DynamicsLab Engine

DynamicsLab is a work in progress real-time simulation tool for multibody dynamics systems. It is designed to help users visualize, analyze, and interact with complex system data efficiently. By combining powerful simulation capabilities with an intuitive interface.

Interface

Features

• Real-time Simulation: Simulates rigid body dynamics with position, velocity, and acceleration updates.
• Interactive Camera: Provides intuitive camera controls for navigation and inspection of the simulation environment.
• Modular Project Structure: Designed for extensibility, making it easier to integrate new features or modify existing functionality.
• User Interface: Built using Dear ImGui, providing real-time control and visualization of simulation and camera data.
• Cross-platform Support: Based on GLFW and OpenGL, the project is compatible with major OS platforms.

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Camera Movement

The application enables camera navigation to explore the simulation from any viewpoint. Users can control the camera position and orientation via:

• Mouse Input:

  • Movement: Adjusts the camera's rotation.
  • Scrolling: Camera movement speed.

• Keyboard Input:

  • W/A/S/D keys for navigation.
  • Left Shift/Control for upward/downward movement.

Important Equations

$$ n_i^* = 2 L_i^T n_i' $$ $$ n_i^* = 2 G_i^T n_i $$ $$ J_i^* = 4 L_i^T J_i' L_i $$ $$ H_i = 4 \dot L_i^T J_i' L_i $$

Unconstrained Body

$$ \begin{bmatrix} M^* & P^T \cr P & 0 \end{bmatrix} \begin{bmatrix} \ddot q \cr \sigma \end{bmatrix} + \begin{bmatrix} b^* \cr c \end{bmatrix} = \begin{bmatrix} g^* \cr 0 \end{bmatrix} $$

$$ M^* = \begin{bmatrix} N_1 \cr & J_1^* \cr & & N_b \cr & & & J_b^* \end{bmatrix} \quad P = \begin{bmatrix} 0^T & p_1^T \cr & & 0^T & p_b^T \end{bmatrix} $$

$$ \ddot q = \begin{bmatrix} \ddot r_1 \cr \ddot p_1 \cr \ddot r_b \cr \ddot p_b \end{bmatrix} \quad \sigma = \begin{bmatrix} \sigma_1 \cr \sigma_b \end{bmatrix} $$

$$ b^* = \begin{bmatrix} 0 \cr 2 H_1 \dot p_1 \cr 0 \cr 2 H_b \dot p_b \end{bmatrix} \quad c = \begin{bmatrix} \dot p_1^T \dot p_1 \cr \dot p_b^T p_b \end{bmatrix} \quad g^* = \begin{bmatrix} f_1 \cr n_1^* \cr f_b \cr n_b^* \end{bmatrix} $$

Constrained Body

$$ \begin{bmatrix} M^* & P^T & B^T \cr P & 0 & 0 \cr B & 0 & 0 \end{bmatrix} \begin{bmatrix} \ddot q \cr \sigma \cr - \lambda \end{bmatrix} + \begin{bmatrix} b^* \cr c \cr 0 \end{bmatrix} = \begin{bmatrix} g^* \cr 0 \cr \gamma \end{bmatrix} $$

Unconstrained Demo Body

Initial Position, Orientation, Velocity & Angular Velocity

1 x 1 x 1 m 10 kilogram cube.

Mass: $ m = 10 kg $

Moment of Inertia: $ I = 60 kgm^2 $

$$ r_1 = \begin{bmatrix} 0 \cr 0 \cr 0 \end{bmatrix} \quad p_1 = \begin{bmatrix} 1 \cr 0 \cr 0 \cr 0 \end{bmatrix} \quad \dot r_1 = \begin{bmatrix} 0 \cr 0 \cr 0 \end{bmatrix} \quad \dot p_1 = \begin{bmatrix} 0 \cr 0 \cr 0 \cr 0 \end{bmatrix} $$

Mass & Inertia Matrix

$$ N_1 = \begin{bmatrix} 10 & 0 & 0 \cr 0 & 10 & 0 \cr 0 & 0 & 10 \end{bmatrix} \quad J_1^* = \begin{bmatrix} 0 & 0 & 0 & 0 \cr 0 & 60 & 0 & 0 \cr 0 & 0 & 60 & 0 \cr 0 & 0 & 0 & 60 \end{bmatrix} \quad $$

Variable Mapping Table

Variable from Book	Description	Corresponding Name in Code	Additional Notes
M*	System mass-inertia matrix	SystemMassInertiaMatrix	Represents the mass and inertia properties of the system.
P	Quaternion constraint matrix	QuaternionConstraintMatrix	Handles constraints related to quaternion representation.
q	Generalized position coordinates of the system	GeneralizedCoordinates	Position variables describing the configuration of the system.
q̇ (q dot)	Generalized velocity coordinates	GeneralizedVelocities	Velocity variables associated with the system coordinates.
q̈ (q double dot)	Generalized accelerations	GeneralizedAccelerations	Acceleration variables for the system's motion.
b*	Velocity-dependent term	VelocityDependentTerm	Represents effects dependent on the velocity, such as Coriolis or damping forces.
c	Quaternion norm squared value	QuaternionNormSquared	Constraint ensuring quaternions remain normalized.
g*	Generalized external forces	GeneralizedExternalForces	External forces acting on the system.

References

1. Nikravesh, Parviz E. Computer-Aided Analysis of Mechanical Systems. Prentice-Hall, Inc., USA, 1988. ISBN: 0131642200.

2. GLFW - A multi-platform library for OpenGL, OpenGL ES, and Vulkan development.

   • Website: https://www.glfw.org

3. GLM - OpenGL Mathematics, a header-only C++ library for graphics software.

   • GitHub: https://github.com/g-truc/glm

4. Dear ImGui - A bloat-free graphical user interface library for C++.

   • GitHub: https://github.com/ocornut/imgui

5. Eigen - A C++ template library for linear algebra.

   • Website: https://eigen.tuxfamily.org