Calc hessian of potential energy

[sadraddini]

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[sadraddini]

@copilot please write a unit test with higher degrees of freedom (feel free to make, for example, N dof chain of pendulums) and use my method, as well as a version that uses AutoDiff for the plant. Compare the times of two methods

[Copilot]

@copilot please write a unit test with higher degrees of freedom (feel free to make, for example, N dof chain of pendulums) and use my method, as well as a version that uses AutoDiff for the plant. Co...

Added a NDOFPendulumHessianTest in commit 0bb91a7 that builds a 10-DOF chain of planar pendulums (z-axis revolute joints, gravity in x-direction) and compares both methods:

Method 1 — CalcHessianOfPotentialEnergy: the bias-acceleration-based analytical method.

Method 2 — AutoDiff: converts the plant to MultibodyPlant<AutoDiffXd>, sets positions as AutoDiff variables with identity gradient (math::InitializeAutoDiff(q)), calls CalcGravityGeneralizedForces, and extracts the Hessian via H = -d(τ_g)/dq (since τ_g = −∂V/∂q).

Both are timed over 50 repetitions and results are printed. The test asserts agreement to 1e-10 tolerance.

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