6-axis articulated robot (R-R-R-R-R-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. With q = 0 for all joints, the arm points straight up.

Kinematic diagram (schéma cinématique)

Frames:

Each joint is drawn with its symbol: a circle when its axis points towards you or away from you, and a thick bar along its axis otherwise. The orange line goes from the base to the tool. Axes: x red, y green, z blue. ⊙ points at you, ⊗ points away. The dashed squares are the pads A and B, and the blue square is the cube.

DH table (standard)

Frame i is attached to the axis of joint i+1. Each row is: Ai = Rz(θ) · Tz(d) · Tx(a) · Rx(α). Orange numbers are the current values. The "+ 90" in rows 2 and 3 is an offset that puts the arm straight up when q = 0.

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 centre point (TCP, between the fingertips) in the base frame {0}.
       The 3×3 block says where the tool points: its third column is the direction of z6, the direction the
       gripper is pointing in. Both tables give exactly this same matrix.


    
The spherical wrist

    
Look at the table: rows 4, 5 and 6 have a = 0. So the axes of joints 4, 5 and 6 meet at one point:
       the wrist centre. Joints 1, 2 and 3 place that point (the position), and joints 4, 5 and 6 turn the tool around
       it (the orientation). This is why almost every industrial robot has this shape.

    

      
Try it: a singularity

      
Set q5 = 0. Now the axes of joints 4 and 6 are on the same line (choose
         Show: Frame {4} only and compare z3 and z5). Move q4, then move q6: the tool
         turns the same way. The robot has lost one degree of freedom. This is a wrist singularity, and
         robot programmers avoid it.

    


    
Inverse kinematics: how the demo finds the joint angles

    
The demo says "put the tool at this point, pointing down" and the page finds the 6 joint angles.
       For 6 joints there is no simple formula to type in, so it uses a numerical method:

    
1. compute where the tool is now (forward kinematics)
2. compute the error between the target and the tool
3. use the Jacobian to find the small joint changes that reduce the error
4. repeat until the error is below 0.01 mm

    
The tool must point straight down, with the fingers along x. These are the angles it found:

    


    

      
Where is a 6-axis articulated robot used?

      
Car factories (spot welding, arc welding, painting, sealing), machine tending
         (loading and unloading CNC machines), palletizing, assembly,
         material handling, foundries and forging, and laboratories. Payloads go from a few kg to more than
         1000 kg, with a repeatability of 0.02 to 0.1 mm. Well-known makers: FANUC, KUKA, ABB, Yaskawa, Stäubli.

      
End effector (tool)

      
Yes, a pinch gripper (pince) works, like the two-finger one in the 3D model. Thanks to the 6 axes
         it can pick an object from any direction, not only from above. Other tools: