Concept 02: Work, Kinetic Energy & Elevator Counterbalancing

Physics in robotics is governed by the conservation of Work, Energy, and Momentum:

1. Kinetic Energy (KE = ½ · m · v²): A 60 kg robot sprinting at 5.5 m/s carries nearly 900 Joules of kinetic energy that must be safely absorbed by bumpers during defense collisions.
2. Gravitational Potential Energy (PE = m · g · h): An elevator lifting a heavy 15 kg carriage must continuously fight gravity with high motor stall current—unless you counterbalance it with Constant-Force Springs.

Open the interactive demo below to adjust elevator mass and constant-force spring assistance, and observe how spring potential energy reduces the motor’s holding voltage to near zero.

1. Kinetic Energy & Momentum in Collisions

When a robot of mass m moves at velocity v:

• Linear Momentum (p = m · v): Governs collision impulse (F · Δt = Δp).
• Kinetic Energy (KE = ½ · m · v²): Because energy scales with velocity squared (v²), a robot driving at 6 m/s carries 4× more destructive energy than one driving at 3 m/s!

2. The Elevator Gravity Problem

To hold a 12 kg elevator carriage stationary at a height of 1.5 meters, gravity constantly pulls downward with a force:

F_gravity = m · g = 12.0 kg · 9.81 m/s² = 117.7 Newtons

Without compensation, the lift motors must continuously apply holding voltage (~2.5 to 3.5 Volts), drawing steady stall current, draining the battery, and heating up motor coils.

Furthermore, moving UP requires fighting F_gravity + F_accel, while moving DOWN is accelerated by gravity (-F_gravity + F_accel), making control asymmetrical.

3. The Counterbalance Solution: Constant-Force Springs

A Constant-Force Spring (made from pre-stressed coiled spring steel) exerts an almost perfectly constant upward pulling force F_spring regardless of how far the elevator extends.

       Top Pulley
          ┌─┐
          │ │
          └┬┘
           │ 
       ┌───┴───┐ ◄── Pulling Force F_spring (Constant ~118 N)
       │Carriage│
       │(12 kg) │
       └───┬───┘ ◄── Gravity F_g = m · g (118 N)

By sizing springs such that F_spring ≈ m_carriage · g:

F_net_gravity = (m · g) - F_spring ≈ 0 Newtons

Benefits of Mechanical Counterbalancing:

1. Holding Current Drops to 0 Amps: The elevator floats in mid-air at any height with zero battery drain.
2. Symmetrical Motion: Lifting up and lowering down feel identical to the control loop.
3. Double the Speed: 100% of motor torque is dedicated to pure acceleration rather than fighting Earth’s gravity.

4. Solving It in Code (Java & WPILib)

public class ElevatorPhysics {
    public static final double GRAVITY = 9.81;

    public static double computeRequiredHoldingVoltage(
            double carriageMassKg, 
            double springAssistN, 
            double spoolRadiusMeters, 
            double gearRatio, 
            double motorKt, 
            double motorR) {
        
        // 1. Net downward force after spring assist
        double netForceDown = (carriageMassKg * GRAVITY) - springAssistN;

        // 2. Torque required at motor shaft
        double spoolTorque = netForceDown * spoolRadiusMeters;
        double motorTorque = spoolTorque / gearRatio;

        // 3. Motor current and required holding voltage
        double currentAmps = motorTorque / motorKt;
        double holdingVolts = currentAmps * motorR;

        return Math.max(0.0, holdingVolts);
    }

    public static void main(String[] args) {
        double mass = 12.0;            // 12 kg carriage
        double spoolRadius = 0.0254;   // 1-inch spool radius
        double gearRatio = 15.0;       // 15:1 gearbox
        double kt = 0.019;             // Kraken motor Kt
        double r = 0.025;              // Kraken internal resistance

        // Case A: Uncompensated Elevator (0 N Spring)
        double voltsUnbalanced = computeRequiredHoldingVoltage(mass, 0.0, spoolRadius, gearRatio, kt, r);

        // Case B: Counterbalanced Elevator (Two 55 N Springs = 110 N)
        double voltsBalanced = computeRequiredHoldingVoltage(mass, 110.0, spoolRadius, gearRatio, kt, r);

        System.out.printf("Holding Voltage WITHOUT Springs: %.2f Volts (High Heat!)%n", voltsUnbalanced);
        System.out.printf("Holding Voltage WITH Springs:    %.2f Volts (Cool & Efficient)%n", voltsBalanced);
        // Output: ~3.28V drops to ~0.21V!
    }
}

5. Math! Translation Sidebar

Work-Energy and Potential Energy formulations:

W = ∫ F · dx = ΔKE + ΔPE
PE_total = m · g · h - F_spring · h ≈ 0

Kinetic Energy in Robot Collisions:

• KE = ½ · m · v²
• A 60 kg robot moving at 5.5 m/s delivers ½ · 60 · (5.5)² = 907.5 Joules. Compressible foam bumpers provide ~0.05m of deceleration distance, resulting in peak impact forces of F_impact = W / d = 907.5 / 0.05 = 18,150 Newtons (~4,000 lbs of force)!