Add dampratio attribute to position and intvelocity actuators.

PiperOrigin-RevId: 639745135
Change-Id: I94ba4c346d2750c6c454a60a568f2e6f6bce19d4

doc/XMLreference.rst

@@ -5045,10 +5045,26 @@ This element has one custom attribute in addition to the common attributes:
    Damping applied by the actuator.
    When using this attribute, it is recommended to use the implicitfast or implicit :ref:`integrators<geIntegration>`.
 
+.. _actuator-position-dampratio:
+
+:at:`dampratio`: :at-val:`real, "0"`
+   Damping applied by the actuator, using damping ratio units.
+   This attribute is exclusive with :at:`kv` and has similar meaning, but instead of units of force/velocity, the units
+   are :math:`2 \sqrt{k_p \cdot m}`, corresponding to a harmonic oscillator's
+   `damping ratio <https://en.wikipedia.org/wiki/Damping#Damping_ratio_definition>`__.
+   A value of 1 corresponds to a *critically damped* oscillator, which often produces desirable behavior.
+   Values smaller or larger than 1 correspond to underdamped and overdamped oscillations, respectively.
+   The mass :math:`m` is computed at the reference configuration ``mjModel.qpos0``, taking into account joint
+   :ref:`armature <body-joint-armature>`.
+   However, passive :ref:`damping <body-joint-damping>` or :ref:`frictionloss <body-joint-frictionloss>` in the affected
+   joints are not taken into account; if they are non-negligible, :at:`dampratio` values smaller than 1 might be
+   required to achieve desirable motion.
+   When using this attribute, it is recommended to use the implicitfast or implicit :ref:`integrators<geIntegration>`.
+
 .. _actuator-position-timeconst:
 
 :at:`timeconst`: :at-val:`real, "0"`
-   Time-constant of the first-order filter.  If larger than zero, the actuator uses the :at:`filterexact`
+   Time-constant of optional first-order filter.  If larger than zero, the actuator uses the :at:`filterexact`
    :ref:`dynamics type<actuator-general-dyntype>`, if zero (the default) no filter is used.
 
 
@@ -5214,6 +5230,11 @@ This element has one custom attribute in addition to the common attributes:
    Damping applied by the actuator.
    When using this attribute, it is recommended to use the implicitfast or implicit :ref:`integrators<geIntegration>`.
 
+.. _actuator-intvelocity-dampratio:
+
+:at:`dampratio`: :at-val:`real, "0"`
+   See :ref:`position/dampratio<actuator-position-dampratio>`.
+
 .. _actuator-intvelocity-inheritrange:
 
 :at:`inheritrange`: :at-val:`real, "0"`
@@ -8097,6 +8118,8 @@ tendon, slidersite, cranksite.
 
 .. _default-position-kv:
 
+.. _default-position-dampratio:
+
 .. _default-position-timeconst:
 
 :el-prefix:`default/` |-| **position** (?)
@@ -8159,6 +8182,8 @@ refsite, tendon, slidersite, cranksite.
 
 .. _default-intvelocity-kv:
 
+.. _default-intvelocity-dampratio:
+
 :el-prefix:`default/` |-| **intvelocity** (?)
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
 

doc/XMLschema.rst

@@ -705,7 +705,7 @@
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
 |                                    |    |    | :ref:`cranksite<actuator-position-cranksite>`                   | :ref:`site<actuator-position-site>`                             | :ref:`refsite<actuator-position-refsite>`                       | :ref:`kp<actuator-position-kp>`                                 | |
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
-|                                    |    |    | :ref:`kv<actuator-position-kv>`                                 | :ref:`timeconst<actuator-position-timeconst>`                   |                                                                 |                                                                 | |
+|                                    |    |    | :ref:`kv<actuator-position-kv>`                                 | :ref:`dampratio<actuator-position-dampratio>`                   | :ref:`timeconst<actuator-position-timeconst>`                   |                                                                 | |
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
 +------------------------------------+----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
 | |_| actuator |br| |_| |L|          |    | .. table::                                                                                                                                                                                                                                                                   |
@@ -737,7 +737,7 @@
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
 |                                    |    |    | :ref:`slidersite<actuator-intvelocity-slidersite>`              | :ref:`cranksite<actuator-intvelocity-cranksite>`                | :ref:`site<actuator-intvelocity-site>`                          | :ref:`refsite<actuator-intvelocity-refsite>`                    | |
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
-|                                    |    |    | :ref:`kp<actuator-intvelocity-kp>`                              | :ref:`kv<actuator-intvelocity-kv>`                              |                                                                 |                                                                 | |
+|                                    |    |    | :ref:`kp<actuator-intvelocity-kp>`                              | :ref:`kv<actuator-intvelocity-kv>`                              | :ref:`dampratio<actuator-intvelocity-dampratio>`                |                                                                 | |
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
 +------------------------------------+----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
 | |_| actuator |br| |_| |L|          |    | .. table::                                                                                                                                                                                                                                                                   |
@@ -1477,7 +1477,9 @@
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
 |                                    |    |    | :ref:`forcerange<default-position-forcerange>`                  | :ref:`gear<default-position-gear>`                              | :ref:`cranklength<default-position-cranklength>`                | :ref:`user<default-position-user>`                              | |
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
-|                                    |    |    | :ref:`group<default-position-group>`                            | :ref:`kp<default-position-kp>`                                  | :ref:`kv<default-position-kv>`                                  | :ref:`timeconst<default-position-timeconst>`                    | |
+|                                    |    |    | :ref:`group<default-position-group>`                            | :ref:`kp<default-position-kp>`                                  | :ref:`kv<default-position-kv>`                                  | :ref:`dampratio<default-position-dampratio>`                    | |
+|                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
+|                                    |    |    | :ref:`timeconst<default-position-timeconst>`                    |                                                                 |                                                                 |                                                                 | |
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
 +------------------------------------+----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
 | |_| default |br| |_| |L|           |    | .. table::                                                                                                                                                                                                                                                                   |
@@ -1501,6 +1503,8 @@
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
 |                                    |    |    | :ref:`user<default-intvelocity-user>`                           | :ref:`group<default-intvelocity-group>`                         | :ref:`kp<default-intvelocity-kp>`                               | :ref:`kv<default-intvelocity-kv>`                               | |
 |                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
+|                                    |    |    | :ref:`dampratio<default-intvelocity-dampratio>`                 |                                                                 |                                                                 |                                                                 | |
+|                                    |    |    +-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+-----------------------------------------------------------------+ |
 +------------------------------------+----+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
 | |_| default |br| |_| |L|           |    | .. table::                                                                                                                                                                                                                                                                   |
 | :ref:`damper                       | ?  |    :class: mjcf-attributes                                                                                                                                                                                                                                                   |

doc/changelog.rst

@@ -11,8 +11,13 @@ General
 1. Added :ref:`mj_geomDistance` for computing the shortest signed distance between two geoms and optionally a segment
    connecting them. Relatedly, added the 3 sensors: :ref:`distance<sensor-distance>`, :ref:`normal<sensor-normal>`,
    :ref:`fromto<sensor-fromto>`. See the function and sensor documentation for details. Fixes :github:issue:`51`.
-2. Added :ref:`timeconst<actuator-position-timeconst>` attribute to the :ref:`position actuator<actuator-position>`.
+2. Improvements to position actuators:
-   When set to a positive value, the actuator is made stateful with :at:`filterexact` dynamics.
+
+   - Added :ref:`timeconst<actuator-position-timeconst>` attribute to the :ref:`position actuator<actuator-position>`.
+     When set to a positive value, the actuator is made stateful with :at:`filterexact` dynamics.
+   - Added :ref:`dampratio<actuator-position-dampratio>` to both :el:`position` and :el:`intvelocity` actuators. An
+     alternative to the :at:`kv` attribute, it provides a convenient way to set actuator damping using natural units.
+     See attribute documentation for details.
 
 MJX
 ^^^

src/engine/engine_setconst.c

@@ -341,12 +341,48 @@ static void set0(mjModel* m, mjData* d) {
 
     // compute positional offsets
     mju_sub3(m->light_pos0+3*i, d->light_xpos+3*i, d->xpos+3*id);
-    mju_sub3(m->light_poscom0+3*i, d->light_xpos+3*i, d->subtree_com+ (id1 >= 0 ? 3*id1 : 3*id));
+    mju_sub3(m->light_poscom0+3*i, d->light_xpos+3*i, d->subtree_com + (id1 >= 0 ? 3*id1 : 3*id));
 
     // copy dir
     mju_copy3(m->light_dir0+3*i, d->light_xdir+3*i);
   }
 
+  // compute actuator damping from dampratio
+  for (int i=0; i < m->nu; i++) {
+    // get bias, gain parameters
+    mjtNum* biasprm = m->actuator_biasprm + i*mjNBIAS;
+    mjtNum* gainprm = m->actuator_gainprm + i*mjNGAIN;
+
+    // not a position-like actuator: skip
+    if (gainprm[0] != -biasprm[1]) {
+      continue;
+    }
+
+    // damping is 0 or negative (interpreted as regular "kv"): skip
+    if (biasprm[2] <= 0) {
+      continue;
+    }
+
+    // === interpret biasprm[2] > 0 as dampratio for position-like actuators
+
+    // "reflected" inertia (inversely scaled by transmission squared)
+    mjtNum* transmission = d->actuator_moment + i*nv;
+    mjtNum mass = 0;
+    for (int j=0; j < nv; j++) {
+      mjtNum trn = mju_abs(transmission[j]);
+      mjtNum trn2 = trn*trn;  // transmission squared
+      if (trn2 > mjMINVAL) {
+        mass += m->dof_M0[j] / trn2;
+      }
+    }
+
+    // damping = dampratio * 2 * sqrt(kp * mass)
+    mjtNum damping = biasprm[2] * 2 * mju_sqrt(gainprm[0] * mass);
+
+    // set biasprm[2] to negative damping
+    biasprm[2] = -damping;
+  }
+
   mj_freeStack(d);
 }
 

src/xml/xml_native_reader.cc

@@ -177,18 +177,16 @@ const char* MJCF[nMJCF][mjXATTRNUM] = {
             "dyntype", "gaintype", "biastype", "dynprm", "gainprm", "biasprm", "actearly"},
         {"motor", "?", "8", "ctrllimited", "forcelimited", "ctrlrange", "forcerange",
             "gear", "cranklength", "user", "group"},
-        {"position", "?", "12", "ctrllimited", "forcelimited", "ctrlrange", "inheritrange",
+        {"position", "?", "13", "ctrllimited", "forcelimited", "ctrlrange", "inheritrange",
-            "forcerange", "gear", "cranklength", "user", "group", "kp", "kv", "timeconst"},
+            "forcerange", "gear", "cranklength", "user", "group", "kp", "kv", "dampratio", "timeconst"},
         {"velocity", "?", "9", "ctrllimited", "forcelimited", "ctrlrange", "forcerange",
-            "gear", "cranklength", "user", "group",
+            "gear", "cranklength", "user", "group", "kv"},
-            "kv"},
+        {"intvelocity", "?", "13", "ctrllimited", "forcelimited",
-        {"intvelocity", "?", "12", "ctrllimited", "forcelimited",
             "ctrlrange", "forcerange", "actrange", "inheritrange",
             "gear", "cranklength", "user", "group",
-            "kp", "kv"},
+            "kp", "kv", "dampratio"},
         {"damper", "?", "8", "forcelimited", "ctrlrange", "forcerange",
-            "gear", "cranklength", "user", "group",
+            "gear", "cranklength", "user", "group", "kv"},
-            "kv"},
         {"cylinder", "?", "12", "ctrllimited", "forcelimited", "ctrlrange", "forcerange",
             "gear", "cranklength", "user", "group",
             "timeconst", "area", "diameter", "bias"},
@@ -389,22 +387,22 @@ const char* MJCF[nMJCF][mjXATTRNUM] = {
             "ctrllimited", "forcelimited", "ctrlrange", "forcerange",
             "lengthrange", "gear", "cranklength", "user",
             "joint", "jointinparent", "tendon", "slidersite", "cranksite", "site", "refsite"},
-        {"position", "*", "22", "name", "class", "group",
+        {"position", "*", "23", "name", "class", "group",
             "ctrllimited", "forcelimited", "ctrlrange", "inheritrange", "forcerange",
             "lengthrange", "gear", "cranklength", "user",
             "joint", "jointinparent", "tendon", "slidersite", "cranksite", "site", "refsite",
-            "kp", "kv", "timeconst"},
+            "kp", "kv", "dampratio", "timeconst"},
         {"velocity", "*", "19", "name", "class", "group",
             "ctrllimited", "forcelimited", "ctrlrange", "forcerange",
             "lengthrange", "gear", "cranklength", "user",
             "joint", "jointinparent", "tendon", "slidersite", "cranksite", "site", "refsite",
             "kv"},
-        {"intvelocity", "*", "22", "name", "class", "group",
+        {"intvelocity", "*", "23", "name", "class", "group",
             "ctrllimited", "forcelimited",
             "ctrlrange", "forcerange", "actrange", "inheritrange", "lengthrange",
             "gear", "cranklength", "user",
             "joint", "jointinparent", "tendon", "slidersite", "cranksite", "site", "refsite",
-            "kp", "kv"},
+            "kp", "kv", "dampratio"},
         {"damper", "*", "18", "name", "class", "group",
             "forcelimited", "ctrlrange", "forcerange",
             "lengthrange", "gear", "cranklength", "user",
@@ -2087,18 +2085,33 @@ void mjXReader::OneActuator(XMLElement* elem, mjsActuator* pact) {
     ReadAttr(elem, "kp", 1, pact->gainprm, text);
     pact->biasprm[1] = -pact->gainprm[0];
 
-    if (ReadAttr(elem, "kv", 1, pact->biasprm + 2, text)) {
+    // read kv
-      if (pact->biasprm[2] < 0)
+    double kv = -1;  // -1: undefined
-        throw mjXError(elem, "kv cannot be negative");
+    if (ReadAttr(elem, "kv", 1, &kv, text)) {
-      pact->biasprm[2] *= -1;
+      if (kv < 0) throw mjXError(elem, "kv cannot be negative");
     }
 
+    // read dampratio
+    double dampratio = -1;  // -1: undefined
+    if (ReadAttr(elem, "dampratio", 1, &dampratio, text)) {
+      if (dampratio < 0) throw mjXError(elem, "dampratio cannot be negative");
+    }
+
+    // set biasprm[2]; negative: regular damping, positive: dampratio
+    if (dampratio > 0 && kv > 0) {
+      throw mjXError(elem, "kv and dampratio cannot both be defined");
+    }
+    if (kv > 0) pact->biasprm[2] = -kv;
+    if (dampratio > 0) pact->biasprm[2] = dampratio;
+
+    // read timeconst, set dyntype
     if (ReadAttr(elem, "timeconst", 1, pact->dynprm, text)) {
       if (pact->dynprm[0] < 0)
         throw mjXError(elem, "timeconst cannot be negative");
       pact->dyntype = pact->dynprm[0] ? mjDYN_FILTEREXACT : mjDYN_NONE;
     }
 
+    // handle inheritrange
     ReadAttr(elem, "inheritrange", 1, &pact->inheritrange, text);
     if (pact->inheritrange > 0) {
       if (type == "position") {

test/engine/engine_forward_test.cc

@@ -18,6 +18,7 @@
 
 #include <cmath>
 #include <cstdlib>
+#include <limits>
 #include <vector>
 #include <string>
 
@@ -839,6 +840,83 @@ TEST_F(ActuatorTest, ActuatorGravcomp) {
   mj_deleteModel(model);
 }
 
+// Check that dampratio works as expected
+TEST_F(ActuatorTest, DampRatio) {
+  static constexpr char xml[] = R"(
+  <mujoco>
+    <option integrator="implicitfast"/>
+
+    <worldbody>
+      <body>
+        <joint name="slide1" axis="1 0 0" type="slide"/>
+        <geom size=".05"/>
+      </body>
+
+      <body pos="0 0 -.15">
+        <joint name="slide2" axis="1 0 0" type="slide"/>
+        <geom size=".05"/>
+      </body>
+    </worldbody>
+
+    <actuator>
+      <position name="slightly underdamped" joint="slide1" kp="10" dampratio="0.99"/>
+      <position name="slightly overdamped"  joint="slide2" kp="10" dampratio="1.01"/>
+    </actuator>
+  </mujoco>
+  )";
+  mjModel* model = LoadModelFromString(xml);
+  mjData* data = mj_makeData(model);
+
+  data->qpos[0] = data->qpos[1] = -0.1;
+
+  mjtNum under_damped = data->qpos[0];
+  mjtNum over_damped = data->qpos[1];
+  while (data->time < 10) {
+    mj_step(model, data);
+    under_damped = mju_max(under_damped, data->qpos[0]);
+    over_damped = mju_max(over_damped, data->qpos[1]);
+  }
+
+  // expect slightly underdamped to slightly overshoot
+  EXPECT_GT(under_damped, 0);
+  EXPECT_LT(under_damped, 1e-6);
+
+  // expect slightly overdamped to slightly undershoot
+  EXPECT_LT(over_damped, 0);
+  EXPECT_GT(over_damped, -1e-6);
+
+  mj_deleteData(data);
+  mj_deleteModel(model);
+}
+
+
+// Check dampratio for actuators with nontrivial transmission
+TEST_F(ActuatorTest, DampRatioTendon) {
+  const std::string xml_path =
+      GetTestDataFilePath("engine/testdata/actuation/tendon_dampratio.xml");
+  char error[1000];
+  mjModel* model = mj_loadXML(xml_path.c_str(), nullptr, error, sizeof(error));
+  ASSERT_THAT(model, NotNull()) << error;
+  mjData* data = mj_makeData(model);
+
+  data->ctrl[0] = 1;
+  data->ctrl[1] = 4;
+
+  while (data->time < 1) {
+    mj_step(model, data);
+  }
+
+  // expect first and second fingers to move together
+  double tol  = 1e-10;
+  EXPECT_THAT(AsVector(data->qpos, 4),
+              Pointwise(DoubleNear(tol), AsVector(data->qpos + 4, 4)));
+  EXPECT_THAT(AsVector(data->qvel, 4),
+              Pointwise(DoubleNear(tol), AsVector(data->qvel + 4, 4)));
+
+  mj_deleteData(data);
+  mj_deleteModel(model);
+}
+
 // ----------------------- filterexact actuators -------------------------------
 
 using FilterExactTest = MujocoTest;

test/engine/testdata/actuation/dampratio.xml

@@ -0,0 +1,44 @@
+<mujoco>
+  <visual>
+    <global elevation="0"/>
+  </visual>
+
+  <default>
+    <position ctrlrange="-.2 .2"/>
+    <joint axis="1 0 0" type="slide"/>
+  </default>
+
+  <option integrator="implicitfast"/>
+
+  <worldbody>
+    <light pos="0 0 2"/>
+
+    <body pos="0 0 .15">
+      <joint name="0"/>
+      <geom size=".05"/>
+    </body>
+
+    <body>
+      <joint name="1"/>
+      <geom mass="1" size=".05"/>
+    </body>
+
+    <body pos="0 0 -.15">
+      <joint name="2"/>
+      <geom mass="1" size=".05"/>
+    </body>
+
+    <body pos="0 0 -.3">
+      <joint name="3"/>
+      <geom mass="1" size=".05"/>
+    </body>
+  </worldbody>
+
+  <actuator>
+    <position name="r=1/3"          joint="0" dampratio=".333" kp="10"/>
+    <position name="r=1"            joint="1" dampratio="1"    kp="10"/>
+    <position name="r=3"            joint="2" dampratio="3"    kp="10"/>
+    <!-- transmission increased by 10: reduce kp by 10^2 -->
+    <position name="r=1, g=10"      joint="3" dampratio="1"    kp=".1" gear="10" ctrlrange="-2 2"/>
+  </actuator>
+</mujoco>

test/engine/testdata/actuation/refsite.xml

@@ -3,7 +3,6 @@
   <!--
   Adding a high fluid viscosity and using implicit integration for extra stabillity.
   Extra stabillity is required because the abstract "arm" model is not very realistic.
-  (e.g. 4 consecutive ball joints is not a realistic kinematic design)
   -->
   <option viscosity="10" integrator="implicit">
     <flag contact="disable"/>
@@ -13,7 +12,7 @@
 
   <default>
     <default class="translation">
-      <position kp="100" kv="10" ctrlrange="-.25 .25"/>
+      <position kp="100" dampratio="1" ctrlrange="-.25 .25"/>
     </default>
     <default class="rotation">
       <!--
@@ -22,7 +21,7 @@
       Increase this range to pi or bigger in order to see the instabillity.
       See here for more details https://mujoco.readthedocs.io/en/latest/XMLreference.html#actuator
       -->
-      <position kp=".5" kv=".05" ctrlrange="-1.571 1.571"/>
+      <position kp=".5" dampratio="1" ctrlrange="-1.571 1.571"/>
     </default>
     <!--
     Clamping the total actuator torque at the joints means that the motion produced by the Cartesian

test/engine/testdata/actuation/tendon_dampratio.xml

@@ -0,0 +1,84 @@
+<mujoco>
+  <visual>
+    <global elevation="-20"/>
+  </visual>
+
+  <default>
+    <joint axis="0 1 0" armature="0.01"/>
+    <geom type="capsule" size="0.02"/>
+  </default>
+
+  <option integrator="implicitfast">
+    <flag gravity="disable"/>
+  </option>
+
+  <worldbody>
+    <light pos="0 0 2"/>
+
+    <body>
+      <joint name="a0"/>
+      <geom fromto="0 0 0 .1 0 0"/>
+      <body pos=".1 0 0">
+        <joint name="b0"/>
+        <geom fromto="0 0 0 .1 0 0"/>
+        <body pos=".1 0 0">
+          <joint name="c0"/>
+          <geom fromto="0 0 0 .1 0 0"/>
+          <body pos=".1 0 0">
+            <joint name="d0"/>
+            <geom fromto="0 0 0 .1 0 0"/>
+          </body>
+        </body>
+      </body>
+    </body>
+
+    <body pos="0 -0.05 0">
+      <joint name="a1"/>
+      <geom fromto="0 0 0 .1 0 0"/>
+      <body pos=".1 0 0">
+        <joint name="b1"/>
+        <geom fromto="0 0 0 .1 0 0"/>
+        <body pos=".1 0 0">
+          <joint name="c1"/>
+          <geom fromto="0 0 0 .1 0 0"/>
+          <body pos=".1 0 0">
+            <joint name="d1"/>
+            <geom fromto="0 0 0 .1 0 0"/>
+          </body>
+        </body>
+      </body>
+    </body>
+  </worldbody>
+
+  <equality>
+    <joint joint1="a0" joint2="b0" polycoef="0 .5"/>
+    <joint joint1="b0" joint2="c0" polycoef="0 .5"/>
+    <joint joint1="c0" joint2="d0" polycoef="0 .5"/>
+
+    <joint joint1="a1" joint2="b1" polycoef="0 .5"/>
+    <joint joint1="b1" joint2="c1" polycoef="0 .5"/>
+    <joint joint1="c1" joint2="d1" polycoef="0 .5"/>
+  </equality>
+
+  <tendon>
+    <fixed name="finger0">
+      <joint joint="a0" coef="1"/>
+      <joint joint="b0" coef=".5"/>
+      <joint joint="c0" coef=".25"/>
+      <joint joint="d0" coef=".125"/>
+    </fixed>
+
+    <fixed name="finger1">
+      <joint joint="a1" coef="4"/>
+      <joint joint="b1" coef="2"/>
+      <joint joint="c1" coef="1"/>
+      <joint joint="d1" coef="0.5"/>
+    </fixed>
+  </tendon>
+
+  <actuator>
+    <position name="finger0" tendon="finger0" dampratio="1" kp="128" ctrlrange="-1 1"/>
+    <!-- transmission increased by 4: reduce kp by 16 -->
+    <position name="finger1" tendon="finger1" dampratio="1" kp="8"   ctrlrange="-4 4"/>
+  </actuator>
+</mujoco>