](classrviz__visual__tools_1_1RvizVisualTools.html#afec243c2b7224c66726fe364272c9b50)
(const [ geometry_msgs::Pose
](../../../api/tf/html/c++/namespacetf.html#af31f272a1879787c24c3decb07cdb361)
&pose, [ colors
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3) color=
[ BLUE
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3a6cd27d693df45662c3ecebdc4dc8b671)
, [ scales
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835) scale=
[ MEDIUM
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835a322895c45564210483089d04f6a07473)
, double [ length
](../../../api/tf2/html/namespacetf2.html#ab20034dc53b40043a702876ff38c78a1)
=0.0, std::size_t id=0)
bool  |  [ publishArrow
](classrviz__visual__tools_1_1RvizVisualTools.html#af2ae779255ecc5735eb8e3efed561a78)
(const geometry_msgs::PoseStamped &pose, [ colors
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3) color=
[ BLUE
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3a6cd27d693df45662c3ecebdc4dc8b671)
, [ scales
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835) scale=
[ MEDIUM
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835a322895c45564210483089d04f6a07473)
, double [ length
](../../../api/tf2/html/namespacetf2.html#ab20034dc53b40043a702876ff38c78a1)
=0.0, std::size_t id=0)
bool  |  [ publishArrow
](classrviz__visual__tools_1_1RvizVisualTools.html#a77bfd8028a00243631f099daf7be5652)
(const [ geometry_msgs::Point
](../../../api/tf/html/c++/namespacetf.html#a55ddfd1e43e4fc81a11d51e0e75e9e2c)
& [ start
](../../../api/roscpp/html/namespaceros.html#add7faea1768dbd48d3fe2970204c3422)
, const [ geometry_msgs::Point
](../../../api/tf/html/c++/namespacetf.html#a55ddfd1e43e4fc81a11d51e0e75e9e2c)
&end, [ colors
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3) color=
[ BLUE
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3a6cd27d693df45662c3ecebdc4dc8b671)
, [ scales


` HalfEdgeTriangleMesh ` _= Type.HalfEdgeTriangleMesh_ Â¶


Definition at line [ 1608 ](rviz__visual__tools_8cpp_source.html#l01608) of
file [ rviz_visual_tools.cpp ](rviz__visual__tools_8cpp_source.html) .


|  visualization_msgs::Marker
rviz_visual_tools::RvizVisualTools::block_marker_
---
protected

Definition at line [ 1091 ](rviz__visual__tools_8h_source.html#l01091) of file
[ rviz_visual_tools.h ](rviz__visual__tools_8h_source.html) .


bool RvizVisualTools::publishSphere(const geometry_msgs::Pose& pose, colors color, double scale, const std::string& ns,
                                    std::size_t id)
{
  geometry_msgs::Vector3 scale_msg;
  scale_msg.x = scale * global_scale_;
  scale_msg.y = scale * global_scale_;
  scale_msg.z = scale * global_scale_;
  return publishSphere(pose, color, scale_msg, ns, id);
}


##  ◆  publishArrow()  [4/4]


Definition at line [ 2492 ](rviz__visual__tools_8cpp_source.html#l02492) of
file [ rviz_visual_tools.cpp ](rviz__visual__tools_8cpp_source.html) .


Parameters


     pose  |  \- the location and orientation of the cone
---|---
color  |  \- color of the cone
scale  |  \- size of the cone

Returns


` get_box_points ` (  _self_ )  Â¶


  return true;
}


##  ◆  publishAxis()  [2/4]


Returns


](../../../api/tf/html/c++/namespacetf.html#af31f272a1879787c24c3decb07cdb361)
&pose, [ colors
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3) color=
[ BLUE
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3a6cd27d693df45662c3ecebdc4dc8b671)
, [ scales
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835) scale=
[ MEDIUM
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835a322895c45564210483089d04f6a07473)
, double [ length
](../../../api/tf2/html/namespacetf2.html#ab20034dc53b40043a702876ff38c78a1)
=0.0)
bool  |  [ publishZArrow
](classrviz__visual__tools_1_1RvizVisualTools.html#a30f58b666ed01a75b75ce5b812105951)
(const geometry_msgs::PoseStamped &pose, [ colors
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3) color=
[ BLUE
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3a6cd27d693df45662c3ecebdc4dc8b671)
, [ scales
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835) scale=
[ MEDIUM
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835a322895c45564210483089d04f6a07473)
, double [ length
](../../../api/tf2/html/namespacetf2.html#ab20034dc53b40043a702876ff38c78a1)
=0.0)
bool  |  [ publishZArrow
](classrviz__visual__tools_1_1RvizVisualTools.html#a943f01cddf7d10f692520380958c835c)
(const geometry_msgs::PoseStamped &pose, [ colors
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3) color=
[ BLUE
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3a6cd27d693df45662c3ecebdc4dc8b671)
, [ scales
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835) scale=
[ MEDIUM
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835a322895c45564210483089d04f6a07473)
, double [ length
](../../../api/tf2/html/namespacetf2.html#ab20034dc53b40043a702876ff38c78a1)
=0.0, std::size_t id=0)
void  |  [ resetMarkerCounts
](classrviz__visual__tools_1_1RvizVisualTools.html#a110da4c039428fb2be5d5c109f3cc227)


|  visualization_msgs::Marker
rviz_visual_tools::RvizVisualTools::cylinder_marker_
---
protected

Definition at line [ 1092 ](rviz__visual__tools_8h_source.html#l01092) of file
[ rviz_visual_tools.h ](rviz__visual__tools_8h_source.html) .


true on success


  if (id == 0)
  {
    mesh_marker_.id++;
  }
  else
  {
    mesh_marker_.id = id;
  }


bool RvizVisualTools::publishWireframeRectangle(const Eigen::Isometry3d& pose, double height, double width,
                                                colors color, scales scale, std::size_t id)
{
  if (id == 0)
  {  // Provide a new id every call to this function
    line_list_marker_.id++;
  }
  else
  {  // allow marker to be overwritten
    line_list_marker_.id = id;
  }


##  ◆  publishCuboid()  [2/4]


|  [ geometry_msgs::Pose
](../../../api/tf/html/c++/namespacetf.html#af31f272a1879787c24c3decb07cdb361)
rviz_visual_tools::RvizVisualTools::getIdentityPose  |  (  |  |  )  |
---|---|---|---|---
static

Definition at line [ 2688 ](rviz__visual__tools_8cpp_source.html#l02688) of
file [ rviz_visual_tools.cpp ](rviz__visual__tools_8cpp_source.html) .


bool rviz_visual_tools::RvizVisualTools::publishCylinder  |  (  |  const
Eigen::Isometry3d & |  _pose_ ,
---|---|---|---
|  |  [ colors
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3) |
_color_ = ` [ BLUE
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3a6cd27d693df45662c3ecebdc4dc8b671)
` ,
|  |  double  |  _height_ = ` 0.1 ` ,
|  |  double  |  _radius_ = ` 0.01 ` ,
|  |  const std::string & |  _ns_ = ` "Cylinder" `
|  )  |  |

Display a marker of a cylinder.


bool result = publishMarkers(markers_);


bool rviz_visual_tools::RvizVisualTools::publishPolygon  |  (  |  const
geometry_msgs::Polygon & |  _polygon_ ,
---|---|---|---
|  |  [ colors
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3) |
_color_ = ` [ RED
](namespacerviz__visual__tools.html#abd3566ac96adb96a35044b029930e6e3a269d2451e2a4b2d6744d2d958a8a8467)
` ,
|  |  [ scales
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835) |
_scale_ = ` [ MEDIUM
](namespacerviz__visual__tools.html#a5b544decae321663ea907bb26338d835a322895c45564210483089d04f6a07473)
` ,
|  |  const std::string & |  _ns_ = ` "Polygon" `
|  )  |  |

Display a marker of a polygon.


  // Rviz marker publisher
  ros::NodeHandle nh(nh_.getNamespace());
  nh.setCallbackQueue(&vis_marker_queue_);
  pub_rviz_markers_ = nh.advertise<visualization_msgs::MarkerArray>(marker_topic_, 10, latched);
  ROS_DEBUG_STREAM_NAMED(LOGNAME, "Publishing Rviz markers on topic " << pub_rviz_markers_.getTopic());


Parameters


Definition at line [ 2463 ](rviz__visual__tools_8cpp_source.html#l02463) of
file [ rviz_visual_tools.cpp ](rviz__visual__tools_8cpp_source.html) .


` get_center ` (  _self_ )  Â¶


Definition at line [ 843 ](rviz__visual__tools_8cpp_source.html#l00843) of
file [ rviz_visual_tools.cpp ](rviz__visual__tools_8cpp_source.html) .


##  ◆  arrow_marker_


void RvizVisualTools::setRandomSeed(unsigned int seed)
{
  mt_random_engine_.seed(seed);
}


    </link>

    <joint name="right_end_joint" type="fixed">
        <parent link="right_link_5"/>
        <child link="end_link"/>
        <origin xyz="-0.074 0.074 0" rpy="0.0 0.0 0.0"/>
    </joint>

    <link name="left_end_link">
        <inertial>
        <origin xyz="0 0 0" rpy="0 0 0"/>
        <mass value="0.001"/>
        <inertia
            ixx="0.0001" ixy="0.0" ixz="0.0"
            iyy="0.0001" iyz="0.0"
            izz="0.0001"/>
        </inertial>
    </link>

    <joint name="left_end_virtual_joint" type="fixed">
        <parent link="end_link"/>
        <child link="left_end_link"/>
        <origin xyz="0 0 0" rpy="0 0 0"/>
    </joint>

    <gazebo>
        <joint name="left_end_joint" type="fixed">
            <parent> left_link_5 </parent>
            <child> left_end_link </child>
            <pose> 0 0 0 0 0 0 </pose>
        </joint>
    </gazebo>

    <gazebo>
        <plugin name="gazebo_ros_control" filename="libgazebo_ros_control.so">
        <robotNamespace></robotNamespace>
        <robotSimType>gazebo_ros_control/DefaultRobotHWSim</robotSimType>
        <legacyModeNS>true</legacyModeNS>
        <gravity>0</gravity>


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G  θ  ˙  =  150  I  7  ×  7  ,  G  ee  =  100  I  7  ×  7


The contact force magnitude was higher in the case of a prismatic joint case
(−0.89 N rather than −0.63 N in baseline case), but also the contact was
shorter: 1.8 s for the prismatic impedance-controlled joint architecture vs.
2.8 s for the baseline, as depicted in  Figure 5  . The disturbances of the
base of the free-floating robot in both cases are compared in  Table 3  . The
disturbance was higher under the Cartesian control with a prismatic impedance-
controlled joint, in the sense that its center of mass came out of the
maneuver moving and rotating faster.


A  =  (  m  s  \+  ∑  i  =  1  n  (  m  i  )  )  I


[ Gazebo ](http://gazebosim.org) | [ Ignition ](https://ignitionrobotics.org)
| [ Community ](https://community.gazebosim.org/)


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[ edit ](/questions/186681/edit/) [ retag ](/s/questions/186681/retag/) flag
offensive  [ close ](/questions/186681/close/) merge  delete


[ **asked 2023-07-10 04:51:47 -0500  ** ](/questions/29263/revisions/)


##  Journals


Q  =  [  F  s  H  s  T  ]


[ Ackermann robot URDF or SDF ](/question/24439/ackermann-robot-urdf-or-sdf/)


![](/bundles/mdpisciprofileslink/img/unknown-user.png) Tomasz Rybus


[ __ Download PDF ](/2226-4310/9/8/406/pdf?version=1658994379)


####  2.7. Free-Floating Platform-Based Space Robot under Cartesian Impedance
Control


* Torque magnitudes in robot joints;


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Tutorial  Viewpoint


Although I have defined the joint in the xacro file, the robot does not move
in the z-axis. What is the reason?


[ Rviz viewpoint transform ](/question/283462/rviz-viewpoint-transform/)


> <robot xmlns:xacro="http://www.ros.org/wiki/xacro" name="anil_robot">

    <xacro:include filename="$(find five_bar_robot)/urdf/right.xacro"/>
    <xacro:include filename="$(find five_bar_robot)/urdf/left.xacro"/>

    <link name="world"/>

    <joint name="world2dummy" type="fixed">
        <parent link="world"/>
        <child link="dummy_link"/>
    </joint>

    <link name="dummy_link">
        <mu1>0.2</mu1>
        <mu2>0.1</mu2>
        <inertial>
            <origin xyz="0.0 0.0 0.0" rpy="0.0 0.0 0.0" />
            <mass value="1" />
            <inertia ixx="20.0" ixy="0.0" ixz="0.0" iyy="20.0" iyz="0.0" izz="20.0" />
        </inertial>
    </link>

    <joint name="dummy2dummy" type="prismatic">
        <parent link="dummy_link"/>
        <child link="dummy_link_2"/>
        <axis xyz="0 0 1"/>
        <limit lower="0" upper="0.1" velocity="0.8" effort="1000"/>
        <origin xyz="0 0 0" rpy="0 0 0"/>
    </joint>

    <link name="dummy_link_2">
        <mu1>0.2</mu1>
        <mu2>0.1</mu2>
        <inertial>
            <origin xyz="0.0 0.0 0.0" rpy="0.0 0.0 0.0" />
            <mass value="1" />
            <inertia ixx="20.0" ixy="0.0" ixz="0.0" iyy="20.0" iyz="0.0" izz="20.0" />
        </inertial>
    </link>


    void Komodo2RobotHWSim::registerJointLimits(const std::string& joint_name,
                                                   const hardware_interface::PosVelJointHandle& joint_handle,
                                                   const ControlMethod ctrl_method,
                                                   const ros::NodeHandle& joint_limit_nh,
                                                   const urdf::Model *const urdf_model,
                                                   int *const joint_type, double *const lower_limit,
                                                   double *const upper_limit, double *const effort_limit)
    {
        *joint_type = urdf::Joint::UNKNOWN;
        *lower_limit = -std::numeric_limits<double>::max();
        *upper_limit = std::numeric_limits<double>::max();
        *effort_limit = std::numeric_limits<double>::max();


[ **asked 2014-07-15 14:39:57 -0500  ** ](/questions/186681/revisions/)


**Figure 4.** Comparison of time histories of the torques in robot joints.
**Left** : model-based full robot impedance control, **right** : impedance
control with single prismatic joint at the end effector.


In this section we derive the dynamics model of the free-floating robot
manipulator. The logical steps we take are the following: First, in  Section
2.1  we start with a brief introduction to the nonholonomic nature of the
robot system. Then, in  Section 2.2  we define the reference frames of the
satellite with a robot manipulator.  Section 2.3  focuses on the description
of the kinematics of such robot. In the  Section 2.4  , the dynamics of the
free-floating robot are derived by taking the following steps: in  Section 2.4
, the dynamical model of a full, free-flying satellite-based robot is
presented. The Free-flying robot is one which is mounted on a base equipped
with actuators, e.g., thrusters [  25  ]. In  Section 2.4  ., we discuss the
constraints of the robot having zero initial momentum and angular momentum
which are then needed to reformulate the dynamical model of the satellite as a
free-floating robot in  Section 2.4.2  . This way, we obtain the free-floating
robot dynamics equations where the state vector contains only the joint
variables and is independent of the base coordinates. The gravity gradient
effects, as well as orbital mechanics, solar wind pressure, residual
atmospheric drag, etc., are ignored in the following analysis.


Zoom  |  Orient  |  As Lines  |  As Sticks  |  As Cartoon  |  As Surface  |
Previous Scene  |  Next Scene


Note, however, that in this case, the impedance control is achieved with
rotational joints only, and the Jacobian does not include any terms other than
rotational. With use of the standard manipulator Jacobian, the transformation
between the Cartesian and joint space is as follows:


Asked: ** 2014-07-15 14:39:57 -0500  **


Θ  s  =  [  ψ  ϕ  θ  ]  T  ,


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* [ interface ](/questions/scope:all/sort:activity-desc/tags:interface/page:1/)


[ I can't move my model in the world ](/question/25569/i-cant-move-my-model-
in-the-world/)


The mass matrix **M** [  10  ,  24  ] for the general case is:


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E  =  1000  [  5  5  1  5  5  1  ]  T  ,  D  =  500  I  7  ×  7  ,


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Author to whom correspondence should be addressed.


        </plugin>
    </gazebo>

    <transmission name="prismatic">
        <type>transmission_interface/SimpleTransmission</type>
        <joint name="dummy2dummy">
            <hardwareInterface>hardware_interface/EffortJointInterface</hardwareInterface>
        </joint>
        <actuator name="prismatic_motor">
            <mechanicalReduction>1.0</mechanicalReduction>
            <hardwareInterface>hardware_interface/EffortJointInterface ...


##  Share and Cite


The site is read-only. Please transition to use Robotics Stack Exchange


Figure 1

Schematic depiction of reference frames of a robotic arm on a satellite base \(left\). Detail showing the end effector with the prismatic joint displacement \(right\).

                                                ") [ ](https://pub.mdpi-res.com/aerospace/aerospace-09-00406/article_deploy/html/images/aerospace-09-00406-g002.png?1658994451 "
                        Figure 2

Various gripping strategies for interception of a space object. Dominant direction in each case is shown by the arrow.

                                                ") [ ](https://pub.mdpi-res.com/aerospace/aerospace-09-00406/article_deploy/html/images/aerospace-09-00406-g003.png?1658994452 "
                        Figure 3

The simulated satellite robot and target body in the initial position \(left\), and during contact \(right\). The line of dominant direction of the final approach is shown by the dashed line.

                                                ") [ ](https://pub.mdpi-res.com/aerospace/aerospace-09-00406/article_deploy/html/images/aerospace-09-00406-g004.png?1658994449 "
                        Figure 4

Comparison of time histories of the torques in robot joints. Left: model-based full robot impedance control, right: impedance control with single prismatic joint at the end effector.

                                                ") [ ](https://pub.mdpi-res.com/aerospace/aerospace-09-00406/article_deploy/html/images/aerospace-09-00406-g005.png?1658994447 "
                        Figure 5

Contact force between the spheres simulating the end effector and the target. Left: model-based full robot impedance control, right: impedance control with a single prismatic joint at the end effector.

                                                ")


        return true;
    }


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In the inertial reference system, the satellite’s center of gravity position
r  s  is:


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##  Question Tools


r  s  =  [  (  r  s  )  x  (  r  s  )  y  (  r  s  )  z  ]  T  ,


As depicted in  Figure 1  . The  k  prism  is the unit vector of the
translation axis of the prismatic joint. By taking the time derivative of the
expression for position vector, we arrive at the end effector velocity in the
Cartesian inertial frame:


Palma, P.; Seweryn, K.; Rybus, T. Impedance Control Using Selected Compliant
Prismatic Joint in a Free-Floating Space Manipulator. _Aerospace_ **2022** ,
_9_ , 406. https://doi.org/10.3390/aerospace9080406


Asked: ** 2023-07-10 04:51:47 -0500  **


####  Variable Impedance


[ Prismatic Joint not working properly with ROS2 & Gazebo 11
](/question/25618/prismatic-joint-not-working-properly-with-ros2-gazebo-11/)


#include "komodo2_sim_interface/komodo2_sim_interface.h"
namespace
{


        for(unsigned int j=0; j < n_dof_; j++)
        {
            switch (joint_control_methods_[j])
            {


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        ej_sat_interface_.enforceLimits(period);
        ej_limits_interface_.enforceLimits(period);
        pj_sat_interface_.enforceLimits(period);
        pj_limits_interface_.enforceLimits(period);
        vj_sat_interface_.enforceLimits(period);
        vj_limits_interface_.enforceLimits(period);


[ ![Arn-O gravatar
image](//www.gravatar.com/avatar/fe66bd7b67c0ce29399ba5b715fbf9cf?s=16&d=identicon&r=PG)
](/users/5387/arn-o/) [ Arn-O ](/users/5387/arn-o/) (  2014-08-02 01:50:42
-0500  )  edit


[ __ ](/cdn-cgi/l/email-
protection#4d622e2329602e2a2462216228202c2421603d3f2239282e392422236e7d7d7c757b2f7d287c2e7d287c747c7f7d787f2f7d757d747d7d79787c2e7d2c7c2e79787c2f7d7a)
and


thank you...it was indeed vague


T  act  =  I  (  θ  )  J  M  −  1  (  θ  )  M  d  −  1  E  [  X  0  −  X  ]
\+  S  (  θ  )  (  position  terms  )  \+  I  (  θ  )  J  M  −  1  (  θ  )  M
d  −  1  D  [  X  ˙  0  −  V  ]  \+  X  ˙  c  (  ω  )  (  velocity  terms  )
\+  I  (  θ  )  J  M  −  1  (  θ  )  M  d  −  1  F  int  −  J  M  T  (  θ  )
F  int  (  force  interaction  terms  )  −  I  (  θ  )  J  M  −  1  (  θ  )  G
(  θ  ,  ω  )  \+  C  c  (  θ  ,  ω  )  (  inertial  coupling  terms  )


[ PR2 stuck picking object ](/question/244207/pr2-stuck-picking-object/)


d  V  dt  =  M  d  −  1  E  (  X  0  −  X  )  \+  M  d  −  1  D  (  X  ˙  0  −
X  ˙  )  \+  M  d  −  1  F  int


[ Accept ](/accept_cookies)


Seen: **322 times**


Space Research Center, Polish Academy of Science (CBK PAN), 00-716 Warsaw,
Poland


        if (!has_limits)
            return;


[ Volume 9 ](/2226-4310/9)


##  Comments


Special Issue


            if (limits.has_position_limits)
            {
                *joint_type = urdf::Joint::REVOLUTE;
            }
            else
            {
                if (limits.angle_wraparound)
                    *joint_type = urdf::Joint::CONTINUOUS;
                else
                    *joint_type = urdf::Joint::PRISMATIC;
            }
        }


        // Initialize values
        for(unsigned int j=0; j < n_dof_; j++)
        {
            // Check that this transmission has one joint
            if(transmissions[j].joints_.size() == 0)
            {
                ROS_WARN_STREAM_NAMED("default_robot_hw_sim","Transmission " << transmissions[j].name_
                                                                             << " has no associated joints.");
                continue;
            }
            else if(transmissions[j].joints_.size() > 1)
            {
                ROS_WARN_STREAM_NAMED("default_robot_hw_sim","Transmission " << transmissions[j].name_
                                                                             << " has more than one joint. Currently the default robot hardware simulation "
                                                                             << " interface only supports one.");
                continue;
            }


Copy


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[ How to check if there is problem in odometry? ](/question/306389/how-to-
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edit


Piotr Palma


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In the model-based impedance control scenario, the following impedance
parameters provided in SI units were used:


(more)


[ blocking screw/prismatic joint ](/question/29074/blocking-screwprismatic-
joint/)


##  Funding


Architecture proposal, distinction of the concept of dominant direction,
methodology, impedance control development and simulation, writing—P.P.;
supervision—K.S.; Cartesian control, supervision—T.R. All authors have read
and agreed to the published version of the manuscript.


All fields  Title  Abstract  Keywords  Authors  Affiliations  Doi  Full Text
References


where  F  s  and  H  s  are the forces and torques acting on the satellite
base center of mass, e.g., from the thrusters. Vector  T  is the torques in
the joints of the robotic arm. The generalized equations of motion for the
satellite equipped with a robotic manipulator, using the Lagrangian equations
of the second kind, take the following form [  11  ]:


Two simulation cases were executed: the model-based impedance controller (40)
and the one where the single axis impedance control of the end effector was
provided by the prismatic impedance-controlled joint between the joint no. 7
and the end effector. The seven rotary joints of the robot arm were under the
control of the Cartesian controller tracking the virtual trajectory.


[ LIDAR model publishes upside-down pointcloud ](/question/22159/lidar-model-
publishes-upside-down-pointcloud/)


For an  i  -th joint,  k  i  represents the unit vector of the rotation axis
and  θ  ˙  i  is its angular rate, while  r  i  represents the position of the
ith kinematic pair. The angular velocity of the end effector is simply:


####  2.9. Basic Cartesian Controller for Free-Floating Robot


T  act  =  M  m  (  θ  )  J  D  −  1  (  θ  )  M  d  −  1  E  [  X  0  −  X  ]
\+  M  m  (  θ  )  J  D  −  1  (  θ  )  M  d  −  1  D  [  X  ˙  0  −  X  ˙  ]
\+  M  m  (  θ  )  J  D  −  1  (  θ  )  M  d  −  1  F  int  −  J  D  T  (  θ
)  F  int  −  M  m  (  θ  )  J  D  −  1  (  θ  )  G  (  q  p  ,  q  v  )  \+
C  (  q  p  ,  q  v  )  q  v


M  d  (  λ  ¨  −  λ  ¨  0  )  \+  D  (  λ  ˙  −  λ  ˙  0  )  \+  E  (  λ  −  λ
0  )  =  F  (  t  )


Search Text


[  v  ee  ω  ee  ]  =  (  J  M  −  J  S  H  2  −  1  H  3  )  θ  ˙


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_[ Aerospace ](/journal/aerospace) _ , EISSN 2226-4310, Published by MDPI


Its orientation  Θ  s  is expressed using Euler angles as follows:


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](/users/18746/lockandfire/)


###  Support


The simulations of both schemes of the free-floating robotic manipulator
impedance control showed that it is possible to achieve the impedance control
behavior along the axis of interest using a simplified architecture with one
compliant prismatic joint at the expense of slightly higher loads and
disturbances. The proposed method was demonstrated in a simulation of a 7-DoF
robot, and can be used with any manipulator with a suitable number of degrees
of freedom to allow the tracking by the end effector the trajectory along the
line of the dominant direction of contact. The room for improving the
performance has not been explored in the frame of this study, and would likely
lead to good performance. The simplification in construction and lower
computational load of the proposed architecture are relevant aspects in the
design of equipment intended to be used in the space environment, making this
kind of architecture potentially worth further detailed development.


##  Related questions


q  p  =  θ  q  v  =  θ  ˙


// Register the limits of the joint specified by joint_name and joint_handle. The limits are
// retrieved from joint_limit_nh. If urdf_model is not NULL, limits are retrieved from it also.
// Return the joint's type, lower position limit, upper position limit, and effort limit.
    void Komodo2RobotHWSim::registerJointLimits(const std::string& joint_name,
                                                   const hardware_interface::JointHandle& joint_handle,
                                                   const ControlMethod ctrl_method,
                                                   const ros::NodeHandle& joint_limit_nh,
                                                   const urdf::Model *const urdf_model,
                                                   int *const joint_type, double *const lower_limit,
                                                   double *const upper_limit, double *const effort_limit)
    {
        *joint_type = urdf::Joint::UNKNOWN;
        *lower_limit = -std::numeric_limits<double>::max();
        *upper_limit = std::numeric_limits<double>::max();
        *effort_limit = std::numeric_limits<double>::max();


* [ ROS ](/questions/scope:all/sort:activity-desc/tags:ROS/page:1/)


[ **answered 2014-07-20 13:58:04 -0500  ** ](/answers/187192/revisions/)


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N  =  ∑  i  =  1  n  (  J  R  i  T  I  i  J  R  i  \+  m  i  J  T  i  T  J  T
i  )


        if (urdf_model != NULL)
        {
            const boost::shared_ptr<const urdf::Joint> urdf_joint = urdf_model->getJoint(joint_name);
            if (urdf_joint != NULL)
            {
                *joint_type = urdf_joint->type;
                // Get limits from the URDF file.
                if (joint_limits_interface::getJointLimits(urdf_joint, limits))
                    has_limits = true;
                if (joint_limits_interface::getSoftJointLimits(urdf_joint, soft_limits))
                    has_soft_limits = true;
            }
        }
        // Get limits from the parameter server.
        if (joint_limits_interface::getJointLimits(joint_name, joint_limit_nh, limits))
            has_limits = true;


[ ![aerospace-logo](https://pub.mdpi-res.com/img/journals/aerospace-
logo.png?8600e93ff98dbf14) ](/journal/aerospace)


    PLUGINLIB_EXPORT_CLASS(gazebo_ros_control::Komodo2RobotHWSim, gazebo_ros_control::RobotHWSim)
}


Volume


[ Numerical Investigation on Cooling Performance of Rectangular Channels
Filled with X-Shaped Truss Array Structures ](/2226-4310/9/8/405)


The manipulator’s angular momentum is described as follows:


* The final linear and angular velocities of the satellite body after the contact event.


Search  for Articles  :


The robot manipulator had the mass and geometrical parameters as listed in
Table 1  . The manipulator mounting point was shifted from the center of mass
of the satellite by [0.2 0.1 0.4] m.


Section


[ ![gvdhoorn gravatar image](/upfiles/avatars/gvdhoorn/resized/16/4550046.png)
](/users/5184/gvdhoorn/) [ gvdhoorn ](/users/5184/gvdhoorn/) (  2014-07-15
14:58:07 -0500  )  edit


[ Rosserial arduino connectivity problem " socket.error: [Errno 111]
Connection refused" ](/question/248280/rosserial-arduino-connectivity-problem-
socketerror-errno-111-connection-refused/)


###  Further Information


with  r  s  _  q  =  r  s  −  r  q  ,  r  i  _  s  =  r  m  ,  i  −  r  s  ,
where the  r  m  ,  i  is the position of the center of mass of the  i  -th
link, while the  J  T  i  ,  J  R  i  are the translational and rotational
components of a standard Jacobian of the manipulator [  28  ,  29  ]:


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**Table 1.** Geometric and mass properties of the simulated robot.


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            }
//ROS_ERROR("LIMIT on: %s     UP: %f    DOWN: %f",joint_names_[j].c_str(),joint_lower_limits_[j],joint_upper_limits_[j]);


##  Acknowledgments


[ ![rnunziata gravatar image](/m/default/media/images/nophoto.png?v=27)
](/users/5119/rnunziata/) [ rnunziata ](/users/5119/rnunziata/) (  2014-07-15
15:31:30 -0500  )  edit


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[ ![rnunziata gravatar image](/m/default/media/images/nophoto.png?v=27)
](/users/5119/rnunziata/) [ rnunziata ](/users/5119/rnunziata/) (  2014-07-20
20:22:53 -0500  )  edit


(This article belongs to the Special Issue [ Space Debris Removal: Challenges
and Opportunities ](
    /journal/aerospace/special_issues/Space_Debris_Removal_Challenges_Opportunities
) )


* [ prismatic_joints ](/questions/scope:all/sort:activity-desc/tags:prismatic_joints/page:1/)


##  Notice


###  Feedback


In case of a free-floating manipulator system, being the scope of this
research, the above formulations of the kinematics and dynamics equations
change.


Aa  Aa  Aa


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**Figure 3.** The simulated satellite robot and target body in the initial
position ( **left** ), and during contact ( **right** ). The line of dominant
direction of the final approach is shown by the dashed line.


In the case of the free-floating satellite platform equipped with a robotic
manipulator, the Jacobian in (34)–(39) is replaced with its dynamic
counterpart given by (25). The mass matrix (29) encodes the configuration-
dependent inertial properties of the manipulator and base. Combining the
above, we arrive at the following expression:


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with **L 0  ** being the initial angular momentum and **P** being the linear
momentum for which the following relationship holds:


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[ edit ](/s/answers/187192/edit/) flag offensive  delete  [ link
](/question/186681/no-valid-hardware-interface-element-found-in-
joint/?answer=187192#post-id-187192 "permanent link") more


Hello. I met the same problem as you. Finally I find out the reason is that in
this version of ros_control/transmission_interface/transimission_parser.cpp ,
the parser will check if there are <hardwareinterface> tags in both <joint>
section and <actuator> section. But in rrbot and other gazebo example, only
<actuator> has <hardwareinterface> tag. Therefore this error occurs.


##  1  Answer


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policy ](/privacy/) |  [ terms of service ](/terms/)


where  G  θ  ˙  is the gain matrix converting the difference between  θ  ˙
contr  and current joint angular velocities  θ  ˙  to the control torque.


d  ω  dt  =  J  M  −  1  (  θ  )  [  d  V  dt  −  G  (  θ  ,  ω  )  ]


<joint name="joint1"/>


* on DOAJ


        // Register interfaces
        registerInterface(&js_interface_);
        registerInterface(&ej_interface_);
        registerInterface(&pj_interface_);
        registerInterface(&vj_interface_);
        registerInterface(&pvj_interface_);


Query controller state at any future time
##  Further information  
  * [ Trajectory Representation ](trajectory.html)
  * [ Trajectory Replacement ](trajectory.html#trajectory-replacement)
  * [ joint_trajectory_controller Parameters ](parameters.html)
  * [ rqt_joint_trajectory_controller ](../../rqt_joint_trajectory_controller/doc/userdoc.html)
Footnote
[  1  ]  (  1  ,  2  ,  3  ,  4  )
Adolfo Rodriguez: [ joint_trajectory_controller
](http://wiki.ros.org/joint_trajectory_controller)
[ Previous ](../../gripper_controllers/doc/userdoc.html "Gripper Action
Controller") [ Next  ](trajectory.html "Trajectory Representation")
* * *
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I  (  θ  )  d  ω  dt  \+  C  c  (  θ  ,  ω  )  \+  V  c  (  ω  )  \+  S  (  θ
)  =  T  act  \+  T  int


M  d  (  x  ¨  −  x  ¨  0  )  \+  D  (  x  ˙  −  x  ˙  0  )  \+  E  (  x  −  x
0  )  =  F  (  t  )


    [ERROR] [1405708119.053840656]: No valid hardware interface element found in joint 'joint1'.
    [ERROR] [1405708119.053997707]: Failed to load joints for transmission 'tran1'.
    [ERROR] [1405708119.054095180]: No valid hardware interface element found in joint 'joint2'.
    [ERROR] [1405708119.054185473]: Failed to load joints for transmission 'tran2'.
    [ INFO] [1405708119.161913723]: Loaded gazebo_ros_control.
    [ERROR] [1405708120.087262628, 0.468000000]: Exception thrown while initializing controller joint1_position_controller.
    Could not find resource 'joint1' in 'hardware_interface::EffortJointInterface'.
    [ERROR] [1405708120.087374119, 0.468000000]: Initializing controller 'joint1_position_controller' failed
    [ERROR] [1405708121.093701156, 1.280000000]: Exception thrown while initializing controller joint2_position_controller.
    Could not find resource 'joint2' in 'hardware_interface::EffortJointInterface'.
    [ERROR] [1405708121.093770162, 1.281000000]: Initializing controller 'joint2_position_controller' failed

    In Gazebo:

        <joint name='joint2' type='revolute'>
            <pose>0 0 0.03 0 1.5707 1.5707</pose>
            <child>right_wheel</child>
            <parent>base_link</parent>
            <axis>
              <xyz>0 1 0</xyz>
              <limit>
                <lower>-1e+16</lower>
                <upper>1e+16</upper>
              </limit>
            </axis>
          </joint>
          <plugin name='ros_control' filename='libgazebo_ros_control.so'>
            <robotNamespace>/rrbot</robotNamespace>
            <robotSimType>gazebo_ros_control/DefaultRobotHWSim</robotSimType>
          </plugin>

    In URDL .xmacro

      <transmission name="tran2">
        <type>transmission_interface/SimpleTransmission</type>


Q  m  =  T  =  [  T  1  ,  T  2  …  T  n  ]  T


####  3.2. Simulation Scenario


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The Jacobian **J** M  is a standard manipulator Jacobian mentioned earlier and
the satellite Jacobian **J** S  is defined as below, with  r  ˜  ees  being
the end effector position with respect to the satellite:


G  (  θ  ,  ω  )  =  d  [  J  (  θ  )  ]  dt  ω


Both use the ` trajectory_msgs/msg/JointTrajectory  ` message to specify
trajectories, and require specifying values for all the controller joints (as
opposed to only a subset) if ` allow_partial_joints_goal  ` is not set to `
True  ` . For further information on the message format, see [ trajectory
representation  ](trajectory.html#joint-trajectory-controller-trajectory-
representation) .
####  Actions  [  1  ]  
<controller_name>/follow_joint_trajectory
[control_msgs::action::FollowJointTrajectory]

Action server for commanding the controller
The primary way to send trajectories is through the action interface, and
should be favored when execution monitoring is desired.
Action goals allow to specify not only the trajectory to execute, but also
(optionally) path and goal tolerances. When no tolerances are specified, the
defaults given in the parameter interface are used (see [ Details about
parameters  ](parameters.html#parameters) ). If tolerances are violated during
trajectory execution, the action goal is aborted, the client is notified, and
the current position is held.
The action server returns success to the client and continues with the last
commanded point after the target is reached within the specified tolerances.
####  Subscriber  [  1  ]  
<controller_name>/joint_trajectory [trajectory_msgs::msg::JointTrajectory]

Topic for commanding the controller
The topic interface is a fire-and-forget alternative. Use this interface if
you don’t care about execution monitoring. The goal tolerance specification is
not used in this case, as there is no mechanism to notify the sender about
tolerance violations. If state tolerances are violated, the trajectory is
aborted and the current position is held. Note that although some degree of
monitoring is available through the ` ~/query_state  ` service and ` ~/state
` topic it is much more cumbersome to realize than with the action interface.
####  Publishers  
<controller_name>/controller_state
[control_msgs::msg::JointTrajectoryControllerState]

Topic publishing internal states with the update-rate of the controller
manager
####  Services  
<controller_name>/query_state [control_msgs::srv::QueryTrajectoryState]


The single-degree-of-freedom impedance control of the prismatic joint can be
expressed as a scalar version of Equation (33), where the displacement ** x
** and virtual trajectory ** x  0  ** are expressed as positions in the
direction collinear with  k  prism  in terms of  λ  in such a way that the
point  λ  0  lies in **x 0  ** and  k  prism  is in line with the dominant
direction of the final approach and contact:


[ Aerospace ](/journal/aerospace)


            if (limits.has_position_limits)
            {
                *joint_type = urdf::Joint::REVOLUTE;
            }
            else
            {
                if (limits.angle_wraparound)
                    *joint_type = urdf::Joint::CONTINUOUS;
                else
                    *joint_type = urdf::Joint::PRISMATIC;
            }
        }


The above basic formulation of a generic impedance control law can be
described from the point of view of the environment as “hiding” the natural
dynamics of the manipulator and exposing only the desired dynamics at the
interaction port. At the implementation level, there are numerous ways by
which the dynamic behavior of the robot’s interaction port can be shaped.
There exist purely mechanical methods such as springs and dampers or
exploiting the redundancies of the manipulator to achieve different inertial
properties by the virtue of their dependence of the manipulator’s
configuration [  5  ,  31  ]. The software-based methods rely on using control
strategies which reproduce the dynamic behavior with robot’s actuators based
on the loop closed via sensors. Three important types of software approaches
to impedance implementation are [  6  ]:


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[ ![ gravatar image]() ]() [ ]() (  2024-04-16 00:59:02 -0500  )  edit


Citations  \-


whereas this work expands it to a three-dimensional case. In the work [  24
], the authors considered stabilization of contact using position-based
impedance control, which relies only on kinematic information about the space
robot. The authors stressed the aspects of low computational complexity of the
method.


►  ▼  Article Menu


The vector of generalized forces becomes:


##  Share


Model-based approach, which fundamentally differs from the previous two,
because it uses the known manipulator dynamics and substitutes into it the
desired dynamics described by the impedance parameters.


Number


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The satellite equipped with robotic arm is modelled as a multibody system. It
is described using the coordinate systems depicted in  Figure 1  .


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image](//www.gravatar.com/avatar/9ca6e6fe619a51425fcb84fbde6ace8a?s=16&d=identicon&r=PG)
](/users/1279/dave-coleman/) [ Dave Coleman ](/users/1279/dave-coleman/) (
2015-01-10 18:20:25 -0500  )  edit


interfaces:
  * With command interface ` position  ` , there are no restrictions for state interfaces.
  * With command interface ` velocity  ` :
    * if command interface ` velocity  ` is the only one, state interfaces must include ` position,  velocity  ` .
  * With command interface ` effort  ` , state interfaces must include ` position,  velocity  ` .
  * With command interface ` acceleration  ` , state interfaces must include ` position,  velocity  ` .
Further restrictions of state interfaces exist:
  * ` velocity  ` state interface cannot be used if ` position  ` interface is missing.
  * ` acceleration  ` state interface cannot be used if ` position  ` and ` velocity  ` interfaces are not present.”
Example controller configurations can be found  below  .
##  Other features  
  * Realtime-safe implementation.
  * Proper handling of wrapping (continuous) joints.
  * Robust to system clock changes: Discontinuous system clock changes do not cause discontinuities in the execution of already queued trajectory segments.
##  Using Joint Trajectory Controller(s)  
The controller expects at least position feedback from the hardware. Joint
velocities and accelerations are optional. Currently the controller does not
internally integrate velocity from acceleration and position from velocity.
Therefore if the hardware provides only acceleration or velocity states they
have to be integrated in the hardware-interface implementation of velocity and
position to use these controllers.
The generic version of Joint Trajectory controller is implemented in this
package. A yaml file for using it could be:
>
>     controller_manager:
>       ros__parameters:
>         joint_trajectory_controller:
>         type: "joint_trajectory_controller/JointTrajectoryController"
>
>     joint_trajectory_controller:
>       ros__parameters:
>         joints:
>           - joint1
>           - joint2
>           - joint3
>           - joint4
>           - joint5
>           - joint6
>
>         command_interfaces:


####  2.4.2. Dynamics of the Free-Floating Manipulator


The parameters which were used for the comparison of the performance of the
two schemes were the following:


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      <transmission name="tran1">
        <type>transmission_interface/SimpleTransmission</type>
        <joint name="joint1">
          <hardwareInterface>EffortJointInterface</hardwareInterface>
        </joint>
      </transmission>


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In this subsection, the standard description of the manipulator kinematic
equations is recalled after [  10  ,  24  ].


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[ ![ gravatar image]() ]() [ ]() (  2024-04-16 01:05:12 -0500  )  edit


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    double clamp(const double val, const double min_val, const double max_val)
    {
        return std::min(std::max(val, min_val), max_val);
    }


![](https://pub.mdpi-res.com/bundles/mdpisciprofileslink/img/unknown-
user.png?1713167732) George Z. H. Zhu


Palma, Piotr, Karol Seweryn, and Tomasz Rybus. 2022\. "Impedance Control Using
Selected Compliant Prismatic Joint in a Free-Floating Space Manipulator"
_Aerospace_ 9, no. 8: 406. https://doi.org/10.3390/aerospace9080406


E  M  =  I  s  \+  ∑  i  =  1  n  (  I  i  \+  m  i  r  ˜  i  _  s  T  r  ˜  i
_  s  )


This seems to work..thanks.. , not sure if this is a gazebo or ros issue. I
can not find documentation of how or why this changed. Any thoughts on this?


θ  ˙  cont  =  J  D  −  1  e  v


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alert)


r  ee  =  r  s  \+  r  q  \+  ∑  i  =  1  n  l  i  \+  k  prism  λ


[ edit ](/questions/29263/edit/) [ retag ](/s/questions/29263/retag/) flag
offensive  [ close ](/questions/29263/close/) merge  delete


add a comment


                    const double effort_limit = joint_effort_limits_[j];
                    const double effort = clamp(pid_controllers_[j].computeCommand(error, period),
                                                -effort_limit, effort_limit);
                    sim_joints_[j]->SetForce(0, effort);
                }
                    break;


break;


[ Hi there! Please sign in ](/account/signin/?next=/question/186681/no-valid-
hardware-interface-element-found-in-joint/) [ help ](/help/ "help")


        joint_limits_interface::JointLimits limits;
        bool has_limits = false;
        joint_limits_interface::SoftJointLimits soft_limits;
        bool has_soft_limits = false;


__ __ __


            // Add data from transmission
            joint_names_[j] = transmissions[j].joints_[0].name_;
            joint_position_[j] = 1.0;
            joint_velocity_[j] = 0.0;
            joint_effort_[j] = 1.0;  // N/m for continuous joints
            joint_effort_command_[j] = 0.0;
            joint_position_command_[j] = 0.0;
            joint_velocity_command_[j] = 0.0;


[ ALL ](/questions/scope:all/sort:activity-desc/page:1/) [ UNANSWERED
](/questions/scope:unanswered/sort:answers-asc/page:1/)


The desired behavior in Cartesian space can be expressed as follows:


####  2.4.1. Free-Floating Space Manipulator


edit


I am getting the following error trying to load simulation interface for
gazebo ros . This worked in hydro but after moving to ubuntu 14 - indigo I now
get these errors. I can not see any thing that has changed in the
documentation for this feature. It there a indigo-devel lib I need? I pull all
software from ubuntu software center. I also compiled gazebo-ros-pkgs from
source from indigo-devel.


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H  2  =  [  A  B  B  T  \+  r  ˜  s  A  E  M  \+  r  ˜  s  B  ]  ,


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**Chicago/Turabian Style**


####  2.3. Kinematics of the Free-Floating Space Robot


        // Resize vectors to our DOF
        n_dof_ = transmissions.size();
        joint_names_.resize(n_dof_);
        joint_types_.resize(n_dof_);
        joint_lower_limits_.resize(n_dof_);
        joint_upper_limits_.resize(n_dof_);
        joint_effort_limits_.resize(n_dof_);
        joint_control_methods_.resize(n_dof_);
        pid_controllers_.resize(n_dof_);
        joint_position_.resize(n_dof_);
        joint_velocity_.resize(n_dof_);
        joint_effort_.resize(n_dof_);
        joint_effort_command_.resize(n_dof_);
        joint_position_command_.resize(n_dof_);
        joint_velocity_command_.resize(n_dof_);


The symbol  I  7  ×  7  denoted an identity matrix of the size given by the
subscript. The matrix **M d  ** , describing the impedance in terms of desired
mass parameters, was chosen such that the end effector behaved similarly to a
rigid body with 10% of the actual system’s mass and inertia. The stiffness
**E** was selected so that it was five times higher in the directions normal
to the dominant contact direction z to keep the end effector on the planned
line.


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© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an
open access article distributed under the terms and conditions of the Creative
Commons Attribution (CC BY) license ( [
https://creativecommons.org/licenses/by/4.0/
](https://creativecommons.org/licenses/by/4.0/) ).


where the  m  i  j  ∈  M  (  q  p  )  and  i  ,  j  ,  k  =  1  …  n  and the
sub-matrices are defined below, with the tilde symbol ~ denoting a skew
symmetric matrix of a vector and **I i  ** denoting the inertia matrices of
each link. **I** is an identity matrix of suitable size.


The contributions of this work are the following: The distinction of the
dominant direction of contact is made as a prerequisite for single-axis
prismatic impedance control for providing control of an end effector in
contact with the target body. The GJM is modified to include the prismatic
stage at the end effector in the mathematical model of the free-floating robot
and the simulation of the space robot with a prismatic impedance-controlled
stage at the end effector is carried out in three dimensions, where both the
robot and the target are free-floating.


The center of mass of the satellite-arm system remains constant under the
assumption of no momentum exchange with the environment and by neglecting the
orbital motion, but the base is free to change its orientation and position in
the inertial reference frame. In such a case, the state vector  q  p  in (4)
contains only the joint positions.


Article Type


[ __ ](/questions/) |


In both cases, the robot controller was tracking a straight-line trajectory
constituting the dominant direction of the final approach. The starting point
was chosen such that the initial distance between the end effector sphere and
the target sphere was 5 mm. In order to assure that the contact between the
end effector and the target took place, the length of the planned trajectory
was 7 mm, implying that the virtual trajectory ended 2 mm into the target
sphere envelope. The 7 mm distance was planned to be travelled from full stop
at the beginning to a full stop at the end over 15 s. The simulation time
lasted 20 s in total to allow the recording of an additional 5 s more of the
simulation data after the planned motion stopped. The impedance parameters
were kept constant throughout the simulation in both cases.


ω  ee  =  ω  s  \+  ∑  i  =  1  n  k  i  θ  i  ˙


##  Information


    void Komodo2RobotHWSim::writeSim(ros::Time time, ros::Duration period)
    {
        // If the E-stop is active, joints controlled by position commands will maintain their positions.
        if (e_stop_active_)
        {
            if (!last_e_stop_active_)
            {
                last_joint_position_command_ = joint_position_;
                last_e_stop_active_ = true;
            }
            joint_position_command_ = last_joint_position_command_;
        }
        else
        {
            last_e_stop_active_ = false;
        }


Link  |  Length [m]  |  Mass [kg]  |  Inertia [kgm  2  ]
(Dominant about the Joint Axis)
---|---|---|---
1  |  0.28  |  1.60  |  0.004
2  |  0.20  |  4.20  |  1.25
3  |  0.60  |  1.60  |  0.004
4  |  0.60  |  4.40  |  1.14
5  |  0.20  |  1.60  |  0.004
6  |  0.10  |  1.60  |  0.008
7  |  0.10  |  0.19  |  1 × 10  −  4

**Table 2.** Comparison of peak torque loads in robot joints.


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Open Access  Article


|  Model-Based
Impedance Ctrl.  |  Single Prismatic Joint
Impedance Ctrl.
---|---|---
Velocity along x [m/s]  |  −0.3 × 10  −4  |  −1.1 × 10  −4
Velocity along y [m/s]  |  5.8 × 10  −4  |  1.8 × 10  −3
Velocity along z [m/s]  |  0.4 × 10  −4  |  1.2 × 10  −4
Angular rate about x [°/s]  |  −3 × 10  −4  |  23 × 10  −3
Angular rate about y [°/s]  |  −6 × 10  −4  |  24 × 10  −3
Angular rate about z [°/s]  |  −9.0  |  −15.1

|


The free-floating satellite equipped with manipulator arm exhibits
nonholonomic properties as the angular momentum of the system consisting of
the satellite and manipulator chain is not integrable [  25  ]. The presented
control method does not address the nonholonomic aspects of the trajectory
planning and tracking, unlike, e.g., [  26  ], but the simulation cases are
selected such that the simulated trajectories are unlikely to evolve into
singular ones.


By using a single prismatic impedance-controlled joint, like the one discussed
in [  32  ], as the last joint before the end effector, a controlled
mechanical impedance along the dominant direction of the contact can be
achieved and the hardware and control laws of the whole free-floating space
manipulator can be substantially simplified. Although this joint provides an
additional degree of freedom and can be incorporated into the GJM by
translational term in a construction of the Jacobian, its primary objective is
to provide space for the compliant behavior of the mechanical impedance, not
to play any major role in the planning or realization of a trajectory.


By adding <hardwareinterface> tags to <joint> section, this error disappears.
For example, in rrbot.xacro, change


  * [ Project Ideas for GSoC 2024 ](../../../project_ideas.html)
  * [ Acknowledgements ](../../../acknowledgements/acknowledgements.html)
__ [ ROS2_Control: Humble ](../../../../index.html)
  * [ ](../../../../index.html)
  * [ ros2_controllers ](../../doc/controllers_index.html)
  * joint_trajectory_controller
  * [ Edit on GitHub ](https://github.com/ros-controls/ros2_controllers/blob/humble/joint_trajectory_controller/doc/userdoc.rst)
* * *
**You're reading the documentation for an older, but still supported, version
of ROS 2. For information on the latest version, please have a look at[ Iron
](../../../../../iron/doc/ros2_controllers/joint_trajectory_controller/doc/userdoc.html)
. **
#  joint_trajectory_controller  
Controller for executing joint-space trajectories on a group of joints. The
controller interpolates in time between the points so that their distance can
be arbitrary. Even trajectories with only one point are accepted. Trajectories
are specified as a set of waypoints to be reached at specific time instants,
which the controller attempts to execute as well as the mechanism allows.
Waypoints consist of positions, and optionally velocities and accelerations.
_Parts of this documentation were originally published in the ROS 1 wiki under
the_ [ CC BY 3.0 license ](https://creativecommons.org/licenses/by/3.0/) .
_Citations are given in the respective section, but were adapted for the ROS 2
implementation._ [  1  ]
##  Hardware interface types  
Currently, joints with hardware interface types ` position  ` , ` velocity  `
, ` acceleration  ` , and ` effort  ` (defined [ here
](https://github.com/ros-
controls/ros2_control/blob/humble/hardware_interface/include/hardware_interface/types/hardware_interface_type_values.hpp)
) are supported in the following combinations as command interfaces:
  * ` position  `
  * ` position  ` , ` velocity  `
  * ` position  ` , ` velocity  ` , ` acceleration  `
  * ` velocity  `
  * ` effort  `
This means that the joints can have one or more command interfaces, where the
following control laws are applied at the same time:
  * For command interfaces ` position  ` , the desired positions are simply forwarded to the joints,
  * For command interfaces ` acceleration  ` , desired accelerations are simply forwarded to the joints.
  * For ` velocity  ` ( ` effort  ` ) command interfaces, the position+velocity trajectory following error is mapped to ` velocity  ` ( ` effort  ` ) commands through a PID loop if it is configured ( [ Details about parameters  ](parameters.html#parameters) ).
This leads to the following allowed combinations of command and state


Last updated: **Jul 10 '23**


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  *[
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             ]: 2023-07-10 04:51:47 -0500
  *[
        2023-07-10 04:51:47 -0500
       ]: 2023-07-10 04:51:47 -0500
  *[
        2024-04-16 01:05:12 -0500
       ]: 2024-04-16 01:05:12 -0500


* [ Google Scholar ](https://scholar.google.com/scholar?q=Impedance%20Control%20Using%20Selected%20Compliant%20Prismatic%20Joint%20in%20a%20Free-Floating%20Space%20Manipulator)


![](/bundles/mdpisciprofileslink/img/unknown-user.png) Karol Seweryn


The Cartesian impedance control law takes the form:


[ ![Arn-O gravatar
image](//www.gravatar.com/avatar/fe66bd7b67c0ce29399ba5b715fbf9cf?s=16&d=identicon&r=PG)
](/users/5387/arn-o/) [ Arn-O ](/users/5387/arn-o/) (  2014-08-02 01:49:18
-0500  )  edit


The velocity-dependent effects are modelled by the Coriolis matrix which has
entries with the following form [  27  ]:


[ Powered by Askbot version 0.7.58 ](http://askbot.com)


Since most robotic tasks involving interaction with an environment are
naturally defined in coordinates relative to the environment, it is useful to
replace the  X  ˙  (  s  )  with relative displacement  s  X  r  (  s  )  .
Specifically, the relative displacement is understood as the difference
between the current actual position of the interaction port  X  relative to
the intended one, often referred to as the “virtual” trajectory  X  0  [  5
].


[ Gazebo on headless simulation server ](/question/5387/gazebo-on-headless-
simulation-server/)


The potential technical benefit of the approach based on a single prismatic
impedance-controlled joint to provide tunable compliance at the end effector
in contact with a target body is twofold: Firstly, the overall computational
effort is lower. Both methods require computation of inverse of the
generalized Jacobian matrix and keeping it updated throughout the maneuver
since it is present in both the model-based control law (40) and in the simple
Cartesian joint controller (43) used to follow the virtual trajectory with the
prismatic impedance-controlled joint. Nonetheless the control law (40) also
requires the computing and inverting of the instantaneous mass matrix, and
four more matrix multiplications. Secondly, the full model-based impedance
control also requires a current, measured value of the forces and torques at
the interface between the end effector and the target  F  i  n  t  , a rather
significant complication for the gripper to be designed for reliable operation
in space. The impedance control of the single axis actuator, however, requires
a positional feedback. This type of a sensor is not free of its own drawbacks,
in particular the noise increased by taking the derivatives of its signal, but
still some type of position feedback is required for all the joints of the
robot in any case.


Tomasz Rybus


[ how to make velocity of turtlebot is smoother?? ](/question/318914/how-to-
make-velocity-of-turtlebot-is-smoother/)


* [ gazebo-all-versions ](/questions/scope:all/sort:activity-desc/tags:gazebo-all-versions/page:1/)


        if (!has_limits)
            return;


Column Width:


Feature papers represent the most advanced research with significant potential
for high impact in the field. A Feature Paper should be a substantial original
Article that involves several techniques or approaches, provides an outlook
for future research directions and describes possible research applications.


The realization of the virtual trajectory  x  0  ,  x  ˙  0  and keeping the
**k** prism  oriented along the dominant direction of the approach can be
conducted using known strategies for the joint control of a free-floating
space robot, which require less calculation effort than the model-based
impedance control. One example of such control methods is the Cartesian joint
control based on the GJM presented in the following subsection.


##  5\. Conclusions


The series-type manipulator with n rotational joints and a prismatic joint
before the end effector is described by a vector of configuration variables
corresponding to the angles of each joint  θ  1  …  θ  n  and the
translational position of the slider in the prismatic joint  λ  :


##  About


In the presented work, we perform a simulation study of the robot with an
impedance-controlled prismatic joint at the end effector in a three-
dimensional simulation of a 7-DoF free-floating robot and a free-floating
target. The robot configuration used in this study is based on the 7-DoF
configuration common in the European Space Agency’s projects [  19  ],
although the method is applicable to any robot with six or more degrees of
freedom. In our previous research we investigated free-floating robot
manipulators where the impedance control law was incorporating the dynamic
model of the manipulator [  20  ] involving all robot joints in the
realization of the impedance control. This kind of control law allows the
independent shaping of the mechanical impedance along each direction xyz in
Cartesian space as well as the corresponding three torsional components.
Observing that in the intercept missions the final approach and contact
establishment between the end effector and the target usually happens along a
single dominant direction of action, we analyze how a controlled impedance can
be achieved along a single translational degree of freedom, while keeping the
rest of the joints under classical position and velocity control, rather than
torque control. In the case of space applications, where reliability and low
complexity are highly valued design features, a simplification of the system
while maintaining its key functionality is a useful gain. In order to account
for the attitude change of the target, the work [  21  ] proposed a 2D contact
model and dual manipulator setup. The presented work addresses the attitude
tracking by separating the function of impedance control and orientation-
position control into two separate control tasks. As a consequence, the
complexity of the method is lowered in terms of computational requirements and
by allowing the limitation of the necessity for high-bandwidth hardware needed
for the impedance control to one prismatic joint. We simulate a free-floating
robotic manipulator with a single impedance-controlled joint at the end
effector coming into contact with a target body in a final approach maneuver
scenario. We compare the resulting loads on the joints and the overall
satellite disturbance with a full model-based impedance-controlled free-
floating space robot in the same scenario. We then discuss the limitations and
possibilities of such architecture in a practical setting. A similar
architecture, i.e., using a prismatic stage at the end effector, was also
proposed for a space manipulator, though one with a controlled base (non-free-
floating) [  22  ], and not with a focus on simplification, but rather on
introducing an additional layer of interaction between the robot pose control
and impedance control. Our earlier work considered a two-dimensional
simulation case of a robot with a prismatic compliant stage at the end
effector focusing on disturbance to the base of the satellite [  23  ],


I have posted an issue in the ros_control bug tracker: [
https://github.com/ros-controls/ros_control/issues/177
](https://github.com/ros-controls/ros_control/issues/177%3C/p%3E)


H  3  =  [  D  M  F  M  \+  r  ˜  s  D  M  ]


[ none ](/questions/scope:all/sort:activity-desc/tags:none/page:1/) ×


**AMA Style**


Export citation file: [ BibTeX ](javascript:window.document.forms\['export-
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Substituting (31) and (32) into (30), rearranging terms and taking the inverse
Laplace transform gives the time-domain differential equation describing the
interaction point forces **F** as a function of the difference between current
and virtual trajectory, parametrized by the desired inertia parameters:


The mechanical impedance control presented in previous sections allows the
shaping of the end effector’s impedance properties along Cartesian axes
independently. The control law involves interface forces  F  int  as seen
Equation (40). In a practical setting, those forces and torques need to be
measured by a hardware sensor and appropriately conditioned. This adds
complexity to the spaceborne hardware of the robot. The whole control loop,
including calculation of the torques and generation of the appropriate
currents in the motors in each joint, needs to be fast to cope with the
dynamics of the contact. Calculation of the torques with the control law (40)
requires numerous multiplications of 6 × 6 and 6 × 7 matrices by the control
computer, which needs to provide adequate processing power to meet the timing
requirements.


During the last part of the intercept maneuver when the gripper is closing,
the controlled impedance of the end effector shall maintain stable contact
with the target feature. Typically, the chaser and target satellite will be
moving with respect to each other, and as a consequence there will be a
dominating direction along which the displacement of the contact point will
take place throughout the time required for closing the gripper. Simplified
illustrations of various gripping scenarios of in-orbit intercept are shown in
Figure 2  . Scenario A is applicable to many defunct satellites equipped with
maneuvering and orbit-correction engines [  5  ], which is typical for large
geostationary communication and meteorological and observational missions.
Scenario B is a simplified illustration of a clamping gripper locking on a
feature from a side, like the docking gripper for locking on a satellite’s
separation ring [  3  ]. The scenario C depicts a symmetrically acting
gripper, similar to the one from e.Deorbit project [  19  ]. In case of a
tumbling target satellite, the dominant direction will be collinear with the
vector of a momentary linear velocity of the feature to be gripped. Assuming
that this dominating direction can be tracked with the end effector
orientation control, so that the gripper center point stays on the line and
the gripper orientation with respect to the feature to be gripped is kept
constant throughout the final approach and gripping phase, it is sufficient
that the compliance due to impedance control is present only in the axis of
the dominant direction.


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M  =  [  A  B  D  M  B  T  E  M  F  M  D  M  T  F  M  T  N  ]


Background:


Joint
No.  |  Peak Torque
---|---
Model-Based
Impedance Ctrl.[Nm]  |  Single Prismatic Joint Impedance Ctrl.[Nm]  |
Comparison
1  |  0.62  |  0.885  |  43%
2  |  0.067  |  0.04  |  −40%
3  |  0.048  |  0.097  |  102%
4  |  0.051  |  0.077  |  51%
5  |  0.047  |  0.076  |  62%
6  |  0.011  |  0.012  |  9%
7  |  0.636  |  0.888  |  40%
|  |  |  Average: 38%

**Table 3.** Velocity and angular rates of the center of mass of the satellite
body at the end of simulation.


The vector of generalized coordinates  q  p  , describing the system of
satellite base and manipulator, is assembled from definitions (1)–(3)


q  p  =  [  r  s  Θ  s  θ  ]


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No valid hardware interface element found in joint


####  3.1. Simulation Parameters and Configuration


**Figure 1.** Schematic depiction of reference frames of a robotic arm on a
satellite base ( **left** ). Detail showing the end effector with the
prismatic joint displacement ( **right** ).


            // Debug
            ROS_DEBUG_STREAM_NAMED("komodo2_hw_sim","Loading joint '" << joint_names_[j]
                                                                            << "' of type '" << hardware_interface << "'");


Torque/force-based approach, where instead of the position, the outer loop
commands the torque or force and the inner loop is the torque/force servo.


[ My Robot Doesn't Move - What am I doing wrong? ](/question/24349/my-robot-
doesnt-move-what-am-i-doing-wrong/)


[ Sign In / Sign Up ](/user/login)


Note that from the first issue of 2016, this journal uses article numbers
instead of page numbers. See further details [ here
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            if (joint_control_methods_[j] != EFFORT)
            {
                // Initialize the PID controller. If no PID gain values are found, use joint->SetAngle() or
                // joint->SetVelocity() to control the joint.
                const ros::NodeHandle nh(model_nh, robot_namespace + "/gazebo_ros_control/pid_gains/" +
                                                   joint_names_[j]);
                if (pid_controllers_[j].init(nh, true))
                {
                    switch (joint_control_methods_[j])
                    {
                        case POSITION:
                            joint_control_methods_[j] = POSITION_PID;
                            break;
                        case VELOCITY:
                            joint_control_methods_[j] = VELOCITY_PID;
                            break;
 			case POS_VEL:
			    joint_control_methods_[j] = POSITION_PID;
                          //  ROS_ERROR("PID on: %s",joint_names_[j].c_str());
                            break;
                    }
                }
                else
                {
                    // joint->SetMaxForce() must be called if joint->SetAngle() or joint->SetVelocity() are
                    // going to be called. joint->SetMaxForce() must *not* be called if joint->SetForce() is
                    // going to be called.
                    #if GAZEBO_MAJOR_VERSION > 2
                                        joint->SetParam("fmax", 0, joint_effort_limits_[j]);
                    #else
                                        joint->SetMaxForce(0, joint_effort_limits_[j]);
                    #endif


        <joint name="joint2"/>
        <actuator name="motor2">
          <hardwareInterface>EffortJointInterface</hardwareInterface>
          <mechanicalReduction>1</mechanicalReduction>
          <motorTorqueConstant>10</motorTorqueConstant>
        </actuator>
      </transmission>

    In Yaml

    rrbot:
      # Publish all joint states -----------------------------------
      joint_state_controller:
        type: joint_state_controller/JointStateController
        publish_rate: 10

      # Position Controllers ---------------------------------------
      joint1_position_controller:
        type: effort_controllers/JointVelocityController
        joint: joint1
        pid: {p: 40, i: 1, d: .1, i_clamp: 1}
      joint2_position_controller:
        type: effort_controllers/JointVelocityController
        joint: joint2
        pid: {p: 40, i: 1, d: .1, i_clamp: 1}


The aim of the simulation was to compare the results from both controllers and
assess the disturbance which the contact forces cause to the free-floating
robot system. The parameters selected for comparison of the performance of the
controllers in this study were: velocities of the base satellite center of
mass after the contact maneuver and peak of the torques about the driven axis
of each joint.


Author / Affiliation / Email


Since the planned virtual trajectory of the end effector,  x  0  **,** is
typically not defined in the velocity space but rather in the position space,
we use the approximation of the velocity error **e** v  by the position error
e  p  =  x  0  −  x  c  **,** where **x** c  is the current EE position,
multiplied by some gain matrix **G ee  ** . The joint control torques defined
by the simple Cartesian trajectory-following controller are given by:


The mechanical connection between the chaser satellite equipped with a robotic
grappler arm and the intercepted object can be achieved by closing the
grappler around a geometric feature of the target, e.g., a rim of separation
ring [  3  ], or interlocking an expandable mechanical penetrator tip inside a
sufficiently stiff opening, e.g., inside a thruster combustion chamber like in
[  4  ]. The chaser and target velocities and angular rates will very likely
vary since in many cases the target will be in an uncontrolled motion. This
difference needs to be accounted for when coming into contact with the target,
since it may cause abrupt forces to act on the gripping robot. A common method
to manage such interaction as contact in robotic applications is the
mechanical impedance control of the end effector with the use of a full robot
dynamical model [  5  ], although other methods exist, most of which achieve
the commanded impedance at the end effector by suitably controlling all the
robot joints [  4  ,  5  ,  6  ,  7  ,  8  ]. The impedance matching to the
target satellite was proposed as a means to shape the dynamic properties of
the capturing robotic arm of the intercepting satellite [  4  ], and in a more
recent study a frequency-independent method was proposed to match the velocity
of the end effector and the target satellite, hence guaranteeing capture [  9
]. The control of free-floating space manipulators using the Generalized
Jacobian Matrix (GJM) has established history [  10  ,  11  ] and numerous
applications, including optimization [  12  ] and control under nonlinear
predictive control [  13  ], as well as evolution to multiple manipulators on
a single satellite [  14  ]. The work on the application of impedance control
to a satellite-based robot [  15  ] employs the GJM in the formulation of
control, and while it approaches formulation of model-based impedance control,
it focuses on the topics of computational complexity and uncertainty of the
intercepted object. The approach to compliance control of the satellite
manipulator using a force controller is proposed in [  16  ], but the concept
of a compliant manipulator wrist is introduced in the work [  17  ]. The work
does not focus on the manipulator architecture, but on the coefficient of
restitution and its relationship with the impedance parameters as well as a
method to establish desired impedance control parameters for attaining a
controlled coefficient of restitution between the target and a robotic arm.
Importantly, the verification work is carried out using a ground-based robot
and a fixed wall as a target. The work [  18  ] discusses a combined method
involving resolved motion rate control and admittance control for providing
compliant contact control and shows simulation results for a 3-DoF (degree of
freedom) robot capturing a rotating target body.


We consider a drift-less case; hence, the initial momentum and angular
momentum above is zero. A different case is discussed in [  30  ]. The system
is free floating; hence, the Equation (21) is equated to zero. The
relationship between the angular velocities of the joints of the space robot
and the linear and angular velocity of the end effector  v  ee  ,  ω  ee  are
given by:


##  4\. Discussion


add a comment


**Chicago/Turabian Style**


####  2.1. The Nonholonomic Nature of the Free-Floating Space Robot


**Figure 2.** Various gripping strategies for interception of a space object.
Dominant direction in each case is shown by the arrow.


In the architecture proposed in this work, the last joint before the end
effector is prismatic; thus, including its effect in the Jacobian requires the
use of the nonzero  J  T  i  term for this last joint, while in our previous
work [  20  ] we considered impedance control of a free-floating manipulator
with all rotational joints only.


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##  Article Metrics


An exhaustive discussion and comparison of the above methods is presented in [
6  ]. In this work, we propose the use of a Cartesian, model-based impedance
controller to a space robot on a free-floating satellite platform.


This research received no external funding.


Previous Article in Special Issue


**Figure 1.** Schematic depiction of reference frames of a robotic arm on a
satellite base ( **left** ). Detail showing the end effector with the
prismatic joint displacement ( **right** ).


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##  Related questions


as:


##  Comments


J  D  =  (  J  M  −  J  S  H  2  −  1  H  3  )


###  Article Access Statistics


It seems to be a ros issue with indigo because this change is found in indigo
version of ros_control pkg. I don't know why ...


####  2.2. Reference Frames in a Space Robot


Disclaimer


[ Spectral Light Curve Simulation for Parameter Estimation from Space Debris
](/2226-4310/9/8/403)


This section introduces the dynamics of a full (non-free-floating) space robot
without nonholonomic constraints. The reaction forces and torques acting on
the system are the following:


Note that from the first issue of 2016, this journal uses article numbers
instead of page numbers. See further details [ here
](https://www.mdpi.com/about/announcements/784) .


The  Table 2  compares the resultant loads in joints. The percentages in the
“Comparison” columns show how much the loads in the single prismatic impedance
control simulation exceeded the loads in the model-based impedance control of
the full robot, for each respective joint.


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The dynamic Jacobian relates the velocity space to joint space, so a simple
controller computing the joint velocities as given by the Equation (24). It
can be used to find joint velocities  θ  ˙  cont  , minimizing the deviation
**e** v  between the intended and actual velocity as follows:


[ ](https://pub.mdpi-
res.com/aerospace/aerospace-09-00406/article_deploy/html/images/aerospace-09-00406-g001.png?1658994450
"


**Table 1.** Geometric and mass properties of the simulated robot.


Palma P, Seweryn K, Rybus T. Impedance Control Using Selected Compliant
Prismatic Joint in a Free-Floating Space Manipulator. _Aerospace_ . 2022;
9(8):406. https://doi.org/10.3390/aerospace9080406


        if (limits.has_position_limits)
        {
            *lower_limit = limits.min_position;
            *upper_limit = limits.max_position;
        }
        if (limits.has_effort_limits)
            *effort_limit = limits.max_effort;


##  Initiatives


AND  OR


Font Size:


//sim_joints_[j]->SetVelocityLimit(0,joint_velocity_command_[j]);
#if GAZEBO_MAJOR_VERSION >= 4
                    sim_joints_[j]->SetPosition(0, joint_position_command_[j]);
#else
                    sim_joints_[j]->SetAngle(0, joint_position_command_[j]);
#endif
                    sim_joints_[j]->SetVelocity(0, e_stop_active_ ? 0 : joint_velocity_command_[j]);


    void Komodo2RobotHWSim::readSim(ros::Time time, ros::Duration period)
    {
        for(unsigned int j=0; j < n_dof_; j++)
        {
            // Gazebo has an interesting API...
            if (joint_types_[j] == urdf::Joint::PRISMATIC)
            {
                joint_position_[j] = sim_joints_[j]->GetAngle(0).Radian();
            }
            else
            {
                joint_position_[j] += angles::shortest_angular_distance(joint_position_[j],
                                                                        sim_joints_[j]->GetAngle(0).Radian());
            }
            joint_velocity_[j] = sim_joints_[j]->GetVelocity(0);
            joint_effort_[j] = sim_joints_[j]->GetForce((unsigned int)(0));
        }
    }


**Figure 3.** The simulated satellite robot and target body in the initial
position ( **left** ), and during contact ( **right** ). The line of dominant
direction of the final approach is shown by the dashed line.


The impedance is fundamentally a relationship between the input “flow
variables”  X  ˙  such as velocity or electric current and output “effort
variables”  F  —force or voltage. In the Laplace domain this can be written as
the following ratio:


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####  2.8. The Simplified Robot Architecture with Controlled Impedance on a
Single Axis


##  Author Contributions


##  2\. Mathematical Model of the Space Robot


##  Stats