Added graph rendering, Actuator force visualization[no sites], Sim reset method(backspace) and window positioning

.gitignore

@@ -4,3 +4,164 @@ dist/
 *.egg-info
 mujoco_viewer/__pycache__
 mujoco_viewer/MUJOCO_LOG.TXT
+
+# Byte-compiled / optimized / DLL files
+__pycache__/
+*.py[cod]
+*$py.class
+
+# C extensions
+*.so
+
+# Distribution / packaging
+.Python
+build/
+develop-eggs/
+dist/
+downloads/
+eggs/
+.eggs/
+lib/
+lib64/
+parts/
+sdist/
+var/
+wheels/
+share/python-wheels/
+*.egg-info/
+.installed.cfg
+*.egg
+MANIFEST
+
+# PyInstaller
+#  Usually these files are written by a python script from a template
+#  before PyInstaller builds the exe, so as to inject date/other infos into it.
+*.manifest
+*.spec
+
+# Installer logs
+pip-log.txt
+pip-delete-this-directory.txt
+
+# Unit test / coverage reports
+htmlcov/
+.tox/
+.nox/
+.coverage
+.coverage.*
+.cache
+nosetests.xml
+coverage.xml
+*.cover
+*.py,cover
+.hypothesis/
+.pytest_cache/
+cover/
+
+# Translations
+*.mo
+*.pot
+
+# Django stuff:
+*.log
+local_settings.py
+db.sqlite3
+db.sqlite3-journal
+
+# Flask stuff:
+instance/
+.webassets-cache
+
+# Scrapy stuff:
+.scrapy
+
+# Sphinx documentation
+docs/_build/
+
+# PyBuilder
+.pybuilder/
+target/
+
+# Jupyter Notebook
+.ipynb_checkpoints
+
+# IPython
+profile_default/
+ipython_config.py
+
+# pyenv
+#   For a library or package, you might want to ignore these files since the code is
+#   intended to run in multiple environments; otherwise, check them in:
+# .python-version
+
+# pipenv
+#   According to pypa/pipenv#598, it is recommended to include Pipfile.lock in version control.
+#   However, in case of collaboration, if having platform-specific dependencies or dependencies
+#   having no cross-platform support, pipenv may install dependencies that don't work, or not
+#   install all needed dependencies.
+#Pipfile.lock
+
+# poetry
+#   Similar to Pipfile.lock, it is generally recommended to include poetry.lock in version control.
+#   This is especially recommended for binary packages to ensure reproducibility, and is more
+#   commonly ignored for libraries.
+#   https://python-poetry.org/docs/basic-usage/#commit-your-poetrylock-file-to-version-control
+#poetry.lock
+
+# pdm
+#   Similar to Pipfile.lock, it is generally recommended to include pdm.lock in version control.
+#pdm.lock
+#   pdm stores project-wide configurations in .pdm.toml, but it is recommended to not include it
+#   in version control.
+#   https://pdm.fming.dev/#use-with-ide
+.pdm.toml
+
+# PEP 582; used by e.g. github.com/David-OConnor/pyflow and github.com/pdm-project/pdm
+__pypackages__/
+
+# Celery stuff
+celerybeat-schedule
+celerybeat.pid
+
+# SageMath parsed files
+*.sage.py
+
+# Environments
+.env
+.venv
+env/
+venv/
+ENV/
+env.bak/
+venv.bak/
+
+# Spyder project settings
+.spyderproject
+.spyproject
+
+# Rope project settings
+.ropeproject
+
+# mkdocs documentation
+/site
+
+# mypy
+.mypy_cache/
+.dmypy.json
+dmypy.json
+
+# Pyre type checker
+.pyre/
+
+# pytype static type analyzer
+.pytype/
+
+# Cython debug symbols
+cython_debug/
+
+# PyCharm
+#  JetBrains specific template is maintained in a separate JetBrains.gitignore that can
+#  be found at https://github.com/github/gitignore/blob/main/Global/JetBrains.gitignore
+#  and can be added to the global gitignore or merged into this file.  For a more nuclear
+#  option (not recommended) you can uncomment the following to ignore the entire idea folder.
+#.idea/

README.md

@@ -72,4 +72,4 @@ img = viewer.read_pixels(camid=2)
 - `title`: set the title of the window, for example: `viewer = mujoco_viewer.MujocoViewer(model, data, title='My Demo')` (defaults to `mujoco-python-viewer`). 
 - `width`: set the window width, for example: `viewer = mujoco_viewer.MujocoViewer(model, data, width=300)` (defaults to full screen's width). 
 - `height`: set the window height, for example: `viewer = mujoco_viewer.MujocoViewer(model, data, height=300)` (defaults to full screen's height). 
-- `hide_menus`: set whether the overlay menus should be hidden or not (defaults to `False`).
+- `hide_menus`: set whether the overlay menus and graph should be hidden or not (defaults to `False`).

examples/actuator_force_rendering.py

@@ -0,0 +1,76 @@
+import mujoco
+import mujoco_viewer
+import math
+
+MODEL_XML = """
+<?xml version="1.0" ?>
+<mujoco>
+    <option gravity="0 0 -9.81" timestep="0.002" />
+    <asset>
+        <material name="blue" rgba="0 0 1 1" />
+        <material name="green" rgba="0 1 0 1" />
+        <material name="red" rgba="1 0 0 1" />
+        <material name="white" rgba="1 1 1 1" />
+    </asset>
+    <worldbody>
+        <geom type="plane" size="10 10 0.1" pos="0 0 -1.6" rgba=".3 .3 .3 1" />
+        <light diffuse=".5 .5 .5" pos="0 0 3" dir="0 0 -1" />
+
+        <body name="link_1" pos="0 0 0" euler="90 0 0">
+            <joint name="hinge_1" type="hinge" axis="1 0 0" />
+            <geom name="g_link_1" type="cylinder" size=".1 .5" pos="0 0 .5" euler="0 0 0" material="red" mass="1" />
+            <body name="link_2" pos=".2 0 1" euler="0 0 0">
+                <joint name="hinge_2" type="hinge" axis="0 1 0" />
+                <geom name="g_link_2" type="cylinder" size=".1 .5" pos="0 0 .5" euler="0 0 0" material="green" mass="1" />
+                <body name="link_3" pos=".2 0 1" euler="0 0 0">
+                    <joint name="hinge_3" type="slide" axis="1 0 0" />
+                    <geom name="g_link_3" type="cylinder" size=".1 .5" pos="0.3 0 0" euler="0 90 0" material="blue" mass="1" />
+                    <geom name="end_mass" type="sphere" size=".25" pos="1 0 0" euler="0 0 0" material="blue" mass="1" />
+                </body>
+            </body>
+        </body>
+
+    </worldbody>
+    <actuator>
+        <position name="pos_servo_1" joint="hinge_1" kp="100" />
+        <position name="pos_servo_2" joint="hinge_2" kp="100" />
+        <position name="pos_servo_3" joint="hinge_3" kp="100" />
+    </actuator>
+    <sensor>
+        <jointpos name="pos_sensor_1" joint="hinge_1" />
+        <jointpos name="pos_sensor_2" joint="hinge_2" />
+        <jointpos name="pos_sensor_3" joint="hinge_3" />
+    </sensor>
+</mujoco>
+"""
+
+model = mujoco.MjModel.from_xml_string(MODEL_XML)
+data = mujoco.MjData(model)
+
+# create the viewer object
+viewer = mujoco_viewer.MujocoViewer(model, data, hide_menus=True)
+
+f_render_list = [
+    ["hinge_1", "pos_servo_1", "force_at_hinge_1"],
+    ["hinge_2", "pos_servo_2", "force_at_hinge_2"],
+    ["hinge_3", "pos_servo_3", "force_at_hinge_3"],
+]
+
+for _ in range(10000):
+    # Render forces
+    viewer.show_actuator_forces(
+        f_render_list=f_render_list,
+        rgba_list=[1, 0.5, 1, 0.5],
+        force_scale=0.05,
+        arrow_radius=0.05,
+        show_force_labels=True,
+    )
+
+    # step and render
+    mujoco.mj_step(model, data)
+    viewer.render()
+    if not viewer.is_alive:
+        break
+
+# close
+viewer.close()

examples/graphs.py

@@ -0,0 +1,90 @@
+import mujoco
+import mujoco_viewer
+import math
+
+MODEL_XML = """
+<?xml version="1.0" ?>
+<mujoco>
+    <option gravity="0 0 -9.81" timestep="0.002" />
+    <default>
+        <site type="cylinder" size="0.01 0.03" zaxis="1 0 0" rgba="0.1 0.1 1 .1" />
+    </default>
+
+    <asset>
+        <material name="blue" rgba="0 0 1 1" />
+        <material name="green" rgba="0 1 0 1" />
+        <material name="red" rgba="1 0 0 1" />
+        <material name="white" rgba="1 1 1 1" />
+    </asset>
+    <worldbody>
+        <geom type="plane" size="10 10 0.1" pos="0 0 -1.6" rgba=".3 .3 .3 1" />
+        <light diffuse=".5 .5 .5" pos="0 0 3" dir="0 0 -1" />
+
+        <body name="link_1" pos="0 0 0" euler="90 0 0">
+            <joint name="hinge_1" type="hinge" axis="1 0 0" />
+            <site name="site_1" />
+            <geom name="g_link_1" type="cylinder" size=".1 .5" pos="0 0 .5" euler="0 0 0" material="red" mass="1" />
+            <body name="link_2" pos=".2 0 1" euler="0 0 0">
+                <joint name="hinge_2" type="hinge" axis="1 0 0" />
+                <site name="site_2" />
+                <geom name="g_link_2" type="cylinder" size=".1 .5" pos="0 0 .5" euler="0 0 0" material="green" mass="1" />
+                <body name="link_3" pos=".2 0 1" euler="0 0 0">
+                    <joint name="hinge_3" type="hinge" axis="1 0 0" />
+                    <site name="site_3" />
+                    <geom name="g_link_3" type="cylinder" size=".1 .5" pos="0 0 .5" euler="0 0 0" material="blue" mass="1" />
+                    <geom name="end_mass" type="sphere" size=".25" pos="0 0 1" euler="0 0 0" material="blue" mass="1" />
+                </body>
+            </body>
+        </body>
+
+    </worldbody>
+    <actuator>
+        <position name="pos_servo_1" joint="hinge_1" kp="100" />
+        <position name="pos_servo_2" joint="hinge_2" kp="100" />
+        <position name="pos_servo_3" joint="hinge_3" kp="100" />
+    </actuator>
+    <sensor>
+        <jointpos name="pos_sensor_1" joint="hinge_1" />
+        <jointpos name="pos_sensor_2" joint="hinge_2" />
+        <jointpos name="pos_sensor_3" joint="hinge_3" />
+    </sensor>
+</mujoco>
+"""
+
+model = mujoco.MjModel.from_xml_string(MODEL_XML)
+data = mujoco.MjData(model)
+
+# create the viewer object
+viewer = mujoco_viewer.MujocoViewer(model, data)
+
+
+# Initialize the graph lines
+viewer.add_graph_line(line_name="line_1")
+viewer.add_graph_line(line_name="downscaled_force_sensor")
+viewer.add_graph_line(line_name="position_sensor")
+
+# Initialize the Legend
+viewer.show_graph_legend(show_legend=True)
+
+# For a time-based graph, 
+# x_axis_time is the total time that you want your viewing window to see
+# It needs to be set to grater than [model.opt.timestep*50]
+# If you want to over ride that, change the "override" parameter to True
+viewer.set_grid_divisions(x_div=5,
+                          y_div=4,
+                          x_axis_time=model.opt.timestep * 1000,
+                          override=True)
+
+for i in range(10000):
+    viewer.update_graph_line(line_name="line_1", line_data=math.sin(i / 10.0))
+    viewer.update_graph_line(line_name="downscaled_force_sensor", line_data=data.actuator_force[0]/100.0)
+    viewer.update_graph_line(line_name="position_sensor", line_data=data.sensordata[0])
+
+    # step and render
+    mujoco.mj_step(model, data)
+    viewer.render()
+    if not viewer.is_alive:
+        break
+
+# close
+viewer.close()

mujoco_viewer/callbacks.py

@@ -18,6 +18,7 @@ def __init__(self, hide_menus):
         self._last_mouse_x = 0
         self._last_mouse_y = 0
         self._paused = False
+        self._hide_graph = False
         self._transparent = False
         self._contacts = False
         self._joints = False
@@ -97,6 +98,9 @@ def _key_callback(self, window, key, scancode, action, mods):
                 self.model.geom_rgba[:, 3] /= 5.0
             else:
                 self.model.geom_rgba[:, 3] *= 5.0
+        # Toggle Graph overlay
+        elif key == glfw.KEY_G:
+            self._hide_graph = not self._hide_graph
         # Display inertia
         elif key == glfw.KEY_I:
             self._inertias = not self._inertias
@@ -115,6 +119,10 @@ def _key_callback(self, window, key, scancode, action, mods):
             self.vopt.flags[
                 mujoco.mjtVisFlag.mjVIS_CONVEXHULL
             ] = self._convex_hull_rendering
+        # Reload Simulation 
+        elif key == glfw.KEY_BACKSPACE:
+            mujoco.mj_resetData(self.model, self.data)
+            mujoco.mj_forward(self.model, self.data)
         # Wireframe Rendering
         elif key == glfw.KEY_W:
             self._wire_frame = not self._wire_frame

mujoco_viewer/mujoco_viewer.py

@@ -6,7 +6,6 @@
 import yaml
 from .callbacks import Callbacks
 
-
 class MujocoViewer(Callbacks):
     def __init__(
             self,
@@ -16,8 +15,12 @@ def __init__(
             title="mujoco-python-viewer",
             width=None,
             height=None,
+            window_start_x_pixel_offset=6,
+            window_start_y_pixel_offset=30,
             hide_menus=False):
         super().__init__(hide_menus)
+        if hide_menus is True:
+            self._hide_graph = True
 
         self.model = model
         self.data = data
@@ -40,14 +43,23 @@ def __init__(
 
         if not height:
             _, height = glfw.get_video_mode(glfw.get_primary_monitor()).size
-
-        if self.render_mode == 'offscreen':
-            glfw.window_hint(glfw.VISIBLE, 0)
-
+
+        # disable rendering, by default
+        glfw.window_hint(glfw.VISIBLE, 0)
         self.window = glfw.create_window(
             width, height, title, None, None)
         glfw.make_context_current(self.window)
+        glfw.set_window_pos(self.window, 
+                            window_start_x_pixel_offset,
+                            window_start_y_pixel_offset)
+
+        # Select if offscreen or on window mode
+        if self.render_mode == 'offscreen':
+            glfw.window_hint(glfw.VISIBLE, 0)
+        else:
+            glfw.show_window(self.window)
         glfw.swap_interval(1)
+
 
         framebuffer_width, framebuffer_height = glfw.get_framebuffer_size(
             self.window)
@@ -70,8 +82,37 @@ def __init__(
         self.cam = mujoco.MjvCamera()
         self.scn = mujoco.MjvScene(self.model, maxgeom=10000)
         self.pert = mujoco.MjvPerturb()
+        self.fig = mujoco.MjvFigure()
+        mujoco.mjv_defaultFigure(self.fig)
+
+        # Points for sampling of sensors... dictates smoothness of graph
+        # Do not exceed: self._num_pnts = 1000
+        self._num_pnts = 1000
+        self._data_graph_line_names = []
+        self._line_datas = []
+
+        for n in range(0, mujoco.mjMAXLINE):
+            for i in range(0, self._num_pnts):
+                self.fig.linedata[n][2 * i] = float(-i)
+
         self.ctx = mujoco.MjrContext(
             self.model, mujoco.mjtFontScale.mjFONTSCALE_150.value)
+
+        # Adjust placement and size of graph
+        width, height = glfw.get_framebuffer_size(self.window)
+        width_adjustment = width % 4
+        self.graph_viewport = mujoco.MjrRect(
+            int(3 * width / 4) + width_adjustment,
+            0,
+            int(width / 4),
+            int(height / 4),
+        )
+        mujoco.mjr_figure(self.graph_viewport, self.fig, self.ctx)
+        self.fig.flg_extend = 1
+        self.fig.flg_symmetric = 0
+
+        # Makes the graph to be in autorange
+        self.axis_autorange()
 
         # load camera from configuration (if available)
         pathlib.Path(
@@ -118,6 +159,155 @@ def __init__(
         self._overlay = {}
         self._markers = []
 
+    def set_grid_divisions(self, x_div: int, y_div: int, x_axis_time: float = 0.0, override=False):
+        if override is False:
+            assert x_axis_time >= self.model.opt.timestep * 50, "Set [x_axis_time] >= [self.model.opt.timestep * 50], inorder to get a suitable sampling rate"  
+        self.fig.gridsize[0] = x_div + 1
+        self.fig.gridsize[1] = y_div + 1
+        if x_axis_time != 0.0:
+            self._num_pnts = x_axis_time / self.model.opt.timestep
+            print("self._num_pnts: ", self._num_pnts)
+            if self._num_pnts > 1000:
+                self._num_pnts = 1000
+                new_x_axis_time = self.model.opt.timestep * self._num_pnts
+                print(
+                    f"Minimum x_axis_time is: {new_x_axis_time}"
+                    + " reduce the x_axis_time"
+                    f" OR Maximum time_step is: "
+                    + f"{self.model.opt.timestep*self._num_pnts}"
+                    + " increase the timestep"
+                )
+                # assert x_axis_time ==
+            assert 1 <= self._num_pnts <= 1000, (
+                "num_pnts should be [1000], it is currently:",
+                f"{self._num_pnts}",
+            )
+            # self._num_pnts = num_pnts
+            self._time_per_div = (self.model.opt.timestep * self._num_pnts) / (
+                x_div
+            )
+            self.set_x_label(
+                xname=f"time/div: {self._time_per_div}s"
+                + f" total: {self.model.opt.timestep * self._num_pnts}"
+            )
+
+    def axis_autorange(self):
+        """
+        Call this function to auto-range the graph 
+        """
+        self.fig.range[0][0] = 1.0
+        self.fig.range[0][1] = -1.0
+        self.fig.range[1][0] = 1.0
+        self.fig.range[1][1] = -1.0
+
+    def set_graph_name(self, name: str):
+        assert type(name) == str, "name is not a string"
+        self.fig.title = name
+
+    def show_graph_legend(self, show_legend: bool = True):
+        if show_legend is True:
+            for i in range(0, len(self._data_graph_line_names)):
+                self.fig.linename[i] = self._data_graph_line_names[i]
+            self.fig.flg_legend = True
+
+    def set_x_label(self, xname: str):
+        assert type(xname) == str, "xname is not a string"
+        self.fig.xlabel = xname
+
+    def add_graph_line(self, line_name, line_data=0.0):
+        assert (
+            type(line_name) == str
+        ), f"Line_name is not a string: {type(line_name)}"
+        if line_name in self._data_graph_line_names:
+            print("line name already exists")
+        else:
+            self._data_graph_line_names.append(line_name)
+            self._line_datas.append(line_data)
+
+    def update_graph_line(self, line_name, line_data):
+        if line_name in self._data_graph_line_names:
+            idx = self._data_graph_line_names.index(line_name)
+            self._line_datas[idx] = line_data
+        else:
+            raise NameError(
+                "line name is not valid, add it to list before calling update"
+            )
+
+    def sensorupdate(self):
+        pnt = int(mujoco.mju_min(self._num_pnts, self.fig.linepnt[0] + 1))
+
+        for n in range(0, len(self._line_datas)):
+            for i in range(pnt - 1, 0, -1):
+                self.fig.linedata[n][2 * i + 1] = self.fig.linedata[n][
+                    2 * i - 1
+                ]
+            self.fig.linepnt[n] = pnt
+            self.fig.linedata[n][1] = self._line_datas[n]
+
+    def update_graph_size(self, size_div_x=None, size_div_y=None):
+        if size_div_x is None and size_div_y is None:
+            width, height = glfw.get_framebuffer_size(self.window)
+            width_adjustment = width % 3
+            self.graph_viewport.left = int(2 * width / 3) + width_adjustment
+            self.graph_viewport.width = int(width / 3)
+            self.graph_viewport.height = int(height / 3)
+
+        else:
+            assert size_div_x is not None and size_div_y is None, ""
+            width, height = glfw.get_framebuffer_size(self.window)
+            width_adjustment = width % size_div_x
+            self.graph_viewport.left = (
+                int((size_div_x - 1) * width / size_div_x) + width_adjustment
+            )
+            self.graph_viewport.width = int(width / size_div_x)
+            self.graph_viewport.height = int(height / size_div_x)
+
+    def show_actuator_forces(
+        self,
+        f_render_list,
+        rgba_list=[1, 0, 1, 1],
+        force_scale=0.05,
+        arrow_radius=0.03,
+        show_force_labels=False,
+    ) -> None:
+        """f_render_list: [  ["jnt_name1","act_name_1","lable1"] ,
+                             ["jnt_name2","act_name_2","lable2"] ]
+        """
+        if show_force_labels is False:
+            for i in range(0, len(f_render_list)):
+                self.add_marker(
+                    pos=self.data.joint(f_render_list[i][0]).xanchor,
+                    mat=self.rotation_matrix_from_vectors(
+                        vec1=[0.0, 0.0, 1.0],
+                        vec2=self.data.joint(f_render_list[i][0]).xaxis),
+                    size=[
+                        arrow_radius,
+                        arrow_radius,
+                        self.data.actuator(f_render_list[i][1]).force* force_scale,
+                    ],
+                    rgba=rgba_list,
+                    type=mujoco.mjtGeom.mjGEOM_ARROW,
+                    label="",
+                )
+        else:
+            for i in range(0, len(f_render_list)):
+                self.add_marker(
+                    pos=self.data.joint(f_render_list[i][0]).xanchor,
+                    mat=self.rotation_matrix_from_vectors(
+                        vec1=[0.0, 0.0, 1.0],
+                        vec2=self.data.joint(f_render_list[i][0]).xaxis),
+                    size=[
+                        arrow_radius,
+                        arrow_radius,
+                        self.data.actuator(f_render_list[i][1]).force* force_scale,
+                    ],
+                    rgba=rgba_list,
+                    type=mujoco.mjtGeom.mjGEOM_ARROW,
+                    label=f_render_list[i][2]
+                    + ":"
+                    + str(self.data.actuator(f_render_list[i][1]).force[0]),
+                )
+
     def add_marker(self, **marker_params):
         self._markers.append(marker_params)
 
@@ -205,6 +395,10 @@ def add_overlay(gridpos, text1, text2):
             topleft,
             "[J]oints",
             "On" if self._joints else "Off")
+        add_overlay(
+            topleft,
+            "[G]raph Viewer",
+            "Off" if self._hide_graph else "On")
         add_overlay(
             topleft,
             "[I]nertia",
@@ -250,6 +444,8 @@ def add_overlay(gridpos, text1, text2):
             add_overlay(topleft, "Cap[t]ure frame", "Saved as %s" % fname)
         else:
             add_overlay(topleft, "Cap[t]ure frame", "")
+        add_overlay(topleft, "[ESC] to Quit Application", "")
+        add_overlay(topleft, "[BACKSPACE] to Reload Sim", "")
 
         add_overlay(
             bottomleft, "FPS", "%d%s" %
@@ -355,6 +551,27 @@ def update():
                         t1,
                         t2,
                         self.ctx)
+
+                # Handle graph and pausing interactions
+                if (
+                    not self._paused
+                    and not self._hide_graph
+                ):
+                    self.sensorupdate()
+                    self.update_graph_size()
+                    mujoco.mjr_figure(
+                        self.graph_viewport, self.fig, self.ctx
+                    )
+                elif self._hide_graph and self._paused:
+                    self.update_graph_size()
+                elif not self._hide_graph and self._paused:
+                    mujoco.mjr_figure(
+                        self.graph_viewport, self.fig, self.ctx
+                    )
+                elif self._hide_graph and not self._paused:
+                    self.sensorupdate()
+                    self.update_graph_size()
+
                 glfw.swap_buffers(self.window)
             glfw.poll_events()
             self._time_per_render = 0.9 * self._time_per_render + \
@@ -391,3 +608,17 @@ def close(self):
         self.is_alive = False
         glfw.terminate()
         self.ctx.free()
+
+    def rotation_matrix_from_vectors(self, vec1, vec2):
+        """ Find the rotation matrix that aligns vec1 to vec2
+        :param vec1: A 3d "source" vector
+        :param vec2: A 3d "destination" vector
+        :return mat: A transform matrix (3x3) which when applied to vec1, aligns it with vec2.
+        """
+        a, b = (vec1 / np.linalg.norm(vec1)).reshape(3), (vec2 / np.linalg.norm(vec2)).reshape(3)
+        v = np.cross(a, b)
+        c = np.dot(a, b)
+        s = np.linalg.norm(v)
+        kmat = np.array([[0, -v[2], v[1]], [v[2], 0, -v[0]], [-v[1], v[0], 0]])
+        rotation_matrix = np.eye(3) + kmat + kmat.dot(kmat) * ((1 - c) / (s ** 2))
+        return rotation_matrix