Fix typos and minor corrections in MuJoCo documentation and comments.

PiperOrigin-RevId: 875163889
Change-Id: Ic604d6235bf9cb999058f191e3ee41e47ec1a567

doc/computation/fluid.rst

@@ -72,7 +72,7 @@ body coordinates are
    g_{V, i} = \quad &- 8 \beta \pi r_{eq}^3 \omega_i \\
    \end{aligned}
 
-One can also affect these forces by specifing a non-zero :ref:`wind<option-wind>`, which is a 3D vector subtracted
+One can also affect these forces by specifying a non-zero :ref:`wind<option-wind>`, which is a 3D vector subtracted
 from the body linear velocity in the fluid dynamics computation.
 
 .. _flEllipsoid:
@@ -357,7 +357,7 @@ of symmetry because under this assumption the kinetic energy greatly simplifies
 
 .. math::
    2 \mathcal{T} = m_{A, x} v_x^2 + m_{A, y} v_y^2 + m_{A, z} v_z^2 +
-                 I_{A, x} \omega_x^2 + I_ {A, y} \omega_y^2 + I_{A, y} \omega_z^2
+                 I_{A, x} \omega_x^2 + I_ {A, y} \omega_y^2 + I_{A, z} \omega_z^2
 
 
 For convenience we introduce the added-mass vector :math:`\mathbf{m}_A = \{m_{A, x}, m_{A, y}, m_{A, z}\}` and added-moment of
@@ -429,7 +429,7 @@ bagheri2016`. See screen capture of the
 We derive a formula for :math:`\mathbf{f}_\text{D}` based on two surfaces :math:`A^\text{proj}_\mathbf{v}` and
 :math:`A_\text{max}`. The first, :math:`A^\text{proj}_\mathbf{v}`, is the cylindrical projection of the body onto a
 plane normal to the velocity :math:`\mathbf{v}`. The second is the maximum projected surface
-:math:`A_\text{max} = 4 \pi r_{max} r_{min}`.
+:math:`A_\text{max} = \pi r_{max} r_{mid}`.
 
 .. math::
    \mathbf{f}_\text{D} = - \rho~ \big[  C_{D, \text{blunt}} ~ A^\text{proj}_\mathbf{v} ~ +
@@ -541,7 +541,7 @@ onto the surface. The lift force per unit length can be computed with the Kutta
 :math:`\mathbf{f}_K / L = \rho \Gamma_\text{K} \times \mathbf{v}`.
 
 In order to extend the lift force equation to three-dimensional motions, we consider the normal
-:math:`\mathbf{n}_{s, \mathbf{v}} = \{\frac{r_y r_z}{r_x}v_x, \frac{r_z r_x}{r_y}v_y, \frac{r_x r_x}{r_z}v_z\}`
+:math:`\mathbf{n}_{s, \mathbf{v}} = \{\frac{r_y r_z}{r_x}v_x, \frac{r_z r_x}{r_y}v_y, \frac{r_x r_y}{r_z}v_z\}`
 to the cross-section of the body which generates the body's projection :math:`A^\text{proj}_\mathbf{v}` onto a plane
 normal to the velocity given in the :ref:`lemma<flProjection>` above and the corresponding unit vector
 :math:`\hat{\mathbf{n}}_{s, \mathbf{v}}`.

doc/computation/index.rst

@@ -531,7 +531,7 @@ All three single-step integrators in MuJoCo use the update :eq:`eq_implicit_upda
 Semi-implicit with implicit joint damping (``Euler``)
    For this method, :math:`D` only includes derivatives of joint damping. Note that in this case :math:`D` is diagonal
    and :math:`\widehat{M}` is symmetric, so :math:`L^TL` decomposition (a variant of Cholesky) can be used. This
-   factorization is stored ``mjData.qLD``. If the model has no joint damping or the
+   factorization is stored in ``mjData.qH``. If the model has no joint damping or the
    :ref:`eulerdamp<option-flag-eulerdamp>` disable-flag is set, implicit damping is disabled and the semi-implicit
    update :eq:`eq_semimplicit` is used, rather than :eq:`eq_implicit_update`, avoiding the additional factorization of
    :math:`\widehat{M}` (*additional* because :math:`M` is already factorized for the acceleration update
@@ -579,11 +579,11 @@ Fast implicit-in-velocity (``implicitfast``)
     step is "just right", but that range is model-dependent.
 
    :ref:`integrator<option-integrator>`
-    Summary: The recommended integrator is ``implicitfast`` which usually has the best tradeoff of stabillity and
+    Summary: The recommended integrator is ``implicitfast`` which usually has the best tradeoff of stability and
     performance.
 
     **Euler**:
-     Use ``Euler`` for compatibillity with older models.
+     Use ``Euler`` for compatibility with older models.
     **implicitfast**:
      The ``implicitfast`` integrator has similar computational cost to ``Euler``, yet provides
      increased stability, and is therefore a strict improvement. It is the recommended integrator for most models.
@@ -839,14 +839,14 @@ as defined later. The ``condim`` parameter determines the contact type, and has
 
 ``condim = 3`` : 3 for elliptic, 4 for pyramidal
    This is a regular frictional contact, which can generate normal force as well as a tangential friction force opposing
-   slip. An interpertation of this number is the slope of a surface above which a flat object will begin to slip
+   slip. An interpretation of this number is the slope of a surface above which a flat object will begin to slip
    under gravity.
 
 ``condim = 4`` : 4 for elliptic, 6 for pyramidal
    In addition to normal and tangential force, this contact can generate torsional friction torque opposing rotation
    around the contact normal, corresponding to a torque generated by a contacting surface patch. This is useful for
    modeling soft fingers, and can substantially improve the stability of simulated grasping. Torsional friction
-   coefficients have **units of length** which can be interperted as the diameter of the surface contact patch.
+   coefficients have **units of length** which can be interpreted as the diameter of the surface contact patch.
 
 ``condim = 6`` : 6 for elliptic, 10 for pyramidal
    This contact can oppose motion in all relative degrees of freedom between the two geoms. In particular it adds