Midterm Sample

David LiuDecember 16, 2024About 1 min

Midterm Sample

Q1

25, Kinematics

out of, out of, into
2*3=6(3 xyz, 3theta), each have 2 in x, y axis
ans
1. $p^2=F^2_1p^1=(F_2^1)^{-1}p^1$
2. $p^3=F^3_2F^2_1p^1=$
3. $q^1=F_2^1q^2$
4. $q^3=F_2^3q^2$

1.5

$p^0_4=F^0_4p^0_4=F^0_1F_2^1F_3^2F_4^3p^4_4$

$d_4^0=R_1^0R_2^1R_3^2d_4^3+R_1^0R_2^1d_3^2+R_1^0d_2^1+d_1^0$

$d_4^0=R_1^0R_2^1R_3^2[0,1,0]+(R_1^0R_2^1+R_1^0)[1,0,0]+d_1^0$

Q2

15

$R_2^1=R_{xyz}=R_xR_yR_z$
$\phi_y=-\arcsin r_{31} \\ \phi_x = \arctan\frac{r_{32}}{r_{33}}\\ \phi_z = \arctan\frac{r_{21}}{r_{11}}$

2.3

No limits X

two, $\theta_y=20.7,180-20.7$

Q3

15

$w=\frac{\sqrt{1+r_{11}+r_{22}+r_{33}}}{2}\\ x=\sqrt\frac{1+r_{11}-2w^2}{2}=\frac{r_{32}-r_{23}}{4w}\\ y=\sqrt\frac{1+r_{22}-2w^2}{2}=\frac{r_{13}-r_{31}}{4w}\\ z=\sqrt\frac{1+r_{33}-2w^2}{2}=\frac{r_{21}-r_{12}}{4w}$
Axis, $\theta$ , around axis by theta
q is a 4x1 vector of unit length that represents an alternative method to Euler angles for computing rotations
s-> $\theta$ , v->Axis

Q4

link a couple of point, form smooth curve
Catmul-Rom: Bezier, Hermite

4.2

Catmul-Rom Spline vs, Hermite Spline, pros & cons

A Catmul-Rom Spline is constructed from individual Bezier curves .

A Bspline uses overlapping basis functions. Both provide local function approximation.

Catmul-Rom Splines are easy to construct using central differences of the control points and the Casteljau algorithm (i.e. recursive linear interpolation), are computationally efficient, but have discontinuous second derivatives at the knot points.

Bsplines provide an optimal interpolant to the control points, have continuous second derivatives at the knot points, but require matrix inverses to compute the spline coefficients from the control points.