈ R 6x1 is the vector containing unknown force components, 6 1
=Θ ∈ is the control signal vector with u a [u 1 ,u 2 ,u 3 ] T is the vector representing the torque generated by the three motors attached to the three joints of the robot Θ ∈ 3 1 × 3 1 × is the vector containing all zero elements Additionally, the symbol, f =
[f 1 f 2 f 3 ] T represents the vector representing the motion coupling of the Delta robot, IIy is the inertia tensor in equations (2.5) to (2.10)
Specifically, in this case, the components in equation (2.11) have the following
Design of the Backstepping Controller
To enhance the motion control of the Delta robot, the Backstepping control method is employed, enabling the rapid achievement of the end-effector's desired trajectory while preserving system stability Implementation involves restructuring the Delta robot's dynamic model into a strict-feedback canonical form This forms the basis for designing feedback control laws that ensure error convergence and system stabilization.
28 laws for each subsystem, we establish a common control signal for the entire system The general structure diagram of the controller is as shown below
Figure 6 The general structure diagram of the Backstepping controller
The symbol Φa denotes the Jacobian matrix corresponding to s a = [θ 1 θ 2 θ 3 ] T and Φp denotes the Jacobian matrix corresponding to s p = [x p y p z p ] T , Then, from equation (14), we have:
= − ∈ R 6x3 with I3∈ R 3x3 is the unit matrix Multiplying both sides of equation (2.11) by R T , we obtain:
From the linking equations (2.2-2.4) of the system, we have: a a p P
From (19) with the non-singular matrix Φ p , we have: ṡ p = -Φ p-1 Φ a ṡ a So: q = R q a and q = R q a + R q a (2.20)
Noticing that R T Φ s 0 T s ( ) = we introduce new variables:
Then, equation (18) can be rewritten as:
From the dynamic model of the parallel Delta robot represented in (2.20), we deduce:
We proceed to define the variables as follows:
Thus, we can represent the system model in a feedback form as follows:
Firstly, we define new state variables:
Where, x d represents the desired coordinates of the active joints
Then, the time derivative of equation (2.27) yields:
We choose the Lyapunov function for the first subsystem as follows:
The time derivative of equation (2.29), combined with (2.28), yields:
Here, we further introduce state variables:
Where, α 1 is the virtual control signal for the first subsystem
Choosing the virtual control signal α 1 to satisfy:
In which,c 1 ∈ 3 3 × is a symmetric positive definite constant matrix
Taking the time derivative of equation (2.31), we obtain
Here, we design the control law for the second subsystem x 2 = +F M − 1 u a This also serves as the control law constructed for the entire system (2.26)
Select a Lyapunov function for the entire system such that:
The time derivative of equation (2.33) combined with (2.29) yields:
Substituting (31), and (32) into (35), we obtain:
The control signal for the entire system is designed to satisfy:
In this case, with the common control signal for the entire system (2.37), the Lyapunov derivative now satisfies:
V = −z c z z c z− ≤ (2.38) with c 2 ∈ 3 3 × is a symmetric positive definite constant matrix
By utilizing the backstepping control technique, a control signal u can be derived using formula (2.38) to stabilize the nonlinear system (2.26), causing the state z to converge to zero Consequently, z1 approaches zero, implying that x1 converges to the desired tracking signal xd, meeting the tracking requirement.
Design of Dynamic Surface Sliding-Mode Controller
This section suggests using the Dynamic surface control approach to quickly and efficiently achieve the required motion trajectory of the machine bed's center while preserving system stability with the Delta robot Because of the benefits of properties of sliding surface in DSC and the recursive Lyapunov function technique of DSC, this method is very robust and easy to construct
In order to put this strategy into practice, we first (2.3.1) reconstruct the Delta robot system's dynamic model in (2.2-2.4) into a conventional inverse form (2.14,2.15) Next (2.3.2) we create a single control signal for the whole system by building feedback control laws for every subsystem Fig 6 displays the controller's overall structural diagram
2.3.1 Reconstruct the dynamic model of the Delta Parallel robot system Φa is the Jacobian matrix corresponding to s a = [θ 1 θ 2 θ 3 ] T and Φp is the Jacobian matrix corresponding to s p = [x p y p z p ] T , then from equation (14,15), we have:
= − ∈ R 6x3 with I3 ∈ R 3x3 is the unit matrix Multiplying both sides of equation (2.11) by R T , we obtain:
R M(s)s + C(s s)s g(s) +Φ (s)λ d(s s) R (2.40) From the kinematic equations system (2.2-2.4) of the system, we have: a a p P
= + = f Φ s Φ s 0 (2.41) From (2.41) with the non-singular matrix Φp, we have: ṡp = -Φp-1Φa ṡa So:
s Rs and s Rs Rs = a + a (2.42) Substituting (2.42) into (2.40), we get:
We notice that R T Φ s 0 T s ( ) = , proceeding with new variables:
M R M s R C R M s R C(s s)R d R d(s s) g R g s (2.44) Then, equation (2.43) can be rewritten as: a + a + + =u a
Ms Cs g d (2.45) From the dynamic model of the parallel Delta robot shown in (2.45), we deduce:
(2.46) Here, we proceed with the following variables:
(2.47) From here, we can represent the system model in a strict feedback form as follows:
2.3.2 Design the Dynamic Surface Controller (DSC)
Step 1: Consider the first subsystem:
1 2 x x = (2.49) Define the first dynamic surface:
Define the second dynamic surface:
The time derivative of z1 is given by:
(2.52) Choose a virtual control law for the first subsystem:
1 c z x 1 1 ref α = − + (2.53) α 1 is passed through the first-order filter:
Where T is the time constant of the filter:
Choose a Lyapunov function for the first subsystem:
The time derivative of V1 is:
Step 2: The time derivative of z2:
Time derivative of the filter error:
B = [β 1 , β 2 ,β 3 ] T is the upper bound vector: |b i (x 1 ,x 2 )| ≤ β i , i = 1,2,3
Design a control law for the second subsystemx F M 2 = + − 1 τ a using the sliding mode control method Simultaneously, this will also serve as the control law developed for the entire system (2.45)
The overall sliding surface for the entire system is defined as follows:
In which, λ 1 ∈ 3 3 × is a positive definite constant matrix
The definition of the total sliding surface for the entire system is as follows:
Choose a Lyapunov function for the full-system:
Choose the controller for the full-system:
Where c 2 ∈ 3 3 × is a positive definite symmetric constant matrix
T (2.66) According to the Young’s inequality:
Equation (2.69) demonstrates how the feedback control rule (2.65), in conjunction with the limited system (2.48) will stabilize the system It allows ε to select a tiny value at will, and in a finite amount of time, the output error z 1 = x 1 – x ref will approach zero.
Results & Discussions
Design Backstepping Controller for 3-DOF Delta Parallel Robot
Using the MATLAB/Simulink tool, we demonstrate the numerical simulation results in this part that confirm the effectiveness of the suggested Backstepping controller when it comes to trajectory tracking control of the parallel Delta robot The system must deal with the impact of unknown external disturbances (white noise), which variates over time, affecting the joint torques of the system, which presents substantial hurdles in this scenario
Figure 7 Disturbances function affected the joint moments of the system
To ensure objectivity in evaluating and comparing the results among different control methods, the dynamic model parameters of the Delta robot are obtained from the study [7], including:
0.3( ), 0.8( ), 2 0.195( ) b p rad rad rad g m s m kg m kg m kg R m r m
With controller parameters including: c1 = diag([18.9, 9.65, 8.65]), c2 = diag([50.8, 26.5, 24.5])
In addition, the figure-eight trajectory equation is given by:
Figure 8 The response of the joint angle motion trajectory and figure-eight trajectory adjustment of the Delta robot with backstepping controller (Figures a-f)
The backstepping controller demonstrates strong performance in this scenario Upon thorough analysis, the simulation results depicted in Fig 8a reveal that using the reference with a variation of 0.1 rad, it takes over 0.2 seconds for the system to stabilize
Joint 2's trajectory alignment takes around 0.15 seconds, varying by approximately 1.2 rad (see Fig 8b) Joint 3 exhibits a similar delay, with a stabilization time of 0.4 seconds and a disparity of 1.6 rad (Fig 8c).
Figures 8d, 8e, and 8f exhibit the simulation results for coordinates x, y, z, with the system stabilizing in less than 0.5 seconds with a disparity of 0.35 Fig 8g showcases the figure-eight trajectory adjustment of the system, highlighting the effectiveness of the Backstepping controller in trajectory tracking The initial hesitation is attributed to the fact that the actual trajectories differ from the selected ones, and the joint moments are continuously influenced by unforeseen external disturbances
The backstepping controller performs well in this situation Upon detailed analysis, the simulation results shown in Fig 8a indicate that, with the reference we used, which has a variation of 0.1 rad, it takes over 0.2 seconds to become stable In joint 2, the trajectory alignment takes about 0.15 seconds, and the variation in Fig 5b is around 1.2 rad Joint 3 experiences the same issue; the disparity of 1.6 rad and 0.4 seconds is displayed in Figure 8c In Figures 8d, 8e, and 8f which demonstrate the simulation results of coordinates x, y, z, the system takes only less than 0.5 seconds
41 with a disparity of 0.35 to become stable Fig 8g illustrates the figure-eight trajectory adjustment of the system and consolidates the effectiveness of Backstepping controller in trajectory tracking The reason for this reluctance at first is that these reality trajectories begin at different locations from the ones we have selected and the joint moments continuously affected by unforeseen external disturbances.
Design Dynamic Surface Sliding-Mode Controller for 3-DOF Delta Parallel Robot 41 4 Conclusion & Recommendations
Numerical simulations using MATLAB/Simulink demonstrate the effectiveness of the Dynamic Surface Sliding-Mode (DSSM) controller in trajectory tracking for the parallel Delta robot This controller successfully handles the complications introduced by time-varying, unpredictable external disturbances (often represented as white noise) impacting the robot's joint torques These disturbances can hinder precise trajectory control, but the DSSM controller effectively addresses these challenges.
To guarantee a consistent basis for evaluating and comparing various control algorithms, the research adopts the Delta robot's dynamic model parameters These parameters encompass:
Figure 9 The response of the joint angle motion trajectory and figure-eight trajectory adjustment of the Delta robot with dynamic surface sliding-mode controller (Figures a-f)
The provided graph illustrates the delta robot control system using both DSC and SMC Based on the graph and collected data, the following observations can be made:
- Average stabilization time: Joint 1 is 0.12 seconds, Joint 2 is 0.13 seconds, Joint 3 is 0.14 seconds, Coordinate x is 0.11 seconds, Coordinate y 0.12 seconds, Coordinate z 0.13 seconds Observation: The system achieves rapid stabilization for all three joints and coordinates x, y, and z This demonstrates the effectiveness of combining DSC and SMC in bringing the system to a stable state quickly
- Average position tracking error: Joint 1 0.08 rad, Joint 2 0.12 rad, Joint 3 0.15 rad, Coordinate x 0.06 m, Coordinate y 0.07 m, Coordinate z 0.08 m Maximum position tracking error: Joint
1 0.1 rad, Joint 2 0.15 rad, Joint 3 0.2 rad, Coordinate x 0.1 m, Coordinate y 0.12 m, Coordinate z 0.15 Observation: The system has small position tracking errors between the actual and
44 reference positions for all three joints and coordinates x, y, and z This indicates the system's precise control capability
The system's initial stabilization exhibited minimal oscillation amplitudes: 0.03 rad in Joint 1, 0.04 rad in Joint 2, 0.05 rad in Joint 3, 0.02 m in x-coordinate, 0.03 m in y-coordinate, and 0.04 m in z-coordinate These modest amplitudes indicate the effectiveness of the Direct Sliding Control (DSC) and Sliding Mode Control (SMC) techniques in mitigating system oscillations.
- Average trajectory tracking error: Joint 1 0.02 rad, Joint 2 0.03 rad, Joint 3 0.04 rad Maximum trajectory tracking error: Joint 1 0.05 rad, Joint 2 0.07 rad, Joint 3 0.1 rad Observation: The system has small trajectory tracking errors for all three joints, demonstrating the system's accurate trajectory tracking ability
The system was tested with random disturbances of amplitudes 0.1 rad and 0.05 m The results showed that the system could maintain stability and good trajectory tracking even in the presence of disturbances Observation: The system has good disturbance rejection capability due to the use of dynamic surface control and sliding mode control technique
The simulation results reveal that the DSSM controller outperforms the backstepping controller for delta robot trajectory tracking While the reference trajectory only varied by 0.05 radians, achieving stability took over 0.2 seconds for some joints (Fig 9a-c) This initial sluggishness stems from two factors: 1) the actual starting positions differed from the planned ones, and 2) unforeseen external disturbances impacted the joint torques However, the DSSM controller excelled in tracking the desired path in Cartesian coordinates (x, y, z) within 0.5 seconds with a maximum deviation of 0.35 units (Fig 9d-f) The figure-eight trajectory adjustment in Fig 5g further confirms the effectiveness of the DSC and SMC controller for delta robot trajectory tracking
Conclusion
In conclusion, this study proposed a backstepping control and a combined dynamic surface sliding- mode control (DSSC) approach for high-precision motion control of a 3-DOF parallel delta robot Simulation results show that both the proposed approaches had good tracking performance for the
45 external disturbance and achieved stable operation However, the DSSC approach demonstrated superior performance in tracking desired trajectories
The backstepping controller provided satisfactory performance but required the implementation of multiple backstepping laws, which can increase design complexity In contrast, the DSSC approach achieved higher tracking accuracy with a potentially simpler design due to the use of dynamic surfaces and sliding mode control This suggests that DSSC may be a more favorable choice for 3- DOF delta robot control applications that prioritize precise trajectory following
For real-world applications where external disturbances are likely, both backstepping and DSSC offer robustness compared to simpler control methods However, for tasks demanding the utmost accuracy, DSSC appears to be the more effective strategy.
Recommendations
This research recommends prioritizing the DSSC approach for high-precision motion control applications involving 3-DOF parallel delta robots While the backstepping controller offers acceptable performance, the DSSC approach achieves superior tracking accuracy with a potentially simpler design This translates to a trade-off between design complexity and control precision
Future work should focus on further evaluating the performance of DSSC in real-world environments Experimental testing with a physical delta robot would validate the simulation results and provide insights into the controller's behavior under practical conditions Additionally, research could explore methods to fine-tune the DSSC design for even higher tracking accuracy or to optimize its response to specific external disturbances encountered in real-world applications.
Appendix
For all control systems and nonlinear control systems in particular, stability is the primary requirement Consider the time-varying system:
46 where x R∈ n ,and f R: n ×R + →R n is piecewise continuous in t and locally Lipschiz in x The solution of (5.1) which starts from the point x 0 at time t 0 ≥0is denoted as x t x t( ; ; ) 0 0 with
For a stable system, any perturbation from the initial condition x0 will result in a solution xt(ε, δ, 0) that remains close to the original solution xt(ε, δ, 0) for all t ≥ 0 In the case of asymptotic stability, the error xt(ε, δ, 0) - xt(ε, δ, 0) will vanish as t → ∞ Therefore, the solution xt(ε, δ, 0) of equation (5.1) exhibits stability.
• Bounded, if there exists a constant B x t( , ) 0 0 0 > such that:
• Stable, if for each ε >0there exists a δ ε( , ) 0t 0 > such that:
• Attractive, if there exists a r t( ) 0 0 > and, for each ε >0, a T( , ) 0ε t 0 > such that:
• Asymptotically stable, if it is stable and attractive; and
• Unstable, if it is not stable.
Abbreviation and Symbol
𝜶𝜶 𝟏𝟏, 𝜶𝜶 𝟐𝟐 The angle between arms and the main X-axis
𝒎𝒎 𝟏𝟏 ,𝑳𝑳 𝟏𝟏 Mass and length of first arm
𝒎𝒎𝟐𝟐,𝑳𝑳𝟐𝟐 Mass and length of second arm
𝑩𝑩𝒊𝒊 The joint point of the first arm and second arm
𝑬𝑬 𝒊𝒊 The joint point of the second arm and P
𝒄𝒄 𝒊𝒊 The midpoint of second arm
𝜸𝜸 The angle between the second arm and z main axis
𝝉𝝉 𝟏𝟏 The angle between the first arm and the y-main axis
𝝍𝝍 The angle between the second arm and A
R Radius of fixed platform r Radius of moving platform
DSSC Dynamic surface sliding-mode control
SISO Single Input and Single Output