chore: added mechanics post

Author: Mummanajagadeesh

assets/jsconfig.json

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content/docs/mechanics/comp.png

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+---
+title: "Mechanics and Kinematics"
+date: 2025-01-10T18:08:19+05:30
+lastmod: 2025-01-10T18:08:19+05:30
+author: "ORIGO"
+cover: "cover.png"
+images:
+  - "cover.png"
+categories:
+  - "category1"
+tags:
+  - "mechanics"
+  - "gears"
+  - "kinematics"
+  - "handout"
+  - "rignitc"
+draft: false
+katex: true
+
+---
+
+<!-- Summary -->
+Mechanics and Kinematics
+<!--more-->
+
+---
+
+## 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](planar.png)
+
+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](dh.png)
+
+---
+
+### Transformation Matrices
+
+Each link’s transformation is defined by a **4×4 homogeneous matrix**, based on its DH parameters:
+
+![Transformation Matrix](transmat.png)
+
+---
+
+### Composite Transformation
+
+To find the end-effector’s pose relative to the base:
+
+$$
+^0T_2 = A_1 × A_2
+$$
+
+![Composite Matrix](comp.png)
+
+This multiplication results in a composite matrix:
+
+![Final Matrix](mat.png)
+
+---
+
+### 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) $
+
+---
+