Inverse and Forward Dynamics in Movement Analysis

Acknowledgement of Country

The University of Sydney acknowledges the Traditional Owners of Australia and recognizes their continuing connection to land, water, and culture.
This presentation was developed on the land of the Gadigal people of the Eora Nation.
Respects are paid to Elders past, present, and emerging, as well as to the Traditional Owners and Elders of the country on which the audience is located.

Learning Objectives

LO1: Understand the principles of data collection for biomechanics, with a particular focus on the principles of filming movements for quantitative analysis.
LO2: Understand how the mechanical properties of biological tissues influence the body's response to loads, which can lead to acute and chronic injuries.
LO6: Be able to conduct a biomechanical assessment of movement technique and effectively communicate the findings to a lay audience.

Introduction to Forward Dynamics

Definition: Forward dynamics is the opposite process of inverse dynamics. It begins with the neural command and ends with the resulting movement.
The Process Flow of Forward Dynamics:
- Variation in neural drive.
- Varying levels of recruitment of agonist and antagonist muscles.
- The net effect of all muscle forces acting at each joint generates a time-varying moment ( $M$ ).
- This moment causes the acceleration or deceleration of the adjacent segment.
- This leads to the displacement of the segment.

Purposes and Applications of Forward Dynamics

Validation: It asks whether the forces estimated from inverse dynamics can successfully reproduce the motion that was actually observed.
Understanding: It helps determine how muscle forces generate motion, identifying "cause and effect" relationships between muscle activation and movement.
Prediction: It allows for "what if" scenarios, such as predicting how performance or movement patterns will change if a specific muscle or joint is altered or injured.
Software Tools: Research and analysis are often conducted using platforms such as OpenSim (simtk.org).

Assumptions and Requirements of Forward Dynamics

General Assumptions: It shares the same foundational assumptions as the inverse dynamics solution.
Kinematic Constraints: There are no kinematic constraints; the model must be permitted to move freely as dictated by the motor (neural/muscle) inputs.
Initial Conditions: The model requires known initial conditions, specifically the position and velocity of every segment.
Inputs: The only inputs to the model are externally applied forces (e.g., Ground Reaction Forces) and internally generated muscle forces or moments.
Degrees of Freedom (DOF): The model must incorporate all important degrees of freedom and constraints relevant to the movement.
External Forces: External reaction forces, such as the Ground Reaction Force (GRF), must be calculated within the simulation.

Clinical Case Study: Hamstring Injuries in Sprinting

Theory of Injury: Hamstrings are traditionally thought to be most susceptible to muscle strain injury during the terminal swing phase of sprinting, during eccentric contraction.
Conflicting Evidence: Some research (Orchard, 2012) suggests hamstrings are more susceptible during the early stance phase.
Dynamics of High-Speed Running (Chumanov et al., 2011):
- Peak hamstring length occurs during the terminal swing phase.
- Two peak hamstring forces are observed: one in the terminal swing and one in the early stance.
- During early stance, the hamstrings transition from eccentric to concentric muscle action.
Injury Modeling (Schache et al., 2010): Forward dynamics are used to compare muscle forces in pre-injury trials versus the actual injury trial during sprinting.

EMG-Driven Musculoskeletal Modeling

Input Data:
- Musculotendon Activation: 16 linear envelopes derived from EMG.
- Kinematics: 3D joint angles (Hip Flexion/Extension, Hip Abduction/Adduction, Hip Rotation, Knee Flexion/Extension, Ankle Flexion/Extension, Ankle Subtalar Flexion).
Process Steps:
- Musculotendon Kinematics: Calculation of musculotendon lengths and moment arms.
- Musculotendon Dynamics: Determining musculotendon force based on activation and kinematics.
- Moment Computation: Calculating the joint moment from force and moment arms.
- Model Calibration: Adjusting parameters off-line by comparing predicted joint moments with experimental joint moments.

HAMI Study: Effects of Game Fatigue

Study Objective: Using forward dynamics to analyze the effects of game fatigue on hamstring and adductor muscle dynamics in elite athletes.
General Findings: Most participants ( $40\text{--}80\%$ ) showed lower muscle forces post-game. Reduced forces in hamstrings and quadriceps are linked to knee instability, electromechanical delay impairments, and anterior tibial displacement.
Individual Variability: Approximately $20\%$ of participants showed significantly increased muscle forces post-game.
Specific Muscle Force Increases (Post- vs. Pre-game):
- Adductor longus ( $addl$ ): Increase of $5\text{--}29\%$ ( $n=2$ ).
- Biceps femoris long head ( $bflh$ ): Increase of $18\text{--}61\%$ ( $n=2$ ).
- Biceps femoris short head ( $bfsh$ ): Increase of $7\text{--}31\%$ ( $n=4$ ).
- Semimembranosus ( $semim$ ): Increase of $190\text{--}205\%$ ( $n=2$ ).
- Semitendinosus ( $semit$ ): Increase of $8\text{--}12\%$ ( $n=2$ ).
Individual Participant Data (Percentage Change: $-(pre-post)/pre$ ):
- P3: Semitendinosus ( $11.7\%$ ), Semimembranosus ( $205.1\%$ ).
- P5: Biceps femoris long head ( $61.9\%$ ), Semimembranosus ( $189.9\%$ ).
- Mean Changes ( $\mu$ ): Adductor longus ( $-7.7\%$ ), Biceps femoris long head ( $-5.8\%$ ), Biceps femoris short head ( $1.3\%$ ), Semimembranosus ( $31.0\%$ ), Semitendinosus ( $-23.0\%$ ).
Technology Note: The HAMI study utilized 3D Motion Capture (Mocap) and subject-specific modeling (EMG + MRI) to capture geometric between-limb differences and unilateral pathologies that generic models miss.

Fundamentals of Inverse Dynamics

Link-Segment Modeling: Combines kinetics and kinematics to calculate internal forces and moments.
Variables Required:
- Kinetic data: Ground Reaction Force (GRF).
- Kinematic data: Segment angles, segment accelerations, moment arms.
- Anthropometric data: Segment masses, segment lengths.
Forces Acting on Model:
- Gravitational Force: Acts at the center of mass (COM) of each segment.
- Ground Reaction Force (GRF) / External Force: Measured by force transducers, acts at the Center of Pressure (COP).
- Muscle & Ligament Forces: Represented as a net muscle moment. Individual contributions (e.g., co-contraction, friction, passive structures) cannot be separated from the net value.

Assumptions of the Link-Segment Model (Inverse Dynamics)

Each segment has a fixed mass located as a point at its center of mass.
The location of the segment's center of mass remains fixed relative to the segment.
Joints are considered either hinge joints (ankle, knee) or ball-and-socket joints (hip).
The mass moment of inertia of each segment about its center of mass is constant during movement.
The length of each segment remains constant during movement.

Interpretation of Joint Data

Joint Reaction Forces (JRF): Calculated for each segment using Newton's 3rd Law (Action and Reaction) at each hinge joint.
Joint Moment: Represents the net internal moment. It indicates the primary muscle group acting (e.g., a knee extension moment indicates the knee extensor group is dominant), though it is a net effect and does not exclude co-contraction.
Joint Power: Determines the type of muscle contraction.
- Formula: $P = M \times \omega$
- $M$ = Joint moment.
- $\omega$ = Joint angular velocity.
Interpretation Criteria:
- Angular velocity/angle: Indicates whether the joint is flexing or extending.
- Moment: Records whether flexors or extensors are performing the work.
- Power: Quantifies if work was positive (concentric contraction) or negative (eccentric contraction).

Applied Calculations: Ankle Joint

Linear Force Equations:
- $Fy_{\text{ankle}} = \text{VGRF} - mg_{\text{foot}} + IF_y$
- $Fx_{\text{ankle}} = \text{HGRF} + IF_x$
Numerical Example (Ankle Forces):
- $VGRF = 600\sin(80^\circ)$ , $HGRF = 600\cos(80^\circ)$ , $m_{\text{foot}} = 1\,kg$ , $a = 0\,m/s^2$ .
- $Fy_{\text{ankle}} = 590.9 - (9.81 \times 1) + 0 = 581\,N$
- $Fx_{\text{ankle}} = -104 + 0 = -104\,N$
Ankle Moment Equation ( $M_{xy, \text{ankle}}$ ):
- $M_{xy,\text{ankle}} = (VGRF \times X_{\text{ankle}}) + (HGRF \times Y_{\text{ankle}}) + (mg_{\text{foot}} \times X_{com}) - I_{com}\alpha_{\text{foot}} + ma_xX_{com} + ma_yY_{com}$
- $M_{xy,\text{ankle}} = (600\sin(80^\circ) \times 0.02) + (600\cos(80^\circ) \times 0.05) + (10 \times 0.06) + (0.007 \times -2) + 0 + 0 = 17.6\,Nm$

Applied Calculations: Knee Joint

Inertial Force (IF) Rule: Inertial forces act in the opposite direction to accelerations; therefore, swap the sign of the acceleration during calculation.
Linear Force Equations:
- $Fy_{\text{knee}} = Fy_{\text{ankle}} - mg_{\text{shank}} + IF_y$
- $Fx_{\text{knee}} = Fx_{\text{ankle}} + IF_x$
Numerical Example (Knee Forces):
- $Fy_{\text{knee}} = 581 - (4 \times 9.81) + (4 \times -2) = 581 - 40 - 8 = 533\,N$
- $Fx_{\text{knee}} = -104 + (4 \times -5) = -104 - 20 = -124\,N$
Knee Moment Equation ( $M_{xy,\text{knee}}$ ):
- Includes contributions from the ankle moment, joint forces at the ankle, weight of the shank, and inertial factors.
- Numerical Result: $M_{xy,\text{knee}} = -2.08\,Nm$
- Note: $Fy_{\text{ankle}}$ and $Fx_{\text{ankle}}$ signs are flipped at the knee due to the opposite definition of the positive axis relative to the joint.

Applied Calculations: Hip Joint

Linear Force Equations:
- $Fy_{\text{hip}} = Fy_{\text{knee}} - mg_{\text{thigh}} + IF_y$
- $Fx_{\text{hip}} = Fx_{\text{knee}} + IF_x$
Numerical Example (Hip Forces):
- $Fy_{\text{hip}} = 533 - (6 \times 10) + (6 \times 2) = 485\,N$
- $Fx_{\text{hip}} = -124 + (6 \times -3) = -142\,N$
Hip Moment Equation ( $M_{xy,\text{hip}}$ ):
- Includes contributions from the knee moment, knee joint forces, weight of the thigh, and inertia.
- Numerical Result: $M_{xy,\text{hip}} = -53.1\,Nm$