Kinematics, Path Planning & Robot Motion

Kinematics: The Math Behind Robot Motion

What is Kinematics?

Kinematics is the study of motion without considering forces. For robots, it answers:

• Where can the robot reach? (Workspace)
• How do joint angles determine end-effector position? (Forward Kinematics)
• What joint angles achieve a desired position? (Inverse Kinematics)

Key Definitions

• Degree of Freedom (DoF): Number of independent movements (e.g., a 6-DoF arm can move in 3D space + rotate).
• Configuration Space (C-space): All possible joint configurations (e.g., a 2-link arm has a 2D C-space).
• Workspace: All reachable positions in Cartesian space.

Forward Kinematics: From Joints to End-Effector

Problem: Given joint angles, compute the end-effector position. Solution: Use Denavit-Hartenberg (DH) parameters to model each joint as a transformation matrix.

DH Parameters Table

Transformation Matrix (T_i): $$T_i=\left[\begin{array}{cccc}\cos {\theta }_i& -\sin {\theta }_i\cos {\alpha }_i& \sin {\theta }_i\sin {\alpha }_i& a_i\cos {\theta }_i\\ \sin {\theta }_i& \cos {\theta }_i\cos {\alpha }_i& -\cos {\theta }_i\sin {\alpha }_i& a_i\sin {\theta }_i\\ 0& \sin {\alpha }_i& \cos {\alpha }_i& d_i\\ 0& 0& 0& 1\end{array}\right]$$

Worked Example: 2-Link Planar Arm Assume:

• Link 1 length = 1 m, Link 2 length = 1 m.
• Joint angles: θ₁ = 30°, θ₂ = 45°. Compute end-effector position (x, y).

graph TD
    A["Base"] --> B["Joint 1 (θ₁)"]
    B --> C["Link 1 (1m)"]
    C --> D["Joint 2 (θ₂)"]
    D --> E["Link 2 (1m)"]
    E --> F["End-Effector (x,y)"]

Step-by-Step Calculation:

1. Link 1 Transformation (T₁): $$T_1=\left[\begin{array}{cccc}\cos 30°& -\sin 30°& 0& 1\cdot \cos 30°\\ \sin 30°& \cos 30°& 0& 1\cdot \sin 30°\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$ End of Link 1: $(x_1,y_1)=(0.866,0.5)$.

2. Link 2 Transformation (T₂): $$T_2=\left[\begin{array}{cccc}\cos 45°& -\sin 45°& 0& 1\cdot \cos 45°\\ \sin 45°& \cos 45°& 0& 1\cdot \sin 45°\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$ End-effector position: $x=x_1+1\cdot \cos (30°+45°)=0.866+0.5176=1.3836$ $y=y_1+1\cdot \sin (30°+45°)=0.5+0.9659=1.4659$.

Inverse Kinematics: Solving for Joint Angles

Problem: Given end-effector position (x, y), find joint angles θ₁, θ₂. Solution: Use geometric or algebraic methods (e.g., Law of Cosines for 2-link arms).

Worked Example: Inverse Kinematics for the 2-Link Arm

Given end-effector at (1.3836, 1.4659), find θ₁, θ₂.

1. Compute distance to end-effector: $d=\sqrt{1.3836^2+1.4659^2}=2$ m (matches link lengths).

2. Law of Cosines: $$\cos {\theta }_2=\frac{1^2+1^2-2^2}{2\cdot 1\cdot 1}=-0.5⟹{\theta }_2=120°$$ $${\theta }_1=\arctan 2(1.4659,1.3836)-{\theta }_2=30°$$

Note: Multiple solutions exist (e.g., θ₂ = 240°). Use constraints (e.g., joint limits) to pick the valid one.

Path Planning: Navigating Obstacles

Path planning finds a collision-free trajectory from start to goal. Key methods:

1. Configuration Space (C-space) Approach

• Obstacle Inflation: Expand obstacles by robot radius to simplify collision checking.
• C-space Obstacles: Regions where the robot collides with obstacles or itself.

graph TD
    A["C-space: Free"] --> B["C-space: Occupied"]
    C["Robot"] --> D["Obstacle"]
    D --> E["Inflated Obstacle"]

2. Common Path Planning Algorithms

Worked Example: A* for Grid Navigation

Scenario: A robot in a 5×5 grid must go from (0,0) to (4,4). Obstacles at (1,2), (2,1).

graph TD
    A["Start (0,0)"] --> B["(0,1)"]
    A --> C["(1,0)"]
    B --> D["(1,1)"]
    D --> E["(1,2) Obstacle"]
    C --> F["(2,0)"]
    F --> G["(2,1) Obstacle"]
    G --> H["(2,2)"]
    H --> I["(3,2)"]
    I --> J["(4,2)"]
    J --> K["(4,3)"]
    K --> L["Goal (4,4)"]

Steps:

1. Heuristic (h(n)): Manhattan distance to goal.
2. Cost (g(n)): Steps taken so far.
3. f(n) = g(n) + h(n): Prioritize nodes with lowest f(n).
4. Path: (0,0) → (1,0) → (2,0) → (3,0) → (4,0) → (4,1) → (4,2) → (4,3) → (4,4).

Real-World Applications

1. eSewa (Nepal)

• Idea Used: Path Planning (RRT)
• How: eSewa’s automated customer service chatbots use decision trees (a form of path planning) to navigate through user queries. For example:
  • If user says “bill payment,” the bot follows a predefined path: Verify account → Select service → Enter amount → Confirm → Redirect to payment gateway.
  • Visual: Think of this as a decision tree where each node is a question (e.g., “Do you want to pay electricity or water bill?”), and edges are possible responses.

2. Pathao (Ride-Hailing App)

• Idea Used: Kinematics + A Path Planning*

• How: When you request a ride, Pathao’s algorithm:

  1. Kinematics: Calculates the driver’s current position and velocity (using GPS data as joint angles in a virtual “vehicle arm”).
  2. Path Planning: Uses A* to find the shortest collision-free route from the driver to your location, avoiding traffic (obstacles) and one-way streets (C-space constraints).

• Worked Example: Suppose you’re at Kathmandu Traffic (Thamel) and the driver is at Naxal (near TU). The app:

  • Models the city as a grid where intersections are nodes.
  • Obstacles = closed roads (e.g., during Dashain).
  • Computes the path: Naxal → Putalisadak → Durbar Marg → Thamel.

  graph TD
      A["Naxal"] --> B["Putalisadak"]
      B --> C["Durbar Marg"]
      C --> D["Thamel (Goal)"]
      E["Obstacle: Closed Road"] --> F["Alternative: Ring Road"]

3. NTC (Nepal Telecommunications)

• Idea Used: Inverse Kinematics for Antenna Alignment
• How: NTC’s 5G towers use pan-tilt mechanisms (2-DoF) to align antennas toward users. For example:
  • If a user at coordinates (x, y) requests service, the antenna’s inverse kinematics solves for the tilt (θ₁) and pan (θ₂) angles to point directly at the user.
  • Real Scenario: During a live concert at Tundikhel, NTC’s system dynamically adjusts antenna angles to maintain signal strength for thousands of attendees.

Exam Tip

1. Kinematics Questions:

   • Always draw the robot arm and label joints/links before calculations.
   • For forward kinematics, show the transformation matrices step-by-step.
   • For inverse kinematics, explain whether you’re using geometric or algebraic methods.

2. Path Planning Questions:

   • Compare A* and RRT in a table (as above) and justify which to use for a given scenario (e.g., “A robot in a cluttered warehouse” → RRT; “A drone in a grid” → A*).
   • Always sketch the C-space when obstacles are involved. Highlight:
     • Free space (white).
     • Occupied space (gray).
     • Inflated obstacles (red).

3. Common Pitfalls:

   • Forgetting to check joint limits in inverse kinematics (e.g., θ₂ = 240° may be invalid if max θ₂ = 180°).
   • Misapplying the heuristic in A* (e.g., using Euclidean distance in a grid where Manhattan distance is correct).
   • Ignoring collision constraints in path planning (e.g., assuming a straight line is valid when obstacles exist).

Summary Table: Kinematics vs. Path Planning