Development of a Wearable Arm Exoskeleton for Teleoperation Featuring with Model-Data Fusion to Gravity Compensation

Abstract

The upper-limb exoskeleton is ergonomically designed to align with human arm motion and can be configured for deployment as a master tool manipulator (MTM) in teleoperation systems. However, existing teleoperated exoskeletons are limited by excessive weight and inadequate force feedback. This study proposes a novel lightweight exoskeleton with optimized shoulder and wrist joint structure, enabling full arm mobility and sufficient force feedback. In practical applications, gravitational forces can lead to muscle fatigue and degrade teleoperation performance, making compensation essential for ergonomic and safety. However, unknown system disturbance caused by unmodeled dynamics (such as internal compliance and cables) pose challenges for compensation precision. A theoretical dynamics model and a Bayesian neural network (BNN) trained on separate datasets to predict joint torques and their corresponding uncertainties were independently developed. Then a Bayesian fusion method was employed to combine model-based and data-driven estimates, using predicted standard deviations to assign fusion weights and produce a refined torque output. Compared to relying solely on the CAD model, the proposed fusion framework combines the physical consistency of model-based approaches with the adaptability of data-driven methods. Experiments ultimately demonstrate that the proposed algorithm effectively reduces modeling errors and enhances the accuracy and robustness of gravity compensation.

1. Introduction

The upper-limb exoskeleton, as a representative electromechanical system, has been developed over decades and widely applied in rehabilitation, assistive devices, and teleoperation. The primary function is to cover the human limb, facilitating physical human–robot interaction (pHRI) through synchronized and coordinated motion with the multi-joint of the upper limb. This unique form of human–robot interaction offers distinct potential applications, including application in post-stroke rehabilitation [1,2,3,4], strength augmentation [5], and teleoperation [6,7]. However, it also presents substantial challenges, such as a profound understanding of human anatomy, motion control, and the necessity to avoid interference with the human body during motion [8].

The design configurations, actuation types, and control methods of upper-limb exoskeletons vary significantly depending on their intended applications [9]. This paper focuses on exoskeletons designed for bilateral teleoperation as the master manipulator, emphasizing lightweight design and force feedback. Existing master teleoperation manipulators such as Phantom from Geomagic Touch (serial configuration) and sigma.7 (parallel configuration) from Force Dimension are widely used. However, their lack of compatibility with human anatomy necessitates extensive training and makes them unsuitable for humanoid robots or large-scale operations due to unintuitive interaction. In contrast, wearable upper-limb exoskeletons provide a highly intuitive interface, facilitating accurate motion data capture and effective force feedback. Motivated by these attributes, a novel lightweight exoskeleton featuring seven active degrees of freedom (DOFs) designed for unilateral arm teleoperation is presented.

Nevertheless, master robots face physical constraints in practical application such as gravity affecting the performance of force feedback, leading to diminished transparency and ergonomic fatigue [10,11]. The inherent gravity of the master manipulator will also lead to fatigue and compromising operational effects. Consequently, gravity compensation is important to preserve the fidelity and usability of the exoskeleton master system during practical deployment.

Feedforward gravity compensation torque is primarily determined by the mass and center of mass of each link, which are typically determined through parameter identification techniques or derived from computer-aided design (CAD) models. However, discrepancies between theoretical models and the actual system dynamics—arising from nonlinear errors such as assembly tolerances, calibration inaccuracies, and the influence of external cabling—can significantly degrade the accuracy of gravity compensation. Furthermore, conventional identification approaches [12] typically require high-precision torque sensors to capture joint torques along complex identification trajectories, thereby increasing both experimental and computational costs. These requirements may also introduce potential experimental safety risk.

To address the constraints of traditional physics-based models, model-free approaches based on machine learning have emerged as compelling alternatives to capture the unknown nonlinearity of system disturbance. However, such methods are inherently sensitive to the quality and quantity of training datasets and often exhibit limited generalization capabilities in unstructured settings [13]. To fill this gap, a gravity compensation framework integrating a theoretical model with a Bayesian neural network (BNN) is proposed which combines physical constraints, analytical accuracy, and adaptive capability. This approach effectively mitigates model inaccuracies and generalization deviation, ensuring stable and robust performance.

Our expected contributions can be summarized as follows:

(1) A 7-DOF upper-limb exoskeleton for teleoperation, in which the double parallelogram mechanism (DPM) is adopted in the shoulder and wrist joints, was developed.

(2) To alleviate operator fatigue induced by prolonged wearing of the exoskeleton, a gravity compensation algorithm that fuses BNN predictions with the theoretical physical model is proposed.

(3) Constrained optimization is employed to confine the fusion deviations of the BNN within the feasible domain of the physical model, thereby ensuring control stability and the reliability of gravity compensation outcomes.

This paper is organized as follows: Section 2 reviews existing work on upper-limb exoskeletons and gravity compensation controllers. Section 3 details the proposed exoskeleton design and its gravity compensation algorithm. Section 4 presents experimental validation, and Section 5 concludes with remarks and future research directions.

2. Related Works

2.1. Upper-Limb Exoskeleton for Teleoperation

Research on upper-limb exoskeletons primarily addresses assistance and rehabilitation, with particular emphasis on coordinated motion [14,15] and compliant control [16] in interaction with the human upper limb. A particular interest in exoskeletons over the past decade has been their use as master haptic devices to capture human arm functions for teleoperation with humanlike workspaces. In addition, a lightweight ALEx exoskeleton was seamlessly integrated with a 3-DOF wrist robot [17] as a telepresence suit for intuitive teleoperation of the CENTAURO robot [18]. A wearable, lightweight, passive upper-limb exoskeleton was proposed for humanoid robots to replicate human operator movements [19]. An exoskeletal master device for dual-arm robot task-space teaching was developed in [20]. T-HR3 robot from Toyota is controlled via a master maneuvering system for intuitive full-body operation [21]. An exoskeleton capable of homogeneous and heterogeneous teleoperation was proposed in [7].

According to the aforementioned study, the joints of exoskeletons designed for both rehabilitation and teleoperation must be anatomically aligned with human joint axes to ensure ergonomic integrity. Nevertheless, teleoperated upper-limb exoskeletons remain distinct from rehabilitation-oriented systems. The following three design concept differences are identified and summarized as follows:

(1) While control algorithms can achieve gravity compensation and enhance transparency, high power-to-weight ratio and portability must be prioritized during the design phase.

(2) To encode the human upper-limb motion and enable Cartesian mapping of the pose of the slave manipulator, a teleoperated exoskeleton system requires at least 7 active DOFs and maximized coverage of the joint space of the human arm.

(3) The range of motion required for activities of daily living (ADL) and sufficient force feedback after compensating for gravitational torques needs to be supported.

2.2. Gravity Compensation for Robotic Manipulator

To ensure precision and stability, teleoperation with a wearable exoskeleton arm is typically performed at low speeds. In such conditions, gravitational torque generated by the mass of rigid links becomes the main source of actuation torque, far exceeding dynamic torques. Therefore, gravity compensation is essential for designing control algorithms in upper-limb exoskeleton teleoperation. Gravity compensation mechanisms are classified as active or passive based on energy use [22,23]. Passive gravity compensation mechanisms achieved by counterweights or springs take advantage of a lack of energy expenditure [24,25]. However, the inclusion of elastic springs or counterweights adds structural inertia, complexity, and nonlinear disturbances to the exoskeleton [26]. Therefore, active compensation is often combined to improve effectiveness [11,27].

PD-based feedback control has been proven to be a practical approach for gravity compensation with unmodeled external disturbances [28]. However, traditional PID control methods generally lack learning capability, often requiring integration with neural networks or fuzzy logic to enable parameter adaptation [29,30]. This limitation hinders their applicability in complex scenarios involving varying gravitational and operational conditions. Moreover, extending this method to multi-input multi-output (MIMO) systems remains nontrivial [30].

Gravity compensation can also be achieved through regression analysis by minimizing the discrepancy between the dynamic model and estimated parameters [31,32]. This relates the condition number of the regressor matrix to the reliability of the collected data. Excitation trajectories were designed to minimize error via the least squares method (LSM) [33]. The gravity compensation approach identifies the model term via geometry and torque, without relying on mass parameters [34]. This requires carefully designed excitation trajectories for accurate parameter estimation, which increases difficulty and potentially reduces experimental safety for complex robot systems [35]. Moreover, machine learning was developed, focusing on the correlation between joint torques and positions, without relying on precise dynamic parameters [36,37]. However, machine learning methods have high demands on both the quality and quantity of the dataset.

Building on prior gravity compensation research, this paper proposes a sensorless algorithm for exoskeleton robotic arms that enhances safety and reliability while reducing complexity. The approach first establishes the dynamic equilibrium equation then employs a BNN to learn the torque–position relationship from multiple joint space datasets, including uncertainty estimates. A Bayesian fusion framework integrates model-based computations with BNN predictions, weighted by their respective standard deviations, to produce a fused torque estimate. To address limited generalization under out-of-distribution (OOD) conditions, a physical model is introduced as a constraint to reduce bias and ensure stability. Compared to conventional approaches, the proposed method combines physical consistency with data-driven adaptability, mitigating modeling errors and significantly improving the precision and robustness of gravity compensation.

3. Methods

To address the deficiencies in the lightweight design of existing upper-limb exoskeletons, this section first delineates a comprehensive design methodology for our exoskeleton, followed by a detailed comparative analysis with the performance of state-of-the-art counterparts. Furthermore, the model-data fusion approach to gravity compensation for the exoskeleton is expounded in meticulous detail.

3.1. Exoskeleton Design and Overall Layout

3.1.1. Design and Comparison to State-of-the-Art Exoskeleton

The upper-limb exoskeleton, designed in accordance with anthropometric principles, features a configuration of 1 passive DOF and 7 active DOF to support sternoclavicular, shoulder, elbow, and wrist joint movements, respectively. Figure 1a illustrates the joint configuration of the right arm, while Figure 1b depicts the wearable exoskeleton on the user. The glenohumeral joint employs a planar six-link mechanism, described in [38,39], to enable internal/external rotation and forearm pronation/supination. However, these studies primarily emphasize shoulder kinematics, restricting detailed representation of wrist joint motions. To address this and ensure full-limb control, all joints are actively driven using motors, planetary gear reducers, and absolute encoders. Furthermore, for enhanced safety, adjustable mechanical limiters are incorporated at the shoulder and elbow joints.

The glenohumeral (GH) joint, commonly referred to as the shoulder joint, functions anatomically as a ball-and-socket joint. Three orthogonally intersecting revolute axes converge at a point coinciding with the anatomical center, as shown in Figure 2b. To avoid mechanical singularities and interference with the human body, a DPM composed of six links is employed so that the motor output axis Z₂₁ can be transmitted to the vertical axis Z₂ responsible for internal/external rotation with a 1:1 ratio. This is in contrast to the traditional capstan gears mechanism, which is lighter in weight and simpler in structure. Meanwhile passive SC joint motion in the horizontal plane is achieved by a parallelogram mechanism shown in green. All joints ensure precise alignment of the shoulder joint’s abduction/adduction and flexion/extension axes with the coronal and sagittal planes.

The elbow and wrist joints are integrated into the forearm segment, as illustrated in Figure 3a. The wrist joint emulates a ball-and-socket joint, employing a DPM akin to the shoulder design, as shown in Figure 3b.

The motor axis, which is posterior to the wrist Z₆₁, is transmitted to the sagittal axis Z₆ for Wr.pro/sup motion with a 1:1 ratio. To counterbalance gravity during Wr.rad/uln deviation, the motor of Z₇ located below the palm and transmits torque by bevel gears. Leveraging the human arm curvature, 3D printed upper arm and forearm cuffs were designed, ensuring a snug fit while preserving joint alignment. Finally, a Bluetooth data glove docking house is integrated at the end of wrist joint, serving as the third fixed connection to human arm.

Table 1. Comparison with state-of-the-art exoskeletons.

Joint	Motion Range (°) [40,41]			Speed (°/s) [7]
	ADL	MAX	Ours	ADL	MAX	Ours
Sh.abd/add	0–100	−30–150	0–90	30–50	130–170	288
Sh.int/ext	−50–65	−70–90	−42–50	30–50	100–140	228
Sh.flex/ext	0–110	−60–180	0–90	30–50	100–140	288
Elb. flex/ext	0–135	0–145	0–135	20–40	140–170	228
Wr.uln/rad	−15–40	−25–55	−25–55	30–60	410–480	207
Wr. pro/supi	−60–60	−90–90	−42–60	2–2.5	180–210	207
Wr. flex/ext	−60–35	−90–70	−90–50	30–60	140–240	207

compares the motion range and speed of activities of daily living (ADL) with the design parameters of our exoskeleton. Designed for teleoperation, the exoskeleton’s joint speed and motion range fully satisfy ADL requirements. A differential parallel mechanism (DPM) is also employed to minimize mass restrictions for Sh.ext rotation and Wr.pro motion ranges. Furthermore, considering the shoulder–humeral rhythm and operation focus below the shoulder horizontal plane and anterior to the coronal plane, Sh.abd and Sh.flex are constrained to 90°.

Table 2. Comparison of our exoskeleton with different exoskeletons [7].

Item	SAM	Harmony	Anyexo2	[7]	Ours
DOF	7	7	9	9	7 + 1
MLF 1	0, 2, 2, 3	2, 3, 1, 1	2, 3, 1, 3	2, 2, 2, 3	1, 3, 1, 3
Mass	7 kg	15.6 kg	12.9 kg	5.9 kg	6.9 kg
TWR 2	3.06	12.10	12.94	14.36	15.58

¹ Motor layout form, ² torque-to-weight ratio.

presents a comparison of key parameters for several state-of-the-art upper-limb exoskeletons. Despite incorporating shoulder–humeral rhythm, Harmony and AnyExo are hindered by excessive weight, compromising portability. In contrast, SAM and the exoskeleton in [7] suffer from limited joint torque, leading to inadequate force feedback after gravity compensation. The Sh.abd/add and Sh.flex/ext joints are actuated by GIM8108-36 motors (SteadyWin, Nanchang, China) with a nominal continuous torque of 35.10 N·m. The Sh.int/ext and Elb.flex/ext joints are actuated by GIM6010-36 motors (SteadyWin, Nanchang, China) delivering a nominal continuous torque of 18.00 N·m. The wrist joints are driven by AM-CL2242MAN brushless DC motors (Assun, Shenzhen, China) with a nominal continuous torque of 15.0 mN·m, coupled with an AM-PB-29:1 planetary gearbox. Concurrently, with a self-weight of 6.9 kg, our exoskeleton achieves the highest continuous torque-to-weight ratio among comparable systems. Human joint torque and state-of-the-art teleoperation exoskeletons are compared in Figure 4 and parts (a)–(c) below, which demonstrate the effects of the wearable limit angles of the exoskeleton manipulator, exhibiting excellent joint comfort and a sufficient workspace.

Beyond gravity compensation, joint torque needs to provide perceptible force feedback. Consequently, shoulder and elbow joints were designed to closely match ADL motion ranges while delivering torque significantly below isometric levels. The wrist joints benefiting from structural gravity balance employ 0.3 Nm motors, sufficient to ensure noticeable force feedback.

3.1.2. Kinematics Modeling

The exoskeleton with the shoulder, elbow, and wrist joints is configured as spherical–revolute–spherical (SRS) kinematic pairs. The shoulder width (l_b) and lengths of the upper arm (l_c), forearm (l_d), and palm length, illustrated in Figure 2 and Figure 3, can be adjustable by passive prismatic joints, ensuring precise alignment with human joint axes for enhanced ergonomic performance. The distance between two SC joints is set at a fixed value of 50 mm, and the distance between the SC joint and GH joint on the sagittal plane, noted as a fixed value, l_a = 42 mm. Referring to [42], the operator’s height range of 165–185 cm is incorporated into the design specifications. The dimensions are specified as l_b = 175 mm, l_c = 265 mm, and l_d = 315 mm in the design specifications according to the operator body dimensions of this study. The kinematic model was developed to the base coordinate Z₀ with modified Denavit–Hartenberg (DH) parameters, as presented in

Table 3. Modified DH parameters of our exoskeleton.

Link i	θ	d	a	α	Offset
1	θ1	la	lb	−pi/2	0
2	θ2	0	0	pi/2	pi/2
3	θ3	0	0	−pi/2	pi/2
4	θ4	0	lc	0	0
5	θ5	0	ld	0	−pi/2
6	θ6	0	0	pi/2	pi/2
7	θ7	0	0	pi/2	0

, and the kinematic equation can be established based on the coordinate homogeneous matrix T:

$$p_{c_i}\left({\theta }_1,..,{\theta }_i\right)=\left({\displaystyle \prod _{k=1}^i{}^{k}T_{k-1}}\left({\theta }_k\right)\right)r_i={\left[x_{c_i}{y}_{c_i}{z}_{c_i}1\right]}^T$$

(1)

where ${}^{k}T_{k-1}\left({\theta }_k\right)$ is the homogeneous transformation matrix from frame i − 1 to frame i, and r_i is the end effector of link i represented in frame i.

3.2. Gravity Compensation

3.2.1. Modeling-Based Gravity Compensation

Through the application of the Euler–Lagrange approach, the compact dynamic equations for the exoskeleton manipulator can be derived as follows:

$${\tau }_j={\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial K}{\partial {\dot{\theta }}_j}}\right)-{\displaystyle \frac{\partial K}{\partial {\theta }_j}}+{\displaystyle \frac{\partial P}{\partial {\theta }_j}}$$

(2)

where K and P represent total kinetic energy and total potential energy, respectively, as follows:

$$\begin{array}{l}K={\displaystyle \sum _{i=1}^3\left(\frac{1}{2}{m}_i{v}_i^T{v}_i+\frac{1}{2}{w}_i^T{R}_i{\tilde{I}}_i{R}_i^T{w}_i\right)}\\ P={\displaystyle \sum _{i=1}^3{m}_i}{g}^T{x}_{cj}\end{array}$$

(3)

where m_i and v_i denote the mass and the velocity of the i-th rigid body. ω_i and R_i denote the angular velocity and rotation matrix of the i-th rigid body. ${\tilde{I}}_i$ denotes the inertia tensor matrix in Global Coordinate System.

Teleoperation systems typically function at low velocities, where the gravitational torque induced by the mass of the rigid link substantially predominates over dynamic torques. Consequently, the dynamic characteristics can be effectively approximated using a static model [35]. Note that gravity compensation for the wrist is omitted as its linkage has minimal mass, and balanced joint motors 6 and 7 counteract gravitational effects in pitching. Meanwhile, to simplify the model, gravitational compensation is considered only for the shoulder and elbow joints. During the experiment, the motors of wrist joints are controlled to maintain at zero position.

The gravity torque on each joint arising from the contributions of subsequent robotic links in the kinematic chain and is modeled by determining centroid position of each link through coordinate transformations. The centroid position of link j relative to the link coordinate Z_j, denoted ^jx_cj, is achieved by applying the homogeneous transformation matrix ⁰T_j to the local frame ⁰x_cj, ^jx_cj = ⁰T_j·⁰x_cj. The relationship of gravity torque of a single joint is implemented as

$$G_j={\displaystyle \sum _{k=j}^4{}^{j}x_{cj}}\times m_{k}g$$

(4)

Joints are categorized as either active or passive depending on whether they are actuated by motor-generated torture. Z₁~Z₇ denote the Z-axis of equivalent active joint coordinates, while the link masses of the passive mechanism DPM are represented by m₂~m₆, and Z₂₁~Z₂₅ correspond to the respective coordinate systems. τ_j as a function of gravity torque is ultimately formulated in terms of the active joint angle θ_j as

$${\tau }_j\delta {\theta }_j-\left({\displaystyle \sum _{j=1}^i{m}_{i}g\frac{d}{d{\theta }_j}\left(z_i\right)\delta {\theta }_j}\right)=0$$

(5)

Leveraging the mass and centroid of each link determined from SolidWorks 2024 SP5.0 3D model is shown in

Table 4. Mass and centroid of each link for the exoskeleton.

Link Index	Link Mass (g)	Relative Coordinate	Centroid of the Link
			xci (mm)	yci (mm)	zci (mm)
1	1206	Z1	26.46	103.30	−173.93
2	88	Z21	−22.593	−103.65	−69.87
3	218	Z22	−23.10	−121.70	−68.03
4	175	Z23	86.29	9.56	−83.72
5	130	Z24	61.87	17.16	−89.28
6	1594	Z25	32.46	43.89	−10.07
7	1697	Z3	284.28	0.04	−143.67
8	1814	Z4	248.192	32.693	−72.03

. The final relationships for each joint have been simplified using trigonometric functions and are shown in Appendix A.

3.2.2. BNN-Based Gravity Compensation

To estimate joint gravity torque with the associated uncertainties, a Bayesian neural network (BNN), which comprises an input layer, hidden layers, and output layers, was trained across multiple datasets. Based on the kinematic model, initial zero positions of each joint were defined as θ_0,j, with max motion range of S_j = {90°, 80°, 90°, 120°} for shoulder and elbow joints. Target angles P_j were discretized in 10° increments for each joint, finally yielding 11,200 datasets. While a smaller dataset simplifies collection process, it may reduce training accuracy and model generalization.

Motors drivers are operated in Cyclic Synchronous Position (CSP) mode, starting from zero position θ_0,j and moving to target position P_j step by step. After reaching the target, a 3 s delay ensures dynamic equilibrium conditions. Then joint angles θ_exp,j and driver currents value I_exp,j are subsequently recorded in real time by the EtherCAT bus over 1000 cycles. Finally, the dataset undergoes mean filtering, normalization, and outlier removal to yield the processed values of measurement joint angle and driving current.

The BNN was trained to predict gravity compensation driving currents and variance uncertainty with 4 active DOFs. The input feature vectors consist of experimentally measured joint angles θ_exp,j and corresponding gravity compensation driving currents I_exp,j, yielding an input dimension of Nin = 2D = 8. The model is optimized to maximize the Evidence Lower Bound (ELBO), which balances the expected data log-likelihood and model regularization. The ELBO is formulated as

$$\mathrm{ELBO}={\mathbb{E}}_{q_{\varphi }(w)}[\log p(\mathcal{D}|w)]-\beta \cdot \mathrm{KL}(q_{\varphi }(w)||p(w))$$

(6)

where the former term represents the expected log-likelihood of the data $\mathcal{D}$ under the variational posterior $q_{\varphi }(w)$, and KL is the Kullback–Leibler divergence between $q_{\varphi }(w)$ and the prior p(w)∼N(0, 1), scaled by a hyperparameter β to balance model complexity and data fidelity. The KL divergence is computed analytically as

$$\mathrm{KL}(q_{\varphi }(w)||p(w))={\displaystyle \sum _i\left(\frac{1}{2}\left({\mu }_i^2+{\sigma }_i^2-1-\log {\sigma }_i^2\right)\right)}$$

(7)

where q_ϕ(w) = N(μ,σ) parameterizes the variational posterior with mean μ and variance σ². p(w)∼N(0, 1) is the standard normal prior. The reparameterization method, w(i) = μ + σ⋅ϵ with ϵ∼N(0, 1), enables sampling of weight w(i) for stable gradient computation. The super parameter β is tuned during training to ensure effective regularization.

Training was performed using the Adam optimizer with a learning rate of 0.001 over 100 epochs and a batch size of 128. An 80:20 train–test split was employed to assess the generalization performance of the model. The output layer consists of two parallel fully connected sub-networks: one predicts the mean current μ_j = I_nn,j for each joint j, and the other estimates the logarithm of the standard deviation logσ_j, with σ_j = exp(logσ_j) ensuring positive standard deviations and stable gradients. The predicted current I_nn,j and its uncertainty σ_nn,j provide both a mean estimate and a confidence measure, facilitating precise gravity compensation in robotic control applications. The detailed architecture of the BNN used in this work is shown in Figure 5.

3.2.3. Model-Data Bayesian Fusion

The theoretical model correlates torque with joint angle, while the BNN fits current values from experimental data requiring normalization of torque and current for consistent analysis. Under static equilibrium conditions, the motor torque and current exhibit a linear correlation with respect to the motor torque constant K_t (Nm/A). Accordingly, the fitted torque for joint j, denoted as τ_nn,j is expressed as τ_nn,j = K_t,j × I_nn,j, where τ_nn,j denotes the fitted torque, K_t,j is the motor torque constant, and I_nn,j is the current value acquired from experiment. Thereby, the fitted outputs from the BNN (expressed in unit current) can be unified with the theoretical model computation values (expressed in unit torque).

Meanwhile, to quantify model uncertainty, the standard deviation σ_m,j of joint j is determined by experimental current measurements I_exp,j. The standard deviation σ_m,j for joint j is derived as

$${\sigma }_{m,j}=\sqrt{{\displaystyle \frac{1}{b}}{\displaystyle \sum _{i=1}^n{(e_{j,i}-{\overline{e}}_j)}^2}}$$

(8)

where b is the total number of samples, ${\overline{e}}_j$ is the mean residual, and e_j,i is the residual of the i-th sample for joint j. Subsequently, the mean torque τ_nn,j and its associated uncertainty σ_nn,j derived from the BNN output layer can be integrated with the theoretical torque τ_m,j and σ_m,j through the Bayesian fusion. The detailed calculation results for σ_m,j are presented in Section 4.3. The fused torque τ_f is then calculated as a weighted average of the BNN outputs and theoretical model, with weights inversely proportional to their respective standard deviations, satisfying w_nn + w_m = 1.

$$\begin{array}{l}{\tau }_{f,i}={\omega }_{nn,i}\cdot {\tau }_{nn,i}+{\omega }_{m,i}\cdot {\tau }_{m,i},\\ {\omega }_{nn}={\displaystyle \frac{1/{\sigma }_{nn}^2}{1/{\sigma }_{nn}^2+1/{\sigma }_m^2}}{\omega }_m={\displaystyle \frac{1/{\sigma }_m^2}{1/{\sigma }_{nn}^2+1/{\sigma }_m^2}}\end{array}$$

(9)

where ω_nn and ω_m denote the computational weight represented by the inverse of the standard deviation of BNN and model calculation, respectively. To ensure the fusion result aligns with physical laws, the final fusion torque is constrained within a range utilizing σ_m as a probabilistic safety margin to prevent excessive deviation that could cause injury to the operator’s arm. The result τ_f defined by the theoretical model deviation is proposed as τ_f ∈ [τ_f − k·σ_m, τ_f + k·σ_m], where k is the variable constraint coefficient. The compensated current is derived from τ_f and compared with the theoretical torque. If error |τ_f − τ_m| ≤ σ_m, τ_f is directly adopted as the final output. The final fused current is formulated as follows.

If the fused estimate torque τ_f exceeds the permissible range, it is restricted to the boundary value. The sign function ensures that the corrected current aligns with the trend predicted by the BNN while remaining within the uncertainty bounds of the theoretical model, thereby optimizing both estimation accuracy and operational reliability.

A logic diagram of the process of the algorithm was constructed, as shown in Figure 6, to present the details and logic of the proposed model-data fusion for gravity compensation of the exoskeleton, and the pseudocode for the proposed algorithm is summarized in Appendix B.

4. Experiment Results

We conducted BNN training on the collected data to validate the effectiveness in fitting data. Subsequently, we calculated the root mean squared error for each joint of the CAD model method to support Bayesian fusion. Finally, trajectory experiments were performed to verify the efficacy of our gravity compensation method and obtain the final experimental results. The test joint trajectory covered most of the exoskeleton operational workspace, defined as all possible configurations within the joint range. The results of Bayesian fusion are compared and contrasted with those obtained from the CAD model and the standalone BNN.

4.1. System Implementation

The control system implementation, as depicted in Figure 7, incorporates an industrial local computer, integrated with a Preempt-RT real-time system for the EtherCAT IGH bus to enable precise real-time motion control. This system interfaces with the active joint motors, which are serially connected and actuated via Elmo drivers over the EtherCAT bus. Operating at a 1 kHz signal acquisition rate for low-level communication (including control word substitution) with a 50 Hz upper-level control (including forward kinematics calculation and Bayes fusion for compensation) enables real-time and reliable gravity compensation control.

The hardware model of the local computer is ARK-3530F from Advantech with an Intel Core i7-7700 CPU @ 3.60 GHz, 16 GB RAM, and NVIDIA GeForce GTX 2070 GPU board. The motor driver from Elmo model POLTWI10/100 employs the full-duplex EtherCAT bus and supports the Service Data Object (SDO) protocol to access and manage the Object Dictionary of CANopen devices. The transmission of configuration data, including torque current values and joint position information, is critical for reconstructing the gravity model. This enables precise real-time and accurate data acquisition.

4.2. Training on the BNN Dataset

The Bayesian approach can enhance stable convergence by modeling weight uncertainty, further effectively balancing expressiveness. The BNN model was trained using a dataset obtained as described in Section 3.2.2 and split into 80% for training and 20% for validation, following standard practice to ensure robust model evaluation. As shown in Figure 8, the training and validation loss curves exhibit a steep decline over the first 50 epochs, indicating rapid learning and effective optimization as the model quickly captures the underlying patterns in the data. This initial phase reflects the BNN’s ability to adjust its weight efficiently, minimizing prediction errors across both datasets.

Over the subsequent 100 epochs, the loss curves transition into a convergence phase with gradual, marginal improvements, ultimately stabilizing at minimal values. The model shows robust optimization and stability, with aligned training and validation losses indicating minimal overfitting.

As shown in Figure 9, the X-axis represents the actual current values, and the Y-axis represents the predicted current means for joint 1–4. Each data point represents the actual value of a test sample plotted against its predicted mean value. The red dashed line denotes the ideal prediction line, where perfect alignment indicates optimal accuracy. Blue points clustered closely around this line signify higher prediction accuracy, thereby demonstrating the robust and reliable predictive performance of the model across the evaluated dataset.

4.3. RMSE Analysis of Compensation Current Variability

To further quantify the variability in compensation currents, the root mean squared error (RMSE) between the measured and model calculation results was computed for each joint. The RMSE is defined as Equation (10), where I_cal,i denotes the CAD model calculation values for subset i, ${\stackrel{\wedge }{I}}_{meas,i}$ denotes the vector of actual observed values, and ||·|| denotes the Euclidean norm. The 11,200 datasets obtained as described in Section 3.2.2 were used for computing the RMSE values of each joint CAD model. Actuator-rated currents are 7 A for joints 1 and 2 and 5 A for joints 3 and 4. Unit current is the thousandth ratio of rated current, at 7‰ (A) for joints 1 and 2 and 5‰ (A) for joints 3 and 4. Joints 1 and 3 and joints 2 and 4 utilize identical motor models, with respective motor torque constants given by K_t = [6.52, 3.75, 6.52, 3.75, 6.02]. Thereby, the motor driving current and output torque can be interchangeably calculated.

Mean current values and standard deviations are compared for each joint in Figure 10. For joints 1 and 2, the RMSE values are relatively high, while joint 3 exhibits a slightly lower RMSE. The elbow joint (joint 4) shows the lowest RMSE, indicating high accuracy in current predictions. The experimental results demonstrate that the RMSE increases toward the base (Z₀), indicating that cumulative errors from assembly imperfections, mass estimate inaccuracies, or unmodeled dynamics cause greater current variability in proximal joints.

$${\sigma }_m={\displaystyle \frac{1}{\mathrm{n}}}{\displaystyle \sum _{i=1}^n\left|\right|I_{cal,i}-{\stackrel{\wedge }{I}}_{meas,i}\left|\right|}$$

(10)

While joints 1 and 2 show higher σ_m, suggesting limited accuracy of the CAD-based compensation method, the calculated values consistently align with the experimental results. This consistency underscores the computational model stability, affirming its suitability for the gravity compensation fusion algorithm, thus supporting reliable control.

4.4. Trajectory Test for Compensation and Evaluation

In the trajectory test experiment, the exoskeleton moved along a joint space trajectory, going through ten randomly chosen joint configurations using a position controller. Figure 11 shows the desired joint trajectory. Ten arbitrary joint combinations within the joint space range were randomly selected. Employing the final cycle as an example, the joint arrives at the desired configuration within 1 s, as delineated by the yellow moving area. Following a 2 s interval, it transitions to the steady state, as illustrated by the gray region, and this steady state is maintained for 5 s while high-precision measurements of joint angle and current values are continuously recorded. Output torques were computed using the CAD, BNN, and fusion methods and compared against the measured torques (see Figure 12).

In Figure 12, the blue curve represents the model-based calculated value, the green curve indicates the BNN fitting values, and the red curve depicts the fused values obtained using the algorithm proposed in this paper. The difference between the BNN and fused values is illustrated by the red shaded region. Figure 12 shows that the fusion algorithm improves the results for joints 1, 2, and 3 using standard-deviation-based refinement, as indicated by the red shaded regions. The absence of deviation in joint 4 further confirms the accuracy and consistency of both the analytical method and the BNN for the elbow joint. The BNN-fitted currents (green curves) are closer to the measured values because the BNN was trained directly on experimental data. However, the fused values are intentionally more conservative. The fusion approach helps prevent overfitting associated with the BNN and ensures that the output currents for gravity compensation remain safe and physically consistent. Consequently, the incorporation of the fusion algorithm proves both methodologically necessary and scientifically valid.

To further compare the three methods while reducing the impact of torque magnitudes on RMSE, the relative root mean square error (RRMSE) was used, as shown in Equation (11), and the results for ${\epsilon }_{RM{S}_i}$ (%) are presented in

Table 5. RRMSE between the measured and joint torques in the trajectory test.

Method	Joint 1	Joint 2	Joint 3	Joint 4
CAD	7.75	8.25	5.04	9.97
BNN	2.80	3.27	2.90	9.45
Fused	5.32	5.60	3.61	9.45

 $W_i\widehat{\beta }$ denotes the predicted values for subset i, and ${\omega }_i$ denotes the vector of actual observed values. Meanwhile, the maximum absolute errors are computed and shown in

Table 6. Maximum absolute error (Nm) between the measured and joint torques in the trajectory test.

Method	Joint 1	Joint 2	Joint 3	Joint 4
CAD	1.4883	0.9676	00.5250	0.4368
BNN	0.3985	0.3989	0.3846	0.3746
Fused	0.8891	0.7165	0.4346	0.3746

. Note that due to significant fluctuations in experimental current values during the motion and waiting phases (black curve in Figure 12), only steady-state torques corresponding to the specified configuration were used for computation.

$${\epsilon }_{RM{S}_i}(\%)=‖W_i\widehat{\beta }-{\omega }_i‖/‖{\omega }_i‖\cdot 100\%$$

(11)

Table 5 and Table 6 show that compared to traditional CAD-model-based methods, both the BNN and fusion approaches achieve higher compensation accuracy. Although the BNN fitted results closely align with true computational outcomes, the fusion method may slightly reduce gravity compensation accuracy. To ensure experimental stability and safety by preventing divergence in BNN fitting, the fusion approach is necessary and effective. Nonetheless, it is difficult to assess the BNN results, which will critically affect the gravity compensation value, unless a smaller dataset is used or the motion range of the exoskeleton exceeds the effective dataset range, causing BNN input values to fall outside the pre-training dataset.

5. Conclusions

In this paper, an upper-limb exoskeleton is proposed for teleoperation, incorporating double parallelogram spherical mechanisms at the shoulder and wrist joints. The exoskeleton kinematic design prioritizes alignment with human joint axes and ensures interaction safety. Furthermore, compared to existing upper-limb exoskeletons, this design achieves the highest continuous torque output-to-weight ratio, enabling sufficient feedback torque after self-gravity compensation. In addition, to address gravity compensation, we developed a novel approach integrating BNN pretraining with a CAD model calculation, which can effectively mitigate the limitations of CAD model accuracy due to nonlinear disturbances and the overfitting of the BNN approach through Bayesian fusion. Experimental validation demonstrates that our method significantly outperforms standalone model-based calculations and feedforward neural network approaches in terms of precision, stability, and reliability.

Experimental validation conducted on 11,200 collected datasets demonstrates that the proposed fused approach significantly outperforms both the standalone CAD-model-based method and the pure BNN method in terms of precision, stability, and reliability. Relative to the CAD-only method, the fused approach reduces the relative root mean square error (RRMSE) by 31.2% (joint 1), 32.1% (joint 2), 28.4% (joint 3), and 5.2% (joint 4), with maximum absolute error (MAE) reduced by an average of 26.9%. Although the pure BNN method achieves the lowest absolute errors (0.3887 Nm versus 0.8544 Nm for CAD and 0.6037 Nm for the fused method), the fused approach intentionally adopting conservative predictions is necessary and effective. This deliberate trade-off effectively prevents overfitting and extrapolation divergence, ensuring that maximum compensation errors remain well below a safe threshold across the entire workspace, thereby significantly enhancing the reliability and operational safety of real-time gravity compensation.

In future work, we aim to deploy the exoskeleton system across a broader range of teleoperation scenarios and leverage data collected from these operations to advance research in imitation learning.