Differential Motion and Statics

Once a trajectory is planned in tool-configuration space, the robot must convert the required tool motion into joint motion.

flowchart LR
    QD["Joint velocity<br/>q-dot"] -->|"V(q)"| XD["Tool velocity<br/>x-dot"]
    FT["Tool wrench<br/>F-tool"] -->|"J-transpose(q)"| TAU["Joint torque or force<br/>tau"]

The direct kinematics relation is:

$$x=w(q)$$

where:

• $q\in {\mathbb{R}}^n$ is the joint-variable vector
• $x\in {\mathbb{R}}^6$ is the tool-configuration vector
• $w(q)$ is the direct kinematics function, that contains transformation $T$ matrix and then extracting the pose of TCP which is then our $x$, but note that $x$'s orientation components 1 2 and 3 are scaled approach-vector representation

Differentiate the direct kinematics equation with respect to time:

$$\boxed{\dot{x}=V(q)\dot{q}}$$

The Tool-configuration Jacobian Matrix $V(q)$ maps joint velocity $\dot{q}$ to tool-configuration velocity $\dot{x}$.

Differential motion

Differential motion uses small changes or instantaneous velocities to relate joint space to tool space.

The main sequence is:

1. Plan a tool trajectory $x(t)$
2. Calculate the required tool velocity $\dot{x}(t)$
3. Use the Jacobian or its pseudoinverse to calculate $\dot{q}(t)$
4. Integrate $\dot{q}(t)$ to obtain the joint trajectory $q(t)$

The related topics are:

• Tool-configuration Jacobian Matrix
• Joint-space Singularities
• Generalised Inverses
• Resolved-motion Rate Control

Statics

Statics studies a robot that is in equilibrium while an external force and moment act at the tool.

The Manipulator Jacobian relates an infinitesimal joint displacement to an infinitesimal tool displacement. Its transpose relates a tool force and moment to the equivalent joint torques and forces:

$$\boxed{\tau =J^T(q)F^{tool}}$$

The related topics are:

• Manipulator Jacobian
• Manipulator Jacobian of a Four-Axis SCARA Robot (Adept One)
• Induced Joint Torques and Forces

Important

Differential motion uses the Jacobian to map motion. Statics uses the Jacobian transpose to map force and moment.