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Abstract

To enhance the manipulator's motion accuracy, a Hertz collision force model for the hinge positions and a dynamic model for the manipulator is established, a novel adaptive clearance compensation algorithm is proposed to counteract the nonlinear effects induced by joint clearance. This article proposes a novel adaptive optimal clearance compensation tracking control method for dynamic manipulator systems with joint clearance. The method simultaneously computes feedforward and feedback control actions through an enhanced system approach based on Adaptive Dynamic Programming (ADP) and a performance index function. To implement the optimal control strategy, a generalized policy learning algorithm is developed, which reduces the dependency on known system dynamics. Additionally, the algorithm enables continuous, synchronous updates of adaptive evaluation and control actions, eliminating the need for iterative steps. Unlike traditional approaches, this method discards the use of behavioral neural networks (ANNs), thereby reducing computational complexity. Simulation results demonstrate the effectiveness of the proposed learning algorithm and control method for manipulator clearance compensation. The effectiveness of the clearance compensation method was further validated through experiments conducted on the robotic arm test platform. By implementing clearance compensation-based optimization control, the Integral Absolute Error (IAE) of manipulator link1 and link2 displacement was reduced by 54.2% and 40.8%, respectively, compared to the uncompensated clearance state.

Keywords

Optimum control manipulator policy learning reinforcement learning clearance compensation

Introduction

A manipulator is a mechanical device that can imitate the movement of the human arm and is widely used in industrial automation, medical care, food processing, warehousing, logistics, and other fields. A manipulator typically consists of multiple joints and linkages, enabling flexible operation via motorized, hydraulic, or pneumatic actuation. Closed-chain linkages are utilized to transmit or transform motion, providing control of precise positioning and orientation. Joint clearances, inevitably introduced by design tolerances, manufacturing imperfections, and assembly errors, cause deviations between the manipulator's actual trajectory and its ideal path, consequently degrading motion precision.¹ Conversely, a manipulator's operation induces changes in joint clearances due to friction, wear, and collisions between moving components. The combined effect of clearance, gravity, and other coupled factors exacerbates the decline in motion accuracy, making the impact of clearance variations on the system dynamics more pronounced.^2–5 Notably, joint clearances induce highly nonlinear disturbances, resulting in output position uncertainty and irregular motion, thereby substantially compromising the manipulator's operational performance in industrial settings.

With technological advancements, the demand for higher precision in manipulators has significantly increased, making joint clearance-induced nonlinear dynamics an important research focus.^6–9 Based on the dynamics model of spatial developable mechanism, the dynamic analysis of the spatial expansion mechanism is carried out, and the influence of the roughness of the contact surface and the size of the clearance on the dynamic characteristics of the system is explored.¹⁰ The dynamic response of a novel high-speed structural redundant parallel manipulator considering multiple spatial revolute joints with radial and axial clearances is studied.¹¹ A planar robotic manipulator system with a revolute clearance joint, mounted on a spacecraft, is used to investigate the effects of joint clearance on the dynamic performance of practical mechanical systems.¹².

To address motion path deviations caused by hinge clearance, Scholars have conducted in-depth analyses of linkage control with joint clearances. To mitigate the adverse effects of clearances on dynamic characteristics, an improved second-order sliding-mode controller (IS-SMC) has been proposed.¹³The optimization of various physical parameters in planar mechanisms with joint clearances is implemented through a neural network-genetic algorithm (NN-GA) hybrid approach.^14,15 A control mechanism based on the Pyragas method is presented to optimize component dynamic performance, enhance fatigue life, and suppress undesirable vibrations caused by mating part impacts in clearance joints.¹⁶ An open-loop preload control method with high computational efficiency has been developed to address backlash elimination in redundantly actuated parallel manipulators.¹⁷. A particle swarm optimization-based algorithm is developed to solve this highly nonlinear optimization problem for a slider-crank mechanism with revolute clearance joints.¹⁸

The clearance compensation method is applied to enhance the manipulator's dynamic performance in the presence of joint clearance. An analysis is conducted on planar four-bar manipulators with joint clearance, where one joint is actuated by either collocated open-loop control or state feedback controllers, including proportional-derivative control, state feedback linearization, and passivity-based control.¹⁹ Advances in control theory have led to the development of advanced manipulator control strategies, including model predictive control (MPC),²⁰ robust control,²¹ adaptive control,²² and SMC.^23,24 In addition, with the development of artificial intelligence technology, the manipulator can achieve autonomous learning and automated control, thus adapting more flexibly to the needs of various complex tasks, including: NN²⁵ and fuzzy logic control (FLC).²⁶ While adaptive NN/FLC controls handle nonlinearities, their high errors and computational cost limit stability. Since energy consumption critically affects autonomous vehicle performance, optimal control methods are developed to maximize efficiency²⁷.

Manipulator dynamics are modeled via nonsmooth contact dynamics (NSCD) for joint clearance and collisions. NSCD captures rigid impacts, while nonsmooth models describe discontinuous behaviors.²⁸ A novel computational methodology based on nonsmooth dynamics algorithms is proposed to simulate spatial frictional contact dynamics in multibody systems.²⁹ To track the trajectories of wheeled-legged balancing robots, a simple LQR-based controller is proposed, taking a considerable step toward closing this gap by presenting a fast trajectory optimizer for generating trajectories over a large class of challenging terrains.³⁰ In order to obtain an online solution to the optimal control problem, adaptive dynamic programming (ADP) has recently been proposed based on reinforcement learning (RL) principles.^31–36 Strategy iteration, an important method in ADP, s often used to approximate the solution of the optimal equation. A simple and effective ADP structure is proposed based on the critic neural network (CNN), which guarantees the convergence of the CNN weights and allows the direct computation of the control actions. As a result, the actor neural network (ANN) used in the traditional ADP structure is avoided, and the stability and convergence of the controlled system can be simply demonstrated.^37–39 However, most of the ADP schemes mentioned above assume that the system dynamics are fully known, an assumption that limits the application of optimal control in manipulator control. Song et al. propose a novel data-driven ADP algorithm for optimal tracking of discrete-time nonlinear systems with unknown dynamics. Using only I/O data, it ensures optimality, robustness, and time efficiency while guaranteeing tracking performance. Theoretical proofs and simulations validate its superiority over existing methods.⁴⁰ Liu et al. present an output-feedback ADP method for adaptive robot impedance control in unknown environments. It optimizes tracking and force interaction while handling unobservable states via a discounted ADP algorithm.⁴¹ Zhou et al. present a decentralized fault-tolerant control scheme for modular manipulators using ADP, the method guarantees stability through Lyapunov analysis and demonstrates its efficacy experimentally.^42,43

Traditional dynamic programming methods often suffer from the curse of dimensionality when applied to nonlinear or high-dimensional systems. While MPC depends heavily on model accuracy and demonstrates limited adaptability to nonlinear/time-varying systems, its computational complexity also escalates with extended prediction horizons. RL, though powerful, demands substantial interaction data, incurs high training costs, and lacks theoretical stability guarantees. In contrast, ADP effectively overcomes these limitations by utilizing function approximators (e.g. NNs) to estimate value functions or control policies, thereby eliminating the need for precise mathematical models. Through iterative interactions with environmental data, ADP progressively refines control strategies, demonstrating notable efficacy in time-varying systems. Leveraging the advantages of ADP, this article proposes a generalized policy learning algorithm for optimal trajectory tracking control of robotic manipulators. In contrast to classical policy iteration algorithms, the proposed method eliminates the need for both an initial stable control policy and partial system dynamics knowledge. Moreover, the convergence of evaluation NN weights is rigorously guaranteed through a novel adaptive training law, replacing conventional gradient-based methods.

To address the trajectory optimal control problem for manipulator systems online, a new generalized policy learning framework is proposed. Furthermore, a novel adaptive optimal tracking control method is developed by adopting the augmented generalized system approach and an ADP-based performance index function to synchronously generate feed-forward and feedback control actions. An augmented system incorporating both tracking error dynamics and the desired trajectory is first constructed. To obtain the optimal control solution, an ADP-based generalized policy learning algorithm is developed, significantly relaxing the prior knowledge requirements on system dynamics. Notably, this algorithm enables continuous co-adaptation between evaluation and control actions without requiring iterative updates, while completely eliminating dependency on traditional methods. Both simulation and experimental results validate the efficacy of the proposed control scheme and learning framework.

During the movement of the manipulators, hinge clearance causes unexpected collisions at the joint connection points. This wear in the clearance leads to additional contact forces, which result in increased deviations in the movement trajectory. The dynamic model that incorporates clearance is highly nonlinear, making it difficult for traditional control algorithms to precisely compensate for these effects. The proposed approach effectively addresses motion accuracy deterioration in worn manipulator systems. Firstly, a contact dynamics model based on Hertz theory is established to capture the transient contact behavior between shafts and bearings with joint clearances. Subsequently, a novel clearance compensation controller is designed to enhance motion precision, with its performance validated through computational simulations.

The main contributions of this work are summarized as follows: First, investigates the degradation in motion trajectory accuracy of robotic manipulators caused by joint clearance. Secondly, by integrating ADP with a clearance collision force model, a compensation method for the additional torque generated by joint clearance and the consequent trajectory deviation is developed. Thirdly, the novel approach for manipulator's joint clearance compensation id validated by numerical and experiment.

The article is organized as follows: the second section formulates the manipulator problem caused by joint clearance, present the optimal controller of clearance compensation; the third section validate the stability of proposed algorithm; the fourth section presents numerical and experiment validation results and the related analysis and discussion; and the fifth section concludes the research work.

Control method of joint clearance compensation for manipulators

Accurate modeling of the gap contact force is essential for designing improved control strategies, thereby ensuring the stability and accuracy of the mechanical motion trajectory. A two-degree-of-freedom manipulator with a fixed base is utilized for this study, to investigate the kinematic and stress changes of the mechanism under clearance conditions, as shown in Figure 1.

Figure 1.

The structure diagram of the manipulator with joint clearance.

In Figure 1, $J_{1}$ is an ideal revolute joint, $J_{2}$ a joint with clearance; $θ_{1}$ and $θ_{2}$ the output angles of joints 1 and 2 of the manipulators; $l_{1}$ and $l_{2}$ the lengths of links 1 and 2; c the radial clearance of the joint; and $δ$ the displacement error due to joint clearance.

Joint clearances induce contact collisions between manipulator links, thereby altering the system dynamics. To mitigate clearance effects in the dynamic model, this article utilizes moment variations from clearance-induced dynamic calculations as control system inputs. This feed-forward compensation strategy effectively improves system kinematics accuracy. Set $Y = {[\begin{matrix} x & y \end{matrix}]}^{T}$ , $Q = {[\begin{matrix} q_{1} & q_{2} \end{matrix}]}^{T}$ , and $Y = J Q + μ (q, \dot{q}, t)$ as the uncertain interference term of the system, J is the Jacobi matrix, the disturbance is mitigated using an optimal controller, and the control block diagram of the system is presented in Figure 2.

Figure 2.

The system control diagram.

In Figure 2, the given end trajectory is denoted as $x_{d}, y_{d}$ , corresponding joint angles derived from the given trajectory via the inverse kinematics solution are represented as $q_{d 1}, q_{d 2}$ . The actual end trajectory is $(x, y)$ , actual joint angles at any given time, the change in the system's topology introduced by the clearance is denoted as $μ (q, \dot{q}, t)$ , while the controller's output torque is $τ_{1}, τ_{2}$ . By optimally controlling the driving torques of each joint, the actual output of the manipulator can effectively track the desired trajectory.

Hertz's collision model is suitable for analyzing continuous contact forces during low-speed movements, accounting for energy loss while maintaining high computational accuracy. To analyze the dynamics of the manipulator's joint clearance state, it is essential to model the collision forces at the joints. This article presents a simplified dynamic model with joint clearance based on Hertz's collision model and designs an optimal controller with clearance compensation to reduce the impact of joint clearance on the manipulator's motion accuracy and improve its dynamic performance.

Three types of motion can be observed during the dynamics of the revolute joint clearance: (1) free flight mode, where the shaft moves freely within the bearing's boundaries without contact (Figure 3(a)), (2) impact mode, which occurs at the end of the free flight mode (Figure 3(b)), and (3) continuous contact mode, where contact is always maintained between the bearing and journal (Figure 3(c)). The $F_{n}$ denotes the normal force and $F_{t}$ the tangential friction force. The pin can move freely within the bearing in the absence of lubrication until contact occurs between the two links. Upon impact with the bearing, both normal and tangential contact forces are generated. In scenarios where friction is negligible, the direction of the clearance vector coincides with the normal direction of the contact interface.

F_{n} = {\begin{matrix} \begin{matrix} K δ^{n} + D δ^{n} \dot{δ} & δ \geq 0 \end{matrix} \\ \begin{matrix} 0 & \begin{matrix}  \end{matrix} δ < 0 \end{matrix} \end{matrix}

(1)

Figure 3.

Types of motion between the shaft and bearing. (a) Free flight mode, (b) impact mode, and (c) continuous contact mode.

where K is the contact stiffness coefficient, $δ$ the deformation, n a nonlinear index, and D the damping coefficient.

D = \frac{3 K}{4 \dot{δ}} (1 - η^{2}) δ^{n}

(2)

where $η$ is the recovery coefficient.

where $\dot{δ}$ is the penetration velocity, $\dot{δ} = \frac{x \dot{x} + y \dot{y}}{\sqrt{x^{2} + y^{2}}}$ .

K = \frac{4}{3 π (σ_{1} + σ_{2})} [\frac{R_{a} R_{b}}{R_{a} - R_{b}}]^{\frac{1}{2}}

(3)

where, $σ_{1} = \frac{1 - v_{1}^{2}}{π E_{1}}, σ_{2} = \frac{1 - v_{2}^{2}}{π E_{2}}; E_{1}, E_{2}$ is elastic modulus; $v_{1}, v_{2}$ the passion ratio; and $R_{a}, R_{b}$ the radius of shaft and bearing, respectively.

The Coulomb friction model is expressed in the form of a piece-wise function:

F_{t} = μ_{d} F_{n} sgn (v_{t})

(4)

where $μ_{d}$ is kinetic friction coefficient and $v_{t}$ the relative tangential velocity.

The contact force in the revolute joint with clearance can be expressed as:

F_{c} = u (δ) (F_{n} + F_{t})

(5)

u (δ) = {\begin{matrix} 0, δ \leq 0 \\ 1, δ > 0 \end{matrix}

(6)

Based on the Newton-Euler method, the inertia forces and moments of each link are derived from their velocities and accelerations. The driving forces and moments at each joint are then determined by considering the collision forces and the effects of gravity. This results in the establishment of the dynamic model for the two-degree-of-freedom manipulator, as shown in the following equation:

M (q) \ddot{q} + C (q, \dot{q}) \dot{q} + G (q) + F_{c} (q) = τ

(7)

where, $M (q) \in R^{2 \times 2}$ is the inertia matrix of a planar two-degree-of-freedom manipulator system, $C (q, \dot{q}) \in R^{2 \times 2}$ the centrifugal force matrix, $G (q) \in R^{2 \times 1}$ the gravitational vector matrix, $q = [q_{1}, q_{2}]^{T} \in R^{2 \times 1}$ the displacement vector of the joint angle, $τ = [τ_{1}, τ_{2}]^{T} \in R^{2 \times 1}$ the driving torque acting on the manipulator system; and $F_{c} (q) \in R^{2 \times 1}$ the vector matrix of forces in the clearance state of the manipulator, $G (q) = [g_{1}, g_{2}]^{T}$ .

{\begin{matrix} g_{1} = m_{3} g (l_{2} + c + δ_{d}) c_{12} + (m_{1} + m_{3}) g l_{1} c_{1} \\ g_{2} = m_{3} g (l_{2} + c + δ_{d}) c_{12} \end{matrix}

(8)

Displacement error due to joint clearance change with time is denoted as $δ_{d} = | δ_{m a x} \cos (q_{1} + q_{2}) |$ . When a given trajectory $(x_{d}, y_{d})$ is known, the inverse kinematics equation of the joint angle $q_{d 1}, q_{d 2}$ is expressed as follows:

q_{d 2} = - \arccos (x_{d}^{2} + y_{d}^{2} - (l_{1}^{2} + l {_{2}^{'}}^{2}) / 2 l_{1} l_{2}^{'})

(9)

β_{1} = {\begin{matrix} \arctan (x_{d} / y_{d}), y_{d} \geq 0 \\ π + \arctan (x_{d} / y_{d}), y_{d} < 0 \end{matrix}

(10)

β_{2} = \arccos ((x_{d}^{2} + y_{d}^{2} + l_{1}^{2} - l {_{2}^{'}}^{2}) / 2 l_{1} \sqrt{x_{d}^{2} + y_{d}^{2}})

(11)

q_{d 1} = β_{2} - β_{1}

(12)

where, $l_{2}^{'}$ represents the change in the length of link 2 in the clearance state, $l_{2}^{'} = l_{2} + c + δ$ .

The dynamic model of the two-degree-of-freedom manipulator is then developed based on the Newton-Euler method. Firstly, starting from the base and moving toward the end of the connecting rod, the velocity and acceleration of each link are recursively determined, and the inertial forces and moments of each link are calculated as follow:

{\begin{matrix} f_{c i + 1} = m_{i + 1} {\dot{v}}_{i + 1} \\ M_{c i + 1} = w_{i + 1} \times (I_{i + 1} w_{i + 1}) + I_{i + 1} {\dot{w}}_{i + 1} \end{matrix}

(13)

Secondly, the joint driving forces and torques are calculated by recursively moving from the end of the connecting rod toward the base:

{\begin{matrix} f_{i} = R^{i + 1} f_{i + 1} + f_{c i} \\ M_{i} = R^{i + 1} M_{i + 1} + M_{c i} + r_{c i} \times f_{c i} + O_{i + 1} \times R^{i + 1} f_{i + 1} \\ T_{i} = M_{c i}^{T} Z_{i} \end{matrix}

(14)

where, $v_{i + 1}, w_{i + 1}$ represent the linear velocity and angular velocity of the connecting rod, respectively; ${\dot{v}}_{i + 1}, {\dot{w}}_{i + 1}$ the linear acceleration and angular acceleration of the connecting rod $i + 1$ , respectively; $f_{i + 1}$ the force of the link i acting on link $i + 1$ ; $M_{i + 1}$ the torque of the link i acting on link $i + 1$ ; $R^{i + 1}$ the rotation transformation matrix between coordinate systems $i + 1$ and i; $O_{i + 1}$ the $i + 1$ position vector relative to the origin of the coordinate system; $r_{c i}$ the center-of-mass offset distance of the connecting rod i from coordinate origin.

A dynamic model was developed based on the structural of the manipulator with joint clearance, assuming that the mass of each link of the two-degree-of-freedom manipulator is $m_{1}, m_{2}, m_{3}$ , respectively. The radius vector of links 1 and 2 mass center is presented as $r_{c 1} = l_{1} X_{1}, r_{c 2} = l_{2} X_{2}$ ; no force acting on the actuator end of the manipulator, because it is free, $f_{3} = 0, M_{3} = 0$ , and the base does not rotate, $ω_{0} = 0, {\dot{ω}}_{0} = 0$ ; under the influence of gravity, ${\dot{v}}_{0} = g Y_{0}$ , g is the gravitational acceleration, and $F (q) = [f_{1}, f_{2}]^{T}$ are presented as follow:

{\begin{matrix} f_{1} = \begin{matrix} m_{2} (c + δ_{d}) ({\ddot{q}}_{1} + {\ddot{q}}_{2}) (c + δ_{d} + 2 l_{2}) + ((l_{2} + c + δ_{d}) c_{12} + l_{1} c_{1}) (k δ_{d} + b {\dot{δ}}_{d}) \\ - s g \dot{n} q_{2} (k δ_{d} + b {\dot{δ}}_{d}) l_{1} (s_{12} + μ s_{1}) - m_{2} l_{1} (c + δ_{d}) ({\dot{q}}_{2} + 2 {\dot{q}}_{1}) {\ddot{q}}_{2} s_{2} \\ + m_{2} l_{1} (c + δ_{d}) (2 {\ddot{q}}_{1} + {\ddot{q}}_{2}) c_{2} \end{matrix} \\ f_{2} = \begin{matrix} - (l_{2} + c + δ_{d}) (k δ_{d} + b {\dot{δ}}_{d}) s g \dot{n} q_{2} μ s_{12} + m_{2} l_{1} (c + δ_{d}) ({\ddot{q}}_{1} + {\ddot{q}}_{2}) (2 l_{2} + c + δ_{d}) \\ + m_{2} l_{1} (c + δ_{d}) ({\dot{q}}_{1}^{2} s_{2} + {\ddot{q}}_{1} c_{2}) + (l_{2} + c + δ_{d}) (k δ_{d} + b {\dot{δ}}_{d}) c_{12} \end{matrix} \end{matrix}

(15)

Optimal tracking controller design

An optimal controller is designed based on the dynamics model of a two-degree-of-freedom manipulator, defining $x_{1} = q, x_{2} = \dot{q}, u = τ$ , the dynamic equations of the manipulator based on the Lagrange equation (7):

{\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = - M^{- 1} C x_{2} - M^{- 1} G - M^{- 1} F + M^{- 1} u \end{matrix}

(16)

where $x_{1} = [\begin{matrix} q_{1} & q_{2} \end{matrix}]$ and $x_{2} = [\begin{matrix} {\dot{q}}_{1} & {\dot{q}}_{2} \end{matrix}]$ are the displacements and velocities of the two axes of the manipulator respectively. $M^{- 1}$ is the inverse of the inertia matrix, the Coriolis/centripetal torque is C and gravity matrix is G. Thus, the system can be in the form of a state-space equation:

\dot{x} (t) = F (x) + B u (t)

(17)

where $x (t) = {[\begin{matrix} q & \dot{q} \end{matrix}]}^{T} \in R^{2}$ denote the angle position of the manipulator, $F (x) = {[\begin{matrix} x_{2} & - M^{- 1} (C x_{2} + G + F) \end{matrix}]}^{T}$ is the unknown dynamics of the system, $B = {[\begin{matrix} 0 & M^{- 1} \end{matrix}]}^{T}$ the control matrix of the system, and $u (t) \in R^{m}$ the input of the manipulator system. In this article, the function $F (x)$ is the Lipschitz continuous and the manipulator system is controllable.

This article aims to achieve optimal tracking control for the aforementioned system by designing a clearance-compensated optimal controller u. The controller will ensure the system output accurately tracks the reference trajectory while near-optimally minimizing the following infinite time-domain performance index function:

V (t) = \int_{0}^{\infty} e^{- 2 γ t} (e {(t)}^{T} Q_{r} e (t) + u {(t)}^{T} R u (t)) d t

(18)

where $Q_{r}$ and R are positive definite matrices, $e^{- 2 γ t} (e {(t)}^{T} Q_{r} e (t) + u {(t)}^{T} R u (t)) : R^{n} \times R^{m} \to R^{n}$ a function of utility, and $e (t)$ the tracking error, which can be defined as:

e (t) = x (t) - {\dot{x}}_{d} (t)

(19)

where $x_{d} (t) \in R^{n}$ is reference trajectory, $x_{d} (t)$ and its derivative ${\dot{x}}_{d} (t)$ in this article is continuous and bounded.

To address the technical challenges posed by the time-varying nature, a clearance-compensated optimal controller is designed.⁴⁴ By constructing an augmented generalized system, the time-varying clearance compensation optimization problem is transformed into a time-invariant formulation. This approach enables the simultaneous derivation of both the steady-state component and feedback component through the minimization of the modified cost function described by Lv et al.⁴⁵ Following the augmented system methodology from Modares and Lewis,⁴⁶ the tracking error and commanded trajectory are incorporated into the following augmented system state:

X (t) ≜ e^{- γ t} {[\begin{matrix} x^{T} (t) & x_{d}^{T} (t) \end{matrix}]}^{T} \in R^{2 n}

(20)

Defining $U_{u} (t) = e^{- γ t} u$ and combining the original system and tracking error derivatives, the augmented system is formulated as:

\dot{X} (t) = A (X) + G_{R} (X) U_{u} + c (t)

(21)

where $A (X) = - γ X + e^{- γ t} [F (x), {\dot{x}}_{d}^{T}]^{T}$ is the augmentation system matrix and $G_{R} (X) = [B, 0]^{T}$ the input matrix, respectively, $γ$ a sufficiently small discount factor to ensure the boundedness of the system state, and $c (t)$ compensation for the clearance.

It can be seen from equation (21) that the system augmentation and generalized dynamics $A (X)$ is unknown due to the inclusion of the unknown dynamics system $F (x)$ , so the use of the augmentation and generalized dynamics $A (X)$ needs to be avoided in the ensuing optimal tracking control algorithm, and an ADP-based generalized policy learning algorithm will be used to implement the clearance compensation optimal controller design for the manipulator's system. The system is defined as:

V (X (t - T)) = \int_{t - T}^{t} r (X (τ), u (τ)) d τ + V (X (t))

(22)

where $r (X (τ), u (τ)) = X^{T} Q_{T} X + u^{T} R u, T > 0$ is the sampling time, $Q_{T} = {[\begin{matrix} 1 & - 1 \end{matrix}]}^{T} Q_{r} [\begin{matrix} 1 & - 1 \end{matrix}]$ , and $R = d i a g (a_{1}, \dots, a_{m})$ are positive definite matrices.

For approximating the performance index function, a single-layer NN is defined:

V (X) = W^{T} α (X) + ε (X)

(23)

where $α (X) : R^{n} \to R^{N}$ is the excitation function, N the number of neurons, W the evaluation NN weights, and $ε (X)$ the NN approximation error, which is assumed to be bounded by the error $ε (X)$ and its derivative $\nabla ε (X)$ .

To enable online updating of the evaluation mechanism, substituting formula (23) into formula (22) yields:

\underset{k (X, u)}{\underset{⏟}{\int_{t - T}^{t} r (X (τ), u (τ)) d τ}} + \underset{W^{T} Δ α (t)}{\underset{⏟}{W^{T} α (X (t)) - W^{T} α (X (t - T))}} = - ε_{X}

(24)

where $k (X, u)$ is the integral reinforcement term, $Δ α (t) = α (X (t)) - α (X (t - T))$ , and the error bounded is $ε_{X} = ε_{X} (X (t)) - ε_{X} (X (t - T))$ .

To design the online algorithm to update the NN weights W, two variables $P_{1} \in R^{N \times N}$ and $Q_{1} \in R^{N}$ are introduced:

{\begin{matrix} {\dot{P}}_{1} = - ℓ_{1} P_{1} + Δ α Δ α^{T} (t), P_{1} (0) = 0 \\ {\dot{Q}}_{1} = - ℓ_{1} Q_{1} + Δ α (t) k (X, u), Q_{1} (0) = 0 \end{matrix}

(25)

where $ℓ_{1} > 0$ is the design parameter, therefore, the solution of equation (19) can be obtained as:

{\begin{matrix} P_{1} = \int_{0}^{t} e^{- ℓ_{1} (t - τ)} Δ α (τ) Δ α^{T} (τ) d τ \\ Q_{1} = \int_{0}^{t} e^{- ℓ_{1} (t - τ)} Δ α (τ) k (τ) d τ \end{matrix}

(26)

According to the evaluation NN, its estimated form is expressed as:

\hat{V} (X) = {\hat{W}}^{T} α (X)

(27)

where $\hat{V}$ and $\hat{W}$ are the estimates of V and W, respectively.

From the above analysis, the finite time adaptive law is derived as:

\dot{\hat{W}} = - Γ_{1} P_{1} \frac{M_{1}}{‖ M_{1} ‖}

(28)

where $M_{1} = P_{1} \hat{W} + Q_{1} \in R^{N}$ , $Γ_{1} > 0$ are the online learning gains.

Algorithm stability validation

Definition 1:

If there is a positive constant $σ > 0$ in the time range $[t - T, t]$ that satisfies the equation as follow:

\int_{t - T}^{t} Δ α (τ) Δ α (τ)^{T} d τ \geq σ I, \forall t > 0

(29)

Then $Δ α$ satisfy the Persistent excitation (PE) condition. This PE condition⁴⁷ is widely required in adaptive control to ensure parameter convergence.

Lemma 1:

If the signal $Δ α$ satisfies the PE condition at $t > 0$ , then the auxiliary variable $P_{1}$ is positive, that is, the minimum characteristic root will satisfy $λ 1_{\min}$ when $\forall t > 0$ .

The convergence of the adaptive law can be summarized as follows:

Theorem 1:

If the signal $Δ α$ satisfies the PE condition, the evaluated weight error $\tilde{W} = W - \hat{W}$ converges to zero in finite time.

Proof: 1

The first verification of the boundedness of $M_{1}$ : for equation (26) and bounded system states $X (t), X (t - T)$ , the upper bound of variable $P_{1}$ satisfies $λ 1_{1 \max}$ when $α_{1} > 0$ . Substituting equation (24) $k (X, u)$ into equation (26) deduces $Q_{1} = - P_{1} W + θ_{1}$ , which is bounded by $θ_{1} = \int_{0}^{t} e^{- ℓ_{1} (t - τ)} Δ α (τ) ε_{X} (τ) d τ$ due to the Bellman error $ε_{X}$ . Therefore, $M_{1}$ can be obtained:

M_{1} = - P_{1} \tilde{W} + θ_{1}

(30)

Since $Δ α$ satisfies the continuous incentive condition and $P_{1}$ is reversible, $P_{1}^{- 1} M_{1} = - \tilde{W} + P_{1}^{- 1} θ_{1}$ . Derivation of $P_{1}^{- 1} M_{1} = - \tilde{W} + P_{1}^{- 1} θ_{1}$ leads to:

\frac{\partial}{\partial t} (P_{1}^{- 1} M_{1}) = - \dot{\tilde{W}} + \frac{\partial P_{1}^{- 1}}{\partial t} θ_{1} + P_{1}^{- 1} {\dot{θ}}_{1} = \dot{\hat{W}} + θ_{s}

(31)

where $θ_{s} = - P_{1}^{- 1} {\dot{P}}_{1} P_{1}^{- 1} θ_{1} + P_{1}^{- 1} {\dot{θ}}_{1}$ is bounded due to $θ_{1}$ is bounded, that is, $θ_{s} \leq α_{2}, α_{2} > 0$ are positive constants. From the above analysis, it is clear that $P_{1}^{- 1}$ is bounded due to $λ_{1 2 \min}^{- 1}$ , and $λ_{1 2 \min}^{- 1}$ , hence, $P_{2}^{- 1}$ is bounded by $λ 1_{1 \min}$ and $λ 1_{\max}$ . Thus the Lyapunov function can be defined as:

V_{1} = \frac{R_{1}}{2} (P_{1}^{- 1} M_{1})^{T} Γ_{1}^{- 1} P_{1}^{- 1} M_{1}

(32)

where $R_{1} > 0$ is design parameter. Derivation $V_{1}$ with respect to time t yields:

{\dot{V}}_{1} = R_{1} M_{1}^{T} P_{1}^{- 1} Γ_{1}^{- 1} (\dot{\hat{W}} + θ_{s}) = R_{1} M_{1}^{T} P_{1}^{- 1} Γ_{1}^{- 1} (- Γ P_{1} \frac{M_{1}}{‖ M_{1} ‖} + θ_{s}) \leq - α_{3} \sqrt{V_{1}}

(33)

where $α_{3} = (σ - R_{1} α_{2} λ_{m a x} Γ_{1}^{- 1} \sqrt{2 / λ_{m a x} Γ_{1}^{- 1}}$ is the positive constant due to $0 < R_{1} < σ / (λ_{m a x} Γ_{1}^{- 1} α_{2})$ .

It is known that $R_{1} = 0, M_{1} = 0$ , in finite time $t_{1} = 2 \sqrt{R_{1} (0)} / α_{3} > 0$ , hence it can be obtained as $ε_{z} (z) \neq 0, M_{1} = 0$ for $ε_{1} (z) \neq 0$ , which shows that $\tilde{W} = P_{1}^{- 1} θ_{s}$ , hence $‖ \tilde{W} ‖ \leq α_{1} / σ$ is bounded in finite time $t_{1}$ . Ideally, when $ε (z) = 0$ , there are $ε_{z} (z) = 0, M_{1} = 0$ and $θ = θ_{s} = 0$ , which shows that $\tilde{W}$ converges to zero in finite time $t_{1}$ . From the above algorithm, it can be seen that the system dynamics $A (X)$ is not used in the implementation of the manipulator tracking control, which further relaxes the requirement of the system dynamics.

Numerical verification of the joint clearance compensation approach for manipulators

The simulation experiment

In this section, a two-degree-of-freedom manipulator model will be used for simulation verification. The dynamics of the manipulator system can be:

\begin{aligned} \dot{x} & = [\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] \\ = [\begin{matrix} x_{2} \\ - M^{- 1} C x_{2} - M^{- 1} G - M^{- 1} U - M^{- 1} F + M^{- 1} u \end{matrix}] \\ + c (t) \end{aligned}

(34)

M^{- 1} = [\begin{matrix} b & - b - c \\ - b - c & a + b + 2 c \end{matrix}] / (a b - c^{2})

(35)

C = [\begin{matrix} - 2 d {\dot{q}}_{2} & - d {\dot{q}}_{2} \\ d {\dot{q}}_{1} & 0 \end{matrix}]

(36)

G = [\begin{matrix} 0 \\ 0 \end{matrix}]

(37)

where $a = 0.613, b = 0.1173, c = 0.1584 \cos (q_{2}),$ $d = 0.1584 \sin (q_{2})$ .

To investigate the impact of joint clearance on manipulator trajectory accuracy, a simulation study was conducted using ADAMS, MATLAB Robotics System Toolbox and Simulink. A two-degree-of-freedom manipulator model with joint clearance was established. In this model:

The Joint Space Motion Model generates desired joint-space trajectories;

The Inverse Kinematics module converts Cartesian-space target positions into corresponding joint angles;

The Joint Actuator drives the manipulator's motion based on these commands;

The Rigid Body Tree constructs the manipulator's dynamic model; and

Tracking control is implemented through the designed clearance compensation controller.

The study focuses on observing the effect of a clearance-compensating controller on the trajectory performance of the manipulator's end-effector output after a specified trajectory is provided. The other simulation parameters are chosen as $ℏ = 9, Γ = 0.1, γ = 0.01, R = 1$ , $Q_{r} = d i a g ([\begin{matrix} 1003 & 1003 & 1003 & 10 \end{matrix} 03])$ . The evaluation network weight matrix W_a is initialized with small random values sampled from a Gaussian distribution N(0, 0.1) and updated online via temporal difference (TD) learning. The execution network weight matrix W_a is initialized to match the expected action range (e.g. corresponding to joint torque limits) and updated through policy gradient backpropagation. Both target networks employ a soft update mechanism with rate τ = 0.001. These parameter configurations ensure stable training convergence.

During the simulation, the regression vector for evaluating the NN is designed as $ϕ (x) = [x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}, x_{1 d}^{2}, x_{2 d}^{2}, x_{3 d}^{2}, x_{4 d}^{2}]^{T} / 2$ , and the reference trajectory is $x_{d} = [q_{1 d}, q_{2 d}, {\dot{q}}_{1 d}, {\dot{q}}_{2 d}]^{T}$ , where $q_{1 d} = 5 \sin (\sqrt{5} t)$ , $q_{2 d} =$ $5 \sqrt{5} \cos (\sqrt{5} t)$ , and ${\dot{q}}_{1 d} = q_{2 d}, {\dot{q}}_{2 d} = - 5 q_{1 d}$ . Parameter of the Manipulator model is shown in Table 1.

Table 1.

Parameter of the manipulator model.

$m_{1} (k g)$	$m_{2} (k g)$	$m_{3} (k g)$	$l_{1} (m)$	$l_{2} (m)$	Clearance (mm)	$μ$	$c (m)$	$b (N m / s)$	$k (N / m)$
9	2	0	0.25	0.25	0.1	0.29	5 × 10⁻⁴	1700	6.5 × 10⁷

To enable Adams-MATLAB co-simulation, the manipulator's mechanical model developed in Adams must be exported to MATLAB/Simulink as a functional subsystem. This integration establishes a co-simulation platform within Simulink, enabling coupled mechanical-control system analysis. The Adams/Control module functions as the bidirectional data interface, with nine critical variables required for parameter synchronization: three control torque variables, three joint position variables, and three angular velocity variables. The Joint simulation system is shown in Figure 4.

Figure 4.

Joint simulation system. (a) Mechanical subsystem and (b) mechanical subsystem structure.

During co-simulation, Adams utilizes its built-in solver to dynamically acquire torque commands computed by the control system. These real-time torque values actuate the manipulator's joints, while corresponding joint state variables (position and angular velocity) are concurrently fed back to the controller. This bidirectional data exchange creates a closed-loop control architecture that maintains precise joint positioning through continuous error compensation.

Figure 5 demonstrates the convergence characteristics of the evaluation NN weights. As shown, the network weights asymptotically approach constant values, which: (1) verifies the convergence properties of the adaptive law and (2) experimentally validates both the effectiveness and practical applicability of the proposed generalized policy learning algorithm.

Figure 5.

Evaluation neural network weight plot.

The manipulator's trajectory tracking performance is demonstrated in Figures 6 and 7. Figure 6 presents the position and velocity tracking results, confirming the effectiveness of the proposed method. Figure 6(a) to (b) compare the ideal trajectories of links 1 and 2 against both the clearance state and the compensated trajectories. The corresponding velocity profiles, with and without clearance compensation, are shown in Figure 6(c) to (d). The results clearly demonstrate that the proposed clearance-compensated control algorithm achieves: significantly reduced tracking errors, and markedly improved transient response speed compared to the joint clearance case.

Figure 6.

Mechanical arms trajectory tracking performance. (a) The position of link 1, (b) the position of link 2, (c) the velocity of link 1, and (d) the velocity of link 2.

Figure 7.

Manipulators tracking error.

Figure 6 demonstrates the effectiveness of the proposed compensation method in addressing frictional dynamics, particularly evident in the reduced tracking errors at maximum amplitude positions. As shown in Figure 7, this approach achieves minimal tracking error. The compensation mechanism effectively counteracts the joint clearance effects in the manipulator dynamics, as visible in Figure 6, enabling rapid convergence of the actual trajectory to the desired reference. This compensation strategy ultimately drives the tracking error to zero.

Figure 8 presents the bounded optimal control action developed in this study, exhibiting smooth trajectory characteristics. The results confirm that the proposed optimal control strategy successfully stabilizes the manipulator system with joint clearance, driving the tracking error to asymptotically converge to zero equilibrium.

Figure 8.

Action of optimal controller.

To further illustrate the superiority of the proposed algorithm, a linear feedback controller is further designed to compare with the optimal controller proposed in this article, and the linear controller is designed as $u = - G_{T}^{+} K X$ , where K is the feedback control gain, and $G_{R}^{+} = (G_{R}^{T} G_{R})^{- 1} G_{R}^{T}$ the generalized inverse matrix of $G_{R}$ .

A comparison between Figures 7 and 8 reveals that the linear feedback controller can only achieve approximate error convergence (to near-zero values), demonstrating its limited capability in handling complex systems. Furthermore, while the linear feedback control attains basic tracking performance, it fails to optimize the transient response. The corresponding joint driving torques for both control schemes are presented in Figure 9.

Figure 9.

Joint driving torque. (a) Driving torque of joint and (b) driving torque of joint with clearance compensation.

The simulation results demonstrate pronounced torque oscillations in the joint driving signals, clearly indicating that joint clearance significantly degrades the manipulator's control performance.

Experiment validation of manipulator

This section presents the experimental validation of the proposed joint clearance compensation control strategy using our laboratory's custom-designed manipulator's platform. The system configuration and experimental setup are illustrated in Figure 10.

Figure 10.

The structure of the manipulator. 1. Base; 2. servo motor of link 1; 3. harmonic reducer of link 1; 4. link 1; 5. joint 2 with clearance; 6. link 2; 7. load; 8. electric actuator of load; 9. harmonic reducer of link 2; and 10. servo motor of link 2.

The experimental platform's software architecture comprises an upper-level control system and a lower-level servo motion control system. The upper control computer performs inverse kinematics calculations to generate motion trajectory instructions for the robotic arm at planned path points, which it then transmits to the controller. Additionally, it receives and stores feedback parameters for manipulator's system error analysis and optimization. The controller serves as an intermediary: it processes instructions from the upper computer while simultaneously monitoring motor motion states relayed by the driver. By integrating these data streams and their associated errors, it dynamically adjusts target parameters and forwards them to the driver. At the lowest level, the driver executes servo motion control by converting the controller's angle and angular velocity requirements into voltage/current signals for motor actuation. Leveraging the ADP control algorithm, it continuously refines motor performance based on real-time encoder feedback.

The manipulator system is anchored through its base assembly. Joint 1's rotation is powered by a servo motor-harmonic drive assembly, with position feedback from a high-resolution integrated encoder. Joint 2 incorporates an internally mounted motor within the link 2, which transmits torque through a hinged shaft configuration and also employs a harmonic drive for precision motion control. The front-end load houses an electric linear actuator, with all cabling (power, encoder signals) routed internally through its hollow channels (Figure 11).

Figure 11.

Control system composition of the manipulator.

The manipulator's controller primarily consists of an STM32F407IG chip, a D/A conversion module, an analog voltage output module, and an encoder signal conversion module. The STM32F407IG chip is based on the ARM architecture with a clock frequency of 168 MHz. The D/A conversion module operates at 5 V and utilizes the AD5689 chip to output a voltage range of −10V～+10 V. This module includes an integrated operational amplifier driver, with a maximum output current of 20 mA, and communicates with the STM32F407IG control chip via SPI. The encoder signal conversion module converts the differential signals from the motor's built-in encoder into single-ended signals, which are then fed back to the STM32F407IG control chip for pulse counting. The TLP2745 optocoupler chip is employed to isolate the input and output signals, effectively preventing electromagnetic interference (EMI) on the encoder signals. Additionally, it enables high-speed conversion between differential and single-ended signals at a frequency of 2 MHz.

The inherent parameters of the manipulator are shown in Table 2. When the load is 1 kg, the learning gain is set as follows: $λ = 0.5, Γ = 0.1, γ = 0.01$ , and the initial position of the manipulator is: $q (0) = [0.1, 0.1]^{T}$ . After installing the experimental manipulator setup, dynamically adjust the joint clearance to 0.1 mm by pretightening the bolts on the conical bushing or eccentric sleeve assembly. Based on the established conditions, the proposed clearance compensation algorithm demonstrates effective trajectory optimization for manipulator systems operating with joint clearance size is 0.1 mm.

Table 2.

The inherent parameters of the manipulator.

Manipulator	Joint 1	Joint 2
Operation range	±135°	±135°
Length of link (m)	0.25	0.25
Center of mass distance from joint (m)	0.05	0.16
Weight of link (kg)	9.35	1.73
Rotational inertia (kg.m²)	0.117	0.065
Maximum load (kg)	/	1
Total weight of manipulator(kg)	/	14
Clearance(mm)	/	0.1

Figure 12 illustrates the manipulator output trajectory generated by the proposed backlash-compensated control algorithm. A comparative analysis between joint clearance state and compensated trajectories reveals that the latter substantially mitigates backlash-induced deviations and high-frequency jitter. This improvement is further quantified in Figure 13, where the compensated trajectory exhibits a reduction in tracking error compared to the joint clearance state baseline. These results empirically validate the efficacy of the proposed algorithm in refining control inputs, thereby enhancing trajectory-tracking precision.

Figure 12.

Manipulator position optimization. (a) The position of link 1 and (b) the position of link 2.

Figure 13.

Errors comparison. (a) The error of link 1 and (b) the error of link 2.

For quantitative assessment of the clearance compensation algorithm's effectiveness, the absolute values of tracking errors were computed to derive the Integral Absolute Error (IAE) metric, as shown in Figure 14. The experimental results demonstrate that the proposed clearance compensation control scheme achieves: a significant reduction in IAE (specify percentage if available) marked improvement in trajectory tracking precision and effective suppression of clearance-induced positioning errors. The IAE analysis conclusively verifies the algorithm's compensation capability, with the compensated system exhibiting superior tracking performance compared to joint clearance state operation.

Figure 14.

Integral absolute error.

As shown in Figure 14, the IAE of the manipulator q₁ and q₂ reduced by 54.2%, and 40.8% with clearance compensation optimization respectively compared to the joint clearance state, the results demonstrate the effectiveness of the proposed method in trajectory optimization for manipulator systems with actual joint clearance conditions.

Analysis and discussion

This study systematically examines the impact of joint clearance on the precision of end-effector trajectories in a two-degree-of-freedom robotic manipulator, using both numerical simulations and experimental validation. The experimental results quantitatively verify that the proposed generalized policy learning-based control framework with integrated clearance compensation: (1) effectively mitigates nonlinear clearance dynamics, (2) implements stable active compensation control, and (3) significantly reduces displacement errors induced by revolute joint clearance disturbances.

Comparative analysis of experimental and simulation results demonstrates that the adaptive clearance compensation control method effectively reduces motion trajectory errors induced by joint clearance in robotic arm articulation. Although both approaches validate the method's efficacy, certain discrepancies in control performance exist between simulation and experimental measurements, as summarized in Table 3. Comparative analysis of simulation and experimental results for the adaptive clearance compensation control method demonstrates consistent reduction in manipulator trajectory errors caused by hinge clearances, confirming the strategy's effectiveness. However, as quantified in Table 3, measurable discrepancies exist between simulated and experimental performance metrics.

Table 3.

Comparation of simulate and experiment results.

Indexes	Simulate values	Experiment values	Source of error
Maximum tracking error (mm)	0.37	0.82	Mechanical vibration
Stable state time (ms)	130	170	Motor response delay
Overshoot (%)	2.3	6.1	Difference of system inertia
RMSE (mm)	0.24	0.67	Actual joint friction is greater than in simulation

RMSE: root mean square error.

As shown in Table 3, after applying the ADP compensation control method, the end position of the manipulator in the clearance state exhibits a maximum tracking error of 0.37 mm in simulation and 0.82 mm in the experiment. The primary reasons for these errors are mechanical vibrations and stable state times of 130 and 170 ms, respectively, mainly due to motor response delays. The overshoot is 2.3% and 6.1%, and the corresponding overshoot distances are 0.61 and 1.83 mm, respectively. These discrepancies are mainly attributed to differences in system inertia, where the ideal values used in simulation do not account for the actual system's inertia. The root mean square error (RMSE) values are 0.24 and 0.67 mm, respectively. The increase in error is primarily due to the limitations of the actual encoder resolution, as well as the fact that the actual friction at the joints is greater than the simulated friction.

Both the experiment and simulation were conducted under the same articulation clearance and with identical size and weight parameters. After applying the ADP clearance compensation, the position error of the manipulator is <5%, and the velocity curves overlap by more than 95%. This fully verifies the effectiveness and robustness of the ADP clearance compensation method.

Conclusion

In this article, the impact of joint clearance on the end trajectory accuracy of a two-degree-of-freedom manipulator is investigated. A clearance compensation method leverages the augmentation and generalization system approach along with an ADP-based performance index function to simultaneously acquire feedforward and feedback control actions. An optimal controller is designed based on generalized policy learning for the manipulator's joint clearance compensation. An ADP-based finite-time online learning algorithm is developed, reducing the need for joint clearance system dynamics by directly updating the Bellman equations online without requiring complete system models. The simulation and experiment results demonstrate that the optimal controller effectively compensates for clearance dynamics, providing active compensation control. Unlike traditional algorithms, which rely on execution NNs, this method continuously updates adaptive evaluation and control actions simultaneously, without the need for iterative steps.

Future work will focus on enhancing the equivalent model of multi-degree-of-freedom manipulators with joint clearance to better address complex dynamic coupling in practical engineering applications. Particular emphasis will be placed on optimal tracking control under clearance-induced perturbations while accounting for hardware and environmental factors such as friction and thermal variations that may affect control performance.

Footnotes

Acknowledgements

The authors thank the editor and referees for the helpful comments and suggestions for improving the paper.

ORCID iDs

Wenting Liu

Zhiwen Wang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Shandong Province, National Natural Science Foundation of China (grant number ZR2024QE206, 52274132, 52474175, U23A20599).

Declaration of conflicting interests

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

Data availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

Jia

Chen

Zhang

, et al. Dynamic characteristics and reliability analysis of parallel mechanism with clearance joints and parameter uncertainties. Meccanica 2023; 58: 813–842.

Jin

Wang

Liang

, et al. Modeling and dynamics characteristics analysis of six-bar rocking feeding mechanism with lubricated clearance joint. Arch Appl Mech 2023; 93: 2831–2854.

Chen

. A nonlinear dynamic characteristic modeling method of shift manipulator for robot driver with multiple clearance joints. Nonlinear Dyn 2022; 110: 219–236.

Erkaya

Uzmay

. Modeling and simulation of joint clearance effects on mechanisms having rigid and flexible links. J Mech Sci Technol Aug 2014; 28: 2979–2986.

Kong

Nie

Wang

, et al. Product carbon emissions estimation method in the early design stage based on multi-perspective similarity matching of design scenarios. Adv Eng Inf 2025; 64: 103094.

Bai

Shi

Wang

. Effects of body flexibility on dynamics of mechanism with clearance joint. Mechaism Machine Sci 2017; 408: 1239–1247.

Chen

Jiang

Wang

, et al. Dynamics analysis of planar multi-DOF mechanism with multiple revolute clearances and chaos identification of revolute clearance joints. Multibody Sys Dyn 2019; 47: 317–345.

Sheng

Chen

. Dynamics behavior analysis of spatial parallel coordinate measuring mechanism with spherical clearance joints. Mechanics Industry 2024; 25: 1–19.

Sun

Wang

. Passive chaos suppression for the planar slider-crank mechanism with a clearance joint by attached vibro-impact oscillator. Mechaism Machine Theory 2022; 174: 1–23.

10.

Wang

Zhang

Wang

, et al. Research on spatial developable mechanism considering revolute clearance joints with irregular rough surfaces. Actuators 2024; 13: 274–283.

11.

Chen

Zhao

. Dynamic response investigation of high-speed structural redundant parallel manipulator considering multiple spatial revolute joints with radial and axial clearances. Mech Based Des Struct Mach 2024; 52: 6633–6659.

12.

Liu

Bai

. A study on clearance effects on dynamic responses of robot manipulator. Mechanika 2021; 27: 130–138.

13.

Jamshidi

Dardel

Daniali

. Clearance control of mechanism with sliding mode controller. Proc Inst Mech Eng Part C-J Mech Eng Sci 2024; 238: 9960–9996.

14.

Erkaya

Uzmay

. Balancing planar mechanisms having imperfect joints using neural networkgenetic algorithm (NN-GA) approach. In: Zhang

Wei

(eds) Dynamic balancing of mechanisms and synthesizing. Berlin: Berlin Springer, 2016, pp.299–317.

15.

Erkaya

Uzmay

. Determining link parameters using genetic algorithm in mechanisms with joint clearance. Mechaism Machine Theory 2009; 44: 222–234.

16.

Olyaei

Ghazavi

. Stabilizing slider–crank mechanism with clearance joints. Mechaism Machine Theory 2012; 53: 17–29.

17.

Muller

. Internal preload control of redundantly actuated parallel manipulators—its application to backlash avoiding control. IEEE Trans Robot 2005; 21: 668–677.

18.

Varedi

Daniali

. Dynamic synthesis of a planar slider–crank mechanism with clearances. Nonlinear Dyn 2015; 79: 1587–1600.

19.

Akhadkar

Acary

Brogliato

. Analysis of collocated feedback controllers for four-bar planar mechanisms with joint clearances. Multibody Sys Dyn 2016; 38: 101–136.

20.

Yang

Chang

Johnson-roberson

, et al. Energy-optimal control for autonomous underwater vehicles using economic model predictive control. IEEE Trans Control Syst Technol 2022; 30: 2377–2390.

21.

Jing

Wang

, et al. Robust H1 output-feedback control for path following of autonomous ground vehicles. Mech Syst Signal Process 2019; 70: 414–427.

22.

Lin

CZF

Zhao

, et al. Distributed drive electric vehicle stability control with mechanical elastic wheels. J Mech Eng 2022; 58: 236–243.

23.

Chen

Shuai

Zhang

, et al. Path following control of autonomous four-wheel-independent-drive electric vehicles via second-order sliding mode and nonlinear disturbance observer techniques. IEEE Trans Ind Electron 2021; 68: 2460–2469.

24.

Chen

Zhang

Pan

, et al. A time-varying second-order sliding mode control of super-twisting based on radial basis function neural networks. J Brazil Soc Mech Sci Eng 2025; 47: 282.

25.

Liu

Wang

Peng

, et al. PNNUAD: perception neural networks uncertainty aware decision-making for autonomous vehicle. IEEE Trans Intell Transp Syst 2022; 23: 24355–24368.

26.

Lakhekar

Waghmare

Roy

. Disturbance observer-based fuzzy adapted s-surface controller for spatial trajectory tracking of autonomous underwater vehicle. IEEE Trans Intell Vehicles 2019; 4: 622–636.

27.

Lewis

Vrabie

. Reinforcement learning and adaptive dynamic programming for feedback control. IEEE Circuits Syst Mag 2009; 9: 32–50.

28.

Song

Zhang

, et al. Dynamic research on winding and capturing of tensegrity flexible manipulator. Mechaism Machine Theory 2024; 193: 105554.

29.

Wang

Tian

. Nonsmooth spatial frictional contact dynamics of multibody systems. Multibody Sys Dyn 2021; 53: 1–27.

30.

Klemm

Viragh

Rohr

, et al. Nonsmooth trajectory optimization for wheeled balancing robots with contact switches and impacts. IEEE Trans Robot 2025; 41: 497–517.

31.

Wang

. Robust policy learning control of nonlinear plants with case studies for a power system application. IEEE Trans Ind Inf 2020; 16: 1733–1741.

32.

Shen

. Policy iteration approach to control residual gas fraction in IC engines under the framework of stochastic logical dynamics. IEEE Trans Control Syst Technol 2017; 25: 1100–1107.

33.

Lee

Park

Choi

. Integral reinforcement learning for continuous-time input-affine nonlinear systems with simultaneous invariant explorations. IEEE Trans Neural Networks Learn Syst 2015; 26: 916–932.

34.

Vrabie

Vamvoudakis

Lewis

. Optimal adaptive control and differential games by reinforcement learning principles. IEEE Control Syst Mag 2014; 34: 80–82.

35.

Vrabie

Lewis

. Neural network approach to continuous-time direct adaptive optimal control for partially unknown nonlinear systems. Neural Netw 2009; 22: 237–246.

36.

Werbos

. Approximate dynamic programming for real-time control and neural modeling . In: White

Sofge

(eds) Handbook of intelligent control neural fuzzy & adaptive approaches. New York: Van Nostrand Reinhold, 1992, pp.493–525.

37.

Vamvoudakis

Lewis

. Online actor-critic algorithm to solve the continuous-time infinite horizon optimal control problem. Automatica 2010; 46: 878–888.

38.

Zhang

, et al. Adaptive resilient event-triggered control design of autonomous vehicles with an iterative single critic learning framework. IEEE Trans Neural Networks Learn Syst 2021; 32: 5502–5511.

39.

Bhasin

Kamalapurkar

Johnson

, et al. A novel actor-critic-identifier architecture for approximate optimal control of uncertain nonlinear systems. Automatica 2013; 49: 82–92.

40.

Song

Gong

Zhu

, et al. Data-driven optimal tracking control for discrete-time nonlinear systems with unknown dynamics using deterministic ADP. IEEE Trans Neural Networks Learn Syst 2025; 36: 1184–1198.

41.

Liu

Zhao

, et al. Optimized impedance adaptation of robot manipulator interacting with unknown environment. IEEE Trans Control Syst Technol 2021; 29: 411–419.

42.

Zhou

Nie

, et al. Decentralized fault tolerant control of modular manipulators system based on adaptive dynamic programming. Int J Control Automat Syst 2022; 20: 3252–3263.

43.

Zhu

Dong

, et al. Adaptive dynamic programming-based sliding mode optimal position-force control for reconfigurable manipulators with uncertain disturbance. In: Presented at the Proceedings Of The 32nd 2020 Chinese Control and Decision Conference. Hefei, CHINA, 2020.

44.

Kamalapurkar

Dinh

Bhasin

, et al. Approximate optimal trajectory tracking for continuous-time nonlinear systems. Automatica 2015; 51: 40–48.

45.

Yang

. Adaptive optimal tracking control of unknown nonlinear systems using system augmentation. In: International Joint Conference on Neural Networks. 2016: 3516–3521.

46.

Modares

Lewis

. Optimal tracking control of nonlinear partially-unknown constrained-input systems using integral reinforcement learning. Automatica 2014; 50: 1780–1792.

47.

Mahyuddin

Herrmann

, et al. Robust adaptive finite-time parameter estimation and control for robotic system. Int J Robust Nonlinear Control 2015; 25: 3045–3071.

Optimal control of manipulator with joint clearance compensation via generalized policy learning

Abstract

Keywords

Introduction

Control method of joint clearance compensation for manipulators

Optimal tracking controller design

Algorithm stability validation

Numerical verification of the joint clearance compensation approach for manipulators

The simulation experiment

Experiment validation of manipulator

Analysis and discussion

Conclusion

Footnotes

Acknowledgements

ORCID iDs

Funding

Declaration of conflicting interests

Data availability

References