Question

Figure 1 shows a SCARA (Selective Compliance Assembly Robot Arm)- type robot. 1. According to preliminary rules, how many frames should we at least assign? Draw the z axis for each frame and then x axis and y axis accordingly based on the following Denavit-Hartenberg (DH) frame rules. (1) The z axis must be the axis of revolution or the direction of motion (2) The x axis must be perpendicular to the z axis of the frame before it (3) The x axis must intersect the z axis of the frame before it (4) The y axis must be drawn so the whole frame follows the right-hand rule d4 s d2 ZU ds ZH YU Figure 1: A SCARA-type Robot 2. Fill out the following DH parameter table (If the space is not enough, you can extend the table.) d a No. O-1 I-2 2-3 3-4 4-5 axis; d is the offset along where 0 is the angle about previous z axis, from old x axis to new x previous z axis to the common normal; a is the length of the common normal; a is the angle about common normal, from old z axis to new z axis. 3. The transformation A, (i- 1, 2,..) between two successive frames represents the preceding four movements, i.e. the product of four matrices. The resultant matrix is shown as follows. [C-SCa Sa a,C -CSa a S Cat d A,SGCa Sa 0 1 0 0 0 where C is cos0, S is sine, Cais cosa, and Sa, is sina, respectively. Write down all the A, (i 1, 2, ..) matrices and finally derive the whole matrix T from the universal frame Fo to the end effector frame FH by multiplying all A, (i 1, 2,..) matrices together. (You can handle Fu to Fo like you did in your project and ignore di+d2.)

Answer 1

We need foamey to fsmul ate DH forometers a ard one Prismatte got Ser SCARA fo hos 3 havihc So Gn degree of fneaelom = j angle Lctwaen om M- to maoywed abou 2 di = t dgtance betwcan di toYemeame adong dstan CO nple Setuean Ze to Z abot Hi .

Homogenous Matrix (T)=Rotation About X-axis(\alpha)* Translation along X-axis(a)* translation along Z-axis(d)* Rotation about Z-axis(\theta)

T=T1*T2*T3*T4*T5

CO,d, a,C, d0, (oprp) -»X? o,1-Same about x-d-drstanca bejuean Z1 0 Z so dc di ( z bap Z4 C d,a, C0,0) otation o & = 0 C4Ca,00) DH-Porameteny 2 ds d2 2-3 dy 3-4 ds -5 Homodomous Trans Jomation 6tlow mathemetrca code for Ra Ta TRz H T

DH2HomTrans[\[Theta]_, d_, a_, \[Alpha]_] :=
Module[{Rz, Td, Ta, Rx, Ti},
Rz = {{Cos[\[Theta]], -Sin[\[Theta]], 0, 0}, {Sin[\[Theta]],
Cos[\[Theta]], 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}};
Td = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, d}, {0, 0, 0, 1}};
Ta = {{1, 0, 0, a}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}};
Rx = {{1, 0, 0, 0}, {0, Cos[\[Alpha]], -Sin[\[Alpha]], 0}, {0,
Sin[\[Alpha]], Cos[\[Alpha]], 0}, {0, 0, 0, 1}};
Ti = Rx.(Ta.(Td.Rz));
Return[Ti];
]

T1 = DH2HomTrans[\[Theta]1, 0, 0, 0]
MatrixForm[T1]

T2 = DH2HomTrans[0, d1, 0, 0];
MatrixForm[T2]

T3 = DH2HomTrans[\[Theta]2, d2, d3, 0];
MatrixForm[T3]

T4 = DH2HomTrans[0, 0, d4, 0];
MatrixForm[T4]

T5 = DH2HomTrans[\[Theta]4, d5, 0, 0];
MatrixForm[T5]

T = T1.T2.T3.T4.T5;
Simplify[MatrixForm[T]]

Final Answer

Cos [e1e2 04] - Sin[el Cos [e1 + e2 +04] e2 + 04] 0 d3 Cos [el) +d4 Cos [el + e2] 0 d3 Sin[e1 +d4 Sin[el e2] Sin[e1 e2+ 04] d1 d2 1 +d5 1 ро

You can use Matlab too