# Velocity Verlet WORSE than Sympletic Euler for simple spring. What gives?

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Hello

I've taken it upon myself to finally start learning how game engines work by making a toy engine myself. I've never done any game programming or anything like that but as an avid gamer and someone with a physics degree I've always wondered how these things are done.

Today I implemented a basic spring force using Hooks Law. I was surprised however when over a short time span, maybe 30s, my default integrator of Velocity Verlet caused the spring's motion to escalate wildly. I tried stepping down to Symplectic Euler and the motion became normal. At first I was thinking there might be a bug in my code, but I can't see one so now I'm wondering if the good people of this forum might help shed some light on this. It does not make any sense to me that Symplectic Euler would be a better integrator than Velocity Verlet. Both are time reversible and symplectic, but Velocity Verlet is second order accurate.

I've been writing up what I learn on a little blog, feel free to check it out to see my current understanding of these methods

Code

Symplectic Euler

void SymplecticEuler::Solve(
const NetForceAccumulator& net_force_accumulator,
const std::vector<std::shared_ptr<PhysicsEntity>> &entity_ptrs,
const std::shared_ptr<PhysicsEntity> entity_ptr)
{
Vector3Gf xi = entity_ptr->GetPosition();
Vector3Gf vi = entity_ptr->GetVelocity();
GLfloat mass = entity_ptr->GetMass();

Vector3Gf F;
F.setZero();
net_force_accumulator.ComputeNetForce(entity_ptrs,entity_ptr,F);

Vector3Gf vf = vi + m_dt*(1/mass)*F;
Vector3Gf xf = xi + m_dt*vf;

entity_ptr->SetNextPosition(xf);
entity_ptr->SetNextVelocity(vf);
};

Velocity Verlet

void Verlet::Solve(
const NetForceAccumulator& net_force_accumulator,
const std::vector<std::shared_ptr<PhysicsEntity>> &entity_ptrs,
const std::shared_ptr<PhysicsEntity> entity_ptr)
{
Vector3Gf xi = entity_ptr->GetPosition();
Vector3Gf vi = entity_ptr->GetVelocity();
GLfloat mass = entity_ptr->GetMass();

Vector3Gf Fi,Ff,ai,xf,af,vf;

Fi.setZero();
net_force_accumulator.ComputeNetForce(entity_ptrs,entity_ptr,Fi);

ai = (1/mass)*Fi;
xf = xi + m_dt*vi + 0.5f*m_dt*m_dt*ai;

entity_ptr->SetPosition(xf); // Load xf into position slot for computing F(x(t+h))

Ff.setZero();
net_force_accumulator.ComputeNetForce(entity_ptrs,entity_ptr,Ff); // Compute F(x(t+h))

af = (1/mass)*Ff;
vf = vi + 0.5f*m_dt*(ai + af);

entity_ptr->SetPosition(xi); // Load xi back into position slot as to not affect force calulations for other entities

entity_ptr->SetNextPosition(xf);
entity_ptr->SetNextVelocity(vf);
};

Spring force

void SpringForceGenerator::AccumulateForce(
const GLfloat k,
const GLfloat l0,
const std::shared_ptr<PhysicsEntity> entity1_ptr,
const std::shared_ptr<PhysicsEntity> entity2_ptr,
Vector3Gf &F) const
{
Vector3Gf x1 = entity1_ptr->GetPosition();
Vector3Gf x2 = entity2_ptr->GetPosition();

Vector3Gf n = x2 - x1;
GLfloat l = n.norm();
n = n/l;

F += k*(l - l0)*n;
}

Thanks!

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I believe that the reason for the enhanced stability of Sympletic/Semi-implicit Euler is that it tends to have a damping effect (loses energy) and undershoots the integral, rather than overshoots. Vertlet might be more accurate but since it doesn't have that damping effect it is less stable. This is why most game physics engines (e.g. Bullet) use Semi-implicit Euler, even though it's not that accurate (it's also faster).

When I once implemented physics for a cannon projectile, Semi-implicit Euler would always undershoot the analytical trajectory, regular Euler would overshoot, and RK4 would be very close to the analytical path. With springs/constraints, RK4 was actually less stable than Semi-implicit Euler. Accuracy != stability.

Edited by Aressera

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Thanks for your reply Aressera. I'm aware of the difference between accuracy and stability (I wrote some articles about it as a slightly high level on the blog I linked). I would have expected the region of stability for Velocity Verlet to be nicer than Symplectic Euler, although to be fair I've never seen it pictured. Since both methods are symplectic they should both nearly conserve energy (see sections 1.1.3 and 1.2.2 here) so I would not have expected the damping of Symplectic Euler to have such a pronounced effect. However I guess the degree to which they (nearly) preserve energy need not be the same. I am new to all of this so I'll have to take your word for it. It's good to know I'm not insane after all!

Thanks!

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Shortcut to sterp implementation.
Shortcut to code used to generate animations in this post.
An Alternative to Slerp
Slerp, spherical linear interpolation, is an operation that interpolates from one orientation to another, using a rotational axis paired with the smallest angle possible.
Quick note: Jonathan Blow explains here how you should avoid using slerp, if normalized quaternion linear interpolation (nlerp) suffices. Long store short, nlerp is faster but does not maintain constant angular velocity, while slerp is slower but maintains constant angular velocity; use nlerp if you’re interpolating across small angles or you don’t care about constant angular velocity; use slerp if you’re interpolating across large angles and you care about constant angular velocity. But for the sake of using a more commonly known and used building block, the remaining post will only mention slerp. Replacing all following occurrences of slerp with nlerp would not change the validity of this post.
In general, slerp is considered superior over interpolating individual components of Euler angles, as the latter method usually yields orientational sways.
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If we slerp between the two orientations, this is what we get:

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This is where swing-twist decomposition comes in.

Swing-Twist Decomposition
Swing-Twist decomposition is an operation that splits a rotation into two concatenated rotations, swing and twist. Given a twist axis, we would like to separate out the portion of a rotation that contributes to the twist around this axis, and what’s left behind is the remaining swing portion.
There are multiple ways to derive the formulas, but this particular one by Michaele Norel seems to be the most elegant and efficient, and it’s the only one I’ve come across that does not involve any use of trigonometry functions. I will first show the formulas now and then paraphrase his proof later:
Given a rotation represented by a quaternion R = [W_R, vec{V_R}] and a twist axis vec{V_T}, combine the scalar part from R the projection of vec{V_R} onto vec{V_T} to form a new quaternion: T = [W_R, proj_{vec{V_T}}(vec{V_R})]. We want to decompose R into a swing component and a twist component. Let the S denote the swing component, so we can write R = ST. The swing component is then calculated by multiplying R with the inverse (conjugate) of T: S= R T^{-1} Beware that S and T are not yet normalized at this point. It's a good idea to normalize them before use, as unit quaternions are just cuter. Below is my code implementation of swing-twist decomposition. Note that it also takes care of the singularity that occurs when the rotation to be decomposed represents a 180-degree rotation. public static void DecomposeSwingTwist ( Quaternion q, Vector3 twistAxis, out Quaternion swing, out Quaternion twist ) { Vector3 r = new Vector3(q.x, q.y, q.z); // singularity: rotation by 180 degree if (r.sqrMagnitude < MathUtil.Epsilon) { Vector3 rotatedTwistAxis = q * twistAxis; Vector3 swingAxis = Vector3.Cross(twistAxis, rotatedTwistAxis); if (swingAxis.sqrMagnitude > MathUtil.Epsilon) { float swingAngle = Vector3.Angle(twistAxis, rotatedTwistAxis); swing = Quaternion.AngleAxis(swingAngle, swingAxis); } else { // more singularity: // rotation axis parallel to twist axis swing = Quaternion.identity; // no swing } // always twist 180 degree on singularity twist = Quaternion.AngleAxis(180.0f, twistAxis); return; } // meat of swing-twist decomposition Vector3 p = Vector3.Project(r, twistAxis); twist = new Quaternion(p.x, p.y, p.z, q.w); twist = Normalize(twist); swing = q * Quaternion.Inverse(twist); } Now that we have the means to decompose a rotation into swing and twist components, we need a way to use them to interpolate the rod’s orientation, replacing slerp.
Swing-Twist Interpolation
Replacing slerp with the swing and twist components is actually pretty straightforward. Let the Q_0 and Q_1 denote the quaternions representing the rod's two orientations we are interpolating between. Given the interpolation parameter t, we use it to find "fractions" of swing and twist components and combine them together. Such fractiona can be obtained by performing slerp from the identity quaternion, Q_I, to the individual components. So we replace: Slerp(Q_0, Q_1, t) with: Slerp(Q_I, S, t) Slerp(Q_I, T, t) From the rod example, we choose the twist axis to align with the rod's longest side. Let's look at the effect of the individual components Slerp(Q_I, S, t) and Slerp(Q_I, T, t) as t varies over time below, swing on left and twist on right:
And as we concatenate these two components together, we get a swing-twist interpolation that rotates the rod such that its moving end travels in the shortest arc in 3D. Again, here is a side-by-side comparison of slerp (left) and swing-twist interpolation (right):

I decided to name my swing-twist interpolation function sterp. I think it’s cool because it sounds like it belongs to the function family of lerp and slerp. Here’s to hoping that this name catches on.
And here’s my code implementation:
public static Quaternion Sterp ( Quaternion a, Quaternion b, Vector3 twistAxis, float t ) { Quaternion deltaRotation = b * Quaternion.Inverse(a); Quaternion swingFull; Quaternion twistFull; QuaternionUtil.DecomposeSwingTwist ( deltaRotation, twistAxis, out swingFull, out twistFull ); Quaternion swing = Quaternion.Slerp(Quaternion.identity, swingFull, t); Quaternion twist = Quaternion.Slerp(Quaternion.identity, twistFull, t); return twist * swing; } Proof
Lastly, let’s look at the proof for the swing-twist decomposition formulas. All that needs to be proven is that the swing component S does not contribute to any rotation around the twist axis, i.e. the rotational axis of S is orthogonal to the twist axis. Let vec{V_{R_para}} denote the parallel component of vec{V_R} to vec{V_T}, which can be obtained by projecting vec{V_R} onto vec{V_T}: vec{V_{R_para}} = proj_{vec{V_T}}(vec{V_R}) Let vec{V_{R_perp}} denote the orthogonal component of vec{V_R} to vec{V_T}: vec{V_{R_perp}} = vec{V_R} - vec{V_{R_para}} So the scalar-vector form of T becomes: T = [W_R, proj_{vec{V_T}}(vec{V_R})] = [W_R, vec{V_{R_para}}] Using the quaternion multiplication formula, here is the scalar-vector form of the swing quaternion: S = R T^{-1} = [W_R, vec{V_R}] [W_R, -vec{V_{R_para}}] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_R} + W_R (-vec{V_{R_para}})] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R (vec{V_R} -vec{V_{R_para}})] = [W_R^2 - vec{V_R} ‧ (-vec{V_{R_para}}), vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}}] Take notice of the vector part of the result: vec{V_R} X (-vec{V_{R_para}}) + W_R vec{V_{R_perp}} This is a vector parallel to the rotational axis of S. Both vec{V_R} X(-vec{V_{R_para}}) and vec{V_{R_perp}} are orthogonal to the twist axis vec{V_T}, so we have shown that the rotational axis of S is orthogonal to the twist axis. Hence, we have proven that the formulas for S and T are valid for swing-twist decomposition. Conclusion
That’s all.
Given a twist axis, I have shown how to decompose a rotation into a swing component and a twist component.
Such decomposition can be used for swing-twist interpolation, an alternative to slerp that interpolates between two orientations, which can be useful if you’d like some point on a rotating object to travel along the shortest arc.
I like to call such interpolation sterp.
Sterp is merely an alternative to slerp, not a replacement. Also, slerp is definitely more efficient than sterp. Most of the time slerp should work just fine, but if you find unwanted orientational sway on an object’s moving end, you might want to give sterp a try.

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