Fig 1.
Residual force resulting from a classical inverse dynamics analysis on the typical example.
Zero percent of the stride coincides with right toe off for all time series. Force was expressed in percentage of body weight. The analysis shows considerable residual forces to enforce consistency between kinematics and measured ground reaction forces. Zero crossings indicate the time nodes where measured ground reaction forces and kinematics were ‘accidentally’ consistent. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot. Right toe off is defined as the onset of the stride.
Fig 2.
Residual torque on the trunk of the typical example.
To show the effect on the residual torque, the residual force (Fres) was assumed to either apply at the shoulder or at the center of mass of the trunk. Clockwise torques were defined positive. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
Fig 3.
Typical example of segment angles during a stride.
The respective segments are denoted by color. Segment angles before and after optimization are denoted by dashed and solid lines respectively. For definitions of the segment angles, see the Materials and Methods section. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
Fig 4.
Typical example of segment angular velocities during a stride.
The respective segments are denoted by color. Angular velocities before and after optimization are denoted by dashed and solid lines respectively. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
Table 1.
Average RMS values of the segment angles with standard deviations.
Fig 5.
Net joint torques of the typical example before and after optimization.
Optimized and classical net joint torques were similar. Thin dashed lines indicate joint torque values prior to optimization. Thick solid lines indicate joint torque values after optimization. Positive values denote plantar flexion, knee flexion and hip extension torques. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
Table 2.
RMS values of the marker positions and net joint torques.
Fig 6.
Net joint and residual power values of the typical example before and after optimization.
Residual power is the sum of the power of the residual torque and force at the shoulder before optimization. Hcr: heel contact right foot, Tol: toe off left foot, Hcl: heel contact left foot.
Fig 7.
Scheme of the proposed algorithm.
The optimization starts by providing an initial guess for the matrix Q0 that contains the values for the degrees of freedom at each time node, calculated from the measured marker coordinates. The optimizer generates a modified version of Q. Using rigid body kinematics and numerical differentiation, the kinematic variables relevant for inverse dynamics are calculated. In the inverse dynamics block, net joint torques and forces (including residual forces and torques) are calculated on the basis of these kinematic variables, in combination with the measured external forces Fext, and their points of application R. The residual forces Fres and torques Tres and the predicted marker positions are fed back to the optimizer, which updates Q such that, ultimately, the residuals are zero and the sum of the weighed squared Euclidian distances (J) between predicted (S) and measured (S’) marker positions is minimal.
Fig 8.
The mechanical model used in the evaluation of the proposed algorithm, considering sagittal plane walking.
The model consists of seven rigid segments connected with pin joints. It has nine degrees of freedom. Angular coordinates used to describe the degrees of freedom are indicated by q1- q7. The remaining two degrees of freedom are described by the position of the hip (q8,q9).
Fig 9.
Mechanical model of the foot as applied in inverse dynamics.
The external force , its point of application
and the markers
and
are input of the analysis. Torques of
, the net ankle force
around the foot’s center of mass, the net ankle torque T1 and the force of gravity m1g are inserted into Newton’s equations of motion and solved for
and T1. These are subsequently reversed according to Newton’s third law and input for the same procedure on the shank.