In the field of advanced robot technology, parallel robots have gained significant attention due to their high load capacity, stiffness, and precision. Among these, the Delta parallel robot stands out for its exceptional speed and efficiency, making it ideal for applications such as pick-and-place operations and electronic assembly. However, achieving high positioning accuracy in parallel robots remains a critical challenge, as errors from manufacturing tolerances, assembly imperfections, joint clearances, and drive system inaccuracies can severely impact performance. Traditional error modeling methods, such as the Denavit-Hartenberg (D-H) parameter method and spatial vector approach, often suffer from issues like singularity and incompleteness. To address these limitations, this paper employs screw theory combined with the local Product of Exponentials (POE) formula to develop a comprehensive error analysis model for Delta robots. By categorizing error sources into structural errors, actuation angle errors, and spherical joint clearance errors, we systematically investigate their effects on end-effector positioning accuracy. Through MATLAB simulations, we demonstrate that actuation angle errors have the most pronounced impact, followed by spherical joint clearances, while structural errors exhibit relatively stable influences. This approach not only avoids singularity problems but also provides a more intuitive and complete framework for error modeling in parallel mechanisms, contributing to advancements in robot technology.
Screw theory offers a robust mathematical foundation for describing rigid body motions in robot technology, leveraging the geometric properties of mechanisms. According to Chasles’ theorem, any rigid body motion can be represented as a screw motion—a combination of rotation about an axis and translation along that axis. This representation avoids singularities and ensures continuity in kinematic models. The local POE method, in particular, defines joint screw coordinates in local frames, enhancing physical clarity. Key concepts include the Lie groups SO(3) and SE(3) for rotations and transformations, respectively, and their corresponding Lie algebras so(3) and se(3). A screw motion is described by a twist vector $\xi = (\omega^T, v^T)^T$, where $\omega$ is the unit direction vector of the rotation axis, and $v$ is the linear velocity component. The exponential map $e^{\hat{\xi} \theta}$ transforms a twist into a homogeneous transformation matrix, enabling concise kinematic modeling. For a revolute joint, the twist coordinates are $(\omega^T, (p \times \omega)^T)^T$, and for a prismatic joint, they are $(0_{1\times3}, v^T)^T$, where $p$ is a point on the axis. The local POE formula for a serial chain is given by $T = T_{i,1} \cdots T_{i,n}$, where each $T_{i,j} = e^{\hat{\xi}_{ij} q_{ij}}$ represents the transformation between consecutive local frames. This method’s elegance lies in its ability to handle complex parallel structures by simplifying them into equivalent serial chains, as demonstrated in the Delta robot’s case.
The Delta robot features a symmetric design with three identical kinematic chains connecting a fixed base to a moving platform. Each chain consists of an active arm driven by a rotary actuator and a passive arm forming a parallelogram structure, which ensures the moving platform maintains constant orientation. To apply the local POE method, we simplify the parallelogram mechanism by equivalencing it to a serial chain, as the constrained motion reduces to a single degree of freedom. For chain $i$, local frames are established at each joint, from the base frame $\{L_0\}$ to the end-effector frame $\{L_{i,11}\}$. The transformation matrices include both structural and joint components, with twists derived from geometric parameters. The key parameters for the Delta robot are the active arm length $L_a$, passive arm length $L_b$, base platform radius $R$, moving platform radius $r$, actuation angles $\theta_i$, and orientation angles $\beta_i$. The forward kinematics for a single chain is expressed as $T = T_{i,1} \cdots T_{i,10} T_{i,11}$, where each $T_{i,j}$ is computed using the exponential map. For instance, the transformation due to a revolute joint with twist $\xi_{ij}$ and variable $q_{ij}$ is $e^{\hat{\xi}_{ij} q_{ij}} = \begin{bmatrix} e^{\hat{\omega}_{ij} q_{ij}} & (I_3 – e^{\hat{\omega}_{ij} q_{ij}})(\omega_{ij} \times v_{ij}) + q_{ij} \omega_{ij} \omega_{ij}^T v_{ij} \\ 0_{1\times3} & 1 \end{bmatrix}$ for $\omega \neq 0$, and for pure translation, it simplifies to $\begin{bmatrix} I_3 & q_{ij} v_{ij} \\ 0_{1\times3} & 1 \end{bmatrix}$. The symmetry of the Delta robot allows deriving twists for all chains from the first chain using rotation matrices, streamlining the model.
To establish the error model, we differentiate the kinematic equation using the exponential map’s differential properties. The differential of a transformation $T = e^{\hat{\xi} q}$ is given by $\frac{d e^{\hat{\xi} q}}{dt} e^{-\hat{\xi} q} = A \frac{d\xi}{dt} + \xi \frac{dq}{dt}$, where $A$ is an adjoint-related matrix that ensures linearity in the error terms. For the entire chain, the end-effector error twist $\delta e = [\delta T T^{-1}]^\vee$ is derived as $\delta e = J_i \delta \xi_i + \Psi_i \xi_i \delta q_i$, where $J_i$ and $\Psi_i$ are Jacobian-like matrices composed of adjoint transformations, $\delta \xi_i$ represents screw coordinate errors, and $\delta q_i$ denotes joint variable errors. This formulation allows decomposing the error into contributions from various sources, enabling detailed analysis. The error model is comprehensive, covering all potential error sources without ideal assumptions, which is crucial for accurate calibration in robot technology.
We categorize error sources into three main types: structural errors, actuation angle errors, and spherical joint clearance errors. Structural errors arise from manufacturing inaccuracies in parameters such as $R$, $r$, $L_a$, $L_b$, $a_1$, and $a_2$. Assuming independent errors of 0.100 mm each, we analyze their impact on the end-effector positioning accuracy along the X, Y, and Z axes within the workspace $X \in [-600, 600]$ mm, $Y \in [-600, 600]$ mm, and $Z \in [650, 1000]$ mm. The composite error $\Delta e = \sqrt{(\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2}$ is used as a metric. The results show that errors in $L_a$ and $L_b$ have the most significant influence, with $\Delta e$ varying between 0.100 mm and 0.117 mm, while other structural errors cause nearly constant $\Delta e \approx 0.100$ mm. This highlights the importance of precise manufacturing for active and passive arms in robot technology.
| Error Source | Effect on $\Delta e$ (mm) | Variation Trend |
|---|---|---|
| $\Delta R$ | 0.100 (constant) | Minimal variation |
| $\Delta r$ | 0.100 (constant) | Stable |
| $\Delta L_a$ | 0.100–0.112 | Increases with distance from origin |
| $\Delta L_b$ | 0.100–0.117 | Decreases with distance from origin |
| $\Delta a_1$ | 0.100 (constant) | Stable |
| $\Delta a_2$ | 0.100 (constant) | Stable |
Actuation angle errors result from clearances in the drive system, such as keyway fits, leading to deviations $\Delta \theta$ in the input angles. Assuming an error of 0.100° per chain, the analysis reveals that these errors dominate the positioning inaccuracy, with $\Delta e$ ranging from 1.500 mm to 2.000 mm. Specifically, for motion along X, errors from chains 2 and 3 cause $\Delta e$ between 1.513 mm and 1.631 mm, while chain 1 contributes to higher variations (1.513–1.891 mm). Along Y, chain 1 errors yield constant $\Delta e = 1.515$ mm, whereas chains 2 and 3 show symmetric variations. Along Z, all chains exhibit proportional increases in $\Delta e$ with Z-coordinate, from 1.167 mm to 1.864 mm. This underscores the critical role of precise actuation in robot technology for high-accuracy tasks.
Spherical joint clearance errors arise from manufacturing tolerances and wear in the ball joints connecting the arms. Modeling the clearance as a displacement $D$ between the ball and socket centers, with angles $\alpha$ and $\gamma$ defining the direction, the error twist $\delta \xi$ is derived. For a clearance of 0.100 mm in all directions, the composite error $\Delta e$ is approximately 0.340 mm, with minor variations depending on the chain and direction. For instance, along X, chains 2 and 3 show $\Delta e$ from 0.337 mm to 0.346 mm, while chain 1 remains constant at 0.346 mm. Along Y, chain 1 errors vary between 0.333 mm and 0.346 mm, and chains 2 and 3 are nearly constant. Along Z, all chains result in $\Delta e = 0.346$ mm. Although less impactful than actuation errors, spherical joint clearances are non-negligible and should be considered in error compensation strategies for robot technology.
To validate the individual analyses, we simulate the combined effect of all error sources, each set to 0.100 units (mm or degrees). The composite error $\Delta e$ at 30 random points in the workspace closely follows the trend of actuation angle errors, ranging from 1.500 mm to 2.000 mm, indicating their dominance. Spherical joint errors contribute around 0.340 mm, and structural errors add approximately 0.100 mm. Notably, the combined error is not a linear sum due to cancellations in X and Y directions from symmetry, but Z-direction errors accumulate. Further, scaling the error sources to 0.025, 0.050, 0.075, and 0.100 units shows proportional increases in $\Delta e$, with actuation errors remaining the most influential. This comprehensive analysis provides insights for prioritizing error compensation in robot technology.
| Error Source Magnitude (units) | Average $\Delta e$ (mm) | Dominant Error Source |
|---|---|---|
| 0.025 | 0.375–0.500 | Actuation angles |
| 0.050 | 0.750–1.000 | Actuation angles |
| 0.075 | 1.125–1.500 | Actuation angles |
| 0.100 | 1.500–2.000 | Actuation angles |
In conclusion, the screw theory-based error model using the local POE method offers a superior approach for analyzing Delta parallel robots in robot technology, overcoming limitations of traditional methods. The analysis reveals that actuation angle errors are the primary contributor to end-effector inaccuracy, followed by spherical joint clearances and structural errors, with active and passive arm lengths being the most sensitive structural parameters. These findings emphasize the need for focused error compensation on drive systems and joint clearances to enhance positioning accuracy. Future work could integrate this model with real-time calibration algorithms, further advancing the capabilities of robot technology in high-precision applications. The methodology presented here is also applicable to other parallel mechanisms, promoting broader adoption in industrial robot technology.