Tangent-space optimization for interactive animation control

Character animation tools are based on a keyframing metaphor where artists pose characters at selected keyframes and the software automatically interpolates the frames inbetween. Although the quality of the interpolation is critical for achieving a fluid and engaging animation, the tools available to adjust the result of the automatic inbetweening are rudimentary and typically require manual editing of spline parameters. As a result, artists spend a tremendous amount of time posing and setting more keyframes. In this pose-centric workflow, animators use combinations of forward and inverse kinematics. While forward kinematics leads to intuitive interpolations, it does not naturally support positional constraints such as fixed contact points. Inverse kinematics can be used to fix certain points in space at keyframes, but can lead to inferior interpolations, is slow to compute, and does not allow for positional contraints at non-keyframe frames. In this paper, we address these problems by formulating the control of interpolations with positional constraints over time as a space-time optimization problem in the tangent space of the animation curves driving the controls. Our method has the key properties that it (1) allows the manipulation of positions and orientations over time, extending inverse kinematics, (2) does not add new keyframes that might conflict with an artist's preferred keyframe style, and (3) works in the space of artist editable animation curves and hence integrates seamlessly with current pipelines. We demonstrate the utility of the technique in practice via various examples and use cases.