In order to better understand the motion of a serial manipulator arm, an example is given below. A 6 DOF arm is shown, and contains three prismatic and three revolute joints. For this arm, the three revolute joints compose a spherical wrist. The forward and inverse kinematics of the arm will be determined and a demonstration of the serial arm's motion will be presented.
The general procedure for deriving the forward kinematics of a serial robot manipulator is given in the following steps:
1. Label all joints i = 1 to n.
Figure 1: Step 1
2. Assign z-axes for joints 0 to n-1 (zo along joint 1, etc.).
Figure 2: Step 2
3. Assign xo normal to zo.
4. Assign x1 through xn-1, which lie at the common normals between zo to zn-1.
Figure 3: Steps 3,4
5. Establish y1 to yn-1 to complete each frame.
6. Assign zn freely (but carefully) and define xn.
Figure 4: Steps 5,6
7. Create the table of DH link parameters, defining $\alpha$i, ai, di, and $\theta$i for each joint.
Link i
$\alpha$i
ai
di
$\Theta$i
Tii-1
1
90
0
d1*
180
T10
2
90
0
d2*
-90
T21
3
90
0
d3*
90
T32
4
90
0
d4
$\Theta$4*
T43
5
90
0
0
$\Theta$5*
T54
6
0
0
d6
$\Theta$6*
T65
(* denotes variable joint parameters)
8. Create Tii-1 for i = 1 to n.
9. Solve Tno = T1o * T21 * … * Tnn-1.
2) Solve the forward kinematics for T3o and R63. Normally R63 is a spherical wrist, so it controls the orientation of the end effector. T3o will work on satisfying the position requirement.
3) Find the location of the wrist center, pc. The vector (x,y,z) would be the end effector location. We need to back it up to the wrist center, (xp , yp , zp).
5) From the forward kinematics, use R3o and R6o. Determine R63 (given) = (R3o)T * R6o.
Remember that R6o = R3o * R63. For this case, it is a spherical wrist.
This video shows how the robot moves through some simple lines in the x, y, and z direction. Next, it shows the changing orientation of the spherical wrist. Last, it moves through a more complex shape by solving the inverse kinematics as shown above.