Design, Implementation, and Experimental Evaluation of a 6-DoF Parallel Manipulator Driven by Pneumatic Muscles

Abstract

This paper presents the design, implementation, and experimental results of a six-degree-of-freedom Delta-type parallel manipulator, in which all actuators were realized using proprietary pneumatic muscles. The objective of the study was to evaluate the suitability of this type of actuator for applications in parallel robotics, with particular attention to their dynamic properties, nonlinearities, and potential limitations. In the first part of the article, the details of the manipulator’s construction and the kinematic model, covering both the forward and inverse kinematics, are presented. The control system was based on antagonistic pairs of pneumatic muscles forming servo drives responsible for the motion of individual arms. The experimental investigations were focused on analyzing trajectory-tracking accuracy and positioning repeatability, both in unloaded conditions and under additional payload applied to the end-effector. The results indicate that positioning errors for simple trajectories were generally below 1 mm, whereas for complex trajectories and under load, they increased, particularly during changes in motion direction, which can be attributed to friction and hysteresis phenomena in the muscles. Repeatability tests confirmed the ability of the manipulator to repeatedly reach the desired positions with small deviations. The analysis carried out confirms that pneumatic muscles can be effectively applied to drive parallel manipulators, offering advantageous features such as high power density and low mass. At the same time, the need for further research on nonlinearity compensation and durability enhancement was demonstrated.

1. Introduction

The six-degree-of-freedom (6DoF) Delta-type parallel manipulator belongs to the group of closed-chain kinematic structures, which are gaining increasing importance in industrial and research applications. Their main advantages include a high load-to-actuator mass ratio, high structural stiffness, and high dynamic performance resulting from the relatively low mass and moments of inertia of the moving components [1,2,3,4,5,6]. Owing to these features, parallel manipulators are applied, among others, in assembly processes, packaging, and medical robotics. A significant limitation of these structures, however, is the smaller workspace compared to serial manipulators of similar dimensions [7,8,9,10,11,12,13].

In recent years, there has been growing interest in the use of pneumatic muscles as alternative actuators in robotics [14,15,16,17]. These muscles, inspired by the McKibben design, are characterized by a simple structure, high power density, and the ability to generate significant forces while maintaining low self-weight [18,19,20,21]. Their flexibility, operational safety, and overload resistance make them attractive for applications where lightweight structures and human–robot interaction are required. At the same time, pneumatic muscles exhibit strong nonlinearity, hysteresis, and limited durability, which necessitate the use of advanced modeling and control methods.

The present article constitutes the third part of a publication series devoted to the design and control of a Delta-type parallel manipulator driven by proprietary pneumatic muscles. In the first study [1,20], the static and dynamic characteristics of the developed muscle were presented, the methodology of their determination was described, and the first mathematical model was proposed. Based on this, a muscle-driven servo actuator was constructed, consisting of a pair of antagonistically arranged muscles capable of generating the required torque. The feasibility of its application for driving a single manipulator arm was demonstrated.

In other publications [22,23], the proprietary muscle was compared with a commercial solution developed by Festo [19], which allowed for the evaluation of the proposed design’s applicability. Dynamic characteristics of the muscle were also presented, and an inverse model used in the control process was proposed. On this basis, a servo actuator capable of achieving a prescribed angular position of a single arm of the parallel manipulator was developed and experimentally verified.

This study is a natural continuation of the above research. Its objective is the design, implementation, and experimental evaluation of a 6DoF parallel manipulator driven by pneumatic muscles. This paper presents a detailed description of the device’s construction, the kinematic model covering both inverse and forward kinematics, the control system based on muscle-driven servo actuators, and the results of accuracy and repeatability tests of positioning. The obtained results make it possible to assess the suitability of pneumatic muscles as actuators for parallel manipulators and indicate directions for further work aimed at improving the quality of control and the reliability of such systems.

Recent studies on Delta-type parallel manipulators have explored various approaches to achieve high positioning accuracy with reduced mass and energy consumption. For example, Creţescu et al. analyzed the dynamic performance of a Delta robot with flexible links and demonstrated that link compliance can lead to direction-dependent positioning errors [24]. Vodovozov et al. investigated energy management strategies for linear Delta robots and reported up to an 18% reduction in energy consumption through trajectory and configuration optimization [25]. A summary comparison of these solutions with the proposed pneumatic-muscle-driven manipulator is presented in

Table 1. Comparison of selected Delta-type manipulators and the proposed pneumatic-muscle-driven design.

Reference	Topology/Actuation	Moving Mass/Weight	Positioning Accuracy/Errors	Additional Remarks
Creţescu et al. (2023)  [24]	Delta with flexible links	Lightweight links	Direction-dependent positioning errors due to compliance	Focus on structural deformation analysis and dynamic response
Vodovozov et al. (2024)  [25]	Linear Delta, electric drives	Not specified	Standard accuracy (tens of μm to mm depending on configuration)	Up to 18% energy reduction through trajectory optimization
Proposed manipulator	6-DOF Delta, pneumatic muscles	2 kg/15 kg	<1.1 mm	Low inertia, inherent compliance, possibility of pre-tensioning, improved safety

. Compared with these solutions, the proposed manipulator achieves a positioning error of <1.1 mm with a total moving mass of only 2 kg, offering a favorable accuracy-to-weight ratio. Furthermore, the use of pneumatic muscles inherently improves safety due to their low inertia and compliance, making the design particularly attractive for applications where human–robot interaction, adaptability, or energy-efficient actuation are required.

The main contribution of this work is the design, implementation, and experimental validation of a fully pneumatic-muscle-driven 6-DoF Delta-type parallel manipulator. Unlike previous studies that typically rely on electric or hydraulic actuators, the proposed solution achieves positioning errors below 1.1 mm while maintaining a lightweight and inherently compliant structure. This combination of high accuracy, low moving mass, and safe actuation makes the manipulator suitable for applications requiring energy-efficient operation and safe human–robot interaction. The presented results bridge the gap between actuator-level research on pneumatic muscles [1,22] and the system-level evaluation of a complete parallel manipulator.

2. Description of the Manipulator Design

The task of the presented six-degree-of-freedom Delta-type parallel manipulator with a closed kinematic chain is to execute the working motions of the moving platform with respect to the fixed base. These motions result from the controlled movement of all six arms simultaneously, while only the active segments of the arms are directly actuated. This design offers a number of advantages, the most important of which is the high load-to-actuator power ratio, particularly in comparison to manipulators with an open kinematic chain. The high load capacity arises from the fact that the device’s end-effector is supported simultaneously by all six arms, and the loads resulting from the manipulator’s operation are distributed across all actuators. Moreover, none of the actuators is burdened with the mass of another actuator or with an arm other than the one it directly drives. Such a configuration also ensures high dynamic performance of the device, since the masses and moments of inertia of the individual moving components are significantly lower than in open-chain manipulators of comparable dimensions. The main drawback of this construction, however, is the substantially smaller workspace compared to serial manipulators. The issue of the workspace of the presented manipulator is discussed in more detail in the following section. Figure 1 shows the solid model developed in 3D CAD software (Solid Works 2022) and the actual view of the device.

The manipulator consists of a fixed base, six identical and independent arms arranged symmetrically in pairs, and a moving platform (2). Each arm is composed of an active part (4) connected to the drive unit (5), a passive part (3) connected to the working platform, and two spherical joints (1a, 1b). Each arm is actuated by a pair of pneumatic muscles (6), supplied with a working medium through pneumatic pressure servo valves (7). The base of the device was made of 6 mm stainless steel sheets from alloy EN1.4301 and aluminum profiles from alloy ENAW6060. Two valve manifolds, each equipped with six Parker Tecno Plus pneumatic pressure servo valves, were mounted on the base column. In addition, the base houses a proprietary electronic system performing the functions of signal concentration and conversion from angular-position transducers, from the SSI serial protocol to the RS232 serial protocol. The manipulator platform was manufactured from 2 mm aluminum sheets from alloy ENAW1050A. The active parts of the arms were made of aluminum tubes with a diameter of 12 mm from alloy ENAW6060, with a mass of 0.1 kg each. The use of third-class kinematic pairs in the arm connections ensures that only compressive and tensile loads along the longitudinal axis occur in the passive parts of the arms. Each manipulator arm is driven by an independent muscle-driven servo actuator, in which the linear motion of the pneumatic muscles is converted into the rotary motion of the active arm link by means of a toothed belt transmission equipped with a toothed pulley with a pitch diameter of $d_p=63.66$ mm and an HTD5m timing belt. A solid model of the drive and its actual view are shown in Figure 2. A single drive consists of two pneumatic muscles working antagonistically (5), a toothed belt transmission (1) and (4), a bearing assembly (2), and an absolute optical angular-position transducer OCD-S101G-1416-S060-CAW with a resolution of 16 bits (3). Each muscle is equipped with a mounting element with a pneumatic fitting (6) and a strain gauge force sensor (7). A single manipulator arm has seven degrees of freedom: two third-class kinematic pairs and one fifth-class kinematic pair. Six of the seven degrees of freedom are active, while the seventh degree of freedom, corresponding to the rotation around the longitudinal axis of the passive arm part, is passive. Motion in this rotation does not change the position of the point located inside the upper spherical joint and, therefore, does not alter the position or orientation of the end-effector.

3. Kinematic Model of the Manipulator

The task of the manipulator is to execute the motion of the working platform with respect to the fixed base. The platform’s motion is the resultant of the simultaneous controlled movements of all six arms. This method of motion realization makes manual control of the manipulator—understood as changing the angular position of each actuator independently of the others, as is the case in open-chain manipulators—very difficult or even impossible in this construction. In order to control the manipulator, knowledge of its kinematic model is required. The kinematic model of the developed device consists of mathematical relationships linking the positions and orientations of the individual moving elements of the manipulator, and in particular, the relationships connecting the position and orientation of the end-effector with the angular positions of the actuators (position kinematics). In the present study, dependencies relating to the velocities and accelerations of the manipulator’s moving elements were not considered. For the development of the kinematic model, the following assumptions and notations were adopted. All coordinate systems used are right-handed Cartesian systems. The base coordinate system $U_0$ was assigned to the base of the device, while the coordinate system $U_P$ was assigned to the end-effector, as shown in Figure 3.

The Cartesian coordinates of the manipulator are denoted by the vector $\mathbf{p}$ = $\left[xyz\alpha \beta \gamma \right]$, where the first three elements represent the position coordinates of the end-effector, and the remaining three define the orientation coordinates of the end-effector. The position of the end-effector is understood as the coordinates of the origin of the $U_p$ frame with respect to the base frame $U_0$. The orientation of the end-effector was defined using three Euler angles in the $XYZ$ convention, meaning that each successive rotation is performed around the axis of the current coordinate system (1).

$${\mathbf{M}}_P^0={\mathbf{T}}_{x,x}{\mathbf{T}}_{y,y}{\mathbf{T}}_{z,z}{\mathbf{R}}_{x,\alpha }{\mathbf{R}}_{y,\beta }{\mathbf{R}}_{z,\gamma }=\left[\begin{array}{cccc}{c}_{\beta }{c}_{\gamma }& -c_{\beta }{s}_{\gamma }& s_{\beta }& x\\ s_{\alpha }{s}_{\beta }{c}_{\gamma }+c_{\alpha }{s}_{\gamma }& c_{\alpha }{c}_{\gamma }-s_{\alpha }{s}_{\beta }{s}_{\gamma }& -s_{\alpha }{c}_{\beta }& y\\ -c_{\alpha }{s}_{\beta }{c}_{\gamma }+s_{\alpha }{s}_{\gamma }& c_{\alpha }{s}_{\beta }{s}_{\gamma }+s_{\alpha }{c}_{\gamma }& c_{\alpha }{c}_{\beta }& z\\ 0& 0& 0& 1\end{array}\right]$$

(1)

where $c_{\beta }$—$cos\left(\beta \right)$, $s_{\beta }$—$sin\left(\beta \right)$.

The configuration coordinates of the manipulator were divided into two sets: primaryand secondary configuration coordinates. The primary configuration coordinates, denoted by the vector $q_{i,1}$, correspond to the angular positions of the actuators, where $i=1,2,\dots ,6$ denotes the arm number. The secondary configuration coordinates are denoted by the vectors $q_{i,2}$ and $q_{i,3}$. They represent two rotation angles around two of the three axes of the lower spherical joint of the i-th arm. It should be emphasized that the primary configuration coordinates fully determine the position and orientation of the end-effector, while the values of the secondary coordinates directly follow from the values of the primary coordinates. The arrangement of the configuration coordinates is shown in Figure 4.

3.1. Inverse Kinematics Problem

The analysis of the manipulator’s kinematic model was reduced to two fundamental problems: the inverse kinematics problem and the forward kinematics problem. The inverse kinematics problem consists of determining the configuration coordinates of the manipulator (angular positions of the actuators) based on the known Cartesian coordinates of the device (position and orientation of the end-effector) and the geometric dimensions of the device, as expressed by the relation $\mathbf{q}=g\left(\mathbf{p}\right)$. The solution of the inverse kinematics problem was carried out by considering each manipulator arm independently as an open kinematic chain with three degrees of freedom, and the same computational procedure was applied to each arm. Two additional coordinate systems, $U_{R_{1,i}}$ and $U_{R_{2,i}}$, were assigned to the i-th arm, and their arrangement is shown in Figure 5.

The coordinate system $U_{R_{l,i}}$ was obtained by rotating the platform system $U_P$ around the $U_{P,z}$ axis by the angle ${\zeta }_i$, translating along the X-axis of the current system by the value r, translating along the Y-axis of the current system by the value d, and translating along the Z-axis of the current system by the value $-h$. These transformations are described by matrix (2).

$${\mathbf{M}}_{R_{l,i}}^P={\mathbf{R}}_{Z,{\zeta }_i}{\mathbf{T}}_{X,r}{\mathbf{T}}_{Y,d}{\mathbf{T}}_{Z,-h}=\left[\begin{array}{cccc}cos{\zeta }_i& -sin{\zeta }_i& 0& rcos{\zeta }_i-dsin{\zeta }_i\\ sin{\zeta }_i& cos{\zeta }_i& 0& dcos{\zeta }_i+rsin{\zeta }_i\\ 0& 0& 1& -h\\ 0& 0& 0& 1\end{array}\right]$$

(2)

Meanwhile, the matrix ${\mathbf{M}}_{R_{l,i}}^0$, representing the position and orientation of the coordinate system $U_{R_{l,i}}$ with respect to the base coordinate system $U_0$, was obtained by multiplying the matrices according to the relation (3).

$${\mathbf{M}}_{R_{l,i}}^0={\mathbf{M}}_P^0{\mathbf{M}}_{R_{l,i}}^P$$

(3)

The first three elements of the fourth column of this matrix define the coordinates of the origin of the system $U_{R_{l,i}}$ with respect to the base coordinate system, and for a known end-effector position and orientation vector $\mathbf{p}$, they can be uniquely determined. The system $U_{R_{2,i}}$, on the other hand, was obtained by rotating the base coordinate system $U_0$ around the $U_{0,z}$ axis by the angle ${\zeta }_i$, translating along the X-axis of the current system by the value R, translating along the Y-axis of the current system by the value a, translating along the Z-axis of the current system by the value H, rotating around the Y-axis of the current system by the value of the primary configuration coordinate $q_{i,1}$, translating along the Y-axis of the current system by the value b, and translating along the X-axis of the current system by the value $l_1$. These transformations are described by Equation (4), while Equation (5) presents the first three elements of the fourth column of this matrix. The first three elements of the fourth column of this matrix define the coordinates of the origin of the system $U_{R_{2,i}}$ with respect to the base coordinate system and are functions of a single variable, $q_{i,1}$.

$${\mathbf{M}}_{R_{2,i}}^0={\mathbf{R}}_{Z,{\zeta }_i}{\mathbf{T}}_{X,R}{\mathbf{T}}_{Y,a}{\mathbf{T}}_{Z,H}{\mathbf{R}}_{Y,q_{i,1}}{\mathbf{T}}_{Y,b}{\mathbf{T}}_{X,l_1}$$

(4)

$$\begin{array}{cc}\hfill {\mathbf{M}}_{R_{2x,i}}^0\left(q_{i,1}\right)& =Rcos{\zeta }_i+l_{1}cos{\zeta }_{i}cos{q}_{i,1}-asin{\zeta }_i+bcos{\zeta }_{i}sin{q}_{i,1}\hfill \\ \hfill {\mathbf{M}}_{R_{2y,i}}^0\left(q_{i,1}\right)& =acos{\zeta }_i+Rsin{\zeta }_i+l_{1}cos{q}_{i,1}sin{\zeta }_i+bsin{\zeta }_{i}sin{q}_{i,1}\hfill \\ \hfill {\mathbf{M}}_{R_{2z,i}}^0\left(q_{i,1}\right)& =H+bcos{q}_{i,1}-l_{1}sin{q}_{i,1}\hfill \end{array}$$

(5)

The vector $\mathbf{v}=\left[U_{R_{1,i}},U_{R_{2,i}}\right]$ was adopted, whose magnitude is equal to $l_2$, i.e., the length of the passive part of the arm. It can therefore be expressed by Equation (6), which, in its expanded form, is a nonlinear function of a single variable $q_{i,1}$.

$$\begin{array}{cc}\hfill l_2^2& =v_{i,x}^2+v_{i,y}^2+v_{i,z}^2\hfill \\ \hfill l_2^2& ={\left(U_{R_{1x,i}}-U_{R_{2x,i}}\left(q_{i,1}\right)\right)}^2+{\left(U_{R_{1y,i}}-U_{R_{2y,i}}\left(q_{i,1}\right)\right)}^2+{\left(U_{R_{1z,i}}-U_{R_{2z,i}}\left(q_{i,1}\right)\right)}^2\hfill \end{array}$$

(6)

Next, the trigonometric identity for the half-angle tangent was applied, which, after transformation, is expressed by Equation (7).

$$sin{q}_{i,1}={\displaystyle \frac{2t_i}{1+t_i^2}},cos{q}_{i,1}={\displaystyle \frac{1-t_i^2}{1+t_i^2}}$$

(7)

where $t_i=tan\left({\displaystyle \frac{q_{i,1}}{2}}\right)$.

By substituting Equation (7) into Equation (6), a single-variable equation in $t_i$ was obtained, the solution of which yields a set of two solutions, $t_{i,1}$ and $t_{i,2}$. Each of these was then substituted as the parameter of the function $2·{tan}^{-1}$, resulting in a set of two final solutions of the inverse kinematics problem (8), expressed as functions of the prescribed position and orientation parameters of the end-effector $\mathbf{p}$.

$$q_{i,1a}=2{tan}^{-1}{t}_{i,1},q_{i,1b}=2{tan}^{-1}{t}_{i,2}$$

(8)

Both solutions are valid and represent two cases in which the arm reaches the target position, as illustrated in Figure 6. They can be considered analogous to the “elbow-up” and “elbow-down” configurations known from open-chain manipulators. The configuration that is desired for the robot control process should be selected. In the case of the manipulator under discussion, this is the configuration with the smaller absolute value.

Next, the secondary configuration coordinates of the manipulator were determined. For this purpose, the coordinate system $U_{R_{l,i}}$, which is expressed with respect to the base system $U_0$ by the matrix ${\mathbf{M}}_{R_{l,i}}^0$, was expressed with respect to the system $U_{R_{2,i}}$ using the matrix ${\mathbf{M}}_{R_{l,i}}^{R_{2,i}}$. This matrix was obtained as the product of the inverse of the matrix ${\mathbf{M}}_{R_{2,i}}^0$ and the matrix ${\mathbf{M}}_{R_{l,i}}^0$, as shown in Equation (9).

$${\mathbf{M}}_{R_{l,i}}^{R_{2,i}}={\left({\mathbf{M}}_{R_{2,i}}^0\right)}^{-1}{\mathbf{M}}_{R_{l,i}}^0$$

(9)

The vector $\mathbf{w}$ was adopted, whose components are formed by the first three elements of the fourth column of the matrix ${\mathbf{M}}_{R_{2,i}}^{R_{l,i}}$. Subsequently, the coordinate system $U_{R_{3,i}}$ was associated with the manipulator arm, as shown in Figure 7.

The coordinate system $U_{R_{3,i}}$ was obtained by rotating the coordinate system $U_{R_{2,i}}$ around the axis $U_{R_{2Y,i}}$ by the angle $q_{i,2}$, which is the first secondary configuration coordinate, then rotating around the Z-axis of the current coordinate system by the angle $q_{i,3}$, which is the second secondary configuration coordinate, and finally translating along the X-axis of the current coordinate system by the value $l_2$. This transformation is described by the matrix ${\mathbf{M}}_{R_{3,i}}^{R_{2,i}}$ shown in Equation (10). The origin of the coordinate system $U_{R_{3,i}}$ coincides with the origin of the system $U_{R_{l,i}}$.

$$\begin{array}{cc}\hfill {\mathbf{M}}_{R_{3,i}}^{R_{2,i}}& ={\mathbf{R}}_{Y,q_{i,2}}{\mathbf{R}}_{Z,q_{i,3}}{\mathbf{T}}_{X,l_2}\hfill \\ \hfill {\mathbf{M}}_{R_{3,i}}^{R_{2,i}}& =\left[\begin{array}{cccc}cos{q}_{i,2}cos{q}_{i,3}& -cos{q}_{i,2}sin{q}_{i,3}& sin{q}_{i,2}& l_{2}cos{q}_{i,2}cos{q}_{i,3}\\ sin{q}_{i,3}& cos{q}_{i,3}& 0& l_{2}sin{q}_{i,3}\\ -cos{q}_{i,3}sin{q}_{i,2}& sin{q}_{i,2}sin{q}_{i,3}& cos{q}_{i,2}& -l_{2}cos{q}_{i,3}sin{q}_{i,2}\\ 0& 0& 0& 1\end{array}\right]\hfill \end{array}$$

(10)

Next, the corresponding values of the vector $\mathbf{w}$ were equated to the first three elements of the fourth column of the matrix ${\mathbf{M}}_{R_{3,i}}^{R_{2,i}}$ according to Equation (11).

$$\begin{array}{cc}\hfill w_x& =l_{2}cos{q}_{i,2}cos{q}_{i,3},\hfill \\ \hfill w_y& =l_{2}sin{q}_{i,3},\hfill \\ \hfill w_z& =-l_{2}cos{q}_{i,3}sin{q}_{i,2}.\hfill \end{array}$$

(11)

From this, relationships were obtained that make it possible to determine the secondary configuration coordinates, as shown in Equation (12). The procedure described above was applied to all arms, yielding the final solution of the inverse kinematics problem.

$$\begin{array}{cc}\hfill q_{i,3}& ={sin}^{-1}\left({\displaystyle \frac{w_y}{l_2}}\right),\hfill \\ \hfill q_{i,2}& =\left\{\begin{array}{cc}-{cos}^{-1}\left({\displaystyle \frac{w_x}{l_{2}cos{q}_{i,3}}}\right),\hfill & w_z\ge 0,\hfill \\ \phantom{-}{cos}^{-1}\left({\displaystyle \frac{w_x}{l_{2}cos{q}_{i,3}}}\right),\hfill & w_z<0.\hfill \end{array}\right.\hfill \end{array}$$

(12)

3.2. Forward Kinematics Problem

The next step was to solve the forward kinematics problem. In this case, all manipulator arms and the working platform are treated as a whole. The known values are the prescribed configuration coordinates $q_{i,1}$, while the unknowns are the desired position and orientation of the platform, expressed in the form of the vector $\mathbf{p}$. For each arm, the coordinates of the system $U_{R_{3,i}}$ with respect to the base coordinate system $U_0$ were determined, described by the matrix ${\mathbf{M}}_{R_{3,i}}^0$, which was constructed using transformation (13).

$${\mathbf{M}}_{R_{3,i}}^0\left(q_{i,2},q_{i,3}\right)={\mathbf{M}}_{R_{2,i}}^0{\mathbf{M}}_{R_{3,i}}^{R_{2,i}}\left(q_{i,2},q_{i,3}\right)$$

(13)

The elements of the matrix ${\mathbf{M}}_{R_{3,i}}^0$ contain two unknowns, $q_{i,2}$ and $q_{i,3}$, which are the secondary configuration coordinates of the i-th arm. The entire model therefore includes 12 unknowns, and in order to determine them, a system of 12 equations must be established. To this end, 12 vectors $\mathbf{u}=\left[\begin{array}{c}{u}_j\end{array}\right],j=1,2,\dots ,12,$ were introduced, linking the origins of the coordinate systems $U_{R_{3,i}}$ of the individual arms, as described by relations (14). The coordinates of these origins are given by the expressions found in the first three elements of the fourth column of the matrices ${\mathbf{M}}_{R_{3,i}}^0$ for the respective arms.

$$\begin{array}{cccccc}\hfill u_1& =U_{R_{3,1}}^0-U_{R_{3,2}}^0,\hfill & \hfill u_2& =U_{R_{3,1}}^0-U_{R_{3,3}}^0,\hfill & \hfill u_3& =U_{R_{3,1}}^0-U_{R_{3,4}}^0,\hfill \\ \hfill u_4& =U_{R_{3,1}}^0-U_{R_{3,5}}^0,\hfill & \hfill u_5& =U_{R_{3,1}}^0-U_{R_{3,6}}^0,\hfill & \hfill u_6& =U_{R_{3,2}}^0-U_{R_{3,3}}^0,\hfill \\ \hfill u_7& =U_{R_{3,2}}^0-U_{R_{3,4}}^0,\hfill & \hfill u_8& =U_{R_{3,2}}^0-U_{R_{3,5}}^0,\hfill & \hfill u_9& =U_{R_{3,2}}^0-U_{R_{3,6}}^0,\hfill \\ \hfill u_{10}& =U_{R_{3,3}}^0-U_{R_{3,4}}^0,\hfill & \hfill u_{11}& =U_{R_{3,3}}^0-U_{R_{3,5}}^0,\hfill & \hfill u_{12}& =U_{R_{3,3}}^0-U_{R_{3,6}}^0\hfill \end{array}$$

(14)

The magnitudes of these vectors are known and result from the geometric dimensions of the platform. Thus, a system of 12 equations with 12 unknowns was formulated, as expressed in Equation (15), and subsequently solved using Wolfram Mathematica software (version 11.1.1.0).

$$u_{j,x}^2+u_{j,y}^2+u_{j,z}^2=k_j^2,j=1,2,\dots ,12$$

(15)

Such a solution of the forward kinematics problem requires providing a vector of 12 initial values for the coordinates $q_{i,2}$, $q_{i,3}$. In the present study, the forward kinematics problem was used to analyze the results of accuracy and repeatability tests of the manipulator end-effector positioning. In this analysis, the desired position and orientation of the end-effector, which the manipulator was supposed to achieve, were known, as well as the actual values of the primary configuration coordinates $q_{i,1}$. The vector of initial values was therefore determined by solving the inverse kinematics problem for the secondary configuration coordinates. Having obtained the values of all 18 configuration coordinates, and thus the coordinates of the origins of the systems $U_{R_{3,i}}$ with respect to the base coordinate system $U_0$, the determination of the position and orientation of the system $U_P$ was carried out, which in fact constitutes the solution of the forward kinematics problem. An additional coordinate system $U_{P4}$ was introduced, obtained by translating the system $U_P$ along the z-axis by the value $-h$. The origin of this system lies on the plane passing through the origins of the coordinate systems $U_{R_{3,i}}$. The coordinates of the origin of the system $U_{P4}$ are the arithmetic means of the respective coordinates of the origins of the systems $U_{R_{3,i}}$, as expressed in Equation (16).

$$U_{P4,x}={\displaystyle \frac{1}{6}}\sum _{i=1}^6{U}_{R3x,i};U_{P4,y}={\displaystyle \frac{1}{6}}\sum _{i=1}^6{U}_{R3y,i};U_{P4,z}={\displaystyle \frac{1}{6}}\sum _{i=1}^6{U}_{R3z,i}$$

(16)

Next, two leading vectors ${\mathbf{m}}_x$ and ${\mathbf{m}}_2$ were defined in the system $U_{P4}$. The first vector points to the midpoint of the segment connecting the origins of the systems $U_{R3,1}$ and $U_{R3,2}$, and thus coincides with the axis $U_{P4,x}$. The second vector, in turn, points to the origin of the system $U_{R3,2}$. These vectors are described by the relations (17).

$${\mathbf{m}}_x={\displaystyle \frac{\left(U_{R,1}+U_{R,2}\right)}{2}}-U_P,{\mathbf{m}}_2=U_{R,2}-U_P$$

(17)

After normalization, the unit vectors ${\widehat{\mathbf{m}}}_x$ and ${\widehat{\mathbf{m}}}_2$ were obtained. By applying the cross product of these unit vectors, the unit vector ${\widehat{\mathbf{m}}}_z$ was determined, coinciding with the axis $U_{P4,z}$. Subsequently, the cross product of the unit vectors ${\widehat{\mathbf{m}}}_z$ and ${\widehat{\mathbf{m}}}_x$ made it possible to determine the unit vector ${\widehat{\mathbf{m}}}_y$, coinciding with the axis $U_{P4,y}$. These relationships are shown in Equation (18).

$${\widehat{\mathbf{m}}}_z={\widehat{\mathbf{m}}}_x\times {\widehat{\mathbf{m}}}_2,{\widehat{\mathbf{m}}}_y={\widehat{\mathbf{m}}}_z\times {\widehat{\mathbf{m}}}_x$$

(18)

The components of the determined unit vectors are the direction cosines of the system $U_{P4}$ with respect to the base system $U_0$. Therefore, the position of the system $U_{P4}$ was defined by the matrix ${\mathbf{M}}_{P4}^0$. Subsequently, the system $U_{P4}$ was translated along the z-axis by the value h, yielding the final system $U_P$, whose position with respect to the base system was described by the matrix ${\mathbf{M}}_P^0$, expressed by the relation ${\mathbf{M}}_P^0\left(\mathbf{q}\right)={\mathbf{M}}_{P4}^0\left(\mathbf{q}\right){\mathbf{T}}_{z,h}$.

The matrix ${\mathbf{M}}_p^0$, which includes the direction cosine submatrix, the position vector, scale, and perspective, fully defines the position and orientation of the end-effector but contains redundant information. Therefore, a robot parametrization procedure must be carried out, i.e., the determination of Euler angles based on the matrix ${\mathbf{M}}_p^0$. In the general case, this problem is ambiguous, since multiple sets of Euler angles in the $XYZ$ convention may correspond to the same direction cosine matrix. However, in the manipulator under consideration, the motion range of the end-effector is limited, which means that the angle $\beta$ is always within the range $\beta \in 〈-\frac{\pi }{2}\mathrm{rad},\frac{\pi }{2}\mathrm{rad}〉$. Consequently, the parametrization problem has a unique solution, which is presented in Equation (19).

$$\begin{array}{cc}\hfill \alpha & ={tan}^{-1}\left(\frac{-{\widehat{m}}_{x,y}}{{\widehat{m}}_{z,z}}\right)\hfill \\ \hfill \beta & ={tan}^{-1}\left({\widehat{m}}_{x,z}\sqrt{{\widehat{m}}_{x,y}^2+{\widehat{m}}_{y,y}^2}\right)\hfill \\ \hfill \gamma & ={tan}^{-1}\left({\widehat{m}}_{x,z}sin\alpha +{\widehat{m}}_{x,y}cos\alpha ,{\widehat{m}}_{y,y}cos\alpha +{\widehat{m}}_{y,z}sin\alpha \right)\hfill \end{array}$$

(19)

Simulation-based verification of the developed kinematic model of the manipulator was then carried out. For this purpose, a set of end-effector trajectories in space was generated, and for each trajectory, the inverse kinematics problem was solved for both the primary and secondary configuration coordinates. In Figure 8 and Figure 9, plot a shows the prescribed trajectory of the end-effector position, while plot b presents the prescribed trajectory of the end-effector orientation. Plot c shows the trajectories of the primary configuration coordinates $q_{i,1}$, whereas plots d and e present the trajectories of the secondary configuration coordinates $q_{i,2}$ and $q_{i,3}$, respectively. Figure 8 presents the results obtained for the end-effector trajectory: $x=100sint$, $y=100sin\left(t+\frac{\pi }{2}\right)$, $z=500+30sint$, $\alpha =0.1sint$, $\beta =-0.1sint$, $\gamma =0$, $t\in \langle 0,2\pi \rangle$.

Figure 9 presents the results obtained for the end-effector trajectory: $x=100sint$, $y=100sin\left(t+\frac{\pi }{2}\right)$, $z=440+10t$, $\alpha =0.1sint$, $\beta =-0.1sin\left(3t\right)$, $\gamma =0$, $t\in \langle 0,4\pi \rangle$.

The simulation studies conducted according to the above scheme provide valuable insights into the operation of the designed manipulator. They can be used to evaluate the working ranges of the individual moving elements, including the required operating range of the actuators for prescribed trajectories. Another type of simulation study that delivers important information about the manipulator’s performance is the determination of its workspace. The workspace is understood as the set of points in Cartesian space that can be reached by the manipulator’s end-effector, taking into account the geometric dimensions of the device and the operating ranges of its individual joints. Determining the complete workspace is a challenging task, as it requires a broader analysis of the kinematic model. This analysis must address the determination of all angle values between the moving elements of the manipulator for a given end-effector position and orientation. It must also take into account that some angular values are mutually dependent. For example, two rotation angles about two perpendicular axes in a single spherical joint are interdependent in such a way that the maximum rotation about one axis is possible only when the rotation about the other axis is zero. For these reasons, this section presents only an approximate shape and size of the workspace of the designed manipulator. Two workspaces were determined by randomly generating, according to a uniform distribution, a set of 500,000 vectors of the prescribed end-effector position and orientation based on relation (20).

$$\begin{array}{cc}\hfill x& \in \langle -500\mathrm{mm},500\mathrm{mm}\rangle ,y\in \langle -500\mathrm{mm},500\mathrm{mm}\rangle ,\hfill \\ \hfill z& \in \langle 300\mathrm{mm},700\mathrm{mm}\rangle ,\alpha =\beta =\gamma =0\mathrm{rad}\hfill \end{array}$$

(20)

Next, for each vector of prescribed values, the inverse kinematics problem was solved, and the obtained configuration coordinates were verified in terms of compliance with the assumptions defined by Equation (21).

$$q_{i,1}\in 〈-\frac{\pi }{2}\mathrm{rad},\frac{\pi }{6}\mathrm{rad}〉,q_{i,3}\in 〈-\frac{\pi }{9}\mathrm{rad},\frac{\pi }{9}\mathrm{rad}〉$$

(21)

Only those prescribed values that satisfied condition (21) were included in the set forming the workspace. Figure 10 presents the shape and size of the workspace for the conditions defined by relation (20).

The most important aspect analyzed in this chapter is the solution of the inverse and forward kinematics problems of the manipulator. The inverse kinematics problem was solved analytically, using matrix notation and geometric relations. This approach made it possible to obtain a set of mathematical dependencies that can be applied in the manipulator control process by implementing them in the control software, which largely allows us to conclude that the specific objective of developing the device’s kinematic model has been achieved. Subsequently, the forward kinematics problem was solved. This problem is significantly more complex than the inverse kinematics problem; however, it is not essential in the control process of the device. For this reason, it was solved numerically, and the algorithm itself was used offline to evaluate the accuracy of the entire system. The simulation results of the kinematic model presented in this chapter, showing the trajectories of individual actuators for prescribed end-effector trajectories, indicate a strong nonlinearity of the computed trajectories. Moreover, these studies made it possible to assess the maximum required motion ranges of the actuators, as well as the dependencies between changes in the position of a given actuator and changes in the end-effector position. This allowed estimating the influence of servo actuator accuracy on the end-effector positioning accuracy. This chapter also presented the shape and size of the workspace. It should be emphasized, however, that this represents only an approximate workspace. Determining the exact workspace is a challenging task, as its size and shape depend, among other factors, on the permissible motion ranges of all joint connections. Nevertheless, even an approximate size and shape of the workspace constitute a valuable source of information about the manipulation capabilities of the developed device.

3.3. Description of the Control System

The manipulator presented in this work is built from six servo drives described in ref. [22]. Each drive consists of two Parker Tecno Plus pneumatic pressure servo valves, each equipped with an analog voltage input of $0÷10$ V, two strain gauge force sensors with a measurement range of $\pm 2$ kN corresponding to an analog voltage output in the range of $\pm 10$ V, and an absolute angular position encoder with a resolution of 16 bits per revolution, equipped with an RS422 interface using the SSI serial protocol. The control system of the manipulator was built using the dSpace real-time system and Matlab/Simulink software (version 2019b). The dSpace system was equipped with a DS2004 extension board containing 16 independent analog voltage inputs with 16-bit resolution, operating within the range of $\pm 10$ V. In the presented control system, 12 inputs operating in single-ended mode were used to measure the voltage signals from the strain gauge force sensors. The system was also equipped with a DS2103 extension board containing 16 multiplexed analog voltage outputs with 16-bit resolution, operating in the range of ±10 V. In the presented control system, 12 outputs operating in single-ended mode were used, each generating a voltage control signal for one of the valves. Since the real-time system used was not equipped with extension boards supporting the SSI protocol, a dedicated electronic system was developed, containing an STM32F407VGT6 microcontroller capable of handling six sensors with the RS422 interface and SSI protocol, as well as two devices with the RS232 interface. The device was equipped with proprietary software designed to aggregate and convert signals from angular position encoders from the SSI protocol to the RS232 protocol. This device is shown in Figure 11. The real-time system was equipped with one RS232 standard port, through which the information about the current angular position of the actuators was received. The maximum transmission speed was 115.2 kbps, which made it possible to transfer the complete information about the sensor states at a frequency of 500 Hz.

Figure 12 presents the block diagram of the manipulator control system. The diagram shows the main blocks of the system performing the key functions and illustrates the flow of individual signals.

The yellow block performs the function of generating the prescribed position and orientation of the end-effector $\mathbf{p}$ in Cartesian coordinates. It provides the capability of generating linear, parabolic, and sinusoidal trajectories with different parameters. The generated values are fed into the orange block, which solves the inverse kinematics problem. The output of the orange block is the vector ${\theta }_{zad}$, containing six prescribed angular position values for the servo drives. The red block performs the function of reading the current angular positions $\theta$ from the serial port of the dSpace system. It implements initialization with initial values, transmission error checking, and linear extrapolation of values, which enables the control system to operate at a higher computation frequency than the data reception frequency from the serial port. The purple block performs the function of reading the current torque values $\mathbf{M}$ acting on the individual pneumatic servo drives. The measurement frequency of each analog input on the DS2004 extension board was set to a value forty times higher than the operating frequency of the entire control system. This allowed the current force value to be determined as the arithmetic mean of the last 40 measurements, reducing the influence of electromagnetic interference on the final value. The blocks marked in blue perform the control of the muscle-driven servo actuators, which is described in greater detail in ref. [1,22]. The inputs of a single servo drive block include the current angular position of the actuator, the prescribed angular position, and the values of the two forces acting on the pneumatic muscles. The output of each servo drive is a vector of two control signals for the valves, ${\mathbf{u}}_{i,i+1}$. The block marked in green performs the function of generating the voltage control signals using the DS2103 extension board.

4. Experimental Studies of the Manipulator

4.1. Tracking Control Experiments

A series of experimental tests was carried out to verify the performance of the presented manipulator. This subsection presents the results of positioning accuracy tests under tracking control. A set of prescribed end-effector trajectories in Cartesian coordinates was generated. During their execution, the values of the primary configuration coordinates $q_{i,1}$ were acquired. Then, by solving the forward kinematics problem, the actual Cartesian coordinates of the end-effector were determined. The tracking control experiments were divided into two groups: without additional load and with an additional end-effector load. In the second case, a cylindrical mass load with a diameter of $d=120$ mm, a height of $h=12$ mm, and a mass of $m=1.1$ kg was mounted on the working platform. Three series of tests were performed for each group. In the first series, the end-effector executed 24 prescribed one-dimensional trajectories, i.e., trajectories in which only one Cartesian coordinate was varied, while the remaining coordinates were kept constant at different values. In the second series, the end-effector executed 24 prescribed two-dimensional trajectories, i.e., trajectories in which two Cartesian coordinates were varied, while the others were held constant. In the third series, the end-effector executed two prescribed three-dimensional trajectories, i.e., trajectories in which three Cartesian coordinates were varied simultaneously, while the others remained constant. The results for eight selected trajectories from the first series, eight selected trajectories from the second series, and two trajectories from the third series are presented in such a way as to allow comparisons between the two groups of experiments, i.e., without additional load and with mass load applied. Figure 13 and Figure 14 present selected results of the first series of experiments for both groups. Plots a and b show spatial views of the prescribed and actual trajectories for the tests without and with load, respectively. Plots c and d present the prescribed and actual values of the varied Cartesian coordinate for the tests without and with load, respectively. Plots e and f show the absolute errors of the end-effector position coordinates for the tests without and with load, respectively. Plots g and h present the absolute errors of the end-effector orientation coordinates for the tests without and with load, respectively. Figure 13 shows the results obtained for the trajectory: $x=100sint,y=0,z=500,\alpha =\pi /12,\beta =0,\gamma =0$.

Figure 14 presents the results obtained for the trajectory: $x=0,y=0,z=500,$ $\alpha =\frac{\pi }{12}sint,\beta =0,\gamma =0$.

In

Table 2. Positioning accuracy test results of the end-effector for selected trajectories.

	${\mathit{e}}_{\mathit{x}}$ (mm)	${\mathit{e}}_{\mathit{y}}$ (mm)	${\mathit{e}}_{\mathit{z}}$ (mm)	${\mathit{e}}_{\mathit{\alpha }}$ (rad)	${\mathit{e}}_{\mathit{\beta }}$ (rad)	${\mathit{e}}_{\mathit{\gamma }}$ (rad)	l (mm)
$x=100sint,y=0,z=500$	0.601	0.168	0.110	0.000	0.001	0.001	0.634
$\alpha =\pi /12,\beta =0,\gamma =0$	0.898	0.287	0.194	0.001	0.003	0.001	0.963
$x=100sint,y=0,z=500$	0.482	0.178	0.096	0.001	0.001	0.001	0.523
$\alpha =-\pi /12,\beta =0,\gamma =0$	1.209	0.597	0.248	0.002	0.005	0.003	1.714
$x=0,y=100sint,z=500$	0.359	0.497	0.065	0.001	0.001	0.001	0.617
$\alpha =0,\beta =\pi /12,\gamma =0$	0.552	4.534	0.274	0.007	0.001	0.002	1.653
$x=0,y=100sint,z=500$	0.370	0.731	0.082	0.001	0.001	0.001	0.824
$\alpha =0,\beta =-\pi /12,\gamma =0$	0.869	1.237	0.254	0.003	0.005	0.002	1.533
$x=y=0,z=500+60sint$	0.163	0.186	0.299	0.001	0.001	0.001	0.389
$\alpha =\beta =0,\gamma =\pi /12$	0.178	0.217	0.439	0.000	0.001	0.002	0.521
$x=y=0,z=500+60sint$	0.159	0.164	0.332	0.000	0.001	0.001	0.403
$\alpha =\beta =0,\gamma =-\pi /12$	0.224	0.149	0.452	0.001	0.002	0.001	0.526
$x=y=0,z=500$	0.277	0.201	0.056	0.002	0.001	0.002	0.347
$\alpha =\frac{\pi }{12}sint,\beta =\gamma =0$	0.518	0.923	0.103	0.007	0.003	0.002	1.063
$x=y=0,z=500$	0.284	0.153	0.054	0.000	0.003	0.001	0.328
$\alpha =0,\beta =\frac{\pi }{12}sint,\gamma =0$	0.979	0.240	0.124	0.001	0.008	0.001	1.016

, for selected trajectories, the mean values of the absolute errors of the individual Cartesian coordinates are summarized, calculated according to relation (22). The last column presents the mean values of the absolute error of the end-effector position, calculated according to relation (23). For each trajectory, the upper row shows the test parameters without additional load, while the lower row shows the test parameters with additional load.

$$e_k=\sum _{i=1}^n\left|e_{i,k}\right||n,k=\left[xyz\alpha \beta \gamma \right]$$

(22)

$$l=\sqrt{e_x^2+e_y^2+e_z^2}$$

(23)

Figure 15 and Figure 16 present the results of selected two-dimensional trajectories for both groups of experiments. Plots a and b show spatial views of the prescribed and actual trajectories for the tests without and with load, respectively. Plots c and d present the two prescribed and two actual values for each of the two Cartesian coordinates for the tests without and with load, respectively. Plots e and f show the absolute errors of the end-effector position coordinates for the tests without and with load, respectively. Plots g and h present the absolute errors of the end-effector orientation coordinates for the tests without and with load, respectively. Figure 15 shows the results obtained for the trajectory: $x=100sint,y=100sin\left(t+\frac{\pi }{2}\right),z=500,\alpha =\pi /12,\beta =0,\gamma =0$.

Figure 16 presents the results obtained for the trajectory: $x=0,y=0,z=500,\beta =0,$ $\alpha =\frac{\pi }{12}sint,\gamma =\frac{\pi }{12}sin\left(t+\frac{\pi }{2}\right)$.

In

Table 3. Evaluation of the results of selected two-dimensional trajectory tests.

	${\mathit{e}}_{\mathit{x}}$ (mm)	${\mathit{e}}_{\mathit{y}}$ (mm)	${\mathit{e}}_{\mathit{z}}$ (mm)	${\mathit{e}}_{\mathit{\alpha }}$ (rad)	${\mathit{e}}_{\mathit{\beta }}$ (rad)	${\mathit{e}}_{\mathit{\gamma }}$ (rad)	l (mm)
$x=100sint,z=500,y=100sin(t+\pi /2)$	0.401	0.589	0.188	0.001	0.000	0.000	0.737
$\alpha =\pi /12,\beta =\gamma =0$	1.200	1.085	0.434	0.006	0.007	0.006	1.675
$x=100sint,z=500,y=100sin(t+\pi /2)$	0.362	0.651	0.173	0.001	0.001	0.001	0.765
$\alpha =-\pi /12,\beta =\gamma =0$	1.470	2.577	0.757	0.007	0.006	0.007	3.062
$x=100sint,y=0,z=500+50sin(t+\pi /2)$	0.413	0.177	0.236	0.001	0.000	0.001	0.508
$\alpha =0,\beta =\pi /12,\gamma =0$	0.934	0.267	0.316	0.001	0.002	0.002	1.022
$x=100sint,y=0,z=500+50sin(t+\pi /2)$	0.489	0.141	0.307	0.001	0.004	0.004	0.595
$\alpha =0,\beta =-\pi /12,\gamma =0$	1.232	0.204	0.488	0.002	0.007	0.007	1.341
$x=0,y=100sint,z=500+50sin(t+\pi /2)$	0.299	0.606	0.240	0.001	0.001	0.001	0.718
$\alpha =\beta =0,\gamma =\pi /12$	0.514	1.193	0.336	0.002	0.003	0.003	1.342
$x=0,y=100sint,z=500+50sin(t+\pi /2)$	0.312	0.656	0.288	0.001	0.002	0.002	0.781
$\alpha =\beta =0,\gamma =-\pi /12$	0.701	1.268	0.315	0.006	0.004	0.004	1.483
$x=y=0,z=500$	0.230	0.204	0.068	0.001	0.002	0.002	0.315
$\alpha =\pi /12sint,\gamma =0,\beta =\pi /12sin(t+\pi /2)$	0.899	0.864	0.189	0.006	0.007	0.003	1.261
$x=y=0,z=500$	0.233	0.234	0.049	0.001	0.001	0.003	0.334
$\alpha =\pi /12sint,\beta =0,\gamma =\pi /12sin(t+\pi /2)$	0.363	0.530	0.168	0.005	0.001	0.007	0.664

, for selected trajectories, the mean values of the absolute errors of the individual Cartesian coordinates are summarized, calculated according to relation (22). The last column presents the mean values of the absolute error of the end-effector position, calculated according to relation (23). For each trajectory, the upper row shows the parameters for the test without additional load, while the lower row shows the parameters for the test with additional load.

Figure 17 and Figure 18 present the results of three-dimensional trajectory tests for both groups. Plots a and b show spatial views of the prescribed and actual trajectories for the tests without and with load, respectively. Plots c and d present the two prescribed and two actual values for two out of the three Cartesian coordinates for the tests without and with load, respectively. Plots e and f show the prescribed and actual values of the third Cartesian coordinate for the tests without and with load, respectively. Plots g and h present the absolute errors of the end-effector position coordinates for the tests without and with load, respectively. Plots i and j present the absolute errors of the end-effector orientation coordinates for the tests without and with load, respectively. Figure 17 shows the results obtained for the trajectory: $x=100sint,y=100sin(t+\pi /2),z=500+50sin\left(0.1t\right),$ $\alpha =\beta =\gamma =0.$

Figure 18 presents the results obtained for the trajectory: $x=100sint,y=100sin(t+\pi /2),$ $z=500+50sin(t-\pi /2),\alpha =\beta =\gamma =0.$

Table 4. Evaluation of the results of selected three-dimensional trajectory tests.

	${\mathit{e}}_{\mathit{x}}$ (mm)	${\mathit{e}}_{\mathit{y}}$ (mm)	${\mathit{e}}_{\mathit{z}}$ (mm)	${\mathit{e}}_{\mathit{\alpha }}$ (rad)	${\mathit{e}}_{\mathit{\beta }}$ (rad)	${\mathit{e}}_{\mathit{\gamma }}$ (rad)	l (mm)
$x=100sint,y=100sin(t+\pi /2)$	0.483	0.581	0.103	0.000	0.001	0.001	0.763
$z=500+50sin\left(0.1t\right),\alpha =\beta =\gamma =0$	0.895	0.962	0.303	0.002	0.003	0.003	1.349
$x=100sint,y=100sin(t+\pi /2)$	0.490	0.844	0.355	0.001	0.002	0.002	1.039
$z=500+50sin(t-\pi /2),\alpha =\beta =\gamma =0$	0.984	1.019	0.520	0.003	0.006	0.004	1.510

summarizes the mean values of the absolute errors of the individual Cartesian coordinates, calculated according to relation (22). The last column presents the mean values of the absolute error of the end-effector position, calculated according to relation (23). For each trajectory, the upper row shows the parameters for the test without additional load, while the lower row shows the parameters for the test with additional load.

Analysis of the presented tracking control results revealed a significant influence of external load on the accuracy of the manipulator’s performance. The maximum absolute error values for the individual Cartesian position coordinates of the end-effector were found to be two to three times higher compared to the unloaded case. This effect was even more pronounced for the Cartesian orientation coordinates of the end-effector, where, for some trajectories, the error values with load were up to four times higher than the corresponding values without additional load. Similar relationships can be observed when analyzing the mean values of the quality indices summarized in Table 2, Table 3 and Table 4, where the considerable impact of external load on the accuracy of end-effector positioning is also evident. It was also observed that the largest absolute errors occurred in those Cartesian coordinates of the end-effector whose prescribed values varied during the experiment. For example, for the one-dimensional trajectory defined as $x=0,y=100sint,z=500,\alpha =0,$$\beta =\pi /12,\gamma =0$ and the one-dimensional trajectory defined as $x=0,y=100sint,$$z=500,\alpha =0,\beta =-\pi /12,\gamma =0$, the maximum absolute error was $e_y=4.6$ mm. In contrast, for one-dimensional trajectories in which the prescribed value of this coordinate remained constant, the error did not exceed 1 mm. Furthermore, two particular cases were identified where the above relationship changes. The first concerns the trajectory defined as $x=0,y=0,z=500,\alpha =\beta =\pi /12sint,\gamma =0$, i.e., a trajectory where the prescribed orientation coordinate involving rotation about the y-axis varied. The results for this trajectory indicated reduced accuracy in achieving the position along the y-axis. The second case refers to situations where the prescribed variable is the Cartesian coordinate along the z-axis, as in the trajectory defined as $x=0,y=0,z=500+60sint,\alpha =0,$$\beta =0,\gamma =\pi /12$. In this situation, the controlled motion of the end-effector requires uniform operation of all actuators, which does not lead to a noticeable deterioration in accuracy with respect to any Cartesian position or orientation coordinate.

4.2. Repeatability Tests

Tests of positioning accuracy and repeatability of the manipulator in steady state were carried out. For this purpose, the Cartesian coordinates of five prescribed points were defined according to relation (24).

$$\begin{array}{cc}\hfill P_1& (x=0,y=0,z=500,\alpha =0,\beta =0,\gamma =0)\hfill \\ \hfill P_2& (x=100,y=100,z=520,\alpha =\pi /12,\beta =0,\gamma =0)\hfill \\ \hfill P_3& (x=-50,y=-50,z=480,\alpha =0,\beta =-\pi /12,\gamma =0)\hfill \\ \hfill P_4& (x=50,y=50,z=530,\alpha =\pi /18,\beta =0,\gamma =-\pi /18)\hfill \\ \hfill P_5& (x=-75,y=75,z=475,\alpha =-\pi /12,\beta =\pi /12,\gamma =0)\hfill \end{array}$$

(24)

Next, a sequence of manipulator motions through all prescribed points was executed. The repositioning motion from one point to another was performed using linear interpolation of each coordinate over 5 s. After reaching a given point, the manipulator was stopped for 5 s. This sequence was then repeated 117 times. During motion execution, the prescribed coordinates were recorded. From the acquired data, two-second fragments consisting of the fourth and fifth seconds of stable manipulator positioning were selected. From each fragment, the arithmetic mean was calculated, yielding one value for each Cartesian coordinate within each stable operating interval of the manipulator. The obtained values were then grouped, producing for each of the five prescribed points a set of 117 values for each coordinate $P_{n,i}(x_{n,i},y_{n,i},z_{n,i},{\alpha }_{n,i},{\beta }_{n,i},{\gamma }_{n,i})$, where $n=1,2,\dots ,5,i=1,2,\dots ,117$. An equivalent analysis was carried out for each of the five points. Figure 19 and Figure 20 present the results of accuracy and repeatability tests for two selected prescribed operating points of the manipulator. Plot a shows the end-effector position coordinates in the three-dimensional $XYZ$ space, while plot b shows the position error in the $XYZ$ space. Plot c presents the orientation of the end-effector in the three-dimensional $\alpha \beta \gamma$ space, while plot d shows the orientation error in the $\alpha \beta \gamma$ space. Plots e and f present the error values of the individual Cartesian position and orientation coordinates of the end-effector for all 117 repetitions.

Table 5. Quantitative evaluation of positioning for point 1.

	x (mm)	y (mm)	z (mm)	$\mathit{\alpha }$ (rad)	$\mathit{\beta }$ (rad)	$\mathit{\gamma }$ (rad)
$\overline{x}$	0.043	0.140	0.014	0.000	0.000	0.001
$x_{\max }$	0.333	0.601	0.133	0.002	0.004	0.003
$\sigma$	0.063	0.113	0.019	0.000	0.001	0.001

and

Table 6. Quantitative evaluation of positioning for point 5.

	x (mm)	y (mm)	z (mm)	$\mathit{\alpha }$ (rad)	$\mathit{\beta }$ (rad)	$\mathit{\gamma }$ (rad)
$\overline{x}$	0.132	0.057	0.090	0.001	0.001	0.000
$x_{\max }$	0.218	0.597	0.168	0.003	0.003	0.003
$\sigma$	0.033	0.095	0.014	0.001	0.000	0.001

provide quantitative indicators describing the manipulator performance for each prescribed position. For each Cartesian coordinate, the mean value of the absolute error $\overline{x}=\frac{1}{n}{\sum }_{k=1}^n\left|e_k\right|$, the maximum value of the absolute error $x_{\max }$, and the mean standard deviation $\sigma$ were determined. Figure 19 and Table 5 present the results of accuracy and repeatability tests of the manipulator for the prescribed end-effector position and orientation: $P_1(x=0,y=0,z=500,\alpha =0,\beta =0,\gamma =0)$.

Figure 20 and Table 6 present the results of accuracy and repeatability tests of the manipulator for the prescribed end-effector position and orientation: $P_5(x=-75,$$y=75,z=475,\alpha =-\pi /12,\beta =\pi /12,\gamma =0)$.

Analysis of the presented results shows that in steady state, the manipulator exhibits significantly higher positioning accuracy than during tracking operation. The maximum absolute error value for a single Cartesian position coordinate is $x_{\max }=1.045$ mm, while for orientation, it is $x_{\max }=0.007$ rad. The mean values of the absolute errors are also much smaller than in the case of tracking control, with their maximum values amounting to $x_{\max }=0.178$ mm for position and $x_{\max }=0.001$ rad for orientation. No significant differences were observed between the analyzed quality indicators for the individual points reached by the end-effector. This leads to the conclusion that the accuracy and repeatability of the device positioning do not depend on the prescribed position and orientation values.

To better emphasize the value of the obtained results, the experimental data have been compared with typical performance metrics reported for motor-driven Delta robots available in the literature. Commercial Delta manipulators such as ABB FlexPicker or FANUC M-1iA typically offer repeatability in the range of $\pm 0.1$–$0.2$ mm under nominal conditions. In our experiments, the proposed pneumatic-muscle-driven manipulator achieved a positioning error of <1.1 mmeven under varying payloads, which is promising given the lightweight structure and the compliance of the actuation system. Although the achieved accuracy is slightly lower than that of high-end motor-driven systems, the design offers advantages in terms of safety, reduced inertia, and potential energy efficiency, which are beneficial for applications requiring human–robot collaboration or operation in weight-constrained environments.

5. Conclusions

The article presents the design, implementation, and experimental results of a six-degree-of-freedom Delta-type parallel manipulator, in which all actuators were realized using proprietary pneumatic muscles. A detailed construction of the manipulator, the kinematic model covering both inverse and forward kinematics, and the developed control system based on muscle-driven servo actuators were presented.

The tracking control experiments demonstrated that the positioning accuracy of the end-effector depends on the type of trajectory executed and on the presence of additional load. The largest errors were observed at moments of direction change, which result from friction between the fibers of the braid and the rubber bladder of the muscle. For one-dimensional and two-dimensional trajectories, the error values were generally below 1 mm, whereas for complex trajectories, the error increased, particularly under additional load conditions. The analysis showed that the largest errors occurred in those coordinates that varied within the given trajectory, which confirms the dependence of control accuracy on the dynamic behavior of pneumatic muscles.

The repeatability tests showed that the manipulator is capable of repeatedly reaching prescribed positions and orientations with small deviations, both in unloaded conditions and when the end-effector is loaded. The obtained results indicate that despite nonlinearities and hysteresis, pneumatic muscles provide sufficient stability and precision of operation to be applied as actuators of a parallel manipulator.

In conclusion, the conducted studies confirmed the feasibility of the practical application of pneumatic muscles for driving a 6DoF parallel manipulator. Pneumatic muscles additionally offer unique features such as high overload capacity and low weight, making them particularly attractive in lightweight robotics and in applications requiring human–robot collaboration. Future research should focus on developing methods for compensating nonlinearities and hysteresis, as well as on increasing the durability of the actuating elements.

Future research will focus on integrating advanced control strategies, such as model predictive or adaptive control, to compensate for the nonlinear and hysteretic behavior of pneumatic muscles and further improve trajectory tracking accuracy [17]. Additionally, the proposed manipulator concept shows strong potential for application in areas where lightweight and safe interaction are critical, including human–robot collaboration, rehabilitation and assistive devices, and deployable robotic arms for aerospace or service robotics. These directions will help bridge the gap between laboratory prototypes and practical, energy-efficient robotic solutions.

6. Patents

Pietrala, D.; Łaski, P.; Błasiak, S.; Takosoglu, J.; Zwierzchowski, J.; Bracha, G.; Borkowski, K.; Janecki, D. Napęd elastyczny, zwłaszcza robota rehabilitacyjnego. Polski Urząd Patentowy, Patent No. PL 233065 B1, 21 May 2019.