An enhanced robust control algorithm with inequality constraints for service robot joint performance optimization

Abstract

The joint module of collaborative robots must simultaneously guarantee accurate trajectory tracking and strict joint-limit compliance during human–robot interaction. Its nonlinear, strongly coupled and time-varying dynamics, however, amplify the influence of parametric uncertainty, external disturbances, and model errors, causing large tracking deviations. We present a robust, model-based PD controller that explicitly enforces state inequality constraints on joint displacement. A diffeomorphic transformation converts the constrained dynamics into an unconstrained equivalent, allowing standard robust-synthesis tools to be applied while preserving the original limits. Uniform ultimate boundedness of the closed-loop error is proven under lumped uncertainties. Experiments on a 1-DoF joint prototype yield a steady-state tracking error within ±0.1° without violating the prescribed angular bounds, outperforming both unconstrained $H_{\infty}$ and PID baselines in accuracy and safety margin.

Keywords

robust control joint module safety modeling uncertainty inequality constraints

1. Introduction

The manufacturing industry is shifting toward human–robot coexistence, where safety and flexibility take precedence over pure operational speed Zhang et al. (2021); Zhu et al. (2023). Collaborative service robots operating in human-populated environments, such as airport VIP lounges (see Figure 1), must therefore achieve accurate trajectory tracking and strict joint-limit compliance simultaneously under significant parametric uncertainties, external disturbances, and unmodeled dynamics. Unlike traditional industrial machines confined to safety cages, these service-oriented robots demand intrinsic compliance and real-time collision detection capabilities Xian et al. (2024); Wang et al. (2022a).

Figure 1.

Collaborative service robot in an airport VIP lounge.

This scenario entails specific joint-level challenges: frequent start-stop operations, varying payloads (e.g., beverage trays of different weights), and the need for smooth motion to ensure guest comfort and safety.

These requirements place stringent demands on the underlying joint modules—integrated electromechanical units comprising servo motors, harmonic reducers, and torque sensors Wang et al. (2024); Chen et al. (2020). As fundamental actuation elements, joint modules serve as the critical interface between high-level motion planning and low-level physical execution.

The performance of robotic joint modules is susceptible to uncertainties including moment of inertia inaccuracies, gear backlash, unmodeled dynamics, and electromagnetic interference Wang et al. (2024). These time-varying, nonlinearly coupled factors complicate precise motion control Ma et al. (2022), necessitating advanced control methodologies that ensure robust tracking performance and disturbance rejection.

Numerous control strategies have been proposed for joint motion control, yet each exhibits distinct limitations. PID control remains widely adopted due to its structural simplicity Shao et al. (2021, 2022), but its fixed linear structure struggles with pronounced nonlinearities in harmonic drive systems, including velocity-dependent friction and position-dependent stiffness. Furthermore, standard PID lacks inherent mechanisms for enforcing physical constraints, typically relying on ad hoc anti-windup techniques that prove inadequate under significant disturbances.

Adaptive control adjusts parameters in response to changing conditions Chen and Hua (2022); Huang et al. (2021); Liu et al. (2021), but requires persistent excitation and substantial computational resources. Neural network controllers provide powerful approximation capabilities Udwadia (2003); Wang et al. (2023a), yet require extensive offline pre-training and lack transparency for safety-critical applications. Fuzzy control offers intuitive rule-based frameworks Zhu et al. (2023), but lacks systematic design procedures and analytical stability guarantees. Model-based robust PD strategies improve tracking through nominal model feedforward Chen (2011), yet primarily focus on asymptotic convergence without explicitly enforcing hard constraints on joint displacement and velocity.

Recent approaches such as sliding mode control and nonlinear model predictive control handle state constraints Zhang et al. (2023a). Sliding mode control confines states within prescribed bounds but inherently induces chattering. Nonlinear model predictive control accounts for constraints through online optimization, but its computational intensity renders it impractical for real-time embedded controllers.

Modern robust control techniques provide a structured framework for managing disturbances, uncertainties, and unmodeled dynamics while maintaining stability guarantees Hale (1969); Sun et al. (2020). These methods offer a promising foundation for achieving precise motion tracking, explicit constraint enforcement, and computational tractability—essential capabilities for safe deployment in demanding service environments.

Safety remains paramount in collaborative robotics. During operation, joint outputs may exceed predefined limits due to unexpected disturbances, leading to potential mechanical damage or human injury. Ensuring that joint angular displacement and velocity remain within strictly defined safe operating ranges constitutes an essential prerequisite for deployment in human-populated environments.

This paper proposes a robust constraint-following control approach based on inequality constraints. The method employs a diffeomorphism transformation to map original state variables into a new coordinate system where they become inherently bounded by construction. This geometric transformation embeds hard constraints into the system dynamics, eliminating the need for explicit constraint checking. Combined with a robust control design compensating for uncertainties characterized by a bounded function ψ, the proposed controller guarantees strict confinement of joint displacement within specified limits. Lyapunov-based analysis demonstrates uniform boundedness and uniform ultimate boundedness Sun et al. (2022).

The main contributions are summarized as follows:

(1) A diffeomorphism-based state transformation to convert joint displacement into an inherently bounded variable space.

(2) A model-based robust PD control strategy ensuring accurate position tracking under uncertainties while respecting displacement constraints.

(3) Comprehensive simulations and hardware-in-the-loop experiments demonstrating steady-state tracking errors within ±0.1° with strict constraint enforcement, outperforming standard robust controllers and PID baselines.

The remainder is organized as follows. Section 2 presents dynamic modeling of the robotic joint module. Section 3 describes the diffeomorphism-based state transformation. Section 4 details the robust controller design and stability analysis. Section 5 presents simulations and experiments. Section 6 concludes with future directions.

2. Dynamic modeling

The joint module of a collaborative robot comprises two primary electromechanical components: a permanent magnet synchronous motor (PMSM) Zhang et al. (2024); Liu et al. (2019) and a harmonic drive reducer Yang et al. (2022a); Qin et al. (2022). The PMSM serves as the primary actuation source, supplying driving torque through precise electromagnetic conversion, while the harmonic reducer introduces notable nonlinearities that significantly complicate transmission characteristics. These nonlinearities include friction torque with strong hysteresis effects—rendering torque transmission direction-dependent and history-dependent—as well as structural compliance originating primarily from flexspline elasticity, which contributes to joint torsional stiffness k. Furthermore, the harmonic drive exhibits kinematic errors due to manufacturing imperfections and assembly misalignments, introducing additional periodic disturbances at the wave generator rotation frequency. These coupled factors, particularly the joint flexibility-induced position error between motor-side and load-side encoders and the hysteresis-related dynamic lag in torque transmission, substantially elevate both modeling complexity and control challenges, necessitating careful consideration of two-inertia resonant dynamics in controller design. The system structure is illustrated in Figure 2.

Figure 2.

The composition of the joint module system.

The dynamic characteristics of the PMSM are captured by

\{\begin{cases} L_{d} {\dot{i}}_{d} = - R_{c} i_{d} + L_{q} n_{c} ω i_{q} + u_{d}, \\ L_{q} {\dot{i}}_{q} = - R_{c} i_{q} - L_{d} n_{c} ω i_{d} - n_{c} ω ψ_{f} + u_{q}, \\ J \dot{ω} = T_{m} - B_{f} ω - T_{n} . \end{cases}

(1)

Here, i_d, i_q, u_d, u_q, L_d, and L_q denote the stator currents, voltages, and inductances on the d and q axes, respectively; n_c is the pole-pair count; ω is the rotor angular velocity; ψ_f is the permanent magnet flux linkage; R_c is the stator resistance; J is the rotor inertia; B_f is the viscous-friction coefficient Zhang et al. (2023b); T_m is the electromagnetic torque; and T_n is the load torque. The electromagnetic torque is given by

T_{m} = \frac{3}{2} n_{c} i_{q} [ψ_{f} + (L_{d} - L_{q}) i_{d}] .

(2)

PMSMs are classified into surface-mounted and interior types based on permanent magnet placement. This paper adopts the surface-mounted PMSM (Wang et al., 2022b), where L_d = L_q = L. According to the field-oriented control (FOC) principle (Zhao et al., 2022), the d-axis current is regulated to zero (i_d = 0), yielding i_q = i_s (the stator current magnitude). This enables direct torque control through i_q regulation. Under these conditions, the system dynamics simplify to

\{\begin{cases} L {\dot{i}}_{q} = - R_{c} i_{q} - n_{c} ω ψ_{f} + u_{q}, \\ J \dot{ω} = T_{m} - B_{f} ω - T_{n} . \end{cases}

(3)

The electromagnetic torque for the surface-mounted PMSM becomes

T_{m} = \frac{3}{2} n_{c} ψ_{f} i_{q} = K_{f} i_{q},

(4)

where K_f is the torque coefficient (Yang et al., 2022b). Substituting (4) into the mechanical dynamics of (3) yields

J \dot{ω} = T_{m} - T_{n} - B_{f} ω .

(5)

The harmonic reducer transmits torque from the motor to the load. Let T_u denote the output torque at the reducer, λ the speed reduction ratio, and η the transmission efficiency. The torque relationship is

T_{u} = λ η T_{n} .

(6)

Let θ_m denote the motor rotor angular displacement, where ${\dot{θ}}_{m} = ω$ and ${\ddot{θ}}_{m} = \dot{ω}$ . Incorporating the friction torque F_f, the dynamic equation of the joint module is

J {\ddot{θ}}_{m} + B_{f} {\dot{θ}}_{m} + T_{n} + F_{f} = T_{m} .

(7)

Substituting (6) into (7) to eliminate T_n gives

J {\ddot{θ}}_{m} + B_{f} {\dot{θ}}_{m} + \frac{T_{u}}{λ η} + F_{f} = T_{m} .

(8)

Friction significantly influences the stability and tracking accuracy of robotic joints (Hu et al., 2022), particularly in high-precision positioning tasks where stick-slip phenomena degrade performance. Various friction models have been developed, ranging from simple static descriptions to dynamic formulations such as the LuGre and Dahl models that capture hysteresis and rate-dependent behaviors (Wang et al., 2023b). For moderate velocity operation, this paper employs the Coulomb–viscous friction model, which provides a balance between fidelity and implementation simplicity. The friction torque F_f comprises Coulomb friction (constant at zero velocity) and viscous friction (proportional to velocity):

F_{f} ({\dot{θ}}_{m}) = f_{c} s g n ({\dot{θ}}_{m}) + f_{v} {\dot{θ}}_{m},

(9)

where f_c is the Coulomb friction coefficient and f_v is the viscous friction coefficient.

3. State transformation

The joint module dynamics derived in the preceding section are governed by

J {\ddot{θ}}_{m} + B_{f} {\dot{θ}}_{m} + \frac{T_{u}}{λ η} + F_{f} = T_{m} .

(10)

where

θ_{m} \in (θ_{m}^{\min}, θ_{m}^{\max})

, with

θ_{m}^{\min}

and

θ_{m}^{\max}

denoting the lower and upper bounds of the angular displacement, respectively. To embed these hard constraints into the control design, we introduce the tangent state transformation:

z = \tan [\frac{π}{θ_{m}^{\max} - θ_{m}^{\min}} (θ_{m} - θ_{m}^{\min}) - \frac{π}{2}] .

(11)

The desired trajectory in the transformed coordinate is

z_{d} = \tan [\frac{π}{θ_{m}^{\max} - θ_{m}^{\min}} (θ_{m}^{d} - θ_{m}^{\min}) - \frac{π}{2}],

(12)

where

θ_{m}^{d}

represents the desired trajectory of the angular displacement.

This transformation ensures that $θ_{m} \to θ_{m}^{\min}$ as z → −∞ and $θ_{m} \to θ_{m}^{\max}$ as z → +∞, thereby guaranteeing θ_m remains strictly within $(θ_{m}^{\min}, θ_{m}^{\max})$ by construction. The inverse mapping follows from (11):

θ_{m} = \frac{θ_{m}^{\max} - θ_{m}^{\min}}{π} \arctan (z) + \frac{θ_{m}^{\max} + θ_{m}^{\min}}{2} .

(13)

Differentiating (13) with respect to time yields

{\dot{θ}}_{m} = \frac{θ_{m}^{\max} - θ_{m}^{\min}}{π} \frac{\dot{z}}{1 + z^{2}},

(14)

{\ddot{θ}}_{m} = \frac{θ_{m}^{\max} - θ_{m}^{\min}}{π} \frac{(1 + z^{2}) \ddot{z} - 2 z {\dot{z}}^{2}}{{(1 + z^{2})}^{2}} .

(15)

Substituting (14) and (15) into (10) and rearranging terms gives the transformed dynamics:

B (z) \ddot{z} + N (z, \dot{z}) \dot{z} + Y (z, \dot{z}) = τ (t),

(16)

where τ(t) = T_m denotes the motor torque, and

B (z) = \frac{J (θ_{m}^{\max} - θ_{m}^{\min})}{π (1 + z^{2})},

(17)

N (z, \dot{z}) = \frac{B_{f} (θ_{m}^{\max} - θ_{m}^{\min})}{π (1 + z^{2})} - \frac{2 J (θ_{m}^{\max} - θ_{m}^{\min}) z \dot{z}}{π {(1 + z^{2})}^{2}},

(18)

Y (z, \dot{z}) = \frac{T_{u}}{λ η} + F_{f} .

(19)

Here, z ∈ (−∞, + ∞) is the transformed state derived from θ_m. The detailed algebraic derivation can be found in our previous work Zhang et al. (2023c). This diffeomorphism maps the constrained physical variable into an unconstrained coordinate while preserving the topological properties of the dynamics, enabling control design without explicit constraint handling.

Assumption 1

On the compact operating region of interest—where θ is strictly confined within (θ_min, θ_max) and consequently z remains bounded—the inertia term B(z) in (17) is uniformly positive definite. That is, there exist constants $\underset{̲}{λ}, \bar{λ} > 0$ such that

0 < \underset{̲}{λ} I \leq B (z) \leq \bar{λ} I for all z \in [- Z_{\max}, Z_{\max}],

(20)

where Z_max < ∞ corresponds to the maximum admissible z under the physical joint limits. In practice, Z_max is finite because θ never reaches θ_min or θ_max due to the safety margin enforced by the controller.

Property 1

(Skew-symmetry-like structure): For the transformed single-degree-of-freedom dynamics (16), the scalar coefficients satisfy

\dot{B} (z) - 2 N (z, \dot{z}) = - \frac{2 B_{f} Δ θ}{π (1 + z^{2})},

which is strictly negative for any B_f > 0. This dissipative term does not affect the stability proof and, in fact, contributes positively to the damping of the closed-loop system. In the Lyapunov analysis, the term

{(\dot{r} + S r)}^{T} (1 / 2 \dot{B} - N) (\dot{r} + S r)

simplifies to

- B_{f} Δ θ / π (1 + z^{2}) {(\dot{r} + S r)}^{2}

, which is non-positive. Therefore, the structure required for the stability proof holds without requiring exact skew-symmetry.

Extension to multi-joint robots: For multi-joint robots, the proposed method applies naturally to each joint independently. Specifically, for joint i with physical limits $(θ_{i}^{\min}, θ_{i}^{\max})$ , the same diffeomorphic transformation (11) is applied using its own limits, yielding transformed dynamics (16) with parameters B_i(z_i) and $N_{i} (z_{i}, {\dot{z}}_{i})$ that scale linearly with $(θ_{i}^{\max} - θ_{i}^{\min})$ . Controller parameters (P_i, D_i, S_i, γ_i) can be tuned separately for each joint following the guidelines in Section 4.3. When the angle limits of a joint change significantly, the factor $(θ_{i}^{\max} - θ_{i}^{\min})$ in (17) and (18) directly scales the transformed inertia, so P_i and D_i should be adjusted proportionally to maintain similar closed-loop behavior. Joint coupling effects appear as additional disturbance torques in each joint’s dynamic equation; these are absorbed into the lumped uncertainty Φ and handled by the robust term in (29) as long as they remain within the prescribed bound ρ. Thus, the method extends to multi-joint systems without fundamental redesign.

4. Robust control

4.1. Robust controller design

Let the desired trajectory be z^d(t), with desired velocity ${\dot{z}}^{d} (t)$ and desired acceleration ${\ddot{z}}^{d} (t)$ . The tracking error is defined as

r (t) = z (t) - z^{d} (t) .

(22)

The error derivatives satisfy

\dot{r} (t) = \dot{z} (t) - {\dot{z}}^{d} (t), \ddot{r} (t) = \ddot{z} (t) - {\ddot{z}}^{d} (t) .

(23)

Define the error state vector

\tilde{r} (t) = {[r (t) \dot{r} (t)]}^{T} .

(24)

The control objective is to ensure that $\tilde{r} (t)$ achieves uniform boundedness and uniform ultimate boundedness.

The dynamics (16) can be rewritten in terms of the error dynamics as

\begin{matrix} B (z) [{\ddot{z}}^{d} (t) + \ddot{r} (t)] + C (z, \dot{z}) [{\dot{z}}^{d} (t) + \dot{r} (t)] \\ + Y (z, \dot{z}) = τ (t) . \end{matrix}

(25)

To address system uncertainties, the dynamics are decomposed into nominal and uncertain components:

\begin{matrix} B (z) & = \bar{B} (z) + Δ B (z), \\ C (z, \dot{z}) & = \bar{C} (z, \dot{z}) + Δ C (z, \dot{z}), \\ Y (z, \dot{z}) & = \bar{Y} (z, \dot{z}) + Δ Y (z, \dot{z}), \end{matrix}

(26)

where

\bar{B} (\cdot)

\bar{C} (\cdot)

, and

\bar{Y} (\cdot)

denote nominal terms, while ΔB(⋅), ΔC(⋅), and ΔY(⋅) represent uncertain components.

For a given positive constant S, the scalar ρ is chosen as the upper bound of the lumped uncertainty function Φ on the reachable set of the system. Since the diffeomorphic transformation (11) guarantees that z remains bounded for all constraint-satisfying trajectories (as discussed in Assumption 1), the reachable set of $(z, \dot{z})$ is compact. On this compact set, the continuous function Φ attains a finite maximum, ensuring the existence of a uniform bound ρ. Specifically, ρ satisfies

‖ Φ (r, \dot{r}, t) ‖ ⩽ ρ (r, \dot{r}, t) for all (z, \dot{z}) in the reachable set .

(27)

Here

\begin{matrix} Φ (r, \dot{r}, t) = & - Δ B (z) ({\ddot{z}}^{d} - S \dot{r}) \\ - Δ C (z, \dot{z}) ({\dot{z}}^{d} - S r) \\ - Δ Y (z, \dot{z}) . \end{matrix}

(28)

The proposed robust controller is designed as

\begin{matrix} τ (t) = & \bar{B} (z) ({\ddot{z}}_{d} - S \dot{r}) + \bar{C} (z, \dot{z}) ({\dot{z}}_{d} - S r) \\ + \bar{Y} (z, \dot{z}) - P r - D \dot{r} - \frac{α}{‖ ϖ ‖ + \hat{η}}, \end{matrix}

(29)

where P and D are proportional and derivative gains,

\hat{η} > 0

is a design constant, and

α = γ^{2} (\dot{r} + S r) ρ^{4} ‖ \dot{r} + S r ‖^{2},

(30)

ϖ = γ ‖ \dot{r} + S r ‖^{2} ρ^{2} .

(31)

The robust term $- α / (‖ ϖ ‖ + \hat{η})$ is designed to dominate the lumped uncertainty Φ bounded by ρ in (27). The gain γ scales the robustness level: larger γ reduces the ultimate bound at the cost of increased control effort, while $\hat{η} > 0$ is a small regularization constant to avoid singularity when ‖ϖ‖ = 0.

The scalars γ and ρ are positive design parameters. The last term in (29) serves as the robust compensation for system uncertainties.

The schematic diagram in Figure 3 illustrates the proposed control structure.

Figure 3.

Robust control block diagram.

Practical estimation of ρ: In practical implementations, the uncertainty bound ρ is rarely known exactly. The theoretical analysis in Section 4.2 gin Section 4.2 guarantees that uniform boundedness and uniform ultimate boundedness are preserved as long as ρ is chosen conservatively, that is, as an overestimate rather than an underestimate of the true uncertainty magnitude. If the actual uncertainty momentarily exceeds the preset ρ, the Lyapunov derivative $\dot{V}$ may become temporarily positive, but the system does not escape to infinity; instead it converges to a slightly larger ultimate bound. Equations (46) and (48) both scale with ρ, meaning a larger ρ yields a larger residual set but does not destabilize the system.

For the joint module considered in this paper, a practical way to estimate ρ is to combine physical reasoning with worst-case experimental observations. The main sources of uncertainty include parameter variations (ΔB, ΔC), unmodeled friction dynamics, and external load disturbances. A straightforward method is to record the tracking error under nominal operation, identify the maximum deviation, and then multiply that value by a safety factor (typically between 1.5 and 2.0) to obtain ρ. Overestimation does not harm stability—it only increases control effort because the robust term becomes more aggressive. This trade-off is discussed in Section 4.3, where we note that larger γ improves precision at the cost of higher energy consumption and potential actuator saturation. Thus, ρ and γ can be tuned jointly to balance robustness against practical constraints such as torque limits and noise sensitivity.

4.2. Theoretical analysis

The stability is analyzed using the Lyapunov method. Select the Lyapunov function candidate

\begin{matrix} V (\tilde{r}, t) = & \frac{1}{2} {(\dot{r} + S r)}^{T} B (z) (\dot{r} + S r) \\ + \frac{1}{2} r^{T} (P + S D) r . \end{matrix}

(32)

By (20), B(z) is bounded, yielding

\begin{matrix} V \geq & \frac{1}{2} \underset{̲}{λ} ‖ \dot{r} + S r ‖^{2} + \frac{1}{2} r^{T} (P + S D) r \\ = & \frac{1}{2} \underset{̲}{λ} ({\dot{r}}^{2} + 2 S \dot{r} r + S^{2} r^{2}) + \frac{1}{2} (P + S D) r^{2} \\ = & \frac{1}{2} {[\begin{matrix} r \\ \dot{r} \end{matrix}]}^{T} \underset{̲}{Ψ} [\begin{matrix} r \\ \dot{r} \end{matrix}], \end{matrix}

(33)

where

\underset{̲}{Ψ} ≔ [\begin{matrix} \underset{̲}{λ} S^{2} + P + S D & \underset{̲}{λ} S \\ \underset{̲}{λ} S & \underset{̲}{λ} \end{matrix}] .

(34)

Since $\underset{̲}{Ψ} > 0$ , V is positive definite, satisfying

V \geq \frac{1}{2} λ_{\min} (\underset{̲}{Ψ}) (r^{2} + {\dot{r}}^{2}) \geq σ_{1} ‖ \tilde{r} ‖^{2},

(35)

where

σ_{1} = 1 / 2 λ_{\min} (\underset{̲}{Ψ}) > 0

Similarly, by the upper bound of B(z) in (20)

\begin{aligned} V & \leq \frac{1}{2} \bar{λ} ‖ \dot{r} + S r ‖^{2} + \frac{1}{2} r^{T} (P + S D) r \\ = \frac{1}{2} {[\begin{matrix} r \\ \dot{r} \end{matrix}]}^{T} \bar{Ψ} [\begin{matrix} r \\ \dot{r} \end{matrix}], \end{aligned}

(36)

where

\bar{Ψ} ≔ [\begin{matrix} \bar{λ} S^{2} + P + S D & \bar{λ} S \\ \bar{λ} S & \bar{λ} \end{matrix}] .

(37)

Thus,

V \leq \frac{1}{2} λ_{\max} (\bar{Ψ}) (r^{2} + {\dot{r}}^{2}) \leq σ_{2} ‖ \tilde{r} ‖^{2},

(38)

where

σ_{2} = 1 / 2 λ_{\max} (\bar{Ψ}) > 0

Differentiating V with respect to time yields

\begin{aligned} \dot{V} = & {(\dot{r} + S r)}^{T} B (z) (\ddot{r} + S \dot{r}) \\ + \frac{1}{2} {(\dot{r} + S r)}^{T} \dot{B} (z) (\dot{r} + S r) \\ + r^{T} (P + S D) \dot{r} . \end{aligned}

(39)

Substituting the error dynamics into the first two terms, denoted as ${\dot{V}}_{s}$ ,

\begin{aligned} {\dot{V}}_{s} = & {(\dot{r} + S r)}^{T} B (z) (\ddot{z} - {\ddot{z}}^{d} + S \dot{r}) \\ + \frac{1}{2} {(\dot{r} + S r)}^{T} \dot{B} (z) (\dot{r} + S r) . \end{aligned}

(40)

Using (17), (30), and (41)

\begin{aligned} {\dot{V}}_{s} = & - {(\dot{r} + S r)}^{T} Δ B (z) ({\ddot{z}}^{d} - S \dot{r}) \\ - {(\dot{r} + S r)}^{T} Δ C (z, \dot{z}) ({\dot{z}}^{d} - S r) \\ - {(\dot{r} + S r)}^{T} Δ Y (z, \dot{z}) \\ - γ {(\dot{r} + S r)}^{T} (\dot{r} + S r) ρ^{2} \\ - {(\dot{r} + S r)}^{T} P r - {(\dot{r} + S r)}^{T} D \dot{r} \\ + {(\dot{r} + S r)}^{T} (\frac{1}{2} \dot{B} - C) (\dot{r} + S r) . \end{aligned}

(41)

By (21), the skew-symmetry property gives

{(\dot{r} + S r)}^{T} (\frac{1}{2} \dot{B} (z) - C (z, \dot{z})) (\dot{r} + S r) = 0 .

(42)

Using (27) and (28)

\begin{matrix} {\dot{V}}_{s} = & {(\dot{r} + S r)}^{T} [Φ - \frac{α}{‖ ϖ ‖ + \hat{η}}] - {(\dot{r} + S r)}^{T} (P r + D \dot{r}) \\ \leq & {(\dot{r} + S r)}^{T} [ρ - \frac{α}{‖ ϖ ‖ + \hat{η}}] - {\dot{r}}^{T} D \dot{r} - r^{T} S P r \\ - r^{T} (P + S D) \dot{r} . \end{matrix}

(43)

To clarify how the robust term leads to the constant 1/(4γ), we expand the treatment of the inequality involving ρ and α. From the definitions (30) and (31), we have $α = γ^{2} (\dot{r} + S r) ρ^{4} ‖ \dot{r} + S r ‖^{2}$ and $ϖ = γ ‖ \dot{r} + S r ‖^{2} ρ^{2}$ . Substituting these into the term ${(\dot{r} + S r)}^{T} α / (‖ ϖ ‖ + \hat{η})$ gives

\frac{α}{‖ ϖ ‖ + \hat{η}} = \frac{γ^{2} (\dot{r} + S r) ρ^{4} ‖ \dot{r} + S r ‖^{2}}{γ ‖ \dot{r} + S r ‖^{2} ρ^{2} + \hat{η}} .

(44)

Since

\hat{η} > 0

is a small design constant, the denominator is always positive. To establish an upper bound for

{(\dot{r} + S r)}^{T} ρ - {(\dot{r} + S r)}^{T} α / (‖ ϖ ‖ + \hat{η})

, we first apply the Cauchy-Schwarz inequality to obtain

{(\dot{r} + S r)}^{T} ρ \leq ‖ \dot{r} + S r ‖ ‖ ρ ‖

. Denote

ξ = ‖ \dot{r} + S r ‖ ‖ ρ ‖ \geq 0

. Then the term of interest satisfies

{(\dot{r} + S r)}^{T} ρ - \frac{α}{‖ ϖ ‖ + \hat{η}} \leq ξ - \frac{γ^{2} ξ^{3}}{γ ξ^{2} + \hat{η}} .

(45)

Consider the function $f (ξ) = ξ - γ^{2} ξ^{3} / (γ ξ^{2} + \hat{η})$ . Its derivative is

f^{'} (ξ) = 1 - \frac{3 γ^{3} ξ^{4} + 3 γ^{2} ξ^{2} \hat{η} - 2 γ^{3} ξ^{4}}{{(γ ξ^{2} + \hat{η})}^{2}} = 1 - \frac{γ^{3} ξ^{4} + 3 γ^{2} ξ^{2} \hat{η}}{{(γ ξ^{2} + \hat{η})}^{2}} .

(46)

Setting f′(ξ) = 0 yields a maximum at $ξ^{2} = \hat{η} / γ$ . Substituting back gives

f_{\max} = \sqrt{\frac{\hat{η}}{γ}} - \frac{γ^{2} {(\hat{η} / γ)}^{3 / 2}}{γ (\hat{η} / γ) + \hat{η}} = \frac{1}{2} \sqrt{\frac{\hat{η}}{γ}} .

(47)

Since

\hat{η}

can be made arbitrarily small, this bound approaches zero. However, a more standard approach in robust control (see e.g., the analysis in Hale (1969)) obtains a conservative but simpler bound that is independent of

\hat{η}

. Noting that

\hat{η} > 0

can be omitted in the denominator for the purpose of obtaining an upper bound (since removing a positive term from the denominator increases the fraction, and we are subtracting it), we have from (44)

\frac{α}{‖ ϖ ‖ + \hat{η}} \geq \frac{γ^{2} (\dot{r} + S r) ρ^{4} ‖ \dot{r} + S r ‖^{2}}{γ ‖ \dot{r} + S r ‖^{2} ρ^{2}} = γ (\dot{r} + S r) ρ^{2} .

(48)

Then

{(\dot{r} + S r)}^{T} ρ - \frac{α}{‖ ϖ ‖ + \hat{η}} \leq ‖ \dot{r} + S r ‖ ‖ ρ ‖ - γ ‖ \dot{r} + S r ‖ ρ^{2} .

(49)

In the worst-case alignment, the direction of ρ coincides with that of

(\dot{r} + S r)

, so the right-hand side becomes ξ − γξ². Treating ξ as a nonnegative variable, the quadratic function ξ − γξ² attains its maximum at ξ = 1/(2γ), yielding the maximum value 1/(4γ). Therefore,

{(\dot{r} + S r)}^{T} ρ - \frac{α}{‖ ϖ ‖ + \hat{η}} \leq \frac{1}{4 γ} .

Substituting the bound (50) into (43) and then into (39) yields

\dot{V} \leq \frac{1}{4 γ} - {\dot{r}}^{T} D \dot{r} - r^{T} S P r - r^{T} (P + S D) \dot{r} .

(51)

The last three terms can be combined into a quadratic form in $\tilde{r} = {[r, \dot{r}]}^{T}$ . Standard PD control analysis shows that

- {\dot{r}}^{T} D \dot{r} - r^{T} S P r - r^{T} (P + S D) \dot{r} \leq - σ_{3} ‖ \tilde{r} ‖^{2},

(52)

where σ₃ = min{λ_min(D), λ_min(SP)}. Finally we obtain

\dot{V} \leq \frac{1}{4 γ} - σ_{3} ‖ \tilde{r} (t) ‖^{2},

(53)

For all

(\tilde{r}, t) \in R^{2} \times R

The uniform boundedness is established as follows. For any a > 0 with $‖ \tilde{r} (t_{0}) ‖ \leq a$ , where t₀ is the initial time,

d (a) = \{\begin{cases} \sqrt{\frac{σ_{2}}{σ_{1}}} a, & if a > R, \\ \sqrt{\frac{σ_{2}}{σ_{1}}} R, & if a \leq R, \end{cases}

(54)

where

R = \sqrt{\frac{1}{4 γ σ_{3}}} .

(55)

This ensures $‖ \tilde{r} (t) ‖ \leq d (a)$ for all t ≥ t₀.

Furthermore, the uniform ultimate boundedness holds with

\bar{d} > \underset{̲}{d} = \sqrt{\frac{σ_{2}}{σ_{1}}} R,

(56)

and the convergence time is

T (\bar{d}, a) = \{\begin{cases} 0, & if a \leq \bar{d} \sqrt{\frac{σ_{1}}{σ_{2}}}, \\ \frac{σ_{2} a^{2} - σ_{1} {\bar{d}}^{2}}{\frac{σ_{3} σ_{1} {\bar{d}}^{2}}{σ_{2}} - \frac{1}{4 γ}}, & if a > \bar{d} \sqrt{\frac{σ_{1}}{σ_{2}}} . \end{cases}

(57)

This expression follows from solving the differential inequality

\dot{V} \leq - (σ_{3} / σ_{2}) V + 1 / (4 γ)

with

V \geq σ_{1} ‖ \tilde{r} ‖^{2}

and

V \leq σ_{2} ‖ \tilde{r} ‖^{2}

. The ultimate bound

\underset{̲}{d} = \sqrt{σ_{2} / σ_{1}} R

is obtained when

\dot{V} = 0

. Note that

\bar{d}

can be made arbitrarily small by increasing γ, P, and D.

4.3. Parameters selection

Control parameter selection determines the trade-off between tracking performance and implementation constraints Li et al. (2022); Ma et al. (2023). Based on the stability analysis, guidelines for tuning the key parameters are as follows.

Gain γ: This parameter governs the ultimate bound of the uniform ultimate-bounded region. Equation (48) indicates that increasing γ asymptotically reduces this bound, thereby improving steady-state precision. However, larger γ increases control effort and energy consumption and may induce saturation or noise amplification. Thus, γ balances tracking accuracy against actuator limitations and disturbance rejection capability.

Gains P and D: The proportional gain P suppresses steady-state error and accelerates response by increasing bandwidth, yet excessive values cause overshoot and oscillations. The derivative gain D dampens oscillations and improves stability margins, but excessive values slow the response and amplify sensor noise. Iterative tuning is recommended to identify suitable P-D combinations.

Filter parameter S: This parameter determines the bandwidth of the error dynamics. Larger S accelerates convergence but increases noise sensitivity; smaller S yields smoother action at the cost of slower transient recovery. The selection should respect the physical bandwidth of the actuator and sensor hardware.

5. Simulation and experimental

5.1. Numerical simulations

In the following figures, x denotes the joint angular displacement θ and e denotes the tracking error (An et al., 2023). In this section, sinusoidal and step signals were chosen to verify the controller with inequality constraint (MPDW) and without inequality constraint (MPDN) Yu et al. (2024) and PID controller, respectively.

Within the controller developed in this work, the disturbance of unstructured and uncertain load torque exerts a more significant impact on the system’s stability. Therefore, for the purposes of simulation, we limit our discussion to the impact of load torque disturbances on the control of motion trajectory tracking for the robotic joint module. The load torque disturbance is given by the following description:

Δ F = 0.2 \sin t .

(50)

Table 1.

Joint-module parameters.

Parameter	Value	Physical meaning
J	0.00165 (kg ⋅ m²)	Equivalent rotational inertia
B	0.008 (Nm/rad/sec)	Viscous-friction constant
η	0.95	Power-transmission efficiency
λ	101	Gear reduction ratio
n _c	6	Number of pole pairs
ψ _f	0.025	Permanent magnet flux linkage
$\hat{F}$	0.32	Rated load torque

The correlation coefficients of the Coulomb viscous friction model are shown below: f_c = 0.24, f_v = 0.06. Table 1 shows the system parameters selected in simulation. The controller related parameters are as follows: MPDN controller: P = 150, D = 10, S = 1, γ = 3; MPDW controller: P = 150, D = 15, S = 1, γ = 3; PID controller: P = 80, I = 4, D = 10. All parameters were obtained through iterative tuning to ensure a fair comparison. We acknowledge that the PID controller was implemented without anti-windup, which may bias the comparison in favor of the proposed method.

5.1.1. Step response

For the step signal, set the desired signal amplitude to 135° and set the upper and lower boundaries of the angular displacement to 135.9° and −135.9°, respectively. The simulation results are presented in Figure 4. Under the inequality constraint, the response trajectory more closely follows the reference signal compared with the MPDN and PID controllers, while also exhibiting faster transient response. In addition, the output angle remains strictly bounded within the prescribed upper limit of 135.9° without compromising tracking accuracy. In contrast, both the MPDN and PID controllers exceed this limit during the response process.

Figure 4.

Simulation results under step signal.

5.1.2. Tracking a sinusoid signal

For the sine signal, the amplitude is 90° and the frequency is 0.2 Hz. Set the upper and lower boundaries of the angular displacement to 91.2° and −91.2°, respectively. The sinusoidal tracking curve and the corresponding steady-state error of the angular displacement of the robotic joint module are illustrated in Figures 5 and 6, respectively. As shown in these figures, the controller with inequality constraints achieves faster dynamic response and more accurate sinusoidal tracking. Moreover, the output trajectory remains strictly confined within the prescribed upper and lower bounds throughout the tracking process, demonstrating effective constraint enforcement capability. When the system is stable, MPDW controller exhibits smaller steady-state error compared to MPDN controller and PID controller. The controller with inequality constraints maintains a steady-state error within 0.03°, whereas the controller without inequality constraints exceeds a steady-state error of 0.5°. The steady-state error for a PID controller exceeds 1.5°. Based on the simulation results, the MPDW controller demonstrates superior dynamic performance.

Figure 5.

Simulation results under sinusoidal signal.

Figure 6.

Sinusoidal-tracking error response under simulation conditions.

5.2. Experimental platform and experiment results

Figure 7 presents the experimental apparatus employed in this work. The configuration integrates a host computer running MATLAB/Simulink, a joint actuator, the CSPACE real-time control board, a rotary encoder, and a magnetic-particle brake. The implementation of our control algorithm on this experimental platform is mainly divided into the following steps:

Step 1: Develop the control algorithm within the MATLAB/Simulink environment.

Step 2: Convert the implemented Simulink model into C code using the automatic code generation feature of MATLAB/Simulink.

Step 3: Transfer the generated C code to the CSPACE platform via Code Composer Studio (CCS).

Step 4: Execute the deployed C code program on the CSPACE platform.

Figure 7.

Collaborative robot joint modular experimental platform.

In order to further verify the effectiveness of the proposed control algorithm, sinusoidal and step signals are used as input signals, and the comparison experiment of different algorithms is completed on the robotic joint module platform. In the experimental results, x represents the angular displacement of the robotic joint module and e represents the angular displacement error.

5.2.1. Step response

As in the numerical simulation, set the desired signal amplitude to 135° and set the upper and lower boundaries of the angular displacement to 135.9° and −135.9°, respectively. The experimental results can be seen in Figure 8. Figure 8 demonstrates that the controller with inequality constraints achieves a faster transient response and maintains the output strictly within the prescribed bounds. In contrast, both the MPDN and PID controllers exceed the predefined limits. Moreover, the proposed method exhibits reduced overshoot and improved steady-state convergence characteristics, indicating enhanced constraint-handling capability and robustness.

Figure 8.

Experimental results under step signal.

5.2.2. Tracking a sinusoid signal

The same amplitude, frequency and upper and lower bounds are used here as in the numerical simulation. The experimental results are presented in Figures 9 and 10. Figure 9 illustrates the sinusoidal position tracking performance, while Figure 10 depicts the corresponding tracking error curves. As shown in Figure 9, the controller with inequality constraints achieves more accurate sinusoidal tracking and faster dynamic response compared with both the controller without inequality constraints and the PID controller. Moreover, the output trajectory remains strictly confined within the predefined upper and lower bounds throughout the entire motion cycle, demonstrating effective constraint enforcement capability. Figure 10 further indicates that the steady-state tracking error under inequality constraints is stabilized within ±0.08°, whereas the steady-state errors of the controller without inequality constraints and the PID controller are approximately ±0.15° and ±0.2°, respectively.

Figure 9.

Experimental results under sinusoidal signal.

Figure 10.

Experimental results of the error tracking under sinusoidal signal.

Experiments show that MPDW controller can not only control the angular displacement within the upper and lower bounds, but also reduce the steady-state error and improve the dynamic performance of the system. The experimental results are consistent with the numerical simulations, confirming the validity and practical applicability of the proposed control framework.

6. Conclusions

This work addresses robust control of an uncertain robotic joint module. The scheme integrates a PD core with a robust layer that upper-bounds all parametric uncertainties and external disturbances. By employing the Lyapunov approach, the algorithm’s uniform boundedness and uniform ultimate boundedness are demonstrated. We transform the state of the desired signal to confine the output within the predetermined boundary, as uncertainties in structural deformation, friction, and external load impact the tracking performance of the output angular displacement in the robotic joint module. Finally, experimental evaluations were conducted on a robotic joint module platform, providing a detailed comparison of the results between three distinct algorithms. The results demonstrate that the proposed robust control strategy with inequality constraints not only enhances transient response characteristics but also guarantees bounded steady-state tracking performance under parameter uncertainties and external disturbances. This confirms the effectiveness of integrating inequality-constrained output shaping within a robust control framework. The control algorithm with inequality constraints can effectively limit the output amplitude to a set upper and lower limit without compromising stability and tracking accuracy. Moreover, the robust control algorithm with inequality constraints can address the stability issue of nonlinear multivariable systems of the same type. In future work, we will extend the proposed controller to multi-joint collaborative robots and validate it on an actual service robot in real-world VIP lounge environments under unpredictable human disturbances and varying service loads.

Footnotes

ORCID iD

Zhengqing Ni

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Wang

Liu

, et al. (2023) Cooperative game-based approximate optimal control of modular robot manipulators for human-robot collaboration. IEEE Transactions on Cybernetics 53(7): 4691–4703. https://doi.org/10.1109/tcyb.2023.3277558

Chen

(2011) A new approach to the control design of fuzzy dynamical systems. Journal of Dynamic Systems, Measurement, and Control 133(6): 5267–5275. https://doi.org/10.1115/1.4004579

Chen

Hua

(2022) Adaptive full-state-constrained control of nonlinear systems with deferred constraints based on nonbarrier lyapunov function method. IEEE Transactions on Cybernetics 52(8): 7634–7642. https://doi.org/10.1109/TCYB.2020.3036646

Chen

Zhao

Zhen

, et al. (2020) Novel optimal adaptive robust control for fuzzy underactuated mechanical systems: a nash game approach. IEEE Transactions on Fuzzy Systems 29(9): 2798–2809. https://doi.org/10.1109/tfuzz.2020.3007452

Hale

(1969) Ordinary Differential Equations. New York. Wiley-Interscience.

Huang

Yang

, et al. (2022) Robust observer design and nash-game-driven fuzzy optimization for uncertain dynamical systems. Fuzzy Sets and Systems 432: 132–151. https://doi.org/10.1016/j.fss.2021.07.003

Huang

Yang

Pan

, et al. (2021) Composite learning enhanced neural control for robot manipulator with output error constraints. IEEE Transactions on Industrial Informatics 17(1): 209–218. https://doi.org/10.1109/tii.2019.2957768

Zhao

Zhen

, et al. (2022) Control design with optimization for fuzzy steering-by-wire system based on nash game theory. IEEE Transactions on Cybernetics 52(8): 7694–7703. https://doi.org/10.1109/tcyb.2021.3050509

Liu

Zhen

Sun

, et al. (2019) A novel model-based robust control for position tracking of permanent magnet linear motor. IEEE Transactions on Industrial Electronics 67(9): 7767–7777. https://doi.org/10.1109/tie.2019.2945281

10.

Liu

Gong

Liu

, et al. (2021) Fuzzy observer constraint based on adaptive control for uncertain nonlinear MIMO systems with time-varying state constraints. IEEE Transactions on Cybernetics 51(3): 1380–1389. https://doi.org/10.1109/tcyb.2019.2933700

11.

Zhu

Tomizuka

, et al. (2022) On robust stability and performance with a fixed-order controller design for uncertain systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems 52(6): 3453–3465. https://doi.org/10.1109/tsmc.2021.3068903

12.

Huang

, et al. (2023) Cooperative game-based optimization of flexible robust constraint following control for spacecraft rendezvous system with uncertainties. IEEE Transactions on Systems, Man, and Cybernetics: Systems 53(11): 6849–6860. https://doi.org/10.1109/tsmc.2023.3288542

13.

Qin

Liu

Yin

, et al. (2022) Constraint-based adaptive robust control for active suspension systems under the sky-hook model. IEEE Transactions on Industrial Electronics 69(5): 5152–5164. https://doi.org/10.1109/tie.2021.3084178

14.

Shao

Zheng

Wang

, et al. (2021) Tracking control of a linear motor positioner based on barrier function adaptive sliding mode. IEEE Transactions on Industrial Informatics 17(11): 7479–7488. https://doi.org/10.1109/tii.2021.3057832

15.

Shao

Zheng

Tang

, et al. (2022) Barrier function based adaptive sliding mode control for uncertain systems with input saturation. IEEE/ASME Transactions on Mechatronics 27(6): 4258–4268. https://doi.org/10.1109/tmech.2022.3153670

16.

Sun

Yang

Wang

, et al. (2020) Designing robust control for mechanical systems: constraint following and multivariable optimization. IEEE Transactions on Industrial Informatics 16(8): 5267–5275. https://doi.org/10.1109/tii.2019.2951842

17.

Sun

Wang

Yang

, et al. (2022) Adaptive robust control for pointing tracking of marching turret-barrel systems: coupling, nonlinearity and uncertainty. IEEE Transactions on Intelligent Transportation Systems 23(9): 16397–16409. https://doi.org/10.1109/tits.2022.3150043

18.

Udwadia

(2003) A new perspective on the tracking control of nonlinear structural and mechanical systems. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences 459(2035): 1783–1800. https://doi.org/10.1098/rspa.2002.1062

19.

Wang

Yang

Wang

, et al. (2022a) Adaptive-adaptive robust boundary control for uncertain mechanical systems with inequality constraints. Nonlinear Dynamics 110(1): 449–466. https://doi.org/10.1007/s11071-022-07666-2

20.

Wang

Zheng

Yin

, et al. (2022b) Toward human-centric smart manufacturing: a human-cyber-physical systems (HCPS) perspective. Journal of Manufacturing Systems 63: 471–490. https://doi.org/10.1016/j.jmsy.2022.05.005

21.

Wang

, et al. (2023a) Hybrid active-passive robust control framework of a flexure-joint dual-drive gantry robot for high-precision contouring tasks. IEEE Transactions on Industrial Electronics 70(2): 1676–1686. Available at: https://doi.org/10.1109/tie.2022.3165305

22.

Wang

Yang

, et al. (2023b) Robust bounded control for fuzzy mechanical systems: fuzzy optimal design and inequality constraint reorganize. IEEE Transactions on Fuzzy Systems 31(8): 2605–2616. Available at: https://doi.org/10.1109/tfuzz.2023.3234585

23.

Wang

Chen

Zhen

, et al. (2024) Prescribed performance adaptive robust control for robotic manipulators with fuzzy uncertainty. IEEE Transactions on Fuzzy Systems 32(3): 1318–1330.

24.

Wang

Chen

Zhen

, et al. (2024) Optimal design of udwadia-kalaba theory-based adaptive robust control for permanent magnet linear synchronous motor. Journal of Vibration and Control 30(3-4): 616–631. https://doi.org/10.1177/10775463221149086

25.

Xian

Huang

, et al. (2024) Adaptive robust control for fuzzy underactuated mechanical systems: a stackelberg game-theoretic optimization approach. Information Sciences 665: 120394. https://doi.org/10.1016/j.ins.2024.120394

26.

Yang

Huang

Yang

, et al. (2022a) Design and optimization of robust path tracking control for autonomous vehicles with fuzzy uncertainty. IEEE Transactions on Fuzzy Systems 30(6): 1788–1800. https://doi.org/10.1109/tfuzz.2021.3067724

27.

Yang

Shi

Zhang

, et al. (2022b) Leader-follower stackelberg game oriented adaptive robust constraint-following control design for fuzzy exoskeleton robot systems. Information Sciences 606: 272–291. https://doi.org/10.1016/j.ins.2022.05.043

28.

Zhao

Wang

, et al. (2024) An intelligent cooperative game approach for adaptive robust control of fuzzy mechanical systems. IEEE Transactions on Fuzzy Systems 32(6): 3594–3607. https://doi.org/10.1109/tfuzz.2024.3376694

29.

Zhang

Cheng

, et al. (2021) Trajectory generation by chance-constrained nonlinear MPC with probabilistic prediction. IEEE Transactions on Cybernetics 51(7): 3616–3629.

30.

Zhang

Zhao

Chen

, et al. (2023a) A novel modeling and control approach considering equality and inequality constraints based on generalized udwadia-kalaba equation. Nonlinear Dynamics 111(18): 17109–17122. https://doi.org/10.1007/s11071-023-08738-7

31.

Zhang

Zhao

Jiao

(2023b) Optimization of robust formation tracking control for traffic cone robots with matching and mismatching uncertainties: a fuzzy-set theory-based approach. Expert Systems with Applications 227: 120233. https://doi.org/10.1016/j.eswa.2023.120233

32.

Zhang

Zhao

, et al. (2023c) Formation tracking control design for uncertain artificial swarm systems under inequality constraints: leakage and dead-zone type. ISA Transactions 135: 325–338. https://doi.org/10.1016/j.isatra.2022.10.013

33.

Zhang

, et al. (2024) A generalized udwadia-kalaba control design for uncertain belt conveyor systems with inequality constraints. ISA Transactions 151: 409–422. https://doi.org/10.1016/j.isatra.2024.05.046

34.

Zhao

Lin

Huang

, et al. (2022) Optimal design of robust control based on compound form: steady-state performance and control cost. Transactions of the Canadian Society for Mechanical Engineering 46(4): 738–748. https://doi.org/10.1139/tcsme-2022-0058

35.

Zhu

Sun

, et al. (2023) Game-theoretic optimization toward diffeomorphism-based robust control of fuzzy dynamical systems with state and input constraints. IEEE Transactions on Fuzzy Systems 32(2): 373–387. https://doi.org/10.1109/tfuzz.2023.3298341