pr.probability – Almost evenly distributed spherical random vectors

Consider $n$ i.i.d uniformly distributed random vectors $z_1 ,cdots , z_n in mathbb{S}^{d-1}$. What is the best lower bound on $n$ for which whp there exists a constant $c>0$ such that the following bound holds for all $vin mathbb{R}^dbackslash {0}$:

begin{equation}
cnleq leftvertleft{i:langle z_i,v rangle>0 right} rightvert
end{equation}

differential equations – Spherical Parallel Manipulator Lagrangian problem

I faced a very serious problem and I urgently need the help of specialists in robotics, mechanics, physics and mathematics.

I am trying to derive equations of motion from the Lagrangian of a spherical parallel manipulator based on work (3.2 Inverse dynamic modeling): https://www.sciencedirect.com/science/article/abs/pii/S0921889014001250

enter image description here

According to this article, the Lagrangian for the corresponding drive axle consists of two parts:

  1. Lagrangian of distal and proximal links.

$L_i=frac{1}{2}I_{l_1}theta_i^2+frac{1}{2}boldsymbol{omega_i}^Tboldsymbol{I}_{l_2}boldsymbol{omega_i}-m_{l_1}chi_1boldsymbol{g}^Tboldsymbol{h_i}-m_{l_2}chi_2boldsymbol{g}^Tboldsymbol{e_{ix}}$

where $boldsymbol{h_i}=frac{u_i+v_i}{||u_i+v_i||}$;$boldsymbol{e_{ix}}=frac{v_i+w_i}{||v_i+w_i||}$; $m_{l_1},m_{l_2},chi_1,chi_2,I_{l_1}$ – masses, center of masses of proximal and distal links and moment of inertia of proximal link; $u_i,v_i,w_i$ – design vectors depending on design parameters of the mechanism, angles of rotation of drives and something else? …

  1. Platform Lagrangian

$L_p=frac{1}{2}boldsymbol{Omega}^Tboldsymbol{I}_{p}boldsymbol{Omega}-m_{p}Rcos(beta)g^Tboldsymbol{p}$

where $boldsymbol{p}$ – unit vector of platform direction; $boldsymbol{I}_{p}$ – tensor of inertia of platform; $boldsymbol{g}=(0;0;-9.81)^T$;$m_p,R,cos(beta)$ – mass, radius of sphere and angle beetwen platform and center of rotation;

In general, the Lagrangian is written as follows:

$L=L_1+L_2+L_3+L_p$

In the article, the authors took as generalized coordinates:

$boldsymbol{q}=(theta_1,theta_2,theta_3,phi,Theta,sigma)$;

where – $theta_1,theta_2,theta_3$ – drive angle of rotation; $phi,Theta,sigma$ – platform angles of rotation;

Further, the authors derive the equations of motion according to the classical formula, but they do not give more visual and additional calculations:

$frac{d}{dt}(frac{dL}{ddot{boldsymbol{q}}})-frac{dL}{dboldsymbol{q}}=0$


From that moment on, I encountered problems of the following nature, which I cannot solve on my own:

  1. Lagrangian structure assumes that the vector of generalized coordinates $q$ proposed by the authors lacks the position and velocity vector for the distal link and platform, i.e. $q≠(theta_1,theta_2,theta_3,phi,Theta,sigma)$, but $q=(theta_1,theta_2,theta_3,boldsymbol{omega},boldsymbol{Omega})$. Their generalized coordinates do not allow deriving the equations of motion, since angular velocity of the distal link $boldsymbol{omega}$ in the article there is no corresponding angular position.
  2. Moments of inertia of the platform $boldsymbol{I}_{p}$ and the distal link $boldsymbol{I}_{l_2}$ are nonstationary and, strictly speaking, depend on the angle of rotation of the drives, i.e. from $theta_1,theta_2,theta_3$. There is absolutely no indication of this in the article, but this should be taken into account when deriving the equations of motion from $L$. But the question is that it is not clear what all these moments of inertia depend on – on the angle of rotation of the drives or on the angle of rotation of the platform, i.e. what to include in the Lagrangian, $boldsymbol{I}_{p}(theta_1,theta_2,theta_3)$ or $boldsymbol{I}_{l_2}(phi,Theta,sigma)$ ?
  3. The third problem follows from the second and is connected with the fact that we can still find the coordinates of the design vectors $u_i$ and $v_i$ using the known design parameters and angles of rotation of the drive $theta_1,theta_2,theta_3$, but the position of the vector $w_i$ is calculated iteratively using the problem of direct kinematics, and one position of the drives $theta_1,theta_2,theta_3$ corresponds to 8 platform positions, i.e. it is impossible to establish a direct analytical connection between the angles of rotation of the drives $theta_1,theta_2,theta_3$ with the position of the platform $phi,Theta,sigma$ and the coordinates of the vector $w_i$, and therefore, it is impossible to differentiate the potential energy component of the Lagrangian $L$ by the generalized coordinate $q$.

Dear experts,
please help me understand the principle of working with this Lagrangian $L$ as soon as possible.
I would be grateful for your help in resolving these issues.
I have Mathematica, Maple and Simulink at hand and I am ready to do the proposed calculations and present the results.


Some excerpts from the article are presented below:

enter image description here
enter image description here
enter image description here

equation solving – Spherical harmonics Y (l,m,theta,phi) for general l, m

I am trying to solve integrals involving spherical harmonics Y(l,m, theta, phi) and their derivatives. I do not have any particular l,m, theta, phi values. I need to solve it for general l,m. When I am putting particular l,m values(forex l=3,m=1), Mathematica evaluates it and gives the answer. However, if I keep l,m as it is, Mathematica is not giving output. I am new to programming and coding. The code I have written is

Y(l_, m_, (Theta)_, (Phi)_) := 
 SphericalHarmonicY(l, m, (Theta), (Phi))

Ybar(l_, m_, (Theta)_, (Phi)_) := 
 SphericalHarmonicY(l, m, (Theta), (Phi)) /. I -> -I

Integrate(
 D(Y(3, 1, (Theta), (Phi)), (Theta)) D(
   Ybar(4, 1, (Theta), (Phi)), (Theta)) Sin((Theta)) 
Cos((Theta)), {(Theta), 0, (Pi)}, {(Phi), 0, 2 (Pi)})

Mathematica is evaluating this as I have put particular l,m values above.

However, If I write

Integrate(
  D(Y(l, m, (Theta), (Phi)), (Theta)) D(
    Ybar(L, m, (Theta), (Phi)), (Theta)) Sin((Theta)) 
Cos((Theta)), {(Theta), 0, (Pi)}, {(Phi), 0, 2 (Pi)})

Mathematica is not giving any output for this. Can anyone help me with this? I need to solve the above integral keeping l,m as it is, without any particular values of l,m.
Can anyone suggest any code?

How to represent Spherical Coordinate System unit vectors?

I have set the coordinate system to Spherical and then tried taking derivative of unit vectors with respect to spherical coordinates, but I didn’t get the result as shown in the textbooks.
i.e., given the input as D[UnitVector[3,2],{Ttheta}] but the output obtained as zero, whereas the textbook result is -r̂. How to quickly represent a Spherical unit vector in Mathematica?

dg.differential geometry – Comparison inequality between Sobolev-seminorm w.r.t spherical uniform distribution and gaussian distribution

Let $d$ be a large positive integer. Let $f:mathbb R^d to mathbb R$ be a continuously differentiable function. Let $X$ be uniformly distributed on the unit-sphere $S_{d-1} := {x in mathbb R^d |x| = 1}$ and let $Z$ be a random vector in $mathbb R^d$ with iid coordinates from $N(0,1/d)$. Let $nabla_{S_{d-1}}f:S_{d-1} to mathbb R^d$ be the spherical gradient of $f$, defined by $nabla_{S_{d-1}} f(x) :=(I_d-xx^top)nabla f(x)$, where $nabla f(x)$ is the usual / euclidean gradient of $f$ at $x$.

Question. Is there any comparison inequality between $mathbb E|nabla_{S_{d-1}} f(X)|^2$ and $mathbb E|nabla f(Z)|^2$, meaning the existence of abolute constants $c,C>0$ independent of $f$ and $d$, such that $c mathbb E|nabla_{S_{d-1}} f(X)|^2 le mathbb E|nabla f(Z)|^2 le C mathbb E|nabla_{S_{d-1}} f(X)|^2$?

Tangential component of a vector field to a spherical surface.

The problem is:
Vector E is given by
$$E = hat{R}5Rcos(theta) – hat{theta}frac{12}{R}sin(theta)cos(phi) +hat{phi}3sin(phi)$$
Determine the component of E tangential to the spherical surface $R = 2$ at point P($2$,$30^circ$,$60^circ$).

The solution to the above problem is (according to the solution manual):

At P, E is given by
$$E = hat{R}5(2)cos(30^circ) – hat{theta}frac{12}{2}sin(30^circ)cos(60^circ) +hat{phi}3sin(60^circ) = hat{R}8.67 – hat{theta}1.5 + hat{phi}2.6 $$

The $hat{R}$ component is normal to the spherical surface while the other two are tangential.
Hence
$$
E_t = -hat{theta}1.5 + hat{phi}2.6
$$

End of solution

Now my question is, how do you know which components are tangential and which are perpendicular.

sphericalharmonicy – Confirming orthogonality of spherical harmonics symbolically

One can confirm the orthogonality of the SphericalHarmonicYs for specific values of their parameters (l, m, ll, mm) as I showed in my solution here, but I have been unable to verify it for the general case, as in:

Assuming({l, ll, m, mm} ∈ Integers && 
   -l <= m <= l && -ll <= mm <= ll , 
 Integrate(
  Conjugate(
    SphericalHarmonicY(l, m, ϑ, φ)) 
    SphericalHarmonicY(ll, mm, ϑ, φ) 
    Sin(ϑ),
  {ϑ, 0, π}, {φ, 0, 2 π}))

Are there any tricks, or assumptions, or other techniques that enable Mathematica to symbolically evaluate that integral (leading to a product of KroneckerDelta functions)? I even tried FunctionExpand to express each SphericalHarmonicY using exponentials and other simpler functions, but the integral was still not evaluated.

dg.differential geometry – Spherical harmonics, $frak{sl}_2$, and algebra gradings

LEt $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $mathrm{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $mathrm{Pol}(S^2)$ into its spherical harmonics, but here I would rather decompose them into simple $frak{sl}_2$-modules:
$$
mathrm{Pol}(S^2) simeq bigoplus_{k in 2mathbb{N}} V_k.
$$

This decomposition must surely not be an algebra grading ($V_k.V_l$ will surely not be contained in $V_{k+l}$). What is an easy way to prove this?

plotting – Plot a figure in cartesian XY coordinates in 2D using spherical angles

I have the following equations in $x$, $y$, $z$ coordinates:

$x = R_0(f(theta,phi)sin(theta)cos(phi) + partial_{theta}f(theta,phi)cos(theta)cos(phi) – partial_{phi}f(theta,phi)sin(phi)/sin(theta)$

$y = R_0(f(theta,phi)sin(theta)sin(phi) + partial_{theta}f(theta,phi)cos(theta)sin(phi) + partial_{phi}f(theta,phi)cos(phi)/sin(theta)$

$z = R_0(f(theta,phi)cos(theta) – partial_{theta}f(theta,phi)sin(theta)$

with $f(theta,phi) = 1 + frac{4epsilon_e}{1-4epsilon_e}(cos^4(theta) + sin^4(theta)(1 -2sin^2(phi)cos^2(phi)))$

where $epsilon_e = 0.047, R_0 = 15 $. Here, $theta,phi$ are spherical angles

Now, I would like to plot $(x,y)$ for $epsilon_e = 0.047$ and $0$

This is what I have tried:

f((Theta)_, (Phi)_) := 
 1 + ((4 Subscript((Epsilon), e))/(
   1 - 3 Subscript((Epsilon), e)))((
     Cos^4)((Theta)) + (
      Sin^4)((Theta)) (1 - 2 (Sin^2)((Phi)) (Cos^2)((Phi)) ))
x = Subscript(R, 0)(
  f((Theta)_, (Phi)_) Sin((Theta)) Cos((Phi))  + 
   Diff(f((Theta)_, (Phi)_), (Theta)) Cos((Theta)) Cos((Phi)) - 
   Diff(f((Theta)_, (Phi)_), (Phi)) Sin((Phi))/Sin((Theta)))

y = Subscript(R, 0)(
  f((Theta)_, (Phi)_) Sin((Theta)) Sin((Phi))  + 
   Diff(f((Theta)_, (Phi)_), (Theta)) Cos((Theta)) Sin((Phi)) + 
   Diff(f((Theta)_, (Phi)_), (Phi)) Cos((Phi))/Sin((Theta)))

z = Subscript(R, 0)(
  f((Theta)_, (Phi)_) Cos((Theta)) - 
   Diff(f((Theta)_, (Phi)_), (Theta)) Sin((Theta)))

Can anyone suggest how to proceed forward to plot in XY-2D plane ? Do I have to use SphericalPlot or ListPlot?