White Paper. OLGA Explained. Lasse Roren. Author:

Transcription

1 White Paper OLGA Explained Author: Lasse Roren Revision: 05/001 - August 2005

2 Introduction OLGA (Optimized Lower-limb Gait Analysis) was introduced in 2003 as a plug-in which works with the Vicon Workstation software. OLGA co-exists with the Conventional Gait Model ([1][2], henceforth called CGM), implemented by Vicon in the VCM and Plug-in Gait software packages, but introduces the additional step of a procedure which optimizes the locations of joint centers and segment orientations prior to the calculation of joint kinematics and kinetics. This paper aims to explain the concepts, methods and processes used by OLGA in plain English. OLGA goes hand in hand with the CGM in that it uses this model to both come up with the initial guesses for the patient s skeletal anatomy and to calculate the kinematic and kinetic results. This paper aims to explain the concepts, methods and processes used by OLGA in plain English. The complexity of the technique is often both a confusing factor and a discouragement to potential testers/users. The challenge is that the advanced mathematical techniques used by OLGA cannot always be explained in simple language a certain amount of background knowledge about motion capture in general and biomechanical modeling in particular must therefore be assumed. Page 1

3 Background OLGA is conceptually simple. The basic idea is: 1. Use the CGM s static gait trial to provide an initial guess for joint center locations and segment orientations. In other words, OLGA s first estimate for joint center locations relative to marker locations correspond exactly to the CGM s. 2. Use a dynamic trial with significant joint movement to calibrate the model. The calibration will optimize the locations of joint centers and orientations of segments based on the recorded movement in the trial. The process is statistically based, and will not change the initial estimates unless there s enough recorded movement to have a high confidence that the location/ orientation can be safely improved. FIGURE 1 OLGA s options dialog 3. Use the calibrated model to do kinematic fitting of movement trials. This operation finds the joint angles that best fit the recorded data on a frameby-frame basis. Furthermore, to ensure smooth results it also uses a Kalman kinematic ([4]) filter. 4. Use the same algorithms as used in Plug-in Gait to calculate the dynamic output variables, i.e. the joint angles, moments and powers. These four steps correspond to the four different individually selectable stages in the OLGA plug-in options box (Figure 1). Step 1 should be done on the ordinary gait static trial, step 2 could be done on a single representative walking trial or a range-of-motion trial, steps 3 & 4 should be done on all trials. The calibration will optimize the locations of joint centers and orientations of segments based on the record movement in the trial. In addition to this, there are several processing parameters that can be changed. These will be discussed in detail later in the document. Page 2

4 The Kinematic Model The key to understanding OLGA is the calibration procedure, and the background to the calibration procedure is the concept of a simplified kinematic model of the lower body of a human being. In this respect it is exactly the same as the conventional gait model in that it is based on: A Pelvis root segment Two femur segments attached to the Pelvis using ball-and-socket (3 degree-of-freedom) joints, at the locations of the hip joint centers (HJC). Two tibia segments, attached to the femur segments using ball-and-socket joints at the locations of the knee joint centers (KJC). Two feet vectors, attached to the tibia segments using ball-and-socket joints at the locations of the ankle joint centers (AJC). In other words, OLGA uses the same kinematic skeletal model as the CGM as far as the hierarchical structure and the joint types of the model is concerned. FIGURE 2 The Conventional Gait Model The Conventional Gait Model (CGM) To understand how OLGA differs it is useful to remind ourselves of how the CGM does the modeling: The pelvis origin is situated half-way between the ASIS points, pelvis orientation is directly defined by LASI, RASI and SACR markers. The CGM can be viewed as being highly constrained with respect to the measured markers - there are direct geometric relationships between the model and the markers. HJCs are located by a regression equation ([1]) and specified in the pelvis local coordinate system. Femur segments and KJCs (positioned at the femur s distal end) are defined by the HJCs, the lateral thigh markers (LTHI/RTHI) and the lateral knee markers (LKNE/RKNE) Tibia segments and AJCs are defined by the KJCs, the lateral tibia markers (LTIB/RTIB) and the lateral ankle markers (LANK/RANK). Foot vectors are defined by the AJCs and the toe markers (LTOE/RTOE), but offset to be parallel to the long axis of the foot according to the static offset angles. Page 3

5 This means that every segment origin and orientation is calculated directly from measured marker positions on a frame-by-frame basis. All the measurement noise and artifacts caused by skin movement relative to the bone will be translated straight to the orientations and locations of segments except the noise that is filtered out by the trajectory filter. The CGM can be viewed as being highly constrained with respect to the measured markers there are direct geometric relationships between the model and the markers. On the other hand, the conventional model does not put any constraints on the anatomical parameters all joint segments are scaled and moved to fit the markers. In other words, the modeled bones will be stretched and twisted to fit their locations to the current marker positions even if the markers have been misplaced or wobble around because of soft tissue artifacts. Kinematic Fitting Kinematic fitting is an optimization algorithm that happens in space only, whereas the calibration algorithm optimizes in both space and time. The calibration will be described in more detail later, but as the kinematic fitting concept is somewhat simpler and is used by the calibrator we shall begin with that. FIGURE 3 OLGA skeleton with model and real markers The kinematic fitting algorithm assumes that we have a skeletal model that is rigid with respect to both bone properties and bone-marker relationships. The model we have contains the following information: 1. A hierarchical structure that describes the skeleton, with the pelvis as the root segment, followed by femura, tibiae and feet segments. Kinematic fitting is the process of finding the segment locations and joint angles that minimise all the distances between the measured markers and the model markers for any given set of measured markers in time. 2. All segments distal to the pelvis are described with their origin at the proximal joint center, and they specify the distal joint center (and therefore the segment length) in local coordinates. 3. All segments specify positions for the attached markers in their own local coordinate systems. 4. All joints are specified as 3 degree of freedom joints, which gives the distal segment the freedom to rotate around all three axes at the joint center. This is illustrated in Figure 2. The blue markers represent the real, measured markers whereas the red markers represent the model s marker positions a line connects the red marker to the bone to symbolize the fact that the model marker is rigidly associated with its bone. Kinematic fitting is the process of finding the segment locations and joint angles that minimize all the distances between the measured markers and the model markers for any given set of measured markers in time. If we did this on a single segment, it would reduce to an ordinary least-square fit algorithm. However, kinematic fitting also allows joints and joint angles to be optimized for the problem. In either case, the objective is the same to find a solution that minimizes the distances between what has been measured and the model. This minimized distance note that unless the measured markers are Page 4

6 in exactly the same locations as the model ones, there will be such a distance is referred to as the kinematic fit residual. If the model contains the correct anatomical distances as well as the correct locations of the markers with respect to the bones, the kinematic fitting process will yield the best possible fit of the skeleton given a single frame of measured markers and thus the most accurate joint kinematics obtainable given the input information. Of course, the problem is that we do not know the correct anatomical distances, nor do we know the accurate locations of the markers with respect to the segments. It is the topic of the subject calibration section below how we can obtain the best possible estimates for these. Before we proceed there is one more point to make. Kinematic fitting can, in addition to performing the least square fit to find the best skeletal pose to fit the measured markers, also put different weights on different markers. If, for example, we know that certain markers are more wobbly than others because of soft tissue artifacts, we can assign lower weights to these. This means that the markers in which we have greater confidence will pull the solution towards them harder than the wobbly ones. The Kinematic Kalman Filter The filtering of trajectories in order to get rid of high-frequency noise has been common procedure since the start of motion capture. OLGA s kinematic fitting procedure introduces new intelligent filtering which uses knowledge not only about the XYZ location of markers but about the entire skeletal model to do the filtering. Put simply, OLGA will, when performing the kinematic fit for frame N, look at the few frames behind and try to predict the marker locations for frame N. The Kalman filter [4] is a statistical kinematic filter, which has been used widely in science since the paper was first published in Then, given the assumption that the measured trajectory at frame N has some measurement noise associated with it, the Kalman filter smoothes the result by combining the measured and predicted body positions. Thus, the Kalman filter [4] is a statistical kinematic filter, which has been used widely in science since the paper was first published in The main benefit when compared to trajectory-only filters such as the Woltring filter is that the Kalman filter recognizes that the markers are connected to the same skeletal structure and therefore the movement of one marker tends to be highly correlated to others. Traditional filters completely disregards this information. Page 5

7 Subject Calibration As mentioned in the previous section, kinematic fitting is the process of taking a skeletal model with a given set of parameters (segment properties and segment-marker relationships) and finding the skeletal location and joint angles that minimize the kinematic fit residual. This happens on a frame-by-frame basis and is thus essentially a least-square optimization in space only. Subject Calibration is the process of finding the set of parameters that minimizes the residual in both space and time. In other words, the calibration algorithm will adjust the model s parameters the segment lengths, the orientations and the model s marker positions until the optimum set of parameters has been found. OLGA uses an iterative procedure for this purpose which step by step guides the solution, i.e. the set of model parameters, towards convergence. For each step in the procedure, a kinematic fit residual is calculated for all the frames used for the calibration, and the difference in residual between two neighboring iterations denotes how much improvement the iteration produced. When the improvement in residual is sufficiently small, it is assumed that subsequent iterations will not yield significant improvements and the calibration algorithm decides that the model has converged. See Figure 4 for a graphical description of OLGA s calibration procedure. Initial Estimate No Converged? Yes Calibrated Subject FIGURE 4 The Calibration Procedure Kinematic Fit Calibrate Model Model Constraints If we study the number of parameters that are available for the calibration algorithm, it soon becomes clear that it is quite a daunting task. The marker set itself contains 15 markers, which means that there are 45 parameters X, Y and Z of each marker position in the segment s local coordinates that fully describe these relationships. In addition, we have 5 skeletal anatomy parameters: the distance between the HJC, the length of the femura and the length of the tibiae. There are two main problems with this: first, the scale of the problem is such that it takes a long time for each iteration; and second, a typical movement trial does not contain enough information to determine many of the parameters with a high enough degree of confidence. Therefore, we need to constrain some of the parameters, i.e. fix them and ban the calibrator from touching them. The calibration algorithm will adjust the model s parameters - the segment lengths, the orientations and the model s marker positions - until the optimum set of parameters has been found. Page 6

8 To understand the necessity and concepts of the model constraints, it is helpful to begin analyzing a much simpler model, for example just the pelvis and the left femur; two rigid bodies that have been modeled as being attached to each other with a ball-and-socket joint at the location of the HJC. The simplest approach is to attach markers to either segment in any location as long as the calibration knows which markers are attached to which segment, it will have something to get going with. At this stage this is quite similar to the functional method approach [5]: as the femur moves, the markers on the femur will all be recorded to lie roughly on a sphere with its centre in the HJC. As the femur moves, the markers on the femur will all be recorded to lie roughly on a sphere with its centre in the HJC. With sufficient appropriate movement, this approach is just enough to determine the HJC s most likely location. However, other crucial parameters are completely undetermined: the orientation of the pelvis and the orientation of the femur. The solution to this problem involves introducing rigid, nonoptimized constraints between the model and the positions of the markers. The introduction of more segments will also help. The first thing to note is that the pelvis has two HJCs. This means that as soon as we attach the second femur and calibrate using the movement of both femura, we will effectively be able to constrain the orientation of the pelvis segment simply by assuming symmetry. This allows us to place the origin of the pelvis half-way between the calibrated HJCs and to align the major axis of the pelvis with the line between the two. However, the direction of the pelvis has only been resolved for the major axis (the tilt axis) and there is no movement that can help us establish a secondary axis to fix the coordinate system. This is where the initial estimate, which OLGA takes straight from the CGM s static model, is used. Moving down to the femura and tibiae, it is clear that the average human being is incapable of producing enough movement in the coronal and transverse planes to fully optimize the KJC. However, we do have a very accurate measure of the width of the knee from the subject measurements, and assuming the KJC lies half-way inside the knee enables us to constrain the position of the knee marker in the mediolateral and anterior-posterior directions. These constraints reduce the scale of the optimization problem, as we allow the KJC position to be optimized with respect to the knee marker only in the proximal-distal direction. The locations of the knee and ankle joint centres are constrained by the joint width in the mediolateral direction. Exactly the same applies to the tibiae and the AJCs: they are constrained in the mediolateral direction by the ankle marker and in the anterior/posterior direction by the ankle and tibia markers. Page 7

9 Full-Body Optimization Lu and O Connor s paper [3] Bone position estimation from skin marker coordinates using global optimization with joint constraints used the term Global Optimization in the sense of optimizing a biomechanical model over more than one joint. OLGA expands on the work done by Lu and O Connor by introducing the space-time optimization calibration stage. Furthermore, the term Global Optimization is somewhat unfortunate in that the same term is used in mathematics for a fairly well-defined but entirely separate problem. This White Paper uses the term Full-Body Optimization (FBO) to describe the generic technique described here - OLGA is the specific implementation of this technique. Therefore, this White Paper uses the term Full-Body Optimization (FBO) to describe the generic technique described here OLGA is the specific implementation of this technique. As mentioned above, the calibration algorithm in OLGA tries to establish the most likely values of the skeletal and marker position parameters given the movement applied to the calibration. So how does that relate to the concept of FBO? The main principle of the FBO is that all parameters are optimized as part of the same procedure. This has clear benefits in that the convergence of the whole procedure is guaranteed OLGA will never end up in a ping-pong scenario where optimizing one parameter will make another worse, which when optimized will make the first one worse and so on. Thus, FBO ensures that the solution will converge towards the statistically most likely values for the different parameters given the measured movement of the markers. Given the limited information we have a set of labeled 3D coordinates in space and time, and an idea of which segment each marker is attached to, the converged FBO will produce the statistically best solution given the limitations: imperfect joint descriptions (mathematical 3-DOF joints), soft tissue artifacts and marker placement inaccuracies. Page 8

10 Processing Parameters This section describes the parameters that can be changed when using OLGA. The most important parameters are the ones controlling how many frames of data OLGA will use when calibrating. These are controlled by the Frame Range and the Maximum Iterations. The frame range allows the user to select which frames to use for calibration: FIGURE 5 OLGA s advanced calibration options FIRST to LAST, uses the range from the first fully labeled frame to the last fully labeled frame. FIRST_HEEL_STRIKE to LAST_HEEL_STRIKE, requires events to be defined prior to OLGA being run and will set the frame range accordingly. This allows the sweet spot of the capture volume to be used if, for example, only one gait cycle for each side has been defined. Optionally, it is possible to enter the actual frame numbers. The Maximum Calibration Frames entry determines the level of sub-sampling. If this entry is 50 and there are 200 frames in the frame range, OLGA will use every 4th frame for its calibration. The lower the number, the faster the calibration will be performed, but the trade-off is less accuracy because the calibration will be based on less data. The convergence tolerance and maximum iterations are used to determine when OLGA will end the calibration. The convergence tolerance and maximum iterations are used to determine when OLGA will end the calibration. The iterative procedure will go on until one of two events have occurred: the maximum number of iterations has been reached OR the difference in the residual between this step and the last falls below the convergence tolerance. Knee Cross-Talk Minimization OLGA also implements an algorithm which adjusts the rotation of the femura and tibiae segments to fi nd the orientations that result in minimum knee cross talk. The cross talk is a well-known phenomenon caused by inaccurate marker placement in the CGM, and the result is that some of the knee fl exion leaks into the valgus/varus angle because the model s knee fl exion/extension axis is not properly aligned with the true one. Page 9

11 It is important to note that this is not equivalent to saying there is no valgus/varus. On the contrary, it is equivalent to saying the valgus/varus is uncorrelated to the knee flexion/extension. It is a statistical procedure that looks at all the knee kinematics for the trial and adjusts the femur and tibia orientations so that the correlation between the angles in the sagittal and coronal planes is minimized. FIGURE 6 OLGA s advanced fitting options Advanced Kinematic Fitting Parameters The frame range for kinematic fitting is equivalent to the calibration one. The really interesting advanced parameters are the next two: the prediction model and the prediction confidence. The prediction model is used by the Kalman filter to predict where a frame of data will be on the basis of the previous frames. If the value of this parameter is 1.0, the model will assume that all segments have a constant velocity and will predict the next frame accordingly. However, if the value is less than 1.0, there will be a certain amount of damping, in other words a tendency to slow down. The lower the value, the more damping will be assumed until, at value 0.0, the prediction will assume zero velocity. The prediction confidence determines how much weight is put on the prediction compared to the actual measured markers. The prediction confidence determines how much weight is put on the prediction compared to the actual measured markers. So, whilst the prediction model affects the location of the prediction, the prediction confidence is used to combine the prediction with the measurement to generate the filtered model. The lower the value, the more relative confidence is put on the markers until, at 0.0, the predictions will not be used at all (and the data will thus be unfiltered). The final tick options determine whether you want to do a frame-by-frame fit only, which is equivalent to turning the Kalman filter off (which will speed up the process considerably) and whether you want to perform the minimization of the knee-angle cross talk (as described above) or not. Page 10

12 References [1] Davis, R.B. et al. (1991) A gait analysis data collection and reduction technique. Human Movement Science, 10, p [2] Kadaba, M.P. et al. (1990) Measurement of lower extremity kinematics during level walking. Journal of Orthopaedic Research, 8, p [3] Lu, T.W and O Connor, J.J. (1999) Bone position estimation from skin marker co-ordinates using global optimisation with joint constraints. Journal of Biomechanics, 32, p [4] Kalman, R.E. (1960) A new approach to linear filtering and prediction problems. Journal of Basic Engineering, 82: [5] Leardini, A., et al., Validation of a functional method for the estimation of hip joint centre location. Journal of Biomechanics, : p Page 11

13 Contact details Web page: United Kingdom Office: Vicon Peak - UK 14 Minns Business Park West Way Oxford OX2 0JB United Kingdom oxford@viconpeak.com Telephone: Fax: United States California Office: Vicon Peak California 9 Spectrum Pointe Drive Lake Forest, CA USA california@viconpeak.com Telephone: Fax: United States Colorado Office: Vicon Peak Colorado 7388 S. Revere Parkway Suite 901 Centennial, CO USA colorado@viconpeak.com Telephone: Fax: Page 12