## Dynamics, Theory and Applications

Thomas R. Kane, David A. Levinson

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I want to know how to calculate rotation angles using inverse kinematics. I am planning on using this for real time 3d animation. Anyone know of some good literature that details a specific solution?

Thomas Kane is one of the world's authorities on dynamics. I'd recommend his "Dynamics" text highly, but it's not for the faint of heart or weak at mathematics.

The following resources survey some popular numerical methods for inverse kinematics problems:

Buss's survey may be particularly interesting, because it explicitly discusses multiple limbs.

IK systems for animation must generally support multiple, possibly conflicting, constraints. For example, one arm can hold on to a rail while the other arm reaches for a target.

6 dof industrial robots generally have closed form IK solutions, as mentioned by Andrew and explained in e.g. Craig: Introduction to Robotics. More useful for figure animation are methods for 7 dof human-like arms and legs:

Let's consider a disk with mass m and radius R on a surface where friction u also involved. When we give this disk a starting velocity v in a direction, the disk will go towards that direction and slow down and stop.

In case the disk has a rotation (or spin with the rotational line perpendicular on the surface) w beside the speed then the disk won't move on a line, instead bend. Both the linear and angular velocity would be 0 at the end.

How can this banding/curving/dragging be calculated? Is it possible to give analytical solution for the X(v,w,t) function, where X would give the position of the disk according to it's initial v w at a given t?

Any simulation hint would be also fine. I imagine, depending on w and m and u there would be an additional velocity which is perpendicular to the linear velocity and so the disk's path would bend from the linear path.

Numerical integration of Newton's laws of motion would be what I'd recommend. Draw the free body diagram of the disk, give the initial conditions for the system, and numerically integrate the equations for acceleration and velocity forward in time. You have three degrees of freedom: x, y translation in the plane and the rotation perpendicular to the plane. So you'll have six simultaneous ODEs to solve: rates of change of linear and angular velocities, rates of change for two positions, and the rate of change of angular rotation.

Be warned: friction and contact make that boundary condition between the disk and the table non-linear. It's not a trivial problem.

There could be some simplifications by treating the disk as a point mass. I'd recommend looking at Kane's Dynamics for a good understanding of the physics and how to best formulate the problem.

I'm wondering if the bending of the path that you're imagining would occur with a perfectly balanced disk. I haven't worked it out, so I'm not certain. But if you took a perfectly balanced disk and spun it about its center there'd be no translation without an imbalance, because there's no net force to cause it to translate. Adding in an initial velocity in a given direction wouldn't change that.

But it's easy to see a force that would cause the disk to deviate from the straight path if there was an imbalance in the disk. If I'm correct, you'll have to add an imbalance to your disk to see bending from a straight line. Perhaps someone who's a better physicist than me could weigh in.