function

LieGroups.SO3.Quat.left_jacobian_exact

Left Jacobian J_l(v) of SO(3), closed form down to the exp_map branch radius

The same matrix as left_jacobian, evaluated with the branch radius and the coefficient forms of right_jacobian_exact. Both coefficients depend on the rotation vector only through t^2, so J_l(v) = J_r(-v) is exact rather than an approximation, and reflecting is the whole implementation.

No rule returns a left-trivialized derivative, so no rule calls this today; Tests.RuleChecks.so3JacobianFamilyResiduals does, against the transpose of right_jacobian_exact. It exists because the reflection is the definition of the left Jacobian and a left-trivialized rule added later must not reach for left_jacobian, whose 0.1 rad radius would put the same |v|^5/720 series truncation back into a derivative.

Public declarations

Explicit source declarations; dimensions, modifications and defaults are shown as written, without evaluation.

Role	Name	Type	Dimensions / attributes / default	Description
input	`v`	`Real`	`[3]`	Rotation vector
output	`J`	`Real`	`[3, 3]`	I + ((1-cos t)/t^2)[v]x + ((t-sin t)/t^3)[v]x^2, t = \|v\|

Modelica declarations and implementation

Show parameters, interfaces and source

Parameters, inputs, outputs, defaults and equations from the source file.

function left_jacobian_exact
  "Left Jacobian J_l(v) of SO(3), closed form down to the exp_map branch radius"
  input Real v[3] "Rotation vector";
  output Real J[3, 3] "I + ((1-cos t)/t^2)[v]x + ((t-sin t)/t^3)[v]x^2, t = |v|";
algorithm
  J := LieGroups.SO3.Quat.right_jacobian_exact(-v);
  annotation(Documentation(info="<html>
    <p>The same matrix as <code>left_jacobian</code>, evaluated with the branch
    radius and the coefficient forms of
    <code>right_jacobian_exact</code>. Both coefficients depend on the rotation
    vector only through t^2, so J_l(v) = J_r(-v) is exact rather than an
    approximation, and reflecting is the whole implementation.</p>
    <p>No rule returns a left-trivialized derivative, so no rule calls this
    today; <code>Tests.RuleChecks.so3JacobianFamilyResiduals</code> does, against
    the transpose of <code>right_jacobian_exact</code>. It exists because the
    reflection is the definition of the left Jacobian and a left-trivialized
    rule added later must not reach for <code>left_jacobian</code>, whose
    0.1 rad radius would put the same |v|^5/720 series truncation back into a
    derivative.</p>
  </html>"));
end left_jacobian_exact;