Cornel Brisan
Technical University of Cluj-Napoca C.Daicoviciu, 15, Cluj-Napoca, Romania
Abstract: This paper presents basic elements concerning modeling and simulation of parallel robots which are used for machine-tools development. The advantages of this new kind of machine–tools consist on their reconfigurability, on the possibility to improve accuracy of those machines etc.
The main point of the paper is that few variants of parallel mechanisms with only one mobile platform and with number of degrees of freedom between 3 and 6 were emphases. It must be also remarked that these variants are developed into modular manner in order to ensure full reconfigurability. The paper presents also numerical results obtained with the virtual models which were developed. Few prototypes which ere developed are also presented.
I. INTRODUCTION
A good dynamic behaviour (high stiffness), a high accuracy and a good ratio between total mass and manipulated mass are just few advantages of parallel robots compared with serial type.
However, the design, trajectory planning and application development of the parallel robot are difficult and tedious because the closed-loop mechanism leads to complex kinemactics. To overcome this drawback, modular design concept is introduced in the development of parallel robots.
Also, during the last period a new type of applications were developed. These new applications are related to the machine tools with parallel topology. Utilisation of parallel topology in the machine tools field creates the possibility for a reconfigurable design which is still an open problem and lacks theoretical base. One of the problems for reconfigurable robots is to determine the topology and geometry of the robot which is the suitable to fulfil a set of criteria. In the following sections we first present the modular topologic synthesis. Then, we describe the kinematics and an example is given.
II. MODULAR TOPOLOGIC SYNTHESIS
The structural synthesis of parallel mechanisms could be made if the relation of the number of degrees of freedom it is considered:
where m is the number of common restrictions for all elements, n is the number of the mobile elements, k is the number of restrictions which define a joint (for example in the case of
prismatic joint k=5), Ck is the number of joints with (6-k) degrees of freedom and MP is the number of identical degrees of freedom.
In the case of parallel mechanisms without common restrictions and also without identical degrees of freedom the relation (1) it becomes:
∑
= number of joints with (6-k) degrees of freedom which directly connect the platforms of the mechanism.With these notations it results:
∑
=where n1 is the number of the elements which compose the loops which connect the platforms of the mechanism.
We can also assume (Fig.1) two types of basic modules (named basic legs) which can connect the platforms of the mechanism.
Let a1 be the number of the loops with prismatic - universal - spherical (PUS) topology, let a2 be the number of the loops with prismatic - rotational - spherical (PRS) topology, and also let a3
be the number of the loops with prismatic – 2 universal - 2 spherical (P2U2S) topology.
In the case of parallel mechanisms which are used in the field of machine tools, it is common to consider:
{
1,...,5}
With these notations, the relation (3) becomes:3
2 3
6 a a
M= − − (5) Also, each loop contains only one degree of freedom. Thus, it results:
T. Sobh et al. (eds.), Innovative Algorithms and Techniques in Automation, Industrial Electronics and Telecommunications, 101–106.
© 2007 Springer.
101
f= 3
f= 2
f= 1
f= 3
f= 1
f= 1
a )1 a )2 a )3
Fig.1. Adopted loops for PARTNER robots
Integer solutions of the equations:
0 , 0 3 6
3 2 1
3 2
= + +
−
= + +
−
a a M a
a a
M (7)
gives all variants of parallel mechanisms with assumed hypothesis.
The system of equations (7) has many solutions. Also, if other parameters are taken into consideration (the order of the joints in the loop, the geometrical parameters of the loops etc) the topology problem becomes very complex.
The relation (7) defines the topology of parallel robots in a modular manner. If other parameters are taken into consideration (the order of the joints in the loop, the geometrical parameters of the loops etc) the topology problem becomes very complex.
Table 1 presents variants of PARTNER robots, solutions of (7), with a3 =0 and 6 ≥ M ≤ 3.
TABLE 1.
SPECIAL VARIANTS OF PARTNER ROBOTS
No M a1 a2 Remarks
1 6 6 0 Stewart Platform
2 5 4 1 3 4 2 2 4 3 0 3
Also, figure 2 presents kinematic loops of those variants.
a)
b)
c)
d)
Fig.2. Special PARTNER robots BRISAN
102
III. KINEMATICS
General algorithms used to solve direct kinematics in the case of parallel mechanisms consider that for each independent loop of the mechanism one vector equation can be write. Thus, a nonlinear system of scalar equations is obtained. Usually, this system of equations can be solved only with numerical methods and for that an accurate initial value of the solution it is required.
Of course, this initial value of the solution is strongly related to the geometric parameters of the mechanism. When the geometric parameters of the mechanism are changed also the initial solution must be changed. According to that, the kinematics of the parallel mechanism will be developed in a modular manner, based on kinematics of the legs which connect the platforms and in order to ensure an analytical value for the initial solution. Each leg is in fact the right (or left) side of one independent closed loop and can be described by two coordinate systems: one attached to the frame and the other one attached to the mobile platform (Fig. 3).
{T }
iml{T }
imr{T }
0q
ilq
irFig. 3 Independent closed loop
The relationship between these coordinate systems is given by: Himl=
∏
Ail(qil), (8)for the left part of the independent loop and :
∏
= ir(qir)
imr A
H , (9) for the right part.
Himl , Himr are absolute transformation matrices and )
(qil
Ail , Air(qir) are relative transformation matrices.
For an independent loop it results:
imr
iml H
H = (10)
Matrix equation (10) leads to six independent scalar equations.
For whole parallel mechanism, a nonlinear system of equations (with 6n independent scalar equations, where n is the number of independent loops) will be obtained. This system of equations can be solved only with numerical methods. Generally, the legs of the parallel component have the same topology. It results that the relative transformation matrices for the left and right part of each loop are formal similar. Therefore, for each topology of the legs, a formal mathematical entity (named LMM - Leg Mathematical Model) can be developed. Similarly a modular kineto-static model can be developed. This mathematical model leads to non-linear system of equations. Classic algorithms of numerical methods, e.g. Newton-Raphson, can be used in order to solve this system of equations.
Usually a virtual model must be designed in order to ensure a friendly way to cooperate with the customer. Related to the virtual parallel mechanisms and in order to ensure this property, the virtual model of LMM must include an automatic way to find an initial solution for the nonlinear system of equations.
Without lose the generality of the problem, a leg with PSU topology is considered (Fig.4a). An analytical solution of the initial values of the angular parameters of the joints of the leg means that a solution of the inverse geometric model for the initial position must be determined. This solution is also the initial solution for the nonlinear system of equations for the whole mechanism.
Thus, for the leg from figure 4a, it results:
1 6
DESIGNING ASPECTS OF RECONFIGURABLE PARALLEL ROBOTS 103
b)
Fig. 4. Virtual models of the modules
where H is the absolute transformation matrix, which describe the absolute position and orientation of the mobile platform (known for the initial position of the mechanism), Ai (i=1,6) is the relative transformation matrix. The elements of the Ai are functions of the joint coordinate (qi for the prismatic joint and "ij
for all other joints of the leg). Using relations (11) a set of initial values for the parameters which describe the leg from figure 4a can be found.
IV. VIRTUAL MODELS
Based on relations (8),…, (11) and using MOBILE software package few kinds of virtual models may be developed. For example the mechanism shown in figure 5 has three degrees of freedom and five independent kinematic loops. This mechanism results for a1 = a2 = 0.
Thus, for each closed independent loop, the closing equations are:
iR iL
j ijR j
ijL
R R
a a
=
=
∑
∑
= =,
4
1 4
1 (12)
where RiL and RiR are the absolute orientation matrices, corresponding to the left and right side respectively of the closed independent loop.
The system of equations describe by (12) contains (in the case of all five loops) 30 unknowns.
These are the angular displacements (θjiL and θjiR ) at the level of universal and spherical joints respectively. This system of equations can be solved with numerical methods. In order to find an initial solution (necessary for numerical methods) classical algorithm of inverse kinematics applied for open loop, which connects the mechanism platforms, can be used:
B2
A1
A2
B1
A6
A5
A4
A3
B3 B6
B5
B4
P {T }n
{T }0
ai1L
ai3L
ai4L
ai1R
ai2R
ai3R
ai4R
q3 q3
q3
ai2L
θ θ θ1iL 2iL 3iL
θ θ4iL 5iL
a)
q
3q
1q
2{T }
0{T }
nb)
Fig. 5. a) Mechanism with 3 dof; b) Virtual model
).
( )
( 6 0
1 0 0 1
1
jiL k
j jiL jiL
k
j
jiL
θ ∏ θ
∏
= = +− H = A
A (13)
where H0 is the absolute transformation matrix and AjiL are the relative transformation matrices.
Also, figure 6 presents virtual models of the robots from figure 2.
BRISAN 104
a)
b)
c)
Fig. 6 PARTNER robots
a) with 6 dof; b) with 5 dof; c)with 4 dof;d) with 3 dof
Using these virtual models it is possible to get out also numerical results, those corresponding to assumed movements.
Thus, Stewart platform, with concrete geometrical data were considered (Fig. 7).
0.6
0.2
0.64
Fig. 7. Geometric data of Stewart platform
For this mechanism a subject of interest is the volume of work space and as consequence the values of the angles at the level of passive joints are object of interest. Figure 8 presents time variation for the angles of the passive joints for this mechanism (“alpha ij” are the angles at the level of universal joint and “beta ij” are the angles at the level of spherical joints).
-60 -40 -20 0 20 40 60
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Time[s]
Angles[°] alpha11
alpha12 beta11 beta12 beta13
-80 -60 -40 -20 0 20 40
1 4 7 10 13 16 19 22 25 28 31 34 37 40
Time[s]
Angles[°] alpha21
alpha22 beta21 beta22 beta23
DESIGNING ASPECTS OF RECONFIGURABLE PARALLEL ROBOTS 105
-10 0 10 20 30 40 50
1 4 7 10 13 16 19 22 25 28 31 34 37 40 Time [s]
Angles [°]
alpha31 alpha32 beta31 beta32 beta33
Fig. 8 Time variation of the passive angles in the case of Stewart platform
In addition, figure 9 presents few prototypes of the parallel robots from figure 6.
a)
b)
c)
Fig. 9. Prototypes of PARTNER robots a) 4 dof; b) 5dof; c) 6 dof
V. CONCLUSIONS
The conclusion can be drawn as follows.
Based on assumed modules and on relation of the number of degrees of freedom for a mechanism, a topologic synthesis can be done.
The kinematics of the whole mechanism can be developed on a modular manner, each module based on the kinematics of one leg.
Solving inverse kinematics of one leg it is possible to find an analytical solution of the initial value of the solution of the system of equations, which solve the direct kinematics of the mechanism.
Analytical solution for the initial value of the solution of the system of equations corresponding to the direct kinematics of the mechanism increases significantly flexibility of the simulation model. Thus, it is possible to change automatically and interactive the geometric parameters of the mechanism during the simulation.
ACKNOWLEDGMENT
This paper was financial supported by Alexander von Humboldt Foundation
REFERENCES
[1] Angeles, J., “Fundamentals of Robotic, Mechanical System”, Springer – Verlag, 1997.
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[3] Carretero, J.A., s.a., “Kinematic Analysis And Optimization of a New Three-Degree-of-Freedom Spatial Parallel Manipulator," ASME Journal of Mechanical Design, Vol. 122, No. 1, 2000, pp. 17-24.
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[5] Kecskemethy, A., “MOBILE -User’s Guide and Reference Manual”
Gerhard-Mervator-Universität-GH Duisburg. 1994.
[6] Parenti, V., Castelli, A, “Classification and kinematic modelling of fully-parrallel manipulators-a review” Parallel Kinematic Machines:
Theoretical Aspects and Industrial Requirements. 1999, pp.51-68.
[7] Pateli, A.J.,.Ehman, K.F, “Volumetric Error Analysis of a Stewart Platform Based Machine Tool” Annals of the CIRP, Vol 46,1997, pp287-290 [8] Pritschow, G., Tran, T.L., “Parallel Kinematics and PC-based Control
System for Machine Tools” In: Proceedings of 37th IEEE Conference on Decision and Control, 16./18. December 1998,Tampa/Florida, USA, pp.2605-2610.
[9] Pritshow, G.,.Wurst, K.-H, “Systematic design of hexapods and other parallel link systems” In: Annals of the CIRP Vol 46, 1997, pp.291-295.
[10] Pritshow, G.,.Wurst, K.-H, “Modular Robots for flexible assembly” In:
Proceedings of the 28th CIRP International Seminar on Manufacturing Systems "Advances in Manufacturing Technology - Focus on Assembly Systems, May 15-17, 1996.
[11] Ryu, S.-J., s.a., “Eclipse: an Over actuated Parallel Mechanism for Rapid Machining” Parallel Kinematic Machines: Theoretical Aspects and Industrial Requirements.1999, pp.441-454.
[12] Song J. s.a., “Error Modeling and Compensation for Parallel Kinematic Machines” Parallel Kinematic Machines: Theoretical Aspects and Industrial Requirements. 1999, pp.171-187.
[13] Wurst, K.-H., “LINAPOD-Machine Tools as Parallel Link Systems Based on a Modular Design” In: Proceedings of 1st European-American Forum on Parallel Kinematic Machines,1998, Mariland.
BRISAN 106
Abstract—Banyan Networks are a major class of Multistage Interconnection Networks (MINs). They have been widely used as efficient interconnection structures for parallel computer systems, as well as switching nodes for high-speed communication networks. Their performance is mainly determined by their communication throughput and their mean packet delay. In this paper we use a performance estimation model that is based on a universal performance factor, which includes the importance aspect of each of the above individual performance factors (throughput and delay) in the design process of a MIN. The model can also uniformly be applied to several representative networks. The complexity of the model requires to be investigated by time-consuming simulations. In this paper we study a typical (8X8) Baseline Banyan Switch that consists of (2X2) Switching Elements (SEs). The objective of this simulation is to determine the optimal buffer size for the MIN stages under different conditions.
Index Terms—Multistage interconnection networks, baseline networks, delta networks, crossbar switches, packet switching, performance analysis.
I. INTRODUCTION
MINs have been recently identified as an efficient interconnection network for a switching fabric of communication structures such as gigabit Ethernet switch, terabit router, and ATM switching. They are also frequently used for connecting processors in parallel computing systems.
They have received considerable interest in the development of networks. A significant advantage of MINs is their low cost, taking into account the overall performance they offer.
The important thing about an interconnection system is that it has the capacity to route many communication tasks concurrently. The situation where more than one packets claim the same communication resource is called a conflict.
When a packet finds the next buffer position already occupied, then it cannot be routed and is thus blocked. The primary purpose of buffers in a SE is to prevent loss of packets due to routing conflicts.
For the estimation of MIN performance, a number of studies and approaches have been published. There are studies assuming uniform arriving traffic on inputs like [1,2]. [3]
addresses non-Markovian processes which are approximated by Markov models. Markov chains are also used in [4] to compare MIN performance under different buffering schemes.
Hot spot traffic performance in MINs is examined by [5], while [6] deals with multicast in Clos networks as a subclass of MINs. [7] uses mathematical methods for investigating group communication in circuit switched MINs, and employs Markov chains as a modeling technique. The throughput of finite and infinite buffered MINs under uniform and non uniform traffic can also be calculated. In the literature, there are also other approaches that focus only on non uniform arriving traffic [8,9]. [10] discusses approaches that examine the case of Poisson traffic on inputs of a MIN. Rehrmann [11], makes an analysis of communication throughput of single-buffered multistage interconnection networks consisting of (2X2) switches with maximum arrivals of packets 100%, using relaxed blocking model. Furthermore, there are studies that deal with self-similar traffic on inputs.
In this paper, we assume that packets are uniformly distributed across all the destinations and each queue uses a FIFO policy for all output ports. We study the performance of a Baseline Banyan Switch with blocking SEs that operates under different conditions. At first we present and analyze a typical (8X8) Baseline Banyan Switch. Then, we explain the performance criteria and parameters of this. Finally we present the results of our simulation experiments and provide the concluding remarks.
II. ANALYSIS OF A (NXN)BANYAN SWITCH
A MIN can be defined as a network used to interconnect a group of N inputs to a group of M outputs using several stages of small size Switching Elements (SEs) followed (or leaded) by link states. It is usually defined by, among others, its topology, routing algorithm, switching strategy and flow control mechanism. A Banyan Network was defined by [12]
and is characterized by the property that there is exactly a unique path from each source (input) to each sink (output).
The path can be encoded as a sequence of labels of the