Screw theory is one of the most elegant mathematical frameworks for analyzing robot kinematics. Instead of dealing with separate translations and rotations, screw theory unifies them into a single concept: the screw motion.
š” KEY INSIGHT
Any rigid body motion in 3D space can be described as a rotation about an axis combined with a translation along that same axis ā just like a screw going into wood.
This tutorial will introduce you to the fundamentals of screw theory, starting with intuitive concepts and building up to practical applications in serial and parallel robots.
1. What is a Screw Motion?
Imagine turning a screw into a piece of wood. As you rotate it, the screw also moves forward along its axis. This combination of rotation and translation is the essence of a screw motion.
A screw motion combines rotation (Ļ) around an axis with translation (v) along that axis
Mathematically, a screw (denoted by $) consists of:
- Direction vector (Å): The axis of the screw
- Angular velocity (Ļ): How fast it rotates
- Linear velocity (v): How fast it translates
- Pitch (h): The ratio h = v/Ļ (translation per rotation)
Screw (Twist) representation
$ = [Ļ; v]įµ = [Ļā, Ļā, Ļā, vā, vā, vā]įµ
6-dimensional vector combining angular and linear velocity
2. Special Cases: Pure Rotation & Pure Translation
The beauty of screw theory is that both pure rotations and pure translations are just special cases of screws:
| Motion Type | Pitch (h) | Description |
|---|---|---|
| Pure Rotation | h = 0 | No translation, only rotation (revolute joint) |
| General Screw | 0 < h < ā | Combined rotation and translation (helical joint) |
| Pure Translation | h = ā | No rotation, only translation (prismatic joint) |
The three fundamental joint types as special cases of screw motion
3. Twists and Wrenches
In screw theory, we use two fundamental 6-dimensional vectors:
Twist ($)
A twist describes the instantaneous velocity of a rigid body. It combines angular velocity (Ļ) and linear velocity (v):
$ = [Ļ; v]įµ ā āā¶
Twist: describes motion (velocity)
Wrench (W)
A wrench describes the forces and torques acting on a rigid body. It combines torque (Ļ) and force (f):
W = [Ļ; f]įµ ā āā¶
Wrench: describes forces
š” DUALITY
Twists and wrenches are dual to each other. The power transmitted equals the dot product: P = $įµW = ĻįµĻ + vįµf
4. Example: 2R Planar Robot
Let's apply screw theory to the simplest serial robot: a 2R planar manipulator (two revolute joints in a plane).
2R planar robot with two revolute joints. Both screw axes point out of the page (įŗ direction).
Step-by-Step: Finding the Screws
Define the reference frame
Place the origin at joint 1. The x-axis points right, y-axis points up, z-axis points out of the page.
Identify screw axis for Joint 1
Joint 1 is a revolute joint at the origin. The rotation axis is įŗ = [0, 0, 1]įµ.
$ā = [0, 0, 1, 0, 0, 0]įµ
Pure rotation about z-axis at origin
Identify screw axis for Joint 2
Joint 2 is also revolute, but located at position rā = [Lā, 0, 0]įµ (assuming Īøā = 0). The screw includes a velocity component from the offset.
$ā = [0, 0, 1, 0, Lā, 0]įµ
Rotation about z-axis at distance Lā from origin
Build the Jacobian matrix
The Jacobian relates joint velocities to end-effector velocity:
J = [$ā | $ā]
6Ć2 Jacobian matrix
5. Mobility: How Many DOF?
One of the most useful applications of screw theory is calculating the mobility (degrees of freedom) of a mechanism using the Chebychev-GrĆ¼bler-Kutzbach formula:
M = d(n - 1) - Ī£įµ¢(d - fįµ¢)
where d = 6 (spatial) or 3 (planar), n = number of links, fįµ¢ = DOF of joint i
š EXAMPLE: 2R PLANAR ROBOT
For our 2R planar robot:
- d = 3 (planar mechanism)
- n = 3 (ground + 2 links)
- Two revolute joints, each with f = 1
M = 3(3 - 1) - 2(3 - 1) = 6 - 4 = 2 DOF
The robot can position its end-effector in 2D (x, y), confirming our intuition!
6. Serial vs Parallel Robots
Screw theory is particularly powerful for analyzing parallel robots, where multiple kinematic chains connect the base to the end-effector.
Serial robots have one chain; parallel robots have multiple chains connecting base to end-effector
| Property | Serial Robot | Parallel Robot |
|---|---|---|
| Structure | Open kinematic chain | Closed kinematic chains |
| Workspace | Large | Limited |
| Stiffness | Lower | Higher |
| Speed | Lower (moving motors) | Higher (fixed motors) |
| Accuracy | Error accumulates | Error averages out |
Summary
šÆ KEY TAKEAWAYS
- Screw motion unifies rotation and translation
- Twist ($) describes velocity; Wrench (W) describes forces
- Joint types are special cases of screws (h = 0, h = ā)
- The Jacobian maps joint velocities to end-effector velocity
- Screw theory works for both serial and parallel robots
š§ Quick Check
A revolute joint has what pitch value?