Inverse Dynamics. OpenSim Workshop

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OpenSim Workshop Inverse Dynamics

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Key Concepts

• Kinematics: coordinates and their velocities and accelerations

• Kinetics: forces and torques • Dynamics: equations of motion

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Kinematics: Coordinates and

their Velocities and Accelerations

• Coordinate

– Joint angle or distance specifying relative orientation or location of two body segments

• Coordinate velocity

– Derivative (rate of change) of a coordinate with respect to time

• Coordinate acceleration

– Time derivative of a coordinate velocity with respect to time

• Kinematics

– Set of all coordinates and their velocities and accelerations

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Kinetics: Forces and Torques

• Kinetics

– Forces and torques cause the model to accelerate

• Force

– Applied to points (e.g., ground reactions) or between points (e.g., muscles)

• Torque

– Applied to a coordinate (e.g., joint torque)

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Dynamics: Equations of Motion

ɺ

knowns unknowns

F

q

G

q

C

q

M

τ

=

₍

₎

−

₍

_,

₎

−

₍

₎

−

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Exercise

23.6°

θ

31.3°

1. For the model shown on the right, what is the value (

θ

) of the knee

coordinate (Note: extension is +)? A. 23.6°

B. -54.9° C. 31.3° D. -125.1°

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Exercise

23.6°

θ

31.3°

2. Given that the model shown on the right is at rest, what is the

velocity of the knee?

A. 23.6°/s B. -54.9°/s C. 3.89°/s D. 0°/s

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C. B. Exercise A. w w w

3. For the model poses shown below at rest and with gravity (g) as the only force acting on the model, which pose requires the largest torque at the knee joint ?

d_A _d_B dC

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The inverse problem

Position Data Moments Accelerations Velocities.

Musculoskeletal Geometry Multi-Joint Dynamics Forces d dt d dt Force Data Angles Position Data Moments Accelerations Velocities.

Musculoskeletal Geometry Multi-Joint Dynamics Forces d dt d dt Force Data Angles

Going from subject motion to joint

kinematics

[Inverse Kinematics]

Dealing with noisy data

t 0 1 2 -5000 0 5000 ( )t x ′′ t 0 1 2 -5000 0 5000 ( )t x ′′

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Major flexors a M Tibialis posterior Soleus Gastrocnemius Major extensors Extensor digitorum Tibialis anterior

Net ankle moment

tp g s ed ta θ r I m, y m ɺɺ x m ɺɺ mg θɺɺ I Fx Fy T

Going from joint kinematics to joint moments

[Inverse Dynamics] Solving the muscle force distribution problem [Static Optimization] x m F_x = ɺɺ Σ θɺɺ I T = Σ x x x, ,ɺ ɺɺ y y y, ,ɺ ɺɺ θ θ θ, ,ɺ ɺɺ y m F_y = ɺɺ Σ

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The Inverse Problem

Position Data

Moments Accelerations Velocities.

Musculoskeletal Geometry Multibody Dynamics Forces d dt d dt Force Data Angles

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Position Data

Musculoskeletal Geometry Multibody Dynamics Forces d dt d dt Force Data Angles F_x F_y F_z M_x M_y M_z 3 Forces 3 Moments

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Position Data

Musculoskeletal Geometry Multibody Dynamics Forces d dt d dt Force Data Angles Video Cameras Reflective Markers

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• Identify research question for the inverse problem • Determine what should be

measured and modeled • Compute joint kinematics • Filter and differentiate joint

kinematics data Position

Data

Musculoskeletal Geometry Multibody Dynamics Forces d dt d dt Force Data Angles Inverse Kinematics

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Example Research Questions

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( )t ( t) ( t) x′ = 40π cos 2π +20π cos 20π t 0 1 2 -200 0 200 ( )t x′ ( )t ( t) ( t) x = 20sin 2π +sin 20π t 0 1 2 -30 0 30 x

Differentiation Amplifies High-Frequency Noise 1 Hz signal ( )t ( t) ( t) x′′ = −80π2 sin 2π −400π2 sin 20π t 0 1 2 -5000 0 5000 ( )t x ′′ 10 Hz noise 20 = SNR 2 = SNR 2 . 0 = SNR

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Position Data

Musculoskeletal Geometry Multibody Dynamics Forces d dt d dt Force Data Angles Inverse Kinematics

• Identified research question for the inverse problem

• Determined what should be measured and modeled

• Computed joint kinematics

• Filtered and differentiated joint kinematics data

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Position Data

Musculoskeletal Geometry Multibody Dynamics Forces d dt d dt Force Data Angles Inverse Dynamics Inverse Kinematics

• Derive equations of motion defining the model

• Solve equations of motion for joint moments

θ r I m, y m ɺɺ x m ɺɺ mg θɺɺ I Fx Fy T x m F_x = ɺɺ Σ θɺɺ I T = Σ x x x, ,ɺ ɺɺ y y y, ,ɺ ɺɺ θ θ θ, ,ɺ ɺɺ y m F_y = ɺɺ Σ

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A Possible Inverse Dynamics Question

What are the sagittal plane moments about the ankle, knee, and hip during a maximum height jump?

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Inverse Dynamics Input: The Experimental Results

Experiments provide • joint angles

• angular velocities

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Inverse Dynamics Equations: Multibody Dynamics

• Planar 3 degrees of freedom • Position (orientation) in

global coordinate system

• Segment length = l_i

• Distance to mass center = r_i

• Moments of inertia about mass center 1

θ

2

θ

3

θ

3 r 2 r 2 l 1 r 1 l x y 3 3, I m 2 2, I m 1 1, I m

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Inverse Dynamics Equations: Multibody Dynamics

Solved algebraically from the ground up

x m F_x = ɺɺ Σ θɺɺ I T = Σ x x x, ,ɺ ɺɺ y y y, ,ɺ ɺɺ θ θ θ, ,ɺ ɺɺ y m F_y = ɺɺ Σ

Joint Moments that generate the motion

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Inverse Dynamics Output: Net Joint Moments 40 0 20 40 60 80 100 80 120 160 200 240 Percent Jump J o in t M o m e n ts ( N -m ) knee ankle hip

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Position Data

Musculoskeletal Geometry Multibody Dynamics Forces d dt d dt Force Data Angles Inverse Dynamics Inverse Kinematics

• Derived equations of motion defining the model

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Inverse Dynamics

TIPS & TRICKS

Filter your raw coordinate data

Make sure GRFs were applied correctly and check residuals on the body connected to ground