CDS 110 L10.2: Motion Control Systems. Motion Control Systems

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CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS CDS L.

Murray 4 December 22 Announcements Final exam available at 3

Control Systems Two degree of freedom design Control design

Systems (optional) Reading: Optional: Motion Control Systems

given path Special features: second order dynamics, input

1 CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS CDS L.2: Motion Control Systems Richard M. Murray 4 December 22 Announcements Final exam available at 3 pm (during break); due 5 pm, Friday, 3 Dec 2 Outline: Motion Control Systems Two degree of freedom design Control design for the Kelly II vehicle Cooperative Control of Multi-Vehicle Systems (optional) Reading: Optional: Motion Control Systems Control objective: move the vehicle to a give point or along a given path Special features: second order dynamics, input constraints are important, often have a human in the loop (someplace) 4 Dec 2 R. M. Murray, Caltech CDS 2

Plant P Local Control output Use real-time trajectory generation to construct (suboptimal) feasible trajectories Use local linear/nonlinear control for tracking & robust performance 4 Dec 2 R. M.

2 CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS 2 Approach: Two Degree of Freedom Design Nonlinear design global nonlinearities input saturation state space constraints Linear design ref Trajectory Generation u d x d noise δ u Plant P Local Control output Use real-time trajectory generation to construct (suboptimal) feasible trajectories Use local linear/nonlinear control for tracking & robust performance 4 Dec 2 R. M. Murray, Caltech CDS 3 Multi-Vehicle Wireless Testbed for Integrated Control, Communications and Computation (DURIP) Cremean et al CDC 22 Testbed features Distributed computation on vehicles + command and control console Point to point networking (bluetooth) + local area networking (82.) Overhead vision system provides global position data (LPS) 4 Dec 2 R. M. Murray, Caltech CDS 4

3 CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS 3 MVWT Block diagram 4 Dec 2 R. M. Murray, Caltech CDS 5 Vehicle Hardware Fan Motor Motor Controller Casters (3) Batteries Safety Screen PIC Board Laptop Hat for vision system not shown. 4 Dec 2 R. M. Murray, Caltech CDS 6

4 CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS 4 Software Architecture Controller Library (LQR,NTG, MPC etc) Vision Module 82.b Vision System OR RHexLib Module Manager Simulator QNX USB Driver PIC software Data Logger Controller Shell RHexComm 82.b Other Vehicles Other Vehicles Other Vehicles Onboard Sensors Motors Master controller /User Interface 4 Dec 2 R. M. Murray, Caltech CDS 7 Dynamics f Vehicle dynamics: mx = f cos f sin dx 2 my = fsin + f2cos dy J = rf Fan dynamics Control algorithm Fan dynamics Vehicle dynamics m ω = τω+ α( i) m ω = τω + α( i ) f f = Nω = Nω 2 2 Nonlinear map Wireless network Vision system Plant Vision System Time delay + noise (pixelation) Wireless network Variable time delay 4 Dec 2 R. M. Murray, Caltech CDS 8

5 CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS 5 y f x Linearization Linearize around a mv mx = = f fcos dx cos f2sin dv constant velocity along y mx my = = f fsin + dy sin + f2cos dx JJ = = rf rf v = v f = dv x =, x = f2 = =, = Shift coordinates to the origin and write in state space form: v = v v u = f 2 u = f dv 2 Remarks Ignores actuator dynamics (assume fast) Ignores time delays (for now) Parameters m = 5.5 b =. J =.47 r =.23 d =.5 4 Dec 2 R. M. Murray, Caltech CDS 9 Control Design: State Space f Choose control to stabilize error, e = x-x d u = K(x-x d )+u d x d = desired state u d = nominal force Pole-zero map x Imag Axis.2 x -.2 x x Real Axis 4 Dec 2 R. M. Murray, Caltech CDS

6 CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS 6 5 f Control Design: Frequency Domain Bode Diagrams From: U() P vu = s + d P yu2 2 Js + bs+ Jdv / m 2 2 = ( ms + ds)( Js + bs) Compute transfer functions using H = C(sI-A) - B Use loop shaping to design compensator Forward velocity: simple proportional gain Lateral position: use lead compensator 5 Bode Diagrams From: U() Loop xfer fcn Phase (deg); Magnitude (db) To: Y() fwd vel fwd vel lateral lateral Phase (deg); Magnitude (db) To: Y() Frequency (rad/sec) Frequency (rad/sec) 4 Dec 2 R. M. Murray, Caltech CDS Control Design: Frequency Domain Phase (deg); Magnitude (db) To: Y() f Bode Diagrams From: U() Pvu = s + d C = K v p 2 Js + bs+ Jdv / m Pyu = ( ms + ds)( Js + bs) s+ a Cl = K s + b Kp = a =. K = b = Loop xfer fcn Amplitude Amplitude To: Y() To: Y() Step Response From: U() Time (sec.) Step Response From: U() Time (sec.) Frequency (rad/sec) 4 Dec 2 R. M. Murray, Caltech CDS 2

7 CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS 7 Time Delay Estimates 8 ms 33 ms 6 ms Camera takes picture LPS Broadcasts data Laptop receives data EVENT No actuator delay included in estimates or measurement Total delay estimate: 65 ms Measured delay: 65.3 ms average ms 6 ms ms PIC sends new PWM signal and lights the LEDs PIC receives data Controller receives data 4 Dec 2 R. M. Murray, Caltech CDS 3 f Unmodeled Dynamics: Sensor Delay Control algorithm Fan dynamics Vehicle dynamics ( z, z2) Wireless network Vision system Plant Phase (deg); Magnitude (db) To: Y() Bode Diagrams From: U() Loop xfer fcn Amplitude To: Y() Amplitude To: Y() Step Response From: U() Time (sec.) Step Response From: U() Frequency (rad/sec) Time (sec.) 4 Dec 2 R. M. Murray, Caltech CDS 4

Operations Applications Cooperative control in dynamic,

flight of micro-satellite clusters (TechSat 2, TPF)

What should we communicate between vehicles and how often?

Murray, Caltech CDS 5 What you should know for the final

summary slides on web for overview of main concepts

Stability/performance Step response Frequency response 4.

8 CDS, Lecture.2 4 Dec 2 R. M. Murray, Caltech CDS 8 Next Steps: Multi-Vehicle Operations Applications Cooperative control in dynamic, uncertain, adversarial environments (RoboCup) Formation flight of micro-satellite clusters (TechSat 2, TPF) Questions How do we coordinate motion between multiple vehicles? How do we provide redundancy and failure tolerance? What should we communicate between vehicles and how often? 4 Dec 2 R. M. Murray, Caltech CDS 5 What you should know for the final Basic concepts + analytical tools from entire course See summary slides on web for overview of main concepts Course Topics. Feedback concepts 2. System modeling 3. Stability/performance Step response Frequency response 4. Linear systems 5. Controllability, state space feedback 6. Transfer functions 7. Loop analysis Nyquist criterion Gain/phase margin 8. Loop shaping Loop xfer specs 9. PID + root locus 4 Dec 2 R. M. Murray, Caltech CDS 6

CDS 101/110a: Lecture 8-1 Frequency Domain Design

CDS 101/110a: Lecture 8-1 Frequency Domain Design CDS 11/11a: Lecture 8-1 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve