Control Variable Bounds¶
In the optimization scheme, upper and lower bounds are specified for the control variables which define the ligament slack lengths. The bounds serve to avoid unstable simulations and to limit the control variable space. The upper bounds are defined from a rigid body forward kinematic simulation of the test cases included in the optimization scheme. The results of this analysis are used to determine the maximum length of the controlled ligament fibers, and these maximum lengths are the upper bounds of the corresponding control variable. The lower bounds are defined as a percentage of the corresponding upper bound.
Forward Kinematics Simulation¶
A rigid body forward kinematics finite element simulation is used to estimate the insertion-to-insertion length for the ligament fibers for every test case that is used in the optimization scheme. The forward kinematics simulation utilizes the same tibia, femur and ligament geometries as the model used to calibrate ligament properties. Effectively, it is the same model but setup to run in full kinematics based control. In this simulation, the tibia is fixed and the femur is kinematically controlled in six degrees of freedom. Experimentally measured kinematics for each test case in the optimization are applied to the femur.
Prestrains are applied to ensure that the ligaments are not slack throughout the simulated test cases. The results of each finite element simulation will be manually inspected to verify each ligament is not slack throughout the simulation. If ligaments are observed to be slack, prestrain for slack ligaments will be decreased until all ligaments are taut.
The results of the forward kinematics simulation are used to determine the ligament fiber lengths that correspond to the control variables. At each of the simulated test points, the node positions of the ligament fibers are used to determine the fiber’s insertion-to-insertion length. The maximum insertion-to-insertion fiber length that are achieved in the simulated test cases are used as the upper bound for the corresponding control variable.
Upper Bounds¶
The upper bounds are intended to limit the size of the control variable space. A forward kinematics is used simulate the test cases included in the optimization, and the results of these simulations are used to determine the maximum insertion-to-insertion length of the controlled ligament fibers. These maximum insertion-to-insertion lengths are the upper bounds for the corresponding control variables.
Lower Bounds¶
The lower bounds are intended to prevent the optimization algorithm from evaluating unrealistically small ligament slack lengths. This serves the practical purpose of avoiding unstable simulations caused by extremely tight ligaments. The lower bound for each control variable is defined as 60% of the corresponding upper bound.