Degrees of Freedom (DOF) and Mobility in Robotics

Definition and Fundamental Concepts of Degrees of Freedom (DOF)

• Degree of Freedom (DOF): This is defined as the total number of independent movements required to completely and accurately describe the position and orientation of a robot in space.

• Conceptual Example: The Door: A standard door serves as a basic example of DOF. It is constrained to rotate about its hinge, providing it with $1$ DOF.

• Conceptual Example: Robotic Arm: A robotic arm can possess multiple independent motions, such as rotation at the base, movement of the shoulder, movement of the elbow, and rotation of the wrist. Each of these independent motions adds to the total DOF of the robot.

• The Accumulation of DOF: In a simple robotic arm, the total DOF is the sum of the movements allowed by each joint:

  • Joint 1: Base rotation provides $1$ DOF.

  • Joint 2: Shoulder rotation provides $1$ DOF.

  • Joint 3: Elbow rotation provides $1$ DOF.

  • Total DOF: $DOF = 1 + 1 + 1 = 3$ DOF. This indicates the robot has three independent movements.

The Importance of DOF in Robotics

• Understanding Capabilities: DOF helps determine how many independent motions a robot is capable of performing.

• Task Specification: It informs the designer on how many joints are required to execute a specific task.

• Complexity Assessment: The number of DOF is a direct indicator of the complexity of the robotic system.

• Control of the End-Effector: DOF determines the robot's ability to precisely position and orient its end-effector within its workspace.

• Workspace and Motion: It defines the robot's reach and kinematic capabilities.

• Task Complexity Examples:

  • Pick-and-place: Simple tasks may require fewer DOF.

  • Welding and Assembly: These require a higher number of DOF to control both the position and the orientation of the tool or component.

Basic Motions of a Rigid Body

• A rigid body can undergo two fundamental types of motion:

• Translation: This involves movement from one location to another along primary axes:

  • Along the X-axis (Surge/Forward-Backward).

  • Along the Y-axis (Sway/Left-Right).

  • Along the Z-axis (Heave/Up-Down).

• Rotation: This involving turning around a specific axis:

  • Rotation about the X-axis (Roll).

  • Rotation about the Y-axis (Pitch).

  • Rotation about the Z-axis (Yaw).

Six Degrees of Freedom (6-DOF) in 3D Space

• A free rigid body moving in three-dimensional (3D) space has a maximum of $6$ DOF.

• Composition of 6-DOF:

  • $3$ Translational DOF: X direction, Y direction, and Z direction.

  • $3$ Rotational DOF: Roll, Pitch, and Yaw.

  • Memory Trick: $3 \text{ Translation} + 3 \text{ Rotation} = 6 \text{ DOF}$.

Robotic Joints and Their Degrees of Freedom

• Joint Definition: A joint is a connection that joins two robot links and allows relative motion between them.

• Common Joint Types:

• Revolute Joint (R):

  • Imposes a rotational motion.

  • Symbol: R.

  • Examples: Robot elbow, robot shoulder, door hinge.

  • DOF: $1$ DOF.

• Prismatic Joint (P):

  • Imposes a translational (linear) motion.

  • Symbol: P.

  • Examples: Sliding robot arm, linear actuator, elevator mechanism.

  • DOF: $1$ DOF.

• Cylindrical Joint (C):

  • Allows both rotation and translation.

  • DOF: $2$ DOF.

• Spherical Joint (S):

  • Allows three different rotations.

  • DOF: $3$ DOF.

Joint DOF vs. Robot DOF

• Joint DOF: Refers specifically to the number of independent movements provided by a single joint (e.g., a Revolute joint provides $1$ DOF).

• Robot DOF: Refers to the total number of independent movements available to the entire robotic system.

• Summation Example: In a robot with three revolute joints, the total DOF is calculated as $1 + 1 + 1 = 3$ DOF.

Mobility in Robotics

• Mobility Definition: Mobility is the number of independent joint variables or movements that a mechanism can make.

• Mobility for Simple Robots: In simple serial manipulators, Mobility is approximately equal to the number of independent joint variables.

• Complexity Variance: While Mobility and DOF are often used interchangeably for simple serial manipulators, complex mechanisms involving constraints or closed loops require more careful Mobility analysis.

Examples of Robotic Configurations and DOF

• 2-DOF Planar Robot:

  • Consists of Link 1 and Link 2 connected by two revolute joints.

  • Joint 1 (Revolute) + Joint 2 (Revolute) = $2$ DOF.

  • The robot controls two independent angles: $\theta_1$ and $\theta_2$.

• 3-DOF Robot:

  • Uses three revolute joints: Joint 1 ($1$ DOF), Joint 2 ($1$ DOF), Joint 3 ($1$ DOF).

  • Total DOF: $DOF = 1 + 1 + 1 = 3$.

  • Independent variables: $\theta_1, \theta_2, \theta_3$.

• 6-DOF Industrial Robot (Articulated):

  • Consists of 6 axes: 1st axis (Base), 2nd axis, 3rd axis, 4th axis, 5th axis, and 6th axis.

  • Position Control (3 DOF): Controlled by base rotation, elbow rotation, etc. ($X, Y, Z$).

  • Orientation Control (3 DOF): Controlled by wrist pitch, wrist rotation, etc. ($\text{Roll, Pitch, Yaw}$).

  • Total: $3 \text{ Position DOF} + 3 \text{ Orientation DOF} = 6 \text{ DOF}$.

  • This allows the end-effector to be positioned and oriented freely in 3D space.

• SCARA Robot:

  • Typically has $4$ DOF.

  • Configuration: Base rotation ($1$), Arm rotation ($1$), Vertical movement ($1$), and Wrist rotation ($1$).

  • Applications: Pick and place, assembly, packaging.

• Cartesian Robot:

  • Typically has $3$ DOF.

  • Configuration: X-axis movement, Y-axis movement, and Z-axis movement.

  • Motion: $X + Y + Z = 3$ translational DOF.

  • Applications: Machining, handling.

• Cylindrical Robot:

  • Typically has $3$ DOF.

  • Configuration: Base rotation ($1$), Vertical translation ($1$), and Radial translation ($1$).

  • Operates within a cylindrical workspace.

• Spherical/Polar Robot:

  • Typically has $3$ DOF.

  • Configuration: Base rotation ($1$), Arm rotation ($1$), and Radial extension ($1$).

  • Workspace is approximately spherical.

Constraints and Their Impact on Mobility

• Constraint Definition: A constraint is a limitation placed on the movement of a mechanism.

• Mathematical Effect of Constraints: Every constraint reduces the available DOF of a system.

• Constraint Example: A free object in 3D space has $6$ DOF. Placing that object on a flat surface restricts motion in at least one direction and may restrict certain rotations, thereby reducing its total DOF.

• The Wheel Example: A wheel free in space has several possible motions. However, once attached to an axle, the axle restricts movements, resulting in fewer independent movements (reduced mobility).

Open-Chain vs. Closed-Chain Mechanisms

• Open-Chain Robot (Serial): The links form a single chain (e.g., Link 1 → Joint → Link 2 → Joint → Link 3). Determining DOF is usually straightforward.

• Closed-Chain Mechanism: The links form one or more loops (e.g., a parallel robot or a four-bar link mechanism). These loops introduce extra constraints, making mobility calculations more complex.

Mobility Calculation: The Kutzbach/Gr61bler Equation

• For a planar mechanism, the mobility $M$ is determined by the number of links, joints, and the DOF provided by those joints.

• Equation:     $M = 3(n - 1) - 2j_1 - j_2$

• Variables:

  • $M$ = Mobility.

  • $n$ = Number of links (including the ground link).

  • $j_1$ = Number of joints with $1$ DOF (e.g., revolute or prismatic).

  • $j_2$ = Number of joints with $2$ DOF.

• Example Calculation: Consider a planar mechanism with $n = 3$ links, $j_1 = 2$ revolute joints, and $j_2 = 0$.

  • Step 1: Substitute values into the formula: $M = 3(3 - 1) - 2(2) - 0$

  • Step 2: Solve: $M = 3(2) - 4 = 6 - 4 = 2$

  • Result: The mechanism has $2$ independent DOF.

Redundancy and Humanoid Comparison

• Redundant Robots: A robot is categorized as redundant when it possesses more DOF than the minimum required to complete a specific task.

• Humanoid Arm: Typically possesses $6$ to $7$ DOF to allow for human-like manipulation.

• Position vs. Orientation in Tasks:

  • Simple Task: Picking an object and moving it from Point A to Point B may only require position control ($3$ DOF).

  • Complex Task: Picking an object, rotating it, and placing it with precision requires both position and orientation control ($6$ DOF).