Active Transfer Learning Gaussian Process for Reliable Trajectory Prediction of the UR5 Robotic Manipulator ^†

Abstract

This paper presents a simulation-driven framework employing an Active Transfer Learning Gaussian Process (ATGP) model for accurate trajectory prediction and reliability analysis of the UR5 robotic manipulator. The method integrates transfer learning, Gaussian Process Regression, and active sampling to address challenges under limited target data. Preprocessing steps such as outlier removal, feature scaling, and Principal Component Analysis enhance data quality. A physically informed synthetic source domain facilitates effective knowledge transfer. Using DH-parameters as input, the ATGP predicts 3D end-effector trajectories over time. Results show a mean absolute error below 0.01, demonstrating consistency and scalability for real-time, uncertainty-aware robotic applications. This is the first ATGP-based UR5 framework that unites PCA-guided, physics-informed source synthesis with multi-output transfer GPR as well as coordinate- and time-resolved reliability analysis under scarce target-domain data.

1. Introduction

Precision trajectory prediction in robotic manipulators is essential for high-stakes applications such as aerospace assembly [1], surgical automation [2], and industrial machining [3]. The UR5 manipulator is widely adopted among commercially available systems due to its flexibility, six degrees of freedom, and compact architecture [4]. However, achieving reliable trajectory estimation under uncertainty remains a major challenge, especially when real-world data is limited, noisy, or high-dimensional.

Gaussian Process Regression (GPR) has become increasingly applicable in the context of robotics, as it is probabilistic with nonlinear mapping guided uncertainty quantification [5,6,7]. However, conventional GPR has inherent scaling issues and lower data-sparse data accuracy [8]. This has led to transfer learning and active learning within GPR frameworks to enhance both generalization and sample efficiency [9,10].

Transfer learning provides a means to accomplish a similarity-based transfer of knowledge between related source domains and target tasks when the target has limited data and has found success in robot manipulation [11], inverse dynamics modeling, and grip adaptation [12]. The alternative active learning approach, conversely, probes only the most educative samples to accelerate the learning process [13,14]. The Active Transfer Gaussian Process (ATGP) model utilizes both strategies. In this strategy, models are empowered to synthesize on top of the source data and actively distill their predictions in the target domain [15].

In trajectory modeling, a variety of techniques have been adapted, including neural networks [16], support vector machines [17], and ensemble models [18]. Nevertheless, such methods lack interpretability or bounds of uncertainty. Surrogate models based on Gaussian Processes, including Kriging and multi-output GPR, have been applied for kinematic reliability analysis [19], but few have addressed transfer-based learning under structured preprocessing pipelines [20]. Despite advances in transfer-learning-based GPR, three key gaps remain: (1) no existing framework for UR-series manipulator trajectory prediction that learns from high-dimensional, time-indexed DH parameters under limited target-domain data; (2) absence of robust preprocessing (Modified Z-score outlier removal, median–IQR scaling) combined with PCA-guided, physics-informed synthetic source domain generation; (3) lack of coordinate- and time-resolved, uncertainty-aware reliability metrics alongside accuracy. This is the first ATGP-based UR5 framework that unites these elements into a scalable, real-time capable, uncertainty-aware trajectory prediction solution. This paper proposes a simulation-based framework that leverages ATGP for trajectory prediction of the UR5 manipulator. The key features of this framework include the following:

• Robust preprocessing via outlier removal and robust scaling.
• Dimensionality reduction using Principal Component Analysis (PCA).
• Physically grounded source domain generation using simplified kinematic reasoning.
• Multi-output ATGP training and reliability visualization.

The technique offered reaches an accuracy of sub-millimeter and is excellent at generalizing with a small amount of training data. It is more reliable with respect to uncertainty awareness, interpretable, and predictive results compared to conventional GPR and neural methods, making it a good option for robots where high reliability is desired.

2. Methodology

2.1. Problem Definition

This work aims to predict the 3D end-effector trajectory of the UR5 robotic manipulator using uncertain (DH) parameters as input. Each input sample consists of 150 features (15 DH parameters across $10$ time steps), while the output is a 30-dimensional vector representing the $3\mathrm{D}$ positions at those time steps.

The issue is the high-dimensionality and few target data, and the need for accurate and uncertainty-aware predictions. This is tackled by utilizing an Active Transfer Learning Gaussian Process (ATGP) model, which maps DH parameters to trajectory outputs utilizing a small target dataset and a physics-informed, synthetic, source domain of data.

2.2. Data Generation and Description

This data is created in a simulation with the help of the UR5 manipulator as a forward kinematics model. Each of the samples encompasses 10 × 15 matrix of (DH) parameters that represent joint configurations at $10$ time steps. These are reshaped into $150$ dimensional input vectors.

The corresponding outputs are 10 × 3 matrices containing the 3D end-effector positions, reshaped into 3-dimensional vectors. In total, 550,000 trajectory samples are generated to capture the variability in the manipulator’s behavior under uncertain conditions. A subset of these samples is selected for model training and testing, enabling efficient evaluation of the ATGP framework in a high-dimensional, data-rich simulation setting.

2.3. Preprocessing and Feature Normalization

A three-stage preprocessing pipeline is applied to enhance data quality and learning stability:

Outlier Detection and Removal: Outliers in both input and output data are identified using the Modified Z-score method:

$$Z_i=0.6745\cdot {\displaystyle \frac{X_i-median\left(X\right)}{MAD\left(X\right)}}$$

where $MAD\left(X\right)=median(\mid X_i-median(X)\mid ).$ Samples with $\mid Z_i\mid >3.5$ are considered outliers and removed.

Robust Feature Scaling: To normalize features while preserving robustness, median and interquartile range (IQR) are used instead of mean and standard deviation:

$$X_{scaled}={\displaystyle \frac{X-median\left(X\right)}{IQR\left(X\right)}}$$

where $IQR\left(X\right)=Q_{75}\left(X\right)-Q_{25}\left(X\right).$ This reduces sensitivity to extreme values and ensures numerical stability across all 150 input features and 30 outputs.

2.4. Dimensionality Reduction (PCA)

Dimensionality Reduction (PCA): Principal Component Analysis (PCA) is applied to reduce the 150-dimensional input space while preserving at least 95% of the total variance.

Retaining 95% variance required 6 principal components out of the original 150 features. The cumulative variance explained curve (Figure 1) shows a rapid rise that plateaus after this point, confirming that most of the variance is captured in a reduced subspace. The reduced data is denoted as follows:

$$X_{PCA}=(X_{scaled}-\mu )\cdot W_k$$

where $\mu$ is the mean of the scaled data and $W_k$ contains the top $k$ eigenvectors.

2.5. Synthetic Source Domain Construction

To enhance learning in the data-scarce target domain, a synthetic source domain is generated using physically informed approximations of the UR5 manipulator’s kinematics.

Sampling in PCA Space: Source inputs are first generated in the reduced PCA space to ensure coverage and diversity. Let $X_{PCA}^{source}\in {\mathbb{R}}^{n_s\times k}$ represent $n_s$ source samples over $k$ principal components. These are uniformly sampled within the observed bounds of the target data’s PCA distribution.

Inverse PCA Transformation: The source samples are then projected back to the scaled feature space:

$$X_{scaled}^{source}=X_{PCA}^{source}\cdot W_k^{\mathrm{\top }}+\mu$$

where $W_k$ and $\mu$ are derived from the PCA on target data.

Back-transformation to Physical DH Space: The scaled source inputs are converted to physical DH parameters using inverse robust scaling:

$$X^{source}=X_{scaled}^{source}\cdot IQR\left(X\right)+median\left(X\right)$$

Each row is reshaped into a 10 × 15 matrix representing synthetic DH parameters over 10 time steps.

Simplified Forward Kinematics for Output Generation: For each synthetic input, a forward kinematics approximation is applied to compute 3D positions using structured DH parameter blocks:

$$\begin{gathered} x=\sum _{i=1}^N{a}_{i}cos\left({\alpha }_i\right) \\ y=\sum _{i=1}^N{a}_{i}sin\left({\alpha }_i\right) \\ z=\sum _{i=1}^N{d}_i, \end{gathered}$$

where $a_i,{\alpha }_i,d_i$ are extracted from each joint’s DH parameters. Random noise is added to simulate realistic variability. The simplified forward kinematics used for synthetic source-domain output generation is formulated as follows:

$$\begin{gathered} G_D={\Pi }_{i=1}^6{T}_{i-1}^i \\ T_{i-1}^i = R_z\left({\theta }_i\right). T_x\left(a_i\right).T_z\left(d_i\right). R_x\left({\propto }_i\right) \\ {G}_D={\Pi }_{i=1}^6\left[\begin{array}{cccc}cos\left({\theta }_i\right)& -sin\left({\theta }_i\right)& 0& a_i\\ sin\left({\theta }_i\right)cos\left({\propto }_i\right)& cos\left({\theta }_i\right)cos\left({\propto }_i\right)& -sin\left({\propto }_i\right)& -\mathit{sin}\left({\propto }_i\right)d_i\\ sin\left({\theta }_i\right)sin\left({\propto }_i\right)& cos\left({\theta }_i\right)sin\left({\propto }_i\right)& cos\left({\propto }_i\right)& \mathit{cos}\left({\propto }_i\right)d_i\\ 0& 0& 0& 1\end{array}\right] \end{gathered}$$

The end-effector position $p={\left[x\hspace{0.25em} y\hspace{0.25em} z\right]}^{\top }$ is taken from the first three elements of the last column of $T_0^6.$ In this study, only ${\theta }_1\left(t\right)$ vary across time steps, with other joint angles fixed at nominal values from

Table 1. DH Parameters for UR−5.

Joint	θi (deg)	αi−1 (deg)	ai−1 (mm)	di (mm)
1	θ1	90	0	89.2
2	−28.49	0	425	0
3	89.82	0	392	0
4	58.67	90	0	109.4
5	θ5	−90	0	94.75
6	−179.99	0	0	82.5

. Gaussian noise is added to $p$ to simulate realistic variability.

2.6. Active Transfer Learning Gaussian Process (ATGP) Modeling

The proposed model leverages Active Transfer Learning Gaussian Process (ATGP) to accurately predict high-dimensional outputs using limited target data and a synthesized source domain. The framework integrates Gaussian Process Regression (GPR) with knowledge transfer and selective refinement.

Gaussian Process Regression (GPR): Given training inputs $X\in {\mathbb{R}}^{n\times d}$ and targets $Y\in {\mathbb{R}}^n$, GPR assumes

$$f\left(x\right)\sim GP\left(m\right(x),k(x,x^{\prime }\left)\right)$$

where $m\left(x\right)$ is the mean function (usually zero) and $k(x,x^{\prime })$ is the covariance function (e.g., squared exponential kernel). The predictive distribution at the test point $x_{*}$ is

$$\begin{gathered} \mu \left(x_{*}\right)=k_{*}^{\mathrm{\top }}{\left(K+{\sigma }_n^{2}I\right)}^{-1}y \\ {\sigma }^2\left(x_{*}\right)=k(x_{*},x_{*})-k_{*}^{\mathrm{\top }}{\left(K+{\sigma }_n^{2}I\right)}^{-1}{k}_{*} \end{gathered}$$

where $K$ is the kernel matrix, $k_{*}$ is the covariance vector, and ${\sigma }_n^2$ is the noise variance.

Transfer Learning Component: ATGP integrates both target domain data $(X_t,y_t)$ and source domain data $(X_s,y_s)$. Each model is trained by combining these datasets while preserving domain relevance. Source and target data are normalized into a common feature space and processed with the same GPR kernel. Domain relevance emerges naturally, as the adaptive enrichment loop continually adds only the highest-error target samples, biasing the model toward the target distribution without explicit weighting. This design choice avoids unnecessary hyperparameter tuning for weight factors, while preserving computational efficiency and scalability. The total training set becomes

$$X_{train}=\left[\begin{array}{c}{X}_t\\ X_s\end{array}\right],y_{train}=\left[\begin{array}{c}{y}_t\\ y_s\end{array}\right]$$

The model benefits from the informative structure of $X_s,$ which guides learning in the underrepresented regions of $X_t$.

Active Component (Implicit): While explicit query-based active learning is not applied, the model implicitly selects and prioritizes high-quality target slices and informative synthetic source samples. Explicit query-based active learning was omitted because the sequential high-error sample selection loop already serves a similar purpose while being computationally lighter and better suited for large-scale, real-time deployment. Additionally, robust preprocessing and dimensionality reduction act as a sample refinement mechanism.

Multi-Output Decomposition: The $30$-dimensional output vector $y\in {\mathbb{R}}^{30}$ represents the $x,y,$ and $z$ positions of the end-effector across 10 time steps. To preserve this physical structure, 30 independent ATGP regressors are trained—each predicting one coordinate ($x,y,$ or $z$) at a specific time step. This decomposition allows scalable training and inference while maintaining the interpretability and consistency of spatial predictions.

The overall methodology followed in this study is summarized in Figure 2. It outlines the key stages from data preprocessing and dimensionality reduction to synthetic source generation and ATGP-based multi-output prediction.

3. Simulation Setup

This section outlines the dataset generation strategy, DH parameter modeling, and simulation procedures used to evaluate the proposed ATGP framework for UR5 trajectory prediction under uncertainty.

3.1. Dataset and Evaluation Strategy

The evaluation phase uses a dataset of $\mathrm{550,000}$ simulated trajectory samples, each consisting of 150 input values (from 10 time steps × 15 DH parameters) and 30 output values (3D end-effector positions across 10-time steps). This dataset was generated with injected uncertainty and is used exclusively for prediction and reliability evaluation. For training, a separate target-domain dataset was independently generated using the same uncertainty modeling but at a smaller scale—at 15% the size of the main evaluation set. The model was trained using the entire training dataset, along with $100$ synthetic source-domain samples generated via PCA-guided sampling and forward kinematics from the source configuration.

Once trained, the model was deployed to predict the entire 550,000-sample evaluation set, which includes random variations and uncertainty. This allowed large-scale estimation of prediction accuracy and system reliability across the full input space. This decoupled setup ensures that the training process remains data-scarce while reliability evaluation is comprehensive and unbiased.

3.2. DH Parameter Modeling and Variation

UR5 robot kinematics are described using standard (DH) parameters. These serve as the input structure for trajectory generation. The nominal joint configuration is listed in Table 1.

In this study, only ${\theta }_5$ are actuated across time. All other joint angles remain fixed. The time-based actuation profile is shown in

Table 2. Time-Based Actuation Scheme for UR-5.

Time Interval [ti − t1]	θ1 (deg)	θ5 Range (deg)
Tracked steps	90	−90 to 90 (step size 20)

.

All motion trajectories are generated by varying θ₁ and θ₅, while maintaining fixed values for other joints. To simulate uncertainty, parameter-level noise is introduced to both translational and rotational DH terms. This allows the model to handle real-world perturbations in joint configurations. The applied distribution strategy is summarized in

Table 3. Distribution Techniques for UR-5 Parameters.

Variable	Mean	Standard Deviation	Distribution Type
a2	425	0.4250	Normal
a3	392	0.3920	Normal
d1	89.2	0.0892	Normal
d4	109.4	0.1094	Normal
d5	94.75	0.09475	Normal
d6	82.5	0.0825	Normal
θ1	θ1	0.0183	Uniform
θ5	θ5	0.0183	Uniform

.

This stochastic modeling helps benchmark the model’s predictive stability and failure tolerance under diverse input distributions.

3.3. Implementation Details

All simulations were conducted in MATLAB 2023a using the following workflow:

3.3.1. Preprocessing

• Modified Z-score for outlier detection (threshold = 3.5)

We selected a modified Z-score threshold of 3.5 as recommended in robust statistics literature [20], since it effectively identifies extreme deviations even in non-Gaussian datasets while minimizing false positives. From the 5.5 × 10⁵ available samples, only 11,175 (≈2.03%) exceeded this threshold and were discarded. This limited removal eliminated high-noise anomalies without impacting on the diversity or representativeness of the retained dataset, ensuring the model was trained on clean yet comprehensive data.

2.

Robust scaling using median and IQR.

3.

PCA applied to retain 95% f the variance.

3.3.2. Model Training

• 30 ATGP regressors trained independently (one for each output value).
• Training is parallelized when supported by hardware.

3.3.3. Prediction and Evaluation

• Model predicts all 550,000 samples from the test dataset.
• Metrics include MAE, RMSE, STD, and reliability estimation based on failure thresholds.

This setup provides a scalable, reproducible, and uncertainty-aware evaluation environment for robotic trajectory prediction.

3.3.4. Incremental Training Strategy for Scalability

Rather than training on the full 5.5 × 10⁵-sample target dataset, the ATGP framework starts with only 10 target domain samples. The trained model predicts outputs for all remaining candidates, performs sample-wise error analysis, and identifies the input with the maximum prediction error. Its true output is computed using the original forward-kinematics-based performance function, and this new pair is added to the training set. The process repeats until the desired accuracy is reached. Using this strategy, training the complete 30-output model with just 27 total samples required ~32 s, while generating predictions and error evaluations for the entire 5.5 × 10⁵-sample dataset took ~13 min. This incremental, error-driven enrichment drastically cuts computation and memory cost compared to full-dataset training yet remains practical for large-scale and near-real-time robotic applications.

4. Results and Discussion

This section evaluates the performance of the proposed ATGP model in terms of trajectory accuracy and error reliability across spatial dimensions and joints. The results are supported by $6$ representative visualizations grouped into two sets: trajectory prediction analysis and reliability assessment.

Trajectory prediction accuracy is illustrated in Figure 3, where four subplots show the close agreement between the predicted and ideal trajectories.

Figure 3a: A 3D trajectory plot indicates that the predicted paths (blue) and the ideal trajectory (red) were closely linked, which validated spatial accuracy by using $10$ time steps. Figure 3b: The X-coordinate trajectory displays a smooth curve with the predicted line closely matching the ideal path. Figure 3c: Y-coordinate predictions follow the ground truth with minimal deviation, indicating consistency in upward progression. Figure 3d: The Z-coordinate prediction tracks the decline in height with high fidelity across all joints. These results confirm that the ATGP model accurately predicts position data in all three spatial dimensions, even under high-dimensional input uncertainty.

Figure 4 shows a detailed analysis of errors across coordinates and joints, highlighting how the model performs on each component.

Figure 4a: A grouped bar chart compares MAE, RMSE, and STD for the X, Y, and Z coordinates. The X- and Z-axes exhibit slightly higher RMSE than Y, but all errors remain under acceptable bounds (below ~2.5 mm). This indicates balanced performance across dimensions. Figure 4b: Sample-wise error plots illustrate how prediction accuracy varies across the $10$-time steps. Samples 1 to 3 show slightly higher errors across all coordinates, while samples 4 to 10 maintain low and stable errors. The average error curve (black dashed line) confirms this trend.

This analysis highlights that the ATGP model performs more consistently in the mid-to-late trajectory phases, possibly due to more structured motion dynamics in those regions.

Visualization and Confidence Evaluation

The predicted trajectories produced by the ATGP model show strong alignment with the ground truth across all spatial dimensions. Visually, the model generalizes well across time steps and joint configurations, with smooth, stable predictions and minimal deviation. Even without displaying explicit confidence intervals, the tight match between predicted and actual paths implies low uncertainty.

To validate these observations, a Monte Carlo Simulation (MCS) was used as a statistical benchmark. Both methods were tested on the same dataset of $\mathrm{550,000}$ trajectories using a uniform failure threshold.

Table 4. Reliability Comparison of ATGP vs. Monte Carlo.

Metric	Monte Carlo (MCS)	ATGP (Proposed)
Total Samples (N)	$\mathrm{550,000}$	$\mathrm{550,000}$
Failure Cases (NF)	$7410$	$\mathrm{11,175}$
Passed Cases (NP)	$\mathrm{542,590}$	$\mathrm{538,825}$
Reliability (R)	$98.65\%$	$97.96\%$
Failure Probability (PF)	$1.35\%$	$2.04\%$
Relative Error	$-$	$0.51\%$
Accuracy vs. MCS	$-$	$99.47\%$

summarizes the reliability results for both approaches. The ATGP model performs competitively, achieving high prediction accuracy and a low relative error when compared to the MCS baseline—highlighting its effectiveness for uncertainty-aware trajectory modeling.

These results confirm that while ATGP trails slightly behind MCS in raw reliability, it maintains strong accuracy and provides valuable uncertainty estimates with less computational effort, making it well-suited for real-time robotic applications.

5. Conclusions

In this research, a simulation framework based on an Active Transfer Learning Gaussian Process (ATGP) model is developed, enabling accurate and reliable prediction of the trajectory of the UR5 manipulator robotic arm. The combination of strong preprocessing, dimensionality-reduction based on PCA, and a generation of the source domain in the target enables the model to handle high-dimensional input space and limited target data. The results are also characterized by good predictive performance and one that matches well the predicted and ideal trajectory in the three spatial dimensions. Coordinates-based error analysis and joints-based error analysis validate the robustness of the model, especially towards a later stage of the trajectory, where the accuracy level is more decisive. Although the other models exhibit low values of the average absolute error, the ATGP model has low values of both the average absolute error and maintains high reliability in all output channels. The suggested solution allows a scaling and data-efficient approach to the modeling of robotic trajectories in the presence of uncertainty. Though the present work is on simulation, the formulation can be embodied in the application of sensor noise, actuation errors, and online learning. This work has so far been validated only in simulation, meaning real-world tests—subject to sensor noise and unmodeled dynamics—are still pending. While not undermining the presented results, addressing these factors in hardware experiments will be the next step toward full deployment.