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PD & SPD Controllers (Stable Physics Tracking)

Controlling physics objects with raw forces often leads to jitter, overshoot, and “explosions.” PD Controllers (Proportional-Derivative) and their advanced cousin SPD (Stable Proportional Derivative) are the industry standard for making physics bodies follow targets—essential for VR hands, active ragdolls, and networked physics.

1. The PD Controller Formula

A PD controller is a feedback loop that calculates the necessary force to reach a target: Force = (PositionError * kP) + (VelocityError * kD)

• kP (Proportional): The “Spring” strength. How hard the object is pulled toward the target.
• kD (Derivative): The “Damper” strength. It resists motion to prevent the object from overshooting and oscillating forever.

2. SPD: The “Black Magic” of Stability

Traditional PD controllers can become unstable at high spring strengths or low framerates. SPD (Stable Proportional Derivative), introduced by Jie Tan, makes the controller implicit. It effectively calculates the force based on the predicted state of the next frame, making it nearly impossible to “explode.”

3. VR Physics Hands

This is the core of high-end VR interaction (like Boneworks or Half-Life: Alyx).

• The Setup: The “Physical Hand” is a Rigidbody driven by a PD controller toward the “Ghost Hand” (the actual VR controller position).
• Two-Way Force: Because it’s physics-based, if the hand is blocked by a wall, the PD controller applies a force to the Body. This gives you “Climbing” and “Physics Presence” for free.

4. Networking: Physics Syncing

Instead of teleporting a player’s position (which breaks physics and causes jitter), you can sync the Target Transform and use a PD controller locally to drive the actual Rigidbody.

• Result: Smooth movement that still reacts to local collisions and physics-based impulses.

5. Implementation (C# Example)

public class PDController : MonoBehaviour {
    public float kP = 1000f; // Spring
    public float kD = 100f;  // Damper
    public Transform target;
    private Rigidbody rb;

    void FixedUpdate() {
        Vector3 posError = target.position - transform.position;
        Vector3 velError = (target.position - lastTargetPos) / Time.fixedDeltaTime - rb.linearVelocity;
        
        Vector3 force = (posError * kP) + (velError * kD);
        rb.AddForce(force);

        lastTargetPos = target.position;
    }
}

6. References & Deep Dives

• Digital Opus: Controlling Objects with Forces
• Stable Proportional Derivative Control (Jie Tan et al.)

Summary

• PD Controllers turn your “Move” commands into “Physics” commands.
• kP gets you there; kD stops you from vibrating.
• Use SPD for high-precision character controllers that must never explode.