Master planar transformation mathematics through SCARA robot programming, covering complex number analysis, homogeneous coordinates, and transformation matrix composition as foundation for 3D spatial mechanics.
By the end of this lesson, you will be able to:
SCARA (Selective Compliance Assembly Robot Arm) robots dominate electronics manufacturing and precision assembly. These 2D planar robots require precise mathematical control of position and orientation to place components with micrometer accuracy while maintaining high-speed operation.
SCARA Robot Architecture:
SCARA programming requires precise control of:
Engineering Question: How do we mathematically represent and program complex 2D trajectories that combine rotations, translations, and tool orientations in a systematic, precise manner?
Consequences of Poor Mathematical Foundation:
Benefits of Systematic Planar Analysis:
Complex numbers provide an elegant mathematical framework for representing 2D rotations and translations. Every point in the plane can be represented as a complex number z = x + iy, and transformations become simple algebraic operations.
🔢 Complex Number Planar Point
Where:
Physical Meaning: Every 2D point corresponds to a unique complex number, enabling algebraic manipulation of geometric transformations.
🔄 Complex Rotation Operator
Rotation by angle
Matrix form:
Physical Meaning: Multiplying by
↔️ Complex Translation Operation
Translation by vector (a, b):
Matrix form (requires homogeneous coordinates):
Physical Meaning: Adding a complex constant shifts all points by the same displacement vector.
🔄↔️ General Planar Transformation
Rotation followed by translation:
Translation followed by rotation:
Physical Meaning: Order matters! Different sequences of rotation and translation produce different final positions.
Homogeneous coordinates solve the fundamental problem that translation cannot be represented as matrix multiplication in Cartesian coordinates. By adding a third coordinate, both rotation and translation become matrix multiplications.
🎯 Homogeneous Coordinate Representation
2D point in homogeneous coordinates:
General transformation matrix:
Where:
Physical Meaning: Homogeneous coordinates enable all 2D transformations to be represented as 3×3 matrix multiplications.
Pure Translation:
Pure Rotation:
Identity Transformation:
Sequential transformations:
Order significance:
Inverse transformations:
Three-step process:
Combined result:
Real robot control requires precise composition of multiple rotations and translations. Understanding the systematic rules for matrix multiplication order is essential for accurate end-effector positioning and complex trajectory programming.
🔧 Transformation Composition Rules
Matrix multiplication is non-commutative - order matters!
For robot positioning with multiple transformations:
General composition:
Where transformations are applied in sequence: H_1 first, H_n last.
Problem: 40° rotation about OX-axis, then 7 units translation along mobile B-axis
Setup: H = H(x,40°) · I · H(B,7)
Matrix composition:
Final result:
Problem: Complex robotic motion sequence
Matrix sequence: H = H(x,4) · H(x,50°) · I · H(c,-6) · H(b,25°)
Step-by-step computation:
Final transformation matrix:
System setup: Robotic work cell with camera and 6-joint robot
Given transformation matrices:
Object position relative to robot base:
Object position relative to gripper:
Complex robotic systems require systematic handling of multiple coordinate frames, object manipulations, and dynamic transformations. Mastering these principles enables precise control of sophisticated manufacturing and assembly operations.
Problem setup: Triangular prism with vertices A(1,3,0), B(-1,3,0), C(-1,3,2), D(1,3,2), E(1,5,2), F(-1,5,2)
Required transformations:
Transformation matrix composition:
Matrix multiplication sequence:
Final transformation matrix:
New vertex coordinates: Apply H to each vertex
Example for vertex A:
🎯 Essential Transformation Guidelines
Critical Success Factors:
Common Programming Errors:
While SCARA robots operate primarily in 2D, understanding 3D rotations is essential for complete spatial mechanics mastery. 3D rotations about individual coordinate axes form the foundation for complex orientation control in 6-DOF industrial robots.
Following the established axis convention where counterclockwise rotation is positive, we can derive rotation matrices for each coordinate axis. Each rotation transforms a mobile frame (A, B, C) relative to a fixed frame (X, Y, Z).
Rotation about X-axis by angle α:
Geometric Analysis:
After rotating axis B:
After rotating axis C:
Transformation Summary:
| Before | After |
|---|---|
| A(1,0,0) | A(1,0,0) |
| B(0,1,0) | B(0, cos α, sin α) |
| C(0,0,1) | C(0, -sin α, cos α) |
🔄 X-Axis Rotation Matrix
Physical Meaning: Rotates vectors around the X-axis, leaving X-coordinates unchanged while rotating Y and Z components in the YZ-plane.
Rotation about Y-axis by angle β:
Geometric Analysis:
After rotating axis A:
After rotating axis C:
Transformation Summary:
| Before | After |
|---|---|
| A(1,0,0) | A(cos β, 0, -sin β) |
| B(0,1,0) | B(0,1,0) |
| C(0,0,1) | C(sin β, 0, cos β) |
🔄 Y-Axis Rotation Matrix
Physical Meaning: Rotates vectors around the Y-axis, leaving Y-coordinates unchanged while rotating X and Z components in the XZ-plane.
Rotation about Z-axis by angle γ:
Geometric Analysis:
After rotating axis A:
After rotating axis B:
Transformation Summary:
| Before | After |
|---|---|
| A(1,0,0) | A(cos γ, sin γ, 0) |
| B(0,1,0) | B(-sin γ, cos γ, 0) |
| C(0,0,1) | C(0,0,1) |
🔄 Z-Axis Rotation Matrix
Physical Meaning: Rotates vectors around the Z-axis, leaving Z-coordinates unchanged while rotating X and Y components in the XY-plane.
Point rotation about X-axis:
Problem: Point q = (3, 7, 5) rotated 60° about X-axis
Solution:
Point rotation about Y-axis:
Problem: Point p = (4, 4, 2√3) in mobile frame rotated 60° about Y-axis
Solution:
Point rotation about Z-axis:
Problem: Point p = (7, 6, 5) rotated 30° about Z-axis
Solution:
Inverse transformations (world to body coordinates):
For Y-axis rotation: Mobile frame rotated 60° about Y-axis
Problem: Points p_xyz = (2, 3, 6) and q_xyz = (4, 2, 5) in fixed frame
Solution:
Since rotation matrices are orthogonal:
Problem: Given rotation matrix, determine axis and angle
Example matrix:
Analysis steps:
Robot joint control application:
Problem: Single-axis robot with mobile frame point p_M = (2, 2, 8)
Find coordinates when:
Solutions:
Coordinate system transformation:
Problem: Robot frame rotated 60° about X-axis Point Q = (4, 2√3, 5) in base coordinates
Find mobile frame coordinates:
To perform rotation about an arbitrary axis through the origin, we extend the individual axis rotation concepts to handle any vector direction. This fundamental capability enables complete 3D orientation control for advanced robotics applications.
Problem Setup: Given a fixed frame OXYZ and an arbitrary rotation axis V = (x,y,z) with components V_x, V_y, V_z, we need to construct the rotation matrix R(V,θ) for rotation angle θ.
🔄 Arbitrary Axis Rotation Strategy
Five-step decomposition process:
Matrix composition: R(V,θ) = R(x,-α) R(y,β) R(z,θ) R(y,-β) R(x,α)
The complete rotation matrix:
Geometric relationships: For unit vector |V| = 1:
⚡ Simplified Arbitrary Axis Formula
Final rotation matrix in compact form:
Where: C = cos θ, S = sin θ, T = (1 - cos θ)
Unit vector components:
Calculate unit vector components:
Calculate trigonometric values:
Compute matrix elements:
Final rotation matrix:
Calculate vector magnitude and unit components:
Calculate trigonometric values:
Compute matrix elements (showing key calculations):
Final rotation matrix:
⚡ Rodrigues' Rotation Formula
Alternative approach using matrix exponentials:
Where:
Note: Rodrigues formula only works for rotations about axes through the origin.
Arbitrary axis rotation capabilities are essential for advanced 6-DOF robot programming, tool orientation control, and complex trajectory planning where rotations cannot be decomposed into simple XYZ sequences.
Key applications:
Let’s program a complete pick-and-place operation for electronics assembly.
System Parameters:
±0.01 mm and ±0.1°5×3 mm with 0.2 mm placement tolerance45° for proper component orientationJoint coordinate representation:
Using complex number approach:
End-effector position:
In Cartesian form:
Matrix representation:
Tool orientation:
Tool angle =
Base to Link 1 transformation:
Link 1 to Link 2 transformation:
Combined base to end-effector:
Tool frame transformation:
Including tool offset and rotation:
Waypoint definition:
Linear interpolation in Cartesian space:
For path from point A to point B:
Position:
Orientation:
Transformation matrix interpolation:
For smooth rotation interpolation:
Translation interpolation:
Velocity profile generation:
Trapezoidal velocity profile:
Geometric approach for 2-DoF SCARA:
Given end-effector position (x, y):
Two solutions for elbow configuration:
Elbow up:
Elbow down:
Shoulder angle calculation:
Solution selection criteria:
Forward Kinematics
Complex method: Elegant rotation representation
Matrix method: Systematic composition
Accuracy: Sub-millimeter precision
Status: Multiple approaches available
Trajectory Planning
Interpolation: Linear and circular paths
Velocity: Trapezoidal profiles
Smoothness: C¹ continuous motion
Status: Optimized for cycle time
Inverse Solutions
Multiple configs: Elbow up/down options
Selection: Optimization criteria
Singularity: Systematic avoidance
Status: Robust solution methods
Understanding transformation composition is critical for complex motion programming. The order of operations fundamentally affects the final result, making systematic matrix composition essential for reliable robot programming.
Rotation then Translation vs Translation then Rotation:
Case 1: Rotate 45° then translate (2, 0)
Case 2: Translate (2, 0) then rotate 45°
Result: Different final positions despite same operations!
Local vs Global transformations:
Local (body-fixed) frame:
Global (world-fixed) frame:
Programming implication: Choose consistent convention
Tool coordinate programming:
Workpiece coordinate systems:
Matrix pre-computation:
Numerical stability:
Trajectory continuity:
Acceleration limiting:
Robustness measures:
Calibration considerations:
In this lesson, you learned to:
Key Mathematical Insights:
Critical Foundation: Homogeneous transformation matrices:
Coming Next: In Lesson 3, we’ll extend these 2D concepts to full 3D spatial transformations, covering rotation matrices, Euler angles, and complex 3D orientation control for 6-DOF industrial robot arms.
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