Three-Dimensional Biomechanical Analysis of Human Movement

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1 Three-Dimensional Biomechanical Analysis of Human Movement Anthropometric Measurements Motion Data Acquisition Force Platform Body Mass & Height Biomechanical Model Moments of Inertia and Locations of COM of Body Segments Marker 3-D Positions 3-D Orientation and Position of Body Segments 3-D Kinematics of Body Segments Ground Reaction Forces and Moments Center of Pressure/Point of Force Application Inverse Dynamics Resultant Joint Forces and Moments Powers/Energy

2 Ground Reaction Forces Vertical Force A/P Force M/L Force Horizontal Shear Twisting Torque (free moment) F=F x +F y +F z F z F y F x

3 Ground Reaction Forces

4 Piezoelectric type Force Platforms A piezoelectric material, quartz crystal, will generate an electric charge when subject to mechanical strain. Quartz crystals are cut into disks that respond to mechanical strain in a single direction. Strain Gauge type Use strain gauge to measure stress in machined aluminum transducers (load cells). Deformation of the material causes a change in the resistance and thus a change in the voltage (Ohms Law: V = I * R).

5 Two Common Types of Force Plates A flat plate supported by four triaxial transducers A flat plate supported by one centrally instrumented pillar

6 AMTI (Strain Gauge) Force Platform Output signals from the platform: F x : the anterior/posterior force F y : the medial/lateral force F z : the vertical force M x : the moment about the anterior/posterior axis M y : the moment about the medial/lateral axis M z : the moment about the vertical axis

7 Kistler (Strain Gauge) Force Platform

8 Center of Pressure (COP) T z F Center of Pressure All the forces acting between the foot and the ground can be summed and yield a single reaction force vector (F) and a twisting torque vector (T z about the vertical axis). Under normal condition there is no physical way to apply T x and T y. The point of application of the ground reaction force on the plate is the center of pressure (COP).

9 Generally, the true origin of the plate is not at the geometric center of the plate surface. The manufacturer usually provides the offset data. Computation of the COP X Y Z r T z F COP (x, y, 0) true origin (a, b, c) The moment measured from the plate is equal to the moment caused by F about the true origin plus T z.

10 Computation of the COP Z T z F X r COP (x, y, 0) Y true origin (a, b, c) M = r F + T z

11 Computation of the COP M = r F + T z r = (x-a, y-b, -c) F = (F x, F y, F z ) T z = (0, 0, T z ) M = (M x, M y, M z ) known: a, b, c; unknown: x, y force values from plate outputs unknown: T z torque values from plate outputs

12 T z r cop COP (a x, a y, a z ) r cop = [a x, a y, a z ] M GRF = r cop F + T z

13 Computation of the COP M x = (y-b) F z + c F y M y = -c F x - (x-a) F z Z Tz F M z = (x-a) F y - (y-b) F x + T z x = -(M y + cf x )/F z +a X r COP (x, y, 0) y = (M x - cf y )/F z +b T z = M z - (x-a)f y + (y-b)f x Y true origin (a, b, c) M = r F + Tz

14 Force Plate Coordinate System Y f X f COP r FPCS O FP Z f r cop r OFP in GCS r cop = R r FPCS + r OFP Z g X g known information during laboratory setup (calibration) Y g

15 GRF in Global Coordinate System GRF Y f Tz X f O FP Z f GRF GCS = R GRF FPCS Z g X g Tz GCS = R Tz FPCS Y g

16 Three-Dimensional Biomechanical Analysis of Human Movement Anthropometric Measurements Motion Data Acquisition Force Platform Body Mass & Height Biomechanical Model Moments of Inertia and Locations of COM of Body Segments Marker 3-D Positions 3-D Orientation and Position of Body Segments 3-D Kinematics of Body Segments Ground Reaction Forces and Moments Center of Pressure/Point of Force Application Inverse Dynamics Resultant Joint Forces and Moments Powers/Energy

17 Determining Body Segment and Joint Kinematics Three-step procedure Three-dimensional marker positions Body segment (limb) positions and orientations (assuming rigid body) Relative orientation and movement of limb segments (joint kinematics)

18 Step #1 Marker Position 3-D reconstruction from several 2-D images Each point seen by at least 2 cameras Vicon system displays reconstructed points (saves you a ton of time) Now, for Step #2

19 Step #2 Segment positions and orientations Defining segment coordinate systems Position described by the segment origin Orientation provides the absolute angles

20 Segment definitions (the need for marker sets) Absolutely necessary for kinematic variables to be measured/calculated Key definitions in 2D and 3D kinematics: Segment endpoints for creation of links Segment dimensions (body segment parameters) Orientation of segments, for angular data

21 Pelvis marker set (general)

22 Helen Hayes marker set

23 Foot marker set (general)

24 Step #3 Relative position and orientations between segments Relationship between the Local c.s. (LCS) and the Global c.s. (GCS) y i x i Linear Rotational z z i x y

25 Linear Kinematics of a Rigid Body Position Vector: A vector starting from the origin of a coordinate system to a point in the space is defined as the position vector of that point. Y r P P r Q r Q/P Q O X Displacement Vector: The vector difference of two position vectors is defined as the displacement vector from the first point (P) to the second point (Q). r Q/P = r Q - r P

26 Linear Transformation x z o r oi x i y o i r p z i x i r pi o i p y i z i Assume that GCS (x,y,z) and LCS (x i,y i,z i ) coincide with each other in the beginning (t = 0). The LCS is only translating, that y i is there is no rotational movement. At time t, the LCS moves to a location which is represented by a position vector of r oi. r p = r pi + r oi

27 Rotational Transformation x x i z i z z r pi p y i y Assume that GCS (x,y,z) and LCS (x i,y i,z i ) coincide with each other in the beginning (t = 0). At time t, the LCS rotates with respect to the GCS and reaches a final orientation. p r p = R r pi z i r p y i R: rotation matrix from LCS to GCS x x i

28 Rotational Matrix z r p = R r pi x z i p r p x i y i If directions of the x i, y i, and z i of the LCS axes can be expressed by unit vectors v 1, v 2, and v 3, respectively, in the GCS, the rotation matrix from the LCS to GCS is defined as R R = & v1 i $ v $ 1 j $ % v1 k v 2 i v2 j v k 2 v v v i # j! =! k! " & v $ v $ $ % v 1x 1y 1z v v v 2x 2y 2z v v v 3x 3y 3z #!!!"

29 Relationship between the LCS and Fixed (Global) coordinate system (GCS) Linear transformation+rotational transformation y i o i x i r p = R r pi + r oi z r oi z i r pi p r p x y

30 Relationship between the LCS and Fixed (Global) coordinate system (GCS) 4 4 Transformation Matrix r p = R r pi + r oi & r $ $ r $ % r px py pz #!!! " = & a $ $ a $ % a a a a a a a #& r! $! $ r!" $ % r pxi pyi pzi #!!! " + & r $ $ r $ % r oix oiy oiz #!!!" & r $ $ r $ r $ % 1 px py pz #!!!! " = & a $ $ a $ a $ % a a a a a a r r r oix oiy oiz 1 #& r! $! $ r! $ r! $ "% 1 pxi pyi pzi #!!!! "

31 Determining Joint Kinematics If the orientation of two local coordinate systems (two adjacent body segments) are known, then the relative orientation between these two segments can be determined. y j r p = R i r pi (1) r p = R j r pj (2) z (1)=(2) z j o z i j P x j r pj r pi o i r x p i yi R i r pi = R j r pj r pi = R i -1 R j r pj x o y

32 Determining Joint Angles 3-D joint angles are concerned about the relative orientation between any two adjacent body segments, therefore, only the rotation matrix is needed for computation. r pi = R i -1 R j r pj =R i/j r pj

33 Basic Rotational Matrices Rotation about the X-axis r p = R r pi z i z θ y i θ x i y x R = $ ' & ) & 0 cosθ sinθ) %& 0 sinθ cosθ ()

34 Basic Rotational Matrices Rotation about the Y-axis r p = R r pi z i x i x z φ φ yi y R = $ cosφ 0 sinφ' & ) & ) %& sinφ 0 cosφ( )

35 Basic Rotational Matrices Rotation about the Z-axis z r p = R r pi z i γ γ y i y x x i R = $ cosγ sinγ 0' & ) & sinγ cosγ 0 ) %& 0 0 1( )

36 Cardan / Euler Angles Cardan/Euler angles are defined as a set of three finite rotations assumes to take place in sequence to achieve the final orientation (x 3,y 3,z 3 ) from a reference frame (x 0,y 0,z 0 ). Cardan angles: all three axes are different Euler angles: the 1 st and last axes are the same z 0 z 1 y 1 z 2 θ 2 y 2 z 1 y 1 z 3 z 2 θ 3 y 2 y 3 x 0 x 1 y 0 θ 1 x 1 x 3 x 2 x 2

37 Three-Dimensional Biomechanical Analysis of Human Movement Anthropometric Measurements Motion Data Acquisition Force Platform Body Mass & Height Biomechanical Model Moments of Inertia and Locations of COM of Body Segments Marker 3-D Positions 3-D Orientation and Position of Body Segments 3-D Kinematics of Body Segments Ground Reaction Forces and Moments Center of Pressure/Point of Force Application Inverse Dynamics Resultant Joint Forces and Moments Powers/Energy

38 Inertial parameters

39 Inertial parameters

40 Kinetic Analysis of Human Movement Anthropometric Measurements Body Mass & Height Biomechanical Model Moments of Inertia and Locations of COM of Body Segments Motion Data Acquisition Marker 3-D Positions 3-D Orientation and Position of Body Segments 3-D Kinematics of Body Segments Force Platform Ground Reaction Forces and Moments Center of Pressure/Point of Force Application Inverse Dynamics Resultant Joint Forces and Moments Powers/Energy

41 Assumptions of the Link-Segment Model Each segment has a point mass located at its individual COM Location of the segmental COM remains fixed (w.r.t. segment endpoints) during the movement Joints are considered as hinge or ball & socket joints (max. 3 DOF each) Segment length and Mass moment of inertia about the COM are constant during movement

42 Forces Acting on the Link-Segment Gravitational Force acting at the COM of the body segment Ground Reaction or External Contact Forces acting at the COP or contact point Net Muscle and/or Ligament Forces acting at the joint

43 Free-Body Diagram A free-body diagram is constructed to help identify the forces and moments acting on individual parts of a system and to ensure the correct use of the equations of motion The parts constituting a system are isolated from their surroundings and the effects of the surroundings are replaced by proper forces and moments In a free-body diagram, all known and unknown forces can be shown

44 Free-Body Diagram (where to draw the line ) com air resistance F air ma F A mg m f a f m f g F R (GRF) F = F R + mg + F air = ma F R F = F R + m f g + F A = m f a f

45 Equations of Motion of a Rigid Body If the resultant force acting on a body is not zero, à the body s acceleration will be proportional to the magnitude and in the direction of this resultant force y y F 3 F 4 I α F 2 COM x COM m a x F 1 F = m a M = I α

Università degli Studi di Bari. mechanics 1. Load system determination. Joint load. Stress-strain distribution. Biological response 2/45 3/45

Università degli Studi di Bari mechanics 1 Load system determination Joint load Stress-strain distribution Biological response 2/45 3/45 ? 4/45 The human body machine Energy transformation Work development