1Angular Kinematics of Human Movement (Ch 9)• Angular Measures Relative versus Absolute Angles• Angular Kinematic QuantitiesAngular Position/DisplacementAngular VelocityAngular Acceleration• Relationship between Linear & Angular MotionCopyright © 2017Brian R. Umberger, Ph.D.University of Massachusetts AmherstAngular or Rotational MotionAngular or rotational motion occurs when an object moves in an arc about an axis. The axis may be real or imaginary, but will always be oriented perpendicular to the plane in which the rotation occurs.real axis imaginary axisAngular vs Linear MotionAngular MotionCurvilinear MotionRectilinear MotionAngular vs Curvilinear MotionAngular Motion Curvilinear MotionP1P3P2P1P3P2lines are different lengthslines are the same length2Absolute & RelativeAnglesAbsolute (Segment) AnglesAngles measured between the longitudinal axis of a body segment and a fixed reference lineExample: thigh angle relative to the horizontalAbsolute & RelativeAnglesRelative (Joint) AnglesAnatomically-based angles measured between the longitudinal axes of adjacent body segmentsExamples: knee joint angle, elbow angleMeasurement UnitsAngles are typically measured in degrees One full circle = 360Angles are measured so that counterclockwisemotion is considered to be positivecircumference = 2r Therefore, there are 2radians in a circleMeasurement UnitsFor some calculations the required unit of measure for angles is the radianSince a radian is a length (of the arc) divided by a length (of the radius), a radian is a dimensionless number3Measuring Body AnglesGoniometer:– One arm fixed to protractor at 0 deg– Other arm free to rotate– Center of goniometer over joint center– Arms aligned over longitudinal axesElectrogoniometer (elgon):– Can be used to record joint angles dynamicallyInclinometers:– Provides absolute angles of body segmentsMeasuring Body AnglesPelvis/TrunkThighFootSegmentsJointsHipKneeAnklesegment endpoints are used to define segments and jointsmarkers placed at segment endpointsLegCompute angle of inclination of a segment relative to a reference frame fixed at the distal endPelvis/TrunkThighFootAngle is measured counter-clockwise from the right horizontaltan = (yprox-ydist)/(xprox-xdist) = tan-1[(yprox-ydist)/(xprox-xdist)]Absolute or Segmental AnglesLegCompute the angle of inclination of the leg segment, relative to the right horizontal, given the XY coordinates of the knee and ankleleg= tan-1[(yprox-ydist)/(xprox-xdist)]leg= tan-1[(0.51-0.09)/(1.42-1.09)]leg= tan-1[0.42/0.33]leg= tan-1[1.2727]leg= 51.8Absolute or Segmental AnglesLeg(1.42, 0.51)(1.09, 0.09)leg4footlegthightrunkAngles are expressed relative to the anatomical positionhip= thigh- trunk(+flexion extension)knee= thigh- leg(+flexion extension)ankle= leg- foot+ 90 (+dorsi plantar)Relative or Joint AnglesJoint angles are computed as the difference between the angles of the segments that form the jointfootlegthightrunkknee= thigh- legknee= 86.1 - 51.8 = 34.3This is a positive values, so it means that the knee is flexed 34.3 from the anatomical positionIt would be equally correct to say that the knee joint angle is 145.7 (180 - 34.3)Relative or Joint AnglesCompute the knee joint angle as the difference between the thigh and leg segment angles51.886.1knee• Sagittal plane joint angles during able-bodied human walking– Solid lines are for normal walking– Dashed and dotted lines are for walking with 20% slower and 20% quicker strides than normalJoint Angles Knee Joint Angle - Stiff Knee GaitSome people with cerebral palsy can’t flex or extend their knee very well; interventions include surgically lengthening the rectus femoris, and/or transferring the location of the RF insertionpre-surgerypost-surgeryknee joint angle (deg)knee joint angle (deg)normal ------patient _________% of gait cycle% of gait cycle5Joint AnglesClinically-relevant joint angles are not limited to the sagittal planerearfoot angle (right limb, rear view)Excessive pronation of the ankle has been linked with knee pain in runningMay be possible to control this using orthoticsJoint AnglesQ-angle (right limb, front view)The Q-angle tends to be larger in women than in men, due to a wider pelvis relative to leg lengthA large Q-angle may predispose a person to develop patellofemoral painClinically-relevant joint angles are not limited to the sagittal planeJoint AnglesNon-contact ACL InjuryA suspected mechanism of non-contact ACL injury is the combination of femur adduction with simultaneous knee abduction and ankle eversionPlaces strain on the ACL(Hewett et a., 2005)Clinically-relevant joint angles are not limited to the sagittal planeAngular DisplacementAngular displacement () is difference in initial& final angular positions– Counterclockwise is positive– Clockwise is negative– Units are degrees, radians, or revolutions = angular position = ang posf− ang posior, = f- iFor human joints, direction can also be indicated with appropriate terminology (flexion, adduction, etc)6Angular Distance & DisplacementSimilar distinction as with linear motion: Angular displacement is final position minus initial position, while angular distance accounts for whole path of motionangular dist = 160angular disp = 40Positive Direction40070 30 10 50 Angular speed () is how fast something is rotatingangular speed = angular distance = change in time tAngular velocity () is rate of change in ang posang vel = angular displacement = change in time tAppropriate Units: deg/s, rad/s, rev/s, & rpmAngular Speed & VelocityTo calculate angular velocity: = = ang posf– ang posittimef–timeiAngular velocity is a vector quantity; directionof motion must be specified– positive/negative (e.g., −6 deg/s)– clockwise/counterclockwise (e.g., 15 rad/s CW)– joint specific (e.g., 90 deg/s into flexion)Angular Speed & VelocityAngular acceleration () is the rate of change of angular velocityang accel = change in ang vel = change in time tTo calculate angular acceleration:ang accel = ang velf– ang velitimef–timeiAppropriate Units: deg/s2, rad/s2, & rev/s2Angular Acceleration7Angular KinematicsDifferentiation*i= (i+1- i-1) / 2ti= (i+1- i-1) /
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