Structure and Synthesis of Robot Motion Topics in Control

1. Structure and Synthesis of Robot MotionTopics 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 Structure and Synthesis of Robot Motion

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: k u(t) m b y(t) Structure and Synthesis of Robot Motion

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, Structure and Synthesis of Robot Motion

5. Properties of a Linear System Structure and Synthesis of Robot Motion

6. Properties of a Linear System, contd. Makes design convenient!t! Structure and Synthesis of Robot Motion

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

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!) Structure and Synthesis of Robot Motion

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 Structure and Synthesis of Robot Motion

10. Factors Influencing the MPC Problem Structure and Synthesis of Robot Motion

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 Structure and Synthesis of Robot Motion

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

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) Structure and Synthesis of Robot Motion

14. MPC – Problem Formulation [Source: ZK Nagy, ECMI 2004] Structure and Synthesis of Robot Motion

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 Structure and Synthesis of Robot Motion

16. Adaptive Control Structure and Synthesis of Robot Motion

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! Structure and Synthesis of Robot Motion

18. Least Squares Estimation Known since Gauss (late 18th century): Structure and Synthesis of Robot Motion

19. Least Squares Example: Cubic Polynomial Structure and Synthesis of Robot Motion

20. Least Squares can be Solved Recursively 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 2 Structure and Synthesis of Robot Motion

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 Structure and Synthesis of Robot Motion

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

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 Structure and Synthesis of Robot Motion

24. Feedback Linearization: Basic Procedure Structure and Synthesis of Robot Motion

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 Structure and Synthesis of Robot Motion