Denavit-Hartenberg Parameters

The link length is the perpendicular distance measured along the common normal from the Zᵢ₋₁ axis to the Zᵢ axis. It represents the shortest distance between the two joint axes.

• Definition: Distance from Zᵢ₋₁ to Zᵢ along Xᵢ.
• Physical Meaning: For a rotary joint, it is the length of the physical link connecting the joints. For parallel axes, it is zero.
• Example: In a simple 2D planar arm, this is the literal length of the link between two revolute joints.

The link twist is the angle between the Zᵢ₋₁ axis and the Zᵢ axis, measured about the Xᵢ axis using the right-hand rule.

• Definition: Angle from Zᵢ₋₁ to Zᵢ about Xᵢ.
• Physical Meaning: It describes the 'twist' or angular offset between the rotational axes of two consecutive joints.
• Example: In a standard industrial robot, if two adjacent revolute joints have axes that are perpendicular, the link twist is 90 degrees (π/2 radians).

The joint offset is the distance measured along the Zᵢ₋₁ axis from the origin of frame {i-1} to the intersection point of the Zᵢ₋₁ axis and the common normal (Xᵢ axis).

• Definition: Distance along Zᵢ₋₁ to the common normal.
• Physical Meaning: For a prismatic (sliding) joint, this is the variable joint parameter. For a revolute joint, it is a constant offset along the joint axis.
• Example: In a SCARA robot, this parameter defines the vertical offset between the horizontal arm links.

The joint angle is the angle between the Xᵢ₋₁ axis and the Xᵢ axis, measured about the Zᵢ₋₁ axis using the right-hand rule.

• Definition: Angle from Xᵢ₋₁ to Xᵢ about Zᵢ₋₁.
• Physical Meaning: For a revolute (rotary) joint, this is the variable joint parameter. For a prismatic joint, it is a constant angle.
• Example: In a simple 1-degree-of-freedom hinge, this angle is the primary control variable for the robot's pose.

Each DH parameter is used to construct an elementary transformation matrix. The complete transformation from frame {i-1} to frame {i} is the product of four sequential transformations:

ⁱ⁻¹Tᵢ = Rot(Z, θᵢ) * Trans(Z, dᵢ) * Trans(X, aᵢ₋₁) * Rot(X, αᵢ₋₁)

• This 4x4 homogeneous transformation matrix contains both the rotation and translation between the two link coordinate frames.
• The forward kinematics of the entire manipulator is found by chaining these matrices: ⁰T_N = ⁰T₁ * ¹T₂ * ... * ᴺ⁻¹T_N.

Two primary conventions exist: Standard (Classic) DH and Modified DH (introduced by Craig). They differ in where the link frame is attached and the order of parameter definitions.

Key Differences:

• Frame Attachment: In Standard DH, frame {i} is attached to link i at the far end of the link. In Modified DH, frame {i} is attached to link i at the near end.
• Parameter Indexing: The associated parameters (a, α) are indexed differently relative to the link number.
• Usage: Standard DH is common in older textbooks. Modified DH is often preferred in modern robotics software (e.g., ROS, MATLAB Robotics Toolbox) for its algorithmic simplicity in deriving dynamics.

Forward kinematics is the process of calculating the position and orientation of a robot's end-effector from its joint angles and the known geometry of its links. The Denavit-Hartenberg (DH) convention provides the standardized method for building the homogeneous transformation matrices between consecutive links, which are then multiplied together to compute the final end-effector pose. This is a fundamental prerequisite for path planning and operational space control.

• Input: Set of joint variables (angles for revolute joints, displacements for prismatic joints).
• Output: Position (x, y, z) and orientation (roll, pitch, yaw) of the end-effector in the base frame.
• Core Computation: Sequential multiplication of DH transformation matrices: (^0T_N = ^0T_1 * ^1T_2 * ... * ^{N-1}T_N)

Inverse kinematics (IK) is the reverse problem of forward kinematics: determining the set of joint angles required to achieve a desired end-effector position and orientation. While DH parameters define the kinematic structure, solving IK is non-trivial and often involves:

• Analytical solutions: Closed-form equations derived from the specific robot geometry (possible for many serial chains with 6 or fewer DOF).
• Numerical solutions: Iterative algorithms (e.g., Jacobian-based methods) used for complex or redundant manipulators.
• Application: Essential for robot control, as high-level tasks are specified in task space (e.g., "move gripper here"), but actuators are controlled in joint space.

The Jacobian matrix is a linear mapping that relates joint velocities to the linear and angular velocity of the end-effector in task space. For a robot defined using DH parameters, the Jacobian is derived from the partial derivatives of the forward kinematics equations.

• Function: ( \dot{x} = J(q) \dot{q} ), where (\dot{x}) is the task-space velocity vector and (\dot{q}) is the joint-space velocity vector.
• Uses: Velocity-based control, singularity analysis, and force transformation (via the transpose of the Jacobian).
• Singularities: Configurations where the Jacobian loses rank, indicating a loss of mobility in certain directions. DH parameterization helps identify these conditions analytically.

The Featherstone Algorithm (Articulated Body Algorithm) is an efficient O(n) recursive method for computing the forward dynamics of articulated rigid body systems, such as robotic manipulators. While DH parameters define the kinematic structure, the Featherstone algorithm computes the dynamics—the joint accelerations resulting from applied forces/torques.

• Efficiency: Avoids constructing and inverting the large, dense mass matrix of the whole system.
• Process: Performs a recursion from the base to the tips to compute velocities and inertias, then a recursion from the tips back to the base to compute accelerations.
• Context: This is the dynamic counterpart to the purely kinematic description provided by DH parameters.

Spatial Vector Algebra is a compact 6D mathematical framework that combines linear and angular components to describe rigid body motion, forces, and inertias. It provides a more elegant and efficient notation for formulating dynamics algorithms (like the Featherstone algorithm) compared to traditional 3D vector approaches.

• Spatial Velocity: Combines angular velocity and linear velocity into a 6D vector.
• Spatial Force: Combines torque and force into a 6D vector.
• Relation to DH: DH parameters define the kinematic transformations between links. Spatial vector algebra uses these transformations (as 6D spatial transforms) to propagate velocities, accelerations, and forces across the kinematic chain during dynamic computation.

Operational Space Control is a robotics control framework where control laws are formulated directly in the task space (e.g., Cartesian coordinates of the end-effector) rather than in joint space. This provides a more intuitive interface for commanding complex behaviors like force-controlled insertion.

• Foundation: Relies heavily on the robot's kinematic model (defined by DH parameters) and its Jacobian.
• Key Equation: The control torque (\tau) is computed as (\tau = J^T F), where (F) is the desired force in task space and (J^T) is the transpose of the Jacobian. This maps task-space forces to joint-space torques.
• Benefit: Allows for natural specification of tasks ("apply a force along this axis") while automatically handling the kinematic coupling between joints.