Delta Robot Simulation: Kinematics, Dynamics & Control in MATLAB/Simulink

Multibody Dynamics studies the dynamics of systems of rigid bodies, extending previous work on particle systems and single rigid bodies. Two fundamental examples are introduced: a single particle under forces (MX = FX, MY = FY) and a cart-pendulum system with cart mass M1, pendulum mass M2, length l, and generalized coordinates X and θ. The cart-pendulum equations are: (M1 + M2)X + M2Lcos(θ)θ̈ - M2Lθ̇²sin(θ) = F and M2L²θ̈ + M2Lcos(θ)Ẋ + M2gLsin(θ) = T. These examples demonstrate how equations of motion are derived for mechanical systems.

This segment covers the complete process of formulating equations of motion using Lagrangian mechanics for multi-rigid body systems. Students learn to compute kinetic energy by summing contributions from each rigid body using T = ½mv_G² + ½I_Gω², and potential energy for conservative forces. The instructor demonstrates applying Lagrange's equations d/dt(∂L/∂q̇_j) - ∂L/∂q_j = Q_j to derive coupled differential equations. Key concepts include recognizing that for mechanical systems, ∂V/∂q̇_j = 0 simplifies calculations, and that generalized coordinates must be chosen consistently whether using Lagrange's method or direct Newton-Euler approaches.

This extensive section covers the mathematical foundations of multibody dynamics. Rigid bodies are described using body reference coordinate systems with translation (rx, ry) and rotation (orthogonal 2x2 matrix). Euler angles describe spatial rotation through three successive rotations (φ, θ, ψ), though quaternions are more commonly used in practice. Generalized coordinates consist of position components and orientation angle, requiring three variables per rigid body. Degrees of freedom classify systems: dynamic systems (positive DOF) require equation of motion derivation; kinematic systems (zero DOF) solve using only constraints; redundant systems (negative DOF) indicate dependent constraints. Kinematic analysis proceeds through position, velocity, and acceleration levels using the Jacobian matrix for substitution. Generalized forces derive from virtual work principles using the partial derivative matrix ∂r/∂q. The mass matrix has specific structure: diagonal blocks for translation contain mass, diagonal block for rotation contains mass moment of inertia, and off-diagonal blocks are zero when reference point is at center of mass. Quadratic velocity vector represents centrifugal forces. Lagrange multipliers represent constraint enforcement forces, not inertia or acceleration. Orthogonal matrices satisfy transpose × original = identity, with columns forming orthonormal sets. Particles have no rotational degrees of freedom as they are treated as point masses.

The augmented formulation of equations of motion for multi-body systems uses Lagrange multipliers to incorporate constraints, resulting in a system of differential-algebraic equations where the unknowns include both the acceleration of generalized coordinates and the Lagrange multipliers (representing reaction forces due to constraints). This approach converts the original equations of motion by adding the constraint equations multiplied by Lagrange multipliers, allowing simultaneous solution of the dynamics while respecting motion limitations.

PID controller tuning methods are empirical techniques used to determine optimal controller parameters (KP, TI, TD) for industrial processes. The Ziegler-Nichols methods (1942) provide systematic approaches: the first method approximates first-order systems with dead time using tangent line analysis to extract plant parameters (gain K, time constant T, dead time L), while the second method determines critical gain and period by inducing sustained oscillations. The Cohen-Coon method (1950s) uses regression-based formulas for similar systems, and the ITAE method (1960s) minimizes time-weighted integral error for both tracking and regulatory modes. These classical methods remain widely used in industry as practical starting points, though they may require fine-tuning for optimal performance.

PID (Proportional-Integral-Derivative) control is the most widely used classical control structure. The control law combines proportional response (immediate error correction), integral action (eliminates steady-state error by integrating error over time), and derivative action (anticipates future error changes to reduce oscillations). PID tuning methods include experimental approaches (Ziegler-Nichols method, step response method) and model-based approaches (Partial Model Matching). Each parameter affects performance differently: increasing Kp speeds response but increases overshoot, increasing Ki eliminates steady-state error but may cause oscillations, and increasing Kd reduces oscillations by providing damping.

PID controller tuning determines how effectively a controller responds to process variations. Proportional action provides immediate response proportional to error magnitude, while integral action eliminates steady-state error through accumulated past error. Derivative action predicts future error based on current rate of change. Tuning involves balancing these actions to achieve desired response characteristics. Increasing proportional gain improves response speed but risks instability. Integral action reduces steady-state error but can cause oscillations. Derivative action dampens oscillations but amplifies noise. The Ziegler-Nichols method provides empirical tuning guidelines by inducing sustained oscillations and deriving parameters from critical gain and oscillation period measurements.

The PID controller is the 'brain' of automatic control systems, using three reasoning types: Proportional (P), Integral (I), and Derivative (D). Proportional action responds proportionally to error magnitude - larger errors produce larger corrections. Integral action accumulates small errors over time to eliminate steady-state deviations that proportional action cannot fix. Derivative action observes how quickly errors change and anticipates future trends, preventing overshoot. Each has parameters (KP, KI, KD) requiring tuning. High KP causes oscillations; high KI creates overshoot; KD smooths rapid responses. Tuning methods include Ziegler-Nichols (1942 experimental technique) and reaction curve methods.

Classical control theory analyzes time-varying systems described by differential equations. The course introduces Laplace transform as a mathematical tool. Control systems aim to achieve three objectives: no overshoot, fast response, and no steady-state error. Proportional control alone cannot eliminate steady-state error and creates trade-offs between overshoot and response speed. PID control combines proportional, integral, and derivative components to overcome these limitations. The transfer function represents the input-output relationship of dynamic systems, enabling analysis without directly solving differential equations.

Simscape is a MATLAB/Simulink extension that enables multi-domain physical system modeling through a physical network approach, where users build models using domain-specific blocks (like resistors, capacitors, and voltage sources) with bidirectional physical connections instead of unidirectional signal lines, eliminating the need to manually derive system-level equations; the workflow involves selecting components from foundation libraries, connecting them while ensuring same-domain compatibility, configuring parameters with units support, adding sensors and converters to interface with Simulink scopes, and using specialized solver configuration blocks with recommended solvers (ODE15s, ODE23t, or ODE14x) for accurate simulation of physical systems such as battery equivalent circuits.

Physical modeling in Simscape and Simulink is a model-based design approach where engineers create mathematical representations of physical systems using graphical block diagrams, enabling simulation and analysis of complex electromechanical systems before physical implementation. The course covers fundamental concepts including mathematical modeling, simulation configurations, solvers, and Stateflow for control logic implementation, with practical applications in mechanical, electrical, and electromechanical systems such as lever mechanisms, gear transmissions, and DC motor-driven fan loads.

This tutorial demonstrates how to model vehicle longitudinal dynamics using Simscape Multibody, covering the creation of vehicle body blocks parameterized with MATLAB variables, implementation of Magic Formula tire models for longitudinal force calculation, incorporation of wind and terrain effects through physical signal inputs, and addition of brake systems using loaded contact rotational friction blocks; the complete workflow includes building sensor systems for speed and distance measurement, automating parameter loading with Simulink callbacks, and validating simulations against real-world vehicle data from the Virginia Tech Formula Student team.

This video demonstrates the complete workflow for modeling and simulating a 2-link planar robotic arm using Solidworks for CAD design, Simscape Multibody Link for exporting the mechanical model to MATLAB/Simulink, and implementing a PID control system with inverse and forward kinematics to achieve precise trajectory tracking. The process involves creating individual CAD parts, assembling them with appropriate mates, exporting to Simscape Multibody, building a control algorithm in Simulink, and tuning PID parameters to minimize tracking error between desired and actual end-effector positions.

Simscape enables engineers to model multi-domain physical systems using physical networks with bidirectional energy flow between components, as opposed to traditional signal-based input/output modeling; the platform provides a MATLAB-based language for creating custom components and add-on libraries for advanced modeling, while its editing modes allow sharing models with users who don't have add-on product licenses for tasks like parameter testing and code generation.