Delta Robot (3D)

The robot

A delta robot (Clavel, 1985) hangs three light carbon upper arms from motors fixed to the base. Each upper arm ends in a pair of parallel rods (a parallelogram) joined to a small effector plate. The parallelograms keep the plate horizontal, so the gripper only translates, in x, y and z. Because the heavy motors do not move, the moving mass is tiny and industrial delta robots reach hundreds of pick-and-place cycles per minute.

Inverse kinematics (closed form)

For arm i the shaft is at B_i and the elbow at E_i = B_i + L (cos th u_i - sin th z). The effector joint is A_i = P + R_e u_i. The forearm has fixed length l: |A_i - E_i| = l. Writing C = A_i - B_i in the arm frame gives one equation per arm,

-2 L c_u cos th + 2 L c_z sin th = l^2 - L^2 - |C|^2

of the form a cos th + b sin th = c, with two solutions; the page takes the one with the elbow up and outward. If |c| > sqrt(a^2 + b^2) the point is out of reach.

Forward kinematics

Given three arm angles, the three elbow points shifted by -R_e u_i are the centres of three spheres of radius l; the effector is the lower intersection (trilateration).

Jacobian, singularities and torque

Differentiating the constraint gives N dP = D dth with rows n_i = (A_i - E_i)/l and D = diag(n_i . de_i/dth_i). Two kinds of singularity appear: when N loses rank (the three forearms are nearly coplanar and the effector loses stiffness), and when some n_i . de_i/dth_i = 0 (an arm is stretched or folded so its motor cannot move the effector). The page shows sigma_min(N)/sigma_max(N) and the arm-angle factor, and blocks poses below an index of 0.02. By virtual work the motor torques are tau = D N^-T F + I_a th'' - G_a cos th with F = M_e (a + g z): the force the actuators apply to the effector, plus arm inertia and arm weight.

Trajectory and cycle time

Each half-cycle is a rest-to-rest arch. A minimum-jerk time law u = 10 t^3 - 15 t^4 + 6 t^5 (t in units of the move time) drives a horizontal move combined with a lift of the chosen height. The planner bisects the move time until the peak motor torque and speed along the sampled path touch the limits you set; the cycle is two half-moves plus two gripper dwells. Heavier payload, weaker motors or longer reach all reduce cycles per minute, and a chart shows the whole curve against motor torque.

Numerical checks on this page

(1) Inverse then forward kinematics on random poses must return the same point (two different algorithms). (2) The forearm length stays l for every arm and the analytic Jacobian matches a numerical derivative of the inverse kinematics. (3) For a rest-to-rest move the work done by the motors, computed from the torque formula, equals the potential-energy change. (4) The sampled peak acceleration of a minimum-jerk move equals the closed form 10 D / (sqrt3 T^2).

Honest simplifications

Masses are lumped (half of each forearm at the elbow, half at the effector), the parallelogram is treated as one rigid link, and there is no friction, gearbox compliance, motor thermal limit, controller delay or vibration. Real cycle rates also depend on the gripper and part handling: the numbers here are a kinematic and torque limit, not a catalogue figure for any commercial robot.