# Algorithms

## Index

`RigidBodyDynamics.DynamicsResult`

`RigidBodyDynamics.Spatial.center_of_mass`

`RigidBodyDynamics.Spatial.center_of_mass`

`RigidBodyDynamics._point_jacobian!`

`RigidBodyDynamics.default_constraint_stabilization_gains`

`RigidBodyDynamics.dynamics!`

`RigidBodyDynamics.dynamics!`

`RigidBodyDynamics.dynamics_bias`

`RigidBodyDynamics.dynamics_bias!`

`RigidBodyDynamics.geometric_jacobian`

`RigidBodyDynamics.geometric_jacobian!`

`RigidBodyDynamics.geometric_jacobian!`

`RigidBodyDynamics.geometric_jacobian!`

`RigidBodyDynamics.inverse_dynamics`

`RigidBodyDynamics.inverse_dynamics!`

`RigidBodyDynamics.mass`

`RigidBodyDynamics.mass_matrix`

`RigidBodyDynamics.mass_matrix!`

`RigidBodyDynamics.momentum_matrix`

`RigidBodyDynamics.momentum_matrix!`

`RigidBodyDynamics.momentum_matrix!`

`RigidBodyDynamics.momentum_matrix!`

`RigidBodyDynamics.point_jacobian`

`RigidBodyDynamics.point_jacobian!`

`RigidBodyDynamics.subtree_mass`

## The `DynamicsResult`

type

`RigidBodyDynamics.DynamicsResult`

— Type`mutable struct DynamicsResult{T, M}`

Stores variables related to the dynamics of a `Mechanism`

, e.g. the `Mechanism`

's mass matrix and joint acceleration vector.

Type parameters:

`T`

: the scalar type of the dynamics-related variables.`M`

: the scalar type of the`Mechanism`

.

## Functions

`RigidBodyDynamics.Spatial.center_of_mass`

— Method```
center_of_mass(state, itr)
```

Compute the center of mass of an iterable subset of a `Mechanism`

's bodies in the given state. Ignores the root body of the mechanism.

`RigidBodyDynamics.Spatial.center_of_mass`

— Method```
center_of_mass(state)
```

Compute the center of mass of the whole `Mechanism`

in the given state.

`RigidBodyDynamics._point_jacobian!`

— Method```
_point_jacobian!(Jp, state, path, point, transformfun)
```

Compute the Jacobian mapping the `Mechanism`

's joint velocity vector $v$ to the velocity of a point fixed to the target of the joint path (the body succeeding the last joint in the path) with respect to the source of the joint path (the body preceding the first joint in the path).

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.default_constraint_stabilization_gains`

— MethodReturn the default Baumgarte constraint stabilization gains. These gains result in critical damping, and correspond to $T_{stab} = 0.1$ in Featherstone (2008), section 8.3.

`RigidBodyDynamics.dynamics!`

— Method```
dynamics!(result, state)
dynamics!(result, state, torques)
dynamics!(result, state, torques, externalwrenches; stabilization_gains)
```

Compute the joint acceleration vector $\dot{v}$ and Lagrange multipliers $\lambda$ that satisfy the joint-space equations of motion

\[M(q) \dot{v} + c(q, v, w_\text{ext}) = \tau - K(q)^{T} \lambda\]

and the constraint equations

\[K(q) \dot{v} = -k\]

given joint configuration vector $q$, joint velocity vector $v$, and (optionally) joint torques $\tau$ and external wrenches $w_\text{ext}$.

The `externalwrenches`

argument can be used to specify additional wrenches that act on the `Mechanism`

's bodies.

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.

`RigidBodyDynamics.dynamics!`

— Method```
dynamics!(ẋ, result, state, x)
dynamics!(ẋ, result, state, x, torques)
dynamics!(ẋ, result, state, x, torques, externalwrenches; stabilization_gains)
```

Convenience function for use with standard ODE integrators that takes a `Vector`

argument

\[x = \left(\begin{array}{c} q\\ v \end{array}\right)\]

and returns a `Vector`

$\dot{x}$.

`RigidBodyDynamics.dynamics_bias!`

— Method```
dynamics_bias!(torques, biasaccelerations, wrenches, state)
dynamics_bias!(torques, biasaccelerations, wrenches, state, externalwrenches)
```

Compute the 'dynamics bias term', i.e. the term

\[c(q, v, w_\text{ext})\]

in the unconstrained joint-space equations of motion

\[M(q) \dot{v} + c(q, v, w_\text{ext}) = \tau\]

given joint configuration vector $q$, joint velocity vector $v$, joint acceleration vector $\dot{v}$ and (optionally) external wrenches $w_\text{ext}$.

The `externalwrenches`

argument can be used to specify additional wrenches that act on the `Mechanism`

's bodies.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.dynamics_bias`

— Method```
dynamics_bias(state)
dynamics_bias(state, externalwrenches)
```

Compute the 'dynamics bias term', i.e. the term

\[c(q, v, w_\text{ext})\]

in the unconstrained joint-space equations of motion

\[M(q) \dot{v} + c(q, v, w_\text{ext}) = \tau\]

given joint configuration vector $q$, joint velocity vector $v$, joint acceleration vector $\dot{v}$ and (optionally) external wrenches $w_\text{ext}$.

The `externalwrenches`

argument can be used to specify additional wrenches that act on the `Mechanism`

's bodies.

`RigidBodyDynamics.geometric_jacobian!`

— Method```
geometric_jacobian!(jac, state, path, transformfun)
```

Compute a geometric Jacobian (also known as a basic, or spatial Jacobian) associated with a directed path in the `Mechanism`

's spanning tree, (a collection of `Joint`

s and traversal directions) in the given state.

A geometric Jacobian maps the `Mechanism`

's joint velocity vector $v$ to the twist of the target of the joint path (the body succeeding the last joint in the path) with respect to the source of the joint path (the body preceding the first joint in the path).

See also `path`

, `GeometricJacobian`

, `Twist`

.

`transformfun`

is a callable that may be used to transform the individual motion subspaces of each of the joints to the frame in which `out`

is expressed.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.geometric_jacobian!`

— Method```
geometric_jacobian!(out, state, path, root_to_desired)
```

Compute a geometric Jacobian (also known as a basic, or spatial Jacobian) associated with a directed path in the `Mechanism`

's spanning tree, (a collection of `Joint`

s and traversal directions) in the given state.

A geometric Jacobian maps the `Mechanism`

's joint velocity vector $v$ to the twist of the target of the joint path (the body succeeding the last joint in the path) with respect to the source of the joint path (the body preceding the first joint in the path).

See also `path`

, `GeometricJacobian`

, `Twist`

.

`root_to_desired`

is the transform from the `Mechanism`

's root frame to the frame in which `out`

is expressed.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.geometric_jacobian!`

— Method```
geometric_jacobian!(out, state, path)
```

Compute a geometric Jacobian (also known as a basic, or spatial Jacobian) associated with a directed path in the `Mechanism`

's spanning tree, (a collection of `Joint`

s and traversal directions) in the given state.

A geometric Jacobian maps the `Mechanism`

's joint velocity vector $v$ to the twist of the target of the joint path (the body succeeding the last joint in the path) with respect to the source of the joint path (the body preceding the first joint in the path).

See also `path`

, `GeometricJacobian`

, `Twist`

.

See `geometric_jacobian!(out, state, path, root_to_desired)`

. Uses `state`

to compute the transform from the `Mechanism`

's root frame to the frame in which `out`

is expressed.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.geometric_jacobian`

— Method```
geometric_jacobian(state, path)
```

`Mechanism`

's spanning tree, (a collection of `Joint`

s and traversal directions) in the given state.

`Mechanism`

's joint velocity vector $v$ to the twist of the target of the joint path (the body succeeding the last joint in the path) with respect to the source of the joint path (the body preceding the first joint in the path).

See also `path`

, `GeometricJacobian`

, `Twist`

.

The Jacobian is computed in the `Mechanism`

's root frame.

`RigidBodyDynamics.inverse_dynamics!`

— Method```
inverse_dynamics!(torquesout, jointwrenchesout, accelerations, state, v̇)
inverse_dynamics!(torquesout, jointwrenchesout, accelerations, state, v̇, externalwrenches)
```

Do inverse dynamics, i.e. compute $\tau$ in the unconstrained joint-space equations of motion

\[M(q) \dot{v} + c(q, v, w_\text{ext}) = \tau\]

given joint configuration vector $q$, joint velocity vector $v$, joint acceleration vector $\dot{v}$ and (optionally) external wrenches $w_\text{ext}$.

`externalwrenches`

argument can be used to specify additional wrenches that act on the `Mechanism`

's bodies.

This method implements the recursive Newton-Euler algorithm.

Currently doesn't support `Mechanism`

s with cycles.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.inverse_dynamics`

— Method```
inverse_dynamics(state, v̇)
inverse_dynamics(state, v̇, externalwrenches)
```

Do inverse dynamics, i.e. compute $\tau$ in the unconstrained joint-space equations of motion

\[M(q) \dot{v} + c(q, v, w_\text{ext}) = \tau\]

`externalwrenches`

argument can be used to specify additional wrenches that act on the `Mechanism`

's bodies.

This method implements the recursive Newton-Euler algorithm.

Currently doesn't support `Mechanism`

s with cycles.

`RigidBodyDynamics.mass`

— Method```
mass(m)
```

Return the total mass of the `Mechanism`

.

`RigidBodyDynamics.mass_matrix!`

— Method```
mass_matrix!(M, state)
```

Compute the joint-space mass matrix (also known as the inertia matrix) of the `Mechanism`

in the given state, i.e., the matrix $M(q)$ in the unconstrained joint-space equations of motion

\[M(q) \dot{v} + c(q, v, w_\text{ext}) = \tau\]

This method implements the composite rigid body algorithm.

This method does its computation in place, performing no dynamic memory allocation.

The `out`

argument must be an $n_v \times n_v$ lower triangular `Symmetric`

matrix, where $n_v$ is the dimension of the `Mechanism`

's joint velocity vector $v$.

`RigidBodyDynamics.mass_matrix`

— MethodCompute the joint-space mass matrix (also known as the inertia matrix) of the `Mechanism`

in the given state, i.e., the matrix $M(q)$ in the unconstrained joint-space equations of motion

\[M(q) \dot{v} + c(q, v, w_\text{ext}) = \tau\]

This method implements the composite rigid body algorithm.

`RigidBodyDynamics.momentum_matrix!`

— Method```
momentum_matrix!(mat, state, transformfun)
```

Compute the momentum matrix $A(q)$ of the `Mechanism`

in the given state.

The momentum matrix maps the `Mechanism`

's joint velocity vector $v$ to its total momentum.

See also `MomentumMatrix`

.

The `out`

argument must be a mutable `MomentumMatrix`

with as many columns as the dimension of the `Mechanism`

's joint velocity vector $v$.

`transformfun`

is a callable that may be used to transform the individual momentum matrix blocks associated with each of the joints to the frame in which `out`

is expressed.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.momentum_matrix!`

— Method```
momentum_matrix!(mat, state, root_to_desired)
```

Compute the momentum matrix $A(q)$ of the `Mechanism`

in the given state.

The momentum matrix maps the `Mechanism`

's joint velocity vector $v$ to its total momentum.

See also `MomentumMatrix`

.

The `out`

argument must be a mutable `MomentumMatrix`

with as many columns as the dimension of the `Mechanism`

's joint velocity vector $v$.

`root_to_desired`

is the transform from the `Mechanism`

's root frame to the frame in which `out`

is expressed.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.momentum_matrix!`

— Method```
momentum_matrix!(out, state)
```

Compute the momentum matrix $A(q)$ of the `Mechanism`

in the given state.

The momentum matrix maps the `Mechanism`

's joint velocity vector $v$ to its total momentum.

See also `MomentumMatrix`

.

The `out`

argument must be a mutable `MomentumMatrix`

with as many columns as the dimension of the `Mechanism`

's joint velocity vector $v$.

See `momentum_matrix!(out, state, root_to_desired)`

. Uses `state`

to compute the transform from the `Mechanism`

's root frame to the frame in which `out`

is expressed.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.momentum_matrix`

— Method```
momentum_matrix(state)
```

Compute the momentum matrix $A(q)$ of the `Mechanism`

in the given state.

The momentum matrix maps the `Mechanism`

's joint velocity vector $v$ to its total momentum.

See also `MomentumMatrix`

.

`RigidBodyDynamics.point_jacobian!`

— Method```
point_jacobian!(out, state, path, point)
```

Compute the Jacobian mapping the `Mechanism`

's joint velocity vector $v$ to the velocity of a point fixed to the target of the joint path (the body succeeding the last joint in the path) with respect to the source of the joint path (the body preceding the first joint in the path).

Uses `state`

to compute the transform from the `Mechanism`

's root frame to the frame in which `out`

is expressed if necessary.

This method does its computation in place, performing no dynamic memory allocation.

`RigidBodyDynamics.point_jacobian`

— Method```
point_jacobian(state, path, point)
```

Compute the Jacobian mapping the `Mechanism`

's joint velocity vector $v$ to the velocity of a point fixed to the target of the joint path (the body succeeding the last joint in the path) with respect to the source of the joint path (the body preceding the first joint in the path).

`RigidBodyDynamics.subtree_mass`

— Method```
subtree_mass(base, mechanism)
```

Return the mass of a subtree of a `Mechanism`

, rooted at `base`

(including the mass of `base`

).