1 Introduction. Systems 2: Simulating Errors. Mobile Robot Systems. System Under. Environment
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1 Systems 2: Simulating Errors Introduction Simulating errors is a great way to test you calibration algorithms, your real-time identification algorithms, and your estimation algorithms. Conceptually, the system under test is placed in a test environment which duplicates the interfaces whose errros are under consideration. Test System Under Test Environment Figure 3 Simulating Errors. The test environment generates erroneous signals with the right characteristics.
2 2 Systematic Errors Systematic errors are easy to model. You tell the test environment the truth (include the error) and don t tell the rest of the system. A great example is time delays. Time delays can be modelled with FIFO queues. Data goes in one side and comes out the other a few iterations later. s s2 s3 s4 s5 s6 s7 time time c c2 c3 c4 Figure 4 FIFO Queues as Delay Simulators. The command which responds to the first state does not reach the actuators until three more states have been read into the system. Well engineered systems cycle faster than their own delays so this is the typical situation. Using a large delay in the test environment FIFO than is used in the system makes it possible to assess the impact of delay 2 Systematic Errors 2 3.The Transformation Method (for Generating Arbitrary PDFs) calibration errors on your roadfollower, for example. Another is wheel radius or other vehicle dimensions. Its easy to have the external test environment use the right value whereas 3 Random Errors Simulating random errors is a good deal harder to do well and correctly. 3. The Transformation Method (for Generating Arbitrary PDFs) The Box-Meller method [2] of generating a simple Gaussian random variable goes like so. Most system supplied random number generators are uniform deviates (i.e. uniform distributions) but you often want a Gaussian random variable. Suppose you have a uniform deviate valued between 0 and and you want one that is twice as likely to return a value between 0.25 and 0.75 Obviously, we could achieve this if we took the numbers we get from rand() and
3 3 Random Errors 3 3.The Transformation Method (for Generating Arbitrary PDFs) p(x) HAVE THIS p2(x) WANT THIS p(x) p(y) y f( x) 0 x 0 x alternately stretched and compressed them so that all numbers between /6 and 5/6 were scaled onto 0.25 to We need to stretch the x axis in the middle and squish it on the ends. But what is the function f() which, when applied to x, will give us a new variable y with the right distribution. First, when y() is an invertible function of x, the likelihood of a<x<b must be the same as the liklihood of y(a) < y < y(b) because when you happen to get a particular x between a and b, the second inequality follows immediately. squish is an obscure technical term from ancient probability theory Transformation Rule x.p( y)dy p( x)dx 2.dx 3.x py ( ) dy px ( ) x py ( ) dy px ( ) For a uniform deviate p(x), so x 4.x p( y) dy This is In general, to convert a uniform deviate into any other distribution, put x rand(), and compute yf - (x) where F - is the inverse of the cumulative distribution function for the required new distribution. This new variable y has the required distribution. 5.x F( y) 6.y F ( x) By defn y This is our squishing function
4 This is why libraries often only give you a uniform deviate. Its easy to convert it to anything else. 3.2 Scaling Distributions If x is a random variable with mean 0 and standard deviation, then: y σx + µ is a new random variable with mean µ and standard deviation σ. 3.3 Computing Probability Ellipses from Gaussian Distributions There is alot to this, so I wrote it all down. A covariance matrix C encodes the first moment of a probability density function. Using it alone is tantamount to assuming a gaussian distribution. In n dimensions, this is: Px ( ) e ( 2π) n C -- [( x Xˆ ) T C 2 ( x Xˆ )] 3 Random Errors 4 3.2Scaling Distributions This is formula for a probability given a random vector x. Contours of constant probability are curves containing all values of x for which P(x) constant. In general, these curves are n - ellipsoids because P(x) is constant when the exponent, called the Mahalanobis distance is constant. An n - ellipsoid can be written in the form: ( x Xˆ ) T C ( x Xˆ ) k 2 ( p) where k 2 ( p) is the squared Mahalanobis distance which corresponds to probability p. See [3] if you need a proof that this is an ellipse. Can only plot one in 2D, so remove all but two corresponding rows and columns, then for an equi-probability ellipse of probability p: k 2 ( p) 2ln( p) A 2D covariance matrix has these internals by definition:
5 C σ xx σ xy σ xy σ yy 3.3. Rotating Covariance Suppose frame X is a counterclockwise rotate version of frame U as indicated below. y v u U Let R R X be the rotation matrix which converts coordinates of points from frame X to frame U thus: r u Rr x Let x be an unbiased random vector of covariance which is expressed in frame X. r X C X θ x 3 Random Errors 5 3.3Computing Probability Ellipses from Gaussian Distributions C U The covariance of the same vector u r U expressed in frame U is: C U Exp[ uu T ] Exp[ Rxx T R T ] RC X R T We can interpret R as either an operator on x producing u or as a conversion of coordinates. Using the latter interpretation, note that C X and C U would generate the same physical uncertainty region in space because we are only converting coordinates. Stated differently, R is an orthonormal matrix Diagonalizing Covariance All symmetric matrices are diagonalizeable via a matrix similiarity transform based on an orthonormal (rotation) matrix R. Covariance matrices are symmetric because of their definition as an outer product. Thus there is always a rotation of coordinates which renders a covariance matrix diagonal. Lets try to find it. Using the last result, require that the covariance in the U frame be diagonal and solve for the rotation matrix R
6 which transforms the covariance in the X frame to this. σ C uu 0 cθ sθ σ xx σ T U xy cθ sθ RC X R T 0 σ vv sθ cθ σ xy σ yy sθ cθ Multiplying out the internal equality: σ uu 0 cθ sθ 0 σ vv sθ cθ σ uu 0 cθ sθ 0 σ vv sθ cθ σ xx σ xy σ xy σ yy cθ sθ sθ cθ ( cθσ xx sθσ xy ) ( sθσ xx + cθσ xy ) ( cθσ xy sθσ yy ) ( sθσ xy + cθσ yy ) From the off diagonal element at (,0) we can write: 0 σ xx sθcθ σ xy s 2 θ+ σ xy c 2 θ σ yy sθcθ 0 σ xy c2θ + -- ( σ 2 xx σ yy )s2θ s2θ 2σ xy c2θ ( σ yy σ xx ) 3 Random Errors 6 3.3Computing Probability Ellipses from Gaussian Distributions Hence, the rotation angle is: θ -- atan22σ [ 2 xy, ( σ yy σ xx )] Note that: if σ xy 0, C X is already diagonal and θ 0 is computed - which is correct. if σ xx σ yy, θ π 4 regardless of σ xy. if both arguments are zero, θ is arbitrary. Detect this case and set θ 0. if σ xx, σ yy, σ xy < ε covariance is basically zero. Detect and exit. Draw a dot if this is for graphics. The values of the diagonal covariances come from the diagonal elements: σ uu σ xx c 2 θ 2σ xy sθcθ + σ yy s 2 θ σ vv σ xx s 2 θ + 2σ xy sθcθ + σ yy c 2 θ
7 3.3.3 Drawing Covariance Now that the coviariance is diagonal in some rotated coordinate system, here is an algorithm for drawing it. Reusing the earlier result, the equation of the ellipse for a diagonal matrix must be: uv u σ uu σ uu 0 0 σ vv v σ vv Dividing by produces the ellipse in standard form: u k 2 σ uu u v + k 2 k 2 v k 2 σ vv u 2 a 2 v 2 k b 2 3 Random Errors 7 The parametric equations of the contour in this frame are: x acosθ y bsinθ The rotation of the model frame with respect to original (x,y) coordinates is given by the negative of θ because we require (for drawing) the angle through which we rotate frame X to bring it into coincidence with U. 3.4 Some Important Caveats on Discrete Random Variables While standard deviation is a linear operator, the variance is not. So, given: y ax We can draw this ellipse by letting the U frame be the model frame of the ellipse sprite.
8 We have: σ y σ yy aσ x a 2 σ xx This fact has big implications when doing simulations of dynamic systems. See two sections below Discretizing Continuous Random Processes - Linear Velocity Be careful when you want to control the integrated behavior of a random process by controlling the noise in the original derivative. A classic case is odometry. Suppose you are computing distance by integrating velocity and the velocity is noisy. Suppose you want the variance in computed distance to be linear in distance: σ ss If distance is computed from the sum of incremental distances: αs () (2) n 3 Random Errors 8 s V t t V i Reusing (), the variance in each term in the sum is: The variance in the sum is the sum of the variances: Speed Measurements Equating this result to (2) gives: So that we must have: n n i σ s s σ vv t 2 (3) σ ss nσ vv t t σ t vv t 2 σ vv t t σ vv σ vv t t αs ---- α s - t α ---- V (4) t t
9 Hence, velocity variance must be proportional to velocity and inversely proportional to time step in order to get distance error which is proportional to distance. Doubling the time step creates four times the variance per sample added and half the computed distance variance because the integral multiplies by t. To corrupt a velocity measurement in order to generate the desired behavior you would use the Gaussian: V meas V true + N 0, αv t Differential Distance Measurements However, if your simulated encoder is returning a differential distance, considering (3), you use the Gaussian: s meas s true + N( 0, σ s ) s meas s true + N( 0, σ vv t) 3 Random Errors 9 Which can be rewritten using (4) as: s meas s true + N 0, αv t t Hence, differential position variance must be proportional to velocity and proportional to time step in order to get distance error which is proportional to distance. In this way, the behavior of the injected error is the same regardless of the update rate used in your application layer and regardless of whether the simulated encode generates differential position or instaneous velocity Discretizing Continuous Random Processes - Angular Velocity Suppose you want to generate a sequence of discrete angular velocity measurements from a simulated gyro and you want to generate a computed heading error which is linear in time.
10 Continuous Time In continuous time, the process is: θ ω (5) This is of the form: x u Where the system and input Jacobians are: F x 0 L x The transition matrix is: x u Ψ( t, τ) exp F( ζ) dζ On substitution, this is identity because the system is entirely forced (no autonomous behavior to be encoded in a transition matrix): t τ Ψ( t, τ) 3 Random Errors 0 For vanishing initial conditions, the solution for the covariance of the state θ is: σ θθ t 0 Φ( t, τ)l( τ)q( τ)l T ( τ)φ T ( t, τ) dτ Let the continuous variance Q in the input ω be: σ ωω σ gg const Since Φ and L are dimensionless here, Q must have units of rads^2/sec to generate rads/ sec upon integration. This is why gyro specs often quote random walk (standard deviation) in units of angle/root(time). On substitution: σ θθ t (6) Q( τ) dτ σ gg t 0
11 Hence, continuous heading variance grows linearly with respect to time when the continuous gyro noise is constant Discrete Time Let a tilde over a symbol denote a function of a discrete variable. Now, suppose we want to simulate this behavior in discrete time. A random sequence of angular velocity errors δω is generated and integrated with respect to time to get the associated heading errors δθ. If the same time period is divided into k steps of size t, the discrete integral is: δθ( k) δω( k) t (7) k Suppose by analogy to the continuous case that the variance in the discrete velocities is constant: var[ δω( k) ] σ γγ ( k) const 3 Random Errors Using (), the variance in the discrete integral is then: σ θθ ( k) k( t) 2 [ σ γγ ( k) ] (8) Thus, the variance as a function of sample number is linear in sample number and quadratic in the time step. Since k corresponds to the time t, and k t t this also means that the discrete variance computed for time t is proportional to that time and the time step: σ θθ () t σ γγ ()t t t The intuition behind this can be generated from equation (8). If: σ γγ ( k) const Then, the number of random numbers added in equation (7) over time period t depends on the magnitude of t. In fact, if t is reduced in size by a factor of 2, the number of random numbers is doubled. However, the variance in
12 the numbers themselves is the variance in δω( k) t which is not halved but quartered. Adding twice as many numbers of one quarter the variance produces half the original result. Therefore if it is required that the discrete integral expressed in equation (7) produce the same results over time as the continuous integral expressed in equation (6), we must equate the variances thus: σ γγ ()t t t σ gg t Solving for the discrete variance leads to: Thus, the discrete time equivalent (in terms of generating the same integrated behavior) to a continuous time variance, is computed by dividing by the time step. σ γγ () t σ gg t 4 Summary 2 4 Summary While systematic erros are pretty easy to model, random ones take some work. Arbitrary distributions can be generated from a uniform one with the transformation method. Simulating noises in sensors in order to generate specific behaviors is quite subtle. The discrete equivalent (in terms of generating the same integrated behavior) of a continuous time variance is computed by dividing by the time step. However, if the measurements are integrated wrt time first, and then added, the discrete standard deviation must be multiplied by the time step.
13 5 Notes Move FIFO queues to simulation section... 5 Notes 3 6 References [] Knuth Art of Computer Programming for pseudorandom number generators. [2] Numerical Recipes for the Box-Meller method of generating a simple Gaussian random variable. [3] R. C. Smith, P. Cheeseman, "On the Representation and Estimation of Spatial Uncertainty", The International Journal of Robotics Research, vol. 5, Number 4, pp Published by MIT Press, 987.
14 6 References 4
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