Manipulator Jacobian of a Four-Axis SCARA Robot (Adept One)

The four-axis SCARA robot has three revolute joints and one prismatic joint in this order:

$$\text{revolute},\text{revolute},\text{prismatic},\text{revolute}$$

flowchart LR
    Q["SCARA joints<br/>R, R, P, R"] --> A["Differentiate p(q)<br/>linear block A(q)"]
    Q --> B["Use joint axes<br/>angular block B(q)"]
    A --> J["Stack A(q) and B(q)<br/>manipulator Jacobian J(q)"]
    B --> J

Use the Manipulator Jacobian relation:

$$J(q)=\left[\begin{array}{c}A(q)\\ B(q)\end{array}\right]$$

Angular block

The angular columns are obtained from each joint axis in base coordinates.

For joint 1:

$$b^1(q)=R_0^0(q)i^3=i^3$$

For joint 2:

$$b^2(q)=R_0^1(q)i^3=-i^3$$

Joint 3 is prismatic:

$$b^3(q)=0$$

For joint 4:

$$b^4(q)=R_0^3(q)i^3=-i^3$$

Therefore:

$$B(q)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& -1& 0& -1\end{array}\right]$$

Tool-tip position

The arm matrix gives:

$$p(q)=\left[\begin{array}{c}{a}_1{C}_1+a_2{C}_{1-2}\\ a_1{S}_1+a_2{S}_{1-2}\\ d_1-q_3-d_4\end{array}\right]$$

where:

$$C_1=\cos q_1,S_1=\sin q_1$$ $$C_{1-2}=\cos (q_1-q_2),S_{1-2}=\sin (q_1-q_2)$$

Take the partial derivative of $p(q)$ with respect to each joint variable:

$$A(q)=\left[\begin{array}{cccc}-a_1{S}_1-a_2{S}_{1-2}& a_2{S}_{1-2}& 0& 0\\ a_1{C}_1+a_2{C}_{1-2}& -a_2{C}_{1-2}& 0& 0\\ 0& 0& -1& 0\end{array}\right]$$

Complete manipulator Jacobian

Combine $A(q)$ and $B(q)$:

$$J(q)=\left[\begin{array}{cccc}-a_1{S}_1-a_2{S}_{1-2}& a_2{S}_{1-2}& 0& 0\\ a_1{C}_1+a_2{C}_{1-2}& -a_2{C}_{1-2}& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& -1& 0& -1\end{array}\right]$$

The row meanings are:

• Rows 1 to 3 give linear tool displacement
• Rows 4 to 6 give angular tool displacement

The first three rows are identical to the first three rows of the Tool-configuration Jacobian Matrix because both matrices use the same tool-tip position $p(q)$.

The last three rows differ because the two matrices represent orientation differently:

• $V(q)$ differentiates the scaled approach-vector representation
• $J(q)$ uses infinitesimal tool rotation

All four SCARA axes are vertical. Therefore, the third joint causes no angular displacement because it is prismatic. The revolute joints contribute only to rotation about the vertical axis.

See Induced Joint Torques and Forces for the static-equilibrium use of $J^T(q)$.