Robotics: Science and Systems

Transcription

1 Robotics: Science and Systems System Identification & Filtering Zhibin (Alex) Li School of Informatics University of Edinburgh 1

2 Outline 1. Introduction 2. Background 3. System identification 4. Signal filtering 2

3 Achieving high-level goals Adversarial actions & other agents Perception High-level goals Environment Adversarial actions & other agents Action Problem: How to generate actions, to achieve high-level goals, using limited perception and incomplete knowledge of environment (including others actions)? 3

4 Types of models Models of motion Own dynamics Object dynamics Other agents Models of environment Space & how I move in space Other navigation considerations Models of self What is the connection between my sensors and actuators? Temperature of computation/actuation system (internal state)? 4

5 Introduction 5

6 Problem solving using different models Decision Perception Action Behaviour 6

7 Types of models Focus: a mathematical model for describing the dynamic behavior of a system or process in the time/frequency domain. In robotics, models usually involve principles of physics. White-box model Grey-box model Black-box model 7

8 White-box and black-box model White-box model: a system with all necessary and a priori information available; we have knowledge of its internal structure, or the process (relationship between inputs and outputs). Black-box model: a system which can only be observed in terms of its inputs and outputs, without any knowledge of the internal mechanism. A typical black-box is the human brain. Picture is from from wikipedia author Krauss. 8

9 Grey-box model Grey-box model: a system that can be described by a partial theoretical structure and only needs data to complete the model. The exact parameter values need to be identified from data in order to be fit into the theoretical model. Most robotic systems can be represented by grey box models, as opposed to black box models (no model at all), or white box models (only theoretically exist or in simulation). Then, the remaining questions are: How to build such models? How to identify the model parameters? 9

10 Grey-box model How to build models? Answer: represent principles of physics, robotics or other subjects by using mathematical symbols (examples will be given in the following lectures). A model? A mathematical function, eg a=f/m, a simple case that relates acceleration with applied force given a known mass. How to identify the model parameters? Answer: system identification. 10

11 Example: what parameters need to be identified? A particular example: this kinematic structure is the model, where each segmental COM is located, in terms of (x, y, z) in each link, are the model parameters to identify. 11

12 Background knowledge 12

13 Linear Time-Invariant (LTI) system Linear Time-invariant (LTI) systems have two properties: 1. Linearity the relationship between the input and the output of the system follows superposition: given pairs of input-output, x1(t) y1(t), x2(t) y2(t), then a new input of c1x 1(t) + c2x2(t) will generate output of c1y1(t) + c2y2(t). 2. Time-invariance the above relationship will not change along the time, ie if we have x1(t) y1(t), then the same input applied at different time will produce the same output, x1(t-t) y1(t-t). An example of LTI is a RCL circuit (resistors, capacitors, and inductors). 13

14 Typical components in LTI systems Typically, a system receives an input signal, x(t) (generally a vector) and transforms it into the output signal, y(t). In modelling and control, this further boils down to the components as follows: u(t) is an input signal that can be manipulated (control signal) w(t) is an input signal that can be measured (measurable disturbance) v(t) is an input signal that cannot be measured (unmeasurable disturbance) y(t) is the output signal Typically, the system may also have hidden internal states 14

15 System identification 15

16 System identification System identification (SI): to build and complete mathematical models of dynamical systems from measured data. System identification involves design of experiments for collecting measurements for identifying such models, ie fitting the model parameters. Generally, we aim to formulate a model of the system from experimental results. A model is a mathematical representation that explains the observed data, and further allows us to predict the future responses of the system given the new inputs. SI is strongly related with data. 16

17 System identification A model is a purposeful simplification and abstraction of a reality. As mentioned in the first lecture, most controls or methods are model based. To build more accurate mathematical models of dynamical systems from real measurements, identification needs: First choose an appropriate model Measure the input and output signals in time or frequency domain Use an estimation method to estimate values of the parameters Evaluate the estimated model, if the model outputs data with reasonable accuracy compared to the new measurement a. b. Explain past observations Predict future observations 17

18 System identification To sum up briefly, constructing a model from data includes 3 steps: Model (first-principle) Data Criterion 18

19 To identify a system To design a controller, generally a model needs to be constructed for control purposes. Model: a representation that maps the transforms from an input to an output ^ y(t) Model Control signal u(t) Reference r(t) u(t) Controller Physical system y(t) A good model ^ matches y(t) and y(t) Disturbance d(t) Often, models are represented by differential or difference equations and used explicitly/implicitly by the control design. 19

20 Representation of a dynamical system In discrete-time form a linear time-varying, deterministic, dynamical system is represented by xt +1 = At xt + Bt ut where xt Rnx1 is a n-dimensional state vector, ut Rrx1 is an input vector, and At, Bt are matrices. Outputs of the system are functions of the state vector and are represented with a l-dimensional vector y Rlx1 yt = Ht xt where Ht Rlxn is an observation matrix. 20

21 Representation of a dynamical system A LTI dynamical system in discrete-time xt +1 = At xt + Bt ut y t = H t xt ut xt+1 Bt + Time delay of a unit xt yt Ht + At 21

22 Basics of parameter estimation Consider a simple scenario of sensor measurement: yt = Ht xt A system outputs a ground truth of variable yd (d stands for desired), but this signal is only available by sensor measurement y, which is mixed with noises. Sensor measurement y is noise-corrupted version of yd. et ut xt+1 Bt + Time delay of a unit xt yt d yt Ht + At 22

23 Basics of parameter estimation Goal: identify parameters of the sensor model Ht. Given the model structure in the form of yt = Ht xt, collect multiple measurements xt, yt, t=1,... N. So we have dataset as: y1 = H1 x1 y2 = H2 x2 yn = HN xn Note: for LTI system, H1=H2=...=HN. Stack the above equations into matrices. [y1, y2,., yn] = H * [x1, x2,., xn] 23

24 Basics of parameter estimation Let Y = [y1, y2,., yn], X=[ x1, x2,., xn], we have Y = H*X H = Y*X+ X+ is the right pseudo inverse of matrix X (X*X+=I, where I is the identity matrix). X+ = XT(X*XT)-1 Note: left pseudo inverse of matrix X is X+=(XT*X)-1XT. 24

25 The Least Square (LS) solution Solving the equation Y = H*X by H = Y*X+ using pseudo inverse X+ is a least square solution that minimizes the sum of squared errors between a model and data. In other words, H = Y*X+ is the solution that minimizes the square of the norm of vector (Y - H*X): (Y - H*X)T*(Y - H*X) See interpretation in case study To construct a model from data: 1. Model 2. Data 3. Criterion *We will learn more about least square optimization in the section of optimization later. 25

26 Case study Goal: identify parameters of the sensor model H, assuming the underlying principle is y =10*x+50, then the ground truth is H=[10, 50]. yt = Ht xt becomes [x y =[k, b]* 1], Ht=[k, b], xt=[x, 1]T; Collect multiple measurements xt, yt, t=1,... N. So we have dataset as: [ x, x,., xn [y1, y2,., yn] = Ht * 1 2 1, 1,., 1 ] Note: [x, 1]T=[x ; 1] = [x 1] 26

27 Case study [ x, x2,., xn Let Y = [y1, y2,., yn], X= 1,1 1,., 1 ], we have Y=H*X H=Y*X+ =Y*XT(X*XT)-1 X+ =XT(X*XT)-1 the right pseudo inverse of matrix X. 27

28 Case study: LS estimation x= [0:0.01:1]; e=randn(1, length(x)); y= 10*x + 50; X = [x; ones(1, length(x))]; measurement = y+e; H = measurement*x'*(x*x')^-1; yest = H*X; figure(1); hold on; set(gcf,'color','w'); plot(x,y,'r','linewidth',3) plot(x,measurement, '.', 'markersize',15) plot(x,measurement) legend('ideal','measurement'); One run of the estimated model is H = [9.8129, ] 28

29 Case study: the geometric interpretation Minimize the sum of squared errors between a model and data. What is still missing as a standard procedure for system identification? 29

30 Case study: the geometric interpretation Evaluation: how well an identified model matches the data. e = yestmeasurement mean(e) e = ydmeasurement e std(e) mean(e.^2) sum(e.^2)

31 Parameter estimation of a dynamical system The previous model of the sensor was static, which does not change with time or other parameters; then we obtained data and estimated constant parameters of the model. Apply similar procedure of estimating parameters of the model of a dynamical system system identification How to model the dynamics of a system? 31

32 Model a dynamical system: car suspension 32

33 Model abstraction A simple spring-mass-damper system M M Fs Fd m 0 (Simpfilication) 33

34 How to build a model The force applied by the spring: Correspondingly, the force by the damper: Equation of motion is (Newton s 2nd law): M M Rewrite in the form of matrix/vector, yields: Fs Fd 34

35 Parameter estimation: 2nd order dynamic system We generate a forced motion of this second order dynamic system M Data M Fs Fd Collect multiple measurements of position, velocity and acceleration Estimate the system parameters from this (hint: least square LS) 35

36 Parameter estimation: 2nd order dynamic system Recall that for each measurement, we have For a series of N measurements, 1, 2, N, construct the least square problem. 36

37 Parameter estimation: 2nd order dynamic system The stiffness k and damping c of the car suspension are identified by: 37 A similar example with real experimental data

38 Summary: model-based and model-free Model-based grey-box, because all models are the approximations of reality. Grey-box modelling: a first-principle model structure and experimental estimation of the model parameters. Model-free approach: a model may exist in reality, but is not used; or a model is too difficult to be constructed, and the system is treated as a black-box. Black-box modelling: the input-output relationship is established purely from experimental data only statistical. 38

39 Summary grey-box model and black-box model Grey-box model Black-box model Knowledge based on laws of physics and others; Physical insight and explainable; Generalization to the same type of problems Possibility of having prior simulation and study 1. High dimension of model parameters Model parameter structure can be complex, nonlinear and coupled (interdependent) Parameter tuning is challenging Represent better the real input-output behavior Convenient structure for parameter tuning Suitable for complex systems for which physical models are too difficult to build No direct implication of physical parameters or meanings No insight of its internal structure meaning no simulation of the real system until it s built. 39

40 Summary of system identification Take the equation of motion as given Perhaps from laws of physics or other domain knowledge Example is in earlier slide (a spring-mass-damper system) System model has open parameters (k, c) Measure inputs and outputs Need to do this carefully (need sufficient excitation) Obtain data of position, velocity and acceleration Estimate parameters that are unknown Least squares (specify errors, e.g., argmin (Y - H*X)T*(Y - H*X) ) 40

41 Signal filtering 41

42 Real world has uncertainties anywhere Sources of uncertainties: Models are approximations of the reality Signals are noisy, drifting Sensors have limited accuracy/resolution Environment is changing Unconsidered factors, eg temperature, humidity, illumination Lab/factory environment is more controllable, while real world is not 42

43 Why filtering? Before state estimation, we need to inspect the quality of signals first. Usually, procedure of signal filtering is required because: 1. A direct let-go of noises will jeopardize state estimation algorithm 2. Direct use of noisy signals cause instability issues in control 3. Necessary pre-filtering can reduce noise to a satisfactory level 43

44 Real world: sensory feedback is imperfect Consider a real world scenario: We have a humanoid robot Valkyrie We wish to design behaviours using visual perception First of all, we need accurate estimates of objects Stereo camera from multi-sense is like this 44

45 Why filters are needed? Signals have noises and the direct use of noisy signals cause instability issues. Necessary pre-filtering can reduce noise to a satisfactory level. Velocity signal 45

46 Common filters used in engineering Low-pass filter low frequencies lower than cut-off frequency are passed High-pass filter high frequencies higher than cut-off frequency are passed Band-pass filter only frequencies within a frequency band are passed Band-stop filter only frequencies within a frequency band are attenuated Notch filter rejects a particular frequency, eg resonance 46 Pictures of filters are adapted from from wikipedia author SpinningSpark

47 Filter effects 47

48 Common filter tools Low-pass filters are the most common ones used in practice, aim is to remove/reduce noise Occasionally, high-pass filters are needed for eliminating low frequency drifts or unwanted offset. Butterworth filter is the most commonly used. function [x_filter]=lowpass(cutoff,x, SamplingTime) samplingf=1/samplingtime; fnorm = cutoff / (samplingf/2); [b,a] = butter(3, fnorm, 'low'); x_filter = filtfilt(b, a, x); end 48

49 Example: filtering of link acceleration Case study: 1. Link encoder resolution of 19-bit; 2. 2nd order Butterworth filter, cut-off frequency of 5 Hz. Acceleration signal 49