SimMER

Simulated Minimalist Engine for Robotics

SimMER is a tiny, educational robot arm simulation engine built from first principles. The goal is to make the core math of robotics (kinematics, Jacobians, inverse kinematics) transparent and hackable.

If you can read ~300 lines of Python, you can understand the whole engine.

What's Inside

• Joint — a revolute joint with an angle (the tunable parameter)
• Link — a rigid link with a fixed length
• Transform — a 3×3 homogeneous transformation matrix encoding rotation + translation in one shot
• Chain — a sequence of joints and links with forward kinematics, Jacobian, and IK solvers
• viz — matplotlib-based visualization: static snapshots and IK animation

Core Idea

A robot arm is a chain of transforms. The end-effector position is:

T_total = T1 @ T2 @ ... @ Tn
EE = T_total @ [0, 0, 1]

Each T_i is a homogeneous transform for joint i:

T = [[cos θ, -sin θ, l·cos θ],
     [sin θ,  cos θ, l·sin θ],
     [0,      0,     1      ]]

Angles are relative — each joint angle is measured from the previous link's direction.

The Jacobian maps joint velocities to end-effector velocities. Column i is the perpendicular to the vector from joint i to the EE:

J[:,i] = [-(y_ee - y_i), (x_ee - x_i)]

IK Solvers

Two iterative solvers are included:

• Gradient descent — J^T @ e, simple and robust
• Damped pseudoinverse — J^T (J J^T + λ²I)^{-1} @ e, faster convergence and stable near singularities

Roadmap

• Forward kinematics
• Geometric Jacobian
• Inverse kinematics (gradient descent)
• Inverse kinematics (damped pseudoinverse)
• Visualization (matplotlib)
• Joint limits
• Null space control (redundant arms)
• 3D extension (SE(3), 4×4 transforms)
• Simple dynamics