20/10/2009 IVR Herrmann IVR:Control Theory OVERVIEW Control problems Kinematics Examples of control in a physical system A simple approach to kinematic control

20/10/2009 IVR Herrmann The control problem For given motor commands, what is the outcome? → Forward model For a desired outcome, what are the motor commands? → Inverse model From observing the outcome, how should we adjust the motor commands to achieve a goal? → Feedback control Motor command Robot in environment Outcome Goal Action

20/10/2009 IVR Herrmann The control problem Forward kinematics is not trivial but usually possible Forward dynamics is hard and at best will be approximate But what we actually need is backwards kinematics and dynamics Difficult! V(t) T(t) C(t) A(t) X(t) command voltage torque force angle position camera

20/10/2009 IVR Herrmann Inverse model Find motor command given desired outcome Solution might not exist Non-linearity of the forward transform Ill-posed problems in redundant systems Robustness, stability, efficiency,... Partial solution and their composition V(t) T(t) C(t) A(t) X(t)

20/10/2009 IVR Herrmann Problem: Non-linearity In general, we have good formal methods for linear systems Reminder: Linear function: In general, most robot systems are non-linear x F(x)

20/10/2009 IVR Herrmann Kinematic (motion) models Differentiating the geometric model provides a motion model (hence sometimes these terms are used interchangeably) This may sometimes be a method for obtaining linearity (i.e. by looking at position change in the limit of very small changes) x y r = (x,y) φ x = l cos φ y = l sin φ φ = atan2(x,y) l 0 Example: A simple arm model

20/10/2009 IVR Herrmann Differential Equations Mathematics: Equation that is to be solved for an unknown function Physics: Description of processes in nature Engineering: Realizability of a goal by a plant by including control terms Informatics: Tool for realistic modeling Using known relations between quantities and their rate of change in order to find out how these quantities change

20/10/2009 IVR Herrmann Differential Equations fast growth starting from initial value x(t 0 )=x 0 decay with time scale -1/a=τ

20/10/2009 IVR Herrmann Dynamic models Kinematic models neglect forces: motor torques, inertia, friction, gravity… To control a system, we need to understand the continuous process Now: An Example for control of a physically realistic model Next: A simple example of control

20/10/2009 IVR Herrmann Dynamic models Kinematic models neglect forces: motor torques, inertia, friction, gravity… To control a system, we need to understand the continuous process Start with simple linear example: Battery voltage V B Vehicle speed s ? VBVB I R e

20/10/2009 IVR Herrmann Example: Electric motor Ohm’s law & Kirchhoff's law Motor generates voltage: proportional to speed Vehicle acceleration: ( M is a motor constant) Torque  is proportional to current: Putting together:

20/10/2009 IVR Herrmann General form V B – Control variable – input s – State variable – output A+Bd/dt – Process dynamics Dynamics determines the process, given an initial state s(t 0 )=s 0. State variable s(t) separates past and future Continuous process models are often differential equations!

20/10/2009 IVR Herrmann Process Characteristics Given the process, how to describe the behaviour? Concise, complete, implicit, obscure … Characteristics: Steady-state: What happens if we wait for the system to settle, given a fixed input? Transient behaviour: What happens if we suddenly change the input? Frequency response: What if we smoothly/regularly change the inputs?

20/10/2009 IVR Herrmann Control theory Control theory provides tools: Steady-state: ds/dt = 0, Transient behaviour (e.g. change in voltage from 0 to 7V) exponential decay towards steady state Half-life of decay: (Solve for using )

20/10/2009 IVR Herrmann

Example Suppose: M :vehicle mass R :setting If robot starts at rest, and apply 7 volts: Steady state speed Half-life: Time taken to cover half the gap between current and steady-state speed

20/10/2009 IVR Herrmann Motor with gears Battery voltage V B s out ? Gear ratio  where more gear-teeth near output means  > 1 s motor s motor =  s out : for  > 1, output velocity is slower torque motor =  -1 torque out : for  > 1, output torque is higher Thus: Same form, different steady-state, time-constant etc.

20/10/2009 IVR Herrmann Motor with gears Steady-state: Half-life: i.e. for γ > 1, reach lower speed in faster time, robot is more responsive, though slower. N.B. we have modified the dynamics by altering the robot morphology.

20/10/2009 IVR Herrmann Electric Motor Over Time Simple dynamic example – We have a process model: Solve to get forward model: Derivation of this and more general cases using e.g. Laplace transformation Battery voltage V B Vehicle speed v ? VBVB IR e

20/10/2009 IVR Herrmann A fairly simple control algorithm Compensator High-frequency oscillator Compensator in order to determine the effector characteristics Effector High pass filter Control of the compensator characteristics Addition N. Wiener: Cybernetics, 1948

20/10/2009 IVR Herrmann a simple choice for prediction: x pred = x old System: System + Controller: What if there is no analytical description of the system? Stabilizing controller for box pushing or wall-following more complex behaviors for more complex predictors A Simple Controller a simple choice for the controller:

20/10/2009 IVR Herrmann How to find better parameters c i in K = Σ c i x i ? Perform “test actions” at both sides of the trajectory works best in 1D (e.g. for steering) A Simple Controller c expl = c + a sin(  t) Δc = short-term average

20/10/2009 IVR Herrmann Summary Forward and inverse models Calculating control is hard … but not impossible for many control problems Controlling by probing Feed back control (next time)

20/10/2009 IVR Herrmann Beyond Inverse Models Feed-back control Dynamical systems Adaptive control Learning control 1788 by James Watt following a suggestion from Matthew Boulton