Novel Kinematically Redundant (3+1)-DOF Delta-Type Parallel Mechanisms

Abstract

Although parallel mechanisms are used in various fields, their application is often limited by singularities and a restricted workspace. Kinematic redundancy is a promising approach for mitigating these issues while also extending the functionality of the mechanisms. This article contributes to this field by introducing two novel Delta-type kinematically redundant parallel mechanisms with linear actuators. The moving platform in these mechanisms has three translational degrees of freedom and consists of two parts connected by a prismatic joint, providing an extra translation between the parts. First, we present closed-form solutions to the inverse and forward kinematic problems, accompanied by numerical examples that validate the theoretical derivations. Next, we analyze singular configurations of the mechanisms with a symmetrical design, focusing on parallel singularities. Using an iterative approach, we identify points within the workspace corresponding to these configurations, including finite-motion singularities. Based on this analysis, we changed the geometrical parameters of one mechanism and presented the design where the singularity-free region of the workspace occupies 95% of the total workspace. This study forms the basis for future research on the proposed mechanisms and their prototyping.

1. Introduction

Modern advanced robotics has been significantly influenced by the development of parallel mechanisms, a class of mechanisms renowned for their high stiffness, superior load-carrying capacity, and exceptional dynamic performance compared to serial counterparts. These performance advantages come from a closed-loop architecture, where the end-effector is connected to the base by multiple independent kinematic chains, thereby distributing the load across several actuators [1,2]. The foundational principles of these mechanisms, including their classification, mobility analysis, and the systematic derivation of their kinematic models, have been detailed in seminal works by Merlet [3] and Tsai [4].

Among various parallel mechanisms, the Delta mechanism, invented by Clavel [5], stands out as a landmark contribution to high-speed automation. Its architecture features three revolute actuators on the base connected to the moving platform through parallelogram linkages, providing three translational degrees of freedom (DOFs). The lightweight design of the mechanism makes it suitable for high-acceleration pick-and-place operations in packaging, electronics assembly, and pharmaceutical manufacturing [6,7]. The success of this architecture has spurred the development of a large number of Delta-type mechanisms [8,9,10,11]. Numerous studies have investigated their kinematic and dynamic modeling [12], workspace analysis [13], and performance optimization [14].

Singularities are a major challenge for parallel mechanisms, as they degrade motion control and force transmission [15,16]. Furthermore, the performance of these mechanisms is highly non-uniform; it is usually optimal only in a small central region of the workspace and deteriorates significantly near the boundaries [17]. While commercial devices based on the Delta architecture are typically designed to mitigate the influence of singularities, certain robots are still affected by them [18,19]. To overcome these limitations, introducing redundancy into parallel mechanisms has emerged as a powerful and widely researched design methodology. There are two main types of this redundancy [20]: kinematic redundancy, where additional chains or joints increase the mechanism mobility, and actuation redundancy, where extra actuators are added to a non-redundant architecture. Kinematic redundancy is often more beneficial, as it not only helps to avoid singularities but also extends the workspace [21,22] or provides additional functionality, such as grasping [23,24,25].

Redundancy is rarely added to Delta-type mechanisms, as the original architecture usually demonstrates exceptional performance for its intended applications. However, several notable examples exist. One of the best-known is the Quattro robot produced by Omron Adept, which is based on the H4 mechanism [26]. This mechanism has four kinematic chains identical to the original Delta, but its moving platform is not rigid and comprises two parts connected by a link with revolute joints. The Quattro robot is kinematically equivalent to the H4 but utilizes two parallel links that connect the two parts of the moving platform, forming a planar four-bar parallelogram linkage. Inspired by these ideas, scholars have developed several variations of the H4 mechanism. For instance, in the I4L [27] and I4R [28] mechanisms, the parts of the moving platform are coupled by a rack-and-pinion and a cable-and-pulley transmission, respectively. Nabat et al. [29] proposed the Par4 mechanism, whose moving platform is a planar parallelogram linkage with a belt-and-pulley system that converts the parallelogram motion into an end-effector rotation. Company et al. [30] described a variation of the H4 where this rotation was achieved by a gear mechanism. All these mechanisms can be considered kinematically redundant, with three translational DOFs and one additional DOF corresponding to the relative motion of the platform parts. Leveziel et al. [31] used this redundant motion for grasping and developed a waste sorting robot based on this concept.

In contrast to the family of H4 mechanisms with articulated moving platforms, Kim [32] proposed a kinematically redundant Delta-type mechanism by introducing an additional link and an actuated revolute joint into one of its kinematic chains. This modification, along with a specific placement of the chains, increases the mechanism workspace. Balmaceda-Santamaría et al. [33] mounted a Delta mechanism on a rig that moves by rotating a screw. This motion reconfigures the mechanism base and changes the location of the actuators, thereby reshaping the workspace. Ma et al. [34] presented an innovative (6+3)-DOF kinematically redundant mechanism that utilizes three Delta mechanisms connected to an articulated moving platform. The kinematic chains of each Delta mechanism have a T-shape arrangement to reduce link interference. The mechanism shows excellent rotational capabilities, with rotation angles exceeding ${100}^{°}$.

Although kinematic redundancy is a promising solution for mitigating the workspace and singularity limitations of parallel mechanisms, the literature reveals a scarcity of research on kinematically redundant Delta-type architectures. Existing mechanisms often include auxiliary transmission units, which complicate their design and kinematic model. Furthermore, these mechanisms remain susceptible to singular configurations that can partition the workspace into disjoint regions. This article aims to solve the aforementioned problems and contributes to the further development of kinematically redundant Delta-type parallel mechanisms. It proposes two novel mechanisms with an articulated moving platform whose parts are coupled by a prismatic joint. Apart from this, the major contributions of this work include:

• The algorithms for solving the inverse and forward kinematic problems for these mechanisms. The algorithms are applicable to mechanisms with a general geometry and provide closed-form solutions to both problems.
• Workspace and singularity analysis of the mechanisms with a symmetrical design. The results show that one mechanism has a finite-motion singularity, but changing its geometry makes most of its workspace singularity-free.

The remainder of this paper is structured as follows. Section 2 details the architecture of the mechanisms and introduces kinematic notations. Section 3 and Section 4 consider the inverse and forward kinematics, respectively, and validate the proposed algorithms with numerical examples. Section 5 analyzes workspaces and singular configurations of the mechanisms. Section 6 discusses the results and indicates potential applications. Finally, Section 7 summarizes the research and outlines directions for future work.

2. Architecture of the Proposed Mechanisms

The proposed mechanisms have four identical kinematic chains (Figure 1). Each chain includes linear guide $M_i{N}_i$ ($i=1,\dots ,4$) with actuated carriage (slider) $A_i$ that moves along it. The guide and the slider form an active prismatic joint. The slider is connected to the output link (moving platform) via a parallelogram sub-chain with spherical joints $B_{i1}$, $B_{i2}$, $C_{i1}$, and $C_{i2}$ in its vertices. The moving platform consists of two parts coupled by a prismatic joint, which enables them to translate relative to each other. The first three chains may connect to one part of the platform while the fourth connects to another (Figure 1a), or the first two chains may connect to one part and the third and fourth to the other (Figure 1b). We will refer to the first mechanism as “3+1” and the second as “2+2.”

To analyze the mobility of these mechanisms, we first assume that the moving platform does not include the prismatic joint. In this case, the mechanisms transform into Delta-type mechanisms with actuation redundancy. Each chain constrains the platform rotation around the axis orthogonal to the plane of its parallelogram sub-chain, i.e., each chain imposes an infinite-pitch constraint wrench [35]. Provided that the mechanisms are assembled with linearly independent wrenches, the moving platform has three translational DOFs in a non-singular configuration. The prismatic joint does not add rotational DOFs and, therefore, does not affect the DOFs of the moving platform. This joint, however, introduces local mobility between the two parts of the moving platform, since the chains do not constrain translational movements. Hence, the mechanisms have four DOFs: three translational DOFs of the moving platform and the relative translation between its parts.

The proposed mechanisms represent kinematically redundant systems, and we can denote them as (3+1)-DOF mechanisms in accordance with other studies [24,36,37]. As discussed in the introduction, kinematic redundancy extends the workspace, helps to avoid singular configurations, and can provide extra functionality such as grasping. Unlike other mechanisms reviewed in the literature, the proposed ones have simpler architecture, as they do not include additional transmission units. Subsequent sections analyze kinematics of these mechanisms and describe their features in more detail.

Before starting the kinematic analysis, we introduce the following notations. Let $Oxyz$ be the fixed reference frame placed at the base of the mechanisms (Figure 1). The spatial configuration of the moving platform is fully described by coordinates $x_D$, $y_D$, and $z_D$ of point D, which is located on one part of the platform, and distance s between the two parts of the platform. The position of the i-th slider on the corresponding linear guide is specified by parameter $h_i$, defined as the distance between points $M_i$ and $A_i$. We refer to parameters $x_D$, $y_D$, $z_D$, and s as generalized coordinates of each mechanism, while $h_1,\dots ,h_4$ are called joint coordinates. Kinematic analysis relates the generalized and joint coordinates, providing a means to compute one set of the coordinates based on the other.

3. Inverse Kinematics

We begin by solving the inverse kinematics of the mechanisms, that is, determining the values of joint coordinates $h_1,\dots ,h_4$ for a given set of generalized coordinates $x_D$, $y_D$, $z_D$, and s. The solution algorithm for the “3+1” and “2+2” mechanisms is nearly the same, and we will analyze the inverse kinematics of both mechanisms simultaneously.

3.1. Solution Algorithm

Let point $C_i$ be the midpoint of side $C_{i1}{C}_{i2}$ of parallelogram $B_{i1}{B}_{i2}{C}_{i2}{C}_{i1}$ in the i-th kinematic chain, and let $x_{Ci}^{\prime }$, $y_{Ci}^{\prime }$, and $z_{Ci}^{\prime }$ be its coordinates in the $D{x}^{\prime }{y}^{\prime }{z}^{\prime }$ reference frame attached to the moving platform. Without loss of generality, we position the $Oxyz$ reference frame such that its $Ox$ axis is parallel to the direction of the prismatic joint connecting the two parts of the moving platform (Figure 1). We also assume that the axes of the $D{x}^{\prime }{y}^{\prime }{z}^{\prime }$ reference frame are parallel to the corresponding axes of the $Oxyz$ frame (this assumption is relevant because the moving platform performs translational motion). With these assumptions, coordinates $y_{Ci}^{\prime }$ and $z_{Ci}^{\prime }$ of all chains are constant and determined by the geometry of the moving platform. Depending on the mechanism architecture, either $x_{C1}^{\prime }$, $x_{C2}^{\prime }$, and $x_{C3}^{\prime }$ (for the “3+1” mechanism) or $x_{C1}^{\prime }$ and $x_{C2}^{\prime }$ (for the “2+2” mechanism) are constant. Consequently, either $x_{C4}^{\prime }$ or both $x_{C3}^{\prime }$ and $x_{C4}^{\prime }$ depend on the value of s. Let subscript 0 indicate the value of $x_{Ci}^{\prime }$ corresponding to $s=0$. Then, the following expressions are valid for the “3+1” mechanism:

$$\left\{\begin{array}{cc}{x}_{Ci}^{\prime }=x_{0Ci}^{\prime },\hfill & \mathrm{if}i=1,2,3,\hfill \\ x_{Ci}^{\prime }=x_{0Ci}^{\prime }+s,\hfill & \mathrm{if}i=4,\hfill \end{array}\right.$$

(1)

and for the “2+2” mechanism:

$$\left\{\begin{array}{cc}{x}_{Ci}^{\prime }=x_{0Ci}^{\prime },\hfill & \mathrm{if}i=1,2,\hfill \\ x_{Ci}^{\prime }=x_{0Ci}^{\prime }+s,\hfill & \mathrm{if}i=3,4.\hfill \end{array}\right.$$

(2)

The solution algorithm to the inverse kinematics of both the “3+1” and “2+2” mechanisms is the same, except for the calculation of $x_{Ci}^{\prime }$, as shown in Equations (1) and (2). First, we compute coordinates $x_{Ci}$, $y_{Ci}$, and $z_{Ci}$ of point $C_i$ in the $Oxyz$ reference frame. Since frames $Oxyz$ and $D{x}^{\prime }{y}^{\prime }{z}^{\prime }$ have the same orientation, we can write:

$$x_{Ci}=x_D+x_{Ci}^{\prime },y_{Ci}=y_D+y_{Ci}^{\prime },z_{Ci}=z_D+z_{Ci}^{\prime }.$$

(3)

Next, we consider the i-th kinematic chain (Figure 2). The parallelogram sub-chain is rigidly coupled to the slider, and midpoint $B_i$ of side $B_{i1}{B}_{i2}$ moves along line $P_i{Q}_i$, which is parallel to linear guide $M_i{N}_i$. Moreover, the distance between points $P_i$ and $B_i$ is equal to the distance between points $M_i$ and $A_i$, i.e., to joint coordinate $h_i$. To find this coordinate, we first look at triangle $C_i{P}_i{Q}_i$. Denoting the angle between sides $C_i{P}_i$ and $P_i{Q}_i$ as ${\alpha }_i$, we apply the law of cosines to this triangle:

$$l_{CQi}^2=l_{CPi}^2+l_{PQi}^2-2l_{CPi}{l}_{PQi}cos{\alpha }_i,$$

(4)

where $l_{CQi}$, $l_{CPi}$, and $l_{PQi}$ are the lengths of sides $C_i{Q}_i$, $C_i{P}_i$, and $P_i{Q}_i$, respectively:

$$\begin{array}{cc}\hfill l_{CQi}& =\sqrt{{\left(x_{Qi}-x_{Ci}\right)}^2+{\left(y_{Qi}-y_{Ci}\right)}^2+{\left(z_{Qi}-z_{Ci}\right)}^2},\hfill \\ \hfill l_{CPi}& =\sqrt{{\left(x_{Pi}-x_{Ci}\right)}^2+{\left(y_{Pi}-y_{Ci}\right)}^2+{\left(z_{Pi}-z_{Ci}\right)}^2},\hfill \\ \hfill l_{PQi}& =\sqrt{{\left(x_{Qi}-x_{Pi}\right)}^2+{\left(y_{Qi}-y_{Pi}\right)}^2+{\left(z_{Qi}-z_{Pi}\right)}^2}.\hfill \end{array}$$

(5)

Coordinates $x_{Pi}$, $y_{Pi}$, and $z_{Pi}$ of point $P_i$ and coordinates $x_{Qi}$, $y_{Qi}$, and $z_{Qi}$ of point $Q_i$ are defined by the mechanism geometry, because line $P_i{Q}_i$ is parallel to $M_i{N}_i$. In general, linear guide $M_i{N}_i$ (and line $P_i{Q}_i$) can be placed anywhere, provided that the chains are arranged correctly, as discussed in Section 2.

We also apply the law of cosines to triangle $B_i{C}_i{P}_i$:

$$l_{BCi}^2=l_{CPi}^2+h_i^2-2l_{CPi}{h}_{i}cos{\alpha }_i,$$

(6)

where $l_{BCi}$ is the length of $B_i{C}_i$, which is equal to the length of parallelogram sides $B_{i1}{C}_{i1}$ and $B_{i2}{C}_{i2}$.

After substituting $cos{\alpha }_i$ from Equation (4) to Equation (6) and rearranging the terms, we obtain a quadratic equation with respect to $h_i$:

$$a_i{h}_i^2+b_i{h}_i+c_i=0,$$

(7)

whose coefficients $a_i$, $b_i$, and $c_i$ are listed in Appendix A in Equation (A1).

Quadratic Equation (7) has two solutions, which we will refer as “+” and “−” solutions depending on the sign before the square root of the equation discriminant. The considered mechanisms have four chains; therefore, there are 16 possible solutions to the inverse kinematic problem in a general case. In practice, the linear guide has the finite length, and the solution is feasible only if the value of $h_i$ is within range $\left[h_{i\min },h_{i\max }\right]$.

3.2. Numerical Example

We apply the proposed algorithm to solve the inverse kinematics of both mechanisms, whose geometrical parameters are specified in

Table 1. Geometrical parameters of the “3+1” mechanism (in millimeters).

Parameter	Chain 1	Chain 2	Chain 3	Chain 4
$x_{Mi};y_{Mi};z_{Mi}$	$0;-403.1;0$	$-403.1;0;0$	$0;403.1;0$	$403.1;0;0$
$x_{Ni};y_{Ni};z_{Ni}$	$0;0;-403.1$	$0;0;-403.1$	$0;0;-403.1$	$0;0;-403.1$
$x_{Pi};y_{Pi};z_{Pi}$	$0;-413.7;-10.6$	$-413.7;0;-10.6$	$0;413.7;-10.6$	$413.7;0;-10.6$
$x_{Qi};y_{Qi};z_{Qi}$	$0;-10.6;-413.7$	$-10.6;0;-413.7$	$0;10.6;-413.7$	$10.6;0;-413.7$
$x_{0Ci}^{\prime };y_{Ci}^{\prime };z_{Ci}^{\prime }$	$0;-58.5;0$	$-58.5;0;0$	$0;58.5;0$	$58.5;0;0$
$x_{0Ci1}^{\prime };y_{Ci1}^{\prime };z_{Ci1}^{\prime }$	$-40;-58.5;0$	$-58.5;40;0$	$40;58.5;0$	$58.5;-40;0$
$x_{0Ci2}^{\prime };y_{Ci2}^{\prime };z_{Ci2}^{\prime }$	$40;-58.5;0$	$-58.5;-40;0$	$-40;58.5;0$	$58.5;40;0$
$l_{BCi}$	320	320	320	320

and

Table 2. Geometrical parameters of the “2+2” mechanism (in millimeters).

Parameter	Chain 1	Chain 2	Chain 3	Chain 4
$x_{Mi};y_{Mi};z_{Mi}$	$-285;-285;0$	$-285;285;0$	$285;285;0$	$285;-285;0$
$x_{Ni};y_{Ni};z_{Ni}$	$0;0;-403.1$	$0;0;-403.1$	$0;0;-403.1$	$0;0;-403.1$
$x_{Pi};y_{Pi};z_{Pi}$	$-292.5;-292.5;-10.6$	$-292.5;292.5;-10.6$	$292.5;292.5;-10.6$	$292.5;-292.5;-10.6$
$x_{Qi};y_{Qi};z_{Qi}$	$-7.5;-7.5;-413;7$	$-7.5;7.5;-413.7$	$7.5;7.5;-413.7$	$7.5;-7.5;-413.7$
$x_{0Ci}^{\prime };y_{Ci}^{\prime };z_{Ci}^{\prime }$	$-41.4;-41.4;0$	$-41.4;41.4;0$	$41.4;41.4;0$	$41.4;-41.4;0$
$x_{0Ci1}^{\prime };y_{Ci1}^{\prime };z_{Ci1}^{\prime }$	$-69.7;-13.1;0$	$-13.1;69.7;0$	$69.7;13.1;0$	$13.1;-69.7;0$
$x_{0Ci2}^{\prime };y_{Ci2}^{\prime };z_{Ci2}^{\prime }$	$-13.1;-69.7;0$	$-69.7;13.1;0$	$13.1;69.7;0$	$69.7;-13.1;0$
$l_{BCi}$	320	320	320	320

. With these parameters, the linear guides form an inverted square pyramid. Its base is in the $Oxy$ plane, and its apex is on the negative side of the $Oz$ axis. All edges of the pyramid have the equal length of 570 mm; thus, the angle between each lateral edge and the $Oxy$ plane is ${45}^{°}$. For both mechanisms, we also limit the values of s and $h_i$: $s_{\min }=0$ mm, $s_{\max }=80$ mm, $h_{i\min }=0$ mm, $h_{i\max }=570$ mm.

As an example, we consider the following set of the generalized coordinates:

$$x_D=100\mathrm{mm},y_D=-150\mathrm{mm},z_D=-450\mathrm{mm},s=50\mathrm{mm}.$$

(8)

Implementing the proposed algorithm in MATLAB R2022a, we obtained “+” and “−” solutions to the inverse kinematic problem.

Table 3. Solutions to the inverse kinematics for the “3+1” mechanism (in millimeters).

Solution to Equation (7)	${\mathit{h}}_1$	${\mathit{h}}_2$	${\mathit{h}}_3$	${\mathit{h}}_4$
“+”	$710.7$	$915.0$	$968.3$	$684.8$
“−”	$200.9$	$350.1$	$367.5$	$226.7$

and

Table 4. Solutions to the inverse kinematics for the “2+2” mechanism (in millimeters).

Solution to Equation (7)	${\mathit{h}}_1$	${\mathit{h}}_2$	${\mathit{h}}_3$	${\mathit{h}}_4$
“+”	$789.8$	$998.1$	$793.9$	$653.7$
“−”	$283.9$	$375.6$	$329.8$	$170.0$

present these solutions for the “3+1” and “2+2” mechanisms, respectively. There are 16 real-number solutions for each mechanism. However, if the “+” solution to Equation (7) is chosen, then the values of all $h_i$ exceed $h_{i\max }=570$ mm. Therefore, only one feasible solution exists for each mechanism, corresponding to the “−” solutions to Equation (7) for all chains. Figure 3 shows these solutions, visualized in MATLAB.

4. Forward Kinematics

The forward kinematic problem involves determining the values of generalized coordinates $x_D$, $y_D$, $z_D$, and s for a given set of joint coordinates $h_1,\dots ,h_4$. Unlike the inverse kinematics, solution algorithms to this problem differ for the “3+1” and “2+2” mechanisms. Therefore, we will consider the forward kinematics of each mechanism separately.

4.1. Solution Algorithm for the “3+1” Mechanism

For the “3+1” mechanism, the algorithm relies on the kinematic analysis of a 3-DOF Delta mechanism, whose forward kinematic problem has a closed-form solution [38]. We start by computing the coordinates of points $B_i$:

$$\left[\begin{array}{c}{x}_{Bi}\\ y_{Bi}\\ z_{Bi}\end{array}\right]=\left[\begin{array}{c}{x}_{Pi}\\ y_{Pi}\\ z_{Pi}\end{array}\right]+{\displaystyle \frac{h_i}{l_{PQi}}}\left[\begin{array}{c}{x}_{Qi}-x_{Pi}\\ y_{Qi}-y_{Pi}\\ z_{Qi}-z_{Pi}\end{array}\right].$$

(9)

We can solve the forward kinematics for one half of the moving platform, which is coupled to the first three chains ($i=1,2,3$), by finding the intersection points of three spheres. The i-th sphere has the radius of $l_{BCi}$ and the center at point $E_i$ with the following coordinates (Figure 4):

$$x_{Ei}=x_{Bi}-x_{0Ci}^{\prime },y_{Ei}=y_{Bi}-y_{0Ci}^{\prime },z_{Ei}=z_{Bi}-z_{0Ci}^{\prime },$$

(10)

where we used the conditions that the orientation of reference frame $D{x}^{\prime }{y}^{\prime }{z}^{\prime }$ remains the same as $Oxyz$ and that $x_{Ci}^{\prime }=x_{0Ci}^{\prime }$, $y_{Ci}^{\prime }=y_{0Ci}^{\prime }$, and $z_{Ci}^{\prime }=z_{0Ci}^{\prime }$ for $i=1,2,3$ according to Equation (1).

The equations of the three spheres are as follows:

$$\left\{\begin{array}{c}{\left(x_D-x_{E1}\right)}^2+{\left(y_D-y_{E1}\right)}^2+{\left(z_D-z_{E1}\right)}^2=l_{BC1}^2,\hfill \\ {\left(x_D-x_{E2}\right)}^2+{\left(y_D-y_{E2}\right)}^2+{\left(z_D-z_{E2}\right)}^2=l_{BC2}^2,\hfill \\ {\left(x_D-x_{E3}\right)}^2+{\left(y_D-y_{E3}\right)}^2+{\left(z_D-z_{E3}\right)}^2=l_{BC3}^2.\hfill \end{array}\right.$$

(11)

Thus, we have a system of three equations with three variables: $x_D$, $y_D$, and $z_D$. After expanding these equations and subtracting the second and third equations from the first one, we obtain a system of two linear equations:

$$\left\{\begin{array}{c}{m}_{12}{x}_D+n_{12}{y}_D+p_{12}{z}_D=t_{12},\hfill \\ m_{13}{x}_D+n_{13}{y}_D+p_{13}{z}_D=t_{13},\hfill \end{array}\right.$$

(12)

whose coefficients $m_{12},\dots ,t_{13}$ are listed in Equation (A2).

Assuming that $p_{12}\ne 0$ and $p_{13}\ne 0$, we can rewrite system (12) as follows:

$$\left\{\begin{array}{c}{z}_D={\displaystyle \frac{t_{12}}{p_{12}}}-{\displaystyle \frac{m_{12}}{p_{12}}}{x}_D-{\displaystyle \frac{n_{12}}{p_{12}}}{y}_D,\hfill \\ z_D={\displaystyle \frac{t_{13}}{p_{13}}}-{\displaystyle \frac{m_{13}}{p_{13}}}{x}_D-{\displaystyle \frac{n_{13}}{p_{13}}}{y}_D.\hfill \end{array}\right.$$

(13)

Subtracting the second equation in system (13) from the first one and rearranging the terms, we express variable $y_D$ in terms of $x_D$:

$$y_D=q_y-r_y{x}_D,$$

(14)

where

$$q_y=\left({\displaystyle \frac{t_{13}}{p_{13}}}-{\displaystyle \frac{t_{12}}{p_{12}}}\right)/\left({\displaystyle \frac{n_{13}}{p_{13}}}-{\displaystyle \frac{n_{12}}{p_{12}}}\right),r_y=\left({\displaystyle \frac{m_{13}}{p_{13}}}-{\displaystyle \frac{m_{12}}{p_{12}}}\right)/\left({\displaystyle \frac{n_{13}}{p_{13}}}-{\displaystyle \frac{n_{12}}{p_{12}}}\right).$$

(15)

Next, we substitute Equation (14) into any equation in system (13). For instance, substituting it into the second equation yields the following:

$$z_D=q_z+r_z{x}_D,$$

(16)

where

$$q_z={\displaystyle \frac{t_{13}}{p_{13}}}-{\displaystyle \frac{n_{13}}{p_{13}}}{q}_y,r_z={\displaystyle \frac{n_{13}}{p_{13}}}{r}_y-{\displaystyle \frac{m_{13}}{p_{13}}}.$$

(17)

Now, we can substitute Equations (14) and (16) into any equation in system (11). Using the first equation, after expanding and rearranging the terms, we obtain a quadratic equation for variable $x_D$:

$$a_x{x}_D^2+b_x{x}_D+c_x=0,$$

(18)

whose coefficients $a_x$, $b_x$, and $c_x$ are listed in Equation (A3).

In a general case, Equation (18) has two solutions, which we substitute into Equations (14) and (16) to calculate $y_D$ and $z_D$.

If $p_{12}=0$, $p_{13}=0$, or $n_{12}=n_{13}=0$, the presented algorithm becomes infeasible because of the division by zero in Equations (13) and (15). However, if either $p_{12}=0$ or $p_{13}=0$, we can derive the relation between $x_D$ and $y_D$ directly from the first or second equation of system (12), respectively, and then substitute it into the other equation of the same system to obtain the relation between $x_D$ and $z_D$. These relations are then substituted into any equation of system (11) to yield an equation similar to Equation (18). If both $p_{12}$ and $p_{13}$ are zero, Equation (12) reduces to a system of two linear equations with two unknowns, $x_D$ and $y_D$, which can be easily solved. Their values are then substituted into any equation of system (11), resulting in a quadratic equation in $z_D$. A similar approach applies when $n_{12}=n_{13}=0$, but variables $x_D$ and $z_D$ are found from linear system (12) and then substituted into any equation of system (11) to obtain a quadratic equation in $y_D$.

Computed parameters $x_D$, $y_D$, and $z_D$ determine the configuration of one part of the platform, and it remains to find displacement s in its prismatic joint. For this purpose, we can write the following equation for the fourth chain:

$${\left(x_{C4}-x_{B4}\right)}^2+{\left(y_{C4}-y_{B4}\right)}^2+{\left(z_{C4}-z_{B4}\right)}^2=l_{BC4}^2.$$

(19)

Using Equations (1) and (3), we obtain a quadratic equation for variable s:

$$a_s{s}^2+b_{s}s+c_s=0,$$

(20)

whose coefficients $a_s$, $b_s$, and $c_s$ are listed in Equation (A4).

Equation (20) has two solutions in a general case; therefore, there can be up to four different solutions to the forward kinematic problem for the “3+1” mechanism. In practice, the displacement in the prismatic joint between the platform parts is limited, and the solution is feasible only if the value of s is within range $\left[s_{\min },s_{\max }\right]$.

4.2. Solution Algorithm for the “2+2” Mechanism

Unlike the “3+1” case, we cannot use system (11) to find coordinates $x_D$, $y_D$, and $z_D$ as the intersection of three spheres because the third equation in this system is not valid anymore. Therefore, we propose another approach in this section.

First, we consider each half of the moving platform independently. Suppose we disassembled the platform at its prismatic joint and locked the actuators of each part of the mechanism. Under these conditions, point D on one half of the platform traces a circle formed by the intersection of two spheres centered at points $E_1$ and $E_2$, with radii $l_{BC1}$ and $l_{BC2}$ (Figure 5a). The first two equations in system (11) are the equations of these spheres. Likewise, we can define point F on the other half of the platform, which traces the second circle. This circle is the intersection of two spheres centered at points $E_3$ and $E_4$, with radii $l_{BC3}$ and $l_{BC4}$. If point F has coordinates $x_F$, $y_F$, and $z_F$ in the $Oxyz$ frame, then these two spheres have the following equations:

$$\left\{\begin{array}{c}{\left(x_F-x_{E3}\right)}^2+{\left(y_F-y_{E3}\right)}^2+{\left(z_F-z_{E3}\right)}^2=l_{BC3}^2,\hfill \\ {\left(x_F-x_{E4}\right)}^2+{\left(y_F-y_{E4}\right)}^2+{\left(z_F-z_{E4}\right)}^2=l_{BC4}^2.\hfill \end{array}\right.$$

(21)

Next, we apply the assembling condition: points D and F stay on a line parallel to the $Ox$ axis at the distance of s:

$$x_F=x_D+s,y_F=y_D,z_F=z_D.$$

(22)

Thus, we have four quadratic equations of the four spheres and three linear equations of the assembling condition with seven variables: $x_D$, $y_D$, $z_D$, $x_F$, $y_F$, $z_F$, and s. Instead of solving these equations directly, we reduce the number of variables by using the parametric equations of the circles.

Each circle is determined by the following parameters: the radius, the coordinates of its center, and the normal to its plane. For example, we consider the first circle for point D. Let $r_D$ be the circle radius; $x_{O12}$, $y_{O12}$, and $z_{O12}$ be the coordinates of its center $O_{12}$ in the $Oxyz$ frame; $n_D^x$, $n_D^y$, and $n_D^z$ be the coordinates of its unit normal in the $Oxyz$ frame (Figure 5b). Appendix B shows how to compute these parameters when we know the parameters of the two spheres. Now, we can write the coordinates of point D as follows:

$$\left[\begin{array}{c}{x}_D\\ y_D\\ z_D\end{array}\right]=\left[\begin{array}{c}{x}_{O12}\\ y_{O12}\\ z_{O12}\end{array}\right]+r_D\left(\left[\begin{array}{c}{u}_D^x\\ u_D^y\\ u_D^z\end{array}\right]cos{\theta }_D+\left[\begin{array}{c}{v}_D^x\\ v_D^y\\ v_D^z\end{array}\right]sin{\theta }_D\right),$$

(23)

where angle ${\theta }_D$ defines the angular position of point D on the circle; ${\left[u_D^x{u}_D^y{u}_D^z\right]}^{\mathrm{T}}$ and ${\left[v_D^x{v}_D^y{v}_D^z\right]}^{\mathrm{T}}$ are two mutually orthogonal unit vectors, orthogonal to the normal of the circle plane (Figure 5b). These unit vectors are determined from the coordinates of the normal, as described in Appendix B.

Now, let $r_F$ be the radius of the second circle, which corresponds to point F; $x_{O34}$, $y_{O34}$, and $z_{O34}$ be the coordinates of its center $O_{34}$ in the $Oxyz$ frame; $n_D^x$, $n_D^y$, and $n_D^z$ be the coordinates of its unit normal in the $Oxyz$ frame. These parameters are computed similarly to the first circle. Using these notations, we can compose an equation like Equation (23) by replacing subscripts D and $O12$ with F and $O34$, respectively. Substituting the obtained expressions into assembling condition (22), we derive a system of three equations with respect to variables s, ${\theta }_D$, and ${\theta }_F$:

$$\left\{\begin{array}{c}{x}_{O34}+r_F{u}_F^{x}cos{\theta }_F+r_F{v}_F^{x}sin{\theta }_F=x_{O12}+r_D{u}_D^{x}cos{\theta }_D+r_D{v}_D^{x}sin{\theta }_D+s,\hfill \\ y_{O34}+r_F{u}_F^{y}cos{\theta }_F+r_F{v}_F^{y}sin{\theta }_F=y_{O12}+r_D{u}_D^{y}cos{\theta }_D+r_D{v}_D^{y}sin{\theta }_D,\hfill \\ z_{O34}+r_F{u}_F^{z}cos{\theta }_F+r_F{v}_F^{z}sin{\theta }_F=z_{O12}+r_D{u}_D^{z}cos{\theta }_D+r_D{v}_D^{z}sin{\theta }_D.\hfill \end{array}\right.$$

(24)

The second and third equations in the system above do not depend on s, and we use these equations to determine variables ${\theta }_D$ and ${\theta }_F$. For this purpose, we apply the tangent half-angle substitution [39] (sect. 4.5) to transform these trigonometric equations into the algebraic ones:

$$\begin{array}{cccccc}\hfill t_D& =tan{\displaystyle \frac{{\theta }_D}{2}},\hfill & \hfill cos{\theta }_D& ={\displaystyle \frac{1-t_D^2}{1+t_D^2}},\hfill & \hfill sin{\theta }_D& ={\displaystyle \frac{2t_D}{1+t_D^2}},\hfill \\ \hfill t_F& =tan{\displaystyle \frac{{\theta }_F}{2}},\hfill & \hfill cos{\theta }_F& ={\displaystyle \frac{1-t_F^2}{1+t_F^2}},\hfill & \hfill sin{\theta }_F& ={\displaystyle \frac{2t_F}{1+t_F^2}}.\hfill \end{array}$$

(25)

Substituting these expressions into the second and third equations of system (24) and eliminating the denominators, we obtain two quartic equations in variables $t_D$ and $t_F$:

$$\left\{\begin{array}{c}(y_{O12}-y_{O34}-r_D{u}_D^y+r_F{u}_F^y)t_D^2{t}_F^2-2r_F{v}_F^y{t}_D^2{t}_F+2r_D{v}_D^y{t}_D{t}_F^2,\hfill \\ +(y_{O12}-y_{O34}-r_D{u}_D^y-r_F{u}_F^y)t_D^2+(y_{O12}-y_{O34}+r_D{u}_D^y+r_F{u}_F^y)t_F^2\hfill \\ +2r_D{v}_D^y{t}_D-2r_F{v}_F^y{t}_F+y_{O12}-y_{O34}+r_D{u}_D^y-r_F{u}_F^y=0,\hfill \\ (z_{O12}-z_{O34}-r_D{u}_D^z+r_F{u}_F^z)t_D^2{t}_F^2-2r_F{v}_F^z{t}_D^2{t}_F+2r_D{v}_D^z{t}_D{t}_F^2,\hfill \\ +(z_{O12}-z_{O34}-r_D{u}_D^z-r_F{u}_F^z)t_D^2+(z_{O12}-z_{O34}+r_D{u}_D^z+r_F{u}_F^z)t_F^2\hfill \\ +2r_D{v}_D^z{t}_D-2r_F{v}_F^z{t}_F+z_{O12}-z_{O34}+r_D{u}_D^z-r_F{u}_F^z=0.\hfill \end{array}\right.$$

(26)

Although the obtained equations are quartic in $t_D$ and $t_F$, each of them is quadratic in either $t_D$ or $t_F$. Therefore, we can consider Equation (26) as a system of two quadratic equations with respect to variable $t_D$ ($t_F$), whose coefficients depend on variable $t_F$ ($t_D$). For example, we can rewrite Equation (26) as follows:

$$\left\{\begin{array}{c}{a}_y{t}_F^2+b_y{t}_F+c_y=0,\hfill \\ a_z{t}_F^2+b_z{t}_F+c_z=0,\hfill \end{array}\right.$$

(27)

whose coefficients $a_y,\dots ,c_z$ are listed in Equation (A5).

Next, we follow the dialytic elimination approach [40] to exclude variable $t_F$ from Equation (27). For this purpose, we multiply both equations in system (27) by $t_F$ and obtain a system of four equations:

$$\left\{\begin{array}{c}{a}_y{t}_F^3+b_y{t}_F^2+c_y{t}_F=0,\hfill \\ a_y{t}_F^2+b_y{t}_F+c_y=0,\hfill \\ a_z{t}_F^3+b_z{t}_F^2+c_z{t}_F=0,\hfill \\ a_z{t}_F^2+b_z{t}_F+c_z=0,\hfill \end{array}\right.$$

(28)

which can be written in a matrix form:

$$\left[\begin{array}{cccc}{a}_y& b_y& c_y& 0\\ 0& a_y& b_y& c_y\\ a_z& b_z& c_z& 0\\ 0& a_z& b_z& c_z\end{array}\right]\left[\begin{array}{c}{t}_F^3\\ t_F^2\\ t_F\\ 1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right].$$

(29)

Equation (29) represents a system of four linear equations with respect to the variables in the column-vector. This system should have a nontrivial solution, which is possible if and only if the determinant of the matrix in Equation (29) equals zero. This condition yields the following equation:

$${(a_y{c}_z-a_z{c}_y)}^2-a_y{b}_y{b}_z{c}_z+a_y{b}_z^2+a_z{b}_y^2{c}_z-a_z{b}_y{b}_z{c}_y=0.$$

(30)

According to Equation (A5), each coefficient in the obtained equation is quadratic in $t_D$. Thus, Equation (30) is an octic polynomial equation in variable $t_D$. Note that coefficients $b_y$ and $b_z$ has common multiplier $2r_F(t_D^2+1)$. Symbolic computations show that difference $a_y{c}_z-a_z{c}_y$ also has this multiplier. Therefore, we can factor the polynomial in Equation (30) and represent this equation as follows:

$$4r_F^2{\left(t_D^2+1\right)}^2\left(a_D{t}_D^4+b_D{t}_D^3+c_D{t}_D^2+e_D{t}_D+f_D\right)=0,$$

(31)

where coefficients $a_D,\dots ,f_D$ are constant and can be deduced from symbolic computations.

Since we are only interested in real solutions, we can ignore the ${(t_D^2+1)}^2$ multiplier and get the final polynomial equation:

$$a_D{t}_D^4+b_D{t}_D^3+c_D{t}_D^2+e_D{t}_D+f_D=0.$$

(32)

Note that the forward kinematics of the “2+2” mechanism cannot have more than four distinct solutions. Indeed, if we project the circles traced by points D and F onto the plane orthogonal to the $Ox$ axis, these projections will be two ellipses in a general case. These ellipses can intersect at up to four points, with each point corresponding to a unique solution to quartic polynomial Equation (32). This equation has a closed-form solution, which can be found using Ferrari’s method or other techniques [41]. These methods usually involve tricky variable substitutions and can be inconvenient for practical use. Alternatively, we can compute all solutions numerically as the eigenvalues of the companion matrix of the obtained polynomial [42]. In any case, we assume that we can solve Equation (32) and find all its solutions.

For each solution to Equation (32), we compute $cos{\theta }_F$ and $sin{\theta }_F$ using Equation (25). After that, we determine $cos{\theta }_D$ and $sin{\theta }_D$ from the second and third equations in system (24), which can be considered as two linear equations with respect to these variables:

$$\left\{\begin{array}{c}{r}_D{u}_D^{y}cos{\theta }_D+r_D{v}_D^{y}sin{\theta }_D=y_{O34}-y_{O12}+r_F{u}_F^{y}cos{\theta }_F+r_F{v}_F^{y}sin{\theta }_F,\hfill \\ r_D{u}_D^{z}cos{\theta }_D+r_D{v}_D^{z}sin{\theta }_D=z_{O34}-z_{O12}+r_F{u}_F^{z}cos{\theta }_F+r_F{v}_F^{z}sin{\theta }_F.\hfill \end{array}\right.$$

(33)

A solution to this linear system must satisfy Pythagorean’s identity. This condition holds if $u_D^y=v_D^z$ and $v_D^y=-u_D^z$ [39] (sect. 4.4), but it may not be true in a general case, including the considered one. On the other hand, solutions to the forward kinematic problem (the values of variables ${\theta }_D$, ${\theta }_F$, and s) should not depend on the reference frame where we establish the kinematic equations. In particular, one can transform system (24) to the frame, whose x axis is parallel to vector ${\left[n_D^x{n}_D^y{n}_D^z\right]}^{\mathrm{T}}$ and where conditions $u_D^y=v_D^z$ and $v_D^y=-u_D^z$ are always true. Therefore, we conclude that system (33) can indeed be used to find $cos{\theta }_F$ and $sin{\theta }_F$, which will be verified in the subsequent section.

Once we found $cos{\theta }_F$, $sin{\theta }_F$, $cos{\theta }_D$, and $sin{\theta }_D$, we can determine variables $x_D$, $y_D$, and $z_D$ using Equation (23) and variable s using the first equation in system (24). This concludes the forward kinematic problem for the “2+2” mechanism and shows that it can have up to four distinct solutions, similar to the “3+1” mechanism. This result also agrees with paper [43]. Like in the “3+1” case, the solution is feasible only if the value of s is within range $\left[s_{\min },s_{\max }\right]$.

Note that we could rewrite Equation (26) as a system of two quadratic equations for variable $t_D$ instead of $t_F$ and then use the dialytic elimination approach to find variable $t_F$ first. Equations (27)–(27) will have the same structure in this case, so either approach can be used. Finally, we should mention three special cases:

• If $r_D=0$, we cannot use Equation (33) to find parameters $cos{\theta }_D$ and $sin{\theta }_D$. In this case, the spheres centered at points $E_1$ and $E_2$ touch each other at a single point, and angle ${\theta }_D$ is undefined. The coordinates of point D are readily obtained from Equation (23), and it remains to compute variable s. For this purpose, we first determine $cos{\theta }_F$ and $sin{\theta }_F$ from the second and third equations of system (24), as we solved Equation (33). After that, we find variable ${\theta }_D$ from the first equation of this system. The forward kinematic problem has a single solution in this case.
• If $u_D^y{v}_D^z-u_D^z{v}_D^y=n_D^x=0$, we also cannot use Equation (33) to find parameters $cos{\theta }_D$ and $sin{\theta }_D$. In this case, the $Ox$ axis belongs to the plane of the circle traced by point D. To find the position of point D on this circle and solve the forward kinematics, we should use the dialytic elimination approach to eliminate varible $t_D$ instead of $t_F$, as discussed earlier.
• If both $n_D^x=0$ and $n_F^x=0$, we cannot use proposed method to solve the forward kinematic problem, which has an infinite number of solutions under these conditions. Indeed, in this case, the $Ox$ axis belongs to the planes of both circles, and the mechanism can be assembled for any values of variable s. This is an example of a finite-motion singular configuration, which will be considered in Section 5.

4.3. Numerical Examples

We apply the proposed algorithms to solve the forward kinematics for the mechanisms examined in Section 3.2. We use the feasible solutions to the inverse kinematics obtained in that section as the input data for solving the forward kinematic problem.

For the “3+1” mechanism, the solutions corresponding to joint coordinates $h_1=200.9$ mm, $h_2=350.1$ mm, $h_3=367.5$ mm, $h_4=226.7$ mm are listed in

Table 5. Solutions to the forward kinematics for the “3+1” mechanism (in millimeters).

Solution #	${\mathit{x}}_{\mathit{D}}$	${\mathit{y}}_{\mathit{D}}$	${\mathit{z}}_{\mathit{D}}$	s
1	$100.0$	$-150.0$	$-450.0$	$139.7$
2	$100.0$	$-150.0$	$-450.0$	$50.0$
3	$-10.7$	$38.8$	$44.3$	$439.1$
4	$-10.7$	$38.8$	$44.3$	$-28.0$

and illustrated in Figure 6. All four solutions to the forward kinematics are real. Solution #2 provides the values of $x_D$, $y_D$, $z_D$, and s specified in Equation (8). Furthermore, only this solution is feasible because solutions #1, #3, and #4 violate condition $s_{\min }\le s\le s_{\max }$.

For the “2+2” mechanism, the solutions corresponding to joint coordinates $h_1=283.9$ mm, $h_2=375.6$ mm, $h_3=329.8$ mm, $h_4=170.0$ mm are listed in

Table 6. Solutions to the forward kinematics for the “2+2” mechanism (in millimeters).

Solution #	${\mathit{x}}_{\mathit{D}}$	${\mathit{y}}_{\mathit{D}}$	${\mathit{z}}_{\mathit{D}}$	s
1	$100.0$	$-150.0$	$-450.0$	$50.0$
2	$-150.8$	$97.1$	$29.8$	$402.6$
3	$-223.8+486.2\mathrm{i}$	$214.1-70.3\mathrm{i}$	$289.3+156.7\mathrm{i}$	$478.1-930.0\mathrm{i}$
4	$-223.8-486.2\mathrm{i}$	$214.1+70.3\mathrm{i}$	$289.3-156.7\mathrm{i}$	$478.1+930.0\mathrm{i}$

and shown in Figure 7. Unlike the “3+1” mechanism, the forward kinematics has only two real solutions. Solution #1 gives us the values of $x_D$, $y_D$, $z_D$, and s specified in Equation (8). This is the only feasible solution because solution #2 violates condition $s\le s_{\max }$.

5. Workspace and Singularity Analysis

The performed kinematic analysis allows us to estimate the workspace and singular configurations of the mechanisms. We apply an iterative approach [44] to solve these problems: we fix the value of s and vary the values of $x_D$, $y_D$, and $z_D$ within a specified range with some step size. For each set of generalized coordinates, we solve the inverse kinematics: if a real-number solution exists and satisfies the displacement limits, the point lies within the workspace. In this case, we compute two matrices, ${\mathbf{J}}_A$ and ${\mathbf{J}}_B$:

$${\mathbf{J}}_A=\left[\begin{array}{cccc}{\displaystyle \frac{\partial F_1}{\partial x_D}}& {\displaystyle \frac{\partial F_1}{\partial y_D}}& {\displaystyle \frac{\partial F_1}{\partial z_D}}& {\displaystyle \frac{\partial F_1}{\partial s}}\\ {\displaystyle \frac{\partial F_2}{\partial x_D}}& {\displaystyle \frac{\partial F_2}{\partial y_D}}& {\displaystyle \frac{\partial F_2}{\partial z_D}}& {\displaystyle \frac{\partial F_2}{\partial s}}\\ {\displaystyle \frac{\partial F_3}{\partial x_D}}& {\displaystyle \frac{\partial F_3}{\partial y_D}}& {\displaystyle \frac{\partial F_3}{\partial z_D}}& {\displaystyle \frac{\partial F_3}{\partial s}}\\ {\displaystyle \frac{\partial F_4}{\partial x_D}}& {\displaystyle \frac{\partial F_4}{\partial y_D}}& {\displaystyle \frac{\partial F_4}{\partial z_D}}& {\displaystyle \frac{\partial F_4}{\partial s}}\end{array}\right],{\mathbf{J}}_B=\left[\begin{array}{cccc}{\displaystyle \frac{\partial F_1}{\partial h_1}}& 0& 0& 0\\ 0& {\displaystyle \frac{\partial F_2}{\partial h_2}}& 0& 0\\ 0& 0& {\displaystyle \frac{\partial F_3}{\partial h_3}}& 0\\ 0& 0& 0& {\displaystyle \frac{\partial F_4}{\partial h_4}}\end{array}\right],$$

(34)

where $F_i$ is the constraint equation of the i-th chain, expressed as an implicit function of four generalized coordinates and the i-th joint coordinate; that is, $F_i\left(x_D,y_D,z_D,s,h_i\right)=0$, as shown in Equation (7).

Next, we follow Gosselin and Angeles’s method [15] to analyze singular configurations. If $det\left({\mathbf{J}}_B\right)=0$, the mechanism experiences a serial (Type I) singularity, and its moving platform loses one or more DOFs. If $det\left({\mathbf{J}}_A\right)=0$, a parallel (Type II) singularity occurs, resulting in uncontrolled movement of the platform. The iterative approach struggles to accurately identify the singular points because of the finite precision of numerical computations. A good approximation of the singularity loci can be achieved by analyzing changes in the sign of the determinants [45]. If the iteration step is small enough, a sign change in the determinant indicates a singularity between the two points. In this study, we focus only on parallel singularities because they have more adverse effects on the mechanism performance and because serial singularities are usually located only at the workspace boundary.

During the simulations, we used the same dimensions of the mechanisms as in the previous examples. The iteration ranges for the coordinates of point D were $[-400,400]$ mm for $x_D$ and $y_D$ and $[-800,200]$ mm for $z_D$. The iteration step for all coordinates was 10 mm, and the total number of iterations was 662,661. We analyzed workspaces and singularities for three different values of s: 0 mm, 40 mm, and 80 mm. No constraints or conditions—such as allowable rotation angles in the spherical joints or link interference—were taken into account, except for condition $h_{i\min }\le h_i\le h_{i\max }$. Under these assumptions, the workspace represents the intersection of the volumes generated by the spheres considered in Section 4 and displaced along the linear guides between the specified limits [46]. The results of the analysis are presented in

Table 7. The number of points with positive or negative value of $det\left({\mathbf{J}}_A\right)$ within the workspace for different values of s.

s/mm	“3+1” Mechanism		“2+2” Mechanism
	$\mathbf{det}\mathbf{\left(}{\mathbf{J}}_{\mathit{A}}\mathbf{\right)}\mathbf{>}\mathbf{0}$	$\mathbf{det}\left({\mathbf{J}}_{\mathit{A}}\mathbf{\right)}\mathbf{<}\mathbf{0}$	$\mathbf{det}\mathbf{\left(}{\mathbf{J}}_{\mathit{A}}\mathbf{\right)}\mathbf{>}\mathbf{0}$	$\mathbf{det}\mathbf{\left(}{\mathbf{J}}_{\mathit{A}}\mathbf{\right)}\mathbf{<}\mathbf{0}$
0	15,559	95,819	53,527	57,514
40	18,346	86,276	49,241	52,717
80	20,156	76,019	44,138	47,423

and illustrated in Figure 8 and Figure 9. In these and subsequent figures, we use the red and blue color to indicate the points where $det\left({\mathbf{J}}_A\right)>0$ and $det\left({\mathbf{J}}_A\right)<0$, respectively.

With the selected geometrical parameters, the “3+1” mechanism suits better for practical applications than the “2+2” mechanism. The central region of the “3+1” mechanism workspace includes points with $det\left({\mathbf{J}}_A\right)<0$, while most points with $det\left({\mathbf{J}}_A\right)>0$ are on the side of the workspace in the positive direction of the $Ox$ axis. Some areas with $det\left({\mathbf{J}}_A\right)>0$ are on the three other sides of the workspace, and few such points are at the top of the workspace. The central region of the workspace takes 86% of the total workspace volume when $s=0$ mm, 82% when $s=40$ mm, and 79% when $s=80$ mm. In contrast, the workspace of the “2+2” mechanism divides into four large regions by two vertical planes: the $Oxz$ plane and a plane parallel to the $Oyz$ plane passing through point $x=-s/2$. The mechanism can only operate in approximately one quarter of the total workspace volume without crossing the parallel singularity. Although this analysis ignores link interference and constraints in the passive joints, the difference between the two mechanisms is still significant.

The analysis of singular configurations with uncontrollable motion between the two parts of the moving platform reveals other differences between the mechanisms. For the “3+1” mechanism, we examine a point near the boundary between the central region of the workspace, which contains points with $det\left({\mathbf{J}}_A\right)<0$, and the largest region with $det\left({\mathbf{J}}_A\right)>0$: $x_D\approx 145.8$ mm, $y_D=0$ mm, $z_D=-500$ mm, and $s=40$ mm. At this point, the plane defined by the parallelogram of the fourth chain becomes orthogonal to the direction of the prismatic joint connecting the two parts of the moving platform (Figure 10), preventing force transmission through this chain. This results in a loss of movement control associated with generalized coordinate s. Theoretically, the relative movement between the two parts of the moving platform is infinitesimal. In a real mechanism, however, this movement can have a finite value because of link elasticity and joint clearances [47].

For the “2+2” mechanism, we first examine a point in the $Oxz$ plane (Figure 11a): $x_D=50$ mm, $y_D=0$ mm, $z_D=-500$ mm, and $s=40$ mm. Since the moving platform maintains its orientation, the first and second chains allow point D to be anywhere on the green circle when the actuators are stopped. Likewise, the third and fourth chains allow point F on the other half of the platform to be anywhere on the orange circle. Because the mechanism is symmetric about the $Oxz$ plane, both circles lie in this plane. Coordinate s represents the distance along the $Ox$ axis between points D and F. Since the circles are in the same plane, infinitely many lines parallel to the $Ox$ axis connect these points. In this case, the forward kinematics has an infinite number of solutions, where $y_D=0$ mm, but $x_D$, $z_D$, and s can take any value. This result implies uncontrollable relative motion between the two parts of the moving platform. Unlike the “3+1” mechanism, the motion is finite and limited only by design constraints $s_{\min }$ and $s_{\max }$.

A similar situation occurs when point D lies in a plane parallel to the $Oyz$ plane and passing through $x=-s/2$ (Figure 11b). As an example, we consider the mechanism configuration with $x_D=-20$ mm, $y_D=-50$ mm, $z_D=-500$ mm, and $s=40$ mm. The two circles are not in the same plane but exhibit mirror symmetry about the $Oyz$ plane. In this case, we also obtain an infinite number of solutions to the forward kinematics, where $x_D$ is always equal to $s/2$, but $y_D$ and $z_D$ can take any values as long as point D stays on the green circle. Similar to the previous example, the uncontrollable relative motion between the two parts of the moving platform is finite and limited only by design constraints $s_{\min }$ and $s_{\max }$. These two examples correspond to the third special case considered at the end of Section 4.2, verifying the theoretical results.

The performed workspace and singularity analysis suggests that the “3+1” mechanism is superior to the “2+2” one. However, if we deny the symmetry of the moving platform and apply the geometry of the “3+1” mechanism to the “2+2” mechanism, the situation improves dramatically. The results of the iterative analysis of the alternative “2+2” mechanism, whose geometrical parameters match Table 1 instead of Table 2, are presented in

Table 8. The number of points with positive or negative value of $det\left({\mathbf{J}}_A\right)$ within the workspace of the alternative “2+2” mechanism for different values of s.

s/mm	$det\left({\mathit{J}}_{\mathit{A}}\right)>0$	$det\left({\mathit{J}}_{\mathit{A}}\right)<0$
0	5,078	106,300
40	5,093	97,528
80	4,792	87,446

and illustrated in Figure 12. The central region of the obtained workspace contains only points with $det\left({\mathbf{J}}_A\right)<0$ and takes 95% of the total workspace volume, which is even greater than that of the “3+1” mechanism. Points with $det\left({\mathbf{J}}_A\right)>0$ are located only in a small volume at the top of the workspace and in several “patches” near its boundary. These results show that the “2+2” mechanism has a larger singularity-free workspace and can be more suitable for practical use, provided that the linear guides are arranged as described.

6. Discussion

The methods we proposed to solve the inverse and forward kinematic problems are suitable for the “3+1” and “2+2” mechanisms with any geometrical parameters and arrangement of the linear guides. In the examples, the linear guides formed a pyramid with a ${45}^{°}$ angle between the guides and the horizontal plane. We considered such geometry as a compromise between two extreme cases: when this angle is $0^{°}$ (all guides are in the same plane) and ${90}^{°}$ (all guides are collinear or parallel). The analysis of these and other cases needs further investigation, but we expect to get similar results for other geometrical parameters too. In particular, we expect that “2+2” mechanisms with a symmetrical arrangement of the linear guides will outperform “3+1” mechanisms in terms of the maximum singularity-free workspace.

This study focused primarily on parallel singularities, which correspond to an uncontrolled motion of the moving platform. We identified workspace regions separated by singular loci; moving from one region to another inevitably requires passing through a singular configuration. From design and motion planning perspectives, it is essential to avoid crossing parallel singularities or even approaching them [35]. Kinematic redundancy allows us to alter the spatial position of singular loci within the workspace, thereby changing the shape and size of singularity-free regions, but it cannot be used to cross the singularities. Therefore, designing a mechanism with a singularity-free workspace is a significant challenge, which we have partially addressed in this paper.

The considered kinematic models assume that a prismatic joint connects the two parts of the moving platform. In a real mechanism, this joint can be designed as a Sarrus linkage [48] or any other feasible alternative. Such design choices would not affect the results obtained in this article. These ideas are reflected in works [49,50], which have presented several similar (3+1)-DOF mechanisms. These studies, however, focused on the type synthesis of such mechanisms but ignored their kinematics, workspaces, and singular configurations. In this regard, we believe our research provides valuable insights for designing and analyzing these and other kinematically redundant mechanisms.

Among various other mechanisms with articulated platforms [51], only a few are based on a Delta-type architecture and have four DOFs. The most notable examples belong to the family of H4 mechanisms: H4 [26], I4R [27], I4L [28], and Par4 [29], which were mentioned in the introduction. These mechanisms have been designed for high-speed pick-and-place tasks, and the internal mobility of their moving platform is converted into end-effector rotation. In the I4R and I4L mechanisms, the parts of the moving platform perform a relative translational motion but require auxiliary transmission units (rack-and-pinion and belt-and-pulley), which complicates their design. The H4 mechanism avoids such transmissions and has a simpler design, where the parts of the moving platform are coupled with a link via two revolute joints. This design, however, is susceptible to link collisions and constraint singularities that partition the workspace [52]. The Par4 mechanism, whose articulated platform represents a parallelogram linkage, does not have these issues. On the other hand, the Par4 mechanism (like the H4) has a more complex kinematic model; its forward kinematic problem reduces to an eighth-degree polynomial equation that lacks a closed-form solution [43]. The “3+1” and “2+2” mechanisms introduced in this paper offer several advantages over the aforementioned designs. They do not require auxiliary transmission units, their forward kinematics admits a closed-form solution, and they are expected to be free of constraint singularities—a hypothesis that we will examine in future work. Although the proposed mechanisms may not be suitable for high-speed pick-and-place tasks, their design is promising for other applications, as discussed in the following paragraphs.

First, kinematic redundancy allows the moving platform to function as a gripper [23,24,25]. Furthermore, replacing the prismatic joint with a Sarrus or parallelogram linkage can alter the direction of the grasping movement [53]. For any such design, this application requires modeling the mechanism’s statics and dynamics, including the analysis of the grasping force. The magnitude of the force will depend on the moving platform’s position, velocity, and external load [54]. This analysis is a subject of future studies. The preliminary design of the mechanism prototype for these studies, with the dimensions matching the preceding simulations, has already been completed, as shown in Figure 13.

Another possible application is to use the proposed mechanisms as a module in a hybrid or multi-robot system. Examples include manufacturing [55,56] or cooperative manipulation [57,58]. The proposed kinematic algorithms would be an essential part of the overall model, and the results of the singularity analysis would be crucial for selecting the proper design and arrangement of the modules and their trajectory planning.

7. Conclusions

In this article, we studied two novel kinematically redundant Delta-type parallel mechanisms with linear actuators. The moving platform of each mechanism comprises two parts coupled by a prismatic joint and has four DOFs. The mechanisms include three translational DOFs of the platform as a whole and one translational DOF between its parts. The two mechanisms differ in the arrangement of the kinematic chains connecting the moving platform to the base. Specifically, the “3+1” and “2+2” architectures were considered.

First, we solved the inverse and forward kinematic problems for both mechanisms. The closed-form solution to the inverse kinematic problem was obtained by analyzing the geometry of the mechanisms. The forward kinematic problem was formulated as a problem of finding the intersection of three or four spheres, depending on the architecture. For both the “3+1” and “2+2” mechanisms, we showed that the forward kinematic problem can have a maximum of four solutions. The proposed algorithms allow us to compute all these solutions analytically, although numerical methods may be preferable for the “2+2” mechanism. Numerical examples verified the correctness of the algorithms.

Next, we used these kinematic models to analyze the workspaces and singular configurations of the mechanisms by applying an iterative approach. Based on the kinematic constraint equations, we computed the Jacobian matrices and examined the changes in the sign of their determinants to identify parallel singularities, where the moving platform gains uncontrollable DOFs. The analysis shows that up to 79–86% of the workspace of the “3+1” mechanism is singularity-free, depending on the distance between the parts of the moving platform. In contrast, the “2+2” mechanism admits a finite-motion singular configuration and can operate in only a quarter of its workspace without encountering parallel singularities. Altering the configuration of the linear guides allows these singular configurations to be excluded and significantly increases the singularity-free workspace of the “2+2” mechanism, which takes 95% of the whole workspace in this case.

The proposed mechanisms can serve as gripping devices or become modules in hybrid and cooperative manipulation systems. Future studies will consider these applications and use the presented results for the static and dynamic analysis of the mechanisms, including the analysis of grasping forces. We will also use the developed algorithms to optimize the geometrical parameters of the mechanisms by evaluating their kinematic and dynamic performance indices.