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ME451 Kinematics and Dynamics of Machine Systems Review of Linear Algebra and Differential Calculus 2.4, 2.5 September 09, 2013 Radu Serban University of Wisconsin-Madison Before we get started… Last time: Discussed geometric and algebraic vectors Brief review of matrix algebra Today: Transformation of coordinates: Rotation Matrix, Rotation + Translation Vector and Matrix Differentiation HW 1 Due on Wednesday, September 11 Problems: 2.2.5, 2.2.8. 2.2.10 (from Haug’s book) Upload a file named “lastName_HW_01.pdf” to the Dropbox Folder “HW_01” at [email protected] Dropbox Folder closes at 12:00PM 2 2.4 Transformation of Coordinates Vectors and Reference Frames (1) Recall that an algebraic vector is just a representation of a geometric vector in a particular reference frame (RF) Question: What if I now want to represent the same geometric vector in a different RF? 4 Vectors and Reference Frames (2) Transforming the representation of a vector from one RF to a different RF is done through (left) multiplication by a so-called “rotation matrix” A: Notes We transform the vector’s representation and not the vector itself. What changes is the RF used to represent the vector. As such, the rotation matrix defines a relationship between RFs. A rotation matrix A is also called “orientation matrix”. 5 6 The Rotation Matrix Rotation matrices are orthogonal: Geometric interpretation of a rotation matrix: Important Relation Expressing a vector given in one reference frame (local) in a different reference frame (global): This is also called a change of base. Since the rotation matrix is orthogonal, we have More acronyms: ′ ′ LRF: local reference frame (𝑂 𝑥 𝑦′) GRF: global reference frame (𝑂𝑥𝑦) 7 Example 1 8 Example 2 https://respond.cc 9 Your answers… 10 Example 2 - solution 11 The Kinematics of a Rigid Body: Handling both Translation and Rotation What we just discussed was re-expressing a vector from one coordinate frame (LRF) to another coordinate frame (GRF). Recall that vectors for us are really “free vectors” and therefore independent of a translation of the reference frame. What about position of a point P? Use the definition of position vector, the vector addition, and the formula for changing base for vectors: 12 More on Body Kinematics Much of ME451 is based on the ability to look at the position of a point P in two different reference frames: a local reference frame (LRF), typically fixed (rigidly attached) to a body that is moving in space a global reference frame (GRF), which is the “world” reference frame and serves as the universal reference frame In the LRF, the position of point 𝑃 is described by 𝐬′𝑃 (sometimes, the notation 𝐬 𝑃 is used) In the GRF, the position of point 𝑃 is described by the position vector 𝐫 𝑃 13 14 ME451 Important Slide The position and orientation of a body (that is, position and orientation of the LRF) is completely defined by 𝐫 , 𝜙 . The position of a point P on the body is specified by: • 𝐬′𝑃 in the LRF • 𝐫 𝑃 in the GRF 15 Example C 2 y1′ 1 2222 O′555 5 % % % φ% Y 1 O X 2 x1′ D 2.5 Vector and Matrix Differentiation 17 Derivatives of Functions GOAL: Understand how to Take time derivatives of vectors and matrices Take partial derivatives of functions with respect to its arguments We will use a matrix-vector notation for computing these partial derivs. Taking partial derivatives might be challenging in the beginning It will be used a lot in this class 17 Derivative, Partial Derivative, Total Derivative The derivative of a function (of a single variable) is a measure of how much the function changes due to a change in its argument. A partial derivative of a function of several variables is the function derivative with respect to one of its variables when all other variables are held fixed. The total derivative of a function of several variables is the derivative of the function when all variables are allowed to change. 18 Derivatives: Examples Derivative Partial derivative Total derivative 19 Time Derivative of a Vector 20 Consider a vector 𝐫 whose components are functions of time: which is represented in a fixed (stationary) Cartesian RF. In other words, the components of r change, but not the reference frame: the basis vectors 𝑖 and 𝑗 are constant. Notation: Then: 20 Time Derivatives Vector-Related Operations 21 Time derivatives of a matrix 22 Partial Derivatives, Warming Up: Scalar Function of Two Variables Consider a scalar function of two variables: To simplify the notation, collect all variables into an array: With this, the derivative of f with respect to q is defined as: 23 24 Partial Derivatives, General Case: Vector Function of Several Variables You have a set of “m” functions each depending on a set of “n” variables: Collect all “m” functions into an array F and collect all “n” variables into an array q: So we can write: Partial Derivatives, General Case: Vector Function of Several Variables Then, in the most general case, we have The result is an m x n matrix! Example 2.5.2: 25 26 Partial Derivatives Compact Notation Let x, y, and be three generalized coordinates Define a (vector) function r of x, y, and as “Verbose” notation Collect the three generalized coordinates into the array q Define the function r of q: “Terse” notation 27 Example [handout] Example (based on Example 2.4.1) Find the partial derivative of the position of P with respect to the array of generalized coordinates q 28 Partial Derivatives: Remember this… In the most general case, you start with “m” functions in “n” variables, and end with an (m x n) matrix of partial derivatives. You start with a column vector of functions and then end up with a matrix Taking a partial derivative leads to a higher dimension quantity Scalar Function – leads to row vector Vector Function – leads to matrix In this class, taking partial derivatives can lead to one of the following: A row vector A full blown matrix If you see something else chances are you made a mistake… 29 Chain Rule of Differentiation Formula for computing the derivative(s) of the composition of two or more functions: We have a function f of a variable q which is itself a function of x. Thus, f is a function of x (implicitly through q) Question: what is the derivative of f with respect to x? Simplest case: real-valued function of a single real variable: 30 31 Case 1 Scalar Function of Vector Variable f is a scalar function of “n” variables: q1, …, qn However, each of these variables qi in turn depends on a set of “k” other variables x1, …, xk. The composition of f and q leads to a new function: Chain Rule Scalar Function of Vector Variable Question: how do you compute x ? Using our notation: Chain Rule: 32 Assignment [due 09/16] 33 34 Case 2 Vector Function of Vector Variable F is a vector function of several variables: q1, …, qn However, each of these variables qi depends in turn on a set of k other variables x1, …, xk. The composition of F and q leads to a new function: Chain Rule Vector Function of Vector Variable Question: how do you compute x ? Using our notation: Chain Rule: 35 Assignment [due 09/16] 36 37 Case 3 Vector Function of Vector Variables F is a vector function of 2 vector variables q and p : Both q and p in turn depend on a set of k other variables 𝐱 = 𝑥1 , ⋯ , 𝑥𝑘 𝑇 : A new function (x) is defined as: Example: a force (which is a vector quantity), depends on the generalized positions and velocities 38 Chain Rule Vector Function of Vector Variables Question: how do you compute 𝚽𝐱 ? Using our notation: Chain Rule: [handout] Example 39 Case 4 Time Derivatives In the previous slides we talked about functions f depending on y, where y in turn depends on another variable x. The most common scenario in ME451 is when the variable x is actually time, t You have a function that depends on the generalized coordinates q, and in turn the generalized coordinates are functions of time (they change in time, since we are talking about kinematics/dynamics here…) Case 1: scalar function that depends on an array of m time-dependent generalized coordinates: Case 2: vector function (of dimension n) that depends on an array of m time-dependent generalized coordinates: 40 Chain Rule Time Derivatives Question: what are the time derivatives of and Applying the chain rule of differentiation, the results in both cases can be written formally in the exact same way, except the dimension of the result will be different Case 1: scalar function Case 2: vector function 41 Example Time Derivatives 42 A Few More Useful Formulas 43