Forward Kinematics (FK) Solver & Visualizer

📝 Problem Input

Enter Problem Statement (exam question, assignment, or research query):

📊 Interactive Robot Visualizer

Manipulate joint angles using sliders to see real-time changes in robot pose and coordinate frames.

🌍 Workspace Plot

Reachable workspace of the robot end-effector based on current configuration.

📚 D-H Convention Reference

1. Denavit-Hartenberg Parameters

The D-H convention uses four parameters to describe the relationship between consecutive coordinate frames attached to robot links:

Parameter	Symbol	Description	Variable For
Link Length	a_i	Distance along x_i from z_i to z_i+1	Geometric constant
Link Twist	α_i	Angle about x_i from z_i to z_i+1	Geometric constant
Link Offset	d_i	Distance along z_i-1 from x_i-1 to x_i	Prismatic joint variable
Joint Angle	θ_i	Angle about z_i-1 from x_i-1 to x_i	Revolute joint variable

2. Standard D-H Transformation Matrix

Equation 1: Individual Link Transformation (Standard D-H)

^i-1T_i = R_z(θ_i) · T_z(d_i) · T_x(a_i) · R_x(α_i)

General D-H Matrix:

cos(θ)-sin(θ)cos(α)sin(θ)sin(α)a·cos(θ)

sin(θ)cos(θ)cos(α)-cos(θ)sin(α)a·sin(θ)

0sin(α)cos(α)d

0001

3. Frame Assignment Rules (Denavit-Hartenberg)

Identify all joint axes z₀, z₁, ..., z_n-1
Establish base frame {0} at joint 1 with z₀ along joint 1 axis
For each joint i, assign z_i along joint axis i+1
Locate origin O_i at intersection of z_i and common normal between z_i-1 and z_i
x_i along common normal from z_i-1 to z_i (pointing away from z_i-1)
y_i completes right-hand coordinate system: y_i = z_i × x_i
Attach end-effector frame {n} to the last link

4. Joint Types

Revolute Joint R

θ_i is the joint variable
d_i is constant (fixed offset)
Rotation about z-axis
Typical range: ±180°

Prismatic Joint P

d_i is the joint variable
θ_i is constant (fixed angle)
Translation along z-axis
Range: physical stroke limits

5. Forward Kinematics

Equation 2: Overall Transformation (Product of Matrices)

⁰T_n = ⁰T₁ · ¹T₂ · … · ^n-1T_n

Equation 3: End-Effector Pose Extraction

⁰T_n = [R_3×3 | p_3×1; 0 0 0 1] where R = rotation matrix, p = position vector

6. Inverse Kinematics Overview

Given desired end-effector pose T, find joint variables q = [q₁, ..., q_n]:

Equation 4: Inverse Kinematics Problem

Given: ⁰T_n(q) = T_desired → Find: q = f^-1(T_desired)

Geometric method: Use geometry of robot structure (preferred for simple robots)
Algebraic method: Solve system of equations from matrix equality
Numerical method: Iterative Jacobian-based methods (for complex robots)

⚖️ Robot Configuration Comparison

Comparison of Robot Configurations

Feature	Cartesian (PPP)	Cylindrical (RPP)	Spherical (RRP)	SCARA (RRPR)	Articulated (RRR)
DOF	3	3	3	4	3–6
Workspace	Rectangular box	Cylindrical	Spherical shell	Cylindrical (selective)	Irregular sphere
Rigidity	Very High	High	Medium	High (horizontal)	Medium
Reach	Limited	Moderate	Good	Good (planar)	Excellent
Accuracy	Highest	High	Moderate	High	Moderate
Applications	CNC, 3D printing	Assembly, welding	Spray painting	PCB assembly	General purpose
D-H Complexity	Simple	Simple	Moderate	Moderate	Complex
IK Solution	Trivial	Simple	Closed-form	Closed-form	Numerical/Closed
Singularities	None	At origin	At boundaries	Elbow	Multiple

Standard vs Modified D-H Convention

Aspect	Standard D-H (Craig)	Modified D-H (Denavit)
Frame attachment	Distal (at joint i+1)	Proximal (at joint i)
Parameter order	θ, d, a, α	α, a, θ, d
Matrix form	R_z·T_z·T_x·R_x	R_x·T_x·R_z·T_z
Usage	Craig, Spong textbooks	Some industrial robots
Singularity handling	Standard	Better for some configs
Base frame	Coincides with frame 1	Separate base frame

D-H Parameters for Common Robots

Robot	Joint Sequence	Typical a values	Typical α values	Notes
2R Planar	R-R	a1, a2 > 0	0°, 0°	All motion in XY plane
SCARA	R-R-P-R	a1, a2, 0, 0	0°, 0°, 0°, 0°	Selective compliance
Stanford Arm	R-R-P-R-R-R	All zero	0°, -90°, 0°, 90°, -90°, 90°	Spherical wrist
PUMA 560	R-R-R-R-R-R	0, a2, a3, 0, 0, 0	90°, 0°, -90°, 90°, -90°, 0°	Classic 6-DOF
Cartesian	P-P-P	0, 0, 0	0°, -90°, 90°	Orthogonal axes

⚠️ D-H Table Error Checker

Enter your D-H parameter table below. The system will validate entries and highlight errors with correction suggestions.

Number of Links/Joints:

🔄 Reverse D-H Problem Solver

Given a transformation matrix or end-effector pose, extract D-H parameters and identify robot configuration.

Input: 4×4 Homogeneous Transformation Matrix

Enter the matrix elements row by row (T[row][col]):

Robot Type Hint:

🏭 Application-Oriented Problems

🔧 Industrial Welding

6-DOF articulated robot for arc welding. PUMA-type with wrist assembly.

📦 Palletizing Robot

4-DOF robot for pick-and-place palletizing in manufacturing.

💻 PCB Assembly

SCARA robot for high-speed PCB component placement.

🏥 Surgical Robot

7-DOF redundant robot for minimally invasive surgery.

🤝 Collaborative Robot

UR5-type 6-DOF cobot for human-robot collaboration.

🎨 Spray Painting

Spherical robot for automotive spray painting.

🤖 Forward Kinematics (FK) Solver & Visualizer