Master planar transformation mathematics through SCARA robot PCB assembly programming. Learn complex number rotations, 2D rotation matrices, forward and inverse kinematics calculations, and precision pick-and-place programming for electronics manufacturing applications.
By the end of this lesson, you will be able to:
Planar motion systems span robotics, manufacturing, and autonomous navigation. From precision assembly robots placing electronic components with micrometer accuracy, to CNC machines cutting complex curves in composite materials, to mobile robots navigating warehouses - all require precise mathematical control of 2D position and orientation.
Planar Motion Applications:
These systems require precise control of:
Engineering Question: How do we mathematically represent and program complex 2D trajectories that combine rotations, translations, and orientations in a systematic, precise manner that works across diverse applications?
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.
🔄 2D Rotation Matrix
Rotation by angle
Applied to a point:
Expanded form:
Physical Meaning: This matrix rotates any 2D vector counterclockwise by angle
A SCARA robot must pick an SMD (Surface Mount Device) component from a component tray and place it on a PCB with precise position and orientation control.
🔧 Equivalent System Model
Given:
Complex number representation of joint positions:
Link 1 endpoint:
Link 2 endpoint (end-effector):
Total end-effector position:
Expanding to Cartesian coordinates:
Separating real and imaginary parts:
Example calculation - Pick position (450, 200) mm:
Assuming
This doesn’t match target (450 mm), so we need inverse kinematics to find correct angles.
Tool orientation:
Total tool angle:
Geometric inverse kinematics for pick position (450, 200):
Distance from base:
Elbow angle
Two solutions:
Shoulder angle
Shoulder angle:
Verification using forward kinematics:
Accuracy: Position error < 0.5 mm ✅ (well within ±0.05 mm after calibration)
Pick orientation (tray at 30°):
Tool must align with component orientation:
Required total angle:
Place orientation (PCB at -45° with component rotated 75°):
Component final orientation:
Wait, that doesn’t seem right. Let me reconsider…
Corrected logic:
Actually, the component must be placed at orientation that results from:
Place position inverse kinematics (-200, 300) mm:
Elbow angle:
Wrist angle for place:
Summary table:
| Position | x (mm) | y (mm) | θ₁ (°) | θ₂ (°) | φ_wrist (°) | Total φ (°) |
|---|---|---|---|---|---|---|
| Pick | 450 | 200 | -4.63 | 70.78 | -36.15 | 30 |
| Place | -200 | 300 | 82.57 | 108.33 | -85.90 | 105 |
✅
Component rotation during transfer:
A CNC router must cut an elliptical slot in a composite panel at a 25° angle with a precise tool offset to account for cutter diameter.
🔧 Equivalent System Model
Given:
Basic parametric ellipse (unrotated, centered at origin):
For parameter
For our ellipse:
Example at t = 0°:
Example at t = 90°:
Rotation by α = 25° using complex multiplication:
Rotation operator:
Rotated ellipse:
Expanding (using complex multiplication rules):
Separating into x and y:
Translation to center (150, 100):
Final parametric equations:
Verification at t = 0°:
This is the rightmost point of the rotated ellipse.
Tangent vector calculation (derivative with respect to t):
Outward normal vector (perpendicular to tangent, rotated 90° clockwise):
For 2D rotation by -90° (clockwise):
Normal vector magnitude:
At
Unit normal vector:
At
Tool center offset (inside the ellipse for cutting, offset by radius r = 6 mm):
For inside cutting (pocket milling):
In complex form:
At
General formula for tool path:
Waypoint parameter spacing:
Waypoint table (with tool offset applied):
| Point | t (°) | t (rad) | x (mm) | y (mm) | x_tool (mm) | y_tool (mm) |
|---|---|---|---|---|---|---|
| 0 | 0.0 | 0.000 | 222.50 | 133.81 | 217.06 | 131.27 |
| 1 | 22.5 | 0.393 | 214.08 | 156.18 | 208.79 | 153.35 |
| 2 | 45.0 | 0.785 | 193.74 | 171.74 | 189.00 | 168.24 |
| 3 | 67.5 | 1.178 | 165.96 | 178.89 | 161.54 | 174.74 |
| 4 | 90.0 | 1.571 | 133.19 | 178.13 | 129.06 | 173.43 |
| 5 | 112.5 | 1.963 | 99.04 | 170.26 | 95.21 | 165.04 |
| 6 | 135.0 | 2.356 | 66.87 | 157.06 | 63.30 | 151.46 |
| 7 | 157.5 | 2.749 | 40.29 | 139.32 | 37.01 | 133.51 |
| 8 | 180.0 | 3.142 | 22.37 | 118.51 | 19.42 | 112.22 |
| 9 | 202.5 | 3.534 | 15.79 | 96.14 | 13.20 | 89.07 |
| 10 | 225.0 | 3.927 | 21.13 | 73.58 | 18.93 | 66.02 |
| 11 | 247.5 | 4.320 | 37.92 | 52.26 | 36.17 | 44.66 |
| 12 | 270.0 | 4.712 | 64.19 | 33.32 | 62.90 | 26.18 |
| 13 | 292.5 | 5.105 | 96.04 | 18.06 | 95.20 | 11.34 |
| 14 | 315.0 | 5.498 | 129.26 | 8.06 | 129.00 | 1.56 |
| 15 | 337.5 | 5.890 | 160.92 | 5.19 | 161.21 | -1.20 |
✅ All 16 waypoints calculated
Verification - ellipse equation check at t = 0°:
For rotated ellipse:
Actually, simpler verification: distance from center at t = 0° before rotation:
After rotation and translation to (150, 100):
Archimedean spiral in polar form:
Where:
Starting point at t = 0° on ellipse (entry point):
Entry at
Distance from center:
Angle from center:
Spiral equation (working inward to center):
In complex form centered at (150, 100):
Spiral waypoints (8 points over 2 revolutions):
| Point | θ (rad) | θ (°) | r (mm) | x (mm) | y (mm) |
|---|---|---|---|---|---|
| 0 | 0.000 | 0.0 | 80.00 | 222.50 | 133.81 |
| 1 | 0.785 | 45.0 | 79.94 | 206.37 | 153.69 |
| 2 | 1.571 | 90.0 | 79.87 | 175.94 | 162.45 |
| 3 | 2.356 | 135.0 | 79.81 | 138.19 | 157.31 |
| 4 | 3.142 | 180.0 | 79.75 | 100.59 | 139.44 |
| 5 | 3.927 | 225.0 | 79.69 | 70.41 | 112.81 |
| 6 | 4.712 | 270.0 | 79.62 | 54.84 | 82.06 |
| 7 | 5.498 | 315.0 | 79.56 | 57.76 | 52.78 |
| 8 | 6.283 | 360.0 | 79.50 | 77.44 | 32.19 |
✅ Spiral reduces radius by 0.5 mm over 2 revolutions (80.00 → 79.50 mm)
Tool offset application to spiral:
Same normal vector method as ellipse, but now normal points radially outward from (150, 100):
Tool center offset (6 mm inward):
For circular/spiral paths, this simplifies to:
An autonomous mobile robot must navigate from a starting position to a goal position while avoiding a circular obstacle using arc trajectories.
🔧 Equivalent System Model
Given:
Path segments:
Key waypoints (manually chosen to avoid obstacle):
| Waypoint | Description | Position (m) | Orientation |
|---|---|---|---|
| P₀ | Start | (0, 0) | 0° |
| P₁ | Pre-arc | (2, 0) | 0° |
| P₂ | Arc start | (3, 1) | 30° |
| P₃ | Arc mid | (4.5, 3.5) | 60° |
| P₄ | Arc end | (6, 5) | 90° |
| P₅ | Goal | (8, 6) | 45° |
Arc center calculation for Segment 2 (P₂ to P₄):
Given P₂ = (3, 1) and desired radius R = 3 m:
Arc center perpendicular to left of robot at P₂ (θ = 30°):
Center offset direction (rotate 90° left from heading):
Arc center in complex form:
Center position:
Verification - distance from center to P₂:
Parametric arc equation (Segment 2: P₂ to P₄):
Arc center:
Starting angle (from center to P₂):
Ending angle (from center to P₄):
Arc sweep:
Position along arc as function of angle α:
In Cartesian form:
Robot orientation along arc (tangent direction):
Tangent vector:
This represents a vector rotated 90° from the radius vector.
Orientation angle:
Verification at α = -60° (P₂):
Arc length calculation:
Number of waypoints along arc (0.5 m spacing):
Obstacle representation in complex form:
Obstacle center:
Distance from any point z to obstacle:
Collision condition:
Check all key waypoints:
| Point | Position z | z - z_obs | Distance (m) | Safe? |
|---|---|---|---|---|
| P₀ | 0 + 0i | -4 - 2i | 4.47 | ✅ Yes |
| P₁ | 2 + 0i | -2 - 2i | 2.83 | ✅ Yes |
| P₂ | 3 + i | -1 - i | 1.41 | ❌ No |
| P₃ | 4.5 + 3.5i | 0.5 + 1.5i | 1.58 | ❌ No |
| P₄ | 6 + 5i | 2 + 3i | 3.61 | ✅ Yes |
| P₅ | 8 + 6i | 4 + 4i | 5.66 | ✅ Yes |
Analysis:
P₂ and P₃ are too close to the obstacle! We need to recalculate the arc with larger radius.
Revised arc calculation with R = 4 m:
New arc center from P₂ = (3, 1) with R = 4 m:
Closest point on arc to obstacle:
The closest point is along the line from arc center to obstacle center.
Direction from arc center to obstacle:
Distance from arc center to obstacle center:
Closest point on arc:
Distance from closest arc point to obstacle:
❌ Still too close! (< 2.0 m required)
Revised arc with R = 5.5 m:
Following same process:
Minimum clearance:
Wait, this means the arc center is inside the avoidance zone. Let me recalculate…
Actually:
Correct minimum distance (from arc to obstacle):
Final revised path - go around from the top with R = 3 m, but adjust P₂:
New P₂:
Arc center:
Distance from center to obstacle:
Minimum clearance:
❌ Still less than 2.0 m!
Simpler approach - straight line segments avoiding obstacle:
Revised waypoints:
Check P₂ clearance:
Segment P₀ to P₁ (straight, 2 m):
| Point | s (m) | x (m) | y (m) | θ (°) |
|---|---|---|---|---|
| 0 | 0.0 | 0.0 | 0.0 | 0 |
| 1 | 0.5 | 0.5 | 0.0 | 0 |
| 2 | 1.0 | 1.0 | 0.0 | 0 |
| 3 | 1.5 | 1.5 | 0.0 | 0 |
| 4 | 2.0 | 2.0 | 0.0 | 53.13 |
✅ 5 waypoints (orientation changes at P₁ for next segment)
Segment P₁ to P₂ (arc):
Vector from P₁ to P₂:
For smooth arc, use radius R = 6 m:
Arc center perpendicular bisector method:
Midpoint:
Direction P₁→P₂:
Perpendicular:
Actually, let’s use direct complex parameterization instead:
Linear interpolation (simpler for this segment):
Orientation along segment:
| Point | t | s (m) | x (m) | y (m) | θ (°) |
|---|---|---|---|---|---|
| 5 | 0.121 | 0.5 | 2.12 | 0.48 | 76 |
| 6 | 0.242 | 1.0 | 2.24 | 0.97 | 76 |
| 7 | 0.364 | 1.5 | 2.36 | 1.46 | 76 |
| 8 | 0.485 | 2.0 | 2.49 | 1.94 | 76 |
| 9 | 0.606 | 2.5 | 2.61 | 2.42 | 76 |
| 10 | 0.727 | 3.0 | 2.73 | 2.91 | 76 |
| 11 | 0.848 | 3.5 | 2.85 | 3.39 | 76 |
| 12 | 0.970 | 4.0 | 2.97 | 3.88 | 76 |
| 13 | 1.000 | 4.12 | 3.0 | 4.0 | 45 |
✅ 9 waypoints (orientation changes to 45° at P₂)
Segment P₂ to P₃:
But we want constant 45° orientation, so use:
| Point | t | s (m) | x (m) | y (m) | θ (°) |
|---|---|---|---|---|---|
| 14 | 0.149 | 0.5 | 3.45 | 4.22 | 45 |
| 15 | 0.299 | 1.0 | 3.90 | 4.45 | 45 |
| 16 | 0.448 | 1.5 | 4.34 | 4.67 | 45 |
| 17 | 0.597 | 2.0 | 4.79 | 4.90 | 45 |
| 18 | 0.746 | 2.5 | 5.24 | 5.12 | 45 |
| 19 | 0.896 | 3.0 | 5.69 | 5.34 | 45 |
| 20 | 1.000 | 3.35 | 6.0 | 5.5 | 45 |
✅ 7 waypoints
Segment P₃ to P₄ (final to goal):
| Point | t | s (m) | x (m) | y (m) | θ (°) |
|---|---|---|---|---|---|
| 21 | 0.243 | 0.5 | 6.49 | 5.62 | 45 |
| 22 | 0.485 | 1.0 | 6.97 | 5.74 | 45 |
| 23 | 0.728 | 1.5 | 7.46 | 5.86 | 45 |
| 24 | 1.000 | 2.06 | 8.0 | 6.0 | 45 |
✅ 4 waypoints
Total waypoints: 5 + 9 + 7 + 4 = 25 waypoints ✅
In this lesson, you mastered planar transformation mathematics through three applications: SCARA robot kinematics, CNC elliptical toolpaths, and mobile robot navigation.
Key Skills Developed:
Critical Formulas:
Key Insights:
Coming Next: Lesson 3 extends to 3D space with homogeneous transformation matrices, Euler angles, and 6-DOF spatial robot control.
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