Video 5.1 Vijay Kumar and Ani Hsieh


 Collin O’Neal’
 4 months ago
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1 Video 5.1 Vijay Kumar and Ani Hsieh Robo3x1.1 1
2 The Purpose of Control Input/Stimulus/ Disturbance System or Plant Output/ Response Understand the Black Box Evaluate the Performance Change the Behavior Robo3x1.1 2
3 Anatomy of a Feedback Control System Disturbance Input Desired Speed +  Controller Actuator Gas Pedal + + Output Vehicle Speed Sensor Robo3x1.1 3
4 Twin Objectives of Control Performance Disturbance Rejection Robo3x1.1 4
5 Learning Objectives for this Week Linear Control Modeling in the frequency domain Transfer Functions Feedback and Feedforward Control Robo3x1.1 5
6 Frequency Domain Modeling Algebraic vs Differential Equations Laplace Transforms Diagrams Robo3x1.1 6
7 Laplace Transforms Integral Transform that maps functions from the time domain to the frequency domain with Robo3x1.1 7
8 Example Let, compute Robo3x1.1 8
9 Inverse Laplace Transforms Integral Transform that maps functions from the frequency domain to the time domain Robo3x1.1 9
10 Example Let, compute Robo3x
11 Laplace Transform Tables Robo3x
12 Video 5.2 Vijay Kumar and Ani Hsieh Robo3x
13 Generalizing Given How do we obtain? Robo3x
14 Partial Fraction Expansion Case 1: Roots of D(s) are Real & Distinct Case 2: Roots of D(s) are Real & Repeated Case 3: Roots of D(s) are Complex or Imaginary Robo3x
15 Case 1: Roots of D(s) are Real & Distinct Compute the Inverse Laplace of Robo3x
16 Case 2: Roots of D(s) are Real & Repeated Compute the Inverse Laplace of Robo3x
17 Case 3: Roots of D(s) are Complex Compute the Inverse Laplace of Robo3x
18 Video 5.3 Vijay Kumar and Ani Hsieh Robo3x
19 Using Laplace Transforms Given Solving for x(t) 1. Convert to frequency domain 2. Solve algebraic equation 3. Convert back to time domain Robo3x
20 Properties of Laplace Transforms Robo3x
21 Summary Laplace Transforms time domain <> frequency domain differential equation <> algebraic equation Partial Fraction Expansion factorizes complicated expressions to simplify computation of inverse Laplace Transforms Robo3x
22 Example: Solving an ODE (1) Given with, and. Solve for. Robo3x
23 Example: Solving an ODE (2) Robo3x
24 Video 5.4 Vijay Kumar and Ani Hsieh Robo3x
25 Controller Design Disturbance Input +  Input Controller + + Output Robo3x
26 Controller Design Disturbance Input Output Controller Robo3x
27 Controller Design Disturbance Input +  Controller + + System Output Robo3x
28 Controller Design Disturbance Input +  Controller + + System Output Robo3x
29 Transfer Function Relate a system s output to its input 1.Easy separation of INPUT, OUTPUT, SYSTEM (PLANT) 2.Algebraic relationships (vs. differential) 3.Easy interconnection of subsystems in a MATHEMATICAL framework Robo3x
30 In General A General NOrder Linear, Time Invariant ODE G(s) =Transfer Function = output/input Furthermore, if we know G(s), then output = G(s)*input Solution given by Robo3x
31 General Procedure Given performance criteria and desired 1. Convert 2. Analyze 3. Design using 4. Validate using 5. Iterate Robo3x
32 Underlying Assumptions Linearity 1. Superposition 2. Homogeneity System System B/c the Laplace Transform is Linear! Robo3x
33 Video 5.5 Vijay Kumar and Ani Hsieh Robo3x
34 Characterizing System Response Given How do we characterize the performance of a system? Special Case 1: 1 st Order Systems Special Case 2: 2 nd Order Systems Robo3x
35 Poles and Zeros Given Poles Zeros Example: Robo3x
36 First Order Systems In general Let U(s) = 1/s, then As such, Therefore, Robo3x
37 Characterizing First Order Systems Given with U(s) = 1/s Robo3x
38 Characterizing First Order Systems Time Constant Rise Time Settling Time Robo3x
39 Second Order Systems Given, and U(s) = 1/s Possible Cases 1. r 1 & r 2 are real & distinct 2. r 1 & r 2 are real & repeated 3. r 1 & r 2 are both imaginary 4. r 1 & r 2 are complex conjugates Robo3x
40 Case 1: Real & Distinct Roots Overdamped response Robo3x
41 Video 5.6 Vijay Kumar and Ani Hsieh Robo3x
42 Case 2: Real & Repeated Roots Critically damped response Robo3x
43 Case 3: All Imaginary Roots Undamped response Robo3x
44 Case 4: Roots Are Complex Underdamped response Robo3x
45 A Closer Look at Case 4 Exponential Decay (Real Part) Sinusoidal Oscillations (Imaginary Part) Robo3x
46 Summary of 2 nd Order Systems Given, and U(s) = 1/s Solution is one of the following: 1. Overdamped: r 1 & r 2 are real & distinct 2. Critically Damped: r 1 & r 2 are real & repeated 3. Undamped: r 1 & r 2 are both imaginary 4. Underdamped: r 1 & r 2 are complex conjugates Robo3x
47 2 nd Order System Parameters Given and U(s) = 1/s Natural Frequency n System s frequency of oscillation with no damping Damping Ratio Robo3x
48 General 2 nd Order System Given and U(s) = 1/s When b = 0 For an underdamped system Robo3x
49 General 2 nd Order Systems Secondorder system transfer functions have the form with poles of the form Example: For Compute, n, and s 1,2? Robo3x
50 Video 5.7 Vijay Kumar and Ani Hsieh Robo3x
51 Characterizing Underdamped Systems Robo3x
52 Peak Time Same Envelope Motion of poles Robo3x
53 Settling Time 2 1 Same Frequency 2 1 Motion of poles 2 1 Robo3x
54 Overshoot 3 Same Overshoot Motion of poles Robo3x
55 In Summary splane Robo3x
56 Video 5.8 Vijay Kumar and Ani Hsieh Robo3x
57 Independent Joint Control In general, nlink Robot Arm generally has n actuators Single Input Single Output (SISO) Single joint control Robo3x
58 Permanent Magnet DC Motor Picture Here Basic Principle Source: Wikimedia Commons Robo3x
59 Electrical Part Armature Current Back EMF Motor Torque Torque Constant Robo3x
60 Mechanical Part Gear ratio Actuator Dynamics Robo3x
61 Combining the Two Correction: the K b terms should be K m Robo3x
62 Two SISO Outcomes Input Voltage Motor Shaft Position Load Torque Motor Shaft Position Robo3x
63 Video 5.9 Vijay Kumar and Ani Hsieh Robo3x
64 Two SISO Outcomes Input Voltage Motor Shaft Position Load Torque Motor Shaft Position Assumption: L/R << J m /B m Robo3x
65 2 nd Order Approximation Divide by R and set L/R = 0 In the time domain Robo3x
66 OpenLoop System Actuator Dynamics Setpoint tracking (feedback) Trajectory tracking (feedforward) Robo3x
67 Our Control Objectives Motion sequence of endeffector positions and orientations (EE poses) EE poses Joint Angles Motor Commands Transfer function Three primary linear controller designs: P (proportional) PD (proportionalderivative) PID (proportionalintegralderivative) Robo3x
68 SetPoint Tracking The Basic PID Controller Robo3x
69 Proportional (P) Control Control input proportional to error K P controller gain Error is amplified by K P to obtain the desired output signal Robo3x
70 P Control of Vehicle Speed Example: Cruise Control X v Desired linear speed q Y vehicle wheel speed Control input proportional to error Robo3x
71 Performance of P Controller K P = 10 K P = 50 K P = 100 Increases the controller gain decreases rise time Excessive gain can result in overshoot Robo3x
72 Video 5.10 Vijay Kumar and Ani Hsieh Robo3x
73 ProportionalDerivative (PD) Control Control input proportional to error AND 1 st derivative of error Including rate of change of error helps mitigates oscillations Robo3x
74 Performance of PD Controller K P = 10 K D = 1 K P = 10 K D = 3 Decreases rise time K P = 500 K D = 10 K P = 500 K D = 50 Reduces overshoot & Settling Time Robo3x
75 PD Control of a Joint Closed loop system given by w/ Robo3x
76 PD Compensated Closed Loop Response (1) Robo3x
77 PD Compensated Closed Loop Response (2) Robo3x
78 Picking K P and K D Closed loop system w/ Design Guidelines Critically damped w/ Pick and Robo3x
79 Performance of the PD Controller Assuming and Tracking error is given by At steadystate Robo3x
80 Video 5.11 Vijay Kumar and Ani Hsieh Robo3x
81 ProportionalIntegralDerivative (PID) Controller Control input proportional to error, 1 st derivative AND an integral of the error The integral term offsets any steadystate errors in the system Robo3x
82 Performance of PID Controller K P = 10 K D = 3 K P = 10 K D = 3 K I = 1 K P = 10 K D = 3 K I = 50 Eliminates SSError Increases overshoot & settling time Robo3x
83 PID Control of a Joint Closedloop system is given by w/ Robo3x
84 PID Compensated Closed Loop Response (1) Robo3x
85 PID Compensated Closed Loop Response (2) Robo3x
86 Picking K P, K D, and K I Closed loop system w/ Design Guidelines System stable if K P, K D, and K I >0 Set K I = 0 and pick K P, K D, then go back to pick K I w/ in mind Robo3x
87 Summary of PID Characteristics CL Response Rise Time % Overshoot K P Decrease Increase K D Small Change Decrease Settling Time Small Change Decrease SS Error Decrease Small Change K I Decrease Increase Increase Eliminate Robo3x
88 Tuning Gains Appropriate gain selection is crucial for optimal controller performance Analytically (RLocus, Frequency Design, Ziegler Nichols, etc) Empirically The case for experimental validation Model fidelity Optimize for specific hardware Saturation and flexibility Robo3x
89 Feedforward Control Motion sequence of endeffector positions and orientations (EE poses) What if instead of, we want? Robo3x
90 Video 5.12 Vijay Kumar and Ani Hsieh Robo3x
91 Closed Loop Transfer Function (1) Robo3x
92 Closed Loop Transfer Function (2) Robo3x
93 Closed Loop Transfer Function (3) Robo3x
94 Picking F(s) Closed loop transfer function given by Behavior of closed loop response, given by roots of H(s) and F(s) be chosen so that Robo3x
95 Will This Work? Let F(s) = 1/G(s), i.e., a(s) = p(s) and b(s) = q(s), then System will track any reference trajectory! Robo3x
96 Caveats Minimum Phase Systems Picking F(s) = 1/G(s), leads to Assume system w/o FF loop is stable By picking F(s) = 1/G(s), we require numerator of G(s) to be Hurwitz (or ) Systems whose numerators have roots with negative real parts are called Minimum Phase Robo3x
97 Feedforward Control w/ Disturbance Assume: D(s) = constant w/ Pick F(s) = 1/G(s) = Note the following: Robo3x
98 Tracking Error Control law in time domain System dynamics w/ control + disturbance Robo3x
99 Overall Performance Robo3x