Stability of an Assembly Project - Revision history

Lynch: /* Program Specification */

2019-05-28T12:25:27Z

Program Specification

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	Unlike the form and force closure linear programming tests, where the elements of your contact vector had to be greater than or equal to a positive value, in this problem the elements of your contact vector only need to be nonnegative. (You are only trying to find one solution to the assembly equilibrium equations; in the form and force closure tests, you needed to show that the contacts could create arbitrary wrenches.)		Unlike the form and force closure linear programming tests, where the elements of your contact vector had to be greater than or equal to a positive value, in this problem the elements of your contact vector only need to be nonnegative. (You are only trying to find one solution to the assembly equilibrium equations; in the form and force closure tests, you needed to show that the contacts could create arbitrary wrenches.)

	Your biggest job in this programming assignment is taking the specifications of the bodies' mass properties and the contacts and turning them into static equilibrium equations of the form <math>Fk = b</math>, similar to Example 12.11. These equations are solved by linear programming with the constraint that the elements of the contact vector <math>k</math> must all be nonnegative. If you have <math>m</math> bodies in the assembly, you will have <math>3m</math> wrench balance equations, and if you have <math>n</math> contacts, you will have <math>2n</math> friction cone edges and therefore <math>2n</math> elements in your contact vector <math>k</math>.		Your biggest job in this programming assignment is taking the specifications of the bodies' mass properties and the contacts and turning them into static equilibrium equations of the form <math>Fk = b</math>, similar to Example 12.11 in the book. These equations are solved by linear programming with the constraint that the elements of the contact vector <math>k</math> must all be nonnegative. If you have <math>m</math> bodies in the assembly, you will have <math>3m</math> wrench balance equations, and if you have <math>n</math> contacts, you will have <math>2n</math> friction cone edges and therefore <math>2n</math> elements in your contact vector <math>k</math>.

	=== Testing Your Program ===		=== Testing Your Program ===

Lynch: /* Program Specification */

2018-07-20T11:04:15Z

Program Specification

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	The output of your program will be binary: it is either possible or impossible for the assembly to remain standing. Your solution method is likely to use linear programming (linprog in MATLAB, LinearProgramming in Mathematica, or scipy.optimize.linprog in Python). If you find a solution to the contact vector <math>k</math> (see Example 12.11) and standing is possible, then it is recommended that you also output <math>k</math>. This represents one set of contact forces that would keep the assembly standing.		The output of your program will be binary: it is either possible or impossible for the assembly to remain standing. Your solution method is likely to use linear programming (linprog in MATLAB, LinearProgramming in Mathematica, or scipy.optimize.linprog in Python). If you find a solution to the contact vector <math>k</math> (see Example 12.11) and standing is possible, then it is recommended that you also output <math>k</math>. This represents one set of contact forces that would keep the assembly standing.

	: '''Important note for Python users:''' A Coursera student mentioned a bug in linprog, apparently occurring when using the 'simplex' method for solving linear programs where the equality constraints are specified by a matrix that is not full rank. One solution is to use the interior point method by adding the option "method='interior-point'". See the discussion at [https://github.com/scipy/scipy/issues/6690 https://github.com/scipy/scipy/issues/6690].		: '''Important note for Python users:''' A Coursera student mentioned a bug in an earlier version of linprog, apparently occurring when using the 'simplex' method for solving linear programs where the equality constraints are specified by a matrix that is not full rank. One solution is to use the interior point method by adding the option "method='interior-point'". See the discussion at [https://github.com/scipy/scipy/issues/6690 https://github.com/scipy/scipy/issues/6690].

	Unlike the form and force closure linear programming tests, where the elements of your contact vector had to be greater than or equal to a positive value, in this problem the elements of your contact vector only need to be nonnegative. (You are only trying to find one solution to the assembly equilibrium equations; in the form and force closure tests, you needed to show that the contacts could create arbitrary wrenches.)		Unlike the form and force closure linear programming tests, where the elements of your contact vector had to be greater than or equal to a positive value, in this problem the elements of your contact vector only need to be nonnegative. (You are only trying to find one solution to the assembly equilibrium equations; in the form and force closure tests, you needed to show that the contacts could create arbitrary wrenches.)

Lynch: /* Program Specification */

2018-07-20T11:00:17Z

Program Specification

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	The output of your program will be binary: it is either possible or impossible for the assembly to remain standing. Your solution method is likely to use linear programming (linprog in MATLAB, LinearProgramming in Mathematica, or scipy.optimize.linprog in Python). If you find a solution to the contact vector <math>k</math> (see Example 12.11) and standing is possible, then it is recommended that you also output <math>k</math>. This represents one set of contact forces that would keep the assembly standing.		The output of your program will be binary: it is either possible or impossible for the assembly to remain standing. Your solution method is likely to use linear programming (linprog in MATLAB, LinearProgramming in Mathematica, or scipy.optimize.linprog in Python). If you find a solution to the contact vector <math>k</math> (see Example 12.11) and standing is possible, then it is recommended that you also output <math>k</math>. This represents one set of contact forces that would keep the assembly standing.

			: '''Important note for Python users:''' A Coursera student mentioned a bug in linprog, apparently occurring when using the 'simplex' method for solving linear programs where the equality constraints are specified by a matrix that is not full rank. One solution is to use the interior point method by adding the option "method='interior-point'". See the discussion at [https://github.com/scipy/scipy/issues/6690 https://github.com/scipy/scipy/issues/6690].

	Unlike the form and force closure linear programming tests, where the elements of your contact vector had to be greater than or equal to a positive value, in this problem the elements of your contact vector only need to be nonnegative. (You are only trying to find one solution to the assembly equilibrium equations; in the form and force closure tests, you needed to show that the contacts could create arbitrary wrenches.)		Unlike the form and force closure linear programming tests, where the elements of your contact vector had to be greater than or equal to a positive value, in this problem the elements of your contact vector only need to be nonnegative. (You are only trying to find one solution to the assembly equilibrium equations; in the form and force closure tests, you needed to show that the contacts could create arbitrary wrenches.)

Lynch: /* Introduction */

2018-05-27T22:44:17Z

Introduction

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	=== Introduction ===		=== Introduction ===

	You will write a program to determine if an assembly of planar rigid bodies, in frictional contact with each other, can remain standing in gravity, or if the assembly must collapse. (See Example 12.11 and Figure 12.27 in Chapter 12.3 of the book.)

	Your assignment will be graded on:		Your assignment will be graded on:

Lynch at 22:43, 27 May 2018

2018-05-27T22:43:44Z

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	This page describes the "Stability of an Assembly" Project from the Coursera course "Modern Robotics, Course 5: Robot Manipulation and Wheeled Mobile Robots."		This page describes the "Stability of an Assembly" Project from the Coursera course "Modern Robotics, Course 5: Robot Manipulation and Wheeled Mobile Robots."

			=== Introduction ===

			You will write a program to determine if an assembly of planar rigid bodies, in frictional contact with each other, can remain standing in gravity, or if the assembly must collapse. (See Example 12.11 and Figure 12.27 in Chapter 12.3 of the book.)

			Your assignment will be graded on:
			* An evaluation of your source code, including its clarity for the reader.
			* The correctness of your code's output on example assemblies, as well as your physical explanation for the results.

	=== Program Specification ===		=== Program Specification ===

Lynch: /* What to Submit */

2018-05-20T20:15:18Z

What to Submit

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	'''2. A directory called "results."''' In this directory you should have		'''2. A directory called "results."''' In this directory you should have
	* An output log showing your program being called and the resulting output for the case of the '''collapsing assembly''' and the '''assembly that can continue to stand''' described in the section "Testing Your Program" above. The figure on this page shows the assembly and the text above gives the body masses and friction coefficients for these two cases.		* An output log showing your program being called and the resulting output for the case of the '''collapsing assembly''' and the '''assembly that can continue to stand''' described in the section "Testing Your Program" above. The figure on this page shows the assembly and the text above gives the body masses and friction coefficients for these two cases.
	* A drawing of another assembly of your own design. This assembly should have at least two bodies in addition to ground. You could use the three-body arch in Figure 12.27 as your assembly if you'd like (choose your own contact and center of mass locations that agree approximately with the figure). Without changing the masses of the bodies, this assembly must begin to collapse for one set of friction coefficients and should be able to stand for another set of friction coefficients.		* A drawing of another assembly of your own design. This assembly should have at least two bodies in addition to ground. This drawing should clearly indicate the center of mass and contact <math>(x,y)</math> locations, as well as the masses of each body. You could use the three-body arch in Figure 12.27 as your assembly if you'd like (choose your own contact and center of mass locations that agree approximately with the figure). Without changing the masses of the bodies, this assembly must begin to collapse for one set of friction coefficients at the contacts and should be able to stand for another set of friction coefficients.
	* An output log showing your program being called with your assembly from the previous bullet, for a choice of friction coefficients where the assembly must begin to collapse and another set of choices where it can continue to stand. For each of the two cases, you should provide a brief explanation of the results. For example, for the case where the assembly can continue to stand, you can use your linear programming solution for <math>k</math> to draw the contact wrenches for each body, graphically showing that the sum of the contact wrenches cancels the gravity wrench.		* An output log showing your program being called with your assembly from the previous bullet, for a choice of friction coefficients where the assembly must begin to collapse and another set of choices where it can continue to stand. For each of the two cases, you should provide a brief explanation of the results. For example, for the case where the assembly can continue to stand, you can use your linear programming solution for <math>k</math> to draw the contact wrenches for each body, graphically showing that the sum of the contact wrenches cancels the gravity wrench.

Lynch: /* What to Submit */

2018-05-20T19:58:15Z

What to Submit

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	* An output log showing your program being called and the resulting output for the case of the '''collapsing assembly''' and the '''assembly that can continue to stand''' described in the section "Testing Your Program" above. The figure on this page shows the assembly and the text above gives the body masses and friction coefficients for these two cases.		* An output log showing your program being called and the resulting output for the case of the '''collapsing assembly''' and the '''assembly that can continue to stand''' described in the section "Testing Your Program" above. The figure on this page shows the assembly and the text above gives the body masses and friction coefficients for these two cases.
	* A drawing of another assembly of your own design. This assembly should have at least two bodies in addition to ground. You could use the three-body arch in Figure 12.27 as your assembly if you'd like (choose your own contact and center of mass locations that agree approximately with the figure). Without changing the masses of the bodies, this assembly must begin to collapse for one set of friction coefficients and should be able to stand for another set of friction coefficients.		* A drawing of another assembly of your own design. This assembly should have at least two bodies in addition to ground. You could use the three-body arch in Figure 12.27 as your assembly if you'd like (choose your own contact and center of mass locations that agree approximately with the figure). Without changing the masses of the bodies, this assembly must begin to collapse for one set of friction coefficients and should be able to stand for another set of friction coefficients.
	* An output log showing your program being called with your assembly from the previous bullet, for a choice friction coefficients where the assembly must begin to collapse and another set of choices where it can continue to stand. For each of the two cases, you should provide a brief explanation of the results. For example, for the case where the assembly can continue to stand, you can use your linear programming solution for <math>k</math> to draw the contact wrenches for each body, graphically showing that the sum of the contact wrenches cancels the gravity wrench.		* An output log showing your program being called with your assembly from the previous bullet, for a choice of friction coefficients where the assembly must begin to collapse and another set of choices where it can continue to stand. For each of the two cases, you should provide a brief explanation of the results. For example, for the case where the assembly can continue to stand, you can use your linear programming solution for <math>k</math> to draw the contact wrenches for each body, graphically showing that the sum of the contact wrenches cancels the gravity wrench.

	'''3. (OPTIONAL).''' For the three-body arch in Figure 12.27 in the book, assume all three bodies are identical, with the same masses and friction coefficients. Use your program to determine the minimum friction coefficient that keeps the assembly standing.		'''3. (OPTIONAL).''' For the three-body arch in Figure 12.27 in the book, assume all three bodies are identical, with the same masses and friction coefficients. Use your program to determine the minimum friction coefficient that keeps the assembly standing.

Lynch at 19:56, 20 May 2018

2018-05-20T19:56:44Z

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	=== Program Specification ===		=== Program Specification ===

	[[image:leaning.png\|thumb\|right\|x300px\|An assembly consisting of two planar rigid bodies in contact with each other and stationary ground. This assembly will begin to collapse if the mass of body 1 is 2, the mass of body 2 is 5, and all friction coefficients are 0.5 except friction between body 1 and the ground, which is 0.1. The assembly can remain standing if the mass of body 1 is 2, the mass of body 2 is 10, and the friction coefficient at all four contacts is 0.5.]]		[[image:leaning.png\|thumb\|right\|x300px\|An assembly consisting of two planar rigid bodies in contact with each other and stationary ground. This assembly will begin to collapse if the mass of body 1 is 2, the mass of body 2 is 5, and all friction coefficients are 0.5 except the friction coefficient between body 1 and the ground, which is 0.1. The assembly can remain standing if the mass of body 1 is 2, the mass of body 2 is 10, and the friction coefficient at all four contacts is 0.5.]]

	You will write a program to determine if an assembly of planar rigid bodies, in frictional contact with each other, can remain standing in gravity (where gravity acts in the <math>-y</math> direction), or if the assembly must collapse. (See Example 12.11 and Figure 12.27 in Chapter 12.3 of '''[http://modernrobotics.org the book]'''.) An example assembly is shown in the image at right.		You will write a program to determine if an assembly of planar rigid bodies, in frictional contact with each other, can remain standing in gravity (where gravity acts in the <math>-y</math> direction), or if the assembly must collapse. (See Example 12.11 and Figure 12.27 in Chapter 12.3 of '''[http://modernrobotics.org the book]'''.) An example assembly is shown in the image at right.

Lynch at 19:56, 20 May 2018

2018-05-20T19:56:06Z

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	=== Program Specification ===		=== Program Specification ===

	[[image:leaning.png\|thumb\|right\|x300px\|An assembly consisting of two planar rigid bodies in contact with each other and stationary ground.]]		[[image:leaning.png\|thumb\|right\|x300px\|An assembly consisting of two planar rigid bodies in contact with each other and stationary ground. This assembly will begin to collapse if the mass of body 1 is 2, the mass of body 2 is 5, and all friction coefficients are 0.5 except friction between body 1 and the ground, which is 0.1. The assembly can remain standing if the mass of body 1 is 2, the mass of body 2 is 10, and the friction coefficient at all four contacts is 0.5.]]

	You will write a program to determine if an assembly of planar rigid bodies, in frictional contact with each other, can remain standing in gravity (where gravity acts in the <math>-y</math> direction), or if the assembly must collapse. (See Example 12.11 and Figure 12.27 in Chapter 12.3 of '''[http://modernrobotics.org the book]'''.) An example assembly is shown in the image at right.		You will write a program to determine if an assembly of planar rigid bodies, in frictional contact with each other, can remain standing in gravity (where gravity acts in the <math>-y</math> direction), or if the assembly must collapse. (See Example 12.11 and Figure 12.27 in Chapter 12.3 of '''[http://modernrobotics.org the book]'''.) An example assembly is shown in the image at right.

Lynch: /* What to Submit */

2018-05-20T19:50:14Z

What to Submit

← Older revision		Revision as of 14:50, 20 May 2018
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	'''2. A directory called "results."''' In this directory you should have		'''2. A directory called "results."''' In this directory you should have
	* An output log showing your program being called and the resulting output for the case of the '''collapsing assembly''' and the '''assembly that can continue to stand''' described in the section "Testing Your Program" above. The figure on this page shows the assembly and the text above gives the body masses and friction coefficients for these two cases.		* An output log showing your program being called and the resulting output for the case of the '''collapsing assembly''' and the '''assembly that can continue to stand''' described in the section "Testing Your Program" above. The figure on this page shows the assembly and the text above gives the body masses and friction coefficients for these two cases.
	* A drawing of another assembly of your own design. This assembly should have at least two bodies in addition to ground. You could use the arch in Figure 12.27 as your assembly if you'd like (choose your own contact and center of mass locations that agree approximately with the figure). Without changing the masses of the bodies, this assembly must begin to collapse for one set of friction coefficients and should be able to stand for another set of friction coefficients.		* A drawing of another assembly of your own design. This assembly should have at least two bodies in addition to ground. You could use the three-body arch in Figure 12.27 as your assembly if you'd like (choose your own contact and center of mass locations that agree approximately with the figure). Without changing the masses of the bodies, this assembly must begin to collapse for one set of friction coefficients and should be able to stand for another set of friction coefficients.
	* An output log showing your program being called with your assembly from the previous bullet, for a choice friction coefficients where the assembly must begin to collapse and another set of choices where it can continue to stand. For each of the two cases, you should provide a brief explanation of the results. For example, for the case where the assembly can continue to stand, you can use your linear programming solution for <math>k</math> to draw the contact wrenches for each body, graphically showing that the sum of the contact wrenches cancels the gravity wrench.		* An output log showing your program being called with your assembly from the previous bullet, for a choice friction coefficients where the assembly must begin to collapse and another set of choices where it can continue to stand. For each of the two cases, you should provide a brief explanation of the results. For example, for the case where the assembly can continue to stand, you can use your linear programming solution for <math>k</math> to draw the contact wrenches for each body, graphically showing that the sum of the contact wrenches cancels the gravity wrench.