Kinematics

 

 

Figure 4: (a) Reconstructed polyhedral model of the knee showing detail of two slices. (b) Sections extracted for the 2D 3-dof kinematics simulation. (c) and (d) show planar views of the two extracted sections from (b) (left and right sections respectively).

 

 

Figure 5: (a) Planar knee model. (b) Configuration space for planar knee model. (c) Configuration space slice .

Kinematic analysis computes the motion constraints imposed by bone contacts. We are developing a fast algorithm for joints with approximately planar motion. Efficient algorithms are available for ball/socket, hinge, and other standard (lower-pair) joints. These cases cover most human and prosthetic joints [10, 8]. The modeling requirements are to reconstruct the bone contours from the clinical data, to separate the bones in a joint, and to construct planar slices parallel to the motion plane. We discuss only the knee, since modeling the wrist and other joints poses analogous problems.

Figure 3 shows a solid model of a knee reconstructed from CT images of the visible male. Kinematic analysis of the knee is difficult. The contacts between the tibia, femur, and patella impose complicated nonlinear constraints on their positions and orientations. The constraints must be computed numerically because analytical solution methods are unavailable or impractical. The constraints induced by the three pairs of bones interact. Computing the interaction requires the simultaneous solution of the contact constraints. The constraints change during knee motion as different portions of the bones come in contact and as the axes of motion shift. In normal walking, the femur rotates in the sagittal plane and the posterior portion of the femoral condyle touches the tibial plateau. In rapid walking or running, the femur rotates around an orthogonal axis at the beginning and end of each step (automatic rotation), which shifts the contact to a portion of the condyle with a different shape. In hyperextension, the proximal portion of the condyle comes in contact with the tibia. Many other interactions are possible.

The knee moves primarily in the saggital plane, so we approximate the joint with a slice through the left lobe of the femur, a second slice through the right lobe, and a third slice through the middle. Figure 4 shows two of these slices extracted from the reconstructed polyhedral model of the knee. We approximate the full kinematics with the planar kinematics of several such slices. We derive the kinematics directly from the bone geometry with our configuration space computation algorithm [15, 16, 19, 18]. This approach complements physical measurement of the kinematics, which we can perform with the Purdue knee simulator [24]. We will validate our computations on cadaveric simulations then use the simulator to obtain custom kinematic models for natural or prosthetic knees.

The configuration space of the knee is a geometric space (manifold) whose points represent the configurations (positions and orientations) of the bones. The dimension of the configuration space equals the number of degrees of freedom of the parts. It is at most six times the number of bones, since each part may translate along and rotate around three independent axes, but is generally much lower due to part contacts. For example, the pin joint in a robot knee allows a single, rotational motion of the femur relative to the tibia, whereas human knees allow six independent motions. Configuration space partitions into free space where the bones do not touch and into blocked space where they overlap. The common boundary, called contact space, contains the configurations where the objects touch. The contact space geometry encodes the motion constraints that we need for dynamical simulation, including contact points, contact normals, and contact changes.

Figure 5(b) shows the configuration space of the planar knee slice shown in Figure 5(a). We hold the tibia fixed and allow the femur to move freely with position (x,y) and orientation . The contact space is grey. The blocked space is the interior of the contact space and the free space is the exterior. Figure 5(c) shows the configuration space slice , which shows the contacts when the bones translate in that fixed orientation. The shaded region is the blocked space where the femur and the tibia overlap. The white region is the free space where they do not touch. The filled circle in free space marks the displayed configuration of (5,-3). The solid boundary between free and blocked space is the contact space where the bones touch. It consists of curves that represent contacts between the features of the tibia and the femur.