Doubts about the size of the Hessian matrix

[Mr-Red331]

Hi, there. Thanks for making this great repo public.

I'm using Pinocchio to compute the partial derivatives of the Jacobian matrix with respect to q, a.k.a. dJ/dq. I have been calculating the Hessian matrix in the following way:

    pinocchio::computeJointJacobians(model, data, q);
    for(int i=0; i<28; i++){
        if(num_joint[i] == END_INPUT_FLAG) break;
        pinocchio::getJointJacobian(model, data, num_joint[i], frame, _wbc_all_jac[num_joint[i]]);
    }
    pinocchio::computeJointKinematicHessians(model, data);
    Tensor<double, 3> H;
    H = pinocchio::getJointKinematicHessian(model, data, Walker::end_point_left_limb, LOCAL_WORLD_ALIGNED);
    std::cout << "\n H size \n" << H.dimension(0) << "x" << H.dimension(1) << "x" << H.dimension(2) << std::endl;

The final output result is 6x32x32, which seems to represent the Hessian matrix for all 32 joints. However, based on my understanding of the Hessian matrix, its dimension should be 6x32, matching the size of the Jacobian matrix. This has led to my confusion about the additional 32 dimensions in the Hessian matrix. How should I select the 6x32x32 H matrix above to populate the Hessian matrix that I need

[jcarpent]

The kinematic Jacobian is the variation of a placement entity (position + orientation) with respect to the joint configuration vector.
Otherwise said, the kinematic Jacobian is the first-order derivative, reading $^0J_i = \frac{\partial ^0M_i(q)}{\partial q}$. The Jacobian evaluated at a given configuration value $q$ is a matrix of dimension $6 x \text{nv}$ where $\text{nv}$ is the size of the velocity vector (given by model.nv in Pinocchio), which can also be interpreted as the number of degrees of freedom.

The kinematic Hessian is one order of derivation more and corresponds to $^0H_i = \frac{{\partial^{2}} \ {^0M_i}(q)}{\partial q^2}$.
So, it is a $6 x \text{nv} x \text{nv}$ tensor, with one dimension more due to the additional order of derivation with respect to the kinematics Jacobian.

I hope this clarifies your issue.

[Mr-Red331]

Thank you for your patient and thorough response. Your answer has resolved my concerns.

However, I still have another question regarding the form of the Hessian matrix: if I need the value of $\frac{\partial ^0J}{\partial q_i} $, where $q_i$ is the ith value of $q$，which matrix do I need, $^0H(:, :, i)$ or $^0H(:, i, :)$ ?