WeB1.4
Proceedings of 2011 8th Asian Control Conference (ASCC) Kaohsiung, Taiwan, May 15-18, 2011
Hybrid PD/PID Controller Design for Two-Link Flexible Manipulators Rasheedat M. Mahamood
Jimoh O. Pedro
School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Johannesburg, South Africa Email:
[email protected]
School of Mechanical, Industrial and Aeronautical Engineering, University of the Witwatersrand, Johannesburg, South Africa Email:
[email protected]
Abstract—This paper investigates the development of a hybrid collocated PD and non-collocated PID controller designed for input tracking and vibration control of two-link flexible manipulator. The two-link robot manipulator was modelled using Lagrange and assumed mode method. The PD controller is used for motion tracking and the PID for vibration control. Effect of changing controller gains on performance is studied using two case studies. Also studied is the effect of payload variation on the performance of the proposed controller. The performance of the designed controllers is evaluated in terms of input tracking capability, energy utilization, deflection suppression and vibration control. Results show that a simple PD-PID controller can be effectively designed for point-to-point motion control and vibration suppression for two link flexible manipulators. Also the study reveals that the controller is robust to payload variation.
locations. Closed loop (feedback) control technique utilizes an accurate real time monitoring of the plant to be controlled for successful implementation of control action. Different methods have been used in closed-loop form to control flexible link manipulator. Examples include ProportionalIntegral-Derivative (PID) [5], end-point acceleration feedback [6], [7], state feedback [8], optimal control technique [9], robust control techniques [10], and singular perturbation method [11], [12].
Advantages of flexible robot manipulators over their rigid counterparts cannot be overemphasised: they require less material, lower power consumption, have higher manipulation speed, can use smaller actuators, are more manoeuvrable and transportable, are safer to operate due to reduced inertia, higher payload to robot weight ratio and most importantly they have less overall cost. Amidst the aforementioned advantages, the control of flexible manipulators to maintain accurate positioning is very challenging. The flexible nature and distributed characteristics of the system makes the dynamics a highly non-linear one [1]. Application of flexible link robot in industry is expected to increase only if their performance is improved.
The most widely used form of industrial controllers is the PID Controller. They constitute more than 90% of feedback controller used today [13]. This is because it is cheap, simple in structure, and robust in performance over a wide range of operating conditions [14]. PID control is also good at dealing with actuator saturation and integrator windup [15]. This is why many authors have designed controller for FMSs based on PID control technique [13], [15], [16], [17], [18], [19]. Tokhi and Azad [5] carried out a comprehensive study on open loop control and a hybrid collocated proportional derivative (PD) and non-collocated PID control strategy for single-link flexible manipulator. Simulation and experimental results showed a better performance in the proposed hybrid PD-PID controllers. Cheong el al. [20] also developed a PID composite controller for single link flexible manipulator. PD and a disturbance observer were proposed to control the slow dynamics while PID for fast dynamics. Experimental results show the effectiveness of the proposed controller.
The control strategies for flexible manipulator systems are classified as: feedforward (open-loop) and feedback (closed-loop) control [1]. Open-loop control (feedforward) [2], [3] [4], which is the simplest method does not require any measurement from the plant for the control action to be implemented. The problem with the open loop control is that exact knowledge of the plant is required. Feedback control strategies for Flexible Manipulator Systems (FMSs) are classified as collocated and non-collocated control. Collocated means the actuators and the sensors are at the same location. It is used to guarantee stable control of rigid-body motion. Non-collocated control on the other hand means that the actuators and the sensors are at different
The literature shows that relatively few PID controllers have been used to control FMSs compared to their rigid counterpart. The reason can be associated to problem of the common tuning methods that shows sluggish responses when applied to a non-minimum phase system like FMSs [17]. In this study, a hybrid PD-PID controller is developed for two-link FMSs. The manipulator is modelled using Lagrange and assumed mode methods. The PD controller is for point to point motion control, while the PID controller is for vibration suppression. Simulation is performed within Matlab/Simulink environment for evaluation of the control strategies. A unit-step response analysis is conducted, and performance evaluation of the control strategies is performed in terms of
I. I NTRODUCTION
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The previous absolute position vectors pi of a point along the deflected ith link, is defined by recursive kinematics equations: pi = ri + Wii pi ,
ri+1 = ri + Wii ri+1
(3)
where i pi =i pi (xi ) = (xi yi (xi ))T is the position of a point along the deflected ith link, with respect to frame (Xi , Yi ), and i ri+1 =i pi (li ) = (li yi (li ))T being the position of the origin of frame (Xi , Yi ). The absolute velocity of this point pi on the links is: Fig. 1.
˙ i pi + W i p˙i , p˙i = r˙i + W i i
Two-link flexible manipulator system
reference tracking, deflection, end-point acceleration, and input torque. Effects of varying payload on the proposed controller is also studied. The results are presented and discussed. The paper ends with concluding remarks.
˙ i ri+1 + W i r˙i+1 (4) r˙i+1 = r˙i + W i i
and r˙i+1 =i p˙i (li ), with i p˙i (xi ) = (0 y˙ i (xi ))T . The links are assumed inextensible in the longitudinal direction. The rates of the recursions take the form of: ˆ˙ i−1 Ai + W ˆ i−1 A˙ i W ˙ i Ei + Wi E˙ i (5) ˆ˙ i = W Wi = W C. Lagrangian formulation The system total kinetic energy T is given by:
II. S YSTEM M ATHEMATICAL M ODEL
T =
A. Modelling of robotic manipulators Lagrangian approach is commonly used to derive the dynamic equations of motion of flexible multi-body systems, although there are three main methods used in the literature: Newton-Euler, Lagrangian approach and Hamilton approach [21]. Assumed mode method is the most used approximation method for reducing partial differential equation (PDE) (equation of motion) into ordinary differential equation (ODE) [22]. The first two modes are adequate to describe the system dynamics [23]. The model of the two-link flexible, planar, manipulators derived by [22] is used in this study. The links are modelled as Euler-Bernoulli beams, with proper clampedmass boundary conditions. It is assumed the beams’ elastic deflections are small and no deflections in the axial direction. B. Formulation of the recursive kinematics equations Figure 1 shows a two-link flexible robot manipulator system, both links are actuated by individual motors at the hubs. ˆ i Yˆi ) are the inertial frame, the rigid ˆ 0 Yˆ0 ), (Xi Yi ), and (X (X body moving frame, and the flexible body moving frame associated with the ith link. θi is the angular position of the ith link, and yˆi (ˆ xi ) is the transversal deflection of the ith link (0 ≤ x ˆi ≤ li ) where li is the length of the ith link. The rigid transformation matrix and the elastic homogenous transformation matrix due to the deflection of the link are defined respectively as: cos θi − sin θi 1 −yie Ai = (1) and Ei = sin θi cos θi yie 1 ∂yi , and tan−1 (yie ) ≈ yie (small where yie = ∂x i xi =li deflections assumption). The global transformation matrix ˆ 0 Yˆ0 to Xi Yi follows a Wi transforming coordinates from X recursion as: ˆ i−1 Ai , Wi = Wi−1 Ei−1 Ai = W
ˆ0 = I W
n
Thi +
i=1
n
T li + T p
(6)
i=1
where Thi , Tli , and Tp are the kinetic energies of the ith hub, ith link, and the payload, respectively. The ith hub kinetic energy is given by: Thi =
1 1 mh r˙ T r˙i + Jhi α˙ i2 2 i i 2
(7)
where mhi is the mass of the ith hub, Jhi is the moment of inertia of the ith hub, and α˙ i is the absolute angular velocity of frame (Xi , Yi ): α˙ i =
i j=1
θ˙j +
i
y˙ k e
(8)
k=1
The kinetic energy of the ith link is given by: 1 li ρi (xi )p˙Ti (xi )p˙i (xi )dxi Tli = 2 0
(9)
where ρi is the linear density of the ith link. The kinetic energy associated with the payload is given by: 1 1 T (10) r˙n+1 + Jp α˙ i2 + y˙ n e Tp = mp r˙n+1 2 2 where mp and Jp are the mass and moment of inertia of the payload located at the end of link n. The total potential energy U is given by: 2 2 n n d yi (xi ) 1 li U= Ui = (EI)i (xi ) dxi (11) 2 0 dx2i i=1 i=1 Ui is the elastic energy stored in the ith link, with (EI)i being its flexural rigidity. Computing the total kinetic energy T and potential energy U , then the Lagrangian L is given by:
(2)
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L=T −U
(12)
D. Assumed mode shapes The links are modelled as Euler-Bernoulli beams and they satisfy the following equation: (EI)i
∂ 4 yi (xi , t) ∂ 2 yi (xi , t) + ρi = 0, 4 ∂xi ∂t2
i = 1, . . . , n (13)
where yi is the deflection of the ith link. Equation (13) is a partial differential equation satisfying the following boundary conditions: yi (0, t) = 0,
yi (0, t) = 0,
i = 1, . . . , n
Fig. 2.
Assuming a nei number of modes, deflection of each link can be obtained by the method of separation of variables as: yi (xi , t) =
nei
φij (xi )δij (t)
(15)
j=1
where δij (t) are the time varying variable associated with the special mode shape function φij (x) of the ith link. Solution of the two variables are as follows: φij (xi )
δij (t)
= C1,ij sin(βij xi ) + C2,ij cos(βij xi ) + C3,ij sinh(βij xi ) + C4,ij cosh(βij xi ) (16) = exp(jωij t) = C5,ij sin(ωij t) + C6,ij cos(ωij t)
(17)
where: 4 2 βij = ωij ρi /(EI)i
(18)
ωij is the natural angular frequency of the ith link, C1,ij . . . C6,ij are constants obtained from the following boundary conditions, Eq. (14). This yields: C2,ij + C4,ij = 0,
C1,ij + C3,ij = 0
PD-PID controller structure for the two-link flexible manipulator
(14)
(19)
E. Dynamic equations of motion The dynamic model is formulated using Lagrange-Euler equation:
d ∂L ∂L = τi , i = 1, . . . , n (20) − dt ∂ q˙i ∂qi Solution of Eq. (20) yields the closed form equation: ˙ B(q(t))¨ q(t) + h(q(t), q(t)) + Kq(t) = τ (t) (21) where q(t) = θ1 , . . . , θn , δ11 , . . . , δ1ne1 , . . . , δn1 , . . . , δnnei is a N -vector generalised coordinates (N = n + i=1 nei ), τ is an n-vector of generalized torques applied at the joints. B is a positive-definite symmetric inertia matrix, h is a vector of Coriolis and centripetal forces, and K is the diagonal stiffness matrix. Detailed derivation of the mathematical model can be found in [22].
III. C ONTROLLER D ESIGN The control objective for the two-link flexible manipulator shown in Fig. 1 is to design PD collocated controllers for each link so that the hub angles follow the reference trajectories.Also to design non-collocated PID controllers so that the vibrations of the end effectors are eliminated simultaneously. There are two stages involved in the controller design. the first stage involves the design of PD controllers for hub angle motion; while the second stage is concerned with the PID controller for the vibration control of the two links. A. Collocated PD Controller Similar PD control structures are used for both links, they only differ in gains. The PD control input is given by:
uP Di (t) = Aci KPi (θid (t) − θi (t)) − Kvi θ˙i (t) i = 1, 2 (22) where: uP Di (t) is the PD control input, θid (t), θi (t) and θ˙i (t) are the desired joint angle, actual joint angle and actual joint velocity of ith -link respectively, KPi , Kvi , and Aci are the PD control proportional and derivative gains, and motor amplifier gain for the ith -link respectively. B. Non-Collocated PID Controller A separate PID controller is designed for the control of endpoint elastic acceleration of each of the links; this is necessary because of the coupling effects. The control input for the ith link is as follows: uP IDi (t) = KPi ei (t) + KIi ei (t)dt + KDi e˙ i (t) i = 1, 2 (23) where ei (t) = αid − αi (t), αid , and αi (t) are the desired and actual tip acceleration of ith -link respectively. One of the control objectives is to achieve zero elastic acceleration which corresponds to zero vibration, hence αid = 0. C. Hybrid Controller The two control schemes are combined as shown in Fig. 2. The motion tracking and the vibration suppression are required to be achieved simultaneously. The total control input τi (t) from each motor is given by adding (22) and (23):
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τi (t) =
2 i=1
[uP Di (t) + uP IDi (t)]
(24)
D. Performance Evaluation The performance of the proposed controller is investigated using the performance index: 1 (J1 + J2 + J3 ) (25) tf ⎧ ⎡ 2 ⎤⎫
2 ˙ 2 ⎬ ⎨ ˙i (t) θ − θ (t) − θ θ i i i d d ⎣ ⎦ dt + ⎭ ⎩ θimax θ˙i J=
J1 =
tf
0
i=1
max
(26) 2
2
2 yid − yi (t) αid − αi (t) J2 = + dt yimax αimax 0 i=1 (27)
2 tf 2 τid (t) − τi (t) dt (28) J3 = τimax 0 i=1
tf
J is the overall performance index; J1 is the performance index related to the rigid body motion tracking; J2 is the performance index related to the deflection and vibration control of the manipulator; J3 is the performance index related to the overall control inputs at the joints; tf , θimax , θ˙imax , yimax , αimax , and τimax are the final simulation time, maximum hub angle, maximum hub angular velocity, maximum deflection, maximum tip acceleration, and maximum actuator torque of the link i respectively. yi (t) and τi (t) are the deflection and torque of the ith link respectively. yid is the desired hub deflection of the ith link.
procedure where the proportional gain is tuned until there is overshoot; two-link flexible manipulator is a highly openloop unstable and highly coupled system. This method does not work well for the gain tuning. According to [24], for optimum performance of PID controllers, the proportional, integral and derivative gains must be simultaneously tuned. Manual tuning is still the most favoured method despite different types of tuning methods available Eriksson and Wikander [25]. As the proportional gain was tune, the derivative gain has to be tune simultaneously in order to achieve satisfactory result. while tuning the PD gains of the fist link, the derivative gain of the second gain has to be tune otherwise improved result cannot be achieved. tuning the three controller gains enable good tracking to be achieved in link 1. This behaviour can be attributed to the coupling in the system. Then the Proportional gain of the second link controller is tuned systematically . As the proportional gain of the second link is tuned the good tracking already obtained in in link 1 is affected therefore the previous three already tuned gains needed to be retuned simultaneously until good tracking in the two links are achieved. To study the effect of controller gains on the performance two sets of controller gains are obtained for two cases as shown in Table II. TABLE II PD
CONTROLLER GAINS FOR THE TWO - LINK FLEXIBLE MANIPULATOR
Cases
1 2
IV. S IMULATION R ESULTS Numerical simulation was carried out on the two-link flexible manipulator in Matlab/Simulink environment to test the performance of the proposed control scheme. The system parameters used in the simulation are presented Table I. TABLE I PARAMETERS OF THE TWO - LINK FLEXIBLE MANIPULATOR [22] Symbol ρ1 = ρ2 EI 1 = EI 2 l1 = l2 Jh1 = Jh2 G m1 = m2 mp Jl1 = Jl2 Jp
Parameter Mass density Flexural rigidity Link’s length Hub mass moment of inertia Gear ratio Mass of the link Mass of the payload Link mass moment of inertia Payload mass moment of inertia
Value 0.2 kgm−3 1.0 Nm2 0.5 m 0.2 kgm2 1 0.1 kg 0.1 kg 0.0083 kgm2 0.0005 kgm2
The manipulator is to track a desired step response while suppressing end-effector vibration. Ziegler-Nichols gain tuning procedure was tried, but very poor performance was achieved. This is because two-link flexible manipulator is a highly open loop unstable system. Gains are tuned manually in two stages.
B. Stage 2 After achieving good motion tracking with the PD controller, then the PID controllers are also tuned systematically as done in stage 1. Tuning the PID tends to degrade the perfect tracking initially obtained. After some few trials of tuning and re-tuning, the gains in Table III are achieved for the two cases. The proposed algorithm is realizable experimentally because PD/PID control algorithm has been successfully implemented for single-link flexible manipulator in the literature [20], [5]. The future work is to carry out experimental validation and also to extend this algorithm to a 4-degree of freedom gantry type two-link flexible manipulator. Results of the effects of two different sets of PD and PID controllers’ gains on the response of the two-link manipulator are shown in Figs.3–5. Their effects on the performance index are also studied using (25). TABLE III PID
A. Stage 1
PD Gains Link 1 Link 2 Kp Kv Kp Kv 0.55 1.5 0.06 0.2 1.1 1.15 0.25 0.42
CONTROLLER GAINS FOR THE TWO - LINK FLEXIBLE MANIPULATOR
Cases
Initially, both PID controllers and the PD controller for link 2 are turned off. The amplifier gains are fixed as ’1’ for both links. The proportional and derivative gains of link 1 are then tuned simultaneously. Unlike the Ziegler-Nichols
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1 2
PID Gains Kp 2 0.2
Link 1 KI 10−4 10−3
KD 2 1.5
Kp 0.1 0.1
Link 2 KI KD 0.1 1.5 0.1 0.5
Fig. 3. Time histories of joints’ angles with hybrid PD-PID controller for cases 1 and 2
Fig. 6. Time histories of joints’ angles with hybrid PD-PID controller for case study 1 with varying payloads.
Fig. 4. Time histories of end-points’ accelerations with hybrid PD-PID controller for cases 1 and 2. Fig. 7. Time histories of end-points’ accelerations with hybrid PD-PID controller for case study 1 with varying payloads.
Figure 3 shows the motion tracking, joint angles of link 1 and 2 in case 1 reaching the steady state at about 5 and 7 seconds respectively. While in case 2, the joint angles reach the steady state in 3 and 5 seconds respectively. Comparing these two cases it is observed that the higher the PD gains the faster the response to the input tracking. Also the higher the PID gains the faster the vibration suppression as shown in Fig. 4. Case 1 with higher PID settles in less than 5 seconds as compared to case 2 which settles at about 7 seconds. The
Fig. 5. Time histories of torque inputs with hybrid PD-PID controller for cases 1 and 2.
faster response achieved in case 2 is traded off for higher torque required (maximum of 0.84 Nm for link 1 and 0.36Nm for link 2) shown in Fig. 5 compared to maximum of 0.52 and 0.145 Nm for links 1 and 2 respectively in case 1. However the overall performance index of case 2 (0.3389) is better than that of case 1 (0.3976) according to the factors penalized in the performance index in Eq. (25). Effect of variation in payload was also studied. Figures 6 - 8 show the response of the two-link flexible manipulator with hybrid PD collocated and PID non-collocated control to a payload of 0.04 kg, 0.08kg and the nominal 0.1 kg for the case 1. Figure 6 shows the joint angles input tracking with the various payloads. It can be observed that there is overlapping in the response. The endpoint accelerations show increase in amplitude of vibration with increase in payload but the ststem settles at the same time (see Fig. 7). Hence variation in payload does not affect the vibration suppression of the proposed controller. The applied torques shown in Fig. 8 also show no significant change . These mean there is no significant effect on controller performance with payload variation. It can be concluded that the proposed controller is robust to payload variation.
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Fig. 8. Time histories of torque inputs with hybrid PD-PID controller for case study 1 with varying payloads.
V. C ONCLUSION This paper developed hybrid collocated PD and noncollocated PID controller for a two-link flexible manipulator. The PD controller is for joint motion control and the PID controller is for endpoint vibration suppression. The proposed hybrid controller was tested within Matlab/Simulink environment. The performance of the proposed controller has been evaluated in terms of input tracking, vibration suppression and control torque. Effects of payload variation on the controller was also studied. Simulation result using the proposed controller has shown that the controller is very effective for input tracking and vibration control for a highly nonlinear and coupled system like two-link flexible manipulator. Payload variation does not have a significant effect on the proposed controller; this shows that the controller is robust. It can therefore be concluded that the proposed hybrid PD/PID controller is capable of tracking the desired joint angle while suppressing vibration simultaneously in the presence of payload uncertainty of the two-link flexible manipulator. ACKNOWLEDGMENT This work is supported by the Advanced Manufacturing Technology Strategy (AMTS), an operating unit of the Council for Scientific and Industrial Research (CSIR) in South Africa. R EFERENCES [1] M. O. Tokhi and A. K. M. Azad, Flexible Robot Manipulators Modelling, Simulation and Control, Institution of Engineering and Technology, London, 2008. [2] M. S. Alam and M. O. Tokhi, Designing feedforward command shapers with multi-objective genetic optimisation for vibration control of a single-link flexible manipulator, Engineering Applications of Artificial Intelligence, Vol. 21, No. 2, 2008, pp. 229 – 246. [3] L. Consolini and A. Piazzi, Generalized Bang-Bang Control for Feedforward Constrained Regulation, Automatica, Vol. 45, No. 10, 2009, pp. 2234 – 2243. [4] Z. Mohamed and M. O. Tokhi, Command Shaping Techniques for Vibration Control of a Flexible Robot Manipulator, Mechatronics, Vol. 14, No. 1, 2004, pp. 69 – 90. [5] M. O. Tokhi and A. K. M. Azad, Control of Flexible Manipulator Systems, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Vol. 210, No. 12, 1996, pp. 113 – 130.
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