J. Cent. South Univ. Technol. (2011) 18: 1953-1961

DOI: 10.1007/s11771-011-0928-7

Modeling and motion control simulation of tendon based parallel manipulator translation mechanism for sensor based high value waste processing

HUANG Jiu(»Æôñ)^{1, 2}, T. Pretz¹, BIAN Zheng-fu(±åÕý¸»)²

1. Department of Processing and Recycling, RWTH Aachen University, Aachen 52062, Germany;

2. School of Environmental Science and Spatial Informatics, China University of Mining and Technology,Xuzhou 221008, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: A novel sorting system based on one degree of freedom (DOF) tendon based parallel manipulator (TBPM) for high value waste processing was presented and designed. In order to control the motion of loads, nonlinear state feed forward control algorithm in the tendon length coordinate was used. Considering the system redundancy and actuation behavior, algorithms of optimal tension distribution and forward kinematics were designed. Then, the simulation experiments of motion control were implemented. The results demonstrate that the proposed TBPM translation system performs robust capacities. It can transfer the loads 1 m away within 1.5 s. With further optimization, the translation duration can be further reduced to be about 1 s and the optimized translation is followed with 43.59 m/s² maximum acceleration. The translation errors at the aim position remain below 0.4 mm.

Key words: municipal solid waste incineration; tendon based parallel manipulator; sensor based sorting; motion control

1 Introduction

Most of the solid wastes in the world are generated by human activities [1]. Although solid wastes are usually considered as a kind of pollutant to the environment, most of the ingredients in solid wastes could be utilized as secondary resource when they get processed and sorted. In some countries, especially in China, the growth rate of solid wastes, especially the municipal solid wastes (MSW), was 11% in the past three decades, greater than that of China¡¯s GDP [2]. Since 1980s, China has disposed solid wastes by the principle of minimization harmlessness and utilization so as to promote the development of environmental industry and improvement of the environment. Incineration has an important place in MSW treatment. Due to its hygienic control, volume reduction, mass reduction and energy recovery advantages, incineration becomes an attractive way for MSW disposal in the industrial countries all over the world [3-4]. In France, around 130 incineration plants treated 10.8 Mt MSW per year and in 2008, the overall capacity of MSW incinerator in Germany achieved 20.4 Mt/a. Especially with the development of economy in China, increasing amounts of MSW have been produced. Till 2006, there were a total of 69 municipal solid waste incinerators (MSWI) with a treatment capacity of 39 966 t/d in China [5-7]. MSWIs not only treat the wastes, but also produce heat and power. However, large amounts of bottom residues are also generated from MSW incineration. For instance, in Japan and Germany, about 6 and 3 million tons of MSWI residues are produced respectively each year [8]. MSWI residues contain pollutants such as heavy metals. The conventional method for disposal of MSWI residues is dumping. The increasing dump of residues not only occupies plenty of land, but also wastes resources and can potentially have an impact on the environment due to water pollution. Therefore, it is desirable to recover metals from residues and to utilize them [9]. MSWI residues contain about 16% metallic phases. All the metallic slags can be divided into two kinds: ferrous slags and non-ferrous slags. Both of them can be sorted by using magnetic and eddy current separator [8, 10].

A solid waste processing facility costs a lot and consumes large amount of energy which also uplift, the price of secondary raw material from wastes processing. Therefore, the sorting of a specific waste mixture is always ordered according to the values of different ingredients, and the separation of the more valuable ingredients always has a higher priority during solid wastes processing. This principle is able to reduce the cost of solid waste disposal and reuse the resources from solid wastes as much as possible [2].

The traditional sorting technologies such as magnetic and eddy current sorting cannot fulfill the requirements of modern solid waste disposal. For instance, non-ferrous metal wastes contain aluminum, copper, brass and some other metallic ingredients. They are not available to be separated by using conventional sorting technologies. In the last decade, the sensor based sorting technology for solid waste processing was developed rapidly. Compared with the conventional sorting technology, the sensor based sorting system can separate different particles according to their characteristics which do not have force field between the separator and particles, such as the colour, the geometrical parameters and X-ray spectra. In the last decade, the sensor based sorting by using optical sensor and near infrared (NIR) sensors made great technical improvement and enabled their industrial implementation [1,11]. Laser triangulation scanning by using line camera and laser beam has also been developed for measurement of particles¡¯ geometries. The accuracy of laser triangulation has achieved 0.005 mm [12-14]. The separation process of sensor based sorting needs additional force field to separate the detected waste pieces. Normally, a compressed air nozzle is used to blow the detected particles out of the main waste stream. With different air pressures, the particles can be blown to different positions and carried away by conveyor belt. The disadvantages of this system are: 1) Only a few kinds of solid wastes are separated because the accuracy of nozzle is not available for long distance transfer. The blowing transfer by compressed air is hard to be controlled; 2) In order to blow different ingredients with different distances, the pre-compressed air should have a high pressure about 600-700 MPa, which consumes too much energy; 3) The conveyor belt needs to keep running in order to transfer the selected pieces away. It consumes too much energy, since the concentration of high value ingredients in solid waste mixtures is low.

The above mentioned problems can be avoided by using a system controlled conveyor sorting belt based on tendon based parallel manipulator (TBPM) system. Compared with the normal transfer system, it is very energy-efficient. Its drivers are fixed and the payload is subdivided between drivers. The moving parts are light but high strength tendons such as nylon cables or belts. Therefore, they are appropriate to handle very heavy loads like cranes and can achieve very high accelerations and velocities. They can be designed in extremely large scale as well as in micro-scale applications. The TBPM translation system can be controlled by computers easily. Because it is easy to change its behavior to do different works, it has the advantages of high rigidity, large output force and torque, high accuracy, low moment of inertia and payload capacity [15-22]. FANG et al [15] proved that the TBPM translation system could achieve high accuracy which satisfied the sorting processes. HUANG et al [19] proved the availability of TBPM translation system in two degrees of freedom (DOF) by using forward kinematics and optimal tension distribution. YANG et al [23] developed many methods of manipulator forward kinematics. Their results show that this one degree of freedom TBPM translation system should have more robust capacities. KILLMANN et al  [1] proved that all the detected pieces can be blown out of the main stream with a constant low nozzle pressure and then get sorted. Hence, by using of a one DOF TBPM translation system, all the selected pieces from nozzle sorting system are able to be separated accurately. Much energy could be saved by changing the nozzle pressure and transfer mechanism. This work concerns with the design and simulation experiments of a one DOF TBPM translation belt for a three-dimensional (3D) optical sensor based sorting system.

2 Setting of tasks

MSWI residues contain specific high-value metallic components such as coins and even jewelries made of precious metals. The mass concentration of the high value pieces in the incineration residues is very low but the value of them cannot be neglected. The aim of this work is to develop a TBPM sorting system as compensation and optimization of conventional sensor based sorting system in order to separate different high value ingredients from metallic MSWI residues.

The metallic residue samples were collected in a MSWI facility in City of Weisweiler, which has a treatment capacity of 360 kt/a. Among the samples, several hundreds of coins and one gold watch frame were found. The average value of coins was about 0.6 €/kg and the average content of gold was 40 mg/kg, as shown in Fig.1. Especially for the euro coins, the costs to produce them are higher than their par values. They could be easily refunded in Bank system when they are purified. It is uneconomical to recycle them as raw materials.

Figure 1 shows that although after incineration, their shape characteristics such as diameters and thicknesses of the coins and the colour of gold are still recognizable. By using existing 3D scanning camera as optical sensor, they are able to be simply detected and located on a conveyor belt and then get sorted by air nozzle blowing.

3 Design of TBPM translation system

During the sampling process, the metallic MSWI residue samples were classified by different ranges of particle size through screening. All the coins and jewelry pieces were concentrated in the screening section of 10- 30 mm. They get classified and selected by an optical sensor based sorting system which is shown in Fig.2.

Figure 2 shows that the proposed TBPM translation system could be set under the normal 3D sensor based sorting system, in which the waste pieces were dropped onto a conveyor belt individually by vibration feeder and then got scanned by a 3D colour scan camera. The appearance frequency of high value pieces were about 0.085 piece/s. The camera measured the 3D shapes and the colour of each waste piece. All recognized pieces were firstly blown away from the main stream into a chamber, in which the selected pieces dropped onto a fixed position on the TBPM translation belt. This belt was designed to be triggered by dropped pieces through an optical laser beam. With the help of the computer, the translation distances of the followed pieces were calculated one by one and then converted to be motor revolutions. Then, the motors could drive the belt to transfer the selected pieces to the corresponding positions where air nozzles were installed. There the pieces were blown out of the translation belt according to their sorts and get collected.

In the whole sorting system, the part of sensor and recognition had been realized [1]. The main task was to design the TBPM translation system. It should have only one DOF and move on a horizontal straight line.

The position where the waste pieces following onto the translation belt surface was set to be the ¡°zero point¡± of referent coordinates. At the zero point, an additional weight was designed to be connected on the back side of the belt as the end-effector. The additional weight had the same width with belt. The conveyor belt was divided into two parts and connected by the additional weight in order to form two tendons. The belts are made of light and thin nylon with high strength. The mass of the additional weight was much higher than the masses of following pieces (m_a >> m_piece) in order to avoid the influence of mass changing which was caused by the following pieces. The initial position of the end-effector was set on the zero point.

Fig.1 Coins and gold piece in incineration residues

Fig.2 Principles of optical sensor based sorting system and TBPM translation

The motion control of this TBPM system was designed to be realized by two high-accuracy servo- motors. The key part of servo-motor was the drum which was set on the shaft of motor. The tendons were rolled on the drums. The belt was designed to be sustained by a big platform which kept the translation horizontal. The model of the drum is shown in Fig.3.

Fig.3 Equivalent model of drum

From Fig.3, J_i is the moment of inertia of the shafts/drums, ¦È_i is the rotation angel of the motors, C_i is the friction coefficient of the shafts/drums, r_i is the equivalent radius of the shafts/drums, f_i is the tension on the tendons, u_i is the output torque of the motors, and L_i is the equivalent length of the belt from the contact point on drum i to the position of additional weight.

Then, the combined motor shaft/drum dynamic model is

 (i=1, 2)                                      (1)

The one DOF TBPM system constituted two motors and two tendons (belts). In the system model, the two motors were set to lay on both sides of the straight line. The two tendons were connected to the mass center of the additional weight. And the zero point was set to be the middle point between two motors, as shown in Fig.4.

Hence the dynamical model of this manipulator system is [20]

 (f_i>0)                                            (2)

where

, , , w=

 and  summarizes the tension matrix of the

tendons, a_p summarizes the acceleration of the end- effector (additional weight), w summarizes all other forces and torques which act on end-effector, m is the mass of the end-effector, and u_m,x, u_m,y, u_m,z are the unit vectors of the tendons.

Matrix A^T is called the structure matrix. The function of this matrix is to transform the forces on tendons into end-effector. In general, for n end-effector- DOF and m tendons, the structure matrix A^T should be a n¡Ám matrix [20].

For this one DOF TBPM system, the Eq.(2) can be written as [19]

                            (3)

The material of the belts is high strength nylon, hence the model of the tendons can be written as

 (i=1, 2)                                            (4)

where  is the spring coefficient, d_i is the damper coefficient, and ¦¤x_i is the elongation of the belts.

The elongations ¦¤x_i of the tendons can be calculated by

                                (5)

where L_0i is the initial length of the belt parts; L_i is the length of belt parts with forces effect on them during the translation of end-effector; ¦È_i is the rotation angle of the motors and ¦È_i0=0. Compared with the drum diameter, the thickness of the nylon belts can be neglected.

And the time derivative of Eq.(5) can be used to get ?x_i, as below:

                                                               (6)

Summarizing Eqs.(1), (2), (4), (5) and (6), the dynamical model of this one DOF manipulator system is written as

                                                   (7)

Fig.4 Structure of one DOF TBPM system

Since the two tendons can only exert positive tensions (they cannot push), in order to express the cable tensions as a function of the motor torques and angular motion, Eq.(1) can be converted to [21]

                                            (8)

The symbol V_pos( ) means the positive value of each vector component would be taken. The original negative value of each vector component is set to be zero. When the torque on each motor is large enough to keep all tendons in tension, Eq.(8) can be written as

                                                  (9)

And static relationship between forces T on the end-effector and the force f on the tendons can be written as [19-21]

                                                                    (10)

In Eq.(10), A^T is the structure matrix [21]:

                                         (11)

where ¦Â_i (i=1, ¡­, n) is the cable angle (see Fig.4) which remains constants in this system, ¦Â₁=0 and ¦Â₂=¦Ð. So the structure matrix A^T can be written as

                                                               (12)

and also , so the Eq.(10) can be written as

                   (13)

4 Motion control of system

4.1 Control scheme and forward kinematics

A tendon-based parallel manipulator is a nonlinear, coupled, and redundant system. According to the theory for decoupled control of nonlinear systems [24], using nonlinear state feedback compensation, a linear decoupled control system could be derived, but the position of the end-effector must be measured in real time during the control process. In this research, a computed torque controller in tendon length coordinates was employed. It contained mainly a feed forward part of inverse dynamics and feedback loop. The control scheme is shown in Fig.5 [15, 19, 25].

And the control law followed

        (14)

where u=[u₁, u₂]^T, summarizes the vector of motor torques. K_p and K_v summarize feedback gain matrices. J=diag[J₁, J₂], summarizes the matrix of inertia. C= diag[C₁, C₂], is the matrix of friction coefficient and f summarizes the desired the tension vector of tendons corresponding to a desired position of end-effector (additional weight).  and  summarize the desired motor angle vector and its time derivatives.

Measuring the position of the end-effector in its operational space and velocity precisely is very difficult and expensive [15, 26]. Moreover, it is not desirable to obtain the position of the end-effector using direct kinematics, because lots of complicated calculation must be performed at run time. In this design, ¦È_d can be measured through motor rotation. Hence, the actual length of the tendons, i.e., the x-coordinate of the end-effector can be calculated by forward kinematics and the solution of equation:

                                                     (15)

where x=r_i ¦È_i, hence , .

Equation (15) is derived from Eq.(5) where the tendon elongation ¦¤x are neglected. By comparing of the high strength of nylon belt and the low mass of end-effector, the actuating forces on tendons were designed to be relatively low. Only small tendon elongations would be generated.

Fig.5 Control scheme of one DOF TBPM system

4.2 Optimal tension distribution

Since this one DOF TBPM translation system was driven by two tendons with redundancy, it was necessary to distribute tensions among the two conveyor belts. An important condition was that the belts must not move over the aim position then come back, because the belts must be always in tension (f_i>0). Once the belt is relaxed (f_i=0), the translation system can become unstable immediately and make some unpredictable dangers. According to the control law, the main task of tension distribution is to determine the vector f in order to make the belts always in tension and the tensions on belts as small as possible.

According to the results in Ref.[27], the structure matrix can be divided into two parts: an n¡Án matrix B, whose rank is n, and a vector h corresponding to the redundant tendon, where n is the DOF of the manipulator system.

                                                              (16)

Then, the desired tension vector on tendons can be calculated by Eq.(2) [19]:

                              (19)

where U = u = [u₁, u₂]^T.

Then define C=-B^-1h, D=B^-1U, then Eq.(17) can be written as

                                    (20)

In order to keep the minimum tensions on the tendons, a minimum force f_min is set [19]:

                      (21)

Assume f₂ to be the minimum force:

                                                       (22)

Here, the value of f₂ could be given:

                                      (23)

Equations (22) and (23) ensure that during the movement of end-effector in operational space, the two belts are always kept in tension and always pulled the additional weight. They were realized by MATLAB programming. The minimum tensions are f_min.

5 Results and discussion

5.1 Experimental results

The operational space of the end-effector is a straight line with a length of 2.4 m. The system parameters of the system J_i, C_i, r_i,  d_i are fixed by actual equipment or desired equipment which are available. The mass of the end-effector together with load piece is about 390 g.

The initial position of the end-effector was set to be: P(x, y)=P(0, 0) and sustained by pallet. The tendons were preloaded and then relaxed in order to set the initial lengths of the two parts of the belt: L₁=L₂=1.2 m, and the initial tensions on the tendons were F₁=F₂=0 N. The test path was from the initial position to the aim position which was set to be P_a(1 m, 0 m). The minimum tension on the tendons was set to be f_min=5 N. The feedback gain matrices K_p and K_v were given by system analysis in MATLAB. And the whole simulation experiments were also implemented in MATLAB.

The results of the simulation experiment obtained are illustrated in Figs.6 and 7.

The motion response of the first simulation is shown in Fig.6. The end-effector achieves the aim position smoothly and directly within about 1.5 s, without any unnecessary manipulation. The motion behavior of the end-effector is fully under control. The aim of motion control is fulfilled.

Fig.6 Motion response of end-effector

Fig.7 More rapid motion response of end-effector

In order to unload the waste piece through air nozzle and avoid unnecessary motion behavior, the end- effector is designed to stay at the aim position until it receives the next signal from trigger. And the current position of the end-effector will be memorized in computation system in order to compute the next transfer distance. In this research, the appearance of selected high value waste pieces is about every 11.8 s. Hence, this result of the end-effector motion response is able to satisfy the sorting tasks. However, for different sorting tasks, the appearance of load objects could be more frequent. Therefore, this one DOF manipulator needs to have the ability to provide more rapid motion responses. This requirement is realized by further optimization of feedback gain matrices K_p and K_v. The optimization process is realized through MATLAB programming. For example in Fig.7, the aim position is still P_a(1 m, 0 m) and the end-effector also remains the same. After optimization, the time to achieve the aim position is reduced to be about 1 s. The end-effector also achieves the aim position smoothly and directly, without any unnecessary manipulation. This optimized motion behavior is also fully controllable and its motion response is much faster than before.

5.2 Discussion

This one DOF TBPM system drives end-effector only by two light and thin nylon conveyor belts as tendons, on which the tension responses corresponding to motion responses are also investigated, since the instantaneous tension loads are not allowed to be too large in order to avoid tendon fractures.

The tension responses on tendons of the first simulation experiment are shown in Figs.8 and 9.

Fig.8 Force response on right tendon (belt)

Fig.9 Force response on left tendon (belt)

Figures 8 and 9 show that at 0.1 s after the starting of motion response, there are corresponding tension impulse responses on tendons. And at about 0.2 s, the tension impulses reach their maximum value simultaneously. The maximum tensions on belts are about 125 N on the right tendon and 110 N on the left tendon, respectively. Finally, when the end-effector has reached its aim position, the tensions on both belts also reach the minimum tension which has been set to be f_min=5 N. The tension impulses are caused by the lack of preload on tendons. According to the tension responses, it can be calculated that the translation is followed with 38.46 m/s? maximum acceleration.

The maximum tensions are safe and acceptable for the high strength nylon belts. Since the strength level of the belts is around 2¡Á10⁵-5¡Á10⁵ N/m, at the aim position, the tensions on both belts reach the minimum value of  5 N and keep in constant.

After the optimization, the motion response becomes faster; hence the instantaneous tensions on the belts also become higher, which corresponds to the requirement of higher acceleration. The tension responses after optimization are shown in Fig.10.

Fig.10 Tension responses after optimization

Figure 10 shows that after the optimization the translation of end-effector has been reduced to be about  1 s, and the maximum tensions on the belts increase to be about 136 N on the right belt and 119 N on the left belt, respectively, which are higher than the results without optimization. The tensions reach their maximum values also at about 0.2 s. This means although the average translation velocity has increased by 33%, the tensions only increase by 11 N on the right belt and 9 N on the left belt. Compared with the strength of the belts, this extent of tension increasing still has no influence on belts¡¯ safety and stability. Finally, when the end-effector has reached its aim position, the tensions on both belts also reach the minimum tension of f_min=5 N. The optimized motion response follows with 43.59 m/s? maximum acceleration. These results show that the system still has potential in further optimization of translation velocity.

The response of end-effector acceleration after optimization is shown in Fig.11.

Fig.11 Acceleration of end-effector along with motion response

Figure 11 shows that the acceleration of end- effector has an impulse up to 43.59 m/s?, which corresponds to the tension responses on both belts, and then drops to zero when the aim position has been achieved. At the aim point, the acceleration of end- effector is not absolute zero; it has errors between   -1.2 m/s? and 0.8 m/s?. These errors are caused by forward kinematics calculation. In Eq.(15), the part of tendon elongation ¦¤x_i is neglected in order to simple the computing process. But, in the dynamical model of the whole system, the elongation ¦¤x_i is not neglected. Hence, the actual aim position can be only approximately achieved with small fluctuant errors, which leads to the continuous tension fluctuations on tendons. This phenomenon means that the motor are always driving the end-effector to move in a small domain around the aim position and trying to achieve it accurately. This behavior leads to the acceleration fluctuant errors which are shown in Fig.11.

The translation error fluctuation of optimized simulation is shown in Fig.12.

Fig.12 Translation error around aim position

Figure 12 shows that according to the simulation results of the control optimized system, the actual translation errors fluctuate between -0.2 mm and 0.2 mm. And at the aim position, the effective diameter of a compressed air nozzle is about 5 mm. For sorting tasks, this accuracy is high enough and fully acceptable. This accuracy has also verified the reliability and availability of Eq.(15), which calculates the actual tendon lengths by neglecting the tendon elongations through forward kinematics. As the results of simulation experiments, the proposed control law with forward kinematics and tension distribution does not decrease the translation accuracy apparently.

6 Conclusions

1) A one DOF TBPM translation system for high value solid waste ingredients processing is proposed as enrichment and optimization of present sensor based sorting facilities. Through this design, the sorting efficiency and accuracy can be improved and the energy consumption during sorting process can be reduced.

2) The control scheme for the translation system is given. The algorithms of tension distribution and forward kinematics corresponding to the control law are also proposed.

3) Through the simulation experiments, the proposed one DOF TBPM translation system has been verified to be able to fulfill the sorting task of selected high value waste pieces. The time of 1 m translation is about 1.5 s, which can be improved to be about 1 s by further optimization.

4) During the translation, the belts transfer the end- effector with high accelerations. The tension impulses on the belts always remain in the acceptable range. In the further work, they can be further optimized by setting preloads on tendons.

5) The translation errors of the proposed TBPM system remain no more than 0.4 mm, which is high enough for the sorting tasks. This also verifies that the design of forward kinematics algorithm is appropriate for the whole system.

Acknowledgement

The China Scholar Council (CSC) of Ministry of Education of China is gratefully acknowledged for financial support of this research work.

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(Edited by DENG L¨¹-xiang)

Foundation item: Project(B07028) supported by ¡°111¡± Introducing Talents of Discipline to University Program through Ministry of Education of China

Received date: 2010-04-22; Accepted date: 2011-06-17

Corresponding author: HUANG Jiu, PhD Candidate; Tel: +49-179-7849642; E-mail: huang@ifa.rwth-aachen.de