This page is part of an article about rotary joint driven by a linear motor. Please, start by reading the introduction.

This article is splitted in three parts:

- Part 1. Introduction
- Part 2. Angle as a function of length
- Part 3. Coordinates of motor
- Part 4. Torque

To compute the torque as a function of the motor length, we'll need the expression of the vector \( \overrightarrow{AB} \). Let's start by calculating the coordinates of \(A=(x_A, y_A)\):

$$ \begin{split} x_A = | \overrightarrow{OA} | \times \cos ( \pi - \alpha ) \\ y_A = | \overrightarrow{OA} | \times \sin ( \pi - \alpha ) \end{split} $$

Since \( \cos(\pi-\theta) = -\cos(\theta) \) and \( \sin(\pi-\theta) = \sin(\theta) \), the previous expression can be simplified:

$$ \begin{split} x_A &=& -| \overrightarrow{OA} | \times \cos ( \alpha ) \\ y_A &=& | \overrightarrow{OA} | \times \sin ( \alpha ) \end{split} $$

To get the coordinates of point \(B\), we will calculate the intersection of circle \(C_1\) and \(C_2\):

- \(C_1\) is a circle of center \(A\) and radius \( | \overrightarrow{AB} | \)
- \(C_2\) is a circle of center \(C\) and radius \( | \overrightarrow{CB} | \)

You'll find a detailed demonstration of the intersections calculation on this page: How to calculate the intersection points of two circles?.

The distance between centers \( | AC | \) is given by:

$$ d = | \overrightarrow{AC} | = \sqrt {| \overrightarrow{BC} |^2 + | \overrightarrow{BA} |^2 } $$

- if \( d > | \overrightarrow{BA} | + | \overrightarrow{BC} | \) the circles are too far apart and do not intersect;
- if \( d < | | \overrightarrow{BA} | - | \overrightarrow{BC} | | \) one circle is inside the other and do not intersect;
- if \( d = 0 \) and \( \left| \overrightarrow{BA} | = | \overrightarrow{BC} \right| \) the circles are merged and there are an infinite number of points of intersection;
- if \( d = | \overrightarrow{BA} | + | \overrightarrow{BC} | \) there is a single intersection point;
- if \( d < | \overrightarrow{BA} | + | \overrightarrow{BC} | \) there are two intersection points.

In the three first cases, there is no solution. The coordonates of point \(B\) is given by:

$$ \begin{split} x_B &=& ~ x_1 ~\pm~ \dfrac{h}{d}(y_C - y_A) \\ y_B &=& ~ y_1 ~\pm~ \dfrac{h}{d}(x_C - x_A) \end{split} $$

where:

- \( a = \dfrac{| \overrightarrow{AB} |^2 - | \overrightarrow{BC} |^2 + d^2 }{2d} \)
- \( h = \sqrt{ | \overrightarrow{AB} |^2 - a^2 } \)
- \( x_1 = x_A + \dfrac{a}{d} \times ( x_C - x_A ) \)
- \( y_1 = y_A + \dfrac{a}{d} \times ( y_C - y_A ) \)

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Last update : 05/20/2022