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:
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 unify rotation and translation into systematic matrix operations
Coming Next: In Lesson 3, we’ll extend these 2D transformation concepts to 3D space, introducing individual axis rotations (X, Y, Z), 3D homogeneous transformations, and systematic methods for 6-DOF robot orientation control.
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