Download M.E. 530.646 Problem Set 1 [REV 1] Rigid Body Transformations

M.E. 530.646 Problem Set 1 [REV 1]
Rigid Body Transformations, Groups, and Rotations
Noah J. Cowan∗
Department of Mechanical Engineering
G.W.C. Whiting School of Engineering
Johns Hopkins University
Due: 16 September 2016 in class
Review your lecture notes and text [1], as required.
1. Prove (or disprove) that the following functions are linear:
a. The definite integral:
Z
f (x(t)) =
1
x(t)2 dt
0
where x(t) is a smooth function on [0, 1].
b. Let A ∈ Rn×n and b ∈ Rn . Then, for all x ∈ Rn , define f by
f (x) = Ax + b
c. The Laplace transform:
Z
L(x(t)) =
∞
e−st x(t)dt
0
d. Let x = [x0 , x1 , . . . , xn−1 ]T ∈ Rn be any vector. Then, define
f (x) = x0 + x1 s + · · · + xn−1 sn−1
2. Polynomials can be conceived as vectors:
a. Show that polynomials of a single variable, s, of order n − 1 or less, form a linear vector space
(LVS), where vector addition is just the addition of two polynomials, scalar multiplication is just
the multiplication of a polynomial by a scalar. For example, let x1 = s2 + 3s + 1, and x2 = s5 , then
x1 + x2 = s5 + s2 + 3s + 1.
b. Show that xi = si−1 , i = 1, . . . , n form a basis for the LVS. (Hint: show linear independence by
making an argument about the expression p0 + p1 s1 + · · · pn−1 sn = 0 is only zero, in general, at a
finite number of points, except in the special case of the zero polynomial.)
c. What is the dimension of the space?
3. A group homomorphism is a mapping between two groups, σ : G → H such that if a, b ∈ G then
σ(a)σ(b) = σ(ab).
In other words, we can first map a and b separately to H, and then “multiply” the results using the
group operation in H, or we can first multiple in G, and then map the result.
This problem considers a homomorphism from the additive group to the multiplicative group.
∗ This
c Noah J. Cowan.
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a. Show real numbers form a group under addition.
b. Show that the positive real numbers (0, ∞), also written as R+ , form a group under multiplication.
c. Show that the exponential operator, exp : R → R+ is a group homomorphism.
d. Show that exp is
Property (i). Surjective (onto), that is for all A ∈ R+ , there exists a real number, a ∈ R, such
that exp{a} = A. In other words, the “range” of exp is all of the positive reals.
Property (ii). Injective (one-to-one), that is exp{a} = exp{b} is true only if a = b. In other words,
the function satisfies the “horizontal line test”.
Note: a homomorphism that is into and onto is a group isomorphism.
e. Let σ : G → H be a group homomorphism. Show that
Property (i). If e and i are the identity elements in G and H, respectively, then σ(e) = i.
Property (ii). Let a ∈ G, with inverse a−1 . Then σ(a−1 ) = σ(a)−1 .
4. Show that any linear vector space (LVS) is a group under vector addition, +.
5. Suppose g : R3 → R3 is a rigid transformation (see Definition 2.1 in Murray, Li and Sastry (MLS)).
Recall that the action induced on vectors is given by g∗ (v) := g(p + v) − g(p), where p ∈ R3 is any
point. Assuming that all rigid transformations can be written as g(p) = Rp + d, where R ∈ SO(3) and
d ∈ E3 , prove that g∗ is linear.
6. Prove that the rotation matrices,
SO(3) := {R ∈ R3×3 : RT R = I, |R| = 1}
form a group under matrix multiplication.
7. Very useful identities. (We are now dropping the bold notation for vectors, now that the distinction is
clear from context).
a. [TYPO CORRECTED IN THS PROBLEM. ] The cross product a × b is bilinear in a and
b, in the sense that it is “linear in the a slot”, since (α1 a1 + α2 a2 ) × b = α1 a1 × b + α2 a2 × b, and
likewise for the “b slot ”. Prove this.
b. This bi-linearity can be fleshed out a bit further to put a × b in terms of a matrix multiplied by a
vector. In particular, show that there is a matrix, b
a, such that a × b = b
a b where


0
−a3 a2
0
−a1  .
b
a =  a3
−a2 a1
0
Prove also that a × b = −b̂a.
c. Here is another interesting fact to prove:
a[
×b=b
a bb − bb b
a
d. And another. Let R ∈ SO(3) and v ∈ E3 . Show that
d
R
v = R vb R−1 .
Hint: write R in terms of its rows, i.e. RT = [r1 , r2 , r3 ], and recall that that the rows are orthonormal.
Moreover, show that , vb RT = [−r1 × v, −r2 × v, −r3 × v], and you can show the desired result by
direct computation.
e. Show that R(v × w) = (Rv) × (Rw).
8. Explain, in your own words, why two points in space cannot be added.
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9. Recall that in class we showed (or WILL show before long!) that IF the linear transformation y = Rx
where y, x ∈ R3×1 and R ∈ R3×3 is a rigid body transformation (i.e. it satisfies the mathematical
definition of rigid body transformation) THEN the matrix R is an element of the set SO(3) = {R :
R ∈ R3×3 ; RT R = I; det(R) = 1} — i.e. (a) RT R = I and (b) det(R) = +1.
Your assignment in this problem is to show that IF R ∈ SO(3) THEN the y = Rx is a rigid-body
transform.
10. Euler Angle Conventions: Recall from notes or the book that the explicit matrix representations for
the rigid body rotations:
• Rx (φ) : R 7→ SO(3) corresponding to a rotation of φ radians about the x-axis — i.e. a 3 × 3
matrix whose elements contain expressions such as sin φ and cos φ.
• Ry (θ) : R 7→ SO(3) corresponding to a rotation of θ radians about the y-axis.
• Rz (ψ) : R 7→ SO(3) corresponding to a rotation of ψ radians about the z-axis.
(a) Construct the explicit representation for the commonly employed Euler-angle rotation convention
Rxyz : R3 7→ SO(3) given by
Rxyz (ψ, θ, φ) = Rz (ψ)Ry (θ)Rx (φ)
(1)
where roll = φ, pitch = θ, and yaw = ψ.
(b) Define and construct an explicit matrix representation for a new Euler-angle convention employing
a different order of operation as follows:
Ryxz (ψ, θ, φ) = Rz (ψ)Rx (φ)Ry (θ).
(2)
Are the rotations computed with the convention Rxyz identical to those obtained with the convention Ryxz . Why or why not? What is going on here?
−1
−1
(R) then
: SO(3) 7→ R3 such that ∀R ∈ SO(3) if y = Rxyz
(c) Construct the inverse function Rxyz
R = Rxyz (y).
11. Consider a matrix R ∈ SO(3) as defined in class.
(a) Show by direct computation that e = [r32 − r23 , r13 − r31 , r21 − r12 ]T is an eigenvector of R with
unity eigenvalue.
(b) Show that if a vector e is an eigenvector of R with unity eigenvalue then Re = RT e.
(c) Show by example example that ∃R ∈ SO(3) such that the equation Rx = RT x holds for any
vector x. R = I is the trivial solution. Can you construct another one?
(d) Show that J = R − RT always has a non-trivial kernel.
(e) R represents a rotation of a certain magnitude around a fixed spatial axis — i.e. ‘rotate 10◦
about the Z axis’. Construct a closed form expression for the magnitude of rotation (in radians)
represented by the 3 × 3 rotation matrix R.
References
[1] R. M. Murray, Z. Li, and S. S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press,
Boca Raton, FL, 1994.
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