1.3 Degrees of freedom

Degrees of freedom (DOF) are crucial in robotics, defining how robots move and interact with their environment. This concept determines a robot's capabilities, from simple linear motions to complex spatial manipulations. Understanding DOF is key to designing effective robotic systems.

Calculating DOF involves considering the number of independent parameters needed to specify a robot's configuration. This impacts everything from joint design to control algorithms. Mastering DOF is essential for creating robots that can perform tasks efficiently and adapt to various scenarios.

Degrees of freedom overview

• Degrees of freedom (DOF) is a fundamental concept in robotics that describes the number of independent parameters required to completely specify the configuration of a robotic system
• Understanding DOF is crucial for designing, analyzing, and controlling robots, as it determines the robot's capabilities and limitations in terms of motion and interaction with the environment

Definition of DOF

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• DOF refers to the minimum number of independent parameters (variables) needed to uniquely define the position and orientation of a rigid body or a robotic system in space
• Each DOF corresponds to a single independent motion, such as translation along an axis or rotation about an axis
• The total DOF of a robotic system is the sum of the DOF of all its components, including links and joints

Importance in robotics

• DOF directly influences a robot's ability to perform tasks and adapt to different environments
• Robots with higher DOF are generally more versatile and capable of executing complex motions and manipulations
• However, increasing DOF also leads to higher complexity in terms of control, planning, and mechanical design
• Striking the right balance between DOF and simplicity is a key consideration in robot design, depending on the specific application requirements

Types of DOF

• DOF can be classified into two main categories: translational and rotational
• Translational DOF describe linear motion along axes (x, y, z), while rotational DOF describe angular motion about axes (roll, pitch, yaw)

Translational DOF

• Translational DOF refers to the ability of a robot or its components to move linearly along one or more axes
• Each translational DOF corresponds to a linear motion along a single axis (x, y, or z) in a Cartesian coordinate system
• Examples of translational DOF include a prismatic joint (sliding joint) or a linear actuator that allows a robot to move along a straight line

Rotational DOF

• Rotational DOF refers to the ability of a robot or its components to rotate about one or more axes
• Each rotational DOF corresponds to an angular motion about a single axis (roll, pitch, or yaw) in a Cartesian coordinate system
• Examples of rotational DOF include a revolute joint (hinge joint) or a servo motor that allows a robot to rotate about an axis
• Rotational DOF are essential for orienting a robot's end-effector (tool) or changing the direction of motion

Calculating DOF

• Determining the DOF of a robotic system is crucial for understanding its motion capabilities and designing appropriate control strategies
• The most common method for calculating DOF is using Gruebler's equation, which takes into account the number of links, joints, and constraints in the system

Gruebler's equation

• Gruebler's equation is a formula used to calculate the DOF of a mechanical system, including robots
• The equation is: $DOF = 6(n - 1) - 5j_1 - 4j_2 - 3j_3 - 2j_4 - j_5$
  • $n$: number of links (including the base)
  • $j_1, j_2, j_3, j_4, j_5$: number of joints with 1, 2, 3, 4, and 5 DOF, respectively
• The equation assumes that the system is a single-loop, planar mechanism with only lower-pair joints (joints with surface contact between links)

Examples of DOF calculation

• A planar 3R robotic arm (3 revolute joints) with 3 links:
  • $DOF = 6(3 - 1) - 5(3) - 4(0) - 3(0) - 2(0) - 0 = 3$
• A spatial 6R robotic arm (6 revolute joints) with 6 links:
  • $DOF = 6(6 - 1) - 5(6) - 4(0) - 3(0) - 2(0) - 0 = 6$
• A Stewart platform (parallel manipulator) with 6 prismatic joints and a base:
  • $DOF = 6(7 - 1) - 5(0) - 4(0) - 3(0) - 2(0) - 6 = 6$

DOF in robotic joints

• Robotic joints are the primary determinants of a robot's DOF, as they enable relative motion between connected links
• The most common types of robotic joints are prismatic, revolute, and spherical joints, each with different DOF

Prismatic joints

• Prismatic joints, also known as sliding or linear joints, allow translational motion along a single axis
• They have 1 DOF, which corresponds to the linear displacement between the connected links
• Examples of prismatic joints include hydraulic or pneumatic cylinders, lead screws, and linear bearings
• Prismatic joints are often used in Cartesian robots, gantry systems, and linear actuators

Revolute joints

• Revolute joints, also known as rotary or hinge joints, allow rotational motion about a single axis
• They have 1 DOF, which corresponds to the angular displacement between the connected links
• Examples of revolute joints include servo motors, stepper motors, and pin hinges
• Revolute joints are the most common type of joint in robotic manipulators, as they enable articulated motion and orientation control

Spherical joints

• Spherical joints, also known as ball-and-socket joints, allow rotational motion about three orthogonal axes
• They have 3 DOF, which correspond to the angular displacements about the axes (roll, pitch, and yaw)
• Examples of spherical joints include ball joints and universal joints (U-joints)
• Spherical joints are often used in parallel manipulators, robotic wrists, and compliant mechanisms to provide multiple rotational DOF

DOF vs mobility

• While DOF and mobility are related concepts, they have distinct meanings in the context of robotic systems
• DOF refers to the number of independent parameters needed to describe the configuration of a system, while mobility refers to the system's ability to move in space

Distinction between DOF and mobility

• DOF is a kinematic property that describes the number of independent motions a system can perform, regardless of the forces acting on it
• Mobility, on the other hand, takes into account the system's dynamics and the constraints imposed by the environment and the task
• A system with high DOF may have low mobility if it is subject to constraints that limit its motion (non-holonomic constraints)
• Conversely, a system with low DOF may have high mobility if it can effectively navigate its environment and complete its tasks

Mobility calculation

• Mobility is often calculated using the Kutzbach-Gruebler equation, which considers the system's DOF and the number of independent constraints
• The equation is: $M = DOF - C$
  • $M$: mobility
  • $DOF$: degrees of freedom (calculated using Gruebler's equation)
  • $C$: number of independent constraints
• Examples of constraints include joint limits, obstacle avoidance, and contact constraints (friction, adhesion)

DOF in robotic manipulators

• Robotic manipulators are a class of robots designed for manipulation tasks, such as grasping, positioning, and assembling objects
• The DOF of a robotic manipulator determines its workspace, dexterity, and ability to perform complex motions

Serial manipulators

• Serial manipulators, also known as open-chain manipulators, consist of a series of links connected by joints in a chain-like structure
• They typically have 6 DOF (3 translational and 3 rotational) to achieve full spatial positioning and orientation of the end-effector
• Examples of serial manipulators include industrial robotic arms (SCARA, articulated, Cartesian) and collaborative robots (cobots)
• Serial manipulators offer a large workspace and high dexterity but may suffer from lower stiffness and accuracy compared to parallel manipulators

Parallel manipulators

• Parallel manipulators, also known as closed-chain manipulators, consist of multiple kinematic chains connecting the base to the end-effector
• They typically have 6 DOF, with each kinematic chain providing one or more DOF
• Examples of parallel manipulators include Stewart platforms, Delta robots, and hexapods
• Parallel manipulators offer high stiffness, accuracy, and load-carrying capacity but have a more limited workspace compared to serial manipulators

Redundant manipulators

• Redundant manipulators have more DOF than necessary to perform a given task, typically more than 6 DOF
• The extra DOF allow for multiple joint configurations to achieve the same end-effector pose, enabling obstacle avoidance and optimization of secondary criteria (energy efficiency, singularity avoidance)
• Examples of redundant manipulators include the 7-DOF KUKA LBR iiwa and the NASA Robonaut 2
• Redundant manipulators offer increased flexibility and adaptability but require more complex control algorithms to manage the redundancy

DOF in mobile robots

• Mobile robots are designed to navigate and operate in various environments, such as on land, in the air, or underwater
• The DOF of a mobile robot determines its ability to move and maneuver in its environment

Wheeled robots

• Wheeled robots use wheels for locomotion and typically have 2-3 DOF (1-2 translational and 1 rotational)
• The most common configurations are differential drive (2 DOF) and omnidirectional drive (3 DOF)
• Examples of wheeled robots include AGVs (Automated Guided Vehicles), mobile manipulators, and planetary rovers
• Wheeled robots are efficient on flat surfaces but may struggle in uneven or cluttered environments

Legged robots

• Legged robots use articulated legs for locomotion and have multiple DOF per leg (3-6) to enable walking, running, and climbing
• The number of legs and their configuration determine the robot's stability, payload capacity, and terrain adaptability
• Examples of legged robots include humanoid robots (bipeds), quadrupeds, and hexapods
• Legged robots can navigate complex terrains but are more mechanically complex and computationally demanding than wheeled robots

Aerial and underwater robots

• Aerial robots, such as drones and UAVs (Unmanned Aerial Vehicles), typically have 6 DOF (3 translational and 3 rotational) to enable flight and maneuverability
• Underwater robots, such as AUVs (Autonomous Underwater Vehicles) and ROVs (Remotely Operated Vehicles), also have 6 DOF for navigation and manipulation in aquatic environments
• These robots require specialized propulsion systems (propellers, thrusters) and control algorithms to account for the dynamics of their respective environments

DOF constraints

• While a robot's DOF determines its potential motion capabilities, these motions may be limited by constraints imposed by the environment or the task
• Constraints can be classified as holonomic or non-holonomic, depending on their nature and impact on the robot's motion

Holonomic vs non-holonomic constraints

• Holonomic constraints are those that can be expressed as algebraic equations relating the robot's configuration variables (position, orientation) and time
• Examples of holonomic constraints include joint limits, obstacle avoidance, and position control
• Non-holonomic constraints cannot be expressed as algebraic equations and typically involve differential relationships between the robot's configuration variables and velocities
• Examples of non-holonomic constraints include rolling without slipping (wheeled robots) and conservation of angular momentum (aerial robots)

Impact on robot motion planning

• Constraints, whether holonomic or non-holonomic, must be considered in robot motion planning to ensure feasible and safe trajectories
• Holonomic constraints can be incorporated into the robot's configuration space representation and used as boundaries or obstacles in planning algorithms (e.g., sampling-based planners)
• Non-holonomic constraints require specialized planning algorithms that consider the differential constraints and generate feasible trajectories (e.g., RRT for non-holonomic systems)
• Dealing with constraints often involves trade-offs between motion efficiency, safety, and computational complexity

DOF and robot control

• The DOF of a robotic system directly impacts its control strategies and algorithms
• Key aspects of robot control that are influenced by DOF include inverse kinematics, Jacobian matrix, and singularity handling

Inverse kinematics

• Inverse kinematics (IK) is the process of determining the joint configurations required to achieve a desired end-effector pose
• For robots with high DOF (redundant manipulators), IK may have multiple solutions, requiring optimization or prioritization techniques to select the best configuration
• Examples of IK algorithms include analytical methods (closed-form solutions), numerical methods (iterative optimization), and learning-based methods (neural networks)

Jacobian matrix

• The Jacobian matrix is a linear mapping between the robot's joint velocities and the end-effector velocities
• It is a crucial tool for robot control, as it enables the computation of joint velocities required to achieve desired end-effector motions
• The Jacobian matrix is also used in singularity analysis, force control, and compliance control
• The dimensions of the Jacobian matrix depend on the robot's DOF and the task space dimensions

Singularities and redundancy resolution

• Singularities are configurations in which the robot loses one or more DOF, leading to a loss of controllability or dexterity
• They occur when the Jacobian matrix becomes rank-deficient, i.e., when certain rows or columns are linearly dependent
• Redundancy resolution techniques are used to handle singularities and optimize the robot's motion in the presence of multiple IK solutions
• Examples of redundancy resolution methods include the pseudoinverse Jacobian, nullspace projection, and task prioritization

DOF in robot design

• The selection of DOF is a critical aspect of robot design, as it determines the robot's capabilities, complexity, and cost
• Designers must consider the trade-offs between DOF and other factors, such as workspace, payload, speed, and accuracy

Determining required DOF

• The required DOF for a robotic system depends on the specific application and task requirements
• For manipulation tasks, 6 DOF (3 translational and 3 rotational) are generally sufficient for full spatial positioning and orientation
• For mobile robots, the required DOF depends on the environment and locomotion method (wheels, legs, propellers)
• Additional DOF may be necessary for redundancy, obstacle avoidance, or secondary tasks

Tradeoffs in DOF selection

• Increasing the DOF of a robotic system offers greater flexibility and adaptability but also increases the complexity and cost of the mechanical design, control system, and motion planning
• Higher DOF also requires more sensors, actuators, and computational resources, which may impact the robot's power consumption, weight, and reliability
• Designers must balance the benefits of higher DOF with the practical limitations and constraints of the application

DOF and robot complexity

• The complexity of a robotic system is directly related to its DOF, as each additional DOF introduces new challenges in terms of mechanical design, control, and planning
• Higher DOF systems require more sophisticated control algorithms, such as advanced inverse kinematics solvers, redundancy resolution schemes, and adaptive control techniques
• The increased complexity also impacts the robot's maintainability, scalability, and ease of use, which are important considerations in real-world applications
• Designers must strive to find the optimal balance between DOF and complexity, based on the specific requirements and constraints of the application