1. Introduction to Control: How Its Done In Robotics R. Lindeke, Ph. D. ME 4135

2. Remembering the Motion Models: As we found using the L-E approach, the Required Joint Torque is: Dynamical Manipulator Inertial Tensor – a function of position and acceleration Coupled joint effects (centrifugal and coriolis) issues due to multiple moving joints Gravitational Effects Frictional Effect due to Joint/Link movement

3. Lets simplify the model (a Bit) This torque model is a 2 nd order one (in position) lets look at it as a velocity model rather than positional one then it becomes a system of highly coupled 1 st order differential equations We will then isolate Acceleration terms (acceleration is the 1 st derivative of velocity)

4. Considering Control: Each Link’s torque is influenced by each other links motion We say that the links are highly coupled Solution then suggests that control should come from a simultaneous solution of these torques We will model the solution as a “State Space” design and try to balance the torque-in with positional control-out – the most common way it is done! But we could also use ‘force control’ to solve the control problem!

5. The State-Space Control Model:

6. State Space Approach The State Space General Model feeds out positional kinematics based on the “torque (power) demand” input Notice: 1/s is the Laplace transform of a unit step (torque) impulse As you remember, Laplace transforms convert linear differential equation sets into algebraic equation sets once solved we need to do inverse Laplace transforms to return to torque/position space In LaPlace models, S is a complex variable!

7. Ultimately how do we know How much Torque to specify – and if we are ‘In Control’ In robots with moving joints, we set targets for motion We sense motion at the joint level (using kinesethic sensors) We study differences between where we are and where we’re going as a “Feedback” (ie. servo) error Control means that we try to minimize error (make error go to zero) when we move toward a new location

8. Setting up a Real Control We will (start) by using positional error to drive our torque devices This simple model is called a PE (proportional error) controller

9. PE Controller: To a 1 st approximation,  = K m * I Torque is proportional to motor current And the Torque required is a function of ‘Inertial’ (Acceleration) and ‘Friction’ (velocity) effects as suggested by our L-E models

10. Setting up a “Control Law” We will use the positional error (as drawn in the state model) to develop our torque control We say then for PE control: Here, k pe is a “gain” term that guarantees sufficient current will be generated to develop appropriate torque based on observed positional error

11. Using this Control Type: It is a representation of the physical system of a mass on a spring! We say afer setting our target as a ‘zero goal’ that:  a is a function of the servo feedback as a function of time!

12. Examining this ‘solution’: The 1 st term is a damping term for the motion The systems ‘natural frequency’ is given by:

13. Studying the solution: If (F 2 /4k pe )> J we are ‘over damped’ and the system will never quite reach its target considering “reasonable time” If (F 2 /4k pe )= J we are ‘critically damped’ giving the system ideal behavior If (F 2 /4k pe )< J we are ‘under damped’ and the system will over-shoot and oscillate about our target at the system’s natural frequency – a dangerous situation in robotics!

14. These behaviors make sense (physically!): Under High Friction Conditions (over damped): A system is hard to start but easy to stop With High Moment of Inertia Conditions (under damped): A system is hard to start and it is hard to stop leading to overshoot and possibly one that oscillates about the target ‘forever’

15. Problems with PE Control: First the so-called “Steady-State” error – the torque goes to zero when the target is hit! Secondly, we may be out of balance – the GAIN is not meeting our Inertial vs. Friction balance leading to overshoot or undershoot Typically we will add a term to our model to react to increasing speed so we minimize overshoot

16. Dealing With Steady State Zero Error This is a gravitational issue: we must add an ‘L’ or gravitational term back to our Torque control model Gravitational input is positionally controlled:  L = -g (M 1 r 1 + M 2 R)* Cos(  ) For a  R Manipulator with a payload on M 2

17. Solving the Overshoot Problem: Lets expand our control law: We should include a term that reacts to the velocity of the link – But velocity is the derivative of the position We will call this a proportional – derivative controller (PD Control)

18. State Space Model of PD:

19. Solving the PD Model: Remembering:

20. Leads to a Solution of the form:

21. Effect of Derivative Term: It is observed to be a form of “Active Friction” It tends to slow down the link as it moves faster when high errors (being far from goal) are observed Thus it can be thought of as a brake on the motion It is a component that anticipates changes and provides very fast response to these changes

22. Taking Care of Trouble: We add an integrator to the model To damp out oscillations from over shoot To control effects due to environmental perturbations To damp out Wild data gyrations – typically due to encoder errors This leads to a model of control:

23. PID State Space Model:

24. Developing Optimal Control PID is most often found in the systems Arm Joints The components of Torque are functions of POSE meaning the J eq and F eq as well as L factors change as one observes the robot moving throughout the work envelope We achieve control with k pe, k d and k i if they are fixed values, we can expect critically damped control at only a single (or very few) position(s)

25. What is done: We operate off critical PID on arm joints and PD w/ gravitational compensation for remote and wrist joints Most controls use a form of adaptive control Tables of gains applicable over certain geometries with automatic changes as the manipulator moves about the work envelope We swap in and out the gain values such that we minimize energy consumed by the drive:

26. Another Idea: Develop a Performance Index (PI) that judges controller stability This PI is an external measurement scheme that using logic and comparisons between desired and actual performance then adjusts the model

27. State Model of Adjustable Controller

28. Thus Ends our Introductory Studies of “Robotics & Controls” This is a rich and deep field of applied Mechanical and Industrial Engineering While we have deeply explored some topics, others have only been scanned I wish you well as you move forward in your lifelong exploration of “AUTOMATION” and its myriad of supporting technologies I sincerely hope that you have all learned something of this fascinating field and that these lessons will prove to be valuable in your careers!