Geometric Error Modeling of 3-DOF Planar Parallel Manipulators Using Conformal Geometric Algebra

Abstract

This paper proposes a geometrically intuitive error modeling method for planar parallel mechanisms (PPMs) based on conformal geometric algebra (CGA). First, the end point of each limb is determined through rotational transformations in geometric algebra. Using this point as the center, the kinematic geometry body (KGB) of each limb is constructed. The end-effector position of the parallel mechanism is then obtained via intersection operations in CGA, while its orientation is derived by constructing the motion plane from the end points. Finally, the error model of the parallel mechanism is established through differential operations. To validate the proposed method, a kinematic calibration simulation was performed using a 3-RPR planar parallel mechanism as an example. The simulation results demonstrate a significant reduction in both position and orientation errors after calibration, indicating a substantial improvement in accuracy and verifying the correctness of the approach.

1. Introduction

Parallel manipulators (PMs) are widely used in various fields because of their strong bearing capacity, high stiffness, and excellent dynamic performance [1]. Among them, 3-DOF planar parallel mechanisms (PPMs) are widely used in high-precision positioning and precise operation fields. Considering symmetry, planar parallel mechanisms can be classified into 10 types [2], as shown in Figure 1, with single and double arrows representing revolute and prismatic joints, respectively. The underlined parts represent the driving joints. During the production process, deviations in manufacturing and assembly can arise, which may compromise the operational precision of PMs. The literature indicates that geometric inaccuracies are among the most significant contributors to these errors [3,4,5]. Consequently, enhancing the accuracy of PMs continues to be a vital focus of research. Although methods like improving machining tolerances can aid in achieving higher precision, kinematic calibration has proven to be a more cost-efficient and practical approach for enhancing the positioning accuracy of such systems.

Figure 1. The basic 3-DOF PPMs with identical legs.

Error modeling is the first step of kinematic calibration. Its principal aim is to formulate a mathematical framework that correlates joint displacement errors with the spatial pose of the end-effector [6]. Although error modeling methods for serial manipulators are now well-established, corresponding methodologies for PMs remain underdeveloped and necessitate considerable advancement in multiple areas. Attaining high-precision calibration performance requires the construction of an error model that ensures completeness, continuity, and independence among parameters, as these properties are fundamental to obtaining accurate and trustworthy calibration results.

Many methods for error modeling have been proposed. Early studies predominantly relied on the Denavit–Hartenberg (D-H) convention [4] to formulate error models for PMs. As an illustration, Wang and Masory [7] utilized this methodology to establish an error model for a Stewart platform. Comparable D-H-based techniques have also been developed for PMs with reduced mobility. A significant drawback of the standard D-H method, however, is its breakdown in cases where two adjacent joint axes are collinear. This limitation has prompted the creation of multiple revised D-H conventions [8,9]. Nevertheless, despite such adaptations, the fundamental D-H parameters and their subsequent modifications continue to be constrained by inherent mathematical deficiencies and singularity problems. Therefore, these approaches remain inadequate for establishing a consistent mathematical structure required for holistic error modeling.

Another classic method is to apply the inverse kinematics or closed-form formulations, primarily due to the significant complexity inherent in forward kinematic computations [10,11]. However, error models developed via inverse kinematic solutions often depend on idealized structural assumptions, which may result in the omission of certain potential error sources. For instance, He et al. [12] conducted a calibration study on a 2UPR-RPU PM using a differential inverse kinematics approach. Despite its utility, this method still fails to incorporate a complete set of error terms.

Two other established methodologies in this domain are the Product of Exponentials (POE) approach and screw theory. Initially introduced by Park and Okamura [13] for calibrating PMs, the POE method was later refined through the Local POE formulation by Chen et al. [14,15], which bears structural similarities to Denavit–Hartenberg parameter-based techniques. The POE method has been applied and improved by many scholars [16,17,18]. However, there is a problem with the POE method: the error term sometimes lacks clear physical meaning. Screw theory offers another rigorous framework for error modeling in PMs [19,20,21]. For instance, Liu et al. [22] developed a systematic strategy for geometric error modeling in limited-DOF PMs, clearly separating joint motion inaccuracies from geometric deviations. Although screw-based methods allow unified error representation, their reliance on instantaneous kinematics often complicates position-level analysis. Sun et al. [19] addressed this by integrating finite and instantaneous screw theory to support intuitive error propagation and identification across both serial and parallel architectures. Alternative strategies, including the multi-loop circuit incremental method [23] and unit dual quaternions [24], have also contributed to the development of error models for such systems.

Conformal geometric algebra (CGA) belongs to 5-dimensional geometric algebras [25,26]. Within the framework of CGA, geometric entities such as points, lines, planes, and spheres can be represented with convenience. By leveraging the operations defined in CGA, rigid body transformations and geometric intersection operations can be effectively carried out. The application scope of CGA is remarkably extensive, spanning multiple disciplines including physics, computer science, and robotics [27,28]. CGA serves as a highly potent computational framework. In the domain of robotics, it is capable of addressing numerous crucial problems. These include the mobility analysis [29], the direct and inverse kinematics [30,31,32], the singularity analysis [33,34], and the configuration synthesis of PMs [35]. Specifically, Lian [36] employed CGA to perform error modeling for PMs. In this method, the finite motion of an open-loop chain is firstly formulated. Through linearization of the finite motion, error propagation of the open-loop chain is analyzed. Then, motions and constraints of PMs are analyzed and geometric error model of PMs are formulated considering the actuations and constraints.

This work proposes a new method for error modeling of PPMs based on CGA. The advantage of this method is that it has geometric intuitiveness. Moreover, it is also universal. For different structures of PPM, there is no need to rederive the modeling process. All that is required is to replace some structural parameters. Section 2 offers a brief introduction on CGA and outlines the theoretical foundations employed. A generalized error model for PPMs is introduced in Section 3. Validation of the proposed framework is conducted in Section 4 via a case study on a 3-RPR parallel manipulator. Finally, some conclusions are presented in the Section 5.

2. Conformal Geometric Algebra

This section introduces fundamental theory of CGA used in this paper briefly. For details about CGA, please refer to the relevant reference books [27,28].

2.1. Fundamental Theory of CGA

CGA contains the three Euclidean basis vectors ${\mathit{e}}_1$, ${\mathit{e}}_2$, ${\mathit{e}}_3$ and two additional basis vectors ${\mathit{e}}_{+}$, ${\mathit{e}}_{-}$ with positive and negative signatures, respectively. The two additional basis vectors satisfy the following equations.

$${\mathit{e}}_{+}^2=1, {\mathit{e}}_{-}^2=-1, {\mathit{e}}_{+}·{\mathit{e}}_{-}=0$$

(1)

3D origin ${\mathit{e}}_0$ and infinity ${\mathit{e}}_{\infty }$ can be defined by

$${\mathit{e}}_0={\displaystyle \frac{1}{2}}({\mathit{e}}_{-}-{\mathit{e}}_{+}), {\mathit{e}}_{\infty }={\mathit{e}}_{-}+{\mathit{e}}_{+}$$

(2)

${\mathit{e}}_0$ and ${\mathit{e}}_{\infty }$ are all null vectors with the following properties:

$${\mathit{e}}_0^2={\mathit{e}}_{\infty }^2=0, {\mathit{e}}_{\infty }·{\mathit{e}}_0=-1$$

(3)

$${\mathit{e}}_{\infty }\wedge {\mathit{e}}_0=\mathit{E}$$

(4)

$${\mathit{e}}_{\infty }{\mathit{e}}_0={\mathit{e}}_{\infty }\wedge {\mathit{e}}_0+{\mathit{e}}_{\infty }·{\mathit{e}}_0={\mathit{e}}_{\infty }\wedge {\mathit{e}}_0-1$$

(5)

where E is a unit pseudoscalar in CGA, and E² = −1.

The main operations in CGA include geometric product “juxtaposition of two vectors”, inner product “$·$” and outer product “$\wedge$”. The outer product can be used to solve the intersection of geometry.

2.2. Conformal Geometric Entities

Table 1 lists basic geometric entities, such as points, spheres, planes, circles, lines, and point pairs. These entities have two algebraic representations: the inner product null space (IPNS) and the outer product null space (OPNS). They are dual to each other.

Table 1. Basic geometric entities in CGA.

Entity	IPNS Representation	OPNS Representation
Point	$\mathit{P}=\mathit{x}+{\displaystyle \frac{1}{2}}{\mathit{x}}^2{\mathit{e}}_{\infty }+{\mathit{e}}_0$	${\mathit{P}}^{*}={\mathit{S}}_1\wedge {\mathit{S}}_2\wedge {\mathit{S}}_3\wedge {\mathit{S}}_4$
Sphere	$\mathit{S}=\mathit{P}-{\displaystyle \frac{1}{2}}{r}^2{\mathit{e}}_{\infty }$	${\mathit{S}}^{*}={\mathit{P}}_1\wedge {\mathit{P}}_2\wedge {\mathit{P}}_3\wedge {\mathit{P}}_4$
Plane	$\mathsf{\pi }=\mathit{n}+d{\mathit{e}}_{\infty }$	${\mathsf{\pi }}^{*}={\mathit{P}}_1\wedge {\mathit{P}}_2\wedge {\mathit{P}}_3\wedge {\mathit{e}}_{\infty }$
Circle	$\mathit{Z}={\mathit{S}}_1\wedge {\mathit{S}}_2$	${\mathit{Z}}^{*}={\mathit{P}}_1\wedge {\mathit{P}}_2\wedge {\mathit{P}}_3$
Line	$\mathit{L}={\mathsf{\pi }}_1\wedge {\mathsf{\pi }}_2$	${\mathit{L}}^{*}={\mathit{P}}_1\wedge {\mathit{P}}_2\wedge {\mathit{e}}_{\infty }$
Point pair	${\mathit{P}}_p={\mathit{S}}_1\wedge {\mathit{S}}_2\wedge {\mathit{S}}_3$	${\mathit{P}}_P^{*}={\mathit{P}}_1\wedge {\mathit{P}}_2$

From Table 1 we can see that a sphere can be constructed with a CGA point P and radius r. A 3D point can be represented in CGA form in the inner product null space. In the outer product null space, the intersection of four spheres gives us an intersection. They have the following relationship:

$${\mathit{P}}^{\ast }=\mathit{P}{\mathit{E}}^{-1}$$

(6)

where ${\mathit{E}}^{-1}$ is the inverse of E, $\mathit{E}{\mathit{E}}^{-1}=1$.

The point can be obtained by the point-pair decomposition

$$\mathit{P}={\displaystyle \frac{\pm \sqrt{{\mathit{P}}_p^{*}·{\mathit{P}}_p^{*}}+{\mathit{P}}_p^{*}}{{\mathit{e}}_{\infty }·{\mathit{P}}_p^{*}}}={\displaystyle \frac{{\mathit{C}}_2}{c{o}_3}}\pm {\displaystyle \frac{\sqrt{c{o}_4}}{c{o}_3}}{\mathit{C}}_1$$

(7)

where ${\mathit{C}}_1={\mathit{e}}_{\infty }·{\mathit{P}}_{\mathit{p}}^{\ast }$, ${\mathit{C}}_2={\mathit{C}}_1·{\mathit{P}}_P^{\ast }$, $c{o}_3={\mathit{C}}_1·{\mathit{C}}_1$, $c{o}_4={\mathit{P}}_p^{*}·{\mathit{P}}_p^{*}$.

In fact, three spheres and a plane can also intersect at one point. Figure 2 shows some possible cases of intersection.

Figure 2. Intersection point of spheres or spheres and plane.

2.3. Conformal Transformations

Transformations can easily be described in CGA. For example, the rotation of a rigid body $o^{\prime }$ can be written as follows:

$$\mathit{o}=\mathrm{Rot}(\varphi ){\mathit{o}}^{\prime }\tilde{\mathrm{R}}\mathrm{ot}(\varphi )$$

(8)

where $\mathrm{Rot}(\varphi )$ and $\tilde{\mathrm{R}}\mathrm{ot}(\varphi )$ are defined by Equation (9)

$$\begin{array}{l}\mathrm{Rot}(\varphi )=\cos ({\displaystyle \frac{\varphi }{2}})-\mathit{n}\sin ({\displaystyle \frac{\varphi }{2}})\\ \tilde{\mathrm{R}}\mathrm{ot}(\varphi )=\cos ({\displaystyle \frac{\varphi }{2}})+\mathit{n}\sin ({\displaystyle \frac{\varphi }{2}})\end{array}$$

(9)

and n is the rotation axis, represented by a normalized bivector.

The translation of a rigid body $o^{\prime }$ can be described as

$$\mathit{o}=\mathrm{Trans}(\mathit{t}) {\mathit{o}}^{\prime }\tilde{\mathrm{T}}\mathrm{rans}(\mathit{t})$$

(10)

where $\mathrm{Trans}(\mathit{t})$ and $\tilde{\mathrm{T}}\mathrm{rans}(\mathit{t})$ are defined by Equation (11)

$$\begin{array}{l}\mathrm{Trans}(\mathit{t})=1-{\displaystyle \frac{1}{2}}\mathit{t}{\mathit{e}}_{\infty }\\ \tilde{\mathrm{T}}\mathrm{rans}(\mathit{t})=1+{\displaystyle \frac{1}{2}}\mathit{t}{\mathit{e}}_{\infty }\end{array}$$

(11)

A rigid motion containing rotation and translation can be described by the following geometric product:

$${\mathit{o}}_2=\mathit{T}{\mathit{o}}_1\tilde{\mathit{T}}=\mathrm{Rot}(\varphi )\mathrm{Trans}(\mathit{t}) {\mathit{o}}_1 \tilde{\mathrm{T}}\mathrm{rans}(\mathit{t})\tilde{\mathrm{R}}\mathrm{ot}(\varphi )$$

(12)

where $o_1$ and $o_2$ are the original and transformed object, respectively. $\mathit{T}$ is a versor and $\tilde{\mathit{T}}$ is its reverse.

$$\begin{array}{l}\mathit{T}=\mathrm{Rot}(\varphi )\mathrm{Trans}(\mathit{t})=(\cos {\displaystyle \frac{\varphi }{2}}-\mathit{l}\sin {\displaystyle \frac{\varphi }{2}})(1-{\displaystyle \frac{1}{2}}\mathit{t}{\mathit{e}}_{\infty })\\ \tilde{\mathit{T}}=\tilde{\mathrm{T}}\mathrm{rans}(\mathit{t})\tilde{\mathrm{R}}\mathrm{ot}(\varphi )=(\cos {\displaystyle \frac{\varphi }{2}}+\mathit{n}\sin {\displaystyle \frac{\varphi }{2}})(1+{\displaystyle \frac{1}{2}}\mathit{t}{\mathit{e}}_{\infty })\end{array}$$

(13)

3. Error Modeling of PPMs Using CGA

The main idea of the method in this paper is to use the meet operation in CGA to deal with the error modeling of PPMs. By means of rotation transformation, the position of the last joint of the limb can be obtained. With this point as the center, a sphere can be constructed. This process is called constructing a limb kinematic geometric body. Similar operations can be performed on other limbs. In this way, we can construct three spheres and intersect them to get the center point of the moving platform. The orientation of the moving platform can also be derived using the endpoints of limbs. By using differential operations, the error model of PPMs can be derived. The error modeling process is shown in Figure 3. The detailed content will be introduced in Section 3.1, Section 3.2 and Section 3.3.

Figure 3. Error modeling process based on CGA.

3.1. Construction of Limb KGB

Figure 4a shows 3-DOF PPMs, and each limb has n (n = 3) joints. For modeling purpose, reference frames are set on the base, moving platform, and joints. The first reference frame $R_{0,i}$ of limb i is located at O, and the last reference frame is located at O′. Frames $R_{j,i}$ associated with jth (j = 1, 2, …, n) joint are located at the center of each joint, and $z_{j,i}$ axis is set along the axis of the joint.

Figure 4. Schematic diagram of a 3-DOF PPM. (a) Establishment of the coordinate system from the x-y perspective. (b) A KGB for limb i with a radius of r viewed from a three-dimensional perspective.

For limb i, the joint reference frame $R_{j-1}$ is transformed to the frame $R_j$ by the following transformation:

$${}_j{}^{j-1}\mathit{T}={\mathrm{Trans}}_x(x_j){\mathrm{Trans}}_x(y_j){\mathrm{Trans}}_z(z_j){\mathrm{Rot}}_x({\alpha }_j){\mathrm{Rot}}_x({\beta }_j){\mathrm{Rot}}_x({\gamma }_j)$$

(14)

where

$$\begin{array}{l}{\mathrm{Trans}}_x(x_j)=1-{\displaystyle \frac{1}{2}}{x}_j{\mathit{e}}_1{\mathit{e}}_{\infty }\\ {\mathrm{Trans}}_y(y_j)=1-{\displaystyle \frac{1}{2}}{y}_j{\mathit{e}}_3{\mathit{e}}_{\infty }\\ {\mathrm{Trans}}_z(z_j)=1-{\displaystyle \frac{1}{2}}{z}_j{\mathit{e}}_3{\mathit{e}}_{\infty }\\ {\mathrm{Rot}}_x({\alpha }_j)=\mathrm{c}({\displaystyle \frac{{\alpha }_j}{2}})-\mathrm{s}({\displaystyle \frac{{\alpha }_j}{2}}){\mathit{e}}_x{\mathit{e}}_{\infty }\\ {\mathrm{Rot}}_y({\beta }_j)=\mathrm{c}({\displaystyle \frac{{\beta }_j}{2}})-\mathrm{s}({\displaystyle \frac{{\beta }_j}{2}}){\mathit{e}}_x{\mathit{e}}_{\infty }\\ {\mathrm{Rot}}_z({\gamma }_j)=\mathrm{c}({\displaystyle \frac{{\gamma }_j}{2}})-\mathrm{s}({\displaystyle \frac{{\gamma }_j}{2}}){\mathit{e}}_3{\mathit{e}}_{\infty }\end{array}$$

x_j, y_j, z_j (${\alpha }_j$, ${\beta }_j$, ${\gamma }_j$) (j = 1, 2, 3) are the geometric error components of R_j relative to R_j−1 along (about) three orthogonal axes of R_j−1.

Equation (14) encompasses both translational motion and rotational motion.

$${}_j{}^{j-1}\mathit{T}=\mathit{T}\left({\mathit{r}}_j\right)\mathit{T}\left({\theta }_j\right)$$

where ${\mathit{r}}_j=\left(x_j,y_j,z_j\right)$, ${\theta }_j=\left({\alpha }_j,{\beta }_j,{\gamma }_j\right)$.(j = 1, 2, 3)

The pose of point B_i (i = 1, 2, 3) is

$${}_j{}^O\mathit{T}_i={}_0{}^O\mathit{T}_i{}_1{}^0\mathit{T}_i\dots {}_j{}^{j-1}\mathit{T}_i$$

(15)

$${\mathit{B}}_i=B_{ix}{\mathit{e}}_1+B_{iy}{\mathit{e}}_2+B_{iz}{\mathit{e}}_3+{\displaystyle \frac{1}{2}}(B_{ix}^2+B_{iy}^2+B_{iz}^2){\mathit{e}}_{\infty }+{\mathit{e}}_0$$

(16)

where $(B_{ix},B_{iy},B_{iz})$ is the coordinates of B_i.

The modeling process shown in Figure 2 is applicable to all planar parallel mechanisms. The only difference is that due to the variations in the limb structures, the transformations included when calculating point B are different.

The KGB of limb i can be constructed as

$${\mathit{S}}_i={\mathit{B}}_i-{\displaystyle \frac{1}{2}}{b}_i^2{\mathit{e}}_{\infty }(i=1, 2, 3)$$

(17)

which is a sphere with its origin at point ${\mathit{B}}_i$. The radius is defined by the distance b_i from ${\mathit{B}}_i$ to the center of the moving platform P. Figure 4b shows the KGB of limb i. KGB is the accumulation of movements. The motion space of the moving platform is the intersection of all the KGBs on the limbs of the PPM.

3.2. Kinematics Analysis of PPMs Using CGA

The intersection of three KGB spheres gives a point pair ${\mathit{P}}_P^{\ast }$

$$\begin{array}{l}{\mathit{P}}_p^{*}=({\mathit{S}}_1\wedge {\mathit{S}}_2\wedge {\mathit{S}}_3){\mathit{E}}^{-1}\\ \mathit{P}={\displaystyle \frac{\pm \sqrt{{\mathit{P}}_p^{*}·{\mathit{P}}_p^{*}}+{\mathit{P}}_p^{*}}{{\mathit{e}}_{\infty }·{\mathit{P}}_p^{*}}}={\displaystyle \frac{{\mathit{C}}_2}{c{o}_3}}\pm {\displaystyle \frac{\sqrt{c{o}_4}}{c{o}_3}}{\mathit{C}}_1\end{array}$$

(18)

where

$${\mathit{C}}_1={\mathit{e}}_{\infty }·{\mathit{P}}_p^{*}, {\mathit{C}}_2={\mathit{C}}_1·{\mathit{P}}_p^{*}, c{o}_3={\mathit{C}}_1·{\mathit{C}}_1, c{o}_4={\mathit{P}}_p^{*}·{\mathit{P}}_p^{*}, \mathit{E}={\mathit{e}}_1\wedge {\mathit{e}}_2\wedge {\mathit{e}}_3\wedge {\mathit{e}}_{\infty }\wedge {\mathit{e}}_0, {\mathit{E}}^{-1}=-\mathit{E}.$$

Note that, ideally, the center point of the moving platform can take a point in a point pair or the midpoint of a point pair; however, the center point of the moving platform can take the midpoint of a point pair if errors are considered.

According to Equation (18), the three-dimensional coordinate of the moving platform center is

$$\begin{array}{l}{x}_o=-(c_{e12}{c}_{e20}+c_{e13}{c}_{e30}-c_{e10}{c}_{e\infty 0})/A\\ y_o=(c_{e10}{c}_{e12}-c_{e23}{c}_{e30 }+c_{e20}{c}_{e\infty 0})/A\\ z_o=(c_{e10}{c}_{e13}+c_{e20}{c}_{e23}+c_{e30}{c}_{e\infty 0})/A\end{array}$$

(19)

where $c_{e12}$, $c_{e13}$, $c_{e23}$, $c_{e10}$, $c_{e20}$, $c_{e30}$, $c_{e\infty 0}$ and A are shown in Appendix A. For planar parallel mechanisms, z_o = 0.

The plane composed by B_i (i = 1, 2, 3), namely the moving platform is

$$\mathsf{\pi }=({\mathit{B}}_1\wedge {\mathit{B}}_2\wedge {\mathit{B}}_3\wedge {\mathit{e}}_{\infty }){\mathit{E}}^{-1}=\mathit{n}+d{\mathit{e}}_{\infty }$$

(20)

where $\mathit{n}$ is the normal vector of $\mathsf{\pi }$, $\mathit{E}={\mathit{e}}_1\wedge {\mathit{e}}_2\wedge {\mathit{e}}_3\wedge {\mathit{e}}_{\infty }\wedge {\mathit{e}}_0$, ${\mathit{E}}^{-1}=-\mathit{E}$, and d is the distance from the plane $\mathsf{\pi }$ to the origin.

The components of the normal line of the plane $\mathsf{\pi }$ in each direction are

$$\begin{array}{l}{n}_{z1}^{\prime }=B_{2y}{B}_{1z}-B_{1y}{B}_{2z}+B_{3z}(B_{1y}-B_{2y})-B_{3y}(B_{1z}-B_{2z})\\ n_{z2}^{\prime }=B_{1x}{B}_{2z}-B_{2x}{B}_{1z}-B_{3z}(B_{1x}-B_{2x})+B_{3x}(B_{1z}-B_{2z})\\ n_{z3}^{\prime }=B_{2x}{B}_{1y}-B_{1x}{B}_{2y}+B_{3y}(B_{1x}-B_{2x})-B_{3x}(B_{1y}-B_{2y})\end{array}$$

(21)

The normalized vector is

$${\mathit{N}}_z={[n_{z1} n_{z2} n_{z3}]}^{\mathrm{T}}$$

(22)

where

$$n_{z1}=n_{z1}^{\prime }/\sqrt{n_{z1}^{\prime 2}+n_{z2}^{\prime 2}+n_{z3}^{\prime 2}}, n_{z2}=n_{z2}^{\prime }/\sqrt{n_{z1}^{\prime 2}+n_{z2}^{\prime 2}+n_{z3}^{\prime 2}}, n_{z3}=n_{z3}^{\prime }/\sqrt{n_{z1}^{\prime 2}+n_{z2}^{\prime 2}+n_{z3}^{\prime 2}}$$

The x axis of the moving platform constructed from B_i (i = 1, 2, 3) is

$${\mathit{N}}_x^{\prime }={[n_{x1}^{\prime } n_{x2}^{\prime } n_{x3}^{\prime }]}^{\mathrm{T}}$$

(23)

The normalized x vector is

$${\mathit{N}}_x={[n_{x1} n_{x2} n_{x3}]}^{\mathrm{T}}$$

(24)

where

$$n_{x1}=n_{x1}^{\prime }/\sqrt{n_{x1}^{\prime 2}+n_{x2}^{\prime 2}+n_{x3}^{\prime 2}}, n_{x2}=n_{x2}^{\prime }/\sqrt{n_{x1}^{\prime 2}+n_{x2}^{\prime 2}+n_{x3}^{\prime 2}}, n_{x3}=n_{x3}^{\prime }/\sqrt{n_{x1}^{\prime 2}+n_{x2}^{\prime 2}+n_{x3}^{\prime 2}}.$$

The y axis of the moving platform constructed from B_i (i = 1, 2, 3) is

$${\mathit{N}}_y^{\prime }={[n_{y1}^{\prime } n_{y2}^{\prime } n_{y3}^{\prime }]}^{\mathrm{T}}$$

(25)

The normalized y vector is

$${\mathit{N}}_y={[n_{y1} n_{y2} n_{y3}]}^{\mathrm{T}}$$

(26)

where

$$n_{y1}=n_{y1}^{\prime }/\sqrt{n_{y1}^{\prime 2}+n_{y2}^{\prime 2}+n_{y3}^{\prime 2}}, n_{y2}=n_{y2}^{\prime }/\sqrt{n_{y1}^{\prime 2}+n_{y2}^{\prime 2}+n_{y3}^{\prime 2}}, n_{y3}=n_{y3}^{\prime }/\sqrt{n_{y1}^{\prime 2}+n_{y2}^{\prime 2}+n_{y3}^{\prime 2}}.$$

According to the above deduction, the orientation matrix of the moving platform is

$${\mathit{T}}_c=[{\mathit{N}}_x {\mathit{N}}_y {\mathit{N}}_z]$$

(27)

Then, the pose matrix of the moving platform is

$$\mathit{T}=\left[\begin{array}{cc}{\mathit{T}}_c& \mathit{P}\\ \mathbf{0}& 1\end{array}\right]$$

(28)

where $\mathit{P}={[x_o,y_o,z_o]}^{\mathrm{T}}$.

3.3. Error Modeling of PPMs

Equation (28) is a function of B_i (i = 1, 2, 3) and the moving platform radius b_i, taking the derivative of Equation (28) with respect to parameters B_i and b_i.

$$\begin{array}{l}\Delta \mathit{T}={\displaystyle \frac{\partial \mathit{T}}{\partial B_{ix}}}\Delta B_{ix}+{\displaystyle \frac{\partial \mathit{T}}{\partial B_{iy}}}\Delta B_{iy}+{\displaystyle \frac{\partial \mathit{T}}{\partial B_{iz}}}\Delta B_{iz}+{\displaystyle \frac{\partial \mathit{T}}{\partial b_i}}\Delta b_i\\ =\Delta T_{B_{ix}}+\Delta T_{B_{iy}}+\Delta T_{B_{iz}}+\Delta T_{b_i}\end{array}$$

(29)

where $\Delta b_i$ is the difference between the actual and the ideal radius of the moving platform, and ΔB_ix, ΔB_iy, ΔB_iz are the differences between the actual and desired coordinates at point B_i (i = 1, 2, 3).

$$\begin{array}{l}\Delta B_{ix}={\displaystyle \sum \frac{\partial B_{ix}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial B_{ix}}{\partial \theta }\Delta \theta }\\ \Delta B_{iy}={\displaystyle \sum \frac{\partial B_{iy}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial B_{iy}}{\partial \theta }\Delta \theta }\\ \Delta B_{iz}={\displaystyle \sum \frac{\partial B_{iz}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial B_{iz}}{\partial \theta }\Delta \theta }\end{array}$$

(30)

where $\Delta r=\left\{\delta x_{i,j},\delta y_{i,j},\delta z_{i,j}\right\}$, $\Delta \theta =\left\{\delta {\alpha }_{i,j},\delta {\beta }_{i,j},\delta {\gamma }_{i,j}\right\}$. (j = 1, 2, 3, j = 1, 2, 3)

The difference between the actual and ideal pose of the mechanism is

$$d({\mathit{T}}_N)={\mathit{T}}_R-{\mathit{T}}_N={\mathit{T}}_N\mathsf{\Delta }$$

(31)

where

$$\mathsf{\Delta }=\left[\begin{array}{cccc}0& -\delta z& \delta y& dx\\ \delta z& 0& -\delta x& dy\\ -\delta y& \delta x& 0& dz\\ 0& 0& 0& 1\end{array}\right]$$

(32)

According to Equations (29) and (30),

$$\begin{array}{l}d({\mathit{T}}_N)={\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{ix}}\frac{\partial B_{ix}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{ix}}\frac{\partial B_{ix}}{\partial \theta }\Delta \theta }+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{iy}}\frac{\partial B_{iy}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{iy}}\frac{\partial B_{iy}}{\partial \theta }\Delta \theta }+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{iz}}\frac{\partial B_{iz}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{iz}}\frac{\partial B_{iz}}{\partial \theta }\Delta \theta }+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial b}\Delta b}\\ \end{array}$$

(33)

where $\Delta r=\left\{\delta x_{i,j},\delta y_{i,j},\delta z_{i,j}\right\}$, $\Delta \theta =\left\{\delta {\alpha }_{i,j},\delta {\beta }_{i,j},\delta {\gamma }_{i,j}\right\}$, $\Delta b=\left\{\delta b_1,\delta b_2,\delta b_3\right\}$ (i or j = 1, 2, 3).

$\mathsf{\Delta }$ can be obtained as

$$\mathsf{\Delta }={({\mathit{T}}_N)}^{-1}d({\mathit{T}}_N)$$

(34)

Based on the above deduction, the error vector can be obtained as

$${\left[dx dy dz \delta x \delta y \delta z\right]}^T=\mathit{J}\Delta \rho$$

(35)

where J is the error matrix of the PM

$$\begin{array}{l}\mathit{J}=\left({\displaystyle \frac{\partial B_{ix}}{\partial r}}{\displaystyle \frac{\partial B_{ix}}{\partial \theta }}{\displaystyle \frac{\partial B_{iy}}{\partial r}}{\displaystyle \frac{\partial B_{iy}}{\partial \theta }}{\displaystyle \frac{\partial B_{iz}}{\partial r}}{\displaystyle \frac{\partial B_{iz}}{\partial \theta }}{\displaystyle \frac{\partial {\mathit{T}}_N}{\partial b}}\right), \Delta \rho =\left\{\Delta r,\Delta \theta ,\Delta b\right\}, \Delta r=\left\{\delta x_{i,j},\delta y_{i,j},\delta z_{i,j}\right\},\\ \Delta \theta =\left\{\delta {\alpha }_{i,j},\delta {\beta }_{i,j},\delta {\gamma }_{i,j}\right\}, \Delta b=\left\{\delta b_1,\delta b_2,\delta b_3\right\} (i \mathrm{or} j = 1, 2, 3).\end{array}$$

4. Case Study

This section takes the 3-DOF planar 3-RPR PM as an example, and verifies the validity of the proposed error model through calibration simulation.

4.1. Architecture of the 3-RPR PPM

As shown in Figure 5, it is the 3-RPR PPM. The moving platform is an equilateral triangle, and it is connected to the fixed platform $\Delta A_1{A}_2{A}_3$ through three rotational joints. The three limbs are symmetrically distributed, and each limb contains two rotational joints R and one prismatic joint P, with the prismatic joint P being the driving joint. The axis of the rotational joint is perpendicular to the fixed platform $\Delta A_1{A}_2{A}_3$ and the prismatic joint P. The distances from the center point of the moving platform to B₁, B₂, and B₃ are b₁, b₂, and b₃ respectively. Ideally, b₁ = b₂ = b₃. The moving distances of the three driving joints are q₁, q₂, and q₃. A fixed reference coordinate system O-xy is established at point A₁ on the fixed platform. Point O is located at A₁, and the x-axis is along the direction of A₁A₂, and the y-axis is determined by the right-hand rule. A moving coordinate system O’-uv is established at the center point of the moving platform. The u-axis is along the direction of B₁B₂, and the v-axis is determined by the right-hand rule.

Figure 5. Structural diagram of 3-RPR parallel mechanism.

The position and orientation of the moving platform are determined by the position of the center point O′(x, y) of the moving platform and the angle $\varphi$ between the x-axis and B₁B₂. The coordinates of point A_i (i = 1, 2, 3) are A₁ = (c₁, d₁, 0), A₂ = (c₂, d₂, 0), A₃ = (c₃, d₃, 0). The coordinates of point B_i (i = 1, 2, 3) in the moving coordinate system are $B_1^{\prime }$ = (−b₁ cos$\varphi$, −b₁ sin$\varphi$), $B_2^{\prime }$ = (b₂ cos$\varphi$, −b₂ sin$\varphi$), $B_3^{\prime }$ = (0, b₃). Ideally, c₁ = d₁ = d₂ = 0.

4.2. 3-RPR Planar PM Limb KGB Construction

In Figure 5, the coordinate systems of each joint are established [22]. The limb coordinate systems $R_{0,i}$ (i = 1~3) are established at point A₁, and coincide with O-xy. The joint coordinate systems $R_{j,i}$ (j = 1~3) are established at joint j, with the axes $z_{j,i}$ along the joint axis direction and $x_{j,i}$ perpendicular to $z_{j,i}$ and $z_{j+1,i}$.

${}_{j-1}{}^j\mathit{T}_i$ (j = 1~3, i = 1~3) represents the homogeneous rotation transformation of joint j-1 relative to joint j on limb i. The three limbs of the 3-RPR mechanism are symmetrically distributed, and the transformation between each joint is as follows.

Move the coordinate system {O} along its x-axis by a distance of c_i, along its y-axis by a distance of d_i, and then rotate it around the z-axis to transform it into the system {R_0,i}, that is

$${}_0{}^O\mathit{T}_i=\mathrm{Trans}(x,c_i)\mathrm{Trans}(y,d_i)Rot(z,{\theta }_{1.i}-{\beta }_i), {\beta }_i=\pi /2$$

(36)

where

$$\begin{array}{l}\mathrm{Trans}(x,c_i)=1-{\displaystyle \frac{1}{2}}{c}_i{\mathit{e}}_1{\mathit{e}}_{\infty }\\ \mathrm{Trans}(y,d_i)=1-{\displaystyle \frac{1}{2}}{d}_i{\mathit{e}}_2{\mathit{e}}_{\infty }\\ \mathrm{Rot}(z,{\theta }_{1,i}-{\beta }_i)=\mathrm{c}({\displaystyle \frac{{\theta }_{1,i}-{\beta }_i}{2}})-\mathrm{s}({\displaystyle \frac{{\theta }_{1,i}-{\beta }_i}{2}}){\mathit{e}}_3{\mathit{e}}_{\infty }\end{array}$$

{R_0,i} moves a distance of q_i along its y-axis, and then rotates around the x-axis by an angle of ${\alpha }_{0,i}$ to transform into {R_1,i}. The transformation is as follows:

$${}_1{}^0\mathit{T}_i=\mathrm{Trans}(y_{0,i},q_i)\mathrm{Rot}(x_{0,i},{\alpha }_{0,i})$$

(37)

where

$$\begin{array}{l}\mathrm{Trans}(y_{0,i},q_i)=1-{\displaystyle \frac{1}{2}}{q}_i{\mathit{e}}_2{\mathit{e}}_{\infty }\\ \mathrm{Rot}(x_{0,i},{\alpha }_{0,i})=\mathrm{c}({\displaystyle \frac{{\alpha }_{0,i}}{2}})-\mathrm{s}({\displaystyle \frac{{\alpha }_{0,i}}{2}}){\mathit{e}}_1{\mathit{e}}_{\infty }\end{array}$$

{R_1,i} rotates around its x-axis by an angle ${\alpha }_{1,i}$ and is transformed into the coordinate system {R_2,i}

$${}_2{}^{1}T_i=\mathrm{Rot}(x_{1,i},{\alpha }_{1,i})$$

(38)

where

$$\mathrm{Rot}(x_{1,i},{\alpha }_{1,i})=\mathrm{c}({\displaystyle \frac{{\alpha }_{1,i}}{2}})-\mathrm{s}({\displaystyle \frac{{\alpha }_{1,i}}{2}}){\mathit{e}}_1{\mathit{e}}_{\infty }$$

The transformation of the entire limb is

$${}_2{}^O\mathit{T}_i = {}_0{}^O\mathit{T}_i{}_1{}^0\mathit{T}_i{}_2{}^1\mathit{T}_i$$

(39)

From Equation (39), the conformal geometric expression form of B_i (i = 1, 2, 3) can be obtained. The Cartesian coordinates extracted from it are

$${\mathit{B}}_1={[c_1+q_1\cos {\theta }_{1,1}, d_1+q_1\sin {\theta }_{1,1}]}^{\mathrm{T}}$$

(40)

$${\mathit{B}}_2={[c_2+q_2\cos {\theta }_{1,2}, d_2+q_2\sin {\theta }_{1,2}]}^{\mathrm{T}}$$

(41)

$${\mathit{B}}_3={[c_3-q_3\cos {\theta }_{1,3}, d_3+q_3\sin {\theta }_{1,3}]}^{\mathrm{T}}$$

(42)

The limb kinematic geometric bodies S_i (i = 1, 2, 3) can be constructed as

$${\mathit{S}}_i={\mathit{B}}_i-{\displaystyle \frac{1}{2}}{b}_i^2{\mathit{e}}_{\infty }(i=1, 2, 3)$$

(43)

4.3. Kinematics Analysis of 3-RPR Planar PM Using CGA

Three limb kinematic geometric bodies intersect to form a point pair, and the center point O’ of the moving platform is the midpoint of this point pair.

$$\begin{array}{l}{\mathit{P}}_p^{*}=({\mathit{S}}_1\wedge {\mathit{S}}_2\wedge {\mathit{S}}_3){\mathit{E}}^{-1}\\ \mathit{P}={\displaystyle \frac{\pm \sqrt{{\mathit{P}}_p^{*}·{\mathit{P}}_p^{*}}+{\mathit{P}}_p^{*}}{{\mathit{e}}_{\infty }·{\mathit{P}}_p^{*}}}={\displaystyle \frac{{\mathit{C}}_2}{c{o}_3}}\pm {\displaystyle \frac{\sqrt{c{o}_4}}{c{o}_3}}{\mathit{C}}_1\end{array}$$

(44)

where

$${\mathit{C}}_1={\mathit{e}}_{\infty }·{\mathit{P}}_p^{*}, {\mathit{C}}_2={\mathit{C}}_1·{\mathit{P}}_p^{*}, c{o}_3={\mathit{C}}_1·{\mathit{C}}_1, c{o}_4={\mathit{P}}_p^{*}·{\mathit{P}}_p^{*}, \mathit{E}={\mathit{e}}_1\wedge {\mathit{e}}_2\wedge {\mathit{e}}_3\wedge {\mathit{e}}_{\infty }\wedge {\mathit{e}}_0, {\mathit{E}}^{-1}=-\mathit{E}.$$

From Equation (44), the ideal position of the moving platform center point of the 3-RPR parallel mechanism is

$$\begin{array}{l}{x}_o=-c_{e13}/c_{e30}\\ y_o=-c_{e23}/c_{e30}\end{array}$$

(45)

where

$$\begin{array}{l}{c}_{e13}=(B_{y1}(B_{x2}^2-B_{x3}^2)-B_{y2}(B_{x1}^2-B_{x3}^2+B_{y1}^2)+B_{y3}(B_{x1}^2-B_{x2}^2+B_{y1}^2-B_{y2}^2)\\ +B_{y1}{B}_{y2}^2-(B_{y3}^2-b_3^2)(B_{y1}-B_{y2})+b_1^2(B_{y2}-B_{y3})-b_2^2(B_{y1}-B_{y3}))/2\end{array}$$

$$\begin{array}{l}{c}_{e23}=(B_{x1}^2{B}_{x2}-B_{x1}{B}_{x2}^2-B_{x3}(B_{x1}^2-B_{x2}^2)+B_{x3}^2(B_{x1}-B_{x2})\\ +(B_{y1}^2-b_1^2)(B_{x2}-B_{x3})-(B_{y2}^2-b_2^2)(B_{x1}-B_{x3})+(B_{y3}^2-b_3^2)(B_{x1}-B_{x2}))/2\end{array}$$

$$c_{e30}=(B_{x2}-B_{x1})B_{y3}+(B_{x1}-B_{x3})B_{y2}+(B_{x3}-B_{x2})B_{y1}$$

The components of the planar normal vector composed of B_i (i = 1, 2, 3) in each direction are

$$\begin{array}{l}{n}_{z1}^{\prime }=0\\ n_{z2}^{\prime }=0\\ n_{z3}^{\prime }=(B_{x2}-B_{x1})B_{y3}+(B_{x1}-B_{x3})B_{y2}+(B_{x3}-B_{x2})B_{y1}\end{array}$$

(46)

The unitized normal vector is

$${\mathit{N}}_z={[0, 0, n_{z3}]}^{\mathrm{T}}$$

(47)

where $n_{z3}=n_{z3}^{\prime }/\sqrt{n_{z1}^{\prime 2}+n_{z2}^{\prime 2}+n_{z3}^{\prime 2}}$.

The x-axis of the mechanism is constructed along the B₁B₂ direction.

$$\begin{array}{l}{n}_{x1}^{\prime }=B_{2x}-B_{1x}\\ n_{x2}^{\prime }=B_{2y}-B_{1y}\\ n_{x3}^{\prime }=0\end{array}$$

(48)

Normalize the x-axis vector as

$${\mathit{N}}_x={[n_{x1}, n_{x2}, 0]}^{\mathrm{T}}$$

(49)

where

$$n_{x1}=n_{x1}^{\prime }/\sqrt{n_{x1}^{\prime 2}+n_{x2}^{\prime 2}}, n_{x2}=n_{x2}^{\prime }/\sqrt{n_{x1}^{\prime 2}+n_{x2}^{\prime 2}}$$

The y-axis of the mechanism constructed from the midpoint of B₂B₁ to the B₃ direction is

$$\begin{array}{l}{n}_{y1}^{\prime }=B_{3x}-(B_{2x}+B_{1x})/2\\ n_{y2}^{\prime }=B_{3y}-(B_{2y}+B_{1y})/2\\ n_{y3}^{\prime }=0\end{array}$$

(50)

Normalize the y-axis vector as

$${\mathit{N}}_y={[n_{y1}, n_{y2}, 0]}^{\mathrm{T}}$$

(51)

where

$$n_{y1}=n_{y1}^{\prime }/\sqrt{n_{y1}^{\prime 2}+n_{y2}^{\prime 2}}, n_{y2}=n_{y2}^{\prime }/\sqrt{n_{y1}^{\prime 2}+n_{y2}^{\prime 2}}$$

From the derivations in Equations (47), (49) and (51), the orientation matrix of the moving platform is

$${\mathit{T}}_c=[{\mathit{N}}_x {\mathit{N}}_y {\mathit{N}}_z]$$

(52)

The pose matrix of the moving platform is

$$\mathit{T}=\left[\begin{array}{cc}{\mathit{T}}_c& \mathit{P}\\ \mathbf{0}& 1\end{array}\right]$$

(53)

where $\mathit{P}={[x_o, y_o]}^{\mathrm{T}}$, 0 is a 0 vector of $1\times 3$.

4.4. Error Modeling of 3-RPR Planar PM

The orientation matrix T is a function of the coordinates B_i (i = 1, 2, 3) and the length of the moving platform centerline b_i. Differentiate with respect to the coordinates B_i and b_i.

$$\Delta T={\displaystyle \frac{\partial T}{\partial B_{ix}}}\Delta B_{ix}+{\displaystyle \frac{\partial T}{\partial B_{iy}}}\Delta B_{iy}+{\displaystyle \frac{\partial T}{\partial b_i}}\Delta b_i$$

(54)

where $\Delta b_i$ represents the difference between the actual length and the ideal length of the centerline of the moving platform. ΔB_ix, ΔB_iy, and ΔB_iz are the differences between the actual and ideal values of the coordinates of B_i (i = 1, 2, 3).

Take the derivative with respect to B_i (i = 1, 2, 3)

$$\begin{array}{l}\Delta B_{ix}={\displaystyle \sum \frac{\partial B_{ix}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial B_{ix}}{\partial \theta }\Delta \theta }\\ =\Delta c_i+\cos {\theta }_{j,i}\Delta q_i-q_i\sin {\theta }_{j,i}\Delta {\theta }_{j,i}\\ \Delta B_{iy}={\displaystyle \sum \frac{\partial B_{iy}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial B_{iy}}{\partial \theta }\Delta \theta }\\ =\Delta d_i+\Delta q_i\sin {\theta }_{j,i}+q_i\cos {\theta }_{j,i}\Delta {\theta }_{j,i}\end{array}$$

(55)

where $\Delta r=\left\{\delta x_{i,j},\delta y_{i,j},\delta z_{i,j}\right\}$, $\Delta \theta =\left\{\delta {\alpha }_{i,j},\delta {\beta }_{i,j},\delta {\gamma }_{i,j}\right\}$ (i or j = 1, 2, 3).

The difference between the actual pose matrix and the ideal pose matrix is

$$d({\mathit{T}}_N)={\mathit{T}}_R-{\mathit{T}}_N={\mathit{T}}_N\mathsf{\Delta }$$

(56)

where

$$\mathsf{\Delta }=\left[\begin{array}{cccc}0& -\delta z& \delta y& dx\\ \delta z& 0& -\delta x& dy\\ -\delta y& \delta x& 0& dz\\ 0& 0& 0& 1\end{array}\right]$$

(57)

From Equations (54) and (56), we can obtain

$$d({\mathit{T}}_N)={\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{ix}}\frac{\partial B_{ix}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{ix}}\frac{\partial B_{ix}}{\partial \theta }\Delta \theta }+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{iy}}\frac{\partial B_{iy}}{\partial r}\Delta r}+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial B_{iy}}\frac{\partial B_{iy}}{\partial \theta }\Delta \theta }+{\displaystyle \sum \frac{\partial {\mathit{T}}_N}{\partial b}\Delta b}$$

(58)

where $\Delta r=\left\{\delta x_{i,j},\delta y_{i,j},\delta z_{i,j}\right\}$, $\Delta \theta =\left\{\delta {\alpha }_{i,j},\delta {\beta }_{i,j},\delta {\gamma }_{i,j}\right\}$, $\Delta b=\left\{\delta b_1,\delta b_2,\delta b_3\right\}$ (i or j = 1, 2, 3).

$\mathsf{\Delta }$ can be solved as

$$\mathsf{\Delta }={({\mathit{T}}_N)}^{-1}d({\mathit{T}}_N)$$

(59)

The error model can be obtained as

$$\left[\begin{array}{c}dx\\ dy\\ \delta z\end{array}\right]=\mathit{J}\Delta \rho$$

(60)

where J represents the error matrix of the 3-RPR parallel mechanism. $\mathit{J}=\left({\displaystyle \frac{\partial B_{ix}}{\partial r}}{\displaystyle \frac{\partial B_{ix}}{\partial \theta }}{\displaystyle \frac{\partial B_{iy}}{\partial r}}{\displaystyle \frac{\partial B_{iy}}{\partial \theta }}{\displaystyle \frac{\partial B_{iz}}{\partial r}}{\displaystyle \frac{\partial B_{iz}}{\partial \theta }}{\displaystyle \frac{\partial {\mathit{T}}_N}{\partial b}}\right)$, $\Delta \rho =[\delta x_{j,i} \delta y_{j,i} \delta {\gamma }_{j,i} \delta b_i]$, there are a total of 15 errors, they are $\delta x_{1,1}$, $\delta y_{1,1}$, $\delta {\gamma }_{1,1}$, $\delta y_{2,1}$, $\delta b_1$, $\delta x_{1,2}$, $\delta y_{1,2}$, $\delta {\gamma }_{1,2}$, $\delta y_{2,3}$, $\delta b_2$, $\delta x_{1,3}$, $\delta y_{1,3}$, $\delta {\gamma }_{1,3}$, $\delta y_{2,3}$, and $\delta b_3$, respectively.

The redundant error parameters were analyzed using QR decomposition. It was found that the redundant error parameters were $\delta b_2$, $\delta b_3$, and $\delta y_{1,3}$. To avoid the influence of the redundant error parameters on the parameter identification results, these redundant error parameters were removed.

4.5. Simulation Verification of Error Model

The structural parameters of the 3-RPR parallel mechanism are set as c₂ = 1 m, c₃ = 0.5 m, d₃ = $\sqrt{3}/2$ m, b₁ = b₂ = b₃ = $\sqrt{3}/10$ m, and $\varphi =1.5$. Considering that the singular configuration has an impact on the stability of parameter identification, experimental points are selected within the non-singular workspace. In this paper, 30 points on a circular trajectory within the workspace are selected as experimental points. The circular trajectory simultaneously encompasses the movements in both the x and y directions, as well as the continuous changes in the end position, which can effectively trigger most or even all of the geometric errors. Each calibration point can provide two valid position equations, and there will be a total of 60 equations. These 60 equations are used to solve 15 error parameters. In terms of quantity, this is sufficient and provides good redundancy. The motion trajectory is shown in Figure 6.

Figure 6. Selection of reference points in the non-singular workspace of 3-RPR planar PM.

After selecting the simulation calibration points, the simulation identification is carried out through the error matrix. The identification process is shown in Figure 7. The error set value and the identified value are as shown in Table 2.

Figure 7. Parameter identification flowchart.

Table 2. The set values for parameter identification and the identified value.

Errors	Set Values (m/rad)	Identified Value (m/rad)	Errors	Set Values (m/rad)	Identified Value (m/rad)
$\delta x_{1,1}$	$3\times {10}^{-4}$	$2.64\times {10}^{-4}$	$\delta y_{1,2}$	$4\times {10}^{-4}$	$3.62\times {10}^{-4}$
$\delta y_{1,1}$	$3\times {10}^{-4}$	$3.80\times {10}^{-4}$	$\delta {\gamma }_{1,2}$	$3\times {10}^{-3}$	$3.03\times {10}^{-3}$
$\delta {\gamma }_{1,1}$	$2\times {10}^{-2}$	$2.10\times {10}^{-2}$	$\delta y_{2,2}$	$2\times {10}^{-4}$	$1.76\times {10}^{-4}$
$\delta y_{2,1}$	$3\times {10}^{-4}$	$2.83\times {10}^{-4}$	$\delta x_{1,3}$	$3\times {10}^{-4}$	$2.13\times {10}^{-4}$
$\delta b_1$	$2\times {10}^{-4}$	$3.24\times {10}^{-4}$	$\delta {\gamma }_{1,3}$	$2\times {10}^{-3}$	$2.07\times {10}^{-3}$
$\delta x_{1,2}$	$4\times {10}^{-4}$	$3.74\times {10}^{-4}$	$\delta y_{2,3}$	$4\times {10}^{-4}$	$4.15\times {10}^{-4}$

(1)

Select the simulation calibration points, and the number of reference points should be greater than the number of error equations.

(2)

Set the initial kinematic error parameters $\Delta \rho$ and calculate the actual pose $P_i^{\prime }$.

(3)

Based on the current kinematic parameters ${\rho }_i$, the 3-RPR PM’s pose P_i is calculated using the CGA kinematic model.

(4)

Calculate the pose error $\Delta e_i$ and determine whether $\Delta e_i$ is less than the set value $\epsilon$.

(5)

If the conditions are not met, then the regularization parameter identification algorithm is used to identify the error parameters, and then return to the third step to update the identified error parameters. The regularization parameter $\alpha$ is set to 0.9.

(6)

When $\Delta e_i$< $\epsilon$, output the identified error parameters.

The parameter errors have been identified by five times iteration calculation and the pose errors are less than 10⁻⁵, which indicates that the error matrix can effectively identify the error parameters of the mechanism. The identified error parameter values in this table are then used to calculate the driving values, and the compensation and calibration simulation for the 3RPR PM is carried out.

The results before and after position error calibration are shown in Figure 8a, with the maximum value decreasing from 0.18 $\times$ 10⁻² m to 0.25 $\times$ 10⁻³ m, and the average value decreasing from 1.6 $\times$ 10⁻³ m to 0.24 $\times$ 10⁻³ m. The results of the orientation error calibration are shown in Figure 8b, with the maximum value decreasing from 0.45 $\times$ 10⁻² m to 0.33 $\times$ 10⁻³ m, and the average value decreasing from 0.4 $\times$ 10⁻² m to 0.28 $\times$ 10⁻³ m. The experimental results indicate that after calibration, the position error and orientation error of the mechanism have been significantly reduced, proving the effectiveness of the error model.

Figure 8. (a) The position errors before and after calibration (b) The orientation errors before and after calibration.

In the simulation, to focus on the geometric error identification and eliminate the influence of dynamic effects, we adopted a quasi-static approach. The motion speed for both the identification path and the validation path was set to a constant low value of 0.05 m/s in terms of the end-effector’s linear velocity. This speed was selected based on the physical scale of the 3-RPR PPM to ensure that the entire calibration process operates in a quasi-static condition, where the positioning accuracy is dominated by geometric errors rather than dynamic forces. Thereby isolating and verifying the contribution of this work.

5. Conclusions

This paper proposes a novel error modeling method for planar parallel mechanisms based on conformal geometric algebra. The 3-RPR planar parallel mechanism is analyzed as an example. The kinematic geometry body of each limb of the 3-RPR mechanism is derived, and the kinematic model is obtained through intersection operations in CGA. By taking the total differential of the kinematic model, an error model is established. To validate the effectiveness of the CGA-based error modeling approach, a calibration simulation was conducted on the 3-RPR parallel mechanism. The simulation results show that the maximum position error was reduced from 0.18 $\times$ 10⁻² m to 0.25 $\times$ 10⁻³ m after calibration, while the average position error decreased from 1.6 $\times$ 10⁻³ m to 0.24 $\times$ 10⁻³ m. Similarly, the maximum orientation error was reduced from 0.45 $\times$ 10⁻² m to 0.33 $\times$ 10⁻³ m, and the average orientation error decreased from 0.4 $\times$ 10⁻² m to 0.28 $\times$ 10⁻³ m. These results verify the effectiveness of the proposed error modeling method. Future work will be focused on the experimental validation of proposed error modeling method and its application to the error modeling and calibration of spatial parallel mechanisms.