Basic HTML Version

|  | Table of Contents |  | View Full Version

Page 65 - Toucpad robotics C11
P. 65

    Think of it as a simplified version of a human arm, with a shoulder and an elbow.
                                                     y

                                                                           final position

                                                                      motor 2

                                              motor 1                            initial condition

                                                   0         Link 1           Link 2               x
                        Description:
                     u
                            It has a base, two links of lengths L   and L  , and two revolute joints.
                           l                              1     2
                            The first joint connects the base to the first link (L  ), and the second joint connects the first link to the second
                           l                                            1
                            link (L  ).
                                 2

                            The joint variables are the angles of rotation,   and   , relative to their respective links.
                           l                                        1     2
                            The goal of forward kinematics is to find the (x,y) coordinates of the end-effector.
                           l
                     u   Mathematical Model: We can use basic trigonometry to determine the end-effector’s position.
                            The first link’s end is at (x  ,y  ). Using the angle   , its coordinates are: x  =L cos(  ) y  =L  sin(  )
                           l                      1  1               1                  1  1    1  1  1  1
                            The second link’s end (the end-effector) is at (x  ,y  ). Its position is relative to the first link’s end. The angle of the
                           l                                         2  2
                            second link relative to the base is (  +  ). Therefore, its coordinates are: x  =x  +L  cos(  +  )=L  cos(  )+L  cos(  +  ) y
                                                             2
                                                                                                                  2
                                                                                                               1
                                                                                                                       1
                                                                                                                            2
                                                           1
                                                                                                                         2
                                                                                                       2
                                                                                             1
                                                                                           2
                                                                                                          1
                                                                                                     1
                                                                                                2
                            =y  +L  sin(  +  )=L  sin(  )+L  sin(  +  )
                                                      1
                                                        2
                                                 2
                                     1
                                       2
                              1
                                2
                                          1
                                              1
                        Example:
                     u
                            Imagine a 2R manipulator with link lengths L  =10 cm and L  =5 cm.
                           l                                      1           2
                            If the joint angles are   =30° and   =60°.
                           l                   1         2
                            We can calculate the end-effector’s position: x
                           l                                        2
                            =10cos(30°)+5cos(30°+60°)=10(0.866)+5(0)=8.66 cm y 2
                            =10sin(30°)+5sin(30°+60°)=10(0.5)+5(1)=5+5=10 cm
                            The end-effector’s position is (8.66,10) cm.
                           l
                     Configuration 2: The RP Planar Manipulator
                     The “RP” configuration refers to a planar manipulator with one revolute joint followed by one prismatic joint.
                                                   z 2
                                                                          r i (t)
                                                           (t)
                                                           i
                                                               q (t)
                                                                i
                                                                                             z 1

                                                                                                                           63
                                                                                                              Mechanical System

| 60 61 62 63 64 65 66 67 68 69 70 |