SCARA robot (R-R-P-R)

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Pick-and-place demo: the robot moves the blue cube between pad A and pad B.

Joints

Move a slider, or click in a number box and type a value with your keyboard.

Kinematic diagram (schéma cinématique), top view

Frames:

Axes: x red, y green, z blue. A circle with a dot ⊙ is an axis pointing at you, a circle with a cross ⊗ is an axis pointing away from you. The square is the sliding joint q3, seen from above.

DH table (standard)

Frame i is attached to the axis of joint i+1. Each row is one transformation: Ai = Rz(θ) · Tz(d) · Tx(a) · Rx(α). Orange numbers are the current values.

DHM table (modified DH)

Frame i is attached to the axis of joint i. Each row is: Ai = Rx(αi−1) · Tx(ai−1) · Rz(θ) · Tz(d).

Result: where is the tool?


    
The last column is the position of the tool in the base frame {0}. The 3×3 block is its
       orientation. Both tables, multiplied row by row, give exactly this same matrix.


    
Forward kinematics of a SCARA

    
x = L1·cos(q1) + L2·cos(q1+q2)
y = L1·sin(q1) + L2·sin(q1+q2)
z = 0.60 + 0.10 − (q3 + 0.30) − 0.06
tool angle = q1 + q2 − q4      (z points DOWN, so q4 turns the other way)


    
Inverse kinematics: how the demo finds the joint angles

    
Forward kinematics goes from the joint angles to the tool position. The demo needs the
       opposite: "go to the point (x, y)", so which q1 and q2? For a two-link arm
       this has a closed-form answer:

    
c2 = (x² + y² − L1² − L2²) / (2·L1·L2)
q2 = ± arccos(c2)                         (± = elbow left or elbow right)
q1 = atan2(y, x) − atan2(L2·sin q2, L1 + L2·cos q2)
q4 = q1 + q2                              (keeps the cube pointing the same way)

    
If |c2| is bigger than 1, the point is out of reach: there is no solution. The demo uses
       the "elbow left" solution (q2 positive):

    


    

      
Where is a SCARA used?

      
Electronics factories (placing and inserting components on circuit boards), screwing
         and small-parts assembly, packaging, pick-and-place of light objects, and dispensing
         glue or sealant. It is fast (a pick-and-place in under a second) and very stiff
         vertically, but soft sideways, which helps to insert a pin into a hole.

      
End effector (tool)

      
Yes, a pinch gripper (pince) works: a two-finger parallel gripper, like
         the one drawn in the 3D model. Other tools:

      
      
The tool always points straight down and can only rotate around the vertical axis (q4).
         So the object must be picked from above.