ECE5463: Introduction to Robotics Lecture Note 1: Background Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 1 / 21
Outline Linear Algebra Linear Differential Equation Linear and Angular Motion of Point Mass Outline Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 21
Free Vector Free Vector: geometric quantify with length and direction free means not necessarily rooted anywhere; only length and direction matter Given a reference frame, v can be moved to a position such that the base of the arrow is at the origin without changing the orientation. Then the vector v can be represented its coordinates v in the reference frame. v denotes the physical quantify while v denote its coordinate wrt some frame. Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 21
Point and Its Coordinate Point: p denotes a point in the physical space A point p can be represented by as a vector from frame origin to p p denotes the coordinate of a point p The coordinate p depends on the choice of reference frame Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 4 / 21
Vector (Math) Vector: p R n : Inner product of two vectors p R n, q R n : Norm of a vector p: Angle between two vectors p, q R n : Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 5 / 21
Matrix A R n m Symmetric matrix Matrix vector multiplication as linear combination of columns Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 6 / 21
Change of Basis Two Bases for R n : {a} ={â 1,..., â n } and {b} = {ˆb 1,..., ˆb n } v a and v b are the corresponding coordinates of v w.r.t. {a} and {b}, how to they relate? Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 7 / 21
Cross Product Cross product or vector product of a R 3, b R 3 is defined as a b = a 2b 3 a 3 b 2 a 3 b 1 a 1 b 3 (1) a 1 b 2 a 2 b 1 Properties: a b = a b sin(θ) a b = b a a a = 0 Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 8 / 21
Skew symmetric representation It can be directly verified from definition that a b = [a]b, where 0 a 3 a 2 [a] a 3 0 a 1 (2) a 2 a 1 0 a = a 1 a 2 a 3 [a] [a] = [a] T (called skew symmetric) Example: Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 9 / 21
Positive Semidefinite Matrix A symmetric square matrix A R n n is called positive semidefinite (p.s.d.), denoted by A 0, if x T Ax 0, x R n A symmetric square matrix A R n n is called positive definite (p.d.), denoted by A 0, if x T Ax > 0 for all nonzero x R n p.d. matrices characterize positive definite quadratic forms: Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 10 / 21
Positive Semidefinite Matrix II Equivalent definitions for p.s.d. matrices: - All eigs of A are nonnegative - There exists a factorization A = B T B Equivalent definitions for p.d. matrices: - All eigs of A are strictly positive - There exists a factorization A = B T B with B square and nonsingular. If A R n n is p.d., then A 1 is also p.d. Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 11 / 21
Ellipsoid in R n Unit sphere in R n : S = {x R n : x x c = 1} Ellipsoid in R n : S = {x R n : (x x c ) T A 1 (x x c ) = 1}, for some p.d. A R n n. Let λ 1,..., λ n be the eigenvalues of A with corresponding eigenvectors v 1,..., v n. - Principal semi-axis lengths are λ 1,..., λ n - Direction of Principal semi-axes are aligned with v 1,..., v n - volume of the ellipsoid is proportional to det(a) Linear Algebra Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 12 / 21
Outline Linear Algebra Linear Differential Equation Linear and Angular Motion of Point Mass Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 13 / 21
Scalar Linear Differential Equation ẋ(t) = ax(t), with initial condition x(0) = x 0 (3) x(t) R, a R is constant The above ODE has a unique solution x(t) = e at x 0 What is the number e? Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 14 / 21
Complex Exponential For real variable x R, Taylor series expansion for e x around x = 0: e x = k=0 This can be extended to complex variables: e z = k=0 x k k! = 1 + x + x2 2! + x3 3! + z k k! = 1 + z + z2 2! + z3 3! + This power series is well defined for all z C In particular, we have e jθ = 1 + jθ θ2 2 j θ3 3! + Comparing with Taylor expansions for cos(θ) and sin(θ) leads to the Euler Identity Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 15 / 21
Matrix Exponential Similar to the real and complex cases, we can define the so-called matrix exponential e A A k = I + A + A2 k! 2! + A3 3! + k=0 This power series is well defined whenever A is finite and constant. One can verify directly from definition: - Ae A = e A A - If A = P DP 1, then e A = P e D P 1 Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 16 / 21
Vector Linear Differential Equation ẋ(t) = Ax(t), with initial condition x(0) = x 0 (4) x(t) R n, A R n n is constant matrix, x 0 R n is given. With the definition of matrix exponential, we can show that the solution to (4) is given by x(t) = e At x 0. Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 17 / 21
Example Find the solution to ẋ(t) = [ 0 1 1 0 ] x(t) Linear ODE Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 18 / 21
Outline Linear Algebra Linear Differential Equation Linear and Angular Motion of Point Mass Point Mass Motion Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 19 / 21
Linear Motion Consider a particle with mass m position velocity/acceleration Force Momentum Newton s Second Law Point Mass Motion Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 20 / 21
Angular Motion Angle Angular velocity Torque (Moment) Angular Momentum Newton s Second Law Point Mass Motion Lecture 1 (ECE5463 Sp18) Wei Zhang(OSU) 21 / 21