# Simulation

## Basic simulation

RigidBodyDynamics.simulateFunction
simulate(state0, final_time)
simulate(state0, final_time, control!; Δt, stabilization_gains)


Basic Mechanism simulation: integrate the state from time $0$ to final_time starting from the initial state state0. Return a Vector of times, as well as Vectors of configuration vectors and velocity vectors at these times.

Optionally, a function (or other callable) can be passed in as the third argument (control!). control! will be called at each time step of the simulation and allows you to specify joint torques given the time and the state of the Mechanism. It should look like this:

function control!(torques::AbstractVector, t, state::MechanismState)
rand!(torques) # for example
end

The integration time step can be specified using the Δt keyword argument (defaults to 1e-4).

The stabilization_gains keyword argument can be used to set PD gains for Baumgarte stabilization, which can be used to prevent separation of non-tree (loop) joints. See Featherstone (2008), section 8.3 for more information. There are several options for specifying gains:

• nothing can be used to completely disable Baumgarte stabilization.
• Gains can be specifed on a per-joint basis using any AbstractDict{JointID, <:RigidBodyDynamics.PDControl.SE3PDGains}, which maps the JointID for the non-tree joints of the mechanism to the gains for that joint.
• As a special case of the second option, the same gains can be used for all joints by passing in a RigidBodyDynamics.CustomCollections.ConstDict{JointID}.

The default_constraint_stabilization_gains function is called to produce the default gains, which use the last option.

Uses MuntheKaasIntegrator. See RigidBodyDynamics.OdeIntegrators.MuntheKaasIntegrator for a lower level interface with more options.

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## Lower level ODE integration interface

RigidBodyDynamics.OdeIntegrators.MuntheKaasIntegratorType
struct MuntheKaasIntegrator{N, T, F, S<:OdeResultsSink, X, L, M<:(RigidBodyDynamics.OdeIntegrators.MuntheKaasStageCache{N,T,Q,V,S} where S<:(AbstractArray{T,1} where T) where V<:(AbstractArray{T,1} where T) where Q<:(AbstractArray{T,1} where T))}

A Lie-group-aware ODE integrator.

MuntheKaasIntegrator is used to properly integrate the dynamics of globally parameterized rigid joints (Duindam, Port-Based Modeling and Control for Efficient Bipedal Walking Robots, 2006, Definition 2.9). Global parameterizations of e.g. $SO(3)$ are needed to avoid singularities, but this leads to the problem that the tangent space no longer has the same dimension as the ambient space of the global parameterization. A Munthe-Kaas integrator solves this problem by converting back and forth between local and global coordinates at every integration time step.

The idea is to do the dynamics and compute the stages of the integration scheme in terms of local coordinates centered around the global parameterization of the configuration at the end of the previous time step (e.g. exponential coordinates), combine the stages into a new set of local coordinates as usual for Runge-Kutta methods, and then convert the local coordinates back to global coordinates.

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RigidBodyDynamics.OdeIntegrators.MuntheKaasIntegratorMethod
MuntheKaasIntegrator(state, dynamics!, tableau, sink)


Create a MuntheKaasIntegrator given:

• a callable dynamics!(vd, t, state) that updates the joint acceleration vector vd at time t and in state state;
• a ButcherTableau tableau, specifying the integrator coefficients;
• an OdeResultsSink sink which processes the results of the integration procedure at each time step.

state must be of a type for which the following functions are defined:

• configuration(state), returns the configuration vector in global coordinates;
• velocity(state), returns the velocity vector;
• additional_state(state), returns the vector of additional states;
• set_velocity!(state, v), sets velocity vector to v;
• set_additional_state!(state, s), sets vector of additional states to s;
• global_coordinates!(state, q0, ϕ), sets global coordinates in state based on local coordinates ϕ centered around global coordinates q0;
• local_coordinates!(ϕ, ϕd, state, q0), converts state's global configuration q and velocity v to local coordinates centered around global coordinates q0.
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RigidBodyDynamics.OdeIntegrators.OdeResultsSinkType
abstract type OdeResultsSink

Does 'something' with the results of an ODE integration (e.g. storing results, visualizing, etc.). Subtypes must implement:

• initialize(sink, state): called with the initial state when integration begins.
• process(sink, t, state): called at every integration time step with the current state and time.
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RigidBodyDynamics.OdeIntegrators.ExpandingStorageType
mutable struct ExpandingStorage{T, Q<:(AbstractArray{T,1} where T), V<:(AbstractArray{T,1} where T)} <: OdeResultsSink

An OdeResultsSink that stores the state at each integration time step in Vectors that may expand.

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