How to use the geometric_jacobian correctly?

[wly2014]

When I use geometric_jacobian to calculate the jacobian of a simple robot, I found that the linear of result is not same to the result of MATLAB, but the angular is same. The robot is as follows. It has a fixed joint and a revolute joint.

<?xml version="1.0" encoding="utf-8"?>
<robot
  name="door">
  <link
    name="base_link">
    <inertial>
      <origin
        xyz="0.0 0.0 0.0"
        rpy="0 0 0" />
      <mass
        value="6000.0" />
      <inertia
        ixx="7.1711"
        ixy="1.6399E-05"
        ixz="1.1295"
        iyy="13.321"
        iyz="-0.00010257"
        izz="12.459" />
    </inertial>
    <visual>
      <origin
        xyz="0.0 0.0 0.0"
        rpy="0 0 0" />
      <geometry>
        <box size="0.1 0.1 0.1" />
      </geometry>
      <material
        name="">
        <color
          rgba="0.37647 0.37647 0.37647 1" />
      </material>
    </visual>
  </link>
  <link
    name="door_base_link">
    <inertial>
      <origin
        xyz="0.0 0.0 0.0"
        rpy="0 0 0" />
      <mass
        value="2.8423" />
      <inertia
        ixx="0.011332"
        ixy="-0.00032807"
        ixz="-0.00025666"
        iyy="0.011319"
        iyz="-0.00020965"
        izz="0.0084278" />
    </inertial>
    <visual>
      <origin
        xyz="0.0 0 0"
        rpy="0 0 0" />
      <geometry>
        <cylinder radius="0.05" length="0.5"/>
      </geometry>
      <material
        name="">
        <color
          rgba="1 1 1 1" />
      </material>
    </visual>
  </link>
  <joint
    name="door_base_joint"
    type="fixed">
    <origin
      xyz="0.4 0.6 0.0"
      rpy="0.0 0.0 0.0" />
    <parent
      link="base_link" />
    <child
      link="door_base_link" />
    <axis
      xyz="0 0 0" />
  </joint>
  <link
    name="door_link">
    <inertial>
      <origin
        xyz="0.0 -0.3 0.0"
        rpy="0 0 0" />
      <mass
        value="20" />
      <inertia
        ixx="7.85"
        ixy="-2.4416E-05"
        ixz="1.5843E-05"
        iyy="7.21"
        iyz="0.0012197"
        izz="0.65" />
    </inertial>
    <visual>
      <origin
        xyz="0.0 -0.3 0.0"
        rpy="0 0 0" />
      <geometry>
        <box size="0.02 0.6 1" />
      </geometry>
      <material
        name="">
        <color
          rgba="1 1 1 1" />
      </material>
    </visual>
  </link>
  <joint
    name="door_joint"
    type="revolute">
    <origin
      xyz="0.0 0.0 0.0"
      rpy="0.0 0.0 0.0" />
    <parent
      link="door_base_link" />
    <child
      link="door_link" />
    <axis
      xyz="0 0 1" />
    <limit
      lower="-3.12"
      upper="3.12"
      effort="0"
      velocity="0" />
  </joint>
</robot>

The robot is shown in MATLAB:

model = parse_urdf(urdf_model_path)
   
    state_dynamics = MechanismState{Float64}(model)
    q = [0.0]
    set_configuration!(state_dynamics, q)

    from_ = RigidBodyDynamics.findbody(model,"world")
    to_ = RigidBodyDynamics.findbody(model,"door_link")
    path_ = RigidBodyDynamics.path(model, from_, to_)
    Jac = RigidBodyDynamics.geometric_jacobian(state_dynamics, path_)
    @show Jac

The result of geometric_jacobian is:

Jac = GeometricJacobian: body: "after_door_joint", base: "world", expressed in "world":
[0.0; 0.0; 1.0; 0.6; -0.4; 0.0;;]

The result of MATLAB is [0.0; 0.0; 1.0; 0.0; 0.0; 0.0].

As I known, the rotation of last joint will not affect the velocity of the frame in the last joint, so the linear should be zeros (the results of MATLAB)?

Thank you.

[wly2014]

I may find where I am wrong. In RigidBodyDynamics.jl, the result of geometric_jacobian is relative to the twist at the root frame, expressed in the root frame. And the result of MATLAB is relative to the twist at the end-effector, expressed in the root frame.
So, if I transform the result of geometric_jacobian as follows and I will get the same results.

where er is the vector from the end-effector frame to the root frame, expressed in the root frame.