Scalar (Dot) Product

Scalar or Dot Product as Vector Operation

• A vector operation defined as something that can be performed between two vectors.

Local vs. Global Coordinate System

• Local Coordinate System:

  • Fixed to a particular body segment.

  • There are many local coordinate systems when quantifying human movement by segmenting the body into rigid bodies.

  • Moves with the body, differentiating it from the inertial (global) coordinate system which remains fixed.

  • Comparisons between local and global coordinate systems enable calculations of joint angles.

Joint Angle Calculations

• Absolute Joint Angle:

  • Defined as the comparison of a local coordinate system of one segment to a global coordinate system.

  • Example: Measuring whether the thigh segment is parallel to the horizontal line of the environment.

• Relative Joint Angle:

  • Involves comparing the orientation between two local coordinate systems (e.g., the thigh segment and the shank segment).

  • Typically described as the anatomical knee joint flexion angle.

Anatomical Position and Angles

• Angles are typically measured as deviations from the anatomical position (where the palm faces forward).

Motion Capture and Local Coordinate Systems

• Marker-Based Motion Capture:

  • Involves placing markers on anatomical landmarks (bony prominences) to track skeletal motion.

  • Markers are tracked using video cameras, recording anatomical movements in a defined local coordinate system based on their positions.

Defining Local Coordinate Systems in Motion Capture

• Assigning directions:

  • Example: Defining the x-direction as the vector pointing from one marker to another, and y and z directions similarly.

  • These coordinates move with the body segment they represent.

Dot Product Calculation

• Calculation of dot product for vectors:

  • For two vectors a and b defined by their coordinates:

  • $a = (a<em>1, a</em>2)$

  • $b = (b<em>1, b</em>2)$

  • The dot product is defined as:

  • $a \bullet b = a<em>1 imes b</em>1 + a<em>2 imes b</em>2$

  • Result: A single scalar number.

• Work Done by Force Example:

  • $ext{Work} = ext{Force} imes ext{Distance}$

  • Example calculation:

  • $ext{Work} = 30 imes 4 + 40 imes 5 = 320 ext{ Joules}$

Relative Joint Angle Calculation

• Calculation of the joint angle between two local coordinates:

  • Compare the vectors from the hip to the knee ($ext{p}<em>{hk}$) and the knee to the ankle ($ext{p}</em>{ka}$).

  • The angle between them is the knee joint angle ($heta_k$).

Mathematical Derivation of Knee Joint Angle

• Using the cosine inverse function:

  • $heta<em>k = ext{cos}^{-1} rac{a</em>1 imes b<em>1 + a</em>2 imes b_2}{||a|| imes ||b||}$

  • Numerical example:

  • $heta = ext{cos}^{-1} rac{1200}{44.72 imes 44.72}$

  • Result: Approximately $53.13^{ ext{o}}$ as the knee flexion angle.

  • This angle is less than 90 degrees and reflects a deviation from the anatomical position, which would be zero when the knee is fully extended.

Applications of Scalar Products

• Isolation of Force Components:

  • Scalar product can help decompose vectors into components for analysis.

  • Example:

  • $ext{Basis Vector}_x = (1, 0)$

  • $ext{Basis Vector}_y = (0, 1)$

  • Result of dot product:

    • $ext{(x,x)} = 1$

    • $ext{(x,y)} = 0$

Conclusion on Scalar Products

• Key applications discussed:

  • Joint angle calculations, work done by force, and isolation of components for analyzing forces in different dimensions.

Example Problem Statement

• Analyzing the medical elbow flexion angle:

  • Given joint center marker locations (shoulder, elbow, wrist) in global coordinates.

  • The positional vector requires alignment and directionality as the elbow moves to a fully extended position, emphasizing the movement toward the anatomical position.

  • Calculate the angle based on the vectors formed by the markers.

  • Example computation involves using 2D video capture techniques to analyze joint movements like in a squat trial with specified coordinates for angles.