Tangent Space Optimization of Controls for Character Animation

We formulate the control of interpolations in animation with positional constraints over time as a space-time optimization problem in the tangent space of the curves driving the animation controls.

Animating characters rests on interpolating keyframes over time. In a contemporary workflow, the animator first uses combinations of forward and inverse kinematics to pose a character at certain keyframes via controls, and adjusts animation curves driving the controls for fine tuning. These keyframed controls are then interpolated to get the animation. While forward kinematics leads to intuitive interpolations, it is not possible to impose positional constraints and hence for example fix contact points. Inverse kinematics can be used to fix certain points in space at keyframes, but can lead to inferior interpolations, and does not allow for positional contraints at non-keyframe frames. In this paper, we solve these problems by formulating control interpolation 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 (1) it works in the space of artist editable animation curves and hence integrates seamlessly with the current pipelines, (2) does not add new keyframes, and (3) allows for positional trajectories of positions over time, extending inverse kinematics. We demonstrate the utililty of the technique in practice via various examples and use cases.