Degrees of Freedom (DOF)

In standard serial manipulators, each actuated joint contributes exactly one DOF. A 6-DOF articulated robot—the industry standard for complex tasks—typically features six revolute joints, enabling its end-effector to achieve any position (x, y, z) and orientation (roll, pitch, yaw) within its workspace.

• Revolute Joints: Provide rotational motion, the most common type.
• Prismatic Joints: Provide linear translation, often found in gantry systems.
• Task Space vs. Joint Space: The 6-DOF task space is mapped to the joint space via Inverse Kinematics.

A robot becomes kinematically redundant when its DOF exceeds the 6 parameters required to define an arbitrary end-effector pose. A 7-DOF arm can achieve a target pose using infinite joint configurations.

• Null-Space Motion: The extra DOF allows the arm to reconfigure its internal structure without moving the tool center point, enabling obstacle avoidance.
• Optimization Criteria: Redundancy is resolved by optimizing secondary objectives like manipulability or joint-limit avoidance.

The inverse concept of DOF is Degrees of Constraint (DOC). A rigid body in free space has 6 DOF; a fixture removes DOF by applying constraints.

• 3-2-1 Principle: A standard machining fixture uses 3 locators on the primary plane, 2 on the secondary, and 1 on the tertiary to constrain all 6 DOF.
• Assembly Planning: Successful robotic assembly requires precisely controlling the part's DOF during the insertion phase.

For Autonomous Mobile Robots (AMRs) and Automated Guided Vehicles (AGVs), DOF describes the platform's ability to move in a plane. A standard differential-drive robot has 3 DOF in its task space (x, y, yaw) but only 2 controllable DOF due to its nonholonomic constraint.

• Holonomic Drives: Mecanum or omni-wheeled robots can independently control all 3 planar DOF simultaneously.
• Path Planning Impact: Nonholonomic constraints require specialized planners like RRT with curvature continuity.

A kinematic singularity occurs when a robot loses one or more DOF in its task space, despite having sufficient joint DOF. At a singularity, the Jacobian matrix becomes rank-deficient.

• Wrist Singularity: Occurs when the axes of joints 4 and 6 align, causing infinite joint velocities for a finite Cartesian motion.
• Workspace Boundary: Singularities define the edge of the reachable workspace where motion in one direction is impossible.
• Damped Least Squares: A numerical method used in Inverse Kinematics solvers to stabilize motion near singularities.

Task and Motion Planning integrates high-level symbolic reasoning with the continuous DOF of manipulation. A TAMP solver must reason about which object grasps are feasible given the arm's kinematic constraints.

• Hybrid Planning: Combines discrete task selection with continuous Trajectory Optimization in the robot's Configuration Space (C-Space).
• Regrasping: Complex tasks often require intermediate placements to change an object's grasp, effectively using the environment to extend the system's effective DOF.