KINEMATICS DELTA ROBOT

Inverse kinematic

For a given position of the end effector, it is necessary to determine the corresponding angles of each of three arms in order to set the motors in proper position for grasping. This is known as inverse kinematics problem

The Figure 2-7 shows the kinematic scheme of delta robot which shows the key parameters of the robot that are:

 The side of the fixed triangle (f)

 The side of the end effector triangle ( e)

 The length of the upper joint (rf)

 the length of the parallelogram joint (re)

The reference frame will be chosen with the origin at the fixed triangle’s center of symmetry, as shown in Figure 2-7, so that the z-coordinate of the end effector will always be negative.

Because of the robot's design, joint 𝐹1𝐽1̅̅̅̅̅ can only rotate in the YZ plane, forming a circle centered in point 𝐹1 with radius 𝑟𝑓. Unlike 𝐹1, 𝐽1 and 𝐸1 are called universal joints, which means that 𝐸1𝐽1̅̅̅̅̅ can rotate freely relative to 𝐸1, forming a sphere centered in point 𝐸1 with radius 𝑟𝑒.

The intersection of sphere centered in point 𝐸1 and YZ plane is a circle centered in point 𝐸1′ with radius 𝐸1′𝐽1̅̅̅̅̅̅, where 𝐸1′ is the projection of 𝐸1 on YZ.

As shown in Figure 2-8, the two circles centered in 𝐹1 and 𝐸1′ respectively intersect in two points so the point 𝐽1 should be chosen as the intersection point with smaller y-coordinate. And if 𝐽1 is known, then theta1 can be obtained.