Forward Kinematics with Trigonometry | Robotics Basics

Homogeneous transformation matrices are 4x4 matrices that combine position and orientation to represent rigid-body configurations. They consist of a 3x3 rotation matrix R and a 3x1 position vector p, with a bottom row of [0 0 0 1] for simplifying calculations. The set of all such matrices forms the special Euclidean group SE(3), which shares properties with rotation matrices: every matrix has an inverse, products yield valid transformations, and multiplication is associative but not commutative. Three primary applications exist: representing configurations, changing frame references using subscript cancellation rules, and displacing points or frames. Points require homogeneous coordinates (appending 1 to 3-vectors) for proper transformation. Any configuration results from first rotating then translating, making T both a configuration state and a transformation operator.

Three-dimensional homogeneous transformation matrices are 4×4 matrices combining 3×3 rotation matrices and 3×1 translation vectors. The matrix structure is [R, t; 0, 0, 0, 1], where R is the rotation matrix and t is the translation vector. Points are represented in homogeneous coordinates as 4×1 vectors with a 1 appended. Composition follows matrix multiplication: T^a_c = T^a_b × T^b_c, with rotation R^a_c = R^a_b × R^b_c and translation t^a_c = t^a_b + R^a_b × t^b_c.

A homogeneous transformation matrix is a mathematical tool used to describe the pose (position and orientation) of a coordinate system. It consists of a rotation matrix and a position vector. The rotation matrix describes the orientation, with its first column representing the X-axis direction and the second column representing the Y-axis direction relative to the fixed coordinate system. The position vector gives the location of the origin of the body-fixed coordinate system relative to the fixed coordinate system. When a body rotates, the values in the rotation matrix change accordingly, allowing calculation of how the body's axes are oriented in space.

A homogeneous transformation matrix is a 4×4 matrix representing coordinate frame poses. It contains a 3×3 rotation submatrix (orientation) and a 3×1 position column (origin location), with the last row always [0 0 0 1]. Pure rotations about x, y, or z axes have specific matrix forms with cos(θ) and sin(θ) terms. Pure translations maintain identity rotation with only translation components changing. Position vectors must be expressed in homogeneous coordinates for proper transformation.

Homogeneous transformation matrices are 4x4 matrices that combine rotation and translation information for describing the relative position and orientation between reference frames. They consist of a 3x3 rotation matrix (from the Special Orthogonal group SO(3)) in the top-left corner, a 3x1 translation vector in the top-right, and a bottom row of [0 0 0 1]. This format enables mathematical operations to combine rotations and translations, which cannot be directly concatenated as a 3x4 matrix. These matrices belong to the Special Euclidean group SE(3) and describe both rotation and translation in three-dimensional Euclidean space.

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.

The Denavit-Hartenberg (D-H) convention provides a systematic method for representing robot kinematics using four parameters per joint: link length (a), link twist (alpha), link offset (d), and joint variable (theta or d). The procedure involves: (1) establishing a functional representation of the robot at its home position, (2) assigning coordinate frames following specific rules (z-axes along joint axes, x-axes perpendicular to adjacent z-axes), (3) filling the D-H parameter table with constant and variable parameters, (4) constructing homogeneous transformation matrices for each joint, and (5) multiplying these matrices to obtain the overall transformation from base to end-effector. This method enables precise mathematical modeling of robot kinematics for both planar and spatial manipulators.

The four DH parameters define the relationship between consecutive coordinate frames: Theta (θ) is the rotation angle about the z-axis to align x-axes; D (d) is the distance along the z-axis between origins; A (a) is the distance along the x-axis between origins; Alpha (α) is the rotation angle about the x-axis to align z-axes. For prismatic joints, the rotation angle θ is zero since no rotation occurs.

The Denavit-Hartenberg (DH) parameters are four parameters (theta, d, a, alpha) used to standardize coordinate systems for robot kinematic chains. Developed in 1955 by Denavit and Hartenberg, this convention remains widely used in robotics for solving manipulator kinematics. A robot manipulator consists of n rigid links connected by joints forming a kinematic chain. Links are numbered from 0 (base) to M (end effector). Each link has two parameters: 'a' (link length along X-axis) and 'alpha' (twist angle between Z-axes). Each joint has two parameters: 'd' (offset along Z-axis) and 'theta' (rotation angle about X-axis). For rotational joints, theta is variable; for prismatic joints, d is variable. The DH method uses homogeneous transformation matrices (4x4) to represent movements between joints. The DH method requires fixing a local coordinate frame on each link. The Z-axis of each frame aligns with the joint axis connecting to the next link, following the right-hand rule based on joint rotation direction. The origin is placed at the intersection of the current Z-axis and previous X-axis. If Z-axes intersect, origin is at intersection; if parallel, origin is at joint articulation. The X-axis must be perpendicular to both current and previous Z-axes. The Y-axis is determined by the right-hand rule.

Differential kinematics computes the end-effector's linear and angular velocities from joint velocities. For revolute joints, velocity is calculated as the cross product of the rotation axis and lever arm vector, scaled by joint velocity. This relationship shows that end-effector velocity is a linear function of joint velocities. The Jacobian matrix J combines these contributions into a 6×n matrix mapping joint velocities to 6D end-effector velocities (V = J × Q dot). The Jacobian has a dual role: mapping velocities forward and torques backward through τ = J^T × W, enabling prediction of how robot motion affects tool tip behavior and how external loads translate to joint torques.

The Jacobian matrix is a fundamental mathematical tool in robot kinematics that transforms joint velocities (q_dot) into end-effector linear velocity (v) and angular velocity (ω) through the relationship v = J × q_dot and ω = J_angular × q_dot, where J is the Jacobian matrix and J_angular is the angular velocity matrix; for a robot with n joints, the Jacobian matrix has n columns, with each column representing the partial derivative of the end-effector position with respect to each joint variable, enabling the calculation of end-effector motion from joint space velocities.

The Jacobian can be calculated using partial derivatives of the forward kinematics equations. If the end effector position (x, y, z) and orientation angles (θx, θy, θz) are expressed as functions of the joint variables (q1 to qn), then taking the time derivative using the chain rule gives: ζ = [∂(f1)/∂q1 ∂(f1)/∂q2 ... ∂(f1)/∂qn; ∂(f2)/∂q1 ∂(f2)/∂q2 ... ∂(f2)/∂qn; ...; ∂(f6)/∂q1 ∂(f6)/∂q2 ... ∂(f6)/∂qn] × q̇. For a three-degree-of-freedom planar arm, this yields: J = [[-a1sin(θ1) -a2sin(θ1+θ2) -a3sin(θ1+θ2+θ3); -a2sin(θ1+θ2) -a3sin(θ1+θ2+θ3); -a3sin(θ1+θ2+θ3)], [a1cos(θ1) + a2cos(θ1+θ2) + a3cos(θ1+θ2+θ3); a2cos(θ1+θ2) + a3cos(θ1+θ2+θ3); a3cos(θ1+θ2+θ3)], [1 1 1]]. Each element ∂(fi)/∂qj represents how much the ith component of end effector velocity changes per unit change in the jth joint velocity.

To compute end-effector velocities for a specific robot configuration, the Jacobian matrix is multiplied by the vector of joint velocities. For example, with q̇₁ = π/2 rad/s, q̇₂ = π/2 rad/s, and q̇₃ = 1 m/s, the end-effector velocities are obtained by J × [q̇₁, q̇₂, q̇₃]ᵀ. The result is a 6×1 vector containing the three linear velocity components (vₓ, vᵧ, v_z) and three angular velocity components (ωₓ, ωᵧ, ω_z). This calculation enables real-time prediction of end-effector motion for any given joint velocity configuration, demonstrating practical application of the Jacobian in robot control.

The Jacobian matrix maps joint velocities to end-effector velocities through the relationship x_dot = J * θ_dot, where x_dot represents the combined linear and angular velocity of the end-effector, and θ_dot represents the joint velocities; to compute the Jacobian, first perform forward kinematics to derive the displacement vector d and orientation angle φ as functions of joint angles, then differentiate these with respect to time to obtain the linear and angular velocities, and finally express the velocity vector as a linear combination of the joint velocity terms to populate the Jacobian matrix.

Robot arms use trajectory planning to move between positions smoothly. Methods include constant acceleration/deceleration phases or breaking movements into multiple steps. Control functions take target angles for all axes and move the arm to those positions. The program continuously monitors current positions and adjusts motor outputs. This allows the arm to move to any position within its range of motion while maintaining smooth, controlled movement throughout the trajectory.

A trajectory consists of a geometric path (sequence of points parameterized without time) and a timing law (assigning time to each point). This separation allows programming paths slowly and executing at fast speeds by scaling timing. Trajectories can be planned in joint space or Cartesian space. Control systems integrate sensors, actuators, and commands through local control loops at each joint that receive velocity references and apply appropriate torques. The course assumes local controllers can accurately follow velocity references, treating the robot as a kinematic system initially before advancing to dynamics.

Trajectory planning is the process of determining smooth time functions that describe robot motion from initial to final positions. It involves two coordinate spaces: joint space (using joint angles, velocities, accelerations) and Cartesian space (using position and orientation). Joint space planning offers advantages in motion control and collision detection, while Cartesian space planning requires inverse kinematics and faces singularity challenges. The planning process involves task sequence determination, point interpolation, and inverse kinematics application to generate smooth motion paths.

Trajectory planning involves generating the time history of position, velocity, and acceleration for robotic manipulators. Users specify desired positions and orientations without computing complex equations. Trajectories are computed at runtime with path update rates of 60-2000 Hz. Smooth motion is essential to prevent vibrations and reduce wear on robot links. Waypoints (wire points) ensure smooth paths by imposing constraints. Two main methodologies exist: joint space schemes using joint angles (easier to compute) and Cartesian schemes using frame positions/orientations (higher computational burden).

Trajectory planning for a four degrees of freedom robot arm involves generating motion paths in Cartesian space using MATLAB's robotics toolbox, with trajectory timing laws and PID control for precise motion execution, while the system model and simulation are conducted using Simulink.