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G-Code Path Smoothing with Bezier Curves

CNC G-Code

Goal -- WPM

Ready
Exercise Algorithm Area
1G00 Z10.0; Rapid move to safe height

2

3FUNCTION CalculateBezierPoint(p0, p1, p2, p3, t)

4// Cubic Bezier curve formula: B(t) = (1-t)^3*P0 + 3*(1-t)^2*t*P1 + 3*(1-t)*t^2*P2 + t^3*P3

5VAR mt = 1.0 - t;

6VAR mt2 = mt * mt;

7VAR mt3 = mt2 * mt;

8VAR t2 = t * t;

9VAR t3 = t2 * t;

10

11VAR x = mt3 * p0.x + 3.0 * mt2 * t * p1.x + 3.0 * mt * t2 * p2.x + t3 * p3.x;

12VAR y = mt3 * p0.y + 3.0 * mt2 * t * p1.y + 3.0 * mt * t2 * p2.y + t3 * p3.y;

13

14RETURN {x: x, y: y};

15END FUNCTION

16

17FUNCTION SmoothPathGCode(points, numSegmentsPerCurve, feedRate)

18VAR smoothedCode = "";

19VAR safeHeight = 10.0;

20VAR approachHeight = 2.0;

21VAR depth = -5.0; // Default depth for cutting

22

23IF ArrayLength(points) < 2 THEN

24RETURN "Error: Need at least two points to define a path.";

25END IF

26

27// Move to safe height

28smoothedCode += "G00 Z" + safeHeight + "\n";

29

30// Move to the first point

31VAR startPoint = points[0];

32smoothedCode += "G00 X" + startPoint.x + " Y" + startPoint.y + "\n";

33

34// Descend to approach height

35smoothedCode += "G01 Z" + approachHeight + " F" + feedRate + "\n";

36

37// Descend to final depth

38smoothedCode += "G01 Z" + depth + " F" + feedRate + "\n";

39

40// Process each segment of the path

41FOR i = 0 TO ArrayLength(points) - 2 DO

42VAR p0 = points[i];

43VAR p1 = points[i+1];

44

45// Determine control points for Bezier curve

46// For simplicity, we'll use a basic approach: P1 is the start, P2 is the end.

47// For more advanced smoothing, P1 and P2 would be calculated based on tangents.

48VAR bezierP0 = p0;

49VAR bezierP1 = p0; // Placeholder, ideally calculated tangent

50VAR bezierP2 = p1; // Placeholder, ideally calculated tangent

51VAR bezierP3 = p1;

52

53// If it's the very first segment, we might need to adjust P1/P2

54IF i == 0 THEN

55// For the first segment, P1 can be the same as P0, and P2 can be an interpolation towards P1

56// A simple approach is to use P0 as P1 and P1 as P2 for a linear segment, then refine.

57// For true Bezier, we'd need tangents. Here, we'll approximate.

58bezierP1 = p0; // Start point of the curve

59bezierP2 = {x: (p0.x + p1.x) / 2.0, y: (p0.y + p1.y) / 2.0}; // Midpoint as a simple control

60bezierP3 = p1; // End point of the curve

61ELSE IF i == ArrayLength(points) - 2 THEN

62// For the last segment, P1 is an interpolation towards P0, P2 is P1

63bezierP0 = p0;

64bezierP1 = {x: (p0.x + p1.x) / 2.0, y: (p0.y + p1.y) / 2.0}; // Midpoint as a simple control

65bezierP2 = p1; // End point of the curve

66bezierP3 = p1;

67ELSE

68// Intermediate segments: use points[i-1], points[i], points[i+1], points[i+2] to define a curve

69// For simplicity in this example, we'll treat each pair of points as a linear segment and then smooth.

70// A proper Bezier requires calculating control points based on tangents.

71// Here, we'll use a simplified approach: P0=points[i], P1=midpoint(points[i],points[i+1]), P2=midpoint(points[i],points[i+1]), P3=points[i+1]

72// This is a very basic approximation and not a true Bezier.

73// For a true cubic Bezier, we'd need to calculate P1 and P2 based on tangents derived from neighboring points.

74// Let's use a simpler linear interpolation for now and then refine.

75// For this exercise, we'll treat each segment as a linear move and then add a smoothing step.

76// A more robust solution would involve calculating control points.

77// For this implementation, we'll generate linear segments and then add a note about smoothing.

78// To actually implement Bezier, we'd need to calculate P1 and P2 based on tangents.

79// Let's redefine the problem to generate linear segments and then add a note about smoothing.

80// The prompt asks for Bezier curves, so we MUST implement that logic.

81// Let's assume we have P0, P1, P2, P3 for each segment.

82// For a path of points P_0, P_1, ..., P_n:

83// Segment 1: P_0, P_1, P_2, P_3 (where P_0, P_3 are actual points, P_1, P_2 are control points)

84// This requires calculating control points. A common method is to use Catmull-Rom splines or similar.

85// For cubic Bezier, we need 4 points per curve. If we have N points, we can define N-1 Bezier curves.

86// For each curve i (from 0 to N-2), the points are P_i, C1_i, C2_i, P_{i+1}.

87// C1_i and C2_i are control points. A simple way to estimate them is using tangents.

88// Tangent at P_i is proportional to P_{i+1} - P_{i-1}.

89// C1_i = P_i + (1/3) * Tangent_i

90// C2_i = P_{i+1} - (1/3) * Tangent_{i+1}

91// This requires handling endpoints carefully.

92

93// Let's simplify for this example: we'll generate N-1 Bezier curves, where each curve uses points[i] and points[i+1] as endpoints.

94// The control points will be calculated based on the midpoint and a simple offset.

95// This is an approximation, not a mathematically rigorous Bezier spline generation.

96VAR cp1x = (p0.x + p1.x) / 2.0 - (p1.y - p0.y) * 0.1; // Simple offset for control point 1

97VAR cp1y = (p0.y + p1.y) / 2.0 + (p1.x - p0.x) * 0.1;

98VAR cp2x = (p0.x + p1.x) / 2.0 + (p1.y - p0.y) * 0.1; // Simple offset for control point 2

99VAR cp2y = (p0.y + p1.y) / 2.0 - (p1.x - p0.x) * 0.1;

100

101bezierP0 = p0;

102bezierP1 = {x: cp1x, y: cp1y};

103bezierP2 = {x: cp2x, y: cp2y};

104bezierP3 = p1;

105END IF

106

107// Generate points along the Bezier curve

108FOR j = 1 TO numSegmentsPerCurve DO

109VAR t = j / numSegmentsPerCurve;

110VAR point = CalculateBezierPoint(bezierP0, bezierP1, bezierP2, bezierP3, t);

111smoothedCode += "G01 X" + point.x + " Y" + point.y + " F" + feedRate + "\n";

112END FOR

113END FOR

114

115// Retract to safe height

116smoothedCode += "G00 Z" + safeHeight + "\n";

117

118RETURN smoothedCode;

119END FUNCTION

120

121FUNCTION Main()

122VAR pathPoints = [

123{x: 10.0, y: 10.0},

124{x: 50.0, y: 20.0},

125{x: 80.0, y: 60.0},

126{x: 100.0, y: 100.0}

127];

128VAR gcode = SmoothPathGCode(pathPoints, 10, 300.0);

129PRINT gcode;

130

131// Edge case: only two points

132VAR twoPoints = [{x: 10.0, y: 10.0}, {x: 50.0, y: 50.0}];

133gcode = SmoothPathGCode(twoPoints, 5, 200.0);

134PRINT "\n--- Two Points Path ---\n";

135PRINT gcode;

136

137// Edge case: insufficient points

138VAR onePoint = [{x: 10.0, y: 10.0}];

139gcode = SmoothPathGCode(onePoint, 5, 200.0);

140PRINT "\n--- One Point Path ---\n";

141PRINT gcode;

142END FUNCTION

143

144Main();
Algorithm description viewbox

G-Code Path Smoothing with Bezier Curves

Algorithm description:

This G-code generator smooths a series of linear path points using cubic Bezier curves. It takes an array of points, the number of segments to use for approximating each Bezier curve, and the feed rate as input. The function calculates intermediate points along the Bezier curves to create a smooth, continuous toolpath. This is essential in CNC machining for reducing vibration, improving surface finish, and increasing machining speed, especially for complex contours.

Algorithm explanation:

The `SmoothPathGCode` function generates G-code for a smoothed toolpath. It first defines a helper function `CalculateBezierPoint` which computes a point on a cubic Bezier curve given four control points and a parameter `t` (0 to 1). The main function validates that there are at least two points. It then moves the tool to a safe height, then to the first point, and descends to the cutting depth. For each segment between consecutive points in the input array, it defines four control points for a cubic Bezier curve. The control points are approximated here by using the start and end points and simple offsets derived from the segment's direction. It then iterates `numSegmentsPerCurve` times, calculating and outputting G-code commands for each intermediate point along the Bezier curve. The time complexity is O(N*S), where N is the number of input points and S is `numSegmentsPerCurve`, due to the nested loops. The space complexity is O(1) as it uses a fixed amount of memory for variables. Edge cases like fewer than two points are handled by returning an error. The approximation of control points is a simplification; a more advanced implementation would calculate tangents more rigorously.

Pseudocode:

FUNCTION CalculateBezierPoint(p0, p1, p2, p3, t):
  Calculate intermediate values for (1-t)^3, 3*(1-t)^2*t, 3*(1-t)*t^2, t^3
  Calculate x and y coordinates using the cubic Bezier formula
  RETURN {x, y}
END FUNCTION

FUNCTION SmoothPathGCode(points, numSegmentsPerCurve, feedRate):
  Initialize smoothedCode string
  Define safeHeight, approachHeight, depth

  IF number of points < 2 THEN
    RETURN "Error: Need at least two points."
  END IF

  Append G00 command to safeHeight
  Append G00 command to first point
  Append G01 command to approachHeight
  Append G01 command to depth

  FOR i from 0 to number of points - 2:
    Define bezierP0 = points[i]
    Define bezierP3 = points[i+1]

    // Calculate bezierP1 and bezierP2 (control points) - simplified approximation
    // This step is crucial for actual Bezier curve generation and involves tangent calculations.
    // For this example, we use a simplified offset-based calculation.
    Calculate cp1x, cp1y based on midpoint and offset
    Calculate cp2x, cp2y based on midpoint and offset
    Define bezierP1 = {x: cp1x, y: cp1y}
    Define bezierP2 = {x: cp2x, y: cp2y}

    FOR j from 1 to numSegmentsPerCurve:
      Calculate t = j / numSegmentsPerCurve
      Calculate point = CalculateBezierPoint(bezierP0, bezierP1, bezierP2, bezierP3, t)
      Append G01 command to point with feedRate
    END FOR
  END FOR

  Append G00 command to safeHeight
  RETURN smoothedCode
END FUNCTION

FUNCTION Main():
  Define a list of path points
  Call SmoothPathGCode with points, segments, feedRate and print result
  Call SmoothPathGCode with two points and print result
  Call SmoothPathGCode with one point and print result
END FUNCTION

Call Main()