Dual Arm Robot Research Report

Transcription

1 Dual Arm Robot Research Report Analytical Inverse Kinematics Solution for Moularize Dual-Arm Robot With offset at shouler an wrist Motivation an Abstract Generally, an inustrial manipulator such as PUMA is equippe with egrees of freeom(dof, it is just enough to reach a position an orientation in -D space. However, to achieve exterous movement like the upper limb of human, ifferent with normal inustrial manipulators, such manipulators equippe with -DoF kinematic structure is esirable. With the reunant joint, such manipulators may accomplish tasks such as obstacle-avoiance an singularity avoiance while reaching target position at the same time. hese tasks are all about manipulator kinematics. However, when one wants to utilize avantages of reunant manipulators, one encounters problem of solving inverse kinematic problem of such manipulators. here is two ways to solve manipulator kinematic problem, one is numerical metho an the other is analytical metho. raitionally, it use numerical metho to eal with kinematics problem is a traitional way. However, solving Jacobian matrix is rather teious, not to mention solving its inverse. Also, the relation between joint space an Cartesian space of an arm is not linear, obtaining joint values by numerical metho is not satisfactory. Since that, we evelop an analytical inverse kinematic solution for moularize -DoF reunant manipulators with offsets at shouler an wrist an erive analytical inverse kinematic solutions base on ifferent joints as reunnt for it. he manipulator shown in Figure is one of the ual arm robot evelope by our laboratory, which is a moularize -DoF manipulator with offsets at shouler an wrist. Fig. he physical harware assembly of the moularize -DoF manipulator with offsets at shouler an wrist Fig. he moularize component

2 Figure : he conceptual structure of the arm Figure : he coorinate system of the arm with Main echnology offset at shouler an wrist. he en-effector is not inclue in this figure. Moularize component Parts of the propose arm where motors, controller, reuction gear, an others installe are moularize component, as shown in Figure. In aition, links can be attache to this part at Link A-sie an Link B-sie of this moularize component. Also, there is a winow for irect access to each controller uring maintenance. Generally, the moularize esign grants it avantages in easy assembly.. Analytical inverse kinematic equation As shown in Figure, since there are offsets in shouler an wrist, we cannot apply fixe-arm-angle metho. Hence, we have to use fixe-joint metho. Firstly, we make erivation with joint value regare as reunancy parameter. hen, solution with value of joint seen as reunancy parameter will be solve afterwars. In the following sections, we enote as a value A of a referre to frame b, while a specifie value c is. Also, as shown in Figure an Figure, the origin of frame an are at, but they have ifferent orientation. he origin of frame is at, while the origin of frame is at. he origin of frame an are at with ifferent orientation. Lastly, the origin of frame an are at an respectively. able. DH parameters Figure : he projection triangle use to solve

3 Figure : he triangle spanne by,,an Figure : he trajectory of virtual offset-wrist forms a circle when all joints are fixe except joint A. CLOSED-FORM KINEMAIC EQUAIONS WIH JOIN AS REDUNDAN JOIN With the knowlege of an, we obtain wrist position, : where is the vector from en-effector position to the wrist position. Next, because is known, we are able to solve the position of the shouler, : then also with the knowlege of, we can transform both an into an : P P R R P P ( Next, fin, i.e. project to. As Figure shows, then we can fin with the ai of projection triangle an : atan( (, atan Y, z X, z ( where is an angle of triangle. inicates the istance from to. So far we obtain an, now we can solve an by the following equation: R R R R R cos cos cos sin sin atan( R atan( R,,, R, R, R,, (8 (9 ( where is the value in row, column of matrix. In this step, since we alreay know,,, an,

4 Figure 8: Projection of the reunancy circle on plane we know an : Poffset w P,,, ( P offset ( Next, with the help of projection triangle on the plane., which is spanne by,, an,, we can fin an : cos atan( offsets cos offsets offsets,cos ( offset s offset offset s atan( Y, X / s offsets where is an is as in able, can be obtaine by simple Eucliean norm calculation. Now only remains unknown. It can be foun with the information of all the other joint value an : ( R R R R R ( cos / R sin / (,, atan ( R, R / ( Finally, we fin all the joint value by the aforementione equations using fixe-joint metho in joint is regare as reunant parameter.

5 B. CLOSED-FORM KINEMAIC EQUAIONS WIH JOIN AS REDUNDAN JOIN Since joint an joint can exchange with each other in assembly, that inicates that joint an joint have similar influence on en-effector position an orientation. hus, to make the analytical equation for this type of arm more complete, we also fin equations when joint value is reunancy parameter. When we know joint value, can be easily foun, then we can fin joint value as ( an( i. However, with the circle rawn by trajectory of when all the joints are fixe except for joint, we are able to fin base on known, then applying from (8 to ( so as to fin all values of all other joints. As shown in Figure, is on the circle whose center is. Base on our coorinate system construction shown in Figure, we consier the offset from to between frame an. However, we can efine a virtual point where is the same as, an. After efining virtual offset-wrist, we can fin the istance between an, : Z Z ( Z sin / cos w,, (8 offsetw ' O c ' offsetw Z w, where is the raius of the circle an is the istance from virtual offset-wrist to base. Next, (9 we are able to fin value of joint with the projection shown in Figure 8: O c Yw, Z sin X w, Z cos ( an can be represente by a quaratic from equation where is the parameter. then by rearranging ( into a quaratic equation, such that we can fin two solutions of both of the caniates by equations from ( to ( to fin out which one is correct.. We then check Figure 9. Extreme posture of manipulator for the acceptable joint value. In (a, is reunant, where the tip position an orientation are specifie by ( an (. is reunant an in (b,

6 Experiment Result o verify the correctness of the propose equations, simulations are implemente as follows. he verification is implemente by applying the kinematic equations on the custom-mae manipulator with offsets at shouler an wrist. he arm can be seen at Figure. In aition, the coorinate construction is the same as shown in Figure. hough all the joints can rotate freely in most of the time, after concerning the environment, we shoul assume that there are limitations for every joints. he limitations are shown in able I. Assume the esire position of the en-effector is: P... ( R ( we check both sets of equations where is treate as reunancy parameter an is treate as reunancy parameter. In figure 9(a, while is maximum, the joint value set is as the following: In contrast, while is minimum, the joint value set is as the following: As shown in figure 9(b, while is maximum, the joint value set is as the following: n contrast, while is minimum, the joint value set is as the following: We can check the joint value solve by propose analytical inverse kinematics equations by performing forwar kinematic, then we confirm that the propose equations are correct. Future application

7 By eriving analytical inverse kinematics solution for the moularize -DoF reunant manipulator, we may solve the kinematics faster an more precisely. Also, with the newly-esigne moularize component for all joint part where motors installe, we may assemble the arm in a simpler way.