Mechanics and Kinematics

1. Forward Kinematics

Concept:

Forward Kinematics (FK) is the process of calculating the position and orientation of a robot’s end-effector (e.g., its gripper or hand) based on known joint angles and link geometry.

  • Input: A set of joint angles (e.g., 45°, 90°)
  • Output: Cartesian coordinates (X, Y, Z) and orientation of the end-effector.

How it Works:

Think of it like giving step-by-step directions:

“From the base, rotate the first joint 45°, extend the second link 30 cm, rotate the next joint 90°, and extend the last link 40 cm.”

Forward kinematics uses mathematics to determine exactly where the end-effector ends up after following these instructions.
It is deterministic — meaning, one input gives one clear output.

2. Inverse Kinematics

Concept:

Inverse Kinematics (IK) does the opposite of FK — it determines the joint angles needed to position the end-effector at a desired target location.

  • Input: Desired Cartesian coordinates (X, Y, Z) and orientation.
  • Output: The joint angles that achieve that pose.

Why It’s More Complex:

While FK gives a single solution, IK can have:

  • Multiple solutions (e.g., “elbow up” or “elbow down”).
  • A single solution, or
  • No solution if the target is outside the robot’s reachable workspace.

IK is vital for robot path planning, since we usually know where we want the robot to go — not how it must move its joints to get there.

3. Denavit–Hartenberg (DH) Convention

The Denavit–Hartenberg (DH) convention provides a standardized method to assign coordinate frames to each link of a robot manipulator.
This method simplifies describing the robot’s geometry for both FK and IK analysis.

Core Idea

The DH convention represents each link using four parameters, applied in a specific order to relate one joint to the next.

The Four DH Parameters

  1. Link Length (a): Distance between joint axes (common normal).
  2. Link Twist (α): Angle between Z-axes of consecutive links, measured about the X-axis.
  3. Link Offset (d): Distance along the Z-axis between links (variable for prismatic joints).
  4. Joint Angle (θ): Angle about the Z-axis between X-axes (variable for revolute joints).

The DH Procedure

  1. Assign Frames: Attach a (X, Y, Z) frame to each joint following DH rules.

  2. Determine Parameters: For each pair of frames, find the four DH parameters (a, α, d, θ).

  3. Create Transformation Matrices: Use these to form 4×4 homogeneous transformation matrices for each link.

  4. Multiply Matrices:
    Combine all transformations to get the total transformation from base to end-effector:

    $$ T_{total} = T_{base}^0 × T_0^1 × T_1^2 × \ldots × T_{n-1}^n $$

The resulting matrix gives the position and orientation of the end-effector relative to the base.

2 Link Planar Arm

The figure shows a 2-Link Planar Robotic Arm.

The goal is to find the mathematical expressions for the end-effector position (x, y), given:

  • Link lengths $ a_1, a_2 $
  • Joint angles $ θ_1, θ_2 $

DH Parameter Table

For the 2-link arm, the robot’s geometry can be represented using a DH table:

DH Table

Transformation Matrices

Each link’s transformation is defined by a 4×4 homogeneous matrix, based on its DH parameters:

Transformation Matrix

Composite Transformation

To find the end-effector’s pose relative to the base:

$$ ^0T_2 = A_1 × A_2 $$

Composite Matrix

This multiplication results in a composite matrix:

Final Matrix

End-Effector Equations

From the composite transformation matrix $ T_2^0 $:

$$ x = a_1 \cos(θ_1) + a_2 \cos(θ_1 + θ_2) $$
$$ y = a_1 \sin(θ_1) + a_2 \sin(θ_1 + θ_2) $$

Notation Used in Images:

  • $ c_1 = \cos(θ_1) $
  • $ s_1 = \sin(θ_1) $
  • $ c_{12} = \cos(θ_1 + θ_2) $
  • $ s_{12} = \sin(θ_1 + θ_2) $