1. Structure and Synthesis of Robot Motion Topics in Control Subramanian Ramamoorthy School of Informatics 19 February, 2009

2. In this Lecture We will survey some issues that occur in many realistic robotic systems and discuss control methods for dealing with them We will look at three ideas: predictive, adaptive and nonlinear control We will only try to get an understanding of what the essential issue is – detailed techniques beyond the scope of this lecture 19/02/2009Structure and Synthesis of Robot Motion2

3. Start from the Basics: A Simple System Consider the mechanical system shown From rest, if you apply force u(t), the position y(t) changes in response In fact, the response is given by an ODE: Define the state as: You get equations of motion: 19/02/2009Structure and Synthesis of Robot Motion3 m b k u(t) y(t)

4. How to Control the Simple System? For any system of the general form (very often D = 0 ), One could use many standard linear control techniques For instance, linear quadratic regulator, u = - Kx So, in general, properties of such a system are essentially that of the canonical system, 19/02/2009Structure and Synthesis of Robot Motion4

5. Properties of a Linear System 19/02/2009Structure and Synthesis of Robot Motion5

6. Properties of a Linear System, contd. 19/02/2009Structure and Synthesis of Robot Motion6 Makes design convenient!t!

7. Is that It? Where are the Complications? In real robotics problems, we face many difficulties that are not captured in the simple model. 19/02/2009Structure and Synthesis of Robot Motion7 m b k u(t) y(t) Workspace may change (e.g., Obstacles and constraints) at “run-time” - No longer possible to find and use global solutions System dynamics or operating conditions might change over time Finally, most robot systems’ dynamics are more complex and non-linear

8. Ways to Deal with these Complications Model Predictive Control Given a basic system dynamics model, pose long-term control problem as a sequence of finite-horizon optimal control problems – constraints can then be easily accommodated Adaptive Control If system dynamics is going to change over time, estimate the changes continually and change controller accordingly Nonlinear Control Finally, if the system dynamics is clearly nonlinear then one needs to do things differently – we will discuss linearization methods (many other possibilities exist!) 19/02/2009Structure and Synthesis of Robot Motion8

9. Model Predictive Control – Basic Idea Use a combination of system (sometimes called plant) model and state/control history to anticipate significant changes in control actions Example: Cruise control system in a car Fuel sent to engine depends on desired cruise velocity and actual velocity of the car Actual velocity depends on slope of the road (varies…) An MPC would try to predict significant changes in slope and correct for it 19/02/2009Structure and Synthesis of Robot Motion9

10. Factors Influencing the MPC Problem 19/02/2009Structure and Synthesis of Robot Motion10

11. How does MPC differ from Simple Feedback? What happens as the slope of the road increases? – Car velocity decreases – Fuel input to engine must be increased by the controller In a traditional feedback controller (u = -Kx), car must first slow down for corresponding fuel increase – Controller catches up, but you have oscillations A predictive controller uses knowledge of a finite-horizon forward in time to make corrections much earlier 19/02/2009Structure and Synthesis of Robot Motion11

12. How does Predictive Control work? 19/02/2009Structure and Synthesis of Robot Motion12 Use measurements to decide on set-points, e.g., is there a bump ahead? Use model and set-point to solve an optimal control problem over some horizon Apply control action over a small horizon – no change in control during remaining interval Sometimes, map from measurements to outputs is not direct - Need to estimate intermediate states

13. Moving the Prediction Horizon After each iteration, the prediction horizon is moved forward and a new finite-horizon optimization problem is started An essential tradeoff: If you use a long prediction horizon, you have a much larger problem on each iteration Short control horizons (w.r.t. prediction horizon) are desirable - keeps the actions tame (aggressive changes get penalized in the ‘uncontrolled’ part of prediction horizon) 19/02/2009Structure and Synthesis of Robot Motion13

14. MPC – Problem Formulation 19/02/2009Structure and Synthesis of Robot Motion14 [Source: ZK Nagy, ECMI 2004]

15. Important Feature of MPC - Constraints Once the basic problem is setup as one of (numerical) optimal control, it becomes possible to incorporate constraints on the state and control equations – Inequality constraints, e.g., actuator bounds – Equality constraints, e.g., end-effector stays on a surface The fact that the problem is repeatedly solved over a finite horizon means that these constraints can come and go 19/02/2009Structure and Synthesis of Robot Motion15

16. Adaptive Control 19/02/2009Structure and Synthesis of Robot Motion16

17. A Core Idea in Adaptive Control Adaptive control is a well-established area with numerous possible architectures – we will not survey them all Instead we will only look at one core idea: on-line determination of process parameters If you can use a batch of data from a moving window to ‘identify’ the currently valid system model then we could adapt controller parameters accordingly Ideally, all this happens on-line – one data point at a time! 19/02/2009Structure and Synthesis of Robot Motion17

18. Least Squares Estimation Known since Gauss (late 18 th century): 19/02/2009Structure and Synthesis of Robot Motion18

19. Least Squares Example: Cubic Polynomial 19/02/2009Structure and Synthesis of Robot Motion19

20. Least Squares can be Solved Recursively 19/02/2009Structure and Synthesis of Robot Motion20 I am not going to show how these equations and quantities like K(t) are derived. If you are curious, please refer to: - K. Astrom, B. Wittenmark, Adaptive Control, Addison-Wesley 1995 (Ch 2) - S. Satry, M. Bodson, Adaptive Control (on-line version), Ch 2on-line version

21. Nonlinear Control Very few robotic systems actually exhibit linear dynamics across the entire state space – Variety of nonlinearity: limit stops, hysteresis, static friction, complex dynamics (e.g., trigonometric terms in equations of motion) There are a number of different ways to cope with this: – One could deal with a specific nonlinearity in a “one-off” sense, e.g., friction compensation – Many techniques (including MPC and Adaptive control) can cope with nonlinearities in a ‘slowly varying’ sense Another alternative is to use systematic linearization tools from nonlinear control theory 19/02/2009Structure and Synthesis of Robot Motion21

22. Simple Idea: Jacobian Linearization 19/02/2009Structure and Synthesis of Robot Motion22 0 at equilibrium point Similar to a linear system, But, this is only true in that small neighbourhood!

23. Better Approach: Feedback Linearization Algebraically transform nonlinear system dynamics into linear ones, so that linear control techniques can be applied Different from using the Jacobian – feedback linearization is achieved by exact state transformation and feedback. Used in many demanding applications ranging from mobile robotics and aircrafts to complex chemical processes Biggest disadvantage is reliance on a good model – need significant sophistication to ensure robustness in practice 19/02/2009Structure and Synthesis of Robot Motion23

24. Feedback Linearization: Basic Procedure 19/02/2009Structure and Synthesis of Robot Motion24

25. Summary This lecture surveyed three major techniques for controlling systems when they deviate from the assumptions of simple linear systems We looked at – Model Predictive Control – Adaptive control – Nonlinear control They all attack closely related issues but the focus of attention of each method is different 19/02/2009Structure and Synthesis of Robot Motion25