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By J. M. Selig
One of many major reasons of this ebook is to offer the kinematics of robots in an easy and transparent demeanour. This contains utilizing one co-ordinate body after which utilizing energetic instead of passive ameliorations to explain the positions of inflexible our bodies. The textual content comprises routines.
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Extra resources for Introductory Robotics
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2 Using Trigonometry to Derive the Kinematic Equations The quantities k, and k, are given by:- k, = ky = 11 11 cos61 sin 61 + + + - + + + + 8 2 ) - (Il + 12) sin(61 + 6 2 + 12 cos(61 12 sin(61 62) (11 12) C O S ( ~ ~ 62 63) 63) This matrix K ( d l , 62,63) = A l(61)A 2(62)A 3(63) encapsulates everything there is to know about the kinematics of the manipulator. So we will refer to it as the kinematic transformation matrix of the manipulator. The angle O is easily seen to be = 61 + 6 2 + 03. If we want to find the position of any point attached to the end-effector, we simply multiply its position vector in the home position by the matrix K .
This is one of the central problems in robotics, since whenever we specify the motion of the robot's gripper we need to know the corresponding joint motions. Essentially we must solve the following matrix equation:- where K is the constant matrix which specifies the position and orientation of the gripper. This constitutes a set of highly non-linear equations for the joint angles e l , 8 2 , . . , 96. In general, very little is known about solving such equations; even the number of solutions is problematic.
So we will refer to it as the kinematic transformation matrix of the manipulator. The angle O is easily seen to be = 61 + 6 2 + 03. If we want to find the position of any point attached to the end-effector, we simply multiply its position vector in the home position by the matrix K . For example, the point (Il l2 13, 0) after a general motion will have x - y co-ordinates given by:- + + This yields the so-called kinematic equations of the manipulator: z = ll cos 81 y = l1 sin 61 + + 82) + + 12 sin(& + 6 2 ) + 12 C O S ( ~ ~ + 82 + sin(& + + 13 C O S ( ~ I 13 62 63) 63) It is a simple matter of trigonometry to check that these equations are correct, see fig.