Inverse Kinematics of Articulated Manipulator: Step-by-Step Guide

Forward kinematics determines the end effector pose (position and orientation) given joint positions Q. The problem maps joint space (N dimensions) to task space (M dimensions) through X = F(Q). Key joint types include revolute joints (rotational, 1 DOF), universal joints (two orthogonal revolute joints, 2 DOF), ball joints (three revolute joints, 3 DOF), and prismatic joints (translational motion). For a 2D planar manipulator with two revolute joints, the forward kinematics equations are X = L1*cos(Q1) + L2*cos(Q1+Q2) and Y = L1*sin(Q1) + L2*sin(Q1+Q2), with orientation φ = Q1+Q2.

This section demonstrates how to compute the end effector's position in the fixed frame using forward kinematics. The mathematical relationship X_big = A₁A₂A₃X_small combines rotation matrices and displacement vectors to transform coordinates. By substituting the origin of the end effector frame (0,0) into this equation and solving with known robot parameters (link lengths a₁, a₂ and joint angles θ₁, θ₂, θ₃), one obtains the Cartesian coordinates [x; y] representing the end effector's position in the fixed frame. This process transforms joint-level parameters into spatial information.

Forward kinematics determines the endeffector position in a fixed coordinate frame based on joint rotations and link translations. The three-step setup involves: (1) defining a fixed global frame F attached to the ground, (2) establishing the endeffector frame M3, and (3) assigning moving frames to each joint with X-axes parallel to attached links. Positive rotation follows the counterclockwise convention. Position transformations use equations like y = R3*x + d3, where R represents rotation matrices and d represents translation vectors. For a two-link robot, translations occur only along the X-axis by link lengths L1 and L2. These individual transformations can be combined sequentially to express the final position in terms of joint angles and link parameters.

Forward kinematics determines the end effector's position and orientation from joint parameters. The process involves expressing the moving frame (end effector) relative to the fixed frame (robot base) using homogeneous transformations. By breaking down complex transformations into sequential steps through intermediate frames, we can compose individual transformations (H1, H2) to achieve the overall transformation H = H1 × H2. This systematic approach allows us to calculate the complete spatial relationship between the robot's base and its end effector.

Forward kinematics is the mathematical process of determining a robot's end-effector position and orientation as a function of its joint variables. The methodology involves assigning coordinate reference frames to each link and joint, starting from a fixed base frame. Each joint receives its own coordinate system, numbered sequentially from 1 to n (where n equals the number of joints). This systematic frame assignment enables complete spatial description of the manipulator by establishing relationships between all bodies relative to the fixed base. The ultimate goal is to derive a mathematical model that predicts the exact location and orientation of the tool at the manipulator's tip based on the angles or displacements of each individual joint.

Denavit-Hartenberg notation is a systematic method for defining robotic actuators through forward kinematics by establishing coordinate frames at each joint using four parameters (d, θ, a, α): d represents the distance from the previous origin along the previous Z-axis, θ is the rotation about the previous X-axis to align the current X-axis, a is the distance between origins along the current X-axis, and α is the angle from the previous Z-axis to the current Z-axis about the current X-axis; the coordinate frames are defined such that the Z-axis aligns with the joint axis of rotation or displacement, the X-axis is obtained by taking the cross product of the previous Z-axis and current Z-axis using the right-hand rule, and the Y-axis is perpendicular to both, with the origin located at the intersection of the common normal between consecutive Z-axes.

The Denavit-Hartenberg (DH) notation is a systematic method for analyzing robotic manipulators by comparing coordinate frames. Key concepts include: (1) Frames are Cartesian coordinate systems with x, y, z axes, denoted by letters like A or B; (2) Position refers to location while orientation refers to rotational alignment; (3) The method uses homogeneous transformation matrices to represent changes between frames; (4) The three-step process involves assigning frames to each link, defining four parameters per joint (α, θ, a, d), and multiplying transformation matrices to find the end-effector position relative to the base.

The Denavit-Hartenberg (DH) parameters consist of four variables: theta (θ), d, a, and alpha (α). A coordinate frame is 'normal' when all angles between consecutive axes are 90 degrees. The DH parameters describe transformations between adjacent coordinate frames: theta is rotation around Z, d is distance along Z to new X, a is distance along new X to new Z, and alpha is rotation around new X. Converting a 3x3 rotation matrix to 4x4 homogeneous matrix requires adding a fourth column with zeros except for the (4,4) element set to 1. The complete transformation from base to end-effector is obtained by multiplying all individual joint transformation matrices in sequence.

The Denavit-Hartenberg method for defining coordinate frames at each joint of a robotic manipulator follows four essential rules: (1) The Z-axis must align with the joint axis (rotation axis for revolute joints, motion direction for prismatic joints); (2) The X-axis must be perpendicular to both the current Z-axis (ZN) and the previous Z-axis (ZN-1), with the exception that when no unique perpendicular direction exists (e.g., when ZN and ZN-1 are parallel), the X-axis is defined as the direction from ZN-1 to ZN; (3) The Y-axis follows the right-hand rule; (4) The X-axis must intersect the previous Z-axis (ZN-1), which may require repositioning the frame center when joint offsets prevent natural intersection.

The articulated/revolute (RRR) robot has three rotational joints, making it the most versatile type. It is anthropomorphic and can contour around obstacles with high operational speed. However, it requires complex programming, has a large working volume (spherical shape), and cannot reach points immediately above the manipulator. The working volume is determined by arm and forearm link lengths.

A robot manipulator is a kinematic system consisting of rigid bodies (links) and joints that enable controlled movement for manipulating objects. The four main components are: (1) Base - the stationary support structure, (2) Joints - connections that enable movement between links, (3) Links - rigid arms that are moved by joints, and (4) End Effectors - devices that perform tasks such as grasping or machining. Joints are classified into two types: prismatic joints (linear movement) and revolute joints (rotational movement). Robot manipulators require 6 degrees of freedom (3 for position in X, Y, Z axes and 3 for orientation) to properly position and orient the end effector in 3D space. The five most common configurations are: (1) Cylindrical (RPR) - one revolute joint for rotation and two prismatic joints for linear movement, (2) Spherical (RPR) - similar to cylindrical but with spherical coordinate system for omnidirectional reach, (3) SCARA (RPR) - selective compliance assembly robot arm with parallel Z-axes for horizontal precision, (4) Cartesian (PPP) - three prismatic joints for direct X, Y, Z movement with the simplest kinematic description, and (5) Revolute (RRR) - three revolute joints for complex rotational reach.

Jointed arm robots (RRR configuration) use three rotary joints for positioning. First two joints control planar motion within a plane, third joint provides out-of-plane motion. This allows wrist positioning anywhere in 3D space. Most industrial robots use this architecture because it mimics human arm movement and provides good workspace-to-size ratio. Workspace shape is complex and depends on joint limits, typically resembling a toroidal region with accessible areas in the forward region optimized for practical tasks.

This video demonstrates how to design a 3-degree-of-freedom articulated RRR (Revolute-Revolute-Revolute) manipulator in MATLAB's SimScape Multibody environment. The process involves creating a base, three serially connected links with revolute joints, and an end effector, using rigid transforms to position and orient each component. The model is then converted to a RigidBodyTree object in the Robotics System Toolbox, where joint angles are provided through Simulink-to-PS converters and the forward kinematics block calculates the end effector position in 3D space. The key principle is that non-parallel z-axes of consecutive joints enable 3D manipulation capabilities.

The Articulated Robot Configuration consists of three revolute joints and one prismatic joint. When the arm rotates about all three major axes, this results in the articulated configuration, which is also called the joint arm configuration. The arm resembles a human arm with two rotary joints (shoulder and elbow) about their respective horizontal axes, and a third rotary joint about the vertical base axis. Applications include manufacturing of steel bridges, cutting steel, flat glass handling, heavy duty robot automation in foundry industries, heat resistance robots, metal casting, and spot welding. Advantages include high speed, large working envelope, and great unique control over welding and painting applications. The disadvantage is that it requires a dedicated robot controller like PLC.