I think it would be nice to make anthropomorphic transitions between a few basis motions. So here is the paper I am referring:

Efficient Generation of Motion Transitions using Spacetime Constraints

Charles Rose, Brian Guenter, Bobby Bodenheimer, Michael F. Cohen

*Note: every symbol presented in this post can be found in the original paper.*

The meat of the idea is described in section 3.1 and 3.2. As what I understood, the joint angles during the transition of support limbs are controlled with inverse kinematics constraints, and other limbs are with spacetime constraints. Here I list the steps to compute the joint angles during the transitions of unsupport limbs.

**Basic definitions**

**Objective function to be minimized:**

**Gradient of the objective function:**

**Pseudo-Hessian Matrix**, **H**, is updated with the gradient ** g**.

**New step direction:**

**Algorithm**

**For each iteration:**

Check whether the iteration converges with the sum of the magnitude of the joint torques.

**If the objective function converges: exit the iteration.**

Else update the joint angles for all the joints.

**For each frame t:**

Compute velocities, accelerations, net torques, and forces at each joint starting from the root node and working out to the tips of all the chains in the tree. **(Appendix A.2)**

Compute the joint torques starting from the tips of the chains back to the root node. **(Appendix A.3)**

Compute gradient. **(Appendix A.4)**

Update pseudo-Hessian. **(TBD)**

Update new step direction.

Update the joint angles in current frame with the new step direction.

Sum up the magnitude of the joint torques.

**End**

Sum up the magnitude of the joint torques.

**End**