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```
```use na::{self, DVectorSliceMut, Isometry3, RealField, Translation3, Unit, Vector3};

use crate::joint::{Joint, RevoluteJoint};
use crate::math::{JacobianSliceMut, Velocity};
use crate::solver::{ConstraintSet, GenericNonlinearConstraint, IntegrationParameters};

/// A joint that allows only two relative rotations between two multibody links.
#[derive(Copy, Clone, Debug)]
pub struct UniversalJoint<N: RealField> {
revo1: RevoluteJoint<N>,
revo2: RevoluteJoint<N>,

coupling_dot: Velocity<N>,
coupling_dot_veldiff: Velocity<N>,
}

impl<N: RealField> UniversalJoint<N> {
/// Creates an universal joint allowing rotations along `axis1` and `axis2`.
///
/// The axii are expressed in the local coordinate systems of the attached multibody links.
pub fn new(axis1: Unit<Vector3<N>>, axis2: Unit<Vector3<N>>, angle1: N, angle2: N) -> Self {
UniversalJoint {
revo1: RevoluteJoint::new(axis1, angle1),
revo2: RevoluteJoint::new(axis2, angle2),
coupling_dot: Velocity::zero(),
coupling_dot_veldiff: Velocity::zero(),
}
}
}

impl<N: RealField> Joint<N> for UniversalJoint<N> {
#[inline]
fn ndofs(&self) -> usize {
2
}

fn body_to_parent(&self, parent_shift: &Vector3<N>, body_shift: &Vector3<N>) -> Isometry3<N> {
Translation3::from(*parent_shift)
* self.revo1.rotation()
* self.revo2.body_to_parent(&na::zero(), body_shift)
}

fn update_jacobians(&mut self, body_shift: &Vector3<N>, vels: &[N]) {
self.revo1.update_jacobians(body_shift, vels);
self.revo2.update_jacobians(body_shift, &[vels[1]]);

let axis1 = self.revo1.axis();
let rot1 = self.revo1.rotation();
let jac2 = self.revo2.local_jacobian().rotated(rot1);

self.coupling_dot_veldiff =
Velocity::new(axis1.cross(&jac2.linear), axis1.cross(&jac2.angular));
self.coupling_dot = self.coupling_dot_veldiff * vels[0];
}

fn jacobian(&self, transform: &Isometry3<N>, out: &mut JacobianSliceMut<N>) {
self.revo1.jacobian(transform, &mut out.columns_mut(0, 1));
self.revo2.jacobian(
&(transform * self.revo1.rotation()),
&mut out.columns_mut(1, 1),
);
}

fn jacobian_dot(&self, transform: &Isometry3<N>, out: &mut JacobianSliceMut<N>) {
self.revo1
.jacobian_dot(transform, &mut out.columns_mut(0, 1));
let rot1 = self.revo1.rotation();
let jac2 = self.revo2.local_jacobian_dot().rotated(rot1) + self.coupling_dot;
out.column_mut(1)
.copy_from(jac2.transformed(transform).as_vector())
}

fn jacobian_dot_veldiff_mul_coordinates(
&self,
transform: &Isometry3<N>,
vels: &[N],
out: &mut JacobianSliceMut<N>,
) {
let jac1 = *self.revo1.local_jacobian_dot_veldiff() + self.coupling_dot_veldiff;
out.column_mut(0)
.copy_from(jac1.transformed(transform).as_vector());

let rot1 = self.revo1.rotation();
self.revo2.jacobian_dot_veldiff_mul_coordinates(
&(transform * rot1),
vels,
&mut out.columns_mut(1, 1),
);
}

fn jacobian_mul_coordinates(&self, vels: &[N]) -> Velocity<N> {
self.revo1.jacobian_mul_coordinates(vels) + self.revo2.jacobian_mul_coordinates(&[vels[1]])
}

fn jacobian_dot_mul_coordinates(&self, vels: &[N]) -> Velocity<N> {
let rot1 = self.revo1.rotation();
self.revo1.jacobian_dot_mul_coordinates(vels)
+ self
.revo2
.jacobian_dot_mul_coordinates(&[vels[1]])
.rotated(&rot1)
+ self.coupling_dot * vels[1]
}

fn default_damping(&self, out: &mut DVectorSliceMut<N>) {
self.revo1.default_damping(&mut out.rows_mut(0, 1));
self.revo2.default_damping(&mut out.rows_mut(1, 1));
}

fn integrate(&mut self, parameters: &IntegrationParameters<N>, vels: &[N]) {
self.revo1.integrate(parameters, vels);
self.revo2.integrate(parameters, &[vels[1]]);
}

fn apply_displacement(&mut self, disp: &[N]) {
self.revo1.apply_displacement(disp);
self.revo2.apply_displacement(&[disp[1]]);
}

#[inline]
fn clone(&self) -> Box<dyn Joint<N>> {
Box::new(*self)
}

fn num_velocity_constraints(&self) -> usize {
self.revo1.num_velocity_constraints() + self.revo2.num_velocity_constraints()
}

fn velocity_constraints(
&self,
parameters: &IntegrationParameters<N>,
multibody: &Multibody<N>,
assembly_id: usize,
dof_id: usize,
ext_vels: &[N],
ground_j_id: &mut usize,
jacobians: &mut [N],
constraints: &mut ConstraintSet<N, (), (), usize>,
) {
self.revo1.velocity_constraints(
parameters,
multibody,
assembly_id,
dof_id,
ext_vels,
ground_j_id,
jacobians,
constraints,
);
self.revo2.velocity_constraints(
parameters,
multibody,
assembly_id,
dof_id + 1,
ext_vels,
ground_j_id,
jacobians,
constraints,
);
}

fn num_position_constraints(&self) -> usize {
// NOTE: we don't test if constraints exist to simplify indexing.
2
}

fn position_constraint(
&self,
i: usize,
multibody: &Multibody<N>,
handle: BodyPartHandle<()>,
dof_id: usize,
jacobians: &mut [N],
) -> Option<GenericNonlinearConstraint<N, ()>> {
if i == 0 {
self.revo1
.position_constraint(0, multibody, link, handle, dof_id, jacobians)
} else {
self.revo2
.position_constraint(0, multibody, link, handle, dof_id + 1, jacobians)
}
}
}

revolute_motor_limit_methods_1!(UniversalJoint, revo1);
revolute_motor_limit_methods_2!(UniversalJoint, revo2);
```