Singularity Management Of 2DOF Planar Manipulator Using Coupled Kinematics

Transcription

1 Singularity Management Of DOF lanar Manipulator Using oupled Kinematics Theingi, huan Li, I-Ming hen, Jorge ngeles* School of Mechanical & roduction Engineering Nanyang Technological University, Singapore 9798 *Department of Mechanical Engineering and enter for Intelligent Machines McGill University, anada bstract: This paper describes a new DOF planar manipulator (M) using coupled kinematics to manage the singularities. It is designed in order to overcome the singularity configuration of planar manipulators. Manipulators are composed of five rigid links, five revolute joints and two actuators each actuating two of input links coordinated fashion, what we term coupled kinematics. oupled kinematics is an actuating arrangement. It may be defined as a pair of aial rotations, which are distributed by an actuator at varieties of parameters, which depend transmission ratios of mechanism. Using the geometrical approach, the position of end-effector is obtained. Matching the velocity of endeffector from two branches, the input and output velocities relation can be derived. onsidering the coupled kinematics effects, Jacobian matri is determined. We show that, by mean of proper tuning of the parameters of coupled kinematics, the singularity manifold can be substantially simplified. Finally, note that kinematic coupling can be implemented using mechanical or electronic hardware.. INTRODUTION arallel manipulators have several benefits because of their structures. However, the eisting singular conditions during operation are potentially dangerous and frequently cause the manipulator inoperable in certain configurations. These are dangerous and undesirable postures. In this paper we introduce the singularity management of DOF planar manipulator using coupled kinematics. DOF planar manipulators can be found in many industrial applications as positing devices, which are improved the positional resolution, stiffness and force control of the manipulator Many researchers have already introduced the singularity analysis of parallel manipulator. Gosselin and ngeles [] introduced the singularity analysis of closed-loop kinematics chains. Tsai [] further reported the eistence of three kinds of singularities - inverse, direct and combined singularities. revious researchers [-8] therefore have tried to overcome the singular configurations within the workspace. ervantes-sanchez et al [8] presented on the kinematic design of the R planar. They showed that the output position could be generalized by using a common set of two quadratic equations. lici [9] also proposed the singularity contour for a class of five-bar planer parallel manipulators. Merlet [] proposed a new method based on Grassmann line geometry. major concern of using the coupled kinematics is that the input angular velocity passes through this mechanism, which can transfer the angular velocity to two joints of input link by several transmission ratios. The following steps can analyze the singularity of a DOF planar manipulator with coupled kinematics (i) formulating the output position in terms of two input rotations by solving two quadratic equations analytically; (ii) using two Jacobian matrices, one incorporated the parameters of coupled kinematics, to relate input velocities to the output one; (iii) drawing the singularity manifolds by finding the zero determinant values of Jacobian matrices. This approach allows us to alter the singularity configurations within the workspace and to control the required path of non-singular configurations by varying parameters of coupled kinematics. We can describe the relation between the actuated joint variables q and the location of the moving platform as f(, q) = () Differentiating equation () with respect to time leads to the relationship between the input and output velocities as follows: J = J q () q where J and J are two Jacobian matrices and ẋ and q q are output and input velocities, respectively. In planar manipulators, direct kinematics singularity occurs when det [ J ] is equal to zero and inverse kinematics singularity occurs when det [J q ] is equal to zero. The combined singularity can only occur when, both J and J become simultaneously singular. q. STRUTURE OF DOF LNR MNIULTORS DOF planar manipulator is shown in Figure where θ and θ are driving angles for input, β and γ are passive joint angles. The length of each link is also shown in the Fig. The output position of can be found from the geometry of manipulators

2 l m ( β ) m γ O θ D θ d Figure : Two-DOF planar Manipulator The coordinates of joint and can be denoted as follows, ( (,, ) = ( l cosθ, l sin θ ) = ( d + l cosθ, ), l sin θ ) l () To determine position of joint (end-effector), we approach the geometric epressions from two branches O-- and D--. ( ) + ( ) = m () ( ) + ( ) = m () From Eq. () and (), the artesian coordinates of the point can be found as two solutions because of quadratic equation nature. On the hand they are symmetric about line [8]. ± = a = N + K where b b ac M L K = L = + ( ) M = + N. OULED KINEMTIS = oupled kinematics can be briefly defined as a mechanism, which can distribute the rotations to two coupled joints outputs from one actuating system. The rotations are transmitted through several parameters, which depend on their transmission ratios of coupled kinematics structure. Thus the output angular velocities of coupled kinematics can be adjusted by controlling the parameters. lanetary gear trains and differential gear drives are simple eamples of coupled kinematics mechanism. differential drive mechanism is here considered as the coupled kinematics to employ to DOF planar manipulator. In this case study, we use the principle of differential mechanism that is it allows two outputs from one input rotation. In our application, we employ two sets of differential mechanisms to actuator link O and D, respectively. Denoting the inputs of two differential drives as ω andω, the relationship of input and output velocities can be written as follow: + () θ β = ω θ + γ = ω (7) Where the,, and are parameters of differential gear drive and they depend on the gear ratios we used. The velocity of end-effector Ṗ can be determined from two different branches, O-- and O-D--. = θ E+ β E(-) (8) = θ E(-d)+ γ E(-) (9) E denotes the orthogonal matri-rotating vector. E, and can be epressed as follows - E =, = and = d From Eq. (8) and (9), we can epress the passive joint rotational rates β and γ in terms of input rotations ω and ω and actuated joint rates θ and θ. Substituting these resulting epressions into Eq. () and (), we obtain ω = θ E+ E(-) θ E(-) () ω = θ E(-d)+ E(-) θ E(-) () Therefore we can have the direct relation between input and output rotations. J=J θ () where Q = Q Q Q J and J W = W

3 Q = {(+ ) } Q = {( + ) + } Q = ( + )( d) Q = ( + ) W = [ ( ){ ( )} + ( ){ ( )} ] W = [ ( ){( d) ( + ( ){ ( )].EVLUTION OF SINGULRITIES )} When Eq. () is satisfied, right side (D) is aligned as like the left branch. oth of sides can occur these conditions at the same time. We show singularity manifolds in the joint space to clearly see the advantages of using coupled kinematics. These singularity manifolds are constructed by using Mathematica. First we draw j det J ( θ, θ ) surface, = θ θ π. Then we look at the j = plane and the surface. Their corresponding singularity manifolds are shown in Fig. () and () for l =., m=. and d =. Eq. () clearly shows that there are three possible singularities in planar manipulators. Three types of singularities will be discussed in the followings. O θ θ D Inverse Kinematics Singularity It occurs when det [ J ] is singular. This corresponds to [ ( ){ ( )} + ( ){ ( )} ] = or [ ( ){( d) ( )} + ( ){ ( )] = () () y specifying the dimensions of DOF planar system, the determinant of J can be evaluated for various parameters of differential mechanisms. The singularities can be found when either Eq. () or () is satisfied. When J is singular and the null space of J is not empty, there eist some non-zero θ vectors that result in zero Ṗ vector. Thus, the planar manipulator loses one or more degree of freedom. In this condition, the manipulator resists forces or moment in some directions with zero actuator forces or torque. Figure : Inverse kinematics singularity configurations of DOF M (with and without differential drive mechanism) O γ β θ θ D Figure : Inverse kinematics singularity configurations of DOF M (with and without differential drive mechanism) Fig. () and () schematically show two singular configurations on the outer and inner boundaries of DOF planar manipulator. hysically, when Eq. () is satisfied, the left side links (O) is aligned and the corresponding configuration is one in which the manipulator reaches either an eternal boundary of its workspace or an internal boundary.

4 posses infinitesimal motion in same directions while all the actuators is completely locked. Thus, the end-effector gains one or more degree of freedom. On the other hand, direct kinematic singular configuration, the manipulator cannot resist force or moment in some direction. θ (rad) β γ θ (rad) Figure : Singularity manifolds of DOF M with and without differential drives θ (rad) θ (rad) Figure : Singularity manifolds of DOF M with and without differential drive mechanism Fig. () and () satisfied the Eq. () and (). In inverse kinematics singularity, we meet the same singularity manifolds between simple DOF planar manipulator and that equipped with differential drive mechanism. Therefore we can say that inverse kinematics singularity could not be managed using coupled kinematics. Direct Kinematics Singularity When determinant value of J is zero, we have the direct singularity. This condition can be found by requiring that σ σ σ σ {( + ) }( + ) σ σ σ σ + + = {( ) }( b ) () When J is singular and the null space of J is not empty, there also eist some non-zero Ṗ vectors that result in zero θ vector. Therefore, the end-effector can O θ D θ Figure : Singularity configuration of DOF M with and without differential drives mechanism This kind of singularities only happens within the workspace. However, it should be shifted within the workspace by introducing the differential drive mechanisms. One eample case is shown in Fig. () where a singular configuration eists when joint and coincide. This configuration is no longer singular if the differential drive mechanisms are used. Eamples The simplification of the evaluation of direct kinematics singularity will be realized by the following eamples. The singularity manifolds of DOF planar manipulator (for l =., m=. and d =. ) with differential drive mechanisms are shown below. In these cases, we take the parameters as = and =. θ (rad) θ (rad) Figure 7: Singularity manifold of DOF M with differential drive mechanism ( =., =.7 )

5 θ (rad) θ (rad) θ (rad) Figure 8: Singularity manifold of DOF M with differential drive mechanism ( =.8, =. ) θ (rad) Figure : Singularity manifold of DOF M with differential drive mechanism =.7, =. θ (rad) θ (rad) θ (rad) Figure 9: Singularity manifold of DOF M with differential drive mechanism ( =., =.9 ) θ (rad) Figure : Singularity manifold of DOF M with differential drive mechanism ( =.9, =.) To compare the singularity manifold of DOF planar manipulator without differential drive mechanism, we show in Fig. (). θ (rad) θ (rad) Figure : Singularity manifold of DOF M with differential drive mechanism ( =., =. ) Fig. (7)-() show the singularity manifolds of DOF planar manipulator with differential drive mechanism by changing of various parameters of differential drive mechanisms. mong these singularity manifolds, we can see that direct singularity configurations can be managed within the workspace. θ (rad) θ (rad) Figure : Singularity manifold of DOF M without differential drive mechanism

6 In this case, we overcome these problems by choosing the suitable differential drive mechanism parameters..singulrit NLSIS OF WORK SE The workspace of the DOF planar manipulator includes all reachable positions of the end-effector using all available input motions. In order to obtain the workspace, the following procedure is proposed. The workspace of the DOF planar manipulator is generated by mean of two families of curves derived from the limit of the operation point in the artesian space. Therefore, we need to investigate both situations. In order to physically identify the singularity conditions of the DOF manipulator with differential drive mechanisms, three equations such as Eq. (), Eq. () or Eq. () will be satisfied..onlusions This paper is investigated for the DOF planar manipulator with coupled kinematics. In present work, simple DOF planar manipulator meets a lot of singular conditions and self-locking within its limited workspace. oupled kinematics could adjust the input joint velocities controlling the sets of parameters. Thus some of direct singular conditions could be altered but could not etend the workspace because of using same link dimensions. There are still inverse singular conditions in the outer and inner boundaries of workspace. y turning the suitable parameters of coupled kinematics, we can shift singularity conditions and avoid self-locking. The result shows that the DOF planar manipulator with coupled kinematics can significantly manage the singular configurations within the workspace in comparison with the simple DOF planar manipulator. cknowledgement Thanks are due to rof. J. ngeles of McGill University for his invaluable suggestion in this work. Figure : Singularity curves in workspace of DOF M without differential mechanism Figure : Singularity curves in workspace of DOF M with differential drive mechanism Fig. () and () show the singularity curves, which represented the singular conditions of the DOF planar parallel-kinematics machine with and with differential drive mechanism within the workspace. We can see the significant advantage of using differential drive mechanism in the DOF planar manipulator. References.. Gosselin, J. ngeles, Singularity nalysis of losed-loop Kinematic hains, IEEE Transactions Robotic and utomation, vol., No., 99, pp L. W. Tsai, Robotic nalysis New ork: Wiley, H. R. Mohammadi, Daniali,. J. Zsombor-Murray and J. ngeles, Singularity nalysis of General lass of lanar arallel Manipulators, International onference on Robotic and utomation, 99, pp 7-.. N. Simaan and M. Shoham, Singularity nalysis of omposite Serial in arallel Robots, IEEE Transactions Robotic and utomation, vol. 7, No.,, pp -.. G. ang, I-M hen, W. K. Lim, S. H. eo, Design and Kinematic nalysis of Modular Reconfigurable arallel Robots, International onference on Robotic and utomation, 999, pp -.. G. ang, I-M. hen, Singularity nalysis of Three Legged Si-DOF latform Manipulators with RRRS Legs, International onference on dvance Intelligent Mechatronics,, pp H. hung and J. W. Lee, Design of a New DOF arallel Mechanism, International onference on dvance Intelligent Mechatronics,, pp J. Jesus ervantes-sanchez, J.. Hernandez- Rodriguez and J. ngeles, On the Kinematic Design of the R lanar, Symmetric Manipulator, Mechanism and Machine Theory, vol.,, pp-. 9. G. lici, Determination of Singularity ontours for Five-ar lanar arallel Manipulator, Robotica, vol.8,, pp J.. Merlet, Singularity onfigurations of arallel Manipulators and Grassman Geometry, International Journal of Robotics Research, vol. 8. No., 989, pp -.