Adaptive constraint-following control for maintenance robotic manipulator with uncertainties

Abstract

To enhance the reliability and motion stability of the maintenance robotic manipulator system for armored vehicles, we propose a novel adaptive robust control methodology based on the Udwadia-Kalaba (UK) dynamic framework. To address the challenges such as parameter uncertainties and external disturbances in the system, a crucial leakage mechanism is integrated into the adaptive law, which not only suppresses the unbounded growth of the adaptive parameters under continuous disturbances, but also automatically attenuates the control gain when the error converges, significantly improving the smoothness of the controller’s transient response. Finally, we validate the effectiveness of the proposed controller via co-simulation in a multibody dynamics environment.

Introduction

Background

The application of the manipulator system in the industrial field has expanded from routine production to some special working conditions¹. Researchers have solved the maintenance, processing, and transportation problems that humans face in environments with high temperature, high pressure, and high vibration through technological innovation^2,3. Maintaining manipulator is a representative application in the field of mechanical manipulator technology, in areas such as underwater maintenance⁴, wind power maintenance⁵ and unmanned space station repairs^6,7, manipulators must ensure the accuracy of the positioning of their ends while overcoming external vibrations, which places highly high performance requirements on the control system.

Formulation of the problem of interest for this investigation

Unlike the manipulator used under normal working conditions, the multifunctional maintenance manipulator integrated into a certain type of unmanned heavy vehicle operates under highly harsh conditions⁸. The manipulator system must counteract the frequent vibrations caused by the vehicle’s movement while ensuring precise control of the manipulator position^9,10. This is a challenging issue in the field of control, as there are always various uncertainties in the system, including the system parameter uncertainties caused by the end of the manipulator grasping different tools and the external disturbances caused by vehicle vibrations.

However, precisely determining the boundaries of these uncertainties is highly difficult, and controllers that rely on known boundaries may not be able to guarantee the performance and stability of the system when the boundaries are unknown or time-varying. Therefore, this paper aims to address a key question: When there are uncertainties in the constraints, dynamic model, and external disturbances of a mechanical system, and the boundaries of these uncertainties are also unknown, can we design a controller to ensure the stability and acceptable tracking performance of the system?

Literature survey

In the past, people have done a lot of research and innovation on the control problem of robotic manipulators, for the control problems of some low-degree-of-freedom robotic manipulator under normal operating conditions, PID controllers are widely used in industry due to their simplicity and model-free tuning. However, their performance degrades in highly nonlinear, coupled, and uncertain systems¹¹. By improving and correcting the conventional PID controller, online self-tuning of control parameters can be achieved¹². However, as a linear control method, the PID controller itself has inherent deficiencies in dealing with system nonlinearity, coupling, and uncertainty¹³. More importantly, if system parameters or external disturbances change rapidly, the integral section of the PID controller may not have enough time to respond, resulting in poor tracking performance. Sliding mode control is favored in the control of nonlinear systems due to its simple implementation and strong robustness^14,15. It should be noted that the traditional SMC control gain design relies on an accurate estimation of the upper bound of system uncertainties, and an overly conservative gain design can exacerbate the chattering problem.

To accurately observe the working status, performance, or parameters in real-time of the robotic manipulator, and to grasp the motion indicators of the system, the addition of adaptive control can ensure excellent tracking performance of the manipulator system under uncertainties and faults^16,17. To address the modeling and control issues of high-degree-of-freedom and flexible manipulators, many researchers have integrated artificial intelligence techniques into adaptive robust control¹⁸. Zhang et al.¹⁹ proposed an adaptive controller that leverages deep neural network architectures to address the unique challenges posed by transporting unknown loads in a space environment. Song et al.²⁰ proposed a Lyapunov-based finite-time convergence control scheme for articulated robotic manipulators operating under vibrational perturbations. Hu et al.²¹ proposed a practical method that combines feedback linearization control with a model based on multi-layer neural networks (MNN) to handle parameter uncertainties and external disturbances. For mechanical systems with uncertainties, the intervention of adaptive laws often leads to satisfactory results^22–24. Notably, the intervention of adaptive laws plays a significant role in improving the system’s accuracy in the control problem of complex robotic manipulator systems, especially under vibration conditions and uncertainty perturbation. Although these traditional robust control methods can handle uncertainties in the system, such uncertainties are limited to those that are constant and slowly changing.

To explore a more effective method for handling rapidly varying uncertainties and to provide stable guaranties for transient processes of system states or outputs, Chen et al.^25,26 first proposed the concept of constrained following error from the perspective of servo constraints, which is based on the Udwadia-Kalaba theory^27,28. This transformed the motion control problem of complex, time-varying uncertain mechanical systems into a problem of how to make the controlled system move according to the given constraints (or inequality constraints)²⁹. By combining all the uncertain terms together, the boundary structure of the uncertainties can be determined and used to suppress the uncertainties. This method has been proven to be capable of conducting a detailed dynamic analysis of the manipulator system and improving the fundamental understanding of constrained motion in multibody dynamics. It has been applied to solve problems such as multi-link systems^30–32, flexible manipulator systems^33,34, as well as trajectory tracking control and collaborative position and attitude control of multiple manipulators^35–37.

In conclusion, regarding the possible uncertainties in the system and the situations where deviations from the initial conditions may occur, it is reasonable to expect that with a class of adaptive robust control laws that can ensure uniform boundedness and uniform ultimate boundedness, the system can approximately follow the constraints. Although existing research efforts have partially addressed the control issues in multi-degree-of-freedom manipulator systems under various working conditions, there is still some room for research in improving the tracking control accuracy of strict uncertain systems under severe disturbance conditions. The adaptive control employed in this paper provides a powerful tool for addressing the rapid changes and uncertainties of the system state.

Scope and contribution of this Study

The subject of this study is a complex mechanical system with constraints (which can also be extended to non-complete constraints), and its key feature is that the system contains unknown but bounded uncertainties. The control objective is to achieve constraint tracking within an acceptable small neighborhood, rather than pursuing perfect and precise tracking in the presence of uncertainties.

The main contributions are as follows. (i) In many robust control methods for practical engineering applications, it is difficult to accurately determine the boundaries of system uncertainties. The core contribution of this study lies in the proposed control framework, which does not require prior knowledge of the exact boundaries of uncertainties. We estimate the comprehensive impact of uncertainties online by designing a new adaptive law, enabling the system to provide rigorous guaranties even when faced with rapidly changing and unknown uncertainties at the boundaries. (ii) Our work is based on the constraint mechanics framework proposed by Udwadia and Kalaba. The novelty of this study lies in systematically integrating adaptive robust control theory with a leakage mechanism into the UK framework and applying it to situations with significant model uncertainties. This enhances the robustness and performance of the controller in preventing parameter drift and avoiding overly aggressive control inputs, enabling the UK theory to move from an ideal model to more challenging practical applications. (iii) We innovatively applied this controller to complex mechanical systems operating in high-vibration environments, enhancing its value in practical applications. We established a complete multibody dynamics model that incorporates the dynamic characteristics of the vehicle platform. This setup reproduces vibration disturbances that are difficult to model directly in the controller design. This allowed us to reveal the actual performance of the controller in suppressing unknown and complex disturbances introduced by the platform motion, and to conduct a more comprehensive and rigorous test of the controller’s robustness.

Organization of the paper

The organization of this paper is as follows. The description of the constraint modeling method for systems with uncertainties is presented in Sect. Constrained mechanical system with uncertainty: The agreement on the constraints. The design proof of the adaptive robust controller and the verification process on a simple example are presented in Sect. Adaptive robust control design. Analytical dynamic modeling of the manipulator system and the definition of constraint errors are presented in Sect. Constrained maintenance manipulator with Uncertainties. Multibody dynamics modeling of the manipulator and vehicle platform is presented in Sect. Multibody dynamic model of the maintenance manipulator system and co-simulation procedure, at the same time, we also elaborated on the overall simulation framework of the co-simulation. In Sect. Simulation, we summarize the model parameters and control system parameters, and analyses the simulation results to verify the performance of the controller.

Constrained mechanical system with uncertainty: the agreement on the constraints

Consider the following mechanical system with uncertainties^31,38:

1

where is the time, is an uncertain vector, denote the general coordinate, velocity and acceleration, denotes the control input, is the inertia matrix, is the Coriolis force matrix, is the gravitation vector. Furthermore, the system equations are all continuous, and the dimensions of , and are designed to match each other.

Consider a group of constraints of the following form:

2

where (2) denotes the first-order derivative of the prescribed constraint with respect to t, k=1,2,⋯ , m, and represents the partial derivative term of ϕ, represents the partial derivative term of t. represents the variable element of the i-th state of the mechanical system. Eq. (2) can be written in matrix form as³⁹

3

where , .

The next step is to take the derivative of (2) with respect to t, then

4

where

5

6

The second-order constraints can be obtained as follows:

7

Equation (7) can be written in matrix form as

8

where O(ϕ ,t) is an m × n matrix, c is an m-dimensional column vector.

Thus, we define a constraint−following error as follows

9

the second−order error is as follows

10

Remark 1

The main research content of this paper is servo constraints, that is, to design a control scheme to generate constraint forces, so that the mechanical system can meet the target constraint conditions. At the same time, the constraint design method considered in this paper is compatible with both holonomic and nonholonomic constraints, and the second-order form of constraints has wide applications. For nonholonomic constraint problems, the core idea is to transform the constraints into a control objective, the controller is designed to actively drive the system to ensure that its position and velocity meet the nonholonomic constraint equation. Even if the initial state of the system does not satisfy the constraints, the controller can gradually reduce the constraint error.

Adaptive robust control design

Dynamic modeling with uncertainties

A common practice in modeling mechanical systems with uncertainties is to decompose the model into a nominal part and an uncertainty part, and the uncertainty part can be further categorized as parametric and dynamic uncertainties. For the matrices H(· ), L(· ) and G(· ) mentioned in (1), the same structured approach can also be applied for its division:

11

Let

12

thus

13

By rearranging the kinematic constraints in (3) and (8), we obtain the ideal constraint force/torque (i.e. Udawadia−Kalaba equation) that enforces the constraints exactly.

Theorem 1

In accordance with d'Alembert's principle, the constraint force not only performs zero virtual work but also satisfies all constraints at any time with minimal control effort, as rigorously formulated in the generalized framework of Udwadia and Kalaba. When the uncertainty of the system is ignored, the constraint force that makes the system comply with the constraints expressed by (8) takes the following form:

14

where “+” denotes the Moore-Penrose pseudoinverse, the input force in (14) in closed format obeys the Gaussian principle and renders the constraint strictly satisfied.

Adaptive robust control design

Before presenting the controller design, several key assumptions that play a crucial role in subsequent development are introduced.

Assumption 1

Let

15

16

where has full rank, Q>0. There exists an unknown constant , such that for any , the inequality

17

can be always satisfied. Here stands for the input matrix uncertainty.

Remark 2

The constant is in general unknown. Assumption 1 stipulates that there exists a lower bound for the influence of a specific matrix caused by uncertainty. In an ideal, uncertainty-free special case (i.e. ), we have E = 0 and ϖ = 0. At this point, Eq. (17) simplifies to , and the assumption is valid. According to the principle of mathematical continuity, as long as the uncertainty remains within a certain threshold, Assumption 1 will continue to hold.

Assumption 2

1. To determine the boundaries of the uncertainty in the mechanical system, assume that there exists an unknown constant vector and a known function , such that

   18

2. For each , the function has the following properties:

• The function is first-order differentiable.

• The function is concave.

• The function is non-decreasing in χ.

Thus for any and :

19

Remark 3

In Assumption 2, we acknowledge that all uncertainties (model errors, external disturbances) in the system will have a combined interference effect. However, this combined effect in the worst-case scenario can be enclosed or contained. Meanwhile, smoothness and monotonicity are realistic, but concavity is a stronger and somewhat conservative mathematical assumption, this assumption can be satisfied through judicious selection of the , thereby providing stronger performance guarantees for practical control applications. It should be noted that physical uncertainties in practical systems may exhibit convex or non-concave characteristics. Alternative approaches based on Lipschitz continuity conditions combined with projection-type adaptive laws may potentially relax this requirement⁴⁰. In some systems, the perturbation effect may linearly or even exponentially increase as the intensity of uncertainty increases, whose main purpose is to simplify and strengthen the mathematical proof of the control algorithm.

Remark 4

Assumptions 1 and 2 serve as the fundamental pillars for constructing the adaptive robust control framework proposed in this paper and for conducting the corresponding stability analysis. Assumption 1 imposes a structural boundary on the uncertainties, ensuring that their influence does not exceed the capability range of the controller. This assumption is a key technical prerequisite for the controller to effectively suppress the uncertainties and for the Lyapunov stability proof to hold. Assumption 2 provides a theoretical basis for the adaptive part of the controller, parameterizing the complex, state-related unknown uncertainty boundaries through a known function Ω and an unknown constant vector χ, and stipulating the mathematical properties that this boundary function Ω must satisfy. This transformation simplifies the control task from dealing with an unknown function boundary to online estimation of a finite unknown parameter vector, making adaptive control possible.

Since the boundary information of the uncertainty parameter δ is unknown, it follows that χ is also unknown. To perform an online estimate of the unknown parameters, we define an adaptive law as follows

20

where , is the w-th element of , w=1,2,⋯ ,u, the design parameters , and

21

For all :

• The function f(· ) is non-increasing ().

• The function f(· ) is bounded.

Remark 5

The second term on the right side of (20) is a leakage term (that is, the adaptive law belongs to the leakage type), and its leakage amount is controlled by a nonlinear gain related to performance, what distinguishes it from traditional leakage items is that the leakage rate is designed to vary with the error . Specifically, when the system error is small, the leakage gain increases, and the control system generates more leakage, which can effectively prevent the adaptive parameters from undergoing unnecessary drift or overestimation during the steady state, thereby reducing the control gain and making the control smooth. When the system error is large, the opposite occurs, which enables the controller to compensate for uncertainties more aggressively, thereby achieving rapid convergence of errors and helping to ensure the rationality of the uncertainty bound estimation.

The robust inputs and are designed as follows:

22

where is the tunable scalar, has already been described in (15), constraint error has already been described in (9).

23

where

24

25

where ε > 0 is the tunable scalar.

With (14), (22) and (23), we propose the following adaptive robust constraint-following controller:

26

Theorem 2

Let . Consider the mechanical system (1), subject to Assumptions 1 and 2, the following robust performances can be guaranteed by (26).

1. Uniform boundedness: For any r>0 and , there always exists a positive constant d(r)<∞ such that for all : .

2. Uniform ultimate boundedness: For any r>0 and , when , there always exists a positive constant , such that for any r>0 and , where .

Proof

To analyze the stability of the non-autonomous system, we select a candidate for the time-invariant Lyapunov function as follows:

27

Although the function itself does not explicitly depend on time, its value evolves along the trajectories of the system, taking derivative of , it yields

28

using (11) and the uncertainty decomposition in (8), we have

29

where

30

by invoking Assumption 1, 2 and (13), it leads to

31

using in (22),

32

by according to (12),

33

By in (20),

34

where

35

Then we have

36

According to (24),

37

as we have

38

as we have

39

By Assumption 2(2) we have

40

note that , , we have

41

Invoking the arguments as in Refs. 41 and 42, the uniform boundedness is analyzed,

42

where

43

and

44

45

and the uniform ultimate boundedness with

46

where is guaranteed. □

Key variables and parameters of the control system are listed in Table 1.

Table 1.

Nomenclature of the maintenance manipulator.

H	Inertia matrix	L	Coriolis force matrix
G	Gravitation vector	τ	Control input
δ	Uncertain vector	O	First-order constraint auxiliary matrix
d	First-order constraint auxiliary vector	c	Second-order constraint auxiliary vector
	Nominal value		Uncertain value
	Nominal constraint force	Ω	boundary function
	Boundary constant	χ	Actual adaptive constant
	Estimated adaptive constant		Adaptive gain
	Leakage gain		Robust control gain
	Lyapunov function candidate		Angle error
	Target angle		Constraint error gain
ϑ	Constraint-following error

Remark 6

In this section, an adaptive robust control strategy with uncertainty based on general mechanical systems is proposed, this setup reproduces vibration disturbances that are difficult to model directly in controller design. For complex nonlinear and non-smooth dynamics such as friction and clearance, although this controller has certain robustness, it is not explicitly handled in the model. The outstanding feature of the algorithm is that the control problem of the manipulator under target disturbance is transformed into a kind of constraint-following control problem, and the constraint error is uniformly bounded and always finally bounded.

Remark 7

We acknowledge that the design premise of controller (26) is to be able to obtain the complete state of the system in real time, namely the generalized coordinates ϕ and their corresponding velocities , which is an idealized assumption. In many engineering applications, usually only the position can be directly measured by sensors such as encoders, while the velocity signal is often estimated through numerical differentiation. Therefore, a key research direction in the future is to integrate this control framework with a state observer^40,43–45, thereby relaxing the requirement for full state feedback and significantly enhancing the practical application value of the controller in real vibration environments.

A basic example

Before conducting the co-simulation experiment for the complex system, we used a simple inverted pendulum example to verify the controller. Its structure is shown in Fig. 1. According to the mechanical system description method in (1), the dynamic equation of the inverted pendulum is

47

where J>0 represents the moment of inertia, m represents the mass of the swinging arm, L>0 represents the distance from the rotation center O to the center of mass, ϕ represents the rotation angle of the inverted pendulum.

Fig. 1.

A simple example of inverted pendulum.

Here, we consider incorporating certain uncertainty factors into the rotational inertia and mass, and model the parameter uncertainty as a time-varying function to simulate the lumped uncertainty and changes in task load. Let , . Consider setting the angle target of the inverted pendulum system to , according to the definition of constraints (3) and (8), the first-order and second-order constraints of the system can be expressed as and .

We selected the characteristic parameters of the inverted pendulum system as: , , , , L = 2, g = 9.8. The definition parameters of the constraints and control system are chosen as follows: ζ = 2.5, Q=1, , ε = 0.05, , , .

The simulation results are shown in Figs. 2 and 3. The results indicate that the inverted pendulum can gradually stabilize at the predetermined angle according to the constraints, and the adaptive parameters gradually converge.

Fig. 2.

Angular displacement.

Fig. 3.

Adaptive parameter.

Constrained maintenance manipulator with Uncertainties

Dynamic equation of maintenance manipulator

In Fig. 4, we consider a 3 DoF maintenance manipulator for maintenance tasks under different operating conditions, the DH parameters of the maintenance manipulator are shown in Table 2. The second kind of Lagrangian equation is used to construct the dynamic equation by solving the kinetic energy and potential energy of the manipulator system.

Fig. 4.

Three-degree-of-freedom manipulator.

Table 2.

DH parameters of the maintenance manipulator.

Joint i
1	0	0
2		-π /2	0
3		0	0

The 3DoF maintenance manipulator system is converted into the form of the aforementioned model (1), taking into account the uncertain parameter δ, where , , , , , and

48

49

50

where , are the associated transformation matrices, ‘ is the Jacobian matrix:

51

denotes the vector of homogeneous coordinates that represent the position of the center of mass of the link j, expressed in the reference frame j. The mass of the link , the center of mass of the rigid body , the moment of inertia of the three axes , and are used to describe the mass distribution of the rigid body. Detailed descriptions of , , and in the dynamic equation can be found in the appendix. The nomenclature of the maintenance manipulator system is shown in Table 3.

Table 3.

Nomenclature of the maintenance manipulator.

	Link mass		Link length
	Link offset		Joint angle
	Moment of inertia		Distance of the center of mass
g	Acceleration of gravity		Inertia matrix components
	Coriolis force matrix components		Gravitation vector components
,	Associated transformation matrices		Jacobian matrix

Remark 8

In the dynamic equation of the maintenance manipulator, the system features a positive-definite inertia matrix H with uncertain mass and moment of inertia: , , where and are defined as nominal values, and the uncertainty is modeled by time-varying components and of unknown magnitude.

Stability constraints on maintenance manipulator

The joint stability control objective of the manipulator is described in the form of constraints (3) and (8). Define the target angle of the joint i as , then the tracking error can be defined as

52

where . Taking the derivative of (52) with respect to t, we have

53

Then, for any tracking error at the initial time, if is true, the errors and gradually converge to 0, where is a constant diagonal matrix.

Proof

Solve the differential equation system in (53) to obtain the expression of the error

54

Therefore, as , , and the convergence rate of the error depends on the constant .

Constraints can be organized into constraint matrix equations in the form of ((3)):

55

where

56

57

By differentiating the above equation with respect to t, the constraint in second-order form can be obtained as

58

where the target angle is defined as a function that varies with time t in this case, and

59

Finally, based on ((9)), the constraint-following error can be constructed as:

60

□

Remark 9

At this point, the angle and posture control problem of the robotic manipulator system has transformed into a type of problem that requires the system to meet the expected constraints (55) and (58). Based on the existing system kinematic model and dynamic model, the multibody dynamic modeling process needs to be described in the virtual prototype environment to verify the applicability of the control system in the real vibration environment.

Multibody dynamic model of the maintenance manipulator system and co-simulation procedure

Multibody dynamic model

To evaluate the performance of the proposed adaptive robust control method in practical systems with complex vibration conditions and strong nonlinear coupling, this study constructed a high-fidelity multibody dynamics model in the RecurDyn environment, including the vehicle part, the manipulator part and the road environment part, as shown in Fig. 5, and finally conducted a comprehensive simulation verification.

Fig. 5.

Multibody dynamic model of the manipulator system.

• (i)Vehicle part: To enhance the vehicle’s passability in off-road terrains, the vehicle system adopts a tracked chassis. The track system generally uses steel torsion bar springs as elastic elements, through the deformation of the elastic elements, the impact energy of the road is consumed, and the vibration of the vehicle is attenuated. In this paper, the nonlinear stiffness and damping characteristics of torsion bar springs are defined by writing subroutines.
• (ii)Manipulator part: In the multibody dynamics environment, the model strictly adheres to the preset DH parameters to define the geometric relationship of the connecting rods and the axes of the rotating joints, endows each connecting rod with mass, center of mass position, and inertia tensor, and adds viscous friction at the joints. The core driving force of the control algorithm is the joint torque, this torque signal will be directly applied to the drive ports of each rotating joint through the Simulink Co-Simulation interface to simulate the motor output.

Modeling of pavement irregularity

To simulate vehicle movement on uneven roads, the harmonic superposition method is used to generate random road irregularities, which has been verified through experiments⁴⁶. The program is strictly designed according to the target power spectral density (PSD) of the ISO 8608 standard, and special consideration is given to the spatial coherence of the road excitation received by the left and right tracks. By introducing a coherence function, one ensures that the excitation on both sides is correlated and has differences. The spatial frequency of the road surface is evenly divided into N intervals, the three-dimensional pavement irregularity function can be expressed as

61

62

where represents the vibration amplitude of the harmonic corresponding to the center frequency , represents the corresponding power spectral density value, Δ n represents the range of the interval. represents the center frequency of the i-th sine function component in the pavement roughness function, y represents the displacement perpendicular to the road surface direction, is a random number. Finally, the generated road elevation data is imported into the multibody dynamics software.

Co-simulation and control procedure of the system

Based on the description of the nonlinear and complex time-varying uncertainty control of the system in the previous sections, this section introduces the issue of the control target being affected by environmental disturbances, proposes a class of adaptive robust constrained following control strategies for the maintenance manipulator, and designs the target constraints. This section constructs the multibody dynamics model of the maintenance manipulator. To achieve the goal of co-simulation control, the general structure and flow of the paper are shown in Fig. 6. It may be concisely articulated as follows:

Fig. 6.

Design procedure.

Step 1: Adaptive robust control architecture based on the constraint following theory. Firstly, establish the analytical dynamic model of the general mechanical system. In the first stage, design the basic constraint following the controller to ensure that the system trajectory meets the ideal constraint conditions. For the transient response overshoot problem caused by the initial pose deviation, in the second stage, introduce the finite-time convergent compensator to achieve rapid error correction. In the third stage, to overcome the uncertainty of the system parameters and external disturbances, design a robust controller with adaptive law . Finally, the synthesized control law can be expressed as .

Step 2: Verify the effectiveness of this control strategy in a real physical environment. In multibody dynamics software, a dynamic model of the vehicle maintenance robotic manipulator is established with the road conditions and the vehicle itself. At the same time, nonlinear factors such as joint friction and transmission clearance are considered.

Step 3: The RecurDyn-Simulink collaborative simulation platform is constructed to achieve the closed-loop verification of the controller with the physical model. This framework consists of three core units: (i) The adaptive robust controller module implemented in Simulink, which calculates the joint control torques in real time, (ii) The multibody dynamics model solution in the RecurDyn environment, (iii) The bidirectional data interaction interface based on the RecurDyn/Controls module. During simulation, the RecurDyn solver outputs the actual joint angles and angular velocities of the manipulator to the controller, and the controller generates torque commands based on the target trajectory and feeds them back to the dynamics model, forming a closed-loop control loop.

Simulation

Controller performance comparison

To verify the accuracy of the adaptive constraint-following controller discussed in this paper more accurately, proportional-integral-derivative control (PID) and sliding-mode control (SMC) are selected as the comparison groups. At the same time, the UK controller was also used as a comparison to verify the effectiveness of the adaptive law designed.

The classic PID controller is designed as

63

where , and are controller parameters. Additionally, the SMC is designed as

64

where and are controller parameters.

Remark 10

(i) The reason for choosing PID control for comparison is as follows: The characteristic of the PID controller that it does not rely on an exact mathematical model is one of its most prominent features. The PID controller can achieve sufficient good control effect by parameter tuning even when the model is unknown or inaccurate. On the other hand, over 90% of control loops in the industrial automation field still use PID controllers or their variants. Therefore, when dealing with strong nonlinearity, time-varying parameters, model uncertainty, and strong external disturbances, using PID controllers as a “standard” is justified. (ii) The reason for choosing SMC for comparison is as follows: SMC is a kind of advanced nonlinear control method, which has the advantages of fast response, corresponding parameter changes, insensitive to disturbances, no need for online system identification, and simple physical implementation. By comparing with it, the effectiveness of the proposed control method can be further verified.

Initial state selection

In this paper, for Co-simulation purposes, the nominal values of the system parameters are selected as follows: kg, kg, kg. , , , , , , , , . , , , , , , , , . , , . The uncertain potions are selected as: , . In the multibody dynamics environment, the overall vehicle mass t, the vehicle speed is set to increase from 0 to 30 km/h in 4 seconds and maintain a stable driving state.

Based on Eq. (61), a three-dimensional pavement roughness calculation program was written, a three-dimensional Grade-E road (length 100 m, width 5 m) was reconstructed; using the node stitching method, a road surface file can be generated and imported into the multibody dynamics model; The parameters are selected as: N=1600, , , for .

To test the controller’s convergence properties, we intentionally select initial conditions that do not satisfy the constraint Eq. (55) as , . The angle of each joint is expected to stabilize in in the vibration environment through control. Sensor noise or delay in the mechanical system is ignored, but the data interaction between the control system and actuator in the co-simulation has a delay of 0.001 s.

Based on the tracking errors of the angles of the manipulator system, the cumulative performance error is defined as

65

To compare the control costs of different controllers, we define the control costs as

66

At the same time, we also use the root mean square error (RMSE), integral absolute error (IAE), and maximum absolute value of the error (MAE) as the evaluation indicators for the controller performance.

When designing the co-simulation, we chose SIMULINK as the Host program, the Sampling Period (Control Time Step) was set to 0.001. The solver in SIMULINK was set to ode4 for fixed-step calculation, and the control system program was compiled using S-Function.

The function and adjustment guide of adaptive parameters

We choose the parameters of the stability constraint in (59) as , the parameters define the expected convergence dynamics of the tracking error, an increase in will cause the tracking error e of the system to converge more rapidly. The constants in the adaptive law (20) are chosen as , is the rate of adaptive parameter learning and compensating for system uncertainties, an increase in can enable to grow more rapidly, thereby more quickly offsetting uncertainties and improving the transient performance of the system. However, although an excessively large can accelerate the adaptive process, it may also cause overshoot and oscillations in the control moment.

The constants in the leakage term (21) are chosen as , When the tracking error is greatlylarge, the leakage effect is relatively weak, allowing the adaptive growth term to take the dominant role, enabling to increase rapidly to quickly compensate for the uncertainty and reduce the error; When the constraint error decreases, the value of approaches 1, and the value of approaches , which means the leakage effect is the strongest. It can effectively prevent unnecessary drift or growth in the system, which is already stable and has greatlysmall errors, thereby ensuring the stability of the control.

Positive weight Q = 1 in (15) and (16) is usually adjusted after the other main parameters have been determined. The thickness of the boundary layer in (24), ε =1, is fundamentally designed to eliminate chattering, allowing the control law to switch to a smooth, proportional control similar to high−gain control within a greatlysmall error range. If significant high−frequency oscillations are observed in the control signal during simulation, increasing the value of ε to thicken the boundary layer can make the control more smooth.

The uncertainty in this paper is ultimately expressed as constants, first-order or second-order terms with respect to angular velocity . Therefore, Assumption 2 can obtain the desired result by choosing the following form of the boundary function :

67

where .

The three joints of the manipulator are equipped with independent PID controllers. The parameters of the PID controllers are selected as follows: , ,. , ,. , ,. The parameters of the SMC are selected as follows: , .

Remark 11

The selection of design parameters , , and ε, while involving some heuristic elements, is guided by the theoretical stability bounds established in the proof of Theorem 2. A promising direction for future research is to explore hybrid frameworks that combine Lyapunov-based stability guarantees with the performance optimization capabilities of learning-based methods, such as reinforcement learning or Bayesian optimization^47,48. These approaches are viewed as complementary, with intelligent tuning reserved for future investigation.

Simulation results

All simulations are run on a laptop with processor and 16GB of memory, the graphics card is NVIDIA GeForce GTX 1650. Due to the contact problem between the tracks and the road surface in the multibody dynamics environment, a complete simulation analysis takes approximately 20 minutes. We found that the average computation time , and the maximum computation time . Considering the overall simulation time of the co-simulation, the majority of the simulation time is spent on the simulation in the multibody dynamics environment. Therefore, the computational efficiency of the algorithm proposed in this paper is acceptable in the upcoming practical experiments, which is also due to the fact that the UKE method itself provides a closed-form analytical solution for solving the motion of the constrained system, allowing for the direct calculation of the generalized acceleration of the system⁴⁹.

The simulation results are shown in Figs. 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 and 28. Figures 7, 8 and 9 show the angle tracking conditions of three joints of the manipulator under influence of different controllers, and Figs. 11, 12 and 13 represent the corresponding tracking errors, and the detailed enlarged illustrations are shown in Figs. 10 and 14. From the simulation results, under the action of the proposed controller, the system can ensure convergence even in the presence of initial errors. Compared to the PID controller and SMC controller, the maintenance manipulator system with the adaptive controller proposed in this paper has stronger disturbance suppression ability and smaller tracking error. Meanwhile, although the UK controller ensures that the tracking error remains within a certain range, it is unable to achieve further convergence of the error.

Fig. 7.

Tracking of the joint rotation angle .

Fig. 8.

Tracking of the joint rotation angle .

Fig. 9.

Tracking of the joint rotation angle .

Fig. 10.

Detailed enlarged illustration of ϕ.

Fig. 11.

Tracking error of the joint rotation angle .

Fig. 12.

Tracking error of the joint rotation angle .

Fig. 13.

Tracking error of the joint rotation angle .

Fig. 14.

Detailed enlarged illustration of e.

Fig. 15.

Tracking error under different loads.

Fig. 16.

Detailed enlarged illustration of e.

Fig. 17.

RMS.

Fig. 18.

IAE.

Fig. 19.

MAE.

Fig. 20.

Control cost .

Fig. 21.

Adaptive robust control .

Fig. 22.

Adaptive robust control .

Fig. 23.

Adaptive robust control .

Fig. 24.

Detailed enlarged illustration of τ.

Fig. 25.

Adaptive parameter history.

Fig. 26.

The max value of the adaptive parameter vs. and .

Fig. 27.

The max value of the adaptive parameter vs. and .

Fig. 28.

The cumulative performance error vs. .

To verify the robustness performance of the controller under different loads, Figs. 15 and 16 show the tracking errors of the third joint under different additional load conditions. The system can rapidly converge to a value close to zero from the initial large error state within about 0.5 seconds, the controller can always maintain the steady-state error within a greatlysmall range. When the system is disturbed or performs specific trajectory tracking, as the load increases, the vibration of the curve becomes more intense, at the same time, the system will produce a slightly larger overshoot or residual oscillation when correcting. Overall, the system has strong robustness to load changes and can effectively overcome the inertial effects caused by the increase in load.

Figure 17 shows the root mean square error information for three joints, and its main purpose is to compare the degree of deviation of the errors. Compared with other control methods, the method proposed in this paper has the smallest RMS value. Figure 18 shows the absolute error of integration, the total error accumulation of the controller proposed in this paper is the smallest, which indicates that the system will not experience severe oscillations under the action of this controller, and the performance of the UK controller without adaptive law follows closely behind. By comparing the extreme values of system errors under different controllers, we know that the controller proposed in this paper has the smallest error extreme value after the system has been running for 5 seconds, which also demonstrates the superiority of the adaptive constraint following controller performance, as shown in Fig. 19.

Figure 20 shows the control costs for each joint under different controllers, indicating that the control consumption of the adaptive robust controller proposed in this paper is basically the lowest, while the control cost of the UK controller is slightly higher, and the control cost of SMC is the highest, leading to significant fluctuations in the dynamic aspect of the system. Figures 21, 22 and 23 depict the dynamic comparison chart of the input torques τ for each joint of the maintenance manipulator, the detailed enlarged illustration is shown in Fig. 24. The control input of the SMC shows significant oscillation, the inputs of the other controllers are relatively stable, this kind of oscillation problem is usually solved by integrating sliding mode control with intelligent technologies such as fuzzy logic, neural networks, and adaptive control. This solution is also worthy of consideration^50–55. However, the maximum value of the input of the controller proposed in this paper is generally the lowest, which also indicates the superiority of the comprehensive performance of the controller proposed in this paper.

Figure 25 shows the historical changes of the adaptive parameter . Under the action of the adaptive robust controller, once the tracking errors of each joint stabilize within 0.02 rad, the control input torque τ and the adaptive parameter exhibit a dynamically stable state. Figures 26 and 27 illustrate the specific patterns of the maximum value of the adaptive parameter as the adaptive law design parameters , and change. The maximum value of the adaptive parameter gradually increases as , and increase. Figure 28 illustrates the specific patterns of the cumulative performance error as the parameters and change, as and increase, the accumulative performance error shows a decreasing trend.

Conclusion

This paper presents a novel position and attitude tracking control method. The proposed method combines the advantages of robust constrained following control and leaky adaptive law. The simulation results demonstrate that end-point attitude trajectory tracking and uncertainty compensation of the vehicle-mounted maintenance robotic manipulator are achieved under extreme vibration environments. To demonstrate the validity and performance of the proposed control method, mechanical-electrical Co-simulation is used to verify that the robotic manipulator can accurately follow the given constraints.

The core contribution and key innovation of this paper lie in the explicit integration of the leakage mechanism into the adaptive law. This is not merely an incremental improvement but a targeted solution, aiming to enhance the robustness and performance of the controller in two crucial aspects: preventing parameter drift and avoiding overly aggressive control inputs. Meanwhile, by innovatively applying this controller to complex mechanical systems operating in highly-vibrational environments, it significantly increases the value in practical applications. Although the proposed control scheme demonstrated good performance in the simulation, it also has certain limitations in practical applications. For instance, the online calculation of the core Moore-Penrose generalized inverse used in the UKE can be highly time-consuming for high-dimensional systems. Secondly, this method relies on the positive definiteness of the inertia matrix. Although these are standard assumptions in the field of robotics, in certain special coordinate selections or singular configurations, some of these assumptions may not hold, thereby limiting the direct application of this method.

Future work will focus on three primary avenues: (i) Consider the complex control issues of mechanical systems, such as joint friction, actuator saturation⁵⁶, and backlash/dead zone⁵⁷, but further expansion and design of the controller structure and adaptive law are required. We will focus on integrating anti-saturation compensation strategies, or developing new adaptive laws that can clearly adhere to these input constraints. (ii) The development of an experimental testbed for hardware validation and the extension of the method to more complex robotic manipulator control problems, for instance, there are issues such as the collaborative control of multiple robotic manipulators, high-degree-of-freedom robotic manipulator systems, and more challenging external interference environments^58,59. At the same time, the challenges posed by issues such as sensor noise, computational delay, and actuator saturation in the actual application process should be considered. (iii) The system performance is significantly influenced by the control system parameters, in future work, optimization mechanisms or intelligent algorithms with stronger learning capabilities can be added on the existing basis to reduce the reliance on manual parameter adjustment^60–63.

Appendix

To be specific, , and can be described as

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Author contributions

Methodology, Yuze Ma and Da xu; Investigation, Yuze Ma and Zhaoyang Wang; Writing-original draft, Yuze Ma and Guoyun Dai; Funding acquisition, Yuze Ma; Revision of the manuscript, Yuze Ma. All authors reviewed the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.