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visual3d:documentation:kinematics_and_kinetics:global_optimization [2024/06/17 18:16] – created sgrangervisual3d:documentation:kinematics_and_kinetics:global_optimization [2024/07/17 15:45] (current) – created sgranger
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 +====== Global Optimization ======
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 J Biomech 32: 129-134 J Biomech 32: 129-134
  
-[[IK_Figure1.png]]\\+{{:IK_Figure1.png}}\\
  
  
-Consider a point [[IK_mi.png{{/images/f/f9/IK_mi.png?19x16}}]] attached to a segment, whose location is represented by a vector IK_ai.png]] in the Segment Coordinate System (SCS).+Consider a point {{:IK_mi.png}} attached to a segment, whose location is represented by a vector {{:IK_ai.png}} in the Segment Coordinate System (SCS).
  
-The location of the same marker [[IK_mi.png{{/images/f/f9/IK_mi.png?19x16}}]] is represented by vector IK_pi.png]] in the Laboratory Coordinate System (LCS)+The location of the same marker {{:IK_mi.png}} is represented by vector {{:IK_pi.png}} in the Laboratory Coordinate System (LCS)
  
-[[IK_ai.png]] is computed by Visual3D model builder +{{:IK_ai.png}} is computed by Visual3D model builder 
-[[IK_pi.png]] is the 3D motion capture data recorded +{{:IK_pi.png}} is the 3D motion capture data recorded 
-The relationship between [[IK_ai.png]] and [[IK_pi.png]] is given by:+The relationship between {{:IK_ai.png}} and {{:IK_pi.png}} is given by:
  
-[[IK_Eqn1.png]]                      (1)\\+{{:IK_Eqn1.png}}                      (1)\\
  
 where: where:
-[[IK_r.png]] is a rotation matrix from SCS to LCS, and +{{:IK_r.png}} is a rotation matrix from SCS to LCS, and 
-[[IK_O.png]]is the translation between SCS and LCS. +{{:IK_O.png}}is the translation between SCS and LCS. 
-If a segment moves, the new orientation matrix [[IK_r.png]] and translation vector [[IK_O.png]] may be computed at any instant, given that at least three non collinear points [[IK_ai.png]] are assumed stationary in the SCS (predetermined for a specific subject) and [[IK_pi.png]] are recorded.+If a segment moves, the new orientation matrix {{:IK_r.png}} and translation vector {{:IK_O.png}} may be computed at any instant, given that at least three non collinear points {{:IK_ai.png}} are assumed stationary in the SCS (predetermined for a specific subject) and {{:IK_pi.png}} are recorded.
  
 \\ \\
-For the [[Visual3D:Documentation:Kinematics_and_Kinetics:Six_Degrees_of_Freedom#6_DOF_Tracking|6 DOF analysis described elsewhere]], the rotation matrix [[IK_r.png]] and origin vector [[IK_O.png]] are computed at each frame of motion capture data by minimizing the sum of squares error expression:+For the [[Visual3D:Documentation:Kinematics_and_Kinetics:Six_Degrees_of_Freedom#6_DOF_Tracking|6 DOF analysis described elsewhere]], the rotation matrix {{:IK_r.png}} and origin vector {{:IK_O.png}} are computed at each frame of motion capture data by minimizing the sum of squares error expression:
  
-[[IK_Eqn2.png]]          (2)+{{:IK_Eqn2.png}}          (2)
 Equation 2 represents a constrained maximum-minimum problem that Visual3D solves using the method of Lagrangian multipliers (adapted from Spoor & Veldpaus, 1980). Equation 2 represents a constrained maximum-minimum problem that Visual3D solves using the method of Lagrangian multipliers (adapted from Spoor & Veldpaus, 1980).
  
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 Lu & O’Connor (1999) described a global optimization process where joint constraints added to the model can minimize the effect of sensor noise and soft tissue artifact. Lu and O’Connor termed this process Global Optimization. Visual3D uses the method of Global Optimization described by Lu and O'Connor, but we and others in the biomechanics community prefer the term Inverse Kinematics (IK). Inverse Kinematics is an extension to the 6 DOF pose estimation because if a joint is ascribed 6 degrees of freedom, the solutions are equivalent. Lu & O’Connor (1999) described a global optimization process where joint constraints added to the model can minimize the effect of sensor noise and soft tissue artifact. Lu and O’Connor termed this process Global Optimization. Visual3D uses the method of Global Optimization described by Lu and O'Connor, but we and others in the biomechanics community prefer the term Inverse Kinematics (IK). Inverse Kinematics is an extension to the 6 DOF pose estimation because if a joint is ascribed 6 degrees of freedom, the solutions are equivalent.
  
-[[IK_UpperArm.png]]+{{:IK_UpperArm.png}}
  
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 The solution to the IK is the pose of a model that best matches the motion capture data in terms of a global criterion (e.g. Least Squares). In the general case there is no analytic solution for the IK problem. Global optimization is the search of an optimal pose of a multi-link model such that the overall differences between the measured and model-estimated marker coordinates are minimized in a least squares sense at a system level. In the Lu and O-Connor approach the global optimization solution is found for each frame of data independent of any prior and later frames of data. Mathematically van den Bogert and Su (2008) described this approach based on the configuration of the total body based on a set of generalized coordinates. In this case, and of equation (1) become a function of all the generalized coordinates. The solution to the IK is the pose of a model that best matches the motion capture data in terms of a global criterion (e.g. Least Squares). In the general case there is no analytic solution for the IK problem. Global optimization is the search of an optimal pose of a multi-link model such that the overall differences between the measured and model-estimated marker coordinates are minimized in a least squares sense at a system level. In the Lu and O-Connor approach the global optimization solution is found for each frame of data independent of any prior and later frames of data. Mathematically van den Bogert and Su (2008) described this approach based on the configuration of the total body based on a set of generalized coordinates. In this case, and of equation (1) become a function of all the generalized coordinates.
  
-[[IK_Eqn3.png]]                (1)+{{:IK_Eqn3.png}}                (1)
  
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 **van den Bogert AT and Su A** (2008) A weighted least squares method for inverse dynamic analysis. Computer Methods in Biomechanics and Biomedical Engineering, 11:1,3-9 **van den Bogert AT and Su A** (2008) A weighted least squares method for inverse dynamic analysis. Computer Methods in Biomechanics and Biomedical Engineering, 11:1,3-9
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visual3d/documentation/kinematics_and_kinetics/global_optimization.1718648164.txt.gz · Last modified: 2024/06/17 18:16 by sgranger