Page 1CS 287: Advanced RoboticsFall 2009Lecture 2: Control 1: Feedforward, feedback, PID, Lyapunov direct methodPieter AbbeelUC Berkeley EECS Office hours: Thursdays 2-3pm + by email arrangement, 746 SDH SDH 7thfloor should be unlocked during office hours on Thursdays Questions about last lecture?AnnouncementsPage 2 Control Estimation Manipulation/Grasping Reinforcement Learning Misc. Topics Case StudiesCS 287 Advanced Robotics Overarching goal: Understand what makes control problems hard What techniques do we have available to tackle the hard (and the easy) problems Any applicability of control outside robotics? Yes, many! Process industry, feedback in nature, networks and computing systems, economics, …  [See, e.g., Chapter 1 of Astrom and Murray, http://www.cds.caltech.edu/~murray/amwiki/Main_Page, for more details---_optional_ reading. Fwiw: Astrom and Murray is a great read on mostly classical feedback control and is freely available at above link.] We will not have time to study these application areas within CS287 [except for perhaps in your final project!]Control in CS287Page 3 Feedforward vs. feedback PID (Proportional Integral Derivative) Lyapunov direct method --- a method that can be helpful in proving guarantees about controllers Reading materials: Astrom and Murray, 10.3 Tedrake, 1.2  Optional: Slotine and Li, Example 3.21.Today’s lectureBased on a survey of over eleven thousand controllers in the refining, chemicals and pulp and paper industries, 97% of regulatory controllers utilize PID feedback.L. Desborough and R. Miller, 2002 [DM02]. [Quote from Astrom and Murray, 2009] Practical result: can build a trajectory controller for a fully actuated robot arm Our abstraction: torque control input to motor, read out angle [in practice: voltages and encoder values]Today’s lecturePage 4Intermezzo: Unconventional (?) robot arm useSingle link manipulator (aka the simple pendulum)θmguI¨θ(t) + b˙θ(t) + mgl sin θ(t) = u(t)lI=ml2Page 5Single link manipulatorI¨θ(t) + b˙θ(t) + mgl sin θ(t) = u(t)θmgulHow to hold arm at θ = 45 degrees?Single link manipulatorSimulation results:I¨θ(t) + c˙θ(t) + mgl sin θ(t) = u(t), u = mgl sinπ4θ(0) =π4,˙θ(0) =0θ(0) = 0,˙θ(0) = 0Can we do better than this?The Matlab code that generated all discussed simulations will be posted on www.Page 6Feedforward controlI¨θ(t) + b˙θ(t) + mgl sin θ(t) = u(t)θmgulHow to make arm follow a trajectory θ*(t) ? θ(0) = 0,˙θ(0) = 0u(t) = I¨θ∗(t) + c˙θ∗(t) + mgl sin θ∗(t)Feedforward controlSimulation results:I¨θ(t) + c˙θ(t) + mgl sin θ(t) = u(t)Can we do better than this?θ(0) = 0,˙θ(0) = 0u(t) = I¨θ∗(t) + c˙θ∗(t) + mgl sin θ∗(t)Page 7 Thus far: n DOF manipulator: standard manipulator equations H : “inertial matrix,” full rank  B : identity matrix if every joint is actuated  Given trajectory q(t), can readily solve for feedforward controls u(t) for all times tn DOF (degrees of freedom) manipulator?I¨θ(t) + b˙θ(t) + mgl sin θ(t) = u(t)θmgulH(q)¨q + C(q, ˙q) + G(q) = B(q)u A system is fully actuated when in a certain state (q,\dot{q},t) if, when in that state, it can be controlled to instantaneously accelerate in any direction.  Many systems of interest are of the form: Defn. Fully actuated: A control system described by Eqn. (1) is fully-actuated in state (q,\dot{q},t) if it is able to command an instantaneous acceleration in an arbitrary direction in q: Defn. Underactuated: A control system described by Eqn. (1) is underactuated in configuration (q,\dot{q},t) if it is not able to command an instantaneous acceleration in an arbitrary direction in q:[See also, Tedrake, Section 1.2.]Fully-Actuated vs. Underactuated¨q = f1(q, ˙q, t) + f2(q, ˙q, t)u (1)rankf2(q, ˙q, t) = dimqrankf2(q, ˙q, t) < dimqPage 8Fully-Actuated vs. Underactuated¨q = f1(q, ˙q, t) + f2(q, ˙q, t)u fully actuated in (q, ˙q, t) iff rankf2(q, ˙q, t) = dimq. Hence, for any fully actuated system, we can follow a trajectory by simply solving for u(t): [We can also transform it into a linear system through a change of variables from u to v:The literature on control for linear systems is very extensive and hence this can be useful. This is an example of feedback linearization. More on this in future lectures.]u(t) = f−12(q, ˙q, t) (¨q−f1(q, ˙q, t))u(t) = f−12(q, ˙q, t) (v(t)−f1(q, ˙q, t))¨q(t) = v(t) n DOF manipulator All joints actuated  rank(B) = n  fully actuated Only p < n joints actuated  rank(B) = p  underactuatedFully-Actuated vs. Underactuated¨q = f1(q, ˙q, t) + f2(q, ˙q, t)u fully actuated in (q, ˙q, t) iff rankf2(q, ˙q, t) = dimq.H(q)¨q + C(q, ˙q) + G(q) = B(q)uf2= H−1B, H full rank, B = I, hence rank(H−1B) = rank(B)Page 9 Car Cart-poleExample underactuated systems Acrobot HelicopterFully actuated systems: is our feedforward control solution sufficient in practice?I¨θ(t) + b˙θ(t) + mgl sin θ(t) = u(t)θmgulTask: hold arm at 45 degrees. What if parameters off? --- by 5%, 10%, 20%, … What is the effect of perturbations?Page 10Fully actuated systems: is our feedforward control solution sufficient in practice?I¨θ(t) + b˙θ(t) + mgl sin θ(t) = u(t)θmgulTask: hold arm at 45 degrees. Mass off by 10%: steady-state errorFully actuated systems: is our feedforward control solution sufficient in practice?I¨θ(t) + b˙θ(t) + mgl sin θ(t) = u(t)θmgulTask: swing arm up to 180 degrees and hold there Perturbation after 1sec: Does **not** recover[θ = 180 is an “unstable” equilibrium point]Page 11 Feedback can provide Robustness to model errors However, still: Overshoot issues --- ignoring momentum/velocity! Steady-state error --- simply crank up the gain?Proportional controlu(t) = ufeedforward(t)Task: hold arm at 45 degreesu(t) = ufeedforward(t) + Kp(qdesired(t)−q(t))Proportional controlu(t) = ufeedforward(t) + Kp(qdesired(t)−q(t))Task: swing arm up to 180degrees and hold thereu(t) = ufeedforward(t)Page 12Current status Feedback can provide Robustness to model errors Stabilization around states which are unstable in open-loop Overshoot issues --- ignoring momentum/velocity! Steady-state error --- simply crank up the gain?PD controlu(t) = Kp(qdesired(t)−q(t)) + Kd( ˙qdesired−˙q(t))Page 13Eliminate steady state error by