Open AccessArticle

Dynamic Modeling and Analysis of Flexible-Joint Robots with Clearance

by Jing Wang 1 , Shisheng Zhou 1,2 , Jimei Wu 1,2,* , Jiajuan Qing 1 , Tuo Kang 1 and Mingyue Shao 2 1 School of Mechanical and Precision Instrument Engineering, Xi'an University of Technology, Xi'an , China 2 Faculty of Printing, Packing and Digital Media Engineering, Xi'an University of Technology, Xi'an , China * Author to whom correspondence should be addressed. Sensors , 24(13), ; https://doi.org/10./s Submission received: 14 June / Revised: 4 July / Accepted: 4 July / Published: 6 July (This article belongs to the Section Sensors and Robotics)

Abstract

: The coupling effects of flexible joints and clearance on the dynamics of a robotic system were investigated. A numerical analysis was undertaken to reveal the coupling effects between flexible joints and clearance. The nonlinear spring-damping model and Coulomb model were applied to characterize the contact characteristics of the clearance, and a model for the flexible joint was formulated using the equivalent spring theory. An accurate robot model was established based on the clearance and joint flexibility characterization. The dynamic equation of a robot was obtained according to the Newton-Euler method. A comparative analysis was performed to assess the impacts of both the joint action of clearance and flexible joints and varying joint clearance values on the performance of the robot. The results showed that the coupling effects of flexible joints and clearance had a negative impact on the system dynamic performance. The amplitudes of the dynamic responses caused by the clearance are weakened by the flexible joint, but it leads to the lag of the system response. This study served as the theoretical foundation for exploring precise control techniques in robotics research.

1. Introduction

With the wide application of robots, the dynamic performance and control accuracy of robots are increasingly required, and developing a more precise dynamic model for robots is essential [1]. In conventional studies, the joints of robots are considered as ideal rigid joints [2]. The effects of flexibility and clearance have not been considered. However, the harmonic reducer is often used to improve the transmission efficiency in the actual robot structure, in which the flexible parts show torsional elasticity, causing the output angle of the motor to deviate from the actual rotation angle of the robot link and then increase the joint flexibility of the robot [3]. The existence of flexibility will change the response characteristics of the system and affect the prediction accuracy of the response to the whole structure. Therefore, the influence of flexible joints should be considered in the modeling process, and the joints cannot be simply set as rigid. In addition, due to factors such as manufacturing and assembly errors, clearances will inevitably occur in robot joints, which cannot be eliminated in precision manufacturing [4]. In an ideal condition, the journal of the joint and the bearing should have the same radius, but the radius of the journal and the bearing in the actual production will be different, resulting in clearance between the journal and the bearing, that is, the joint clearance. The joint clearance can lead to significant contact forces and unwanted vibration and noise, adversely affecting system dynamic behaviors. To predict the robot dynamics more accurately and improve the control accuracy of the robot, considering the flexible joints and clearances in the robot dynamic modeling and analysis is important.Many scholars have studied the dynamic model of robots. Zhang and Wu [2] established the dynamic equation of a robot and studied the anti-interference control. Duan et al. [5] compared the dynamic and kinematic responses of the robot under different friction models and analyzed the interaction between various joints. Madsen et al. [6] considered the friction of the system in the dynamic modeling of a robot, and parameters in the model were identified, and the influence of load temperature and other factors on the characteristics of the robot was analyzed. Furthermore, there is a large amount of research on hybrid robots [7,8,9,10]. In the above research, the system is assumed to be rigid, and the influence of flexibility is ignored. Structural flexibility generally includes flexible links and joints in the analysis of multibody systems. Flexible links are links of a mechanism that have elastic deformation, which makes the system have flexible characteristics. Zhang and Yuan [11] examined the dynamic modeling and incorporated feedback control mechanisms for flexible-link robots. Ban et al. [12] investigated the influence of link flexibility on chaotic and bifurcation behaviors exhibited by robots. Based on the flexible-link model, Peza-Solis et al. [13] studied the trajectory tracking of the robot. The rigidity of the link in industrial robots is strong, and the length is short, so it is not necessary to consider the flexible link in modeling. However, the harmonic drive mechanism is generally used at the joint, and the harmonic rotation brings flexibility to the joint, which reduces the action accuracy of the system [14]. Spong [15] first proposed the torsion spring model to characterize the flexible joint and studied the feed-back linearization control of flexible-joint robots. Ruderman et al. [16,17,18,19] conducted a systematic study on flexible-joint robots and analyzed the problems of system modeling, control, and stability. Fateh [20] considered flexible joints in the investigation of the nonlinear control of robots. Spyrakos Papastavridis and Dai [21] also studied the control and trajectory tracking of flexible-joint robots. Farah et al. [22] explored vibration control strategies using a flexible-joint robot model, demonstrating that accounting for joint flexibility in the model enhanced the effectiveness of the implemented control measures. In addition, dynamic linearization analysis and modal studies of flexible-joint robots have been investigated by Do et al. [23]. Jing et al. [24] introduced a recursive approach for examining the inverse dynamics of robots with flexible joints, providing a basis for control optimization. Clearance is typical in mechanical systems, and there have been a lot of studies on clearance. Based on establishing the representation model, Flores et al. [25,26,27,28,29] investigated the kinematic properties of mechanisms incorporating clearance and verified its effectiveness through experiments. Wang et al. [30,31] formulated a contact force model for clearance. Gao et al. [32] examined the effects of clearance in a four-bar mechanism and utilized the coordinate partitioning technique to address the governing equation and found that friction reduces the extent of collision. Clearance as an uncertain factor increases the sensitivity of the system. Xiang and Yan [33] treated clearance as an uncertain parameter and quantified the influences on the dynamic behavior of space robots. Tang et al. [34] discussed the implications of uncertain clearance joints on the performance of robots and applied nonprobability theory to analyze the motion reliability of the system. Recently, Chen and Xu [35] modeled and simulated a driving robot with multiple clearances, evaluating its nonlinear response and affirming the model efficacy through rigorous performance evaluations. Wang et al. [36] delved into the three-dimensional representation of an assembly robot that accounts for clearance. Therefore, large clearance increases the motion amplitude of the robot and affects the work efficiency.Flexible joints and clearances during robot operation cannot be ignored. In previous studies, the effects of flexible joints and clearances on system dynamics were studied, and the coupling between them has not been considered. In robot dynamic modeling, the joint action of flexible joints and clearances should be taken into account, and an improved dynamic model should be established to obtain more accurate dynamic output, which is the basis of accurate control. Therefore, the coupling action of flexible joint and clearance is fully considered. A model for a flexible-joint robot with clearance is formulated, from which the multi-degree-of-freedom dynamic equations are derived. Subsequently, numerical simulations are employed to ascertain the system responses. The characteristics of rigid systems without clearance, rigid systems with clearance, and flexible systems with clearance are compared and analyzed.

2. Modeling of Joint Clearance

Figure 1 depicts a representation of the mathematical framework for modeling joint clearance. The center of bearing is O i , the radius is R i , the center of the journal is O j , the radius is R j , and the position vectors of the bearing and journal center in the global coordinate X Y are r i and r j , respectively. Radial clearance of the joint is c = R i ' R j . Based on the varying relative positions of bearings and journals during motion, clearance joints are classified into three distinct categories: non-contact mode (Figure 1a), critical contact mode (Figure 1b), and contact mode (Figure 1c). In the continuous contact mode, the contact points on the bearing and journal are C i and C j , respectively, and the contact point position vectors are r i C and r j C respectively.The offset between the bearing and journal is referred to as the eccentric distance, which can be formulated as follows: e k = r j ' r i (1) The magnitude of the eccentric distance is as follows: e k = e k T e k = e k x 2 + e k y 2 (2) Then, the contact normal vector is as follows: n = e k e k (3) According to the diagram of the contact model, the contact depth is as follows: δ = e k ' c (4) According to the geometric relationship in Figure 1c, the expression can be formulated to depict the position and velocity of the point of contact as follows: r i C = r i + R i n r j C = r j + R j n (5) r ˙ i C = r ˙ i + R i n ˙ r ˙ j C = r ˙ j + R j n ˙ (6) Then, the normal collision velocity and tangential collision velocity are as follows: v n = ( r ˙ j C ' r ˙ i C ) T n (7) v t = ( r ˙ j C ' r ˙ i C ) T t (8) where t is the tangential unit vector, which is perpendicular to the normal vector n .The tangential force F t and normal force F n during the collision process are shown in Figure 2. There has been some research on the characterization of contact forces during collision [25,26,27,37,38]. Hertz [39] laid the foundation for the characterization of elastic contact forces. However, the energy dissipation during collision is ignored in this study. Energy dissipation is considered in the study of Kelvin and Voigt, while the model cannot represent the nonlinearity and is only applicable to contacts with highimpact velocity [40]. The prevalent L-N model [41] captures contact nonlinearity, enabling its application for characterizing low-speed collisions.Utilizing the L-N model, the normal contact force at the joint can be derived as follows: F n = K δ n + η δ n δ ˙ (9) where K is generalized stiffness, n represents a nonlinear index that correlates with material properties, usually n = 1.5 for metal materials [32,42]; the damping coefficient is η ; and δ ˙ is the relative contact speed.According to the Coulomb model, the tangential force during collision is expressed as follows: F t = ' μ F n sgn ( v t ) (10) where μ is the friction coefficient and sgn ( · ) is the sign function.When contact occurs, the resultant force at the contact point is as follows: F c = F n + F t (11)

3. Modeling of the Flexible-Joint Robots with Clearance

The model of the flexible-joint robots with clearances is demonstrated in Figure 3. Each link is driven by an independent motor, and the torsion spring between the rotor of the motor and the link is given to characterize the reducer, which reflects the joint flexibility. The representation of joint clearance in the figure is an exaggeration. The joint angle θ i , the rotor angle θ r i , and the contact angle due to clearance φ i are the parameters considered in the modeling. The center of mass of the link is S i ; F B x and F B y are the components of the external force.Common approaches for deriving dynamic equations in multi-body systems include the Lagrange method and the Newton'Euler method. In this paper, the influence of joint space is considered, and the internal action between components should be analyzed, so it is more appropriate to establish the robot dynamic equation using the Newton'Euler method. For a multi-link system with clearances, the centroid position of the link k ( k = 1 , 2 , ' ) is formulated as follows: [ x S k y S k ] = [ e 1 x e 1 y ] + ' + L k ' 1 [ cos θ k ' 1 sin θ k ' 1 ] + L S k [ cos θ k sin θ k ] + [ e k x e k y ] (12) where e k x and e k y are the components of the eccentricity. e 1 x and e 1 y are the components of the eccentric distance e 1 in X and Y, respectively. L k ' 1 is the length of link k ' 1 , and L S k is the length between the center of gravity and the end of link k .Then, the velocity and acceleration of the center of mass of the connecting rod is derived as follows: [ x ˙ S k y ˙ S k ] = [ e ˙ 1 x e ˙ 1 y ] + ' + L k ' 1 θ ˙ k ' 1 [ ' sin θ k ' 1 cos θ k ' 1 ] + L S k θ ˙ k [ ' sin θ k cos θ k ] + [ e ˙ k x e ˙ k y ] (13) [ x ¨ S k y ¨ S k ] = [ e ¨ 1 x e ¨ 1 y ] + ' + L k ' 1 θ ¨ k ' 1 [ ' sin θ k ' 1 cos θ k ' 1 ] + L k ' 1 θ ˙ k ' 1 2 [ ' cos θ k ' 1 ' sin θ k ' 1 ] + L S k θ ¨ k [ ' sin θ k cos θ k ] + L S k θ ˙ k 2 [ ' cos θ k ' sin θ k ] + [ e ¨ k x e ¨ k y ] (14) According to the different motion modes of the joints, the calculation of the contact force is as follows: [ F c k x F c k y ] = Q [ cos φ k sin φ k sin φ k ' cos φ k ] [ F n k F t k ] Q = { 1 , δ ' 0 contact 0 , δ < 0 free flight (15) If the link k is an end link, the balance equation of the link can be written as follows: [ F B x F B y ] ' [ F c k x F c k y ] ' [ 0 m k g ] = m k [ x ¨ S k y ¨ S k ] (16) From the Euler equation N = J S ω ˙ + ω × ( J S ω ) , the moment equation of the end link is derived as follows: K r k ( θ r k ' θ k ) ' F B x L k sin θ k + F B y L k cos θ k + F c k x R k 2 sin φ k ' F c k y R k 2 cos φ k ' m k g L S k cos θ k = J k θ ¨ k (17) where K r k is the stiffness of the torsional spring, θ r k ' θ k is the angular deformation between rotor and link, and R k 2 is the radius of the journal in the joint.If the link k is not an end link, the equation is expressed as follows: [ F c ( k + 1 ) x F c ( k + 1 ) y ] ' [ F c k x F c k y ] ' [ 0 m k g ] = m k [ x ¨ S k y ¨ S k ] (18) K r k ( θ r k ' θ k ) ' F c ( k + 1 ) x ( L k sin θ k + e ( k + 1 ) y + R ( k + 1 ) 2 sin φ k + 1 ) + F c ( k + 1 ) y ( L k cos θ k + e ( k + 1 ) x + R ( k + 1 ) 2 cos φ k + 1 ) + F c k x R k 2 sin φ k ' F c k y R k 2 cos φ k ' m k g L S k cos θ k = J k θ ¨ k (19) The equation of rotor k in a flexible-joint robot is established as follows: J r k N k 2 θ ¨ r k + K r k ( θ r k ' θ k ) = T k (20) where J r k is the rotational inertia of the rotor, N k is the deceleration ratio of the joint, and T k is the driving torque generated by the motor.In this paper, the two-link flexible-joint robot with clearance is analyzed. Figure 4 presents the simplified model of the system.Utilizing Equations (14)'(18), one can derive the dynamic equations for a multi-degree-of-freedom flexible-joint robot with joint clearance as follows: { F 2 x ' F c 1 x = m 1 x ¨ S 1 F 2 y ' F c 1 y ' m 1 g = m 1 y ¨ S 1 K r 1 ( θ r 1 ' θ 1 ) ' F 2 x L 1 sin θ 1 + F 2 y L 1 cos θ 1 + F c 1 x R 12 sin φ 1 ' F c 1 y R 12 cos φ 1 ' m 1 g L S 1 cos θ 1 = J 1 θ ¨ 1 J r 1 N 1 2 θ ¨ r 1 + K r 1 ( θ r 1 ' θ 1 ) = T 1 F B x ' F 2 x = m 2 x ¨ S 2 F B y ' F 2 y ' m 2 g = m 2 y ¨ S 2 K r 2 ( θ r 2 ' θ 2 ) ' F B x L 2 sin θ 2 + F B y L 2 cos θ 2 ' m 2 g L S 2 cos θ 2 = J 2 θ ¨ 2 J r 2 N 2 2 θ ¨ r 2 + K r 2 ( θ r 2 ' θ 2 ) = T 2 (21)

4. Simulation and Analysis

In this paper, a two-link flexible-joint robot with clearance is investigated in parameter analysis, and the dynamic equation of the robot system is established. The dynamic equations are simulated in this section to analyze the influences of factors such as the presence of the flexible joint and clearance, and the influence of the size of the joint clearance on the dynamic characteristics of the robot.

4.1. Simulation Parameters and Model

The links of the robot studied in this paper are rigid, joint clearance exists at joint 1, and joints 1 and 2 are considered to be flexible joints. Therefore, the mathematical model of the robot contains six variables. In this paper, MATLAB is employed to simulate the robot model. The simulation parameters and the flow chart of the simulation are given in Table 1 and Figure 5, respectively. The specific simulation calculation flow of dynamic analysis of a flexible-joint robot with clearance is as follows:

(1)

Define parameters required for simulation, set radius of journal R 12 = 0.05 m, size of clearance c = 5 × 10 ' 5 m.

(2)

Define the initial value of the system at t 0 and set the initial variables [ e 1 x , e 1 y , θ 1 , θ 2 , θ r 1 , θ r 2 ] t 0 = [ 1 × 10 ' 5 , 1 × 10 ' 5 , 0 , 0 , 0 , 0 ] .

(3)

Calculate the contact depth according to the parameters and the variable values to further judge the contact state.

(4)

Establish the contact model of the clearance and calculate the contact force F c .

(5)

Establish the system dynamic equation considering joint clearance and joint flexibility.

(6)

Use DAEs solver to solve the system equation to obtain the variable values, and set the solver parameters: integral step is 1 × 10'3 and integral tolerance is 1 × 10'7.

(7)

Repeat steps (3)'(6) until the simulation is over.

4.2. Influences of Flexible Joint and Clearance

The dynamic behaviors of robots under different model parameters are explored in this research, and the main models are as follows: rigid-joint robot without clearance model (rigid-ideal, R-I), rigid-joint robot with clearance model (rigid-clearance, R-C), and flexible-joint robot with clearance model (flexible-clearance, F-C). The locus of the center of joint 1 is drawn in Figure 6a. The red line in the figure is the clearance circle, and the black is the locus of the center. Figure 6b,c show the motion locus in the X and Y direction, respectively. As seen from Figure 6, the motion mode between the bearing and the journal differs at different times. The locus outside the clearance circle indicates the continuous contact mode.As shown in Figure 7, the locus of center of joint 1 and its time-varying trajectories in the X and Y directions are demonstrated when flexible joints are taken into account. Compared with Figure 6, it can be observed that the flexible characteristics of the joint affect the contact state, and the non-contact and impact increase.The curve of contact-force change at robot joint 1 over time is given in Figure 8. The black line depicts the variation in the contact force of a rigid-joint robot with clearance, while the red line depicts the variation in a robot with the flexible joint and clearance. Within the clearance, a significant contact force is observed, exhibiting a pronounced pulse-like characteristic. The flexibility of joints increases the frequency of contact force, but the values decrease. This indicates that the flexible joint affects the characteristics of the joint clearance. There is a coupling between the flexible joint and the clearance.The comparison of the angular displacement of joint 1 for different robot models is shown in Figure 9. The deviation between different models is given in Figure 10. The black-solid line represents the deviation of angular displacement between the rigid-clearance model and the rigid-ideal model (C-I), the red-dashed line represents the difference between the rigid-clearance model and the flexible-clearance model (R-F), and the blue-dotted line represents the deviation between the flexible-clearance model and the rigid-ideal model (F-I). It can be clearly observed that the angular displacement of joint 1 is large when considering the clearance, and the deviation increases gradually with time. When considering flexible joints, the angular displacement of joint 1 also increases. The angular displacement of joint 1 is larger than that of the ideal model when the clearance and flexible joint are considered at the same time, but the angular displacement deviation at the same time is smaller than that of the rigid-clearance model. This is because the deviation caused by flexible joints neutralizes some caused by clearance.Figure 11 and Figure 12 show the comparison of angular displacement for joint 2 and the deviation between different models, respectively. The clearance does not cause much change about angle of joint 2. However, when considering the influence of joint flexibility, the angle of joint 2 changes greatly. The angular displacement of the flexible joint lags notably in comparison to that of the rigid joint.Figure 13 and Figure 14 illustrate the variations in angular velocity at joint 1 for various robot models, highlighting the discrepancies among them. The angular velocity is larger in the flexible-clearance model, and the lag phenomenon is also obvious. Taking into account the flexible joint and clearance, the angular velocity curve of the joint lags significantly and exhibits a diminished amplitude when compared to that of the rigid-ideal model.Figure 15 and Figure 16 show the change in the angular acceleration and the deviation, respectively. As seen from Figure 15 and Figure 16, the angular acceleration curve of joint 1 for the rigid-ideal model is relatively smooth. When the influence of clearance is taken into account, the amplitude of angular acceleration increases, and the curve shows impact peaks. The appearance of a peak means that there is a large contact force, which affects the stability of the robot. The value of the peak is relatively reduced when the flexible joint is considered. These things considered, the curve has a significant difference and noticeable lag when the flexible joint and clearance are coupled.Flexible joints and clearance of the robot not only affect the motion of the joint but also greatly influence the dynamic characteristics of the end-effector. In Figure 17 and Figure 18, the acceleration curves of the end-effector are depicted. Clearance and flexibility exert a great influence on the acceleration of the end-effector. When clearance is taken into account, peaks and an augmentation in acceleration emerge, resulting in a considerable inertial force exerted on the robot and subsequently affecting its stability and accuracy.

4.3. Influences of Clearance Size

Clearance and flexible joints exert a notable impact on the robot performance. A detailed numerical analysis further reveals the influences of clearance size, coupled with the joint clearance and flexibility, on the robot functionality.As depicted in Figure 19, the contact force at the joint exhibits a general upward trend, indicating an overall increase. Especially, the contact force at the peak increases significantly, as shown in Table 2. When c = 0.5, the frequency of the contact force at 1.41'1.72 s increases significantly. This indicates a rapid change in the motion state of the bearing and journal, which will decrease the stability of the system.The variation in the joint contact force exerts a discernible influence on the motion state of the joint. As shown in Figure 20, Figure 21 and Figure 22, the increase in the clearance value corresponds directly to an augmentation in both angular displacement and angular velocity, and the change between 0 and 0.8 s is not apparent. There is a significant increase after 0.8 s. As the clearance size increases, the angular acceleration of the joint and the magnitude of the acceleration peak undergo a corresponding augmentation. When c = 0.5, the frequency of the angular acceleration within 1.41'1.72 s increases. This observation aligns with the varying force trend and further indicates the direct correlation between changes in contact force and the stability of the joint. Larger clearance leads to poor system performance.The acceleration of the end-effector with different clearance sizes is given in Figure 23 and Figure 24. The larger joint clearance leads to a larger acceleration and a value of peak. However, compared with the acceleration of the joint, the change in the end-effector is small, and the increase in the value of peak is small. The frequency of acceleration increases, but the degree of growth is smaller than that of joint 1. The reason behind this phenomenon lies in the cumulative effect of flexible joints within the system, which effectively mitigates the acceleration impact resulting from clearance. Additionally, the flexible joint exhibits a significant damping effect on the robot, further contributing to this reduction.

5. Conclusions

The joint clearance and flexibility caused by the reducer are comprehensively considered to research the dynamics of the robot in this paper. The dynamic equation of the robot is derived by applying the Newton'Euler method combined with the Spong model, Coulomb friction model, and L-N model. The effects of clearance and flexible joints on the system are analyzed through numerical simulation.The presence of clearance enhances the degree of freedom of the system, thereby introducing unpredictability in its movement. Numerical simulations reveal a notable fluctuation in the contact force within the clearance. In comparison to an ideal-joint robot, the angular acceleration of the joint and the acceleration of the end-effector exhibit an increase, and the angular acceleration and acceleration of the end-effector are greatly affected. The contact force increases with increasing clearance size, the frequency of the force also increases, and the system uncertainty becomes more significant. In addition, the amplitude and frequency of the angular acceleration and the acceleration of end also increase, which results in greater noise and lowers the reliability.The dynamic characteristics undergo alterations when considering the flexibility of the robot joint. Compared with the rigid system, the dynamic behaviors of the flexible-joint robot all decrease to an extent and have a lag, in which the accelerations are obviously affected. This demonstrates the significant influence that joint flexibility exerts on the robot performance. Specifically, the flexible joint acts as a damping mechanism in the system, having a profound impact on the precision analysis and vibration control of the robot. At present, the research in this paper takes into account the flexibility of the robot joints and the clearances existing in the joints, but compared with the actual robot, the degree of freedom of the robot is simplified and has not been extended to the three-dimensional space. In addition, the model in this paper is mainly aimed at series robot modeling and is not applicable to parallel robots, but the modeling ideas can provide reference, and the inclusiveness and extensibility of the model in this paper need to be developed. In the future, further research will be conducted based on the proposed model. This work could be extended to three-dimensional space.

Author Contributions

Conceptualization, J.W. (Jing Wang); methodology, J.W. (Jing Wang); software, J.Q. and T.K.; formal analysis, J.W. (Jing Wang); investigation, J.W. (Jing Wang); writing'original draft, J.W. (Jing Wang); writing'review and editing, S.Z., J.W. (Jimei Wu), J.Q. and M.S.; visualization, T.K.; project administration, S.Z.; funding acquisition, J.W. (Jimei Wu) and M.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China, grant number: No. and No. .

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The mathematical model of clearance: (a) non-contact mode, (b) critical contact mode, (c) contact mode. Figure 2. The tangential force and normal force during the collision process. Figure 3. Mechanical model of flexible-joint robots with clearances. Figure 4. Simplified model of flexible-joint robot with clearance. Figure 5. Flow chart of dynamic simulation. Figure 6. Locus of journal center of rigid joint: (a) locus of center, (b) locus in X direction with time, (c) locus in Y direction with time. Figure 7. Locus of journal center of flexible joint: (a) locus of center, (b) locus in X direction with time, (c) locus in Y direction with time. Figure 8. Contact force in joints. Figure 9. Comparative analysis of the angular displacement at joint 1. Figure 10. Deviation in angular displacement at joint 1. Figure 11. Comparative analysis of the angular displacement at joint 2. Figure 12. Deviation in angular displacement at joint 2. Figure 13. Comparative analysis of the angular velocity at joint 1. Figure 14. Deviation in angular velocity at joint 1. Figure 15. Comparative analysis of the angular acceleration at joint 1. Figure 16. Deviation in angular acceleration at joint 1. Figure 17. Comparative analysis of acceleration at end-effector in X direction. Figure 18. Comparative analysis of acceleration at end-effector in Y direction. Figure 19. Comparison of contact force with different clearance sizes. Figure 20. Angular displacement of joint 1 with different clearance sizes. Figure 21. Angular velocity of joint 1 with different clearance sizes. Figure 22. Angular acceleration of joint 1 with different clearance sizes. Figure 23. Acceleration of end-effector with different clearance sizes in X direction. Figure 24. Acceleration of end-effector with different clearance sizes in Y direction. Table 1. Simulation parameters. DescriptionValueUnitLength of link 10.61mLength of link 20.66mDensitykg/m3Restitution coefficient0.46-Friction coefficient0.01-Poisson's ratio0.33-Young's modulus7.17 × N/m2Nonlinear index1.5-Torsional stiffness of spring Nm/radTorsional stiffness of spring Nm/rad Table 2. The contact force at the peak. t / s 0.06-0.32-0.74-0.78-1.05-1.21- F c / N ( c = 0.5 mm)69,649.,131.,781.,830.,785.,705.91 F c / N ( c = 0.3 mm)27,106.,742.,968.,575.,351.,631.73 F c / N ( c = 0.1 mm)23,114.,496.,556.,103.,974.,565.43 t / s 1.37-1.48-1.58-1.63-1.81-1.97- F c / N ( c = 0.5 mm)49,662.,175.,652.,669.,513.,366.82 F c / N ( c = 0.3 mm)37,232.,555.,532.,338.,881.,187.64 F c / N ( c = 0.1 mm)20,992.,598..,292.,553.,334.22 Disclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
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Open AccessArticle

Comparative Study of Methods for Robot Control with Flexible Joints

1 Institute of Industrial Control Systems and Computing (ai2), Universitat Politècnica de València, Valencia, Spain 2 Mechanical Engineering Department, Universidad de Antofagasta, Antofagasta , Chile 3 ITACA Institute, Universitat Politècnica de València, Valencia, Spain * Author to whom correspondence should be addressed. Actuators , 13(8), 299; https://doi.org/10./act Submission received: 3 June / Revised: 18 July / Accepted: 3 August / Published: 6 August (This article belongs to the Special Issue Actuators in Robotic Control'2nd Edition)

Abstract

: Robots with flexible joints are gaining importance in areas such as collaborative robots (cobots), exoskeletons, and prostheses. They are meant to directly interact with humans, and the emphasis in their construction is not on precision but rather on weight reduction and soft interaction with humans. Well-known rigid robot control strategies are not valid in this area, so new control methods have been proposed to deal with the complexity introduced by elasticity. Some of these methods are seldom used and are unknown to most of the academic community. After selecting the methods, we carried out a comprehensive comparative study of algorithms: simple gravity compensation (Sgc), the singular perturbation method (Spm), the passivity-based approach (Pba), backstepping control design (Bcd), and exact gravity cancellation (Egc). We modeled these algorithms using MATLAB and simulated them for different stiffness levels. Furthermore, their practical implementation was analyzed from the perspective of the magnitudes to be measured and the computational costs of their implementation. In conclusion, the Sgc method is a fast and affordable solution if joint stiffness is relatively high. If good performance is necessary, the Pba is the best option.

1. Introduction

Robots with flexible joints are becoming increasingly relevant. New types of robots are gaining importance on the market, such as collaborative robots (cobots), exoskeletons, and prostheses. They are meant to directly interact with humans. In this new generation of robots, the emphasis in their construction is not on precision (such as for rigid robot counterparts) but rather on weight reduction (collaborative robots) and/or soft interaction with humans (exoskeletons and prostheses). Thus, these new robots use more elastic mechanical transmissions.Cobots typically have harmonic drive transmissions instead of classical gears [1] due to their light weight, high reduction ratio, and relatively good back-driveability. Wearable robotics mostly use series elastic actuators (SEAs) [2,3] for transmission. SEAs are added to some cobots to increase the compliance of their harmonic drives [4], such as those produced by the Rethink company.Before the advent of flexible robots, most robots were rigid to achieve high precision. Controlling rigid manipulators is well covered and included in robotics textbooks [5,6,7]. In these cases, the best performance is obtained using inverse dynamics control methods, also called computed torque. This involves compensating all nonlinear forces that act on the robot, such as gravity, inertia, and centrifugal and Coriolis forces.When approaching the problem of controlling flexible robots, the first idea that comes to mind is adapting the well-known inverse dynamics method. However, in a rigid robot, the actuators are directly connected to the links, compensating for external forces. In a robot with flexible links, the motor acts on an elastic element, causing its torsion, which causes the link to move. Thus, the dynamic between the actuator and the link does not directly compensate for external forces.Many applications involve a wide range of compliances in their joints. According to [8], stiffnesses may vary from 5 to 10 kNm/rad down to 0.2 to 1 kNm/rad. This wide elasticity range complicates control considerably. In addition, stability analysis is much more difficult.For example, oscillations may occur, possibly prohibiting many robotics tasks.To achieve a task, the trajectory of the link ( q , q ˙ , e t c . ) must be controlled, but it is only possible to act on the motor ( θ , θ ˙ , e t c . ).Another complication in flexible robots vs. rigid ones is the higher order of the system. While the former is second-order, the latter is fourth [9,10]. Thus, it may be necessary to measure and include higher-order derivatives.Several control strategies have been proposed to deal with this wide range of elasticity. The late s and early s were prolific regarding contributions in this field; researchers aimed to control motor position and velocity to achieve good trajectory tracking with links.In [11,12], some less conventional control methods, like the singular perturbation method (Spm) or backstepping control design (Bcd), were proposed. Tomei [13] introduced an extremely simple PD with the simple gravity compensation (Sgc) method and demonstrated its stability criteria. The authors of [14] improved the previous method, proposing exact gravity cancellation (Egc) while introducing less restrictive criteria with better trajectory tracking. The authors of [15,16] introduced the passivity-based approach (Pba) to determine the control action.For each case, it is difficult to decide which method is appropriate and which constraints to use for its practical application, such as computational costs and expensive sensor requirements. Although some of these methods have been described in previous work [11,12,17,18], this study models and simulates a selection of methods to provide a clearer picture of the performance of each for different stiffness levels.This study is dedicated to applying 'classical' methods to control robots with flexible joints. A few recent strategies have not been included since they have several versions. Their analysis would be very extensive and has been left for future work. However, they are briefly mentioned below.One approach is model predictive control (MPC) [19,20,21]. This method includes constraints such as maximum motor torques and velocities in the controller design.Another strategy is sliding mode control [22,23,24]. It achieves good and robust trajectory tracking, but it may need a very fast sampling period.Several authors have dedicated their research to robustly controlling robots with elastic joints [25,26,27]. This is a wide area, and there are many very different contributions.This paper is organized as follows: the Section 2 presents the approach used to model the selected control algorithms. It then briefly describes the basis of each control algorithm and, finally, the simulation parameters. Then, Section 3 presents the output of the simulations for different stiffness levels. Next, Section 4 provides an interpretation of the simulation results, the requirements of each method for its practical implementation, and the pros and cons. Finally, Section 5 discusses the advantages and disadvantages of each controller. At the end of the article, Appendix A describes the first and second derivatives of the inertia, gravity, centrifugal, and Coriolis matrices.

2. Materials and Methods

2.1. Approaches for Modeling Robots with Flexible Joints

The dynamic model of the rigid robot is well known and can be found in textbooks [5,6,7]. It can be represented by the following expression: τ = M q q ¨ + C q , q ˙ q ˙ + G ( q ) (1) where τ is the vector of the motor torque; q ,   q ˙ , and q ¨ are the vectors of the motor position, velocity, and acceleration, respectively; M q is the inertia matrix of the robot; C q , q ˙ is the matrix of the centrifugal and Coriolis forces; and G ( q ) is the vector of gravity torques on the motors.The following subsections describe the two possible ways to model the dynamics of robots with elastic joints: conventional modeling and the singularly perturbed model.

2.1.1. Conventional Elastic Modeling

The main difference between modeling a rigid robot and a flexible robot is an elastic element between the motor rotor and the link (see Figure 1).The dynamics can be separated into two parts: the motor side and the link side. We can directly actuate the former, but we need to control the latter to achieve tasks, for example, as in [17,18]. This fact can be determined by assuming three conditions:

• A1: Joint deflections are small, so flexibility effects are limited to the linear elasticity domain.
• A2: Actuator rotors are modeled as uniform bodies with their centers of mass on the rotation axis.
• A3: Each motor is located in the robot arm before the driven link. This can be generalized to the case of multiple motors simultaneously driving multiple distal links.

In this case, the complete model can be represented by the following expression: M ( q ) S ( q ) S T ( q ) J m q ¨ θ ¨ + c q , q ˙ + c 1 q , q ˙ , θ ˙ c 2 q , q ˙ + G q + K ( q ' θ ) K ( θ ' q ) = 0 τ (2) where τ is the vector of the motor torque; q , q ˙ , and q ¨ are the vectors of the link position, velocity, and acceleration, respectively; θ , θ ˙ , and θ ¨ are the vectors of the rotor position, velocity, and acceleration, respectively; M q is the inertia matrix of the robot; c q , q ˙ , c 1 q , q ˙ , θ ˙ a n d c 2 q , q ˙ a r e the matrices of the centrifugal and Coriolis forces; G ( q ) is the vector of gravity torque on the motors; and K ( θ ' q ) is the elastic torque. The matrix, S, represents the inertial coupling between the rotors and the links.S is smaller than the other terms and is neglected by most authors, as are the c 1 q , q ˙ , θ ˙ and c 2 q , q ˙ components, providing a reduced model: M ( q ) 0 0 J m q ¨ θ ¨ + c q , q ˙ 0 + G q + K ( q ' θ ) K ( θ ' q ) = 0 τ (3) This model is used for stability analysis in all of the studies mentioned in this article and, in general, by most authors. It will also be used in this study.

2.1.2. Singularly Perturbed Model

Another approach is using a singular perturbation model, that is, to refer to a situation in which a system exhibits two or more distinct time scales of motion. In these systems, one of the time scales is much slower than the others, separating the fast and slow dynamics. It was used in [6,11,12,13].With a flexible joint, the elastic torque is much faster than the link. This separates the fast dynamics (elastic torque) from the slow dynamics (motion of the link). A singularly perturbated model can be obtained using a new coordinate space: q z = I 0 ' K K q θ = q K ( q ' θ ) (4) where z = K ( q ' θ ) is the elastic torque.From Equation (3), θ ¨ = J m ' 1 ( τ + z ) (5) and q ¨ = M q ' 1 ( ' c q , q ˙ ' G q ' z ) (6) From Equation (4), z ¨ = K θ ¨ ' q ¨ = K ( J m ' 1 ( τ + z ) ' M q ' 1 ' c q , q ˙ ' G q ' z (7) z ¨ = K ( J m + M q ' 1 z + J m ' 1 τ + M q ' 1 c q , q ˙ + G q (8) If we assume that the matrix, K , has large and similar elements, it is possible to extract a large common scale factor, K ^ ' 1 , from K : K = 1 ' 2 K ^ = 1 ' 2 d i a g k ^ 1 , k ^ 2 , ' , k ^ n , 0 < ε ' 1 .Thus, Equation (8) can be rewritten as ' 2 z ¨ = K ^ ( J m + M q ' 1 z ) + J m ' 1 τ + M q ' 1 c q , q ˙ + G q (9) Higher stiffness values mean lower ε values.

2.2. Control Strategies

This subsection briefly explains the control methods used in this study.

2.2.1. Singular Perturbation Method

The singular perturbation method [28] control strategy is used for processes that have one part that is much faster than the other. This method treats the slow and the fast parts separately, making control much easier. Two control actions are generated: one for the slow part and another one for the fast one.The output of the slow loop is used as the input for the fast loop. To obtain the final control action, slow and fast control actions are added.For the slow part, the control action (torque) can be generated according to the laws of control for rigid robots, which have been known for decades, for example, the inverse dynamic method provided by Equation (4).The fast control receives the slow control action as a reference value and must ensure that it will be tracked. According to [18], a possible control law is τ f a s t = K p τ τ s l o w ' τ e l a s t i c ' ' K d τ τ ˙ e l a s t i c (10) This is a PD control law for the elastic torque, and K p τ and K d τ are the proportional and derivative constants, respectively.The final motor torque should be τ = τ f a s t + τ s l o w (11) Notably, the stability criteria for this control method are established according to the Tikhonov theorem [28]. This states that if both the slow and fast loops are separately asymptotically stable and ' tends toward zero, their respective errors also tend toward zero. However, since ' = 1 K 2 , it is greater than 0. Consequently, the convergence and stability cannot be determined analytically.Since the convergence criteria assume that ε tends toward zero, the singular perturbation control will work better for robots with stiffer joints than robots with elastic ones.This unclear stability criteria definition limits the singular perturbation method. It cannot be used for applications such as robust or adaptive control.

2.2.2. Backstepping Control Design

Backstepping control design [29] is a control technique that stabilizes systems with nonlinear dynamics. It involves transforming the nonlinear dynamics into a series of intermediate systems with linear or linearizable dynamics and then applying a sequence of feedback controllers to each intermediate system, from top to bottom. The goal is to design a feedback control law that drives the system to its desired trajectory.This system must be expressed so that each state variable derivative depends on this state, the next, and the previous ones. Only the last state derivative depends on the control action and all the previous states: x ˙ 1 = f 1 x 1 + g 1 x 1 , x 2 x 2 x ˙ 2 = f 2 x 1 , x 2 x 2 + g 2 x 1 , x 2 , x 3 x 3 ' x ˙ n = f n x 1 , x 2 , ' , x n x 2 + g n x 1 , x 2 , ' , x n u (12) x 2 is a virtual input to guarantee the stability of x 1 . Then, x 3 is used as a virtual input to guarantee the stability of x 2 . This is repeated iteratively until the last state, which is stabilized by the control action, u.This control method was first used to control elastic joints in [11]. As will be demonstrated in simulations, this method works well. Nonetheless, it needs higher derivatives for the link position, and the system must be represented in a chained form, as in Equation (14).

2.2.3. Simple Gravity Compensation

Simple gravity compensation [13] proposes a PD controller with gravity compensation. The control law is τ = K p θ d ' θ ' K d θ ˙ + G q d (13) where θ d = q d + K ' 1 G ( q d ) (14) q d is the reference position of the link.Asymptotic stability is demonstrated for this case if λ m i n K ' K ' K K + K p > α , where α is a number that fulfills the following condition for the given robot: G q 1 ' G q 2 ' α q 1 ' q 2 .This method is very simple. The reference position of the motors is necessary to compensate for the gravity torque of the links; it does not need the feedback of the link position. Only the motor position and velocity are used in the loop, helping to assure its stability.

2.2.4. Exact Gravity Cancellation

Exact gravity cancellation was proposed in [14]. Its control action consists of two parts: τ m = τ g + τ 0 (15) The first component dynamically compensates for gravity: τ g = G q + J K ' 1 G ¨ ( q ) (16) The second component is a PD-type law: τ 0 = K p q d ' θ + K ' 1 G q ' K d ( θ ˙ ' K ' 1 G ˙ q ) (17) The global asymptotical stability can be shown via Lyapunov analysis. There are no constraints on the proportional constant.This method is an improvement over the previous one.

2.2.5. Passivity-Based Approach

The passivity-based approach was proposed by the German Aerospace Center group and the Kuka company [15,16].The final control law can be expressed as τ m = J J θ ' 1 u + ( I ' J J θ ' 1 ) τ e l a s t i c (18) u = J θ θ ¨ r e f + K θ r e f ' q r e f ' K θ θ ~ ' K D θ θ ~ ˙ (19) θ r e f = q r e f + K ' 1 M q q ¨ r e f + C q , q ˙ q ˙ r e f + G q (20) J θ is introduced for inertia shaping of the rotor since control is easier if the rotor and link inertias are similar orders of magnitude. The passivity of the system is thus demonstrated.However, to obtain θ r e f , this method must compensate for the elastic torque and the feedback of the link velocity and acceleration to compute the inertia and centrifugal matrices. Regarding the elastic torque, the authors of [16] proposed a lowpass filter with a cut-off frequency of 250 Hz.Thus, it is necessary to compute up to the second derivatives of the inertia, centrifugal, and gravity matrices to obtain θ ¨ r e f , which has a very high computational cost.

2.3. Modeling Robot Dynamics

The described control methods were modeled with MATLAB. The model assumes a two-degrees-of-freedom robot with revolute joints. The MATLAB files needed for the simulations are included in the Supplementary Materials. There are five files, one for each controller. There is also a file called gentray5 that contains the fifth-order trajectory generator used by the other files.The dynamics equations were obtained from [17,18]. For simplicity, the S inertia coupling matrix was set to 0 and the gear ratios to 1.The links were modeled as uniform thin rods, with the following characteristics according to suggestions from experts in the field:

• Their lengths are L 1 = L 2 = 0.5 m .
• Their masses are m 1 = 10 k g and m 2 = 0.5 k g .
• The distances of the centers of gravity from the rotation axes are both d 1 = d 2 = 0.25 m .
• Moments of inertia: I 1 = m 1 L 1 2 12 k g m 2 and I 2 = m 2 L 2 2 12 k g m 2 .
• Gear ratios: r 1 = r 2 = 1 .
• Weight of the rotor of the second motor: m r 2 = 2 k g .
• Inertia carried by the second motor: J m 2 = ( I 2 + m 2 d 2 2 ) / r 2 2 .
• Inertia carried by the first motor: J m 1 = ( I 1 + m 1 d 1 2 + m r 2 L 1 2 ) / r 2 2 .
• The stiffnesses are set to K 1 = K 2 = 200 , K 1 = K 2 = 10 3 , and K 1 = K 2 = 10 4 Nm/rad in different simulations.

Some intermediate variables were introduced: a 1 = I 1 + m 1 d 1 2 + m r 2 + m 2 L 1 2 + I 2 + m 2 d 2 2 a 2 = I 2 + m 2 d 2 2 a 3 = m 2 L 1 d 2 (21) For the dynamics expressed in Equation (2), the following matrix values were obtained: B q = a 1 + a 3 cos ' ( q 2 ) a 2 + a 3 cos ' ( q 2 ) a 2 + a 3 cos ' ( q 2 ) a 2 (22) J = J m 1 0 0 J m 2 (23) S = 0 0 0 0 (24) M = B S S T J (25) c q , q ˙ = ' a 3 s i n ( q 2 ) ( q ˙ 1 q ˙ 2 + q ˙ 2 2 ) a 3 s i n ( q 2 ) q ˙ 1 2 (26) c 1 q , q ˙ , θ ˙ = c 2 q , q ˙ = 0 0 (27) G ( q ) = m 1 g d 1 cos ' q 1 + m r 2 g L 1 cos ' q 1 + m 2 g L 1 cos ' q 1 + d 2 cos ' q 1 + q 2 m 2 g d 2 cos ' q 1 + q 2 (28)

2.4. Adjusting the Gains for the Controllers

All the controllers use some sort of feedback, typically proportional'derivative. Their performance will depend on their gains.To control a single joint [6], there are generally several (or infinite) combinations of proportional and derivative constants that work very well. They are computed based on the desired dynamics of the system, i.e., the natural frequency and damping ratio. To compute the proportional and derivative constants, it is necessary to know the inertia moment and the viscous friction coefficient of the system. Generally, better trajectory tracking is achieved with higher proportional and derivative gains. However, there is a point when increasing the gains practically does not improve the controller.A multiple-degrees-of-freedom robot is much more complicated. The inertia carried by a motor is variable. Furthermore, centrifugal and Coriolis forces and gravity act on the links.Most robot controllers (for rigid robots) compensate for the external forces and add a proportional'derivative controller for feedback [5,6,7]. If all the dynamics (inertia, gravity, centrifugal forces) is compensated, the values of the proportional and derivative gains may be computed for the required natural frequency and damping ratio. However, when, for example, only gravity is compensated, the optimal values of the gains vary as the robot moves.Usually, authors do not explain how these gains are obtained. One option is to adjust them through trial and error. Another is computing the value of the gains for each motor in real time, as in the case of a single joint, for the desired natural frequency and damping ratio of the system. However, this is time-consuming and not frequently used. Another method [5] is gain scheduling. This involves reading the best gains for the actual robot configuration from a database in every sampling period.In this study, the trial-and-error method was used. For every controller, many combinations were simulated. The simulations stopped when no more important improvements could be obtained.

3. Results

The simulations were conducted for a fifth-order polynomial trajectory generator. The first joint went from 0 to 2 π and the second from 0 to ' π in five seconds.The simulation was repeated for stiffnesses of K 1 = K 2 = 200 , K 1 = K 2 = 10 3 , and K 1 = K 2 = 10 4 Nm/rad for both joints. The first two values are typical for elastic mechanical transmissions like harmonic drives. A value of 200 is almost the most elastic found in the bibliographic research we conducted for this article [30].Before comparing the different controllers, simulations were conducted, controlling the robot as if it was rigid, i.e., directly compensating for the inertia, gravity, centrifugal, and Coriolis terms. For cases K 1 = K 2 = 200 and K 1 = K 2 = 10 3 , the system became unstable. For case K 1 = K 2 = 10 4 , it worked acceptably. Of course, this result also depends on the other dynamic parameters of the robot, such as its mass and moments of inertia. Figure 2 shows the results of the simulation for K 1 = K 2 = 10 4 . The mean quadratic errors for both joints were 0. and 0..Then, the simulations were run for the different control strategies and stiffnesses. The simulation results for stiffness K 1 = K 2 = 200 are shown in Figure 3, and the mean quadratic errors are shown in Table 1.The simulation results for stiffness K 1 = K 2 = 10 3 are shown in Figure 4, and the mean quadratic errors are shown in Table 2.Finally, the simulation results for stiffness K 1 = K 2 = 10 4 are shown in Figure 5, and the mean quadratic errors are shown in Table 3.Figure 6 and Figure 7 summarize the mean quadratic errors of the different methods for joint 1 and joint 2 for the three stiffness values.The results show that the simple gravity control method presents the highest position errors with many oscillations when the rigidity is 200. A specific analysis was carried out for this control method: First, the position error was evaluated for various levels of rigidity with values between 200 and . In turn, the proportionality and derivative gains of this controller were modified to observe their influence. Figure 8 and Figure 9 show the position errors of each joint for different stiffness and controller gain levels.In joint 1, the decreased error is more significant when stiffness increases. The controller gains may provide a minor error, but this is insignificant. For all controllers with low stiffnesses, the system oscillates. When the stiffness value reaches 600'800, the oscillations begin to disappear.In joint 2, like joint 1, the error decreases as the stiffness increases; however, when low proportional gain and high derivative gain are used, a steady state error occurs with stiffnesses greater than 600.

4. Discussion

As expected, the simulations show that the errors are higher when the joint stiffness is lower. In addition, oscillations appear with low stiffness values for the Sgc, Spm, and Egc controllers.Regarding mean quadratic errors, the Pba exhibits better results than the other controllers independently of the stiffness value.The error in the first joint is one order of magnitude higher than that in the second joint, probably because the higher load carried by the first motor increases the nonlinearities. The error is much higher for K = 200 than for the other cases. The worst results are obtained with Sgc and then Egc. Spb is comparable to the Pba and Bcd for the second joint but not for the first one.Although some controllers work very well in simulations, they need to measure or estimate certain magnitudes, such as the high derivatives of the position or torque. For example, position and velocity may be fed back using low-cost sensors and computer interfaces. However, many authors are reluctant to feed back the acceleration because of the significant effect of the noise. Few have used the first and second derivatives of acceleration (jerk and snap, respectively). Thus, the feasibility of these controllers in the real world is doubtful. The same problem occurs with torque feedback because torque measurements are noisy, and its derivative may be impossible to determine.To summarize the requirements of each method, Table 4 enumerates the necessary sensors and dynamic parameters.More than one factor influences the complexity of the controller. One aspect is the necessary amount of computation. Another is the set of dynamic parameters that must be known. Some of these factors are not easy to identify, like moments of inertia. The dynamical parameters may vary from one robot to another even if they are the same model.Another point to be considered is the necessary sampling period. A too-short sampling time may cause problems with real-time calculus and necessitate a more powerful computer. The necessary sampling time depends on the rate of change in the measured magnitude. Thus, typically, methods (e.g., Pba) that need to control the elastic torque need faster sampling than those that require only the positions and their derivatives.From the cost perspective, position sensors are cheap, and they are necessary for all the control methods described in this article. However, adding torque sensors greatly increases the cost of the system.The Spm works well for the stiffest case; however, it worsens as elasticity increases. For a stiffness of 200 Nm/rad, oscillations appear. This is logical since the initial supposition of this strategy is that the fast part is much faster than the slow one. On the other hand, the stability of this technique is determined by Tikhonov's theorem [28], which does not provide exact criteria for stability. This affects the robustness of the controller. In addition, the singular perturbation method requires a torque sensor and the first derivative of the elastic torque. Generally, this method is the third best regarding trajectory tracking.The Bcd method has the second-best performance regarding trajectory tracking in simulations. However, it is hard to implement it in real applications since it requires feeding back higher-order derivatives.The Sgc method is extremely simple and cheap (only sensors for the rotor position are required). It has the worst trajectory following, and its performance worsens as joint elasticity increases. For stiffnesses of 200 Nm/rad and Nm/rad, oscillations appear.The Egc method is the second worst regarding trajectory tracking. It requires measuring the acceleration of the link to fully compensate the gravity. For a stiffness of 200 Nm/rad, oscillations appear.The Pba has good performance regardless of joint stiffness. However, it requires an expensive torque sensor for each joint. The sampling period must be faster.Since adjusting gains is an important part of controller design, a few words will be dedicated to this topic.The Sgc and Egc methods have no feedback for the link position end velocity'just the motor side. Thus, the link works in an open loop. Varying the gains on the motor side cannot control the link side well for robots with relatively high elasticity.The Spm has two sets of proportional'derivative gains: one for the fast part and another for the slow part. The fast part is very sensitive, and system stability can be easily lost with small variations in gains.Regarding the Pba and Bcd methods, all relevant variables are fed back. These controllers are not approximative but exact methods. For these reasons, good trajectory tracking may be achieved with several gain combinations.

5. Conclusions

All the methods performed well for joints with small elasticity; however, oscillations appeared in Sgc for medium and high elasticity and Egc and the Spm for low stiffness.Considering all the drawbacks and the advantages of the Spm, it is not the most advisable.Egc is an improvement over simple gravity compensation. However, its small trajectory tracking upgrade does not justify the high derivative requirement or the increased computational cost.The backstepping method has very good performance in simulations. However, its implementation in the real world is problematic.In conclusion, the Sgc method is a fast and affordable solution if joint stiffness is relatively high. If good performance is necessary, the Pba is the best option.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10./act/s1, MATLAB files; SingularPerturbationMethod.m, SimpleGravityCompensation.m, BacksteppingControlDesign.m, PassivityBasedApproach.m, ExactGravityCancellation.m and gentray5.m.

Author Contributions

Conceptualization, R.Z.-S.; methodology, R.Z.-S. and A.P.; software, R.Z.-S. and R.P.-U.; validation, R.Z.-S., R.P.-U., and A.P.; formal analysis, R.Z.-S. and R.P.-U.; investigation, R.Z.-S.; resources, R.Z.-S., R.P.-U. and A.P.; data curation, R.Z.-S. and R.P.-U.; writing'original draft preparation, R.Z.-S.; writing'review and editing, R.Z.-S., R.P.-U. and A.P.; visualization, R.Z.-S., R.P.-U. and A.P.; supervision, R.Z.-S., R.P.-U. and A.P.; project administration, R.Z.-S.; funding acquisition, A.P. All authors have read and agreed to the published version of the manuscript.

Funding

Grant PID-RB-I00 funded by MCIN/AEI/10./.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Appendix A

Appendix A.1. Determining the Derivatives of the Dynamic Model

According to Equation (2), this term can be expressed as τ i n e r t i a = B ( q ) S ( q ) S T ( q ) J m q ¨ θ ¨ (A1) Since J and S are constant, their derivatives are zero.Deriving Equation (A1) gives τ ˙ i n e r t i a = B ˙ ( q ) 0 0 0 q ¨ θ ¨ + B ( q ) 0 0 0 q ' θ ' (A2) Deriving it again gives τ ¨ i n e r t i a = B ¨ q 0 0 0 q ¨ θ ¨ + 2 B ˙ q 0 0 0 q ¨ θ ¨ + B q 0 0 0 q 4 θ 4 (A3) The first and second derivatives of the matrix can be obtained for Equation (22): B ˙ q = ' 2 a 3 s i n ( q 2 ) q ˙ 2 ' a 3 s i n ( q 2 ) q ˙ 2 ' a 3 s i n ( q 2 ) q ˙ 2 0 (A4) B ¨ q = ' 2 a 3 ( c o s q 2 q ˙ 2 + s i n q 2 q ¨ 2 ) ' a 3 ( c o s q 2 q ˙ 2 + s i n q 2 q ¨ 2 ) ' a 3 ( c o s q 2 q ˙ 2 + s i n q 2 q ¨ 2 ) 0 (A5) In summary, the first derivative of the inertia matrix depends on the positions and velocities of the joints. Its second derivative also depends on acceleration. The total inertia torques depend on up to the fourth derivative of the position.

Appendix A.2. The Centrifugal and Coriolis Terms

Given Equations (21) and (22), for a two-degrees-of-freedom robot, the torque related to centrifugal and Coriolis forces can be expressed as τ C = c ( q , q ˙ ) q ˙ (A6) By deriving, we obtain τ ˙ C = c ˙ q , q ˙ , q ¨ q ˙ + c q , q ˙ q ¨ (A7) The first derivative of the matrix c is c ˙ = ' a 3 ( c o s q 2 q ˙ 2 2 + s i n q 2 q ¨ 2 ) ' a 3 ( c o s q 2 q ˙ 2 2 + s i n q 2 q ¨ 2 ) a 3 ( cos ' q 2 q ˙ 1 q ˙ 2 + s i n q 2 q ¨ 1 ) 0 (A8) We then introduce the following: a u x 1 = ( ' s i n q 2 q ˙ 2 3 + 3 c o s q 2 q ˙ 2 q ¨ 2 + s i n q 2 q ' 2 ) (A9) a u x 2 = ( s i n q 2 q ˙ 1 q ˙ 2 2 ' c o s q 2 ( q ˙ 1 q ¨ 2 + q ˙ 2 q ¨ 1 ) ' c o s q 2 q ˙ 2 q ¨ 1 + s i n q 2 q ' 1 ) (A10) a u x 2 = ( s i n q 2 q ˙ 1 q ˙ 2 2 ' c o s q 2 ( q ˙ 1 q ¨ 2 + 2 q ˙ 2 q ¨ 1 ) + s i n q 2 q ' 1 ) (A11) By deriving (A8) and introducing (A9) into (A11), we obtain c ¨ = ' a 3 ' a u x 1 a u x 1 a u x 2 0 (A12) The first derivative of the centrifugal and Coriolis terms depends on the positions, velocities, and accelerations of the joints. Its second derivative depends on the jerks.

Appendix A.3. The Gravity Term

The first derivative is obtained by deriving the gravity term using Equation (28): G ˙ = g ' m 1 d 1 sin ' q 1 q ˙ 1 ' m r 2 L 1 sin ' q 1 q ˙ 1 ' m 2 ( L 1 sin ' q 1 q ˙ 1 ' d 2 sin ' q 1 + q 2 ( q ˙ 1 + q ˙ 2 ) ) ' m 2 d 2 sin ' q 1 + q 2 ( q ˙ 1 + q ˙ 2 ) 0 0 (A13) We then introduce the intermediate variables: g 11 = m 1 d 1 + m r 2 L 1 sin ' q 1 q ¨ 1 + cos ' q 1 q ˙ 1 2 g 12 = m 2 L 1 ( sin ' q 1 q ¨ 1 + cos ' q 1 q ˙ 1 2 ) g 13 = m 2 d 2 ( sin ' q 1 + q 2 ( q ¨ 1 + q ¨ 2 ) + cos ' q 1 + q 2 ( q ˙ 1 2 + q ˙ 2 2 + 2 q ˙ 1 q ˙ 2 ) ) (A14) The second derivative of the gravity torque is obtained by deriving Equation (A13) and substituting with Equation (A14): G ¨ = g ' g 11 ' g 12 ' g 13 ' g 13 0 0 (A15) The first derivative of the gravity term depends on the positions and the velocities of the joints. Its second derivative depends on acceleration.

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Figure 1. Schema of an elastic joint. θ is the position of the motor rotor, q is the position of the link, and τ e l a s t i c = K ( θ ' q ) is the elastic torque. K is the stiffness of the joint in this figure. Figure 2. The positions of the links when both stiffnesses are K 1 = K 2 = 10 4 . The blue (first joint) and red (second joint) lines represent the reference positions (first link in blue and second link in red), while the yellow (first joint) and purple (second joint) lines represent the real positions. Figure 3. The positions of the links when both stiffnesses are K 1 = K 2 = 200 . The blue (first joint) and red (second joint) lines represent the reference positions, while the other lines represent the real positions for Sgc, Spm, Pba, Bcd, and Egc. Figure 4. The positions of the links when both stiffnesses are K 1 = K 2 = 10 3 . The blue (first joint) and red (second joint) lines represent the reference positions (first link in blue and second link in red), while the other lines represent the real positions for Sgc, Spm, Pba, Bcd, and Egc. Figure 5. The positions of the links when both stiffnesses are K 1 = K 2 = 10 4 . The blue (first joint) and red (second joint) lines represent the reference positions (first link in blue and second link in red), while the other lines represent the real positions for Sgc, Spm, Pba, Bcd, and Egc. Figure 6. The mean quadratic error of joint 1 when the stiffnesses are 200, , and 10,000. Figure 7. The mean quadratic error of joint 2 when the stiffnesses are 200, , and 10,000. Figure 8. The mean quadratic error of joint 1 with simple gravity compensation with various stiffness values and control gains. Figure 9. The mean quadratic error of joint 2 with simple gravity compensation with various stiffness values and control gains. Table 1. Mean quadratic errors with a stiffness of K = 200. ModelError J1Error J2Simple gravity compensation0..Singular perturbation method0..Passivity-based approach0..Backstepping control design0..Exact gravity cancellation0.. Table 2. The mean quadratic errors with a stiffness of K = . ModelError J1Error J2Simple gravity compensation0..Singular perturbation method0..Passivity-based approach0..Backstepping control design0..Exact gravity cancellation0.. Table 3. The mean quadratic errors with a stiffness of K = 10,000. ModelError J1Error J2Simple gravity compensation0..Singular perturbation method0..Passivity-based approach0..Backstepping control design0..Exact gravity cancellation0.. Table 4. Magnitudes necessary to be measured and parameters to be known for each controller. MethodMagnitudes to Be MeasuredDynamic Parameters to Be KnownSgcJoint positions and velocities.Stiffnesses, masses, and positions of the c.o.g.SpmElastic torques and their first derivatives, positions, and velocities of the links.All the inertia moments (links and motors), masses, and centers of gravity of the links.PbaPosition and velocity of the rotor, up to the third derivative of the link position.AllBcdUp to the fourth derivative of the position.AllEgcPosition and velocity of the rotor, up to the second derivative of the link.Stiffness, rotor inertia, mass, and position of the c.o.g. Disclaimer/Publisher's Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
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Zotovic-Stanisic, R.; Perez-Ubeda, R.; Perles, A. Comparative Study of Methods for Robot Control with Flexible Joints. Actuators , 13, 299. https://doi.org/10./act

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Zotovic-Stanisic, Ranko, Rodrigo Perez-Ubeda, and Angel Perles. . "Comparative Study of Methods for Robot Control with Flexible Joints" Actuators 13, no. 8: 299. https://doi.org/10./act

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Zotovic-Stanisic, R., Perez-Ubeda, R., & Perles, A. (). Comparative Study of Methods for Robot Control with Flexible Joints. Actuators, 13(8), 299. https://doi.org/10./act

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