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Hybrid PD/PID Controller Design for Two-Link

Flexible Manipulators

Rasheedat M. MahamoodSchool of Mechanical, Industrial and Aeronautical

Engineering, University of the Witwatersrand,

Johannesburg, South Africa

Email: Mahamoodmr@unilorin.edu.ng

Jimoh O. PedroSchool of Mechanical, Industrial and Aeronautical

Engineering, University of the Witwatersrand,

Johannesburg, South Africa

Email: Jimoh.Pedro@wits.ac.za

AbstractThis paper investigates the development of a hybridcollocated PD and non-collocated PID controller designed forinput tracking and vibration control of two-link flexible ma-nipulator. The two-link robot manipulator was modelled using

Lagrange and assumed mode method. The PD controller is usedfor motion tracking and the PID for vibration control. Effect ofchanging controller gains on performance is studied using twocase studies. Also studied is the effect of payload variation onthe performance of the proposed controller. The performance ofthe designed controllers is evaluated in terms of input trackingcapability, energy utilization, deflection suppression and vibrationcontrol. Results show that a simple PD-PID controller canbe effectively designed for point-to-point motion control andvibration suppression for two link flexible manipulators. Also thestudy reveals that the controller is robust to payload variation.

I. INTRODUCTION

Advantages of flexible robot manipulators over their rigid

counterparts cannot be overemphasised: they require lessmaterial, 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 control strategies for flexible manipulator systemsare 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

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 controlflexible link manipulator. Examples include Proportional-

Integral-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].

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 FMSsbased 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 literature shows that relatively few PID controllershave 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

Proceedings of 2011 8th Asian Control Conference (ASCC) WeB1.4

Kaohsiung, Taiwan, May 15-18, 2011

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Fig. 1. 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 I . SYSTEM MATHEMATICAL MODEL

A. Modelling of robotic manipulators

Lagrangian approach is commonly used to derive the dy-

namic 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 approxima-

tion 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 linksare modelled as Euler-Bernoulli beams, with proper clamped-

mass 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 sys-

tem, both links are actuated by individual motors at the hubs.

(X0Y0), (XiYi), and (XiYi) are the inertial frame, the rigidbody moving frame, and the flexible body moving frame

associated with the ith link. i is the angular position of theith link, and yi(xi) is the transversal deflection of the ith

link (0 xi 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:

Ai =

cos i sin isin i cos i

and Ei =

1 yie

yie 1

(1)

where yie =yixi

xi=li

, and tan1(yie) y

ie(small

deflections assumption). The global transformation matrix

Wi transforming coordinates from X0Y0 to XiYi follows arecursion as:

Wi = Wi1Ei1Ai = Wi1Ai, W0 = I (2)

The previous absolute position vectors pi of a point along thedeflected ith link, is defined by recursive kinematics equations:

pi = ri + Wiipi, ri+1 = ri + W

ii ri+1 (3)

where

i

pi =i

pi(xi) = (xiyi(xi))T

is the position of a pointalong the deflected ith link, with respect to frame (Xi, Yi),and iri+1 =

i pi(li) = (liyi(li))T being the position of the

origin of frame (Xi, Yi). The absolute velocity of this pointpi on the links is:

pi = ri +Wiipi + W

ii pi, ri+1 = ri + W

ii ri+1+ W

ii ri+1 (4)

and ri+1 =i pi(li), with

i pi(xi) = (0 yi(xi))T. The links are

assumed inextensible in the longitudinal direction. The rates

of the recursions take the form of:

Wi =Wi1Ai + Wi1Ai

Wi = WiEi + WiEi (5)

C. Lagrangian formulation

The system total kinetic energy T is given by:

T =n

i=1

Thi +n

i=1

Tli + Tp (6)

where Thi , Tli , and Tp are the kinetic energies of the ith hub,

ith link, and the payload, respectively. The ith hub kineticenergy is given by:

Thi =1

2mhi r

Ti ri +

1

2Jhi

2i (7)

where mhi is the mass of the ith hub, Jhi is the moment of

inertia of the ith hub, and i is the absolute angular velocity

of frame (Xi, Yi):

i =i

j=1

j +i

k=1

yke (8)

The kinetic energy of the ith link is given by:

Tli =1

2

li0

i(xi) pTi (xi) pi(xi)dxi (9)

where i is the linear density of the ith link. The kinetic energy

associated with the payload is given by:

Tp =1

2mpr

Tn+1rn+1 +

1

2Jp

2i + y

ne(10)

where mp and Jp are the mass and moment of inertia of thepayload located at the end of linkn. The total potential energyU is given by:

U =n

i=1

Ui =n

i=1

1

2

li0

(EI)i(xi)

d2yi(xi)

dx2i

2dxi (11)

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:

L = T U (12)

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D. Assumed mode shapes

The links are modelled as Euler-Bernoulli beams and they

satisfy the following equation:

(EI)i 4

yi(xi, t)x4i

+ i 2

yi(xi, t)t2

= 0, 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, y

i(0, t) = 0, i = 1, . . . , n (14)

Assuming a nei number of modes, deflection of each link canbe obtained by the method of separation of variables as:

yi(xi, t) =

nei

j=1

ij(xi)ij(t) (15)

where ij(t) are the time varying variable associated with thespecial mode shape function ij(x) of the i

th link. Solution

of the two variables are as follows:

ij(xi) = C1,ij sin(ijxi) + C2,ij cos(ijxi)

+ C3,ij sinh(ijxi) + C4,ij cosh(ijxi) (16)

ij(t) = exp(jijt)

= C5,ij sin(ijt) + C6,ij cos(ijt) (17)

where:4ij =

2iji/(EI)i (18)

ij is the natural angular frequency of the ith link,

C1,ij . . . C 6,ij are constants obtained from the followingboundary conditions, Eq. (14). This yields:

C2,ij + C4,ij = 0, C1,ij + C3,ij = 0 (19)

E. Dynamic equations of motion

The dynamic model is formulated using Lagrange-Euler

equation:

ddt

Lqi

L

qi= i, i = 1, . . . , n (20)

Solution of Eq. (20) yields the closed form equation:

B(q(t))q(t) + h(q(t), q(t)) +Kq(t) = (t) (21)

where q(t) =

1, . . . , n, 11, . . . , 1ne1 , . . . , n1, . . . , nnei

is a N-vector generalised coordinates (N = n +

i=1 nei), 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].

Fig. 2. PD-PID controller structure for the two-link flexible manipulator

III. CONTROLLER DESIGN

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 trajecto-

ries.Also to design non-collocated PID controllers so that thevibrations 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:

uPDi(t) = Aci

KPi (id(t) i(t))Kvi i(t)

i = 1, 2

(22)

where: uPDi(t) is the PD control input, id(t), i(t) andi(t)are the desired joint angle, actual joint angle and actual joint

velocity ofith-link respectively, KPi , Kvi , and Aci are the PDcontrol 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 end-

point 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:

uPIDi(t) = KPiei(t) + KIi

ei(t)dt + KDi ei(t) i = 1, 2

(23)

where ei(t) = id i(t), id , and i(t) are the desiredand actual tip acceleration of ith-link respectively. One of thecontrol objectives is to achieve zero elastic acceleration which

corresponds to zero vibration, hence id = 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 i(t)from each motor is given by adding (22) and (23):

i(t) =2

i=1[uPDi(t) + uPIDi(t)] (24)

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D. Performance Evaluation

The performance of the proposed controller is investigated

using the performance index:

J =

1

tf(J1 + J2 + J3) (25)

J1 =

tf0

2i=1

id i(t)

imax

2+

id i(t)

imax

2 dt

(26)

J2 =

tf0

2

i=1

yid yi(t)

yimax

2+

id i(t)

imax

2dt

(27)

J3 =

tf0

2

i=1

id(t) i(t)

imax

2dt (28)

J is the overall performance index; J1 is the performance

index related to the rigid body motion tracking; J2 is the per-formance index related to the deflection and vibration control

of the manipulator; J3 is the performance index related to theoverall control inputs at the joints; tf, imax, imax, yimax,imax, and imax are the final simulation time, maximum hubangle, maximum hub angular velocity, maximum deflection,

maximum tip acceleration, and maximum actuator torque of

the link i respectively. yi(t) and i(t) are the deflection andtorque of the ith link respectively. yid is the desired hubdeflection of the ith link.

IV. SIMULATION RESULTS

Numerical simulation was carried out on the two-link

flexible manipulator in Matlab/Simulink environment to testthe performance of the proposed control scheme. The system

parameters used in the simulation are presented Table I.

TABLE IPARAMETERS OF THE TWO-LINK FLEXIBLE MANIPULATOR [22]

Symbol Parameter Value

1 = 2 Mass density 0.2 kgm3

EI1 = EI2 Flexural rigidity 1.0 Nm2

l1 = l2 Links length 0.5 mJh1 = Jh2 Hub mass moment of inertia 0.2 kgm

2

G Gear ratio 1m1 =m2 Mass of the link 0.1 kg

mp Mass of the payload 0.1 kgJl1 = Jl2 Link mass moment of inertia 0.0083 kgm

2

Jp Payload mass moment of inertia 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.

A. Stage 1

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

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 thereforethe previous three already tuned gains needed to be re-

tuned 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 IIPD CONTROLLER GAINS FOR THE TWO-LINK FLEXIBLE MANIPULATOR

Cases PD Gains

Link 1 Link 2Kp Kv Kp Kv

1 0.55 1.5 0.06 0.22 1.1 1.15 0.25 0.42

B. Stage 2

After achieving good motion tracking with the PD con-

troller, 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 oftwo different sets of PD and PID controllers gains on the

response of the two-link manipulator are shown in Figs.35.

Their effects on the performance index are also studied using

(25).

TABLE IIIPID CONTROLLER GAINS FOR THE TWO-LINK FLEXIBLE MANIPULATOR

Cases PID Gains

Link 1 Link 2Kp KI KD Kp KI KD

1 2 104 2 0.1 0.1 1.5

2 0.2 103 1.5 0.1 0.1 0.5

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Fig. 3. Time histories of joints angles with hybrid PD-PID controller forcases 1 and 2

Fig. 4. Time histories of end-points accelerations with hybrid PD-PIDcontroller for cases 1 and 2.

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 forcases 1 and 2.

Fig. 6. Time histories of joints angles with hybrid PD-PID controller forcase study 1 with varying payloads.

Fig. 7. Time histories of end-points accelerations with hybrid PD-PIDcontroller for case study 1 with varying payloads.

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 forcase study 1 with varying payloads.

V. CONCLUSION

This paper developed hybrid collocated PD and non-collocated 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 environ-

ment. 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 con-

troller 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 pay-

load 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.

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