Biomechanics of Movement | Lecture 8.2: Leaping into Inverse Dynamics Analysis of a Vertical Jump

Inverse Dynamic Analysis in Maximum Height Jump

Introduction to Inverse Dynamic Analysis

• The video introduces the concept of inverse dynamic analysis, focusing on calculating sagittal plane moments about the ankle, knee, and hip during a maximum height jump.

• A research question is posed: "What are the sagittal plane moments about the ankle, knee, and hip during a maximum height jump?"

Data Collection for Analysis

• Data collection involves using a force plate to gather ground reaction forces and motion capture sensors placed at key joints (hip, knee, ankle).

• Additional data includes joint angles, angular velocities, angular accelerations, and EMG data for muscle force estimation validation.

Experimental Setup

• The experimental setup features participants crouching before takeoff; only projectile motion needs simulation post-takeoff.

• A model with four body segments (foot, shank, thigh, hat segment for head/arms/trunk) is utilized for calculations.

Required Model Parameters

• Essential parameters include masses and inertias of body segments along with distances between joints (e.g., distance from ankle to knee).

• Calculations will be performed at each time point leading up to takeoff; joint moments are plotted against time.

Analyzing Joint Moments

• Observations indicate that hip moments peak before ankle moments—similar patterns observed in sports like baseball pitching.

Steps in Inverse Dynamic Analysis

Free Body Diagram Creation

• The first step involves creating free body diagrams isolating each body segment while including all relevant forces and moments.

Motion Equations Formation

• Next steps involve forming motion equations by differentiating position expressions to compute velocities and accelerations.

Application of Newton's Laws

• Finally, Newton's second law is applied: F = ma , considering both translational and rotational dynamics without additional terms found in textbooks.

Simplified Equation Overview

• The simplified equation focuses on taking moments about the center of mass without extra terms present in standard equations.

Visual Representation of Forces

• A visual representation shows reaction forces (Fx, Fy), torque (T), weight due to gravity acting at the center of mass alongside known geometry.

Assumptions in Modeling

• Assumptions include a planar system with three degrees of freedom related to angles at the ankle, knee, and hip relative to horizontal positions.

Free Body Diagrams and Forces in Biomechanics

Creating a Free Body Diagram for the Foot

• The process begins with drawing a free body diagram of the foot, including reaction forces and moments. A ground reaction force is assumed to act at the toe during a jump.

• The components of the ground reaction force are identified as f_xg and f_yg , drawn in positive directions for clarity, regardless of their actual direction.

• Reaction forces at the ankle are denoted as f_x1 (negative x-direction) and f_y1 (negative y-direction), along with a negative moment to maintain consistency across diagrams.

• Newton's third law dictates that corresponding forces on different segments must be consistent; thus, similar variables will have opposite directions in subsequent diagrams.

• The speaker encourages viewers to derive equations based on the free body diagram before revealing them, promoting active engagement with the material.

Applying Newton's Laws

• The sum of forces in the x-direction is established: f_xg - f_x1 = 0 . Assuming no mass or acceleration simplifies this equation to zero.

• In the y-direction, we have f_yg - f_y1 = 0 , reinforcing that all forces balance out under static conditions.

• Moments about the ankle are calculated by considering contributions from various forces. Choosing this point simplifies calculations since certain terms drop out due to their line of action through it.

• Contributions from ground reaction forces are included: f_xgh - f_ygl - t_1 = 0 . This equation also equals zero due to no mass or inertia present.

• The system can be solved if ground reaction forces (f_xg, f_yg) are known, leading to three equations with three unknowns (f_x1, f_y1, t_1).

Transitioning to Shank Analysis

• Moving on from foot analysis, a new free body diagram for the shank segment is created using similar steps as before while ensuring clarity in representation by placing it in quadrant one.

• Reaction forces at both ankle and knee joints are represented consistently. Forces at these points follow Newton’s third law but are drawn oppositely compared to previous diagrams for clarity.

• Gravity's effect is incorporated into the shank diagram as an additional downward force (m_1 g), which must be accounted for when analyzing motion dynamics.

Kinematic Equations for Shank Motion

• Kinematic equations focus on determining positions, velocities, and accelerations relative to an origin set at the ankle joint.

• Position coordinates for center of mass are defined:

• x_1 = r_1 cos(theta_1)

• y_1 = r_1 sin(theta_1)

• Derivatives of position expressions yield velocity equations; specifically, velocity in x-direction involves both radius change and angle change over time.

Understanding Joint Moments and Forces in Biomechanics

Derivatives and Acceleration of Body Segments

• The derivative of sine is cosine, leading to the expression for velocity: x_1 dot = -r_1 theta_1 dot sin(theta_1) , where r_1 is constant.

• The second derivative gives acceleration: x_1 ddot = r_1 (theta_1 ddot sin(theta_1) - (theta_1 dot)^2 cos(theta_1)).

• Expressions for y_1 dot and y_1 ddot are derived similarly; refer to the textbook for details.

Applying Newton's Second Law

• After deriving expressions, forces in the x and y directions are summed, with known ground reaction forces aiding in solving unknowns.

• Moments about the center of mass are calculated using forces acting at joints (ankle and knee), following a similar summation approach.

Expanding the System of Equations

• With measured ground reaction forces, only three unknowns remain (forces at knee), allowing for solvability of equations.

• Moving to analyze the thigh segment involves creating a free body diagram and computing positions, velocities, and accelerations.

Thigh Segment Analysis

• The same procedure applies as before; expressions become longer but follow established methods.

• A system with six equations emerges from applying Newton’s laws to both forces and moments.

Final Body Segment: Head/Arms/Torso

• For this segment, repeat previous steps: create diagrams, compute forces/moments. This leads to nine equations with nine unknown variables if ground reaction forces are not measured.

• Even without precise measurements of ground reaction forces, inverse dynamic analysis can be performed across time points to derive joint moments throughout motion trajectories.

Recap on Joint Moments