[ODE] Other integrators (was: ODE implementors guide)

[Russ Smith]

> First, the vector of Lagrange multipliers for the constraint forces
> is calculated in dInternalStepIsland_x2 with the call to "dSolveLCP",
> handling both "normal" constraint forces (unbounded) as well as
> LCP constraint forces (contact forces) (but what are the mixed
> unbounded+LCP forces?)

here's the story: each individual element of lambda (the lagrange
multipliers) has a low and high limit. if the limits are -infinity
to +infinity then that component is called 'unbounded'. knowing the
unbounded components is important because the LCP solver deals with
them more quickly. each joint has a number of lambda entries. if
all those entries are unbounded, the joint is called unbounded.
if some of those entries are not unbounded, the constraint is
called 'mixed'.

> to change this to use a higher
> order integrator you would need to:
> 1. In dInternalStepIsland_x2, don't merely multiply the computed
>    acceleration by the stepsize, but instead use the currently
>    computed acceleration along with previously computed accelerations,
>    saved from previous timesteps, to use a higher order integration
>    scheme

higher order schemes genererally dont use results computed in
previous iterations, at least not for these kinds of highly
impulsive systems. you would have to compute multiple 'acceleration'
values in a single time step. the entire simulation runs N times
slower because of this.

> The problem with all of this ... is the interplay of all of this
> with the time-stepping scheme.

actually i'm not sure what a higher order time stepping scheme
will look like in this case. i'd have to sit down and read some
papers and write some equations. it's not a problem i have
considered yet.

> This appears to imply that the LCP which must be solved at
> each timestep depends on the integration scheme

yes. a higher order integration scheme will require a number of
different LCPs to be solved at each step.

russ.

--
Russ Smith
http://www.q12.org