- Stiff harmonic bonds - uneconomical
- SHAKE method by Ryckaert et al. [RYCKAERT 77] - better

SHAKE: Consider the smallest non-trivial

with etc.

Lagrange constraint forces: keep the bond lengths constant.
The constraint force on atom is parallel to
.
Atom is subject to two constraint forces along
and
. Atom is kept in line by
a force along
:

where are the physical accelerations due to Lennard-Jones or other pair potentials.

Procedure:

- Let the positions
be given at time .
Integrate the equations of motion for one time step with all
set to zero, i. e.
*without considering*the constraint forces; denote the resulting preliminary positions (at time ) as . These will*not*yet fulfill the constraint equations; instead, the values of and will have some nonzero values , . - Now we make the correction
*ansatz*

with undetermined , requiring that the corrected positions fulfill the constraint equations:

with ; the terms are quadratic in .Instead of solving these two quadratic equations for the unknowns we ignore, for the time being, the small quadratic terms. The remaining

*linear*equations are solved iteratively, meaning that this system of linear equation is solved to arrive at an improved estimate for which is again inserted in 4.1-4.3 leading to a new set of linearized equations etc., until the absolute values of are negligible; generally, this will occur after a very few iterations.

Solving the linearized equations involves a matrix inversion. To avoid this we introduce one more simplification:

In passing through the chain from one end to the other we consider only*one*constraint per atom:- First the bond is repaired by displacing and .
- By repairing the next bond we disrupt the first bond again; this is accepted in view of further iterations.
- By going through the chain several times we reduce both the error
introduced by neglecting the quadratic terms
*and*the error due to considering only one constraint at a time.

In our case the procedure is:

insert this in 4.1-4.3 and iterate until are negligible.

Robot arms made up of several successive links and joints bear some resemblance to chain molecules. A standard problem of robotics, the

The principle of the so-called

- Define a required ``world trajectory''for the final element in the chain.
- If the robot is
*redundant*, meaning that it has more degrees of freedom (links and joints) than necessary, the problem is underdetermined: different combinations of movements by the individual joints produce identical paths of the final member. - This redundancy may be exploited to fulfill additional requirements, such as an overall minimum of angular accelerations (i. e. mechanical wear) in the joints, avoidance of obstacles, etc.
- The last element is now moved, one time step at a time, along its requested trajectory, and SHAKE is invoked to have the preceding joints follow.
- The additional requirements are combined into a
*cost function*which is minimized, at each time step, using a*stochastic search*(simulated annealing or simple random search).

F. J. Vesely / University of Vienna