Lecture Notes: Angular Kinematics and Joint Angle Calculation

Introduction

• Follow-up from last lecture on angular kinematics.
• Focus on calculating and understanding angles.
• Examples provided using a coordinate plane.

Key Concepts

Types of Angles

• Relative Angle: Joint angle; the angle between segments.
• Absolute Angle: Segmental angle relative to the horizontal.

Example Problem

• Objective: Calculate the knee joint relative angle.
• Given: Coordinates for hip, knee, and ankle on a plane.

Calculating the Knee Joint Angle

Method 1: Using Right Triangles

• Concept: Split the non-right triangle into two right triangles.
• Steps:
  1. Create an imaginary horizontal line through the knee joint angle.
  2. Identify two triangles (upper and lower) using coordinates.
  3. Upper Triangle (Thigh):
     • Use coordinate points to establish a right triangle.
     • Calculate the vertical drop (4 units) and horizontal distance (6 units).
  4. Lower Triangle (Shank):
     • Create a right triangle using given coordinates (11,6), (9,1).
     • Calculate distances: vertical (5 units), horizontal (2 units).

Trigonometric Calculations

• Use tangent function (opposite/adjacent) to find angles:
  • Theta (upper triangle): 33.69 degrees
  • Alpha (lower triangle): 68.2 degrees
• Combine angles to find knee joint angle (Beta): 101.89 degrees

Method 2: Law of Cosines

• Equation: ( a^2 = b^2 + c^2 - 2bc \cos(\alpha) )
• Steps:
  • Identify sides of the triangle (A, B, C)
  • Use distance formula to find side lengths.
  • Apply law of cosines to find angle.
  • Result matches Method 1: 101.88 degrees

Calculating Thigh Segment Angle (Absolute Angle)

• Objective: Find the angle relative to the right horizontal.
• Procedure:
  • Use previously found angle (Theta = 33.69 degrees).
  • Calculate remaining angle using 180 degrees as full flat line.
  • Subtract to find absolute angle: 146.31 degrees

Practice Problem

• Objective: Calculate shank segment angle.
• Task: Similar method as thigh segment angle.
• Recommendation: Practice with different triangles and coordinates.

Conclusion

• Multiple methods available to solve angular kinematics problems.
• Importance of practice and familiarity with concepts and formulas.
• Next lecture will cover more practice problems.