Application of Barycentric Coordinates and the Jacobian Matrix to the Analysis of a Closed Structure Robot

Abstract

A new approach is presented to study the kinematic properties of stationary robots with a closed structure. It combines the application of conventional methods from kinematics with geometric parameters represented in a barycentric coordinate system. This allows examining the influence of the proportions of the robot’s links on its basic mechanical characteristics. Each point from the newly introduced barycentric space corresponds to a set of robots with the same link proportions. The proposed approach is used to study three aspects: the link proportions for which the robot can exist; the shape of the robot’s workspace; and the possible singular configurations. This is valuable when evaluating the qualities of existing robots and could be applied to the design of new mechanical systems. An example of a 5-link robot with a closed structure is considered. The conditions for the existence of the mechanism and the conditions under which certain types of singular configurations can occur are defined. The example reveals the great potential of combining barycentric coordinates and Jacobian properties. The barycentric coordinates of 10 robots with a 5-link closed structure known from the literature are determined, and their properties are analyzed. The results are presented graphically. An extension of the application area of the approach is discussed.

1. Introduction

The same proportions occur repeatedly in nature. The arts skillfully apply ratios between sizes, colors, rhythms, and frequencies. Since ancient times, people have sought the “golden ratios” that give perfect qualities to objects and events. This work is devoted to the search for ideal proportions in robotics. The study of the mechanical properties of robots is important, both for the design stage and for the creation of control algorithms. Robots perform complex movements in space, with their mechanical structure driven by multiple independently controlled motors. The qualities of the robot depend on the structure, dimensions, and configuration of its links and joints. This paper presents a new approach to the design and research of stationary robots that combines known methods of kinematics and kinetostatics with the use of barycentric coordinates to represent the proportions of the robot’s links. This allows us to focus on the ratio between the lengths of the robot’s links, which is essential in the analysis of its mechanical system. The approach is aimed at studying stationary robots with a closed structure, but it can be useful for some walking and mobile robot systems. A general concept of the approach and, in more detail, its specific application to 5-link planar robots with two degrees of freedom are presented.

In the following literature review, we are interested in the general methods for robot design and synthesis, the scope of application of barycentric coordinates, and the design and research of 5-link planar robots.

1.1. Literature Review

In the initial stage of robot design, it is important to determine the structure, dimensions, and joint constraints. These elements are fundamental to determining the workspace and mechanical properties of stationary robots with open [1] and closed [2,3] structures. The degrees of freedom (DoF) required to perform the intended tasks are at the heart of robot design. Papers [1,2] propose an approach that combines structural and dimensional synthesis to find the most suitable structure for a given task through optimization. In [3], the design and synthesis of a 3D-printed DELTA robot are presented. A method for optimizing its size using a genetic algorithm is proposed. The aim is to minimize the robot’s dimensions while maximizing the workspace of the parallel robot. Article [4] introduces an approach to optimizing the dimensions of an anthropomorphic robot that considers multiple characteristics, such as performance, stiffness, and dexterity. The Jacobian matrix for the robot is determined, which is the basis for finding the manipulability coefficient and the stiffness of the system.

The search for the optimal dimensions of a robot’s links is important not only for the common industrial and stationary robots. In [5], the synthesis of the dimensions of a walking robot’s leg is performed analytically to develop a parametric equation and model the geometry of the mechanism. Publication [6] compares three walking robot mechanisms and analyzes their geometric syntheses. Based on this, analytical and numerical methods are adopted to reproduce a trajectory similar to the human gait. In [7], an optimal ratio between the links’ lengths of a walking robot is sought in order to minimize energy loss when moving on flat terrain and when overcoming high obstacles. Optimizing the links’ lengths for remotely controlled robots is an important problem and is considered in [8].

Some of the studies, for example [1,3,5,7], deal with purely geometric optimization. In many cases, however, criteria such as dexterity, feasibility, and transformation of forces are important [4]. To consider these, it is necessary to use differential kinematics and statics based on studying the Jacobian matrix for the robot [9,10,11]. The possible movements and degrees of freedom (DoF) of a mechanical system of bodies are determined by the velocities of the links that make it up. The Jacobian matrix is foundational to almost all research, and its fast and correct calculations are important [12].

A barycentric coordinate system is one in which the location of a point is defined by reference to a simplex. The barycentric coordinates of a point are usually interpreted as masses placed at the vertices of the simplex so that the point is the center of mass (or barycenter). These coordinates were introduced by Augustus Möbius in 1827. They are often used to represent a point inside a polygon as an affine combination of the vertices of the polygon and to interpolate data given at those vertices [13,14]. Unlike other common coordinate systems, such as Cartesian, cylindrical, spherical, etc., for which all coordinates are independent of each other, barycentric coordinates always include one redundant coordinate that can be derived from the others. In robotics, barycentric coordinates and barycentric parameters have mainly been applied in two ways. The first is related to motion planning algorithms, where barycentric coordinates are used to detect collisions between objects and the robot [15]. The second way relates to determining the center of gravity, usually in mobile robots [16], and the reactions in the base in stationary robots. In [17], in order to improve the stability of a mobile robot, a barycenter adjustment mechanism was designed and installed in its chassis, and a counterweight control method was proposed to achieve static stability. Paper [18] presents a procedure for estimating the barycentric parameters of a stationary robot that requires the processing of measurements provided by an external experimental setup (base reactions, position, velocities, and accelerations). The procedure is based on the property that the relationship between the robot’s motion and the reaction on its base is independent of the internal joint reactions. Barycentric coordinates are widely used in computer graphics. A major advantage of barycentric coordinate systems is that they are symmetric with respect to n + 1 defining points. Therefore, they are often useful for studying properties that are symmetric with respect to n + 1 points [14].

The planar 5-link kinematic mechanism is known and studied by many authors [19,20,21,22,23,24,25] and is usually considered as a two-degree-of-freedom robot. Ref. [20] proposes placing motors in all 5 joints so that the robot becomes redundant with respect to the control signals and its stiffness is studied. It is assumed that the lengths of all links are equal. Article [21] investigates the computer control of a 5-link robot with two degrees of freedom. The inverse kinematics is solved for the given endpoint positions. The software then sends a dataset containing the joint angles to an Arduino microcontroller, which sets the positions of the two servo motors according to the calculated angles. A kinematic analysis of a 5-link planar mechanism intended for industrial applications was performed in [22]. A method is proposed that uses a geometric approach to find the joint angles, while the velocities are determined by the Jacobian of the mechanism. Ref. [23] presents the design and development process of a double SCARA robot, which in practice has the same 5-link structure. Solutions to the direct and inverse kinematics of the robot are sought, and then the accuracy and effectiveness of the kinematic solutions are tested by simulation. It is assumed that all links are 1 m long. Ref. [24] begins with an overview of 5-link planar parallel robots for pick-and-place tasks, designed to avoid singular configurations. In all these robots, the proximal and distal links are of different lengths. As a consequence, the workspace of each robot is different and often significantly limited, as there are gaps within it. Ref. [24] then proposes a design in which all four links have the same length. Since such a design leads to more singularities, a strategy to avoid them by switching between the modes of operation is implemented. As a result, the usable workspace of the robot increases significantly. Article [25] presents the stages of design, 3D modeling, and kinematics of a 5-link planar robot. A control system for the robot is developed, and experiments to determine the repeatability of the movements are carried out.

1.2. Background and Related Works

A few foundational robotics concepts are first presented here in order to make the exposition clear for a larger number of readers. This is followed by an overview of previous related work that is the basis of the present article.

The configuration of a robot uniquely determines the mutual arrangement of its links. It depends on the joint coordinates of the robot and is set by a configuration vector $q={\left[q_1 q_2 \dots q_n\right]}^T$. The dimension of this vector is usually equal to the number of independently controllable motors. There are several important configurations:

–

Home configuration—the execution of tasks begins from this position. It is determined by the manufacturer, and its exact location is reported by sensors.

–

End configurations—they are determined by the variation limits of the joint coordinates. They usually depend on the physical limitations, but it is possible to set software restrictions.

–

Singular configurations—these are configurations in which the robot qualitatively changes its properties. Physically, the number of DoF of the robot in such a configuration decreases. Mathematically, the rank of the Jacobian matrix decreases with them. These configurations are usually undesirable because controlling the robot near them becomes very difficult or impossible (see [9] (p. 116)).

It is accepted that the set of all points in real space that can be occupied by a certain point of the robot’s end effector is called the robot’s workspace. This definition is not very clear because it does not specify which point should be chosen to represent the end effector. Moreover, the possible orientations of the end effector are not taken into account. However, this definition has gained a lot of popularity in robotics. The shape of the robot’s workspace depends on the proportions of the link lengths, the joint types, and joint constraints. Article [26] presents an approach for optimal kinematic synthesis of parallel manipulators based on a random search technique. The concept of an effective regular workspace is proposed, which reflects both the requirements for the shape and the quality of the workspace. The effectiveness is characterized by the dexterity of the mechanism at every point of the workspace. Article [27] presents a kinematic method for the design of symmetric 5-link robots. Examples of such robots are “DexTAR” and the industrial “RP” by Mitsubishi Electric [28]. The design method proposed in [27] is performed in two steps. First, an optimal configuration is achieved, resulting in a set of closed chains. Second, an algorithm for finding the optimal lengths of the robot’s links is proposed. The algorithm searches for the maximum usable workspace, in which there are no singular configurations.

The linear ${\dot{p}}_e^T$ and angular ${\omega }_e^T$ velocities of the robot’s end effector are represented by the vector $v_e={[ {\dot{p}}_e^T {\omega }_e^T]}^T={[v_x, v_y,v_z{,\omega }_x{,\omega }_y, {\omega }_z]}^T.$ They are determined by the control signals and the mechanical system and are a function of the configuration $q={\left[q_1 q_2\dots q_n\right]}^T$. The relationship between the joint velocities q and the end effector velocities is represented by the equation [9] (pp. 104–106):

$$v_e=J\left(q\right)\dot{q},$$

(1)

Here, $J\left(q\right)$ is the Jacobian of the robot. The number of controllable joints $q$, respectively, speeds $\dot{q}$ may be different from the dimension of the vector $v_e$, which means that the Jacobian matrix can be square or rectangular. On the other hand, the relationship between the forces $F$ applied on the end effector and the torques τ in the joints of the robot is also determined by the Jacobian [9] (pp. 147–149):

$$\tau =J^T\left(q\right)F,$$

(2)

Here, $\tau ={\left[{\tau }_1 {\tau }_2\dots {\tau }_n\right]}^T$ is a vector of the torques (forces at translational joints) applied at the joints of the robot, which are created by the motors, and $F={\left[F_x F_y F_z M_x M_y M_z\right]}^T$ is a vector that reflects external force impacts on the robot’s end effector. Formulas (1) and (2) are foundational to the design, study, and control of robots. Both equations involve the Jacobian $J\left(q\right)$ of the robot, which is a function of the configuration $q$. However, there are two factors that are often overlooked in research. Firstly, the Jacobian also depends on the lengths of the robot’s links, which are assumed to be constants, but at the design stage, they can change and have an impact on $J\left(q\right)$. Secondly, the possible configurations $q$ in many cases also depend on the lengths of the robot links. This becomes evident when moving a serial robot in an environment with obstacles, especially for robots with a closed structure.

Finding the solution to Formula (1) is known as the forward kinematics in robotics. However, to control a robot, the joint coordinate velocities $\dot{q}=J^{-1}{v}_e$, must be found, which is known as the inverse kinematics. When $\mathit{det}\left(J\right)=0$ (for rectangular matrices $\mathit{det}\left(J{J}^T\right)=0$) solving the inverse kinematics becomes problematic since this happens when the robot is in a singular configuration. Various ways of classifying singular configurations have been proposed. Often they are divided into arm (positional) singularities and wrist (orientational) singularities [9]. Another approach classifies them according to whether they appear on the periphery of the workspace or are inside it. Mathematically, these configurations lead to a change in the rank of the Jacobian, which is used as a way of identifying them. This article looks at a simple example that is only interested in the change of the linear velocities of the end effector.

In [19], the kinematics of a 5-link planar robot (Figure 1) is investigated. The driving motors M₁ and M₂ are located on the fixed base, and the end effector is located at joint B. It is assumed that the links $a_i (i=\mathrm{0,1},..4)$ of the robot move in parallel planes without obstructing each other. The example only considers the position of point B of the end effector without accounting for its orientation. In other words, we can imagine that there is a rotary joint at point B that allows the gripper to rotate by $2\pi \left[rad\right]$. In [19], an analytical expression for the Jacobian matrix $J$ of the robot was determined by mentally dividing the system into two open chains.

Under the above assumptions and using the notation from Figure 1a, two assertions were proved in [19] (p. 5):

Assertion 1. 

The determinant of the matrix $J$ or the robot of Figure 1a is zero if and only if either the determinant of $J_1$ or $J_2$ is equal to zero.

$$\det \left(J\right)=0⟺\left|\begin{array}{c}\det \left(J_1\right)=0\\ \det \left(J_2\right)=0\end{array}\right.,$$

(3)

where $J_1$ and $J_2$ are the Jacobian matrices for the corresponding imaginary left and right open chains of the 5-link robot (see Figure 1b).

We know from [9] (p. 117) that the determinants of $J_1$ or $J_2$ are zeroed when ${\theta }_2=0$ or ${\theta }_3=0$, because $\det \left(J_1\right)=a_1{a}_2\sin \left({\theta }_2\right),\mathrm{a}\mathrm{n}\mathrm{d}\det \left(J_2\right)=a_3{a}_4\sin \left({\theta }_3\right)$.

Assertion 2. 

The determinant of the matrix $J$ (for the robot in Figure 1) tends to infinity when:

$${\theta }_1+{\theta }_2={\theta }_3+{\theta }_4.$$

(4)

The angles ${\theta }_i \left(i=\mathrm{1,2},\mathrm{3,4}\right)$ are according to Figure 1a.

The determinant of $J$ was defined in [19] (p. 5) as follows:

$$\det \left(J\right)={\displaystyle \frac{\det \left(J_1\right)\det \left(J_2\right)}{a_2{a}_{3}sin({\theta }_3+{\theta }_4-{\theta }_1-{\theta }_2)}}.$$

(5)

Article [19] provides examples and commentary on the physical interpretation of the spaces from the Jacobian matrix for this type of robot. Numerous articles [20,21,22,23,24,25] show that robots with such a structure (Figure 1) are important from a scientific and practical point of view. The Mitsubishi company offers a range of 3 models of such a robot on the market (Figure 2).

All studies found in the literature deal with specific problems related to design optimization, workspace definition, kinematic modeling, and control without taking into account the proportions of the links. The reviewed works do not define the conditions under which the mechanical system of the robot can exist or consider in detail the conditions under which certain types of singular configurations can occur.

The aim of the present work is to introduce barycentric coordinates to determine the conditions of existence of the closed mechanical system and the different types of singular configurations, and to investigate the kinematic properties of the robot from Figure 1a. To achieve these goals, we will test the hypothesis that a barycentric coordinate system can represent the proportions between the robot’s links. Regions of this barycentric space must satisfy kinematic requirements based on geometric conditions and properties of the Jacobian matrix to fulfill the objective.

2. Materials and Methods

As can be seen from the literature review, the qualities of the mechanical system of a robot depend significantly on several factors: structure, geometric dimensions, joint constraints, mass-inertia characteristics, materials, etc. This work examines the first three factors in more detail. The research method is structured in 6 steps, first giving general recommendations applicable to a large class of robots and then providing instructions for a 5-link, two-degree-of-freedom robot.

2.1. Determining the Relevant Dimensions of the Links and the Barycentric Coordinates

The first thing to consider is which dimensions of the links are relevant to the study of movements. Previous experience can be applied, together with the recommendations given in this section, to set the dimensions $a_i$ and calculate the corresponding barycentric coordinates $u_i$ for the robot. The position of the active links must be determined if not already known. After this, a normalized space, representing the links’ lengths with barycentric coordinates, is defined.

The lengths of the robot’s links have a significant influence on the robot’s behavior. For the purposes of this paper, we are using Denavit and Hartenberg’s definition of link length [9] (pp. 61–63).

When solving for the optimal lengths of a robot’s links, often one or more links are treated as constant and the remaining lengths change within certain limits [1,2]. In some cases it is easy to identify which links to treat as constant and which to vary, but there are many situations where this is difficult to determine. For example, in a closed mechanism (Figure 3a), we have no reason to prefer one particular link over another.

Thus we arrive to the idea of treating the sum of all link lengths as constant $L_{sum}$:

$$a_0+a_1+\dots +a_n=\sum _{i=0}^n{a}_i=L_{sum}=const$$

(6)

This allows the mechanism to be studied while varying the length of each link independently. A large part of the kinematic properties of a robot depend more on the ratios between the lengths of the links than their absolute size. By normalizing (dividing both sides by $L_{sum}$) the sum (6) becomes the following:

$${\displaystyle \frac{a_0}{L_{sum}}}+{\displaystyle \frac{a_1}{L_{sum}}}+\dots +{\displaystyle \frac{a_n}{L_{sum}}}=\sum _{i=0}^n{\displaystyle \frac{a_i}{L_{sum}}}=1$$

(7)

Represented with normalized (or generalized) barycentric coordinates [14] (p. 2) $u_i$, (7) looks like this:

$$u_0+u_1+\dots +u_n=\sum _{i=0}^n{u}_i=1$$

(8)

If we know one of the barycentric coordinates $u_i$ and the corresponding absolute link length $a_i$ for a specific robot, then from (7) and (8) we can calculate the total length $L_{sum}$:

$${\displaystyle \frac{a_i}{L_{sum}}}=u_i \Rightarrow k=L_{sum}={\displaystyle \frac{a_i}{u_i}}$$

(9)

Consequently, if we have determined the ratios between all links and we know the absolute length of any one link, we can calculate all lengths using the scale factor $k$ determined by (9). On the other hand, if we know the total length $L_{sum}$ (perhaps because the robot arm has to reach a certain maximum distance), then we can find the remaining dimensions so that they have the desired proportions. It can be seen from (7) that the barycentric coordinates $u_i$ of the links’ lengths are adimensional. The link $a_0$ (and its respective coordinate $u_0$) is a special case, as it is commonly associated with the base. If the robot has an open structure (Figure 3b), we will assume that $a_0=0$. If the robot has a closed structure (Figure 3a), then $a_0$ will be one of the distances between two links that are connected to the base.

Definition 1. 

The space in which the proportions of the links’ lengths $u_i$ of a mechanism can be changed, and for which condition (8) is fulfilled is called the Robot Links’ Barycentric Space (RLBS).

Open-structure robots can exist regardless of the ratio between the links’ lengths. In contrast, closed mechanisms are structurally possible only when certain conditions are met. Therefore, most of the following reasoning applies only to closed mechanisms.

2.2. Existence of a Mechanism

Conditions under which it is possible to assemble and move the links of the robot are defined. Here, the triangle inequality and the condition for a common workspace of the two imaginary arms of the 5-link robot must be satisfied. For spatial mechanisms, conditions for crossing two or more spheres, a sphere with a circle, etc. are used.

Let us consider the existence of the 5-link closed planar robot from Figure 1. In the following reasoning, it is assumed that the links $a_i$ are located in parallel planes and can move without colliding with each other. By $a_i$ we mean both the link and its absolute length. If the distance $a_0$ is significantly larger than the dimensions of the movable links, such a mechanism does not exist since it cannot be assembled—see Figure 4a.

In order to check whether the robot can be assembled and can execute movements, the workspaces of the two 2-link mechanisms (the left and right hands) must intersect. In general, these workspaces are planar rings. From Figure 4a, the following condition must be fulfilled:

$$a_1+a_2+a_3+a_4\ge a_0$$

(10)

Then, for the barycentric coordinates, from (7) we get:

$$u_1+u_2+u_3+u_4\ge u_0$$

(11)

From Figure 4b, it can be seen that if $a_1-a_2>a_0+a_3+a_4$, then the workspaces don’t intersect. Analogously, from Figure 4c: $a_2-a_1>a_0+a_3+a_4$. Therefore, the barycentric coordinates must satisfy:

$$u_1-u_2\le u_0+u_3+u_4$$

(12)

$$u_2-u_1\le u_0+u_3+u_4$$

(13)

Analogously, for the relationship between $u_3$ and $u_4$, we can consider the mirror images of those shown in Figure 4b,c and two more conditions are obtained:

$$u_3-u_4\le u_0+u_1+u_2$$

(14)

$$u_4-u_3\le u_0+u_1+u_2$$

(15)

From Formulas (8) and (11), when we exclude $u_0$, we get:

$$u_1+u_2+u_3+u_4\ge 0.5$$

(16)

Substituting (8) into (12) we get $u_1\le 0.5$. Analogously, (8) can be substituted into (13), (14), and (15) in turn. The result can be summarized as:

$$u_i\le 0.5 \mathrm{for} i=\mathrm{1,2},\mathrm{3,4}$$

(17)

Additionally, we know that the link lengths are always non-negative, i.e.,

$$u_i\ge 0 \mathrm{for} i=\mathrm{0,1},\mathrm{2,3},4$$

(18)

Thus, a 5-link mechanical system of the type presented in Figure 1 can exist when inequalities: (16), (17), and (18) are simultaneously fulfilled for the RLBS. Equality (8) is an equation of a hyperplane with intersections along the axes $u_i=1$ in the barycentric space. In this hyperplane, inequalities (16), (17), and (18) limit the domain D_b of link ratios for which the robot can exist. A representation of this idea applied to a concrete example (in three-dimensional space) can be seen on page 13 of this article. For now, there is no way to adequately illustrate a hyperplane in five-dimensional space.

2.3. Analysis of the Joint Angles

Ratio conditions are sought that define areas in the RLBS where it is possible to have no joint constraints imposed by the robot’s mechanical components. In other words, for what ratios between the links’ lengths can an active or passive link be rotated more than $2\pi \left[rad\right]$? This is convenient for performing certain movements and controlling the robot. Similar conditions can be found for translational joints in some closed robot systems.

Consider the 5-link mechanism from Figure 1. Since $u_1$ and $u_4$ are active links, we will look for the conditions in which they can be rotated by more than $2\pi \left[rad\right]$. We will talk about the normalized length of a link $u_i$ knowing that it is proportional to $a_i$ from a real robot. In the following figures, the links and their normalized lengths are denoted by $u_i$. From Figure 5a, it can be concluded that for link $u_1$ to rotate more than $2\pi \left[rad\right]$ it is necessary that:

$$u_0+u_1\le u_2+u_3+u_4$$

(19)

Of course, the link ratios must continue to satisfy conditions (16–18) for the mechanism to exist. Taking into account (8) and excluding $u_0$ from (19), we get:

$$u_2+u_3\ge 0.5-u_4$$

(20)

Similarly, in order to rotate the second active link $u_4$ more than $2\pi \left[rad\right]$ it is necessary that:

$$u_0+u_4\le u_1+u_2+u_3$$

(21)

Analogously, from (8) it follows:

$$u_2+u_3\ge 0.5-u_1$$

(22)

Equations (21) and (22) correspond to a mirror image of that given in Figure 5a. From Figure 5b we obtain the condition under which both active links can simultaneously rotate more than $2\pi \left[rad\right]$:

$$u_0+u_1+u_4\le u_2+u_3$$

(23)

From (8) and (23) it follows that in this case:

$$u_2+u_3\ge 0.5$$

(24)

2.4. Solving the Determinant of the Jacobian Matrix Analytically to Define Different Types of Singular Configurations and Exploring How the Rank of the Matrix Changes

The Jacobian matrix J of a 5-link robot is a 2 x2 matrix, as described in previous work [19] (pp. 2–4). The determinant of J is shown in (5). Depending on the ratios between the links’ lengths, the robots can realize different movements and enter into different types of singular configurations. The following classification of 5-link planar robots is proposed:

• $s_1$—Robots that can move in the plane and that could only fall into singular configurations, for which $\det \left(J\right)=0$ and the rank of J decreases by one (i.e., $rank\left(J\right)=1$ or $rank\left(J\right)=2$);
• $s_2$—Robots that can move in the plane and that could fall into singular configurations, for which $\det \left(J\right)=0$ and the rank of J decreases by one or two (i.e., $rank\left(J\right)=2$, $rank\left(J\right)=1$ or $rank\left(J\right)=0$);
• $s_3$—Robots that could fall into singular configurations as in $s_1$ and $s_2$, and that could enter into a configuration, for which $\det \left(J\right)={\displaystyle \frac{const}{0}}$;
• $s_4$—Robots that could fall into singular configurations as in $s_1$, $s_2$ and $s_3$, and that could enter into a configuration, for which $\det \left(J\right)={\displaystyle \frac{0}{0}}$.

From (3) and (5) we can see that the numerator of the Jacobian is zero if $\det \left(J_1\right)=0$ or $\det \left(J_2\right)=0$, which occurs when ${\theta }_2=0, \pi \dots$ and ${\theta }_3=0, \pi \dots$ respectively. Figure 6a,b depicts two such cases for the right-hand side of the robot (i.e., $\det \left(J_1\right)=0$). This occurs when the robot is of type $s_1$.

2.5. Defining the Conditions for $u_i$ in RLBS, for Which Different Types of Singular Configurations Occur

Figure 6a shows a case for which ${\theta }_2=0$. This type of singular configuration is inevitable and will appear for any ratio between the links’ lengths, assuming the 5-link robot exists.

This is because the workspace of the robot is the intersection of the workspaces of the left and right arms, which have Jacobians $J_1$ and $J_2$, respectively. Figure 6b shows a case where ${\theta }_2=\pi$. To reach such a configuration, it is necessary to fulfill the condition:

$$u_1-u_2\le u_0+u_3+u_4$$

(25)

From (8), it follows that:

$$u_1\le 0.5$$

(26)

Analogously, for the case where $\det \left(J_2\right)=0$, the configurations will mirror those given in Figure 6 and $u_4\le 0.5$ must be fulfilled. However, these conditions are part of the requirements defined by Formula (17). Therefore, the singularities corresponding to Figure 6b are inevitable if the mechanism exists.

From the analysis of the cross-sections of the workspaces of the two open planar mechanisms, we can conclude that robots of type $s_1$ are a limited case. With them, it is necessary that the workspace of one manipulator lies entirely within the workspace of the second, which is rarely a practical solution. Based on the reasoning so far, we will consider cases of types $s_2$, $s_3$, and $s_4$.

Article [19] shows that when both $\det \left(J_1\right)=0$ and $\det \left(J_2\right)=0$, the rank of the Jacobian decreases by two. Such a configuration is depicted in Figure 6c. The denominator of the Jacobian determinant (5) is reset when the links $u_2$ and $u_3$ are collinear (Figure 7). Such robots are of type $s_3$ and must fulfill:

$$\left|\begin{array}{c}{u}_0-u_1-u_4\le u_2+u_3\\ u_0+u_1+u_4\ge u_3+u_2\end{array}\right.$$

(27)

By applying (8) and excluding $u_0$ we get:

$$\left|\begin{array}{c}{u}_1+u_2+u_3+u_4\ge 0.5\\ u_2+u_3\le 0.5\end{array}\right.$$

(28)

A singularity of type $s_4$ for which $\det \left(J\right)={\displaystyle \frac{0}{0}}$ is shown in Figure 8. This is possible when:

$$u_0+u_4\ge u_1+u_2+u_3$$

(29)

From (8), it follows:

$$u_1+u_2+u_3\le 0.5$$

(30)

2.6. Workspace Analysis

Each point in the RLBS corresponds to a set of robots that have the same link ratios. Consequently, their workspaces will have the same shape.

Definition 2. 

The volume (or area) of the workspace of a robot calculated with the barycentric coordinates $u_i$, rather than with the real lengths of the links $a_i$, is called the normalized workspace $W_n$.

$W_n$ is a numerical value of the volume (or area) in reference to the sum of the lengths of the robot’s links. If $W_n$ for a robot with certain proportions in RLBS is calculated and the scale factor $k$ (see Formula (9)) that scales it to a real robot is known, then the area $A$ or volume $V$ of the robot’s workspace can easily be calculated:

$$\begin{gathered} A= k^2{W}_n \\ V= k^3{W}_n \end{gathered}$$

(31)

Later, we will analyze the areas of the RLBS in which the normalized workspace has the same type of shape.

For our 5-link robot, each point $P_i[u_0;u_1;u_2;u_3;u_4]$ in the barycentric space of the domain $D_b$ corresponds to a set of possible robots. A normalized workspace $W_n$ can be determined for $P_i$. In order to analytically calculate $W_n$, the formula for finding the intersection area $f$ of two circles is repeatedly used:

$$\begin{gathered} f={\rho }_1^{2}arcos\left({\displaystyle \frac{d^2+{\rho }_1^2-{\rho }_2^2}{2d{\rho }_1}}\right)+{\rho }_2^{2}arcos\left({\displaystyle \frac{d^2+{\rho }_2^2-{\rho }_1^2}{2d{\rho }_1}}\right)- \\ {\displaystyle \frac{1}{2}}\sqrt{(-d+{\rho }_1+{\rho }_2)(d+{\rho }_1-{\rho }_2)(d-{\rho }_1+{\rho }_2)(d+{\rho }_1+{\rho }_2)}, \end{gathered}$$

(32)

where $d$ is the distance between the centers of the circles, ${\rho }_1$ and ${\rho }_2$ are the radii of the small and large circles, respectively (Figure 9). The circles intersect only if ${\rho }_1+{\rho }_2<d$.

3. Results

3.1. Example 1: Defining the Domain $D_b$, the Geometric Properties, and the Singular Configurations of a Closed 5-Link Asymmetric Mechanism

Let’s consider the case of a 5-link robot, for which $u_1=0.2$ and $u_4=0.1$. From (16), it follows:

$$u_2+u_3\ge 0.2$$

(33)

Since $u_0\ge 0$ and $u_0+u_1+u_2+u_3+u_4=1$, we can conclude that:

$$u_2+u_3\le 0.7$$

(34)

The projection of this space on axes $u_2$ and $u_3$ is given in Figure 10b. The domain $D_b$ bounded by the polygon $P_1, P_2, P_3,P_4,P_5, P_6$ defines the set of barycentric points that correspond to possible 5-link robots. Along the boundary segments $P_1{P}_2$ and $P_5{P}_6$, the mechanism becomes a 4-link, because either $u_2=0$ or $u_3=0$. For example: point $P_7$ lies on the segment $P_1{P}_2$ (analogously, $P_8$ on $P_5{P}_6$) and the mechanism corresponding to it is shown in Figure 11. Similarly, the mechanisms corresponding to the remaining boundary points from $P_9$ to $P_{12}$ are shown. It can be seen (Figure 11) that at points $P_9$, $P_{10}$ and $P_{12}$, which belong to sections $P_2{P}_3$, $P_4{P}_5$, and $P_1{P}_6$, respectively, even if assembled, the mechanism would not be able to move, as it could only exist in a singular configuration where all links are in one line.

The case of point $P_{11}$, which belongs to segment $P_3{P}_4$, is interesting. The resulting mechanism can be moved by two independently controlled motors that are coaxially located, since the base link $u_0=0$. A real-world example of such a mechanism is presented in [29,30]–an assistive device for the rehabilitation of an upper limb. The device is a lever system that is driven by a 5-link mechanism with two motors and a link $u_0=0$ (the structural diagram is as shown for point $P_{11}$ of Figure 11). The kinematic analysis in [29] demonstrates that the device has a large workspace.

Figure 10a is created under the assumption that $u_1+u_4=0.3$. If this sum changes, then the plane defined by the segments $Q_1$,$Q_2, Q_3$ will move towards or away from the origin of the coordinate system. The plane will always intersect axes $u_i$ between 0 and 1.

Since we assumed $u_1=0.2$, $u_4=0.1$, from Formula (20) we get $u_2+u_3\ge 0.4$. From (22) it follows that $u_2+u_3\ge 0.3$, and Formula (24) does not change. This result is given in Figure 12.

Robots that fall in the region $D_{rl}-P_2,P_3,P_4,P_5,P_{14},P_{13}$ can rotate their left-hand link $u_1$ more than $2\pi \left[rad\right]$ (see Figure 5a). Region $D_{rr}-P_2,P_3,P_4,P_5,P_{16},P_{15}$ defines mechanisms for which the right-hand link $u_4$ can rotate more than $2\pi \left[rad\right]$. Finally, if a robot falls into region $D_r-$ $P_2,P_3,P_4,P_5$, both links can rotate more than $2\pi \left[rad\right]$ (see Figure 5b and Figure 12). It could appear confusing that $D_r$ is not an intersection of $D_{rl}$ and $D_{rr}$, but this is due to the fact that at the intersection of $D_{rl}$ and $D_{rr}$ there are certain configurations of the robot for which only the left link or only the right link can be rotated, while in $D_r$ the $2\pi \left[rad\right]$ rotation condition is always satisfied.

Since we have assumed that $u_1=0.2$ and $u_4=0.1$, singularities at which the numerator of the Jacobian determinant is zero are possible for all mechanisms within the domain $D_b\equiv D_{s2}$-$P_1,P_2, P_3,P_4, P_5,P_6$ (Figure 13). To find out when the denominator of the Jacobian is zero, from Formula (28) we get:

$$\left|\begin{array}{c}{u}_2+u_3\ge 0.2\\ u_2+u_3\le 0.5\end{array}\right.$$

(35)

This is the area defined by the polygon $D_{s3}$−$P_1,P_2, P_5,P_6$–Figure 13.

Formula (30) determines the condition for possible singular configurations of type $\det \left(J\right)={\displaystyle \frac{0}{0}}$. For the current example it follows:

$$u_2+u_3\le 0.3$$

(36)

In Figure 13, the corresponding region is $D_{s4}$−$P_1,P_7, P_8,P_6$.

Example 1 shows that if the number of unknown parameters is reduced, it is possible to graphically represent regions in the RLBS in which mechanisms share the common properties. This motivates us to investigate the 5-link mechanisms known from the literature by finding their place in RLBS and analyzing the possibilities to reduce the number of unknown parameters.

3.2. Example 2: Examining the RLBS Coordinates of Robots from the Literature

Table 1. Barycentric coordinates of known robots.

n	Literary Source	Robot Link Lengths [m] $\left[{\mathit{a}}_0;{\mathit{a}}_1;{\mathit{a}}_2;{\mathit{a}}_3;{\mathit{a}}_4\right]$	${\mathit{L}}_{\mathit{s}\mathit{u}\mathit{m}}$[m]	Point in RLBS (Calculated from Equation (7)) $\left[{\mathit{u}}_0;{\mathit{u}}_1;{\mathit{u}}_2;{\mathit{u}}_3;{\mathit{u}}_4\right]$	${\mathit{W}}_{\mathit{n}}$
$r_1$	[24,31]	$\left[0.275;0.23;0.23;0.23;0.23\right]$	1.195	$\left[0.23;0.195;0.195;0.195;0.195\right]$	0.31
$r_2$	[20,23]	$\left[1;1;1;1;1\right]$	5	$\left[0.2;0.2;0.2;0.2;0.2\right]$	0.34
$r_3$	[25]	$\left[0.17;0.2;0.2;0.2;0.2\right]$	0.97	$\left[0.176;0.206;0.206;0.206;0.206\right]$	0.39
$r_4$	[29,30]	$\left[0;0.18;0.18;0.18;0.18\right]$	0.72	$\left[0;0.25;0.25;0.25;0.25\right]$	0.79
$r_5$	[21]	$\left[0.019;0.02;0.03;0.03;0.02\right]$	0.119	$\left[0.16;0.17;0.25;0.25;0.17\right]$	0.38
$r_6$	RP-1AH [28]	$\left[0.085;0.1;0.14;0.14;0.1\right]$	0.565	$\left[0.15;0.175;0.25;0.25;0.175\right]$	0.41
$r_7$	RP-3AH [28]	$\left[0.085;0.14;0.2;0.2;0.14\right]$	0.765	$\left[0.11;0.185;0.26;0.26;0.185\right]$	0.49
$r_8$	RP-5AH [28]	$\left[0.085;0.2;0.26;0.26;0.2\right]$	1.005	$\left[0.08;0.2;0.26;0.26;0.2\right]$	0.57
$r_9$	[27]	[0.68; 1;1.9721;1.9721;1]	6.6242	$\left[0.104;0.15;0.298;0.298;0.15\right]$	0.44
$r_{10}$		$\left[0.08;0.04;0.05;0.05;0.04\right]$	0.26	$\left[0.31;0.155;0.19;0.19;0.155\right]$	0.16
$r_{11}$	[32]	[0.222; 0.189; 0.165; 0.164; 0.196]	0.936	$\left[0.237;0.203;0.176;0.175;0.209\right]$	0.27

shows the dimensions $\left[a_0;a_1;a_2;a_3;a_4\right]$ of 11 robots—found in the literature or industrially produced—with a 5-link structure as in Figure 1. The sum $L_{sum}$, their barycentric coordinates $\left[u_0;u_1;u_2;u_3;u_4\right],$ and the normalized workspace $W_n$ derived from the absolute dimensions of the robots are displayed.

Almost all examples found in the literature (with the exception of robot $r_{11}$) have a symmetric structure. For this reason, the symmetric mechanisms are examined in more detail.

3.3. Example 3: Symmetric Mechanisms

Again we are looking at the 5-link robot from Figure 1, however, this time the mechanism is symmetrical, meaning that $u_1=u_4$ and $u_2=u_3$.

To define the domain in which a mechanism of this type can exist, from (16) we obtain:

$$u_1+u_2\ge 0.25$$

(37)

Since $u_0\ge 0$ and $u_0+u_1+u_2+u_3+u_4=1$, it follows that:

$$u_1+u_2\le 0.5$$

(38)

The domain $D_{bs}$ of barycentric coordinates, in which a symmetric 5-link mechanism can exist is the trapezoid $P_{1s},P_{2s}, P_{3s},P_{4s}$, The projection of the trapezoid on the plane $u_1$, $u_2$ is shown in Figure 14.

Since the active links $u_1$ and $u_4$ have equal lengths, we obtain the same results from Formula (20) as from (22), namely:

$$u_1+{2u}_2\ge 0.5$$

(39)

This means that within region $D_{lrs}\equiv D_{rrs}$ between points $P_{2s},P_{3s}, P_{4s}$ (Figure 15) either the left link $u_1$ or the right link $u_4$ but not both simultaneously, could rotate more than $2\pi \left[rad\right]$ (see example for point $r_{s3}$—Video S1).

From (24), the condition for simultaneous independent rotation of both active links becomes:

$$u_2\ge 0.25$$

(40)

Only for mechanisms within the triangle $D_{rs}$-$P_{3s},P_{4s}, P_{5s}$ (Figure 14) both active links $u_1$ and $u_4$ can rotate more than $2\pi \left[rad\right]$ (see example for point $r_{s2}$—Video S2). Robot $r_{s4}$ falls within the region $P_{1s},P_{2s},P_{4s},$ where the motion of the links is significantly restricted—Video S3.

For a symmetric mechanism, the region with singularities of type $s_3$ where the denominator of the Jacobian determinant is zero is obtained from (28):

$$\left|\begin{array}{c}{u}_1+u_2\ge 0.25\\ u_2\le 0.25\end{array}\right.$$

(41)

This is shown in Figure 15 as the region $D_{s3s}$-$P_{1s},P_{2s},P_{5s},P_{4s}$

Singularities in which the Jacobian numerator becomes zero are possible for all mechanisms from the domain of symmetric mechanisms $D_{s2s}\equiv D_{bs}$—$P_{1s},P_{2s}, P_{3s},P_{4s}$. (Figure 15, Video S4). Singularities of type $s_4$ where $\det \left(J\right)={\displaystyle \frac{0}{0}}$ (from Equation (30)) are possible only in the region $D_{s4s}$-$P_{1s},P_{2s},P_{4s}$ (Video S5).

In Figure 14 and Figure 15, points $r_{s2}$,$r_{s3}$,$r_{s4}$ are chosen to fall in different regions of the RLBS. The coordinates of these points and a visualization of their corresponding robot schemes are given in Figure 16. From Figure 16 and Video S6 it can be seen that a robot represented by point $r_{s4}$ could fall into a singular configuration of every type $s_1$,$s_2,s_3$ and $s_4$. Moreover, it is clear that such a robot cannot rotate links $u_1$ and $u_4$ by a large angle, which confirms the geometrical analysis in Figure 14 (Video S3) from the motion map.

A robot represented by point $r_{s3}$ could fall into singular configurations of types $s_1$, $s_2$, and $s_3$. Such a robot could rotate link $u_1$ more than $2\pi \left[rad\right]$ at certain positions of link $u_4$. Similarly, if $u_1$ is at a convenient angle, it is possible to rotate link $u_4$ $2\pi \left[rad\right]$.

Finally, point $r_{s2}$ can fall into singularities of types $s_1$ and $s_2$ only. This robot has the best motor capabilities in terms of links $u_1$ and $u_4$.

If we plot the barycentric coordinates of the studied 10 robots in the diagram from Figure 15, we obtain Figure 17. All robots represented as points in the RLBS fall into the region $P_{2s},P_{3s},P_{4s}$, where singular configurations of type $s_1,s_2$ and $s_3$ are possible, but not of type $s_4$. Furthermore, from Figure 15, it can be seen that this is a region where the left and right links can be rotated by more than $2\pi \left[rad\right]$. For robots $r_6, r_7, r_8$ and $r_9$ these links can rotate independently and only singularities of type $s_1$ and $s_2$ are possible.

The points corresponding to the robots can be divided into three groups. Those that fall on the diagonal line with equation $u_1=u_2$ (marked in black), points that are on the line $u_2=0.25$ (marked in blue or very close to this line marked in light blue), and others marked in green.

We will examine how the shape of the workspace changes with the barycentric coordinates in the RLBS. The workspace of symmetric 5-link robots is obtained as a section of two identical rings with outer radius $R=u_1+u_2$, inner radius $r=u_1-u_2$ (or $r=u_2-u_1$, if $u_2>u_1$) and distance between the centers $u_0$.

The points on the diagonal $P_{5s}{P}_{6s}$ (Figure 17) define robots that have equal lengths of the links except for link $u_0$. Figure 18 shows the shape of the workspace of such robots.

Points on the horizontal line $P_{4s}{P}_{5s}$ (Figure 17) are considered next. Again, the workspace expands from zero at point $P_{4s}$ to a full circle at point $P_{5s}$. How the shape of the workspace changes as we move along this line is shown in Figure 19.

It is worth investigating how the workspace shape behaves along the boundary line $P_{2s}{P}_{4s}$ between the types of singular configurations from Figure 16. For points on the line $P_{2s}{P}_{4s}$–the shape of the workspace has the form given in Figure 20 (Video S9). In this case, the inner circles with radius $r$ always touches the outer circles with radius R. This is because this line is defined by the equation $u_1+{2u}_2=0.5$, which follows from (31), and to have such a workspace shape it is necessary that ${R=r+u}_0$. This can be written as: $u_1+u_2=u_1-u_2+u_0$ which is equivalent to $u_1+{2u}_2=0.5$.

In the RLBS, we can create a new distribution map of the different types of workspaces $W_1-W_5$. Such a map is given in Figure 21.

We have already discussed what happens on the boundaries of these 5 regions. The examples in Figure 22 (Video S10) are of robots that do not lie on the boundaries but fall approximately in the middle of areas $W_1, W_2$,$W_3$,$W_4$, and $W_5$. The diagonal $P_{5s}{P}_{6s}$ is an axis of symmetry for the map in Figure 21.

4. Discussion

4.1. Discussion of the Method and the Definitions

The RLBS defines a space in which a point $r_i$ has n coordinates representing the normalized link lengths of a certain type of robot. Thus, $r_i$ defines a set of robots that have the same link ratios and very similar kinematic properties. The RLBS can be thought of as similar to a robot’s configuration space, which has found many practical applications. In

Table 2. Comparison of the RLBS and the configuration space of a robot.

	Configuration (Point in Configuration Space)	Designation ${\mathit{q}}_{\mathit{i}}$	Point in RLBS	Designation ${\mathit{r}}_{\mathit{i}}$$\mathbf{or} {\mathit{P}}_{\mathit{i}}$
Description	Defines the relative position of the robot’s links. Represented by a vector.		Defines a set of robots with the same ratio between the lengths of their links. Represented by a vector.
	Configuration space	Designation C	RLBS	Designation D
Description	n—dimensional parallelepiped, where n is equal to the number of DoFs of the robot (usually coincides with the number of its motors). If there are obstacles in the workspace C is divided into free workspace—$C_{free}$ and workspace, occupied by obstacles—$C_{obst}$. It is useful for controlling robots.		Part of an n-dimensional plane in which the mechanism is defined. n is equal to the number of robot links. D is divided into regions in which link ratios with similar kinematic properties are defined. It is useful for designing robots.
	Limits of the configuration space		Limits of the RLBS
Description	Determined by the joints’ limits.		Determined by conditions for assembling the mechanism and the requirement for positive dimensions of the links.

, we compare the configuration space with the RLBS.

The method allows for the independent modification of each coordinate within the interval $0\le u_i\le 1$, thus examining its influence on the kinematic characteristics of the robot. It is aimed at designing or researching closed-structure robots.

The method could be applied to open-structure robots with the following adjustments: Step 2.2 should be skipped since open-structure robots do not need to satisfy geometric conditions, such as the triangle inequality, to exist; Step 2.3 should be modified to take into account the joint constraints imposed by the mechanical construction. The method can be used to study the singular configurations of open-structure robots when these depend on the ratio between the robot’s links. This dependence is deduced from the determinant of the Jacobian of the respective robot. In terms of workspace analysis, the method will deliver results that are expected to align with previous research on open-structure robots.

Many previous studies have obtained results related to size optimization for specific goals: achieving a large workspace, good mobility, avoiding singular configurations, etc. The proposed method is different in that it summarizes and visualizes the properties of robots as a function of their link proportions.

4.2. Discussion of Examples

4.2.1. Example 1

This example illustrates the concept of representing different regions in the RLBS. It helps us understand the domain $D_b$ and the boundaries within which the mechanism is defined. A motion map and a singularity map are introduced. The motion map divides $D_b$ into 3 regions with different restrictions on the links’ movements. The singularity map also divides $D_b$ into three parts, and it is notable that some of the boundary lines are the same as in the motion map. Region $D_r$, in which arbitrary movements of the active links can be realized, coincides with an area in which only singularities of type $s_2$ can appear. The region with the smallest possible movements—$P_1,P_6, P_{13},P_{14}$—is also the region where all types of singularities are possible.

4.2.2. Example 2

Each point in the RLBS corresponds to a robot $r_i$ with different link proportions and normalized workspace.

Robots $r_1$ through $r_3$ have equal lengths of all links except $a_0$ and fall on the diagonal $P_{5s}{P}_{6s}$. Their workspace is the intersection of two circles with radius R, and its area grows as it approaches point $P_{5s}$. Singularities of types $s_2$ and $s_3$ are possible.

Robots of type $r_4$ have equal lengths of all links and $a_0=0$. Point $r_4$ coincides with $P_{5s}$ in the RLBS. These robots have the largest normalized workspace ($W_{max}=\pi 0.25$) and can rotate both their active links by more than $2\pi \left[rad\right]$. Singularities of types $s_2$ and $s_3$ are possible.

Robots $r_5$ and $r_6$ lie on the horizontal line in RLBS that both separates the types of singular configurations and the types of movement restrictions for links $u_1$ and $u_4$. The normalized workspace is large and has the shape of $P_{h3}$ from Figure 19.

Robots $r_7$ and $r_8$ fall completely within regions $D_{s2s}$ and $D_{rs}$. Their active links can rotate independently, and only singularities of type $s_2$ are possible. The normalized workspace is large and has a shape similar to $P_{i2}$ from Figure 22. Robots of types $r_6$, $r_7$ and $r_8$ are being mass produced by Mitsubishi Electric.

Robots of type $r_9$ have similar characteristics as $r_7$ and $r_8$, but their normalized workspace is smaller.

Robot $r_{10}$ have similar characteristics as $r_1$, but falls on the segment $P_{1s}{P}_{3s}$, rather than the diagonal $P_{5s}{P}_{6s}$. Its workspace has the shape of $P_{L3}$ from Figure 20. Singularities of types $s_2$ and $s_3$ are possible.

The examples show that the majority of the researched robots are designed to be symmetrical. This is very similar to examples from nature where animals have symmetrical limbs. The robot of type $r_{11}$ has an asymmetric mechanical structure. Such a robot was designed and studied in [32]. Three different optimization techniques were used: particle swarm optimization, genetic algorithm, and differential evolution by varying the lengths of the links. The goal of the optimization was to minimize instabilities and the energy required by the robot to perform a certain task. The final result is a robot with link proportions represented by point $r_{11}$. This shows that when designing robots for specific optimization problems, the optimal mechanisms will fall into different places in the RLBS.

4.2.3. Example 3

If we compare Figure 12 and Figure 13 with Figure 14 and Figure 15, we will notice that the symmetry condition significantly changes the appearance of the motion map and the singularity map. The graphical representation of the mechanisms in Figure 16 confirms and illustrates the results for the regions with different active link restrictions and possible singularities. Singularities for which $\det \left(J\right)=0$ and $\det \left(J\right)={\displaystyle \frac{0}{0}}$ occur at the boundaries of the workspace, while configurations for which $\det \left(J\right)={\displaystyle \frac{const}{0}}$ are within the robot’s workspace (see Videos S4–S6). More examples and a detailed physical interpretation of the singularities for a 5-link robot are given in previous work [19]. In most cases, singularities are undesirable because they complicate the control of the robot. However, [33] showed that in order to pass from one type of solution of the inverse kinematics to another, it is useful because it expands the usable working space; it is necessary to pass through a singularity. Since the mechanism is symmetrical, the shapes of the workspace and the values of $W_n$ are symmetrical with respect to the diagonal $P_{5s}{P}_{6s}$ given in Figure 21. This symmetry does not apply to the singularities and the motion map (see Video S11).

The workspace of robots from $P_{5s}{P}_{6s}$ is an intersection of the two outer circles because in this case $u_1=u_2$ and the radius of the inner circles $r=0$. At point $P_{5s}$ the workspace becomes a circle with radius $R=0.5$, $u_0=0$, and the normalized area of the circle is $W_{max}=\pi {0.5}^2$. Robots with $P_{5s}$ proportions have the largest normalized workspace area. As we move away from $P_{5s}$ the $W_n$ decreases. At the boundaries of the domain $D_{bs}$ the values of $W_n$ approach zero. An exception is the boundary segment $P_{2s}{P}_{3s}$. As we move along the diagonal from $P_{5s}$ to $P_{6s}$, the normalized workspace area decreases, the distance $u_0$ increases, and the workspace shape changes as shown in Figure 20 (Video S7)—points $P_{d3}$; $P_{d2}$; $P_{d1}$. At point $P_{6s}$, the area becomes zero, and the two-dimensional workspace is reduced to a point.

Along the horizontal line $P_{4s}{P}_{5s}$ (analogously the vertical $P_{1s}{P}_{5s}$) the workspace is a section of two identical rings where the inner circles always touch each other. Moreover, their radius r continuously decreases to the point $P_{5s}$, where $r=0$. At point $P_{4s}$ (respectively, $P_{1s}$) $r=R$ and since $u_0=u_1+u_2$, then at $P_{4s}$ (respectively, $P_{1s}$) the workspace becomes a point again. At point $P_{2s}$, the workspace is reduced to a circle with radius $R=u_1=0.5$, i.e., for this point of the RLBS we have a one-dimensional workspace.

Within region $W_1$ the workspace is topologically divided into two regions that do not intersect (Figure 22, Video S10). As a result, the robot can only operate in one of the two regions and cannot move to the other. This further halves the value of $W$. Areas $W_2$ and $W_3$ are very favorable for robots. In them, the workspace is simply connected, and the movements of the robot can be realized by avoiding singular configurations. Moreover, in these areas $W_n$ has relatively large values. All 10 robots from Table 1 with a symmetrical structure fall into these two areas.

In regions $W_4$ and $W_5$ the workspace is simply connected, but the W values are significantly smaller. These regions are also unfavorable with respect to the types of possible singularities

Figure 18, Figure 19 and Figure 20 illustrate how the shape of the workspace changes along the boundaries of areas $W_i$. Point $P_{8s}$ (respectively $P_{9s}$ for the vertical line) intersects the horizontal line $P_{4s}{P}_{5s}$ with the diagonal $P_{1s}{P}_{3s}$. The shape of the workspace there matches that of $P_{h2}$ in Figure 19. Point $P_{7s}$ intersects the three diagonal lines (Figure 21) and the corresponding workspace is like that of point $P_{L2}$ from Figure 22.

The shape of the workspace at points from the segment $P_{4s}{P}_{8s}$ is divided into two regions that touch only at two points (Figure 19). To pass from one region to the other, the robot will necessarily have to pass through a singular configuration. This is also the case for segment $P_{9s}{P}_{2s}$ and its symmetrical counterpart $P_{8s}{P}_{3s}$ (Figure 20).

4.3. Advantages and Disadvantages

The method offers a scalable way of assessing the kinematic properties of every possible closed-structure robot. It allows us to study the influence of the proportions of a robot’s links on its properties. This means that, once generated, results can be applied to all robots with the same proportions. A complete and systematic view of the kinematics of a mechanical system is obtained, in contrast to the study of specific structures with fixed dimensions or variation of specific lengths only, as has previously been done in the reviewed literature.

The mathematically determined areas of the RLBS correspond to qualitative changes in the examined kinematic properties, such as the topology of the workspace and the number and type of singular configurations. All sources found in the literature that examine singular configurations do not take into account the conditions for their existence caused by the proportions of the lengths of the robot’s links. Instead, they only study the influence of joint angles.

In addition to offering theoretical analysis, the method could be integrated in the process of robot design to offer solutions to real world problems. For example, it can be used in combination with other methods to provide manufacturers with a set of robot designs (RLBS points) that meet their needs, i.e., offer the desired workspace, avoid problematic singular configurations, etc. This is particularly valuable in the current context of robot design, when technology, such as 3D printing, allows the production of links with customizable dimensions. Elements of the method have been used in previous work [7] to optimize the main dimensions of a walking robot in order to reduce energy losses during movement on flat terrain.

For robots with a small number of links, the approach allows for a clear and understandable visualization of the results, as can be seen from the examples. This also makes it suitable for teaching purposes.

The method enables us to compare qualitative kinematic characteristics of robots with different scales (see Section 3.2). The scope of the method can be extended to study other parameters affected by the link proportions, such as the service index, the manipulability, the condition number, and others.

A disadvantage is that constructive joint limitations are not taken into account. Not all properties of a mechanical system depend solely on the link proportions. The method is applicable for kinematic and kineto-static studies based on Formulas (1) and (2), but not for dynamic models. For robots with a large number of links, the results are more difficult to obtain and cannot be represented graphically.

5. Conclusions

The paper introduced two new concepts: the RLBS and the corresponding normalized workspace. It presented an algorithm for studying the kinematic properties of robots based on their link proportions. In addition, a geometric representation of the results and their mapping in the RLBS is offered. We introduced maps of the motions; the singular configurations; and how the workspace shape changes with the link proportions. The proposed approach is useful in the design and analysis of closed-structure robots. Once created, the maps for a particular type of robot can be used by researchers, users, and manufacturers for a clear, visual assessment of its qualities.

Future work will include studying the applicability of the method to spatial robots with a closed structure. Furthermore, it is worth applying the method to the study of a robot’s manipulator orientation, orientation-related singular configurations, service index, condition number, etc. The application of the approach to represent regions in the workspace and configuration space that are only reachable by singular configurations will be explored. These are expected to scale while maintaining the proportions of the robot’s links. It would be interesting to investigate the behavior of different types of inverse kinematics solutions in the barycentric space. Future work could examine the behavior of the mobility coefficient as the proportions of the robot’s links change. The method could also be adjusted for the design of open-structure robots that have to operate in an environment with multiple known obstacles.

Through the presented new perspective, future research is expected to reveal new understudied properties of robot mechanisms that have practical and theoretical implications.