IROS2019国际学术会议论文集0646

Centralized Control Architecture for Cooperative Object Transportation using Multiple Omnidirectional AGVs Firhan Huzaefa and Yen-Chen Liu AbstractThis paper addresses the problem of coopera- tive transportation using multiple omnidirectional Automated Guided Vehicles (AGVs). To enhance fl exibility and appli- cation potentials, mecanum-wheeled platform is considered for the proposed multi-AGV system while the cooperative transportation is executed without physical link/gripper to fi x the object on the AGVs. Therefore, the position and number of AGV is adjustable depending on the size and weight of the transported object. Analysis of force distribution to each AGV during cooperative transportation is presented. Furthermore, the gradient projection method is exploited to regulate internal force according to the operational capability of each AGV. Moreover, an adaptive sliding mode controller is designed for AGV to cope with dynamic uncertainty during cooperative transportation. Stability of the proposed controller is proven by using Lyapunov Theorem. Finally, numerical simulation is presented to demonstrate the performance of the proposed control system. Keywords: Cooperative transportation, multi-AGV system, force distribution, mecanum-wheeled platform. I. INTRODUCTION The automation of material handling process in ware- house and factory is important because it provides signifi - cant benefi ts of cost reduction and increasing productivity. Automated Guided Vehicle (AGV) are especially suitable for Material Handling System (MHS) where space is at a premium and fl exibility is critical 1. In practice, various loading weight and merchandise sizes lead to diffi culties in storages and transportation within warehouses or in industrial environments. Therefore, different sizes of AGV are needed for each load types, which is expensive and not fl exible in the case where the industry needs to change the load types periodically. In this particular case, cooperative transporta- tion using multiple relatively small AGV with high capacity can provide a better solution. In this paper, we develop the control architecture for cooperative object transportation using multiple AGVs as shown in Fig. 1. This system enable AGVs to contract or expand the different formation depending on object sizes and weights. Consequently, the variation of transporting different types of objects can be solved by adjusting the number of AGV on the system. The research regarding cooperative transportation using multiple robot has been studied previously 25. However, there have been relatively a few studies for the case of unit This work was partially supported by the Ministry of Science and Technology, Taiwan, under grants MOST 108-2636-E-006-007, MOST 106- 2221-E-006-010-MY3, and MOST 105-2221-E-006-160-MY3. Firhan Huzaefa and Yen-Chen Liu are with the Department of MechanicalEngineering,NationalChengKungUniversity,Tainan 70101,Taiwan.Email:.tw, .tw. Fig. 1: An example of the proposed cooperative transporta- tion for a 3-AGV system load AGV, where the load is placed on top of the AGV. A theoretical analysis and experimental results of cooperative transportation using leader-helper architecture are presented in 6. Cooperative transportation using the reinforcement learning method was introduced in 7. In both studies, the system are limited into two robots and exclude the analysis of force distribution and internal force during transportation. The development of a mathematical model of transported object and force distribution to each AGV is signifi cant and crucial to the studies of cooperative transportation. Many researches studied load distribution and internal force analysis have been done in cooperative manipulation for robot manipulators. These can be classifi ed as a control allocation problem for an over-actuated mechanical system. The authors in 8 described the decomposition of manipu- lator forces into internal and external force. Unique solution that avoids internal loading inside the objects can be found using moore-penrose pseudo inverse presented in 9. In 10, parameterized pseudo-inverse is used for force allocation and their result is the equivalent with equal weight distribution for manipulators 9. Similar concepts can be used for multiple unit-load AGV, where the load is placed on top of the platform, and the formulation of force distribution and internal force analysis will be addressed. In this paper, we consider a multiple AGV system driving by four-mecanum omnidirectional wheels and mounted with a rotary platform for the transported objects. The derivation of the kinematic and dynamic model for mecanum wheeled 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 978-1-7281-4003-2/19/$31.00 2019 IEEE6526 Fig. 2: Design of the proposed AGV with omnidirectional mobile base and the rotary platform. platform have been developed by 1113. The dynamic model is further improved by including mass eccentricity on the system in 14, which will be adapted into our model. During cooperative transportation, AGV need to adapt into different load and operating condition; therefore, we propose a control algorithm for the AGVs to be resilient to dynamic uncertainty during operation. The major contribution of this paper includes that we 1) propose a novel AGV system with an omnidirectional four-mecanum-wheeled platform to ensure higher fl ex- ible and resilient cooperative transportation, 2) analyse the force distribution to the proposed AGVs during cooperative transportation, and the application of gradient projection method to regulate internal force in transported object, and 3) develop an adaptive tracking control algorithm for the proposed AGV system to accomplish cooperative transportation with the consideration of dynamic un- certainties from the AGV and the object. The rest of the paper is organized as follows. In Section II, the problem formulation and AGV modeling are addressed. Force distribution and adaptive sliding mode control are exploited and developed in Section III to achieve coopera- tive transportation. Numerical simulations are presented in Section IV to show the effi cacy and performance of the proposed system and controller. Finally, Section V concludes this paper. II. PROBLEM FORMULATIONANDMODELING A. AGV System The illustration of AGV platform is shown in Fig.2, where the AGV consists of an omnidirectional mobile base and a rotary platform 15. Four mecanum wheels confi guration is used in mobile base, and this provides omnidirectional movement for AGV. Omnidirectional wheeled mobile robots has an advantage over conventional confi guration such as better maneuverability, ability to turn in any direction with zero turning radius, and capability to move in confi ned space 11. During operation, the transported object is placed on top of the rotary platform, and the rotary platform is designed such that the AGV can rotate freely relative to transported object. (a) Friction between platform and transported object (b) Friction on each wheels Fig. 3: Weight distribution and friction force of an AGV B. Cooperative Transportation Consider a system with n AGVs to cooperatively transport a common object on the top of AGV as illustrated in Fig. 1 (for n = 3). Let o denotes the coordinate frame fi xed to center of the object, and Iis the inertial frame. All quantities will be expressed with respect to the inertial frame I otherwise stated. The position of AGV relative to the center mass of the object is expressed by two variable Liand i, where Lidenotes the distance between the ithAGV to the center of the object, and idenotes the angle between the line from x-axis of object local frame to the ithAGV. The position of the AGV can be freely adjusted as long as the transported object is in stable condition. The transported object is assumed to be rigid, and the desired trajectory for each AGV can be determined using rigid body kinematic relation. Since the AGV is omnidirec- tional, there is no need to determine the desired orientation for each AGV. However, the orientation can be utilized to achieve various secondary objective, e.g. energy effi ciency and maximum driving force, which are considered as the future work of this research. The cooperative transportation is fulfi lled if each AGV can follow the desired trajectory under the effect of dynamic uncertainty from external force and friction force from transported object. The external force to drive the transported object is pro- vided by each AGV, as illustrated in Fig. 3a. Since there is no grasping mechanism to fi x the object, the force magnitude is equal to the static friction between rotary platform and transported object. Each AGV will receive the external force from the object but in opposite direction, and the amount of force depends on the center of mass of the object and the formation of the AGVs. Slippage between the object and the rotary platform will not happen if the driving force is lower than the maximum static friction force in the contact 6527 Fig. 4: The geometry of AGV platform point between object and rotary platform. Thus, in Fig. 3a, Fei sNoi, where Noi, the distribution of object weight applied on the ithAGV, can be expressed as Noi= imog,(1) where iis the portion of the object weight distributed to the ithAGV, with n i=1i= 1. Here, we consider an assumption that Assumption 1: The coordinate and orientation of the transported object are known and available to all AGV on the system and rigid body kinematic relation can be used to determine the position of each AGV relative from object center. C. AGV Dynamic Model The geometry and wheel confi guration of four-mecanum wheeled mobile robot is illustrated in Fig. 4. For the dy- namic model of the platform, we follow the derivation step proposed in 14 and modify the equation to match our wheel confi guration. The dynamics of the ithAGV in inertial frame can be expressed as Mi(qi) qi+Ci(qi, qi) qi= 1 rw BT iiBTiHifiFei (2) where qi= xi,yi,iTis the state of the ithAGV, i= i1,i2,i3,i4 is the torque generated in each wheel by electric motors, and Hi= diagsgn(ij) R44for j = 1,.,4 denotes a diagonal matrix for the direction of wheel rotation. The elements of the mass matrix Mi= mij3x3are given as m11= mb+ ? mw+ 4Iw r2 w ? ,m12= mb+ ? mw+ 4Iw r2 w ? , m13= Ib+mb(d2 1+d22)+4 ? mw+ 4Iw r2 w ? (a+b)2, m12= m21= 0;m13= m31= mb(d2cos +d1sin), m23= m32= mb(d1cos +d2sin). Fig. 5: The FBD of external force acting on transported object The matrix C and dynamic transformation matrix B are described by Ci= 00mbi(d1cosid2cosi) 00mbi(d2cosi+d1sini) 000 , and Bi= (cosi+sini)(cosisini)(a+b) (cosisini)(cosi+sini)(a+b) (cosi+sini)(cosisini)(a+b) (cosisini)(cosi+sini)(a+b) , where mbis the mass of the platform, Ibis the platform inertia, (d1,d2) are center mass position relative to platform center, mwis mecanum wheel mass, Iwis mecanum wheel inertia respect to platform center, rwis the wheel radius. Based on the freebody diagram (FBD) in Fig. 3b, the friction force on each wheel is fi= (mbg+Noi)r 4 ,(3) where Ni= is the total weight supported by four wheels and r is the friction coeffi cient between wheel and ground. The dynamic equation satisfi es following properties 16: Property 1: The matrix Mi(qi)2Ci(qi, qi) is skew sym- metric such that T(Mi(q)2Ci) = 0, R3 Property 2: For any differentiable vector R3, the Lagrangian dynamic are linearly parameterizable. Therefore Mi(qi)i+Ci(qi, qi)i=Yi(qi, qi,i, i)iwhere iis a con- stant vector of unknown parameters and Yi(qi, qi,i, i) is the matrix of known functions of the generalized coordinates and its higher derivatives. III. FORCE DISTRIBUTION ANALYSISAND CONTROLLER DESIGN A. Force Distribution and Internal Force Analysis The dynamic equation of the transported object can be expressed as M qo= Fo,(4) 6528 whereMo= diagmo,mo,Ioismassmatrix,qo= xo,yo,oTis the position and orientation of the object, and Fo= Fox,Foy,oTis the external force the applied on the object. The mass of the object is mo, and Iois the object moment of inertia. Force distribution analysis is calculated with respect to the inertial frame. The required force for transported object to follow a desired trajectory, qod(t), can be modeled using impedance control for a linearized second order object model 17 that Fo= Mo qod+Kd( qod qo)+Kp(qodqo),(5) where Kdand Kpare positive control gain and qodis the desired position for the transported object. The freebody diagram (FBD) of the external force from each AGV to the transported object is illustrated in Fig. 5. The combination of force provided by each AGV must meet the required force for the transported object to follow the trajectory. Thus, the effective force acting on the origin of the object frame Fo given the external forces from each AGV is expressed as Fox= Fe1x+.+Fenx, Foy= Fe1y+.+Feny,(6) o= (Fe1xL1sin(1+o)+Fe1yL1cos(1+o)+. +(FenxLnsin(n+o)+FenyLncos(n+o). To simplify the equation, the stacked vector contain- ing all external forces from AGV expressed as Fe= Fe1x,Fe1y,.,Fenx,FenyT. The relation between Foand Fe can be written in form of Fo= GFe.(7) For an n AGV system, the matrix GR3x2ncan be expressed as G = ?I 2 .I2 T1.TN ? (8) where Ti=Lisin(i+o),Licos(i+o) and I2is a 22 identical matrix. The issue of the force distribution is that the system is redundant and there are infi nite combinations to distribute external force to each AGV. Since G is not a square matrix, there is no unique solution to fi nd Fegiven Fo. One of the solution to solve the equation is by using Moore-Penrose in- verse. By using this method we can obtain force distribution that will produce no internal force for the object 9 10. This distribution is expressed as Fe= G+Fo(9) where G+=G(GTG)1is Moore-Penrose inverse of G. The defi nition of internal force in this case are the force that will not produce any work towards object movement. Usually we want the internal force in the AGV to be zero since it requires less energy for transportation. However, the internal force can be used to adjust force distribution into an AGV according to the operational range of the AGV. This means that the force distribution to AGVs with low operational capability will be reduced, while the force distribution to other AGVs with higher operational capabilities will be increased. To achieve this, the redundancy of multi-AGV system can be used to include additional constraint on the force distribution. We can model the force confi guration by including internal force by using Gradient Projection (GP) method 18 that Fe= Fext+Fint(10) Fe= G+Fo+IG+G(11) the gradient function is expressed as follows: = d dFe h(Fe),(12) where h(Fe) is a function that has local minima in the desired confi guration 19, and is a positive constant to adjust the rate of change for force distribution. For the case where each AGV have a different capacity of producing driving force to the object, h(Fe) can be expressed as h(Fe) = (1)n+1 n i=1 ? 1 kFeik Fmax ei ?kF eik Fmin ei 1 ? ,(13) where kFeik = q F2 eix+F2eiy. This function keeps the trans- porting force in Fmin ei ,Fmax ei . The more AGV included in the system, the more fl exible the distribution become. B. AGV Control Design The individual AGV in the system needs to follow the desired trajectory under uncertainty in the dynamic caused by friction force and external force from transported object. In this paper, an adaptive sliding mode control is exploited and developed for the control of AGVs. Sliding mode control is used to handle the uncertainty of external force from transported object. Boundary layer method is applied on control law to minimize chattering on the control input. To reduce chattering on the system, we apply boundary layer method into sliding mode control 20. The goal is to keep sliding variable inside boundary layer range i. The dynamics (2) for the ithAGV can be expressed as Mi(qi) qi+Ci(qi, qi) qi+BT iHifi= uiFei (14) where generalized control input is given as ui= 1 rw BT ii. (15) Since the rotary platform does not transmit any torque into mobile base, the external force received by ithAGV is expressed as Fei= Feix,Feiy,0T. The adaptive sliding mode control for AGV is proposed as ui= Mi qri+Ci qri+BT iHi fi+FeiKsiKrsat ? si i ? , (16) where the sliding variable siand qri error vector is defi ned as si= ei+ei, and qri= qdieiwith ei= qi(t)qd(t) is position tracking error for the ithAGV. The saturation function is described as sat ? si i ? = sgn ? si i ? if ksik i sat ? si i ? = ? si i ? otherwise.(17) 6529 By substituting the control law (16) into the dynamic equa- tion, we obtain the closed-loop dynamic that Mi si+Cisi= Mi( qriqi)+Ci( qri qi)+BT iHi( fi fi) +(FeiFei)KsiKrsat ? si i ? .(18) The mass and damping coeffi cient can be simplifi ed in term of linearized dynamics by Property 2 and expressed as Mi si+Cisi=Yi( qri, qri, qi,qi)()+BT iHi( fi fi) +(FeiFei)KsiKrsat ? si i ? (19) with linearized dynamic parameters formulated as Yi( qri, qri, qi,qi) = xri0 risini+ricosi ricosirisini yri0ricosi+risini risini+ricosi 0 ri xrisini yricosi xrisini+ yricosi i= mb+4(mw+ Iw r2) Ib+mb(d2 1+d22)+4(mw+ Iw r2)(a+b) 2 mbd1 mbd2 . Subsequently, the adaptation law is applied into estimated parameter i= PYT i si, fi= HT i Bisi,(20) where P and are adaptation rate for parameter and fi respectively. The difference between estimated and actual parameter value explained