ARM Equation & Robot Kinematics

• The ARM equation, or Advanced Robotics for Manufacturing equation, mathematically relates a robot's joint variables to its end-effector position and orientation.
• Kinematics describes the motion of bodies without considering the forces involved.
• In robotics, kinematics defines the relationship between joint space (all possible joint configurations) and Cartesian space (all possible end-effector positions/orientations).
• The ARM equation is a mathematical formulation for solving forward and inverse kinematics problems.
• The forward kinematics problem determines the end-effector's position and orientation from given joint variables.
• The inverse kinematics problem determines the necessary joint variables to achieve a desired end-effector position and orientation.

Key Concepts

• Denavit-Hartenberg (DH) parameters establish a standard for assigning coordinate frames to robot links.
• DH parameters define the geometric relationship between the links.
• Transformation matrices represent the rotation and translation between coordinate frames.
• Joint variables define the position or orientation of each robot joint.
• End-effector pose refers to the end-effector's position and orientation in Cartesian space.
• Forward kinematics computes the end-effector pose from joint variables.
• Inverse kinematics computes the joint variables needed for a desired end-effector pose.

Denavit-Hartenberg (DH) Parameters

• The DH parameter convention systematically describes the geometry of a robot manipulator, attaching coordinate frames to each link.
• Four parameters define the relationship between successive frames:
  • θ (theta): Angle about the previous frame's z-axis to align the x-axes.
  • d: Distance along the previous frame's z-axis between the previous frame's origin and the intersection of the current frame's x-axis with the previous frame's z-axis.
  • a: Distance along the previous frame's x-axis between the intersection of the current frame's x-axis with the previous frame's z-axis and the current frame's origin.
  • α (alpha): Angle about the previous frame's x-axis to align the z-axes.

Transformation Matrices

• Transformation matrices represent rotation and translation between coordinate frames.
• A 4x4 homogeneous transformation matrix represents a coordinate frame's pose relative to another.
• Its upper-left 3x3 submatrix represents rotation and the rightmost column represents translation.
• The transformation matrix from frame i-1 to frame i ( ^i-1T_i ) depends on the DH parameters:
  • ^i-1T_i = Rot(z, θ_i) Trans(0, 0, d_i) Trans(a_i, 0, 0) Rot(x, α_i)
• Rot(axis, angle) is rotation about the specified axis, and Trans(x, y, z) is translation along the specified axes.

Forward Kinematics

• Forward kinematics computes the end-effector pose from the joint variables.
• It involves multiplying the transformation matrices between successive links.
• The overall transformation matrix from the base frame (0) to the end-effector frame (n) is ⁰T_n:
  • ⁰T_n = ⁰T₁ ¹T₂ ²T₃ ... ^n-1T_n
• The end-effector's position and orientation can be extracted from the resulting transformation matrix ⁰T_n.

Inverse Kinematics

• Inverse kinematics computes the joint variables for a desired end-effector pose.
• It is more complex than forward kinematics, with potentially multiple, infinite, or no solutions.
• Approaches to solving the inverse kinematics problem include analytical and numerical solutions.
• Analytical solutions involve closed-form expressions for joint variables based on end-effector pose and are suitable for simple kinematic structures.
• Numerical solutions use iterative algorithms to minimize the error between desired and actual end-effector poses, and can be used for complex robots.

Analytical Solutions

• Analytical solutions are derived by manipulating the forward kinematics equations to solve for the joint variables.
• The specific steps depend on the robot's kinematic structure.
• For a 2R planar robot with two revolute joints, the forward kinematics equations are:
  • x = l₁cos(θ₁) + l₂cos(θ₁ + θ₂)
  • y = l₁sin(θ₁) + l₂sin(θ₁ + θ₂)
• (x, y) is the end-effector position, l₁ and l₂ are the link lengths, and θ₁ and θ₂ are the joint angles.
• By manipulating these equations, one can solve for θ₁ and θ₂ in terms of x and y.

Numerical Solutions

• Numerical solutions find joint variables that minimize the error between desired and actual end-effector poses using iterative algorithms.
• A common approach is to use the Jacobian matrix, relating joint velocities to end-effector velocity:
  • J(θ) = ∂f(θ) / ∂θ, where J(θ) is the Jacobian matrix, f(θ) is the forward kinematics function, and θ is the vector of joint variables.
• The inverse kinematics problem can be solved iteratively:
  • Δθ = J⁺(θ) Δx, where Δθ is the change in joint variables, J⁺(θ) is the pseudo-inverse of the Jacobian matrix, and Δx is the difference between the desired end-effector pose and the current end-effector pose.
• Iteration continues until the error is below a threshold.

Singularities

• Singularities are robot configurations where the Jacobian matrix loses rank.
• At singularities, the robot loses the ability to move in certain directions as the inverse kinematics problem admits infinite solutions or no solution.
• Singularities occur when two or more joint axes align or when the robot reaches its workspace boundary.
• It is important to avoid singularities when planning robot trajectories.

Applications of the ARM Equation

• The ARM equation is used in robot control for real-time motion.
• The ARM equation is used in trajectory planning for efficient trajectories.
• The ARM equation is used in robot simulation in virtual environments.
• The ARM equation is used in robot design for desired kinematic properties.

The ARM equation, crucial in robot kinematics, mathematically connects a robot's joint variables to its end-effector's position and orientation. Kinematics describes motion, and in robotics, it links joint space and Cartesian space. This equation solves forward and inverse kinematics problems.