Robotic Systems: Joints, Links, and Degrees of Freedom Explained

The Denavit-Hartenberg (DH) representation is a systematic method for describing the position and orientation of robotic joints using four parameters: a_i-1 (link length, distance between joint axes along the common perpendicular), alpha_i-1 (link twist, angle between joint axes about the common perpendicular), d_i (link offset, distance between origins along the previous joint axis), and theta_i (joint angle, rotation about the current joint axis). For prismatic joints, d_i is non-zero while theta_i is zero; for revolute joints, theta_i is non-zero while d_i is zero. These four parameters are organized into a DH table, which enables the calculation of transformation matrices that describe the spatial relationship between consecutive frames in a robotic manipulator.

This comprehensive section covers the Denavit-Hartenberg (DH) convention for modeling serial robots. The robot Stainsforts has 6 degrees of freedom: two revolute joints, one prismatic joint, and three spherical joints with concurrent axes. The DH convention provides a systematic method for placing reference frames on robot links using four parameters: a (link length), α (link twist), d (link offset), and θ (joint angle). The Z axes are aligned with joint axes, while X axes are defined as common perpendiculars between consecutive Z axes. For concurrent axes (spherical joints), the same Z axis is used for multiple joints. The complete parameter calculation demonstrates how to determine a, α, d, and θ for each joint by isolating consecutive frames and measuring geometric relationships. This systematic approach enables consistent kinematic modeling of serial robots.

The Denavit-Hartenberg (DH) convention is a standardized set of rules adopted by the robotics community for modeling the direct kinematics of serial open-chain robots. It enables associating a reference frame to each robot link regardless of its geometry, applicable to revolute, prismatic, cylindrical, and spherical joints. The convention uses four parameters (theta, d, a, alpha) per link to define transformations between consecutive reference frames, where theta and alpha represent rotation angles and d and a represent displacements. The base frame's z-axis aligns with the first joint's rotation direction, while subsequent z-axes align with each joint's axis. Origins are placed at the intersection of the normal common between consecutive z-axes and the respective z-axis. The final transformation of the end-effector relative to the base is obtained by post-multiplying the individual link transformations, providing a unique solution for direct kinematics when joint dimensions and rotation directions are known.

The Denavit-Hartenberg (DH) convention is a standard method for selecting reference frames in robotic applications. Each homogeneous transformation is represented as a product of four basic transformations. The four parameters are: link length (a), link twist (alpha), link offset (d), and joint angle (theta). Three of these parameters are constant for a given link, while the fourth is the variable joint parameter.

Robot dynamics studies the relationship between generalized forces (torques and forces) and robot motion, governed by second-order differential equations similar to Newton's second law, where the dynamic model consists of the mass matrix, centrifugal and Coriolis forces, and gravity forces; the two fundamental problems are direct dynamics (calculating motion from applied torques for simulation) and inverse dynamics (calculating required torques for desired motion), with the Euler-Lagrange method using energy analysis and the Newton-Euler method using recursive force/moment propagation.

Robot dynamics involves modeling the relationship between a robot's motion and the forces/torques that cause it, using either Newton-Euler formulation (force-based balance of forces and torques) or Lagrange formulation (energy-based approach using kinetic and potential energies); unlike kinematic models which only describe motion, dynamic models incorporate inertia, centripetal/coriolis forces, gravity, and friction to enable accurate simulation and controller design for robotic systems.

Robot dynamics is governed by Newton's second law (F=ma), where applied force is consumed in three ways: accelerating links, overcoming Coriolis/centrifugal forces, and countering gravity. This creates n coupled nonlinear second-order differential equations for n-jointed robots. Inverse dynamics converts desired trajectories to torque commands for position control, while forward dynamics predicts trajectories from torques for torque-controlled applications. These concepts form the mathematical foundation for robotic motion control systems.

Robot dynamics is the study of the relationship between forces and torques applied to robot joints and the resulting motion. Unlike kinematics, which only considers geometric relationships between joint positions and end effector position, dynamics addresses why motion occurs by considering the forces and torques required to produce that motion. The motion variables include position (q), velocity (q_dot), and acceleration (q_ddot) of all joints. The robot dynamics equation describes the relationship between forces and torques applied to the robot's joints (M) and the motion variables. Two primary methods exist for deriving robot dynamics equations: Newton's method (based on force and torque balance on individual bodies) and Lagrange's method (based on energy balance of the entire system). The total energy of a robot system consists of kinetic energy (K) and potential energy (P), where kinetic energy is related to velocity and potential energy is related to position.

Robot dynamics: (1) Studies the forces and torques that cause robot motion, (2) Considers both internal and external forces, (3) Relates joint torques to end-effector motion. It is essential for robot control, especially for dynamic tasks.

This video explains how to plan collision-free paths for robotic manipulators using MATLAB/Simulink, covering RRT and RRT* algorithms for path planning, collision detection techniques including convex decomposition and occupancy grids, and trajectory generation methods such as trapezoidal velocity profiles and polynomial interpolation for smooth motion control.

Path planning subdivides continuous workspace paths into discrete segments for sequential execution. Trajectory generation adds temporal information, creating acceleration-deceleration profiles. Tasks defined in workspace coordinates must be transformed into configuration space for robot execution, with obstacles similarly mapped. Two vision paradigms exist: camera-on-top provides consistent views but risks occlusion, while camera-in-hand eliminates occlusion but introduces perspective distortion. These transformations enable robots to understand and execute workspace-level tasks using joint-level controls.

Robot trajectory planning involves determining the time-stamped path (trajectory) that a robot manipulator's end effector follows, distinguishing it from a simple geometric path (which lacks timing information). The planning process considers whether to work in joint space (where motion is defined by joint angles and velocities) or operational/Cartesian space (where motion is defined by end-effector position and orientation in 3D space). Joint space planning is computationally efficient but produces unpredictable Cartesian paths due to non-linear forward kinematics, while operational space planning enables visualization and collision avoidance but requires solving inverse kinematics for many points. Common trajectory planning algorithms include uniform velocity (constant speed with sudden accelerations), triangular velocity (accelerate then decelerate symmetrically), trapezoidal velocity (with a middle constant-velocity phase for smooth operations), and polynomial trajectories (cubic polynomials for basic smooth motion, fifth-order polynomials for complete control of position, velocity, and acceleration boundaries).

This lecture provides a comprehensive overview of robot path planning methods. The potential field method uses attractive and repulsive potentials to guide robots toward goals while avoiding obstacles. Two candidate functions exist: conic well potential (discontinuous at goal) and parabolic well potential (unbounded away from goal), with hybrid approaches combining both. Forces are derived as negative gradients of potentials and transformed to joint torques using the Jacobian relationship τ = J_v^T × F. For a two-link manipulator, this yields specific torque values. Advanced algorithms include gradient descent, which iteratively computes next configurations but risks local minima entrapment—addressed through random walk perturbations. The D* algorithm extends A* for minimum-cost path finding in grid-based environments. The probabilistic roadmaps method generates random nodes, connects them using neighborhood functions, refines connections, and queries paths from start to finish. These methods provide systematic approaches for robotic manipulators to navigate complex workspaces while avoiding obstacles.

Robot trajectory planning involves creating time-based paths for robot motion through four stages: task planning, path planning, trajectory generation, and trajectory tracking. The process can be performed in workspace (Cartesian coordinates) or joint space (joint angles), with workspace planning offering better obstacle avoidance but requiring inverse kinematics calculations, while joint space planning is computationally faster. Common trajectory generation methods include trapezoidal velocity profiles, cubic and quintic polynomials, and Bezier curves, each providing different trade-offs between smoothness, computational complexity, and adherence to motion constraints.

This section demonstrates programming multiple axes simultaneously on a Cartesian robot. The instructor shows how to configure three axes (rotational, vertical, horizontal) with identical block structures: MC Power for activation, MC Home for origin definition, and MC Absolute for target positioning. Through live camera feeds and HMI interfaces, the instructor demonstrates real-time robot operation—assigning home positions, entering absolute coordinates, and observing physical robot movements. The instructor explains how completion signals enable sequential operation and how different axes may require different velocity settings based on mechanical characteristics like gear ratios. This practical demonstration bridges theory and application in industrial automation.

Industrial robots are classified by their degrees of freedom, which represent axes of movement including position axes (waist, shoulder, elbow) for locating tools in the work area and orientation axes (pitch, roll, yaw) for tool alignment; online programming methods include teach pendant programming (point-by-point recording with coordinate adjustment) and lead-through programming (physical guidance for continuous path applications), both requiring the technician to be in the work envelope during operation.

Multi-axis robot arms can be effectively controlled through puppeteering rigs that translate human body movements into robotic motion. The rigs typically provide control over shoulder rotation, elbow bending, wrist rotation, and sometimes waist movement. Each human movement axis maps directly to a corresponding robot joint, creating an intuitive control experience. This approach allows operators to perform complex tasks like picking up objects and placing them in specific locations without requiring extensive training in robotics programming or control systems.

A multi-axis robot arm is a robotic device controlled by a microcontroller that uses multiple servomotors (typically 5-6 axes) to achieve complex movements, where each axis corresponds to a joint that enables different types of motion (base rotation, arm extension, wrist rotation, and gripper control); the programming involves sending angle commands to each motor through serial communication, with smooth movement requiring gradual position changes rather than abrupt movements to prevent mechanical stress and ensure fluid operation.

This video demonstrates practical industrial robot programming, showing how to control a robot's multiple axes (two-axis movement and three-axis rotation) and establish communication between robot components for automated tasks.