thowell
commented
1 month ago Great questions!
The MuJoCo sensor array is populated by standard sensors (eg, framepos) and user sensors that are defined by the user via a callback. For MJPC, we leverage custom sensors to implement a residual vector that is utilized by the cost function.
A call to mj_step processes the current state $x_t$ and action $ut$ and returns the next state $x{t+1}$ and sensor values $y_t$:
$$x_{t+1}, y_t \leftarrow \text{step}(x_t, u_t)$$
The sensor values are computed prior to integration and are a function of the current state $x_t$ and action $u_t$, see engine_forward.c.
In the MJPC task xml we require that the custom sensors for the residual are specified first, before any measurement sensors (eg, cartpole). As a result, we consider the measurements:
$$y= (r, m) \in \mathbf{R}^{n_r + n_m}$$
comprising the residual $r$ and measurements $m$. As a result of this specification, the residual Jacobians ($r_x$, $r_u$) are the first rows of the sensor Jacobians:
$$C_t = \partial y_t / \partial x_t = [r_x; m_x]\in \mathbf{R}^{(n_r + n_m) \times n_x}$$
$$D_t = \partial y_t / \partial u_t = [r_u; m_u] \in \mathbf{R}^{(n_r + n_m) \times n_u}$$
DMackRus
Hello, I have a question regarding how cost derivatives are computed in this repository, I have read the paper as well as the overview document and I understand the mathematical theory about how the residuals are used with a normal function and then a scalar weight to compute costs.
You detail in the paper how the cost derivatives with respect to the state and control vector are computed, the first and second order derivative of the norm functions are done analytically, and the first order residual derivatives are computed via finite-differencing. Second order derivatives of the residuals are computed using the Gauss-newton approximation, the theory behind all this makes sense.
I have two questions;
The first question is regarding the use of the $C$ and $D$ matrices as the $r_x$ and $r_u$ matrices.
This line in the iLQG planner makes a function call to compute the cost derivatives, and it passes in the $C$ and $D$ matrices in which then become the variables $r_x$ and $r_u$, this implies to me that the $C$ and $D$ matrices are already the residual derivatives, however from my understanding, the $C$ and $D$ matrices are how the sensor outputs change with respect to the state and control vector.
I know the sensors are used to compute the residuals, but I didnt think they are exactly the same, i.e they don't necessarily have the same dimension, as one residual can be a combination of multiple sensor values? So I would have though that to compute $r_x$ the chain rule would be needed to convert $C$ to $r_x$ by multiplying by how the residuals change with respect to the sensors. I.e:
$rx = r{sensor} * C$
Secondly, assuming I am missing something with regards to the first point, is there not an issue with how the $C$ and $D$ matrices is computed, when you compute the $A$ and $B$ matrices, it is required to integrate the simulator as we are evaluating the system dynamics, but the $C$ and $D$ matrices are only meant to tell us how the outputs of the system change with respect to the state and control vector( at least I thought so). i.e $C_t = \delta y_t / \delta x_t$ and not $Ct = \delta y{t+1} / \delta x_t$. When I look into mjd_transitionFD it appears that the system is integrated when computing the sensor outputs as well?
Any help understanding these points would be greatly appreciated!