Responsive Characters from Motion Fragments

James McCann and Nancy Pollard
SIGGRAPH 2007 [html]
Control bins discretize continuous control.
Control model is the conditional probabilities from the previous control signal to the next control signal.
Control quality and motion quality capture the responsiveness and the naturalness of the generated motion.

Motion Graphs

Keith Grochow, Steven L. Martin, Aaron Hertzmann, Zoran Popovic
SIGGRAPH 2004 [html]
The model is represented as a probability distribution over the space of all possible poses. This means that our IK system can generate any pose, but prefers poses that are most similar to the space of poses in the training data.

Andrew Witkin, Michael Kass
SIGGRAPH 1988
minimize \( h \sum | f_i | ^2 \) subject to
- \( m (x_{i+1} - 2x_{i} + x_{i-1}) / h^2 - f_i - mg = 0\)
- \(x_1 - a = 0\)
- \(x_n - b = 0\)
Lagrange interpolation [html]
Sequential Quadratic Programming [pdf] [html]
- Newton's Method

Zoran Popović, Andrew Witkin
SIGGRAPH 1999
Simplify -> Spacetime edit -> Reconstruct
Since all dynamics computations are performed on the simplified model, there is no guarantee that the reconstruction stage of the algorithm would preserve the dynamics properties.

C. Karen Liu, Aaron Hertzmann, Zoran Popović
SIGGRAPH 2005 [html] [ppt]
Minimize \(E(X_T; \theta) - \min_{X \in C} E(X; \theta)\) using SNOPT, with spacetime footstep constraints.
Similar to minimize \(v(s_0) - \sum \gamma^t R(s_t, a_t)\)
Why extracting the parameters from a single short motion? Is it robust?

Lynne Shapiro Brotman, Arun N. Netravali
SIGGRAPH 1988
The first approach using optimal control. A good tutorial for the one who's not familiar with optimal control.
state \(s(t) = [r(t), \theta(t), \frac{dr}{dt}, \frac{d\theta}{dt}]^T\)
physical constraints \(\frac{ds(t)}{dt} = F(t)s(t) + G(t)u(t)\)
constraints \(\Phi_i = M_i s(t_i)\)
objective \(J = \int_{0}^{t_N} [ s^T(t) A s(t) + u^T(t) B u(t) ] dt\)
A and B are appropriately chosen positive semidefinite matrices. By properly choosing A and B, different combinations of control energy and roughness of the trajectory can be minimized.
Introduction to Linear Dynamical Systems
Linear Dynamical Systems