visual3d:documentation:kinematics_and_kinetics:six_degrees_of_freedom
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| visual3d:documentation:kinematics_and_kinetics:six_degrees_of_freedom [2024/07/12 14:00] – created sgranger | visual3d:documentation:kinematics_and_kinetics:six_degrees_of_freedom [2024/07/17 15:45] (current) – created sgranger | ||
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| === Pose Estimation with 6 Degree of Freedom Segments === | === Pose Estimation with 6 Degree of Freedom Segments === | ||
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| A least squares procedure is used by Visual3D to determine the position and orientation. To understand how this procedure works, consider a point located on a segment at position A in the SCS. The location of the point in the LCS (P) is given by: | A least squares procedure is used by Visual3D to determine the position and orientation. To understand how this procedure works, consider a point located on a segment at position A in the SCS. The location of the point in the LCS (P) is given by: | ||
| - | {{ConstructEquation1.gif}} | + | {{:ConstructEquation1.gif}} |
| where T is the rotation matrix from the SCS to the LCS and O is the translation between coordinate systems. | where T is the rotation matrix from the SCS to the LCS and O is the translation between coordinate systems. | ||
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| If the position O and orientation T are defined for some reference position, then the fixed SCS coordinates of a target A can be determined from measurement of P at this position | If the position O and orientation T are defined for some reference position, then the fixed SCS coordinates of a target A can be determined from measurement of P at this position | ||
| - | {{ConstructEquation2.gif}} | + | {{:ConstructEquation2.gif}} |
| If the segment undergoes motion, the new orientation matrix T and translation vector O may be computed at any instant, provided that for at least three noncolinear points A is predetermined and P is measured. The matrix T and origin vector O are found by minimizing the sum of squares error expression: | If the segment undergoes motion, the new orientation matrix T and translation vector O may be computed at any instant, provided that for at least three noncolinear points A is predetermined and P is measured. The matrix T and origin vector O are found by minimizing the sum of squares error expression: | ||
| - | {{ConstructEquation3.gif}} | + | {{:ConstructEquation3.gif}} |
| under the orthonormal constraint | under the orthonormal constraint | ||
| - | {{ConstructEquation4.gif}} | + | {{:ConstructEquation4.gif}} |
| where m is equal to the number of targets on the segment ( m > 2). | where m is equal to the number of targets on the segment ( m > 2). | ||
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| Since the above system of equations represents a constrained maximum -minimum problem, the method of Lagrangian multipliers can be used to obtain the solutions. A function | Since the above system of equations represents a constrained maximum -minimum problem, the method of Lagrangian multipliers can be used to obtain the solutions. A function | ||
| - | {{ConstructEquation5.gif}} | + | {{:ConstructEquation5.gif}} |
| is used to supply the boundary conditions (This solution is adapted from the solution outlined by Spoor & Veldpaus in the Journal of Biomechanics, | is used to supply the boundary conditions (This solution is adapted from the solution outlined by Spoor & Veldpaus in the Journal of Biomechanics, | ||
visual3d/documentation/kinematics_and_kinetics/six_degrees_of_freedom.1720792834.txt.gz · Last modified: 2024/07/12 14:00 by sgranger