Is this not a repetition of the previous section 8.1.1 ? Or is there a difference when you talk about it from a perspective of trajectory?

Why would you want to convert between Joint Impedance and Carteisan Impedance at runtime. What benefit does that have?

How do you command eef forces in this paradigm? I see that you can set a reference acceleration, does it boil down to converting the desired eef force to an accleration? I which case, would this fall within the spectrum of direct force controllers?

Was this demonstrated somehwere else in the notes? (maybe chapter 3)?

Hi, does $$u_{k}$$ here represents the generic decision variable here (for both direct transcription and direct shooting)? I was initially confused thinking that the derivative being estimated here was only w.r.t the control input, in which case the second term inside the sigma does not make sense no?

Should the $$C_{E}$$ term be multiplied by $$\dot{p^{E}}$$

What happends to the $$\ddot{q_{d}}$$ term in the eqation below?

Shouldn't this be $$ \ddot{q} - \ddot{q_{d}} + K_{p}(\cdot) + K_{d}(\cdot) = M^{-1}(\cdot)\tau_{ext} $$

For purposes of clarity, It would be nice to see what exactly the control law is for all the different types of controllers discussed in this chapter, at the outset of each section. For position and direct force control this is mentioned explicitly. However, in the case of indirect force control the fact that:

$$u = -mg + k_{p}(\cdot) + k_{d}(\cdot)$$

needs to be inferred from the other equations. This is not really a problem, but can defintely improve readability.

Thanks for your work Prof. Tedrake.

I'm having trouble understanding how this is a controller. Shouldn't there be u term somewhere in this equation?

The rotation described in the equation below is between frames B and C. It is unclear what the B frame here is. Can you elaborate?

Is this graph supposed to be between the generalized vectors $$x = [\theta, \dot{\theta}]^{T}, \dot{x} = [\dot{\theta}, \ddot{\theta}]$$

I ask because, the notation seems to be a little confusing. if $$u, b = 0$$, isn't the graph that you are constructing here between $$\theta, \ddot{\theta}$$

With respect to your main thesis in your underactuated course, is it fair to say that the loss of rank in a singular configuration equates to the system becoming underactuated? Or am I mixing two different ideas here?

If I were to have a cable rack around a position controlled manipulation arm (no joint torque sensing), would it be fair to say that the cable rack imposes further position-velocity constraints on the C-Space (joint configuration) of the robot? If so, how would one go about writing these constraints in the language of this section? I can imagine the v_range and q_range can be extracted from an assumed model of the cable rack (maybe a rope model).

What does this mean, in an mxn matrix, where m<n, is there such a thing as a column rank exceeding the row rank? In the row reduced form are there not only a certain set of pivots and the column rank is equal to the row rank. It may not be full row rank/ full column rank, what am I missing here?

Is there a relation between $$\delta$$ and $$\epsilon$$ ? Or are they just indications of the initial set of states and the final set of states of the system?

Could you explain what this means? The formulation is least squares, and my naive understanding is that objective functions that are formulated as least squares cost is always parabolic and hence qualifies as convex optimization.

First Assumption. Just for my clarity. No doubt here.

What does high-gain feedback mean in this context? I am new to control so the only formulation I have of this is designing a PID controller without taking into account the dynamics of the robot. In which case, does it mean the estimated proportional/derivative/integral constants that you arrive at after testing the system out?