Manipulator Jacobian

The manipulator Jacobian relates infinitesimal joint displacement to infinitesimal tool translation and rotation.

flowchart LR
    DQ["Joint displacement<br/>dq"] --> A["Linear block<br/>A(q)"] --> DP["Tool translation<br/>dp"]
    DQ --> B["Angular block<br/>B(q)"] --> DPHI["Tool rotation<br/>d-phi"]
    DP --> DU["Tool displacement<br/>du"]
    DPHI --> DU

Let:

• $dp\in {\mathbb{R}}^3$ be an infinitesimal tool-tip translation in base coordinates
• $d\varphi \in {\mathbb{R}}^3$ be an infinitesimal tool rotation in base coordinates

Combine them into the infinitesimal tool displacement vector:

$$du=\left[\begin{array}{c}dp\\ d\varphi \end{array}\right]$$

For each joint configuration $q$, there is a matrix $J(q)\in {\mathbb{R}}^{6\times n}$ such that:

$$\boxed{du=J(q)dq}$$

This gives the displacement mapping:

$$dq\stackrel{J(q)}{\to }du$$

Linear and angular blocks

Partition the manipulator Jacobian into two $3\times n$ blocks:

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

Then:

$$dp=A(q)dq$$ $$d\varphi =B(q)dq$$

Divide the linear relation by $dt$:

$$\dot{p}=A(q)\dot{q}$$

Differentiate the tool-tip position function $p(q)$:

$$\boxed{A(q)=\frac{\partial p(q)}{\partial q}}$$

The $k$th row and $j$th column are:

$$A_{kj}(q)=\frac{\partial p_k(q)}{\partial q_j}$$

Angular block

Write the angular block by columns:

$$B(q)=\left[\begin{array}{cccc}{b}^1(q)& b^2(q)& \cdots & b^n(q)\end{array}\right]$$

Then:

$$d\varphi =\sum _{k=1}^{n}d{q}_k{b}^k(q)$$

The column $b^k(q)$ gives the angular tool displacement caused by joint $k$ when all other joints stay fixed.

For a prismatic joint, motion along $z^{k-1}$ causes no angular tool displacement:

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

For a revolute joint, rotation about $z^{k-1}$ gives:

$$d\varphi =(d{q}_k)z^{k-1}$$

Therefore:

$$b^k(q)=z^{k-1}(q)$$

Define the joint-type parameter:

$${\xi }_k=\{\begin{array}{ll}0,& \text{joint }k\text{ is prismatic}\\ 1,& \text{joint }k\text{ is revolute}\end{array}$$

The two cases combine into:

$$\boxed{b^k(q)={\xi }_k{z}^{k-1}(q)}$$

The joint axis in base coordinates is:

$$\boxed{z^{k-1}(q)=R_0^{k-1}(q)i^3}$$

where $i^3=[0,0,1{]}^T$ is the third unit vector.

Complete form

The manipulator Jacobian is:

$$\boxed{J(q)=\left[\begin{array}{cccc}\frac{\partial p(q)}{\partial q_1}& \frac{\partial p(q)}{\partial q_2}& \cdots & \frac{\partial p(q)}{\partial q_n}\\ {\xi }_1{z}^0(q)& {\xi }_2{z}^1(q)& \cdots & {\xi }_n{z}^{n-1}(q)\end{array}\right]}$$

Calculation procedure

1. Set $T_0^0=I$ and $k=1$
2. Calculate $b^k(q)={\xi }_k{R}_0^{k-1}(q)i^3$
3. Update $T_0^k(q)=T_0^{k-1}(q)T_{k-1}^k(q)$
4. Set $k=k+1$ and repeat steps 2 to 3 until $k>n$
5. Calculate $p(q)=H_1{T}_0^n(q)i^4$ and $a^k=\partial p(q)/\partial q_k$
6. Assemble $J(q)$ from the columns $a^k$ and $b^k$

Here, $H_1=[I0]$ selects the position coordinates from the homogeneous coordinate vector.

Important

$A(q)$ maps joint velocity to linear tool velocity. $B(q)$ maps joint velocity to angular tool velocity.

See Link Coordinates, Denavit-Hartenberg (D-H) Representation, and Manipulator Jacobian of a Four-Axis SCARA Robot (Adept One).