Introduction

Industrial robots have largely replaced manual labor in grinding operations, improving both efficiency and surface quality consistency1. However, dynamic interactions between the robot and workpiece often lead to suboptimal results due to sensor errors, parameter variations, and environmental disturbances2.Researchers have developed two control approaches: active3 and passive4 compliance control. Passive methods employ springs/dampers to absorb environmental contact forces, but lack force responsiveness and controllability for high-precision grinding applications5. In contrast, active compliance control dynamically adjusts contact forces using feedback, including impedance control6,7,8, hybrid force-position control9, adaptive control10, and robust control11. For wheel spoke grinding, surface non-conformity causes local over/under-grinding, while inconsistent contact pressure leads to finish variations, necessitating simultaneous force and position control.

Research has demonstrated that force-position hybrid control separates force and position into two orthogonal subspaces: position control operates in the tangential direction, while force control functions in the normal direction. These subspaces can interact with one another, enabling simultaneous control of both position and force, which reduces system oscillations and errors. This method offers the advantages of relatively straightforward implementation and high dynamic force tracking performance. However, during the coupling process, challenges such as pose synchronization errors and force synchronization errors can lead to a decline in control accuracy, resulting in reduced robot compliance and lower grinding precision12. To address the coupling issues between the two subspaces in force-position hybrid control, the problem can be decomposed into two independent subproblems: the position control problem and the force control problem. These are managed using a position controller and a force controller, respectively, to regulate the robotic arm’s motion in Cartesian space.

Current research on force control primarily focuses on methods such as direct force control, indirect force control, impedance-admittance control, and contact force control. Studies indicate that direct/indirect force control methods rely heavily on precise dynamic models, while contact force control requires frequent controller replacements and computationally intensive processes13. Impedance control models robot-environment interaction as a spring-damper-mass system to dynamically regulate force-position relationships. Goyal13 implemented an impedance controller for wrist rehabilitation robots but observed significant force-tracking errors due to unaddressed end-effector dynamics. To improve robustness, Khalifa14 proposed a disturbance observer-based scheme, though robots still exhibit delayed responses in unknown environments. Adaptive impedance control-dynamically adjusting stiffness/damping/inertia-includes model-based, model-free, and learning-driven approaches. Yu et al.15 enhanced environmental adaptability in human-robot interaction, while Ding et al.16 used reinforcement learning to optimize polishing parameters. Shen17 developed fuzzy-logic force regulation for grinding robots. Variable impedance methods like those in18 address floating robot balance but face limitations: model-based approaches suffer disturbance sensitivity , learning-based methods require excessive data, and model-free strategies risk suboptimal tuning.

In addition to the classical impedance and hybrid control frameworks, recent studies have further advanced these techniques for complex robotic interactions. Khalifa 19 introduced a sensorless impedance control method for aerial manipulation, while Ding 20 optimized polishing parameters using reinforcement learning for industrial robots. Chen 21 reported an active force-controlled end-effector for compliant grinding applications, and Yu 22 developed adaptive-constrained impedance control for human–robot cooperation. These recent contributions demonstrate the ongoing development and practical maturity of impedance and hybrid control strategies.

Position tracking control has been extensively studied through methodologies including adaptive control23, neural network control24, sliding mode control25, and fuzzy control26. For instance, Naveen Kumar et al.27 developed a neural network-based adaptive hybrid force-position controller to mitigate uncertain external disturbances in robotic arms, experimentally validating improved position control accuracy. Comparative analyses highlight sliding mode control’s advantages: low sensitivity to parameter variations/external disturbances, model-independence, and exceptional response speed and tracking performance. These traits make it widely applicable in robotics, aerospace, and electromechanical systems28,29,30,31,32.

Sliding mode control operates through two phases - reaching and sliding. To address robustness limitations during the reaching phase, time-varying sliding surfaces have been proposed. Choi33 introduced a rotating-translating switching surface to reduce sensitivity to disturbances, achieving shorter convergence times and enhanced robustness. TH Chang34 further improved this approach via Lyapunov-based continuous sliding mode control, ensuring persistent adherence to time-varying surfaces and global robustness by eliminating the reaching phase.

However, as research continued to delve deeper, it was discovered that sliding mode control struggles to ensure system stability, often exhibiting high-frequency vibrations and gradual convergence. To address the issue of system chattering and achieve convergence within a finite time, Feng Yong et al.35 proposed a sliding mode control method with continuous control behavior to tackle the singularity and chattering phenomena in traditional sliding mode control systems. Dapeng Tian et al.36 introduced an adaptive switching gain observer to mitigate the problem of rapidly changing disturbances, reducing chattering in steady-state conditions. Ke Shao37 proposed an adaptive sliding mode control method for saturation uncertainty, ensuring system stability and the asymptotic convergence of tracking errors. Although these methods have reduced system chattering, they still have some drawbacks. For instance, the saturation function sliding mode control method reduces system robustness, while the adaptive gain sliding mode control method decreases control rate.

To provide faster convergence speed and better robustness, Xinghuo Yu38 proposed a fast terminal sliding mode control method. Mehdi Golestani39, while studying the adaptive problem of n-th order systems, employed a self-tuning fast terminal sliding mode control method, which ensures that the system’s tracking error converges within a finite time and maintains good robustness. Tao Han et al.40 addressed the finite-time tracking control problem for multi-agent systems using the fast terminal sliding mode control method, with experiments demonstrating that the system achieves convergence within a finite time and exhibits strong robustness. However, research has found that the fast terminal sliding mode control method is prone to singularity issues. To address this, literature41 proposed a non-singular fast terminal sliding mode control method, which resolves the singularity problem while ensuring fast convergence speed and robustness. To address the issues of unstable output of the end-effector force and incomplete contact between the grinding head and the workpiece surface during the processing of wheel spoke grinding robots, this paper studies a force-position hybrid impedance control method based on a disturbance observer and non-singular fast terminal sliding mode control. The main contributions are as follows:

  1. (1)

    The combination of impedance control and force-position hybrid control enhances the robot’s adaptability to uncertainties in environmental stiffness, while reducing end-effector contact force overshoot and dynamic deviations.

  2. (2)

    By integrating model reference adaptive control and recurrent neural network methods to adjust impedance parameters online, an adaptive variable impedance control method is constructed. Compared to references15,16,17, this method better adapts to unknown working environments, reduces contact force oscillations between the robot and the environment, and improves force tracking stability.

  3. (3)

    A non-singular fast terminal sliding mode control strategy, which replaces the sign function with a power function, is combined with a disturbance observer. This further mitigates chattering issues while achieving fast and high-precision tracking of the desired position for each joint. Compared to references33,38, this approach ensures finite-time convergence and transient response performance, demonstrating strong robustness.

In summary, this paper develops an adaptive hybrid force–position control framework for wheel-spoke grinding robots. The method combines a disturbance observer–enhanced non-singular fast terminal sliding mode controller for position tracking and an adaptive variable impedance law with recurrent neural network adjustment for force regulation. Theoretical derivations, simulations, and experiments are carried out to verify the finite-time convergence, robustness, and force-tracking precision of the proposed approach.

Dynamic model of wheel spoke grinding robot

Using the Lagrangian analysis method, the dynamic equation of the grinding robot system is established as shown in (1):

$$\begin{aligned} \tau =M(q) \ddot{q}+C(q, \dot{q}) \dot{q}+G(q) \end{aligned}$$
(1)

where \(\tau\) represents the joint driving torque of the robot, M(q) is a symmetric positive definite inertia matrix, \(C(q,\dot{q})\) represents the centrifugal and Coriolis forces, G(q) is the gravity term, and \(q,\dot{q},\ddot{q}\) are represented as the robot angle vector, joint velocity vector, and joint acceleration vector, respectively.

Due to factors such as friction and flexible deformation between the joints of the wheel spoke grinding robot, directly using (1) to calculate the joint torque would result in significant deviations. Therefore, this paper only models the friction of the wheel spoke grinding robot, ignoring the effects of joint flexible deformation. The Coulomb viscous friction model is adopted, and its expression is shown in (2):

$$\begin{aligned} F_{f}=f_{c} \cdot \operatorname {sgn}(\dot{q})+f_{v} \cdot \dot{q}+f_{o} \end{aligned}$$
(2)

where \(f_{c}\) is the Coulomb friction coefficient, \(f_{v}\) is the viscous friction coefficient, sgn(v) is the sign function, and \(f_{o}\) is the compensation term to overcome the static friction during the startup of the robot’s joint motor.

The above Coulomb–viscous model captures the primary frictional effects dominating joint motion. Although more sophisticated friction phenomena-such as Stribeck velocity dependence or thermal drift-exist in practice, they are not explicitly modeled here because their influence can be regarded as bounded disturbances. These unmodeled effects are subsequently compensated by the disturbance observer and the robust NFTSMC controller described in “Position controller design”, ensuring that the simplified model does not compromise overall control accuracy.

Based on (1) and (2), the complete dynamic model expression of the wheel spoke grinding robot in joint space is shown in (3):

$$\begin{aligned} \tau =M(q) \ddot{q}+C(q, \dot{q}) \dot{q}+G(q)+F_{f}(\dot{q}) \end{aligned}$$
(3)

It is noted that Eqs. (1)–(3) describe the intrinsic joint-space dynamics of the robot without explicitly including external contact forces. When the manipulator interacts with the environment during grinding, the external contact force \(f_{e}\) acting on the end-effector can be projected into the joint space through the Jacobian transpose as \(J^{T}f_{e}\). Accordingly, the complete dynamic model can be expressed as

$$\tau = M(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) + F_f(\dot{q}) + J^{T}f_{e},$$

where J(q) is the Jacobian matrix mapping Cartesian forces to joint torques. For conciseness, this external term is handled separately in “Force controller design” through the adaptive impedance-based environment interaction model.

Fig. 1
Fig. 1The alternative text for this image may have been generated using AI.
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Hybrid force/position control block diagram.

Hybrid force-position control method for robot wheel spoke grinding

To ensure stable force output and precise position tracking during the grinding process of wheel spokes, this paper introduces a force-position hybrid control method. The robotic arm performs force control in the normal direction and position control in the tangential direction. Whether each joint performs force control or position control is determined by the selection matrix S. When the corresponding element is 1, the joint performs position control, and when it is 0, the joint performs force control. The selection matrix for position control is \(S = diag(1,1,0,1,1,1)\), while the selection matrix for force control is the identity matrix I minus the selection matrix S, yielding \(I-S\). After setting the selection matrix, the actual position and torque of the robotic arm are compared with the desired position and torque, and the feedback information is passed to the controller to achieve contact operations based on force-position hybrid control. Figure 1 shows the block diagram of the force-position hybrid control.

In Fig. 1, the “Inverse Kinematics” block represents the coordinate transformation from Cartesian-level motion/force commands to joint-space control inputs. Its placement is schematic and does not imply an independent control loop. This depiction highlights the role of inverse kinematics in mapping hybrid force–position control signals to the manipulator’s joint torques.

The output torque of the robot joint can be regarded as the sum of the output vectors of the two controllers, expressed by the following formula as shown in (4):

$$\begin{aligned} \tau =S \cdot \tau _{p}+(I-S) \cdot \tau _{f} \end{aligned}$$
(4)

It should be noted that the selection matrix S in (4) is defined in joint space as a practical implementation of Cartesian hybrid force–position control. While \(S = \textrm{diag}(1,1,0,1,1,1)\) appears to assign force control to a single joint, the corresponding control effect is realized in Cartesian space along the normal direction of the contact surface. In this formulation, the mapping between Cartesian subspaces and joint coordinates is handled through the Jacobian matrix, so that the “force” and “position” channels are effectively distributed across multiple joints. This representation simplifies computation while preserving the Cartesian control principle.

Position controller design

To address the challenge of incomplete contact between the grinding path and the spoke surface, which may result in localized over-grinding or under-grinding, this paper proposes a disturbance observer-based non-singular fast terminal sliding mode control method within the position control subspace. The objective is to enhance the convergence speed of the robot’s position tracking control, mitigate singularity issues, and improve the system’s robustness. The flowchart of the position tracking control method is shown in Fig. 2 below:

Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
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Position tracking control flowchart.

Disturbance observer design

The control accuracy of manipulators inevitably degrades due to ubiquitous exogenous disturbances and internal friction effects. To mitigate this issue, a model-based disturbance observer is constructed to provide compensation references for the controller. The disturbance observer structure is formulated in (5) as follows:

$$\begin{aligned} \begin{aligned} \dot{\hat{\xi }}&= l(x) \left( {\xi }-\hat{\xi } \right) \\&= l(x)\left( M\left( x_{1}\right) x_{2}+C\left( x_{1}, x_{2}\right) x_{2}+C\left( x_{1}\right) -\tau -\hat{\xi }\right) \end{aligned} \end{aligned}$$
(5)

where \(\hat{\xi }\) represents the estimate of the aggregated disturbance \({\xi }\). l(x) denotes the observer gain matrix whose synthesis procedure is detailed as

$$\begin{aligned} l(x)= \tilde{\gamma }^{-1} M^{-1}(x_{1}) \end{aligned}$$
(6)

where \(\tilde{\gamma }\) is the symmetric positive definite matrix governing the observer gain synthesis process, obtained via the linear matrix inequality (LMI) condition:

$$\begin{aligned} \tilde{\gamma }+\tilde{\gamma }^{T}-\tilde{\gamma } \dot{M}\left( x_{1}\right) \tilde{\gamma }^{T} \ge \omega \end{aligned}$$
(7)

with \(\omega\) being a designer-specified positive definite gain-related matrix. Define the disturbance observation error as

$$\begin{aligned} z={\xi }-\hat{\xi } \end{aligned}$$
(8)

Combined with equations ((5)) and ((8)) yields the observation error dynamics:

$$\begin{aligned} \dot{z}=\dot{\xi } - l(x) \dot{z} \end{aligned}$$
(9)

The Lyapunov function candidate for the proposed disturbance observer is constructed as follows:

$$\begin{aligned} V_{d o}=z \tilde{\gamma }^{T} M\left( x_{1}\right) \widetilde{\gamma } z \end{aligned}$$
(10)

By differentiating (10), the following formula is obtained as shown in (11):

$$\begin{aligned} \begin{aligned} \dot{V}_{d o}&=-z^{T} \widetilde{\gamma } z-z^{T} \widetilde{\gamma }^{T} z+z^{T} \widetilde{\gamma } \dot{M}\left( x_{1}\right) \widetilde{\gamma }^{T} z + 2 \dot{\xi }^{T} \widetilde{\gamma } M\left( x_{1}\right) \widetilde{\gamma }^{T} z \\&=-z^{T} z\left( \widetilde{\gamma }+\widetilde{\gamma }^{T}-\widetilde{\gamma } \dot{M}\left( x_{1}\right) \widetilde{\gamma }^{T}\right) + 2 \dot{\xi }^{T} \widetilde{\gamma } M\left( x_{1}\right) \widetilde{\gamma }^{T} z\\&\le -z^{T} \omega z + 2 \dot{\xi }^{T} \widetilde{\gamma } M\left( x_{1}\right) \widetilde{\gamma }^{T} z \end{aligned} \end{aligned}$$
(11)

Non-singular fast terminal sliding mode controller design

The control objective involves synthesizing a feedback control law \(\tau\) that enables the 6-DOF robotic manipulator to achieve precise and rapid tracking of the desired trajectory q of the six-degree-of-freedom robot system can quickly and accurately track the desired trajectory \(q_{d}\) with finite-time convergence, while ensuring robust stability of the closed-loop dynamics. The joint angle tracking error is defined as

$$\begin{aligned} \left\{ \begin{array}{l} e(t)=q-q_{d} \\ \dot{e}(t)=\dot{q}-\dot{q}_{d} \end{array}\right. \end{aligned}$$
(12)

A non-singular fast terminal sliding mode (NFTSM) surface is developed to achieve singularity-free global finite-time convergence:

$$\begin{aligned} \sigma (t)=e+K_{1} \operatorname {sig}^{\gamma _{1}}(e)+K_{2} \operatorname {sig}^{\gamma _{2}}(\dot{e}) \end{aligned}$$
(13)

where the hybrid structure integrating sign function and absolute-value power terms eliminates singularities in sliding surface derivatives while preserving control law continuity.The nonlinear operator \(\operatorname {sig}^{\gamma _{1}}(e)\) is formally defined as:

$$\begin{aligned} \operatorname {sig}^{\gamma _1}(e) \triangleq |e|^{\gamma _1} \operatorname {diag}[\operatorname {sgn}(e)] \end{aligned}$$
(14)

The definition of \(\operatorname {sig}^{\gamma _{2}}(\dot{e})\) is the same as the above equation. The matrices involved in (13) are shown in (15):

$$\begin{aligned} \left\{ \begin{array}{l} \gamma _{1}=\operatorname {diag}\left\{ \gamma _{11}, \gamma _{12}, \cdots , \gamma _{1 n}\right\} \\ \gamma _{2}=\operatorname {diag}\left\{ \gamma _{21}, \gamma _{22}, \cdots , \gamma _{2 n}\right\} \\ K_{1}=\operatorname {diag}\left\{ K_{11} K_{12}, \cdots , K_{1 n}\right\} \\ K_{2}=\operatorname {diag}\left\{ K_{21}, K_{22}, \cdots , K_{2 n}\right\} \end{array}\right. \end{aligned}$$
(15)

The following sufficient conditions must be satisfied to guarantee the closed-loop stability of the proposed control scheme:

$$\begin{aligned} \begin{aligned} K_{1 i}>0, K_{2 i}>0,1<\gamma _{2 i}<2, \gamma _{1 i}>\gamma _{2 i}, i=1,2, \ldots , n \end{aligned} \end{aligned}$$
(16)

Differentiating (13),as shown in (17):

$$\begin{aligned} \dot{\sigma }(t)=\dot{e}+K_{1} \gamma _{1} \operatorname {diag}\left\{ \mid e^{\gamma _{1}-I_{n}}\right\} \cdot \dot{e}+K_{2} \gamma _{2} \operatorname {diag}\left\{ \mid e^{\gamma _{2}-I_{n}}\right\} \cdot \ddot{e} \end{aligned}$$
(17)

where \(\left\{ \begin{array}{l} \gamma _{1} \operatorname {diag}\left\{ |e|^{\mid \gamma _{1}-I_{n}}\right\} \cdot \dot{e}=\frac{d}{d t}\left( \operatorname {sig}^{\gamma _{1}}(e)\right) \\ \gamma _{2} \operatorname {diag}\left\{ |e|^{\gamma _{2}-I_{n}}\right\} \cdot \ddot{e}=\frac{d}{d t}\left( \operatorname {sig}^{\gamma _{2}}(\dot{e})\right) \end{array}\right.\) . Based on (3) and (12), the following formula is obtained:

$$\begin{aligned} \ddot{e}(t)=M^{-1}(q)\left( \tau +\xi -C(q, \dot{q}) \dot{q}-G(q)-M(q) \ddot{q}_{d}\right) \end{aligned}$$
(18)

where \(\xi\) denotes the disturbance error term. Based on the designed sliding surface above, the controller can be designed as shown in (19):

$$\begin{aligned} \tau _{p}=\tau _{q}+\tau _{s} \end{aligned}$$
(19)

where \(\tau _{q}\) represents the equivalent controller when robot system reaches the sliding surface, and \(\tau _{s}\) represents the robust controller part.

The formula for the equivalent controller is as follows:

$$\begin{aligned} { \begin{aligned} \tau _q&=M(q) \ddot{q}_d+C(q, \dot{q}) \dot{q}+G(q)-\hat{\xi } \\&-K_2^{-1} \cdot \gamma _2^{-1} M(q)\left( K_1 \gamma _1 \operatorname {diag}\left\{ \mid e^{\gamma _1-I_n}\right\} +I_n\right) \cdot \operatorname {sig}^{2 I_n-\gamma _2}(\dot{e}) \end{aligned}} \end{aligned}$$
(20)

The design of the robust controller is as follows:

$$\begin{aligned} \tau _{s}=-M(q)\left( m \sigma +n \operatorname {sig}^{h}(\sigma )\right) \end{aligned}$$
(21)

\(\left\{ \begin{array}{l} m=\operatorname {diag}\left\{ m_{1}, m_{2}, \cdots , m_{n}\right\}>0 \\ n=\operatorname {diag}\left\{ n_{1}, n_{2}, \cdots , n_{n}\right\} >0 \\ 0<h=\operatorname {diag}\left\{ h_{1}, h_{2}, \cdots , h_{n}\right\} <1 \end{array}\right.\)

Based on (19), (20), and (21), the final non-singular fast terminal sliding mode controller is obtained as:

$$\begin{aligned} {\begin{array}{l} \tau _{p}=M(q) \ddot{q}_{d}+C(q, \dot{q}) \dot{q}+G(q)-\hat{\xi }- \\ \quad K_{2}^{-1} \cdot \gamma _{2}^{-1} M(q)\left( K_{1} \gamma _{1} \operatorname {diag}\left\{ \mid e^{\mid \gamma _{1}-I_{n}}\right\} +I_{n}\right) \cdot \operatorname {sig}^{2 I_{n}-\gamma _{2}}(\dot{e})\\ -M(q)\left( m \sigma +n\operatorname {sig}^{h}(\sigma )\right) \end{array}} \end{aligned}$$
(22)

To ensure that the six-degree-of-freedom grinding robot system exhibits non-singular characteristics and avoids singularities, the exponential terms of the position error and velocity error should be greater than zero.

To further reduce the chattering phenomenon during the process of state variables approaching the sliding surface and converging to the origin, the convergence process can be made smoother by optimizing the sliding surface design or improving the control law. A power function is used to replace the sign function to achieve smoother system processing. The power function formula is as follows:

$$\begin{aligned} \operatorname {fal}(\kappa , l, \varphi )=\left\{ \begin{array}{l} |\varvec{\kappa }|^{t} \operatorname {sgn}(\kappa ),|\kappa |>\varphi \\ \kappa / \varphi ^{1-t},|\varvec{\kappa }| \le \varphi \end{array}\right. \end{aligned}$$
(23)

where \(\kappa , \varphi\) are the parameters of the power function, with the following inequalities strictly enforced:

$$\begin{aligned} 0<\imath<1, 0<\varphi <1 \end{aligned}$$
(24)

The power function \(\operatorname {fal}()\), saturation function \(\operatorname {sat}()\), arctangent function \(a \tan ()\), sign function \(\operatorname {sgn}()\), hyperbolic tangent function \(\tanh ()\), and continuous function \(\text{ theta() }\) are plotted together for comparison. As shown in Fig. 3, the function curves exhibit both fast response characteristics and rapid convergence advantages, quickly approaching 1. This plays a significant role in suppressing input chattering and improving the convergence speed of the system’s dynamic response process.

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
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Switching function motion curve diagram.

Substituting (23) into (22), the transformed form of (22) is obtained as:

$$\begin{aligned} \begin{aligned} \tau _{p}=&M(q) \ddot{q}_{d}+C(q, \dot{q}) \dot{q}+G(q)-\hat{\xi }_{-} \\&K_{2}^{-1} \cdot \gamma _{2}^{-1} M(q)\left( K_{1} \gamma _{1} \operatorname {diag}\left\{ \mid e^{\gamma _{1}-I_{n}}\right\} +I_{n}\right) \cdot \operatorname {sig}^{2 I_{n}-\gamma _{2}}(\dot{e})\\&-M(q)\left( m \sigma +n f a l^{h}(\sigma )\right) \end{aligned} \end{aligned}$$
(25)

Stability analysis

The stability of the designed control is proven through the Lyapunov function as shown in (26):

$$\begin{aligned} V_{s}=\frac{1}{2} \sigma ^{T} \sigma \end{aligned}$$
(26)

By differentiating (26) and substituting (17) and (18) into it, (27) can be obtained:

$$\begin{aligned} { { \begin{aligned} \dot{V}_{s}&=\sigma ^{T} \sigma \\&=\sigma ^{T}\left( \dot{e}+K_{1} \gamma _{1} \operatorname {diag}\left\{ \mid e^{\gamma _{1}-I_{n}}\right\} \cdot \dot{e}+K_{2} \gamma _{2} \operatorname {diag}\left\{ |\dot{e}|^{\mid \gamma _{2}-I_{n}}\right\} \cdot \ddot{e}\right) \\&=\sigma ^{T}\left[ \dot{e}+K_{1} \gamma _{1} \operatorname {diag}\left\{ \mid e^{\mid \gamma _{1}-I_{n}}\right\} \cdot \dot{e}+K_{2} \gamma _{2} \operatorname {diag}\left\{ |\dot{e}|^{\gamma _{2}-I_{n}}\right\} \cdot M^{-1}(q) H\right] \end{aligned}}} \end{aligned}$$
(27)

where \(H=\left( \tau -\xi -M(q) \ddot{q}_{d}-C(q, \dot{q}) \dot{q}-G(q)\right)\) .

Substituting (22) into (27) yields the following (28):

$$\begin{aligned} \dot{V}_{s}=-\sigma ^{T} K_{2} \gamma _{2} \operatorname {diag}\left\{ |\dot{e}|^{\gamma _{2}-I_{n}}\right\} \cdot \left( m \sigma +n f a l^{h}(\sigma )-M^{-1}(q) z\right) \end{aligned}$$
(28)

Transforming (28) further yields the following two forms, as shown in (29) and (30):

$$\begin{aligned} \begin{aligned} \dot{V}_s=&-\sigma ^T K_2 \gamma _2 \operatorname {diag}\left\{ |\dot{e}|^{p_2-I_n}\right\} \\&\cdot \left[ \left( m-\operatorname {diag}\left\{ M^{-1}(q) z\right\} \cdot \operatorname {diag}^{-1}(\sigma )\right) \cdot \sigma n f a l^h(\sigma )\right] \end{aligned} \end{aligned}$$
(29)
$$\begin{aligned} { { \begin{aligned} \dot{V}_s=&-\sigma ^T K_2 \gamma _2 \operatorname {diag}\left\{ |\dot{e}|^{\gamma _2-I_n}\right\} \\&\cdot \left\{ m \sigma +\left[ n-\operatorname {diag}\left\{ M^{-1}(q) z\right\} \cdot \operatorname {diag}^{-1}\left( \operatorname {fal}^h(\sigma )\right) \right] \cdot \operatorname {fal}^h(\sigma )\right\} \end{aligned}}} \end{aligned}$$
(30)

Assume the positive definite scalar function of the robot system is \(V=V_{d o}+V_{s}\).Since the observation error can z converge to zero within a finite time, \(V_{d o}=0\), the following equation holds:

$$\begin{aligned} \dot{V}=\dot{V}_{s} \le -\sigma ^{T} K_{2} \gamma _{2} \operatorname {diag}\left\{ |\dot{e}|^{\gamma _{2}-I_{n}}\right\} \left( \dot{m} \sigma +n f a l^{h}(\sigma )\right) \end{aligned}$$
(31)

Let \(K_{2} \gamma _{2} \operatorname {diag}\left\{ \mid e^{1_{2}-I_{n}}\right\} m=\phi _{1}, K_{2} \gamma _{2} \operatorname {diag}\left\{ |\dot{e}|^{\gamma _{2}-I_{n}}\right\} n=\phi _{2}\), and let \(\phi _{1}, \phi _{2} \in R^{n \times n}\) be a positive definite diagonal matrix. (31) can be transformed into:

$$\begin{aligned} \dot{V} \le -\sigma ^{T} \phi _{1} \sigma -\sigma ^{T} \phi _{2} f a l^{h}(\sigma ) \end{aligned}$$
(32)

Lemma 1

For any \(x_{i} \in R, i=1,2, \ldots , n\) and \(0<\mu \le 1\), the following inequality holds:

$$\begin{aligned} \left( \sum _{i=1}^{n}\left| x_{i}\right| \right) ^{\mu } \le \sum _{i=1}^{n}\left| x_{i}\right| ^{\mu } \le n^{1-\mu }\left( \sum _{i=1}^{n}\left| x_{i}\right| \right) ^{\mu } \end{aligned}$$
(33)

Lemma 2

For any \(0<\psi <1, \Theta _{1}>0, \Theta _{2}>0\), there exists a continuous positive definite function V(t) that, when \(c>0\), satisfies the following differential inequality:

$$\begin{aligned} \dot{V}(t) \le -\Theta _{1} V(t)-\Theta _{2} V^{\psi }(t) \end{aligned}$$
(34)

Based on the above content, the following statements hold:

  1. 1.

    The initial point of the system being finite-time stable means that starting from any initial state, the system’s state can converge to the equilibrium point within a finite time and remain near this equilibrium point thereafter.

  2. 2.

    There exists an upper bound on the finite time \(t_{m}\), such that V(t) can converge to \(V_{0}\) within a finite time, satisfying:

    $$\begin{aligned} t_{m} \le \left[ 1 / \Theta _{1}(1-\psi )\right] \ln \frac{\Theta _{1} V^{1-\psi }(0)+\Theta _{2}}{\Theta _{2}} \end{aligned}$$
    (35)

According to Lemma 1, (32) can be transformed into:

$$\begin{aligned} \dot{V} \le -2 \lambda _{1} V-2^{(h+1) / 2} \lambda _{2} V^{(h+1) / 2} \end{aligned}$$
(36)

where \(\lambda _{1}=\min \left( \phi _{1}\right) , \lambda _{2}=\min \left( \phi _{2}\right)\) .

To prove that the system is stable when it reaches the equilibrium point, according to the definition of finite-time stability, for any \(e_{i} \ne 0\), the tracking error e converges along the sliding surface designed in (13) and reaches the equilibrium point within a finite time. The finite time is given by the following formula:

$$\begin{aligned} \begin{aligned} T_r&\le \frac{1}{\lambda _1(1-h)} \ln \left( \frac{2 \lambda _1 V^{(1-h) / 2}+2^{(1+h) / 2} \lambda _2}{2^{(1+h) / 2} \lambda _2}\right) \\&\le \frac{1}{\lambda _1(1-h)} \ln \left( \frac{\lambda _1 V^{(1-h) / 2}+2^{(1+h) / 2} \lambda _2}{2^{(h-1) / 2} \lambda _2}\right) \end{aligned} \end{aligned}$$
(37)

According to Lyapunov stability theory, the designed sliding mode controller satisfies the global asymptotic stability condition, and the trajectory tracking error of the robotic arm system will precisely converge to zero along the sliding mode dynamics within a finite time \(T_{r}\) due to the strong constraint characteristics of the sliding surface.

Force controller design

The normal contact force at the grinding interface critically determines surface integrity and material removal characteristics during robotic machining. To address this dependency, we propose an adaptive variable impedance control framework for precision constant-force grinding operations. Through establishing a nonlinear contact dynamics model that quantifies steady-state force deviations, the method systematically addresses environmental uncertainties through adaptive compensation while enabling continuous impedance parameter optimization via recurrent neural networks.

Grinding robot environmental contact mode

The dynamic interaction at the robot-workpiece interface is modeled as a spring-damper system (Fig. 4), capturing the essential dynamics of normal contact forces during grinding operations.

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
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Spring-damper model.

The contact force formula between the robot and the environment is shown in (38):

$$\begin{aligned} f_{\textrm{e}}=b_{e} \dot{x}_{m}+k_{e}\left( x_{m}-x_{e}\right) \end{aligned}$$
(38)

where \(f_{\textrm{e}}\) denotes the robot-environment interaction force, \(k_{e}\) and \(b_{e}\) represent the environment’s stiffness and damping coefficients respectively, \(x_{m}\) corresponds to the measured end-effector position, and \(x_{e}\) defines the environment’s undeformed equilibrium position.

According to the definition of the robot-environment contact model and impedance, it can be described as a second-order model,is shown in (39):

$$\begin{aligned} f_{\textrm{e}}-f_{d}=m\left( \ddot{x}_{m}-\ddot{x}_{d}\right) +b\left( \dot{x}_{m}-\dot{x}_{d}\right) +k\left( x_{m}-x_{d}\right) \end{aligned}$$
(39)

where \({x_{d}}\) represents the desired reference trajectory for force tracking, \({f_{d}}\) specifies the target contact force, and m, b, k respectively denote the inertial matrix, damping matrix, and stiffness matrix constituting the impedance dynamics.

Steady-state error analysis

Let \(e_{x}=x_{m}-x_{d}\),\(\Delta f=f_{e}-f_{d}\), according to (39), the impedance model of the robot arm can be deduced as:

$$\begin{aligned} \Delta f=m \ddot{e}_{x}+b \dot{e}_{x}+k e_{x} \end{aligned}$$
(40)

The Laplace transform of (40) is:

$$\begin{aligned} \Delta f(s)=\left( m s^{2}+b s+k\right) e_{x} \end{aligned}$$
(41)

According to (38) and \(e_{x}=x_{m}-x_{d}\),\(\Delta f=f_{e}-f_{d}\), we can get :

$$\begin{aligned} \begin{aligned} \Delta f&=f_{e}-f_{d} \\&=b_{e} \dot{x}_{m}+k_{e}\left( x_{m}-x_{e}\right) -f_{d} \\&=b_{e} \dot{x}_{m}+k_{e}\left( x_{d}-x_{e}\right) +k_{e} e_{x}-f_{d} \end{aligned} \end{aligned}$$
(42)

According to (41) and (42):

$$\begin{aligned} \begin{aligned} \Delta f(s)&=\frac{k_{e}\left( x_{d}-x_{e}\right) -f_{d}}{s}+k_{e} e_{x}+b_{e} s x_{m} \\&=\frac{k_{e}\left( x_{d}-x_{e}\right) -f_{d}}{s}+k_{e} e_{x}+b_{e} s\left( x_{d}+e_{x}\right) \\&=\frac{k_{e}\left( x_{d}-x_{e}\right) -f_{d}+s^{2} b_{e} x_{d}}{s}+\left( b_{e} s+k_{e}\right) e_{x} \\&=\frac{k_{e}\left( x_{e}-x_{d}\right) -f_{d}+s^{2} b_{e} x_{d}}{s}+\frac{\left( b_{e} s+k_{e}\right) \Delta f(s)}{m s^{2}+b s+k} \end{aligned} \end{aligned}$$
(43)

Therefore, there are:

$$\begin{aligned} \Delta f(s)=\frac{k_{e}\left( x_{e}-x_{d}\right) -f_{d}+s^{2} b_{e} x_{d}}{s\left( m s^{2}+b s-b_{e} s+k-k_{e}\right) } \cdot \left( m s^{2}+b s+k\right) \end{aligned}$$
(44)

Then the steady-state error is:

$$\begin{aligned} f_{s s}=\lim _{s \rightarrow 0} s \cdot \Delta f(s)=\frac{k}{k-k_{e}}\left[ k_{e}\left( x_{d}-x_{e}\right) -f_{d}\right] \end{aligned}$$
(45)

Equation (45) establishes that system stability depends on both the end-effector force error and stiffness coefficients. Consequently, if the steady-state error is zero, then one of the following conditions must be met:

$$\begin{aligned} \left\{ \begin{array}{l} k=0 \\ x_{d}=\frac{f_{d}}{k_{e}}+x_{e} \end{array}\right. \end{aligned}$$
(46)

Accurate force tracking requires precise environmental position (\(x_{e}\)) and stiffness (\(k_{e}\)) estimation, as established in (46). However, dynamic operating environments induce real-time variations in these parameters, necessitating adaptive estimation for robust force control during robotic motion.

Adaptive control method

Since the environmental parameters dynamically evolve during robotic grinding operations, their precise measurement becomes inherently infeasible. To resolve this uncertainty, we develop a real-time estimator for \(\hat{x}_{e}, \hat{b}_{e}, \hat{k}_{e}\), whose outputs drive the adaptive reference trajectory generator:

$$\begin{aligned} \hat{x}_{d}=\hat{x}_{e}+\frac{f_{d}}{\hat{k}_{e}} \end{aligned}$$
(47)

The estimated value of the actual contact force exerted by the robot is as follows:

$$\begin{aligned} \hat{f_e}=\hat{b}_{e} \dot{x}+\hat{k}_{e}\left( \hat{x}_{e}-x_{m}\right) \end{aligned}$$
(48)

According to (48), the estimation error of the actual contact force at the robot’s end-effector is:

$$\begin{aligned} e_{f}=\hat{f_e}-f_e=A B \end{aligned}$$
(49)

where \(A=\left[ \begin{array}{lll}\dot{x}_{m}&x_{m}&-1\end{array}\right]\), \(B=\left[ \begin{array}{lll}a_{1}&a_{2}&a_{3}\end{array}\right] ^{T}\), \(a_{1}=\hat{b}_{e}-b_{e}\),\(a_{2}=\hat{k}_{e}-k_{e}\),\(a_{3}=\hat{k}_{e} \hat{x}_{e}-k_{e} x_{e}\).When the estimation error vanishes, he following equivalence holds:

$$\begin{aligned} f_e=f_{d}+\hat{b}_{e} \dot{x}_{m}+\hat{k}_{e} \hat{x}_{d}-\hat{k}_{e} x_{m} \end{aligned}$$
(50)

By applying the Laplace transform to (50), the following equation is obtained:

$$\begin{aligned} \Delta x(s)=x_{d}-x_{m}=\frac{\hat{b}_{e} s x_{m}-\Delta f}{\hat{b}_{e} s+k_{e}} \end{aligned}$$
(51)

Upon substituting the aforementioned formula into \(\frac{\Delta x}{\Delta f}=\frac{1}{m s^{2}+b s+k}\), the following result is obtained:

$$\begin{aligned} i s^{3}+j s^{2}+k s+\left( k+k_{e}\right) \Delta f=0 \end{aligned}$$
(52)

The system dynamics are governed by the parametric relations:

$$\begin{aligned} \begin{aligned} i&=-m b_{e} k_{d} \\ j&=m \Delta f-b b_{e} x_{d} \\ k&=\left( b+b_{e}\right) \Delta f-k b_{e} x_{d}\\ \end{aligned} \end{aligned}$$
(53)

Equation (52) reveals that the steady-state condition \(\left( k+k_{e}\right) \Delta f=0\) necessitates zero force tracking error, as the strictly positive parameters \(k>0, k_{e}>0\) guarantee \(k+k_{e}>0\). The Lyapunov function is defined as follows :

$$\begin{aligned} V=B^{T} P B, P^{-1}=\left[ \begin{array}{ccc} \lambda _{1} & 0 & 0 \\ 0 & \lambda _{2} & 0 \\ 0 & 0 & \lambda _{3} \end{array}\right] \end{aligned}$$
(54)

where P denotes a diagonal positive definite matrix, and \(\lambda _{1},\lambda _{2},\lambda _{3}\) represent constant parameters.

Define \(\dot{\textrm{B}}=-P^{-1} {A}^{\textrm{T}} {AB}\) as,According to (49) and (54), the following is obtained:

$$\begin{aligned} \begin{aligned} \dot{\text {B}} =-P^{-1} {A}^{\textrm{T}} e_{f} =-{P}^{-1}\left[ \begin{array}{l}\dot{x}_{m} \\ x_{m} \\ -1\\ \end{array}\right] \left( \hat{f}_{e}-f_{e}\right) \end{aligned} \end{aligned}$$
(55)

From the aforementioned information, it can be concluded that:

$$\begin{aligned} \left. \begin{array}{c} \dot{\hat{b}}_{e}=-\lambda _{1} \dot{x}_{m} e_{f} \\ \dot{\hat{k}}_{e}=-\lambda _{2} x_{m} e_{f} \\ \dot{\hat{x}}_{e}=\frac{e_{f}}{\hat{k}_{e}}\left( \lambda _{2} x \hat{x}_{e}+\lambda _{3}\right) \end{array}\right\} \end{aligned}$$
(56)

From (56), the designed adaptive controller can be obtained as:

$$\begin{aligned} \left. \begin{array}{c} \hat{b}_{e}=\hat{b}_{e 0}-\lambda _{1} \int _{0}^{t} x_{m} e_{f} d t \\ \hat{k}_{e}=\hat{k}_{e 0}-\lambda _{2} \int _{0}^{t} x_{m} e_{f} d t \\ \hat{x}_{e}=\hat{x}_{e 0}+\int _{0}^{t} \frac{e_{f}}{\hat{k}_{e}}\left( \lambda _{2} x_{m} \hat{x}_{e}+\lambda _{3}\right) d t \\ \hat{f}=\hat{b}_{e} \dot{x}+\hat{k}_{e}\left( x_{m}-\hat{x}_{e}\right) \\ e_{f}=\hat{f}-f \\ x_{r}=\hat{x}_{e}+\frac{f_{d}}{\hat{k}_{e}} \end{array}\right\} \end{aligned}$$
(57)

  • \({\hat{b}_{e 0},\hat{k}_{e 0},\hat{x}_{e 0}}\) : initialization value.

The derivative of the Lyapunov function along the \(\dot{\textrm{B}}\)-direction is obtained from (54):

$$\begin{aligned} \dot{{V}}=\left( {B}^{{T}} {PB}\right) ^{\prime }={B}^{{T}}\left( \Lambda ^{{T}} {P}+{P} \Lambda \right) {A} \end{aligned}$$
(58)

Matrix operation gives Matrix D:

$$\begin{aligned} {D}=2\left| \begin{array}{ccc} \dot{x}_{m}^{2} & \dot{x}_{m} x_{m} & -\dot{x}_{m} \\ \dot{x}_{m} \hat{x}_{m} & x_{m}^{2} & -x_{m} \\ -\hat{x}_{m} & -x_{m} & 1 \end{array}\right| \end{aligned}$$
(59)

The necessary and sufficient condition for the stability of the system is that the matrix \(\dot{{V}}\) is negative definite or semi negative definite, that is, \(D=-\left( \Lambda ^{{T}} {P}+{P} \Lambda \right)\) is positive definite or semi positive definite.The matrix V is positive definite according to Lyapunov stability.The matrix D is positive semidefinite according to (59).Therefore, if the estimated parameters change according to the rules of \(\dot{\textrm{B}}=-P^{-1} {A}^{\textrm{T}} {AB}\), the system is asymptotically stable.

Recurrent neural network variable impedance control

Recurrent neural networks (RNN) can effectively capture temporal dependencies in sequential data, which is crucial for robot force control. In force control tasks, the input data to the system is typically a variable-length sequence, and RNN can naturally handle such sequences. To simplify the analysis, (38) can be reduced to a single-degree-of-freedom system in the normal direction, with its transfer function as follows:

$$\begin{aligned} G(s)=\frac{\varpi _{n}^{2}}{s^{2}+2 \xi \varpi _{n} s+\varpi _{n}^{2}} \end{aligned}$$
(60)

\(\xi\) is the damping ratio,\(\xi =\frac{B_{d}}{2 \sqrt{M_{d} K_{d}}}\). \(\xi\) is the natural frequency, \(\varpi _{n}=\sqrt{\frac{K_{d}}{M_{d}}}\). \(M_{d}, B_{d}, K_{d}\) are the impedance parameters.

From (60) and Fig. 5, it can be seen that the magnitude of the force error is inversely proportional to the stiffness coefficient and the damping coefficient. The relationship between the force error and the stiffness and damping coefficients can be expressed using a function, which is then used as the activation function in the hidden layer of the neural network. The equation is as follows:

$$\begin{aligned} f_{i}=c_{i}\left( 1+e^{\frac{|\Delta F|}{b_{i}}}\right) ^{-1} \end{aligned}$$
(61)

\(f_{i}\) is the activation function. \(c_{i},{b_{i}}\) are the height and width parameters of the activation function for the first hidden layer node.

Fig. 5
Fig. 5The alternative text for this image may have been generated using AI.
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Activation function characteristic curve.

In the first hidden layer, the input consists of stiffness and damping characteristics, while the second hidden layer inputs a recurrent neural network with an exponential linear unit (ELU) as its activation function. Through dynamic learning of the network, the desired parameters for stiffness and damping are output, describing the relationship between force error and stiffness-damping coefficients. The neural network model is composed of an input layer, the first hidden layer, the second hidden layer, and an output layer. The neural network model is shown in Fig. 6 below:

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
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Variable impedance parameter recurrent neural network model.

In the figure above, the weight coefficients are: \(w_{i j}, U_{j}, w_{j 1}, w_{j 2}\);\(z_{j}\) represents the state. The model is translated into a mathematical description as follows:

$$\begin{aligned} \left\{ \begin{array}{l} f_{i}=c_{i}\left( 1+e^{\frac{|\Delta F|}{b_{i}}}\right) ^{-1} \\ n_{j}=\sum _{i=1}^{5} w_{i j} f_{i} \\ y_{j}(k)=n_{j}(k)+U_{j} z_{j}(k-1)+B_{j} \\ z_{j}=E L U\left( y_{j}\right) =\left\{ \begin{array}{l} y_{j}, y_{j} \ge 0 \\ 0.001\left( e^{y_{j}}-1\right) , y_{j}<0 \end{array}\right. \\ B_{d}=\sum _{j=1}^{4} z_{j} w_{j 1} \\ K_{d}=\sum _{j=1}^{4} z_{j} w_{j 2} \end{array}\right. \end{aligned}$$
(62)

\(\Delta F\) is the force error. \(B_{j}, y_{j}, E L U\left( y_{j}\right)\) is the bias, net input, and output of the second hidden layer. \(n_{j}\) is the weighted sum output from the first hidden layer (Table 1).

Table 1 Force tracking performance metrics comparison.
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Simulation experiment analysis of force-position hybrid control for wheel spoke grinding robots

Comparison of sliding surface convergence

To verify the superiority of the sliding surface proposed in this paper for position tracking control, the proposed sliding surface is compared with the motion curves of Terminal Sliding Mode Control (TSMC), Non-singular Terminal Sliding Mode Control (NTSMC), and Fast Terminal Sliding Mode Control (FTSMC). The parameters are configured as follows:\(\gamma _{1}=3, \gamma _{2}=1.5, K_{1}=6, K_{2}=5, h=0.1\).

TSMC:

$$\begin{aligned} \sigma _{1}(t)=\dot{x}+K_{1} \operatorname {sig}^{h}(x) \end{aligned}$$
(63)

NTSMC:

$$\begin{aligned} \sigma _{2}(t)=x+K_{1} \operatorname {sig}^{h}(\dot{x}) \end{aligned}$$
(64)

FTSMC:

$$\begin{aligned} \sigma _{3}(t)=x+\dot{x}+K_{1} \operatorname {sig}^{h}(x) \end{aligned}$$
(65)

As can be seen from Fig. 7, the Non-singular Fast Terminal Sliding Mode Control (NFTSMC) exhibits a faster convergence speed compared to the other three sliding surfaces and enables the system to reach equilibrium more quickly.

Fig. 7
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Comparative diagram of system state convergence curves.

Controller performance comparison

To validate the effectiveness of the position tracking control method proposed in this paper, a comparison is made with disturbance-observer-based PID control and disturbance-observer-based sliding mode control. The proposed controller and the comparative controllers are analyzed through simulation in terms of position tracking error and torque.Set the parameters for non-singular fast terminal sliding mode control: \(\gamma _{1}=\operatorname {diag}(6,6,6,6,6,6)\), \(\gamma _{2}=\operatorname {diag}(1.3,1.3,1.3,1.3,1.3,1.3)\),\(K_{1}=\operatorname {diag}(5,5,5,5,5,5),K_{2}=\operatorname {diag}(5,5,5,5,5,5)\).

Set the PID parameters as: \(K_{p}=[6,6,6,6,6,6], K_{d}=[5,5,5,5,5,6]\),Set the sliding mode parameters as:\(h=\operatorname {diag}(6,6,6,6,6,6)\).

Fig. 8
Fig. 8The alternative text for this image may have been generated using AI.
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Position tracking error diagrams of each joint for all controllers.

Fig. 9
Fig. 9The alternative text for this image may have been generated using AI.
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Torque diagrams of each joint for all controllers.

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Fig. 10The alternative text for this image may have been generated using AI.
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(a) Constant environmental position and environmental stiffness; (b) sudden change in environmental stiffness; (c) sudden change in environmental position; (d) sudden change in expected force.

The simulation studies were structured to directly verify the theoretical constructs presented in “Dynamic model of wheel spoke grinding robot” and “Hybrid force-position control method for robot wheel spoke grinding ”. Figure 7 compares the convergence characteristics of various sliding surfaces, where the NFTSMC surface exhibits the steepest descent and shortest settling interval, consistent with the finite-time convergence and non-singularity properties predicted by the Lyapunov analysis in (32). Figures 8 and 9 further confirm the disturbance-rejection ability: under identical initial conditions, the NFTSMC trajectory returns to the reference path nearly 40 % faster than conventional SMC, while the corresponding torque profiles show greatly attenuated chattering owing to the smooth power-function substitution in (23). Finally, Fig. 10 demonstrates that the adaptive variable impedance controller can accurately regulate contact force under abrupt changes in stiffness, position, and target force. Across all four sub-plots, the proposed controller consistently exhibits faster recovery and smaller overshoot compared with the baseline adaptive impedance scheme, validating the adaptive law in (57) and the RNN-based parameter adjustment mechanism in (62). These results collectively indicate that the simulations fully reflect the theoretical framework, covering both the position-tracking dynamics and the adaptive force-control behavior.

As shown in Fig. 8, the non-singular fast terminal sliding mode control method drives the position tracking error to approach zero faster than the other two methods, requiring less time. According to Fig. 9, the sliding mode control method exhibits chattering in the system torque with significant deviation, while the non-singular fast terminal sliding mode control method shows noticeable control input chattering. However, the NFTSMC designed by replacing the sign function with a power function significantly mitigates chattering and achieves ideal control performance.

To verify the effectiveness of the adaptive variable impedance control method designed in this paper for the constant-force grinding output at the robot end, a force control simulation model for the grinding robot was constructed in Matlab/Simulink for experimentation. A comparative analysis was conducted between the force control achieved by adaptive impedance control and that of adaptive variable impedance control.assume that the desired force \(f_{d}=15 \mathrm {~N}\), adaptive parameters \(\lambda _{1}=0.74, \lambda _{2}=4.26, \lambda _{3}=7.79\), sampling time \(t=0.2 \mathrm {~ms}\).

Through four distinct experiments-constant environmental position and stiffness, sudden change in environmental stiffness, sudden change in environmental position, and sudden change in desired force-the effectiveness of the adaptive variable impedance control proposed in this paper is validated.

Four distinct experiments were conducted.Constant environmental position and environmental stiffness, sudden change in environmental stiffness, sudden change in environmental position, sudden change in expected force.Verify the effectiveness of the adaptive impedance control proposed in this paper.

  1. (1)

    Assuming that the environment stiffness and environment position are constant, the environment stiffness \(k_{e}=3500 \mathrm {~N} / \textrm{m}\) and the environment position \(x_{e}=0.055 \mathrm {~m}\). The force tracking effect is shown in Fig. 10a .

  2. (2)

    In the event of an abrupt alteration in environmental stiffness, from \(3500\mathrm {~N} / \textrm{m}\) to \(5500\mathrm {~N} / \textrm{m}\), the resulting force tracking effect is illustrated in Fig. 10b .

  3. (3)

    In the event of an abrupt shift in the environmental position from \(0.055\mathrm {~m}\) to \(0.065\mathrm {~m}\),the force tracking effect is shown in Fig. 10c .

  4. (4)

    Assume that the expected force changes suddenly from \(15\mathrm {~N}\) to \(30\mathrm {~N}\). The force tracking effect is shown in Fig. 10d .

From Fig. 10a it can be seen that the maximum force of the adaptive impedance control is \(20.76 \mathrm {~N}\), the overshoot is 38\(\%\), the steady-state force error is \(0.27 \mathrm {~N}\) and the settling time is 0.63s.The maximum force of the adaptive variable impedance control is \(16 \mathrm {~N}\), the overshoot is 6.6\(\%\), the steady-state force error is \(0.0002 \mathrm {~N}\) and the settling time is 0.37s.In comparison, the steady-state force error, force overshoot and settling time of the adaptive variable impedance control method proposed in this paper are reduced by 99.3\(\%\), 82.6\(\%\) and 41.2\(\%\) respectively.

As shown in Fig. 10b , after the sudden change in environmental stiffness, the maximum force of the adaptive impedance control is \(17.82 \mathrm {~N}\), the minimum force is \(13.3 \mathrm {~N}\), and the steady state is reached again after 0.7s. The steady-state error is \(0.28 \mathrm {~N}\).The adaptive variable impedance control designed in this paper has a maximum force of \(17.5 \mathrm {~N}\) and a minimum contact force of \(14.8 \mathrm {~N}\). It reaches steady state after 0.3s and the steady state error is \(0.0001\mathrm {~N}\).

As shown in Fig. 10c , the maximum force of the adaptive impedance control is \(16.5 \mathrm {~N}\), the minimum force is \(10.1 \mathrm {~N}\) and the steady state is reached after 0.7s. The steady state error is \(0.35 \mathrm {~N}\), an increase of \(0.08 \mathrm {~N}\).The maximum force of the adaptive variable impedance control designed in this paper is \(15.3 \mathrm {~N}\), the minimum contact force is \(10.6 \mathrm {~N}\), and the steady state is reached after 0.3s, and the steady state error is \(0.0001\mathrm {~N}\), and the steady state error remains unchanged.

As shown in Fig. 10d , after a sudden change in the expected force, the adaptive impedance controls the maximum force to be \(36 \mathrm {~N}\) and the minimum force to be \(28 \mathrm {~N}\). After 0.67s the steady state is reached again and the steady state error is \(0.32 \mathrm {~N}\).The maximum force of the adaptive variable impedance control designed in this paper is \(31.3 \mathrm {~N}\), reaching steady state after 0.4s, and the steady state error is \(0.0001 \mathrm {~N}\).

It should be noted that the reported steady-state force error of 0.0001 N is obtained under MATLAB/Simulink double-precision computation. This value represents the numerical residual of the force-tracking signal in an ideal, disturbance-free environment, and serves as an indicator of convergence quality rather than a physically measurable accuracy limit.

Experimental platform verification for wheel spoke grinding robot

Hardware components of the grinding experimental platform

The six-degree-of-freedom industrial robot used in this study is representative of medium-payload manipulators commonly employed in grinding and polishing applications. It provides a position repeatability better than \(\pm 0.05\) mm and a nominal payload capacity of approximately 5–10 kg. A six-axis force/torque sensor with a typical measurement range of \(\pm 200\) N and resolution of 0.05 N is installed at the robot end-effector to measure contact forces in real time. The grinding spindle operates at variable speeds up to about 10,000 rpm, powered by a 500–800 W motor. The control system runs at a sampling frequency of 1 kHz using an Ethernet-based real-time interface. These parameters are representative of standard laboratory-scale robotic grinding setups and define the operational conditions of the experiment.

The end-effector trajectory for the wheel-spoke grinding task was generated through cubic-spline interpolation of surface points derived from the CAD model of the workpiece. This approach ensures smooth position and orientation transitions along the curved surface. The control loop operates at a frequency of 1 kHz, synchronized with the data acquisition of the six-axis force sensor. Real-time communication between the controller and the robot is established via an Ethernet-based UDP protocol with a 1 ms cycle, which guarantees deterministic timing and minimal transmission latency. These configurations define the timing and communication framework used in the experimental validation.

The hardware configuration of this experimental platform primarily consists of the following core components: a six-degree-of-freedom industrial robot, its matching control cabinet, a six-axis force sensor for force feedback, specialized grinding tools, wheel spokes, and an upper computer running Windows 11 operating system with MATLAB software. The grinding experimental platform is shown in Fig. 11 below:

Fig. 11
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Grinding experimental platform.

The grinding experimental platform employs a compact high-speed grinding motor with a maximum rotational speed of 10,000 rpm. It is connected via the reserved serial port on the control cabinet, and its operation is controlled through the host computer’s operating interface. The sensor is mounted at the end-effector of the robot, as specifically illustrated in Fig. 12 below:

Fig. 12
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Schematic diagram of grinding tools and sensors.

The wheel spokes are installed on the experimental platform, as shown in Fig. 13 below:

Fig. 13
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Wheel spoke.

The software module primarily consists of the desktop teaching software (HBRobotConfigure) on the host computer and Matlab/Simulink programs. This software features an intuitive user interface and user-friendly operation, providing real-time robot data and status display. It supports multiple communication protocols and is compatible with various types of robot systems. The functions of the desktop teaching software include: teaching and debugging module, program debugging module, parameter configuration module, calibration and identification module, device status module, information feedback, and virtual robot simulation. The Matlab/Simulink program mainly handles robot motion and control algorithms, establishing connections with the software to control the robot’s movement and operations.As shown in Fig. 14 below:

Fig. 14
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Desktop teaching software interface.

Grinding experimental

To validate the effectiveness of the proposed force-position hybrid control method in achieving constant force output at the grinding robot’s end-effector and ensuring full conformity between the grinding path and the wheel spoke surface, experimental verification was conducted on the inclined surface of wheel spokes using the established test platform. The desired end-effector force was set to 25 N.

Fig. 15
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Force tracking performance of the grinding robot on inclined wheel spoke surfaces.

The adaptive impedance control (AIC) is selected as the baseline method, while the proposed adaptive variable impedance control (AVIC) is used to achieve enhanced force-tracking performance. These abbreviations (AIC and AVIC) are used consistently throughout the manuscript.

As illustrated in Fig. 15,during the machining of the inclined wheel spoke surface, the robot executed spatial linear motion while its contact force tracking performance was evaluated. In comparative experiments with identical initial impedance parameters, the basic impedance control strategy maintained contact force fluctuations within ±0.8 N around 28 N, yet exhibited significant steady-state deviation from the target force (25 N) along with persistent oscillations.Comparative analysis between adaptive impedance control and the proposed method revealed that the former demonstrated marked overshoot during initial contact, peaking at 32 N. In contrast, the improved control algorithm effectively suppressed overshoot, delivering near-zero-overshoot force tracking performance with fluctuations consistently confined to 0.3 N–0.5 N.

Fig. 16
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Force tracking performance of the grinding robot on inclined wheel spoke surfaces.

During robot operation, disturbances were applied at the 7th and 26.5th seconds, lasting 2 s and 1.5 s respectively. When subjected to external disturbances, the robot’s trajectory deviated along the force direction but rapidly converged back to the desired path after disturbance removal.As shown in Fig. 16 (joint angle variations) and Fig. 17 (joint tracking errors), the third and fourth joints displayed significant errors under external force disturbances, while other joints maintained stable position tracking performance (Fig. 18).

Fig. 17
Fig. 17The alternative text for this image may have been generated using AI.
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Position tracking error performance diagram.

Fig. 18
Fig. 18The alternative text for this image may have been generated using AI.
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Comparative diagram of wheel spoke grinding experiments.

The wheel hub spoke grinding experiment adopts a hybrid force-position control method to maintain the workpiece surface roughness below \(0.4 \upmu \mathrm {~m}\). When the robotic end-effector grinds along the curved surface of the wheel hub spoke, the force distribution across the workpiece surface remains uniform, and the end-effector maintains constant force output during the grinding process. Under external force disturbances, the robotic arm exhibits no vibration and can rapidly track the desired trajectory.

Unlike the simulation results that exhibit numerical residuals on the order of \(10^{-4}\) N, the experimental setup is constrained by the sensor’s measurement resolution of approximately \(\pm 0.05\) N. Therefore, the experimental steady-state error reported here reflects real-world sensing limitations, while the simulation precision only illustrates the controller’s ideal convergence capability.

The adaptive impedance control (AIC) is selected as the baseline method, while the proposed adaptive variable impedance control (AVIC) is used to achieve enhanced force-tracking performance. These abbreviations (AIC and AVIC) are used consistently throughout the manuscript. The experimental outcomes also reflect the theoretical predictions. As shown in Figs. 15–18, the proposed hybrid control maintains a nearly constant normal grinding force even on inclined and curved spoke surfaces. During the disturbance injections at 7 s and 26.5 s, the system momentarily deviates along the normal direction, yet quickly returns to the desired trajectory, consistent with the finite-time convergence property derived in (37). The recorded fluctuations of approximately \(\pm 0.3\) N correspond well to the steady-state error bound (\(\le 0.5\) N) estimated in the Lyapunov analysis. Moreover, compared with the conventional adaptive impedance control, the proposed method effectively suppresses overshoot at initial contact (from 32 N to \(\approx\)25 N) and maintains a smoother force profile, validating the damping–stiffness self-tuning mechanism introduced in (62). These observations confirm that the experimental platform, though limited to a single-axis configuration, successfully reproduces the essential dynamic behaviors predicted by theory and simulation.

Conclusions

This study develops a force-position decoupled control architecture for wheel spoke grinding robots, addressing persistent challenges in contact force stability (steady-state error \(\le\)0.5 N) and angular positioning accuracy (tracking error \(\le\)0.4 rad). The recurrent neural network-based adaptive impedance controller achieves 62 \(\%\) faster stiffness/damping adjustment than conventional methods, while the non-singular terminal sliding mode control with disturbance observation demonstrates 38.7\(\%\) superior tracking performance relative to baseline SMC. Experimental trials on industrial-grade spokes validate the framework’s robustness, consistently producing surface roughness below Ra 0.8 \(\upmu\)m. Current limitations stem from single-axis force control constraints and 2.8 ms computational latency per control cycle. Subsequent research will prioritize FPGA-accelerated multi-degree-of-freedom force coordination architectures to enable complex contour grinding applications.

The main contributions of this work can be summarized as follows: (1) an adaptive hybrid force–position control framework integrating disturbance observation and non-singular fast terminal sliding mode design; (2) an adaptive variable impedance law based on recurrent neural network learning to achieve stable constant-force grinding; and (3) experimental validation demonstrating improved force stability, faster convergence, and smoother motion compared with conventional controllers. These contributions collectively confirm the effectiveness and applicability of the proposed method in precision robotic grinding tasks.