1. Introduction
At present, collaborative robots have found extensive applications in sectors including manufacturing [Reference Sathyan and Ma1], healthcare [Reference Lanza, Seidita and Chella2], logistics and warehousing [Reference Rani, Sarkar, Smith and Kirby3], as well as agriculture [Reference Colucci, Tagliavini, Botta, Baglieri and Quaglia4]. Their primary characteristic lies in their ability to operate in conjunction with human workers, thereby enhancing productivity, flexibility, and safety. The joint module serves as a critical component within collaborative robots, as shown in Figure 1, boasting advantages such as compact structure, modularity, lightweight design, high precision, and superior safety features. Through the control of components such as permanent magnet synchronous motor (PMSM), reducers, sensors, and controllers, the joint module facilitates the articulation of collaborative robots. The PMSM, acting as the core power driver within the joint module, plays a pivotal role in the motion system of collaborative robots. Leveraging its advantages in efficient energy conversion, precise motion control, high-speed motion capabilities, stability, and reliability, the PMSM provides robust power support to the joint module, enabling collaborative robots to achieve precise and efficient motion, thereby serving as a key enabler for enhancing operational efficiency and system performance.
Figure 1. Joint module of a collaborative robot.
Control, as a method for enhancing the precision and stability of PMSM, has garnered substantial research interest. Vector control, introduced in the early stages, leverages motor current and rotor position information to precisely manipulate current vector direction and magnitude, thereby enabling accurate control of motor torque and speed [Reference Liu, Zhen, Liu, Zheng, Gao and Chen5]. However, at low speeds and low torque operating conditions, torque and speed fluctuations may arise, leading to a decline in control performance [Reference Wu, Ye, Ma, Zhen, Qiang and Chen6]. Direct Torque Control (DTC), while capable of delivering faster dynamic response, higher torque accuracy, and an extended speed range, imposes precise requirements on the accuracy of motor parameters and models, necessitating parameter calibration and calibration procedures [Reference Wei, Chen, Chai and Fu7]. Model predictive control (MPC) demands accurate motor models and robust knowledge of system dynamic characteristics, potentially encountering computational time delays that impede control performance and response speed [Reference Quiñones, Paterna, De Benedictis, Maffiodo, Franco and Ferraresi8]. Sliding mode control exhibits robustness and disturbance rejection capabilities, facilitating the handling of external uncertainties and load variations. Nevertheless, sliding mode control may introduce elevated levels of noise and oscillations [Reference Zhu, Zhao, Xian, Sun, Chen and Ma9].
As a conventional approach in motor control systems, the establishment of a highly accurate dynamic model and the design of control schemes with model compensation have been recognized as effective means for achieving precise control [Reference Yu, Liu, Zhu, Chen and Zhang10–Reference Zhen, Meng, Xiao, Liu and Chen12]. PID control, the most widely employed method in industrial control, relies on accurate and unchanging dynamic models of the system. However, in the case of an intricate system like a joint module, characterized by inherent uncertainties and vulnerability to external disturbances, the standalone application of PID control proves inadequate in ensuring stable, precise, and dependable control performance [Reference Halder, Das and Gupta13]. To tackle this issue, scholarly works have suggested integrating PID control with fuzzy control [Reference Zhu, Zhao, Xian, Chen, Sun and Ma14], leading to the development of an adaptive fuzzy PID controller that effectively handles uncertainties in the speed loop. In the context of complex dynamic networks with partially unknown mechanical system dynamics, literature [Reference Sathyan and Ma1] introduces a novel online iterative algorithm for synthesizing optimal controllers. To enhance disturbance rejection and robustness against model uncertainties, an adaptive robust control approach that integrates adaptive control and deterministic robust control has been proposed [Reference Varga, Tar and Horváth15]. For complex human–machine cooperative systems with significant uncertainties, literature [Reference Zhang, Huang, Su, Wang, Chen, Yang and Zhong16, Reference Yu, Lu, Zhao, Chen and Li17] presents an adaptive impedance-based control strategy that compensates for dynamic uncertainties in robot trajectory tracking using neural network compensation. Notably, our proposed control approach exhibits significant differences from conventional methods. Compared to the adaptive integral robust controller proposed by ref. [Reference Li, Zhao, Yu, Chen and Lin18], our proposed controller offers a well-defined structure, reduced hardware requirements, and substantial cost savings.
The Udwadia-Kalaba equation is utilized to construct a dynamic model for a constrained mechanical system in this research [Reference Chen19–Reference Yu, Lu, Zhu, Wei, Chen and Li21]. An adaptive robust controller is proposed for a constrained joint module system, taking into account uncertainties and disturbances [Reference Liu, Yin, Xia, Yu and Chen22]. We initiate by formulating a dynamic model for the system with uncertain parameters, followed by the development of a second-order representation for the system’s constraint equations [Reference Chen23]. In situations where uncertainties are either explicitly known or the mechanical system is characterized by the absence of uncertainties, it is advisable to utilize a nominal controller to effectively address the control demands of the system. However, in situations where the mechanical system’s characteristics are unknown, a two-segment approach is proposed. The mechanical system is divided into an ideal nominal segment and an uncertain nominal segment, enabling separate control strategies for each segment [Reference Chen and Zhang24]. By incorporating control terms that account for the uncertain parameters within the controller, effective control of the uncertain portion of the system is achieved.
The main contributions of this study are threefold. First, a comprehensive joint modeling framework is developed, which integrates the dynamic model of the joint module and characterizes the constraint forces through the Udwadia-Kalaba equation. This approach provides a robust theoretical foundation for modeling second-order systems subject to joint constraints. Second, a leakage-type adaptive robust controller is designed within this framework, effectively addressing external disturbances and internal uncertainties while ensuring precise trajectory tracking and robust steady-state performance. Third, the proposed controller is validated through both simulation and experimental studies, demonstrating enhanced performance in terms of tracking accuracy and robustness when compared to conventional control methods, thus underscoring its practical relevance in the domain of collaborative robotics.
The second section of this paper presents the development of the system’s dynamic model. It subsequently formulates the associated constraint equations. The section also derives the second-order derivative representation of these constraints. Following this, the procedure for designing an adaptive robust controller is detailed. Finally, the methodology utilized to establish the system’s stability is discussed. The third section consists of comprehensive simulations and experimental analyses conducted to validate the control effectiveness of the proposed controller with the joint module. Finally, the concluding section provides a concise synthesis of the proposed controller and delineates potential directions for future research.
Figure 2. Joint module structure.
2. Joint modeling and constraints
2.1. Dynamic model of joint module
The joint module comprises a PMSM and a harmonic drive reducer, as shown in Figure 2. The PMSM is an intricate nonlinear system distinguished by notable coupling phenomena. By employing the methodology elucidated in ref. [Reference Zhen, Li, Liu, Wang, Chen and Chen25] to disentangle the interdependent three-phase AC components, the mathematical formulation of the PMSM can be delineated as follows:
\begin{equation} \left \{ \begin{aligned} &\dot {i_{d}}=-\frac {R}{L_{d}}i_{d}+\frac {L_{q}}{L_{d}}n_{p}\dot {q}i_{q}+\frac {u_{d}}{L_{d}},\\ &\dot {i_{q}}=-\frac {R}{L_{d}}i_{q}+\frac {L_{d}}{L_{q}}n_{p}\dot {q}i_{d}-n_{p}\dot {q}\frac {\Psi _{f}}{L_{q}}+\frac {u_{q}}{L_{q}},\\ &\ddot {q}=\frac {\tau }{J}-\dot {q}\frac {B}{J}-\frac {T_{lp}}{J}. \end{aligned} \right . \end{equation}
Here,
$u_d$
and
$u_q$
are the d- and q-axis components of the stator voltage, respectively.
$i_d$
and
$i_q$
represent the d- and q-axis components of the stator current.
$R$
denotes the stator resistance.
$\Psi _f$
represents the permanent magnet flux linkage, while
$L_d$
and
$L_q$
denote the inductance components along the d-axis and q-axis, respectively.
$\tau$
represents the electromagnetic torque.
$B$
is the damping coefficient.
$J$
represents the moment of inertia.
$T_{lp}$
denotes the load torque of the motor.
$q$
,
$\dot {q}$
, and
$\ddot {q}$
denote the generalized coordinates associated with the motor’s rotor, specifically representing the angular displacement, mechanical angular velocity, and mechanical angular acceleration, respectively.
By utilizing the principles of electromechanical energy conversion and the decoupling of three-phase currents in PMSM, the mathematical model, as expressed in (1), enables the formulation of the electromagnetic torque equation subsequently:
\begin{equation} \tau =\dfrac {3}{2}n_{p}i_{p}[\Psi _{f}+(L_{d}-L_{q})i_{d}]. \end{equation}
Vector control provides an effective means of regulating three-phase AC motors, a technique facilitated by employing a variable frequency drive (VFD) [Reference Han, Gao, Huang, Xu and He26, Reference Maciel, Pinto, da S. Júnior, Coelho, Marcato and Cruzeiro27]. By manipulating the VFD’s output frequency, voltage, and phase angle, precise control over the motor’s output characteristics can be achieved. Notably, vector control enables independent regulation of the motor’s magnetic field and torque, akin to other field-oriented control methodologies. Leveraging coordinate transformation theory, control of the stator current’s magnitude and direction within a synchronous reference frame facilitates the decoupling of direct and quadrature axis components. The application of this decoupling control strategy facilitates the independent regulation of the magnetic field and torque in AC motors, resulting in control effectiveness comparable to that of DC motors. In the realm of vector control methodologies, a commonly adopted control strategy involves setting
$i_d$
to zero, where variations in
$i_q$
directly impact the electromagnetic torque of the PMSM [Reference Izadbakhsh, Khorashadizadeh and Kheirkhahan28]. Moreover, for surface-mounted three-phase permanent magnet synchronous motors, a simplified mathematical model can be obtained by specifying a model object, as illustrated below:
\begin{equation} \left \{ \begin{aligned} &\dot {i_{d}}=0,\\ &\dot {i_{q}}=-\frac {R}{L}i_{q}-n_{p}\dot {q}\frac {\Psi _{f}}{L}+\frac {u_{q}}{L},\\ &\ddot {q}=\frac {\tau }{J}-\frac {T_{lp}}{J}-\dot {q}\frac {B}{J}. \end{aligned} \right . \end{equation}
Subsequently,
$\tau$
is transformed into the following expression:
\begin{equation} \tau =\dfrac {3}{2}n_{p}\Psi _{f}i_{q}=k_{t}i_{q}. \end{equation}
The coefficient
$k_{t}$
represents the electromagnetic torque coefficient. By referencing the mathematical model provided in (3), the dynamic equation of PMSM can be derived as follows:
\begin{equation} J\ddot {q}+B\dot {q}+T_{lp}=\tau . \end{equation}
The dynamic equation of a harmonic reducer can be written as follow:
\begin{equation} T_{l}=\varsigma \upsilon T_{lp}. \end{equation}
Here,
$ \varsigma$
and
$ \upsilon$
mean the transmission efficiency of the harmonic reducer.
$ T_{lp}$
is the load torque applied to the joint module.
When taking into account the practical operation of the motor and considering the frictional effects between the rotor and the bearings, the resulting dynamic equation can be formulated as follows:
\begin{equation} J\ddot {q}+B\dot {q}+\dfrac {T_{l}}{\varsigma \upsilon }+T_{f}=\tau . \end{equation}
The frictional torque of the motor, denoted as
$T_{f}$
, encompasses inherent uncertainty and nonlinearity. This frictional component plays a significant role in influencing the dynamic behavior of the three-phase PMSM.
In the academic community, numerous researchers have dedicated efforts to studying the impact of different types of friction on the control performance of mechanical systems. They have achieved this by developing mathematical models that accurately capture frictional phenomena and proposing notable friction models. In this study, we introduce two distinct friction models. The first model is an improved enhanced Stribeck friction model variant, which comprehensively considers the significant influences of velocity inversion, static friction, and Coulomb friction. The second model is the classic Coulomb viscous friction model. We employ these models to effectively simulate the frictional torque experienced in the motor.
Subsequently, we present these two models [Reference Lu, Khonsari and Gelinck29]:
\begin{align} T_{f} &=f_{v}\dot {q}+f_{c}sgn(\dot {q}), \end{align}
\begin{align} T_{f}& =\left[f_{c}+(f_{s}-f_{c})e^{{-\big(\frac {\dot {q}}{\dot {q}_{s}}\big)^{2}}}\right]sgn(\dot {q})+f_{v}\dot {q}-f_{g}. \end{align}
In the provided equation,
$f_{s}$
represents the coefficient of static friction, while
$f_{c}$
and
$f_{v}$
represent the coefficients of Coulomb friction and viscous friction, sequentially. Additionally,
$\dot {q}_{s}$
denotes the lubrication factor. The term
$f_{g}$
, as introduced in (9), is employed to eliminate the variation of the friction force during velocity reversal. Its calculation can be performed as follows:
\begin{equation} \left \{ \begin{aligned} &f_{g}=se^{-\frac {\dot {q}^{2}}{v}},\\ &s=sgn\frac {\dot {q}}{\ddot {q}}f_{s}+f_{c}l,\\ &v=\ddot {q}\dot {q}_{s}(k\times sgn(\ddot {q})+b\times sgn(\dot {q})), \end{aligned} \right . \end{equation}
here
$l$
,
$k$
, and
$b$
are the correlation coefficients.
In order to enhance comprehension and facilitate computational analysis, we shall express (7) in the framework of Lagrangian mechanics.
\begin{equation} M(q(t), \xi (t),t)\ddot {q}(t)+C(q(t),\dot {q}(t),\xi (t),t)\dot {q}(t)+F(q(t),\xi (t),t)=\tau (t), \end{equation}
where
$M=J,C=B,F=\dfrac{T_{l}}{\varsigma \upsilon }+T_{f}$
, the variables
$q\in \mathbb{R}^{n}$
,
$\dot {q}\in \mathbb{R}^{n}$
,
$\ddot {q}\in \mathbb{R}^{n}$
, and
$t\in \mathbb{R}$
. Here,
$q$
,
$\dot {q}$
, and
$\ddot {q}$
represent the generalized coordinates, namely position, velocity, and acceleration, respectively. The parameter
$\xi \in \Sigma$
represents the uncertain parameters of the system, which may exhibit fast time-varying behavior. The control input torque applied to the controlled object is represented by the symbol
$\tau$
.
2.2. Constraint description
Assuming the mechanical system exhibits adequate levels of smoothness, a set of constraint equations can be derived and represented in matrix form. The constraint equations are formulated as follows:
\begin{equation} K(q,t)=d(q,t), \end{equation}
where
$K=[K_{li}]_{m\times n}$
and
$d=[d_{1},d_{2},\dots ,d_{m}]^{m\times n}$
. To analyze the second-order performance characteristics related to the servo constraint, a second-order derivation of the constraint is conducted. The step-by-step derivation is provided below:
\begin{equation} \sum ^{n}_{i=1}K_{li}(q,t)=d_{l}(q,t), l=1,\dots ,m. \end{equation}
By taking the derivative of the equation mentioned above with respect to time (
$t$
), we obtain:
\begin{equation} \sum ^{n}_{i=1}\dfrac {d}{dt}K_{li}(q,t)=\dfrac {d}{dt}d_{l}(q,t). \end{equation}
By formulating it in a generalized manner, we can derive the following expression:
\begin{equation} \sum ^{n}_{i=1}A_{li}(q,t)\dot {q}_{i}=b_{l}(q,t). \end{equation}
By deriving the general form of the constraint condition, we can further differentiate the above equation with respect to time (
$t$
), resulting in:
\begin{equation} \sum ^{n}_{i=1}\left(\dfrac {d}{dt}A_{li}(q,t)\right)\dot {q}_{i}+\sum ^{n}_{i=1}A_{li}(q,t)\ddot {q}_{i}=\dfrac {d}{dt}b_{l}(q,t). \end{equation}
In a generalized form, the expression can be written as follows:
\begin{equation} \sum ^{n}_{i=1}A_{li}(q,t)\ddot {q}_{i}=c_{l}(q,\dot {q},t). \end{equation}
Based on the preceding derivation, we can establish the following first-order and second-order constraint equations:
\begin{align} A(q,t)\dot {q} &=b(q,t), \\[-10pt] \nonumber \end{align}
\begin{align} A(q,\dot {q},t)\ddot {q}&=c(q,\dot {q},t), \end{align}
where
$A=[A_{li}]_{m\times n}$
,
$b=[b_{1},b_{2},\dots ,b_{m}]^{T}$
, and
$c=[c_{1},c_{2},\dotsc _{m}]^{T}$
. In the current context, when there is either explicit acknowledgment or absence of uncertainty within the mechanical system, it becomes feasible to explicitly determine the constraints imposed on the system.
Remark 1. The second-order representation of constraint ( 19 ) encompasses a range of constraint types, such as non-holonomic constraints, holonomic constraints, rigid constraints, and viscoelastic constraints. These constraints affect the system’s dynamics by influencing the interplay between active forces and constraint forces. As a result, the system undergoes dynamic transformations.
As a consequence of the imposed constraints, it becomes imperative to exert generalized constraint forces for system control. The equations of motion for an unconstrained mechanical system can be formulated as follows:
\begin{equation} M(q(t),\xi (t),t)\ddot {q}(t)=Q(q(t),\dot {q}(t),\xi (t),t). \end{equation}
To release the constraints in the unconstrained mechanical system, a generalized active force
$Q(q(t),\dot {q}(t),\xi (t),t)\in \mathbb{R}^{n}$
is exerted on the system.
Consequently, (20) can be reconfigured as follows:
\begin{equation} M(q,\xi ,t)\ddot {q}=Q(q,\dot {q},\xi ,t)+Q^{c}(q,\dot {q},\xi ,t). \end{equation}
When uncertainty is not present, the constraint force
$Q^{c}$
is responsible for ensuring the satisfaction of the expected constraints.
2.3. Solution of the constraint force
Upon gaining a comprehensive understanding of the mechanical system, it becomes imperative to delve into the exploration of constraint forces.
Assumption 1.
Assume that
$(q,t)\in \mathbb{R}^{n}\times \mathbb{R}, \xi \in \sum$
, and the matrix
$M(q,\xi ,t)$
is positive definite.
Remark 2.
When a generalized coordinate
$q$
is selected for the system, the positive definiteness of the inertia matrix is guaranteed due to the inherent positive nature of the system’s kinetic energy.
Definition 1.
The consistency of the constraint equation (
19
) is determined by the existence of a solution
$\ddot {q}$
within a given set of
$A$
and
$c$
.
Assumption 2. According to Definition 1 , we ascertain that the constraint equation ( 19 ) meets the criterion of consistency.
Theorem 1.
By combining (
11
) and (
19
), and considering Assumptions
1
and
2
, as well as Gauss’s principle and Lagrange’s form of d’Alembert’s principle [Reference Izadi and Sanyal30, Reference Cardin and Zanzotto31], we can derive the explicit Lagrangian form of the constraint force. Furthermore, the application of the Udwadia-Kalaba equation enables a direct and efficient computation of the constraint forces in this system. By reformulating the constraint dynamics using the Udwadia-Kalaba equation, we explicitly express the generalized reaction forces resulting from the system’s constraints. This approach simplifies the computation of the constraint force
$Q^{c}$
by directly utilizing the relationship between the system’s generalized velocities and the constraint forces.
\begin{equation} Q^{c} =\bar {M}^{1/2}(q,t)(A(q,t)\bar {M}^{1/2}(q,t))^{+}\times \left[c(q,\dot {q},t) +A(q,t)\bar {M}^{-1}(q,t)\times (\bar {C}(q,\dot {q},t)\dot {q}+\bar {G}(q,t))+\bar {F}(q,t)\right]\!. \end{equation}
Remark 3. In scenarios where the mechanical system is either devoid of uncertainty or where uncertainty is well-characterized, the controller formulated in the preceding equation effectively addresses the control problem. The application of the Udwadia-Kalaba equation facilitates the explicit computation of constraint forces, thereby ensuring accurate constraint-following behavior. However, in practical systems, uncertainty often manifests in an unknown form, such as unmodeled dynamics or external disturbances, which complicates the control challenge. To mitigate this, the subsequent section will focus on robust control strategies and constraint-following control methods tailored to uncertain dynamics. By extending the Udwadia-Kalaba equation to incorporate such uncertainties, the controller can be refined to maintain system stability and performance, even in the presence of disturbances, thereby ensuring robust operation in realistic scenarios.
3. Design of an adaptive robust controller
To account for uncertainty in the design of the control input
$\tau$
, we decompose the terms
$M$
,
$C$
, and
$F$
in (11) as follows:
\begin{equation} \begin{aligned} M(q,\xi ,t)&=\bar {M}(q,t)+\triangle M(q,\xi ,t),\\[2pt] C(q,\dot {q},\xi ,t)&=\bar {C}(q,\dot {q},t)+\triangle C(q,\dot {q},\xi ,t),\\[2pt] F(q,\xi ,t)&=\bar {F}(q,t)+\triangle F(q,\xi ,t). \end{aligned} \end{equation}
Here, the terms
$\bar {M}, \bar {C}, \bar {F}$
represent the nominal components, while
$\triangle M, \triangle C, \triangle F$
represent the uncertainties portions. It is important to note that
$\bar {M}, \bar {C}, \bar {F},$
and
$\triangle M, \triangle C, \triangle F$
are continuous. In light of this, we introduce the following definitions:
\begin{equation} \begin{aligned} B(q,t) \, : \, &=\bar {M}^{-1}(q,t),\\[2pt] \triangle B(q,\xi ,t) \, : \, &=M^{-1}(q,\xi ,t)-\bar {M}^{-1}(q,t),\\[2pt] E(q,\xi ,t) \, : \, &=\bar {M}(q,t)M^{-1}(q,\xi ,t)-I. \end{aligned} \end{equation}
Therefore, we have
\begin{equation} \begin{aligned} M^{-1}(q,\xi ,t)&=B(q,t)+\triangle B(q,\xi ,t),\\[2pt] \triangle B(q,\xi ,t)&=B(q,t)E(q,\xi ,t). \end{aligned} \end{equation}
Assumption 3.
For any
$(q,t)\in \mathbb{R}^{n}\times \mathbb{R}$
,
$A(q,t)$
is full rank, which tells that
$A(q,t)A^{T}(q,t)$
is invertible.
Assumption 4.
Considering the condition of Assumption
3
, for a given
$P\in \mathbb{R}^{m\times m},P\gt 0$
, we make that
\begin{equation} \begin{aligned} W(q,\xi ,t) \, :\!= \, &PA(q,t)B(q,t)E(q,\xi ,t)\bar {M}(q,t)A^{T}(q,t)(A(q,t)A^{T}(q,t))^{-1}P^{-1}. \end{aligned} \end{equation}
For all
$(q,t)\in \mathbb{R}^{n}\times \mathbb{R}$
, there exists a (potentially unknown) constant
$\rho _{E}\gt -1$
such that,
\begin{equation} \frac {1}{2}\underset {\xi \in \sum }{min}\lambda _{m}(W^{T}(q,\xi ,t)+W(q,\xi ,t))\ge \rho _{E}. \end{equation}
The notation
$ {min}(\! \bullet \!)$
denotes the smallest number, while
$\lambda _{m}(\! \bullet \!)$
represents the smallest eigenvalue.
Remark 4.
The constant
$\rho _{E}$
is generally unknown since it relies on the uncertain bound
$\sum$
, which is also unknown. In the specific scenario where
$M=\bar {M}$
,
$E=0$
, and
$W=0$
, it is possible to select
$\rho _{E}=0$
. This assumption implies that, based on continuity, the presence of uncertainty permits the deviation of
$M$
from
$\bar {M}$
within a certain threshold. It is crucial to emphasize that this threshold is unidirectional and not limited to only one direction.
By (22), let
\begin{align} p_{1}(q,\dot {q},t) &=Q^{c},\end{align}
\begin{align} p_{2}(q,\dot {q},t) =-\eta \bar {M}(q,t)A^{T}(q,t)(A(q,t)& A^{T}(q,t))^{-1}P^{-1}(A(q,t)\dot {q}-b(q,t)). \end{align}
Assumption 5.
(1) For all
$(q,\dot {q},t)\in \mathbb{R}^{n}\times \mathbb{R}^{n}\times \mathbb{R}$
and
$\xi \in \sum$
, there exists an unknown constant vector
$\iota \in (0,\infty )^{\eta }$
, which is relevant to the bounding set
$\sum$
, and a known function
$\Pi (\! \bullet \!) \, : \, (0,\infty )^{\eta }\times \mathbb{R}^{N}\times \mathbb{R}^{n}\times \mathbb{R}\to \mathbb{R}_{+}$
which is assumed to be the boundary of uncertainty and external disturbance such that,
\begin{equation} \begin{aligned} &(1+\rho _{E})^{-1}\underset {\xi \in \sum }{max}[\parallel PA(q,t)\triangle B(q,\xi ,t)(-C(q,\dot {q},\xi ,t)\dot {q}\\[2pt] &+p_{1}(q,\dot {q},t)+p_{2}(q,\dot {q},t))-G(q,\xi ,t)-F(q,\xi ,t)\\[2pt] &-PA(q,t)B(q,t)(\triangle C(q,\dot {q},\xi ,t)\dot {q}\\[2pt] &+\triangle G(q,\xi ,t)+\triangle F(q,\xi ,t))\parallel ]\le \Pi (\iota ,q,\dot {q},t). \end{aligned} \end{equation}
(2) Under the premise of Assumption
5
(1), for every
$(\iota ,q,\dot {q},t)$
,
$\Pi (\iota ,q,\dot {q},t)$
can be linearly decomposed in terms of
$\iota$
. There exists a function
$\tilde {\Pi }(\! \bullet \!)\,:\, \mathbb{R}^{n}\times \mathbb{R}^{n}\times \mathbb{R}\to \mathbb{R}_{+}^{\eta }$
such that,
\begin{equation} \Pi (\iota ,q,\dot {q},t)=\iota ^{T}\tilde {\Pi }(q,\dot {q},t). \end{equation}
Let
\begin{align} \gamma (\tilde {\iota },q,\dot {q},t) &=\left \{ \begin{aligned} \begin{array}{lr} \frac {1}{\parallel \Phi (\tilde {\iota },q,\dot {q},t)\parallel }& \quad \text{if} \quad \parallel \Phi (\tilde {\iota },q,\dot {q},t)\parallel \gt \hat {\varepsilon },\\\frac {1}{\hat {\varepsilon }} & \quad \text{if} \quad \parallel \Phi (\tilde {\iota },q,\dot {q},t)\parallel \le \hat {\varepsilon }, \end{array} \end{aligned} \right . \\[-10pt] \nonumber \end{align}
\begin{align} \Phi (\tilde {\iota },q,\dot {q},t) & =(A(q,t)\dot {q}-b(q,t))\Pi (\tilde {\iota },q,\dot {q},t), \end{align}
where
$\hat {\varepsilon }\gt 0$
is a scalar constant. We propose the following control approach:
\begin{equation} \tau (t)=p_{1}(q,\dot {q},t)+p_{2}(q,\dot {q},t)+p_{3}(\tilde {\iota },q,\dot {q},t), \end{equation}
where
\begin{equation} \begin{aligned} p_{3}(\tilde {\iota },q,\dot {q},t) &=-\left[\bar {M}(q,t)A^{T}(q,t)(A(q,t)A^{T}(q,t))^{-1}P^{-1}\right]\\[2pt] &\times \gamma (\tilde {\iota },q,\dot {q},t)\Phi (\tilde {\iota },q,\dot {q},t)\Pi (\tilde {\iota },q,\dot {q},t). \end{aligned} \end{equation}
Let
$\varrho (q,\dot {q},t) \, :\!= \, A(q,t)\dot {q}-b(q,t)$
. To adapt to the uncertain nature of the system, the evolution of the adaptive parameter vector
$\tilde {\iota }\in \mathbb{R}^{\eta }$
is regulated by the following adaptive law:
\begin{equation} \dot {\tilde {\iota }}=\eta _{1}\tilde {\Pi }(q,\dot {q},t)\parallel \varrho (q,\dot {q},t)\parallel -\eta _{2}\tilde {\iota }. \end{equation}
Here,
$\tilde {\iota }_{i}(\text{t}_{0})\gt 0$
(where
$\tilde {\iota }_{i}$
is the
$i-th$
component of the vector
$\tilde {\iota }$
),
$i=1,2,\dots \eta ,\eta _{1,2}\in \mathbb{R},\eta _{1,2}\gt 0$
.
Remark 5.
The control action
$p_{3}$
embodies the adaptive robust behavior, where it relies on the adaptive parameter
$\tilde {\iota }$
. Essentially, the purpose of the parameter
$\tilde {\iota }$
is to emulate the unknown parameter
$\iota$
.
Remark 6.
The adaptive law represented by (
36
) follows a leakage-type approach, where the second term on the right-hand side acts as the leak. It is worth noting that by selecting the initial condition
$\tilde {\iota }(t_{0})$
to be strictly positive, it follows that
$\tilde {\iota }(t)\gt 0$
for all
$t\geq t_{0}$
. This is due to the non-negativity of the first term on the right-hand side and the exponential decay (approaching zero) induced solely by the second term.
Remark 7.
In this particular case, the presence of negligible oscillations is expected due to the finite and constant value of
$\hat {\varepsilon }$
.
Theorem 2.
Let
$\delta :=[\varrho ^{T}(\tilde {\iota }-\iota )^{T}]^{T}\in \mathbb{R}^{m+\eta }$
. Subject to Assumption
1
-
5
, consider the system (
11
). The control (
34
) renders the following performance: (i) Uniform boundedness: For any
$v\gt 0$
, there is a
$d(v)\lt \infty$
such that if
$\parallel \delta (t_{0})\parallel \le v$
, then
$\parallel \delta (t)\parallel \ \le d(v)$
for all
$t\ge t_{0}$
; (ii) Uniform ultimate bounded: For any
$v\gt 0$
with
$\parallel \delta (t_{0})\parallel \ \le v$
, there exists a
$\underline {d}\gt 0$
such that
$\parallel \delta (t)\parallel \ \le \bar {d}$
for any
$\bar {d}\gt \underline {d}$
as
$t\ge t_{0}+T(\bar {d},v)$
, where
$T(\bar {d},v)\lt \infty$
.
Proof. Let
\begin{equation} V(\varrho ,\tilde {\iota })=\varrho ^{T}P\varrho +\eta _{1}^{-1}(1+\rho _{E})(\tilde {\iota }-\iota )^{T}(\tilde {\iota }-\iota ). \end{equation}
Given an uncertainty
$\xi (\! \bullet \!)$
and the associated trajectory
$q(\! \bullet \!)$
,
$\dot {q}(\! \bullet \!)$
, and
$\tilde {\iota }(\! \bullet \!)$
of the controlled system, the derivative of
$V$
can be evaluated as follows (arguments are omitted for brevity, except in critical cases where clarity is required):
\begin{equation} \dot {V}=2\varrho ^{T}P\dot {\varrho }+2\eta _{1}^{-1}(1+\rho _{E})(\tilde {\iota }-\iota )^{T}\dot {\tilde {\iota }}. \end{equation}
Let us analyze each term separately. First,
\begin{equation} \begin{aligned} 2\varrho ^{T}P\dot {\varrho } & =2\varrho ^{T}P(A\ddot {q}-c)\\[2pt] & =2\varrho ^{T}P\{A[M^{-1}(-C\dot {q}-F-G)+M^{-1}(p_{1}+p_{2}+p_{3})]-c\}\\[2pt] & =2\varrho ^{T}P\{A[B(-\bar {C}\dot {q}-\bar {F}-\bar {G})+B(p_{1}+p_{2})+B(-\triangle C\dot {q}-\triangle G-\triangle F)\\[2pt] &\quad+\triangle B(-C\dot {q}-G-F+p_{1}+p_{2})+(B+\triangle B)p_{3}]-c\}. \end{aligned} \end{equation}
Using (28) with the special case where
$\xi \equiv 0$
(resulting in
$Q^{c}=p_{1}$
), we have
\begin{equation} A\big[B\big(-\bar {C}\dot {q}-\bar {G}-\bar {F}\big)+Bp_{1}\big]-c=0. \end{equation}
Next, by (30), we get
\begin{equation} \begin{aligned} 2\varrho ^{T} & PA[\triangle B(-G-C\dot {q}-F+p_{1}+p_{2})+B(-\triangle G-\triangle C\dot {q}-\triangle F)]\\[2pt] \le & 2\parallel \varrho \parallel \parallel PA[\triangle B(-G-C\dot {q}-F+p_{1}+p_{2})+B(-\triangle G-\triangle C\dot {q}-\triangle F)]\parallel \\[2pt] \le & 2\parallel \varrho \parallel (1+\rho _{E})\Pi (\iota ,q,\dot {q},t). \end{aligned} \end{equation}
In (29), after performing matrix cancelation, we obtain
\begin{equation} \begin{aligned} 2\varrho ^{T}PAp_{2}&=2\varrho ^{T}PAB[-\eta \bar {M}A^{T}(AA^{T})^{-1}P^{-1}(A\dot {q}-b)]\\[2pt] &=2\varrho ^{T}(-\eta )(A\dot {q}-b)\\[2pt] &=-2\eta \parallel \varrho \parallel ^{2}. \end{aligned} \end{equation}
By (34) and with
$\triangle B=BE$
, we have
\begin{equation} 2\varrho ^{T}PA(B+\triangle B)p_{3} =2\varrho ^{T}PABp_{3}+2\varrho ^{T}PABEp_{3}. \end{equation}
By direct algebraic manipulation, we can demonstrate that this holds true by using the equation
$\varrho \Pi (\tilde {\iota },q,\dot {q},t)=\Phi$
. Then, we get
\begin{equation} \begin{aligned} &2\varrho ^{T}PABp_{3}\\[2pt] &\quad =-2(\varrho \Pi (\tilde {\iota },q,\dot {q},t))^{T}\gamma \Phi \\[2pt] &\quad =-2\Phi ^{T}\gamma \Phi \\[2pt] &\quad =-2\gamma \parallel \Phi \parallel ^{2}. \end{aligned} \end{equation}
Then, we have
\begin{equation} \begin{aligned} 2&\varrho ^{T}PABp_{3}\\[2pt] & =-2\Phi ^{T}[PABE\bar {M}A^{T}(AA^{T})^{-1}P^{-1}\gamma \Phi ]\\[2pt] & =-2\gamma \dfrac {1}{2}\Phi [PABE\bar {M}A^{T}(AA^{T})^{-1}P^{-1}+P^{-1}(AA^{T})^{-T}A\bar {M}E^{T}BA^{T}P]\Phi \\[2pt] & \le -2\gamma \dfrac {1}{2}\lambda _{m}(W^{T}+W)\parallel \Phi \parallel ^{2}\\[2pt] & \le -2\gamma \rho _{E}\parallel \Phi \parallel ^{2}. \end{aligned} \end{equation}
Combining (44) and (45), we get
\begin{equation} 2\varrho ^{T}PA(B+\triangle B)p_{3}\le -2\gamma (\rho _{E}+1)\parallel \Phi \parallel ^{2}. \end{equation}
If
$\parallel \Phi \parallel \le \varepsilon$
, then by (32), we have
\begin{equation} \begin{aligned} -2\gamma (\rho _{E}+1)\parallel \Phi \parallel ^{2}&=-2(\rho _{E}+1)\dfrac {1}{\varepsilon }\parallel \Phi \parallel ^{2}\\[2pt] &=-2(\rho _{E}+1)\frac {\parallel \Phi \parallel ^{2}}{\varepsilon }. \end{aligned} \end{equation}
With (40)-(47), we have, for
$\parallel \Phi \parallel \gt \varepsilon$
,
\begin{equation} 2\varrho ^{T}P\dot {\varrho }\le -2\eta \parallel \varrho \parallel ^{2}+(\rho _{E}+1)\{-2\parallel \varrho \parallel \Pi (\tilde {\iota },q,\dot {q},t)+2\parallel \varrho \parallel \Pi (\tilde {\iota },q,\dot {q},t)\}. \end{equation}
As
$\parallel \Phi \parallel \le \varepsilon$
,
\begin{equation} \begin{aligned} 2\varrho ^{T}P\dot {\varrho } & \le -2\eta \parallel \varrho \parallel ^{2}-2(\rho _{E}+1)\dfrac {\parallel \Phi \parallel ^{2}}{\varepsilon }+2\parallel \varrho \parallel (\rho _{E}+1) \Pi (\iota ,q,\dot {q},t)\\[2pt] & =-2\eta \parallel \varrho \parallel ^{2}+(\rho _{E}+1) \left[-2\dfrac {\parallel \Phi \parallel ^{2}}{\varepsilon }+2\parallel \varrho \parallel \Pi (\tilde {\iota },q,\dot {q},t) \right]\\[2pt] &\quad +\, (\rho _{E}+1)[-2\parallel \varrho \parallel \Pi (\tilde {\iota },q,\dot {q},t)]+2\parallel \varrho \parallel \Pi (\iota ,q,\dot {q},t)\\[2pt] & \le -2\eta \parallel \varrho \parallel ^{2}+2(\rho _{E}+1)\dfrac {1}{4/\varepsilon }+(1+\rho _{E}) [-2\parallel \varrho \parallel \Pi (\tilde {\iota },q,\dot {q},t)+2\parallel \varrho \parallel \Pi (\iota ,q,\dot {q},t)]\\[2pt] & =-2\eta \parallel \varrho \parallel ^{2}+(\rho _{E}+1)\dfrac {\varepsilon }{2}+(\rho _{E}+1)[-2\parallel \varrho \parallel \Pi (\tilde {\iota },q,\dot {q},t)+2\parallel \varrho \parallel \Pi (\iota ,q,\dot {q},t)]. \end{aligned} \end{equation}
The initial equivalence is established through the process of addition and subtraction of identical terms. Additionally, by invoking Assumption 5, the progression of the derivation can be extended as follows:
\begin{equation} \begin{aligned} &2\parallel \varrho \parallel \Pi (\iota ,q,\dot {q},t)-2\parallel \varrho \parallel \Pi (\tilde {\iota },q,\dot {q},t)\\[2pt] &\quad=2\parallel \varrho \parallel \iota ^{T}\tilde {\Pi }(q,\dot {q},t)-2\parallel \varrho \parallel \tilde {\iota }^{T}\tilde {\Pi }(q,\dot {q},t)\\[2pt] &\quad=2\parallel \varrho \parallel (-\tilde {\iota }+\iota )^{T}\tilde {\Pi }(q,\dot {q},t). \end{aligned} \end{equation}
By utilizing the equation referenced as (50) within the equations referenced as (49) and (48), it can be deduced that for all values of
$\parallel \Phi \parallel$
, the following holds:
\begin{equation} 2\varrho ^{T}P\dot {\varrho }\le -2\eta \parallel \varrho \parallel ^{2}+(\rho _{E}+1)\dfrac {\varepsilon }{2}+2\parallel \varrho \parallel (\rho _{E}+1)(-\tilde {\iota }+\iota )^{T}\tilde {\Pi }(q,\dot {q},t). \end{equation}
Through the employment of the adaptive law represented as (36),
\begin{equation} \begin{aligned} 2&\eta _{1}^{-1}(\tilde {\iota }-\iota )^{T}\dot {\tilde {\iota }}=2\eta _{1}^{-1}(\tilde {\iota }-\iota )^{T}\big(\eta _{1}\tilde {\Pi }(q,\dot {q},t)\parallel \varrho \parallel -\eta _{2}\tilde {\iota }\big)\\[2pt] & =2(\tilde {\iota }-\iota )^{T}\tilde {\Pi }(q,\dot {q},t)\parallel \varrho \parallel -2\eta _{1}^{-1}(\tilde {\iota }-\iota )^{T}\eta _{2}\tilde {\iota }\\[2pt] & =2(\tilde {\iota }-\iota )^{T}\tilde {\Pi }(q,\dot {q},t)\parallel \varrho \parallel -2\eta _{1}^{-1}\eta _{2}(\tilde {\iota }-\iota )^{T}(\tilde {\iota }-\iota +\iota )\\[2pt] & =2(\tilde {\iota }-\iota )^{T}\tilde {\Pi }(q,\dot {q},t)\parallel \varrho \parallel -2\eta _{1}^{-1}\eta _{2}(\tilde {\iota }-\iota )^{T}\iota -2\eta _{1}^{-1}\eta _{2}(\tilde {\iota }-\iota )^{T}(\tilde {\iota }-\iota )\\[2pt] & \le 2(\tilde {\iota }-\iota )^{T}\tilde {\Pi }(q,\dot {q},t)\parallel \varrho \parallel -2\eta _{1}^{-1}\eta _{2}\parallel \tilde {\iota }-\iota \parallel ^{2}+2\eta _{1}^{-1}\eta _{2}\parallel \tilde {\iota }-\iota \parallel \parallel \iota \parallel . \end{aligned} \end{equation}
Multiplying (52) by
$1+\rho _{E}$
and with (51), (38) becomes (note that
$\parallel \delta \parallel ^{2}=\parallel \tilde {\iota }-\iota \parallel ^{2}+\parallel \varrho \parallel ^{2}, \parallel \tilde {\iota }-\iota \parallel \le \parallel \delta \parallel$
)
\begin{equation} \begin{aligned} \dot {V} & \le-2\eta \parallel \varrho \parallel ^{2}-2\eta _{1}^{-1}\eta _{2}(\rho _{E}+1)\parallel \tilde {\iota }-\iota \parallel ^{2}+(\rho _{E}+1)\frac {\varepsilon }{2}\\[2pt] &\quad +\, 2\eta _{1}^{-1}\eta _{2}(\rho _{E}+1)\parallel \tilde {\iota }-\iota \parallel \parallel \iota \parallel \\[2pt] & \le -\underline {\eta _{1}}\parallel \delta \parallel ^{2}+\underline {\eta _{2}}\parallel \delta \parallel +\underline {\eta _{3}}, \end{aligned} \end{equation}
where
$\underline {\eta _{1}}=\text{min}\{2\eta ,2\eta _{1}^{-1}\eta _{2}(\rho _{E}+1)\}$
,
$\underline {\eta _{2}}=2\eta _{1}^{-1}\eta _{2}(\rho _{E}+1)\parallel \iota \parallel$
, and
$\underline {\eta _{3}}=(\rho _{E}+1)\varepsilon /2$
.
In conclusion, we establish the uniform boundedness by
\begin{align} d(r) &=\left \{ \begin{aligned} \begin{array}{lr} \sqrt {\dfrac {\gamma _{2}}{\gamma _{1}}}R& \quad \text{if} \, r\le R,\\[10pt] \sqrt {\dfrac {\gamma _{2}}{\gamma _{1}}}r& \quad \text{if} \, r\gt R, \end{array} \end{aligned} \right . \end{align}
\begin{align} R &=\frac {1}{2\underline {\eta _{1}}}\left(\underline {\eta _{2}}+\sqrt {\underline {\eta _{2}}^{2}+4\underline {\eta _{1}}\underline {\eta _{3}}}\right)\!, \end{align}
where
$\gamma _{1}=\text{min}\{\lambda _{\text{min}}(P), \eta ^{-1}_{1}(\rho _{E}+1)\}, \gamma _{2}=\text{max}\{\lambda _{\text{max}}(P), \eta _{1}^{-1}(\rho _{E}+1)\}$
.
Moreover, uniform ultimate boundedness ensues consequently with
\begin{align} \underline {d} & =\sqrt {\dfrac {\gamma _{2}}{\gamma _{1}}}R, \end{align}
\begin{align} T(\bar {d},r) & =\left \{ \begin{aligned} \begin{array}{lr} 0& \quad \text{if} \, \ r\le \bar {d}\sqrt {\dfrac {\gamma _{1}}{\gamma _{2}}}, \\ \dfrac {\gamma _{2}r^{2}-(\gamma _{1}^{2}/\gamma _{2})\bar {d}^{2}}{\underline {\eta _{1}}\bar {d}^{2}(\gamma _{1}/\gamma _{2})-\underline {\eta _{2}}\bar {d}(\gamma _{1}/\gamma _{2})^{1/2}-\underline {\eta _{3}}}&\text{otherwise}. \end{array} \end{aligned} \right . \end{align}
Remark 8.
Within our designed controller,
$p_1$
represents the desired constraint force in an ideal scenario, where the system’s initial conditions satisfy the expected constraints, and no uncertainty is present. In cases where the system’s initial conditions are incompatible with the desired constraints, the inclusion of the
$p_2$
term becomes necessary to ensure conformity to the constraint requirements. To address potential uncertainty-related disturbances within the system, the incorporated
$p_3$
term serves as a means of mitigation. Through the judicious adjustment of controller parameters, the combination of
$p_1$
,
$p_2$
, and
$p_3$
provides a comprehensive framework for resolving general control problems encountered in mechanical systems.
4. Simulation and experiment validation
4.1. Simulation results
Based on the previous analysis, an adaptive robust controller is developed for the control system of a joint module with inherent uncertainties. The main objective of this controller is to achieve enhanced trajectory tracking precision. To validate the effectiveness of the proposed robust controller, comprehensive simulations are conducted using the Matlab 2020b/Simulink platform. The simulations are performed to assess the viability and precision of the controller. Table I presents the relevant parameters of the PMSM used in the simulations. Throughout the simulation process, a constrained trajectory is imposed to ensure adherence to specified constraints.
Table I. Parameters and variables within the joint module.
4.1.1. Control of the step signal
In the comparative analysis, a reference trajectory is chosen as a step signal with the function
$x^{d}(t)=\pi /3$
, starting from an initial condition of
$x^{d}(0)=0$
. To assess the performance of the proposed controller, a conventional PID controller is implemented for comparison purposes. The main objective of this analysis is to demonstrate the superior trajectory-tracking capability of the proposed controller.
Figure 3. Step response simulation results.
We present the simulation results of the step response, showcasing the behavior of the system when the permanent magnet synchronous motor is incorporated. Figure 3 provides a visual representation of the system’s response, illustrating its dynamic behavior. This figure effectively demonstrates the effectiveness of the proposed controller in achieving precise trajectory tracking.
As depicted in Figure 3, the
$p_1$
controller cannot adequately react to step signals. In contrast, the others demonstrate effective performance, with the
$p_1$
+
$p_2$
+
$p_3$
controller showcasing superior characteristics, notably minimal error and reduced response time. Based on the analysis of Figure 3, the comparison between curves
$p_1$
and
$p_1$
+
$p_2$
suggests that including the proposed
$p_2$
control term reveals an unresolved initial condition incompatibility. However, the absence of the
$p_3$
control term limits the capacity of the
$p_1$
+
$p_2$
curve to address uncertainty, consequently leading to an overshoot. This observation underscores the significance of incorporating the
$p_3$
control term to enhance the system’s robustness, thereby mitigating uncertainties and improving the overall dynamic performance.
To validate the effectiveness of the proposed
$p_1$
+
$p_2$
+
$p_3$
control scheme, a series of simulation experiments is conducted in comparison with several classical controllers, including Sliding Mode Control (SMC) [Reference Gao, Ma, Zhang and Zhou32], Adaptive Sliding Mode Control (ASMC) [Reference Shao, Zheng and Fu33], and PID. The comparative results are presented in Figure 3. Under a step reference input, the
$p_1$
+
$p_2$
+
$p_3$
controller achieves a well-balanced transient response, characterized by a short rise time, limited overshoot, and rapid convergence to the desired steady-state value.
In comparison, the PID controller exhibits a noticeably slower dynamic response, although its tracking error eventually converges after the system reaches steady state, reflecting its limited transient regulation capability. The SMC demonstrates a slower response accompanied by more pronounced overshoot. The enlarged views further reveal the existence of residual chattering and a relatively long settling process in the SMC response, which adversely affects tracking smoothness. In contrast, the
$p_1$
+
$p_2$
+
$p_3$
controller effectively suppresses such oscillatory behavior through its smooth and robustness-enhancing control action, resulting in improved transient quality.
It is further observed that the ASMC exhibits a slightly faster initial response than the
$p_1$
+
$p_2$
+
$p_3$
controller, indicating its effectiveness in accelerating the early-stage system dynamics. However, a closer examination shows that the steady-state tracking error of the ASMC remains larger than that achieved by the proposed controller, although it is smaller than that of the SMC. This result suggests that, while the adaptive mechanism of ASMC enhances transient responsiveness, its overall error regulation capability is still limited by its control structure, particularly in steady-state conditions.
By contrast, the
$p_1$
+
$p_2$
+
$p_3$
controller consistently maintains lower steady-state error while preserving fast convergence, thereby achieving a more favorable trade-off between transient performance and steady-state accuracy. These results demonstrate that the proposed controller provides superior overall tracking performance and robustness compared with the PID, SMC, and ASMC controllers under identical operating conditions.
4.1.2. Control of sinusoidal signal
Subsequently, a sinusoidal signal is employed to assess the trajectory-tracking ability of the joint module. This step involves a comprehensive investigation of the dynamic tracking error exhibited by the system. A sinusoidal signal with an amplitude of
$\pi /3$
and a period of 4s is utilized as the reference signal for the input of the joint module. The response of the system comprising the joint module is depicted in Figure 4(a). Figure 4(b) displays the trajectory error.
In Figure 4(a), with displacement =
$20^\circ$
denoting the input signal within the controller system and serving as the initial incompatibility condition, it is apparent that the
$p_1$
curve deviates from the intended trajectory. Upon the integration of the
$p_1$
+
$p_2$
controller, the initial incompatibility challenge is mitigated; however, due to insufficient robustness, residual tracking errors persist. In contrast, both the
$p_1$
+
$p_2$
+
$p_3$
controller and the other controllers demonstrate heightened precision in attaining the desired trajectory. The
$p_1$
+
$p_2$
+
$p_3$
controller maintains precise phase alignment and amplitude consistency throughout the oscillatory reference trajectory.
Compared with the PID controller, which is primarily limited by a slow dynamic response, the SMC exhibits inherent advantages in terms of faster convergence and stronger disturbance rejection capability. Nevertheless, the SMC still presents observable phase lag and amplitude degradation when tracking time-varying reference inputs, indicating that its tracking accuracy under dynamic conditions remains limited. In contrast, the proposed
$p_1$
+
$p_2$
+
$p_3$
controller maintains more accurate trajectory tracking with improved dynamic fidelity.
The tracking error profiles shown in Figure 4(b) further highlight the advantages of the proposed control strategy. Among all the compared schemes, the
$p_1$
+
$p_2$
+
$p_3$
controller achieves the fastest error attenuation and consistently exhibits the smallest steady-state tracking error. Although the ASMC demonstrates relatively smooth error convergence, its steady-state tracking error remains slightly larger than that of the proposed controller. Moreover, the absence of oscillatory behavior in the error response of the controller confirms its strong robustness and high-precision regulation capability. The ASMC exhibits an overall tracking response that is close to that of
$p_1$
+
$p_2$
+
$p_3$
controller and successfully follows the sinusoidal reference without instability. However, a more detailed examination reveals that the ASMC produces slightly larger tracking errors, particularly near the peaks of the sinusoidal signal. This limitation is mainly attributed to the fixed-gain structure of the ASMC, which restricts its adaptability in the presence of modeling uncertainties and time-varying dynamics. By contrast, the proposed controller achieves more effective error suppression through its robustness-enhancing control design.
In this secton, simulation results are presented to demonstrate the suitability of the proposed control method for the joint module, and a comprehensive comparison is conducted to evaluate the influence of different controllers on system performance. To further validate the effectiveness of the proposed controller, it is subsequently implemented on a physical experimental platform. The following section presents the experimental results, providing real-world verification of the controller’s performance.
Figure 4. Simulation results of a sinusoidal signal. (a) Tracking sinusoidal signal simulation results, (b) Tracking sinusoidal signal error simulation results.
Figure 5. Experimental platform and flowchart for control design via CSPACE.
4.1.3. Experimental evaluation and comparative analysis
To provide a clear overview of the experimental implementation, the overall experimental procedure is summarized in Figure 5. The flowchart illustrates the sequential steps of system initialization, controller execution, data acquisition, and performance evaluation, thereby clarifying the logical structure and execution process of the experimental validation.
Figure 6. Tracking sinusoidal signal error experimental results. (a) Step response experimental results, (b) tracking sinusoidal signal experimental results, (c) tracking sinusoidal signal current curves experimental results without a load, and (d) tracking sinusoidal signal error experimental results without a load.
Figure 6 presents a comprehensive comparison of the proposed
$p_1$
+
$p_2$
+
$p_3$
controller, the conventional PID controller, and a SMC across various experimental conditions, including step response, sinusoidal trajectory tracking, and control input behavior.
In Figure 6(a), the step response displacement results indicate that the proposed controller achieves a significantly faster rise time and reduced settling time compared to the PID controller. The zoomed-in insets highlight the rapid convergence of the
$p_1$
+
$p_2$
+
$p_3$
response to the reference trajectory, with minimal overshoot and negligible steady-state error. In contrast, the PID controller exhibits slower convergence and a larger deviation from the target value, reflecting its limited responsiveness. The SMC exhibits a faster transient than the PID controller and a smaller steady-state deviation, but noticeable high-frequency ripple (chattering) remains in the vicinity of the setpoint; its overshoot and residual oscillations are higher than those of the
$p_1$
+
$p_2$
+
$p_3$
controller.
Figure 6(b) illustrates the tracking performance under a sinusoidal reference input. The proposed controller demonstrates superior phase alignment and amplitude tracking, closely following the reference trajectory throughout the duration. The magnified segments further reveal that the PID controller exhibits noticeable phase lag and amplitude attenuation, particularly at turning points, whereas the
$p_1$
+
$p_2$
+
$p_3$
controller maintains higher fidelity. The SMC reduces the phase lag observed with the PID controller and better preserves the commanded amplitude; however, a small ripple appears around the zero crossings and turning points due to the discontinuous control action, leading to a slightly larger tracking distortion than the
$p_1$
+
$p_2$
+
$p_3$
controller.
The control effort, represented by the current input in Figure 6(c), shows that the proposed controller produces more structured and stable current profiles. Although both controllers operate under no-load conditions, the PID controller introduces relatively larger fluctuations and noise, which may result in increased energy consumption and potential mechanical stress over prolonged operation. The SMC yields higher-frequency control activity with distinct switching ripple. While this improves disturbance rejection compared with the PID controller, it results in a larger control bandwidth and a slightly higher current than that of the
$p_1$
+
$p_2$
+
$p_3$
controller, which achieves comparable accuracy with smoother actuation.
Finally, Figure 6(d) presents the tracking error corresponding to the sinusoidal reference. The
$p_1$
+
$p_2$
+
$p_3$
controller consistently maintains lower error magnitudes across the entire time span, while the PID controller demonstrates periodic deviations, particularly around the peaks and troughs of the trajectory. The SMC error envelope lies between those of the PID controller and the proposed controller: it effectively suppresses large excursions but retains a residual ripple at the switching frequency, resulting in slightly higher peak-to-peak error than the
$p_1$
+
$p_2$
+
$p_3$
controller.
Overall, the experimental results validate the effectiveness of the
$p_1$
+
$p_2$
+
$p_3$
controller in achieving faster transient responses, improved tracking accuracy, and more stable control input characteristics, thereby confirming its superiority over the conventional PID controller strategy and indicating an advantage over SMC in terms of smoothness and steady-state precision while retaining strong disturbance rejection in dynamic control scenarios.
4.1.4. Resilience to load variations
Figure 7 presents the experimental results of sinusoidal trajectory tracking under different load conditions (0.5 kg and 1 kg), comparing the performance of the proposed
$p_1$
+
$p_2$
+
$p_3$
controller with that of the conventional PID controller and SMC. These results aim to evaluate the robustness and adaptability of the proposed control strategy in response to external load variations.
Figure 7. Experimental results of a sinusoidal signal with different loads. (a) Tracking sinusoidal signal error experimental results with a 0.5 kg load, (b) tracking sinusoidal signal current curves experimental results with a 0.5 kg load, (c) tracking sinusoidal signal error experimental results with a 1 kg load, and (d) tracking sinusoidal signal current curves experimental results with a 1 kg load.
Figures 7(a) and (c) illustrate the tracking error profiles under 0.5 kg and 1 kg loads, respectively. It is evident that the
$p_1$
+
$p_2$
+
$p_3$
controller consistently maintains a smaller error amplitude compared to the PID controller, even as the load increases. Under the 0.5 kg load condition, the tracking error remains within
$\pm 0.2^\circ$
for the proposed controller, whereas the PID controller exhibits noticeable deviations. As the load increases to 1 kg, the error range slightly expands for both controllers. However, the proposed controller still demonstrates significantly reduced fluctuations and superior stability. Across both loads, the SMC outperforms the PID controller in limiting the error amplitude – reflecting its inherent robustness to matched disturbances – but its peak-to-peak error and residual ripple remain higher than those of the
$p_1$
+
$p_2$
+
$p_3$
controller, particularly near the extrema of the sinusoid.
Figures 7(b) and (d) present the corresponding control current signals. The proposed controller yields smoother and more structured current waveforms, while the PID controller exhibits pronounced oscillations and noise, particularly under the heavier 1 kg load. These oscillations not only imply potential energy inefficiency but may also lead to mechanical wear and reduced system longevity over time. The SMC maintains tracking under load with increased actuation effort. Its control currents display characteristic switching ripple and a slightly larger level than those of the proposed controller, highlighting a robustness-smoothness trade-off not observed with
$p_1$
+
$p_2$
+
$p_3$
.
The comparative analysis indicates that the
$p_1$
+
$p_2$
+
$p_3$
controller demonstrates enhanced robustness against external disturbances introduced by varying load conditions. Its ability to preserve precise tracking performance and maintain stable actuation effort under increased system demand further validates its effectiveness for real-world applications where external load conditions are not constant. While SMC provides stronger disturbance rejection than a PID controller, the proposed controller achieves comparable robustness with reduced chattering and lower control effort.
In summary, the experimental results confirm that the proposed control strategy outperforms the PID controller in terms of both tracking accuracy and actuation efficiency across different loading scenarios and surpasses SMC by delivering smoother control with smaller steady-state and peak-to-peak errors, thereby exhibiting excellent robustness and reliability.
5. Conclusions
The primary aim of this research is to optimize the trajectory tracking performance of joint modules by effectively mitigating external disturbances and internal uncertainties. To this end, we introduce a leakage-type adaptive robust controller engineered for uncertain systems and implement it in trajectory-tracking tasks involving a joint module. System stability is rigorously assessed through the lenses of uniform boundedness and uniform ultimate boundedness. Extensive simulations and experimental evaluations are conducted to empirically corroborate the theoretical analyses, showcasing the heightened precision of the proposed controller in comparison to traditional robust control methods. By harnessing the Udwadia-Kalaba equation, which integrates constraints, we can achieve commendable steady-state performance at a reduced control expenditure. Notably, the analytical framework and controller design methodology adopted in this study, which entails modeling the system as a constrained Udwadia-Kalaba equation, are transferrable to tackle analogous dynamics and design intricacies encountered in other second-order systems.
Author contributions
Shengchao Zhen and Ye-Hwa Chen conceived and designed the study. Jian Chen conducted data gathering. Shengchao Zhen and Jian Chen performed statistical analyses. Chen Jian and Xiaoli Liu wrote the article. Xiaolong Chen contributed to manuscript revision.
Financial support
This work was supported in part by the National Natural Science Foundation of China (52175083), in part by the University Synergy Innovation Program of Anhui Province (GXXT-2021-010), in part by the Key Research and Development Program of AnHui Province (Grant No. 2022a05020014), in part by the University Synergy Innovation Program of Anhui Province(GXXT-2023-108), and in part by the Fundamental Research Funds for the Central Universities (JZ2024HGTG0300).
Competing interests
The authors declare no conflicts of interest exist.
Ethical approval
Not applicable.
