Kjartan Halvorsen

Using an Extended Kalman Filter for Rigid Body Pose Estimation

Rigid body pose is commonly represented as the rigid body transformation from one (often reference) pose to another This is usually computed for each frame of data without any assumptions or restrictions on the temporal change of the pose. The most common algorithm was proposed by Söderkvist and Wedin (1993, Determining the Movements of the Skeleton Using Well-configured Markers, J. Biomech., 26, pp. 1473-1477), and implies the assumption that measurement errors are isotropic and homogenous. This paper describes an alternative method based on a state space formulation and the application of an extended Kalman filter (EKF). State space models are formulated, which describe the kinematics of the rigid body. The state vector consists of six generalized coordinates (corresponding to the 6 degrees of freedom), and their first time derivatives. The state space models have linear dynamics, while the measurement function is a non-linear relation between the state vector and the observations (marker positions). An analytical expression for the linearized measurement function is derived. Tracking the rigid body motion using an EKF enables the use of a priori information on the measurement noise and type of motion to tune the filter. The EKF is time variant, which allows for a natural way of handling temporarily missing marker data. State updates are based on all the information available at each time step, even when data from fewer than three markers are available. Comparison with the method of Söderkvist and Wedin on simulated data showed a considerable improvement in accuracy with the proposed EKF method when marker data was temporarily missing. The proposed method offers an improvement in accuracy of rigid body pose estimation by incorporating knowledge of the characteristics of the movement and the measurement errors. Analytical expressions for the linearized system equations are provided, which eliminate the need for approximate discrete differentiation and which facilitate a fast implementation.

A method for determining minimal sets of markers for the estimation of center of mass, linear and angular momentum

A new method is proposed for finding small sets of points on the body giving sufficient information for estimating the whole body center of mass (CoM), as well as the linear momenta (LM) and angular momenta (AM). In the underlying model each point (whose trajectory is tracked by a marker) is a point mass: Hence the body is represented by a simple system of point masses. The first step is to determine the appropriate set of points and the mass of each point, which is assumed to be specific for the movement performed. The distribution of the mass to each marker is determined from training data for which the true (or reference) trajectories of the CoM, LM or AM are known. This leads to a quadratic optimization problem with inequality constraints. The use of the method is demonstrated on data from discus throw. Results indicate reasonably small errors, considering the magnitude of other error sources, in CoM position (average magnitude of estimation error 1-2cm), and moderate errors in AM (13-20% of peak value).

Tracking the Motion of Hidden Segments Using Kinematic Constraints and Kalman Filtering

Motion capture for biomechanical applications involves in almost all cases sensors or markers that are applied to the skin of the body segments of interest. This paper deals with the problem of estimating the movement of connected skeletal segments from 3D position data of markers attached to the skin. The use of kinematic constraints has been shown previously to reduce the error in estimated segment movement that are due to skin and muscles moving with respect to the underlying segment. A kinematic constraint reduces the number of degrees of freedom between two articulating segments. Moreover, kinematic constraints can help reveal the movement of some segments when the 3D marker data otherwise are insufficient. Important cases include the human ankle complex and the phalangeal segments of the horse, where the movement of small segments is almost completely hidden from external observation by joint capsules and ligaments. This paper discusses the use of an extended Kalman filter for tracking a system of connected segments. The system is modeled using rigid segments connected by simplified joint models. The position and orientation of the mechanism are specified by a set of generalized coordinates corresponding to the mechanisms degrees of motion. The generalized coordinates together with their first time derivatives can be used as the state vector of a state space model governing the kinematics of the mechanism. The data collected are marker trajectories from skin-mounted markers, and the state vector is related to the position of the markers through a nonlinear function. The Jacobian of this function is derived. The practical use of the method is demonstrated on a model of the distal part of the limb of the horse. Monte Carlo simulations of marker data for a two-segment system connected by a joint with three degrees of freedom indicate that the proposed method gives significant improvement over a method, which does not make use of the joint constraint, but the method requires that the model is a good approximation of the true mechanism. Applying the method to data on the movement of the four distal-most segments of the horses limb shows good between trial consistency and small differences between measured marker positions and marker positions predicted by the model.

ESTIMATING THE DIRECTIONS OF THE TALAR AND SUBTALAR AXES

METHODS We consider a situation when the movements of the lower leg (Segment 1) and the mid-foot (Segment 2) are measured using a motion capture system. The movement between the two segments is described by a two-degrees of freedom joint consisting of two hinge joints. A consequence of the model assumption is that there exists a direction vector fixed in Segment 1, ω1, and a direction vector fixed in Segment 2, ω2, such that the angle between them is constant during the movement. The two direction vectors correspond to the directions of the two axes, respectively. Let the two sequences of rotation matrices R1ue09ek ue09f and R2ue09ek ue09f represent the orientation of Segment 1 and Segment 2, respectively. Then the expression