Jacobian of chain of transformation matrices

[Subodh Mishra]

Hi,

I intend to determine the Jacobian of a chain of homogeneous transformation matrix multiplications w.r.t individual matrices and I am wondering if what I have implemented is correct or not. Here is an example:

In this example I intend to determine the Jacobian of T wrt T1, T2 and T3 and T = T1^{-1}*T2^{-1}*T3. Following is a mock code snippet. Can anyone please tell me if my approach is correct?

***********************************************************************************

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Pose3 T1, T2, T3;
Matrix J1, J2, J3, J4, J5, J6;
Pose3 T = T1.inverse(J1).compose(T2.inverse(J2), J3, J4).compose(T3, J5, J6);
// I intend to determine the Jacobians of T wrt T1, T2 and T3

Matrix J_T_T1 = J5*J3*J1;
Matrix J_T_T2 = J5*J4*J2;
Matrix J_T_T3 = J6;

// where J_T_T1 = Jacobian of T wrt T1
// J_T_T2 = Jacobian of T wrt T2
// J_T_T3 = Jacobian of T wrt T3

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************************************************************************************

I want to clarify this before I move on to my implementation.

Regards,

Subodh.

[Subodh Mishra]

My understanding (and assumption) here is:

J1 is the Jacobian of T1.inverse() wrt T1

J2 is the Jacobian of T2.inverse() wrt T2

J3 is the Jacobian of T1.inverse()*T2.inverse() wrt T1.inverse()

J4 is the Jacobian of T1.inverse()*T2.inverse() wrt T2.inverse()

J5 is the Jacobian of T wrt T1.inverse()*T2.inverse()

J6 is the Jacobian of T wrt T3

So by chain rule. we have

J_T_T1 = J5*J3*J1;

J_T_T2 = J5*J4*J2;

J_T_T3 = J6;