Section 1: Links and Joints

Estimated Reading Time: 8 minutes Bloom's Level: Understand, Apply Key Learning: URDF structure, kinematic relationships

What Are Links and Joints?

Every robot is made of two fundamental building blocks: links and joints.

Link = Rigid Body

A link is a rigid body—a solid part of the robot that doesn't bend or deform.

Examples:

Robot arm segment (metal rod or pipe)
Robot torso (main body)
Wheel (can spin but doesn't change shape)
Gripper finger (solid piece)

In URDF: A link has:

Mass (weight in kg)
Inertia (resistance to spinning)
Visual geometry (how it looks in RViz)
Collision geometry (shape for physics)

Joint = Connection Between Links

A joint is a connection between two links that allows relative motion.

Examples:

Revolute joint: Robot arm elbow (rotates around an axis)
Prismatic joint: Robot gripper (slides/extends)
Fixed joint: Mounted camera (no motion)
Continuous joint: Robot wheel (rotates freely)

In URDF: A joint has:

Type (revolute, prismatic, fixed, continuous)
Parent link (fixed link)
Child link (moving link)
Axis (direction of motion)
Limits (minimum and maximum angles/distances)

The Kinematic Tree

A robot's structure is described as a kinematic tree: each link has parents and children connected by joints.

Example: Simple 2-Link Robot

        Joint 1 (Revolute)
           ↙↖ (Z-axis rotation)

      base_link
           ↓
      (mass=2.0 kg)
           ↓
      link_1
           ↓
      Joint 2 (Revolute)
      (Y-axis rotation)
           ↓
      link_2
           ↓
  end_effector (fixed)

Read as:

base_link is the root (doesn't move)
link_1 connects to base_link via joint_1
link_2 connects to link_1 via joint_2
end_effector is a fixed reference frame on link_2

URDF Syntax Basics

URDF files are XML (eXtensible Markup Language). If you've seen HTML, XML is similar:

<tag attribute="value">
  Content here
</tag>

URDF Structure

<?xml version="1.0"?>
<robot name="my_robot">

  <!-- Define links -->
  <link name="base_link">
    <!-- link properties -->
  </link>

  <link name="link_1">
    <!-- link properties -->
  </link>

  <!-- Define joints (connections) -->
  <joint name="joint_1" type="revolute">
    <parent link="base_link"/>
    <child link="link_1"/>
    <!-- joint properties -->
  </joint>

</robot>

Key XML Rules

Every <tag> must have a closing </tag>
Attributes use quotes: name="value"
Nesting must be proper (no crossing)
XML is case-sensitive: <link> ≠ <Link>

Code Example 1: Links in URDF

Let's look at a link from simple_robot.urdf:

<link name="base_link">
  <inertial>
    <!-- Mass of this link (kg) -->
    <mass value="2.0"/>

    <!-- Center of mass location -->
    <origin xyz="0 0 0.05" rpy="0 0 0"/>

    <!-- Inertia matrix (how hard to spin) -->
    <inertia
      ixx="0.0167" ixy="0.0" ixz="0.0"
      iyy="0.0167" iyz="0.0"
      izz="0.0167"/>
  </inertial>

  <!-- Visual appearance (how RViz shows it) -->
  <visual>
    <origin xyz="0 0 0.05" rpy="0 0 0"/>
    <geometry>
      <!-- Box: width=0.2m, depth=0.2m, height=0.1m -->
      <box size="0.2 0.2 0.1"/>
    </geometry>
    <material name="gray">
      <color rgba="0.5 0.5 0.5 1.0"/>
    </material>
  </visual>

  <!-- Collision shape (for physics simulation) -->
  <collision>
    <origin xyz="0 0 0.05" rpy="0 0 0"/>
    <geometry>
      <box size="0.2 0.2 0.1"/>
    </geometry>
  </collision>
</link>

What this describes:

A box-shaped link named base_link
Mass: 2.0 kg
Dimensions: 0.2m × 0.2m × 0.1m (width × depth × height)
Color: Gray (in RViz)
Can collide in simulation (Gazebo)

Code Example 2: Joints in URDF

Now let's see how joints connect links:

<joint name="joint_1" type="revolute">
  <!-- Parent link (doesn't move with this joint) -->
  <parent link="base_link"/>

  <!-- Child link (moves with this joint) -->
  <child link="link_1"/>

  <!-- Position of joint relative to parent -->
  <origin xyz="0 0 0.1" rpy="0 0 0"/>

  <!-- Axis of rotation: X=1,0,0 (rotate around X-axis) -->
  <axis xyz="0 0 1"/>

  <!-- Joint limits: min angle, max angle, effort, velocity -->
  <limit lower="-3.14159" upper="3.14159" effort="10" velocity="1.0"/>

  <!-- Friction and damping -->
  <dynamics damping="0.1" friction="0.1"/>
</joint>

What this describes:

A revolute (rotational) joint named joint_1
Connects base_link (parent) to link_1 (child)
Rotates around the Z-axis (xyz="0 0 1")
Can rotate from -180° to +180° (radians)
Max force: 10 Nm (torque)
Max speed: 1.0 rad/s

Joint Types

Revolute Joint

Rotates around an axis with limits (like a hinge).

<joint name="elbow" type="revolute">
  <axis xyz="1 0 0"/>  <!-- Rotate around X-axis -->
  <limit lower="0" upper="2.0" effort="10" velocity="1.5"/>
</joint>

Real examples:

Robot arm elbow
Robot neck (yaw rotation)
Door hinge

Continuous Joint

Rotates freely (no angle limits, like a wheel).

<joint name="wheel" type="continuous">
  <axis xyz="0 0 1"/>  <!-- Spin around Z-axis -->
</joint>

Real examples:

Robot wheel
Robot joint that spins 360°
Turntable

Prismatic Joint

Slides (translates) along an axis with limits (like a drawer).

<joint name="elevator" type="prismatic">
  <axis xyz="0 0 1"/>  <!-- Move up/down (Z-axis) -->
  <limit lower="0" upper="1.0" effort="100" velocity="0.5"/>
</joint>

Real examples:

Robot gripper that extends
Elevator lift
Sliding drawer

Fixed Joint

No motion allowed (rigid connection).

<joint name="mount" type="fixed">
  <parent link="arm"/>
  <child link="camera"/>
</joint>

Real examples:

Camera mounted on robot
Gripper tool mounted on arm
Sensor bolted to frame

Coordinate Frames: XYZ and RPY

Every link and joint has an origin—a position and orientation in 3D space.

XYZ: Position

X: Left (-) / Right (+) [red axis]
Y: Back (-) / Forward (+) [green axis]
Z: Down (-) / Up (+) [blue axis]

Origin xyz="0.5 0.2 0.1"
  means: 0.5m right, 0.2m forward, 0.1m up

RPY: Rotation

R (Roll): Rotate around X-axis (like rolling a ball forward)
P (Pitch): Rotate around Y-axis (like nodding your head)
Y (Yaw): Rotate around Z-axis (like turning left/right)

All angles in radians:

π (pi) ≈ 3.14159 = 180°
π/2 ≈ 1.5708 = 90°

Origin rpy="0 1.5708 0"
  means: 0° roll, 90° pitch (nodding), 0° yaw

Full Origin Example

<origin xyz="0.5 0 0" rpy="0 0 0"/>
  Position: 0.5m to the right, no height change
  Rotation: No rotation (aligned with parent frame)

<origin xyz="0 0 0.1" rpy="0 1.5708 0"/>
  Position: 0.1m upward
  Rotation: 90° pitch (tilted forward)

Mental Model: Kinematic Chain

Think of a robot like a chain:

Each link is a link of the chain
Each joint is a connection point
Motion in one joint affects all children

Head (can look left/right)
  ↓ (neck joint)
Torso (main body)
  ↙       ↘
L-Shoulder  R-Shoulder
  ↓          ↓
L-Arm      R-Arm
  ↓          ↓
L-Hand     R-Hand

Rule: When you move a parent, all children move too.

Rotate the torso → head and arms rotate with it
Rotate the shoulder → arm and hand move
Rotate the wrist → only the hand moves

This is called forward kinematics: parent motion affects children.

Common Mistakes

❌ Mistake 1: Circular Kinematic Tree

<joint name="joint_1">
  <parent link="A"/>
  <child link="B"/>
</joint>

<joint name="joint_2">
  <parent link="B"/>
  <child link="A"/>  <!-- ERROR: A is now child of B AND parent! -->
</joint>

Error: Circular reference. Links should form a tree, not a loop.

Fix: Keep parent-child relationships strict. One parent per link.

❌ Mistake 2: Missing Links

<joint name="joint_1">
  <parent link="base_link"/>
  <child link="link_1"/>  <!-- OK -->
</joint>

<joint name="joint_2">
  <parent link="link_2"/>  <!-- ERROR: link_2 was never defined! -->
  <child link="link_3"/>
</joint>

Error: link_2 doesn't exist. Every parent and child in a joint must be defined.

Fix: Use check_urdf simple_robot.urdf to catch these errors:

ERROR: link [link_2] not found

❌ Mistake 3: Wrong Axis Direction

<joint name="elbow" type="revolute">
  <axis xyz="0 0 1"/>  <!-- Z-axis (up/down rotation) -->
  <!-- But this is supposed to be left/right rotation! -->
</joint>

Error: Robot rotates the wrong way.

Fix: Think about which direction the joint should rotate:

Left/Right (yaw): Use Z-axis xyz="0 0 1"
Up/Down (pitch): Use Y-axis xyz="0 1 0"
Forward/Back (roll): Use X-axis xyz="1 0 0"

Try It Yourself

Open simple_robot.urdf and:

Find all <link> definitions. How many are there?
Find all <joint> definitions. What types are they?
Trace the kinematic tree: base_link → link_1 → link_2 → end_effector
Change a link's color in the visual material
Change a joint's rotation axis (then validate with check_urdf)

Key Takeaways

✅ Links are rigid bodies with mass and geometry ✅ Joints connect links and define how they move ✅ Kinematic trees show parent-child relationships ✅ XYZ describes position; RPY describes rotation ✅ Use check_urdf to validate URDF syntax ✅ Every parent and child link must be defined ✅ Circular references cause errors

Next: Gazebo Properties

Now that you understand URDF structure, let's add physical properties for simulation!

👉 Next: Section 2: Gazebo Properties

Or review:

Section Status: Complete ✅ Skills Used: Explanation, diagrams, mental models

What Are Links and Joints?​

Link = Rigid Body​

Joint = Connection Between Links​

The Kinematic Tree​

Example: Simple 2-Link Robot​

URDF Syntax Basics​

URDF Structure​

Key XML Rules​

Code Example 1: Links in URDF​

Code Example 2: Joints in URDF​

Joint Types​

Revolute Joint​

Continuous Joint​

Prismatic Joint​

Fixed Joint​

Coordinate Frames: XYZ and RPY​

XYZ: Position​

RPY: Rotation​

Full Origin Example​

Mental Model: Kinematic Chain​

Common Mistakes​

❌ Mistake 1: Circular Kinematic Tree​

❌ Mistake 2: Missing Links​

❌ Mistake 3: Wrong Axis Direction​

Try It Yourself​

Key Takeaways​

Next: Gazebo Properties​

What Are Links and Joints?

Link = Rigid Body

Joint = Connection Between Links

The Kinematic Tree

Example: Simple 2-Link Robot

URDF Syntax Basics

URDF Structure

Key XML Rules

Code Example 1: Links in URDF

Code Example 2: Joints in URDF

Joint Types

Revolute Joint

Continuous Joint

Prismatic Joint

Fixed Joint

Coordinate Frames: XYZ and RPY

XYZ: Position

RPY: Rotation

Full Origin Example

Mental Model: Kinematic Chain

Common Mistakes

❌ Mistake 1: Circular Kinematic Tree

❌ Mistake 2: Missing Links

❌ Mistake 3: Wrong Axis Direction

Try It Yourself

Key Takeaways

Next: Gazebo Properties