applied

: Delta robot is a lightweight parallel manipulator capable of accurately moving heavy loads at high speed and acceleration along a spatial trajectory. This intensive dynamic process may have a signiﬁcant impact on the end-effector trajectory precision and motor behavior. The paper highlights the inﬂuence on the dynamic behavior of a Delta robot by considering individual and combined effects of clearances and friction in the spherical joints, as well as the ﬂexibility of the rod elements. The CAD modeling of the Delta robot and its motion simulation on a representative spatial trajectory where the maximum allowed values of speed and acceleration are reached were performed using the Catia and Adams software packages. The obtained results show that the methods used were successfully applied and the effects are mutually interconnected, but


Introduction
The Delta parallel robot (DPR) is a three-degree-of-freedom (3-DOF) translational manipulator that consists of a fixed base linked to a mobile platform by three arms. The first model of the Delta parallel robot was invented in 1987 by Reymond Clavel [1] as a suitable structure for high-speed and high-acceleration tasks, specially used for pick and place operations, but also for packaging, sorting, precision positioning, and other applications.
In the industry, parallel robots have light structures and usually operate at high speeds and accelerations with heavy payload; as a result, negative kinematic and dynamic effects may intervene in the operation due to joint clearances and frictions or link flexibility. Preliminary knowledge of the behavior of these robots represents a critical asset for their optimal design.
Various studies related to the analytical modeling (both kinematic and dynamic approaches) of parallel robots but also their CAD modelling and simulation can be found in the literature. To the best of our knowledge, no relevant works have been identified that address the idea of analyzing the cumulative effect of the flexibility of elastic elements in combination with the clearances and friction from the spherical joints. These three parameters can have a major role in the dynamic behavior of parallel robots. The analysis of the effects of these factors is exemplified in the paper on the case study of a Delta parallel robot (DPR).
A new DPR is proposed and developed in [2], along with its dynamic optimization. A direct and inverse pose modeling method for a DPR is addressed in [3] based on ADAMS, and a DPR kinematic model is presented in [4], completed by a closed-form inverse dynamic model using the Newton laws, a formulation called "in two spaces". An analytical approach for the dimensional synthesis of a Delta parallel robot is presented in [5]. The analytical solution presented, with dimensional optimization for the link length, aims to find the The numerical simulation of parallel robots is an attractive topic, mainly using MAT-LAB software (https://en.wikipedia.org/wiki/MATLAB) [23,24] or MATLAB simulation validated by experimental research [25,26].
In dynamic studies, various assumptions can be considered, such as rigid vs. flexible links (analytical or numerical), ideal condition vs. friction and clearance in joints. The effect of link flexibility is analyzed for different parallel robots by using ADAMS software (https://hexagon.com/products/product-groups/computer-aided-engineeringsoftware/adams) compared with rigid link case [27][28][29]. An alternative approach to obtaining the analytical model of a DPR with flexible links is presented in [30].
An optimal trajectory planning for a DPR is carried out in [31] aiming to suppress robot vibration by developing an elasto-dynamic model assuming flexible links, and a polynomial function in the operating space was considered. The dynamical model of a parallel robot considering link flexibility was developed in [32] based on co-rotational and rigid finite elements.
Several scholars have analyzed the phenomenon of friction in spherical joints. The stability analysis of a ball joint based on Coulomb and Stribeck-type model was addressed in [33]. Nonlinear periodic solutions were obtained depending on the ball joint friction parameters. The error modeling for a DPR has been analyzed in [34] by considering joint clearances.
Thus, there are works that separately deal with the influence of these factors on the kinematic and dynamic behavior of the DPR, some through analytical modeling and most through numerical simulation using specialized software, but without identifying relevant results regarding the cumulative effect of these three factors (link flexibility, friction and joint clearance).
The main problems of DPR highlighted in the literature are systematized in the Introduction: This paper addresses the following gap identified in the literature: to the best of the authors' knowledge, there is a lack of significant scientific works dealing with the cumulative effects of link flexibility, joint friction and joint clearance on the dynamic Appl. Sci. 2023, 13, 6693 3 of 18 behavior of parallel robots. Therefore, the proposed research is conducted on a Delta parallel robot using a CAD model obtained in the CATIA software (https://en.wikipedia. org/wiki/CATIA) and deriving specific simulations in the ADAMS software.
The rest of the paper is organized as follows: Section 2 presents the problem formulation; Section 3 proposes seven simulation scenarios and discusses the obtained results; and Section 4 draws final conclusions.

Problem Formulation
The effect of element flexibilities, frictions and clearances can be studied numerically by developing a CAD model of the analyzed robot in the first stage, for example, using the CATIA software, followed in the second stage by the ADAMS analysis. Since ADAMS does not easily allow the creation of elements with complex shapes, it was decided to develop the CAD model in CATIA and then export the 3D bodies (in an IGES format) to ADAMS. Thus, a CAD model can be obtained in ADAMS that reflects the properties of the existing physical robot as accurately as possible.
In this analysis, we consider the case study of a Delta Siax D3-1600 parallel robot ( Figure 1a, [35]). Its simplified CAD model (without motors) at a 1:1 scale was represented in CATIA ( Figure 1b) and then transferred to ADAMS (Figure 1c). The Delta Siax D3-1600 is a three degree-of-freedom (3-DOF) robot (the end-effector performs three independent translations). It is composed of a fixed platform (0) and a mobile platform (4), interconnected by three arms, A, B and C, each of them with a driving element (1) connected to the base by a motor drive (R). Each arm has a parallelogram-type kinematic chain with two flexible elements (2 and 3) and four passive spherical couples each (S 2k1 , S 2k2 and S 3k1 , S 3k2 , where k = A, B, C- Figure 1d). The three arms are equiangularly distributed (Figure 1e) in relation to the global coordinate system (X 0 Y 0 Z 0 ) of the robot, with the origin located at point O ( Figure 1d). The Delta robot has attached to the end-effector (4) a payload in the form of a cylinder (5) with a mass of 5 kg. The characteristic point P is the origin of the mobile coordinate system of the end-effector and it travels a spatial trajectory established so as to reach the maximum velocity and acceleration according to the values specified in Table 1. This paper addresses the following gap identified in the literature: to the best of the authors' knowledge, there is a lack of significant scientific works dealing with the cumulative effects of link flexibility, joint friction and joint clearance on the dynamic behavior of parallel robots. Therefore, the proposed research is conducted on a Delta parallel robot using a CAD model obtained in the CATIA software (https://en.wikipedia.org/wiki/CATIA) and deriving specific simulations in the ADAMS software.
The rest of the paper is organized as follows: Section 2 presents the problem formulation; Section 3 proposes seven simulation scenarios and discusses the obtained results; and Section 4 draws final conclusions.

Problem Formulation
The effect of element flexibilities, frictions and clearances can be studied numerically by developing a CAD model of the analyzed robot in the first stage, for example, using the CATIA software, followed in the second stage by the ADAMS analysis. Since ADAMS does not easily allow the creation of elements with complex shapes, it was decided to develop the CAD model in CATIA and then export the 3D bodies (in an IGES format) to ADAMS. Thus, a CAD model can be obtained in ADAMS that reflects the properties of the existing physical robot as accurately as possible.
In this analysis, we consider the case study of a Delta Siax D3-1600 parallel robot (Figure 1a, [35]). Its simplified CAD model (without motors) at a 1:1 scale was represented in CATIA ( Figure 1b) and then transferred to ADAMS (Figure 1c). The Delta Siax D3-1600 is a three degree-of-freedom (3-DOF) robot (the end-effector performs three independent translations). It is composed of a fixed platform (0) and a mobile platform (4), interconnected by three arms, A, B and C, each of them with a driving element (1) connected to the base by a motor drive (R). Each arm has a parallelogram-type kinematic chain with two flexible elements (2 and 3) and four passive spherical couples each (S2k1, S2k2 and S3k1, S3k2, where k = A, B, C- Figure 1d). The three arms are equiangularly distributed ( Figure  1e) in relation to the global coordinate system (X0Y0Z0) of the robot, with the origin located at point O ( Figure 1d). The Delta robot has attached to the end-effector (4) a payload in the form of a cylinder (5) with a mass of 5 kg. The characteristic point P is the origin of the mobile coordinate system of the end-effector and it travels a spatial trajectory established so as to reach the maximum velocity and acceleration according to the values specified in Table 1.    The geometrical, mass and material properties of the robot's component bodies are key factors in determining its dynamic behavior. In the proposed analysis, the following assumptions have been made: -Elements 0, 1, 4 and 5 are rigid solid bodies; -Rod Elements 2 and 3 have higher elastic characteristics than the other elements due their dimensions, see Table 2; -All bodies are made of steel.

Body CAD Model Parameters
Fixed platform (0) l0 = 125 mm Crank (1) l1 = 500 mm m1 = 1.58 kg d1 = 62 mm a1 = 106 mm The geometrical, mass and material properties of the robot's component bodies are key factors in determining its dynamic behavior. In the proposed analysis, the following assumptions have been made: -Elements 0, 1, 4 and 5 are rigid solid bodies; -Rod Elements 2 and 3 have higher elastic characteristics than the other elements due their dimensions, see Table 2; -All bodies are made of steel. Table 2. Geometrical and mass details of the Delta robot bodies.

CAD Model Parameters
Fixed platform (0) (d) (e)  The geometrical, mass and material properties of the robot's component bodies are key factors in determining its dynamic behavior. In the proposed analysis, the following assumptions have been made: -Elements 0, 1, 4 and 5 are rigid solid bodies; -Rod Elements 2 and 3 have higher elastic characteristics than the other elements due their dimensions, see Table 2; -All bodies are made of steel.   The geometrical, mass and material properties of the robot's component bodies are key factors in determining its dynamic behavior. In the proposed analysis, the following assumptions have been made: -Elements 0, 1, 4 and 5 are rigid solid bodies; -Rod Elements 2 and 3 have higher elastic characteristics than the other elements due their dimensions, see Table 2; -All bodies are made of steel.   The reference model of the Delta robot is based on the assumptions of an ideal mechanism, where all bodies are rigid solids, all kinematic joints are ideal (no clearance, no friction), and the characteristic point P follows a trajectory that reaches maximum allowed values of speed and acceleration. Thus, the fifth degree polynomial function was chosen to generate the movement trajectory in the joint space as well as a short trajectory travel time of 0.2 s, the time resulting from the simultaneous provision of the conditions for the robot to touch the P0P1 trajectory (Figure 2), the maximum speed of the end-effector vPmax = 8 m/s and the maximum acceleration aPmax = 120 m/s 2 (see Table 1).  The reference model of the Delta robot is based on the assumptions of an ideal mechanism, where all bodies are rigid solids, all kinematic joints are ideal (no clearance, no friction), and the characteristic point P follows a trajectory that reaches maximum allowed values of speed and acceleration. Thus, the fifth degree polynomial function was chosen to generate the movement trajectory in the joint space as well as a short trajectory travel time of 0.2 s, the time resulting from the simultaneous provision of the conditions for the robot to touch the P0P1 trajectory (Figure 2), the maximum speed of the end-effector vPmax = 8 m/s and the maximum acceleration aPmax = 120 m/s 2 (see Table 1).  The reference model of the Delta robot is based on the assumptions of an ideal mechanism, where all bodies are rigid solids, all kinematic joints are ideal (no clearance, no friction), and the characteristic point P follows a trajectory that reaches maximum allowed values of speed and acceleration. Thus, the fifth degree polynomial function was chosen to generate the movement trajectory in the joint space as well as a short trajectory travel time of 0.2 s, the time resulting from the simultaneous provision of the conditions for the robot to touch the P 0 P 1 trajectory (Figure 2), the maximum speed of the end-effector v Pmax = 8 m/s and the maximum acceleration a Pmax = 120 m/s 2 (see Table 1).

Rod element
(2 and 3) The reference model of the Delta robot is based on the assumptions of an ideal mechanism, where all bodies are rigid solids, all kinematic joints are ideal (no clearance, no friction), and the characteristic point P follows a trajectory that reaches maximum allowed values of speed and acceleration. Thus, the fifth degree polynomial function was chosen to generate the movement trajectory in the joint space as well as a short trajectory travel time of 0.2 s, the time resulting from the simultaneous provision of the conditions for the robot to touch the P0P1 trajectory ( Figure 2), the maximum speed of the end-effector vPmax = 8 m/s and the maximum acceleration aPmax = 120 m/s 2 (see Table 1).  The starting position of the Cartesian trajectory (P 0 ) corresponds to the initial position of the robot where all three motor torques R A , R B and R C are in the zero position and Elements 1 are arranged in a horizontal plane (parallel to X 0 Y 0 , see Figure 1d). The trajectory in the Cartesian space P 0 P 1 is a spatial curve obtained by applying an angular displacement of 70 • in the positive direction of the joint axis R A , 41 • in the positive direction of the coupling axis R B and of 36 • in the negative direction of the joint axis R C (Figure 3a). Along this trajectory, the maximum angular velocity of 656 • /s in joint A, 384 • /s in joint B and 292 • /s in joint C is reached (Figure 3b), as well as the maximum angular accelerations of 10,103 • /s 2 (engine A), 5124 • /s 2 (engine B) and 4500 • /s 2 (engine C), as shown in Figure 3c.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 of 18 The starting position of the Cartesian trajectory (P0) corresponds to the initial position of the robot where all three motor torques RA, RB and RC are in the zero position and Elements 1 are arranged in a horizontal plane (parallel to X0Y0, see Figure 1d). The trajectory in the Cartesian space P0P1 is a spatial curve obtained by applying an angular displacement of 70° in the positive direction of the joint axis RA, 41° in the positive direction of the coupling axis RB and of 36° in the negative direction of the joint axis RC (Figure 3a). Along this trajectory, the maximum angular velocity of 656°/s in joint A, 384°/s in joint B and 292°/s in joint C is reached (Figure 3b), as well as the maximum angular accelerations of 10,103°/s 2 (engine A), 5124°/s 2 (engine B) and 4500°/s 2 (engine C), as shown in Figure 3c.   of the robot where all three motor torques RA, RB and RC are in the zero position and Ele ments 1 are arranged in a horizontal plane (parallel to X0Y0, see Figure 1d). The trajectory in the Cartesian space P0P1 is a spatial curve obtained by applying an angular displace ment of 70° in the positive direction of the joint axis RA, 41° in the positive direction of the coupling axis RB and of 36° in the negative direction of the joint axis RC (Figure 3a). Along this trajectory, the maximum angular velocity of 656°/s in joint A, 384°/s in joint B and 292°/s in joint C is reached (Figure 3b), as well as the maximum angular accelerations o 10,103°/s 2 (engine A), 5124°/s 2 (engine B) and 4500°/s 2 (engine C), as shown in Figure 3c.   The torque in the active joints ( Figure 6) on the stated trajectory has the allure of angular acceleration (see Figure 3c); higher values of the moment TA are observed due to the higher angular accelerations (and consequently higher values of angular speeds and displacements) compared to the other two active torques. Thus, the comparative analysis of the Delta robotic structure is presented in seven dynamic simulation scenarios, taking as reference the previously defined ideal model. The study makes the following assumptions:  The torque in the active joints ( Figure 6) on the stated trajectory has the allure of angular acceleration (see Figure 3c); higher values of the moment TA are observed due to the higher angular accelerations (and consequently higher values of angular speeds and displacements) compared to the other two active torques. Under these considerations, the aim of this study is to analyze the kinematic and dynamic effects of these three factors, both individually and in combination: • Friction on spherical joints; • Clearances on spherical joints; • Elasticity of the flexible rod elements (2 and 3).
Thus, the comparative analysis of the Delta robotic structure is presented in seven dynamic simulation scenarios, taking as reference the previously defined ideal model. The study makes the following assumptions:  Under these considerations, the aim of this study is to analyze the kinematic and dynamic effects of these three factors, both individually and in combination: • Friction on spherical joints; • Clearances on spherical joints; • Elasticity of the flexible rod elements (2 and 3).
Thus, the comparative analysis of the Delta robotic structure is presented in seven dynamic simulation scenarios, taking as reference the previously defined ideal model. The study makes the following assumptions:

Results and Discussions
The influence of each of the three factors considered (Scenarios 1-3) as well as their combination (Scenarios 4-7) was analyzed by comparison with the reference model (ideal case) in order to identify (a) kinematic (displacements, speeds and accelerations of the characteristic point) and dynamic (driving torques) deviations generated by these factors. These deviations are denoted generically with e_X_p = X_p − X, where X = r P , v P , a P , TA, TB, TC, p is the considered parameter (µ-friction, e-elasticity, c-clearance), and X_p is the value of the X variable in the assumption of considering the p factor, X obtained in the ideal case; (b) the coupling effect of the factors, i.e., the extent to which they are independent variables and whether their effects can be considered additive phenomena.

Scenario 1
In this scenario, we start from the ideal case of the robot structure, to which the friction in the spherical joints S 2k1 , S 2k2 and S 3k1 , S 3k2 , k = A, B, C is added, taking into account steel/steel friction with lard oil lubricant with the 0.11 static friction coefficient and the 0.084 dynamic coefficient [36].
As is known, friction in kinematic joints does not influence the motion transmission function but has an effect on the dynamic behavior of the robot. The friction from the spherical joints has a moderate effect on the diving torques (about 0.007%, Figure 7), resulting in deviations of up to 0.508 N·m for TA (Figure 7a), 0.207 N·m for TB (Figure 7b) and 0.241 N·m for TC (Figure 7c).

Results and Discussions
The influence of each of the three factors considered (Scenarios 1-3) as well as their combination (Scenarios 4-7) was analyzed by comparison with the reference model (ideal case) in order to identify (a) kinematic (displacements, speeds and accelerations of the characteristic point) and dynamic (driving torques) deviations generated by these factors. These deviations are denoted generically with e_X_p = X_p − X, where X = rP, vP, aP, TA, TB, TC, p is the considered parameter (µ-friction, e-elasticity, c-clearance), and X_p is the value of the X variable in the assumption of considering the p factor, X obtained in the ideal case; (b) the coupling effect of the factors, i.e., the extent to which they are independent variables and whether their effects can be considered additive phenomena.

Scenario 1
In this scenario, we start from the ideal case of the robot structure, to which the friction in the spherical joints S2k1, S2k2 and S3k1, S3k2, k = A, B, C is added, taking into account steel/steel friction with lard oil lubricant with the 0.11 static friction coefficient and the 0.084 dynamic coefficient [36].
As is known, friction in kinematic joints does not influence the motion transmission function but has an effect on the dynamic behavior of the robot. The friction from the spherical joints has a moderate effect on the diving torques (about 0.007%, Figure 7), resulting in deviations of up to 0.508 N·m for TA (Figure 7a A variation of these deviations is noted for all three driving torques, with a profile similar to the acceleration ap (see Figure 5c) and in correlation with the moment variation ( Figure 6):  A variation of these deviations is noted for all three driving torques, with a profile similar to the acceleration a p (see Figure 5c) and in correlation with the moment variation ( Figure 6): • the deviation values e_Tk_µ, k = A, B, C are directly proportional to the absolute values of the moments Tk; • friction leads to an increase in the driving torques value during the acceleration phase (0.0-0.1 s interval) and helps the motors to brake during the deceleration phase (0.1-0.2 s).

Scenario 2
In the hypothesis of considering the flexibility of the flexible elements of the Delta parallel robot (Elements 2 and 3 on each arm, see Figure 8) and limiting the analysis to the first 10 vibration modes (with natural frequencies lower than 250 Hz, as the effect of Appl. Sci. 2023, 13, 6693 9 of 18 higher frequencies is negligible-the principal characteristics are presented in Figure 9), the results represented in Figure 10 (motion deviations) and Figure 11 (torque deviations) are obtained.

Scenario 2
In the hypothesis of considering the flexibility of the flexible elements of the Delta parallel robot (Elements 2 and 3 on each arm, see Figure 8) and limiting the analysis to the first 10 vibration modes (with natural frequencies lower than 250 Hz, as the effect of higher frequencies is negligible-the principal characteristics are presented in Figure 9), the results represented in Figure 10   Taking into account the natural properties of the rod elements (Table 2), the ADAMS dynamic simulation leads to low deviations from the theoretical trajectory of the effector (e_rp < 4.2·10 −6 m, Figure 10a), the speed deviation of up to 3.1·10 −4 m/s for (Figure 10b) and deviations of up to 5 m/s 2 for acceleration (i.e., max. 4.2%, Figure 10c). The largest

Scenario 2
In the hypothesis of considering the flexibility of the flexible elements of the Delt parallel robot (Elements 2 and 3 on each arm, see Figure 8) and limiting the analysis to th first 10 vibration modes (with natural frequencies lower than 250 Hz, as the effect of highe frequencies is negligible-the principal characteristics are presented in Figure 9), the re sults represented in Figure 10   Taking into account the natural properties of the rod elements (Table 2), the ADAM dynamic simulation leads to low deviations from the theoretical trajectory of the effecto (e_rp < 4.2·10 −6 m, Figure 10a), the speed deviation of up to 3.1·10 −4 m/s for (Figure 10b and deviations of up to 5 m/s 2 for acceleration (i.e., max. 4.2%, Figure 10c). The larges Taking into account the natural properties of the rod elements (Table 2), the ADAMS dynamic simulation leads to low deviations from the theoretical trajectory of the effector (e_rp < 4.2·10 −6 m, Figure 10a), the speed deviation of up to 3.1·10 −4 m/s for (Figure 10b) and deviations of up to 5 m/s 2 for acceleration (i.e., max. 4.2%, Figure 10c). The largest deviations e_vp and e_ap occur at around 0.042 s and 0.158 s, respectively, the moments of time at which the acceleration a p is at its maximum (see Figure 5c).

Scenario 3
In this subsection, we analyze the influence of the clearances in the spherical, using a single value of 0.1 mm for all 12 joints S2k1, S2k2 and S3k1, S3k2, k = A, B, C. Under these conditions, the deviation of the characteristic point in the initial position is 0.2 mm.
The deviation from the characteristic point trajectory is up to 1.18·10 −4 m (Figure 12a), with a velocity deviation of up to 0.0035 m/s (Figure 12b) and an acceleration of up to 0.068 m/s 2 (Figure 12c). Compared to Scenario 2, the displacement deviation on the trajectory is significantly higher (~20 times higher), but the deviation of the acceleration on the trajectory is much lower (~70 times lower).

Scenario 3
In this subsection, we analyze the influence of the clearances in the spherical, using a single value of 0.1 mm for all 12 joints S2k1, S2k2 and S3k1, S3k2, k = A, B, C. Under these conditions, the deviation of the characteristic point in the initial position is 0.2 mm.
The deviation from the characteristic point trajectory is up to 1.18·10 −4 m (Figure 12a), with a velocity deviation of up to 0.0035 m/s (Figure 12b) and an acceleration of up to 0.068 m/s 2 (Figure 12c). Compared to Scenario 2, the displacement deviation on the trajectory is significantly higher (~20 times higher), but the deviation of the acceleration on the trajectory is much lower (~70 times lower).

Scenario 3
In this subsection, we analyze the influence of the clearances in the spherical, using a single value of 0.1 mm for all 12 joints S 2k1 , S 2k2 and S 3k1 , S 3k2 , k = A, B, C. Under these conditions, the deviation of the characteristic point in the initial position is 0.2 mm.
The deviation from the characteristic point trajectory is up to 1.18·10 −4 m (Figure 12a), with a velocity deviation of up to 0.0035 m/s (Figure 12b) and an acceleration of up to 0.068 m/s 2 (Figure 12c). Compared to Scenario 2, the displacement deviation on the trajectory is significantly higher (~20 times higher), but the deviation of the acceleration on the trajectory is much lower (~70 times lower). Appl. Sci. 2023, 13

Scenario 4
In this subsection, we analyze the cumulative influence of friction in the spherical joints and elasticity of the rod elements. We compare and analyze the cumulative resulting deviations with the sum of the deviations identified in Scenarios 1 and 2 to identify the coupling effect between these two factors. Figure 14 shows the kinematic behavior of the Delta robot with the elastic elements and the joint friction, highlighting both the deviations of the effector motion from the ideal case (red, solid line) and the differences from the case of summing the separate effects of the two factors (blue, dashed line). For the characteristic point displacement (Figure 14a

Scenario 4
In this subsection, we analyze the cumulative influence of friction in the spherical joints and elasticity of the rod elements. We compare and analyze the cumulative resulting deviations with the sum of the deviations identified in Scenarios 1 and 2 to identify the coupling effect between these two factors. Figure 14 shows the kinematic behavior of the Delta robot with the elastic elements and the joint friction, highlighting both the deviations of the effector motion from the ideal case (red, solid line) and the differences from the case of summing the separate effects of the two factors (blue, dashed line). For the characteristic point displacement (Figure 14a

Scenario 4
In this subsection, we analyze the cumulative influence of friction in the spherical joints and elasticity of the rod elements. We compare and analyze the cumulative resulting deviations with the sum of the deviations identified in Scenarios 1 and 2 to identify the coupling effect between these two factors. Figure 14 shows the kinematic behavior of the Delta robot with the elastic elements and the joint friction, highlighting both the deviations of the effector motion from the ideal case (red, solid line) and the differences from the case of summing the separate effects of the two factors (blue, dashed line). For the characteristic point displacement (Figure 14a Similar to the motion case, the cumulative effect of the two factors results in a decrease in the maximum values of the driving torque deviations compared to the additive case ( Figure 15). Therefore, it can be concluded that these factors have no significant coupling effect. Friction (with less significance) does not affect the shape of the deviation curve, but rather contributes to better curve shapes for the torque deviation values.

Scenario 5
The kinematic deviations from the theoretical movement trajectory are shown in Figure 16. In the case of displacement, it can be seen that the cumulative effect of these two factors leads to a deviation similar to that observed in the simulative case. As a result, the effects of the two factors on the kinematic behavior of the Delta parallel robot are not cumulative and their coupling results in the same deviations. Similar to the motion case, the cumulative effect of the two factors results in a decrease in the maximum values of the driving torque deviations compared to the additive case ( Figure 15). Therefore, it can be concluded that these factors have no significant coupling effect. Friction (with less significance) does not affect the shape of the deviation curve, but rather contributes to better curve shapes for the torque deviation values. Similar to the motion case, the cumulative effect of the two factors results in a decrease in the maximum values of the driving torque deviations compared to the additive case ( Figure 15). Therefore, it can be concluded that these factors have no significant coupling effect. Friction (with less significance) does not affect the shape of the deviation curve, but rather contributes to better curve shapes for the torque deviation values.

Scenario 5
The kinematic deviations from the theoretical movement trajectory are shown in Figure 16. In the case of displacement, it can be seen that the cumulative effect of these two factors leads to a deviation similar to that observed in the simulative case. As a result, the effects of the two factors on the kinematic behavior of the Delta parallel robot are not cumulative and their coupling results in the same deviations.

Scenario 5
The kinematic deviations from the theoretical movement trajectory are shown in Figure 16. In the case of displacement, it can be seen that the cumulative effect of these two factors leads to a deviation similar to that observed in the simulative case. As a result, the effects of the two factors on the kinematic behavior of the Delta parallel robot are not cumulative and their coupling results in the same deviations.

Scenario 6
The combination between the flexibility of the rod elements and the clearances in the spherical joints (0.1 mm) has almost no effect in the displacement and velocity of the characteristic point. However, the "picks" in the acceleration are considered reduced and the cumulative effect is taken into consideration ( Figure 18).

Scenario 6
The combination between the flexibility of the rod elements and the clearances in the spherical joints (0.1 mm) has almost no effect in the displacement and velocity of the characteristic point. However, the "picks" in the acceleration are considered reduced and the cumulative effect is taken into consideration ( Figure 18).

Scenario 6
The combination between the flexibility of the rod elements and the clearances in the spherical joints (0.1 mm) has almost no effect in the displacement and velocity of the characteristic point. However, the "picks" in the acceleration are considered reduced and the cumulative effect is taken into consideration (Figure 18). In this scenario, the moment deviations have a relatively constant harmonic pitch and amplitude variation, the maximum values reaching ~50 N·m vs. ~20 N·m (in the additive case) for TA (Figure 19a), to ~30 N·m vs. ~10 N·m for TB (Figure 19b), 20 N·m vs. 22 N·m for TC (Figure 19c).

Scenario 7
In this scenario, the cumulative effects of the three factors (friction, flexibility and clearance) on the kinematic and dynamic behavior of the Delta robot are not considered. ADAMS simulations that consider the simultaneous action of those three factors resulted in values of kinematic deviations that are approximately the same values for displacement and speed ( Figure 20). However, for acceleration, smaller values are emphasized due to the better numerical integration in ADAMS compared to summing the deviations generated individually by each factor (Figure 20c).

Scenario 7
In this scenario, the cumulative effects of the three factors (friction, flexibility and clearance) on the kinematic and dynamic behavior of the Delta robot are not considered. ADAMS simulations that consider the simultaneous action of those three factors resulted in values of kinematic deviations that are approximately the same values for displacement and speed ( Figure 20). However, for acceleration, smaller values are emphasized due to the better numerical integration in ADAMS compared to summing the deviations generated individually by each factor (Figure 20c).

Scenario 7
In this scenario, the cumulative effects of the three factors (friction, flexibility and clearance) on the kinematic and dynamic behavior of the Delta robot are not considered. ADAMS simulations that consider the simultaneous action of those three factors resulted in values of kinematic deviations that are approximately the same values for displacement and speed ( Figure 20). However, for acceleration, smaller values are emphasized due to the better numerical integration in ADAMS compared to summing the deviations generated individually by each factor (Figure 20c). The maximum deviation of the driving torques shows the same type of harmonic variation as in the scenarios where the elasticity factor is considered. It reaches ~120 N·m vs. ~20 N·m (in the additive case) for TA (Figure 21a), ~80 N·m vs. ~10 N·m for TB ( Figure  21b) and ~70 N·m vs. ~10 N·m for TC (Figure 21c).

Conclusions
A new approach is employed in this paper by analyzing the impact of three factors on the kinematic and dynamic behavior of the Delta parallel robot: the elasticity of the robot's supple elements (rod elements), friction and clearance in the spherical joints. For this purpose, the analysis was carried out for the case study of a Delta SIAX 3-1600-type robot based on the 3D models developed in CATIA and simulated in the ADAMS software.
The following summarizes the effects of these factors' actions on the movement trajectory: The maximum deviation of the driving torques shows the same type of harmonic variation as in the scenarios where the elasticity factor is considered. It reaches ~120 N·m vs. ~20 N·m (in the additive case) for TA (Figure 21a), ~80 N·m vs. ~10 N·m for TB ( Figure  21b) and ~70 N·m vs. ~10 N·m for TC (Figure 21c).

Conclusions
A new approach is employed in this paper by analyzing the impact of three factors on the kinematic and dynamic behavior of the Delta parallel robot: the elasticity of the robot's supple elements (rod elements), friction and clearance in the spherical joints. For this purpose, the analysis was carried out for the case study of a Delta SIAX 3-1600-type robot based on the 3D models developed in CATIA and simulated in the ADAMS software.
The following summarizes the effects of these factors' actions on the movement trajectory:

Conclusions
A new approach is employed in this paper by analyzing the impact of three factors on the kinematic and dynamic behavior of the Delta parallel robot: the elasticity of the robot's supple elements (rod elements), friction and clearance in the spherical joints. For this purpose, the analysis was carried out for the case study of a Delta SIAX 3-1600-type robot based on the 3D models developed in CATIA and simulated in the ADAMS software.
The following summarizes the effects of these factors' actions on the movement trajectory: For all factor coupling scenarios (S4-S7), the study observed that individual effects are not always cumulative. The coupling of factors can increase deviation values when the clearances and elasticities in the joints are considered simultaneously. Consequently, it is not recommended to simulate these factors separately and sum their effects. Since the phenomena are not linear, a combined approach of the factors is necessary to obtain relevant results. The authors propose to validate the conclusions of this theoretical study resulting from numerical simulations in the ADAMS software through experimental means in the future.