A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control

Ain Shams Engineering Journal xxx (xxxx) xxx

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A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control Khalil Ibrahim, Abdel Badie Sharkawy ⇑ Faculty of Engineering, Assiut University, Egypt

a r t i c l e

i n f o

Article history: Received 15 March 2017 Accepted 24 January 2018 Available online xxxx Keywords: Flexible joint manipulators (FJM) Adaptive PID controller Sliding mode control (SMC) Parameter bounds Trajectory tracking

a b s t r a c t The article focuses on the issues of dynamic modelling and control of manipulators which have elastic joints. An adaptive hybrid PID (proportional, integral, derivative) control scheme is proposed. For each joint two PID controllers have been implemented. The first is to cope with the joint elasticity and the second is used to deal with the rigid links motion. Simple adaptive mechanism is introduced to eliminate the need of continual tuning the PID gains and a comparison is made with a previously published sliding mode controller (SMC). For both controllers, the desired joint trajectories are computed online. This approach is necessary to overcome the influence of joint flexibility on the tracking performance. The proposed hybrid PID controller and the SMC are compared based on simulation tests. Results demonstrate the effectiveness and usefulness of the proposed hybrid PID control method. Ó 2018 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Flexible-joint manipulators (FJM) offer several advantages with respect to their rigid counterpart, such as light weight, lower cost, smaller actuators, larger work volume, better maneuverability and transportability, higher operational speed, power efficiency, and larger number of applications. Thus, they are often required to operate at high speed to yield high productivity. However, when the control in joint space operates using the motor drive feedback only, that is the most common case in robotic practice, the relative joint torsion remains uncompensated and leads to the link position errors at heavy loads and large joint torques [1–3]. In some cases, joint flexibility can lead to instability when neglected in the control design, as explained in [4]. Thus, the conflicting requirements between high speed and high accuracy make the control task a challenging research problem. Research on the dynamic modelling and control of flexible robots has received increased attention in the last decades. A first step towards designing an efficient control strategy for manipula⇑ Corresponding author. E-mail addresses: [email protected] (K. Ibrahim), [email protected] (A.B. Sharkawy). Peer review under responsibility of Ain Shams University.

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tors with flexible joints must be aimed at developing dynamic models that can characterize the flexibility of the joints accurately. The controller design that minimizes the effects of the flexible displacements in lightweight robots is highly demanded in many industrial and space applications that require accurate trajectory control. In control applications of robot, manipulators with flexible joints are required either to reach a target position or to follow a prescribed trajectory. In the first case to reach a target position, a short settling time is desired while the robot arm displacement is planned in the second case to follow a prescribed trajectory. In both cases, proper control actions have to be applied to the robot arm in order to avoid imprecise motion. Despite the considerable research work that has been done on the dynamics and control of FJM, most existing flexible-joint adaptive control strategies reported in the literature are model-based techniques. A survey for the control of FJM can be found in [5], more recently in [2,6] and the references included. These control algorithms have reasonably good tracking performance only when substantial knowledge of the plant mathematical model and its parameters is available. Consequently, if significant or unpredictable plant parameter variations arise as a result of joint mechanism degradation, or if there are modelling errors due to complex flexible dynamics behaviors, model-based control approaches might perform inadequately. As model free approach, the conventional PID controller for automated machines is widely accepted by industry. This is because PID controllers are easy to understand (has clear physical meanings i.e. present, past and predictive), easy to explain to

https://doi.org/10.1016/j.asej.2018.01.004 2090-4479/Ó 2018 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article as: K. Ibrahim and A. B. Sharkawy, A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2018.01.004

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others, and easy to implement. However, performance of fixedgain PID controllers is limited in real time operations of robot motion. With fixed-gain controller, steady state error will continue to be present if the system is suffering from disturbances. This fact implies the need of continual update of the gains which gives the motivation for tremendous efforts which have been done to develop adaptive PID control schemes. In [7], the author has developed a several methods for tuning the PID gains with application to the feedback control of antilock braking systems. The tuning systems, however, are designed offline which may not work properly if significant changes took place for the system parameters. Model reference-based adaptive for PID have been discussed in [8] with application to rigid manipulators. The authors in [9] have presented a neural network-based selftuning PID control for underwater vehicles. Despite that the algorithms don’t need the exact knowledge of the system parameters, high number of numerical calculation has to be carried online due to their complexity. This paper aims to develop a comprehensive and simple control strategy for FJM. The controller takes into account the flexibility in the joints, while it provides a precise motion for the manipulator’s end-effector. Two coupled PID controllers are implemented in the closed loop system. The first one is designed to produce a control signal to overcome the joint flexibility. The second is designed to control the motion of the rigid part of the manipulator. Adaptivity of the two PID controllers is achieved via simple mechanism. The combined signals are then used to derive the FJM. Results show that this approach is robust against initial position errors and parameter variation and can effectively compensate for the dynamic nonlinearities of the FJR. The rest of the paper is organized as follows: the dynamics of FJM is presented in Section 2. Section 3 makes a review for a sliding mode controller presented in [10], for comparison purposes. The proposed control scheme is demonstrated in Section 4. Simulation results and discussion are presented in Section 5. The concluding remarks is given in Section 6.

q

K Motor

Link

2. Dynamic model A proper choice of mathematical model for a control system design is a crucial stage in the development of control strategies for any system. This is particularly true for robotic manipulators due to their complicated dynamics. FJM are assumed to have rigid links and flexible joints. In [2] the joint flexibility was modeled as a linear spring and showed that as the stiffness goes to infinity, the model behaves like a rigid manipulator. See Fig. 1. The resulting torque stiffness equation at joints along with equations of motion of rigid links is a highly coupled nonlinear ordinary differential equations. The dynamics for an n-link manipulator with flexible joints as proposed by M. Spong [11] can be stated as follows:

€ þ Cðq; qÞ _ þ GðqÞ ¼ Kðh  qÞ DðqÞq

ð1Þ

J €h þ Kðh  qÞ ¼ u

where D is the n  n link inertia matrix, symmetric and positive definite; C is the n  n centrifugal and Coriolis matrix; G is the n  n gravitational vector, K is the n  n drive shaft stiffness matrix, diagonal, J is the n  n motor inertia matrix, diagonal and positive definite, q and h represent the n  1 vectors of link angles, and motor angles respectively. Despite that some energy terms have been neglected in the above model, it has been widely accepted in robotic literature [2,4–6]. It worth noting that when the actuators are mounted to the base of the robot, or generally for 1 DOF robots, this model is the exact one. 2.1. Bounded parameters In most real world problems, one cannot always exactly determine the parameters in an assumed model. It is, however, reasonable to assume that bounds on these parameters do exist. In this work, modelling errors and external disturbances are given the following form. Due to parametric uncertainties and the environ_ Go ðqÞ; K o ; J o of ment, only average values Do ðqÞ, C o ðq; qÞ; _ GðqÞ; K; Jin (1) are available. These average matrices DðqÞ; Cðq; qÞ; represent the normal part of the uncertain continuous system. The uncertainties (modelling errors) satisfy the matching conditions given (bounded) by:

_ j 6 dC ðq; qÞ; _ jDDðqÞj 6 dD ðqÞ; jDCðq; qÞ jDGðqÞj 6 dG ðqÞ; jDK j 6 dK ; jDJ j 6 dJ ð3Þ

Flexible Transmission

ð2Þ

where DA represents the modelling imprecision of matrix (or vector) A defined by:

DA ¼ A  A o

Fig. 1. Flexible joint model.

ð3Þ

Compute desired joint angles d , (4) PID Controller for joint position, (12)

qd Desired Link angles

+ _

PID Controller for link position (13)

+

+

The adaptive gain tuner, (14)

u PID

FJM

q, Measured signals

Fig. 2. The proposed hybrid adaptive PID control scheme for FJM.

Please cite this article as: K. Ibrahim and A. B. Sharkawy, A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2018.01.004

K. Ibrahim, A.B. Sharkawy / Ain Shams Engineering Journal xxx (xxxx) xxx

3

Fig. 3. Time history for the SMC signals when the reference trajectory is sinusoidal input.

where

Table 1 Parameter values and the control gains used in simulation. Parameter

Value

r1 ¼ r2 m1 ¼ m2 g J K Jo Ko

0.5 3 9.81 0.3 I 75 I 0.05 I 75 I 0.7 500 I 350 I 50 I 15

W Kp Ki Kd

l

   _ €d  2kq  k2 q þ C o ðq_ d  k qÞ þ Go uR ¼ Do ½q

k = positive constant, 

q ¼ q  qd ; are the tracking error for the linkages. The control signal uR in (4) and (5), is designed for the rigid part of the robot dynamics. To unify this presentation, we use the normal (average) values for the matrices K, D, C, G, i.e. K o ; Do ; C o ; Go instead of the estimated ones in Slotine and Li’ algorithm [13]. The joints reference trajectories defined by (4) and (5) are computed online (each time step) since they are needed by the control law in order to take into account joint flexibility.

The positive scalar bounds dD ; dC ; dG ; dK are assumed to be known. 2.2. Trajectory planning Let qd ðtÞ 2 C 2 denote a desired link trajectory which is continuously differentiable up to two times. To define the desired joint angles hd , we have adopted an approach similar to that proposed by Brogliato et al. [12], in which the right hand side of the first equation in (1) is replaced by the control law of Slotine and Li [13] uR as follows.

hd ¼ K o1 uR þ qd ;

ð5Þ

ð4Þ

3. Sliding mode control In SMC, a high-speed switching control is designed to derive the non-linear plant’s trajectory onto a specific and user-chosen surface in the state space, called sliding or switching surface sðxÞ, and to maintain the plant’s state trajectory on this surface for all subsequent time. The plant’s dynamics restricted to this surface represent the controlled system behavior and the motion of the state trajectory is described by the so called sliding mode. This section, demonstrates a SMC law without derivation. For detailed mathematical proof, the reader is referred to [10], where a set of controllers are derived for flexible joint robots using the variable structure theory. Only bounds of parameters and the feed-

Please cite this article as: K. Ibrahim and A. B. Sharkawy, A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2018.01.004

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back measurement of position and velocity, are needed to derive the controller for a FJM with n-number of links. According to the SMC theory, if the control ensures:

sTi s_ i 6 0;

i ¼ 1; 2; :::::n

where

T 1i ¼ sgnðs2i Þ

ð6Þ

( )     n  X _        dK i;j ðqj  þ hj Þ þ dJ i;j Ki;j hj  þ dJ i;j €hdj   s2i k1i j¼1

ð10Þ

n

then a sliding mode exists on the sliding surfaces for all x 2 R [14,15]. It means that sufficient conditions for reachability/existence of sliding mode are met if for any point in the state space and for all t P t o , s and s_ are of opposite sign. In designing the control law for FJM, the following sliding surfaces have been chosen: 



s1 ¼ q_ þ K1 q and s2 ¼ h_ þ K2 h

ð7Þ

where s1 ; s2 ¼ 0 are two column vectors each of them has length n, the first is related to link dynamics and the second to joint dynamics, K1 ; K2 are diagonal positive constant matrices, 



q ¼ q  qd ; h ¼ h  hd are the tracking errors for the joints. Using the following Lyapunov function:

V¼

1 T 1 s Ds1 þ sT2 Js2 2 1 2

ð8Þ

it can be shown that the coming control law in (9) achieves asymptotic stability.  _  _ u ¼ T 1 þ T 2 þ Go þ C o q_  J o K2 h  Do K1 q

ð9Þ

T 2i ¼sgnðs1i Þ

( n h X

)  i       €        _ dK i;j ð qj þ hj ÞþdGi þdCi;j qj þdDi;j qdj  s1i k1i

j¼1

ð11Þ in which dK; dJ; dD; dC; dG are bounds of modelling errors as defined in (2) and (3), and assumed to be known. The control law (9) is discontinuous across sliding surface which leads to control chattering. Chattering is undesirable in practice because it involves high control activity. Furthermore, it may excite high frequency dynamics neglected in the course of modelling such as unmodelled structural modes, neglected time delays. Here, the boundary layer technique is used as a possible method to smoothen the control law. By this method, we replace the signum non-linearity by a suitable saturation non-linearity, which is defined as follows

satðsÞ ¼ signðsÞ if jsj > l satðsÞ ¼ s=l if jsj 6 l where l is a positive constant. Consequently, chattering due to data sampling is eliminating. While such replacement leads to small

Fig. 4. Time history for the SMC signals when the reference trajectory is step input.

Please cite this article as: K. Ibrahim and A. B. Sharkawy, A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2018.01.004

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Fig. 5. Time history for the adaptive PID controller signals when the reference trajectory is sinusoidal.

terminal error [16], the practical advantages of having smooth control input may be significant. 4. The control design The importance of PID controllers can be proved not only by their industrial application, but also by the large amount of research documentation available in technical and scientific annals and journals. The proposed control scheme presented in this work involves the use of PID-PID controller for FJM. The scheme is designed so as to be able to deal with the dynamics of both links and joints. The first PID controller forces the links to follow the reference trajectory while the second PID controller compensates the errors resulted from joint flexibilities and parameter variations. This combined PID controllers ensure that joint and link errors are eliminated simultaneously. The equation of the PID controller for the links is given by:

 Z   _ uPIDl ¼ K p q þK d q þ K i

t





q dt

ð12Þ

0

and for the joints is given by:

  Z  _ uPIDj ¼ K p h þK d h þ K i

t 

h dt



continuous gain tuning of the controller gains. This work is inspired by continuous gain tuning as a way of solving this persistent problem of the most widely used industrial controller which makes tuning and retuning a part and parcel of the control process. This adaptive algorithm will eliminate the need for frequent tuning and retuning of the controller gains [17]. The gains auto-tuning equation is driven by both links and joints errors as follows:

k¼

W  2

ð14Þ

 2

1 þ ðqÞ þ ðhÞ

where W is a positive constant-gain. The square of errors is utilized to ensure stability of the adaptive parameter. The overall control law, is then becomes:

 Z    _ _ uPID ¼ k K p ðq þ hÞ þ K d ðq þ hÞ þ K i

t





ðq þ hÞdt

 ð15Þ

0

The control structure is shown in Fig. 2. In comparison with the SMC in (9), this adaptive PID controller (15) requires mush less number of calculations to be carried out online. It means that low-cost processor can be utilized in the hardware installation.

ð13Þ

0

where K p , K d and K i are the proportional, derivative and integral gain matrices. To be able to reject unanticipated disturbances on a dynamic system using conventional PID controller, there is a need for

5. Simulation results and discussion Simulation tests were carried out on a two link planar manipulator. The dynamic model for such manipulator is given in [3], as follows:

Please cite this article as: K. Ibrahim and A. B. Sharkawy, A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2018.01.004

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Fig. 6. Time history for the adaptive PID controller signals when the reference trajectory is step input.

Table 2 The performance index IAE of the two controllers.

Sinusoidal input Step input

Adaptive PID

SMC

0.4216 0.9232

0.5517 0.9388





a þ b þ 2gcosq2 b þ gcosq2 ; b þ gcosq2 b   gq_ 2 sinq2 - gðq_ 1 þ q_ 2 Þsinq2 C¼ gq_ 1 sinq2 0

D¼

 G¼

aecosq1 þ gecosðq1 þ q2 Þ gecosðq1 þ q2 Þ



in which a ¼ r 21 ðm1 þ m2 Þ; b ¼ m2 r22 ; g ¼ m2 r 1 r2 ; e ¼ g=r 1 . In this model, r i is the distance between the base of the link and its centre of gravity, mi is the mass of link i, i ¼ 1; 2, g is acceleration of gravity. Average value for the dynamic model matrices, Do ; C o ; Go ; K o , are chosen as follows:

 Do ¼ Ko ¼

 o       00 aþb b ge J 0 ; ; Jo ¼ ; Go ¼ ; Co ¼ 00 bb ge 0 Jo ! o o k k o

k k

o

They have been utilized to compute the SMC law (9) and the desired joint trajectory (4) which is utilized by both controllers. The bounds dD; dC; dG; dJ; dK are selected as follows:

 dD ¼

     00 2g g ae ; dC ¼ ; dG ¼ ; dJ ¼ 0:1J o ; dK ¼ 0:1K o ; 00 g0 0

so that the sum of Do and Go and their bounds dD and dG cover the full range of configuration dependent terms. This condition is needed by the SMC law, (9). As it could be noticed, both Ao and dA are chosen as time independent, which facilitates the computation burden. The centrifugal and Coriols effects have been neglected. A sampling rate of 1000 Hz is incorporated in simulations. The second order Runge-Kutta method is used for integrating the dynamic equation of the robot in order to compute its response to the examined control laws; (9) and (15). The numerical data included in Table 1 are used in simulation tests, where all units are in SIU and I is the identity matrix. Two types of links reference trajectory have been tested. The first is sinusoidal one with the following equations:

q1d ðtÞ ¼ A1 sinðxtÞ ; q2d ðtÞ ¼ A2 sinðx2 tÞ; with A1 ¼ 0:34 rad,A2 ¼ 0:62 rad,x1 ¼ 0:3p rad s1 x2 ¼ 0:7p rad s1 . This trajectory is chosen because it generates sufficient non-linear coupling between the robot links. The second trajectory is step input with amplitude equal to 1 for link 1 and 0.5 for link 2. To check robustness of the proposed scheme, the initial conditions are chosen such that the robotic manipulator is at rest, i. e.

Please cite this article as: K. Ibrahim and A. B. Sharkawy, A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2018.01.004

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q_ ¼ h_ ¼ ½0; 0T and an initial position q ¼ h ¼ ½0:3p; 0:3pT which is resulted in initial position and velocity errors for both trajectories. Simulation results are demonstrated in Figs. 3–6. As it could be noticed, tracking errors have been converged to a narrow region around zero for the two tested controllers. However, the proposed adaptive PID controller exhibits faster convergence. To quantify the performance, the following closed-loop performance index has been computed; the integrated absolute error (IAE). It is defined as follows:

Z J IAE ¼ IAE ¼

T



j q j dt ¼ 0



M X 2 X



jqi jDt

k¼0 i¼1 

where q and h are the link and joint errors respectively, Mis the number of samples, ii is the number of links and Dt is the sampling time. Results are given in Table 2. It can be concluded that the proposed control scheme outperforms the SMC.

6. Conclusion In this paper, an adaptive hybrid PID control algorithm is proposed for FJM which is robust against parameters uncertainties. Simulation tests include fixed and time varying desired linktrajectories. The control system needs only position and velocities measurements. In literature, higher-order derivative signals may be also required for some control algorithms. The performance of the proposed controller is compared with sliding mode technique. Despite its relative simplicity, results prove that the proposed control scheme is effective in overcoming the influence of the joint flexibility in the tracking performance.

7

[8] Wei SU. A model reference-based adaptive PID controller for robot motion control of not explicitly known systems. Int J Intell Control Syst 2007;12 (3):237–44. [9] Alvarado RH, Valdovinos LG, Jimenez TS, Espinosa AG, Navarro FF. Neural network-based self-tuning PID control for underwater vehicles. Sensors 2016;16(9):1–18. [10] Vitko, Sharkawy AB. A novel variable structure controller for flexible joint robots. In: Proceedings of the ISMCR’96, Brussels. p. 91–7. [11] Spong M. Modelling and control of elastic joint robots. Trans ASME J Dyn Syst Meas Contr 1987;109:310–9. [12] Brogliato B, Ortega R, Lozano R. Global tracking controllers for flexible-joint manipulators: a comparative study. Automatica 1995;31(7):941–56. [13] Slotine J-J, Li W. Adaptive manipulator control: a case study. IEEE Trans Automatic Control 1988;33(11):995–1003. [14] Zhao F, Utkin V. Adaptive simulation and control of variable-structure control systems in sliding regimes. Automatica 1996;32(7):1037–42. [15] Shendge PD, Suryawanshi PV. Sliding mode control for flexible joint using uncertainty and disturbance estimation. Proceedings of the world congress on engineering and computer science 2011, WCECS, San Francisco, USA, October 19–21, 2011, 2011. [16] Slotine J, Sasatry S. Tracking control of nonlinear systems using sliding surfaces, with applications to robot manipulator. Int J Control 1983;38:465–92. [17] Mahmood RM. Direct adaptive hybrid PD-PID controller for two-link flexible robotic manipulator. Proceedings of the world congress on engineering and computer science 2012, WCECS 2012, October 24–26, San Francisco, USA, 2012.

Khalil Ibrahim received the B.E. and M.Sc. degrees in mechanical engineering from the Mechanical Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt, in 2001 and 2006, respectively. Then he worked toward the Ph.D. degree from late 2010 till September, 2013 at Mechatronics and Robotics Engineering, Graduate School of Innovation, Egypt-Japan University of Science and Technology EJUST, Egypt. Starting from September, 2013 he is a Lecturer/Assistant professor in the mechanical department, Faculty of Engineering, and Assiut University. His major research interests include robotics engineering, automatic control, and artificial intelligence techniques.

References [1] Readman M. Flexible joint robots. CRC Press; 1994. [2] Spong MW, Hutchinson S, Vidyasagar M. Robot modeling and control. Wiley; 2006. [3] Siciliano B, Sciavicco L, Villani L, Oriolo G. Robotics: modelling, planning and control. Springer; 2009. [4] Talole SE, Kolhe JP, Phadke SB. Extended-state observer-based control of flexible-joint system with experimental validation. IEEE Trans Ind Electron 2010;57(4):1411–9. [5] Ozgoli S, Taghirad HD. A survey on the control of flexible joint robots. Asian J Control 2006;8(4):1–15. [6] Ulrich S, Sasiadek JZ. On the simple adaptive control of flexible-joint space manipulators with uncertainties. In: Book chapter: aerospace robotics II. Springer; 2014. p. 13–23. [7] Sharkawy AB. Genetic fuzzy self-tuning PID controllers for antilock braking systems. Eng Appl Artif Intell 2010;23:1041–52.

Abdel Badie Sharkawy: received the B.Sc. degree in mechanical engineering, the M.Sc. in production engineering from Assiut University, Egypt, in 1981 and 1990 respectively. He received the Ph.D. degree in control engineering from the Slovak Technical University (STU) in Bratislava in 1999. He was a senior lecturer within the department of mechatronics engineering, the Hashemite University, Jordan; 2001–2004. Now, he is a full professor at mechanical engineering department, Assiut University, Egypt. His research interests include adaptive fuzzy control, automotive control systems, robotics and the use of neural networks in the control of mechanical systems.

Please cite this article as: K. Ibrahim and A. B. Sharkawy, A hybrid PID control scheme for flexible joint manipulators and a comparison with sliding mode control, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2018.01.004