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1Three-Dimensional KinematicsEXS 587Dr. MoranFall 2007Lecture Outline• Three-Dimensional Reconstruction• Coordinate Systems• Marker Systems• Transformation Matrices2Three-Dimensional KinematicsIntroduction• Camera Minimum: 2• Location:• Each camera only provides a set of coordinates of marker locations• How do we go from sets of 2D coordinates to 3D coordinates of markers?Photogrammetry is “the art, science, and technology of obtaining reliable information about physical objects and the environment through the processes of recording, measuring, and interpreting photographic images and patterns of recorded radiant electromagnetic energy and other phenomena” (Wolf, 1983)3D Reconstruction Techniques• Direct Linear Transformation (DLT)• Abdel-Aziz & Karara» ASP Symposium on Close-Range Photogrammetry, 1971, p1-18• Operating Principles– Control Points (n>=6)» Coordinates known precisely in real 3D space»» Usually up to 20 control points is used– 2D Camera Capture»» Control points digitized to get 2D coordinates of each point33D Reconstruction Techniques(Continued)Global CSControl point (x1, y1, z1)UVDigitized Coordinates(U1, V1)CAMERAKNOWN:(U1, V1)(x1, y1, z1)UNKNOWN:11 camera constants A-LDefine camera orientation, distance, and optics3D Reconstruction Techniques(Continued)Global CSControl point (x1, y1, z1)UVDigitized Coordinates(U1, V1)CAMERAKNOWN:(Ui, Vi), (xi, yi, zi)For i=1-6UNKNOWN:11 camera constants A-LDefine camera orientation, distance, and opticsADD MORE CONTROL POINTS(x2, y2,z2)(x6, y6,z6)(x3, y3,z3)(x5, y5,z5)(x4, y4,z4)43D Reconstruction Techniques(Continued)• Two equations are generated for each control point yielding 12 equations with 11 unknowns:ax1 + by1 + cz1 + d – eU1x1 – fU1y1 – gU1z1 = U1....ax6 + by6 + cz6 + d – eU6x6 – fU6y6 – gU6z6 = U6ANDhx1 + jy1 + kz1 + 1 – eV1x1 – fV1y1 – gV1z1 = V1....hx6 + jy6 + kz6 + 1 – eV6x6 – fV6y6 – gV6z6 = V63D Reconstruction Techniques(Continued)• Now repeat for each camera• DO NOT TOUCH CAMERAS!!!!»• How many control points?5Control Point ConfigurationsTypical ExamplesOther techniques include a dynamic wanding were many frames of data are collected and an optimization scheme is used to best fit camera constants to the collected control points.Coordinate Systems (CS)• Global (Lab) CS•• Orthogonal• Local CS•• Assume NO movement between markers and between markers and underlying bone6Marker Systems• Necessary Requirement:» At least THREE points on each rigid segment• Marker Configurations1.) Mounted via bone pinsREADING: Effect of skin movement on the analysis of skeletal knee joint motion during running. Reinschmidt et al. (1997)2.) Skin-mounted on anatomical landmarks3.) Marker arrays on rigid surface attached to body4.) abcHelen-Hayes Marker Set• Typically used in gaithttp://www.sportsci.com/apasgait/Afbeelding1.2.jpg7Finding Local AxesThree Point MethodabcUQStep 1: define unit vector along ab = UStep 2: define unit vector along ac = QWStep 3: define a unit vector perpendicular to the plane containing ab and acW = cross(U,Q)VStep 4: define a unit vector at right angles to both U and W as the third component of the axis triadV = cross(W,U)The triad V, U, W represents an orthogonal body fixed CS [the body reference frame]Body & Reference FramesGLOBALBODYTBGThe transformation of the body segment with respect to global….“Where is the body with respect to global?”XYZTranslations can be LINEAR and/or ROTATIONAL!8Some Matrix MathGLOBALBODYPgPbPG= [TBG] PBPB= [TBG]-1PGTBGWhat is the Transformation Matrix?• 4x4 matrix• Explains where the origin and rotations of the Body CS relative to the Global CS– Displacement (3 degrees of freedom)• X-coordinate• Y-coordinate• Z-coordinate– Rotation (3 degrees of freedom)• Rotation about x-axis• Rotation about y-axis• Rotation about z-axisBODYGLOBAL9Matrix MultiplicationA Refresher• ONLY defined when the inner dimensions of the two matrices are the SAME– Eg. A 2x3 matrix multiplied by a 3x4 yields a 2x4 matrix• To perform multiplication, each element of the result is obtained by multiplying every element in the jth row of A by the corresponding entry in the kth column of B and then adding the result which becomes the jk element of the product matrixMatrix MultiplicationA Refresher (con’t)4 3• AB = 7 29 02 51 6Back to this transformation equation. If we know that the transformation matrix is a square 4x4 matrix what are the dimensions of PB?PG= [TBG] PB10Transformation MatrixTBG=1 0 0 0XoYoZo3x3Rotation MatrixSome QuestionsP1P2P3A rigid segment has three points firmly attached to its surface. At one instance in time the lab coordinates of the points are:P1 (0.1, 3.4, -1.2)P2 (1.1, 3.8, -0.2)P3 (1.3, 3.9, -0.9)1.) What is the distance between P1 and P2?2.) Should this distance remain constant throughout a movement sequence?3.) Find the transformation from lab CS to body-fixed CS4.) Calculate the local coordinates of P1, P2, and


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SUNY Cortland EXS 587 - Three-Dimensional Kinematics

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