From c83c5a19fe3c8d76250fabc2d4b5cc9652f25d0c Mon Sep 17 00:00:00 2001 From: Niel Mistry Date: Sun, 12 Nov 2023 17:48:00 -0500 Subject: [PATCH] finish up position math --- .../{velocityMath.md => position_math.md} | 26 ++++++++++++++---- static/images/math/diagram.png | Bin 76716 -> 480054 bytes static/images/math/updated_diagram.png | Bin 384022 -> 0 bytes 3 files changed, 21 insertions(+), 5 deletions(-) rename content/posts/{velocityMath.md => position_math.md} (59%) delete mode 100644 static/images/math/updated_diagram.png diff --git a/content/posts/velocityMath.md b/content/posts/position_math.md similarity index 59% rename from content/posts/velocityMath.md rename to content/posts/position_math.md index ad39862..1a26bbc 100644 --- a/content/posts/velocityMath.md +++ b/content/posts/position_math.md @@ -1,5 +1,5 @@ --- -title: "Velocity Math" +title: "Finding End Effector Position" date: 2022-03-04T22:15:51-05:00 draft: false math: true @@ -8,9 +8,9 @@ math: true This math allows the robot to translate velocity requirements in the global coordinate frame of x and y to velocities of the ropes. We first defined two coordinate systems and gave names to the various names and angles in the diagram: -![diagram](/blog/images/math/updated_diagram.png) +![diagram](/blog/images/math/diagram.png) -Here, $X^S$ and $Y^S$ make up the canvas coordinate frame. This is intentionally chosen to be the same orientation as the [coordinate system in images](https://www.mathworks.com/help/images/image-coordinate-systems.html). $Y^S$ is chosen so its parallel to the gravity vector. $X^R$ and $Y^R$ define the coordinate system that moves with the robot, located at the robot's centre of gravity. +Here, $X^C$ and $Y^C$ make up the canvas coordinate frame. This is intentionally chosen to be the same orientation as the [coordinate system in images](https://www.mathworks.com/help/images/image-coordinate-systems.html). $Y^C$ is chosen so its parallel to the gravity vector. $X^R$ and $Y^R$ define the coordinate system that moves with the robot, located at the robot's centre of gravity. There are a few values that are constant and known beforehand: - $r_1$: The distance between the pegs that Opus is hanging off of @@ -20,7 +20,7 @@ There are a few values that are constant and known beforehand: $\alpha$ is the angle of the robot's frame relative to the wall's frame. $\alpha$ is typically not known, but we designed Opus with an IMU. This allowed us to use an Attitude Heading Reference System Kalman Filter (AHRS) to get stable $\alpha$ values from the device. We use the output of the AHRS Kalman Filter as a _known_ $\alpha$ value. -For a given location in the $X^S$, $Y^S$ frame, we need to find the lengths $r_2$ and $r_4$ that result in it. +For a given location in the $X^C$, $Y^C$ frame, we need to find the lengths $r_2$ and $r_4$ that result in it. This looks suspiciously like a four bar linkage, with the added constraint that beams $r_2$ and $r_4$ _cannot_ be in compression, since they are cables. Therefore, the four-bar linkage governing formula from _Fundamentals of Machine Theory and Mechanisms_ was used as a starting point, and we came to the following formulae: @@ -30,6 +30,8 @@ Due to the $y$ position being the same on the anchors: $\theta_1 = 0$ **Formula:** +To compute $\theta_2$ and $\theta_4$, we use the AHRS determined angle $\theta_1$ and the current lengths of the cables ($r_2$, $r_4$). + $$ \begin{align} x_3 &= r_1\cos{\theta_1} - r_3\cos{\alpha} \\\\ @@ -37,5 +39,19 @@ y_3 &= r_1\sin{\theta_1} - r_3\sin{\alpha} \end{align} $$ +$$ +% \DelcareMathOperator{\atantwo}{atan2} +\begin{align} +\theta_4 &= \text{atan2}(y_3, x_3) + \arccos\left({\frac{x_3^2 + y_3^2 + r_4^2 - r_2^2}{-2r_4\sqrt{x_3^2 + y_3^2}}}\right) \\\\ +\theta_2 &= \text{atan2}\left(\frac{y_3 + r_4\sin\theta_4}{r_2}, \frac{x_3 + r_4\cos\theta_4}{r_2}\right) +\end{align} +$$ + + +With $\theta_2$, we can get the position of the left spool exit point quite easily: +$$\vec{S_L} = r_2\begin{bmatrix} \cos{\theta_2} \\\\ -\sin{\theta_2} \end{bmatrix} $$ + +Finally, the end effector position can be determined by rotating the vector $\vec{V_E^R}$ that is defined in the robot's coordinate frame by $-\alpha$ radians, and adding it to $\vec{S_L}$: + +$$ \vec{V_E^C} = \vec{S_L} + \begin{bmatrix} \cos\left({-\alpha}\right) & -\sin\left({-\alpha}\right) \\\\ \sin\left({-\alpha}\right) & \cos\left({-\alpha}\right) \end{bmatrix} \vec{V_E^R} $$ -A transformation matrix is used to change bases from the x, y coordinate frame to the robot wire coordinates. diff --git a/static/images/math/diagram.png b/static/images/math/diagram.png index 108297fe36a6921a17697b83c851e1e42d68735e..bf5fa0f8596803edb5266033f9b086f8efd2d6d0 100644 GIT binary patch literal 480054 zcmeFZcRbbq|2{6M6eStQCKTBtGplrnL}VSa?2OFpkrZ(pBeRGiSx2(>D2{XNl)baF z4~{*)k7w`sdcR-2KfnKfFWtJG<2cTFp2xWE*ZsP#htRv~%4g3ooFO72I;(Q)rWO$q zsR8sHnEW3F1tH;6uVF#QDIJhRkSvAT1IhzHy!B|35J0?~2kDc~Q` 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