Kinematics of a delta-2 robot
A delta-2 robot is a parallel robot composed by two legs, each one has three rational joints but only the one attached to the fixed-frame (or top-plate) is not a passive joint. Therefore, to move the end-effector position (TCP-0) of the robot the two active joints must be controlled.
The delta-2 robot can be seen as a simplification of the delta-3 robot, and it is usually used in the packaging industry for pick products on a conveyor belt. You can see some models here.
Due to its mechanical configuration this robot can only move its end-effector on plane XZ, see figure belowt. And its TCP-0 is defined by (x,0,z).
The kinematics of the delta-2 robot can be solved using a geometric method. For doing this, we model the robot (see figure abowe) using four kinematics parameters: rf,lf, le, re. The parameter rf is the distance between the center of the fixed-frame to the position of the active joint, re is the distance between the center of the end-effector (E) and the position of the passive joint (F), and lf and le are the lengths of the links of a leg. And the link 1 and link 2 of a robot leg are connected in point G.
The algorithms for solving the Kinematics Problem of the delta-2 robot have been developed using Scilab.
Inverse kinematics
For knowing the joins (j1,j2) from the end-effector position (TCP-0) we must solve the inverse kinematics problem.
(le) as radius. And the second circle (red) has point F as center and the length link 1 (lf) as radius. Once you know position of G, the value of the joint is calculated as the angle defined by the axis X and the line that connect F with G, that’s the real position of the link 1.
This method is applied independently for each leg of the robot.
Direct kinematics
And we can calculated the end-effector position (TCP-0) from the joint values (j1,j2) solving the kinematics problem.
The end-effector position (E) is obtained from the intersection of the two circles (cyan) defined by the link 2 of each robot leg. The center of each circles is defined by G and the radius by the length of the link (le). And the point G is calculated by trigonometry using the value of the joint, the length of the link 1 (lf) and the position of F.