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 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 66, 2018 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Songying Zhao, Yougang Sun, Ye Zhou 
Copyright © 2018, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-63-1; ISSN 2283-9216 

Study on the Position Control System Framework of 3R 

Underactuated Robot based on Fuzzy Control 

Guan He 

Mechanical and Electrical Engineering school, Anhui Wenda University of Information Engineering, Hefei 231201, China 

guanhe27381@126.com 

Fuzzy control theory is adopted to study the position control system framework of 3R underactuated robot. 

Lagrange equation is used to establish the block matrix to solve the trajectory tracking control issue. Later, 

fuzzy control strategy is used to further plot the control flow chart and establish the simulation model. Based 

on the comparison between expected value and experiment value, the maximum of related error is 2.22% and 

the maximum absolute error is 1.43°. The system framework designed in this article may achieve better 

control over the position of 3D underactuated robot.  

1. Introduction 

A mechanical system not only has passive joints, but also is provided with control input less than the degree of 

freedom of the system. Such system is called 3R underactuated robot. On the one hand, the system has less 

actuation systems, thus lowering the weight of the robot significantly and saving the cost. On the other, less 

actuation devices result in reduced energy consumption of the robot, which is beneficial to drive the reform of 

the industry. Moreover, if the actuation device of any joint fails, we may convert it into the passive joint to 

continue the operation. It can be seen that 3Runderactuated robot has better theoretical study value and 

application prospective, and is a new trend for future robot research. However, the underactuated system is 

subject to two-order non-holonomic constraint, resulting in a great difference between a non-linear structure 

and general system. Since the underacturated system meets Brock-ettcondition, i.e. no smooth status 

feedback control law, it is impossible to control the passive joint directly with a kinetic method.  

Based on domestic and international literature, this article uses the fuzzy control theory to study the position 

control system framework of 3R underactuated robot. This article establishes the block matrix with Lagrange 

equation to solve the trajectory tracking control issue. Later, the article utilizes the fuzzy control strategy to 

further plot the control flow chart, and establishes the simulation model. The model as established in this 

article is featured by simple rule, less calculation, low operation difficulty and simple process.  

2. Literature review 

Before the mid-1990s, all the robots studied were all driven, that is, the joints of these robots were controlled 

by their motors respectively. The characteristic of such a robot is that the dimension of the input space (i.e. 

control space) is equal to the structure of the space dimension and is easy to realize control; the disadvantage 

is that because of the many driving devices, it causes much energy, high cost, and heavy robot, especially 

when the robot is a series structure, the quality of the latter drive is the former drive. Compared with the 

robot's terminal drive, the load of the actuator will be very large. However, the drive motor cannot be done 

very lightly at present. To meet the requirements of light and low consumption, some of the driving devices in 

the robot have been omitted in recent years. These joints degenerate into passive joints with free motion, and 

an underactuated robot with passive joints has emerged. The robot is a kind of mechanical system which 

contains passive joints and control input number is less than system freedom. It has the advantages of low 

cost, tight structure, good flexibility and low energy consumption. Therefore, it has important theoretical 

significance and broad application prospects. As the underactuated robot belongs to the two order 

nonholonomic constraint system, the control difficulty is greatly increased compared with the full drive robot. At 

present, this kind of robot has attracted more and more attention and become a new hot spot in robot research. 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1866120 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: He G., 2018, Study on the position control system framework of 3r underactuated robot based on fuzzy control, 
Chemical Engineering Transactions, 66, 715-720  DOI:10.3303/CET1866120   

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The underactuated robot system is a kind of special nonlinear system, which receives non-complete 

constraints and has special nonlinear structure. Because of the existence of non-driving joints, the system is 

not completely controllable. With the development of science and technology, while studying the 

nonholonomic system mechanics gradually, people began to apply the theory of flying integrity mechanics to 

many aspects. For the researchers in the field of control theory and engineering applications, the most 

concerned problem is how to design and apply control based on fully understanding of the characteristics of 

the nonholonomic system, so that the system has achieved the desired control effect according to the 

practicable trajectory under the inherent constraints. 

Research on underactuated robots has been done home and abroad, and these research results have played 

a positive role in practice. Ding and others analysed the model of a kind of underactuated system, established 

coordinate transformation, represented it as a special chain form, and then used recursive method to stabilize 

the system. The proposed form is simpler than the standard recursive technique, because it avoids the use of 

analytical differentiation, and finally applies the results to the Acrobot system (Ding et al., 2017). Xiong uses a 

stable hybrid scheme to control underactuated mechanical systems. The hybrid controller includes state 

feedback controller and discrete event monitor. When the continuous state collides the switch boundary, a 

new controller is applied to the equipment. The system uses the Lyapunov theory to determine the switch 

boundary and ensures the stability of the closed loop system (Xiong et al., 2017). The position control of the 

two rotating joint plane underactuated robot is considered in the work, and the dynamic model of the system 

and the fuzzy control strategy are proposed. The stability of the proposed method is verified by the simulation 

and the experimental results. But this result cannot be widely applied to other systems, because it is only 

derived from practical considerations, and there is no analytical proof of stability. But it also shows that, in a 

case, the fuzzy logic can solve the control problem in the case that the demand for the work solution is more 

important than the verification of the theorem (Liu et al., 2017). Urakubo studies the stability of a class of 

standard chain two order nonholonomic systems. First, two typical two order nonholonomic systems, the third 

joint underactuated three bar planar operating arm and all joint free motion redundant manipulators, are 

proposed by the force / torque driven model applied on the terminal operator and converted from coordinate 

and input to the two ordered chain forms. Then the law of discontinuous control is proposed. When the initial 

state is or is driven into a specific region of the state space, all the states of the system will be exponentially 

stabilized to the predetermined equilibrium point. The effectiveness of the proposed control scheme is verified 

by simulation of a planar 3R manipulator with underactuated joints (Urakubo, 2017). Aiming at the problems in 

Acrobot control, a functional integrated control method based on fuzzy control, variable structure control and 

LQR control is proposed by Hayashi. Finally, the control strategy of Acrobot is extended to the control of the 

multi degree of freedom underactuated arm robot, and a control strategy of the multi degree of freedom 

underactuated arm robot is proposed, which provides a method for the control of the multi degree of freedom 

underactuated mechanical system (Hayashi et al., 2017). Based on the stability of the non-driving arm at the 

highest point and the lowest point, a neural network based open loop control method for a non-driven joint 

robot is proposed by Cerkala and Jadlovska (Cerkala and Jadlovska, 2017). Li has studied the stabilization 

and tracking control of uncertain nonholonomic dynamic systems. A variety of adaptive and robust feedback 

positive definite and trajectory tracking control strategies are proposed for uncertain nonholonomic dynamic 

systems and non-holonomic wheeled mobile robots. The stability of the closed loop system is proved and 

simulated. The control problem of this kind of system is solved well (Li et al., 2017). The underactuated two 

connecting rod robot is studied by Massou and Boumhidi. The hardware improvement of the robot system, the 

derivation and analysis of the traditional dynamic model and the simulation demonstration under the different 

operating system are studied (Massou and Boumhidi, 2017). 

To sum up, the main work of the above research is to study the model of the underactuated robot system and 

the different methods applied, but there are still some problems in the field of under actuated robot control. 

Therefore, based on the above research status, this paper takes the underactuated 3R robot as the research 

object, establishes the dynamic model and analyses it, and uses the intelligent control theory to study the new 

method of using the robot position control in the operating space, and carries out the simulation analysis. To 

further the feasibility and effectiveness of the inflammatory control strategy, the underactuated experimental 

platform was designed and built, and the experimental research was carried out, and the control of the 

underactuated 3R robot was carried out in the experiment.  

3. Method 

3.1 Underactuated robot kinetic model 

The schematic diagram of a horizontal 3R underactuated robot is shown in Figure.1, in which the 1st joint and 

the 2nd joint are of active joint with an actuation device and the 3rd joint is of passive joint (or free joint) 

without an actuation device. Parameters are specified as follows: for joint angle, angular acceleration and joint 

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control torque (or control voltage), any of them in the counterclockwise direction is positive while that in the 

clockwise direction is negative; θ1, θ2 and θ3 represents three rotation angles; α1 is the included angle 

between the connection line of joint 1 and joint 3 and the rod 3; α2 is the included angle between the 

connection line of joint 2 and joint 3 and the rod 3; and α1 and α2 represents the configuration parameter of 

rod 3 relative toactive rod 1 and active rod 2. The curve S refers to the any curve in the motion plane of the 

underactuated system. d is the normal distance from the robot end point to the curve S. It is specified that the 

normal distance from a point on the side where the origin O is to the curve S is negative while the normal 

distance from a point on the other side to the curve S is positive.  

 

Figure 1: Schematic diagram of a horizontal 3R underactuated robot 

Since the robot moves in the horizontal plane, gravity potential energy item is not taken into account. The 

kinetic equation established based on Lagrange equation of the second kind is:  

 
(1) 

Where, M is mass inertia matrix, F is an item relating to angular velocity including Coriolis, centrifugal force 

and friction damping, θ is joint angle matrix, and τ is joint actuation torque matrix. The equation (1) is 

expanded into a block matrix:  

 

(2) 

It can be seen from the equation (2) that the coupling relationship exists between the actuation joint 

acceleration and the passive joint acceleration and the motion of the passive joint may be coupled-controlled 

through control over various active joint. So, the coordinates of the end point of the passive joint is controlled 

indirectly by the input torques of two active joints, which means the trajectory tracking control issue of the 

underactuated robot may be solved kinetically through an appropriate control strategy.  

3.2 Fuzzy control strategy 

The motion of the underactuated robot is broken down into the rotation motion of the active joint and the 

extension or contraction motion of the passive joint. The extension or contraction of the passive joint is the 

response of the rotation motion of two active joints to the kinetic coupling of the passive joint. To realize the 

trajectory tracking control of end point of 3R underactuatedrobot, it is necessary to control the rotation of the 

active joint and the expansion or contraction of the passive joint to ensure that the normal distance d from the 

end point to the target trajectory S and the velocity at the end point is greater than zero. Here, we only 

consider the formation of the end point trajectory while have no quantitative requirement for the velocity and 

acceleration of the end point.  

In order not to lose the generality, it is necessary to install the brake at the free joint. The trajectory tracking 

control process is divided into two stages. In the first stage, the brake is locked and the underactuated robot is 

degenerated into fully actuated to control the end point of active joint robot to approach the target trajectory 

from any initial position; in the second stage, the brake is unlocked when the end point reaches the target 

trajectory for the first time, the passive joint is completely free, and the underactuated control stage begins.  

 

 

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Figure 2: trajectory tracking control block diagram 

The control flow chart of the underactuated control stage is shown in Figure 2. Two active joints adjust and 

control the deviation of both ends, respectively, and two control layers are independent from each other. 

Below is the specific process: when τ1 and τ2are consideredseparately and τ1is calculated separately, the rod 

1 and the rod 2 are relatively still and considered a whole, d and α1 are fed back as the input variable to the 

fuzzy controller 1 to get the input torque τ1 of the joint 1; when τ2 is calculated separately, d and α2 are fed 

back as the input variable to the fuzzy controller 2 to get the input torque τ2 of the joint 2; finally, the motion of 

the end rod is controlled jointly through the kinetic coupling to enable the coordinates of the end point to track 

the target trajectory curve.  

For Mamdani2D fuzzy controller, the input variables are the normal distance d from the robot end to the target 

curve and its change rate η, while the output variables are the control torques of two active joints (or control 

voltage), i.e. τ1 and τ2.  

Based on the kinetic coupling law and control experience, the fuzzy control rule may be summarized as 

follows: when the end error and the error change rate are positive, impose a large reverse torque on the 

actuated joint and allow the passive joint to generate a reverse acceleration to compensate the positive error; 

when the end error and the error change rate are negative, impose a large torque on the actuated joint to 

compensate the negative error. Other intermediary circumstances may be based on the following principles: if 

the error is great, mainly focus on eliminating the error by adjusting and controlling the variables; if the error is 

small, mainly focus on controlling the system stability by adjusting the control to avoid over-adjustment.The 

experiment system of the underactuated robot is shown in Figure.3. 3Runderactuated robot mainly consists of 

two parts, i.e. mechanical structure and electric part. Since only the motion of three joints is taken into account 

in this experiment, the rod that is connected to the pedestal during the motion is kept still. Active joint 1 and 

active joint 2 adopt Maxon brushless DC servo motor and planetary reducer, while passive joint 3 is equipped 

with an electromagnetic clutch. Each joint is attached with a high-accuracy incremental coder (initially 

10000P/R, 40000P/R following segmentation) which is used to detect the joint angle in a real-time manner. 

The limit switch on each rod is used to limit the rod position and play a role of limit protection. When the rod 

contacts the limit switch, the motor will stop. Structural parameters of the robot (rod length and mass) are 

same as the simulation setting values.  

 

 

Figure 3: Experiment system of underactuated robot 

4. Results and Discussions 

Figure.4 shows the angular velocity curve of joint 1. During 3 s~4 s, there is a great difference between the 

actual angular velocity of the joint and the theoretical angular velocity of the joint mainly because: on one 

hand, the issue of zero drift exists in the control card itself, i.e. the motion status generated by the same 

control voltage when the motor rotates clockwise is inconsistent with that when the motor rotates 

counterclockwise; on the other one, the motion of the rod 2 and the rod 3 has certain influence on the rod 1, 

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and the delay of the control system may cause the experiment curve to be lagged behind the theoretical curve 

generally.  

 

 

Figure 4: The angular velocity ofjoint l 

When the motor control voltage of the joint 2 is higher than 0.72V, the end rod has higher velocity and 

acceleration and moves rapidly to contact the limit switch to cause the motor to stop and result in an 

uncontrollable situation. Therefore, the maximum voltage is limited in the experiment, the voltage curve initially 

is a straight line in Figure.5. It can be seen from Figure.4 that the value of each joint becomes stable, and the 

curve is similar to that obtained from simulation. Compared with the simulation curve, the experiment curve is 

relatively lagged behind and also caused by the delay of the control system. The difference at about 3s is 

greatest mainly due to the influence of velocity and acceleration of the motor at 3s~4s in the active joint I. The 

end position of the joint converges to the expected value, and similar to the shape of the simulation curve. 

Parameters and errors at the final position of the experiment are shown in Table 1.  

 

 

Figure 5: The control voltage ofjoint 2 

Table 1: The experiment parameters and error analysis 

 Expected value Experimental value Relative error 

Θ1 45° 45.09° 0.22% 

Θ2 56° 56.03° 0.05% 

Θ3 -52° -50.57° 2.74% 

α 30° 30.73° 2.60% 

r 360 361.64 0.46% 

x 270 276.00 2.22% 

y 520 523.60 0.69% 

 

It can be seen from Table 1 that compared with the expected value, the maximum relative error at the end 

position of the robot in the operation space is 2.22%, the maximum relative error of the joint angle in the joint 

space is 2.74%, the maximum relative error of the position control parameter (01, r) during the motion 

is2.60%, and the relative error on the whole is within 3%. All these values indicate that the experiment 

accomplishes the operation task of the robot very well and verify the validity and reliability of the theoretical 

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method. Compared with the simulation values, the maximum relative error at the end position of the robot in 

the operation space is 2.31%, the maximum relative error of the joint angle in the joint space is 3.71%, the 

maximum relative error of the position control parameter (01, a, r) during the motion is 1.31%, and the relative 

error on the whole is within 4%.  

It can be also seen from the value simulation and the experiment results that, based on the same control 

principle, an experiment curve similar to the simulation curve is achieved by adjusting structural parameters 

and control parameters of the robot in value simulation and controlling necessary parameters of the robot. It is 

meaningful in two aspects: on one hand, the validity of the theoretical method is verified through the 

experiment; on the other, the results from theoretical simulation and related parameters of the robot are 

fundamental to future value analysis and offer guidance to future experimental research.  

5. Conclusion 

Based on a lot of references, this article adopts the fuzzy control theory to study the position control system 

framework of 3R underactuated robot. In this article, Lagrange equation is used to establish the block matrix to 

solve the trajectory tracking control issue. Later, fuzzy control strategy is used to further plot the control flow 

chart and establish the simulation model. The model as established in this article is featured by simple rule, 

less calculation, low operation difficulty and simple process. Results show that, based on the comparison 

between the expected value and the experiment value, the maximum of related error is 2.22% and the 

maximum absolute error is 1.43°. Results also show that the system framework designed in this article may 

achieve better control over the position of 3D underactuated robot. 

Such factors as gravity needs to be taken into account for activities of 3R underactuated robot in this article 

since under the gravity condition, the position of 3R underacutated robot can be better controlled with lower 

difficulty in operation. Therefore, the control principle and simulation model in this article is more suitable for a 

3D space environment.  

Reference  

Cerkala J., Jadlovska A., 2017, Application of neural models as controllers in mobile robot velocity control 

loop, Journal of Electrical Engineering, 68(1), 39-46, DOI: 10.1515/jee-2017-0005 

Ding F., Huang J., Wang Y., 2017, Sliding mode control with an extended disturbance observer for a class of 

underactuated system in cascaded form, Nonlinear Dynamics, 90(4), 2571-2582, DOI: 10.1007/s11071-

017-3824-3 

Hayashi N., Segawa K., Takai S., 2017, 2D Voronoi Coverage Control with Gaussian Density Functions by 

Line Integration, Sice Journal of Control Measurement & System Integration, 10(2), 110-116, DOI: 

10.9746/jcmsi.10.110 

Li H., Yan W., Shi Y., 2017, A receding horizon stabilization approach to constrained nonholonomic systems in 

power form, Systems & Control Letters, 99, 47-56, DOI: 10.1016/j.sysconle.2016.11.005 

Liu Y., Wu F., Ban X., 2017, Dynamic Output Feedback Control for Continuous-Time T–S Fuzzy Systems 

Using Fuzzy Lyapunov Functions, IEEE Transactions on Fuzzy Systems, 25(5), 1155-1167, DOI: 

10.1109/tfuzz.2016.2598852 

Massou S., Boumhidi I., 2017, Optimal neural network-based sliding mode adaptive control for two-link robot, 

International Journal of Systems Control & Communications, 8(3), 204, DOI: 10.1504/ijscc.2017.085498 

Urakubo T., 2017, Stability Analysis and Control of Nonholonomic Systems with Potential Fields, Journal of 

Intelligent & Robotic Systems, (1), 1-17, DOI: 10.1007/s10846-017-0473-1 

Xiong P., Lai X., Wu M., 2017, A stable control for second-order nonholonomic planar underactuated 

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